NASA Astronautic Structural Manual Volume 2

NASA

TECHNICAL

MEMORANDUM NASA TMX- 73306

ASTRONAUTIC STRUCTURES MANUAL VOLUMEII

(NASA-T MANUAl,

M-X-7330 VOLUME

6) _SI_ONA[JTIC 2 (NASA) 975

N76-Tb167

STEUCTURES

Unclas _)_/9_ Structures

August

and

Propulsion

Laboratory

197 5

/

.

/

I

NASA

Su\ /

.J

George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama

MSFC

- Form

3190

(Rev

Jtme

1971)

.,q_.l

f

TECHNiCAl. t.

REPORT

NO.

NASA 4

J2.

TM

TITLE

l

X-73306

AND

GOVERNMENT

ACCESSION

REPORT

NO.

9|

SUBTITLE

5.

ASTRONAUTIC VOLUME

STRUCTURES

AUTHOR(S)

9.

PERFORMING

REPORT

MANUAL

I

ORGANIZATION

C.

NAME

Marshall

Marshall

Space

AND

Space

Flight

ADDRESS

Flight

Center,

10.

Center

WORK

I.

F;EPORr

ORGANIZATION

UNIT.

CONTRACT

tt

NO.

OR

GRANT

NO.

35812

Alabama

13.

12

PAGE

NO.

DATE

8. PERFORMING

George

TITLE

CATALOG

August 197 5 /;ERFORM1NG ORGANIZATIONCODE

II

7.

STANDARD

RECIFIENT'S

TYPE

OF

REPOR',

&

PERIOD

COVEREC

SPONSORING AGENCY NAME AND ADDRESS Technical

National

Aeronautics

and Space

1

Washington,

D.C.

Memorandum

I

Administration

4.

_PCNSORING

AGENCY

CODE

20546

15. SUPPLEMENTARY NOTES Prepared 16,

by

Structures

and

Propulsion

Laboratory,

Science

and

Engineering

ABSTRACT

This

document

aerospace

strength

cover

most

of the

actual

stress

analysis

structures

An and

for

overview includes

to methods

Section

D is

Section

These

17.

KEY

can

three

only

the

background

of the

manual

of

on thermal

composites;

II,

on

as

be

carried that

is

as

out are

Section rotating

supersede

a compilation by

hand,

methods

Section stresses,

on

and X-60041

18.

A is

devoted

fatigue

TM

_ORDS

not

and

C is

machinery; NASA

are

general

for

enough

to give the

usually

methods in scope

accurate

elastic

and

available,

in to

estimates

inelastic

but

also

as

a

themselves.

follows:

Section

of industry-wide

enough

techniques

of methods

E is

that

sophisticated

analysis

combined

analysis;

on

presents

and

of the

stresses;

volumes

III)

a catalog

loads,

strength

G is

and

It provides

not

sections

devoted

on

that

expected.

It serves

source

I,

encountered,

strength

ranges.

reference

used

(Volumes

and

a general

of methods

interaction

curves;

to

of structural

the

topic

fracture

Section and

introduction

mechanics;

H is

B is stability;

Section

F is

on statistics.

NASA

DISTRIBUTION

Section

TM

X-60042.

ST AT[ZMENI

Unclassified

-- Unlimited

_r

19

SECURTTY

CL ASSI

F, (of

thll

Unclassified MSFC

- Form

3292

1.

report;

20.

(Rev

l)ecember

1972)

Unclassified _ECURITY CLASC }'c_r _ale

IF, l,_

(of

lhi.b

Naliom_!

I

page) fochnlcal

!t(}.

OF

pA_FS

974

[2

22,

PRICE

NTIS

I

Inl',_r,,

,l,,)ll

_crvic,,,

¢il_ringfh'hl,

Virginia

,"}I

_,I

APPROVAL

ASTRONAUTIC STRUCTURES MANUAL VOLUME II

The cation. Atomic

information

Review Energy

Officer.

document

report

information

Commission

Classification be unclassified.

This

of any

in this

has

report,

also

been

concerning

programs This

has

been

has in its

reviewed

for

Department been

made

entirety,

reviewed

security

o; Defense

by the has

and approved

classifi-

MSFC

been

for

or Security

determined

to

technical

accuracy.

A.

A.

Director,

McCOOL Structures

and

Propulsion

'_"U.S,

Laboratory

GOVERNMENT

PRINTING

OFFICE

1976-641-255]447

REGION

NO.4

v

TABLE

OF

CONTENTS Page

f

BIO

HOLES

AND

I0. I

SMALL

CUTOUTS

10. i. I

HOLES

IN PLATES

...................

I0. I.I. i

10.1.1.2

Io

Biaxial

If.

Bending

Elliptical

.............

Tension

Holes

Multiple

Holes

with ............

..............

9

Single

iII.

Double

Row of Holes

IV.

Arrays

of Holes

Constant I.

7 8

II.

10. 1.2.1

9

............

Row of Holes

......

9

......

9

.........

Reinforcement

Asymmetrically Reinforced

10 11

....................

Bl0-iii

6

..............

Two Holes

Holes

6

7

I.

Reinforced

3

7

..........

..............

Rect_mgular Holes Rounded Corners

3

7

.............

Loading

Bending

Oblique

.........

..............

Holes

Axial

II.

10.1.2

..................

Circular Holes

I.

10.1.1.3

3

...........................

Unreinforced Holes

1

............

.......

12

13

TABLE

OF CONTENTS

(Concluded) Page

10.1.2.2 10.2

LARGE 10.2.1

Bending

10,2,2

Holes

BIBLIOGRAPHIES REFERENCES

HOLES

Variable AND

Reinforcement

CUTOUTS

of Plates in Beam

..................

14 15

................

with Circular

Webs

........

Holes

......

15 15 46

..................................

47

.....................................

B10-iv

LIST

OF ILLUSTRATIONS Title

Figure B10-1.

STRESS

CONCENTRATION

FOR AXIAL LOADING WIDTH PLATE WITH Bi0-2.

STRESS FOR

B i0-3.

THE

TENSION PLATE

THE

STRESS FOR

B i0-4.

STRESS FOR

FACTOR,

CASE

.......

OF A SEMIHOLE 2O

FACTOR,

OF A FLAT

HOLE

DISPLACED

K

t' BAR WITH

FROM

...........................

CONCENTRATION

AN ELLIPTICAL

19

t'

.........................

CASF

LINE

K

A CIRCULAR

CONCENTRATION

A CIRCULAR

Kt ,

CASE OF A FINITEA TRANSVERSE HOLE

WITH

EDGE

TENSION

CENTER

FACTOR,

CONCENTRATION

INFINITE NEAR

Page

2i

FACTOR, HOLE

b -

= 2

Kt , AND

a

FOR

A CIRCULAR

TO BIAXIAL BI0-5.

STRESS

STRESS

STRESS FOR

BI0-7.

BiO-8.

FACTOR,

CASE

Kt ,

FACTOR,

K

23

t'

OF FINITE-WIDTH

WITH

STRESS

CONCENTRATION

A TRANSVERSE

AN ELLIPTICAL

HOLE FACTOR,

HOLE

............ K

IN TENSION

t'

.......................

STRESS

CONCENTI:tATION

FACTOR

POINTS

UNDER

TENSION

FINITE

PLATE

MAXIMUM WITH

24

IN AN INFINITE

PLATE

HOLE

22

CASE OF AN INFINITELY WITH A TRANSVERSE HOLE

CONCENTRATION

BENDING

SUBJECTED

.......................

PLATE

FOR

IN A PLATE

CONCENTRATION

FOR BENDING WIDE PLATE Bi0-6.

ItOLE

25 FOR IN A

AN ELLIPTICAL

................................. B10-v

26

LIST

OF ILLUSTRATIONS

Figure B10-9.

Title

STRESS

CONCENTRATION

AN

ELLIPTICAL

VARIATION

Kt

b - WITH TENSILE P OPEN THE SLOT

BI0-11.

Bi0-i2.

27

.....................

WITH

aP

LOADING

FOR

CONSTANT

TENDING

TO 28

.........................

STRESS

CONCENTRATION

ON

NET

AREA

OF

OBLIQUITY

STRESS

Kt ,

BENDING CASE OF SHEET CONTAINING

HOLE

OF

Page

FACTOR,

FOR THE TRANSVERSE AN INFINITELY WIDE

B10-10.

{Continued)

FACTOR

AS A FUNCTION

BASED

OF

ANGLE 29

/3 ........................

CONCENTRATION

FACTOR,

K

t'

FOR TENSION CASE OF AN INFINITE PLATE WITH TWO CIRCULAR HOLES {TENSION PERPENDICULAR

BI0-13.

STRESS

STRESS

LINE

CONCENTRATION

FOR BIAXIAL PLATE WITH

B10-14.

TO

OF

HOLES)

FACTOR,

TENSION CASE TWO CIRCULAR

CONCENTRATION

HOLE

Bi0-i6.

STRESS

CONCENTRATION

POINT

A

STRESS

CONCENTRATION

Bi0-i7.

FOR

WITH

OF AN INFINITE HOLES ..........

FACTORS

CIRCULAR

UNDER

TENSION

SINGLE

ROW

FOR

ON

TO NET

32

LINE

SECTION Bl0-vi

33

AT

IN Y-DIRECTION

A SHEET

OF HOLES.

TWO

FIELD

FACTOR,

OF

3i

...............

FACTOR

TENSION

CASE

PERPENDICULAR K t BASED

NOTCH

3O

Kt ,

UNEQUAL-SIZED HOLE q IN BIAXIAL OF STRESS ............................. Bt0-i5.

.........

Kt WITH

.....

33

t

A

(TENSION OF

HOLES.

)

................

34

LIST

OF

ILLUSTRATIONS

Title

Figure B10-i8.

STRESS FOR

A BIAXIALLY PLATE

STRESS

FACTOR.

STRESSED

TENSION

ROW

OF

NET

SECTION

CASE

HOLES.

OF

K

t

STRESS CONCENTRATION UNIAXIAL TENSION AND

HOLE

CONFIGURATION

STRESS

AND

B 10-26.

STRESS

TENSION

FACTORS

AND

38

FOR

39 FOR

CONFIGURATION

AND

CONCENTRATION

FACTORS

FACTORS

CO::CENTRATION

FOR

FOR

4i

FOR

..............

42

FOR

PARALLEL-

CONFIGURATION

B10-vii

4O

DIAGONAL-

FACTORS FOR

..........

DIAGONAL-SQUARE ....................

CONFIGURATION

HOLE

..........

PERPENDICULAR-

SQUARE

SQUARE

FOR

FACTORS

CONCENTRATION

TENSION

37

FACTORS PARALLEL-

TENSION

UNIAXIAL

DOUBLE

MINIMUM 36

HYDROSTATIC

STRESS

35

t'

WITH

....................

HOLE

HOLE

.....

PARALLEL-TRIANGULAR

UNIAXIAL TENSION HOLE CONFIGURATION

BI0-25.

K

CONFIGURATION

CONCENTRATION

TRIANGULAR B10-24.

ON

CONCENTRATION SHEAR

UNIAXIAL

ItOLES

...................

HOLE

PURE

STRESS

t'

............................

B10-2i.

STRESS

A SHEET

BASED

CONFIGURATIONS

TRIANGULAR

OF

FACTOR.

HOLE

B10-23.

A ROW

CONCENTRATION

FOR

K

INFINITELY

CONTAINING

B10-20.

B i0-22.

Page

CONCENTRATION

WIDE BI0-19.

(Continued)

..............

43

LIST OF ILLUSTRATIONS Title

Figure Bi0-27.

(Concluded)

STRESS FOR

CONCENTRATION

A TENSION

HOLE

PLATE

FACTOR, WITH

KtB ,

A BEADED 44

..................................

Bi0-28.

SQUARE

B 10-29.

WIDE-FLANGE HOLE

Page

PLATE

WITH BEAM

A CIRCULAR WITH

..................................

Bl0-viii

HOLE

........

45

A WEB 45

DEFINITION

Symbol A

OF

SYMBOLS

Definition Cross-sectional area of plate without hole - in.2 Diameter of hole; one-half length of side of rounded rectangular hole; minor diameter of ellipticalhole - in.

a

Ab

Diameter

of bead reinforcement

- in.

'B

Bead factor (Fig. B10-25)

b

Major diameter of ellipticalhole; one-half length of side of rounded

rectangular hole; one-half length of side of square

plate - in. Distance from center of hole to edge of plate; distance between holes; distance between rows in a double row of holes - in. D

Plate flexural rigidity -psi

e

Displacement

F

Ratio of bead cross-sectional area to hole cross-sectional area

h

Thick_ess of plate - in.

of hole from center line of plate - in.

Height of bead reinforcement

- in.

K

Stress concentration factor

Kt

Theoretical stress concentration factor

K

Effective or significantstress concentration factor e

KtB

Stress concentration factor (Fig..BI0-27)

DEFINITION

OF SYMBOLS

(Continued}

Definition

Symbol #

KtB

Stress

L

One-half

span

M

Bending

moment

M o _ Mn

Bending

moments

P

Axial

P

One-half

distance

q

Uniform

normal

R

Radius

of large

hole

R1,R2

Radius

of holes

(Fig.

r

Radius

of hole

- in.

s!

Distance from notch - in.

edge

w

Width

- in.

W

max

concentration

Angle

of a beam

tensile

load

of obliquity

applied

intensity

- multiple on plate

in plate

hole

of large

between

hole

or beam

patterns - psi

- in. or lb/in.

- in.

BI0-15)

of hole

BI0-15)

Stress

holes

- in.

hole

in place

Sl/R 1 (Fig.

(T

reinforced

- in.-lb/in.

between

deflection

of stagger

with

- lb

B10-16)

Angle

plate

- in.

in plates

R1/R 2 (Fig.

0

for

- in.-lb

load

of plate

Maximum

factor

to center

of small

circular

- in.

- deg

holes

to semi-infinite

in double plate

- psi

row

of holes

- deg

DEFINITION

OF

2a/w b/a

(Fig. (Fig.

P

Radius

(7

Maximum

max

_net

%' %om

(Concluded_

Definition

Symbol k

SYMBOLS

Stress

B10-8) B10-8)

of rounded localized

based

Nominal

stress

Largest

value

Smallest

corner;

value

on net

radius

stress

at edge

section

in plate

of stress of stress

of hole of hole

- in. - psi

- psi

without

hole

in a biaxial in a biaxial

- psi

stress stress

field field

- psi - psi

-..,.4

"-.--I"

TABLE

OF

CONTENTS Page

B7.0

Thin

Sh_.lls

1

.........................................

7.0.1

Thin

SheLl

7. 0. 2

Thi.n

Shell

Theori_,s

2

..........................

Theories

Basc_d

on

Linear

Elasticity

...................................

7.0.2.[

First-Order

Approximation

Theory. 7.0.2.2

Second-Order

Approximation

The(_r3

.............................

Shear

D,,forn_ation

Thc<)ri('s

Revolution

.........................

G(,nc'ral

Shells

Spherical

IiI

Circular

IV

V

7.O. I{EFE

RE?3C}£S

Nonlinear

Sc.cond-Order Th(:ories

M(tlnbrane

Sh('ll

'Fh,_ory

..............................................

Theories

h)r

Sh_.lls

Loaded

Shells

of

.........

..................

Cylindrical

for

....

(,f Revoluti,,n

Axisymmetrically

II

Shell

Shell

Specialized

[

Shell

...........................

Ill ........

11

Approximation Shells of Rc.volution...

12

of Shells

II

The(_ry

Shells

9

........

........................

l 1_

B7-iii

SECTION B7 THIN SHELLS

__.i--

Section

!

31

B7

December

Page B7.0

Thin

principal

radii

thinness

admits

state.

The

depends which

relationships small relative of

the

is

approximatethin-

analyst quations

are with

shape, made.

are

various discussed the based.

summarized

foundations

the

in to

approximation

shell The

analyses

are

the behavior of surface dimensions

approximations

of

shell

shell

governing to their

curvature various

de_ree

on the

1

S aells

Basic are

thicknesses

1968

best type

of

the

this

section.

whose to their This

three-dimensional

suited

for

a

loading,

and

there

exists

Consequently,

shells and

stress

particular

the

analysis

material

of

a variety

of

theories. thin

shell

below.

theories The

upon

to

purpose

which

be

used

is

to

commonly

in

subsequent

familiarize

the

employed

shell

e-

Section B7 31 December 1968 Page 2 B7.0. 1

Thin

Shell

Theories to

the

to The

elastic more

shells

broadly

classified

according

approximate:

{classical)

Elasticity

with solve

in

the

adequately small

of of

shallow

those

based

predict

elastic

Elasticity

shells.

on

linear

stresses

deflections;

and they

the

not

are

problems,

equations are more limited

will

basis

theories

buckling

shell are

range

forms

These

shells,

The nonlinear and therefore

inelastic

are

are

problems.

Theory theories

dealing

theories

theories

buckling

Nonlinear

membranes. difficult to

shell

exhibiting

some

deflection

Shells section.

These

for

when

Linear

common

concepts. adaptable

quired

be

they

Elasticity

most

deformations

large

of

may

Inelasticity The

also

shells which

Theory

Nonlinear

III

elasticity

thin

theories

The

II

and

of

fundamental

I

Theories

be

for often and

highly

considerably in use.

discussed

in

finite re-

this

Section 31

Page

B7.0.2

city

are

Thin

Shell

The

classical

based

Theories

on

Linear

three-dimensional

upon

1.

Based

the

following

Displacement

B7

December

1968

3

Elasticity

equations

of

Linear

Elasti-

assumptions:

gradients

ui

are

<<

small;

i.e.

,

1

xj where

u i = generalized

displacement,

x

coordinate,

O

= generalized

2.

Products

negligible compared strains and rotations functions

i = 1, j = l,

Z,

of displacement

e 1 =

e

;

3

are

themselves. small and

gradients,

_ ul

3

gradients

to the gradients are necessarily

of the displacement

2,

therefore

By this assumption, they become linear

i.e.,

2. =

0 u2

8 x 1

_'

;

1Z

=

0Ul

8 x2

Ou Z

+

8 xz

8 x1

etc. 3. holds, strain six

It

is

an assumption condition. components of

at

being

to

the

of

little

three-dimensional accurate

The has

be,.'n

the.

inw.-stigators number im the

thin

framework

thin

of

the

As

and

functions

the

proper

form

be

of

elasticity.

result, thin

The

using

equations. approximations

in

the

many

existence

theories, most

normal

complete

among is

in

performed

these

there shell

six

Sufficiently

Elasticity

controversy a

the a

warranted. can

the

stresses

Hence,

shells

Law

small that the

of

importance,

Linear

specialized

linear

prime

Hooke's with the states

shells,

or

not

considerable

field. and

linear

generally

general of

of

the

of

Gc. neraIizcd

significance.

plates

the

s(,lection

general

are is

are

plates

practical

of

s,_bject in

of

with

the

compatible general form,

point

point.

surface

of

ve. rsions

any

that

solution

analyses

simplified

that

is naturally Law, in its at

dealing

parallel

stresses

stress

strain

When

assumed

which Hooke's

of

components

planes

further

commonly

a large

developed

with-

encounter-

Section B7 31 December 1968 Page 4 ed theories will be discussed in the subsequent sections and classified according to the assumptions upon which they are based. The various linear shell theories will be classified basic categories: 1.

First-Order

2.

Second-Order

3.

Shear

4.

Specialized

5.

Membrane

In shells

are

the

(in

as

Love's

for

thereof

and

Theory

Shells

of

Revolution

Theory shells, on

the

simplified

Love's

theories. original

Theory

bending

Although

some

be

and

theories

approximations, will

theories

first-approximation

categorized

do

they

can

as

either

not be

of second-

adhere

considered a first

or

Shell

theories

approximation.

Although separately

are

zero-order

the

based

because

Linear a

thin

based

two

modifications

presented

to

Shell

Theory

Shell

Shell

Theories

of

shell to

second

Deformation

case

Shell

Approximation

general}

approximation strictly

Approximation

into five

Shear

on of

Love's their

membrane approximation,

Deformation

and

Specialized

first-approximation, particular

physical

theory

is the limiting momentless

or

they

are

classified

significance.

case state.

corresponding

Section B7 31 December 1968 Page 5 B7.0. 2. 1

First-Order

Love approximate

Approximation

was

shell

the

strain-displacement relations,

Love

introduced and

1. the

least

t Rmin

<4

on

the

The

shell of

1 (therefore,

remain

Linear

t,

during

simplify

the

known

Kirchhoff-Love

is

negligibly of

as

first

hypothesis: small

the

the

stress-strain

in

middle

comparison

surface;

i.

e.

,

z <
elements

straight

successful

assumptions, the

Rmin,

terms

a To

consequently,

following

thickness,

present

elasticity.

termed

curvature,

Theory

to

linear and,

commonly

radius

2. face

investigator

based

relationships

approximations

with

first

theory

Shell

normal

to" the

deformation,

and

unstrained their

middle

extensions

surare

negligible. 3. normal

to

Normals

the

4. small ed

the

with

5. with

Strains

ratios

to

tion,

the

NO,

Qd_,

in

of

the

are terms

the

strain

remain

the

middle

and

may

surface be

is

neglect-

linear

By

the

thickness

and

z

are

small

are

so

that

neglected

quantities

in

comparison

equations. is

consistent

elasticity.

elasticity

with The

other

the

f_rmulation

,,f

ass_mlpti_ms

ar_,

relations. condition,

negligible

assumption relative

to

(1) unity.

above, From

the this

condi-

r 2 tc'n

stress

QO,

Q 'b 0

resultants

Q0qb,

= Q 0dp

t,_ Navicr's

hypothesis plane

dllring

that

Q00, and

MO,

M 0 0

Assumption

remain

to

stress,

displacements

assumption

theory

rl

since

last

simplify

z

surface

normal of

higher-order

terms

classical

stress

components

and

second-and

first-order

used

of

other

middle

surface.

relationships.

The the

undeformed

component

stress-strain

containing

the

middle

The

compared

in

to

deformed

(2) in bending.

M0,

act

on MOO,

an

infinitesimal

element

andM0qb)reduces

(N O , to

eight,

= M 0 _of

L_Jve's

elen_cntary

first beam

approximation theory,

is i.e.,

analogous

plane

sections

Secti_n 31

B7

Dt'c_.mbcr

Page

The tion

(3),

by

strain

which

consequence,

normals

remain

normal

degree

of

error

edge

shear

such

shell

are

small.

are

fz, are ratios

by

be

the

be

to

to be shell.

onLy

in

the

of

loaded

case

geneous

a

rare

eous

the

equals

of

surface

of

stresses the

thickness,

the

simultaneous theory

is

approximations.

a

of

the

general

case

of

an

solution

In

the

particular the

general

solution

the

solution

problem.

system the

of

solution

and In

the

nonhomogeneous

membrane)

at

onLy

the

imposed

radius-to-thickness

additional

consists

of

to

shear

approximation

generaL

equations

flexibly

nonhomo-

solution

of

solution

of

unloaded of

the

the the

struct-

homogen-

equations.

approximation, brating

with the

equations.

consists

differentia[

(pure

or

equations

differential solution

The

tions

cases,

axial

shells

through

large

the

shell

and

transverse

stresses

order

on

having

applied

solution

first

and

directly

the

but

a

bending

a

The

around

however,

negligible

that

to

straight

depends

to

forces

the

Love's

differential

homogeneous

have

As

surface.

areas

forces,

normal

structure,

differential

nonhomogeneous ure

of

Local

general,

distributed

speaking,

equations

possible

to

due

remain middle

comparable In

direct

Practically

the

assump-

neglected.

naturally

surface

insignificant

only

as

in

be

ignored.

so

not

6

through

are

assumption forces;

assumption, be

Furthermore,

differential

this

assumed

fourth

stated

taken of the

plane

distributed can

simplified

rotation

may

cannot

edges,

By the

middle same

deformations and

further deformations

shearing

continuously

supported deformations.

the the

transverse

deformations, by

have

introduced

of

are shear

to

and

magnitude

loaded

equations

transverse

1968

of

stress

edges

of

Thus,

there

The

of

shelL

the

Love's

(edge

equations,to

corresponding

homogeneous

resultants

the

of

extensional

solution

which

satisfy

effect)

and

in

a first

is

a self-equili-

compatibility

c_mdi-

other

of

regions

dis-

continuity.

first which

approximation: middle

straills,

plane

and

i,lar,,,

strains

t.llest,

|w_

are (1)

(2) are

extremes.

strains

the

extensional

cc,nsidered.

two

the

extreme

inextensional are

neglected or

membrane

The

general

cases or

possible pure

compared case or

mixed

within

bending with in

case

in

flexural

which case

the

only lies

middle between

Section 31

December

Page

B7.0.2.2

Second-Order

In ratios

are

relaxed

flexure

and

gible.

By

second to

such

the

nonlinear

the

vary

in

the

strain

linearly

across

nonlinearly

of

strains thickness

and

(3)

to

the

of

the

of

the

by

longer

negli-

normal

dis-

middle

surface

be-

changes.

first-order

theory

displacements shell,

t/r

induced no

such

curvature

Thus,

thickness

of

the

whereas,

are are

resaid

strains

to are

distributed.

It is characteristic constitutive relations

and

coordinate,

The curved

(2)

are

parallel

middle-plane

on

stresses

effects

approximation. the

normal

displacements

components

functions

second

that

1968

7

Theory

restrictions

second-order

Assumptions tained

extent

normal

considering

Shell

approximation, an

corresponding

placements, come

Love's

Approximation

B7

shells

of

second contain

approximation second-order

theories terms

in

z.

theory subjected

is to

applicable predominantly

for

small flexural

deflections strains.

of

highly

that the

Section B7 31 December 1968 Page 8 B7.0. Z. 3

theories This

Shear

Deformation

Shell

In

development

of

the neglect

normals

the

effects

of

resulted

remain

the

and,

these

therefore, When

shear

strains

sions

are

no no

be

eliminated

conditions

can

are

is

the

effects

must of

vanish

for

strains

shear and,

some

can

included

a

result,

that

loads

no

in

longer

the

deformation as

neglected.

assumptions

that

be

shell

were

geometrical

shear

effects

longer

the

second-order

deformation

possible

transverse

longer Since

of It

first-and

shear

because

configurations,

the

transverse

normal.

Theories

are the

or be

shell neglected

theory. included, rotation

the expres-

determinate. the from

necessary

she.ar

forces

equilibrium at

each

are

now

equations. boundary

related

to Thus,

(Reference

deformations, five 1).

boundary

they

Section

B7

31 December Page

BT.0.

2.4

Specialized

simplified

considerably

The

ing.

Some

bending

of the

of revolution

section.

These

ever,

particular

General

in

the

components

_ and

manner,

performed shells of into

a

leading constant

single

troduced

by

E. a

H.

shells

Meissner

to

satisfy

of

certain

are the

than

in

dis-

displace-

symmetry,

obtain

equations

in

variables

terms

can

for

all

then

the

values

first

to

by

equations

for

Reissner-Meissner r 1 both

qb,

in-

generalized

the

radius of

for combine

curvature

that

transformable t and

was

and

(1915)

be

a hypergeometric

meridional

showed

also

of

variables

shells

constant

thickness

relationship

the of

From

dependent

spherical

nc:xt

revolution

Loaded

derivatives

of

and

Meissner

theory,

pair of equations which, and constant thickness,

solvable

thickness

provided a

of

for

theories,

zero.

transformation

(1913)

constant

shell

equations

a

of

vanish. Two secondtwo unknown displacement

Rather

manageable curvature

In

shells

Donnell's

all

also Mqb 0 in the

equation

(1914).

general

type

@ are

how-

Axisymmetrically

and

obtained.

such

for

deformation,

zero,

Q0' and equations

be

Reissner

of

is

in this

separately.

presented

theory,

a transformation

fourth-order

Historically,

all

can

consideration

the Reissner-Meissner

to '

to a more meridional

series. to

_

classified are

load-

approximation;

axisymmetrical

(_)

respect

however,

are

from

be and

presented

first-order

of Revolution

of

forces Qqb 0 differential

components

will be

are

case

with

the resultant order ordinary

resulting

theories

can

of geometry

theories

shallow-shell

O direction

discussed

conditions

they

shell

Shells

the

previously

on

9

of Revolution

geometry

based

Included

For

this

are

approximations,

placement ment

shell

of illustration

interest.

I.

theories

specialized

the simplified

Geckeler's and others.

for Shells

of specific

theories

for purposes section,

shell for

simplified

of shells

this

Theories

1968

(the

vary

so

"Meissner

as

con-

dition"). Rcissner-Meissner ient

and

widely

employed

axisymmetrically follow

exactly

meridional

genera[ and

forms

of

the

shells

of

revolution.

the

relations

from

conical,

i urth_rmore,

geometry

and

provided are

follow

and directly

special satisfied.

of

It

are

first constant,

toroidat from

restraints

are

the

rn_Jst

first-approximation

Love's

thickness

spheric_l, they

case

equations

loaded curvature

!indric,d,

type

shells Love's on

the

conven-

theory is

seen

that

approximation as

they

of

uniform

for

they when

are

for

cy-

thickness.

equations

in

the

variation

of

thickness

more

the

Section

B7

31 December Page

Using (Reference

equations vestigated

by

Clark

thickness

by

Meissner

type

stant,

(Reference

Naghdi

and

not

thickness,

tion for

two

are

of

loaded differential

+ A

d¢4

d3Q+

o

3

= i-3

--dq_4

csc4_b

A 1 = cot

qb(2+3

A 2

= 1-3

csc2O_

A

= 2 cot

3

-

csc2¢)

cb

and

>4

= 3(1-

yields

.2)

R2 t2

versions

of

interest,

radius

case,

r I to be

the con-

that assuming

a justifiable

shells

and

spherical equation

+ A2

2

d¢2

spherical the

namely

approximation

shells

of

con-

Reissner-Meissner Geckeler's

the

Esslinger

shells

of

constant

approximaapproximation

thickness,

is:

dZQ_b

where A

latter

however,

loaded

simplified spherical

fourth-order

d4Q_0

of constant

In the

the

in-

Shells

engineering

axisymmetrically

the

require

satisfied

shells

4).

It is shown,

axisymmetrically

for nonshallow shallow shells.

For

to be

Spherical

For stant

would

Reissner-Meissner thickness were

shells.

II.

equations

which

ellipsoidal

(Reference

satisfied.

condition

ellipsoidaI

3) and

DeSilva

condition,

is obviously

the Meissner for

a more recent version of the Z), toroidal shells of constant

1968

i0

dQ, + A1

dq_

+ A°Q_

+ 4

k 4Q

° = 0

SECTION BT. I MEMBRANEANALYSI S OF

i

J

TABLE

F

OF CONTENTS

' B7.1.0.0

Page

Membrane 7.1.1.0

Analysis

General

of Thin

Notations

7.1.1.2

Sign

7.1.1.3

Limitations

7.1.1.4

Equations

........

1 3

...........................

Conventions

12

......................

of Analysis

15

..................

16

...........................

I.

General

II.

Equilibrium

III.

Stress

Resultants

IV.

Stress,

Strain,

V.

Summary

Dome

of Revolution

.........................................

7.1.1.1

7.1.2.0

Shells

18

............................ Equations

Analysis

18 ..................

19

...................... and

Displacement

22 ..........

24

...........................

2_

...........................

30

7. 1.2.1

Spherical

Domes

......................

30

7.1.2.2

Elliptical

Domes

......................

41

7.1.2.3

Cassini

Domes

.......................

47

7.1.2.4

Conical

Domes

.......................

51

7.1.2.5

Parabolic

Domes

......................

63

7.1.2.6

Cyeloidal

Domes

......................

68

7.1.2.7

Toroidal

7.1.3.0

Cylinder

7. 1.3.1

Domes

Analysis

Circular

(Circular

Cross

.........................

Cylinder

.....................

B7.1-iii

Section)

.....

71 82 _:_

....1

Section

B7.1

31 May 1968 Page 1 B7.1.0.0

MEMBRANE In engineering

revolution

find

domes.

ANALYSIS

OF THIN

applications,

extensive

shells

application

Furthermore,

this

type

SHELLS

that

have

in various of shell

OF REVOLUTION the

kinds

offers

form

of surfaces

of containers,

a convenient

of

tanks,

selection

and

of coor-

dinates. Thin subjected

shells,

in general,

to relatively

small

thin shells,

the

the

equilibrium

the

resulting

is referred shells

that

so that

ofsuch

theory

_ith

are

physically

tion

of bending

having

equations

comply

flexurally

membrane

of bending

stiff

but

momentless

state

However,

noncritical

2.

and

is avoided moment

"membrane

supported

and

There (1)

of resisting

in the design

expressions

of stress.

when of

or minimized.

tkeory,"

theory:

incapable

deflections

Therefore,

all

membrane

and

are

the

are

shells

bending,

stato

of stress of

sufficiently

that

(2)

in

neglected,

two types

and

in a manner

If,

flexible shells

avoids

the

that introduc-

strains.

analysis

1.

moments.

state

loaded

stresses

stresses

is called

this

large

shells,

to as a "momentless"

are

to achieve.

bending

condition

shell

they

The

display

The

with for

stresses

and

to the

boundary.

Except

for

twisting

bending

solution.

narrow

along and

of the

the

vertical

solution

the

built

strip

shears

shell

is usually

are

to be practically

its edges

identical

all

analysis

conclusions

of the

boundary,

is difficult

bending

in along

ahnost

strip

problems

complete

following

are

for a narrow

strip

'_oments,

entire

deformations

This

shell

of revolution loading,

except

the

in practical

comparison

axisymmetric

shell

brane

the

a thin shell

on the

the

of stress

and

can for all

surface

no wider

and

be made: locations

adjacent than

bending

moments,

negligible;

this

identical

to the

_/ Rt

causes mere-

.

Section B7.1 31 May 1968 Page 2 3.

Boundary conditions along the supporting edge are however,

local

boundary For

cases

analysis

forces referred cation

where

and

bending

is desired,

of revolution

having

the

same

may

are

B7.3

for

frequently as

loading

decrease

negligible

cannot

symmetry

methods

shear

become

Section

to as axisymmetric of tile analysis

and

stresses

see

Shells

bending

outside

be neglected

the

analysis.

loaded

internally

shell

and

contributes in this

itself.

This

narrow

from

a more

loading

the

strip.

or externally

significantly

section.

significant;

away

or when

bending

the

presented

rapidly

very

complete

by

condition to the

is

simplifi-

Section B7.1 31 May 1968 Page 3 B7.1.1.0

GENERAL

Before we must

investigating

examine

the geometry

generated

by the

generating

curve

face

with

called and

the

rotation

For

of such

a surface.

curve

a meridian.

perpendicular

parallels.

and

of a plane

is called

planes

stresses

such

axis

surfaces,

an axis

lines

of revolution,

of revolution

in its

intersections

of rotation

the

of a shell

A surface

about The

to the

deflections

plane.

is

This

of the generated are

parallel

of curvature

sur-

circles

are

its

and

are

meridians

parallels. A convenient

nate

system

_ and

and

the

the

corresponding

Figure are

selection

axis

where

of rotation

and

- 1.)

complement

to the

is the

with

If the

coordinates

_

r) is the

parallel,

B7.1.1

spherical

0,

of surface

latitude;

angle

is the

between

angle

determining

reference

to some

surface used

coordinates

of revolution

in geography;

hence,

the

the

normal

datum

coordi-

to the

the position

surface

of a point

meridian.

is a sphere,

0 is the

we have

curvilinear

(See

these

longitude

coordinates

and

nomenclature

on

0

is the

of meridians

and

parallels. Figure

B7. 1.1

be the distance

of one

radius

of curvature. on a normal

ured

rotation

and the

of revolution one

of the

- 1 shows of its

shell

meridian

surface.

is completely curvilinear

points

In future to the

a meridian normal

its

that

of revolution. of rotation

also

need

intersection

R

by R 1 and _ .

axis

we will

between

described

coordinates,

to the

equations,

Noting

of a surface

the with

and

R 1 its

length

R2,

the

axis

R 2 sin_b,

the

surface

_hich

are

functions

R 2

R 0 will be the

radius

Let

meas-

of

of the

of curvature

R

shell

of only when

Section B7. 1 31 May 1968 Page4

Datum Meridian (Generating Curvep

Arbitrary Parallel Arbitrary on Surface

Arbitrary

Fig.

B7.1.1

- 1.

Geometry

of Surfaces

of Revolution

Point

Meridian

Section

/

31

B7.1

May

Page The the face" the cally.

thickness or

surface

of

of the

"reference

thickness Figure

"t"

revolution

shell

and

thus will

surface." of the

B7.1.1

described

will

henceforth

be

By specifying

shell

at

any

- 2 shows

an

f

point, element

Axis

referred

the the

be

form

shell

of the

that

surface

to as of the

middle

which

the

dO

Shell

shell.

of Meridians

f=

- 2.

geometri-

of the

Pair

B7.1.1

and

of Revolution

Pair

Fig.

sur-

surface

defined

surface

bisects

"middle

middle

is entirely

1968

5

Element,

Middle

Surface

of Parallels

Section

B7.1

31 May

1 96 8

Page When stresses,

radius

R

be used tion.

= R 2

as (See

be negligible in the

thickness

it becomes

principal when

the

of the

apparent

that

of curvature;

in the

special

the principal Figure

radius

radius

of curvature

- 3.)

for analyzing

of curvature

it is not normal

of circular

The

in all calculations.

meridional

is considered

the

e.g., case

B7.1.1

shell

Note

error that

"R"

to the

cylinders).

introduced

by this

R 1 is a princil)al

R

- :_.

l)rincipal

Radii

be a

surface R 2

in the parallel

do

B7.1.1

internal

cannot

shell

direction.

Fig.

the

Henceforth,

of an element

6

of Curvature

assumption

radius

(except will direcw;ill

of cur_,ature

Section

B7.1

31 May

1968

F

Page Any faces the

of the

element

the

section

work

resulting _)

may

These

stresses

element.

analytical

into

of a shell

that forces

constant

is by definition N

will and

(Figure 0

ds c5

have

follow,

are it is

moments B7.

It is

the

the

usual

internal

indicated

acting

on

l. 1 - 4),

tile

resultant

of

stresses

in Figure

convenient

to the

total

middle

acting

B7.

convert

force

7

I. 1 - 4.

these

_r stresses 0

acting

In

ds

For

tile

to this

section

on this

area.

Middle

z

the

stresses

surface. normal

on

Surface

t/2

<5 t/2 dz

0

Constant : Constant

T

Fig. Because

ds 0

of tile

(R 1R1- z)

cum'ature

,

and

the

T

137. 1.1

of the

force

- 4.

shell,

tr',msmitted

its

Shell

Stresses

xxidth

is

through

m)t

it is

simply

,_odso

ds

, but

(1

- _z

)dz

.

Section

B7.1

31 May 1968 Page 8 The

total

normal

force

for the element

ds

t is found

by integrating

from-

t tO

_

2

.

t

÷--

N

__ f2

ds

o

When

t 2

ds

related

is dropped

¢

to the

to obtain

_0ds_b(1-

t 2

from

normal

N 0g) and

both

stress.

_ll)dz

sides,

we have

In a like

manner,

Q0 " Altogether,

the

resulting

TO_ and

normal

_'0z must

force

be integrated

we have

t +

--

=

a

NO

t

z)

10

dz

_11

2 t +--

=

f2 t

N0(b

z

( 1-

0¢

z)

dz

-R1

2 t =

1Oz

t

1-

dz _'l

-- I

2 Applying

the

same

reasoning

t +

N

_f2 t 2

m

oo(,-

Z

to the

section

_

- constant,

we have

Section

B7.

1

"1 May

1.()(;8

l)ag( _ i)

t

¢" A

N

i:

"

d7

rc_)i _ 1 -

)

c_q

t 0

and

t

Q

I' 2

_

: &Z

0 [

I

dz

'

.)

Note

the

(Refer

(lilfercnt t_) Fi_ur(' If the

and bcnd

1"3(1[i ,)f curv:ttur(_

l_xisting

n re

moments

may

not

distributed

result.

From

i:-_

[ _ t

*'r,

>'(It

I -

.)

and

the

txvis/ir,_

E,_,)mt:nt

is

t M ( 'P(¢)

0

unif()rmly

t Mt)

sec'tit)llS

c'()nstant

and

c_J

construct.

1_7. 1. 1 - 2. )

stresses

it_.,k mt)mcnt

for

j •4

--'_ t

'L

r i_c',

(

i ....

Z II 1

i

Z(IZ

0

constant

across (Fiouvc'

the

thickness, B7.1.

I - 3),

l)cnding the

Section

B7.1

31 May

1968

Page In like

manner,

when _b = constant, t

Me

=

t

t

1 -

0

zdz

and

M

00

t

2

N0_b'

N_b0'

moments

acting

on the sides

element

the

equations

all

results

middle

of stresses,

Z

M0'

exist.

MOO'

in Section

- 5 shows

for

the

Also,

in th,e assumption

small

compared

Since

group

these

of thin

ten

resultants

resultant

forces that

and

the

writing

quantities

are

is "stress acting

on the

theory

transverse shell

fact

when

as a _hole

to membrane

and

the

The

be considered

stress

According moments

element.

i. 4.

these

element.

resultant

will

B7.1.

name

and MOO describe

shell

rectangular

B7.1.1

chapter,

M0'

of a rectangular

a common

of the shell

cannot

theory,

being

con-

shearing the

quantities

Z

R-'_ and three

Figure

in the

forces

Q0'

of equilibrium

surface

sidered

Q0'

is not necessarily

resultants."

zdz

1 -

00

2

NO' N_b'

shell

10

_22 are

quantities

be written determinate

for

very

N o , N0, these if the

three forces

and

N0_

unknowns; acting

to unity; N00 hence,

on the

shell

thus, Three

the

only

unknowns

equilibrium

the

problem

are

known.

becomes

are

equations statically

the can

.

Section

B7

31 December Page

In the and

last in the

Geckeler

equation

approximation

above

are

d4Q%b.

Geckeler's the beam

This high

angles

is particularly

it is considered

to be

For be

0 _) The

small

cos

solution

by


form

for

_,

the

for

the

the

equation

values

of

shells.

first

low

angles

angle

considered

is in terms

for

k and The

of _p = 90 ° , however, as

small

Reissner-Meissner

the usual

equations

large

vicinity

accurate

angles

making

as

spherical

in the

a simplification

of these

same

is valid

good

except

= 0

non-shallow

sufficiently

approximated

and

thin,

11

leaving:

k 4Q o

of the

approximation

_); that is, for

approximation

can

+ 4

equation is seen to be an elastic foundation.

on

all terms

neglected,

1968

as

¢ =20 °.

equations

assumption

in detail

by

of derivatives

that sin

Esslinger.

of Schleicher

functions.

on

Another the transformation:

This

involves

and

was

by

Het6nyi

a

slightly

introduced

more

by

O.

(Reference IfI.

two

due

to

between f_r

flat

approximate

moments

Geckeler's,

solutions

were

given

of

the

twisting and

cylindricaZ are

its

prime

simplified the

original

due

this

curvature

and

equations

are

specially

twist

Donnell.

relations on

the

by

deformations

approximation

in

Love's

to

displacement curvature

By

arbitrarily

importance:

version

strain shell

moment.

change

of

shells

become

the

relations

the

same

as

plates.

problems

(Refers:nee

homogeneous

circular cd!ze

circular

and

Donnell's their

is based

Shells

simplified

influence and

than

Complete

theories

theory,

bending

stability in

of

case

Donnell the

shells

approximation

Cylindrical

the

first-approximation

ignoring

accurate

Blumenthal.

Circular

first

for non-shaliow

5).

For loaded,

approximation

cylinders loads.

A

form under

review

6 and they line

of

such

have loads, solutions

section

on

been

widely

applicable shell

concentrated is

presented

to

stability),

used

for

loads, in

shell however,

problems and

Reference

of

arbitrary 7.

Section

B7

31 December Page

IV.

Second-Order Revolution

The erence unity

8) and in the

relations. cussed

by

second-order

Byrne

resultant

Fliigge

- Byrne

9) retain

equations type

I0) who

theory.

the

and

z/r

for

reference

standards

the

with

Fliigge

obtains

which

type

simplified

of

of FKigge

terms

with

for a general

Applications

- Byrne

Shells

theory

them

of this

shell

(Ref-

respect

to

as

are

dis-

a special

case

second

tion theory have generally been restricted to circular for which case solutions are obtained in Reference_9 latter

IZ

in the strain-displacement

equations

(Reference

thin-shell

Theories

approximation

(Reference

stress

Kempner

of a unified

Approximation

1968

equations

approxima-

cylindrical shapes, and ii. In the

are

considered

first-approximation

as

theories

are

compared.

Second-approximation directly for

from

a thick

tions

made

the

general

shell by

(Reference Vlasov

V.

is

Membrane The

generally

referred

development If,

in

neglected, shells.

the

includes study the Membrane

of

12-). given

An

by

are

as the

theories

is of

discussion

in

the of

of a

sheI1 the

shells

Vlasov equations

of

(Reference

theories of

by

Elasticity the

assump-

sections

are

13).

Shells

consideration theory

analysis

excellent

studied

"bending"

equilibrium

resulting

of

derived

Linear

Novozhilov

Theory

shell to

equations

three-dimensional

the

all

because

flexura[

presented

this

behavior

moment

so-called is

previous shells

of

expressions "membrane" in

Section

shells. are

theory B7.

of 1.

Section

B7.1

31 May

1968

Page

11

0

4, Qo

(b Q4,

Fig.

B7.1.1

- 5.

Stress

Resultants

N

4,

Section

B7.1

31 May 1968 Page 12 NOTATIONS Angle

in vertical

rotation)

plane

defining

(measured

the

location

from

axis

of a point

of

on the

meridian Angle

in horizontal

location R

of a point

Radius

of a point

dicular

to axis

al

Radius

of curvature

R2

Radial

Ro

Shell

N

0

o_ Qo N

¢

shell

measured

of meridian

between

point

when

in direction normal

Inplane

perpen-

at any

point

on the shell

6

¢o Q_

the

0

shear

of surface

normal

stresses inplane

force

per

unit

length

conskmt.

Shear

per

unit length

Transverse

shear

Meridional

inplane

acting

at () force

at 0

constant

constant per

unit

length

constant N

and

stresses

Circumferential 0

N

on the

the

thickness

Internal

, T_O

shell

of curvature

Coordinate

T0Z ' _0_

on the

controls

of rotation

Radius

Z

that

of rotation

distance

axis

plane

Shear

per

Transverse

unit

length

shear

acting

at 0

at _ - constant

constant

at

at

Section

B7.1

31 May Page

M

Bending

0

moment

per

unit

length

196 8

13

at section

0 = constant M

Twisting

moment

per

unit

leng_th at section

0 = constant M

Bending

moment

per

unit

leng/h

at section

qS:: constant M

Twisting _=

P z

, Po , p

moment

per

unit

length

at section

constant

Loading and

components

meridional

in radial,

directions,

P

Vertical

load

C

Ccnstant

of integration

_0

Angle

U

Displacement

defining

opening

circumferential,

respectively

in shell

in the

direction

vertical

of revolution of the tangent

to

the meridian u

Displacement

in the

direction

V

Displacement

in the direction

tangent

to parallel

W

Displacement

in the direction

normal

to surface

W

Displacement

in the horizontal

E

Young's

P

Poisson's

I

modulus ratio

C

Strain

component

in circumferential

E

Strain

component

in mcridional

a

Radius

0

direction

of sphere

or major

axis

direction direction length

of ellipsoid

Section

B7.1

31 May

1968

Page P

Specific

h

Height

of liquid

b

Minor

axis

n

Constant

x

Coordinate

along

generatrix

of cone

Xo

weight

Distance measured Cone

s

Arc

angle length

of liquid head

length

of ellipsoid

defining

from along

14

the

shape

length

apex

of a Cassini

of cylinder

dome

or along

surface of cone

generatrix

to upper

edgc

of cone

Section

B7

1 April

F

1971

Page

B7.

0.

3

Nonlinear The

sections It the

known and

case.

In

of

is

not

may

incorrect the

In

and

It

essential

a

is

shell

to

on

of

the

may This

as

theory of

will

be

for

problems

upon

oi

shells.

cannot d_,tail

In be in

of

nonlinear

Section

The be

the

in

explained shell

the

equa-

expressions

could

be

[inearizstates

hypothesis

and

can

requires

the

basis

the

the

stress-strain of

be

formulation

Additionally, to

nonlinear

shell

described

by

elasticity.

shell relations.

inelastic

shell

Starting

assumptions. of

the on

The principle

d_formation the basis

the

of

stability,

The 0.

stability

with

theory

the of

the

on

a

general

14) strain-

are

of be

are

potential

energy.

required

in

shells. "Large" the investigation

will

of

equations

stationary

effects

shells

based (Reference

strain-displacement by the introduction

elasticity of for

is

equilibrium of

nonlinear

form

case

theory Novozhilov

nonlinear are derived

"large" theori,s

C3.

of

this

theory.

appro×imate equations

based

ignored.

of

respect

forms

application

so-called shell

is

with

approach

Theories ing the flection

can

multiple-equilibrium

nonlinear"

development

simplifying

obtain(.'d

free

and

equilibrium.

equations

the

problem

loading

here.

relations, equilibrium

and

that elasticity

discussed

of

appropriate

shells

shell of

neglected

of

position

constraints.

leads

the

compo-

terms.

nonlinearity

mathematical

displacement relations

that

rotation

of

and

every

a unique and

development

investigation

nonlinear"

The general

the

nonlinear

of

the

order

in

a physical

theory

Law

strain

conditions

were

"geometrically

the

type not

in

of

being

in rotations

these

"physically

latter

and

in

include

basis be

introduced equilibrium

A thought

shell

previous

elasticity.

for

load

positions

linear

of

solution

determines

of

the

Hooke's

equations

identical

possible

which

on

a unique

solution

under

development,

strains

ed. of

this

the

in

theory

based

prescribed the

shell

several to

approximations

tions. for

have

inference

in

theory

with

however, a

linear are

have

shell

shell

unique,

constraints

both

linear

every

discussed

classical

equations,

reality,

always

theories

which

terms

words,

In

by

nonlinear

for

the

operations,

equilibrium

other

equilibrium

from

these

the

of

field

formulated that

omission

nents

Tt_eory

small-deflection

were

is

Shell

1

deformation considered

or of

analyz-

finite dethe stability on

equilibrium

in

greater

Section B7 31 December 1968 Page 14 REFERENCES I

Baker,

,

Verette, Inc.,

.

E.

H., R. M., 66-398,

SlD

Reissner,

E.

e

Clarke,

R.

Journal

of

Naghdi,

P.

A.,

"Spherical

Intern. (1938),

Assoc. of pp. 173-185.

L. TR

Pohle,

F.

V.,

FHigge,

W.

s

Elastic

,

(1949),

Kempner, Mechanics

Kcmpner, Mechanics,

and

Aviation,

H.

Reissner

Toroidal

29

(1950),

"Deformation Proc.

2-nd

to

Struct.

S.

146-178.

Elastic

National

333-343.

Axial

Symmetrical

Engr.,

Thin-Walled

pp. of

U.

pp.

Shells,"

Pub.,

Tubes

Bending,"

Vol.

Under

5,

Zurich

Torsion.

for on

Simplified

the

Circular

Mechanics

Associates,

Polytechnic

und

der

Dynamik

of

Cylindrical Plate

and

Institute

Schalen,"

Shell

Shells of

for

Brooklyn,

Berlin,

Germany:

(1957).

Jr.,

"Theory

Seminar at

Los

of

Small

Reports

Angeles,

in

Deformations

of

Mathematics,

published

in

a

Thin

University

Math,

N.

S.

of

Vol.

2.,

No.

103-152.

J.,

Unified

of

Polytechnic 1960). II.

L.,

(1933).

"Statik

Ralph,

pp.

Vol.

Subjected

of

F.

American

Shells.

Elastic

(1954),

and

"Solutions

Shell,"

California,

10.

479

Thin

DeSilva,

Stability

Verlog

of

Revolution,"

Bridge

Rtsh,

North

Elastic

Physics,

N.

L.,

331-247.

Mechanics

Research 1960.

Springer-

Byrne

C.

" Symposium

Industry March

,

,

Thin

Theory

Shells

NO.

Equations,

.

H.

of pp.

and

of

Hetgnyi,

Donneli,

the

Applied

Kovalevsky, Manual,"

Theory (1949),

and

Shells of

NACA,

°

the

"On

M.,

P.,

Analysis 1966.

Mathematics

Congress

.

On

Volume

Ellipsoidal

.

A.

"Shell June

,

Anniversary

.

Cappelli,

Thin-Shells

Plates

and

Shells

InStitute

of

Brooklyn,

J., Vol.

"Remarks 22, No.

Theory, for

on Donell's l (March

Symposium

Industry PIBAL

on

Research No.

Equations, 1955).

566

the

Associates, (March

"Journal

9-11,

of

Applied

Section B7 31 December 1_)68 Page 15

12.

Vlasov,

V.

Engineering,

13.

Novozhilov, Netherlands:

14.

Novozhilov, Elasticity,

Z.,

General

NASA

V.

V., P.

V.

Theory

Technical

The

Theory

Noordhoff

V.,

Rochester,

of Shells

Ltd.,

Founcatlons New

and

Translations,

York;

of

Thin

Its Applications NASA

Shells.

TTF

- 99

Graningen,

The

(1959)

of the Nonlinear Graylock

Press

Theory (1953).

in (1949).

of

Section B7. i 31 May Page B7. I. i. 2

SIGN

CONVENTIONS

Ingeneral, coordinates, following

196_ 15

the sign conventions

etc., are given

for stresses,

in the various

is a list of appropriate

figures

in Section

BT.l.

loads, 1.0.

The

figures.

Coordinates

Figure

B7.1.

Stress

Figure

137. 1.1

Stresses

Figure

137. 1. 1 - 4

Loads

Figure

B7.

Displacements

Figure

B7.1.

Resultants

displacements,

1 - 1 - 5

1. 1.4

- 4

1.4

- 4,

Figure

B7.1.1.4-

5

Section

B7.1

31 May Page B7.1.1.3

LIMITATIONS The

limitations

1.

The

OF ANALYSIS and

analysis

defined

is limited

comparison

error

monocoque

2.

t/R

-< 1/3.

Flexural

3.

The

The

6.

great

majority

< 1/50

range.

shells

are

can

zero

for

rotations,

detailed

shell

definition.

is homogeneous,

is usually

can be neglected

are

is artificial

negligible

then

the

be dictated of shells means

that

range

commonly

used

they

with

or negligible

caution

the

usual

of thin relation

belong

of 20 to 30 percent be used

and

in comparison

by the

that

in

are

in the

to the

is permissible, even

compared

when

to direct

and

strains

are

small.

(See

Section

) isotropic,

and

monocoque

Law

holds

(stress

are

within

and

is a

of revolution.

It is assumed

that

of strain)

the stresses

The

shell

if it is assumed

This

If an error

deflections,

shell 5.

The

as follows:

definition

is permissible, generally

strains

this

example,

will

of thin

A thin

relation

which

are

strain.

B7.0 4.

values For

family.

theory

t/R

shells

< t/R

the

axial

those

percent

< 1/20.

thin-shell

the

B7.1

shells.

However,

defined.

of five

1/1000

to thin

to unity.

are

of Section

where

unless

to unity

t/R

assumptions

as a shell

arbitrary

1968

16

and

boundaries

normal

to the

Hooke's

of the shell

shell

middle

must surface.

is a linear

the elastic

be free

to rotate

function

range. and

to deflect

Section

B7.1

3t May

1968

f

Page 7.

Abrupt

discontinuities

elastic constants, 8.

Linear

9.

during

Transverse

i0.

Surface

ii.

Only

normal

nonshallow

to the unstrained

deformation,

shear

stresses

not be present

in shell shape,

thickness,

middle

remain

or load distribution.

elements

straight

must

17

strains and body

and their extensions

are zero forces

throughout

are

shells are considered

surface

are negligible.

the thickness.

negligible. (See Section

B7.0.

)

Section

B7. l

31 May 1968 Page 18 B7.1.1.4

EQUATIONS I

GENERAL

The

equations

primary

solution

presented

of the

solutions)

not compatible

on bending

theory.

the

same

moments

results and

and

can

will

be almost

shell.

The

with

Because except

shears

near

be superimposed

in this

to those

the

for

the

of boundary

and

adjacent

boundaries

are

theory

bending

a strip

over

identical

effects

membrane

the

for

section

will

conditions be treated

membrane

to the boundary,

membrane

by using

the

in Section give the

by using

solution.

The

complete,

Boundary Conditions Not Compatible with

Boundary

Conditions

Compatible

Fig.

B7.

1. 1.

4-

1.

Theory

with

Boundary

Membrane

Conditions

Theory

B7.3

practically

effects

of

bending

results

theory.

Membrane

or (secondary

theories

can be calculated

obtained

membrane

theory

thus

exact

obtained

bending

f

Section

B7.1

31 May Page B7.1.1.4

1968 19

EQUATIONS

middle

II

EQUILIBRIUM

The

membrane

surface

(Figure

EQUATIONS solution is begun

of the shell element,

B7. I. i. 4 - 2a).

equations

in three

The

by considering

cut by two meridians

conditions

unknowns,

the equilibrium

adequate

and two parallels

of its equilibrium

to determine

the

of the

will furnish three

three

unknown

stress

resultants : the N

meridional

force

N o

, the

hoop

force

N O , and

the

shear

=N

o_

_o Beginning

transmitted

the

by one

( N0_b

it is

with

{"

the

edge

_N0_ a 0

equilibrium

forces of the

dO)

parallel element

Rid _

condition.

to a tangent is N0_

Only

In the

same

to the

Rid _ , and

their

difference,

way,

we

meridian,

on

the

_N0_ _) 0

have

the

the

shear

opposite

edge

Rid0d¢5

' enters

difference

in the

two

length

Rd0

f

meridional

forces.

Bearing

in mind

that

both

the

force

N

and

the

a vary The

with two

parallel force

_,

we

have

_3-'-;

forces

NoRld_b

circle

where

NoRldSd0

on they

situated

Resolving

this

NoRid

cos_b (Figure

_d0

Finally,

force

into

considering

equilibrium

(RN)d0d0 either

side

include

an

in that

plane

normal

and

The

of the angle and

hoop

dO .

tangential

lie

They,

pointing

of some

forces

element

also in the

contribute. plane

therefore,

towards

the

components

B7. I. I. 4 - 2b) enters

the component

equation

.

of a have

axis

shows

resultant

of the

shell.

that

the condition of equilibrium.

external

force,

P

RRld0d_b,

the

reads:

_N 0 q5Rld0dq O 0

._r

5

O -_(RNb)dqSd0

- NoRld_bd0

cos¢5

+ P_b

RRld0d_b

=

0

Section

B7.1

31 May

1968

Page Noting

that

all

terms

contain

d0dO

20

gives:

aN _--_---(RN b) + R, 3-_00

-

RIN 0 cos _ + P bRRI

0

=

II

Noq _

(I)

_

NCJ P Rj coso

dO dO

No

0N 0 No + _

I

dO 0

2

v+d__¢

!

(a)

Fig.

By

similar

tion of the tangent

B7. I. I. 4 - 2.

reasoning,

we

Equilibrium

(b)

of Shell Element

obtain an equation

fox-the forces

in thc direc-

to a parallel circle.

+

Section

_,

B7.1

31 May 1968 Page 21 0 u_ The surface

ON

third

of the

equation

by RR 1 and

the

equation

(2)

perpendicular

to the

middle

the

=

0

geometric

relation

R

:

R 2

sin q), we arrive

at

of equilibrium.

R2 problem

of determining

solution

load

, P0'

and

However, be considered

stresses

of equations

( 1),

under (2),

and

unsymmetrical (3)

for

given

loading values

of the

Pz it was

in this

of 0 and

(3)

z

to the

reduce

- Pz RR1

using

reduces

pendent

forces

= 0

o _ p

R1

P)

from

P0 RRI

N

_2+

The

is derived

+ N bR

Dividing

N

+ RiN0 q) cos_+

shell.

N0R isin$

third

0

+ R10-- 0

(RNq)0)

NO

stated

section. =

N

previously For

O

0.

this

that type

Therefore,

only axisymmetric of loading, the

the

equations

loading

stresses

are

would inde-

of equilibrium

to: d -_(RNq)) • N

N

+ R1

-R1N

0

R2

0cos¢_

_p z

:= -P(f)RR

1

(4)

(3)

Section

B7.1

31 May

1968

Page

B7.1.1.4

EQUATIONS

III

STRESS

RESULTANTS

By solving (4),

22

we obtain

a first N

integration.

equation order

can

0

(3)

then

N o and

for

differential

substituting

equation

be obtained

the

for N

by equation

results that

into

may

equation

be solved

by

(3).

1 R 2 sinZ0

N

If

R1R2( P z cos0

- P0

sin0)

sin0d0

+ C]

:.:

0 The above shell P

z

constant

a parallel

circle

is closed, at the

loading

the

vertex

If the

of integration

shell

(lantern

6) : 0 0.

loading

of the

2uC

represents is the

(See

an opening, loading,

Figure the

Figure

N

--

These loads may

2_R2sinZ_b

to the

effect of these

concentrated

B7.1.1.4-

angle B7.1.1.4

P

the

resultant

will degenerate

shell.

has type

"C"

of loads

apl)lied

forces. radial

If the force

3a.)

¢ 0 defines - 3b)

the results

opening in the

and

the

following:

P '

NO

2_ R1sin2_

be treated as additive loads because of the loaded opening at

the vertex of the shell. Bending stresses _villbe introduced at Cp0 but xxilltend to dissipate rapidly with increasing 0

•

f

Section B7.1 31 May 19(} Page 23 P

(a)

(b)

Z

P

P

Closed as 40 Approaches Approaches

p

P

Shell Zero Loading P

Open Shell (Lantern Loading)

Z

Fig.

B7.1.1.4

- 3.

Loading

above

_

: 0 0

Section B7.1 31 May 1968 Page 24 B7. I. I. 4 IV

EQUATIONS STRESSj

Once and

the

displacements

rically

loaded

STRAIN

stress

resultants,

are

readily

membrane

circumferential

I AND DISPLACEMENT N obtained

shell,

direction

the

are

zero

N¢

and

N 0,

by the

loading (Figure

are

usual

component B7.1.1.4

N0

obtained,

stresses,

methods.

For

and

strains,

the

symmet-

displacement

in the

- 4).

N_

No

No

4, ¢

Fig. P

P

z

=

Radial

=

Component

P0 = 0 w

B7.1.1.4

=

component

= Component Small

- 4.

Loads

of loading

of loading

acting

of loading

displacement

acting

of a point

and

Displacements

acting

on differential

in X direction(_gential

clement

to meridian)

in Y direction(_gcnaal in the

Z direction

to parallel) (normal

to surface)

Section

r'-

B7.1

31 May 1968 Page 25 u

=

v

Small

t/R

stresses

in X direction

= 0 = Displacement

Because (e.g.,

displacement

can

in Y direction(tangentiM_to

of assumptions

<< 1, all

of membrane

moments

be expressed

_0,

simply

The

strain

-

and

theory

all

shearing

'

aO=

components

can

_

be found

the

-_

(N 0

either

from

eO -

t

-- Thickness

of shell

_t

-- Poisson's

ratio

forces

of the

a

¢ and cr0 or N O

Et(No

-PN

shell

if

problem.

are The

of integTation

Rle 0 _ R2eo

displacement w

= ucot_

and

N

o

:

computed

next,

general

solution

thereby for

completing

u is

sin0

Et The

normal

0)

to be determined

from

and

:

the

.

components

C is a constant

f(O)

_0)

loading

modulus

displacement

u = sin0 where

-#No)

Young's

solution

axisymmetric

1

E

The

and

"

1

where

parallel)

N

t

e0 :

meridian)

as:

N a0

(_ugential!to

-

/_t 1

0 (R1 +pR2) w can then -

R2e 0

be found

[RI(No

-

N0(R2 from

-

PNo)

+pR1) the equation

"

support

conditions

Section

B7.1

31 May Page Because

the

displacements

are

zontal

displacements.

u=u

and

solution

w =w

for u.

u

:

interaction often

f(¢)

from

support w

calculated (See

Figure

of two or more

The general

Re

cot_

-

= R 1 e¢

-

solution

/'f--_d_ a sine

R2E 0

conditions.

= Re 0

The

and

shells

in terms

of u and

w,

B7.1.1.4

- 5. )

Note

. The displacement

0 where

process

u can be found for

the

ina

1968

26

is often vertical

that when manner

required, and

hori-

R 2 = R,

similar

to the

u is

+C

C is again

horizontal

a constant

displacement

of integration

determined

is simply

.

W

(u and w fi

= wsin4_ =-wcos_

w known) + ucos_ + usin4_

(u and w u

= wsinq5 = _cos_

Fig. B7. i. I.4 - 5. Displacements

w known) - ucos_ + fisin_

Section

F

B7.1

31 May 1968 Page 27 These form

formulas

of displacement

requirements

of the

( Figure to the user.

B7.1.1..

other,

4 - 5) can

depending

on the

be used given

to convert solution

from and

the

one

Section

B7.1

31 May 1968 Page 28 B7.1.1.4

EQUATIONS V

SUMMARY

Application conveniently general

for the two following

axisymmetric

to uniform these

of the solutions

load

pressure.

general

shell

general

section

shell

and general

B7.1.1.4

of this types

solutions

displacement

relationships.

the

in Section

equations

in this

- 1 presents

can be classified

of revolution

shell

with

of revolution

a summary

subjected

of solutions

cases.

with various

geometries,

cases:

distribution,

Table

The remainder problems

presented

section

presents

of axisymmetric for N

and

The stresses B7.1.1.4

- IV.

practical loading.

N0 are

shell Based

presented

can be calculated

of revolution on the various

with the forcedirectly

using

for

Section

B7.1

31 May 1968 Page 29

ii

rp

N

_i

0

0

e_ o

0

5

0 +

Z -o

0

I

171

z_

,'--4

o

_o

_Q

°

0 o

_1_ _1 _

0

_-

I

I

!

o z o

o

0 o Z. _

Z

9 %

0

,

w

(J o -o o ffl

0 o

j_

._.

_

_

Z _

0

,

+

o

z_

0 r_

_9

_

o

\

I

_E

N

I

li_

o

o

0

N

o

0

o z I_, _

I_N

_

0

0

'_

_

Section

B7.1

31 May 1968 Page 30 B7.1.2.0

DOME

BT. 1.2.1

SPHERICAL This

exposed The

shell),

presents

to axisymmetric of the

surface. that

and

radial

- 1) ; uniform

pressure

(Table (Table

must

be free

of the

loading load

BT. 1.2.1 B7.1.2.1

conditions over

base

- 3) ; uniform - 5).

These

and

to rotate in shell

geometry,

w = A a.

for nonshallow

closed

discontinuities

deflection

B7.1.2.1

solutions Both

shell

because

following

the

loading.

No abrupt

Note

The

lantern

DOMES

subsection

boundaries

middle

ANALYSIS

open

shells

and deflect thickness

area

(Table

pressure tables

begin

on page

be considered. to the

shell

be present.

of spherical u = w cos

dead

B7.1.2.1 (Table

will

shall

w sin ¢ and

will be considered:

shells

normal

R I = R 2 = a (radius

Therefore,

spherical

weight

_) . (Table

- 2) ; hydrostatic B7.1.2.1 31.

- 4) ; and

Section

B7.1

31

1968

Page

May 31

© o i

/ E

÷

0

0

_= ¢) °,.4

0

I

o,.d 0 q)

h

L

0 0

_z ¢. o

b_ 0 f-------'3 _D

0

_9 ¢) i 0 o

i 0 c_ o

+ Z_-

I _"

0 o o

0 ©

Q

Ii

pl

I1 b_

0 Z

I_

i_

li

Section

B7.1

31 May 1968 Page 32

© I _J

O I

0

D © I

÷

C, v$

"3

Z

d

II

rl

Z

"9

PI

o

"0 0

I I e_

-o 0 v

_

o

+

%

-_

*

,

Z_J

''_

e_ !

Z

i

Z

"0

0

Section

B7.1

31 May 1968 Page 33

o

°_

¢'L

o

_

o o o o

II

o o

I

[I

-0-

N

O9 m

% o 0 I

II

II

II

b

b

II

II

II

Section

B7.1

31 May Page

1968

34

I

¢D

m_

o

÷

0 |

;> 0

o *,,_

li

tl

II

o

II

II

IL

b_ Z

Z

od o

•.w, r_ C_l ¢D

e'_ • --, _._.=

,.Q h_

,

o

II

II

II

b_ b

II

II

II

Section 31

May

Page

35

I----"--1 !

I N

E 0

_i!

# _

L'I:I •,,,,d

0

P-,

I

_

+

_

-.2._--.2_

t

,-,-I

I:.. L_

I

I

"_

_

ml__I_ _°Iml_:

I

;

I

I

i

I

CJ

Ill fill'

P..I

III '",.="' ,-_

.,_

I

-_

',"

_

_ :-

-

-2,

'O

,

,

_ +

}

i ,.

._.,

i,.l,lill l,ltlII ,

_

,.,,

_3

N

,', ,_:l'.;!'Li_ II_.N

Ill.l,I

oil, I

u]


Ill n a.n',_ /

I

I

b

r._

!

_D

o

[.-,

I

N

I

g_

_°

,

v

-_1_

_1_ + ='1-

II

II

¢I

+

IE

II

B7.1

II

1968

Section 31

B7.1

May

Page

1968

36

r-------n

:

_i _ -_

?_j

S

5

ZI5

_=_ ,==.l

,

S

'

-_

'

-

!

iIII

b_

m G}

jl

, Ili_

I

Ill&

<

.5=

II!:I_I!B.

,

_

"-"

_L

I

,£-.

ililiJN .2.

O

b

2.. @

°"_ 4=m

-

C ,/]

".q

k,

O I

m ..2,

'

-c -2 I

'"_

i

G}

_9 c_

I

v_

0

-_

t.{ '

z

r_

_

"d. _

C

I

I

I

_i¢

r: m "ST.

"3

"

Section 31 Page

•

o

I

I

@

cD

I o ©

llllil, i.-,

ll,_

rl

c9

I

_m

4=a

b

B7.1

May 37

1968

Section

B7.1

31 May 1968 Page 38 I nL I

I

v

Z

"_

,

.=. 0

II

II

II

II

II

CD b ¢25

2; 0

q_

b

o

0

_

o

_ _

o .,.d 4_

II

_,i

4_ i

O

_

m

O

L

_m nL !

_I v

,.Q

Z N

**

._

.E

*_

,

,

,

II

I_

II

N I

!

II

II

Zl

II

b

Z

II b

_

I_.

I_

/

Section

B7.1

31

1968

May

Page

II

39

I

_D

0

N _

+

c_

• ,-_

_-_ .,-4

0

_I_ II o

m

II

II

II

II

II

o b Cl_

o

o

•_

m_ o

m

_ o ._,,i

2 _ !

I

m m •

I

-_-_

\

I

%

• r._ t'--

_

!

_._

I

+

_'_

"

_N c_

I I

_,1

I

+

I

I II

I II

_ II

b

b

,

_

II

II

II

Section

B7.1

31 May

1968

Page

_D _P

ct

_}

40

I

II

+

+

I

i

II

Z

Z

l_

l_l

b

0

*_ _

'_ ._

_

o°

_

°_ il

II

Z_

Z_

_

_

II

<> b

+

_T__

I,

_

+

11

_

il

,_

Section

I""

B7.1

31 May Page B7.1.2.2

ELLIPTICAL This

subsection

axisymmetric boundaries the

shell

present.

loadings. of the middle

elliptical surface.

1968

41

DOMES presents Only

the closed

shell Abrupt

solutions

for

elliptical

must

be free

chan_2/es

elliptical

shells to rotate

in the

shell

are and

shells

exposed

considered. deflect

thickness

The following loading conditions will be considered:

The normal

must

to

not be

uniform pressure

(Table B7. i.2.2 - i); stress resultant :tJi(l dislflacemcnt parameters

(Figs.

B7. I.2.2 - i and -2) ;dcad weight (T:tble B7. i.2.2 - 2) ;and uniform load over base area (Table B7. 1.2.2 - 3}.

to

These tables begin on page 42.

Section

B7.1

31 May

196 8

Page

_

J

2_,i

M

r_

9

42

-,

:_

2

z

5

5

_

v

O N I

O

+

¢e-

i

o

,-a._

_

II

1

h

'

2

+

N

II

,-,-t

_ r.r.l Z m

_

°9

o

"_

_:

©

¢"

•

•

+

@

o_

o

,,

u_'

=

=

_

_l_ ,.Q

,

=o]

..._.._

''

_

'

_

o

Section

B7.1

31 May 1968 Page 43

¢O

_ii. _

e_

iii:®

O

O

•

l_

d

6 I

,,

t,D

¢xl

I

!

I

I .--I

,-.--I

©

,--i

I _,]

t-

ca

4

Section

B7.1

31 May Page

It 'u

Figure

B7.1.2.2

- 2.

Displacement Parameters, Uniform Pressure

1968

44

f{ ':t

Ellipsoidal

Shells,

Section

f

31

B7.1

May

Page Table Membrane

t37.1.2.2 Stress

- 2.

Dead

Resultants,

Weight

Closed

1968

45

Loading

Ellipsoidal

Dome Dead P

Weight =

Loading

0

0 P#_= Pt

Let

K

N4;

::

- _

a_ tan_, ') + a2sin(Sl,;md)

;! _ b 2 + a'_tan2¢

+ "_

Ln(1

psin_b =

pcos¢

4 K) J b(K

1 Ln NO

:-

P [ 2mn2(p_/-/+ (b2 -f a2tan2' tan_4) ''_2

- b 2 + a z tnn_

(54) For

o deflections,

_Jlj2

N

N

t

t refer

(\ KI--_

to Section

+G (i b(K + K) +4b2dl

+ a 2 l.an_-(])

B7.1.1.4.

++ a2tan2qb tan2qb)

b 2 + a2tan 2_b,

+ _/1

+ tan20;

J

Section

B7.1

31 May 1968 Page 46 Table

B7.1.2.2

Membrane

Stress

- 3.

Uniform

Resultants, P

Load Closed

over

Base

Area

Ellipsoidal

Dome

Uniform Loading Base Area P

0

P Z

P

aZ_]l

+ tan2_

N

2

S

_]b 2 + a 2 tan2¢

_b

b

L b2 - a2tan2_ db 2 + a2tan26 _] 1 + tan2_

N a_b'

For

_0

-

deflections,

t

N '

refer

0

t

to Section

B7.1.1.4.

= 0 = psin_

P

over

= p cos2_b

cos_

Section

B7.1

31

1968

May

Page B7.1.2.3 This Cassinian

I_MES

family

of shells

curve (r2+

preserved

as

useful

2a2(r

(see

2-z

2)

the

substitution

bulkheads.

Table

=

of this

as

B7.1.2.3

The

equation

- 1)

is

of the

3a 4 shell

(zero nz

curvature

for

z,

at

where

z=

0,

n > 1 but

not

r

=a)

is

much

greater

2.

R1

n2z2) 2 +

:

2[

2a2(r

r2(a

2a[r2(a

of

subsection loading.

tile

must

surface.

No

Because

abrupt

following

+ n4z2(a

Only be

3a 4

free

2 _ r2)]

for

a closed

dome

to rotate

usefulness

Nondimensional

r2)]

solutions

discontinuities

of the limited 2.

:

3/2

1/2

n2z 2

+

presents

pressure shell

n2z 2)

2 + n2z 2) a2 + r 2

This uniform

2-

2+n2z 2) + n4z2(a 2:_,n_a_(a z - r z + n2z 2)

R2

for n

useful

property

by making

(r2+

aries

is

a meridian

z2) 2 +

The

than

CASSINI

47

and

the

Cassini

will

be

to deflect

dome

subjected

considered. normal

in the shell thickness

to the

to

The

bound-

shell

middle

shall be present.

of this shell, :Ldetailed solution is presented

plots are presented

for N

and N

0

according

to the

equations :

N

A all

5(4K

+ 3)

[5(16K

4 + 24K 3 - 7K 2 + 8K

,t(64K

5 + 144K 4 + 44K 3 - 85K 2 - 36K

_ 3)]

1

N r_ ap

where

K

Nondimensional

(4K

plots

+ 3)2]5(16K

+ 23)

4 + 24K 3 - 7K 2 -_ ,_K • 3) ] 1/2

_ are

also

provided

for

w and

u for

t = constant

and

p = 0.3

.

Section

B7.1

31 May 1968 Page 48 Table Membrane

B7.1.2.3 Stresses

- 1. Uniform and Deflections,

Pressure Closed

Loading Cassini Dome

Special

Case,

b

-

Uniform _!

b

= p Z

/ 2 2

_

___

- R1

N a_b'

¢YO-

w

--

_

N

t

__2_ o '

t

2Et

2 - # -

_

:

wcot_

- j

w

=

wsin_

- ucos_

u

=

wcos¢

+ usin$

Equations

R1

Ri(N

u

for R t and

R 2 are

- PNo) - R2(N 0 -#Ng) Et sin_b

given

in Section

_

Pressure

:P

p

-

a 2

Loading P

NO

n :2

B7.1.2.3

dO + C

.

See

Figure

B7.1.2.3

- 1 for

nondimensional

plots

of N

and

See

Figure

B7.1.2.3

- 2 for

nondimensional

plots

of w and

N _ 0 u .

o

:0

r

Section

B7.1

:_1 May

1968

Pa_e

49

1.0

0.9

0.8

0.7

0.6 N ap

0.5

No ap

0.4

0.3

0.2

O. 1

0

0. 2

0. 4

0. t;

(). R

I_/a I,'i_.

B7.1.2.:1 CassiniShells

- 1. (n

Stress 2),

Resultant Uniform

Parameters Pressure

1.0

Section B7.1 31 May 1968 Page 50 0.35

0.30

0.25

'2.0

0.20

1.5

0.15

/

\

[

-%

b] /

0. 10

,0.5

J

!

Lf

f

/

o. o5

\

r

J

I

-0.05

A -0.1o

\

) w

-o.

15

0

0.2

Fig.

0.4

B7.1.2.3

Cassini

Shells

- 2. (n

0. (;

Displacement = 2),

Uniform

0. 8

Parameters Pressure

1.0

Section I37.1 31 May 1968 Page 51 B7.1.2.4

CONICAL

This exposed

DOMES

subsection

presents

to axisymmetric

The

boundaries

shell

middle

of the surface.

the

loading. shell No

solutions Both

must

be

abrupt

for

closed

free

nonshallow

and

to rotate

()pen and

discontinuities

in the

conical

shells

will

deflect shell

shells be

normal thickness

considered. to the shall

be

present. Note

the

cb R For

= o_ -

convenience,

notations

special

are

geometry

constant,

xcos_b

R1 =

(Figure solutions

standard

for

of the

shells

1)

l)resented of

Fig.

137.1.2.4-

in terms

revolution

Meridian Straight I,ine

shell:

oo

B7.1.2.4are

conical

of x instead

as

used

in this

x/_Xx _/

1.

"\_ \ _

C(micnl

\ _-r-

Shell

Geometry

of R. chapter.

All

other

Section

B7.1

31 May Page The (Table

B7.1.2.4

hydrostatic (Table begin

following

- 1) ; uniform

pressure B7.1.2.4-

on page

loading

loading 4) ; and

53.

conditions

will

loading

over

(Table lantern

be considered: base

BT. 1.2.4 loading

area

- 3);

(Table

(Table uniform

B7.1.2.4-

dead

1968

52

weight

loading

B7.1.2.4 pressure 5) .

- 2) ; loading

These

tables

Section 31

B7.1

May

Page

1968

53

::kl_ I ::[I m

v?l ,,-.

:_I _N

,

,

0

0

_

o!o o, _:

© ,J

°j

_. ,m

ii

_

h

_01

,-*

I

"D

"z_ Z

0 L

"E

.o ¢9 2.:

Cq

2-:t

._.__T__.

._ :xl

¢n i

:::1.1v.n

c_

£'

:Ol

r]

"_ o

c_ I

0

0

0

0

o,

0

o

_1

I

0

o

0

o_

I_

0

I

,--,

rl

*D 0 0

I

I

I

II

II

II

Section 31

May

Page

!

II

_J ,.-4

"G

"G-

•..._

0

_J

if

Z

11

'_ "1

II

_r

Z b

!

©

]

_L | © 0

O,

_

_

_i _

If

lb

°1 II

-o t_

C,l

:1

b_

1968

54

¢9 II

B7.1

l]

Section 31

B7.1

May

Page

1968

55

!

Z

I o

_

_-_

_.)

_3

0

0

0

r_

r_

o

0 II

PF

il

II

II

H

0

Z

_o 0

© b

0

0 0

.,.-i

.._ %)

0

ii 0 .,.-i

0

0

a.,

o

ii q:_

ii -o _

_

",..3

! 0 0

Z

[_

_

©

r-"

o 0

_

E (D

o °,

o° ._

N

N

b_ 0

m

I

I

II

II

o

©

• --4

0

I!

tl

b

_

Z

o

t_

im

Section

B7. l

31 May

1968

Page !

56 1

_2

? t_ o

< o

_Q o O

.__

gg _'N

tl

tl

II

]1

Ir

II

r/l 0 o

Z

_o

II o

0

"_

o

II

II "O-

Z

<>

tq

!

!

r/1

_

o

:::k I

•

O

0 I

_I _,

_Io

<>

oN

© ¢q I

I

II

II

2;

2;

Z

_ II

II

-o

II

II

Section

B7.1

31 May 1968 Page 57

n

I

i

Jr

I

oo I]

II

t_.__2___

f

7

I

'-5

'_

I

I

t

v_

O9 _9

O9 D

I ?:

[/)

.,_

i _-I 7t

• -_ ;4,

t

I

2 =

,,..-4

l

1

2x

+

5 ')

'i,

:-,

5

N

r k

I

Section

B7.1

31 May 1968 Page 58

I

4

g ._

8

tl

:i

2

II

II

II

Z

_._ Q

Z

"_ b

O C,) ',_

o

_

C'_

°

_. , '

II tiJll I

I

•

Z I

,-3

,,-]

,-3

,

I

-1 o

o F

t_ , I_

° 0

_;-" _

'?

---_

_o

ZO

,

I:

I!

j

°

I_ 0

H

_u o

o Z

_ Z

_ Z

0

Section 31

B7.1

May

1968

I

Page

4-

I

_0 ..-i

'l::s

_B ©

JllI!f, ',-+ 0

°N

Ii

O

"O

Ii

ii

_, z_] -+ ,t

I

II

¢./

CI

m.

mO

Z

O b

o

2,

N._ t

•

_ I t

© _1_ U3

_

d

_

c-

•

_

,._

h

e_

tz_ _aB ,.Q

O

L

,

? _°1_ _l_

ii

ii

II

H

O

59

Section B7.1 31 May 1968 Page60

::tl_ ! !

_1_

O

0 0

_'_ 0

II

II

II

II

It

II

I

I

0

"0

o'_ *J o o(9 "0 o Q) ,-4

% 0

o °,-I

ID

I

Z_ I:1:1_r_

I

oo r_

-o o

0

!

I

II

II

2;

m

0

It

II

z It

II

-o

Section

B7.1

31

1968

May

Page

61 I--1

'% x '_

×

.,-+

_x

%

Z_

C.It)

i "_

i "-+ I__J

_

_1o_

_lo_

_-(3 0

__0

,

¢.9

++ B o

o

_3 4-1

o ¢9 >¢

0

0

0

II

II

II

o II

II

II

b_ o

,,,-i

b

II

II

,..--I

,-_

_)

II -0

N

,--t

o r..) _

_ I

Q) m

mLl_

. N2 _

t___L

o,_ c

Z

I

s °J

l 0

-0 o o

I

+ 0

o o 0

"_

o

II

II

I II

II

b -(3 -0 b

Section

B7.1

31 May Page

c

1968

62

.,.._ O

II

II

I

!

!

II

II

II

b O

2; b

0 0

o II

"_o

tl

.,_

II

.

_

•

rll

0

j-O-

e_ b_

0

I II

II

tl

b

-o Z

Z b

il

II

II

Section

B7.1

31 May

1968

Page B7.1.2.5

PARABOLIC This

exposed

middle

DOMES

subsection

surface.

No

abrupt

The (Table

following

pressure

of the geometry

parabolic

0, R I

R2

loading These

loading (Table

in the shell thickness

at the vertex

R0

R 0 where

The

to the shell shall be present.

mcridian, ¢5

the solutions 0 .

For

twice the focal distance.

will be considered: over

shells

shells _villbe considered.

of the parabolic

of curvature

loading conditions

BT. i. 2.5 - 4).

closed

discontinuities

137. i. 2.5 - I); uniform

hydrostatic (Table

shell at c_

Only

for nonshallow

be free to rotate and deflect normal

by use of the radius

the parabolic

the solutions

loading.

of the shell must

Note that because simplify

presents

to axisymmetric

boundaries

63

base area

(Table

B7. i. 2.5 - 3) ; and

tables bc_in on pa_e

_;4.

dead

_vcight loading

BT. I. 2.5 - 2);

uniform

pressure

loa(ling

Section

B7.1

31 May

1968

Page Table B7. I. 2.5 - 1. Dead Weight Loading Stress Resultants for Closed Parabolic

Membrane

P

R 0 = z ( Focal

P0 = 0,

N

No

For

z

= pcos_,

P

22_ [\sinZ¢1-cos% 3 cosZ(b )

= -

3 =

P

Distance)

N --_ t

Deflections,

\

sin2_

/

NO '

see

t

Section

B7.1.1.4

- IV .

= psin_

Domes

64

Section

B7.1

31

1968

May

Page Table

B7:

Membrane

1. 2.5

- 2.

Stress

Uniform

Resultants

Loading for

Closed

over

Base

Parabolic

Domes

P

Ro

P0

N _._

PRo 2 cos 4)

(J

ldl 0 cos 2

N

N ac_ ' (f()

For

I)cflc'ctions,

t

0

,

z (Focal

P

z

DistanccI

pcos2qb

N '

see

t

Section

B7.1.1.4

-IV.

,

P

c_

pcosq5

Area

sin4_

65

Section

B7.1

31 May

1968

Page Table Membrane

B7.1.2.5 Stress

66

- 3. Hydrostatic Pressure Loading Resultants for Closed Parabolic Domes

p = Specific

Weight

of Liquid

P0:P0:°pz N_b

-

2 cosq_

NO

--- _ pRQcos_2 N

o For

'

a0

__3_t

Deflections,

[h(

2tan2q5

+ 1) + Rotan2_b (tan2¢

N ' see

__£.0 t Section

B7.1.1.4

- IV .

=

Section 31

May

Page Table

B7.1.2.5

Membrane

Stress

- 4.

P0

N

=

-PRo 2 cos_b

-

pR__ 2

_b

NO

(1

N

+cossin2_ q_

For

t

Deflections,

=-

P

for

z

0

,

]

0 '

see

t

Section

B7.1.1.4

Pressure Closed

N

__¢_ <_b '

Uniform

Resultants

- IV

P z :P

Loading Parabolic

B7.1

Domes

67

1968

Section

B7.1

31 May Page BT. 1.2.6

CYCLOIDAL This

exposed

of the surface. The

(Table These

DOMES presents

to axisymmetric

boundaries middle

subsection

following

begin

the

loading. must

No abrupt

BT. t.2.6 tables

shell

1968

68

on page

Only

be free

to rotate

conditions

uniform 69.

for

closed

discontinuities

loading

- 1) and

solutions

loading

nonshallow

shells and

deflect

in the shell will

will

base

thickness

area

shells

be considered. normal

be considered:

over

cycloidal

dead {Table

to the shall

The shell be present.

weight 137.1.2.6

loading - 2).

Section B7.1 31 May 1968 Page Table Membrane

B7.1.2.6 Stress

- 1. Dead Weight Loading Resultants for Cycloidal Domes

P

/ PO = O,

sins

P

= psin_b,

+ cos¢,

- _cos 1

N(b

--

= __.0I}l_-cos_ _sin2ct> COS_b )

#

N (_¢ ' _0

For

3

2pR 0

_tan(b 2

N

--9t

Deflections,

-

t0

see

Section

B7.1.

1.4

- IV.

P z = pcos¢

69

Section B7.1 31 May 1968 Page70 Table B7.1.2.6 - 2. Uniform Loading over Base Area Membrane Stress Resultants for Cycloidal Domes

P

P0= 0, P

N0

=

-

8

NO

= -

16

.

N

_

'

°0 = -_t

sine5

/

sin_

]

= psin@cos@,

°s2_b

-

sin2_b

N '

t

For Deflections, see Section B7. i. I.4 - IV.

Pz = peos2_

Section

B7.1

31 May

1968

Page B7.1.2.7

TOROIDALDOMES This

subsection

axisymmetric

loading.

boundaries middle

presents Both

of the shell must

surface. Note

No

in Figure

abrupt

and

discontinuities

in the shell thickness

- i that, because (b is changed

Range

_b Definition (Section

B7.

shells will be considered.

of the geometry

(for this subsection

of the solutions.

Useful

Fig.

open

be free to rotate and deflect normal

BT. I. 2.7

the useful range

the solutions for toroidal shells exposed

closed

shell, the definition of the angle increase

71

I. 2. 7. -I.

35°_-_: _) <

90 °

B7.7. I. 2.7 - 1 only)

Toroidal

Shell Geometry

to The

to the shell shall be present. of the toroidal only) to

Section

B7.1

31 May 1968 Page 72 There bisects and

the

is also cross

the angle

a special

section.

This

¢ is defined

Fig. The shells

following

and pointed Regular

loading

Dead

Weight

of shell

in Figure

B7.1.2.7

B7.1.2.7

- 2.

conditions

Loading

Pressure

Lantern

Loading

Domes

where

is referred

the

axis

(Table Loading

(Table

Loading

Loading begin

Over on page

Base 73.

dome,

- 2.

Pointed

will

Dome

be considered

B7.1.2.7

for

regular

- 1)

B7.1.2.7

- 2)

(Table

B7.1.2.7

(Table

of revolution

to as a pointed

B7.1.2.7

- 3)

- 4)

:

Weight

tables

(Table

Loading

Uniform

Uniform These

type

shell

Shells:

Hydrostatic

Dead

of toroidal

domes. Toroidal

Pointed

type

B7.1.2.7 Area

(Table

- 5) B7.1.2.7

- 6)

toroidal

Section 31

May

Page Table

B7.1.2.7

Membrane

- 1.

Stresses,

Dead Regular

Weight Toroidal

Shells

Pz

= = P

F

ab(_,

- _0) + a2(cosq',0(b + asim)) sin_5

P

c5

cosg)

N 0

sine5 P

[(b+asinO)c°sq5-

N cr d

, cr0

t

N '

t

b(4_-00)-

1968

73

Loading

P_

N

B7.1

a(e°s
P0 =

0

Section

B7.1

31 May

1968

Page Table

B7.1.2.7

- 1 (Concluded).

Membrane For

Loading

Same

P

¢ N

_

Cross

Toroidal

Section

(b + asin¢)

sine N

N

___

__0

t

'

t

cos¢

- be

- a(1

Loading

Shells

( ¢0 = -¢1)

+ a2(l-cos_) (b + asin¢) sine

_ _..E_

0

=

Symmetrical

Regular

Weight

as Above

ba_

N

Stresses,

Dead

- cos¢)]

74

Section B7.1 31 May 1968 Page 75 Table B7.1.2.7 - 2. Hydrostatic Pressure Loading Membrane Stresses, Regular Toroidal Shells

..._____..

..-- _ X'L

_t

_,

I

b

___

p=SpecificWeight of Liquid

a_--_--__-_--_

Pc=

_l_ -_----_7--t7--

P

PO = 0

= p(h

- a cos_)

v N

:: -

(b+

asin¢)sin_b pa

ba + _-(sing)

N

0

-

_

[-bh(sin_0 L

0 cos(b 0 -

h-

ah 2 (c°s2_ a 2

+ -5- ( cos%

sing)

- since)

cosg)

+

- (b + _0)

ah (cos20

-7

0 - e°s2_)

:_ (cosa4 _0 - eos3g) )

a cosg) ) (b + asin4) )sin(b+ bh(sinc5 0 - sing)) ba

0 - cos2_b)

-

-7(sin(b0cos(b0

- sin(pcosg) - q_ + qS0)

Section

B7.1

31 May

1968

Page Table

B7. t. 2.7

- 2 {Concluded).

Membrane For

Stresses,

Symmetrical

Hydrostatic Regular

Cross

Pressure

Toroidal

Section

76

Loading

Shells

( _ 0 = _ t)

_o = -¢, or Symmetrical

Same

N

Loading

as

( b + asin pa o ) sinq5

p 0

['ah

sinO -a2(

Section

Above

_b h sin_

ah 2 + -_-sin _

- _-( a2 1 - cos3_ )1

N

Cross

. 2

ba

L'_ sm

O -

c°s_bsin2_b

-

-'z(sinqScos_ 1 -c°s3_ )]3

-_)

ba 2 (sin_cosO

+ _)

Section

B7.1

31 May

1968

Page Table

B7.1.2.7

Membrane

- 3.

Uniform

Stresses,

Regular

Pressure

77

Loading

Toroidal

Shells

P

f

f P

/

q_ or

:P

o P

= p z

r

N

N

_b

2(b

0

_

P + a sin_5)

sinq5

[2b sin_o

[(b + a sin_b)

+ a(sin2_o

2 -

+ sin2O)

(b+

I

a sin(_o) 2 ]

=0

Section B7.1 31 May 1968 Page 78 Table B7.1.2.7 - 3 (Concluded). Uniform Pressure Loading Membrane Stresses, Regular Toroidal Shells For Symmetrical Cross Section (_0 : -_1)

SameLoading as Above

pa N

2

N

pa

0

2

2b+asin(_ b + a sin(b

Section B7.1 31 May 1968 Page 79 Table B7.1.2.7 - 4. Lantern Loading Membrane Stresses, Regular Toroidal Shells

P f

\ I

P(b + a sing)) (b t a sin(p)sin(fl

N

N

P {b + a sinq)Q_ 0

a _

sin2g_

N (7

N

__9_ o

t

'

t

0

= P¢_

/

/

|

I

L

b

=

P

z

--0

Section

B7.1

31 May 1968 Page 80 Table B7.1.2.7 - 5. Dead Membrane Stresses, Pointed

Weight Loading Toroidal Dome

P

P= 0 P_b

0

psinqb , Pz=

pcos_b

f= -pal

N

-1 cOs_O-

L N

0

[

= - __

(4_ - 4_0)sin4_0 - (eos_0

N

N

_____. a¢

, g0

t

cos_b - (4) - q_0) sin_b0/ (sin_b -sinO 0) sins

o '

t

3

- cos4_)

+ (sin4_-

sin4_0)sin4_cos

oj

Section 31 Page Table

B7.

1.2.7

Membrane

- 6.

Uniform

Stresses,

Loading Pointed

over

Toroidal

P

P

==

0

0 PC

p sinc_

cos4)

Pz

= p c°s2¢

N

N 0

2

os2<5

- 2sine5

sin_5 0

s i n 2e/Lq_) J

N ___9_ t

N _ '

t

Base Dome

Area

B7.1

May

1968 81

Section

B7. l

31 May

1968

Page B7. I. 3.0

CYLINDER

Many

types

only the circular discussed

82

ANALYSIS of cylinders

cylinder

in this chapter.

can be analyzed

falls into the category

using membrane

theory.

of shells of revolution

However, being

Section 31

May

Page B7.1.3.1

CIRCULAR The

standard

circular

cylinder

B7.1.3.1

-

1968

83

CYLINDERS is

shell-of-revolution

(Figure

B7.1

a special

type

nomenclature 1) analysis

is

of surface

is

applied

to the

cylinder

geometry_

straightforward.

P Axis

If the

of revolution.

= p

u=(]

o!

I) [)

u t l/:ltlill

I) l)lil"l,]:("l'l_

}NS

IA _,,\ 1),_

',,,/

N

= N _,')

X

'\

/

l/adius

J

<)t ('ylill,:l,,:r

1¢ = I{ o -

r

1_, SIIEf,L

Fig. The Linear

following Loading

Trigonometric Dead

Weight

B7.1.3.1 loading

Loading

These

tables

B7.

Loading Loading

begin

Circular

(Table

1.3

-

B7.1.3

B7.1.3

considered

- 2) - 3)

B7.1.3 - 5)

84.

be

l)

B7.1.3

(Table

on page

Cylinder

will

(Table

Loading (Table

1.

conditions

(Table

Circumferential Axial

-

.

- 4)

l(l.K_ll,]Nr

Geometry for

circular

cylinders:

Section

B7.1

31 May 1968 Page 84 Table

B7. I. 3 - I.

Linear

P

h v

= P

z where

coefficient defining linear load

P

N

0

N N

:

P R(I+_,

--

0

v

p

ratio

tl

-_)

t

V

W

=

1f

Et

_PvRL_(1

+ )_p -

0 1 rPvR2(l Et L

+ _

- _)] P

J

(l+h

:

p

_

v

-4)

X

L

of uniform

No 0

v _

= 0

(Y

P p

P

h

Loading

load

to maximum

value

of

Section

B7.1

31

1968

May

Page Table

B7,

L

1.3

- 2,

Trigonometric

85

Loading

+

/ P0 sin

a_

)_PP0

P

z

:

-Po(sin_

+ _

p

cosfl_ )

x

where

=

coefficients

=

coefficients

P N

=

_

-

defining

shape

defining

ratio

of sin

amplitude

of sin

P¢l(sinot_

+ _

4)

P

cosfl

loading

0

N :

N

u

0

P

_:t

pcdl i, (Cosc¢

\

)t

_

_

p

sirg3

l_

V

W

1 poR2(sinez Ft

_

+

k

p

and

of maximum

maximum

0

L

cosfl_)

( ,)

cos

curves

amplitude

of cos

loading

to

Section

B7.1

31 May 1968 Page 86 Table

B7.1.3

- 3.

Dead

I-_ Wall

Weight

Thickness

Distribution

£

p

N

=

0

=

0

0 N N

_o I U

V

Et =

0 /

W

aL

= P

Loading

o(i - _)

of P

Section 31 Page Table

B7.1.3

- 4.

Circumferential

p

-:

y

N

=

N

0 0

N

p U

0

o 1

V

W

Et 0

L(1

- _)

p

0

Loading

B7.1

May 87

1968

Section

B7.1

31 May Page Table

B7.1.3

- 5.

Axial

P

t

N N

0

=

0

=

0

=

"-"-¢_+

_

u _ w

N

x

Et

#aN Et

v

=

0

C

Load

88

1968

SECTION B7.2 LOCAL LOADS ON

TABLE

OF CONTENTS Page

B7.2.0.0

Local 7.2.

t. 0

Loads Local

7.2.1.1

Loads

II III 7.2.1.2

II

..................

Convention

Limitations

3

..........................

of Analysis

5

....................

General

10

...............................

Parameters

10

............................

14

Resulting

from

Radial

IV

Stresses

Resulting

from

Overturning

V Stresses

Resulting

from

Shear

Stresses

Resulting

from

Twisting

Stresses

Resulting

from

Arbitrary

I II

Calculation Location

of Stresses and

7.2.1.4

Ellipsoidal

7.2.1.5

Nondimensional I II

7.2.1.6

List

7.2.2.1 I II HI

Shells

...... Stress

Load

Load

Loads

Moment Loading

.....

.......

20

.......

21 21

of Maximum

Stresses

. ................... Resultant

17 19

.....................

Curves

.....

22 23

.........

24 24 25

........................

on Cylindrical

15

...........

...........................

Problem

........... Moment

................................

Example Local

Magnitude

of Curves

Curves

8

.............................

Stresses

7.2.1.3

2 3

III

VI

7.2.2.0

Shells

1

..............................

Stresses I

........................

...............................

Notation. Sign

Shells

on Spherical

General I

-

on Thin

Shells

48 .................

52

General

...............................

53

Notation

...............................

53

Sign

Convention

Limitations

..........................

of Analysis

B7.2-iii

.....................

55 59

TABLE

OF CONTENTS

(Concluded) Page

7.2.2.2

Stresses

...............................

63

I

General

...............................

63

II

Stresses

Resulting

from

Radial

III

Stresses

Resulting

from

Overturning

IV

Stresses

Resulting

from

Shear

V Stresses

Resulting

from

Twisting

Stresses

Resulting

from

Arbitrary

7.2.2.3 I II 7.2.2.4

References Bibliography

Calculation Location

of Stresses and

Displacements

Magnitude

Load ............

65

Moment

Load

......

............

Moment Loading

69 ........ .......

..................... of Maximum

...........................

..........................................

67

70 71 71

Stress

.......

72 73

74

.........................................

74

B7.2-iv

_-_

Section

B7.2

15 April Page B7.2.0.0

LOCAL The

method

placements These

ON

contained

THIN

of a double

represent

Fourier

the necessary

equilibrium

for stresses The

The

equations

considerations

can be superimposed

equations

local loads have

P.

radial displacement

developed

solved

dis-

Bijlaard.

[I]

in the form

using these series

are readily

and displacements

calculated

upon

caused

the stresses

been

for determining evaluated

stresses

and

by numerical

by the methods

by other loadings

plotted and are contained

stresses.

A direct method

cal shells caused is provided The

equations

shells caused

tech-

of this section if the specified

the stresses

displacements

stresses

moment.

in cylindrical

twisting

moment.

cal shells

caused

No by

locally

programmed

A direct method

shells caused

method

is provided applied

and

shear

the stresses

the

in spheriNo

method

shells caused

by local loads.

displacements

in cylindrical

in Fortran is presented

IV for radial

for determining

by a locally applied

shear

to calculate displacements loads

vehicle

for use in determining

load or twisting moment.

of spherical

for determining

of space

by

results of this evaluation

for determining

shear

shells caused

ranges

The

in this section

by local loads have been

load and overturning

moment.

is presented

by locally applied

to calculate

in spherical

within the parametric

interest for radial load and overturning been

by P.

and

are observed.

The

have

performed

stresses

and displacements.

stresses

limitations

on analyses

the local loads and

series.

1

SHELLS

in this section for determining

in thin shells is based

analyses

niques

LOADS

1970

or twisting

moments.

load or of cylindri-

Section

B7.2

15 April Page 2 BT. 2.1.0

LOCAL This

section

and

bending

the

attachment-to-shell

obtain

loads

and

load

where

Shell

can

attachment

from

curves

values

rapidly

caused

by local

membrane

attachments

for

the

radial

stress

are

used

load

and

resultants caused

at to

are

by shearing

directly.

in Figure

loads

rigid

stresses

at points

areas

shell

parameters

of the

Shear

be calculated

shaded

through

and

The

reduce

spherical

induced

components.

The

stresses

loads

condition.

stresses

juncture.

SHELI_

of obtaining

resultants

stress moment

load

ment-to-shell region

stress

to calculate

Local

from

juncture.

moment

twisting

a method

resulting

nondimensional

used

ON SPHERICAL

presents

stresses

overturning then

LOADS

1970

are

removed

from

B7.2.1.0-1

the attachlocate

the

considered.

gid Attachment

"- 6

•

10

20

40

60

80

R

I00

200

400

600

1000

2000

/T (in./in.) m

Fig. * This section "Local Stresses

B7.2.1.0-1

Local

Loads

is adapted from the Welding in Spherical and Cylindrical

Area

of Influence

Research Council Bulletin. No. 107, Shells Due to External Loadings" [ 5].

Section

B7.2

31 December Page

B7.2.1.1

GENERAL

I

NOTATION a

-

fillet

c

-

half

d

-

distance

defined

E

-

modulus

of elasticity,

f

-

normal

meridional

-

normal

circumferential

-

shear

x

f Y f xy

Kn,

-

radius width

at attachment-to-shell of square

stress,

stress

attachment,

by Figure

stress,

psi

parameters

Applied

overturning

MT

-

applied

twisting

M. J

-

internal bending shell, in. - lb/in.

moment

N. J

-

internal

force

P

-

applied

R

-

radius

of the

r

-

radius

of the attachment-to-shell,

T

-

thickness

of the

t

thickness

of hollow

U

shell

moment, moment,

normal

for

normal

stresses

in.-lb.

in.-lb. stress

stress

concentrated

radial

shell,

resultant

resultant

per

per

unit

P

-

hollow

0

-

circumferential

T

-

hollow

4_

-

meridional

load,

lb.

in.

shell,

parameter,

applied

and

respectively

lb/in.

-

a

psi

psi

-

V

in.

stress,

concentration

shell,

in.

psi

stresses,

a

3

B7.2.1.0-1

bending M

juncture,

1966

in.

in.

attachment-to-shell,

in.

in./in.

concentrated

shear

attachment-to-shell angular

lb.

parameter, coordinate,

attachment-to-shell angular

load,

coordinate,

parameter, rad.

in./in. tad. in./in.

unit

length

lenglh

of

of

Section

B7.2

31 December Page GENERAL

(Cont'd)

NOTATION

(Cont'd)

Subscripts a

-

applied

b i

-

bending inside

(a=

I ora=

2)

j

-

internal

m n

-

mean (average normal

o

-

outside

x

-

meridional

y z

-

circumferential coordinate radial coordinate

(j = x or j = y) of outside

coordinate

1 - applied

load coordinate

2 - applied

load coordinate

and inside)

4

1966

Section

B7.2

31 December Page B7.2.1.1

5

GENERAL II

stress

SIGN

CONVENTION

Local

loads

on the

inside

(fx),

circumferential

of the

applied

are

1966

applied

outside

stress

loads

indicated

and

at an attachment-to-shell

(Ma,

in Figure

surfaces

(f), y

MT,

shear

Va,

and

of the

shell.

stress

(f

P),

and

xy

the

induce

a biaxiat

The

meridional

the

positive

),

stress

state

of

stress directions

resultants

(M.

J

and

N.) J

B7. 2.1.1-1.

P

(IIOLLOW RIGID

ATTACI_NT

ATTACI_NT SPHERICAL

•

_-_'-_

_]

Y

xy

_..¢_1..

_

SHOWN)

SIIELL

I

Ma=

M I

Va = V2

Fig. The tem

(1-2-3)

sign

of

ing

the

the

B7.2.1.1-1

geometry are

of the

indicated

induced

deflection

Stresses, shell

in Figure

stresses, of the

and

shell

tensile resulting

Stress attachment, B7.2.1.1-2. (+)or from

Resultants, and

and the

local

Loads coordinate

It is possible compressive various

to predict

(-), modes

sys-

by of

consider-

loading.

the

Section B7.2 31 December 1966 Page 6 I _r

-"

RIGID ATTACHMENT

_

a rad._

(HOLLOW ATTACHMENT

_

_j

_P_-_o

SHOWN)

SQUARE ATTACHMENT

Jo__--L_ D_

_SPHERICAL

SHELL T

2

i

Fig. Mode to the

shell

stresses

and

local

local

shell

and

are

compressive

The

stresses

load

to the

result

bending

(P)

radial causes

to the

stresses

in tensile stresses

Geometry

a positive

adjacent

similar

stresses

and Attachment

shows

attachment.

bending

bending

Shell

B7.2.1.1-3

stresses

The

(P)

induced

membrane

The

compressive

by an external

stresses

outside

transmitted

compressive

attachment.

bending

on the

load

on the

of the

shell

pressure.

inside

of the

at points

C

A. Mode

transmitted causes adjacent C

I, Figure by a rigid

membrane

and

B7.2.1.1-2

are

membrane

II,

Figure

to the

shell

compressive

and

B7.2.1.1-3

shows

by a rigid

attachment.

tensile

to the attachment. similar

to the

stresses

by an external stresses

in the

pressive

bending

Tensile

stresses induced

pressure. shell

in the

stresses

local

C on the in the

The stresses

membrane

caused

The at

membrane

a negative

at A

bending outside

shell

at

bending

induced

pressure.

are

A

moment

local

stresses

similar

stresses and

moment

overturning and

by an internal

shell

overturning

on the

A on the

a

stresses shell

at

Compressive stresses

tensile

inside,and

outside

(M)

in the

to the

cause

(Ma)

and

bending cause

at

caused

com-

C on the inside.

'

(

SPHERICAL

SHELL

3 MODE

3 I

MODE

Fig.

B7.2.1.1-3

Loading

Modes

II

Section B7.2 31 December Page B7.2.1.1

LIMITATIONS

Four and

general

shell

location,

stresses

Size

shaded

analysis

to thin area

areas

must

shells.

of Figure

be considered

attachment caused

of Attachment

The and

OF ANALYSIS

thickness,

and A

size

8

GENERAL HI

size

1966

location,

by shear with

Respect

is applicable The

for

in maximum

to Shell

attachments

on these

relative

conditions

A

!

4oo

Fig.

m

/T

stress

Size

B7. 2.1.1-4.

a

attachment

loads.

to small

limitations

shift

limitations:

(in.//in.)

B7.2.1.1-4

are

to the shown

shell

by the

Section B7.2 15 April 1970 Page 9 B

Location The

in Figure

analysis

may

change

in section C

material

certain from

Stress

should

than

any

any

part

ment

opening

shell

and

of the area

stress

of influence

perturbations.

thermal

loading,

of Shell

These

liquid-level

shown pertur-

loading,

Location the

stresses

in the

are

than

may

be higher

at the

juncture.

at The

considered:

stresses

will

be higher

shell.

This

is most

when

reinforcement

in the

is not reinforced,

not on the

shell

juncture

be carefully

instances,

they

Conditions

change.

conditions

In some

wall

to Boundary

the attachment-to-shell

conditions 1.

when

not contain

in Maximum

removed

Respect

by discontinuity,

and

Under

following

does

be caused

Shift

with

is applicable

B7.2.1.0-1

bations

points

of Attachment

attachment,

and when

in the hollow likely

very

when

attachment

the attach-

is placed

on the

thin attachments

are

used. 2.

For

slightly mum

some

removed value

analysis,

Appendix

are

Caused

An accurate shell

varying

and

membrane

here

assumes

If this

assumption

is resisted and

shear

totally stresses.

stress and

certain

resultants

that

2 should

is not available.

appears

juncture.

by dashed deviate

at points

The

from

maxi-

the

curves

lines.

from

the

limitations

of the

be consulted.

caused The

stresses

shell

stresses

by a shear

actual

around

unreasonable,

by membrane

peak

Loads

distribution

the

resultants

is determined

is indicated

by Shear

that

stress

attachment-to-shell

encountered

stress

to a spherical

presented

the

A of Reference

D Stresses

[2]

from

B7.2.1.5

conditions

shear

conditions

of these

in Section When

load

stress the

resists it can

rigid the

(Va)

distribution

shear

applied

consists

attachment.

be assumed

or by some

load

The

of

method

load

by shear

only.

that

the

load

combination

shear

of membrane

Section B7.2 15 April 1970 Page 10 BT.2. l. 2

STRESSES

I

GENERAL

Stress

resultants

nondimensional plots

shell

(M.

J

attachments

parameter

and

Additional B7.2.1.5-3

The in terms

the

ratio

stress

tration

Figure

to shell for

Fatigue

concentration

from

These

form

(T and

in a shell

curves

are

stress

re-

of the

are

used

are

used

p) are

the

for for

required

solid hollow to use

at a rigid

attachment

juncture

is: 2) . parameters

thickness

(K

(a/T).

and

IL)

are

functions

value

except

of the

in the

stress

used

concen-

following

cases:

material;

at attachment-to-shell are

of the

O

is brittle

is necessary parameters

n

The

to unity

juncture

analysis

they

can

juncture.

be determined

from

I. 2-i. value

of the

stress

nondimensional

a stress-resultant

indicated

on the

incorrect

but

at the

parameters

R >> r is equal

(b)

on the

a nondimensional

_- K b (6M./T j

Attachment-to-shell

B7.2.

l. 5.

versus

stresses

resultants

(a)

The

for

radius

stress

for

concentration

parameters

When

line

stress

of fillet

B7.2.

in section

obtained

]37.2.1.5-22.

equation

f. = K (N./W) j n j The

are

curves

attachment

through

general

of

(U)

junctures

N.). Figures B7.2.1.5-1 and B7.2.1.5-2 J Figures BT. 2.1.5-3 through B7. 2.1. 5-22

and

attachments. Figures

at attachment-to-shell

stress-resultant

of the

sultants

+

juncture.

resultant

stress does

conservative

resultant

not occur

nondimensional

at the

juncture

curves.

at the

would

When

the

attachment-to-shell

stress-resultant analysis

is indicated

assume

curves this

by a solid

maximum juncture,

by dashed maximum

lines. stress

value it is An to be

Section B7.2 31 December 1966 Page11

Kn (Axial Load) Kb (Overturning

used

-04 Radius

stress

the

convention

used

obtained

to record from

proper

the

(Figs. stresses

stresses

B7.2.1.1-1. loads,

step-by-step

geometry, procedures

.4,

.8

Ratio

Parameters

at four The when

The

1,, (a/T)

B7.2.1.2-2

attachment.

of the

.4

Thickness

Concentration

outside the

sign

-2

sheets

and

in Figure applied

.I Shell

Stress

BT. 2.1.1-2)around

determine

place

inside

_ To

calculation

to calculate

Figure

.O&

B7.2.1.2-1

Fig. The

• 02 Fillet

Moment)

and points

stress the

stress

for B7.2. (A,

loads

calculation

in paragraphs

1.2-3)

B,

calculation

applied

parameters

R>>r

and

C,

D on

sheets follow sheets

all values III-VI

can be

below.

also the

provide calculated

sign a or

Section B7.2 31 December 1966 Page 12 I

i

i

m

i

i

I

II

]

I

I

STRESS CALCULATION SHEET FOR STRESSES IN SPHERICAL SHELLS CAUSED BY LOCAL LOADS (HOLLOW ATTACHMENT) APPLIEDLOADS

SHELL GEOMETRY

p

II

T

•

V 1

=__

t

=

VZ

=__

_'n

=__

MT

= __

r rn

=

ro

M

I

MI

_________

=_

PARAJ_'_S U

m

J)

=__

= --

K n

=

=--

K b

=

•

STRESSES*

_ON_DIMENSIONAI ; TRESS

LOAD

ST

ADJUSTING

RF.._

STRESS

FACTOR

COMPONENT

RESULTANT

i

II

M

I

M

z

z

9 1:1 _ I!

TO

A.

_

p

i/1 ,J

NyTq]_T= MI

MI

=

Kn

N_

•

T

I--, Z

ta u_

_ u

"z

TOTAL

"_

_

CIRCUMFERENTIAL

STRESS

(f

Vl

_'roT

V 1

V z

_roT

V t

MT

TOTAL

a._

fx

*

SHEAR

STRESS

fy-

(fxy)

)i

fxyl

_E s_ s_ u_ m

f•ymaX

•

LI e

**

SEE

LOAD

*_

CHANGE

18

OPPOSITE

SECTION SIGN

TO

A3.

I.

OF

THE

Fig.

THAT

SHOWN

IN

FIGURE

BT.

7

i

I-

|

THEN

REVEIUSE

THE

SIGN

SHOWN.

0. RADICAL

B7.2.1.2-2

tlr(f

x

+ f

Stress

) iS

NEGATIVE.

Calculation

Sheet

(Hollow

Attachment)

Co

Section 31

B7.2

December

Page i

1966

13

ii i

STRESS CALCULATION

SHEET FOR STRESSES

IN SPHERICAL

(SOLID APPLIED LOADS

i

SHELLS CAUSED

BY LOCAL LOADS

ATTACHMENT)

SHELLGEOMETRY

PARAMETERS

P

T

U

=

VI

R m

K n

=_

V2

r °

K b

=

MT a Ml

MZ

STRESSES* NON

-DIME

NSIONAL USTING

ADJ

STRESS

STRESS

STRESS

.,OAD

FA

C TOR

COM

PONE

N T A

RESULTANT

I NxT

i

A

o

B i

Bo

Ci

D

C o

l

n o

K n l,

--5--

_

T

P "_X

Mx

_--_

_I

6KbP

6KbMx +

T

NxTy_"r

2

Kt,

=

+

*

-

÷

T 2

M 1

V.

Nx

=

¢

--MI

M

Jm

,1z glV;TT

Mx

1

vt_-.l,,

6KbM

M I

< Y

'l' 2 _'71'

NxT,j_'7"r M ,,e ba

T

1

},_ g

-

T

MZ

Kn

=

Nx

"I a _'_

T

MZ 6K

•......._,, TOTAl.

MERIDIONAL

(Ix)

STRESSES

NyT --.-.

"\'\%'>\X5

M x

T--': .q_-a7

I'

Kn

I'

t':r_

N1/ T

[ Z

6KhMy _.

7x

_

I

N_ Iflg77T

K

I

M 'J

_

T z

MI

_,, N/.

1 Z _g_7_

'F

.KbMl .r_i-n-_

Te

M I

My ¢E,:,T M_

d

NyT,et"l_r777' M Z

I.,.I 7.

M

M

TOTAL

14 ,¢

Kn M_. F2'4"[_q_

=

K n T

1' :

:

Z

=

T 2 _'_

STRESS

/r

V Z

71" r .. T

r,,i"

2_"

,1.

< M :r: u]

IO'IAL

,

/(,

m

l

,

,=%

2_

'

IF"

• ,

SI':I-:

LOAD

• '*

CilAN(;E

: IS

SECTION SiGN

:\ ,_\\\\\N"

y,,cv:, ,(%',',', ,7,

I

,,, ,,h_,,: ¸ :,

!_7_r

:

L ,"C,",'<(\,, ,1

,,, ,>>\\,<

V2

'l '

r.T

M T Z/l" r ,,'7 Z T

--

+

t

--

Cfxy}

%'IRESS

)z

, 4

Iy

,/

1

/{t

,L'

,,,, .....

%\\\,<,,',

Ify)

v

I.Z

:ilII':AII

_ , l

4_

_-_

\'_ ,X\\\ "q

=

7'[

,.

'<',,\N\\"",

-

_

CIRCUMFERENTIAL

V I

M

Ny

-

Z

,.,3 n_

/ :2

r

" ,,

)Z

1

I

d

:

4

(I

x

()I'P()SITF: AJ. ()F"

Fig.

,

.1

TO

I

_

12

THAT

y

I

StlOWN

Z

iN

E'i(;LIRE

H7

Z.

I

1

i

T}II.:',_

RI..VI'I,USK

F_IE

Sir,N

S}|l[)_'N.

1.0. rile

RADICAL

B7.2.1.2-3

IF(f

x

_ f

g

) IS

Stress

NEGATIVE,

Calculation

Sheet

(Solid

Attachment)

_

I

Section

B7.2

31 December Page B7.2.1.2

1966

14

STRESSES II

PARAMETERS The A

following

applicable

Geometric 1.

must

be evaluated:

Parameters

Shell a.

parameters

Parameters round

(U)

attachment 1

U = r0/(RmT)2 b.

square

attachment 1

U = 1. 413c/(RmT .

Attachment a.

) _

Parameters

hollow

round

(T

and

p)

attachment

T = rm/t p b.

= W/t

hollow

square

attachment

T = 1.143c/t P B

Stress 1.

= T/t

Concentration Membrane

Parameters stress-stress

concentration

parameter

(K)* n

K = 1 +(T/

5.6a)

o.65

n

2.

Bending Kb=

K rl and

Kb values

stress-stress

1 +(W/9.4a)°.

can be determined

concentration

parameters

(Kb)

80

from

Figure

B7.2

•

1.2-t

with

a/T

values

°

Section B7.2

f

31 December

1966

Page 15 B7.2. i.2 STRESSES III STRESSES A radial both

the

RESULTING load

meridional A

will

and

1.

cause

Stresses

2.

the

the

applicable

membrane

3.

Using

components

parameters

parameters

attachment

from

from

Figures

P and

stress

stress

geometric

in

as defined

T values

resultant

component

calculated

nondimensional

attachment Step

bending

II above.

the geometric

a solid

and

(fx }

Calculate

Using

LOAD

directions.

in paragraph Step

RADIAL

membrane

circumferential

Meridional Step

FROM

Figure

resultant

and

(N

B7.2.1.5-1

or for

B7.2.1.1.5-3 the

(NxT/P),

N /T

stress

in step

through

membrane

calculate

1, obtain x

T/P)

for

a hollow

B7.2.1.5-12.

nondimensional

the

membrane

stress

from:

X

N /T=

(N

X

Step

4.

Using the

Step

5.

T/P)

- (p/T2).

X

the

same

geometric figures

parameters as step

sional

stress

resultant

Using

P and

T values

resultant 6M

(M

X

/P),

2, (M

and

x

calculated obtain

the

in step

bending

1 and

nondimen-

/P).

the

calculate

bending the

nondimensional

stress

bending

stress

component

I, obtain

values

for

(Kn

Kb).

/T 2 from: x

6M

/T 2 = (M X

Step

6.

Using

the

stress Step

7.

Using the

criteria

T2).

in paragraph

concentration the

stress

determine fx =Kn

/P)'(6P/ X

stress

parameters components

concentration the (Nx/T)-

parameters

meridional + %

calculated

stress

(6Mx / T2)"

and

in steps calculated

(fx)

from

:

the

3 and in step

5 and the 6,

Section B7.2 31 December 1966 Page Proper

consideration

meridional the

stress

on the

sign

inside

will and

give

values

outside

for

surfaces

the of

shell.

Circumferential

Stresses

(f) m

B

of the

16

The steps

outlined

obtain the

the

N /T= Y

stress

above

equations

stress to calculate

stress: (N

Y

T/P).

T 2 (M Y/P).

fy = Kn(Ny/T)

+

can

in paragraph

nondimensional

following

ferential

6my/

circumferential

(P/T 2)

(6P/2

_)

DK' (6 My/T2).

be determined A and

by using

resultants the

by following

stress

(N

the

same

T/P and Y components

the

curves

seven to

M /P) and Y and circum-

Section

f

B7.2

15 April Page B7.2.1.2

1970

17

STRESSES IV STRESSES

RESULTING

An overturning components A

moment

in both

Meridional Step

1.

FROM

the

will

cause

meridional

Stresses Calculate

MOMENT

membrane

and

and

bending

circumferential

stress

directions.

(fx) the

applicable

in paragraph Step 2.

OVERTURNING

geometric

parameters

as defined

II above.

Using

the

geometric

parameters

obtain

the

membrane

calculated

nondimensional

in step

stress

1,

resultant

1

[ NxT(RmT)

Step

3.

Z/Ma]

for a solid

B7.2.1.5-2,

or for a hollow

B7.2.1.5-13

through

Using

M , R a

sional

attachment attachment

B7. 2.1.

and

T values

stress

and

the

membrane

IN

X

T(R

component

T} _/5I

m

N /T

a

Step

4.

Using the

geometric

same

figures

the

from:

x

a ]

[Ma/T2(R

parameters as

calculate

1

EN x T(R m T)Z/M the

nondimen-

],

1

N x /T=

Figures

1

resultant

membrane

from

Figure

5-22.

m

stress

from

step

m T)2].

calculated

2, obtain

the

in step

bending

1 and

nondimen-

1

Step

5.

sional

stress

Using

M a,

resultant R m and

stress

resultant

stress

component

[ M x (RmT)

T values

[Mx

and

(RmT)

6Mx/T2

_/M a]. the bending

_/Ma],

nondimcnsional

calculate

Step

6.

Using

the

stress Step

7.

6,

criteria

the the

stress

stress

determine

l

/M a]

in paragraph

concentration

Using and

= EMx(RmW)_

[6Ma/T2(RmT)'_]. I, obtain

parameters components

concentration the

bending

from:

l

6Mx/T2

the

meridional

values

( Kn and calculated

the

K b ).

in steps

parameters

calculated

stress

from:

(fx)

for

3 and

5

in step

Section

B7.2

15 April Page fx = Kn(Nx/W) Proper

consideration

meridional the B

steps

of the

stress

Stress

circumferential

seven

2)

on the

sign

inside

will give and

outside

values

for

surfaces

the of

shell.

Circumferential The

+ Kb(6Mx/T

1970

18

outlined

(fy) stress

above

can be determined

in paragraph

A and

by following by using

the

the same 1

figures

to obtain

the

and

following

equations

the

circumferential

Ny/T

nondimensioaal

stress

to calculate

the

= [Ny T( MmW)½/Ma]

2 = [M x(RmT)

fy = Kn(Nx/W)

stress

stress:

[Ma/T2(RmT)½

1

6My/W

resultants

_/M a]

+ Kb(6Mx/2¢).

] 1

[6Ma/W2(RmW)

_]

[N

y components

T(R

T)_Ma] m and

Section B7.2 31 December 1966 Page 19 STRESSES STRESSES A shear shell

load

at the

as follows A

RESULTING (Va)

FROM

will

cause

attachment-to-shell

SHEAR

LOAD

a membrane juncture.

shear The

stress

shear

stress

:

Round

Attachment V a

fxy-

roT

sin

O

for

Va=

for

V

Vi

V a

orf

xy

-

r0T

cos

O

a

-- V2

B Square Attachment fxy

Va/4eT

(at

O = 90 ° and

270 _

0

(at

O

0 ° and

180 ° )

Va/4cT

(at

O = 0 ° and

180 ° )

for f

xy

or

fxy

} for

fxy -

0

( at O

90 ° and

270 ° )

V

V

a

a

= V1

= V2

(f

) in the xy is determined

Section 3i

December

Page B7.2.

BT. 2 1966

20

i. 2 STRESSES VI

STRESSES A

Round

RESULTING

f

xy

B

moment

stress

The shear The

(fxy)

stress

shear

Square

MOMENT

in the

shell

shear

to a round

at the and

is determined

attachment

attachment-to-shell

is constant

around

will

cause

juncture. the

juncture.

as follows:

1"20T.

Attachment

A twisting

this

(M T) applied

is pure

stress

= MT/27r

complex

TWISTING

Attachment

A twisting a shear

FROM

moment

stress

loading

are

field

applied

to a square

in the shell.

available.

attachment

No acceptable

will methods

cause

a

for analyzing

Section 31

B7.2

December

Page BT. 2. I. 3 I

STRESSES CALCULATION Most

arbitrary

RESULTING

nature. Step

OF

loadings

that

Stresses i.

FROM

induce

Resolve

local

axial

applied

moment

Example

and

interface)

D for

3.

Obtain the uated inside signs

forces,

point

each

and

of the

mcridional,

is

stresses

of the

outside

necessary.

B7.2.

I. 6,

B7.2. at

applied

force

points

com-

components attachmentI. I-i. A,

arbitrary

B,

load

C and by

the

BT. 2.1.2. for

2 for

and

of the

with

in Figure

the

arbitrary

circumferential

step

and

of attachment

moments

moments

paragraph

of application

component

stresses

by

overturning

indicated

of an

and/or

directions

in paragraph the

(forces

positive

outside

are

procedure:

(See

The

are

inside

methods Step

shear

shells

following load

of centerline

Evaluate

spherical

components.

the

(intersection

2.

LOADING

arbitrary

Problem.)

ponents

on

by the

forces,

twisting

Step

loads

determined the

shell

21

STRESSES

are

into

ARBITRARY

1966

each

of the

of the

shell.

loading and

points Proper

shear A,

B,

by

combining

stresses

eval-

Cand

Donthe

consideration

of

Section BT. 2 3t December 1966 Page B7.2.1.3

STRESSES

RESULTING

II LOCATION

AND

The location arbitrary

load A

require

MAGNITUDE

and magnitude

max)

LOADING

OF MAXIMUM

STRESSES

of the maximum

a consideration

The determination fxy=

FROM ARBITRARY

22

stresses

caused

by an

of the following:

of principal

for the calculated

stresses

(fmax'

stresses

(f,

fmin

and f xy = 0 or

fy and fxy ) at a specific

point. B

The orientation B7.2.1.1-1 load

may

ferent

the

coordinate

and B7.2.1.1-2 give

values

C Whether

of the

different are

caused

or not the value

dashed

BT. 2.1.5-22.

lines

or solid

system

(1,

with respect values

for a stress lines

set

arbitrary

stresses.

These

dif-

of components.

resultant

in Figures

3) in Figures

to an applied

for principal

by a different

2,

is obtained

BT. 2.1.5-3

from

through

Section B7.2 31 December 1966 Page23 B7.2.1.4

ELLIPSOIDAL The

ellipsoidal are

equal.

the

analysis

distances

analysis

presented

with

attachment

shells For

SHELLS

attachments

is incorrect, from

the

apex.

in this at the

not located and

the

error

section apex

(B7.2.1.0)

because

at the increases

apex

can

the radii (points

be applied

of curvature

of unequal

for attachments

radii),

at greater

to

Section B7.2 31 December

NONDIMENSIONAL

STRESS

RESULTANT

Page

24

Load

(P) B7.2.

1966

CURVES

LIST OF CURVES A

Solid

Attachments

1.

Nondimensional

2.

NondimenstonaI (Ma)

B

Hollow 1.

,

Stress Stress

Resultants Resultants

for Radial for

Overturning

l..5-1

Moment

BT. 2.1.5-2

Attachments

Nondimensional

Stress

Resultants

for Radial

Load

(P)

T =

5

p=

0.25

B7.2.1.5-3

T =

5

p=

1.0

B7.2.1.5-4

T =

5

p=

2.0

B7.2.

T =

5

p=

4.0

B7.2.1.5-6

T =15

p=

1.0

B7.2.1.5-7

T =15

p=

2.0

B7.2.1.5-8

T = 15

p.=

4.0

B7.2.1.5-9

T = 15

p = 10.0

B7.2.1.5-10

T = 50

p =

4.0

B7.2.1.5-11

T = 50

p = 10.0

B7.2.1.5-12

Nondimensional

Stress

Resultants

for

Overturning

t.5-5

Moment

T =

5

p=

0.25

B7.2.1.5-13

T =

5

p=

1.0

B7.2.1.5-14

T =

5

p=

2.0

B7.2.1.5-15

T =

5

p=

4.0

B7.2.1.5-16

T =15

p=

1.0

B7.2.1.5-17

T = 15

p=

2.0

B7.2.1.5-18

T =15

p=

4.0

B7.2.1.5-19

T = 15

p=

10.0

B7.2.1.5-20

T = 50

p=

4.0

B7.2.1.5-21

T = 50

p = 10.0

B7.2.1.5-22

(Ma)

Section B7.2 3i December

1966

Page 25 B7.2. I.5 NONDIMENSIONAL

STRESS

RESULTANT

CURVES

II CURVES The following curves (Figs. B7.2. I. 5-I -- BT. 2. I. 5-22) are plots of nondimensional

stress resultants versus a shell parameter

and overturning moment

for the axial load

loadings and for various attachment parameters.

Seqtion B7.2 31 December Page

1966

26

i E

z

i m m

m

I

I

I

o0

I

i

i

.5

I

I 1.0

I

i

i

i 1.5

l

l

i

I 2.0

lllJ

5hell Parameter (U) Figure

B7.2.1.5-1

Non-Dimensional for Radial

Load

Stress (P)

Solid

Resultants Attachment

2.5

Section B7.2 31 December 1966 Page

27

oz m

1 .0

l

I

1

l .5

I

1

i

I

I

1.0

1

l

I

1

1.5

1

I

1111

2.0

2

Shell Parameter (U) Figure

B7.2.1.5-2

Non-Dimensional Overturning

Stress Moment

(Ma)

Resultants Solid

for Attachment

Section 31

B7.2

December

Page

1966

28

T=5

p.o.25

.N

i E

L

m m

1

!

I

I

I

I

.S

.0

[

1 1.O

1

I

I

1 1.5

1

1

1 J 2.0

i

l

i

1 2.5

Shell Parameter (U) Figure

B7.2.

I. 5-3

Non-Dimensional Hollow

Attachment

Stress T

Resultants = 5 and

for

p = 0.25

Radial

Load

(P)

Section

F

31

B7.2

December

Page

1966

29

T=5 p=l.0

•

A

A

• ,..=._

. .==._

Z

m

,.=4 I,,.,

m

E o Z c o r-

s._E I

eo z

tO Z

_ m

My D

p

I

F, Om

,.=. m

!

l

I

I

i

I

.0

i

! 1.0

i

I

1

I 1.5

i

i

1

! 2.0

I

i

1 2.5

Shell Parameter (U) Figure

B7.2.1.5-4

Non-Dimensional Hollow

AttachmentT

Stress

Resultants = 5andp=

for 1.0

Radial

Load

(P)

SectionB7.2 31 December 1966 Page

30

m

•

A

A

"_

e_

0.. 0,,m_

Z

e" m

t_

r_ t_ vq t,,. t/')

o Z m

Nx

e--

._o t_ e"

._e

i

e_

!

!

C: 0 Z

tO Z

-

I

t

t

.0

t

I

I

.5

t

I 1.0

I

I

I

I 1.5

l

i

1

I 2.0

I

I

I

I 2.5

Shell Parameter(U) Figure

B7.2.1.5-5

Non-Dimensional Hollow

Attachment

Stress T

Resultants = 5 and

p -_ 2.0

for

Radial

Load

(P)

SectionB7.2 31 December 1966 Page31

T=5 p=

z,.O

•

A

°_

Z v

1::

f_

m

E 0

Z t_

¢c:) °--

t-

._E !

c o Z

I

I

I

!

I

1

1 1.0

Shell Figure

B7.2.1.5-6

I

I

1

! 1.5

Parameter

Non-Dimensional Hollow Attachment

1

1

I

I 2.0

1

I

I

I 2.

(U)

Stress Resultants for T = 5 and p = 4.0

Radial

Load

(P)

Section B7.2 31 December 1966 Page 32

I

I

I

I

I

.5

1

I

I

1.0

I

I

!

1

1

1.5

I

I

1

l

I

2.0

1 2.5

Shell Parameter (U) Figure

B7.2.1.5-7

Non-Dimensional Hollow Attachment

Stress Resultants for T = 15 and p = 1.0

Radial

Load

(P)

Section B7.2 31 December 1966 Page 33

T=I5

p=2.o

"E

t_ _Ny

Z

o o _N

x

_Mx

g _My

Z

Z

m

w

i .0

I

I

I

I .5

I

t

I ].0

l

t

I

1 l.

I

I

1

1 2.0

I

I

I

l 2.5

Shell Parameter (U) Figure

BT. 2. i. 5-8 Non-Dimensional Stress Resultants for Radial Hollow AttachmentT = 15 andp = 2.0

Load

(P)

Section

B7.2

31 December Page

1966

34

!T = 15 p=4.0

"E

.N

m

E

Z

C"

i

I

.0

I

I .5

I

i

l 1.0

I

I

I

I 1.5

1 l

!

I 2.0

I

I

I

I 2.5

Shell Parameter (U) Figure

B7.2.1.5-9

Non-Dimensional Hollow Attachment

Stress Resultants for T = 15 and p = 4.0

Radial

Load

(P)

Section 31

B7.2

December

Page

1966

35

T=I5

p

•

I0.0

I

t_ w

i °_

f-

¢-

E

!

I

_-_

,l.

-

My

tO

_

Z

M.¢-._)

"_,

w

1

l

i

1 !

1

.5

.0

1

I 1.0

Shell Figure

BT. 2.1.5-10

I

I

1

I 1.5

Parameter

Non-Dimensional Hollow Attachment

l

J

i

I 2,0

!

!

! 2.5

(U)

Stress Resultants for T = 15 and p = 10.0

Radial

Load

(P)

SectionB7.2 31 December 1966 Page36

T= 50

!

Z

"

-

_

I

I

I

.0

I

I

i

.5

1

f 1.0

1

I

I 1.5

1

I

I

l

I 2.0

Mx

i

I

i

1 2,5

Shell Parameter (U) Figure

B7.2.1.5-11

Non-Dimensional Hollow

Attachment

Stress T

Resultants = 50 and

for

p = 4.0

Radial

Load

( P )

Section B7.2 31 December Page

1966

37

•

A

o_

Z

e--

¢1¢

•

L_

t_

E O

Z

,i

e" O t-

C, ._E ¢:3 I

tO Z

"Mx O

m

1

I

I

,

I

.5

.0

,

,

l ] .0

l

I

,

I 1.5

t

i

I

l

1 I 2.0

I

l 2.5

Shell Parameter (U) Figure

B7.2.1.5-12

Non-Dimensional Hollow Attachment

Stress Resultants for T = 50 and p = 10.0

Radial

Load

(P)

SectionB7.2 31 December 1966 Page38

i

i

I

.0

i

i

.5

I

I

I

I

i

I

1.0

I

l

i

1.5

l

I

I

I

I

2.O

I 2 5

Shell Parameter (U) Figure

B7.2.

L. 5-13

Non-Dimensional Stress (M) Hollow Attachment a

Resultants for Overturning T = 5 and p = 0.25

Moment

SectionB7.2 31 December 1966 Page39

f--

T=5 A

p

o_

=1.0

I-Z

C::

t'vt/')

•

h.. t/3

h...

0 Z i

C

-

cO

"-----Nx

.m t,,,'l

._o

r_

!

E I C 0

_E

Ny

_ -

_Mx

!

co Z

-

D

l

J

.0

I

l

1

l

1

I

.5

i

1

I

1.0

[

[

I

I

1.

I

I

I

I

l 2.5

2.0

Shell Parameter (t]) Figure

B7.2.1.5-14

Non-Dimensional (Ma)

Hollow

Stress AttachmentT

Resultants = 5andp=

for

Overturning 1.0

Moment

Section B7.2 31 December Page

1966

40

i

My

"_,

- 5

T

"_Mx

p=2.0

."

_ I

i

Mx

I .0

1

i

1

I

I

1

.5

I 1.0

i

I

J

I 1.5

1

I

I

I 2.0

i

1

I

! 2.5

Shell Parameter (U) Figure

B7.2.

I. 6-15 Non-Dimensional (Ma)

Hollow

Stress

Attachment

Resultants T

for Overturning

= 5 and p = 2.0

Moment

Section 31

B7.2

December

Page

1966

41

T'5 A

p -4.0

l,-Z v

e.-

.m

•

11,,..

o_

lb. 0 Z

-

My

m

t_

m

t-

tO

-

,m

to)

E

t0 Z

! tO Z

Mx

g_

I1|1

1111

111

.0

Shell Figure

B7.2.1.5-16

I

1.0

Non-Dimensional (Ma)

Hollow

1

1

1

1.5

Parameter Stress AttachmentT

1

i

I

I

I 2.5

2.0

(U) Resultants = 5andp

for

Overturning

= 4.0

Moment

Section B7.2 31 December Page

1966

42

D

I

I

i

.0

I

I

i

.5

i

i i 1 0

I

l

! 1,5

i

i

I

i 2.0

i

g I

i 2.5

Shell Parameter (U) Figure

B7.2.1.5-17

Non-Dimensional (M a)

Hollow

Stress Attachment

Resultants

for

T = 15 and

Overturning

p = 1.0

Moment

Section 31

B7.2

December

Page

1966

43

|

T -]5

•

N

_ _

Ny

--_

i

_

Nx

E

-

Mx

r-

-

My

0D

C_

-

I

I

tO

Z

7

0 w

m

J

J

i

l

i

i

1

I

l

I

l

1.0

.0

I

l

l

I

1.5

!

I

1

!

2,0

l 2

5

Shell Parameter (U) Figure

B7.2.1.5-18

Non-Dimensional (Ma)

Hollow

Stress Attachment

Resultants T

= 15 and

for Overturning p = 2.0

Moment

SectionB7.2 31 December 1966 Page44

I

,

.0

i

i

I

*

_

.5

i 1.0

I

Shell Figure

B7.2.1.5-19

Non-Dimensional (M a)

Hollow

I

!

! 1.5

Parameter Stress

Attachment

i

I

I

I

I

1

I 2.5

(U)

Resultants T

I 2.0

= 15 and

for

Overturning

p = 4.0

Moment

Section

B7.2

31 December Page

1966

45

m

l

T-15

{

• "

"%

p -lo.o

.

g_

"%

.

g:

g_

_

E

g

g

Z

__

• I

l

I

.0

l

I

I

1

.5

1 1.0

Shell Figure

B7.

2.1.

5-20

Non-Dimensional (Ma)

Hollow

1

I

1

I , 1.5

Parameter Stress

Attachment

I

1

I

1

1 2.5

(U)

Resultants T

1 _ 2.0

= 15 and

for

Overtirning

p = 10.0

Moment

Section

B7.2

31 December Page

-_

1966

46

-4.0 ii

""

T - 50

_Ny

N

$

E Z

"

m

[

i

!

[

[

!

1

I

.5

.0

1

!

I

1.0

1

1

1

1.5

I

1

2.0

2 5

Shell Parameter (U) Figure

B7. 2.1.

5-21

Non-Dimensional (M a)

Hollow

Stress Attachment

Resultants

for Overturning

T = 50 and p = 4.0

Moment

SectionB7.2 31 December 1966 Page47

T -50

"

p -io.o

_

"l

g_

g_

O

_

E

_,

-

¢-

_

o Z

&

\

"

"° O

Z

My

."

J

J I

.0

1

! .5

l

1 1 1.0

1 I

I

1 1.5

i

I

I

1 2.0

_Mx

1 1 2.5

Shell Parameter (U) Figure

B7. 2.1.

5-22

Non-Dimensional (Ma)

Hollow

Stress Attachment

Resultants T = 50 and

for

Overturning

p = i0.0

Moment

B7.2.1.6

Section B7.2 31 December 1966 Page 48 EXAMPLE

PROBLEM

A spherical

bulkhead

with

shown

in Figure

the

force

and

moments

etry

are

shown

in Figures

a welded

hollow

attachment

B7.2.1.6-1.

B7. 2.1.6-1

and

Shell

B7. 2.1.

is subjected

and attachment

to geom-

6-2.

__

i0.0 In.-Kips

f_

.566

Kips

//

Fig.

B7.2.1.6-1

Spherical

Bulkhead

• 3. 60" . -_--4.40"DIA. I

M ATTACtI_NT-

SHELL I

_

0"

INTERFACE •062

" Rad.

,,_

_

.10,.

xx \ \\\\\\\\\\\\\\\_

Fig.

B7.2.1.6-2

Welded

Hollow

Attachment

(Detail

A)

Section 31

December

Page 1. B7.2.1.6-3) face,

so

Establish on

that

the

the

a local center

coordinate

line

loading

of the

in Figure

system

attachment

(Figs. at

B7.2.1.6-1

the

is in the

B7.2

B7.2.

1.1-1

and

attachment-shell 2-3

1966

49

inter-

plane.

ATTACHMENT

i

INTERFACE

Fig. 2. B7.2.

1.6-4)

Resolve

B7.2.1.6-3

and

the

enter

results

(Figs.

B7.2.1.2-2

for

Figure

B7.2.1.6-5

shows

3. B7.2.1.1-2) are

Establish and

enter

R

= 100.0

T

=0.10

r 0

--- 2.20

r

=2.00

system

stress

appropriate

results

on

= 0.40

a

= 0. 0625

and

the

shell stress

System

components

calculation

the

t

Coordinate

appropriate

attachments

the

m

into

on the

hollow

in inches.

m

load

Local

(Figs.

stress

calculation

B7.2.1.2-3 sheet geometric

calculation

B7.2.

for for

the

solid example

properties sheet.

1.1-1 sheet

attachments). problem. (Figure

All

and

dimensions

Section B7.2 31 December Page 50 3 _ POSITIVE

1966

MT = i0.0 In.-Kips

!

LOAD DIRECTIONS

/

;

_

"

V2

=

.4

Kips II M2

Fig. 4. B7.2.1.2-II,

B7.2.1.6-4

Determine and

enter

Arbitrary

the

appropriate

results

on the

Load

System

stress

U= For

hollow

r0/(RmT)2

p a brittle

=2.20/

/t

m

calculation

=T/t

material

Kb= 5. enter

p = 0.25)

(100x0.1)2

=2.00/0.40

= 5.0

=0.10/0.40

=0.25

(weld)

at the

to paragraph

sheet.

=0.695

1 + (T/9.4

Determine results

are for

the

the

on the

obtained radial

T = 5 and p = 0.25)

for

attachment-to-shell

°.65

n

resultants

according

!

K =l+(T/5.6a)

VI and

In.-Kips

attachment: T =r

For

1.0

Components

parameters

!

=

from load the

a)°-

8°

stresses stress

and

=1+(0.1/5.6x.0625)°'66=1

calculation

from

overturning

44

=1+(0.1/9.4x.0625)°.8°=1.70 according

Figure

juncture:

to paragraph sheet.

BT. 2.1.5-3 Figure

The

(Hollow B7.2.1.5-13

moments.

B7.2.1.2-IH nondimensional

Attachment(Hollow

through stress T = 5 and

Attachment

-

Section

B7.2

:31 December Paoje

STRESS

CALCULATION

SHEET FOR STRESSES

IN

(HOLLOW APPLIEDLOADS

e v: v_

.-.4. 0 .4

MT . JO.O Ml |.I -z

|TRESS

A

CAUSED

BY LOCAL LOADS

ATTACHMENT) PAR_S

K_pJ " "

"r , _

. .tO ...40 .Ioo.

u "r p

_','_f_ ,

,_, "o

. 2. oo .Z._O

_<, . |.4,4r x b . 1.70

.

..oi, tS

,,

NSIONA1 RX.5,5

ADJUSTING

COMPONENT A t

_.ob(

K_P

..¢,,_5 .6,o . .Z5

S T R F-.$S

FACTOR

RESULTANT

..r

SHELLS

51

SHEll GEOMETRY

I.o

,ON-DIME ST

LOAD

SPHERICAL

1966

STRESSES" B o

B t

A o

C t

,

C O

D|

D O

!

-ST.@,___L -+11.1.

-.Pdl,t

-÷J.'ll.

4-$:.o

+,g.o

--PII.¢

-÷J.¢

-4,-11.11.

:-,.P._.I

-÷J.Z

i

P

_KbP -_o_

Mx

x

-f-

_.o¢_ _

MI

_:

bKbM_

:.06¥

T2

:

TzTpT;_

-----7--- :.o_7_

--'i----

"

:

°

T

M z

'_

M,,,/I_"I'

TOTAL

6K_z

.IrRIDIONAL

_

T

p

:.o6"r

,p-

_,K M

STRESSES

nyT

_

31_.13

: .O,_q

(f

x )

-6L6,

TZ _

"

T

:

Tz 77.B, " yZ8

,,-g..o

_.!,-$.o --5:o

tS.o

,,-5".o

_,,.-.._:oI

77. B

+Z8

77.0

+7.8

i;;

_*Z_

x'.

:\,:5.\x\,:

4,.O

,4.,0

II.O

-/I.8

8.7

-7.7

-_g.(,

37,4-

14

:'.

- z

'_

:.o_'t

Tz_----'r :

"

T

4-0

_4.o

,.

_:

M z

U

MZ

U TOTAL

_-

CIRCUMFERENTIAL

v=

0

vz

._O

=

T

M

tO-O SHEAR

STRESS

(fxy)

"'" z f

ix

i_

.

ly

-

.

_(ix,

[y)Z

"

lily

Z

"'"

vT

il.,_...._,,.

- ,

,z

TO

THAT

,.

.

z

IF

LOAJD

e*

SEE

eo*

CHANGE

L_ OPPOSITE

SECTION

A_.

SIGN

-3'._

"3._

'a-_

-,_.a

-_._

*3.:S

.3._

-3.9

I-3.9

3.S

s._

-2.7

-Z.7

J._

_._

-_.Sle/.4

3Z.$

-3_o

za;.2

-ZZ'.f-77.=_

9/.(,

-04.o

3_.o

10.5

-//.3

_._

-7.o

-/6._1.

37.Z

Z4..Z

IO.9

/0.9

z_

B.o

Jo.5

Z7.£]

R_VI_:F(S_

THE

ZT.Z

t •

_. -&3

_

mLn

Zl _

3_,$

T

ZX"Z'T"3,0'1' _:3.

TOTAL

_1

-/_L.3

,,,....

< _a u_

,_l_

STRESS

¢)t

Figure

I

SHOWN

IN

FIGURF-

B7

Z

1

I - I THEN

SIGN

SHOWN.

0.

THE

RADICAL

B7.2.1.6-5

IF(l_

* ly)

LS NEGATIVE.

Stress

Calculation

Sheet

(Example

Problem)

Section

B7.2

31 December Page B7.2.2.0 This bending

LOCAL

LOADS

section

presents

stresses induced

through

Shell

geometry

and

resultants

loads and

(P and twisting

rigid

Ma}.

are

used

caused

be calculated

directly

not calculated

for

from

normal

program

stresses

membrane

and

for

shearing

loads

shells. bending

radial-type

by shearing without

and

arbitrary

or unpressurized

to obtain

a computer

shear

(M T) can

Deflections

are

shell

resulting

on pressurized

from

Membrane

cylindrical

juncture

conditions

and deflections

SHELLS

to obtain

attachments

loading

moments

program.

a method

at an attachment-to-shell

loads

stress

ON CYLINDRICAL

1966

52

loads

the and

(V a)

computer

twisting

moments. Local to-shell

load

stresses

juncture.

reduce

Boundaries

loads

can

be determined

gram

by investigation

rapidly

of that

for

those

of the

region

load

stresses

at points of the

cases

removed shell

calculated

and

deflections

caused

from

the

influenced with

at points

attachment-

by the

the computer

local pro-

removed

from

pressure

(pressure

the attachment. The

additional

coupling)

is taken

local

stress

the

load shell

results lated

by the

into

stiffness

of the

shell

account

by the

computer

resultants internal

and pressure

and must

be superimposed

by the method

contained

deflections. are

in this

program The

not included

upon

the local

section.

by internal

stress

for

determination

resultants

in the computer load

stress

of

induced program

resultants

calcu-

in

Section B7.2 15 April 1970 Page 53 B7.2.2.1

puter

GENERAL

I

NOTATION

The

notations

program.

Computer

The

presented

in this

computer

pro_,q'am

Utilization a

b

section

fillet

radius

half

diameter

c2

circumferential

E

mo_lulus

f

normal

longitudinal

normal

circumferential

xy

Kn , K b -

length

shear

of square half

in the

com-

Astronautics

bending

applied

twisting

Mj

inte.rnal of shell,

shell, uniform

p

radial

q

internal

in.

attachment,

in.

attachment,

in.

psi

stress,

psi

parameters

for

normal

stresses

respectively

of cylinder

MT

internal

L/2) pad,

in.

of rectangular

stress,

stresses,

owerturning

number direetiml

(bload

psi

concentration

length

of attachment of elliptical

of rectangular

length

or longitudinal

in.

psi

applied

p

length

of elasticity,

M a

n

pad,

attachment,

half

stress,

stress and

L

defined

_) the

juncture

load

x-coordinate distance to center or circumferential half diameter

longitudinal

f

are

of elliptical

Cl

Y

variables

at attachment-to-shell

half

f

not applicable

Handbook.

C

x

are

moment, moment,

in. -lb.

bending moment in.-lb/in. of equally

normal

stress

spaced

force

in. -lb.

resultant

attaehments

stress

per

in the

resultant

per

radial

load,

unit

circumferential

unit length

lb/in. load load

intz, nsity,

or total

pressure,

psi

distributed psi

length

lb.

of

Section

B7.2

15 April Page B7.2.2.1

GENERAL I

1970

54

(Concluded)

NOTATION

(Concluded)

r

-

radius

of circular

attachment,

R

-

radius

of cylindrical

s

-

circumferential

T

-

thickness

u

-

longitudinal

v

-

circumferential

Va

-

applied concentrated load, lb.

shear

w

-

radial

in.

x

-

longitudinal

y

-

circumferential

z

-

radial

O

-

polar

v

-

Poisson's

-

circumferential

arc

in.

shell,

in.

length;

in.

of cylindrical

shell,

displacement,

in.

in.

displacement,

displacement, coordinate,

in. load

distributed

shear

in.

coordinate,

coordinate,

or total

in.

in.

coordinate ratio cylindrical

coordinate

Subscripts a

-

applied

b i

-

bending inside

(a=

lora=2)

j m n -

internal (j = x or j = y) mean (average of outside normal

o

-

outside

x

-

longitudinal

y z-

circumferential radial

1 -

longitudinally direction

2

circumferentially tial direction

-

directed

and

applied

directed

inside)

load

applied

vector load

or vector

longitudinal or

circumferen-

Section

B7.2

31 December Page B7.2.2.1

SIGN

loads

on

inside

the

circumferential

of

applied

Nj),

and

cated

CONVENTION

Local

(fx), the

55

GENERAL II

stress

t966

the

applied and

(Ma"",

positive B7.

an

outside

stress

loads

in Fixture

at

attachment-to-shell surfaces

(fy), M T,

directions

of

shear P,

q,

of

the

induce

the

shell.

stress and

(fxy),

Va),

the

a biaxial

The

longitudinal

the

positive

directions

resultants

(Mj

stress

displacements

state

(u,

v,

and

w)

of

stress

are

and

indi-

2. 2.1-1.

p

Vl

I

V2

l

M25

I

v

\ £x

Stresses,

Fig.

_:"The

applied

overturning

(circumferentially)

Stress

moment directed

(longitudinal)

overturning

(longitudinal)

direction.

vector moment

Hesultants,

M 1 (M 2) is but since

is

Loads,

and

represented defined

its

\

effect

by

as

an

is

in the

Displacements

a longitudinally

applied

circumferential

circumferential

Section

B7.2

31 December Page The at the

geometry

attachment,

Figure

and

of the

shell

and

the

coordinate

attachment, system

the

of the

local

shell

1966

56

coordinate are

system

indicated

in

BT. 2.2.1-2.

b "

L/2

c

/

I I

C

i \ \ x

Fig.

B7.2.2.

It is possible compressive various

(-),

loading Mode

(P)

1-2

to predict

Shell the

by considering

and

sign

N

Rad

Attachment

of the

Geometry

induced

the deflections

stress,

tensile

shell

resulting

of the

(+) or from

modes. I (radial

transmitted

pressive

a

membrane

load),

to the

shell

stresses

Figure by a rigid and

local

B7. 2. 2.1-3,

shows

attachment.

The

bending

stresses

a positive load

(P)

adjacent

radial causes to the

load com-

Section

f

B7.2

31 December Page attachment.

The

compressive

induced

by an

external

bending

stresses

on

the

outside

of the

Modes Figure

membrane

pressure. the

shell

The

inside

of the

at points

BT. 2.2. I-3,

show

local shell

A,

II (circumferential

stresses

B,

negative

C and

The

pressive

stresses

to the attachment. B or C,

similar

membrane caused

Tensile

to the stresses

stresses

are induced

by an external

bending

stresses

inside,

and cause

stresses

The

inside and at D or A

(Ma)

in tensile stresses

and local bending are

cause

stresses

induced

on

local bending

stresses

,

com-

adjacent

Compressive

similar

stresses

to

in the shell at

by an internal pressure.

bending

stresses

(M a) transmitted

moments

in the shell at B or C on the outside compressive

result bending

moments

in the shell at D or A,

pressure.

to the

and III (longitudinal moment)

stresses

caused

57

D.

overturning

membrane

similar

compressive

overturning

the shell by rigid attachments. and tensile membrane

bending and

moment)

are

t966

to the stresses

cause

and at D or A

tensile on the

in the shell at B or C on the

on the outside.

p M2*

,_

MODE

RADIAL

_

I

MODE

LOAD

applied overturning

(circumferentially)

II

CIRCUMFERENTIAL

Fig.

':-" The

MI*

B7.2.2.

moment

directed

(longitudinal)

overturning

(longitudinal)

direction.

MOMENT

1-3

M 1 (M2)

vector

moment

MODE

Loading

LONGITUDINAL

MOMENT

Modes

is represented

but is defined

III

by a longitudinally

as an applied

circumferential

since its effect is in the circumferential

Section

B7.2

31 December Page The by positive "Stress used

signs applied

Calculation as calculation

of the

stresses

loads Sheet". sheets.

for

rigid

induced

in the

attachments

The figure

or parts

shell are

adjacent shown

thereof

to the

in Figure

1966

58 attachment B7.2.2.2-1

can be reproduced

and

Section B7.2 15 April 1970 Page 59 B7.2.2. 1

GENERAL

III

LIMITATIONS

Considerable of this the

section

OF ANALYSIS

judgment

mu_t

be used

and in the establishment

in the

of the

interpretation

geometry

of the

and loadings

results

used

in

analysis. Six general

shell

size,

caused

attachment

by shear A

size shaded

and

areas

loads,

must location,

of Attachment

The

analysis

area

shells.

of Figure

shift

geometry

Size

to thin

be considered

B7.2.2.

limitations: stress

attachment location,

to Shell

to small

limitations

relative

conditions

arc

10

20

1-4.

40

6@

80

=

100

N/T Fig.

B7.2.2.1-4

200

400

to the shown

I

|

stresses

Size

attachments

on these

rm

•

and

loading.

Respect

is applicable The

in maximum

and

with

for

600

shell

by the

Section B7.2 15 April 1970 Page 60 B

Location The

caused

change can

analysis

by other

turbations

of Attachment

from

the

Shift

The

following 1.

from

the

by the

local

perturbations loads.

loading,

area

These

liquid

influenced

stresses

when

2.

For

level

by the

perloading,

local

and deflections

thin load

slightly

loading

at points

load

shells.

cases

The

t-ion of the

that

stresses

shell

and

are

used.

the

cause

cause

be higher

at

at the juncture.

than

in the

not on the

stress

resultants

peaking

peaking

for can

deflections

shell.

are

local

peak

at

juncture. in most

loads

cases

the

on spherical

be evaluated

at points

when

attachment,

attachment-to-shell

this

of the peaking and

may

is not reinforced,

certain

from

than

attachment

attachment

conditions

that

extent

the

attachments

conditions

shell

considered:

on the

removed

in the juncture

in the

when

is placed

some

stresses

be carefully

likely

very

load

the

attachment-to-shell

reinforcement

same

no stress

to Shell

Locations

can be higher

is most

and

Stress

should

Stresses

The

The of the

conditions

conditions

points

are

thermal

change.

Conditions

attachment.

certain

This

there

by discontinuity,

in Maximum

removed

to Boundary

influenced

by an investigation

Under points

Respect

when

area

and material

be determined

C

in the

be caused

in section

removed

is applicable

loadings

can

with

by an investiga-

slightly

removed

from

the attachment. 3.

Comparison

of analytical

membrane

stresses

shows

calculated

at the point

analytical

and

where

experimental

sultants

at loaded

sultants

must

attachments

be calculated

and that

experimental

membrane

stresses results

at the

stress are

[2,

shows

that

center

results

resultants

desired.

3] for the

[3]

of the

can

Comparison

bending bending

for

stress stress

attachment

of re-

reand

be

Section B7,2 15 April 1970 Page 61 then shifted to the edgefor the determination of stresses at the edge of the attachment. The determination of bending stresses at

D

applied

lated

at a distance

Stresses

Caused

[2]

combination E

If this load

of membrane Shell

and

The

analysis

and

to the

attachment,

shear

The

that

appears totally

by a shear

stresses

assumes

Attachment

resultants

caused

is not available.

assumption

stress

be calcu-

Loads

distribution

here

is resisted

bending

C2/2 closer

and membrane

presented

only. shear

shear

the

by Shear

shell

of varying

that C_/2 or

to a cylindrical

method

the

requires

stress

by shear that

points

An accurate

consists The

other

the

actual around

shell

(V a)

stress the

the

it can

stresses

distribution

rigid

resists

unreasonable,

by membrane

load

attachment. shear

load

be assumed

or by some

stresses.

Geometry

assumes

that

the

cylindrical

shell

has

simply

supported

end conditions or is of sufficientlength that simply supported end conditions can be assumed. The computer

program

requires that circular and ellipticalattach-

ments be converted to equivalent square and rectangular attachments, respectively.

The equivalent attachment must have an area equal to the area

of the actual attachment for a radially applied force. ment must

have a moment

The equivalent attach-

of inertia about the bending axis equal to the moment

of inertia about the bending axis of the actual attachment for bending loads. In both cases the aspect ratios (a/b and ci/c2) of the attachments (actual and equivalent, respectively) mustbe

equal.

If the attachment is welded,

weld size must be added to the attachment when determining ments.

the

equivalent attach-

Section

B7.2

15 April Page 62 L/R m is a secondary 1.0 <- L/R m <-- 5.0. B7. 2. 2.1-2, F

The

must Shell The

computer

(q} must

be positive pressure

loads The

cylindrical position

and has coordinate

little

effect

system,

on the

defined

solution

of

by Figure

at x = L/2.

Loading

in shell

local

attachment

be located

increase

internal

parameter

1970

stresses

program

stiffness or must

caused

by internal

a positive

differential.

be calculated

calculated

shell

deflections

shell

thickness,

of stresses.

accounts

separately

by the method must for

for

be small,

the analysis

pressure

coupling

pressure}.

The

The

stresses

and

superimposed

presented

is,

internal caused

equal and

by the upon

the

to the

to allow

the

pressure

here.

approximately to be valid

(that

super-

k_

Section

B7.2

15 April Page 63 B7.2.2.2

STRESSES I

GENERAL

Stress overturning the

resultants moment

Computer

moment

and

(M a)

displacements

are

Handbook.

(M T) are The

stress

obtained

directly

from

resultants

(Mj

Nj)

program

specified

by x and

values

z) defined The

for

in Figure

computer

eonfigurations

The juncture

0 input

Computer

2 - "n"

Equally

Case

3 - One

Concentrated

Case

4 - "n"

Equally

Case

5 - Longitudinal

Case

6 - Circumferential equation of the

stress from

Figure

accounts

load

location.

v,

twisting

loading. and

The

to the

in

and

and

(u,

and given

(Va)

geometry

according

stress

w) de-

location

coordinate

Radial

Radial

Load

Distributed

Concentrated

Radial

Radial

Loads

Loads

Moment Moment

stresses

in a shell

resultants

is of the

at a rigid

attachment

form:

2)

B7.2.

signs.

displacements

Load

parameters

caused

and

(K n and

I. 2, Sections Calculation by an arbitrary

I and Sheet" local

K b) are

defined

and can

IIB. can

be used

loading.

for The

is

system

Handbook):

Overturning

for

resultants

Uniformly

Overturning

"Stress

for

(P)

program

shear

a specific

calculate

Spaced

concentration

of all stresses

load

displacements

Distributed

• Kb (6Mj/T

B7.2.2.2-1

and

for

Spaced

stress

Paragraph

by

Utilization

Case

be evaluated

automatically

will

Uniformly

general

computer

attachment

determined

1 - One

fj = K n (Nj/T)

calculation

are

Case

in terms

The

and

by radial

B7. 2. 2.1-2.

program (see

the

caused

calculated

by the computer

0,

caused from

Stresses

termined

(x,

1970

the sheet

Section B7.2 15 April Page 64

STRESS

CALCULATION

APPLIED P

SHEET

LOAD_

FOR

SHELL

•

T

STRESSES

IN

CYLINDRICAL

GEOMETRY

CO

SHELLS

CAUSED

BY LOCAL

1970

LOADS

PArAmeters

ORDINATI_,;

• •

9

K n

P Itm" Vi

=

V

L

-

•

A

-

•

K

MT MI |

¢

1

•

B

"-'

¢

I

•

C

m

a

•

•

!i

STR.E_

ADJUIIT.

RESULT,

FACTOR.

D

o

-"'l=.o o,

CALC,

n

!STRE_

ADJUST.

UIru

' 1_

Mx(O

)-

÷

;T_.

_

_

FACTOR

!

N

D

MxCA )" clme

w,J&J

V

N

).

._.

+

+

(A)"

NI[B)-

_ T

+

Mz(B)=

(A) I*

(f_)

_

_-

W

LONGITUDINAL

$TRE_E$ Ny

•

INy

+

mB -i!i t.t tJ

(B)_*

p 6K +

•

M

5

(A)=

My

U i_OTAL

6_=

STRESSES

[

JM_,(o)=

SIGNS

AR.E

(f_,)

_ v v

_.

vz

_----.rot

_l

i MT'T" _" TOTAL

i_

,mmm

C_CUMF_R.ENT_'-

vz.

•

5K

_.

TOTAL

_EAR.

(t_f

STR.E_

)

!_,-i_-_"" C_E _-rTTER. BTR.JI_8 8T_8 _ULTAN_ am*

C_E

8/GN

OF

CALCULATED

IN P_N_B LIr,,SULTANTS FOR.

$_'M

_ US_ OF

+ . .1.1 i!_ .1-1

"

?_1

>K "x

I I

6K

Mx(O

:ALC. 5TRE_

.

Ai

Lc&se

*

•

m__

II

_"

b

• Z

M

0

_T_

D_IGNAT_ SHOULD BE _, THE TH3_

Fig.

B, S_N8

_DI_

C.

_E_

THE CO%,(PUTED

AND D) CAN INDICATED. IF

NEGATIVE POINT BY RE

(A. THE

_DICATED.

B. OR O) AT WHICH COMPUTER PROGRA.M.

OBTA_ED

FROM

_ESE

THE THE ST_

(Ix + Lf ) .st NEGATIVE.

B7.2.2.2-1

Stress

Calculation

Sheet

Section B7.2 15 April 1970 Page 65 B7.2.2.2

STRESSES

II

STRESSES

Radial cause

load

membrane

circumferential A

RESULTING

FROM

configuration and

bending

Cases stress

A RADIAL

I and

LOAD

II (Computer

components

in both

the

Utilization

Handbook)

longitudinal

and

directions. Longitudinal Step

1.

Stress Determine the

Step

2.

the

the

the the

4.

the

obtain

values

resultant

load

A,

for

the

See B and

stress

(M×)

input

for

at point

O

A and

B

(N x) at points Figure

B7.2.2.1-2

for

O.

in Paragraph

B7.2.

1.2,

concentration

Section

I,

parameters

Kb).

the bending

stress

stress

2, and

determined

the

stress

determine the

at point

A as determined

parameters

the

following

O and

longitudinal

as stresses

equation:

± Kb(6Mx/T2)

consideration

longitudinal

(N×)

(M x) at point

concentration

3,

A using

fx = Kn(Nx/T)

resultant

resultant

in Step

at point

Proper

geometric

resultant

program.

criteria

the normal

(fx)

stress

of points

Using

in Step

and

stress

computer

location

Using

load

bending

normal

(K n and Step

required program.

Determine

with

3.

the

computer

and

Step

(fx)

stress

of the

sign

at the inside

will and

give outside

the values surfaces

for

of the

shell. Step

5.

Repeat

Step

4, but use

as determined (Mx) tudinal

for

point

as determined stresses

the normal

for (fx)

B and point

at point

stress

the bending O to determine B.

resultant stress the

the

(N x) resultant longi-

Section B7.2 15 April 1970 Page66 B Circumferential Stresses The the

five

steps

Ny instead Step

circumferential outlined

of M x and

(fy)

stresses

in Paragraph N x in Step

(fy)

A, 2 and

can

above, using

be determined

except

for

the following

by following

determining stress

My and

equation

in

4. fy = Kn(Ny/T) C

Concentrated Points

Cases

III and

stress

caused

stress

resultants

Paragraph

+ Kb( 6My/T

A,

IV.

A,

Load B,

by concentrated calculated above,

Stresses

C and

Longitudinal

after

2)

D in Figure

B7.2.2.

and circumferential loads at point applying

(Cases O. proper

The

I-2

do not exist

membrane

III and IV) stresses modifications

and

is determined are

calculated to the

for

Load

bending from using

equations.

Section B7.2 15 April 1970 Page 67 B7.2.2.2

STRESSES

III

STRESSES

Overturning Utilization in both

the A

RESULTING moment

load

FROM

AN OVERTURNING

configurations,

Handbook},

will

cause

longitudinal

and

circumferential

Longitudinal Step

1.

Stress

Cases

membrane

MOMENT

V and

and

VI (Computer

bending

stress

components

directions.

(fx)

Determine

load

and

geometric

the

bending

resultant

(N x)

input

for

the

computer

program. Step

2.

Determine stress for

Step

3.

Load

Case

Using

the

obtain

the values

(K n and Step

4.

Using

A for

the computer

in Paragraph for

the

(M x) Load

Case

V and

B

program.

B7.2.1.2,

stress

and normal

Section

concentration

I,

parameters

K b) bending

resultant

and

the

stress

Step

3,

determine

using

resultant

at points

VI with

criteria

the

stress

stress

the

stress

resultant

(N x) at point concentration

following

(M x) and

A as determined parameters

the longitudinal

stress

the

normal in Step 2

as determined (fx)

at point

in iX

equation:

f x = K n (N x/T) • Kb(6M

x /T

2)

Proper consideration of the sign will give the values for longitudinal stresses at the inside and outside surfaces of the shell. Step 5.

Repeat Step 4, but use stress resultants as determined point B to determine

for

the longitudinal stresses at point B.

Section 15 April Page 68 B

Circumferential The

the

five

Ny instead Step

steps

Stress

circumferential outlined

of M x and

in Paragraph

A, 2, and

(f) above, using

4: f y =K n (Ny/T)

1970

(fy) stresses

N x in Step

B7.2

+ K b(6M

y /T 2)

can

be determined

except the

for

following

by following

determining stress

My and

equation

in

Section B7.2 31 December Page B7.2.2.2

69

STRESSES IV

STRESSES A shear

RESULTING load

the attachment-to-shell A

(V a) will juncture.

Round

FROM cause The

A SHEAR a shear

shear

stress stress

Attachment V a

f

xy

-

_r0T

sin

0

for

V

cos

0

for

Va = V 2

for

V

for

V

a

= V1

V a

f

xy

-

_r0T

B Rectangular

Attachment

V a

f

xy

-

4clT

a

= V1

V a

f

xy

-

4c2T

a

= V2

LOAD (fxy)

in the

is determined

shell

at

as follows:

1966

Section B7.2 3t December 1966 Page 70 B7.2.2.2

STRESSES

V

STRESSES A

Round

RESULTING

stress

moment

stress

(fxy)

in the

is pure

shear

and

determined

A TWISTING

MOMENT

Attachment

A twisting a shear

FROM

shell

(M T)

applied

at the

attachment-to-shell

is constant

around

to a round

attachment juncture.

the juncture.

The

shear

will

cause

The

shear

stress

is

as follows:

fxy = MT/2_r°T B Square

Attachment

A twisting complex loading

stress are

field

available.

moment

in the shell.

applied

to a square

No acceptable

attachment methods

will

for analyzing

cause the

a

Section B7.2 31 December

_

Page B7.2.2.3

STRESS I

CALCULATION Most

arbitrary

RESULTING

Step

1.

that

induce are

Resolve

the arbitrary

and

the

(intersection shell

Evaluate

as A,

component

Obtain the

the

indicated

and

outside

of the

moments

Paragraph

directions

stresses

D) around

arbitrary

load

com-

components

with

attachment-

B7.2.2.

1-1.

at desired

points

the attachment

applied

and

B7.2.1.6 of the

of the

in Figure

moments)

loading

for

each

by the methods

B7.2.2.2. stresses

longitudinal,

evalUated

and/or

of attachment

are

of an

procedure:

(forces

See

of application

B, C and

in Paragraph 3.

point

are

overturning

positive

of centerline

inside

load

forces,

The

shells

following

components.

interface)

(such

applied

Problem.

ponents

by the

shear

moment

Example

Step

LOADING

on cylindrical

determined

force,

twisting

2.

loads

Stresses

into axial

Step

ARBITRARY

71

OF STRESSES

loadings

nature.

FROM

t966

by Step

inside

and outside

signs

is necessary.

for

the

arbitrary

circumferential 2 for of the

each

loading and

of the

shell.

shear

points

Prosper

by combining stresses

selected cot_sideration

on the of

Section B7.2 31 December

1966

Page 72 B7.2.2.3

STRESSES

II LOCATION

RESULTING AND

FROM

MAGNITUDE

The location and magnitude

ARBITRARY OF

LOADING

MAXIMUM

of the maximum

STRESS stresses caused by an

arbitrary load require a consideration of the following: A. The determination of principal stresses (fmax' fmin' fxy = 0, or f

xy = max) point.

for the determined

B., The

proper

selection

stresses (f'x fY ' and xfy) at a specific

of points

for

determining

the

stresses.

Section B7.2 15 April 1970 Page 73 B7.2.2.4

DISPLACEMENTS

Shell moments

are

Handbook. loads

displacements obtained Shell

are

not

general, order

from

the

by radial

computer

displacements

load

program

caused

configurations

and overturning

described

by twisting

in the moment

Computer and

shear

determined.

Comparison deflections

caused

are

of experimental

sensitive

however, of magnitude.

to the

and detailed

the experimental

and

theoretical conditions theoretical

deflections

indicate

of the attachment. values

are

of the

that In same

Section B7.2 15 April 1970 Page 74 REFERENCES: Bijlaard, Pressure

le

,

e

P. P., Vessels,"

Wichman, in Spherical Research Lowry, Shells,

"Stresses From Transactions

14.

R. ; Hopper, and Cylindrical Council Bulletin,

September

805-816.

Deflections in Pressurized Cylindrical of Asymmetric Local Loadings, "

Dynamics/Astronautics,

GD/A-DDG64-019,

Loadings in Cylindrical Vol. 77, 1955, PP.

A. G. ; and Mershon, J. L., "Local Stresses Shells due to External Loadings, " Welding Bulletin No. 107, August 1965.

J. K., "Stresses and Due to the Application

General

Local ASME,

San 1,

Diego,

California,

Report

No.

1964.

BIBLIOGRAPHY: 1.

Timoshenko, McGraw-Hill

2.

The

Astronautic

S., Woinowsky-Krieger, Book Co., Inc., Second Computer

Utilization

S., Ed.,

"Theory of Plates and Shells," New York, N. Y., 1959.

Handbook,

NASA.

SECTION B7.3 BENDING ANALYSI S OF

TABLE

OF

CONTENTS

Page B7.3.0

Bending 7.3.1

General 7.3. 7.3.

7.3.2

1.1

Equations

1.2

Unit-Loading

Interaction

7.3.2.2

Interaction Elements

Edge

Shells

.....................

1

2

Method

9

.....................

.......................... Between

Two

General

11

Shell

Between Three ..............................

Influence 1

1

.............................

Analysis

7.3.2.1

7.3.3.

Elements

or More

.........

13

Shell 16

Coefficients

......................

20

Discussion

......................

20

7.3.3.2

Definition

of F-Factors

7.3.3.3

Spherical

Shells

7.3.3.4

Cylindrical

7.3.3.5

Conical

Shells

..........................

44

7.3.3.6

Circular

Plate

..........................

54

7.3.3.7

Circular

Ring

..........................

79

7.3.4

Stiffened

Shells

7.3.4.

1

General

7.3.4.

2

Sandwich

7.3.

References

of Thin

..................................

Interaction

7.3.3

7.3.5

Analysis

4. 3

7.3.5.2

Barrel

.......................

23 26 42

..............................

80

..............................

80

Shells

Unsymmetrically Shells

.........................

Shells

Orthotropic

7.3.5.1

....................

.........................

Shells Loaded

of Revolution Vaults

84

.......................

87

Shells

98

...................

......................

..........................

.........................................

98 99 107

B7.3-iii

Section B7.3 31 January 1969 Page 1 B7.3.0

BENDING In this

ANALYSIS

section

OF

some

of the

applied to solve shell problems. and several shell theories, with pointed

out that

the

THIN

theories

they

section,

edge

differential

reduced satisfy are

restraints,

B7.0

defined the structural and ramifications. ratio,

material

sandwich or ring-stiffened a role in establishing which

equations

that

the

and

their

solutions

behavior,

shells), types theory is

must

satisfy.

The

These

influence

lems that involve determining and complex geometries. The

procedure

coefficients, stresses,

for bending

etc., strains,

analysis

are and

of thin

then

edge

used

will

be tabulated to arbiconditions,

restraints

are

equations geometry

to solve

displacements

shells

different

will

symmetric geometries subjected There are certain restraining

solution

be

shell It was

to unit loads and, by making the solution of the differential these unit edge restraints, the influence coefficients lor the

obtained.

will

shallow versus nonshallow shells required fell into the same thin shell theory.

for simple and complex rotationally trary rotationally symmetric loads. called

in Section

thickness-to-radius-of-curvature

Furthermore, even though

In this

discussed

Section B7.0 their limitations

type of construction (e. g., honeycomb of loading, and other factors all play applicable. approaches

SHELLS

prob-

in simple

be as follows:

The

surface loads, inertia loads, and thermally induced loads are included in the equilibrium equations and will be part of the "membrane solution" using Section B7.1. The solution due to edge restraints alone is then found and the results superimposed

over

essentially

the

identical

membrane

to those

solution.

obtained

The

by using

results

obtained

the complete,

exact

will

be

bending

theory. B7.3.1

GENERAL The

of revolution tions and B7.1.1.1

geometry, are

the

coordinates, same

sign conventions and Paragraph

as given

stresses,

and

in Paragraph

are generally the same B7.1.1.2, respectively.

stress B7.1.1.0.

resultants Also,

as those given The limitations

[or the

a shell nota-

in Paragraph of analysis

are the same as given in Paragr,_ph B7.1.1.3, except that in this section flexural strains, stresses, and stress resultants are no longer zero. Boundaries of the shell need not be free to rotate and deflect normal to the shell middle surface.

SectionB7.3 31 January 1969 Page2 B7.3.1.1 I

Equations

Equilibrium

Equations

A shell will

element

now be considered

with

the stress

and the

resultants

conditions

for

as given

in Section

its equilibrium

under

B7. 1.1.0 the

influence

of all external and internal loads will be determined. The equations arising by virtue of the demands of equilibrium and the compatibility of deformations will be derived by considering an individual differential element.

and

The surface

element,

external forces

which

internal

force8

loads are (stresses)

are

sections

will

be stress

comprised of body forces that act on the that act on the upper and lower boundaries of the

curved

surfaces

resultants

acting

For the following equations, external equivalent stresses distributed at the middle thus loaded by forces as well as moments.

may

faces

forces are surfaces.

the shell. of the

shell

The element.

replaced by statically The middle surface is

Now, instead of considering the equilibrium of an element of a shell one study the equilibrium of the corresponding element of the middle surface.

The stresses, in general, the mtress resultants will

of the

bounding

on the

element of the

vary from also vary.

point

to point

in the

Consider the stress resultants of concern applied shell as shown in Figures B7.3.1-1 and B7.3.1-2. The equilibrium

respectively,

of the shell,

is given

ao O°tlN_b

a_

+

by the

+ NO _

o_

aa2N0 + ....

ao

following

_

+ N

a_ _

_,o oo

in the

0,

_b, and

shell,

and

to the

middle

z coordinate

as a result

surface

directions

equations:

-

N_b

N

ao

_

+ Q0 -- R_

+ a la2Pl=

0

+

o a_

Q(_ al_ R2

+ _lCV2P2 = 0 (la)

+ OLlo_2q = 0

Section 31 Page

N

B7.3

January

1969

3

I

,b

1

FIGURE

()(I

B7.3.1-1

NI:: '

+

._.:._._1._ 11 d

TYPICAL

SHELL

AND

IN-PLANE

I

REFERENCE SHEAR

ELEMENT

\_ITH

AXIAL

FORCES

z

FIGURE

B7.3.1-2

TRANSVERSE

TYPICAL SHEAR,

SHELL BENDING,

REFERENCE AND

TWISTING

ELEMENT ELEMENTS

WITH

Section

B7.3

31 January P_ge

where applied

Pl,

P2,

The in

the

and

to the

q are

middle

components

surface

equilibrium

following

moment

of

of

the

the

effective

external

1969

4

force

per

unit

area

shell.

of moments

about

equilibrium

expressions:

the

O,

gS, and

z coordinates

results

Dc_1hi d) 80

+ __.___._ 8_b

Mocp

c3c_l

_ MO _i 0(>

0cx2 M 0 Mc_ 0 +

+ M 6pO

-

O0

Q,,_(_ 1 c_2 = 0

_)cv2 -M

+

M

=

0

{ib)

0 M N0gb

- N._0

The warping

+

M

11t

-

force components

of the faces,

Now,

for

moments

(M_0)

equations

beeom(;:

1t2

-

o

of the last equilibrium

and result from

in-plane

shells

of revolution

the

vanish

and

a 2 = II 2 sin

cz 1 = R1;

resultant

+/_.

I

-

NO Ill

cos

_

+ (_,t. I/1 + l_l)

d(Qqll) d_

0

=0 - Ncl)RI - NOIII ,-;in r) + II1 llq

-

M OR 1 cos

dp - R 1RQ_b

=0

are due to

and twistinffmoments.

forces

d(NdR) d_

expression

shears

+ Na, 0,

Thc,refore,

QO) the

and equilibrium

r

Section

B7.3

31 January Page 5

where the satisfied.

and

second,

fourth,

and

IV)

Strain

This

simplification

For the particular case is zero, and all derivatives In this

o Q

case

have

here, surface

changes arising

arises

from

o

1

= e_,_, -

R1

o

Yl2

of axisymmetric of displacement

the middle

du

been

identically

in the dimensions from its deformathe

assumption

of

surface

deformations, components

strain-displacement

the displacement with respect to equations

be-

w

d+p + lIT

o o _ _ _2 = c o R1

the

of (1)

Displacement

0 vanish. come:

and

equations

In the equilibrium equations presented in the shape of the element of the middle

tion have been neglected. small deformations. II

sixth

1969

udR2 w__ + l_lR2d_/_ + R2

(3)

O

= Y epO = 0

curvature

and

K1 = K¢ = -Ri

K2=KO

--_1

t(12 = K_p0 = 0

twist

expressions

dc/_

dch

R2

become

-

d,,jLd+I ' -u

("U

:_l .Tamm _/ p_-e

for for

general surface R =H 2 sin 0:

of revolution,

the

dR -_

expressions

and

J 989

6

dR2 are d,_-'---_

as

follows

dR dO

R 1 cos

0

dH__2 = d_p ( Ri - R2) cot _

inserting

equation

E2 -

(5)

ucot_ i_2

Ill

remaining

For

an and

(3)

yields

l{2

__

[.dw

ul

strain-displacement

Stress-Strain

resultants

equation

+

cot

while

into

(5)

shell,

couples

I-_-E

N2-

i-p2 Et

of {3) and { 4) are unchanged.

Equations

isotropic

Nil

equations

the

to components

following

constitutive

M 1 =DIK

el +_

M2

( _2o + Pel o)

equations

relate

stress

of strain:

l+pK2]

--Dlg2 + Pgl] (7)

NI2 = N21-

Et

o

2(l+p)

TI2

Ml2 = M21 -

(l-p) 2

D KI2

where Et 3 m

_

12( 1 - p_)

Section B7.3 31 January 1969 Page

O

and where change

IV

(middle

surface)

in curvature

O_

(el, e2 , 3,12, are given in equation

(3) and

(KI, K2, KI2) are given in equation

(4).

and twist terms

Solution of Equations

By

eliminating

determining

obtained.

Q_

from

the first and last equilibrium

the force resultants

differential equations Rather

variables

pair of equations, thickness,

which

combine

hypergeometric

The

from

equation

in the two unknown

than obtain equations

tion of dependent

equations

(7), two second-order

displacement

components

in this manner,

however,

can be performed

leading

for shells of constant

into a single fourth-order

to a more

meridional equation

transformation two new

to the Reissner-Meissner

variables,

the angular

a transformamanageable and constant

in terms

of a

equations

is accomplished

rotation

the, quantity

This

substitution of variables

leads to two

second-order

equations in U and V replacing the corresponding two equations details of this transformation are illustrated in Reference I.

For in fact,

for

formed

pair

the

ordinary

u and w are

curvature

solvable

(2) and

series.

by introducing,

and

strains

7

solution

equation. represented

shells any

of constant shell

of equations of which For

shells

in the

thickness

of revolution can

be

combined

is determined

from

of the simplified

and

constant

satisfying

description form:

the into the

above,

meridional

Meissner

a single solution the

differential in u and w.

curvature

condition, fourth-order equations

or,

the

trans-

equation,

of a second-order shell

The

complex can

be

Section

B7.3

31 January Page 8

L (_

+ ___u R1 _)

= Et

-_11 where

the

--D

operator

L(

of second Following

) = R_ll

F4 -

+ r4U=

Et D

-

The solution of two second-order

L(U)

first

and

_11

I-_

R(-_I)

R---_2 + R 1 cot¢l

d(d_p ) -

Rlc°t2_b R 2R 1

_" i

0

u2 1_

of the fourth-order complex equations

FZU=0

equation can be considered of the form

the

solution

.

Reissner-Meissner type equations are the most convenient and most employed forms of the first approximation theory for axisymmetrically shells of revolution. They follow exactly from the relations of Lovers approximation

as they are thickness. more

d2( d(_ 2 )+

From the system shown above of two simultaneous differential equations order, an equation of fourth order is obtained for each unknown. operations described in Reference 1 yield an equation of the form

LL(U)

widely loaded

1969

general geometry

when

the meridional

for cylindrical, Furthermore, case, are

provided satisfied.

curvature

conical, spherical, they follow directly that

special

and

thickness

and toroidal from Lovels

restraints

are

constant,

shells of uniform equations in the

on the variation

of thickness

Section B7.3 31 January

1969

Page 9

B7.3. I. 2

forces

Unit Loading Method

Generally a shell is a statically indeterminate of the shell are determined from six equations

derived

from

the

three-force

and

three-moment

structure. The internal of equilibrium, which are

equilibrium

conditions.

There are ten unknowns that make the problem internally statically indeterminate because determination of the unknowns does not depend on the supports. The situationis similar to one that occurs in a truss which, as used in practice, is a highly staticallyindeterminate system. If reactions to the applied loading can be found with the help of known equations of statical equilibrium, the system is externally indeterminate system internally (which gether. tional values,

introduces hinges This introduces

calculations simplified

main

at the joints), all joints are the moment into the members.

objective

by replacing but accurate

accomplished displacement,

of the following

sections

and Bending

spherical the results

shell are

1. of the shell

the

To compare

with some compared,

two theories,

axisymmetrical the following

loading conclusions

and deformations

are

The

stresses

with

the exception

adjacent to the boundary. where R is the radius and

the

B7.1, the it permits

of a narrow

strip

elaborate with the This is

compatibility

for

toaddi-

indeterminate

is to bypass

Theories

As discussed in Sections B7.0 and than the membrane theory because conditions.

statically

the conditions of equilibrium, at the junctions.

of Membrane

boundary

the

welded or riveted However, this

the classical methods of elasticity theory procedure called the unit loading method.

by enforcing and rotations

Comparison

general

however, a truss is a statically instead of the assumed simplification

influence is usually negligible. To find deformations must be considered. The

I

determinate; because,

Nonshallow

in

Shells

bending theory is more use of all possible assume

a nonshallow

built in along the edges. can be made:

almost on the

identical shell

for

surface

This narrow strip is generally no wider t is the thickness of the spherical shell.

all

locations

which than

When

is

SectionB7, 3 3! J_nu_rv 19(;_: P_gc 10 2. Except for the strip twisting solution

the boundarT,

all bending

moments, and vertical shears are negligible; to be practically identical to the membrane 3.

Disturbances

along

ever, the local bending may become negligible

results,

along

the

supporting

edge

this causes solution. are

and shear decrease rapidly outside of the narrow strip,

moments,

very

the

entire

significant;

how-

along the meridian, and as described in item 1.

Since the bending and membrane theories give practically except for a strip adjacent to the boundary, the simple

the same membrane

theory can be used; then, at the edges, the influence of moment and shear can be applied to bring the displaced edge of the shell into the position prescribed by boundary conditton_. The bending theory is used for this operation. Consequently, once the special derivation.

solutions are obtained, they can be used later The results obtained from application of both

be superimposed, which will lead to the those obtained by using the exact bending H

Unit-Loading

Method

Applied

to the

final results theory. Combined

being

almost

without theories identical

Obtain Section

Assume

that

the

shell

under

consideration

is a free

a solution for the overall stresses and distortions B7.1. This is the primary solution. 2.

Apply

the following

a.

Moment

b.

Horizontal

c.

Vertical

edge

shear shear

of the

per

in pounds in pounds

inch along per

per

inch

inch

the along

along

edges

edge the edge

the edge

can

membrane.

loadings:

in inch-pounds

to

Theory

The solution of a shell of revolution under axisymmetrical loading be conducted in a simplified way, known as the unit-loading method. i.

any c.')n

by using

Section

B7.3

31 January Page 11

distorted

These loadings should edge of the membrane

supports

(edge

be of such magnitude as to be able into a position prescribed by the

condition).

The

third

edge

loading

not necessary. The amount of applied corrective magnitude of edge deformations due to the primary magnitude will Section B7.3.2. necessary

for

be determined However, deformations

due

moment:

b.

Unit-edge

horizontal

c.

Unit-edge

vertical

shear:

will

be referred

solutions

Having

4. Superposition and corrective

B7.3.2

INTERACTION Missiles,

structural elements.

space

configurations For analysis, down

into

simple

elements a discontinuity exists; that is, unknown conical,

Q,

can consequently

solution

shapes

M = I pound shear:

the primary

into the interaction process. of corrective loadings (M,

broken

of cases

is

on the exact

to be explained in formulas will be

per

inch

Q = 1 pound V = 1 pound

per

to as unit-edge

per

inch

inch. influences,

or as

solutions. 3.

loadings

majority

to the following:

Unit-edge

These

to return the nature of

loadings depends solution. The

by the interaction procedure to start the interaction process,

a.

secondary

in the

1969

that

a complex

toroidal;

these

and

unit-edge

This process and V); all

solutions,

one

can

enter

these

will determine the correct amount stresses and distortions due to these

be determined. of stresses loadings lead

and distortions obtained to the final solution.

by primary

ANALYSIS vehicles,

and pressure

vessels

are

examples

usually consisting of various combinations such complex shell configurations generally elements. (point shears shell

may

shapes

However,

at the

intersection

of abrupt change in geometry and moments are introduced. be broken also

occur

down

to include

in compound

of of shell can be

of these or loading) usually The common

spherical,

bulkheads.

elliptical, Figure

SectionB7. ,_ 31 January Page 12

B7.3.2-1,

for

example,

spherical

transition

of such accuracy

a shell, the required:

illustrates

between

a compound conical

analyst must choose (l) he can consider

use some approximation, the method of interaction. In this

the

section,

not only to monocoque interacting elements

or (2)

the

and

bulkhead

cylindrical

method

it as

and

the thrust

For

a compound

is presented

shells but also to sandwich are often constructed from

which

used loadings load, thermally

loads.

N,)h_': Shell

Not

I)_FI

"l')w't_l'y

Apply

Ilore.

_

'|'l_roidal

Cylimh'i(:;l]

FIGURE

B7.3.2-1

of the analysis

shell,

COMPOUND

BULKHEAD

usiL_g

is applicable

and orthotropic shells. different materials. The

ing can also vary considerably. The most frequently or external pressure, axial tension or compression loads,

consists

between two methods, dependiJ_g on the such a system as an irregular one :,nd

he can calculate

interaction

which sections.

191:9

The load-

are i_ter_ml induced

Section I37.3 31 January 1969 Page 13 B7.3.2. t For

Interaction

Between

described

simplicity, first. The

is usually

the

case

the more

when

Two Shell

Elements

interaction between two structural elements will general case of interaction of several elements,

the combined

bulkhead

is under

consideration,

will

be as be

described second. For the purpose of presentation, a system consisting of a bulkhead and cylinder, pressurized internally, is selected. The bulkhead can be considered as a unit element of some defined shape and will not be subdivided

into separate

assume

the pressurized

portions

in the

container

great

parts, the cylindrical shell and deformations introduced

and dome, by internal

can be determined

part

FIGURE

for

each

B7.3.2-2

majority

of cases.

to be theoretically as shown pressure

separated

For

in Figure B7.3.2-2. (or another external

separately.

CYLINDRICAL

example,

into two main

SHELL

AND DOME

Stresses loading)

Seeticm

_B'_. :;

._1 ,Lnnuarv P:lgo

Assume radial

that

displacement

discontinuity into

¢ 6d

fie

¢ fld

Consequently,

These Figure

_- 6 c ) and

&r

= 5 d and

rotation

exists

the

(b}

in

tic

To

close

slope

this

to hold

gap, the

for

the

supplied

cylinder

Since

the

along

the the

structure

is

forces together.

Q6

M fi

and

M6

C

C'

corresponding

values

Qd6,

M d' 5

Qfldand

deformations

follows:

- fld

pieces

rotation

2-3.

solution)

dome.

as

unknown

two

Displacements and defined as follows:

B7.3.

the

discontinuity

6c - 6d

unit

(_e)

fld for

in displacement

Qfl

{primary

14

elements,

there

C'

The

analysis

(a)

juncture

M are

and

two

6c

membrane

(Ar

line

separated

the

the

19G,'_

(Q

of the

and

M)

cylinder

will

due

be

to unit

introduced

values

around

of Q and

('

for

the

dome

for

the

same

unit

loadings

will

be:

Mdfl

and

unit

loadings

a.t the

junctions

are

presented

in

Section

B7.3

31 January Page

1969

15 5 Mj fl

(I6 fl _d

Q= l

_d(

Around .lun(:ti_m)

M=!

:'N

f_ 5

( Around

c

the

M

,/unction)

{.

M 1'

FIGURE

B7.3.2-3

To

the

Q

Thus,

with

with

two

close

+Qd,

the

UNIT

gap,

Q+

two

It is noted unknowns.

the

M

The

following

1.

Horizontal

2.

Shears

3.

Moments the shell

one

sign

following

+M

equations,

that

DEFORMATIONS

cut

positive

can

UNIT

LOADINGS

be written:

d

unknowns

the

convention

positive are

two

through

deflection, are

equations

M=Se-5

the

AND

(Q

shell

and

leads

M)

can

to two

be

algebraic

determined.

equations

is adopted: 5,

is positive

if they if they

cause cause

outward

deflection tension

outward on

the

inside

fibers

of

Section

B7.3

31 January Page

4.

Rotations

In general, adopted

if it does

are this

positive

sign

not conflict

Observe

that

if they

convention with

and

to M and

around the junction between the cylinder the effect of this force on the displacement B7.3.

2.2

Interaction

described to consider

Between

Three

to a positive

is arbitrary.

logic

in addition

correspond

is used Q,

and

there

Any rule

1969

16

moment. of signs

may

consistently. is an axial

force

distribution

dome (reaction of bulkhead), due to M and Q is negligible.

or More

Shell

be

but

Elements

In practice, most cases are similar to the two-member interaction in the previous paragraph. However, at times it may be convenient interaction of more than two elements. This can be performed in

two ways: 1. solved,

Interact

interact

first

it with

Simultaneously

The

first

a compound Figure

the

2.

method

further explanation. not be approximated bulkhead

the two elements; third

element,

interact

can

combination

is

The

be approximated

second

method

requires

is such that its meridian cancurve, such a bulkhead is called with many

curves

as shown

in

B7.3.2-1. In this

compound

this

elements.

If the shape of the bulkhead with one definite analytical and

when

etc.

is self-explanatory.

case,

two or more

required to separate the compound This is shown in Figure B7.3.2-4, cuts separating the three elementary Figure belong

all

then,

imaginary

B7.3.2-4 also illustrates the to each cut. The discontinuity shell.

cuts

through

the

shell

bulkhead into component shells where the compound shell has shells (spherical, toroidal, loading and influences

discontinuity will restore

will

be

of basic shape. two imaginary and cylindrical}.

influences that the continuity of the

SectionB7.3 31 January 1969 Page 17

Qi

M!

p

.!y indric:il

FIGURE

Figure

The symbols B7.3.2-4.

M_

nn'

used

( 1 itl,r

l;

P" ,,','

+,

IZin.21

Sh_ql

B7.3.2-4 for

Qnn = rotation horizontal

5 = horizontal M n ' Qnn application

the

DISCONTINUITY two successive

at point shear

cuts

n due to a unit Q at point n

LOADS m and

moment

displacement due to the same points as above

n are

also

M or

unit

loading

in

shown

in

SectionB7.3 31 January 1969 Page 18 B Mnlrl

_

= rotation

at point

horizontal M 5 am'

proper

indices

Figure

Additional B7.3.2-4

n = 1 and

for

shell

are

Due

to the

(T)

needed

at point

and

to cover

(_) shear

due

and rotations

by using

subscript

primary

loading

to the

Q

c (cylinder) (internal

M or unit

m same

above

loading

in Figure

the spherical

same

shell

shell

B7.3.2-4.

portion

due

to a unit

same

of point

(_)

instead

conditions

on the

equations

Spherical

Shell:

5 S = M_M2+Q_Q2+A ,

for

the

S

p

total

rotation

and displacement

stated

cylindrical

of subscript the

as

rotations

s (sphere). and

placements will be indicated with fl and Ar = A. As before, the subscripts s refer to the cylinder and sphere. The subscripts It and 2t will be used denote the toroidal shell at the edges (_) and _ . Now the

of

point

to the

pressure),

in

can be considered

as shown

on the spherical at the

displacements

displacements

defined

at point

the nomenclature

portion

or unit

5 Qs = horizontal above.

Similarly,

Q acting

m = 2,

nomenclature is as follows:

moment

to a unit moment

displacement due points as above.

the toroidal

rotations

M5 s'

shear

Q5 nm = horizontal application

Designating

n due

disc and to

can be formed.

r

Section

B7.3

31 January Page 19

Toroidal

Shell:

(52t= M_

M 2 + Q282 Q2 + M261Mi + Qz61Q1 + A2tP

1969

fl2t =Mfl2 M2+Q_22 Qz_Mfl M1+_IQi+fl2t p

Cylindrical Shell: 5c =MS cMI+QSQ1 c

+ AcP

tic =M_Ml+
The following compatibility equations must be satisfied:

5s = 52t

fls = fl2t

5 c = 5It

tic = tilt

Following rearrangements, be obtained.

considerations

of the

a system of four linear In matrix form they are:

relations equations

above

and

with

some

four

mathematical

unknowns

will

finally

M1

M2 +

Alt

- _c p=O

B2t

-/_s

tilt

- tic

SectionB7.3 31 January 1969 Page 20 It is notedthat two imaginary cuts lead to four equationswith four unknowns: M1, M2, Q1, andQ2. Previously, when considering only one imaginary cut, only two equations with two unknownswere obtained. Consequently, if n imaginary cuts are introduced simultaneously, 2n linear equations with 2n unknownscan be obtained. It can be concludedthat the problem of interaction is reduced to the problem of finding rotation (fl) and displacements (Ar = 5) of interacting structural

elements

due

(M = Q = 1) (around introduced into a set and

Q) will

B7.3.3

primary

loadings

and

the

secondary

loadings

displacements indeterminate

then will be values (M

be found.

EDGE The

to the

the junction). The rotations and of linear equations and statically

INFLUENCE

shells

COEFFICIENTS

considered

in this

section

are

homogeneous

isotropic

monocoque shells of revolution. Thin shells are considered and all loadings are axisymmetrical. Par'_graph B7.3.4 will present necessary modifications of derived formulas for nonhomogeneous material and nonmonocoque shells. B7.3.3.

i

General

Unit on upper

Discussion

loadings

or lower

(defined edge

in Paragraph

B7.3.1.2)

are

the

forces

in a shell

loadings

acting

of shell:

M = I lb-in./in. Q = 1 lb/in. Unit influences

are

deformations

to unit loadings. Influences progress very far into the

of this nature shell from the

shaped

at this

represents

shells

are

covered

a bulkhead,

which

and

of revolution

due

are of load character and do not disturbed edge. Various differently

location.

is characterized

Of special with

interest is a shell that = qSma x 90 ° such bulkheads

+

Section

B7.3

31 January Page 21

are very common shells are tangent When

th_ values

the deformations, mine discontinuity The unit loadings. It has loadings

are

in aerospace vehicles and pressure vessels. to the cylindrical body of the vehicle. of deformations

along with the primary stresses (Paragraph

bending

theory

The been

is used

fundamentals mentioned

of local

importance.

to the unit

deflections

and

N

UNIT-EDGE

N

M

explained

internal

loads

that

are

available, to deter-

due

0

due

to unit

disturbances

due become shell.

SOLUTIONS

Q

& r

s

i

q

A ?

:,

q

to

previously.

_ _: 20 ° and will for a spherical

LOADING

M

bulkhead

coefficients

were

It can be concluded

B7. 3.3-1

The

can be used

the influence

procedure

to edge-unit loadings will disappear completely for negligible for a > 10 °, as shown in T_,ble B7.3.3-1, TABLE

loadings

deformations, B7.3.2).

to obtain of this

that

due

19(;9

D

q,...

Section B7.3 31 January 1969 Page 22

edge

Table loadings,

B7.3.3-1 illustrates a very important conclusion: practically all parts of the shell satisfying the

will remain unstressed and undisturbed. These parts satisfying equilibrium. They do not affect the stresses disturbed zone 0 < _ < 20 ° in any way. The material deleted

because

this

material

does

not contribute

due to the unitcondition _ _> 20 °

will not be needed and deformations above _ = 20 ° can

to the

stresses

or

for in the be

strains,

which are computed for the zone defined by 0 < (_ < 20 °. No values of stresses or deformations will be changed in the zone 0 < c_ < 20 ° if we replace the removed

material

imaginary B7.3.3-1 conclusions:

with

any

operations. are statically

1.

The

shape

Consequently, equivalent.

spherical

shell

( M = Q = 1), acts as a lower ment defined with _ = 20 °) . rest of the shell has ( Figure 2.

If any

of shell

shell

(Figure

B7.3.3-1)

which

illustrates

cases (A), (B), and (C) of Figure This discussion leads to the following

of revolution,

loaded

with

the unit

loadings

segment would act under the same loading ( segConsequently, it does not matter what shape the B7.3.3-2).

at the lower

portion

(which

is adjusted

to the

load

edge)

can be approximated with the spherical shell to a satisfactory degree, the solution obtained for the spherical shell which is loaded with M = Q = 1 all around the edges (Figures B7.3.3-2 and B7.3.3-3) can be used for the actual shell.

I

Unhmde,

d,

Unstressed,

Und(ffornmd

(A)

FIGURE

and

Part

!

(j_)

B7.3.3-I

STATICALLY

(c)

ANALOGICAL

SHELLS

Section

B7.3

31 January Page

1969

23

I

I

FIGURE B7.3.3-2 VARIANTS FOR

DIFFERENT UNSTRESSED

FIGURE

B7.3.3-3 APPROXIMATION WITH THE SPHERE

PORTION

place

3. When accuracy of a = 20 ° .

requirements

are

relaxed,

a = 10 ° may

Another approximation, known as Geckeler's assumption, i. e. , if the thickness of the shell (t) is small in comparison with radius (r 1 = the edge may mends a/t > cylinder for shaped graphs.

shells

B7.3.3.2

w'"'

in

may be useful; equatorial

a) and limited by the relation (a/t > 50}, the bending stresses at be determined by cylindrical shell theory. Meissner even recom30. This means that the bulkhead shell can be approximated with a finding unit influences. Many solutions can be presented for various due

to the

Definition The

be used

general + 4Klw

unit loading

action.

This

is done

in the

following

of F-Factors solution

of the homogeneous

differential

equation

= 0

can be represented with hyperbolic functions:

the

following

combination

of trigonometric

and

para-

SectionB7.3 31 January 1969 Page 24

cosh kL_ cos kL_

sinh kL_ cos kL_

cosh kL_ sin kL_

sinh kL_ sin kL_

where kL is a dimensionless parameter and _ is a dimensionless ordinate. In the sections which follow, F-factors will be used that simplify the analysis. Definitions of the F-factors in Table B7.3.3-2 are taken from Reference 2. As a special parameter for determining the F-factors, _?is considered as follows: F = F(_7) i. e., For

a cylindrical

F 1 = sinh2_7 + sin2_

shell

_?=kLor

7?=kL_

andk= _Rt

For

a conical

shell _] 3( i _#2)

_?=kLor

7?=kLandk= _/tx

cotcy0

m

For

a spherical 77 = k for

shell F.;

7? = k(_ for

1

Graphs

of the functions

F.(a)

and

k = _]3(1-p

2) (R/t)

1

are

presented

in Reference

2.

2

Section

B7.3

31 January Page

TABLE

!

B7.3.3-2

F.(_) 1

AND

F.

1

1969

25

FACTORS

Fl{ 0

Fi sinh ] kL - sia I kL

1

stab 2 kL + sia z kL

2

sinh 2 kL_

3

sinh kL_ cosh

kL_ + sin kL_

cos

kL_

siah

4

shah kL_ cosh

kL_ - sin kL_

cos

kL_

slnh kL cosh

5

sin 2 kL_

sin 2 kL

6

sinh 2 kL_

sinh 2 kL

7

cosh

kL_ cos kL_

cosh

kL cos

8

sinh

kL_

sin kL_

siah

kL sin kL

9

cosh

kL_

sin kL_

- sinh

kL_

cos

kL_

cosh

kL sin kL - ainh

kl. cos kl,

10

cosh

kL_

sin kL¢

+ siah

kLt

cos

kL_

cosh

kL sin kL + sinh

kl, cos kl.

11

sin kL_

12

sitffl kL(

cosh

13

cosh

kL_

cos kI.,¢ - sinh

kL_

14

cosh

kL¢

cos

kL¢

15

cosh

16

sinh

17

exp (-kL_

cos

kL_)

exp(-kL

cos

18

expt-kLt

sin kL 0

exp(-kl,

sin klJ

19

exp[-kL_(cos

kLt

+ sin kLO|

exp[-kL(cos

kL + sin klJ

2o

exp[-kL¢(cos

kL_

- sin kLO]

exp[-kL(cos

kL

+ sin 2 kL_

cos

kL_

sin

kL cosh

kL cos

kL + sin kL cos

kL

kL - sin kL co8 kL

kL

kl,

sinh

kl, cosh

sin kL¢

cosh

ki, cos kl, - sixth kL sin kl,

sin kL¢

cosh

kl, cos

kI,_ sin kLt

cosh

kI, sin kL

kL¢ cos

ainh

kL cos

kL¢

kL_

+ siah

kL¢

kl,

kL + sinh

kl, sin kl,

kl, kI.)

- sin kL)]

]

Section

B7.3

31 January Page

B7.3.

3.3

Spherical

1969

26

Shells

The boundaries

of the

shells

considered

herein

must

be free

to rotate

and deflect vertically and horizontally because of the action of unit loadings. Abrupt discontinuities in the shell thickness must not be present. Thickness the shell must be uniform in the range in which the stresses exist. I

Nonshallow

Spherical

Formulas are shells

shells Unit-edge bending

may

was

Tables II

_(R/t)

which this

B7.3.3-4,

section

presents,

relation

cot

_ -

not decay

cally opposite edge around the apex. Tables

upper

edge

shells. Open at the apex.

of the open

of the formulas

shell.

Linear

presented.

will be used:

_)

• and

for shallow

for the

that for shallow

will

or

spherical opening

B7.3.3-6

spherical

are

shells,

presented.

the

solutions

which

i --

is characteristic

loadings

lower

and open circular

derivation

B7.3.3-5,

Shells

the

means

for

2) ; _ = _1-

Spherical

This satisfy

at the

designations

2 3(l-/a

B7.3.3-3,

Shallow

act

employed

The following

k=

Shells

will be tabulated for closed that have an axisymmetrical

loadings theory

of

B7.3.

before

loadings,

3-7

category

shells

of shallow

the disturbances

reaching

the

disturbances

and B7.3.3-8

are

apex.

spherical

shells.

resulting

from

Consequently,

will be superimposed

presented.

Physically, unit-edge

from

diametri-

in some

area

Section 31

B7.3

January

Page

TABLE

Q¢

-x/2Q

B7.3.3-3

sin

cos

- Q0

NO

2Qk

sin

mle

M0

RQ k

sin

¢1

e_°

sin

Ol(cot

RQ k 2,_

2Mk + _ a

s +

-ka e

sin

- QO cot

-ks

cos

sin

ks

2_-2

ks

+ 4)

+.M(_

E_

-2 _2Qk

2 sin

_le

]::t(Ar)

RQ

01 e-kS

_° (_ _) sin

[2k

_Mk2

e -ks

sin

sin

M k

4) cos

96 e

e

R

R sin

EtZ

Et(Ar)

-2Qk

QR

2 sin

sin

_ = 0,

01(2k

R O1 - #cos

For

01)

01::

2Mk 2 sin

90*,

kc_ + pM_

(b(N 0 -pN(b

4k3M

sin

cos

cos

0 =
01

)

c_ + ,T

-_o

oos_ _o_(_o+ _)] For

(.

4)

(p

- _

ks-'_p

cos (k c_ +

-k(_ cot

_

c_ +

ka

_b

,_Me-kS

4)) e -ks

27

SHELL

0

sin(kc_

sin

SPHERICAL

_° (_ _)

NO

M

cot

91 e

CLOSED

1969

c_ = 0

Et_

-2k2Q

-

4k---'_3M R

Et(Ar)

2 RkQ

2 k2M

01

k(_

)

Section

B7.3

31 January Page 28 TABLE

B7.3.3-4

OPEN

Boundary ot

:: 0(0

=-'"1):

M

SPHERICAL

SHELL

Conditions

= 0 ('2

-Ilik

2_ I{

,3 I

sin

";((_'J

-

---i: 1 |"ll((_J

,31 [ -l.'l_l{_)

+

_in '."1 -|

M(j

_

_

_

['i_(_)

2 --1.1 1",1 I"_L_)

I:_.,)

l"_((_)

r

]"1

_,

+

]'

sin

-l.'u((,,)

l:l_]t(L)

ll'

Ilik

(5 t

}"t

I),'lo]'nl;itiol]._

((')

(_),,_

sin

-

and

1'1

k

"-((1),

'

-Ilik

,)

--'

N j

,.', ('()t

--

I11_.: Sill

4) 1

Fol'ces

|tllx.'rna[

N+5

cos

_

+ Nik

+

'"1

1

sin

'

: Qik

l._._l,

tlik

-

--i.,1 1:2 I":_L(_ )]

- _F'1,j_.

+

cot ,:,

I"2 ["1 I{ k -llik

-Iiik_

_

ICk

" sia

sin

_,)sin

01

01

Fe((_,

_-F9(_)

)

_

-2

F_ F1

Fg((g)

c5

'

_

l",,,(_) +

+

F_?,. F1

FT(_)

_

1"_o(_7

1969

Section

B7.3

31 January Page TABLE

B7.3.3-4

OPEN

SPHERICAL

®

Boundary _=0

Me

SHELL

1969

29

(Continued)

I

Conditions

= -Mik

Q¢, = o Ot =_

0

M_b=0

Internal

Forces

and

Deformations

F15(°t)

+ FI Fie

N_

Mik

No

-Mik 2K---_21F-_I:I4((_)+FSFI3(c_)R F I F I - F3F I F1°(c_']

cot

(b _-

_1

(o_)

-

FS(O_

F1

r

Mik

"

LF1

F1

FI6 (c_)

- FI

°

M_

1 - FI

k

FIS((_) -/_Fi4(_)I_

F1

M0

-Mik

FI-F-'IFI3(_)

Ar

-Mik

2k2 'Fs -_ sin (b£ FI4(_) +

/3

-Mik

4k 3 _

_ - Fi

i

- _Fi FI4((_)+ F--ALFg(a)]F t _ Fi

Fia(ot)

F__ Fi0((_} - FI

FI6(oQ _ .__ Fls(Ot) _ F$ Fi _i FT(a)

Section 31

Page TABLE

B7.3.3-4

OPEN

SPHERICAL

NQki

/

SHELL

(_

IIki

l_ounda ry Conditions

(_O = 0 (_ = ao(O

M

--- 0,)

,, : 0

llki

= -Qki

sin o., - Nki cos

Internal

and l)cform:ltion.s

NO

llki cot O sin 0 2

N

llki

2k sm 02 [- _FI I"-. t _) + ]_1,,1FI0_ (_ _]

Qo

ttki

sin

3I o

o 2 IF, f_

I:_

Forces

02

r,0

l.lot(_)

t.,

-2

- _.q2F8

Hki _- sin ¢:'2 FI I"sLe_) - F!

1:_,)]

1_

l"_t_

]

F.qto')

Mo _r

j3

Rk -Hki

sin 02 _

Hki sin 02

_-

F8

ri% 2 sin 0L--_l FT(_

F9(_)

+ FI

- 71

I"10(o

'l

B7.3

January

(Continued)

30

1969

SectionB7.3 31 January 1969 Page31 TABLE B7.3.3-4

OPENSPttERICAL SHELL (Concluded)

l_()tlll(f;Iry

oe = oe0((')

= 02)

:

hl

,\I

,/)

('()nllilioll._;

ki

(..)¢, : 0

Int(,rn:ll

2k

NO

Mki _--

:)rl(l [)('f()r)nati_))]s

Fol'(,es

"I"A

-2

_

I"1o

l.'T((t.) +

r 2kl_ t.'u T[]71 I"1o("

-Mki

M_

Mki

M0

"[F--a - M kiIF t --_)t-L9-k F_t(,,)

Ar

2k2 Mki-V-_--

sin

4,

--

[,'lr,{(_

)

-I ],_ + lCl

Q¢)

2 _FI 1,8(_ )

---].' 1

-_FI° _7,

]2{:1

J'_ ,,:

((tr

0

- pXl.'a((_ _ + 2_I"1 Footk

1"8 1"10 -2--FI l.'.,.(,_) + 177- 1.',o((_)

Mki _ 4k3 IF8 V t l.'.q((vj + I. !:'lt_ 1 i,,¢,1_ ,

J

Section

B7.3

31 January Page

_C i

CD s.

_0

2_

_Q

i

m 0 _Z (9

..

c

\ d

_

/

jt

/__

_/_

.._

/ i _._-I

-S

__.

32

1969

Section

B7.3

31 January Page

TABLE B7.3.3-6

OPEN UNIT

(OR

CLOSED)

SPHERICAL

DISTORTIONS Rotalhm

AT

LOWER

SHELL

196b

33

EXPOSED

TO

EDGE

[J ik

, Boundary

a=0(_b=01)

Conditions

I

: Ar=0:fl=/Jik

a =cr0(q) = qb2) : Ar = 0:fl= 0

Internal Forces Et "F 2 F_(_) _ ,2k---2 .F 1

N

flik

NO

-fiik

Q_

Et flik 2k2

Me

-flik

REt --_

M0

/3ik

REt 4k 3

Ar

-flik

-_ik

cot

Et _ F 2 Fg(_ )+ 2-k - F 1

and Deformations

+ F__ Fg(Ol) FI 2

Fa F8(_ ) _ Flo{_ F1

F2 FT(a ) + F4 F1 _ F_(_)

R 2-k sin

F2 Fi0(o_) FI

0

F21 Fs(a - F

_ Fs(oZ)

+ 2 Fa Fl

) + _I F4 FI°(_}

]

- FS(_}

+ 2 F-A F?(_) F1

F2 Fg(o_) - F1

1

-

+ Fg(c_ 1

Fs((P)

FT( )i"!

J

- FI0((_)

Section 31

B7.3

January

Page

TABLE

BT.

3.3-6 UNIT

OPEN

(OR

CLOS]_D)

DISTORTIONS

AT

SPHERICAL LOWER

Boundary o ,.(_ = _bi)

: _=

a=ao(Cp=cp2):

0. AV=

_=0,

EDGE

SHELL

1969

34

EXPOSED

TO

(Continued)

Condition8

AVik

AV=0

Internal Forces

and Deformations

AVik Fitcot gb N_

R[(

1÷_}

sin

g)l F3 + k cos

_1FI]

[-

F3FT(a)

- FsFIs(_)

+ FsFI6(_)

]

- AVik Et k N_

R[(I+_)

sin

Qc_

]_[(l+p)

M4)

2k[(l+p)

sin

M0

2k[(l+p)

sinOtF3+kcoscPlI,'t]

AVik sin

-/'aVik

4)1F3+kcos

_)IFI]

[F3Fs(c_)

- FsFt4(_)

+ F_FIs((_)]

Et g)i F3'+k

cos

g)IF1] [- F3Fz(c_) - FsF15(c_)

+ F'6FI6(c_)]

Et _1

F_ + kcos

011"I]

[F3F1°(_)

F3

- FsF13(c_)

Fs(c_)

- F6F14(c_)]

-pFl0(c_

-F 5

Ar

2k 2 AVik R[_I+/_)

sin _biF 3 +kcos

_btF'i] [F,F,(.)

- FsF,s(a)

- F, FB(-

_

Section

B7.3

31 January Page TABLE

B7.3.3-6

OPEN

UNIT

(OR

CLOSED)

DISTORTIONS

AT

SPHERICAL

LOWER

SHELL

EDGE

Displacement

A rik

I

"_ ®

1969

35

EXPOSED

TO

(Concluded)

& rik

Boundary

a =0(0

=01):

Ar=

=0(0

=02):

Ar=0,

Arik

, fl=0 B =0

Internal

No

Conditions

0

Et Rk sin

Arik

cot

Arik

Et R sin 01

QO

Arik

Et Rk sin

0t

M0

Arik

Et 2k 2 sin

0t'Ft

Mo

&rik

2k 2 EI sin

0

Forces

F3

0

FT(a

and

) +

Deformations

F5 F1

Fls(a)

-

F__ Fls(a) F1

1

0

l

_F N

_. Fg(_ F1

)

"F__ ! FT(c_) FI "F3

-

___ Ft._(_ FI

) + _ FI

F 5 Fis(+) Ft

F10(o_ ) -

-

F13(_)

F s F16(c_ ) Fl

F 5 Ft3(_) F 1

-

F 6 F14(a) F t

+

-_r

Arik

_ sin

Arik

2k R sin 01

01

F F3t

-cot_k

F,(_)

F6 _Fls(O_) F 1 k

"F 3 F9(¢_) FI

_ F_ Fi

_#Ft0(c_)

F F_1 [COk._

+

- pF14(u)-

Ft4(_)

"F 3 FS(_ ) __FI¢(_} FI - FI

+ F_ FI

1

Ft3(o_ )

__ - F1

FIS(_)"

F16 (

--

)

_

"l

{ -

_F13(ot)

!

Section

B7.3

31 January Page TABLE

B7.3.3-7

EDGE-LOADED

SPHERICAL

Complete

(or

Spherical

Cap,

1969

36

SHALLOW

SHELLS

"Long") rl -

rz - a

[

!

Shallow, Constant Edge

t, Moment

M l

/--o°-.t / _soi8

Esslingerts

2

d4Q_

Differential

de 4 Equation

Approximation

+

3

d3Q_

_b

d2Q_

¢2

d_3

3

d_bZ

d_

- _"_ Q_

and

az

Boundary Conditions

where

k 4 = 3(1-_z)

Q¢)

Solution

Forces

=0

CIM k

M

Be r,(

k._/_qb ) -

= M

C2M ---_--Bei'(k_'.,_)

Q0

-

NO

= (nl CIM

+ n2 C2M)

M t

NO

-: (ql CIM

+ _72 C2M)

M t

M(

= (m i CIM

M 0 : pM(_

+ m 2 C2M)

+ (k I CIM

CVM

Approximate

useful range:

=-

+ k2 C2M)

Et_O

Ber'(k'4-2¢}

, Bet' (k _/'2¢) see Reference

For

CIM,

Xa,

For

n i, n_ ....

and %

M

Xa CwM

= Etqb---_

_0 < 20°.

For

C2M,

M

M

Xb Edge Influence Coefficients Notes:

3

dQ_

+ ¢_

3.

as functions of _2"@0 see Table

etc. as function of k _42"@ see Table

BT. 3.3-8.

B7.3. 3-8

+ 4kaQ0

= 0

Section

B7.3

31 January Page TABLE

t_'olllplult'

( oI" " 1,ollt_")

Spherical

Cap.

B7.3.3-7 SPHERICAL

r I

r 2

EDGE-LOADED SHALLOW SHELLS (Continued)

.'1

5ha I 1o,,_,, ('CIIIS|

:1ZI( t,

tkl_e

I"_rCC

f II

/"

\°< I'i ,_,'.,1i II I._t I" ?_ A p|_l'a_.\

h11:ltiol|

I)it_crt,_tia[ l-:q ua tionl |:_ouT]da

:13lt]

rv

C_miitions

a 2

I_

('211

N °

::

tn 1 Cll

N9

=

(_1

M

b=

M 0 =

I + n 2 (.'21t)

CIH

(m

I

+ _/2

CIH

pM

+

tl

II

C21I)

+ m 2 C2H)

(k 1 C1H

1

+

tit

k2

C2H)

lit

X

Y¢ a

Edge

Influence

CVH

Coefficients

Notes:

Approximate For

Ber'(k

For

CIH,

For

hi,

useful ',]_b)

C2H

a 8 ....

range:

, Bei_(k

, W a,

etc.,

=

and

as

a

Et_---'-O

CwH

=-

E_b-"_

_b o < 20". n/2"@)

X a

functions

as

see

Reference

functions

of

of

k _/2"_b

3.

k

$_

_r2_b o see

Table

.BT.

Table

3.3-8.

B7.3.3-8.

37

1969

Section

B7.3

31 January Page TABLE

B7.3.3-7

EDGE-LOADED

SPHERICAL

Complete Spherical Shallow Constant Edge

SHELLS

SHALLOW

(Continued}

r I = r 2 ==a

(or "Long,) Shell,

M

I

M

Opening, t, M

Moment

Basic

Esslinger's

Dillt'rential

Approximation 3

Equ:Jtion

:rod Dound:l Conditions

d++,

ry

+p d_+

where

3( 1 -#z)

k£

Q¢ I 4'= I'o

-

0

_++ de:

7,3

dl+

t:-;--

M,],

I'

+1'_

M

.

+

("

_¢dtllil>l]

(_,

7

Nq, :IH3

N

i-'7--

C3M

M

L-7-

t- n 4 C4M j

= (r_:_ C:+M + _l+!CtMJ

,M _-M _-

IX|_ _ Ira:+ C:I M + m4 C4M j M

M

I':dge Influvtlce Coelflcients

Notes:

38

= CVM

Approximate For For For

useful

Ker I (k ',/2"_), C3M, %,

phl]j

C4M, n 4 ....

X c, etc.

range: KeU

+ I,k 3 C M , k I C4M _ ,",1

Xd i.:t2¢,o

CwM

X c = Eta0

_o • 20". (k _J2¢J

see

Reference

and X d as funetio-a am functions

of k',/2_b

3.

of k'4r'2_l see

Table

see

Table

B7.3.3-8.

B7.3.

3-8.

1969

Section 31 Page TABLE

B7.3.3-7

EDGE-LOADED

SPHERICAL

SHELLS

SHALLOW

(Concluded)

r I _ r2 _ a Complete Spherical

I

(or "Long") Shell,

Shallow Opening, Constant t, Edge

Force

H

Basis

Esslinger's

Approximation

Differential

de 3

and

Equation Bounda_, Conditb,ns

,_" de 2

_b3

:t 2

_ nere

k4

- II ,/L> _'11

d,/_

'f

,xt,j, /

'/' rl

2

,,

C411

["I_ r c'(,,'-,

N/_

X

Izlt

C31 ! * n 4 ('411 _ tl

: (n: I C311 -

_4 C4111 tl

Mq_

{m 3 C311 + m 4 C41t J lit

hl

#Mb

_- (k 3 C311 + k 4 C411J

lit

X

Noles:

CVH

Approximate For

Ker'

For

C3H,

For

n_,

W c

l':d ge hffl ut.,nc,c C_efficicnts

useful

[k

,4'_),

C4H , W,

n4 ....

range: Kel'(k

etc.

e

= Etq_----_"

C_'ll

E'_b--_

_O < 20. ',J'2_)

see

Reference

and X e as functions as functions

of k ',_

3

of k _/'2_b$ see

see

Table

Table

B7.3.3-8.

B7.3.3-8.

B7.3

January 39

1969

Section

B7.3

31 January Page TABLE

SHALLOW

B7.3.3-8

Equations

ClH

=

K'_

C2H

=-

2

for

[k.q_$o_2-(l-kt)_'t'

the

SPHERICAL

EMllnger

Coefficients

C2M

_z ,J_"

Table

B7.3.3-7

- K,.

71' "]_

l-P

_J

+ h2_ _ 2(k 45_e 2 _ _'_ - _' _._ _ _l-,-_k _,,_ _;-_ + _-';_1

L

a

,,j

for

COEFFICIENTS

]

ClM= - K,,

K,2 =

SHELL

J

_ 0,'* ....

_1'_"" _-':J

___

k'_q,

k\'_'l'

k",_,I,J

'

k.]_-',f

! -

1

k, •t/

_1,

_,

related (Reference

_,

1 • _:

"

1_

_'

are

Seh|eieher

to Bcssel-Kelvtn 1, pages

1969

40

functions functions

491-494,

and

their

by _tl = ber,

and 6-17,

6-20,

derivatives _

- -bei,

and 6-32)

fur

_

the

¢1 = ber_,

III

argument and

_

k x,"2,,_ :lnrl _.'e = -bei',

Section

B7.3

31 January Page TABLE

B7. 3.3-8

SHALLOW Equations

Kit __

C3H =

=

__

C3M

=

-K34_;

W

=

Xc

=

-K"[

the

Essllnger

SHELL

Coefficients

for

COEFFICIENTS Table

(Concluded)

BT. 3.3-7

]

_i-.2)

K3t 1 - V})

_f_'(

c

for

[t.J"20o_4-(l-#)_s'

C4H

SPHERICAL

[(k

',,f2"_ o) 3

(k_J_*°)2(_q_'+_'_,]

[

]

t-g

k#-24,o

., =4-_" _,' kJ_¢

L

k4 : - 4"_"-

@_, _4,

(Referenoe

_t

_,

1,

are

8chleicher

pal_es

491-494,

fuactloas

sad

6-t7,

and

their

6-S0,

derivatives

aml

6-32)

for

the

argument

n$

k _]2"0 a_d

1969

41

are

Section

B7.3

31 January Page 42

B7.3.3.4

Cylindrical

Shells

This paragraph presents the solutions loaded along the boundary with the unit-edge horizontal displacement, and forced rotation in the poses,

cylindrical negligible

x, the analyst conservative, a.

b. the

general

constant

case

for long and short cylinders, loadings (moments, shear, forced at the boundary). All disturbances

by edge loading will become, x = _/'R't. If the height of the

is dealing with a circular the following precautions

If kL _ 5, the

more

The

wall caused at distance

If kL -< 5, the more as short cylinders.

designated

1969

ring instead of a shell. should be taken.

exact

simplified

for practical purcylinder is less than

theory

formula

Further,

is used,

and

such

cylinders

is used;

this

is a special

to be

are

case

a.

k is defined

as follows:

k 4= 3(I -p2)/R 2t2 The length cause

primary

solutions

(membrane

theory)

of the cylinder. The boundaries must of the action of the unit-edge loadings.

uniform

in the range

I

Cylinders

Long

The presented

D-

where

formulas for the in Table B7.3.3-9.

Et3 12( 1 -

the

stresses

are

will

not be affected

by the

be free to rotate and deflect beThe shell thickness must be present.

disturbances caused In this table,

by unit-edge

loadings

are

of

Section

B7.3

31 January Page 43

TABLE

B7.3.3-9

LONG CYLINDRICAL SHELLS, LOADING SOLUTIONS

Q=I

1969

UNIT-EDGE

M=I

'7 l t

,

L

I

4

I NX

NO MX

M0

_.__R

Q Ar

Nx

NO

Mx

M0

Q

Ar

¢

t N X

Oak N_

-k_

• l,k 2 c -k_ cos (k]} +

z_

L

Mx

L -k_, e sink_

M 0

_M

|-

uM x

X

2k -_

Q e

sin _ +

ok/3 e

L

- _

_.k:D

sin

k_

-k,_

_

co_ k_

&r

=

- E't

For L_

the

Case

_ = 0 -L Dk

L3

_L 2

0 - PNx

-

_

x/_Dk2

e

COS

_

+

Section

B7.3

31 January Page 44

12

Short

Cylinders The

following

k 4 = 3( 1 p

constants

are

for

Tables

B7.3.3-10

and

B7.3.3-11:

= sinh 2 flL - sin 2 flL

K 2 = (sin

flL cosh flL cosh

K s = (sinh

2( sin

flL - cos

flL cosh

A summary B7.3.3-12. Conical This shells angle

of (_0 >-- 45°-

flL sinh

ilL)/p

flL + sinh flL cosh flL + cos

The formulas for unit-edge Tables B7.3.3-10 and B7.3.3-11. must be satisfied.

B7.3.3.5

flL)/p

flL sin flL/p

K 5 = 2( sin flL cos K_

flL - sin fiL cos

2 flL + sin 2 flL)/p

K 4 = 2 sinh

conical limiting

used

t2

p2)/R2

K 1 = (sinh

Table

1969

paragraph

of edge

flL siah loading To use

distortions

flL)/p ilL)/p

disturbances are presented these formulas the relation

resulting

from

edge

loadings

for nonshallow

open

in flL -< 5

is given

in

Shells presents

the

solutions

in which _0 is not small. _0 • It is recommended If _0 = 90°,

the

cone

or closed

There is no exact information about that consideration be limited to the range

degenerates

into

a cylinder.

Section

B7.3

31 January Page

1969

45

_g

0

5 N 0 r_

'

r,.)

_1_

'

_ _Zl_

_

-_ _

_z _Zt_:

,

_l_Z ,

CY

n

M r_ _

_

r_l

.< r,.) ,.a

Z !

o !

c4

t

t}

a7

_"

=

'

¢1ff

,<

x;

_

_

<1

,:o.

Section 31

January

Page

8

®

v

--T

_

3

)y

0

a._

B7.3 1969

46

aa.p

! i

I--1

,d +

z¢

i

b_

_

[_

I

fv

_

-T

_21_2

'

A

,__.t

I

®

.-D

+

--& _

M

,_, _1=

z

,

+ _

+

, _

_

L _T"I ._J

L

I

_

Section 31

B7.3

January

Page

19,'_;9

47

_1_ o,,._

2;

z_

(Y

(_

I

o0

Z O

J.d •' b-_

._1_

0

rd

?,

2_

z_

(Y

IV

0 Z

P_ '-_ I r_

,.td I

Z

(.9 ,!d

d

(_

-,"4

I

I

I

I

3'

0_

®@

®

(_

,

®

®

Section

B7.3

31 January Page 48

Another case short

limitation

of the sphere, distance from

at _f_00t). (or apex)

must

be applied

the disturbances the disturbed

to the

height

due to unit-edge edge (for practical

boundaries

must

be free

to rotate

be uniform The

in the

formulas

shells

are

circumference

range

are

assembled

characterized

in the plane

Linear

in which

bending

for

h , _Rot

vertically

closed

and

open

of tile upper

conical part

was

used

to derive

the

Conical

the

Open

some

,

R 0=

formulas.

is practically constants are

MAX

If

dealing important:

R

12( 1 - p2)

Since opentng

above

following

are

indicated

in Fil,mre

B7.3.3-4.

R is variable and is perpendicular to the meridian.

cone,

shells.

sin _b

designations

Open

and hori-

to the base.

_]'3(1-p2)

stant. Table B7.3.3-13

I

edge edge.

Et 3

D-

Additional

will decay at a approximately

exist.

the height of the segment is less than _{-Rt, the analyst with a circular ring instead of a shell. The following

k-

in the

loadings. Abrupt discontinThe thickness of the shell

stresses

by removal

parallel

theory

the

As

by an undisturbed respective opposite

and deflect

zontally as a result of the action of the unit-edge uities in the shell thickness must not be present.

conical

loadings purposes,

Consequently, a '_igh" cone is characterized as a result of unit-loading influences on the

The

must

of the cone.

1969

unit

formulas

at vertex

Shell,

Angle (qS) is con-

presents the formulas for a closed conical shell.

Unit

influences presented ( Figure

Loading are

at Lower

not progressing

in Table B7.3.3-5).

B7.3.3-13

Edge very

far

from

can be used

the e(li4e into tim [or

the

c,_,Lc wiLh

Section

]2,7.3

31 Janu:_ry Page

Xm

_.

FIGURE

B7.3.3-4

CONE

NOMENCLATURE

( )l_!njng

I

/"i

x

I

FIGURE

B7.3.3-5

OPEN AT

CONICAL

LOWER

EDGE

SHELL

LOADING

49

1969

Section

B7. 3

31 January Page

TABLE

B7.3.3-13

CONICAL

SHEIcL,

UNIT-EDGE

1969

50

LOADING

SOLUTIONS

£ llorizontai

Unit Unit

M,)m('nt

lJoading_

r

N N

-,]:-.','os ¢ '21{:k

sin':

e o

cos

t'

_ +

2kcos

_

_in

k.

cos

k_t

h2

Q

h: ,4r_2Hk:

_ sin

0

-

-\'2

sia

0

z-kn '

• e

sin

cos

+ z "4 )

(k(v

kc_

-k_ e

(

ko

h? e

0

cos

:)

c

cos

uM,/,

-k_

h l()l'llI;lli¢)tl>

h

_I)I< -I<(_ ? sin

l'

I vo_

-

k.

21)_

,_nk

_Ul 0

h

:'l_

* ; I]

,')

"_:l)k"

sin

-k() h ('

('()_ I)k

_)

[,or

(_

_in

k¢i ,.)

II

112 ,.._ I"

2Dk 3 sin

_

1 -

2Hk

.uin

,_

2l)k ° .'-;in ,.')

112

- ½'])k: sino

Dk

sitl

_Jll

k(¥ 1

k_ln 2 _', j

T

h2 -k,, .i,, ( .-4 )

,,')

+

k(,

ninny

:-'I< '
+ -_

k(_

r

Ilk

¢" vM4'

]}i

.-.%1"

sin

_,f:_-2 c -l_,,v

h ('(,t .M

-kcr

2xf_2Rk 2 sin 2 (5

-k_

[I

L< t'

e

h

Section

B7.3

31 January Page

II

Open

Conical

Shell_

If it is imagined

Unit

Loading

that

the

at Upper

shell,

loaded

51

Edge as shown

in Figure

B7.3.3-6(A),

is replaced with shell as shown in Figure B7.3.3-6(B), the result shell loaded with unit loading at the lower edge. The same formulas for determining edge influence, but it is noted that _ > 90*. An additional used

for

closed

expressed B7.3.3.2.

set

cone)

of formulas

is presented

for

open

in Table

conical

_3(

shells

B7.3.3-14.

with the functions F. and F.( 4), which l The following constant is used for k:

k-

are

(that

These

is a conical are used

can be also formulas

tabulated

in Paragraph

1 _/_2)

_] tXm cot

o_0

¢

/\ /

I

\

/

I

\

' M

(A)

FIGURE

BT. 3.3-6

OPEN AT

CONICAL

UPPER

1969

EDGE

SHELL

(!_)

LOADING

M

are

Sectioa

B7.3

31 January Page

v

@

r_

Z 0

0 r._

_

_aj

Z

I

_o r_ e,o

®

0

/

Z

I

0 I I

',4

_

_

".7

d

52

1968

Section

B7.3

31 January Page

1969

53

_9

O

o

Jr

®

O

z •t-

+

-%

+

I

O

J

0 oO

O o

..,.,

_

g

ga

0 o

G'

izl

M

_

,

_I_

_I_

M I

Z

!

+

+

.-I-

a.k.0

O 7:

.tz I

O

I

!

+ +

A

!

g

r/l 0 o

® O o

,__ r_

2_ °_,.i

°_-,i

I h

!

I

_n

_n

Section

B7.3

31 January Page

In connection stresses

with

some

problems,

and displacements'in

placements

at the

edges

the are

it may

conical

acting

shell

instead

be of interest

(closed

of M and

54

to know

or open),

1969

the

if unit dis-

Q:

direction At lower

boundary

i i

i At upper

Table

boundary

B7. 3. 3-15(

presents

a)

conical

segment

B7.3.3.6

free

Circular

t

flik

= unit

rotation.

to this

problem.

answer

resulting

of solutions

for

circular

is presented in this hole are considered.

= deflection

= Youngts

modulus

= Poisson_s

ratio

= thickness

of plate

Et 3

D

in horizontal

Arik

unit displacement = direction

from

Table secondary

B7. 3. 3-15( loading

b)

of a

edges.

= rotation E

in horizontal

distortions

or in the process of interaction nomenclature will be used: w

displacement rotation.

Plates

A collection loading conditions a central circular

the

of edge

with

= unit = unit

k

supplies

a summary

Arik _Bik

- 12(I_ 2)

with

plates

section. These

more

with different

Circular solutions

complicated

axisymmetrical

plates with can be used structures.

and without individually The

following

Section

BT. 3

31 January Page 55

O

-%


O G]

*

£t£

--_

+

+

_D

_£

o

e_

1

8

1969

file I

i

I.i

g

M O

g _

g •_

i

._

,_

._

r'-2"n t""Z"n

+

o_

_1_. o

8 o Z M

.%

_

_z

! + "_'_

"t-

¢.o

I .J

e

e-

I

0

i o

212 I

i1_I %

g

g

o

"

;i

M M z N

_ l._

_ z

It N

.E

Section 31

B7.3

January

Pa_e

1969

56

®

-

_l_Z _ ca.

.-,

_

_

_

_

-4,

cq

¢4

_

,

!

Z co..

¢,i

0

0

ZO 0 _ ._

_._

Z_ _0 _1_

O_

"_

Z

"

_ +

_

' _

,_

t_

"4-

®

¢o

¢g +

G

_

_

_

z_

_*

_

"_

_:

..1 r_

zx

,_

,

Section 31

B7.3

January

Page

57

.:It Z 0

O +

M

gll

+

I

g_

..

0

z

°..._ (D ..=1

= +

,.a

4-

Z

!

= +

+

!

= +

+

©

<

2_

0 (.)

I= I

I

,.Q

I

/

.._ °. m

I

1969

Section

B7.3

31 January Page

M

= radial

19_;9

58

moment

r

Other

Mt

= tangential

Qr

= radial

designations The

The

presented

I

are

presented

in tables were

is presented way will

as for

presented

derived first;

the

in this

by using then

shells.

section.

the linear

"secondary" Finally,

bending

solutions

special

cases

theory. are

(fixed

be given.

Solutions

Primary II

same

conditions)

Primary

indicated

solution

in the

boundary

shear.

formulas

"primary"

moment

solutions

are

assembled

in Tables

B7.3.3-16

and

B7.3.3-17.

Secondary Solutions The

only

unit-edge

loading

of importance

is a unit

moment

loading

along

the edgel (Figure BT. 3.3-7). Table B7.3.3-18 presents solutions for this loading for different cases of circular plate with and without the circular opening at the center. Table B7.3.3-19 presents the stresses in circular plates resulting from III

edge

elongation.

Special

Cases

Special cases and solutions for circular plates that occur commonly in practice are presented in this paragraph. The geometry, boundary conditions, and loadings for special circular plates (with and without a central hole) are shown in Tables B7.3.3-20, B7.3. 3-21, and B7.3.3-22.

Section

]37.3

31 January Page

TABLE

B7.3.3-16

Loading

SUPPORTED

CIRCULAR

Constant

p :- const.;

A=

SIMPLY

Parabolic

p = pa_w

P

p

2a-"_ ;

Qr

:

Distribution

p

v

2aw A ::

Poa2._

pp( 2 - p2)

2a"_

; Qr

=

2wa

Po

fP II Illlllll|lllllllllllJlJll

59

PLATES

_p2);

Po(l

P--£-

1969

P

Illln

N

Vlr

_It

_r

W

Pa2(1 -p) 64DTr

(5+_ 1 + #

(3+:_e

16DTr \1

+ p -

p2 ]

288DTr

1 + p

Pap(13+ i + #5p48D_r

1 +/_

6p2+p4

)

P M r

Mt

]-_

T_

(3 + _)(1-pb

+ p" - ( 1 +, 3#)p 2

+ 5p.-

6(3+#)p2

+ (5+p)p4]

+ 5#-

6(1+

2 + (1

3#)p

+ 5p) p4J

Section

B7.3

31 January Page

TABLE

B7.3.

3-16

Conical

SIMPLY

SUPPORTED

CIRCULAR

60

PLATES ll(:verse

Distribution

(Continued)

Parabo]

it' w

W

p = po ( l - p)

h

P = 2.--_w;

;

Qr

v

P _ Po a2

P

.-_

_

l),)(l

-p)

2;

p

Pp(:_ - 2p t 2aw

=

1969

p

I)oa

- x/> ) :I I,

l)p((;

2w:---{ ; Qr

.:_

-

2:,w

v W

M I"

•

I

i

Mt Mt

_t_

Qr

Pa 2 4800D1r

i+ p

-O 2 + 225p

Pap 240D_

pa 2

"3( 183 + 43tD. i0(71 + 29t0

W

4-

I+ p

6405

(71+29# i+ p

45p 2 + 16p 3)

r

323+83/_

2400D_

_ 5(89+41p)

l+p

+ 225p

4-

Pap 240D_

(89+41_ \ 1+#

p2

l+p

128p

5 + 25p

_

6

90p2

+ G4p3

_ 15p4)

P

M r

240D_

71+29p-

45p2(3+P)

P 2 40D_

189 + 41it - 90p2(3+p)

°

+ 16p3(4+_)

+ 64p3(4+t_)-

!

M r

240DTr P L71+29#

+ 16p3( i+ 4it} 1

- 45p2(i+3#)

15p4(5+_)

1

89 + 41p - 90 p2( i + 3p) +

240DTr

+ 64p3(1+4#)

-

15p4(l+5p)

Section

B7.3

31 January Page

1969

61

_k +

C_ r_l

-'_

I --_

+

CD _L _L +

.p..4

o

O !

I

_k I

I

.,,-.I

+

0q vI

-

I

I

I

II

o

0,_ :::k I

O_

::::k I

O t

J

_"

O !

M 0

LJi_@ +

r-------

vr vi

_D

!

+ Ii

+

+

I

i

I

:::k

I

!

R

_

_

,N

_k

_"

+

+

I I

_q

<

I

P_ oo

I

(y

Section

B7.3

31 January Page

62

! A

=k +

!

I +

O

C9 :::k I

:::k !

+ A

+ I

:::k I

_1 .=-

-.2..

:::k I

(9

• i

X VI

_9 r_

+

"5"

%.

::k I X +

I

:::k I

=£ +

I

::k I

+

+

+

+

+ +

,.a

!

1969

P

Section

B7.3

31 January Page

÷

4-

O

(9 +

vJ ÷

i +

If ,r'X'--, ÷ 4

÷ c_

L.Z._._

-,2 rJ

_>"_ _'

_1_

4

f

d

N

i.

+

[.-,

,.. 0

+

63

1969

Section

B7.3

3 t January Page

TABLE

SIMPLY

B7.3.3-16

SUPPORTED

CIRCULAR

i

Concentrated

Loading

Q A :

P 2wa

;

For

p ->- X

For

p = 0

P

P r

Qr

2awp :0

_t

_

W

M

V3

16DTr

r

M t

L l+p

Forp>X,

-

(1

-

- p2) + 2p2

P -(1+_) 47r

In

lnp

--_-P [lr -(1+ 4_ L

p) ln

For

p = 0,

For

p -> X,

---P47r11-p-(I+/_)lnpl

For

p = 0,

P [1 4-'_

-(1

Mr

+/_)ln

X1

×]

PLATES

64

(Concluded)

J 969

SectionB7.3 31 January 1969 Page 65 TABLE B7.3.3-17 SIMPLY SUPPORTEDCIRCULAR PLATES WITH CENTRAL HOLE Equally

Distributed

2

A=

;

Pa(x''i)'2

,

X>

( lb/in.

Loading

A=

(1

l;k,=x'[3+_4

2)

-X2);

X <

4(1+#)1

+_X

I

:1'

I

,n][]

IIIIIII k

I b=: ax__ r=ap

a

1;

,

.::

,

:aP_l

a:

rU_

t

xtllr

V

IIL_

W

Pa4 { 64---E

16D

r

16

Mt

pa2 16

2

C(3+p)

(3+#-

(1-2X

2) +k2]

4x2+k2)p-p3+

- p--_"

-/_)

(

(l-p2)

-

+ (1÷3p)

k2 1-p

- (l_p

1-p

1 p

+4(l+p)×

(l-p2)

4) _

+ 4)_2plnp

r

2 In

+k2

+_

+ 4(l+p)

X2 in

Section

B7.3

31 January Page

TABLE B7.3. PLATES

3-17 WITH

SIMPLY CENTRAL

Concentrated

Qr

=

Px/P;

A

=

Edge

Px

for

SUPPORTED CIRCULAR HOLE (Concluded)

Loading

X<

I

;

(lb/in.)

A

== -Px

|)

a k

for

X >

1

2

k4=l+

P l+-"_X

lax

Ir' J:,,,il :4.

j_;TL

g

,,It,-

4-

W

8D

P

8D

"3 + # - 2ka I +#

(l-p2)

"t-t_ p-

+41k__p

lnp+2p2

• - - plnp

t +p

p

o

M r

Mt

Pax2

t-v-M.

+t

-(l+v)

lnpj

lnp]

66

1969

Section

B7.3

31 January Page

67

q.

,_

°_

"_1

q.

+1

I

m_ _

+

nn

o

_"

_

o

"_

i I

+

!

bP

q.

I

+

c9 roll

+

+

_Z >_

r_

mM o _ q.

I v

m_

II

¢xl I _.t

I

II ! q

J_

_+

+

I +

A_

+

1969

Section 31

B7.3

January

Page

i969

68

I

I

*i

I

_

Z

V_

)'4 CD

50 _J _L

_Z :::k + M M

7 ¢q

A

rJ

c_

m M

c.,

f-

z

Section

B7.3

31 January Page

TABLE

B7.3.3-20 Equally

CIRCULAR

Distributed

k|=

X2

'

'

PLATE

Loading

_#)X2+(l+/_) l-_+

over

WITH the

Surface

(1+4_ (I+_)X 2

P

CENTRAL

_ la x )

p

-_ .-r-_ -

!

--

1 _--

i

64D

+2(1-k

1-

J

2X 2) (1-

Mt

W

Qr

I

-

p2) +p4_

(l - k 1) p-pS+kt.

+/_)

Qr

pa -_

(1-k

_)

Cp _ ×2 1

_

4k 1 lnp,.,.

a_ 16D

HOLE

Area

b=ax!r._ap

W

1969

69

8X 2 p2 lnp]

,

1P + 4×2plnp]

1) +4X:-(3+/_)pZ-(i-_)k

,

1._'+4X

_ (1 +p)

lnp

]

Section

B7.3

31 January Page

TABLE

B7.3.3-20 Equally

CIRCULAR Distributed

PLATE

Loading

k:

,,S A

WITH

over

the

CENTRAL Edge

HOLE

(Continued)

Circumference

1 + (i+/_) In)_ I - _ + { I+/_)X2

p Ib_Sin.

P

I

i

i

u

W

8D

r

_4_

Mt

(')

xs p- _

2

I 1

Y

i

[c_-_> _-_

2D

M

W

Mr

+._. _,+ _,__] -!

-plnp_

- I + (I+_) ks + (l-_)ks

-_- -(l+p)

In w •

m

Mt

2

-p+(l+p)ks-(l-p) .

1 Qr

-P'X

"

P

70

p: -

1969

Section

B7.3

31 January Page

TABLE

B7.3. 3-20

CIRCULAR

Equally

PLATE

Distributed

WITH

CENTRAL

Edge

Moment

lbs-tn. M-- in. M

M

Mr

--

_

Mt

X2 k5

l-p+

w

=

fl

-

M =r

(i +U)×

2

Ma 2 2-'-D" kS(- I + p2 _ 2 • inp)

D

-P

-MksII+U+(I-U}_-

M t

=-Mks[l

Qr

=

0

1

HOLE

1969

71

(Concluded)

Section

B7.3

31 January Page

1969

72

Q. :::k .t-

u_ -t-

•

O

.tQ. !

::k

::k

-t",,,,4

tt_

!

!

:::k

::k

+

÷

!

_p

!

! t_

t,.c¢1

u_ I

I

5

_J

!

:::k :::k -t-

-t-

! !

I

!

:::k

::k -t-

!

Section

B7.3

31 January Page

1969

73

%. :::k

A

'13

+

+

+ illll

-

I

+

O

_J

+

.2

%.

! I

_q L_

%.

$.

_k +

+

_D +

+

u_ I

I

#el_

!

I

:::k

:::k +

+

I ::k u_

ao

_9

+

+

+

Q.

I

!

= E_ u_ CD

I

Q.

+

I

%.

[J

:::k :::k

{J

+

+

+ H

I

I

%.

+ I u_

I

I

I A

I

::k +

+ ::k +

+

+

+

Q

E_

o

Section 31 Ps_.e

TABLE

B7.3.3-21

CIRCULAR

CLAMPED Partially

EDGES Equally

a

PLATES

,q 19_9

74

WITH

(Contin,md)

DiAtrit)utod

_j_

B7.

January

I_mdinl_

a

M r

M

Q

w

64.or L

16Dr.

' ]

- :_\:*- 4x: lnX - ztX: - 4 h_ \_¢,: + _T P_

X_ - 41 n _ -

1+1.0

-M r

_,l_

-

p"

41n

X)

--..'7--X"

--lfi"J)

i [-T )

,o

I-

4-11-

hi t

(_r

p 2ax-_

r

":I I_;I)Tr

-

t

P 1 2aTr p

-

_)'_'_ X2

4 Inf_

+1,1

+i_i

(_,?-

_.lt

,

Section

B7.3

31 January Page 75 TABLE

B7.3.3-21 CLAMPED

CIRCULAR EDGES

Concentrated

PLATES

WITH

(Continued)

Loading

W

M r

Mt

P

Pa2

W

( 1 -

p2

->X

p=0

Qr

+ 2p 2 In p)

Pa

-

M r

4-"_

- _P

plnp

[1+(1+

.) in p]

P NI t

Qr

-"_

[p+(1 P

1

2a_r

p

+p)

lnp]

P 4_

(1 +#)lnx

P 4_" (l+#)lnx

1969

Section

B7.3

31 January Page

TABLE

B7.3.3-21 CLAMPED Circumferential

CIRCULAR EDGES

PLATES

WITH

(Continued) (I,inear)

Loading

w

M

Nlllllli llllll

l"

M t

Lrr_ I1111 I_IIIIIIIV "_ Q}f

P'n_ _D

-

[1

41)

-

_:+

".'2 2 lnX+

(I'+)C

(1

-

i

I 'Z ln_)p:']

_ 2 LnX)

fl)

1 -

l'a M I'

M l

I_

-

]'_

1 -l)

-

'21n;.,

76

1969

Section 31 January Page 77

TABLE

B7.3.3-21

CIRCULAR

CLAMPED

EDGES

PLATES

WITH

(Concluded)

M

M

a

ILL

a

W

Ibs-in. M---v-----In,

M r

m

"_

X-
W

.al

<_ I

O_
(1 - p2+21np

4D

Ma2 [2X 2 ln×

2D 1

M r

+#+(1-/_)

-_-_j i

Mt

Qr

2

.

{i-

x

.

p2(1

(i - x_)

M _- (1+/_)

(I-x

2)

M (l+p)

(I-x 2)

0

X2)]

B7.3 1969

Section

B7.3

31 January Page 78

O Z

(D

U

(9

-._]'

i!!

_'_ _-'__Z -_

____'_ _ _-

1969

Section

B7.3

31 January Page

79

M

al

M

w -

(t .pt)

2D(I +p)

a

•

;

a

Ma D(J

=

M r =

Qr

.p

+p)

M t =

M

llJllllllJlll] llJlllllllJlllllll

=0

FIGURE

B7.3.3-7

SUPPORTED

FORMULAS

OF

CIRCULAR

PLATE

LOADED

DISTRIBUTED

B7.3.

3.7

with

shells.

Circular

Circular about

circular

presented

are

theory

rings.

for

In this

I2

--

Table

B7.3.

3-22

section,

employed

area

of the

moment normal

J =

WITH

A SIMPLY

EQUALLY

MOMENT

torsional presents

structural would

loading

Nomenclature

II,

important of shells

symmetrical

A =

END

FOR

Rings

rings The

INFLUENCES

M

cross

of inertia to the

as

be

elements

which

complete

without

information

respect

to

the

often

interact

information

is

summarized

center

of

the

and ring.

follows:

section

for

plane

rigidity the

such with

is

not

the

centroidal

of the factor

solutions

axis

in the

loads

on

plane

ring of the

for

section.

different

rings.

or

1969

Section

B7.3

31 January Page

B7.3.4

STIFFENED Up to this considered.

been

1969

80

SHELLS

point,

only

homogeneous,

isotropic,

monocoque

shells

have

It is known that certain rearrangements of the material in the section increase the rigidity; consequently, less material is needed, and this affects the efficiency of design. Therefore, to obtain a more efficient and economical structure, the material in the section most resistant to certain predominant stiffened structures were developed. B7.3.4.

fields.

1

to make the section on this premise,

General

Stiffened shells are The shell functions

ferential meridional

should be arranged stresses. Based

commonly used more efficiently

system, or a combination stiffeners usually have

in the aerospace if the meridional

and civil system,

engineering circum-

of both systems of stiffeners is used. The all the characteristics of beams and are

designed to take compressional and bending influences more effectively than the monocoque section. The circumferential stiffeners provided most of the lateral support for the meridional stiffeners, tIowever, circumferential stiffeners are capable

of withstanding If the

to replace corresponding be analyzed in later conical I

stiffeners

the

stiffened

This

are

The

shears,

located

section

ideal modulus as a monocoque

sections. shells.

Cylindrical

moments,

relatively

with

axial close

an equivalent

of elasticity. shell. More

geometry

and

included

Then details is for

stresses. together,

monocoque

it appears section

the shell under on this approach cylindrical,

logical

having

the

discussion can will be given

spherical,

and

Shell shell

may

have

longitudinal

or both. Stiffening may be placed or it may be located on both sides. be located between the stiffeners.

stiffening,

circumferential

on the internal or external If cut-outs are needed,

stiffening,

side of the surface, they will usually

SectionB7.3 31 January 1969 Page 81

II

Spherical

Shell

This

shell,

circumferential the meridional this direction uniform ideal III

Conical

will The

usually

problem

be stiffened may

in both

be slightly

more

meridional

and

complicated

in

direction because, obviously, the section that corresponds to will decrease in size toward the apex. This leads to the nonthickness. Shell

This IV

if stiffened,

directions.

configuration

Approach

for

structurally

lies

between

cases

I and II.

Analysis

ferential

The approach for analysis is similar for all shells. If only circumstiffening exists, the structure can be cut into simple elements con-

sisting

of cylindrical,

conical,

or spherical

Figure B7.3.4-1 and, considering performed as given in Paragraph present, interaction of cylindrical will

be performed,

panel

as shown

longitudinal loaded with

distances pressure

circumferential

If the

There stiffeners

and four

between (external

rings

as shown

in

B7.3.4-2.

longitudinal sides. The

the stiffeners or internal)

and longitudinal are

and

the primary loading, the interaction will be B7.3.2. If only longitudinal stiffeners are panels with longitudinal beams (stiffeners)

in Figure

If both circumferential will be supported on all

elements

stiffeners are present, ratio of circumferential

is very important. will transmit the

These reactions

the to panels to the

stiffeners.

no fixed formulas are close together,

in existence the structure

for stiffened shells in general. can be analyzed as a shell.

Then the stiffened section, for the purpose of analysis, the equivalent monocoque section, which is characterized modulus of elasticity. This replacement has to be done

should be replaced with with the equivalent for both meridional and

circumferential

ideal

the idea

same

of orthotropic

in a later

directions.

thickness, section,

Both

sections

but different material. and a proper

ideal The

concept

analysis

will

moduli

possess of elasticity.

of orthotropy procedure

monocoque This

will

leads

be studied

will be suggested.

properties, to the in detail

Section

B7.3

31 January Page

!

1969

82

J

Stiffener (Circumf_.re nt ial)

FIGURE

FIGURE

V

Method

should

monocoque be

determined

CIRCUMFERENTIALLY

B7.3.4-2

STIFFENED

LONGITUDINALLY

STIFFENED

SHELL

SItELL

Section

approximate

construction, method shows

equivalent

4-1

of Transformed

This wich) This

B7.3.

method

covers

regardless of the how the combined "section for

the

of the

all

variations

kind of elements section can be same

circumferential

of stiffened

(and

that make up substituted with

stiffness.

This

and

meridional

idealized directions

the an

sandscction.

section of the

Section

B7.3

31 January Page

shell.

Then

analysis discussed at that

the analyst

of orthotropic previously, time.

neglected sections.

The

of different characterized First

analysis

is more

Assume

select

In this

for

layers

by a modulus

mined. It should (t.) of individual

shells

section

of material,

where

monocoque

(stiffened,

as shown

modulus

the

shear

be explained

of elasticity

convenient

manner,

an orthotropic,

and will

a composite

one

with

complicated

lent monocoque section be modified and reduced E':'.

deals

shells is similar to the analysis if certain corrections are entered

of elasticity

ideal

transformed

cannot

etc.

) which Each

(E".')

section

be noted that, for the convenience layers was not changed, but areas (E':')

now corresponds

consists

area

as a basis

for

(Figure

the equiva-

B7.3.4-4)

of design, A. become

to every

A_',

is deter-

1

thus

making

the

tt

tt

t_

f

I !

B7.3.4-3

ORIGINAL

COMPOSITE

will with

the thickness A.*. The same

,\. X'QN\\\\\ i/" l,.

FIGURE

(i) (Ai).

Accordingly, all layers which is characterized

1

of elasticity homogeneous.

be

layer

a cross-sectional

1

modulus section

The

in the following

B7.3.4-3.

which is to be established. to one equivalent material the

shell.

distortions

sandwich,

(E i) and

83

of monocoque shells into the formulas cited

in detail

in Figure

1969

SECTION

entire

is

SectionB7.3 31 January 19a9 Page 84

tt t_

ts " t4 t5

FIGURE The tions

are

necessary

given

The equivalent

B7.3.4-4

TRANSFORMED

computations

on the

sketch

computations section.

lead

The

are

included

ideal

presented

SECTION in Table

B7.3.4-1.

Designa-

in the table.

to the determination monocoque

rectangular

of the

moment

section

of inertia

of an

can be determined

as having the same bending resistance as the original section. For example, if the section is symmetrical about the neutral axis, the thickness (t) can be found for the new monocoque rectangular section of the same resistance as follows:

I where

bt 3 i2

b is the

B7.3.4.

2

The is precisely

I*

;

selected

Sandwich

3[. _---_/ b

t = 2.29 width

of the

new

section.

Shells

basic philosophy which the analyst applies to a sandwich the same as he would apply to any structural element.

procedure

consists

of applied

loads

of determining is compared.

a set

of design

allowables

structure This

with which

the set

Section B7.3 31 January 1969 Page 85 TABLE B7.3.4-1

TRANSFORMEDSECTIONMETHOD

5.L//// _:Y)'///,I

_,--H VY////,

,---F-_-/ /

E.

1

Element

ni =

A #. = 1

n.A. 1 1

Yi !

i Yi

2 A_' _i

A ':' t2. i ,

Z A "' 2 i_i

EA., t.2 , ,

i

® ® ®

:_A:.' I

_

EA_Y i

Y-

EA. 1

1

I

1

Section

B7.3

31 January Page

1969

86

Generally, two types of "allowables" data exist. The first type is determined by simple material tests and is associated with material more with geometry, and the ment. If, in a sandwich

second type construction,

is dependent upon the geometry the materials of construction

than

of the eleare con-

sidered to be the core, facings, and bonding media, the basic material properties would be associated with the properties of these three independent elements. The upon bond

second

class

configuration media. This

of allowables

as well as upon class of failure

of failure that include the entire to a portion of the structure but The

most

important

data

is distinguished

by being

dependent

the basic properties of tile facings, core, and modes may be further subdivided into modes

configuration, and those modes that are localized still limit the overall load-carrying capacity.

local

modes

of failure

are

dimpling,

wrinkling,

crimping. These modes of failure are dependent upon the local geometry upon the basic properties of the materials of the sandwich. The general of failure generally are associated with the buckling strength of sandwich structural elements. This will be discussed in Section B7.4. In this paragraph, the general conditions will be presented.

static

t. 2. the

core.

deformations

can be neglected.

Shear

deformations

are

theory.

Once

analysis

is complete. The

orthotropic

the

first

shear

logical

material.

the large family shells, etc. To

design of sandwich shells under pure Two fund,_mental cases will be recognized:

Shear

No new basic

give

theories

extensive;

are

deformation

concepts

of sandwiches,

a systematical

however,

required,

only

is properly

approximation The

and and modes

would

included

be to replace

of orthotropy

but also

description

other

the

actually materials

of orthotropic

shear

can be taken

application

in the analysis,

actual may such

by

of established the

sandwich cover

with

not only

as corrugated

analysis,

attention

will

again be directed to the mathematical structure of the analytical formulas for the monocoque shells presented earlier in this section. This will make clear what kind of modifications can be made to apply the same formulas (that were derived

for

monocoque

material)

to the

orthotropic

shells.

SectionB7.3 31 January 1969 Page 87

B7.3. 4.3 Orthotropic Shells A material is orthotropic if the characteristics of the materials are not the same in two mutually orthogonal directions (two-dimensional space). Such material has different values for E, G, and # for each direction. Poisson's ratio, #, also may be different majority of cases this difference

in the case of bending and axial is negligible, but to distinguish

stresses. one from

In the the

other, # will be designated for Poisson's ratio which corresponds to bending stresses, and p' for axial stresses. The behavior of the shell under loading is a function of certain constants that depend upon the previously mentioned material constants

and

geometry.

The

special

case

of orthotropy

is isotropy

characteristics in both directions of two-dimensional space are see the dependence of stresses and deformations in shells upon mentioned These I

constants,

constants

Extensional

are and

In the past, numerous as follows:

formulas

Extensional

a short

review

designated Bending only

with

extensional

monocoque

presented.

The

of shells

and bending

rigidities.

shells

were

definitions

for

considered, isotropic

and shells

Et

rigidity

E÷ D

12(1 - p2) B and The symmetric

D have

To

is provided.

rigidity B-

Bending

concepts

material

Rigidities

isotropic

were

of isotropic

(the

the same). previously

appeared

following characteristic thin shells:

in many

previous

stress

formulas.

formulas

apply

for

rotationally

are

Section

B7.3

31 January Page

N_

=

B

@+/_O

NO -

B

0+_

The bending

loads

Me

The

final

D

are

fl

stresses

0._-

1969

88

cos

_ +

can be obtained

1 _'_2

+

as follows:

D (J)

%-1 For

--if"

a moaocoque

-

section

t

of rectangular

shape

t_._.3 z 12 (2)

0"0 :

Ne

Me

t

t3 12

The physical compared.

meaning

of D and B is obvious

if equations

(1)

and

(2)

are

SectionB7.3 31 January 1969 Page 89 The componentalstresses due to membrane forces are

lag-

lcr0

(j.__2)B

- _--pz

EN0 = (1-p2)B

"N

E = 1---'_"

Et

- lx t

1 - p2 Et

N0

NO = lxt

=

A

NO =-'A-

where A=lxt It is convenient

to choose

the width

of the

section

strip

that

is equal

to

unity. The

componental

stresses

due

E 1- g2" = M

M0z 2cr0 = ---if-

to bending:

E 12(I -_.2) = _12M@z = z (i - p2) Et3 i x t3

E

E

" 1 -p2

-

M0z

(1 - p2)

12(1

- #2)

Et 3

I

12M0z -

1 × t3

I

where 1-

i× t3 12

Evidently, if stiffened or sandwich be used in the equations, then all shells may "transformed

B-

be used for stiffened section" are used,

A'E* 1-p2

;

shell is being used, a modified previously derived equations for and sandwich then

D-

I'E* 1-_2

shells.

If the values

B and D shall monocoque from

the

SectionB7.3 31 January 1969 Page

In the preceding II

formulas

Orthotropic

p' _/_

is assumed.

Characteristics

Now

the orthotropy

directions,

90

1 and

2,

the

is defined following

if,

for

two mutually

constants

are

known

orthogonal

main

or determined:

!

D1, B1,

/_t and

shear

rigidity

DQ1

D2, B2, P2, P_ and

shear

rigidity

DQ2

To use analysis

Pl,

the previously

of the orthotropic

given

formulas

structures,

the

this purpose, a systematical modification will be provided in the following paragraph edge loading method for the shells, the shear distortions

(for formulas

A.

Analysis,

be modified.

for

the For

of the primary and secondary solutions to make possible the use of the unit-

The previously shear distortion.

analyses which neglect the shear distortions an additional study will be presented which shear. Orthotropic

must

case)

orthotropic case. In the analysis of monocoque usually are neglected. With sandwich, in most

cases, such neglect is justified. isotropic case do not include the

IH

the isotropic

If Shear

collected formulas Consequently,

will be examined first. considers the distortions

Distortions

Are

for the orthotropic Later, due to the

Neglected

Prin]ary Solutions It was

previously

stated

that

in most

cases,

are membrane solutions. For the purpose of interaction, values is needed (considering axisymmetrically loaded Ne -

membrane

load

in circumferential

N¢

membrane

load

in meridional

-

direction direction

the primary

solutions

the following set shells of revolution).

of

Section B7.3 31 January Page 91

u w Actually, the

pure

displacement

in the direction

of tangent

in the direction of the normal-to-the-middle

having

any

Consequently,

u and w, for

relations this

componental

displacements

if only the

purpose,

axisymmetrical

it will

be adequate

To determine N 0 and N , all formulas that case can be used, because the membrane

system

and

independent

When

of the

N o and N_

First determine the correspondent

case,

obtained,

the strains formulas

1

For

are

Et (Ngb - pNO)

_0-

1 Et (N0

orthotropic

-#'N case

same

1

-

c_

e0

B_)(i__O

=

Note:

i

D=B

u and

w can be obtained

components are

t2 12

formulas

(_

may

!

)

(N

from

considered. u and

w.

for the determinate

properties.

and

cO).

) the

are

were presented is a statically

!

c_b -

the

material

surface.

can be obtained cases

to investigate

isotropic

manner.

to the meridian

- displacement

geometric

1969

-#oN

O)

be written

in the

For

the

following

isotropic

SectionB7.3 31 January 1969 Page 92 Displacement can nowbe obtainedfrom the following differential equation: du

¢Pd-7 The

where the

ucot

_) =RI_

solution

C is the

support.

of the

constant Then,

for

of integration

To obtain isotropic case

constants,

they

Generally, solutions

due for

in the form

is

to be determined (w)

every symmetrically are determined for

Secondary

NS,

above

is obtained

from from

the

condition

at

the following

equation:

cot

Consequently, and deformations

the

equation

W

E0=

for

R 2c 0=f((_)

the displacement

U

B.

_-

NO, MS,

loaded shell orthotropic

of revolution case.

the stresses

Solutions the secondary can be used

solutions, the formulas that and then, using the substitution

can be transformed to any

the

edge

M 0,

into formulas

disturbance

(unit

for

loading)

the

orthotropic

the

formulas

were derived of proper case. give

Q, fl, and Ar

of:

Solution

= (edge

disturbance)

x (function

x (function

of geometry).

of significant

constant)

direct

SectionB7.3 31 January 1969 Page _Cylindrical B7.3.

3-10

Shell

-

All formulas

can be modified

k=_L

_f3(l_p2)

__. L

2--fi

Et s D=

12(1

B--

Et l_--_p

E

IV

X

Y

--* D x = 12(1-

BY_-PxPO_

D-X

PIP2)

--

By = i

-#x

tp,

y.

12 = the

modulus

of elasticity

in longitudinal

= the

modulus

of elasticity

in circumferential

this

Analysisj more

If Shear

complicated

Cylindrical

Distortions

case, basis for herein.

the

rically

an analysis loaded

is presented orthotropic

which

sandwich

following

Dx,

stiffnesses D Y = Flexural of orthotropic shell

nomenclature

respectively = Shear

direction.

Included

solution

may

Paragraphs

be found

IV and

V.

in Reference

Cylinders

and

Shell

The

DQ x

direction.

Are

In the case of a cylinder constructed low traverse shear rigidity, the shear distortion fore,

and

E t y

4, which was considered as the spheres only are are considered A.

J

B7.3.3-9 is made:

Ex t3

- p2)

Orthotropic For

in Tables

replacement

t2

D=B--

E

given

if the following

93

stiffness

includes

from a sandwich with may not be negligible; shear

distortion

for

relatively there-

a symmet-

cylinder. is used: of the shell wall per inch in axial and circumferential

of width directions,

( in. -lb). in x-z

plane

per

inch

of width

(lb/in.)

SectionB7.3 31 January 1969 Page 94

Bx,

B

Y

M

= Extensional circumferential = Moment

stiffnesses of orthotropic directions, respectively

acting

in x direction

shell wall (lb/in.).

(in. -lb/in.

in axial

and

).

X

= Transverse

Qx

shear

= PoissonVs

force

acting

in x-z

ratio

associated

with bending

ratio

associated

with

plane

(lb/in.).

in x and y directions,

respectively. !

!

_x'_y

The

solutions Half

V

for

in x and y directions,

are

on Geckler's

cylinders

Influence

in Tables

assumption

can be adapted

of Axial

Usually

presented

B7.3.4-2

and

B7.3.4-3.

Spheres

Based derived

extension

respectively.

derived B.

= PoissonVs

Forces

for

on Bending

it is assumed

that

the

for the

the half

spherical

sphere, shell

all formulas

as well.

in Cylinder

contribution

of the

axial

force

(N 0) to the

bending deflection is negligible; however, for a cylinder with a relatively large radius, the axial force may significantly contribute to the bending deflection. Therefore, the preceding analysis was extended by the same author (Reference to include the effect of the axial force on the deflections. cation of the formulas (Tables B7.3.4-2 and BT. 3.4-3)

This leads in the manner

in Table

as follows:

B7.3.4-4.

The

constants

R2 x Y/ L DQx O_ 2

+

_-

4

÷ --ff-Qx

are

slightly

modified

to modifishown

4)

Section

B7.3

31 January Page 95

1969

f-

Z m,,

"2"

X II

% +

+

_

...

":L _ .

ff

.%

%

%

II e_

L_

z

_

t_

N o

2' lzl

I

+

N

T

tl

a

n % z

.

g

+

{.9 X

¢

i

i

i

÷

I

"%i

i i< i

X :s]U_I_UO:)

Section B7.3 31 January Page

1969

96

>

Z

>

I x

X

.< II

_a

C_

J

_ e_ I o

+

0 ,-1 D..1 0

I

II

II e_

II

0

o --1 o

E I-

+_ '

+

O' I il

% 0

© e..

÷ N_

-r I

+

Z I r

<,

I

I

+

I

II

I|

+ _t

0, °

e4 t.: m il II

m

s_,u_guoD

Section

B7.3

31 January Page TABLE

B7.3.

INCLUDE

4-4

MODIFICATION

THE

EFFECTS

OF TABLES

OF AXIAL

B7.3.4-2

FORCES

1969

97

AND -3 TO

ON BENDING

Quantities: Formula

Formulas

in Tables

B7.3.4-2

Formula

and

-31

Substitute

emXM0 [(4T2 3m21+(m2 4321mx] _m ---_TF_

4

Whole

6

2(_ 2 M0.\

m2hl0

7

\_ hole

F_rmula

w -- 2VD M0e -sx

w

":

[( $2 + ]2)

l}x + _S t7 2 + I¢)

c,_s

V 9

V

lO

XVhoh:

FormuLil

11

Whole

Formuh_

Whole

l,'m'rn u|;l

x

0

_-M_

dw

2VI)I

= I hld_,:/

_2_"

+_i _ - 2_')

_,'I)l

( _,: + i::') t

12

15

W

:

1),i111-

"" [ _I'_"

-

Whole

F'oz'nlula

19

V

2{)

V

21

VChole

Formula

22

Whole

Formula

2VI}

(-QJ2VD}

=0

Zlll-

{Ill

_

-

'|)_)

)

1} sin

l}x

(c_ 2 + fl2}l/Z

2VD

- S cos Px

,]

_in

Px ]

Section

B7.3

31 January Page /3_ =

.. By (l

1969

98

- _'x _'y/_

4DRZII+ lx

72 =

D_x N°

x y/ R2

4DQ

V = 4T4- 4_2 _2 +f14

X

s = D

= D X

B7.3.5

UNSYMMETRICALLY Until

now,

the axisymmetrical

the geometry, used for the

material, solution.

to the shells simplification

without of the

shell

unsymmetrically.

loaded The

derivations, will

scope but

be presented The

These loading B7.3.5.

The

first

axlsymmetric

cases

symmetry, loaded complex procedures

of this

manual

does

for

in the

remaining

are

assumed

tables of solutions and geometry. Shells

SHELLS have

been

treated

with

respect

and loading. The "unit-load method" was It was shown that the most complex solutions

solutions

shells

I

LOADED

also

the most

unsymmetrically. would be the

not permit

exclusively are applied

The first level of usage of axisymmetric

presentation

commonly

to

appearing

of the actual cases

in engineering

tables. to be thin enough

provide

to use

the necessary

the membrane

information

theory.

about

the

of Revolution level

shells

of simplification loaded

of the complex

unsymmetrically.

may have unsymmetrical boundaries, to be no longer symmetrical.

which

Similarly, will

cause

procedures

would

symmetrical the symmetrical

be shells loading

Section

B7.3

31 January Page

Table

B7.3.5-1

presents

spherical,

conical,

and

B7.3.5.2

Barrel

Vaults

This panels

paragraph

of simple

beam

elliptical, cycloidal, for different loadings The shells the derived

some

cylindrical

presents system. parabolic, are given

under consideration formulas.

solutions

shells

loaded

the collection The

geometry

catenary, in Tables arc

thin,

for

certain

loadings

of different of curved

linear

for

unsymmetrically.

solutions panels

and special shape. BT. 3. 5-2, B7.3.5-3, and

1969

99

theory

was

for

curved

is circular, The solutions and BT. 3.5-4. the basis

for

Section B7.3 31 January Page

1969

i00

.5

I_T

z--IL_]l--_ _ =. I

!

7

5

J _a

I

{ 3

I?,

i <)

H Iw 0

_

*i o

_@"

z

2

o

• i

.

j

$

_,

.,

v

7

-'-'1_

,a m

_

_l_

Section

B7.3

31 January Page

II

1969

101

o

II

Z ×

Z ×

O

< > _

0

i I

"T-

< 0

L_ Z < O,

o"

m

L

II 0

,...1

X _9 o o

o4

5

_

o

C) 0

2t

> 0

• o II

-

0 O_

O_

II

II

o o

X II

I

l,

Z

_

.j

>_ z

! u_

m L_ m <

bO

_'----:11

I

z

Z

Z

I .---4.___41--'I __

I'---

O

I

g

I _i I!_l

,,=

-o-

II -0_

0

._

o

'a 0 o

,_,,4

o

I

I "O-

sooaO_lI_namul Z

Z

Z

Section 31

B7.3

January

Page

1969

102

Ill,Ill =_ K

r_ O o O

O o

d

o II

=

II

O

II

O _

A

O

O O o o O

I

(9 O

0 o

41TI']T =_* A

Q

% II

O o

O o

II

+ _

I

"1"

_

+

O N

_4

°

_

o

r/l O o

O !

i

2;

Z

I

Section

B7.3

31 January Page 103 g3 0 0

09 0 o ;<

I

I

I

I Z

_

o

_

Z

-

z

Zx

o

_

-e-

z

"

_.

c

.o

0 o g)

z

¢m I

N

r_ I

I

-0-

-6)-

0 o

N1

ul

03

>

I N

I

I

¢xz m !

tm !

!

N

"0-

-o-

0

0 o

0

o

._

II

II

II

o

II

<

II

b_

V-

_9

0

1969

Section

B7.3

31 January Page

1969

104

tb-

! ! 4-

! I

i

Q,

i

Q, Z o

o o

!

I

i

X

tt

0

N

x

i

o

_.=

z I N i I

J l

r_ "O-

! fl

I

g

U

II

N

tl

N

Ii

ii N

,-y

N

/

w I m I I m m L

WlOqe_ed

IIIII

Section

B7.3

31 January Page

105

x l

o z

x

N

i

! i

v _L

O Z

O. I

n_ cD

i

i 4"

x

o

_9 ×

I

M

z

i i

_1_,

__

i

> I

i

i

,<

i m I

I L_

fl

mH

N

fl _

It

_

N

¢g

,Ill

zu! _ zq *

_ -7,, S_ i ?

'*"el"*

_._

\'

'.

i

,

\ _aUUOle;:)

N

1969

Section

B7.3

31January1969 Page

._1 _

q_

106

Z;-Z

÷

-I i

.=l:J I /

=,

S

I" 1,7" "_" It.

-= +

k,

+

I:1 0

E_

.1

÷

Z

•

t

I

-.?

i

N

7

.=

J

;

I

,i

III

I

q-e

"°I q q

Z

_,_

fill

_- Z/q._l l

L

_---_

.__1

\

=_\

ill (_

um+

_.)

)'/"

+_ "8

=.I

_'1111

Section

B7.3

31 January Page 107

1969

REFERENCES Timoshenko,

le

and a

Baker, Verette, Inc.,

m

P.,

and

New York,

E. R.

Woinowsky-Krieger, McGraw-Hill

Book

S., Co.,

Theory Inc.,

SID 66-398,

June

Technical

Report

of Plates

1959.

H., Cappelli, A. P., Kovalevsky, L., Rish, M., Shell Analysis Manual. North American

F. L., and Aviation,

1966.

Lowell, H. H., Tables of the Bessel-Kelvin Kei, and their Derivatives for the Argument NASA

o

S.

Shells.

R-32,

Functions Ber, Range 0(0.01)

Bei, Ker, 107.50.

1959.

Baker, E. H., Analysis of Symmetrically Loaded American Institute of Aeronautics and Astronautics.

Sandwich

Cylinders.

SECTION B8 TORSION

Section

B8.0

1 March

1969

Page

B8.0.

0

TORSION Sections

strLlctural their

under

elements

that

cross-sectional

for

the

in

Section

3;

be

B8.2;

the

the

which

stated;

the

deal

with

each

Section which

are basic

pertinent theory,

to and

the

the

approach,

limitations,

is

solid

if any,

ground

section,

treated

in Section

section

under

in

treated

such

than

a bar.

a common

represented.

straight

greater

called

cross

section

cross

of

much

provides

section,

pictorially

the

1,

cross

cross

and

analysis

element

B8.

follow:

divisions,

described,

an

closed

open the

torsional

dimensions

Such

thin-walled

of

the

longitudinal

division,

thin-walled

defined,

ditions

have

divisions

In will

first

analytical

and

B8

dimensions.

The

]38.

1

treated Section

B8.4. consideration

Particular

as will

restraints, be

given.

con-

will

be

SECTION B8. 1 GENERAL

TABLE

OF

CONTENTS l);t go

BS.

1.0

General

......................................

8.1.1

Notation

8.1.2

Sign

Convention

............................

Local

II.

Applied

Twisting

III.

Internal

Resisting

IV.

Stresses

VI.

2

.................................

I.

V.

!

Coordinate

Deformations Derivatives

System

5

.....................

Moments Moments

....................

8

....................

9 10

................................

11

............................. of Angle

of

6

Twist

B8.1-iii

..................

12

Section

B8.

1 March Page B8.

I. 0

B8.

ordinate

systems,

stresses,

deformations,

will be

torsion

is considered

that no

relative

on

any

superimposed

ing

if the

not exceed

for

the

upon

deformations

the yield

cross

by

and

are

small

cross

cross

and

of the material.

conven-

are

considered

and

torsion

between

two

is restrained.

sections

is not

rapidly. in this

caused

by

for

unrestrained

Warping

the maximum

local

moments,

These

Restrained

contained

for

B8.4.

occur

attenuates

deformations

resisting

sections,

section.

sections.

and

and

shall

the methods

stresses

stress

cross

convention

of twist.

torsion

closed

of solid

condition

determined

Z, B8.3,

unrestrained

solid

sign

internal

of angle

displacement

torsion

stress

B8.

thin-walled

similar

Restrained

deformations

and

and

longitudinal

two

it is a localized

in Sections

and

moments,

derivations

torsion

open

the notation

twisting

and

followed

the thin-walled

1 presents

applied

Restrained

points

1

GENERAL

Section

tions

I

1969

similar

considered

The

other

ccJmbincd

because

stresses

section

requires

can

types

str(.ss

and be

of load-

d,Jes

co-

bc'ctit,,_ 28 dune Page

B8.1.1

2

NO TA TIO N All

terms

_8. 1 196b

are

general

terms

defined

in the

used

in this

text

a

Width

A

Enclosed section,

as

area in. 2

Length

b'

Width

C

Length

d

Total

D

Diameter

E

Young's

G

Shear

h

Distance

I

Moment

J

Polar

K

Torsional

L

Length X

m t

M. 1

they

of mean

width minus

wall

centerline depth,

j

closed

web,

m.

(circumference),

bar, lb/in.

modulus, modulus

of

in.

in.

in.

of circular

in. 2

of elasticity, Ib/in. 2

between

flange eenterlines,

in.

of inertia, in.4 moment

of inertia, in.4

constant,

of bar,

Arbitrary

in.4

in.

distance

along x-axis

Applied

uniform

varying

applied twisting moment,

concentrated

M

Internal twisting moment as function of x lb/in.

from

origin,

or

_]laximum

twisting moment

Internal twisting moment,

Pressure,

of thin-walled

of flm_ge,

thickness

Applied

P

Special

in.

periphery

Mt

ix)

herein.

J

of flange

section

defined

section,

of element,

of

are

occur.

of rectangular

b

L

section

2

m. ValUe

ol

in.-ib/in.

in.-lb.

twisting moment,

in.-lb.

at point x along bar,

written

Section

B8.1

28 June

1968

Page P

Arbitrary

q

Shear

r

Radius

R

Radius

s

Distance

point

on cross

section

of circular

cross

section,

of circular

fillet,

in.

flow,

3

lb/in.

measured

along

in. t

thin-walled

section

from

origin,

in.

St

Torsional

modulus,

Sw(s)

Warping

t

Thickness

of element,

Thickness

of web,

t w

statical

in. 3

force

moment,

per

in. 4 in.

in.

T

Tensile

unit

u

Displacement

in the x direction,

in.

v

Displacement

in the y direction,

in.

W

Displacement

in the z direction,

in.

Wn(s)

Normalized

V

Volume,

ot

Defined

in Section

B8.4.

Defined

in Section

B8.4.1-IV

Warping

constant,

1"

Sllear 0

arbitrary PO

Radial shear

2

in. 2

i-IV

in.8

strain

Unit twist,

Radial

function,

lb/in.

in. 3

Poisson_s P

warping

length,

rad/in.

(e = d_/dx

= ¢')

ratio distance point distance

center,

from P,

the

to tangent in.

centroid

of the cross

section

to

in. line

of arbitrary

point

P from

Secti¢_n 28 Jun¢' l),'agc 4 Longitudinal

X

Tot'a/

T

shear

_'t

Torsional

r_

Longitudinal Warping

T W

4_

Angle

normal

stress,

stress, shear

lb/ia, stress,

shear shear

of twist,

lb/in. z

lb/in,

stress,

stress, rad

z

lb/in.

lb/in. (_b =

2

fLx

2

2

0dx)

O (_t

O

,4_

f!

)_lf!

First, second, and Lhird derivatives with respect to x, respectively Saint-Venant

stress

Subscripts: i

inside

1

longitudin,_l

n

ll()rllla

o

outside

s

point

t

torsional

W

warping

X

longitudinal

].

s or transvcrsc

dircction

function

of angle

(ff twist

B8.1 19_8

Section B8.2 31 December Page The equation

=

"

for/3

333 -

in terms

0.21 (h/d)

.0

of (b/d)

-

TABLE

b/d

is

0.0833 (b/d) 4

B8.2.2-1

100

2.0

2.5

3.0

4.0

6.0

0.208

0.238

0. 256

0. 269

0.278

0.290

0.303

0.314

0.331

0.333

0. 141

0. 195

0. 229

0. 249

0.263

0.281

0.298

0.312

0.331

0.333

distributions

on different

and the resulting at distance

Elliptical

L

x

from

warping the origin

radial

deformation is shown

lines

are

10.0

oo

1.5

B8.2.2-2A,

IV

11

1.0

The stress

located

1967

shown

at an arbitrary in Figure

in Figure cross

section

B8.2.2-2B.

Section

The maximum and is determined

torsional

shear

stress

occurs

at point

A (Fig.

B8.2.2-3)

by

Tt(max)

= Mt/S t

where bd _16

St =

"

The torsional shear stress at point B is determined

Tt(B) = vt(max)

(-_)

.

The total angle of twist is determined MtL _b(max)

=

KG

by

by the following equation:

Section

B8.2

31 December Page

L

1967

12

Y

X

FIGURE

B8.2.2-3

ELLIPTICAL

CROSS

SECTION

L

FIGURE

BS. 2.2-4

EQUILATERAL

TRIANGULAR

CROSS

SECTION

4"

Section B8.2 31 December Page

1967

13

where

s ds K

._.

16 (b2 + d2) V

..Equilateral Triangular Section The maximum

(Figure B8.2.2-4)

torsional shear stress occurs at points A,. B, and C and is determined

by

where

S t = b3/20.

The total

angle

of twist

is determined

by

MtL _) max

=

KG

where

K

8O

VI

Regular

Hexagonal

The

approximate

rt(max)

Section maximum

=

where

S t = 0.217Ad

Mt/S t

torsional

shear

stress

is determined

by

Section

B8.2

31 December Page and is located

at the midpoints

sectional

and d is the

area

The approximate

(max)

of the _sides

diameter

total

angle

( Fig.

B8.2.2--'5).

of the inscribed

of twist

1967

14

A is the cross-

circle.

is determined

by

MtL

ffi

KG

where K = 0. 133Ad _ .

VII

Regular

Octagonal

Section

The approximate

_t(max)

maximum

=

torsional

shear

stress

is determined

by

Mt/S t

where

St and is located sectional

= 0. 223Ad at the

area

and

midpoints d is the

The approximate

(max)

diameter

total

=

of the sides

angle

MtL K"--_

where K = 0. 130Ad 2 .

of the of twist

( Fig.

A is the

B8.2.2-6).

inscribed

circle.

is determined

by

cross-

r"

31 December Section Page 15 B8.2 ¥

t

FIGURE

B8.2.2-5

REGULAR

HEXAGONAL

CROSS

SECTION

FIGURE

B8.2.2-6

REGULAR

OCTAGONAL

CROSS

SECTION

¥

1967

f-.

Section B8.2 31 December Page VIH

Isosceles The

twist

can

Trapezoidal approximate

be determined

placed

by an equivalent

drawing

perpendiculars

centroid

C and then

maximum for

torsional

an isosceles The

to the

of the

sides rectangle

shear

trapezoid

rectangle.

forming

equivalent trapezoid

EFGH

using

stress when

and

rectangle (CB points

ISOSCELES

TRAPEZOIDAL

total

angle

the trapezoid is obtained and CD) B and

¥

B8.2.2-7

17

Section

BS. 2.2-7).

FIGURE

1967

SECTION

from D ( Fig.

of

is reby the

Section B8.2 31 December Page

B8.2.3

EXAMPLE I

PROBLEMS

Example

Problem

Find the maximum

FOR

TORSION

2.0.

From

Table B8.2.2-I,

in Figure B8.2.3-I.

(_ =-

0.256 and/3 =0.229for

(St) is

St = c_bd2

The torsional vt

=

(5)(2.5)

2

in 3

shear

stress

(T t)

at point A is

M/S t J

= loo, 000/8.00 Tt

=

12,500

The torsional

K=

psi.

constant

(K)

is

_bd 2

= 0.229

(5)(2.5)2

= 7.156

in' .

The total

angle

of twist

(_)

is

_b = MtL/GK 100,000 ( 32)/4,000,000 0. 1118

b/d =

stress will occur at point A in Figure B8.2.3-I.

The torsion section modulus

= 8.00

SECTIONS

torsional shear stress and the total angle of twist

The maximum

= 0.256

SOLID

18

1

for the solid rectangular cross section shown Solution:

OF

1967

rad.

(7.156)

Section

B8.2

31 December Page II

Example

Find

the

at point

B for

tributed

torque.

Problem

maximum

the

2

torsional

tapered

1967

19

shear

stress

bar

shown

in Figure

of the

tapered

bar

moment

M(x)

(Tt) and BS. 2.3-2

the with

angle

of twist

a constant

(¢)

dis-

Solution: The

radius r

The

= 2.5-

twisting

-- mtx=

torsional

modulus

= 0.5

maximum

=

Since with

of the

x-coordinate

is:

(S t) of the

bar

as a function

of the

7rr 3 [2.5-

torsional

0.005x]

shear

3

stress

(T t)

at point

B is

St(x )

200x (2.50

1.5708

= L

Tt

as a function

M(x)

1"t

Forx

is:

is

= 1.5708

The

of the x-coordinate

200x.

section

x-coordinate

St(x)

as a function

0.005x

internal

M(x) The

(r)

x

the x-coordinate,

3

= 300,

= 38,197

both

- 0.005x)

the

psi

.

internal the

angle

twisting of twist

moment

and

is obtained

the

torsional

from

stiffness

vary

Section

B8.2

31 December Page

i

L

A*]

!

I

AJ

SECTION

FIGURE

L " 400"

!

M!

M t = I00,000 in. lb. G = 4,000,000 psi.

L : 300"_ x_

1967

20

L = 3Z"

B8.2.3-1

d 1 = 1" d 2 = 5"

FIGURE

A-A

m t = 200 in. lb. /in. G = 4,000,000 psi.

B8.2.3-2

Section

B8.2

31 December Page

1967

2t

L

_,=

The

_c¢ 1

JoX

M_x_ KCx) ctx

torsional

stiffness

K=

_r 4

(K)

of the

bar

as a function

of the x-coordinate

is

0.5

= 1.5708(2.5 The angle obtained

- 0.005x)

of twist

by evaluating

(qb) in radians

the

1 4x10 s

III

Example

loaded

as shown

following f J0

origin

and

dx

internal

(¢)

at points

A,

B,

C,

and

D are:

and

B8.2.3-3.

moments

M(D)

at points

A,

B,

C,

= 0 5O

M (c)=

f

(6,670-

x 7500

35

)

dx

= 15.0

inrKips

35 M (B)

= 15,000+

f

3O. 0 inrKips

1000(ix= 2O 2O

M (A)

The

= 30,000

equation

for L

,=

fx O

angle M t GK

B can

D for

the bar

.

Solution: The

point

3

of twist

in Figure

the

200x 0.005x)

(2.5-

angle

between

integral.

300

Problem

Find the total

4 .

+

f 0

3000

of twist dx '

is

x

_-_

(ix = 60.0

in:-Kips

be

Section

B8.2

31 December

1967

Page 22

ment

which

is also

Using

the moment-area

can

diagram

under

the

analogy

Mt/GK from

diagram beam

( Fig.

theory,

the

BS. 2.3-4). following

state-

of a bar

which

be made.

"The is loaded

the area

total

with

of twist

an arbitrary

between Using

angle

the

the

torsional

two cross

moment-area

_b (A) = 0

between

fixed

_b (B)

= 30,000(20)

(C)

= 1,000,000+

any two cross

load

is equal

sections.

"

principle

above,

= 1,337,500+

to the

the

angles

area

under

of twist

the

Mt/GK

are

end. + 2/3(30,000)(20)

15,000(15)

1/3

=

+ I/2

= 0. 13375

(D)

sections

(15,000)(15)

1,000,000 GK

(15,000)(15)

= 0.100

rad

= 1,337_500 GK

rad

=

1,412,500 GK

= 0.14125rad.

C

Section 31 Y

3.0

__ ......

__

in.kips/in.

B8.2

December

Page

1967

23

_0 _n_/_n. 1.0 In.kips/in. ,Jji!llUld UI_ X

20"

_ _

A

15"

15"

B K : G

2.0

in.

C

D

4

: 5,000,000

psi.

FIGURE

B8.2.3-3

60.0 GK

30 .____0 GK /

1 20"

15"

J

_

15"

J I

FIGURE

B8.2.3-4

SECTION B8. 3 TORSION OF THIN-WALLED CLOSEDSECTIONS

-...,..b4

TABLE

OF

CONTENTS Page

B8.3.0

Torsion 8.3.1

of Thin-Walled

General I

Basic

Closed

Theory

HI

Membrane

IV

Basic Torsion Closed Sections Sections

I

Constant

Thickness

II

Varying

III 8.3.4

Unrestrained Restrained Stress

4

........................

Circular Circular

Sections Torsion Torsion

Concentration

Example Problems Closed Sections

6 9

.........................

Thickness

Noncircular

II

4

Equations for Thin-Walled ...........................

Circular

I

1

............................. Analogy

1 1

............................

Limitations

8.3.3

..............

................................

II

8.3.2

Sections

Sections Sections

............

9

............

9

.......................

10

.......................

10 18

........................ Factors for Torsion ..........................

.................

18

of Thin-Walled 20

I

Example

Problem

1 ........................

20

II

Example

Problem

2 ........................

22

III

Example

Problem

3 ........................

26

B8.3-iii

Section

BS. 3.0

31 December Page

B8.3.0

TORSION A closed

a closed

THIN-WALLED

section

is any

CLOSED

section

where

1

SECTIONS

the center

line

of the

wall

forms

curve. The

and

OF

1967

torsional

restrained

torsion

and warping stress,

analyses are

stresses

angle

included.

are

of twist,

of thin-walled

closed

Torsional

determined

shear

for restrained

and warping

sections stress,

angle

torsion.

deformations

are

for

unrestrained

of twist,

Torsional

considered

for

shear unrestrained

torsion. Analysis The

analysis

B8.3.1

of multicell of multicell

I.

Basic

sections

is beyond can

the

be found

scope

of this

in References

analysis.

11 and

13.

Theory

The

torsional

analysis

and

deformations

warping

normal

stress

any point

(P)

The

section

and

the

coefficients

exist

coefficients

are (S t)

emanating

case that

called

and

are

torsional from

the

for

(_) plus

section

for solid

geometric

stress

be determined

torsion. ( see

geometry

constant

dimensions

distribution

centroid

(K)

of the

stress

(w)

cross

and

torsional

varies thin-walled

cross

along

( L ) from x

at any

of each

of the

at

an arbitrary

BS. 2. l-I),

the

(r t) , plus

distance between

Section

that

be determined

at an arbitrary

unrestrained

of the

shear

should

deformation

torsional

requires

torsional

the warping

the

functions shear

should

sections

torsion,

section

sections

characterize the

The

restrained

closed

origin

cross

As was

The

for

of twist

the

closed

be determined. (_w)

angle

on an arbitrary

modulus

of thin-walled

on a thin-walled

origin.

cross

closed

sections

GENERAL

stresses

the

closed

point

(P)

two unique

section.

These

section

section. any

closed

radial section.

line Since

Section

B8.3.0

31 December Page

the thickness

of the thin-walled

mtress

very

varies

to be constant

through

Figure BS. 3.1-1B

B8.3.

|bows

_L

since

through

I-IA

shows

X1

shear

tlAX

=

stresses

r

of this

will

give

t2AX LX2

are equal

compared of the cross

with

2

the radius,

section

the

and is assumed

at that point.

a typical

element

(longitudinal)

is small

the thickness

the thickness

a typical

in the x direction

or,

little

section

1967

thin-walled cross the

cross

section.

following

section

and

Equilibrium

Figure of forces

equation:

*

in the longitudinal

and circumferential

directions,

_tl

This the tkicknsss is called

tl =

equation

the

"shear

q=

_t t

"

indicates

at any point

The internal following

_t2 ts

around

flow"

(q).

that

the product

the cross

of the torsional

section

shear

is constant.

stress

This

constant

moment

by the

Therefore:

•

forces

are

related

to the applied

twisting

equation: Mt rt

Mt

= 2A'-'_ =

S-_-

where S t = 2At and A is the enclosed section.

area

of the

mean

periphery

of the thin-walled

closed

and

Section

B8.3.0

31 December Page

Ae

Stress Distribution and Internal Thin-Walled Cross Section

/

Moment

3

for

/

/ / / / /

B. Stresses FIGURE

B8.3.1-1

on Element TYPICAL

A

THIN-WALLED

CLOSED

CROSS

SECTION

1967

Section B8.3.0 31 December 1967 Page4 Written

in terms

of shear

flow,

this

equation

becomes:

Mt 2A

II.

"

Limitations

The torsional to the following

anslysis

of thin-walled

closed

cross

sections

is subject

limitations:

A.

The material

B.

The cross

is homogeneous section

must

and isotropic.

be thin-walled,

but not necessarily

of constant

thickness. C.

Variations

in thickness

corners

( see

D.

No buckling

E.

The stresses

Section

twisting

The applied

G.

The bar cannot

H.

The shear

twisting

face

ABDE

a thin-walled These

uses A.

( Fig.

at reentrant

and at abrupt

changes

not exact.

cannot

abrupt

does

of constraint

changes

not exceed

be an impact in cross

the shearing

to the shear

strain

(elastic

load.

section. proportional

limit

analysis).

Analogy can be made

B8.3.1-2)

tube as was are

are

moment

have

stress

Membrane

at points

moment

and is proportional

use

except

B8.3.3-llI).

calculated

F.

The same

not be abrupt

occurs.

of applied

HI.

must

of the stress

in solving made

function

the problem

of the function

represented

of the torsional

in Section

BS. 2.1-III

by the surresistance for

the solid

as follows:

The twisting

moment

is equal

to twice

therefore,

given

Mt = 2AH

( Mt) to which

the volume approximately

the thin-walled

underneath

the surface

by the equation

tube is subjected ABDE

of

and

is,

bar.

Section

B8.3.0

31 December Page

1967

5

1

i

/

*J

I

I

I i I

-"

i

_-_._

I ,%

It

i

I

FIGURE

B8.3.1-2

ANALOGY

CLOSED

A is defined

plane BQ

The

BD above slope

taken.

any point rt It can Section

=

surface

the

as H/t. is,

cross

FOR TORSION

at any

slope The

OF THIN-WALLED

SECTION

B8.3.1-I

and

H is the height

of the

section.

perpendicular

Hence,

be taken

CROSS

as in Section

the

of the

in a direction

E

I

MEMBRANE

where

D

point

to the

direction

at any point

maximum

is equal

along

shearing

to the in which

the stress

arcs

stress the

in the slope

AB or DE

in a hollow

bar

bar

is may at

therefore,

H/t be seen B8.3.1-I.

that

H is the

same

quantity

as

"shear

flow,"

defined

in

Section

BS. 3.0

31 December Page

IV.

Basic Torsion Equations For Thln-Walled A.

1967

6

Closed Sections

Torsional Shear Stress

The basic equation for determining the torsional shear stress at an arbitrary cross section is: M(x_ _t

=

_

M(x)

St(x,s)

q(x)

2A(x)t(x,s)

-

t(x,s)

where St(x,s) = 2A(x) t (x,s) Q(x)

and A is defined M(x) the torsional arbitrary cross

M(x) 2A (x)

as in Section

or q(x) shear

cross

=

B8.3.1-I.

is evaluated stress

section

for x = L

x

is to be determined, at the point

(s)

at the arbitrary and t(x,

cross

section

s) is evaluated

on the circumference

where

at the

of the arbitrary

section.

If a constant

torque

is applied

to the end of the bar

and the cross

section

is constant along the length of the bar, the equations reduce to: M t rt = S_

M __.t_ =

2At(s)

=

q t(s)

where Mt q=

2A

In the equations for torsional shear stress in Sections B8.3.2 and B8.3.3, A.

which follow, M(x)

is equal to M t and A(x)

is constant and equal to

The equations in these sections determine the shear stress at any point

(s) around the cross section.

Section

BS. 3.0

31 December Page

B.

Angle

any

cross

equation

section

for

determining

located

at a distance

M(x) K(X,

dx=

the angle L

L I

f

C is equal

X

of twist

from

between

the origin

the origin

is:

L x

0

where

7

of Twist

The basic and

1967

to the

s)

1

4-5 f

x

length

of the

wall

(x)

A2(x)

0 center

line

dx

(circumference)

and

C K(X,S)

When

M(x)

constant,

/Io d=

= 4A2(x)

t(s)

is a constant and

torsional

t is not a function

moment of x,

the

applied equation

M

-

i t GK(s)

where

C K(s)

When

t(s) d_s

= 4A2/f

t is a constant,

the

equation

Mtl GK

_bwhere

4A2t

KThe

total

C twist

of the

(max)

-

where K-

4A2t C

bar M1 t GK

is:

reduces

to"

at the reduces

end to:

of the

bar,

A is a

Section B8.3.0 31 December 1967 Page

C.

Warping

The basic point

(P)

origin

Deformation

equation

on an arbitrary

w(s)

plane

section

Mt s 2A-"G f 0

is the warping

through plane

normal

the

through

distance

Example

as the

origin

located

the

Problem

2,

that

lie

deformation

at a distance

(w)

x = L

origin

at any

from

the

by evaluating

measured

system;

w

from

the x-y

is the distance

o

displacement

to the increment

plane,

section,

which

will

of symmetry

pass ( see

the point

which

the following

(s)

ds

plane; of arc

from

and r(s)

length

the

is the

ds (see

BS. 3.4-II).

sections,

through

at point

s coordinate

parallel

t( s---_ 2A

r(s)

of the mean

Section

on axes

unsymmetrical

length

origin

cross

-

deformation

displacement

undeformed

section

1

of the

to a line

The mean

For

cross

the warping

C

w(s)-Wo=

x-y

for determining

Is:

where

8

the mean integral

is located through

those

Example

(s),

at the

displacement

points

Problem

measured

2,

from

plane

same

z coordinate

on the cross Section

BS. 3.4-I1).

an assumed

passes,

arc

is determined

for s.

C /" J 0

1

D.

Warping

a generalized

ds

= 0

Stresses

stress form.

calculations Techniques

are for

very

complicated

evaluating

these

and cannot

stresses

be put into

can be found

1. Warping

B8.3.3-II.

t(s) 2A

Warping

Reference

0

stresses

for a rectangular

section

are

included

in section

in

Section 31

December

Page

B8.3.2

CIRCULAR I.

Circular

A constant-thickness, for unrestrained

for restrained The

1967

9

SECTIONS

Constant-Thickness

no warping

B8.3.0

circular, torsion

Sections thin-walled

and develops

closed

section experiences

no warping

normal

stresses

torsion.

torsional

shear

stress

is determined

by the following

equation:

Mt Tt

-

S t

where S t = 2At

The

.

torsional

shear

stress

defined in terms

of shear

flow is determined

by the following equation:

_'t

=

where

q/t M t

q

2A

The

total

angle

of twist

is

determined

by

the

following

equation:

ML t _b (m._x)

GK

where 4A2t K

_

II.

----

C

Va_g

A circular for unrestrained strained

torsion.

negligible gradual.

Thickness thin-wa|led torsion, The

Circular closed

and warping

warping

and can be neglected

normal when

Sections

section with varying normal

stresses

stresses the change

thickness

are developed

will warp for re-

and warping

deformations

in thickness

is small

and

are

Section

B8.3.0

31 December Page The torsion thickness

circular

stress

sections,

The total varying

shear

angle

thickness

is calculated

except

of twist

in the

manner

that t is now a function

for a circular

is determined

same

as for constant-

of s.

thin-waUed

by the following

1967

10

closed

section

with

equation:

MtL

(max) =

G--"K-

where 4A 2 C

K= f

d..s_s

o t(s) B8.3.3

NONCIRCULAR I.

SECTIONS

Unrestrained Torsion

Noncircular sections experience warping for unrestrained torsion, except for the case noted below, and develop no warping normal

stresses.

Note that no warping occurs in a cross section that has a constant value for the product rt around the circumference Longitudinal not evaluated. mationa

The use

for closed

Problem

II (see A.

Section

Elliptical

deformations

of the basic

sections

The torsional following

warping

( see

Section

usually

not of concern

for determining

BS. 3.1-IVC)

is used

warping

and

are

defor-

in Example

B8.3.4-III). Section

shear

stress

for constant

where S t = 2At and Ir[a b-

are

equation

equation:

A =

of the cross section.

t (a+ b)+ _-

t__2 42 }

thickness

is determined

by the

Section BS. 3.0 31 December 1967 Page

The

values

of a, b,

and

The torsional by the

following

t are

shear

defined stress

in Figure defined

BS. 3.3.

in terms

11

-1.

of shear

flow

is determined

equation:

q "rt

t

where

M

t 2A

q-

and A is defined

as above.

.m

FIGURE The same

manner The

following

torsional

shear

as constant total

,angle

of twist

ML t GK

,tA2t K

z

C

stress thickness,

equ:_tion:

where

B8 3.3-1

ELLIPTICAL for

varying except

for constant

SECTION thickness

that

is calculated

it is now a function

thickness

is determined

in the of s, by the

t(s).

Section

B8.3.0

31 December Page A is defined

above,

C =

for C is,

approximately,

J-1÷ 0.27 Ca - b)_']

,r(a+b-t)

The total following

and the equation

1967

12

L

angle

of twist

(a+ b)2J

for

varying

thickness

is determined

by the

equation: MtL

¢ (max) =

G'-"K

where 4A z C

f 0

and the area B.

d...__ tls)

is as defined Rectangular

The torsional points

A and

B ( Fig.

in Section Section

shear

B8.3.1-I.

(Constant

stress

BS. 3.3-2)

Thickness)

for constant

thickness

by the following

is determined

equation:

Mt ft

=

_

where

S t = 2At and

A = ab - t(a+b) The values

of a,

+t 2

.

b, and t are defined

in Figure

BS. 3.3-2.

at

Section

B8.3.0

31 December Page

P

1967

13

i |

8

V///////////////////A FIGURE

The by the

BS. 3.3-2

torsional

following

rt=

RECTANGULAR

shear

stress

SECTION

defined

(CONSTANT

in terms

of shear

THICKNESS)

flow

is determined

equation:

qt

where M

t 2A

q

and A is defined

as above.

The

stresses

higher

than

the

of the

fillet

section

following

corners

(points

calculated

at points

to thickness

is greater

than

see

total

Section

angle

equation: ML t GK

(max) where K-

inner

stresses

stresses The

at the

4A2t C

of twist

1.5.

C on Fig.

A and For

B unless small

B8.3.3-2) the

radius

ratio

will of the

rectangular

B8.3.3-III. for

constant

thickness

is determined

by the

be radius

Section

B8.3.0

31 December Page

A is defined

above,

C.

and

Rectangular

The torsional determined

at points M

1967

14

C = 2( a + b - 2t). Sections

shear

stress

A and

B ( Fig.

(Different

Thickness)

for different

but nonvarying

BS. 3.3-3)

by the following

thickness

is

equation:

= ....t_ Tt

St

where St = 2At 1

for

point

A, S t = 2At z

for point

B, and

A = (a-t2)

(b-t i)

a

.@

FIGURE

B8.3.3-3

RECTANGULAR

SECTION

(DIFFERENT

THICKNESSES)

Section 31

B8.3.0

December

Page

The by the

torsional

following

shear

stress

defined

in terms

of shear

1967

15

flow

is

determined

equation:

r t = qtl for

point

A,

Tt

for

point

=

B,

q t2

and M

where

q-

t 2A

A is

defined

as

The

stress

at

calculated

at

thickness see

is

Section

greater

total

following

the

A and than

B8.3.

The by the

points

above. inner

corners

B unless 1.5.

For

(points the

ratio

small

C) of the

radius

will radius

rectangular

(max)

K

:

angle

of twist

for

different

but

nonvarying

=

ML t GK

where

D.

of the

than fillet

section

the

stresses

to the stresses

:3-III.

equation:

0

be higher

+-2ttt2(:') - t2) z (h _t_ + bt? - t, 2 - t 2

Arbitrary

Section

tO 2

(Constant

Thickness

I

thickness

is

determined

Section

BS. 3.0

31 December Page

The torsional ( Fig.

B8.3.3-4)

shear

stress

is determined

for an arbitrary by the following

section

with

1967

16

constant

thickness

equation:

Mt 7t=

_t

where St = 2At

and A is defined

in Section

BS. 3.1-I. s.0

FIGURE

B8.3.3-4

The

ARBITRARY

torsional

by the following

shear

stress

SECTION defined

(CONSTANT

in terms

THICKNESS)

of shear

flow

is determined

equation:

where Mt qand

2A

A is defined

asabove.

The

angle

total

of twist

MtL

(max)

-

GK

where 4AZt K

-

C

°

is determined

by the following

equation:

Section B8.3.0 31 December 1967 Page 17

A is defined

as above,

and

C,

the circumference,

is defined

as follows:

C C = f

ds 0

ness

E.

Arbitrary

The

torsional

( Fig.

B8.3.3-5)

Section shear

(Varying stress

for

is determined

Thickness) an arbitrary by the

section

following

with

varying

thick-

equation:

Mt 1"t

-

St

where

S t = 2At

and A is defined following

in Section

B8.3.1-I.

The

shear

flow is determined

by the

equation:,

t

t

t(s)

FIGURE

B8.:3.3-5

where q

t 2A

and A is defined

:_._ :1bore.

AI_BITRARY

SECTION

(VARYING

THICKNESS)

Section

B8.3.0

31 December Page

Note ness.

that the maximum

The total

angle

of twist

shear

stress

occurs

is determined

at the point

by the following

t967

18

of least

thick-

equation:

L (max)

= Mt

where C

and II.

A is defined

Restrained

of abrupt

torsion

of noncircular

change

in torque.

The warping attenuate

rapidly,

shown

w

shear

]38.3.3-I.

in Figure

o

normal

stresses

and their

Torsional as in Section

as above.

Torsion

Restrained at points

ds

(max)

=

sections

associated

analytical

stresses

associated normal

is determined

occurs

at fixed

with the restrained

determination

The warping

BS. 3.3-2

closed

is extremely

difficult.

restraints

stress

the rectangular

by the following

and

torsion

with these for

ends

are

calculated section

equation:

__m K

where

K

tG 4M t

and m is obtained m.

Stress

in Figure

b) 2 (_)

corners B8.3.3-2.

( E

from

Figure

Concentration

The curve entrant

(a+

2 1 -

)½

B8.3.3-6.

Factors

in Figure

BS. 3.3-7

gives

the ratio

to the stress

along

the straight

of the stress

sections

at points

at the reA and B shown

Section 31

B8.3.0

December

Page

19

4.0

3.0 \ 2.0 m

1.0

d

_

c_ a

FIGURE

BS.

3.3-6

VALUE

OF

o

o

_ b

m

FOR

RECTANGULAR

SECTION

3.0

2.5

L

N

\

1.5

\

1.0 0

o. 5

1.0

1.5

r/t

FIGURE

B8.3.3-7

STRESS

CONCENTRATION

ITEENTRANT

CORNERS

FACTORS

AT

1967

Section

B8.3.0

31 December Page

B8.3.4

I.

EXAMPLE SE C TIONS

Example

PROBLEMS

Problem

For

the

FOR

TORSION

OF THIN-WALLED

20

CLOSED

1

problem

shown

in Figure

B8.3.4-1,

it is required

to find the

following: A. mum

Shear

angle

stresses

of twist

B.

Local

at points

caused

A,

B,

by the torsional

normal

stresses

caused

Section

B8.3.3-IB,

and C on the cross

section

and maxi-

load. by restraint

at the fixed

end.

Solution: A.

From

the

shear

stress

at points

A and B is

Mt Tt = where

2A--'t

A = ab-t(a+b)

Therefore,

A=

6(3)

+t 2.

-0.2

(6+

3) + (0.2)

2

A = 16.24in.2 and

lOOmO00

_t =

_t

For

2(16.24)

15,394

psi

the maximum

ratio

of the

(0.2)

.

shear

stress

stress

at point

at point

C to the stress

C,

refer

to Figure

at points

B8.3.3-7

for the

A and B.

rt(max) =

1.7

Tt(A)

• t (max) _t(max) The maximum Section

= 1.7 = 26,170 angle

BS. 3.3-IB:

(t5,394) psi

of twist

is determined

by the

following

equation

from

1967

Section BS. 3.0 31 December 1967 Page

21

MtL

_(max)

=

GK

where 4A2t

K-

and

C

C = 2(a+

b-

2t)

C = 2(6+

Therefore,

3-

2 (0.2))

C = 17.2

K

K = 12.27 _(max)

=

100,000(60) 4x 106 (12.27)

¢(max

I?Al" / _[

Mt

_/-__J_ %. b

=

/

/ /

t

_"_

= 0.2

in..

_ - o 33

3 in.

E M

= t

FIGURE

B8.3.4-1

10.3

-

1

x x

10 6

10 5

[n.-ib.

Section

B8.3.0

31 December Page B. Section

To find

the normal

stress

at the fixed

end,

use

22

the equations

in

BS. 3.3-II. From

Figure

B8.3.3-6,

m=0.3

K=

tG 4M t

(a+b)2

(i_) E

( _ )_ 1 -/_

K

4x

105

- 0.33

1oo)

\10.3x

l0s

K = 24. 217 x I0-s 0.33 24.217x I0-s

o = II.

Example For

the warping

(dimensions

Problem

the cross

= 13,620 psi

2 section

deformations

shown

in Figure

of the points

BS. 3.4-2,

N E

B

0.048 O. 048 0

sin a -

the distribution 0.2425

FIGURE

to find

shown.

are in inches)

Determine

it is required

of

warpin8

.'. ON ffi 0.97

B8.3-4-2

M t

=

G

=

I

x

10 5

4.0

x

in.-Ib.

10 6

1967

From

Section

Section B8.3.0 31 December 1967

B8.3.1-IVC,

Page

23

C M w(s)

-Wo

=

t2AG

I

s f0

1

(

f0

t(s)

-r(s)

2

ds C

G__G_[w(s)

] =

Mt

_

o

fs

1

- r(s)

0

t(s)

2A

ds

2A

A = enclosed area of section = 6.0 in.2

C d_______sI _0 t(s)

=2

KD

+

i 0.048

DC

CF

0.064 = 296.8

Choose

+

! 2' 06------22 + 4 0.048

0.048

For

s from point D.

t = 0.064

2AG

[ wo] Wc

Mt

-

/ 0

"1 I

0.064

0.5( 296.8)| 2(6)

ds

J

= 3.26(4) I For

sector

CF:

r = 0.97

13.04 t = 0.048

2. 062 IO•048 1

= - (3.16)(2.062) =

+

0. 064

- 6.52

.

O. 97(296.8) 2(6)

sector

2. 062 0.048

.

point D as the origin and measure

r = 0.5

= 2

DC:

Section

B8.3.0

31 December Page

For

sector

FB:

ra0.97

Mt2A"-'GG IWB_

For

sector

BA:

t=0.048

WFl

(same

(Same

as sector

1967

24

CF)

= _6.52

as sector

DC)

wol For

sector

AK:

(same

as sector

CF)

2AG Mt

For

sector

w K-

KD:

wA

(same

2A----_G [w D Mt Hence,

wK ]

deflections

around

the section

from

D back

to

zero.

symmetry,

plane.and

CF)

=-6.52

of warping

Now it is desired From

6.52

as sector

the summation

D equals

=-

will

points not deform

to find El,

F1,

from

the mean H1, and their

displacement K 1 will

original

lie

plane

(see

on the mean

positions.

Therefore

Fig. B8.3.4-3). displacement the distance

Section BS. 30 31 December 1967 Page

from

point

D to the mean

displacement

plane

is

2 ol l =_____ w_-w0 Therefore

the warping

deformations

are

6.52 W

=

W

D

=

C

W

=

B

Mt

W

A

2AG 6.52( 1 x 1057 2(6)(4x 106)

=

0.0136

inch

_m

Mean

Displacement

FI

D

C

II

FIGURE

B8.3.4-3

Plane

25

Section B8.3.0 31 December 1967 Page

Example

me

For

Problem

+

26

3

the problem

shown

in Figure

B8.3.4-4,

it is required

to find the

following: Maximum

A.

torsional

shear

stress

L

at L X

L

=0,

Lx

=

-- 2,and

=L. X

B.

Maximum

angle

of twist.

Solution: A.

The formula

_t =

2A(x)

for shear

stress

as obtained

by Section

B8.3.1-IVA

iS

M(x)

Therefore,

t(x,s)

at Lx = O, since

The shear

stress

(max)

_99 4

will

=

be maximum

at thickness

of t o.

-

100, 000 7r(5) 2 0. 1

9

1,414

=L x

M(x)

= M0

A(x)

= 4 7rr02

T (max)

' M(x)

Therefore

4M° 2(2)1r ro2 9 t o MA

L

L 2

"

9 _ ro 2to

Where

= O, _t = 0 at L X -

i )2

A(x) = _ro 2(1+-_" = _ro2

M(x)

M o

2(4)

7fro 2t o

-

I, 591 psi

psi

=

--M° 2

Section 31

B8.3.0

December

Page

A

_--C-ro __

]

i __..__

......

=

:

Xo

I

_

_o_o.1 _.

I

F/_I

ro

7

....

I

_

27

x

in"

" i X

105

in.-lb.

t(S)

= to(l

¢ sin _)

I

r(x)

_

-_

A(x)

-- _rr 2 (x)

_

---M°

_

A

L

SECTION

= 5.0

A-A

FIGURE

B8.3.4-4

ro (I + x/L)

1967

Section

BS. 3.0

31 December

Pa_e28 B.

Tim formula

C

for angle

ds

(x)

of twist

= 4

Is obtained

to(l+

from

Section

stnO)

JO

to

tan

to Therefore,

for maximum

angle

of twist

4 SL M°(-_)r°(l÷(-_ -) '_=_0 _o' (_ _)' dx

M

S L

x dx

L

•_o_ '- (,÷__) ,.(,÷ _=_ =

ML

MnL 8_2r_to G

(max)

(_) -

1 x 10 s x 30 "8_2(5)s(0.1)4x

= 0. 76 x 10 -8 radians

.

101

0

B8.3.1-IVB.

1967

SECTION B8. 4 TORSION OF THIN-WALLED OPEN SECTIONS

TABLE

OF

CONTENTS

Page B8.4.0

Torsion 8.4.1 I. II.

of

Thin-Walled

General Basic

Theory

Limitations Membrane

IV.

Torsional

I. If.

Angle

IV.

Stress

8.4.3 I. II. Ill.

................................ Analogy

Torsion

Concentration Torsion

of Twist

and

April

1970

3 8 9

.......................

24 25 27

.......................

29

Factors

31

...............

......................... Derivatives

..............

33 33

Stresses

...................................

35

Warping

Deformations

37

......................

B8.4-iii 15

2

12

.............................

Deformation

1

......................

..................................

Restrained Angle

.........................

Coefficients

of Twist

Stresses Warping

............

..............................

Unrestrained

III.

Sections

...................................

III.

8.4.2

Open

Section 28 June Page B8.4.0

TORSION An open

section

not form

a closed

sections

are

binations

OF THIN-WALLED is a section

curve.

among

many

rectangular

is used

in aircraft

of these

sections

is that

and

with the other torsional

restrained

angle torsion.

the

of twist,

stress,

angle

of twist

are

shear

of twist, determined

characterized

structures. of the

wall does

and wide-flange

a variety

by com-

of thin-walled

The

component

of thin-walled in this

deformations stress,

open section. are

warping

and the first, for

of the

basic

curved

characteristic

element

is small

in

dimensions.

is included

and warping

Torsional

thickness

analysis

torsion

shapes

elements;

and missile

centerline

I-beams,

structural

1

SECTIONS the

angles,

common

sections

The

in which

Channels,

of thin-walled

comparison

OPEN

BS. 4 1968

restrained

torsion.

for

Torsional

determined

shear

second,

sections

for

both shear

unrestrained stress,

unrestrained

stress,

warping

and third

dcr_vatives

normal of angle

Section B8.4 28 June 1968 Page B8.4.1

GENERAL The

section

stresses

and deformations

can be superimposed

if the limitations of stress

2

of Section

and deformation

determined

with bending BS. 4.1-II sign

and axial

by the equations load

are not exceeded

convention

is taken

stress

and deformations

and proper

into account.

in this

consideration

Section

B8.4

28 June

1968

Page B8.4.

I

GENERAL

I.

BASIC

THEORY

If a member ends

in the plane

warp,

we have

of open

cross

perpendicular the case

A.

of unrestrained

Rotated

However, the

accompanied of change case

of the

of the

angle

of the

torsion

B8.4.1-1. are

warping

along

is called

restrained

torsion.

These

two types

of torsion

by couples

at the

are

free

to

BS. 4.1-1}.

Warped

Section

Warping

varies

to warp

along

the

the bar's

and

fibers.

longitudinal

be discussed

or if the torque

bar

of longitudinal

will

applied

and the ends

(Fig.

not free

or compression of twist

bar

B.

sections

bar,

by tension

is twisted

Section

if cross

length

section to the axis

Figure

along

3

axis

separately

torsion

varies

is

Also,

the rate

varies.

This

in the following

sections. A. The the

torsional

Unrestrained twisting shear

Torsion moment

stress

on thin-walled for

unrestrained

open

sections

torsion.

is resisted

However,

only

the manner

by in

Section

B8.4

1 July Page which

a thin-walled

manner

in which

difference

t-IA

equal

by comparison

flow

goes

that flow

B8.4.1-2,

which

to the negative

section

moment

carries

of Figures

goes

the section, around

that

is linear

and that

of the maximum

_f

I-IA,

closed while

stress

the

moment.

section

carries section

distribution stress

on the other

This

the carries

of the section.

the maximum

stress

4

from

B8.2.2-2A,

the open

the perimeter

the shear

differs

a torsional

B8.2.

The thin-walled

around

it can be seen

of the section

a torsional

closed

8.4.1-2.

the load by a shear

thickness

carries

to Figure

by a shear

Figure

section

the thin-walled

can be seen

and B8.3. load

open

1969

From

across

the

on one edge

edge.

(Ref.

is

1).

_tf

t2

b2

-._

...- t2

b

_w t1 I_ I: I

.I_ W1-I-

bl-I

Figure

B8.4.1-2.

The torsional torsion point

on the

the torsional throughout

shear the

Pure

analysis

will require (P)

bl

_1

that

Shear

of thin-walled

open

the torsional

section. stress

length

Torsion

of the

Because

shear

Stress sections

stress

of the definition

at any point on the section member.

(r t)

Distribution for unrestrained be determined

of unrestrained will

remain

at any

torsion, constant

Section 28 June Page The plus

angle

the warping

unique

that called

(S t)

are

functions

and

are

B.

discussed

Restrained

When

cannot

with

be evaluated

open

sections,

slowly

from

their

a system

warping.

twisting rate

moment

of twist,

produced

sections, cross

two

section.

and

the torsional

cross

section.

These

B8.4.1-IV).

they

developed

and final

by unrestrained

are

on each

torsion.

and may

the

to serious

errors.

warping

from

cannot

and In thin-

diminish

very stress

a way that

it cannot

to eliminate

this

point

by a nonuniform

is no longer

that

the primary

Lwist of the section.

section

stresses

assumption

constitute

vary

is developed

valid,

be developed

stresses

is restrained

is no longer

in such

accompanied

alters

shearing

lead

The

by restrained

must

normal

in turn,

may

is restrained

stresses

section stresses

theories.

produced

section

cross

of longitudinal

principle

member.

these

open

deformation

in the

hence,

This,

section.

of each

of the

(Section

elementary

of application

of normal

distribution.

below

be determined,

closed

(K)

of the dimensions

points

if one

and,

constant

stresses

In general,

the member

the torsional

during

walled

warp,

and thin-walled

a thin-wailed

plane

of Saint-VenantTs

Obviously,

also

on the cross

(P)

the geometry

using

remain

developed

should

characterize

distribution

applications

system

(q))

5

Torsion

a complex

sections

sections

in detail

a member

warping

section

at any point

solid

are

coefficients

against

for

exist

cross

(w)

coefficients

modulus

plane

the case

coefficients

These

of the

deformation

As was

that

of twist

B8.4 1968

to point

shearing

stress

As a result,

proportional

be obtained

along

by those

the

to the that

were

Section B8.4 I July 1969 Page 6 Therefore, three types of stresses must be evaluated for the case of restrained torsion. These are: (I) pure torsional shear stress, (2) warping shear stress, and (3) warping normal stress. are shown for several common

These stress distributions

sections in Figures BS. 4.1-3, B8.4.1-4,

and

B8.4. i-5. It will be required to evaluate these stresses at any point (P) on the cross section and at any arbitrary distance (L) x the angle of twist (¢) should be determined

between an arbitrary cross section

and the origin along with the warping deformation arbitrary cross section. (Ref.

from the origin. Also,

(w) at any point (P) on an

2).

It was shown previously that two coefficientswere necessary to characterize the geometry of the cross section for unrestrained torsion. These were the torsional constant (K) and the torsional section modulus

(St). For

restrained torsion, three additionalcoefficientsare required to characterize fully

the

geometry

be determined. a function

of the cross These

section

coefficients

of the dimensions (Wn) , and

functions

of both the dimensions

B8.4.1-IV.

section.

warping

These

the point

are called

of the cross

function

the cross

and

statical

of the cross

coefficients

the warping

section,

moment

are

where

the

(S w) . section

discussed

the

stresses

constant

normalized The latter and

--

warping two are

a specific

in detail

(r)

are

point on

in Section

to

Tt4

_I

Section

B8.4

1 July

1969

4-------4. qb-

91-

ewO

91-4-,O-

tJ_

*_Z * Locetion

of

Shear

,... *-- ..o ,, ¢ cemp,es,ie,

I J

Center

,t=Gt_

TwO I_._...._

I_teO

T__..._

_1 Figure

B8.4.1-3.

Restrained

_.3

/

_s Warping

Stress

- EWns_

in I-Section

_.s = - E _-_ *'" U

* Locatien

of SkeQr Center

ac_

_rt = G t f

Figure

"rol

B8.4.

l-4.

Restrained

aw$ = E Was #"

Warping

Stresses

in Channel

Section

T_

• Leceolenel SkNr C_,_

Figure

B8.4.1-5.

Restrained

Warping

Stresses

in Z-Section

"

Section B8.4 28 June 1968 Page B8.4.

I

GENERAL

II.

LIMITATIONS The

following

changes

8

torsional

analysis

of thin-waUed

open

sections

is subject

to the

limitations: A.

Homogeneous

B.

Thin-walled

cross

C,

No abrupt

variations

D.

No buckling

E.

Inexact

of applied

calculations twisting

F.

Applied

G.

No abrupt

H.

Shear

to the shear I.

strain Points

and isotropic

not necessarily

in thickness

of stresses

except

of constant at reentrant

at points

thickness corners

of constraint

and at abrupt

moment

twist}ng changes

stress

section

material

moment

can occur

is within

(elastic of constraint

cannot

shearing

be impact

in cross

load

section

proportional

limit

and proportional

analysis). are fully

fixed,

and no partial

fixity

is allowed.

Section

B8.4

1 July 1969 Page 9 GENERAL MEMBRANE In the analogy effect slightly

case

gives of the

ANA LOGY of a narrow

a very short

deflected

simple

sides

rectangular solution

of the

membrane

cross

to the

rectangle

and

is cylindrical

section,

torsional assuming ( Figure

the

membrane

problem. that B8.4.1-6),

Neglecting the

surface the

the of the

deflection

is

8T and

the

membrane

maximum and

slope the

is

xy plane

pt 2T" is

The (Ref.

volume

bounded

by the

deflected

3}:

H A

_] =X

T.

'TI f4X

z

A-A

Figure

B8.4.1-6.

Membrane

Analogy

for

Torsion

of Thin

Rectangular

Section

Section

B8.4

1 5uly 1969 Page i 0 Now using

membrane

analogy

equations,

the twisting

and substituting

moment

(Mt)

2C_

is given

for

p/T

in the previous

by

I M t = -_- bt 3 C_

or

M

t

0 =M(1/3bt3G-sGt t and

the maximum

shearing M

T

= tG0 max

stress M

t

-

__1 bt 2 3

is

t

St

where

St = 1/3bt 2 .

The

equations

used adding small general

for

cross

for

M t and

sections,

the expression error case

Tmax, such

of a section

or points with

n

K-i=1

as those

1/3bt 3 for

at corners

b.tS.. 11

obtained

each

for a thin rectangle

shown

in Figure

element

of the

of intersection

N elements:

can also

BS. 4.1-2, section

be

by simply

(neglecting

of the elements).

In the

a

f-. Section

B8.4

28 June

1968

Page The

maximum

shearing

Mt (Tmax}

.

maximum

stress

element

i is

given

max

on the t

t

max

=

max

any

t

M T

(ti)

on

S

1

The

stress

St

entire

section

is

given

by

by

11

Section 28 June Page

B8.4 1968

12

GENERAL TORSIO_IA In the cross

development

sections,

coefficients entire

L COEFFICIENTS

it is convenient

for

the cross

cross

section,

on the cross

section.

Constant

The

torsional

coefficient

depends

upon

the thin

terms

the certain

terms St,

torsional terms

K and

Sw,

analysis

and

of open

as torsional

F are

properties

Wn apply

of the

to specific

points

(K) (K)

the geometry

constants

for

is called

the

of the cross

thin-walled

open

torsional

constant,

and

based

on formulas

section. sections

are

rectangle.

Section for a general section,

The

the

Torsional

Torsional

ten,

while

for

to designate

section.

A.

its value

for

of the formulas

B8.2.2-III rectangular

it can

be seen

the value

contains

an expression

section.

Since

that

when

of the torsion

we are

for

the

concerned

the length-to-thickness

constant

torsional

constant

with a thin-walled ratio

is approximately

is

K ~ 1/3bt a

This

value

The

torsional

with

is also

constant

b denoting Therefore,

curved

elements,

expression:

verified for

the length for

by the membrane curved

elements

of the curved

a section

the torsional

is the element,

composed constant

analogy

of many can

in Section

same

B8.4.1-HI.

as that for

as shown

in Figure

thin rectangular,

be evaluated

a rectangle BS. 4.1-2. or thin

by the following

F

Section

B8.4

1 July n

K = 1/3 i If a section value

has

of K for

1969

Page

13

than

ten,

b11t

any element that

with a length-to-thickness

element

should

ratio

be determined

by the

less

equations

the

in Section

B8.2.2-IU. More standard fillets

accurate

sections

torsional

constant

by considering

expressions

the junctions

are

of the

determined

rectangles

and

at the junctions. Some 1.

K values

For

for frequently

I sections

K=2/3bt_+

with uniform

1/3(d-2tf)t

used

sections

flanges

_W + 2c_ I:fl-

are

( Fig.

(Ref,

2):

B8.4.1-7A)

:

0.42016t_

where

= 0.094+

0.07--

R tf

(tf +R)2+tw

+

D= 2R + tf 2.

For

I sections

with sloping

flanges

(Fig.

B8.4.1-7B)

b-t W K

__

+2c_

(tf+

D4-Et_

a)(t_+

a 2) + 2/3

t W a3+

1/3(d-

2a) taW

:

for

some

rounded

Section 28 June Page

where (F+c)2+t D

and for

R+

w

F+R+c

5-percent

flange

slope t

(_ = 0. 066 + 0. 021_

+ 0.072

_

a

a

E = 0. 44104

R (_ 9.0250 s-- 2--5

and

for

2-percent

t

flange

)

slope t

=0.084+

0.007_

l:3

+ 0.071 a

_ a

E = 0.42828

R F=5-- _

(

3.

channels

with

tawd +-_-b'

(a + tf)

For

K = 1/3

49.01

-

t) sloping

(a2+

flanges

t_)

( Fig.

B8.4.

i-7C)

:

14

B8.4 1968

/

F

Section

B8.4

28 June

1968

Page 4.

For

Tee

bt_ K=_+----_W 3

section

(Fig.

B8.4.1-7D)

15

:

ht 3 +(_

D4

3

where

= 0. 094

(t+

+ 0.07

R)2+

R/tf

t

(t) R+

W D

=

2R + tf

5.

Angle

section

K = 1/3bt_

(Fig.

+ 1/3

B8.4.1-7E)

:

dt32 + c_ D 4

where

_ = t-2-/0"07+0"076-_)tl

(tl+

R)2+

tl>t2

t_ (R+_)

D_

2R+

6.

Zee

section

t1

and

channel

section

with

uniform

flanges

(Fig.

B8.4.1-7F). K values the

constituent

for angle

these sections

sections

can

computed

be calculated in case

5.

by summing

the

K's

of

S_ction BS. 4 28 June 1968 Page

B.

Sloping

16

Flanges

b1

D k

_-" f W

fw

C.

Channel with Sloping Flanges D.

E.

Angle

Figure

F.

B8.4.1-7.

Frequently

Tee

Section

Zee and Uniform

Used

Sections

Channel

SectionB8.4 28 June 1968 Page 17 It shouldbe notedthat the K formulas for these frequently used sections are basedon membraneanalogyand on reasonably close approximations giving results that are rarely as much as 10 percent in error. B. Torsional Modulus (St) The torsional coefficient (St) is called the torsional modulus. Its value for any point on the section dependsupon abegeometryof the cross section. The basic equationfor determining the torsional modulusat an arbitrary point (s) on a cross section is: K

St(s)

where

,

- t(s)

K is as defined

section

in Section

B8.4.1-IVA

at point

(s).

Because

the torsional

modulus

stress

equation

torsional

shear

in the

and

is necessary

t(s)

for

is the thickness

the calculation

of the

of the

M (x) t - -St(x , s)

it is often

required

to find

the minimum

value

of St(s)

in order

a maximum. Therefore:

K St(min)

-

t max

where

t

max

is the maximum

thickness

in the cross

section.

to make

rt

Section B8.4 28 June 1968 Page C.

Normalized

Warping

The torsional function. specific

Its value points For

equation

coefficient depends

on the cross

the generalized

is used

for

1 =-_-

Wn(S)

Function (Wn)

18

(Wn)

is called

upon the geometry

the normalized of the cross

warping

section

and upon

section. section

calculating

shown

Wn(S)

in Figure

B8.4.1-8,

at any point

(s)

the following

on the section:

b / o

WOS

tds-

w OS

where b A = _ 0

tds

S

0 Some 1.

W

For

For

W

no

values

for frequently

symmetrical

wide flange

used

sections

include:

and I-shapes

bh 4

no

2.

n

-

channel

ah 2 Eh O

Wn2 -

2

sections

(Fig.

]38.4.1-9B)

:

(Fig.

B8.4.

I-9A)

:

p Po cg sc z,y

Section

Perpendicular distance to tangent line from centroid Perpendicular distance to tangent line from shear center Centroid of cross section Shear center of cross section Coordinates referred to the principal centroidal axes Angle of twist

Page

['All directions ore shown positive, p andPo are positive if they are on the left side of an observer at P (z,y) facing the posi. tire direction of s.']

t,_qm_nM

P (_,r)

- ///

,\

b (re,y)

St

• (l,V)

Figure

B8.4.1-8.

General

Thin-Walled

Open

where

(b')2 t

E o

2b't

+ h tw/3

and

u=b'-E o

3.

For

W

no

Wn2

-

zee

uh 2 u'h 2

sections

(Fig.

B8.4.1-9C)

:

B8.4

28 June

Cross

Section

1968 19

Section

B8.4

28 June Page

20

Swl

W $wo _

A.

Symmetrical

H-and

Swo

I-Sections

Swl

$wo

Sw2

Sw3 t Wno_"_,, $w2 Wn2

Swo_

B.

Channel

Sections Swl Sw2

Wn2

w._

A

Wne

s,.2 L.Lj s,. Sw2

C.

Figure

B8.4.1-9.

Distribution

ZeeSec_ons

of

W

n

and

S

w

for

Standard

Sections

1968

Section

B8.4

28 June

1968

Page

21

where

u = b' - u'

u' - (b')2 t h + 2b't tw In the

foregoing

expressions:

h = distance b = flange

between width,

b' = distance

D.

Warping

The

torsional

Its value points

depends

on the For

equation

toe of flange

thickness

= thickness

w

of flange,

of web, Statical

cross

Moments

the

and centerline

of web,

in.

in.

(S)

(Sw)

geometry

w

is called of the

cross

the warping section

statical

and upon

moment. specific

section.

the generalized

is used

in.

in.

coefficient

upon

of flanges,

in.

between

t = average t

centerlines

for

section

calculating

shown

Sw(S)

in Figure

at any point

B8.4.1-8,

the following

(s)

section:

on the

S

s (s) = f W

Wn(s)

tds.

0

The Some

value S

w

of Wn(s)

is determined

values

frequently

for

from used

the previous

sections

include:

subsection

(B8.4.1-WC).

Section BS. 4 28 June 1968 Page For. symmetrical

,

For

Swl

flange

channel

(Figure

BS. 4. I-9A)

:

sections

(Figure

BS. 4.1-9B)

:

u2ht 4

-

(b'-

2E

Sw2 =

)hbWt

o

4 (b' - 2E

)hb't

E h2t

o

Sw3 =

3.

and I-shapes

hbRt 16

Swl -

2.

wide

22

For

zee

O

,

4

_

sections

(ht

W

8

(Fig.

B8.4.1-9C)

:

are defined

in the previous

+ b't) 2 h(b')2t W

Swl

-

4(ht

+ 2b_t) 2 a

h2tw(b') Sw2 - 4(ht

where

u, h,

E.

t,

W

Warping

depends

section

shown

culating

F:

+ 2b't)

b, b t, E

The torsional value

_t

only

O'

and

Constant

t

W

(r)

coefficient on the geometry

in Figure

section.

B8.4.1-8,

(r)

is called of the cross the following

the warping section. equation

constant. For

Its

the generalized

is used for cal-

Section 28 June Page

B8.4 1968

23

b F

= f

Wn(s)

t2ds .

O

The value

of Wn(S)

frequently

used

sections

For

symmetrical

1.

is determined

h2b3 t F

24

For

channel

Some

B8.4.1-IVC.

values

for

are: wide

sections

r = 1/6

(b'3Eo)h2(b')

3.

zee

For

Section

flange

and

I-shapes

(Fig.

B8.4.1-9A)

:

I h2 Y__ 4

__

2.

from

(Fig.

t+

B8.4.1-9B)

:

F 2O IX

sections:

(b,) 3th2

b't+

12

nt

2ht w + 2b_t

W

where moment of inertia

h, b,

t, tw,

of inertia

b',

and

E O are

of the entire

of the entire

section

defined

section about

about the

yy

in Section the axis.

xx

B8.4.1-IVC, axis,

and

I x = the I

Y

= the moment

Section

B8.4

28 June Page B8.4.2

UNRESTRAINED The

formulas

cross

section

to the

longitudinal

Figure

BS. 4.1-1.

TORSION

given

twisted axis

1968

24

in this

by couples of the bar,

section applied

apply

only

at the ends

and the

ends

are

to members

of open

in the plane

perpendicular

free

as shown

to warp

in

SectionB8.4 28 June 1968 Page25 B8.4.2

UNRESTRAINED

I.

ANGLE For

OF TWIST

the case

by the cross

TORSION

of unrestrained

section

torsion,

the torsional

moment

resisted

is

M I = GK _'

where M 1 = resisting

moment

of unrestrained

cross

section,

in.-lb

= Mt G = shear

modulus

K = torsional _)' = _ dX This X,

the

(Fig.

constant

= angle

is the first

distance

of elasticity, for

the cross

of twist

per

derivative

measured

along

psi section,

in. 4

unit of length.

of the angle the length

of rotation

_

with respect

from

the left

the angle

of twist

of the member

to

B8.4.2-1). Therefore,

between

the basic

the origin

equation

and an arbitrary

for determining cross

section

at a distance

L

X

from

the

origin is:

¢(x)

1

M(x) S(x) dx

=--_-

o

where and

K(x) M(x)

is determined

is the is taken from

torsion as

constant

M t applied

the equation:

at

L . x

If the

at the end of the

cross

section

member,

the

does

not vary

angle

of twist

Section

B8.4

28 June

1968

Page

26

MtL x _(x)

The total

=

twist

GK

of the bar

is:

Ay

Ay

l

o

DIRECTION

•

OF VIEWING

__--1

z

x "_

MT

APPLIED

/

-

_ IS

TORQUE

POSITIVE

Figure

B8.4.2-1.

General

Orientation

ANGLE

OF

ROTATION

SectionB8.4 28 June 1968 Page27 BS.4.2

UNRESTRAINED

II.

only

TORSION

STRESSES The

twisting

moment

by the

torsional

shear

shear

stress

at the edge

CM t) on thin-walled stress

for

open

unrestrained

of an element

sections

torsion.

is determined

is resisted The

torsional

by the formula:

r t = Gt _' .

M(X) _b = GK

Because

the basic equation for determining the torsional shear stress

at an arbitrary point (s) on an arbitrary cross section is:

M(x) rt - St(x,s) where

K(x k -t(x,s)

St(x's)

K(x)

is evaluated

determined, point(s)

and

at x = L where x t(x,

on the arbitrary If the member

applied

s)

is evaluated cross

has

Mt

st(s)

torsional

shear

at the arbitrary

stress cross

is to be section

and at the

section.

uniform

to the end of the bar,

_'t =

the

cross

the equation

section reduces

and to:

M(x)

is taken

as

Mt,

Section 28 June Page

B8.4 1968

28

where

K

St(s) = t( s---_

The

maximum

stress

r t will occur

on the thickest

elementl

t(s)

is maximum.

SectionB8.4 28June 1968 Page B8.4.2

UNRESTRAINED

III.

WARPING The

any point

basic

29

TORSION

DEFORMATION equation

on an arbitrary

for

determining

cross

section

the warping

deformation

at a distance

x = L

X

from

w(s) the

at

origin

is

S

w(s)

- w O =_)'

f

rds

O

where

w(s)

cross

section

the point length

the

axis

from

ds

ing inihe

is the warping in the which

from

point

direction of rotation

deformation

x direction;

w

at point

taken

of increasing ( Fig.

Figure

r(s)

positive s

BS. 4.2-2)

B8.4.2-2.

on the middle

is the displacement

o

s is measured; o,

(s)

along

gives

a positive

; _'

is determined

General

of the tangent the

moment

Section

of the

in the x-direction

is the distance

if a vector

line

from

tangent

of of arc

with

and point-

Section

respect

to

B8.4.2-I.

Section 28 June Page For

the case

of unrestrained

torsion,

the point

BS. 4 1968

30

0 can be located

arbitrarily. The warping w has

been found

w(s)

where

Wn(S)

of the cross

section

with respect

to the plane

of average

in Section

B8.4.1-IVC.

to be

= _, Wn(S )

is the normalized

warping

function

found

SectionB8.4 1 July 1969 Page31 B8.4.2 IV.

UNRESTRAINED STRESS Stress

reentrant flanges

CONCENTRATION

concentrations

corner;

that

section.

in composite

at the intersection

or at the interior

Exact

difficult

FACTORS

occur

is,

in the Iosection

angle very

TORSION

analysis

and must

of stress

be carried

cross

sections

of the web angle

joining

concentrations

at any

and either

the

two legs

at these

out experimentally,

of the

usually

of the

points

is

by membrane

analogy. For reentrant

many point

_" max

where

common

sections,

the maximum

stress

is

4)

K 3-

o

_r2D4

1+_

D = diameter

1+

p = radius

.118

of largest

A = cross-sectional

In

1 +

inscribed

circle

+0.238

( Section

angles

in fillets

of concave

with legs is shown

tanh

7_

B8.4.1-IVA)

area boundary

at the point

(positive)

O = angle through which a tangent to the boundary around the concave portion, tad.

stress

or

=K 3 G¢'

(Ref.

For

at the concave

of equal

on Figure

thickness, B8.4.2-3.

rotates

the percentage

in rolling

increase

of

200

+,.-,+i I Fo w

\

120

\

< IM

imp i,.) z -114,1 o .,< l,z w uI

%.

80

,10

LU a

0 0

0.4

0.8

1.2

RATIO

Figure

B8.4

1 July

1969

Page

240

I-.. IM .,.I ..I Ii.

Section

BS. 4.2-3.

Stress

Increase

1.6

2.0

_-

Ln Fillets

of Angles

32

Section

B8.4

15 August Page B8.4.4

EXAMPLE

TORSION

A member

with

by an end

moment

determine

an unsymmetrical

the maximum

and

section

is free

angle

to warp.

of twist,

shown If

torsional

in Figure

M t=

100in.-lbs

shear

stress,

B8.4.4-1 and and

L

0.940

I

_J

t I I

¢

t

D t =

t

,_.---

1.380

t

0.12

in.

A =

0,6912

G =

3 X 106

In. 2 psi

t

B .124

s

S.C. + 3.380

Figure Wos

B8.4.4-i.

= fs

Cross

Section

for

Example

Pods:

O

W

= 0.124s

for

s < 3. 380

OS

W

= 0.419+

0.541s

for

4.760

> s > 3.380

= 1.166-

1.504s

for

s > 4.760

OS

W OS

Problem

L=41

warping

deformations.

Evaluate

1970 1

PROBLEMS

I. UNRESTRAINED

loaded

32.

I

is in.,

Section

B8.4

15 August Page Evaluate

1970

32.2

Wn(S): i

b

Wn(S)=T f

wOS m,,- wOS

0

38

T1 ofb

WostdS_

O. 124sds

0-0"12 69i'2 {of3"38 + f0.94

(1.166

+

l. 0

- 1.504s)ds

= 0.388.

O

Then:

Wn(S)

= 0.388

Wn(S)

= - 0.031

Wn(S)

= - 0.778

Therefore,

at points

n

w (B) =-0.031 n

=-

Wn(D)

= 0.636.

s < 3.380

- 0.541s

for

4.760

+ 1. 504s

for

s>4.76

on the cross

w (A) = 0.388

Wn(C)

for

- 0.. 124s

0.778

section:

> s > 3.380

.

(0.419

+ 0.541s)

ds

Section

B8.4

15 August Page The

distribution

of

W

(s)

is

n

shown

in

Figure

1970

32. 3

B8.4.4-2.

Wn (C) D C

Wn (O)

Wn (A)

B

Wn (B) Figure Warping

Deformations

B8.4.4-2,

(measured

W(s)

=

¢'

W(s)

-

M GK

W(A)

-

W(A)

= 0.0039

W(B)

=-

W(C)

= - 0. 0079

W(D)

= 0. 0065

Wn(

Distribution from

s)

Wn(S)

100 (0. 388) 3 x 106x 3. 285x

in.

0.0003

in. in. in.

10 -3

of

mean

displacement

(Section

B8.4.2-III)

W

n

(s) plane)

:

Section

B8.4

15 August Evaluate

Page

K:

K = 1/3

bt 3

K=1/3(3.38+

1.38+

K=3.285x

Maximum

Torsional

0.94)

(0.12)

3

10 -3 in.4

angle

of twist:

¢ (max)

MtL = G---K

¢ (max)

-

¢ (max)

= O. 416 radian .

Shear

(Section

B8.4.2-I)

I00 x 41 3x106x3.285xlO

-_J

Stress:

Mt ( Section

l"t =S_ where

K St(s) = 100 x 0.12 _t =3.285x10 "s

=3655psi

B8.4.2-II)

32.4

1970

W" Section 28 June Page B8.4.3

RESTRAINED I.

ANGLE It was

resisted

OF TWIST

shown

cross

section,

according

AND DERIVATIVES

that for

Longitudinal This

is

unrestrained

bending

torsion,

M 1 = GK ¢'

bending

and

33

TORSION

by the section

warping.

B8.4 1968

occurs

(Section

when

is accompanied

these

stresses

to the following

the

by shear the

moment

BS. 4.2-I).

a section

resist

torsional

is restrained stresses

external

from

free

in the plane

applied

of the

torsional

moment

relationship:

M 2 = - E F¢"'

where M 2 = resisting section,

caused

E = modulus

of elasticity,

F = warping

constant

4"'

= third

Therefore, sum

moment in.-lb

of M_ and

M 2.

on the

resistance

by M,

the following M=

M I+

of the

the total

torsional

expression

M 2=GK_b'

or

1

a-_- ¢' - ¢'"

the cross

derivative

to warping.

M

- Er

warping

of the cross

psi

for

The first

by restrained

angle

of these Denoting

with

resisted

is always the

(Section

of rotation

moment

is obtained. - EFt"'

section

total

B8.4.1-IVB), respect

to x.

by the section

present;

the

torsional

resisting

in. 6

second

is the depends moment

Section

BS. 4

15 April Page

1970

34

where

EF a 2

_

GK

The applied

solution

torque

Numerical obtained

(M)

and

a computer

Handbook

for

of twist

distribution

and and

program many

can

end

the

distribution

_',

of the

_",

and

Astronautics

and

end

foregoing

defined.

the

restraints

_,

in the

before be

upon

for

loading

derivatives

warping

or

equation

to evaluate its

depends

boundary

of this

It is necessary angle

equation

the

evaluation from

Utilization

of this

of member.

9_'''

is

Computer

conditions. expressions

a complete

picture

for

the

of stress

Section 15

B8.4

April

Page RESTRAINED

1970 35

TORSION

STRESSES

in

A.

Pure

The

equation

Section

Torsional for

B8.4.2-II;

member

and

must

be

torsional

This

stress

For

will

1-4,

loading

and

end

the

but

(Figs.

the

stress This

cross

These

stresses

the

magnitude

B8.4.

1-3,

of

twist

at

same

as

varies

previous

given

along

the

section.

reentrant

corners,

element

common

stress

of

sections, can

be

Computer

the

the

cross

see

Figures

calculated

by

Utilization

section.

a

B8.4. computer

Handbook

for

Stress

section of

the are

varies B8.4. the

1-4, equation:

WS

t

(/)

is

t t l

restrained

member, essentially at

ES --

TW S

the

thickest

for

Shear

length

from

angle

the

conditions.

Warping

determined

is

is

the

Astronautics

entire

induced.

in

1-5.

the

When along

this

B8.4.

B.

from

equation

largest

of

from

the

concentrations

stress

be

and

program

now

stress

P'.

distribution

B8.4.

(t),

stress

Gt

=

shear

determined

shear

Tt

Stress

torsional

however,

Neglecting pure

Shear

and

different B8.4.

from

warping

warping

shear

stresses

uniform

over

locations

of

1-5).

These

freely

the the stresses

are thickness

cross

section are

many

1-3,

Section

B8.4

15 April Page

1970

36

where rws E

= warping

shear

= modulus

Sws

of

= warping

of

= third

moment

can

Computer

point

angle

of

s (Section

B8.4.

I-IVD),

in. 4

in.

of

be

psi

s,

at

element, the

measured

stress

Astronautic

the

point

psi

derivative

distance This

at

elasticity, statical

t = thickness 9 _'''

stress

along

calculated

the by

Utilization

twist

with

length

of

a computer

Handbook

respect

the

to

program

for

x,

member.

many

from

the

and

end

section

is

loading

conditions. C.

Warping

Warping

normal

restrained

from

These and

are

whe

act

constant along

the of

the

equation:

by

aws

= EWns

aws

= warping

caused

along

perpendicular

length

determined

are

freely

across

the

Stress

stresses

warping

stresses

nitude is

Normal

the

to thickness

when

the

entire

the

surface

of

element.

the

an The

cross

length

of

of

cross

the

element

but

magnitude

the

member. section

vary of

in

mag-

these

stresses

_b"

re

E

normal

= modulus

Wns _'

of elasticity,

= normalized = record distance

This Astronautic conditions.

stress

stress

warping

function

of the

measured

along

be

s, psi

psi

derivative

can

Computer

at point

calculated

Utilization

at point

angle the

of twist length

s (Sec.B8.4. with

respect

to x,

of the member.

by a computer Handbook

I-IVC),

for many

program loading

from

the

and

end

in.2

Section

B8.4

15 April Page B8.4.

3

RESTRAINED

Ill.

vary

DEFORMATIONS

the

obtained

from

program

given be noted

corresponding displacements, evident.

deformations

that was

along

should

TORSION

WARPING Warping

equation

1970

37

given

length Section in the

in Section

be

B8.4.

The

3-I or

Astronautic warping

values

normal

of distribution

by using

except expression

can

Computer

displacements;

a picture

calculated

B8.4.2-111,

of the member.

that the warping

can

for

Utilization

hence, of the

by

are

95' will

95' can from

be

a

Handbook.

It

proportional

knowing

warping

same

that now

be obtained

stresses

the

the

to

warping

stresses

is

Section 15

]38.4

April

Page

1970

38

References Oden,

I.

Hill

.

J.

Heins, of

C.

4.

File

Jr.,

and

Design

Seaburg,

Structures.

D.

File/Design

Handbook

Co., R.

Book

Elastic

McGraw-

1967.

A.:

Torsion Bethlehem

Data.

Analysis Steel

13-A-1.

W.: Book

Roark, Hill

P., No.

of

Inc.,

Steel

Flklgge, Hill

Mechanics

Co.,

Rolled

AIA

3.

T.:

Book

of

Inc.,

J.:

Formulas

Co.,

Inc.,

Engineering

Mechanics.

McGraw-

1962. for

Stress

and

Strain.

McGraw-

1965.

Bibliography The Files,

Astronautics J.

IN-P&VE-S-67-1,

H.

Computer :

Restrained August

Utilization Torsion 20,

Handbook, of

1967.

Thin-Walled

NASA. Open

Sections.

SECTION B9 PLATES

TABLE

OF CONTENTS Page

B9

PLATES 9.1

INTRODUCT_N

9.2

PLATE

..............................

THEORY

9.2.1

.............................

SmMIDeflection 9.2.1.1

9.3

9.3.1

3

................... Plates

Theory

9

................

.......................

Deflection

MEDIUM-THICK THEORY)

3

Theory

Orthotropic

Membrane Large

1

Theory

PLATES

i0

...................

(SMALL

13

DEFLECTION 17

.................................. Circular

Plates

.........................

9.3.1.1

Solid,

9.3.1.2

Annular,

9.3.1.3

Solid, I.

II. 9.3.1.4

9.3.1.5

Rectangular

9.3.3

Elliptical

Uniform-Thickness

Plates

Uniform-Thickness

Varying

Nonlinear

Plates

Thickness

Varying

........

Thickness

.......

....

22

....

22 22 31

Annular

Plates

Varying

Thickness

................

Sector

of a Circular

Plate

...........

34

Plate

...........

36

Annular Plates

Plates

with

17

......

Plates

Nonuniform-Thickness

Linearly

I. 9.3.2

17

Linearly

Sectored ......................

........................

B9-iii

32

37 47

TABLE

OF

CONTENTS

(Continued) Page

9.4

9.3.4

Triangular

9.3.5

Skew

ISOTROPIC ANALYSIS

...........................

47

THIN PLATES - LARGE .................................

1

Circular

Plates

-- Uniformly

9.4.

2

Circular

Plates

-- Loaded

Rectangular

ORTHOTROPIC 9.5.1

9.6

Plates

Rectangular

t

Plate

Deflection i. 1

9.6.1.2

DEFLECTION 51 Distributed

at Center

-- Uniformly

Load

...........

Loaded

....

51 58

........

58

........................

65

........................

65

SANDWICH

Small 9.6.

Plates PLATES

STRUCTURAL 9.6.

47

........................

9.4.

9.4.3 9.5

Plates

PLATES Theory.

.................

71

...................

71

Basic Principles for Design of Flat Sandwich Panels Under Uniformly Distributed Normal Load ............ Determining Thickness,

Facing Thickness, Core and Core Shear Modulus

for Simply Supported Panels ........................ 9.6.1.3

Use

9.6 t.4

Determining

9.6.1.5

Checking

72

of Design

Charts Core

Procedure

B9-iv

Flat

Rectangular 72

..............

Shear

Stress

...............

77 ........

84 84

_j

TABLE

OF CONTENTS

(Concluded) Page

9.6.1.6

9.6.2

Determining

Facing

Thickness,

Core Thickness, and Core Shear Modulus for Simply Supported Flat Circular Panels ..................

84

9.6.1.7

Use

93

9.6.1.8

Determining

9.6.1.9

Checking

Large

Deflection

9.6.2.

t

Charts Core

..............

Shear

Procedure Theory

Rectangular Edge

9.6.2.2

of Design

...............

Sandwich

Movable,

and Clamped Conditions

96

Plate

Plate

with

Fixed 96

with

Clamped

Immovable .....................

94 95

..................

Sandwich

Supported

.......

...................

Conditions

Circular

Stress

Simply Movable,

Boundary 101

References

........................................

103

Bibliography

........................................

103

B9-v

Section B9 15 September 1971 Page 1 B9

PLATES

B9.1

INTRODUCTION Plate analysis is important in aerospace applications for both lateral

applied loads and also for sheet buckling problems. The plate can be considered as a two-dimensional counterpart of the beam except that the plate bends in all planes normal to the plate, whereas the beam bendsin one plane only. Becauseof the varied behavior of plates, they have been classified into four types, as followsThick plates

Plates

--

Thick

plate

as a three-dimensional

sequently,

quite

particular

cases.

to short,

deep

and

In thick

entirely

are

compared

problem.

problem

plates,

Plates

the The

stress

analysis

analysis

is completely

shearing

m

by bending to its

become

of

becomes,

solved

stresses

following

i.

There

is no in-plane

2.

Points

of the

of the

In medium-thick

stresses.

Also,

thickness,

the

3.

the

considers

only

for

cona few

important,

similar

beams.

supported

making

elasticity

involved

Medium-Thick

small

theory

t,

plates,

the

lateral

the deflections,

(w < t/3J.

Theory

w,

load

is

of the plate

is developed

by

assumptions:

plate

plate

after

The

normal

disregarded.

plate remain

deformation lying

initially

in the

middle

plane

of the plate.

on a normal-to-the-middle

on the normal-to-the-middle

surface

plane of the

bending. stresses

in the

direction

transverse

to the plate

can

be

Section B9 15 September 1971 Page 2 Thin and direct tions

Plates

tension

of the

bending

of the

load

membrane the large

are

plate

action

of the

of the

problem

deflections, to move

upon

the

one

in the plane

magnitude

--

on the

not present.

For

stretching

Very

large

In the literature able

shapes

in the

aerospace

Some

approximate

shapes

and

thin

D3.0.7.

and

of the deflections

the

Many

different

loading

industry,

thin

methods

greatest

solutions and

plates

of analysis

the

are

type

edges

and

case

of

and edges bearing

to lateral hence,

load

bending

depends action

obtained

conditions most

[1,

frequently

for

is

(w > 10t).

of information

been

available

and the

by a

a considerable

amount have

load

In the

in a membrane

boundary

are

lateral

plate.

and,

occur

and

can be obtained

resistance

plane

would

plates.

in the

the

middle

have

deflec-

These

and partly

immovable

stresses

membranes,

rigidity

may

The

thin plates

is avail-

for

plates

2].

of

However,

encountered. for

common

loads.

This and

with

which

given

bending

< w < 10t)

complicated.

between

plate,

on plates,

on medium-thick

various

of the

plane.

surface.

equations

more

by both

(1/3t

middle

flexural

much

middle

to the

nonlinear

distinguish

of deflections

Membranes exclusively

Thus,

load

thickness

in the

by the

becomes

must

to the

in opposition

partly

applied

of the

by strain act

plate.

the

stretching

compared

stresses

transmitted

supports

the

is accompanied

is now

free

plate

not small

tensile

solution

thin

accompanying

plate

supplementary given

-- The

section

plates. Plates

includes

Plates

some

subjected

constructed

from

of the

solutions

for both

to thermal

loadings

composite

materials

are are

medium-thick covered covered

plates

in Section in Section

F.

Section

B9

15 September Page B9.2

PLATE

THEORY

section

contains

This (small

for

deflection),

thick

the

theoretical

membranes,

plates

will

not

be

given

solutions

and

thin

here

as

plates

this

for

(large

type

medium-thick

plates

deflection).

plate

1971

3

is

Solutions

seldom

used

in the

industry. B9.2.1

Small

Deflection

Technical many for

literature

excellent instance).

Figure upon

by

Theory the

dcrivations

of the

Therefore,

only

B9-1

bending

on

shows

moments

the (per

small

deflection

plate key

bending

analysis equations

equations

will

be

y and act

on

x directions, the

respectively.

presented

element

of an

unit

M

M

and x

Sets

of twisting

contains

(References

differential length)

of plates

flat

axes

plate

acted

parallel

to the

y couples

Mxy(=

-Myx)

also

element.

S

TWISTING

MOMENTS

SHOWN BY RIGHT VECTOR RULE

r

r j

r

P

r My

X

w

Myx

FIGURE

B9-1.

DIFFERENTIAL

PLATE

2,

here.

initially

about

1 and

ELEMENT

HAND

Section

B9

i5 September Page

t971

4

_w As portional

in the case to the moment

reciprocal case

of a beam, M x applied.

of the bending

of a plate,

(negative)

due

in the x,

The constant For

a unit width

Poisson

effect,

the moment

in the x,

z plane.

Thus,

z plane,

_y

with

ts I -- --. 12

of beam, M

Y

, is pro1 is _,

of proportionality

stiffness.

to the

curvature

the curvature

also

In the

produces

both moments

the

a

acting,

one

has 02W 8x 2 where

12 Et 3 (M x-_My)

# is Poisson's 8Zw _y2 -

Rearranging

ratio.

Likewise,

the curvature

in the y,

z plane

is

12 Et 3 (My-_Mx) these

two equations

in terms

of curvature

yields

(i) D/82w

82w

(2)

My= _0y2 + , 0--;r/ where Et 3 D

-

The slope

per

twist unit

of the

distance

the twisting couple M relation

•

12(1 - ,2) element, in the

xy

82w/SxOy y-direction

A careful

analysis

(=O2w/0ySx) (and

vice

(see

is the

versa). References

change

in x-direction

It is proportional 1 and 2) gives

to the

as

Mxy

=

D(1

-

82w #,_-_

(3)

Section

B9

15 September Page Equations couples

to the

M = EId2y/dx

(1), distortion

2 for

Figure but

with

the

"v"

of

the

presence

the

plate

beam

and

(3)

relate

the

of the

plate

in much

the

same

plate

applied the

bending

same

5

and

way

as

the

one

twisting

does

a beam.

B9-2

shows

addition

of internal

theory)

and

of

as

(2),

1971

these

indicated

forces

a distributed

shears,

in

shear

the

Fig.

elements Q

transverse bending

and

x

as and

Q

y

B9-1,

(corresponding

to the

load

With

pressure twisting

in Fig.

moments

q(psi). now

vary

along

B9-2a.

z

.4

y

d

Myx + dMyx

+ dMxy

\M v I_,yx (a)

(b)

FIGURE

B9-2.

DIFFERENTIAL

PLATE

ELEMENT

WITH

LATERAL

LOAD

Section B9 15 September Page By summing about

the

moments

y axis,

one

by dxdy

and

Q

_

of Figs.

B9-2a

and

B9-2b

+ -(Qx + dQx)dxdy

discarding

the

term

= (Mx

of higher

+ dMx)dY

order

+ MyxdX

yields

aM

x (_x

-

sets

+ dMyx)dX

aM Qx

two loading

6

obtains

MxdY + (Myx Dividing

of the

1971

_

_yx 8y

(4)

'

or,

OM x

8x

In a similar

aM

x

•

manner,

xy

.

a moment

summation

ay

aM

[Equations

(4) One

the

+

and

final

about

the

x-axis

yields

aM

= _....2. ay

Qy

(4a)

_

(5)

(5)

8x

correspond

equation

to V --

is obtained

dM/dx

by summing

in beam forces

theory. in the

]

z-direction

on

element:

aQ q

aQ

x by

-

+

Equations

three

additional

pletely

(4),

engineering

(5),

quantities

defined.

presented

...._2 by

in Table theory

A summary B9-1. of beams

and ,

Qx

(6)

Qy

of the For are

provide

, and

q.

quantities

comparison, also

listed.

three The

additional plate

problem

and equations the corresponding

equations is,

in the

thus,

obtained items

corn-

above

are

from

the

Section B9 15 September 1971 Page7 Table B9-1. Tabulation of Class

Item

Plate

Plate

Coordinates Geometry

Deflections

a2w

Bending Stiffness

M

Relation

Y a2w

x

EI

12(i ,M

y

M

,M

xy V

Qy q

x

M

37

=

Y

qorw

+/_

+_ D [azw \3Y 2

aa--_)

D(1-_)--

=

xy

aM

EI d2y dx 2

ax ay aM

x

"_x

+

ax

xy ay

Moments

dM V

Qy

3M

aM

_ay

+ .......__ ax

aQ Forces

M

a_w M

Cb

Equilibrium

dx 2

3xay

Et ._

Qx'

Moment

Law

X

ahv

D==

Shears

Distortion

Theory

y

3x 2 ' 3y 2 '

Lateral

H o oke ' s

Beam

W

Couples Loadings

Theory

x

Distortions

Structural Characteristic

Equations

q

=

--

=

m

dx

0Q

x Ox

+

_..y. 3y

E

q

dV dx

Section B9 15 September 1971 Page 8 Finally, hal

forces

(Mx,

is a relation q/EI

My,

between

important

equation

Mxy , Qx'

%)

the

loading

lateral

is obtained

between

the

q and

by eliminating

six equations

the

all inter-

above.

deflections

The

w (for

result

a beam,

= d4y/dx 4): 04w 0x--7

+

The

plate

bending

problem

a given

lateral

loading

(7).

For

which

s:,tisfies

found,

w(x,y)

nal

one very

forces

equation

can

and

that

Often,

One of the

most

no stresses

stresses).

membrane

stresses

is thus

and

that

acted

specified (1)

function

(5)

approximate

is sought

conditions.

Once

to determine methods

finite

of equation

w(x,y)

boundary

through

is the

Thus, were majority

present

a flat

sheet

stretching

middle result,

some surface

middle

in summing

surface,

without

in deriving

in the

is a nondevelopable

derived.

the

powerful

found

will

to an integration

a deflection

various

to the great

stresses

reduced

q(x,y),

tions

appreciable

(7)

the

are

used

interto solve

difference

technique,

plate-bending

equations

pre-

1.

be emphasized

membrane

q

in equations

stresses.

It must

-5

(7)

be used

in Reference

assumed

34w

+

both equation

(7).

sented

_:w 2 Ox_Oy 2

to help

strains

invalidating

of the must the

(neutral)

forces

plane

to derive

support

of all plate-bending i.e.,

the

the

a surface sheet's occur,

assumption

middle

from

(6),

load.

In the

the deflection

which

then

plate

equation

lateral

problems,

of the

cannot

which

middle

(no no solusurface

be formed

surface.

large

it was

But,

from if

surface

equation

(6)

was

Section

B9

15 September Page Thus, some

practically

middle

tude

of these

more

severe

rately

only

surface very

all loaded stresses.

powerful

plates

deform

It is the

necessity

middle

rule-of-thumb

surface

restriction

to problems

in which

Orthotropic

Plates

into surfaces for

stretching

that

plate

deflections

are

forces

bending

9

which

holding

down that

induce the

magni-

results

formulae

a few tenths

1971

of the

in the

apply

accu-

plate'

s

thickness. B9.2.1.1 In the the

previous

material

that

the

material

elastic

for

with

the

case

it was the

has

plates

more

same three

are

general

assumed in all planes

the

directions.

elastic

properties

It will

now be assumed

of symmetry

generally elastic

that

called

with

respect

to the

plates.

The

orthotropic

properties

of

(anisotropic

plates)

is

F.

orthotropic

plates of plane

the

relationship

stress

in the x,

between y plane

stress

and

is presented

strain

com-

by the

fol-

equations: cr = x = y

E'( xx

+

E'e yy

+ E"e

T xy Following ing

plate

Such

in Section For

were

of the

of plates

considered

lowing

plate

properties.

bending

ponents

of the

discussion

and twisting

E"£

y

x

GTxy

( 8)

procedures moments

outlined are

in Reference

1, the

expression

for

bend-

Section

B9

15 September Page

1971

10

a2w M

M

x

y

=

D

=

D

M

x&-_

okv yO-7

+

(9)

+

(10)

= 2D xy

xy

(11)

Ox_)y

in which E' t3 E' t3 x ._.Z._ i-'-_' D y = 12

D x = The

relationship

between

Dx_ 04w

various

types

(12)

the limiting

lateral

load

develop

has

version

the

whose

Gt 3 12

the deflections

w becomes:

DyWOCw = q

which

have

Specific

.

of plate

bending

different

(12) for

flexural

solutions

many

rigidities

will

be given

one

should

in

Plates.

deflection case

theory

of the

flat

stresses

necessary

of plates

membrane and,

curvatures

difficult

and

retains

the

[3]. desired

is discussed, which

hence,

membrane

to be very solution

q and

in the investigation

directions.

two-dimensional

proven

+

E''t 3 , D xy 12

Theory

by bending

both The

tion

large

loading

construction

Orthotropic

Membrane Before

sider

be used

perpendicular

B9.5,

B9.2.2

can

lateral

04w 2Dxy ) Ox_3y2

of orthotropic

in two mutually Subsection

2 (13 i +

+

Equation

the

, Dt -

cannot

has to deflect membrane

problem However, general

support and

con-

any of the stretch

to

stresses.

is a nonlinear

one

we can study features.

The

whose

solu-

a simplified one-dimensional

Section

B9

15 September Page analysis

the

of a narrow

y-direction

is

strip

very

cut

large

from

an

(Fig.

originally

flat

membrane

1971

11 whose

length

in

B9-3).

Y

z

(a)

st q I

t.

q

t- .,

x

(b)

resembles

by

B9-3

a loaded

summing

st--

vertical

x

x+dx

'1 x+dx

(el

FIGURE

Figure

st

B9-3.

shows cable.

forces

ONE-DIMENSIONAL

the The

on

desired

one-dimensional

differential

the

MEMBRANE

element

equation

of

Fig.

problem of equilibrium

which is

now obtained

B9-3c.

x

or

d_v dx 2

-

q - st

(13)

Section

B9

15 September Page where tion

s is the

of a parabola.

W = The

of the

Substitutil_g

Its

stress

in psi.

solution

Equation

(13)

is the

12

differential

equa-

is

(a-x)

2st

unknown

length

and

membrane

1971

stress

(14)

in equation

strip

(14)

as it deflects.

through

consideration

the

use

of the

can

From

of equation

stress-strain

be found

by computing

Reference

(14)

3, this

stretch

and integrating

relationship

the change

in

5 is

yields

yields

5 S

=

By equating

-a

E

and

solving

for

s one

finds

s0 If equation x

=, a/2

have cal

(15)

is substituted

into equation

(14),

the

maximum

deflection

at

is

Wma x = 0.360

a

Solutions

complete

been to those

of the

obtained obtained

3

in Reference above

for

. two-dimensional 4, the results

the

one-dimensional

(16) nonlinear being

expressed problem,

membrane in forms

problem identiv

Section B9 15 September 1971 Page 13 w

=

nla

(qa_+ \Et]

max

(17)

'oo'° • Here

a is the length

n 1 and n 2 are

given

Table

in Table

B9-2.

1.5

of the

as functions Stress

of the

membrane,

panel

aspect

ratio

and a/b.

and

Deflection

Coefficients

2.5

3.0

4.0

5.0

O. 318

0.228

0.16

O. 125

O. 10

O. 068

O. 052

n2

O. 356

0.37

0. 336

O. 304

0.272

0.23

O. 205

maximum

2.0

rectangular

nl

membrane

stress

(Smax)

occurs

at the

middle

of the

long

panel.

theory

for

B9.2.3 The

square

results

reported

panels

in the

Large

Deflection

theory

has

panels

under

stiffness

is great

slightly)

may

other

extreme,

large

deflections,

ignored.

side

1.0

of the

of sheet

B9-2

Membrane

Experimental the

long

a/b

The side

of the

outlined

lateral

loads. to the

be analyzed very

for

good

agreement

with

range.

sheets,

be treated

the

analysis

of the

At one extreme,

loads

satisfactorily

thin

may

elastic

4 show

Theory

been

relative

in Reference

under

applied

(and plate

lateral

loads

as membranes

sheets which

by the

two extreme

bending

therefore

bending great

whose

whose

deflect

solutions. enough

bending

cases

only

At the

to cause

stiffness

is

:

_

Section B9 15 September 1971 Page 14 As it happens, the most efficient, plate designs generally fall between these two extremes. On the onehand, if the designer is to take advantageof the presence of the interior stiffening (rings, bulkheads, stringers, etc. ), which is usually present for other reasons anyway, then it is not necessary to make the skin so heavy that it behaveslike a '_)ure" plate. On the other hand, if the skin is made so thin that it requires supporting of all pressure loads by stretching and developing membrane stresses, then permanent deformation results, producing "quilting" or "washboarding.,t The exact analysis of the two-dimensional plate which undergoes large deflections and thereby supports the lateral loading partly by its bending resistance and partly by membrane action is very involved. As shown in Reference 1, the investigation of large deflections of plates reduces to the solution of two nonlinear differential equations. The solution of these equations in the general case is unknown, but some approximate solutions of the problem are known and are discussed in Reference 1. An approximate solution of the large deflection plate problem can be obtained by adding the small deflection membrane solutions in the following way: The expression relating deflection anduniform lateral load for small deflection of a plate can be found to be

wmax = where the

(_ is a coefficient

plate.

(19)

Et 3 dependent

upon

the

geometry

and boundary

conditions

of

r

Section B9 15 September Page The similar expression for membrane

w

Solving

=

max

equations q =

"a_ .I, 3 \-E-}'-]

ni a

(19)

q'+

1 q - a

and

(20)

equation

two extreme

behavior

No interaction

system

is nonlinear,

the

Equation

is best

(21)

qa4=Et 4

al

Figure

B9-4

a deflection The

supported panel

and

be seen,

midpoint.

yields

which

the

stress

maximum B9-5

the

a flat

systems

sheet

ean

is assumed

stiffnesses

support and,

a

since

the

only.

.

plotted

Also

plotted

(22)

is somewhat for

for

are

a given

plots

results

plate

using

values

of an exact

analysis

inasmuch

as it

conservative pressure.

of stresses

combined

(22)

a square

the

method

prediction

shows

individual

as

(22)

is too large

as the

summing

be an approximation

large-deflection

Figure

upon

nl 3

equation 0. 318.

(21)

by which

rewritten

equation

insofar edges,

can

a 4 +

shows n1 =

is based

between result

Wmax

approximate

shortcomings

results

max

mechanisms

load.

gives

the

W

(21)

lateral

As may

adding

3

+

Obviously,

[ 1].

q' and q" and

1 Et nl3 a4

max

a4

=0. 0443,

(20)

q" Et3

of a

plates is equation (17)

" for

W

of the

1971

15

outlined

above

has

is concerned.

stresses

are

known

of these

stresses

serious For

to occur for

a square

simply at the panel

Section

B9

15 September Page a.:; .redieted strcs'_

by

the

approximate

method

(substituting

q'

and

q"

1971

16

into

appropriate

equations).

350

/

300

/

250 20O U.J

','r

150

810o 50

N_E_RIPL_AXE...---. 0 0

0.5

1.0

1.5

2.0

w/t

FIGURE

B9-4.

DEFLECTIONS

SQUARE

PANE

AT

TIIE

L BY TWO

30

MIDPOINT

LA RG E-DEF

OF

A SIMPLY

L EC ]'ION

SUPPORTED

THEORIES

I

EXACT

---,,'-----

APPROX. _

----

2O 04

Y=

10

f

50

100

150

200

_50

qa4/ Et 4

FIGURE

B9-5.

LARGE

DEFLECTION SIMPLY

TItEORIES

SUPI)OI_TED

v MIDPANEL

PANEL

STRESSES;

Section

B9

15 September Page B9.3

MEDIUM-THICK This

various this

section

shapes

section

includes

for different

are

based

PLATES

(SMALL

solutions

for

loading

on small

DEFLECTION

stress

and

boundary

deflection

theory

and

1971

17

THEORY)

deflections

for

plates

of

conditions.

All solutions

in

as described

in Paragraph

B9.2.1. B9.3.

1

Circular

For

Plates

a circular

differential

plate

equations

laterally

loaded

plate

it is naturally

convenient

in polar

coordinate

form.

in polar

coordinate

form

The

load

is symmetrically

w is independent

of 0 and

distributed

with

equation

becomes

the

deflection

the governing surface

1 _f]02w_; __ D

+ "_

respect

to the

center

dr

r_rr

The

bending

Mr

=

Mt

=

\r

(1-p)Dtl Solid,

Solutions ---

and

boundary

Or

"

moments

_

+ r2

+

solid

conditions.

circular The

_-_

(25)

(26)

(27)

_-_ _-_)

_'_'_ -a2w

plate,

are

+ " a--_- /

002

Uniform-Thickness for

r2

(23)

(24)

D

and twisting

D[or_ +

Mrt= B9. 3.1.1

_rr

"

of the

1d{d[d(rd :r)J}

r

of a

is

( 02 + -r1 --Or0 + r12 a_]\a-'_" 02_[0_ + r1 _Or If the

to express

Plates plates

results

are

have

been

presented

tabulated in Table

for

many

B9-3.

loadings

Section B9 15 September 1971 Page 18 Table B9-3. Solutions for Circular Solid Plates

Case

Supported Uniform

Formulas

Edges, Load

w

=

For

Deflection

16i)(1+_)

And

max

Moments

64(1+_)

D

q Mr

=

1_6 (3÷g)(:'2-r2)

Mt

=

1_61a_(3+_)

(Mr)

-

max

=

(Mt)max

=

3+_ 16

r2(1+3_)1

At Edge 0

Clamped Uniform

Edges, Load

|Jit

-

w

=

= a_L

c-Al-- (a2- r2) 64D " "

Wmax

64D

q

41ill1

Mr

f

=

1"_-_[aZ(l

(Mr)ma x"

=

at

Mt

+U)

r'=a

=

r2(3+U)l

-_a

--q16 [a2 (1 +U)

r_(l+3g)l

.L2

(Mr)r=

Supported Load Circular

Edges, Uniform Over Concentric Area

of

Radius,

=

_6"(1+g)

(

P w

-

_'q-L_(a2 r 2_ li;_r D (l+p ' 1

+

2r 2 log

r a --

+

2(l+p)

c

°

q

--_-P[_+-:_a2

Wr=0

V-V-I At P -

0

c2

16_DLX+.

c loga

+

-

7+___c, ] 4(1+_) J

Center

7r ¢2q

I) M

At

max

-

47r

Edge Pa

0

4zr (l+p)

1 +/_) log

_

+

1

-

4a 2

J

qa 2

Section

B9

15 September Page Table

B9-3.

(Continued)

Case

Formulas

Simply Supported, Uniform Load On Concentric Circular

Ring Of Radius,

8_D

max(W)r=0

And

Moments

2b 2 log

12 l+u 1-. a_-h_ a_ ] +

+

P b'log _ + (a'b_)_] 2(I+u) J

-

I

8rD

(i+_)i,log_b (l+tt)

Fixed Edges, Uniform Load On Concentric Ring

Deflection

p / (a2-b_) ( 1

-

Mr=b

Circular

For

b

P= 2_bq q

I

(W)r=b

2)

a -

_. 8rD\2a P (a4-b 24

4_

2b2 log b)

÷

b max(w)

p- _rbq

P(a2-b 8va 2

=

(W)r=b

Of Radius,

r=0-

=

)2 log -a

+

q M

H Simply Supported, Concentrated Load At Center

r--'a

-

w

=

=

M r

Mt

Fixed Edges, Concentrated Load

At

w

aZ-b 2 a2

4n[-_'_ (l+/J)

=

=

Pr2D log 8n

1_a----_22 l)

log

l+g)

•

r )

_ 16r(l+g)

max

P

p 4ff

167rD Ll+gi'"

w

'± r

log

r a

+

1

-

P 2 16_D(a-r)

+

g

2

Center w

M

Clamped Uniform Concentric Area

Edges, Load Over Circular

Of Radius,

=

r

Wmax

e

=_

max

3

I)a _

487

D

"[

-4%

(r=0)

l÷bt)

=

h)g

:

"(

64_1)

]

-

1

la2

4c2l°gae

:lc2

At r=a

q

VT3

M

=

r

_-_(1

2_2)

Mt

At r=0 p = 1)" c2q M r

"

Mt

19

=

I)(1+_) 47r

og--

c

+

=

tiM r

)

b a

1971

Section

B9

15 September Page Table

B9-3.

( C ont inued)

Case

Formulas

Supported

By

Pressure

Over

I_wer

Over

Circular

l_,f]t,

etion

And

Moments

r=O

Uniform

W e 2

Concentric

Area

For

Entire

Surlace,

Load

At

Uniform

20

Of

Radius,

4

c M

P = n ¢2Q

M

r

_

4+ (t,_)

t

q

I t _]-]

t t

4(t-_,

c2

If c_O w

|)a 2

=

_

64rid

(i+.)

No Sulq_ort, "Lnilorm

Fdgc

Moment

M

21)(L*O}

=

w

Ma 2 2D(I+U)

Wr=0

M

(-

)

Edge

Rotation M:t 0

Edges

At

Supported,

Central

D(1+p)

['=c

Couph,

(Trunnion

Loading)

M

2nc 9.__m.m I1

_

+

(l+ta)

log

(l*p)

log

Ka

J

where

x

[.j

z

m

Edgc"_

K

Coul_ h.'

(Trunnion

l,oading)

U

M

f

Edges Eccentric Of

z(,)..:5 a-_) ] +

0.45

ka

J

0.1

a_

k 2_ .)_

(c+,).

At

Supported. 1,end

=

I" 9m 2no" [ 1

wh(, re

m

Uniform

.u_ 7a)2

AI r=c

('lampe
Centv':d

0.49 ((: _),

Over

Circul3v Radius,

Point

Small

M

Area

of

I,oad: M

r

t

1

*

(I+u)

log

r a_zp_ o

r 0

At

"_"-

"

i

Point

W

2_"0

q: K_(r_-h_ar2÷eCa

=

3)

+

K,(r4-hlarJ+eta:Jr}

÷

K2(r4-b2ar3+c_a2r

cos

0

° 2) cos

¢

where

LOAD AT !

K,)

-

2(1 +_) P( p'l-bt,al)'/+ 9(5 +V) Kna;

e,)a :_)

K 1

-:

2(3 +_) l)(p4-1)laP_+t:,a:li) 3(9 _u) K "as

}

K2

=

(4 +,u) ':I'(p4-khap:l+c,aZo_ (9+p)(5*/a)Kra _

)

K

=

Et_ 12 (l-u:)

b°

=

3(2 +p) 2(1+/_)

bl

=

2(3+#)

fq " rl lU - °1

})2

_

CO

=

'

¢I

C_

1971

Section

B9

15 September Page Table

B9-3.

(Concluded)

Case

Formulas

Edges Fixed, Uniform Load Over Small Eccentric Circular

Area

of Radius,

At

Point

of

Load:.

-

P 47r

M r

r0

W

At

=

Point

For

Deflection

and

Moments

"l (1+#)

a-p r0

log

(1 +g)

+

=

max

31+'(l-l'£2){a2-p+)2 4_Et3a _

q:

[lCe w At

=

Supported Points

Along

At The

Several

2_E, ta

L2\

a'

Edge:

M

=

Supported

1

At Two

-

=

P at

= ,_

r_O w

r=a,

Loaded

Wr= 0

=

Y2

_)

=

=

0. 269

Supported

pa 2 -D

qa---_4 D =

r = a,

0.371

qa----_4 D

0 = _'/2

At Three

Points

120

O. 0670

pa t -D

Deg

Apart:

=

0.577

P at Center

Load w

= r_ 0

Uniformly

Loaded

w

=

0. It:;7

r=0 Linearly Load

O. 118

Plate:

w

Symmetrical

0,

=

r 0 > 0.6(a-p)

pa z -D

0. 116

O= _/2

Uniformly

Supported, Di strihut¢<]

M when

Center:

w

Edge

(Yt

Points:

max

Boundary Load

max

M

_ D

=

_

r

at

r

a

72xf_-

About

Diameter

ma.xM tl

=

t

qa 2 (5 +_) ( 1+ 3 U)

at r

72(J +,)

=

v max

maxw

edge

=

1971

21

reaction

0.042

per

qa4 Et:+

linear

at

r

inch

--

0.503a

=

0.675a 1 _ qa

(g

=

0.3)

M when

r0<0.6(a-p)

Section

B9

15 September Page f29. _. 1.2

Anr ular,

Solutions TaMe

Uniform-Thickness

for ap.nular

1971

22

Plates

circular

plates

with

a central

hole

are

tabulated

in

B9-4.

B3.3.1.3

Solid, For

tance,

Nonuniform-Thickness

the plates

mid the

t_eated

ncting

I.

Linearly

The

plate

load

here,

the

thickness

is symmetrical

Varying

of this

Plates

with

is a function respect

to the

of the center

radial

dis-

of the plate.

Thiclmess:

type

is shown

in l.'ig.

B9-6.

+8J "_ , _////.(/_////z/_ __ -

j

•

b

I-

-F, -]_

a

-16.

(a) P

(b) FIGURE

B9-6.

Tables moments

CIRCULAR B9-5

of the

and

plate

PLATE

B9-6

give

in two cases

center

in the

bution

of that

load

over

a small

=

Ocan

be expressed

r

=

M

M

t

at r

max

=

P(1

the deflection

of a eentral

+ _) 47r

LINEARLY

of loading.

at the

M

ease

WITlt

load

circular

one

area

e

+

may

values the

assume

of a radius

e.

TItlCKNESS of bending

bending

moment

a uniform

distri-

The

moment

form

(,. c÷) og-

and

max

To calculate

P,

in the

w

VARYING

+

TiP

(28)

Section

B9

15 September Page Table

B9-4.

Solutions

For

Annular,

Case

Outer

Edge

Uniform Entire

Uniform-Thickness

Formulas

Sul)ported, Load

Actual

At

Inner

For

I_,flection

And

1971

23 Plates

Moments

Edge:

Over Surface maxM

When

=

b Is

Ver)'

_L' q

:

Mt

r.o(

:l

4- p)

+

b4(l_p)

_

-

z(t+.)

4a2b 2

_

4(l+p)aZl?log

a] _)

Small

p = q_t(i2.b2l =

ffrN

maxM

r_

qa 2 ( -_-3+g)

Mt

=

A

_I)

_(I+.)

2(I-u)

),

'i +#)

2a_b_(1 (a _-

Outer

Edge

Uniform Entire

Clamped. Load

Actual

At

OuO.'r

')_')(1

/

I

-_)(l°g

"_ ,

b)

}:dg(':

Ovor Surface max

=

M

(J 8

r

a2

-

2b _

I)4( I - _) ,

+

-

41)4( ] + _)

a2(1

u_ p - q_rla2.b 2)

max

w

_

_

+

-t')

lt)_

a E

+

a2b2(

I + u)

I?(I+/_)

a a 4 ._

51)4

_

6a2))1_

.¢

_))4 log

V) a

,_,:(l..) _ .i,,,,,,(:,,.) _ ,.,,.,(I+.)),og_,. i,)u,,.,(, +.)(,og_,)" ,,'(I -u) ' Outer

I.'dge

Unilorm

Supported, I,oad

At

Inner

.'tabu' - za't?(_ < v)o ' 'zh' ;( l :V) a_(i-v) _ u.(l +u)

+ )?(I..) )

E{Ig(':

Along

Inn('r Edg(' max

M

=

Mt

,'i¥

L

IIlUX

W

_

l, [{_,'-'-,,-')(:,4v) ,;.,--'; _,,.)

i,)g

_

(I

-p)

P

,

A o. _,)'] i'i,,aqt(I,,,) _0)¢; _..)v"g

Section

B9

15 September Page Table

B9-4.

1971

24

( C ontinued)

r Case

Outer

Formulas

Edge

Uniform

Clamped, load

Inner

At

Outer

For

Deflection

And

Moments

Edge:

Along

Edge

maxM

_

r

P At

Inner

_b'- 2b'(,_., log_ I

4-_

=

1

-

a2(1_p

) +

bZ(l+#)

maxMwhen

i =

b <

2.4

Edge: i-

maxM

maxw

Supported

Along

Concentric

Circle

Near

Outer

Uniform

Inner

max

Mwhen

P

a_ _ b2

=

Inner

Supported, Load

_

- b_(l+p) 2(1 -pZJaZ

log

>2.4

2b_(a

2-

,.

b2)

-

*

M t

=

_L

=

At

Inner

Edge:

a

8a2b 2log

_

+

'+ b'(i+_')

a;r(_l-.)

a'-b'

log

c'-d'l

_

4

(1-/*)_j

maxM

=

Surface

At

Outer

Mt

at(1-_/*)

Edge

Fixed

log

_

+

4a2b _

+

b_(1-/*)

W

-R--64D eL 4(7

=

At Outer

+ 3/*)

+

16a4b2(1

b4(5

+/,)2/.

+ /*)

-

a2b2(12

+ 4/a)

-

2l

'°g 5)J

Edge:

Supported, Edge

Uniform

Fixed, Load

Actual

Over

max

M r

_

a _ - 3b :_)

+

a-I=z_[log

Surface At Inner

Edge:

p = q,_a2.b 2)

M r

maxw

-

al(l+3#)

Edge:

+ Inner

a b

Edge:

maxM

max

Entire

+p) +

Over

p = cra.ia2.b 2)

And

a _

-P, .a*(].-p) - bz(1

Edge

Actual

Outer

a2(l

+

Circle

Edge

Entire

=

1

e

Along

Inner

Uniform

_" P/*

Edge,

Load

Concentric Near

At

=

t

=

_8

=

[

(a 2+b

2)

_64D

4 +

-

,a',' ,.

(a__)_Iog

3b 4

-

4a2b 2

-

4a_h2

log

_

*

a--T-_Iog

4a2bz(3 (1 +/*)(1 - #)

+/*)

log

a

Section

B9

15 September Page Table

B9-4.

Case Outer

Edge

I-'ormulas Fixed

Supported,

And

Inner

Fixed.

At

Load

Inner

Outer

Edge

Supllur

M

Edge

Fixed

ted,

Load

And

Mo_ndhts

r

=

-,InI'[

And

At

1

-

a-'-'_-_" 2b2

(/ h'g

a)]

EdgE:

max

Mr

max

w

Inner

-:

1

=

_

-

a2

a_-T--'_(log

_

4a21)2 /Uog _ __

h2

Edgt':

Uniform

Over

Actual

I)efh'ction

Edge:

At Inner

Inner

For

Edge

Uniform

Along

( C ontinued)

I':ntire

a

log

.ta*(l+_)

Su rlacc

p - _a2.h21

At

Outer

_

HI.

r

-

a1(1,

;_)

a

bt(l-p)

-_

4a2b_-p

a'_( '__) + ,;([ _p)

Edge: f

maxw

:J(l ._,)

_

_

b2(I-_)

(

4a;'h:!la2(5

- p)

I,'_( I + _)1

+

log

_a

+

16a4b'_(

1 ÷ ,_)(Iog

;(J+.)+ l,_(l-.) Inner

Edge

l,'ixcd

Supp(wtcd, Ltmd

And

At Inner

Fdgc:

('nilorm

Along

O_ter

}:dge max

-,ur

M r

At ()tltt'r

:,':(1

n

a:!(I

"_ p)

,

I):_( I -IQ

-_)

17(1

-P)

I':dg_,:

a

p max

w

:

a'(3+U)

-

h'(l-p)

-

a_( I _ /a)

16ni)

.,a".,:'(I • .)(..g _)_ Out(,r

I':dgr

Uni ftn'nl Along

l,hxrd,

At

Inner

a2(l+p)

I"dge:

Mon.,nt

lnnH"

l':dg('

M ra_l;M N )

m;l.x w ":

M f k

E r

At Outt,r

max lnnrr

I':dg_'

('tfi fornx Along

I"ixod,

h'

-

L a°(,-.)

2:,'l,;'l,,g[_

,,=(,..)

Edge:

M

AI Ildl(,r

,.=

M [

(l+.). _2h2+ (I-_):,: 1

_

(l

I':dg(':

Momont

(hdt, r I':(Ige nl{L'_

M

-_ I):(]-u)

M

r

r

[

' u)a:

+

(1 -v)h

2a2

z1

M AI ()uter

Edge.*

M

F at

-

a2h 2

-

2:,_I/log

'

2a_"h_(l+_) *

h,Z( 1 - p)

-

Ha2b21og

25

1971

Section

B9

15 September Page Table

B9-4.

Outer

Edge

Supported,

Unequal

M

Moments

Along

=

1

r

Uniform

F 'a2M ,

_

"

a

For

b2Mb

Outer

Uniform Entl

Moments

1 -

Mb)

!

At

Inner

b_Mb

a

_,7_

}+

lOgr

(1-.)

Edge:

Fixed,

I,oad

re

aZb2 -_r (Ma

a -

w= _

Ma

SupporO_d,

Edge

and

Edges

Mb

Edge

Inner

Deflection

-

i Ma M b

26

( C ontinued)

Formulas

Case

1971

Actual

Over Su rftLce

max

M

=

q

r maxw

4a2b2(l+_)

log

s L =

_

-

6_

a:(/+u) a4

-

31/

+

b2

+ ff_(1-u)

2a_b 2

-

8a_b

2 log

.[ ab

a 16(l+p)a2b

a

_ log 2 _

+

[4('/+'l#)a_b

_(1+_,) 4(4

+,)a4b

2

-

Both

Edges

Balanced

Fixed,

At

Inner

-

2(3

+p)a

aZ(1+/_)

÷

4

-

4(5

+ 3,_)1

log

÷ b_(1-.)

G

2(5

-

+p)a2.b. ' (

b;_(l_p)

Edge:

Loading

(Piston) max

p - q,n.(a2.b 2 )

=

Mr

maxw

=

q/ 8_ka

2-4a 4b 2

q 64D

3a 4

log

-

a_-

-

3a 2

2

+

b 4

4a2b

b2)

+

+

4a2b

2 log

_a

16a4b2 [ a-T-__(log

-

b)

2 .i

Outer

Edge

Inner

Supported,

Edge

Uniform

Load

Concentric Ring

At

Inner

On max

Circular of

Edge:

Free,

Radius,

Mt

=

_ P

[_-

(l-p)

+

(l+p)

log

a ro

-

(1-p):2a-'_r

max

w

=

P

--

.

(a'-b')(:_+ .)

8_D

2(I+.) c

"

a -

(b 2+

b2

r02)

2a2b

log

_

_

(."_- b2)(1 -u) log I

r0

a r2' -

(a z _ b z)

r 0

I- r0 -I where

l-U)

+

2(l+#)log

ar0

-

(1-p)_-]

r_(a z-b2)(l -

2aZ(l+p)

a

,#)

_

Section

B9

15 September Page Table

B9-4.

Case

Edge

Fixed,

Inner

Edge

Free,

Uniform

Load

Concentric Ring

(Concluded)

Formulas

Outer

At

Inner

Of

For

Deflection

and Moments

Edge:

,,=Mt

Circular

1971

27

_. (_+

ep/1

a

- h_(l+.)]

r o

Of Radius,

r 0 At

Outer

Edge:

P

_t

I

tf

Mr = _7 nllLX

W

=

-

+ (, _,_(I-.)

+ b_(l+.)

_' (_ + d)(a_ - t,_)

(_ + d) "'_ _

,:,a2

_

_;L t,_(, _.) + a_(_ -.)'

where

c

Central Balanct,d Dist

Couple By

ribut('d

At

:

_"

Inner

1+_)

(

2log

:1

+

_

I'll

" /] -

:1

1

Edge:

I,inearly t) r('_surc

max

M r

M fll'_a

where'

1.25

L-I I) q = 4M / Tra3

=

fl

1.5o

O. 1625

2

0.456

3

I. 105

2.25

.l

5

:L :|85

4. ,170

(_ :: ,. :_) Concc*ntratcd Applied

At

I,oad Outer

At Inner

Edge:

Edge P P

m:_x

M

b

[_

wht'r('

= [_

3.7

1.51)

'2

:|

,I. 25

5.2

6.7

for #

=

O, :1

4

7. !l

5

8.8

J

Section

B9

15 September Page Table

B9-5. Deflections Plates Loaded

and Bending Moments Uniformly (Fig. B9-6a) M

of Clamped (p = 0.25)

= fl qa 2

r

M

1971

28

Circular

t

=/3 lqa 2

qa _ W max

--0_

r=0

0

r=b

r=O

r=b

r_a

b a

0.2

0. 008

0. 0122

0. 0040

-0. 161

O. 0122

0. 0078

-0.040

0.4

0. 042

0. 0332

0. 0007

-0. 156

O. 0332

0. 0157

-0.039

0.6

0.094

0. 0543

-0. 0188

-0. 149

O. 0543

0. 0149

-0.037

0.8

0. 148

0. 0709

-0. 0591

-0. 140

O. 0709

0. 0009

-0.035

1.0

0.176

0. 0781

-0.

-0. 125

O. 0781

Table

B9-6. Plates

Deflections and Bending Under a Central Load M pa

r

=M

-0. 031

-0. 031

Moments of Clamped Circular (Fig. B9-6b)(p = 0.25) M

t

= tiP r

2

r=0

max

125

r=b

r=b

r=a

b a

o.21

0. 031

-0. 114

-0.034

-0. 129

-0.028

-0. 032

0.4 i

0. 093

-0.051

-0.040

-0. 112

-0.034

-0. 028

0.6

0. 155

-0. 021

-0.050

-0. 096

-0.044

-0. 024

0.8

0. 203

-0. 005

-0.063

-0. 084

-0.057

-0.

0. 224

0

-0. 080

-0. 080

-0.020

-0. 020

1.0

021

Section 15

September

Page The

last

term

coefficient

Yl

is due

to

is given

in

Symmetrical been

the

nonuniformity

Table

and

some

thickness

of the

1971

29

plate

and

the

B9-6.

deformation

investigated

of the

B9

of plates

results

are

such given

as

those

in Tables

shown

in Fig.

B9-7,

B9-8,

B9-7 and

have

B9-9.

q

l illilll ]ll lilll± -vhl lal P

,U////////////2

7.,

(bl

P

--F

1

P //ra 2 I"

a

"1 (c)

FIGURE

,For

bending

is true

moments

(Y2 is M

max

given -

B9-7.

under

in Table

4n

(1

+

TAPERED

central

load

B9-8) p)log-

CIRCULAR

l) (Fig.

B9-7b)

PLATE

the

following

equation

:

c

+

1

+

y2P

•

(29)

Section

B9

15 September Page 30 Table

B9-7.

Deflections

Plates

Under

and

Bending

Uniform

Moments

Load M

(Fig.

of Simply

B9-7a)(tz

Supported

= 0.25)

= _qa 2

r

1971

M

= _lqa 2

t

qa 4 w max

=

o_

E-_h 0

a

r=0

r

a

r=O

_-

2

r

_--

2

hi

1.00

0.738

0. 203

O. 152

0.203

O. 176

0. 094

1.50

1.26

0. 257

O. 176

0.257

0. 173

0.054

2.33

2.04

0. 304

O. 195

0. 304

0. 167

0.029

Table

B9-8. Circular

Deflections and Bending Moments Plates Under Central Load (Fig. M pa

h0

w max

2

r

=M

M

t

of Simply Supported B9-7b)(tz = 0.25)

= tiP

M t = fliP

r a

--old0 r

-_

a r

2

_-

m

2

ht

1.00

0.582

0

0.069

0. 129

0. 060

1.50

0.93

0. 029

0.088

0. 123

0.033

2.33

1.39

0.059

0. 102

0.116

0.016

Table B9-9. And

Bending Moments

M

_= M

r

ho

of a Circular Plate With Central Load

Uniformly Distributed Reacting Pressure

t

r=0

M

r

= tip

r

_

(Fig. B9-7e)

•

Mt

a

a

2

2

(/_= 0.25)

_tp

r=a

ht Y2 1.00

-0. 065

O. 021

0.073

0.03O

1.50

-0. 053

O. 032

0. 068

0.016

2.33

-0.038

0. 040

0. 063

0.007

Section B9 15 September 1971 Page 31 Of practical interest is a combination of loadings shown in Figs. B9-7a and b. For this case the _2to be used in equation (29) is given in Table B9-9. II.

Nonlinear

In many with

sufficient

Varying

Thickness:

cases

the variation

accuracy

by the

of the

plate

thickness

can be represented

equation

y = e"fl x2/6 in which

fl is a constant

approximates variation values

as closely of thickness

of the

constant

(30) that

must

be chosen

as possible along

the

a diameter

fl is shown

in each

actual

case

proportions

of a plate

in Fig.

particular

of the

corresponding

so that

plate.

it

The

to various

B9-8.

0

I

I

0.5

1.0

X

FIGURE

B9-8.

VARIATION

OF PLATE

THICKNESS

FOR

CIRCULAR

PLATES

Section B9 15 September 1971 Page 32 Solutions for this type of variation for uniformly loadedplates with both clamped edges and simply supported edges are given in Reference 1, pages 301-302. B9.3.1.4

Annular Plates with Linearly Varying Thickness

Consider the case of a circular plate with a concentric hole and a thickness varying as shownin Fig. B9-9. P

1 A_

I FIGURE The p =

B9-9. plate

P/2rb Table

lowing

ANNULAR carries

uniformly B9-10

expressions

PLATE

a uniformly distributed

gives for

the

values

along

WITH

LINEARLY

distributed the

surface

edge

of the

of coefficients

numerically

VARYING

k and

largest

load

q and

a line

load

in the

fol-

hole. kt,

stress

THICKNESS

and

to be used the

largest

deflection

of the plate: qa 2 ((rr)

max

=

k _

p or

(err) max

q a4 w max

= klE--_l

=

k h-_ pa 2

or

w max

=

kt_l

3

(31)

Section

Table

B9-10.

Values

of Coefficients of the

in Equations

(31)

ag (rig. B9-9)(

Ratio

for

B9

15 September Page 33 Various Values

1 -) 3"

CoefCasc

Boundary

ficiezlt

k

q

1971

O.

1.25

1.5

249

O. 638

2

:|.

9fi

:l

4

13.64

5

26.0

4 o.

Conditions

I;

P

=

Ob

7rq (a s _

=

Ij _)

o

t k I

1).00:_72

0.045:_

I).4o|

2.

12

4.25

6.2_

M

= 0 a

k

q

O.

149

O. 991

2.2:;

5.57

7.7_

.9.

lq;

17 _

01)

t

0

_

[)

t k_

0.0O551

1_.o564

(_.112

I.ti73

k

o.

0.515

2.

7.

1275

o5

!)7

2.79

:|.57

17.:_5

_._

30.0

_

0

I ) = rq(a

q

g'l)

=

[)

It;

,,_

=

1)

78

(1 :

_71;

q_a

51

q

t kt

0.1)1)105

0.0115

().o_J'.;I

0.5'A7

1.2(il

k

o.

159

(). 39(;

I. o91

:'.

:| I

[i.

55

lO.

kl

o.

O0174

o.

01

o,

olitN;

o.

21i l

o.

546

o.

k

0.

:15:_

o.

9:1:|

2.

_;:_

6.8_

!2

t 1.47

2.

lB.

= 0

=

_bb

t

_)

0

=

11

t ki

0.00_16

o.

05_:1

11.:115

l.:|5s

2.:19

:L27

M

_

k

o.

o.

2_t8

o.

I. 27

1.94

2.52

P =

0

ki

07s5

I1.001192

o.

(_l_x

o.

52

_1195

(I.

1.9:;

0.

"346

0.4_2

#b

=O

_b a

=

0

0

2 -

I) _)

Section

B9

15 September

B9.3.

1.5

Sector

The for

general

a plate

simply

in the

solution form

supported.

straight

and

moments lowing

circular

developed

point

the can

for

(Fig.

a uniformly

edges

circular

B9-10),

loaded

expressions

also

be adapted

edges

of which

supported

deflections

in each

= (l,_,

_, _, taken

Mr

and f_l are

on the

axis

= fl qa2 '

Mt

numerical

factors.

nf symmetry

= fllq a2

along and

particular

are

the

bending

case

by the

fol-

of a sector

B9-10.

SECTOR

CIRCULAR

of the when

w

=

bending M

t

=

circular

0.0633

moment 0.1331qa

and

in Table

coefficients along

simply

the

factors

the case

circular

supported are

given

It can

be seen

that

in this

the

in Table case

of a boun-

along

edges

for

B9-11.

for

straight

mum

bending

The

following

edge.

= 7/2

max

given

of these

the

B9-12. maxi-

PLATE

unsupported n/k

values

clamped

dary

OF A

(32)

are The

B

FIGURE

,

Several

sector

The

the

can

straight

simply

for

be represented

plates the

plate

0

case

34

formulas:

in which

point

Page Plate

of a sector

For

at a given

w

points

of a Circular

1971

qa4 D

at the 2

same

point

is

stress equation

occurs

at the

is used

for

midthe

section B9 15 September 1971 Page 35

r

7T

Table

B9-11.

Values

of a Sector

of the Simply

Factors

_,

Supported

fl,

at the

and

B 1 for Various

Boundary

Angles

(p = 0.3)

r

1

r

1

r

3

r

a

4

a

2

a

4

a

I

v. 4

o. 00006

-0. 0015

o. 0093

o. 00033

o. 0069

o. 0183

o. 00049

(). 01(;1

0. 0169

0

0

o. ()025

0. 00019

-0.

0025

0. 0177

0. 000_0

0. 0149

0. 0255

0. 00092

0. 0243

0. 0213

0

0

0. 0044

3

m 2

0. 00092

0. 00:16

0. 0319

0. 00225

0. 035:|

0. 0:_52

0. l)020:|

0. 0381

0. 02H6

0

0

0. 008H

Tr

o. 00589

0. 0692

0. 0357

0. 00811

0. 0868

0. 0515

0.0()5(;0

0.0617

0.0,t(;,_

0

0

0. 0221

Values

of the

ff

Table

B9-12.

Coefficients

(_ and

fl for

Various

of a Scctor Clamped Along the Circular Boundary and Supported Along the Straight Edges (g = 0.3)

7r

Angles Simply

r

1

r

1

r

3

r

a

4

a

2

a

4

a

O/

?T

4 7_

ff

ff

0.00005

-0.0008

0.00026

0.0087

O. OOO28

0. 0107

0.00017

-0. 0006

0.00057

0.0143

0.00047

O. 0123

-0.034

0. 00063

0. 0068

O. 00132

0.0272

0.00082

O. 0113

-0.0488

0.00293

0. 0472

O. 0():;37

O. 0446

O. 00153

O. 0016

-0.0756

0

-0.025

Section

B9

15 September Page In the

general

case

edges

clamped

which

allows

cular

arcs.

of a plate

or free,

one

an exact D_ta

having must

solution

regarding

the form

apply

is that the

of a circular

approximate of bending

clamped

sector

methods. of a plate

semicircular

36 with

Another

clamped

plnte

are

1971

radial problem

along

given

two cir-

in Table

Bg- 13. Table B9-13. Values of the Factors Semi.Arcular Plate Clamped Along the

r

--=

r

0

a

Load

a, fl, and fll for a Boundary (_ = 0.3 )

r

-- = 0.486

- = 0.525

a

a

a

Distribution

(£

/3 max

Uniform

Load

Hydrostatic

sectored oorted

surface tion

Annular

For

a semicircular

load

-0. 0584 -0. 0355

edge

edges the

Plate:

SIMPLY SUPPORTED

free,

entire

in Fig.

supFREE

with

FIGURE B9-11. SECTORED

actual

B9-11,

the

equations

for

maximum

are: At A

M t =

0. 0194

annular

outer

other

as shown

/3 lmax

-0. O276

Sectored

with

over

0. 00202

l

a

I.

and the

uniform

q Y

Load

plate

0.0355

-0. 0731

q

max

r --a

r

-- = 0. 483

c (1

ANNULAR PLATE moment

and

deflec-

Section

B9

15 September Page At

1971

37

B

24cIc2b2 Et 3

W

- i

'I cosh _2

+ e 2 cosh

T2 _ +

whe re 1

1

cl

L - T12

T

)t - I) cosh

/

"/1

J

_

Yl

2 -- - Y2

o

4b 2

],,2 + X

- T2 _

:kl =

400"625t_G 22e

]_

K

a function

bee

tanh

TJ

J

following

values:

_1

ff

k - 1

cosh T22

4b__

Yl

(l+b)

U-C

is

of

and

has

the

b-c

b+c

K=

B9.3.2

0.05

0.10

0.2

0.3

0.,t

0.5

0.6

0.7

0.8

0.9

1.0

2.33

2.20

1.95

1.75

1.58

1.44

1.32

1.22

1. 13

1.06

1.0

Rectan_gular

Solutions

for

Plates

many

rectangular

boundary

conditions

are

given

boundary

conditions

not

covered

here,

approximate,

or

various

theoretical,

in

plate

Tables

B9-14 solutions

complete

problems

with

through can

various

18. be found

solutimm

For

loadings loads

by applying

discussed

in

and

and the

Section B9 February Pa ge 38 Table

09-14.

Solutions

for

Rectangular

15,

Plates

! ba P is load

All

Edges

Load

l"nti

At

Supported,

Unif,>r[,l re

Sur

Center:

Over .M

face

_

(0.0377)

'

(1

+

0.

Oii37a

_

O.(lS:13¢ll)qlJ

.

l

a

q

[lllll

Mb

I

+l.ili_7)

:

maiM

O. 1.122

All

Edges

Uniform ('llnct.llt Area

Over

rie Of

A(

Support¢_l,

I,oad

qh i

('_'ntvr:

Small

CI rc'ular

Radius,

r 0

M b

_

] _ Ix)

h,g

2r--i

wh (.' r(,

0,914

m/:l

I). 2fl:lPb rnlL_

W

171i(

1

I + ll.467¢_

t}

b#2

All

Edges

Uni[orm Central Area

})

Supl)orted, Load

Over

AtCt'uter:

nlaxa

=

¢11

=

fti -T

wherel'l

ta

foundinthe

following

H(.etangul:lr Shown Sl_tdcd -

a

0

O. 2

O. 4

b

O. 6

O. 8

I. 0

1. H2

{. :it4

I. 12

fL 93

II. 76

0.2

1.8_

1. _H

1. OH

o. _.}0

II. 76

O. 6:1

0.4

I. :19

1.07

II. H4

o. 77

II. fi2

II. 52

0.6

l.

I). 9it

O. 72

o. iili

O.52

0.43

0. II

(I..q2

0.76

(I.69

0.51

0.42

0.36

I. II

0.71;

If. t;3

0.52

0.42

O. 35

O. 30

0

17

b

a

li

0

1.4b

:

11.2

0.4

ll._l

1.2

1.4

Z, O

i. 55

I, 12

O. _14

O, 75

[1.2

1.714

1.43

1.23

0.95

0.74

0.54

O. 4

I. 39

I. 1:1

]. (l(t

o. HIi

O. 62

0.55

0._

l. lO

0.91

O. H2

i).68

0.53

IL47

0,90 0.75

0,76 II. it2

O, IIH [). [,7

0.57 IL47

0.45 0.38

0.40 O.

O. H I. o

ti

blfll

\

II

tl.4 I. G4

o '

=

0.14

2ti

I.Z

L6

2.0

1.20

O..97

O. 78

O. 64

0.2

1.73

1.:11

l, _:I

O. B4

O. 88

O. 57

0.4

1.32

I. OH

o. HH

O. 74

O. 60

O. 50

{I,rl

1.04

(). Ill)

O, 7fi

(l.(14

0.54

{i,44

O. 8

O. 87

O. Ill

O.G:I

0.54

0.44

O.:IH

i

I.(i

I).7[

II. tl I

0.53

0.45

0.:IN

0.30

I

(ti

=

(I.:1}:

1976

Section

B9

15 September Page Table

All

Edges

B9-14.

Load

Linearly

max

Var,.ing

Along

(Continued)

(11)_

Supported,

Distributed

39

w

=

6

_1>_

/:: anti

where

_

are

found

in

the

following:

Ix'ngth a b

a

i'

'

1

fl

1, 5

16

O.

All

Edges

Supported, Load

Ltnearl,,

max

Varying

Along

n

I

Edges

Fixed. l,¢)ad

Entire

3.0

3.5

4.0

O. 34

0.3H

0.43

0,47

0.49

O. 043

O. (160

0.

0.07_

O. 086

0.09

max

w

070

ql) 4 ._

= 6

where

[_ and

5

1

art,

found

as

folh)ws:

Breadth

i

All

2.5

O. 26

qh 2 fl t-- T-

=

1

b

Uniform

(I

,I (I. 022

Distributed

2.

At

()vt,

Sur

1.5

2.0

2.5

3.0

3.5

4.0

(_

O. 16

(J. 26

O. 32

0.35

O. 37

0.38

O. 38

8

(}, (}22

O. I).l 2

0.

o,

0.

(I.

o.

Centers

I,ong

of

(156

I)fi3

067

069

070

Edges:

r

f:tce

: Mb

12(

q bz + O. (i23¢v

1

c')

-:

max

M

q

4111ILL

At

Centers

nf

M

Edges:

qb 2 -2.1

:: a

At

Short

Center

: ______qK_ X(:I

Mb

0. max

w

One

Long

Edge

Free,

Supported, Over

Fixed,

Short

Center

of

Load

m_tx

Center

Jormula.'-:

Edge

Over

for

M

I:l

p

=

(I. 3;

b'

Edge:

MI)

2(1+:1,2¢v_)

=

l,:dge:

l. :17ql) 4

Hqa2

,-77-,.

(l+

0.2H5_

......

w

:"

m:tx

w

l':t:'(l+

oh2 M:tx

Edges

Uniform Entire

Free

[:lamped,

Three

Supported,

r')

lib'W)

(_ = o.:0

l I

FREE

Long

:

of

a

Other

tv 4)

02H4 I. DS(;_

Fixed

M

M

One

I + 2tv z -

,'-;urface At

,

(I. O0(,kll)_(

Edges

Uniform Entire

At

=

a

_: ( 1 4

Other

M

+ ,tt_ 1)

Stress

0"

=

tv(ll)

fl -_

4

:,

l,o;ul

Surface

where

/l

and

_v

may

[K:

found

Irom

Ihe

Iol]owing:

//////////////////./ b SS [

l.O

1.5

2.0

2.5

3.0

3.5

4.0

SS It

0.50

O. 67

O. 73

0.74

(I. 74

0.75

o.

75

,_

O. 03

(I. 046

I). {}5l

O. ()Stl

(I.

(I. 05x

I). 058

ss

(. = ".:0

057

others

p

=

0

1971

Section

B9

15 September Page Table

O_.,:

Short

Edge

(Aner

rib2 Max

/cktges

Uniform

Over

(Continued)

Clamped,

Thre_'

Sul:;>oeted,

B9-14.

Entire

Stressa

=

fl

t-T"

,

maxw

=

_Et

+

Load

Surlace

where

/3 and

a

may

be

lound

from

the

following:

a b

1.0

L.5

/3

0.50

0.67

ct

0.03

0.071

2.0

2.5

3.0

3.5

4.0

0.73

0,74

0.75

0.75

0.75

0.101

0.122

0.132

0.137

0.139

3.5

4.0

S$ $$

(_ One

Short

E-ge

Other

Feee,

Tbr,_e

Supportod,

Edges

Entire

Surface

r +]

and

fl t]b_ _

=

,

-

Short

Edge

Other

tv

art,

from

found

b

1.0

1.5

fl

0.67

0.77

I_

0,

0.

14

Free,

Three

Supported,

w

=

Et

the

following:

2.0

4.0

O. 79

0.80

dqb

16

0.

O. 167

]65

z


Edges

Varying

Liuearl_'

Along

where

It

and

¢_ are

found

from

q

b

I.O

/3

o.

c_

0.04

1,5

2.0

Long

Edge

O. 28

0.32

0.05

0.

2

Free.

Three

Supported,

flqb maxa

bklges

Uniform

Over

following:

2.5

058

(U

Other

the

I,ength

_

Entire

=

_

3,

0

0.35

0.36

0.37

O. ;17

0.

O. 067

0.069

_0. o71)

(h54

0.:l)

2

crqb

-7--

,

maxw

=

Load

Surface

where

_

and

a

are

found

from

the

following:

q FRIEEI

1

|

1

l

1

1

Lo

L5

2.0

O. 67

0. ,I 5

0,

30

0.

0.

0.

080

EDGE

o,

14

1 IX;

(u = o._) One

Long

Other

Edge Throe

Supported, Load

Free,

/3 qh _

._+

Edges

= --iv-

,

_._w_

Distrihuted Varying

Along

Lintrarly

where

fl

and

,_

are

found

from

the

following:

Lengih a

1, o FREE

J

Dlstrihutofl

l_md

One

max

,_

1

FREE

i:

where

=RE=:

()ue

o"

o. 3)

[_)',._d

Unflorm

Ovc'r

mlL_

=

1.5

2.o

_q fl

O. 2

O, 16

O. 11

a

O. 04

O. 033

0.

b

(/a

_

0.3)

026

ctql_

4

40

1971

Section

B9

15 September Page Table

All

Edges

Fixed,

Load

At

(Continued)

Center:

Small

Concentric Area

Uniform

Over

B9-14°

,

Circular

Of

Radius,

Mb

r 0

where

=

fl

_-

has

l+/a)

values

log-

as

+

2r 0

5(1-(_

max

=

M

follows:

4

2

0. 072

b

41

1

0. 0816

0. 0624

q b

Long

Edges

Fixed,

Short

Edges

Supported,

Uniform

Load

Entire

At

Over

Centers

max

)

At

M

I

Edges Edges

Fixed,

At

+ 0.2a

_)

=

qb 2 + ()._i(r t)

24(1

Centers

of Short

M

1 + O.:le_ 2) 80

=

qb2(

=

0.015qb2(l + 3c_ 2) (z + _')

a

Edges:

Supported,

Uniform

Load

Entire

Over

max

M

=

M

Surface

]

At

•

•

qb2 (). _)

H(I+

Center:

"l Mb

Edges

= a

q

Distributed

qb2 12(1

Center:

Mb

All

_

Edges:

b

b

Short

M

Long

Surface q

Lortg

of

8(1

+O,

qb _ Herz+

M flo 4)

a

Supported, Load

of Triangular

in

Form

max

M

=

flqb

_

2

max

W

c_qb 4

-

D

Prism [J and

_r found

from

the

following:

q

1.0

[,

•

i]

1.5

2.0

3.0

0.03,1

0. 0548

0. 0707

0. 0922

O. 1250

O. 00263

0. 00308

0. 006:46

0.0086M

0. 01302

(u = o.:))

1971

Section

B9

15 September Page Table

B9-14.

42

( C ontinued)

1

.D N

c_ c_

_lco

It

®

....

°

°

o

¢q =

II

_¢_Q

_I

t'_

o

x_|_

o

i

L9

5_

i! _E

_._

1971

Section

B9

15 September Page 43

O e_

197i

%

%

3_ %

%

%

%

S

S

s

d

S

S

¢;

_T

¢T

0

,r._

cxl

%%%%%%% ¢T

CT

_

¢T

S

_

S

S

S

S

S

_

d

d

_

d

d

_

S

_

d

_;

_;

_

d

d

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s

o_

No

L !

r Ex ._-,_

_4

I

i

°l-'L I

I_

,I

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I,

m

•

S

_

_

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d

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d

d

d

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i

q

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O _

S

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Section B9 15 September 1971 Page 44

0 0

0 0

0 0

0 0

0 0

0 0

0 0

%

%

%

_

%

%

%

d

¢;

d

_;

¢_

d

d

0 0

0 0

0 0

0 0

0 0

0 0

0 0

Jl

I =I. ....._

0

.Q

_M l O

A

ii

L

_

ml

_ O"

d

d

_;

d

d

¢;

d I

O

_m

.,.4 !

"_ [--I .p..4 0

0

0

0

0

0

0

I

I

I

,ID tO

0

0

[',-

_

_D

0

I

Section

BS. 1

28 June

1968

Page B8.1.2

SIGN The

moment

and

resisting are this

CONVENTION

local

coordinate

the sign

moments,

defined section.

5

so that

system

conventions stresses, there

for

for

a bar

applied

subjected twisting

displacements,

is continuity

throughout

and

to an applied moments,

derivatives the equations

twisting

internal of displacements presented

in

St'Ctl_,l_BS.1 28 ,]mLe 1968 Page B8. 1.2

SIGN CONVENTION

I.

LOCAL The

specific bar.

5

COORDINATE

local

limitations The

y and

coordinate are z axes

section

is ttnsymmetrical,

Figure

BS. 1.2-1.

to thin-walled

The

opc_l and

SYSTEM system

stated. are

is applied

The x axis the axes

as may coordinate thin-walled

to either

is placed

of maximum

end

along inertia

be seen

in the

system

and

closed

cross

of a bar

solid

the

length

when

the

cross

si._n convention sections

also.

secti,,_

u,_lcss of O_,: cro-:.'= '__h,,_',

_h(,wl_ :_}i_I:,'

,_

_

L

__

_

Section

B8.1

28 June

1968

Page7

MI A.

Local

Positive

Coordinate Applied

System.and

Twisting

Moments

/-SEGMENT Lx _

_Jrz

/

OF BAR

SHOWN ABOVE

__BITRARY

POINT

' B.

Positive Moment

Internal and

y

Shear

Resisting Stresses

L

C.

Figure

Positive

B8. i. 2-I. and Positive

x

Angle

Local

of

Twist

Coordinate

Sign Convention

System

P

Section

B5.1

2b

1968

Jtme

Page

B8. I. 2

SIGN

U.

the when

x axis. viewed

vectorially.

CONVENTION

APPLIED The

b

applied The

TWISTING twisting applied

from

the

(See

Fig.

MOMENTS moments

twisting origin

or

B8.1.2-1A.

(mt

monmnts are

in the )

or

M t)

are

positive

positive

are

x

twisting if they direction

moments are

about

clockwise

when

represented

7"-

Section

BS. 1

28 June

1968

Page B8. 1.2 III.

SIGN CONVENTION INTERNAL The

same

9

sign

internal convention

oa the y-z

plane

BS. 1.2-lB.

)

RESISTING resisting

MOMENTS moments

as the applied

of a bar

segment

(M i) twisting

that

are

about

moment

is farthest

from

the × axis

when

they

the origin.

are

and

have

evaluated (See

Fig.

the

SIGN

S,_cti,m

BS. l

28 Ju,lc

1968

CONVENTION

STRESSES Tensile stresses equivalent

normal

are negative. to positive

stresses Shear internal

(ax) are positive,

stresses

(r)

leslsting" " "

moments.

and compressive

are positive when (See

Fig.

nvrmal

they am Bb.

1.2-1

]3 )

Section

BS.

28 Juac

19(;5

i)t_gc B8.1.2

SIGN V.

the

CONVENTION

DEFORMATIONS An

about

the origin.

applied x axis. (See

a longitudinal See

Fig.

direction

11

twisting The Fig.

moment

rotation B8.1.2-IC.

displacement BS.

2.2-2B.

of the

positive

is

The x axis.

positive )

(u) )

induces

An

in the

longitudinal

a rotation if it is applied

or

clockwise

when

twisting

x direction displacement

for

angle

moment unrestrained is

positive

of twist

(_b)

viewed also

from induces

torsion. when

in

the

1

Section

B8.1

28 June Page B8.1.2 VI.

SIGN CONVENTION DERIVATIVES The

angle

first

of twist

and negative, applied

1968

12

twisting

OF ANGLE

(¢'},

with

respect

respectively, moment

second

OF

(¢"),

and

to the positive when (Mr)

the

TWIST third

derivatives

x coordinate

rotation

is applied

(¢t,,)

are

is positive

at the

ends

positive,

and

of the

of the positive,

a concentrated

bar.

r

SECTION B8.2 TORSION OF SOLID SECTIONS

-.m..--

TABLE

OF

CONTENTS Page

,f

Torsion

B8.2.0 8.2.1

of Solid

General I

Basic

..........................

Theory

III

Membrane

IV

Basic

1 1

................................

3

................................. Analogy

Torsion

Circular

Equations

II Hollow

Section Circular

III

Rectangular

IV

Elliptical

for Solid

V

Equilateral

Sections

............

of Twist

for

Section

4

7

................

Section Section

3

............................

Torsional Shear Stress and Angle Solid Sections ............................... I

1

...................................

II Limitations

8.2.2

Sections

: .............

7

.........................

8 9

...........................

11

.............................

Triangular

Section

13

....................

VI

Regular

Hexagonal

Section

.......................

13

VII

Regular

Octagonal

Section

.......................

14

VIII

Isosceles

8.2.3

Trapezoidal

Section

Problems

I Example

Problem

1 ...........................

18

II Example

Problem

2 ...........................

19

Problem

3 ...........................

21

Example

Torsion

B8.2-iii

of Solid

Sections

........

18

Example

HI

for

17

.....................

Section B8.2 31 December

1967

Page 1 B8.2.0

TORSION

OF

SOLID

SECTIONS

The torsional analysis of solid sections is restricted to un_'estrained torsion and does not consider warping deformations. B8.2. I

General

I

Theory.

Basic

The formations

(P)

angle

and

the origin.

and

the

when

applied

efficients are

are

used

section, the they

functions

for

calculating

the

torsional

torsional are

more

vary

linearly

and

will

have

the

on adjacent sections,

radial the

and

cross

to J/p;

cross

moment

but for

all

These For

cross

co-

constants

of inertia

other

section,

These

respectively.

polar

of twist of the

(St).

section.

origin.

cross

geometry

of each

at

the

angles

modulus

stresses,

to the

reduces

any

distribution radial

lines

cross

( Figs. shear

line

on all

(Vx) , which

of the

torsional

section of the

reduces

distribution

stress

no warping

torsional

dimensions

stress

along

same

shear

the

bar,

geometry

de-

a circular (J),

and

sections

functions. shear

will

the

deformations

complex

bar

produces

of the

modulus

torsional

longitudinal

and

constant

section

The

(K)

resulting

of the

characterize

from

an arbitrary

and the

known.

and

is determined

(L)

between

are

stresses

(Tt)

distance

properties

moment

that

stress

stresses

material

coefficients

constant

shear

is determined shear

twisting

Two unique the torsional

the

requires

at an arbitrary

(_)

These

sections

torsional

section

of twist

be determined

bar,

of solid The

on a cross

resulting

section can

analysis

be determined.

any point The

torsional

stress

emanating radial

is equal

section B8.2.

on any cross

when I-IB

and

distribution

from

lines to the the

section the

( Fig.

stress

B8.2.1-IC). is nonlinear

geometric

B8.2.

torsional

of a circular

I-IA).

shear

The

stress

distribution For

centroid,

is the non circular

(except

(Tt) , same

__

Section B8.2 31 December L_

Page

2

Y A

"__ i"_ ;':o':" __

A.

Circular

Bar

Shear

Stress

Distribution

_Lx Y

x

C.

Differential

FIGURE

B8.2.1-1

Element

SHEAR

STRESS

DISTRIBUTION

1967

Section

B8.2

31 December Page

f" along

lines

line)

and

torsional warping

of symmetry

where

will

be different

on adjacent

and

longitudinal

shear

of the cross

section

When the warping restrained, normal they

normal stresses

are

small,

are

II

there

stress

(a)

neglected

to the

is an abrupt

radial

warping change

to the radial

( Fig.

B8.2.2-2A).

When

are

and

radial

by longitudinal

induced

to maintain

torsional

analysis

have

little

deformation in the

on adjacent

lines,

applied

of solid on the

at fixed

twisting

stresses

equilibrium.

effect

occur

shear

These

sections angle

ends

since

of twist.

and at points

moment.

Limitations The

torsional

analysis

of solid

cross

sections

is subject

to the

following

limitations. A.

The material is homogeneous

B.

The shear stress does not exceed the shearing proportional

and isotropic

limit and is proportional to the shear strain (elastic analysis). C.

The stresses calculated at points of constraint and at abrupt changes of applied twisting moment

III

D.

The applied twisting moment

E.

The

Membrane

bar

sections

is usually

analogy

can be used

terms

of the St.

have

are not exact.

cannot be an impact load.

an abrupt

change

in cross

section.*

with irregularly

shaped

Analogy

The torsional

The basic

cannot

analysis complex,

of solid and for

to visualize

differential Venant's

bars some

cases

the solution equation

stress

for

function,

for

unsolvable. these

a torsional

cross analysis,

cross

The membrane sections. written

in

is:

Stress concentration factors must be used at abrupt changes in the cross section.

the

B8.2.2-2B).

induced

in the

rapidly,

is normal

lines

( Fig.

3

contour

is different

deformation

attenuate

section

will occur

stresses

Restraints where

the cross

1967

is

Section

B8.2

31 December Page a2 _

analysis

+

equation

the

membrane

to the basic

membrane,

a y2

when

- - 2G0.

a z2

is similar

of a deflected

The following

4

0_

_}y2 This

1967

which

_ z2

-

analogies

exist

has the same

differential

equation

used

for

the

is:

between

boundaries

the

solutions

as the cross

of these section

two analyses of a twisted

bar.

A.

The volume

under the deflected membrane

for any pressure

is equal to one-half the applied twisting moment 2G0 B.

= p/T

numerically.

is in the same

C.

(M t), when

The tangent to a contour line of deflected membrane

at the same

(p)

direction as the maximum

at any point

torsional shear stress

point on the cross section.

The slope at any point in the deflected membrane

normal to the

contour at that point is proportional to the magnitude of the torsional shear stress at that point on the cross section. IV

Basic Torsion Equations for Solid Sections A.

Torsional Shear Stress

The basic equation for determining the torsional shear stress at an arbitrary point (P) on an arbitrary cross section is:

Tt=

M(x) St

Section

B8.2

31 December Page where

M(x)

torsional

is evaluated

shear

at x = L

for the x is to be determined.

stress

If a constant

torque

is applied

arbitrary

to the

end

cross

of the

5

section

bar,

1967

where

the

the equation

reduces

to:

Mt T t

=

_

•

t

St will in this

case,

vary the

along

equation

Tt

For St(x)

the case must

the

=

length

for

a varying

cross'

both

section

and,

"

moment

be evaluated

bar

is:

M(x) St(x)

of varying

of the

at the

and cross

varying

cross

section,

section

where

the torsional

M(x)

and

shear

stress

is to be determined. In the B8.2.2-III

through

at the point mine

and

equations

Angle

The

basic

the

equation

stress

is equal

shear

stress.

for determining

the

stress

determinations to M t and the The

in sections stress

resulting

is determined

equations

deter-

only.

of Twist equation

section

(_ -

When

shear

M(x)

torsional

shear

B.

cross

B8.2.2-VIII,

of maximum

maximum

any

for torsional

M(x) reduces

located

1 GK

at a distance L

f

x

0

is a constant to:

i(x)

angle

L from

of twist the

between

origin

the

origin

is:

dx.

torsional

moment

applied

at the

end

of the

bar,

Section B8.2 31 December

1967

Page 6 1 _b = GK

and the total

twist

L f x Mtdx 0

=

MtL GK

X

of the bar is: MtL

(max) =

When

GK

the cross section varies along the length of the bar, the torsional

constant becomes

a function of x and must be included within the integral as

follows:

L G

The moment-area

[ J

x 0

M(x) Ktx)

(ix "

technique (numerical

integration) is very useful in

calculating angle of twist between any two sections when a M(x)/GK(x) Diagram

is uaed.

See Section

B8.2.3,

example

problem

3.

-

Section

B8.2

31 December Page

B8.2.2

TORSIONAL

SHEAR

STRESS

AND ANGLE

OF TWIST

1967

7

FOR

SOLID

SECTIONS The equations for

points

torsional

presented

shear

stress

developed

I

torsional

shear

stress.

equations

are

presented

over

the full

The applied applied

section

of maximum

The equations twist

in this

load

angle

torsional

For

presented

for length

for

for

of twist

shear

some more

cross than

are for

stress

are

sections,

one location.

the total

angle

of

of the bar.

in all cases

is a concentrated

twisting

moment

(Mt)

at the end of the bar.

Circular

Section

The maximum the circular

cross

torsional section

shear

( Fig.

stress

occurs

B8.2.1-1A)

at the outside

and is determined

surface

of

by

Mt rt(max)

-

St

where

Since the

section,

the the

torsional stress

shear at any

stress

point

(P)

varies

linearly

on the

cross

MtP Tt(P ) = The total

angle

_j

•

of twist MtL

(max)

=

GK

is determined

by

from section

the centroid is determined

of by

Section

B8.2

31 December Page

1967

8

where 2

The located

H

angle

at a distance

Hollow The

cylinder

Circular

Section

torsional

following

equations.

and

shear

be determined

constant

0

from

x

MtL x GK

can

r

L

between

-

torsional

where

of twist

and

K=

_-

St

-_-

r. are 1

FIGURE

the

r

defined

BS. 2.2-1

the origin

the origin

and an arbitrary

is determined

cross

section

by

"

stress

and

angle

from

the equations

torsional

section

of twist

for

in Section modulus

are

a thick-walled B8.2.2. determined

- r

by Figure

HOLLOW

B8.2.2-1.

CIRCULAR

CROSS

SECTION

-I

hollow when by the

the

Section

B8.2

31 December Page

III

Rectangular The

and

Section

maximum

is determined

torsional

by the

r t (max)

shear

following

stress

occurs

at point

A (Fig.

equation.

= Mt/S t

where

St Some

= _ b d2 .

typical

values

The

equation

O/

of a for

are

in Table of (b/d)

B8.2.2-1. is

1

=[3 ,=] 1.8

torsional

Tt(B)

The

shown

_ in terms

)

.0+

The

total

shear

stress

at point

(B)

is determined

= ,t(max)(b).

angle

of twist

is determined

by

MtL ¢ (max)

-

K = flbd

a .

GK

where

Some

typical

1967

9

values

of fl are

shown

in Table

BS. 2.2-1.

by

B8.2.2-2A)

Section

B8.2

31 December Page

¥

b

A.

B.

Stress

Warping

FIGURE

Distribution

on Rectangular

Deformation

of a Rectangular

BS. 2.2-2

RECTANGULAR

Cross

Section

Cross

CROSS

Section

SECTION

10

1967

Section

B9

15 September Page

45

b_ I

I

!

I

I

0 b_ C_'

0

0

0

0

0

0

0

<

n

o

o

o .Q

*1

_t

t_

> I

llc

I"

o

_

_

_

0

0

0

0

0

d

d

_

c_

_

c_

O

_'_

.¢'1

['--

,--4

tr_

_

_

o

_

_

_

_-

d

d

c_

c_

d

d

0 '_

_

_'_

_

_'_

0

0

0

0

0

0

0

c_

c_

d

_

c_

c_

d

d

d

c_

c_

c_

S

c_

0 _

_"

0"

0"

_

_

_"

0

0

0

0

0

c_

_

_,

c_

c_

c_

0 _

_'

0 '_

0"

0"

d

d

c_

c_

d

_

d

c_

_

c_

_

c_

c_

c_

_

_'_

,I

N< E_

or

_

[JJJ m

o

0

c_

!

o

.,-.,

o

0

0

c_

_

o r..)

t-: I

t'

.IlllI

,I

!

e_

I

I

1971

Section

B9

15 September Page

.D

-n_

¢q

¢r

.o f_ |_

f_

.i

o¢-4

¢r

0

_0

t_ ¢r w

_

°

E

0

.o ¢4

J_

.D

oe*eo,

46

1971

Section

B9

15 September Page B9.3.3

Elliptical For

been

found

solutions tion

for for

to the

B9.3.4

plates

in Table B9.3.5

boundary

common

differential

for

is the

loadings.

plates.

Triangular

For

shape Table

additional

equations

see

of an ellipse, B9-19

solutions

presents

information

Reference

the

have

available

as to method

of solu-

1.

Plates several

loadings

on triangular

shaped

plates

are

presented

B9-20. Skew Solutions

significant

whose

elliptical

Solutions

47

Plates

some

plate

1971

results

Plates have from

been

obtained

these

for skew

references

are

plates

in References

presented

in Table

1 and B9-21.

5.

The

Section

B9

15 September

Table

B9-19.

Solutions

For

Elliptical,

Page Plates

Solid

1971

48

v

b/a" a

Edge Uniform

Supported, Load Over

Entire

At

Center:

Surface

-0. max

stress

maxw

Edge Uniform

Supported, Load Over

Small

=

(0.146

Mb

3125(2

- cQc_b _

ab =

t2

Et3-0.1ot)qb

4

(for_

Area

of Radius,

Fixed,

Load

Over

=_)1

At Center:

Concentric

Circular

Edge

=

max

M

:

max

w

,_

=

1 + .)log

_-_

b

_r0

+

6.57/_

-

2.57ap

r0

Uniform

At

pb 2 . E----Tt(0.

1 19

-

= _-)

0. 045G)

Edge:

Entire

Surface

qb2c_2

=

M a

=

4 (3 + 2a z + 3a _)

Mb

qb2 (0'2 + #) 80+2a z+3a')

Mb

qb _ 4(3 + 2_' + 3a 4)

At Center:

M

= a

.

qb'(l+ 8(3+

2_ z+

qb 4 max

Edge Fixed, Load Over Concentric Area

of

Uniform Small

At

w

ffi

64D(6

+ 4a z + 6ct 4)

Center:

\

Circular Radius,

r0

Mb

maxw

47r

=

.Pb'(0.

log-

ro

-

0.3170t

0815 - O. 026a)(p Et _

-

=

0.378/

0.25)

3_ _)

]

Section

B9

15 Septcmbor Page Table

B9-20.

Solutions

For

Triangular

49

Plates

X

2/3 a i

.¥

a/3

Equilateral

Triangle,

Edges Distributed l_:nti

max

o"

_

0.

14_8

q;)'_ 7-

x

Supported, Load

at

v

_

at

_r

{h

x=-0.062a

(),

X =- 0.

Over qa 2

r( , SU rfri(!c' max

-

_"

0. 155t

V

w

max

:--

129a

t'_

=

_ :1421}

at

poin{

I)

r Edges

Supported,

I,oad

Concentrated On

Small Ol

At

()

Circular

Dislributed

max

er

=-

_

p)PL

_

o.:lT_la Ll(ig

x/ I.Ih_

I':, :_ (I

rl,

Isosc¢'les

Edges

3(1

t 2

,

-

{).(;75t

Area

Radius,

[tight-Angle Triangle,

I *

- p_)

_

r

0.

max

•

().

131

7

=

0. 0095

_

;it

i)oinl

0

(1:12 _

Supl)ort(.d, Load

()I;M,_)2

I_'I_iXW

x

([;t _ re;ix

(r

:

_'

O. I (25

?-

()vcr

]': nti re Su riat't! max

Equilateral With

"]',a.o

]%1 =

Triangle Or

Clamped, Hydrostatic

w

/'J(I al

(ir

M

(p

,:

:

/tlihla

I).:l)

2

whcr('

Etl_es

Three Uniform

Or E(ll4c

Load

v ,-

0 Supl'iortcd

l JIJLIII DieltrJliution

M

M x I

(lnl[orm

Ilylli'ost;itic

/I

fli

M v)

ili

I';dtr('

IL I)12f;

(I. {_|47

-0. 021_5

(I, 005:1

II. (10:>,5

-0.

I)I00

(p

,

_l. z)

V

i) (+la

rnpl,d

1971

Section

B9

15 September Page Table

All Edges Distributed Entire

Supported. Load Over Surface

m,'_

0'

Solutions

B9-21.

=

f_qb_ := t-_

°'b

where

for

Skew

50

Plates

_ is

t01 ,,de,I 'ooegI 'Sdo

60 deg

]

0.40

I

0.16deg 75

]

(._ o. 2)

Edges

b Supported, Edgen a Fret', Uniform

Distributed Entire

Load

max

_

=

Grb

:

_

where

IJ is

Over

Surface

fl

[

0.762

[

0.615

[

0.437

[

]

0.250

FREE ? _FREE

_/A

All Edges Clamped, Uniform Distrllmted Load

Over

Enttrc

b

At Center:

M ffi flqa }

w

=

where

_ and /31 are a

Ske_ Angle

I*

i)

Surface

I

-_ b

_ =1

a -_1.5 b

1.25

& -=2.0 b

0

15

0.024

0.00112:1

ft. 019

O. 00065

o. 015

0,00_8

0.0097

0.00014

30

0.020

0.00077

0.016

0.0(}045

0.0125

0.00026

0.0075

0.00009

45 60

0.015 0.0085

0.00038 _ O. 00011

0.01 ] 0.0O62

0. 00022 O. 00006

O. 014 O. 0048

O. 00012 O. 00003

0.005 0.0025

0.0OO04 0. OO001

75

0.0025

0.000009

o. 0027

0. 000005

0.00125

0.000002

0.00125

Along

Flx_l

Edge:

The coefficient _ edge at a distance

(M

_ f/_qa 2

Skew

for

for maximum bending moment [a from the acute corner is

a _ :

])

Angle

(deg)

f_

[

15

-0. 0475

O. 6

30

-0.0400

O. 69

45

-0. 0299

0.80

along

the

1971

Section

B9

15 September Page B9.4

ISOTROPIC

Large

It was

resistance

PLATES

deflection theory

determined

' tion analysis

THIN

was

--

covered

for available

plate geometries

section.

Figure

gave

membrane

plates and for medium-thick

the regime

of thin plates, which

and pressures

B9.4.1

encountered

Circular A

placement

circular

by a uniformly

tive to the thickness

diametral

per

and at thc center

is clamped

is shown

cut from

moving

B9-12

for

these two regimes

is

of the plate dimensions

Load

B9-13.

B9-13c.

the plate shows

which

radially, thereby

if the plate is not clampcd

the edge

causing

plate, loaded

is large rela-

In Fig.

B9-13d

the bending

a

moments

act in this strip at the edge

direct tensile forces

prevents

The

deflection which

in Fig.

The

strip from

plates,

so that rotation and radial dis-

in Fig.

of the plate.

at the edge

most

Distributed

tensile forces

metral

Second,

includes

and the direct

the fixed support

tensile stress.

design.

of the plate as shown

First,

the load

of membrane

Between

load, has a maximum

strip of one unit width

unit of width

deflects.

edge

of large deflec-

are given in Fig.

plates.

Uniformly

at the edge

distributed

Curves

generally

--

plate whose

are prevented

and direct

B9.2.3.

and loads will be given in this sub-

in aerospace

Plates

in Paragraph

In this region,

a guide as to the regimes

and thin plates.

51

ANALYSIS

by the classification

of bending

Solutions

plates,

discussed

1 _ t < w/t > 10t .

approximately

of plates is a combination

medium-thick

DEFLECTION

of plates was

that the region

B9-7

LARGE

1971

arise from

at opposite

two sources.

ends

the strip to stretch

at its edge

but is simply

of a dia-

as it

supported

Section

B9

15 September Page

1971

52

©

N R

5 a Z

m

m

< Z

X

iljl

m

t

r..)

j/ji l

i

..

q

dr-

I

'

i

I

©

i

i

_4

H

!

r/ Q

t

I

I

I

I

I

i

I

I

I

t

t

i

Section

B9

15 September Page

1971

53

(el

(a) t

I

t

l

(b)

(fl q

q(e)_

Wmlx

Wmax (g)

(d)

_

(h)

FIGURE as

shown

in

Figs.

outer

concentric

their

original

ring

the acting eter

at the

rings

outer

outside the

to decrease,

inside

BEHAVIOR

B9-13e

and

of the

diameter

original on

B9-13.

as

edge

is

and

radial

plate plate

shown

cut

in doing

as so

as

from

In

the

unloaded shown

they

plate.

plate; in Fig.

introduce

CIRCULAR arise

shown

deflects.

of the ring,

THIN

stresses

(such

the

diameter of the

f,

OF

PLATE

out

of the

in Fig. Fig.

B9-13h)

B9-13h This

the

ring

radial

B9-13h, compressive

tendency

cause

for

to retain

the

concentric

tends

tensile the stresses

to

retain

stresses ring

diamon

every

Section B9 15 September 1971 Page 54 diametral sectiol_ such as xx.

These compressive stresses in the circum-

ferential dircction sometimes cause the plate to wrinkle or buckle near the edge, particularly if the plate is simply supported. The radial stresses are usually larger in the central portion of the plate than they are near the edge. Stresses have been determined for a thin circular plate with clamped edgesand the rem_lts are plotted in Fig. B9-14, where abe

and

abc

are

50

ot c °t e Obc

0 0

1

!

|

v

|

2

3

4

5

MAX DEFLECTION PLATE THICKNESS

FIGURE

B9-14.

DEFLECTIONS,

STRESSES CIRCULAR

IN THIN PLATE

--- Wmax /t

PLATES WITH

HAVING CLAMPED

LARGE EDGES

the

bending

Section

B9

15 September Page stresses atc

in a radial

are

corresponding

stress

abe

tensile

stresses

at the

between

load,

edges.

For

plate

and

the

value

of

curve

on the

w

load /t

left. a

max

Figure whose

Table

of deflections simply

and

supported.

w0 t + A

Also,

the

ar

stresses

= CVr E

and

largest

relatively

q

are

corresponding By projecting r2/Et

2 are

presents

for

are

simply

B9-22

The

in the

and

the

across

to stress

at the

center

similar

the

bending The

with

direct

clamped

of elasticity

qr4/Et

of the

be computed.

_ is found

curves,

The

from

the

corresponding

at the

to those

the

and

relationship

plate

4 can

and

ate

increases.

modulus

of

55

stresses.

show

circular

value

edge

of Fig.

of the B9-15

stress plate. for

a

supported. data

in uniformly deflection

n

a thin

to this

presents

stresses

which

qr4/Et

curves

four

and that

as the deflection

the quantity

read

plate,

It is noted

of curves

stress

given,

of the

of these

larger

a set

and

center

stresses.

if the dimensions

B9-16 edges

Also,

is the

deflection,

max

parameters

edge

presents

example,

edge

tensile

fixed

B9-15

at the

direct

become

Figure

plate

plane

1971

at the

for

the

loaded center

calculation circular

middle

,

plane

plates,

w 0 is given

(t)

q

of approximate both

by the

clamped

values and

equation,

(33)

are

a t = t_l. E

given

by

,

(34)

Section

B9

15 September Page

56

1971

Section

B9

15 September Page

57

1971

Section

B9

15 September Page Table

B9-22.

Data for and Stresses

A

B

_J Otr=_

Clamped

Plate

w0

Edge

Center Conditions

Plate

58

Calculation of Approximate Values of Deflections in Uniformly Loaded Plates (_ = 0.3)

I Boundary

1971

t

Ol

_r=/_t

at

r

Edge Immovable

0.471

0.171

0. 976

2.86

Edge Frce To Move

0. 146

0. 171

0.500

2.86

Edge Immovable

1. 852

0. 696

0. 905

1. 778

0.610

Edge Free To Move

0.262

0.696

0.295

1. 778

0

/] r

Bt

0.143

-4.40

-1.32

-0.333

-4.40

-1.32

0.183

0

0.755

-0.427

0

0.755

0.476

Simply Supported

and

the

extreme

fiber

ar ' =fl

B9.4.2

bending

Wot E "_

r

Circular

center

with

either 1.

the

deflection

B9.4.3

been for b/a= the

the

obtained various

[ 1].

--

of a plate

with

Numerical

2/3,

of the

of the and

plate

b/a= are

the

load 1/2

graphically

of a circular edges

coefficients

for

been

necessary

loaded

obtained for

at the in

solution

of

Loaded

clamped

q

has

plate

(33), (34), and (35).

U_formly

values

(35)

.

at the Center

equations

Plates

intensities

1, b/a= center

case

by

supported

contains

w 0 from

given

of the problem

or simply

B9-23

Rectangular For

Loaded

solution

Table

are

Wot = f3t E-_

at

-

clamped

Reference center

'

Plates

An approximate

stresses

edges,

of all and _=

an approximate

the parameters

for

three

0.3.

represented

have

different The

been

shapes

maximum

in Fig.

solution

B9-17,

has

computed

of the

deflections in which

plate at

Section

B9

February Page 59 Table

B9-23.

Data and

for

Calculation

Stresses

of Approximate

in Centrally

Loaded

Values Plates

of

Plate

Conditions

A

c_ r

_t

0. 357

0. 107

0.443

0. 217

1. 232

Edge Free To Move

0. 200

0. 217

0. 875

1. 430

0.552

0. 895

0.272

0. 552

0.407

Wo

Edge

r=Oet

Edge Immovable

Deflections

(p = 0.3)

Center Boundary

! 976

15,

fir

fit

-2.

198

-0.

659

-2.

198

-0.

659

Clamped

Edge Plate

Immovable

-0.

0.488

250

O. 147

0

0. 606

0

0. 606

Simply Supported

Edge Free To Move

_,

-0.

//'/'/

/

!/I/ / I_I / /

341

\_,.=,

:Lf..L

/

I

I

I

I

|

I

I

I

•

100

_

•

200

qb4/Dt 4

FIGUIIE

B9-17.

MAXIMUM

RECTANGULAR

t)LATE

DEFLECTIONS WITlt

AT

CLAMPED

CENTER EDGES

FOR

•

Section B9 February 15, Page 60 w

/t

max

is plotted

the

use

of the

b/a

= 0,

which

that

the

deflections

obtained

for

membrane They

brane and

represents

supported

w 0 , at the

q -

plates

long

and

are form

a solution

extreme

center

1.37+

b/a

The

1.94

the

included

at the

< 2/3

middle

in Fig.

is the

also curve

are

very

values

of the

includes for

It can

be seen

close

to those

of the

combined

long sides

of the plate.

B9-18.

plate, has

uniformly

been

bending

in terms

figure

long plate.

maximum

equation of the

.

loaded

obtained

stresses

An approximate

of the plate

w0 t[ a4

fiber

Also

with

of a rectangular

respectively.

comparison,

of an infinitely

plate.

stress

edges,

stresses

For

deflections

in graphical

the case

4.

deflections.

of finite

bending

given

B9-20,

qb4/Dt

of small

an infinitely

For simply

theory

and

are

against

1976

load

[ 1].

with Values

are

given

for

maximum

q

is given

immovable, for

in Figs.

memB9-19

deflection,

by:

(36)

Section AP,-

B9

February Page 61

1976

15,

/ .

f.---bl=-

•

z/_

/

.,/

16,

,.,,, //

//-/ ,(;'I/

.

"

,"--,

n

//

/

,:,J/,'/

"11,'/

--- ,-" o,,',<,.-,,,<'., _.o, o.<,<.,-+.,o._

_llll'l #l/l]

6-

/

,Jl,,i; Bm

0 qb4/Dt 4

FIGURE

B9-18. FOR

MAXIMUM

RECTANGULAI_

STRESS PLATE

AT WITIt

CENTER CLAMPED

OF

LONG EDGES

EDGE

Section

B9

15 September Page

1971

62

12

_-

|

•,

/(o,)c.

. A1 A

,,$

__/o

I°v

,.

A

(ox) - (or)

//"

_/-

o_

TENSION

_

O

,

100

Io,(),-

200

300

w4/Et 4

FIGURE

B9-19.

MEMBRANE UNIFORMLY

STRESSES LOADED

IN SQUARE

(°Y)A,

PLATE,

Section 15 Page

EXTREME-FIBER IMMOVABLE

BENDING EDGES/4-

B9

September

1971

63

STRESSES 0.316

i

' F

(O')B = (°')C

0 0

100

3O0

200 qa4 / Et 4

FIGURE

B9-20.

BENDING UNIFORMLY

STRESSES LOADED

IN

SQUARE

PLATE,

Section

B9

15 September

B9.5

ORTHOTROPIC

B9.5.1

Rectangular The

plate

Reference

Page

65

of an

orthotropic

PLATES

Plate

deflection

rectangular

1971

and can

be

the

bending

moments

calculated

from

the

at

the

center

following

equations

obtained

from

1.

q0 b4

w--

,

D

(37)

Y

(38) Mx

where

=

a,

ill,

e

-

fll+fl2"yT'T ,qDylE x

and_2

b_

are

numerical

coefficients

given

in

Table

B9-24

and

(40)

D X

The

(39)

are

of plane

four

needed

B9.2.1.1.

and

subject

geometry

E'

to charactcrize

stress.

Section

are

constants

These

four

Equations

to modifications

of the

E'

x'

y'

the

E"

elastic

constants (41)

'

are through

according

and

G

in equations

properties defined (44)

to the

of a material by

are

(37),

equations expressions

nature

of

the

(8) for

material

(38),

and

in the

case

of rigidities

and

the

stiffening.

EVxh3 Dx

-

Dy

--

12

(41)

E ' h :_ 12

(42)

Section

B9

15 September Page

1971

66

El, h3 D1

Dx'y

be regarded

Gh 3 12

-

Table

All

(43)

12

B9-24. Constants a, /3 l, and f12 for A Simply Rectangular Orthotropic Plate with H = _/D

values

a

1.0

0. 004 07

0. 0368

O. 0368

I.i

0. 004'38

0. O35 9

O. 0447

1.2

0.00565

0. 0344

O. 0524

1.3

0. 00639

0. 0324

O. 0597

1.4

0. 00709

0. 0303

O. 0665

1.5

0.00772

0. 0280

O. 0728

1.6

0. OO831

0. 0257

O. 0785

1.7

0.0O884

0. 0235

O. 0837

1.8

0.00932

0.0214

O. 0884

1.9

0.00974

0. 0191

O. 0929

2.0

0. 01013

0. 0174

O. 0964

2.5

O. 01150

0. 0099

O. 1100

3.0

O. 01223

0. 0055

O. 1172

4.0

O. 01282

0. 0015

O. 1230

5.0

O. 01297

0. 0004

O. 1245

O. 01302

0.0

O. 125 0

of rigidities

values.

interest

are

given

based

approximation

Usual below.

values

of the

91

Supported D xy

e

as a first

reliable

(44)

P2

on purely

theoretical

and

are

tests

rigidities

for

considerations

recommended three

cases

to obtain of practical

should more

Section

B9

15 September Page 1.

sider

in

a plate

Fig.

v ,

Plate

E'

with

reinforced

B9-21. the

elastic

stated

D

-

Equidistant

middle

and axis

I

(45)

respect

of the the

of the

by equations

Stiffeners

with

constants

modulus,

to the

are

By

symmetrically

The

Young's

respect

values

Reinforced

cross

and

to

material

moment

In

One

its

of the

Con-

plane

as

are

E

plating

of inertia

section

Direction:

middle

of the

1971

67

shown

and

of a stiffener, plate.

The

taken rigidity

(46):

Eh 3

x

=

12(1Eh 3

Dy

-

tt

(45)

E' I

12(1 - _2) +

al

,

(46)

0

!

I

I I

FIGURE

B9-21.

2. the

Plate

reinforcement

OIITHOTIIOPIC

Cross-Stiffened to

remain

PLATE

By Two s.vmm(.'trieal

WITII

Sets about

EQUIDISTANT

Of Equidistant the

plating.

STIFFENERS

Stiffeners: The

moment

Ass/_me of

"

Section

B9

15 September Page inertia

of one

stiffener

x-direction. and

is

The corresponding

a1 .

The

rigidity

b I is the spacing

I I , and

values

values

for this

for case

of the

stiffeners are

1971

68

stiffeners

in the

in the y-direction

stated

by equations

are

(47),

12

(48),

and (49)D

x

12(1

D

-

Eh3 E'It. - p2) + bl

(47) ,

Eha E__ 12(Y--- v 2) + a 1

y

(48) '

Eh a H

12(1-

3. and

let

a t , and (51),

and

D

= x

D

DI effect

The

torsional

=

D' xy rigidity

I the moment

the rigidities

are

Refer

to Fig.

of inertia

expressed

B9-22

of a T-section

by equations

'

(50)

,

(51)

•

(52)

transverse

rigidity

xy in which

Then,

Eath3 12(a 1 - t + _3t)

of the

D

of the material,

Ribs:

(52):

= 0

The

(49)

By A Set Of Equidistant

(_ = h/H.

"- E I a!

y

torsional

Reinforced

E be the modulus

of width (50),

Plate

"

vz)

D' xy is the

contraction

may

be calculated

+--c 2a t torsional

of one

rib.

is neglected by means

in the

foregoing

of equation

equations.

(53):

(53)

,

rigidity

of the

plate

without

the

ribs

and

C the -_

Section

B9

15 September Page

1971

69

I _--t

i

I

I

I

I I

I I

I I I

i I I

I I

FIGURE

B9-22.

Formulas

stiffening

can

ORTttOTROPIC

for

be

found

the

in

elastic

Reference

PLATE

constants

6.

WITH

of plates

STIFFENERS

with

integral

ON

ONE

waffle-like

SIDE

Section B9 15 September 1971 Page 71

STRUCTURAL SANDWICttPLATES Small The from

information

Reference

facings

effects

to a thick

core.

elements of core

shear

conditions

as

core

deflection

theory

was

obtained

so

shear

The

core

wieh

be

deflection,

thick

have great

that

is

of the

facings

bonding

procedures

is

buckling,

for

the

can

stress

be

thin

sandwich

inclusion

and

a sandwich

two

of the for

the

summarized

in-

least

and

sandwich

high

wrinkling

have

tmtl('r

enough

sufficient

chosen

flatwise

shear

excessive

design

moduli

of either

to withstand

loads.

bucMing,

occur

enough

enough

ultimate

enough

not

thick

of elasticity

facing

will

loads.

and

and

and

deflection,

ultimate

tensile

rigidity

the

sand-

compressive

not

occur

under

loads. made

is

small

will

at

design

will

ultimate core

be

overall

hav(.' so

be

by

material

for

under

shall

If the

spaces

on

failure

strelNth

in design

of homogeneous

shall

that

shall

design

difference

formed

follows:

shall

and

shall

those

stresses

The

main

facings

strength

4.

small

composite

designprinciph's

Sandwich

design

3.

the

a layered

properties

basic

2.

is The

and

The

1.

for

7.

sandwich

sandwich.

Theory

presented

Structural

structural

to four

Deflection

not

enough

not

occur

of cellular

or

permissible,

so

that

under

corrugated

the

dimpling

design

cell

material

size

or

of either

ultimate

loads.

and

dimpling

corrugation

facing

into

spacing

the

core

Section

B9

15 September B9.6.1.1

Basic

Principles

Uniformly Assuming and

a given

construction with

the

basic

design

ties,

for

formulas

are

materials facing

and of the

and

stress

tion

of use;

erties

same

one

is,

B9.6.1.2

The

thicknesses allowable by bending

panel

following

core

formula

thickness.

shall

of sandwich

be designed

be used

are

Thickness,

Simply

Supported

procedures modulus will maximum

to comply

core

with

isotropic

facings

for

sandwich

with

moduli

For

Thickness

Rectangular

many

of different isotropic El,2,

prop-

combinations

elastic

, and

condi-

facing

such

Core

that behavior. Shear

Panels sandwich

design The

Double

at the

then

to linear

chosen

proper-

each

values

restricted

determining

deter-

of elasticity,

thicknesses

center

for

paragraphs.

to choose

not be exceeded. at the

as necessary

in design.

Core

so that

graphs

temperature,

Flat for

and

or tension

is at elevated

Facing

shear

are

and deflections

panel

following

Facing

be compression

procedures

gives

deflections

moment,

another

and

in the

sandwich

be advantageous

section and

for

it will

for

load

as well

given

shall

Modulus

stresses

formulas

core,

temperature

Determining

This

normal

are

if application

materials

Elt 1 = E2t 2 .

and

F1, 2, shall

that

and

formula

material

values,

design or circular

theoretical

panels

thicknesses

at elevated

of facing

facings

supported

given,

chosen

rectangular

giving

of the

simply

Load

with

distributed

Page 72 Panels under

Sandwich

principles.

procedures

dimensions

Normal

a flat

uniformly

of Flat

begins

to transmit,

Detailed mining

a design

under four

Design

Distributed

that

load

for

1971

facing

of a simply

facing

facing

and core

stresses

stresses, supported

and produced panel

Section B9 15 September 1971 Page 73 under uniformly distributed normal load. If restraint exists at panel edges, a redistribution of stresses may cause higher stresses near panel edges. The procedures given apply only to panels with simply supported edges. Because facing stresses are causedby bending moment, they dependnot only upon facing thickness but also upon the distance the facings are spaced, hence core thickness. Panel stiffness, hencedeflection, is core

dependent

upon

facing

and

thickness. If the

levels, core

the

panel

panel

or facings

deflection which,

is designed deflection

must

by iterative

so that may

and the

A solution

process,

the

facing

be larger

be thickened

requirements.

modulus

stresses

than

allow_qble,

design

facing

is presented

facing

and

are

core

in the

at chosen in which

design case

stress

lowered

form

of charts

thicknesses

and

core

the to meet with shear

can be determined. The

the

also

average

theoretical

facing

stress,

F(st_'ess

at facingcentroid)

, is given

by

formulas:

= K p_b! 2 ht 1,

(for

unequal

FI, 2

F

_b2 K 2 ht

(for

equal

f:,cings)

,

(54)

and

=

facings)

,

(55)

Section

B9

15 September Page 74

where p the

is the

distance

intensity

between

subscripts

denoting

dependent

on panel

If the

core

facing

orthotropic

also

upon

sandwich

v2D V - b2 U

can

be written

C

EltlE2t

sandwich

shear

panel

thickness;

bending

alike

in the

ratio.

dependent

and

is the

not only

h 2

is

are

coefficient

and

shear

rigidities.

two principal The

rigidities

width;

1 and

K 2 is a theoretical

aspect

bending

which

values

directions), of

on panel

K 2 for

aspect

ratio

as incorporated

in the

sandbut param-

as:

2

= hb2Gc(Eltl+

v2t

panel

b

is facing

2; and

and

are

load; t

moduli

upon core

_2t V

ratio (shear

only

with

1 and

aspect

depend

wich

centroids;

facings

is isotropic

K 2 values

eter

of the distributed

1971

E2t2 )

(56)

Et C

V

-

2),b2G

(for

equal

facings)

,

(57)

c where

U

X = 1 -/_2

is sandwich ; #

# = #2 = #2]; to panel shear

side

modulus

G

Solving

c

ratio

is the

of length associated

to the

h _

stiffness;

is Poissonts and

pendicular

shear

plane

equations

a

core and with

is modulus

of facings shear

axes

and

[in

(55)

for

plane

to panel

is denoted

by h/b

(56)

associated

to the

parallel

of elasticity

formula

modulus

perpendicular

of the panel (54)

E

side

(RGc)

it is assumed

with of the

of facing;

axes

parallel The core

panel.

of width

that

b

and per-

.

gives

(55)

Section 15

B9

September

Page

1971

75

h (for

b - 4-E2

A chart

Fig.

B9-23.

unknown, The

facings)

(59)

Z

for

The

but

equal

solving

formulas

by

formulas

_'md charts

iteration

deflection,

(58)

satisfactory 6 , of the

and

(59)

include

ratios panel

graphically

the ratio

of

center

t/h is

given

t/h , which

and

given

is

h/b by

the

can

in

is usually

be

found.

theoretical

fo rmul a:

KI

XFI,2

_l

E,,2t,,2_b 2 (6O

K1

XF

b 2

K2

E

h

5

where of

K 1 is

a coefficient

Solving

equations

(for

dependent

upon

panel

equal

aspect

(61)

facings)

ratio

and

the

value

V.

h

(60)

/-_1 _-ITl'2Jl+El'2tl'2 _] _22_1 F-_i,2

and

(61)

for

h/b

gives

E,,,t2,1 (62)

b

(for

equal

facings)

(63)

Section

B9

15 September Page

P/F1,2

/P

1971

76

x

I r'/_

q<,_ o:_I.--

0.6

0.8

1.0

b/=

0.02

0.04

O.OI

0.10

FIGURE

B9-23.

CtL_RT

FOR

DETERMINING

RECTANGULAR SANDWICH PANEL, UNIFORMLY DISTRIBUTED FACING

STRESS

h/b

WITLI ISOTROPIC NORMAL LOAD WILL

BE

Fls

2

RATIO

FOR

FACINGS, SO THAT

FLAT UNDER

Section

B9

15 September Charts B9-25,

and

for

solving

B9-26.

Use

formulas of the

(62)

and

equations

(63)

are

and charts

Page 77 in Figs.

given

beyond

5/h

1971

B9-24,

= 0.5

is

not recommended. B9.6.1.3

Use The

enable

of Design

sandwich

rapid

Charts

must

determination

be designed of the

by iterative

various

procedures;

quantities

sought.

these The

charts

charts

were

derived for a Poisson's ratio of the facings of 0.3 and can be used with small error for facings having other values of Poisson's As a first approximation,

ratio.

it will be assumed

that V = 0 . If the design

is controlled by facing stress criteria, as may be determined,

this assumption

will lead to an exact value of h if the core is isotropic, to a minimum of h if the core is orthotropic with a greater core shear modulus

value

across the

panel width than across the length, and to too large a value of h if the core is orthotropic with a smaller core shear modulus

across the panel width than

across the length. If the design is controlled by deflection requirements, assumption that V = 0 will produce a minimum is rrnnimum because

value of h . The value of h

V = 0 if the core shear modulus

actu'flcore, the shear modulus

tL, e

is infinite. For any

is not infinite;hence a thicker core must be

used.

P/F1,2,

The

following

1.

Enter using

the

procedure

Fig. curve

B9-23 for

is suggested: with

desired

V = 0 .

values

Assume

for the

a value

for

parameters t_,2/h

b/a

and

and determine

Section

]39

15 September Page

1971

78

V 1.0

0,4 0,2

Gc

Gc

I

1 0.2

0.4

0.6

0.8

1.0

b/a

0.0100 0.0040 0.0020 0.0010 0.0006 iN

uJ 6.0004

0.0002

0.10

0.08

0.06

0.04

0.02

0

h/b

FIGURE

B9-24.

CIIAWI'

RECTANGULAR ISOTROPIC

SANDWICII CORE,

UN1)EF_

PRODUCING

POll

I)I':'I'I,:ILMINING 1)ANI.:I,,

IJNIF()II_II,Y I)EFI,I':CTI()N

WITII

h/b

RATIO

ISOTROPIC

DISTIiI13UTED IiA'HO

5/h

FOR

FLAT

FACINGS

AND

NORMAL

LOAD

Section

B9

15 September Page 1

0

0.2

0.4

E2t 2 / Elt 1 2

0.6

3

0.8

b/a

79

1.0

0.0100 0.0040

V = "/r2D/ b2U

0.0020 0.0010

¢_

. _ _..___

_
i

/

\ \

/

0.08

o, ,.oo_ .oo,

\

/ -- o/ooo, 0.10

_/h

\

j

oooo_ /

:.o._ o,

_'-_

0.06

0.04

\oo_

0.02

0

h/b FIGURE

B9-25.

RECTANGU

CIIAllT

I"OR

1JAR SANDWICI[

ORTHOTI_OPIC NOI{MAI,

(SI'F

SKI,'TCII)

I,OAD

I/I']TEIIMINING

PANI':IJ,

WITII

C()RI,:,

UNDER

PI{OI)UCING

h/b

RATIO

ISOTROPIC UNIFORMLY

I)I,'I,'Id2C'IION

RATIO

FOR

FACINGS

FLAT AND

DISTRIBUTED 6/h

1971

Section

B9

15 September E2t2 / E1 tl

1

2

3

Page

4

1971

80

/'/_:/ V -- 1.0

_o2.5_o I b+ ORTHOTROPIC 0

0.2

0.4

0.6

0.8

CORE

1.0

b/a

0.0100

V= 7r2Dl

0.0040 0.0020 0.4

0.0010

g,I

0.0002

J

/

/

\ \

jr 0.10

0.08

0.06

0.04

_o, . o.oe

\

-v 1

0.02

0.06

0

h/b FIGURE

B9-26.

RECTANGULAR

CttART SANDWICtt

FOR

DETERMINING PANEL,

ORTHOTROPIC (SEE SKETCH) CORE, NORMAL LOAD PRODUCING

WITH

h/b

RATIO

ISOTROPIC

U NDER UNIFORMLY DEI.'LECTION RATIO

FOR FACINGS

FLAT AND

DISTRIBUTED 5/h

Section

B9

15 September Page h/b.

Compute

more

suitable 2.

E2tz/Elt and

and

values Enter

for

suitable

Compute

with

if necessary

•

desired

h

values for

for the

V:

0.

and

5 .

Modify

h

and

6.

for

2 using

2 is equal core

tj,2/h

the curve

1 and

the

ratio

tl, 2

values

steps

by step

4.

and

Compute

Repeat

determined

Modify

2 , using

h/b.

more

h

B9-24

kF_/E

determine

3.

t_, 2 .

Fig.

1 , and

determine

h

h

to,

1971

81 and determine

lower

thickzmss,

t

Assume

ratio

chosen

or a bit less

than,

,

of

5/h

if necessary

facing

h

the

b/a

a value

6/h

design

, using

e

parameters

and

stresses

determined

following

until

by step

1.

formulas:

t1+ t2 tc

= h

tc

:

h - t

This ulus.

entering puted

first

Since

somewhat

(64)

2

(for

approximation

actual

larger

core must

Figs.

B9-23

by equations

(56)

and certain

with

isotropic

core

shear

modulus

to mndwich

which

with

shear

and

isotropic

for

the

was

(R shear

associated orthotropic

facings)

basc_l

modulus

be used.

with

cores

equal

modulus the

cores

or

B9-26

B9-23

for

B9-25

large,

with

the

can vMues

includes

applies

length which

with

l.'i,_ure

associated panel

an infinite

are not very

cores,

Figure

with

approximations

Figure

orthotropic

with

values

B9-25,

and (57).

= 1) ,

on a core

Successive

B9-24,

(65)

be made of

applies

to sandwich the

panel

(R = 0.4) shear

a value

curves

B9-24

modulus

Figure

modof

t

c

by

V

as com-

for

sandwich to sandwich

with

width .

shear

orthotropic

is 0.4 of the B9-26

associated

applies with

Section

B9

15 September Page the

panel

_idth

(n : 2.

is

Gc = GTI,

honeycomb the

tile

For

honeycomb

and

the

cores

shear

wilh

modulus In

iter:)te

times

shear

modulus

associated

with

82 the

panel

length

. NOTE:

a ,

2.5

1971

using

shear

parallel

V

parallel

t() p:mel

length

through

directly

core

ribbon

parallel

to panel

width

parallel

ribbons

B9-23 is

with

modulus

core

Figs.

because

cores

to panel a

B9-26

is

GTL

for

prOlx)rti()lml

width

b

b,

is

length

GTW

.

Ge = GTW

For

and

.

V ¢ 0,

to the

to panel

it is

core

necessary

to

thickness

t

As

an

e

aid

10 tinally

(tt_termine

',

and

G

(:

representing

G

C

V

ranging

for

from

variou,_

1000

1.

Determine

2.

Compute

Io

values

1 000

a core the

,

Fig.

of

G

B9-27

presents

a number

of lines

c

000

psi.

thicMless

constant

with

c

V

rmlging

The

folh)wing

using

a value

relating

V

to

G

from

procedure of

0.01

0.01

is for

to

2 and

suggested:

V .

: e

7r2tcE1tlE2t'kb2(Eltl + l'i2t,:)

.

With

this

l

or

constant,

_ 7r_tcI':t) \ ,,_b2

enter

( for

Fig.

B9-27

equal

and

facings)

=

determine

VG

c

necessary

G C

4.

If the

shear

available,

follow

the

V , for

reasomLble

value

5.

Reenter

Figs.

all

previous

steps.

repeat

modulus

appropriate of

core

B9-23

is outside

line shear

through

of

the

Fig.

range

of values

B9-27

and

pick

the

new

for

a new

materials

value

of

modulus.

B9--26

with

value

of

V

and

Section B9 15 September 1971 Page 83

25 < (D 25 < 0

N _ m 25

O Z M

Z Z

_ N

_

O

d

_

!

Section

B9

15 September Page B9.6.1.4

Determining This

shear

section

stress

Core gives

of a fiat

normal

load.

The

length

of each

edge.

Shear

core

Stress

the procedure

rectangular shear

The

1971

84

for determining

sandwich stress

maximum

panel

under

is maximum shear

the

uniformly

at the

stress,

F

maximum

panel

distributed

edges,

, is given

cs

core

at mid-

by the

formula:

F cs where the

=

b K._p _

K_ is a theoretical parameter

shear

V .

coefficient

If the

core

dependent

is isotropic,

upon

panel

values

of

aspect

V

ratio

and

do not affect

the

core

stress. The

The

chart

chart

of Fig.

should

determined.

B9.6.1.5

Checking The

design

B9-31

facing B9.6.1.6

thicknesses allowable

shall

section

panel

solution

of thicknesses

bc checked

by using

theoretical

coefficients

deflection,

and

a graphical

with values

and

Determining Facing Modulus for Simply This

presents

of formula

and other

(66).

parameters

Procedure

to determine

stresses,

B9-28

be entered

previously

and

(66)

core

gives shear

deflections

core

Thickness, Supported

procedures modulus will

shear

for

the graphs

of Figs.

K 2 , K_ , and

B9-29,

B9-30,

K 3 to compute

stresses. Core Thickness, and Core Fiat Circular Panels

Shear

determining

and core

so that

chosen

not be exceeded.

sandwich design The

facing

facing

facing

stresses

stresses,

and produced r

Section

B9

15 September Page

1971

85

hlb 0.10

0.05

0.03

0.02

0.016

0.014

0.012

[/i 0

0.2

0.,

bla

0.010

,

0.6

0.8

I

I

1.0

II

3O

[

441

1

t

5O

t

V - _?'D I b2U

I t

;

f I

FIGURE

B9-28.

CItART

FOI_

DI':'['I':RMINING

CORE

,

SIIEAR

I 1

,

, ,

STRESS

F

RATIO ISOTROPIC

s__.._cFOR P FACINGS,

FLAT UNDER

RECTANf;ULAR UNII"ORMLY

SANDWICH I)ISTRIBUTED

PANEL,

WITIt

NORMAL

LOAD

Section B9 15 September 1971 Page 86

0.14

[

VI m

0.12

0.10

0.08 N ,,.,.

0.06

0.04

0.02

0 0

0.2

0.4

0.6

0.8

1.0

b/o

FIGURE B9-29. RECTANGULAR ISOTROPIC

K 2 FOR DETERMINING SANDWICH PANELS

FACING STRESS_ WITH ISOTROPIC

OR ORTHOTROPIC CORE (SEE SKETCH) UNIFORMLY DISTRIBUTED NORMAL LOAD

F, OF FACINGS UNDER

FLAT AND

Section B9 15 September 1971 Page 87

\

0.04

\

R "0.4

0.03

\

0.02 1.0 0.01

_ ""-__ ._

o.4 o.2 I

_o 0

I 0

0.2

0.4

0.6

b/a

0.8

_

mm_

q,-

,,,

,!')

1.0

0.02 n

I

[

R-2s

0.01

-'_

0 0

0.2

0.4

0.6

0.8

b/a

0.03

v

I

0.2

V ,, _12D/ b2U

1.0 i

6" K 1 11+llE1,2t ,2/E2,1t2,1 ) pb4/h2E1,2t1,2 _,_"

0.02

1.0

0.01

_0

o

0.2

I 0

0.2

0.4

0.6

0.8

1.0

b/a FIGURE FLAT AND

B9-30. K 1 FOR RECTANGULAR

DETI"RMINING MAXIMUM SANDWICIt PANELS WITtt

ISOTROPIC OR ORTHOTROPIC UNIFORMLY DISTRIBUTED

CORE (SEE NORMAL

DEFLECTION, ISOTROPIC SKETCtl) LOAD

5, OF FACINGS UNDER

Section

B9

15 September Page

0.5

! m

Immmmmd

m

="-" 0.4 mmm-,m-,-_

m

m

_

"--'-

"-"-_

_

]

I

ALSO ANY V FOR

--.

t

I R = 1

' :!

_

=0.

__m

"--"

0.3

__

1971

88

2R _

"_

0.4"

_'_1.0

R = 2.5

FOR SHEAR AT EDGE OF LENGTH b

o,

If

0

0.2

0.4

0.6

0.8

1.0

bla

V = _2D / b2U

Gc

c

b

0.5

0.4 1.0 | 0.4 IR = 0 0.21

,,e

.5

i

°:,'1_=o,

0.3

1.0 / 1 ' 1 ! I "-'-ALSO ANY V FOR R = 1 1 I 1 I 1.0

0.2 0

0.2

0.4

0.6

0.8

b/a

FIGURE Fsc,

B9-31. FOR

FLAT

FACINGS

AND UNDER

K 3 FOR

DETERMINING

RECTANGULAR ISOTROPIC UNIFORMLY

MAXIMUM SANDWICH

OR

ORTHOTROPIC DISTRIBUTED

CORE

PANELS CORE NORMAL

SHEAR

WITH

STRESS,

ISOTROPIC

(SEE SKETCII) LOAD

Section

B9

15 September by bending panel

under

edges, The pic

moment, uniformly

distributed

and

which,

modulus

apply

isotropic

The

of a simply

load.

If restraint

only

may

cause

to panels

higher

with

A solution

process,

the

facing

stresses

simply

core

in the

thicknesses

circular

at panel

near

supported

is presented and

exists

1971

panel

edges, form and

edges. is otro-

of charts core

shear

be determined. average

theoretical

center

normal

cores.

by iterative

can

at the

of stresses

given

facings,

facing

stress,

F (stress

at facing

centroid),

is given

by the

formula:

FI'2 -

F =

where

maximum

a redistribution procedures

with

are

Page 89 supported

3 + __ pr 2 1G tl,2h

3+_ 16

pr 2 th

p is Poisson's

previously

defined Solving

(for

ratio

P = Pl = _2] ; r is the

radius

(see

equations

(67)

cqual

of facings

facings)

[in formula

of the circular

Section (67)

(68)

panel;

(67), and

it is assunmd other

quantitics

that are

as

B9.6.1.2). and

(68)

for

h -- gives r

h

(69)

r

h (for r

equal

facings)

(7o)

Section

B9

15 September Page A chart The

for

solving

formulas

and

iteration

formulas chart

satisfactory

(69)

include

ratios

and

the

of t/h

(70)

ratio

and

¢_.

graphically t/h,

h/r

¢_.

is

which

can

be

¢_.

is

given

usually

1971

90 in Fig.

B9-32.

unknown,

but

found.

¢_.

_.

¢_.

0.00280 i 0.00260

0.0o240 0.00220 i 0.00200

¢b'_'_¢>

0.OO180 0.00160

0 "_

0.00140 0.00120

(b'_

O.OO1OO O.OO090 0.00080 0.00070 0.0O060

O_e, •

0.00050 O.OO040

0 .00_"

0.00030

0.0oo2o

a

0.00010 0.00005 0.00001 0.04

0.08

0.12

0.16

0.20

h/r

FIGURE

B9-32.

CIRCULAR CORE,

CHART SANDWICH

UNDER THAT

FOR

DETERMINING

PANEL,

UNIFORMLY FACING

STRESS

WITH

h/r

ISOTROPIC

DISTRIBUTED WILL

BE

RATIO

NORMAL Fl,

FOR

FACINGS

2;/_ = 0.3

LOAD

FLAT AND SO

by

Section B9 15 September 1971 Page 91 The deflection, 5, of the panel center is given by the theoretical formula:

where

5=

K4l1 + E1'2tl'2 _ XFI' m-- 2r2 E2,it2, t / El,2h

5 =

2K 4

r 2 h

kF E

K 4 depends

in the parameter

(for

on

the

V

=

equal

facings)

sandwich _2D _ _/,rJ-u

7r2t EltlE2t

(71)

,

bending

which

and

(72)

shear

rigidities

can be written

as

incorporated

as

2

c

(73) V

=

4Xr2Gc(Elt

I +

E2t2 )

7r2t Et C

V

-

8Xr-_G

(for

equal

facings)

,

(74)

C

where

r is

panel

radius

and

all

other

terms

are

as

previously

defined

in Section

B9.6.1.2 h Solving

equations

(71)

f XFI'2

h

- =

A chart

equations

J

J

for--,

gives

r

El'2ti'2 E2' It2'I

hF -g"

=

(for

for solving

and

(72)

Jh

h --

and

charts

formulas

beyond

equal

(75) and

5/h

=

0.5

facings)

(76)

.

is given in Fig.

is not

recommended.

(76)

B9-33.

Use

of the

Section B9 15 September 1971 Page 92 E2t2/Elt 0.50

1

>:::

2

/

1 3 ,

4

i ,!.,,/

=o

v : 2D/4,2u

0.20

._

0.10

0.0010

/

0.06

o._i 0.24

0.20

o.. 0.16

0.12

0.08

0.04

0

h/r FIGURE

,,o

B9-33.

CIRCULAR

CHART SANDWICH

UNDER

FOR I WITH

UNIFORMLY

PRODUCING

DETERMINING

CENTI:,R

ISOTROPIC

DISTRIBUTED DEFLECTION

h/r

RATIO

FOR

FACINGS

AND

NORMAL

LOAD

RATIO

5/h

FLAT

CORE,

Section

B9

15 September Page B9.6.1.7

Use The

sandwich

enable

rapid

derived

for

error

for

of Design

will ments,

lead

s ratio

by facing

value

assumption

that

because

actual

shear

the

V = 0 if the modulus

as may

that

V = 0.

were small

this

design

assumption

by deflection

a minimum

is not infinite;

with

If the

be determined,

shear

charts

charts

be used

is controlled

produce core

can

The

the

ratio.

be assumed

If the design

V = 0 will

and

of Poisson's

criteria,

and

sought.

of 0.3

it will

of h.

procedures

quantities

facings

values

stress

of h is minimum core,

of the

other

by iterative

various

approximation,

to an exact the

of the

having

As a first is controlled

be designed

determination

facings

93

Charts

must

Poisson'

1971

value

modulus

of h.

The

is infinite.

hence

a thicker

value

for

core

require-

For must

value any be

used. The .

following Enter

procedure

Fig.

is suggested:

B9-28

with

the

desired

the

P

parameter

F1,2 Assume

a value

for

tl'2 h

and determine

h/r.

Compute

h and t I 2.

tl,2 Modify for

ratio

h and

_

if necessary

and

determine

more

suitable

values

tl, 2E2t2

.

Enter

Fig.

B9-33

with

desire(]

values

of the

parameters

and

XF 2 E2

and assume

Compute more

h and

suitable

V = 0. 5. values

Assume

Modify

ratio

for

h and

a value 6/h 5.

for

if necessary

5/h

and

determine

and

determine

h/r.

Section

B9

15 September

1971 j

Page Repeat

1

steps

until

by step

Compute t

c

t

modulus.

the

core

-

h+

= h

-

t

of t

somewhat

C

by entering

(for

directly

B9-33

proportional

t C and Gc,

Fig.

be computed

equal

shear

this

in Section

B9.6.1.8

shear

stress

the

following

based

on a core

values

values

are

Successive of V as

with

h deter-

formulas:

a value

can

be made

by equations

for

V ¢ 0 it is necessary

to iterate

to the

core

thickness

t .

As an aid

can

again

be used.

The

constant

shear

large,

approximations

computed

e

an infinite

not very

B9-33

Fig.

(73) because

to finally relating

and

section of a flat

B9-27

may

] or

(_2tcEt \_/

be entered.

(for

/

Use

equal

of the

Core

gives

Shear

determine V to G C may

facings)

figure

is as

Stress

the procedure

circular

sandwich

for determining panel

under

the uniformly

(74).

V is

B9.6.1.3.

Determining This

than,

the formula

constant,

described

using

modulus

_r2tcEltlE2t2 VCc=t/ 4)'r2(--'_lt/ + E2t2)

With

or a bit less

stresses,

facings)

be used.

with

B9-27

from

tc,

was

must

Fig.

to,

facing

t2

core

larger

In using

design

2

actual

Fig.

2 is equal

thickness

approximation

Since

chosen

1.

= h

first

lower

by step

C

This

2, using

h determined

mined 1

1 and

94

maximum distributed

core

Section B9 15 September 1971 Page 95 normal load, The core shear stress is maximum at the panel edge. The maximum shear stress, F as' F

= pr 2-_

cs

B9.6.1.9

(77)

Checking The

equation

design

(67)

in equation

Procedure shall

and

the

(71)

which

reducesK

puted

using

deflection

is given

equation

by computing

using

equation

the

facing

(71).

The

stresses value

using

of K 4 to be used

by

[ (5 + +.)_2 [64(1 .) + V ]

4= 0.309+

,

0.491Vwhen_=

0.3.

(78) Values

of V may

becom-

(73).

An alternate by the

be checked

n2(3 16+ _)

K4-

given

is given by the formula

method

for

computing

the

deflection

at the panel

center

is

formula

6 =

Ks(1

8 =

2K 5

+

El'2tl'2) E2, lt2, 1

(79)

_

xpr4

(80)

where

(5 K5 = which

reduces The

a core

shear

+ _)_2

64(1

+ V

+ _)

toK 5 = 0.629 core

selected

modulus

for

value,

+ the

V when panel

should

G , at least c

_ =

0.3. be checked

as high as

that

to be sure assumed

that

it has

in computing

Section

B9

15 September the deflection witlmtand

in equation

(71)

the maximum

B9.6.2

Large Most

core

and that shear

Deflection

stress

classifies

deflection-to-plate

thickness

ratio

B9-36,

a small

difference

theory

for

deflection-to-plate

B9.6.2.1

Rectangular The

lowing edge

curves

Sandwich

corresponding conditions W O

Center

B9-34

Thickness

a,b

Half

than

Elastic

P

External

of panel

are

obtained

t

Thickness

Q

12a3(1

not included):

in x and y directions

of the per

of the -

(77).

unit face

v 2) p/th2E

face core area layer

layers layer

as having

In Figs.

B9-34,

the linear

Edge

from

layer

of the

ratio load

equation

to

a B9-35,

and nonlinear

0.50.

for a rectangular

core

constant

PoissonVs

Page 96 is sufficient

theory

between

with Fixed

a/b E

from

deflection

of plate

of the

length

less

were

deformations deflection

h

strength

than 0.50.

is noted

Plate

nomenclature

(shear

shear

calculated

large

greater

ratios

of Fig.

core

Theory

of the literature

and B9-37,

the

1971

Conditions

Reference sandwich

8 with plate

the fol-

with

fixed

Section

B9

15 September Page

97 O

II

II

II

I t_

¢q

q

I

I/1

0

(N

0

q/OM

NOIJ.:3g'h:lgO

1971

Section

B9

15 September Page

II

LgJ

98

1971

Section B9 15 September 1971 Page 99

Section

B9

15 September Pa ge

1971

100

m 0

H

0

,-1

,-1 ,,_ b" 0 0

N

Z © (9

N r_

Z

!

I

I

I

! 0

Section

B9

15 September Page

1971

101

Y

"b •

t

i

b

f

f

"I//Z,

B9.6.2.2

Reference and

///,,5,

Circular Movable, The

curves 9 for

boundary 1.

3.

Sandwich Plate with Simply Supported Movable, and Clamped Immovable Boundary Conditions

of Figs. a circular

B9-35,

B9-36,

sandwich

plate

and for

B9-37

the

were

obtained

following

states

Clamped

from of loading

conditions: Moments radially

2.

,_k

uniformly

distributed

movable

Uniformly

loaded

boundary,

and

Uniformly

loaded

around

a simply

supported,

boundary, plate

with

a clamped,

radially

movable

plate

with

a clamped,

radially

immovable

boundary. The shear H = shear

equations

are

deformation 0, then distortion

shear

nondimensionalizcd is characterized deformation in the

core.

for by the is neglected;

Nomenclature

each

state

of loading.

nondimensional a nonzero of the

The

parameter value

symbols

effect H.

of H signifies is as follows:

of If

Section

B9

15 September Page W o

t

Normal

deflection

Thickness

at the plate

center

of the core

C

H

Measure

M O

R

Radius

C D q

shear

deformation

moment

of a circular

In-plane

rigidity

=

Bending

rigidity =

plate

2Eftf2 Eftftc/2(I -

v/)

PoissonVs ratio of face sheet

B

Transverse

G C

tf

edge

of core

Applied, transverse load

vf

Ef

Applied

of effect

shear rigidity =

Shear modulus Modulus Thickness

G

t cc

of the core

of elasticity of the face

of the face sheet

sheet

=

D

102

1971

Section

B9

15 September Page

1971

103

REFERENCES

.

o

.

Timoshenko, S. and Shells. McGraw-Hill, Den

Hart,g,

.

1952.

Bruhn,

F. : Analysis

E.

Offset

Iyengar,

K.

Uniform

Normal

T.

February

1967.

Defense,

.

Company,

Design

of Flight

Cincinnati,

and Srinivasan, Loading.

R.

Ohio,

D.

C.,

Shell NACA

Smith, Georgia

C.

Plate

Under Society,

the Elastic Constants NACA Report 1195,

Department

of

Deflection University,

of Rectangular Cookeville,

1967.

V. : Large

Institute

Skew

Uni-

1968.

Kan, Han-Pin and Huang, Ju-Chin: Large Sandwich Plates. Tennessee Technological February

Plating Under TM 965.

Aeronautical

Composites.

July

Structures.

1965.

of the Royal

Sandwich

and

McGraw-Hill,

Vehicle

S. : Clamped

Journal

Structural

Washington,

Tennessee,

.

and

of Plates

of Materials.

F. and Hubka, R. F. : Formulas for with Integral Waffle-Like Stiffening.

MIL°HDBK-23_

.

Strength

S. : Theory

Heuhert, M. and Sommer, A. : Rectangular formly Distributed Hydrostatic Pressure.

Dow, N. of Plates 1954.

.

J. P. : Advanced

New York,

Tri-State

e

Woinowsky-Krieger, New York, 1959.

Deflections

of Technology,

of Circular Atlanta,

Sandwich

Georgia,

Plates.

October

1967.

BIBLIOGRAPHY Roark, Third

R. J. : Formulas Edition, 1954.

for

Stress

and Strain.

McGraw-Hill9

New

York,

.,._,.4"

SECTION B10 HOLES& CUTOUTS IN PLATES

-...,J

BI0

HOLES

AND

CUTOUTS

Section

BI0

15 May

1972

IN PLATES

/

The

magnitude

components The

of structures

localized

nominal

stress

stress

P

around

stress

hole,

has

at the

long

called

or stress

center

concentrations been

a hole

by a factor

the localized

small

of stress

around

an important

is usually a stress

plate

design

obtained

cT

used

factor.

resists

an axial

as

consideration. the For

at the edge

max

that

in plates

by multiplying

concentration

concentration

of a wide

holes

example,

of a relatively

tensile

load

is

arnax = K

_P

= Kao

'

{1)

or

O

! max

K -

,

(2)

(/ O

in which

stress

K

that

is the

would

stress

occur

concentration

at the

same

that

is,

in this

illustration

tile cross

the

area

which

is removed

at the

tively

large,

the

nominal

for

a given

stress.

the net stress

area

discontinuity

O

will

point

the

depend

and

if the

section

hole.

of cross

c_ , hence

factor

a

bar

area

value

is the

is frequently of the

on the

stress

method

is tile

did not contain

If the diameter

section

P = -A

o

gross

used

the hole ;

area

of the

hole

(nominal)

including is rela-

in calculating

concentration

of calculating

factor

tile nominal

Section 15

May

Page

If o

in a member

max

as found

from

method,

etc.,

stress from the

K

of the

subscript

tration

the mathematical

is given

concentration tests

factor.

is the

e,

factor. actual and

theory

the

subscript If,

matei:ial

K6

theoretical

value

of elasticity,

t,

and

Kt

other

hand,

under

the conditions

the effective

1972

2

localized

stress,

or the photoelasticity

on the

is called

of the

B 10

or

is called the value of use,

significant

the

theoretical K

of K

is found

is given

stress

concen-

Section B10 15 May 1972 Page 3 10.1

SMALL HOLES Solutions in this subsectionwill be limited to small holes in plates,

that is, holes which are relatively small in comparison to the plate size such that boundary conditions do not affect the results.

An exception to this

is the case when holes are near a free edge of a plate. 10.1.1

Unreinforced This

plates

with

rings,

Holes

paragraph

contains

no reinforcement

information around

the

on holes hole,

such

of various

shapes

as increased

in

thickness,

or doublers.

10.1.1.1

Circular The

solved

case

by many

Holes

of a circular

hole

investigators.

in an infinite

plate

in tension

stress

concentration

factor,

applies

only

The

has

been

based

on

O"

gross

area,

is

Strictly thin

relative

order,

less

Kt than

to hole the

t

-

max

speaking, to th(, hole

is somewhat 3 at the surface

diameter

surface.

K

-

(_ Kt

3.

= 3

diameter.

greater of the

of three-fourths,

When

than

3

for a plate

these

dimensions

at the

midplane

plate

[ i].

Kt

= 3.1

For

a ratio

at the

which

are

is very

of the

of the

plate

of plate

midplane

and

same

and

is

thickness 2.8

at

Section B10 15 May 1972 Page4 For

has

been

the

case

obtained

B10-1.

These The

has

been

the

section

[ 2] and

values

case

are

and

between

=

the

plate

near

the edge

are

shown

the

and

the

,

stress

applied

to semi-infinite

hole,

are

of the

plate

The has

in Fig.

results. plate

B10-2.

a solution

given

photoelastic

of a semi-infinite

edge

1-

values

with

in Fig.

crch

a transverse

Kt

agreement

results hole

with

corresponding

in good

of a hole

solved

p

of a finite-width

in tension

load

been

carried

shown

by to be

(3)

where

distance

C

r

h

-_-

radius

Although

the

of hole

to edge

of plate.

of plate.

B10-2,

maximum

center

of hole.

= thickness

In Fig.

the

from

plate.

stress

factor

the

at the

aB

upper

curve

gives

edge

of the

hole

may

be used

values

of

nearest

directly

B --a ' where

the edge

in design,

aB

is

of the plate.

it was

thought

on the

basis

{7

desirable

the

load

comparable

to compute

carried with

also

by the the

a "stress

minimum

stress

concentration

net

concentration

section. factors

factor"

This for

Kt

factor

other

will

cases;

of

then this

be is

A

Section

B10

15 May 1972 Page 5 important the

in analysis

minimum

of experimental

section,

data.

the average

stress

Based on the

on the

net

actual

section

load

A-B

carried

is:

. j c_l anetA-B

(c

-

r)

h

I

-

by

(4)

r c

Kt

=

The located

(_-

=

anet

case

hole

o°(,-I)

A-B

_ J 1 - (_'

=

of a tension

is shown

i in Fig.

is increased

o=

in Fig.

B10-3),

the

to infinity,

plate

the load

Assuming

a linear

relation

following

expression

for

of finite

B10-3.

load

load

stress

ff

on the net

net

A-B

A-B

hole

A-B

carried

by Section

A-B

the

foregoing

carried

by section

]

end

an eccentrically

is centrally

by section

_

section

the

having

carried

=

The

width

When

between the

(5)

is

is

conditions A-B

located

ach.

ach

As

#, (r)_ results

-

r)

in the

:

.

(6)

is

, h (c

ec

(7)

/

Section

B10

15 May

/

Page

/

/

1972

6

O" max

! /

Kt

/

-

o

_net

(8) It is

the

Kt

curve

seen

curves

for

the

Biaxial

given

the

together,

(e

equal

of a hole

3 o"1

_2

all

BI0-3

so

that

=

1)is,

that

for

this

all

relation

practical

under

these

brings

purposes

a infinitely

circumstances,

-

are

plate

a circular

o"2

stressed

biaxially

results

hole,

.

to

equal

(9)

_1,

but

Kt

=

2,

of opposite

when

sign,

both

Kt

=

are

of the

same

bending wide

of a plate

plate

with

(Fig.

B10-5).

the

a hole,

of

_

a

_-

are

sign.

4.

following

results

mathematical

have results

been

obtained.

have

been

For

obtained

a

of

the

Bending:

For

in terms

all

eccentricities.

in an infinite

For

in magnitude

o 1 and

II.

for

B10-4.

=

max

of Fig.

Tension:

case

in Fig. cr

When

closely

approximation

For

_2

part

centrallylocatedhole

I.

For

lower

rather

a reasonable

are

in the

shown

in Fig.

B10-6.

Values

for

finite

widths

and

various

valves

Section

Ill0

15 May

1972

Page 10.1.1.2

Elliptical

I.

Axial

The wide the

plate

(b)

Holes

Loading:

stress

distribution

subjected

stress to the

Kt

to

minor

(a)

1

located

elliptical

loading

has

width

in Fig.

(b

=

2),

are

the

results

hole

load

of the

ratio

has

in an

been

of the

infinitely

obtained, major

and

width

B10-7.

distribution

of finite

in Rcf.

3,

of a biaxially

are

of

width

and

the

and

stresses subjected

stress

around to

concentration

a centrally

uniform

axial

factors

arc

stressed

given

concentration in Fig.

concentration

infinite

sheet

plate

in ].'i_.

with

an elliptical

hole

B10-4.

a2

(11)

:

I{eetan,,mlar

in an

of the

a

given

Stress

in Fig.

axial

hole

B10-8.

Bending

10.1.1.3

_iven

in a plate

max

plate

is

a function

an elliptical

(10)

obtained

case

Stress

distributed as

solution

the

II.

with

2b _a

+

hole

been

For

uniformly factor

A photoelastic

sented

associated

concentration

=

7

values

for

an

elliptical

hole

in an

infinitely

wide

B10-9. Iloles

with

[actors in tension

l_oundc(I

for have,

Cm'ners

;tn unk'einlbrced been

cvaluate(I

rectangular

l/Otlllde(I

in llef.

4.

Variation

pre-

Section

B10

15 May 1972 Page of the

stress

radius

concentration

is shown

10.1.1.4

in Fig.

Oblique An oblique

angle

with

surface

may Stress

in fiat are

tive

be very

sharp.

in Fig.

the

radius

because

of hole holes

of side

length

to corner

as one having

factors

method along

At the intersection

to an elliptical

angles

have

its axis

trace

of obliquity

been

5.

with theoretical

for

Results curves

with a plane and produces

with respect

determined

in Ref.

at an

to the

oblique

holes

of their for

analysis

elliptical

holes

widths.

of the

maximum

stress

be noted

diameter in plates

factor

acute-angled

of the loss

It should

rise

large

stress-concentration

plate,

oblique

gives

for

Bl0-1i

be defined

to a surface.

by a photoelasticity

to the radius

ratio

hole

which,

and finite

The

may

to the normal

edge

presented

in infinite

hole

concentration

plates

values

Holes

cylindrical

an acute-angled

for various

B10-10.

or skew

respect

a skew

normal,

factor

8

d

may

based

tip.

actually

However,

the results

to plate

can also

width

be found

is relatively

in a relatively

be increased

of load-carrying that

on net area

insensinarrow

by the addition

of a

area. of Ref. w

5 apply

of 0. I.

in Ref.

6.

only

Additional

to plates

with

information

a on

Section

Bi0

15 May Page i 0. l. 1.5

Multiple

I.

Two

Stress infinite

Holes:

has

perpendicular case

been

to the

of biaxial The

diameters

by Ilef.

with

load are

is in the the

values

For B10-18

give

of holes

y

stress

and

loaded

as

for

solution

shown

The

smaller

a double B10-19.

the

of uniaxial factors;

factors

of two holes

ill Fig.

stress

for the

of different

has

been

B10-14. containing

B10-15.

of notch

tension

and

B10-I3.

biaxial

in an

Figure

(point

or (lecl)er

the

a circular BI0-16

A) when notch

is,

shows

the

the tension the greater

Row of tIolcs: :,_w of holes

biaxially

For

case

diameter

stress.

in an infinite

concentration

Double

in Fig.

same

in Fig.

for a plate

at the bottom

of maximum

stress

given

the

gives

by an equal are

the

direction.

a sihgl,

given

of the

For

B10-12

are

results

factor

III.

in Fig.

the

Fig.

2.

concentration

plate

notch,

Single

in Ref.

results

8 contains

concentration

II,

given

for

a circular

for two holes

of holes, the

7 and

Reference

stress

line

tension,

solution

factors

documented

in an infinite

obtained

hole

Holes

concentration

plate

1972

9

factors

stressed

holes

for

l)late, tension

Figures

B10-17

perpendicul,'tr

and to the

line

respectively.

Row of ttoles: row

of staggered For

the

staggered

holes,

the

holes,

stress

concentration

a problem

arises

factor in basing

is

Section B10 15 May 1972 Page 10 b the relation of net sections A-A and Kt on net section, since for a given -a B-B

depends

following

on

formula

Kt A

for net

0 .

0 < 60 deg,

° [

_

maXa

B-B

in the

1

-

2

is the

uniformly biaxial

spaced

configurations

loading

ratio

of the

o

section

investigated

(12)

and

the

formula

the

of

for

is based

A ,

or square

as shown

A' ,

B'

arrays

in Fig.

through

factor

number

in Ref.

combinations

BI0-21

concentration

at point

a large

methods

several

p in Figs. P stress

a_

triangular

considered

factors

figures,

containing

by photoelastic were

against

value

in a plate

in regular

plotted

peak

,

9.

under Four

B10-20. of configuration

B10-26.

is defined

on the

of

hole

as the

boundary

0 max

namely

K =

on the

(13)

factors

concentration

In these

the

of Holes:

perforations was

are

and

-

of perforation

Stress

and

i

concentration

loading

section

a

Arran's

Stress

minimum

row:

= _

IV.

is the

]

cos0

minimum

max KtB

A-A

is used:

O > 60 deg, section

For

(rI

is the

algebraically

larger

one of the

principal

to

al

Section

B10

15 May

1972

Page p

stresses loads

that

would

applied

angular

if there

positions

the

four

types

type

square

for

triangular

gives

given

in Table

B10-1.

Where

strength

parallel-square

when

P P

is large.

parallel,

wh ich

10.1.2

thickness

around

0.92

wilt cross

the

hole.

factor,

is the

the

main

of the biaxial

concentrations

factors

to

K .

is the

of

K with

rise

the diagonal-square

highest

stress

the

hole

of

perpendicular a list

and

of

configuration

consideration,

of hole

configm'ation

The

triangular

hole

used

is un[avorable

K

and

the parallel-

parallel-

biaxial-load

results

desirable,

configuration,

both

contrary

[o1"

for a comparatively

the above is most

is ahvays

factor

/3 , while P

at

p P

type

concentration

range

and the

type

is

show

that

esl)ecially

perpendicular

and

to expectation.

Holes contains hole

be divided section

each

combination

by subscripts

Fox" example,

for

is u._ually

paragraph

around

Information constant

lowest

Reinforced This

configuration,

in between.

-

stress

tendency

throughout

value

by the

shown

primary

loading

large

the

p P

are

the

lie

of

the

plate The

it produces

the

types

no holes.

of hole

of biaxial

type

in the

boundary

figures,

of disadvantage

every

were

on the

On these

Among

be produced

11

information

in order

to reduce

on holes the

into two categories:

and when

the

reinforcement

which

stress when

lmve

an increased

concentration the

cross

factors.

reinforcement section

varies

is of

Section BI0 15 May Page 10.1.2.1

of increased

areas

12

Constant Reinforcement Stress

Values

1972

concentration thickness

of stress

for

a plate

at a hole

around

can be reduced

the hole,

concentration

factor

having

diameter

a hole

sometimes for

beads

by providing calleJ

a "boss"

of various

one-fifth

the

a region or "beaJ."

cross-sectional

plate

a

width

--

=0.2

W

are

obtained

in Ref.

2.

Also,

the

stress

concentration

ab

on the basis of the radial dimension

were

obtained

- a being

2 hole

factors

small

compared

to the

diameter.

To account

for

other

values

a

of

, an approximate

method

has

been

w

obtained:

(14)

KtB where

KtB

stress

the

concentration

particular

factor

value

of

for

a

plate

with

hole

and

bead,

hole

and without

desired,

W

B

= Bead

Kt

= stress

bead

factor

=

concentration

(Fig.

B10-1)

KtB

-

1.51

factor

1 (Fig.

for

B10-27).

plate

for the particular

with

value

of

a W

desired.

for

Section

B10

15 May Page It is

pointed

ab of a bead,

from

-

out

the

Studies

openings

Dhir

on

appears

area

replaced

to area

ment

did not produce

edges

the

be about

concentration

reinforcement

inner

, should

maximum

stresses

shown

holes was

this

obtained equal

area

area

Any

value

no

longer

replacement

lowered

concentration

factor from

3.0,

of

of reinfoccc-

The

inter-

bet_,,,(m the

Fo,

a 40 percent

of

(percent

of the stresses.

or greater.

to the line of the two holes,

holds.

effective amount

not significant if the distance

load perpendicular

theory

circular

additional amount

reduction

one diameter

the

reinforced

that the most

by hole).

the two holes was

the stress

cross-sectional

Above

two

40 percent

a proportionate

of the two

.

were

around

to be near

removed

_

values

in a thin plate [ ii] have

action between

that

ah hb

stress by

l0

a

2

which

in l=tef,

1972 13

, nsion

,,ca

replaccnacnt

fo' :m unreinforced

hole,

to i. 75.

Additional

_ork

in the area of reinforced

Hcfs.

12 and 13; hovvver,

form,

as a solution must

I.

The

these

be obtained

Asymmetricall.

previous

reinforcement,

from

y Reinforced

discussion

however,

be kept smooth

been done

do not give design

a computerized

in

data in usable

analysis.

:

has only concerned

that is, with reinforcement

In practice,

the plate must

references

holes has

holes with symmetrical

on both sides of the plate.

these are frequent

cases

and the reinforcement

where

one surface

can bc attached

of

to the

Section

B10

15 May Page other

surface

because

of the

highly

that

plate

the

and

the

Ref.

t5 shows

be chosen

have

been

the

forcement

may

then

has

and arrived

cross

the

disadvantage a manufacturing

limiting

problem

is

cases.

bending

eccentricity

a stress

14; however,

the

introduces

of the

It is stresses

as in some

concentration

factor

condition

were a size

in

reinforcement.

be employed,

concentration

of the

of the

added.

cases

greater

The

than

work

of reinforcement

of can

factor.

the

the

stresses

variable of view.

the

must section

stress

reinforcement

for has

around

around

the

reinforced

different a given

concentration is that

a hole

may

be as low as possible.

of variably

reinforcement

is to reduce

point

cross

problem for

reinforcement

structure

be of variable

when

of the

certain

must

loading

stress

at expressions

section

stretching,

if no reinforcement

design

weight

considered

that

for

13 and

Reinforcement

cases

when

its

parameters

for a given

important

and

of the

causes

the

in Refs.

reinforcement

because

present

Variable

shown

solved

of the

that

In some

has

only been of the

to minimize

10.1.2.2

17]

bending

of reinforcement

would

is treated

between

reinforcement

what

[ 16,

has

consideration

addition

problem

asymmetry

and

Careful

This

interaction

nonlinear

found the

only.

1972

14

weight,

holes

systems. the effect

in the

Rein-

Hicks

circular

loading

it may

hole.

be

plate.

be undesirable

He of varying One from

Section

Bl0

15 May 1972 Page 15 i0.2

Some

relatively web

LARGE

IIOLES

designs

may

large

holes

of a beam

available

or

for

discussed

it is

in a plate,

in a plate

of this

Bending

plate either

type;

circular

bending

moments

10.2.2

necessary

such

structure.

as

or

desirable

lightening

holes

A limited

however,

available

with Circular

obtained

for each

amount

to have

in the

of data

solutions

will

is

be

Holes

18 for a uniformly

loaded

along the outer boundary

in Fig. BI0-28.

boundary

in Beam

in Ref.

or clamped

hole as shown

condition

Results

square

with a

for deflections

are given in Tables

BI0-2

and

and

-3.

Webs

are frequently

of pipes and ducts, saving.

been

supported

Holes

Holes

of Plates

have

simply

central

cut in the webs

for access

Little information

of beams

to provide

to the inside of a box beam,

is available on the stress

for passage

or for weight

distribution around

holes

webs.

An

anal_¢ical method

flange beams

shown

tude of the moment-shear

applicable

to circular

technique

were

for calculating

in Fig.

applicability of the analysis

analysis

in which

in this section.

Solutions

of wide

occur

CUTOUTS

cutouts

cutouts

problems

10.2. i

in beam

or

AND

depends

BI0-29

stresses

around

is presented

holes in the web

in Ref.

19.

The

on the size of the hole and on the mal,mi-

ratio at the hole.

The

holes; as for elliptic holes, not established.

analysis

is primarily

limits of applicability of the

Section

BI0

15 May 1972 Page 16 An empirical holes

having

formed

technique 45 deg

for the flanges

analysis

is presented

of webs in Ref.

with 20,

round

lightening

Section

B10

15 May

1972

Page

C)

_4

- 000

i

-

0 0

2

4

.51

,-f

.4

tt_

d

z_

,,-4

¢o

¢t

o

'-00

0

#

d

z_

d

.4

>_ I ©

0 io 0

I t_

.4

O O °,.._

17

Section

B10

15 May Page Table

B10-2.

Supported

Maximum Square

Plate

Deflections

and

With a Circular Uniform Load

Moments Hole

W

in Simply

Subjected

Max

max

1972

18

to

M0

R/b qb 2

D 0 1/6

0.649 0.0719

0. 192 0.344

i/3 1/2

0.0697

0.276 0.207

2/3

0.0303 0.0119

0.143

0.00268

0.036

0.0530

5/6 1

Table BI0-3.

Maximum

0.085

Deflections and Moments

in a Clamped

Plate With a Circular Hole Subjected to Uniform

W max

Max

M 0 along

the hole

Square

Load

Max

M

n

along

the edge

R/b D

qb 2

qb 2

0 1/6

0.02025 0.02148

0.0916 0.i45i

-0.2055 -0.2032

1/3

0.01648

1/2

O.OO858

0.0907 0.0522

-0.1837 -0. i374

2/3 5/6

O.00307

0.0310

-0.0780

0.00081

0.0176

0.00025

0.0067

-0.04i0 -0.0215

t

Section

B10

15 May 1972 Page i 9 3.0

2.9

2.8

2.7

2.6

2.5

2.4

2.3

2.2

2.1 *-

Kt BASED

2.0,

L

ON

NET

SECTION

I

P

1.9

1.8

•

1.7

t

h w

_

._.-*- v

1.6

1.4

1.3,

1.2

1.1 W

I

1.0 0

I 0.1

FIGURE

B10-1.

LOADING

CASE

I 0.2

STRESS

I 0.3

I 0.4

CONCENTRATION

OF A FINITE-WIDTH

PLATE

I 0.5

FACTOR, WITH

[ 0.6

Kt , FOR

A TRANSVERSE

0.7

AXIAL HOLE.

Section

B10

15 May Page

1972

20

r_

<

_.

_1o

Z

\ Z 0

\

i o

Q

_. 0'_

,_1 o

Z,,"

m<

o u

_0

_ L¢)

_Z

Z ©

/

<

M d

Z © _

_

d

_

M

i

_1°

i !

ci

_IILLI I

I o. w

I 0

_

I

o.

I 0

o.

0.

0

M

Section

B10

15 May 1972 Page 21

°max o

BASED

ON NET

SECTION

A-B

K t

2,4 oo

2.0

1.5

|

1.0 0

0.1

]

i 0.2

0.3

l 0.4

0.5

¢

FIGURE

B10-3.

STRESS

CONCENTRATION

TENSION CikSE OF A FLAT BAR WITH HOLE DISPLACED FROM CENTER

FACTOR,

K t , FOR

A CIRCULAR LINE.

Section

B10

15 May 1972 Page 22 8 II

¢q I-

II

#+t 0

ill N

tit

al z

o _

o

_

-0=_ Ul

ttt 0

UI

='1o II

"7

o! #

_.__.....1_

_

Section

B10

15 May

1972

Page 3.0

2.9

2.8

2.7

\ \

2.8

2.5

2.4

2.3 Kt = °ma'_xo WHERE 2.2

"

a = APPLIED BENDING STRESS DUE TO M

2.1

2.0 Kt 1.9

1.8

1.7

I

1.6

1.5

J

M

/)M

1.4

1.3

1.2

1.1 a

I

1.0 0

J 1

FIGURE

I

I

2

B10-5. FOR

3

STRESS

BENDING PLATE

I

I

4

5

CONCENTRATION

CASE WITH

IT

FACTOR,

OF AN INFINITELY

A TRANSVERSE

HOLE.

I 6

WIDE

Kt ,

23

Section

B10

15 _lay 1972 Page 24 3.0

\

2.9

2.8 2.7

\

K t = °max °nom

2.6

_'x

\ 2.5

Ono m BASED

\

ON NET SECTION

.

\

\\

2.4

\

\

\°

2.3

x,_

\% \

\ 2.2

6M

°nom

\ \

_'x

\ 2,1

'_,

\

\.

o.# ,.

\\ 2.0

\

",,,

\ 1.9

\

\

\$.

\

1.8i\,,, \ \ 1.7

•

_.

1.6 Kt 1.5

1.4

,.,

1.3

1.2

i, h

1.1 a_ w

I

1.0 0

0.1

I 0.2

FIGURE FOR

B10-6.

I 0.3

STRESS

BENDING WITH

CASE

0.4

0.5

CONCENTRATION OF FINITE-WIDTH

A TRANSVERSE

HOLE.

I

t 0.6

FACTOR, PLATE

0.7

Kt ,

Section

BI(J

15 May Page

197Z

25

< b_ -I, -2

;d

2",

r,.)

:_

.J

z 0

<

5rj

2/

M o

M ©

tI

z_ L_ 0

Section

B 10

15 May

1972

Page

26

I 21p am°!

_

°msx

i"

_1

t

'-

I - 0.25

¢¢

_,.b

K

"$g

,

°max °net

4

/J-1

2

I

I

|

0

0.2

I

0.4

I

0.6

0.8

1.0

X - 2_ FIGURE

B10-8.

MAXIMUM

STRESS TENSION

CONCENTRATION IN A FINITE

PLATE

FACTOR WITH

AN

FOR

POINTS

ELLIPTICAL

UNDER HOLE.

Section O

B10

15 May 1972 Page 27 O

z

._le

i

z

.<

0

z

0

_

_

0

z

_

z 0

M

\

\ \

\

i

I

t

0

I

M

m

,j I 1.-t

M

i,,,,,,I

2;

Section

B10

15 May

1972

Page

10

---.

" .....

,

-

t.,

i!

5

28

'

!//,

/

•

//

e _.;I

III I o o.2s o.s 1 I

I

1

,,

1

2

s

_o

J

J

J

1

2

FIGURE b P

B10-100 WITH

_-

I

3

VARIATION

TENSILE

LOADING

OF

2o

I 4

K

t

WITH

TENDING

a P

TO OPEN

6

FOR

CONSTANT

THE

SLOT.

be,r

ti,)n

15

_\13y

Pa-,:e

J;lO 1!)7_ 2 (J

1

/ 1_.\\\\\'_.2\\\\\\\_1 /I

/ /

/ <

/

lu er

<{ k-

/

UJ

z z 0 ¢3

/

/ __Z /

1

//'

t.IJ u_

<

/

nO

I-¢J <

//

ix

z O I.<

ELLIPTICAL

HOLE

OF INFINITE

rr I--

IN A PLATE

WIDTH

/71

(THEORETICAL)

\

/J_

/

z .i

u z 0 (..1

' EXPERIMENTAL

UJ {Z:

ELLIPTICAL

I--

'OF FINITE

I 0 o

10

I 20

NET

B10-11. AREA

S'I'I{I'SS AS

I

A

IN A PLATE

40

()I.'

l 60

_] -

DEGREES

AN(;I,E

()1,"

C(,I','CI,;N'I'IIATI()N

I.'UNCTI()N

I 50

OF OBLIQUITY

---

(THEORETICAL)

I

30 ANGLE

FIGURE

HOLE WIDTH

RESULTS

I.AC'IOI_. ()BI,I(_UITY

70

I_ASI';I)

()N i;

,

75

)

Section

B10

15 May

1972

Page

oe4

30

¢0 w Z

,( O

,i(

5 L | d_

Z

,.q 0

0

0

W

0 0 Aim

-

_5

o_

I_

,",z°-z u|l_

_

Z

©

Z

_ <

m Z

Z ©

_

©

_5 Z r_

I

:0"I

o

o.

_

m

_

m

_

d

_

_5

Section

B10

15 May

1972

Page

31

ot 8 o I'q

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B10

15 May Page

1972

33

y

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FIGURE

B10-15.

HOLE

WITH

x

CIRCULAR

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14

13

12

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B10-16.

STRESS UNDER

0

0.5

CONCENTRATION TENSION

FACTOR

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1.0

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B10

15 May

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Page

34

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B10

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36

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15 May

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Section B10 15 May Page

1972

38

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1972

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(A)

SIMPLY

(B)

SUPPORTED

1,'IGUI{

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BEAM

WITH

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May

J i,

IlOI.E.

45

Section

B 10

15 May 1972 Page 46 BIB LIO GRA PHI ES Davies,

G. A.

Journal

R. A.

Holes. 3,

Stresses Journal

Plate

Society,

June

in an Infinite of Mechanics

Having

Circular

Hole.

1965.

Plate

and

a Central

With

Applied

Two

Unequal

Mathematics,

Circular Vol.

20,

1967.

Circular E.

C.

Talmor,

F.,

Chang,

and

Society

of Civil N.:

April

June

Stresses

Division, 1966.

F.:

Reduction

Without Stress

Mechanics

Engineers, Elastic

R.:

Field

W.

Engineering

of the Structural Engineers,

and

in a Tension

of the

G.

H.,

Cutouts

Journal

Savin,

in a Square

Aeronautical

W.:

Quarterly

Lerchenthal,

Low,

Stresses

of the Royal

Hoddon,

Part

O.:

Sheet

of Stress

Near

Reinforcement.

Concentration Division,

Concentration

Around

Proceedings

Shaped

Holes.

of the American

1967. Around

Proceedings

Holes

in Wide-Flange

of the American

Society

Beams. of Civil

Journal

Section

BI0

15 May

1972

Page

47

REFERENCES

i.

Sternberg,

the Stress

E.,

2.

3.

R.

E.:

Inc., New

Durelli,

A.

Applied

Holes

5.

A.

292,

McKenzie,

H.

Round

Ellyin,

a Circular

Hole

Solution for

in a Plate of Arbitrary

of the ASME,

Vol. 71, Applied

Mechanics

Concentration

Design

John\Viley

V. J., and

Transactions

Stress

V.

W.,

and

Vol. 3, No.

Aircraft

D. J.:

Journal

November

2,

1968.

A.

N.:

Journal

Stresses

Around

March

Journal

of

1966.

for Rounded

Establishment

an

Rectangular

Report

No.

Stress

Concentration

Uniaxial

of Engineering

Tension.

Effect of Skew

Caused

Journal

Penetration

Mechanics

by an

Stress

Circular

Division,

Solution for an Infinite

Holes

for Industry,

Under

of

on

1968.

J. B.:

Arbitrary

1968.

C.:

of the Engineering

December

and Mahoney,

Two

and

1963.

White,

of the ASCE,

L.,

Factors.

to Axial Loading.

Factors

Hole in a Flat Plate Under

Plate Containing

Stresses.

H.

of theASME,

Royal

November

Concentration.

Salerno,

Feng,

Concentration

F., and Sherbourne,

Proceedings

ASME,

Three-Dimensional

in a Finite Plate Subjected

J.:

Structures

Stress

A.:

1959.

J., Parks,

Strain Analysis,

7.

York,

in Infinite Sheets.

Oblique

6.

Stress

Mechanics,

Sobey,

Around

Transactions

Elliptical Hole

4.

M.

1949, p. 27.

Peterson,

Sons,

Sadowsky,

Concentration

Thickness.

Section,

and

Equal

Biaxial

Transactions

of the

Section

Bl0

15 51ay 1972 Page REFERENCES

8.

9.

Miyao,

K.:

Journal

of the JSME,

Nisida,

M.,

Uniformly

i0.

Stresses

National

Congress

Davies,

G. A.

Quarterly,

II.

and Shirota,

Spaced

Dhir,

13, No.

Y.:

Circular

Holes,

O.:

Concentration

Proceedings

Mechanics,

Plate-Reinforced

Hole

With

a Notch.

in a Plate due

of the Fourteenth

to

Japan

1965.

Holes.

The

Aeronautical

1967.

Stresses

in a Thin Plate.

a Circular

58, 1970.

Stress

for Applied

February

S. K.:

(Continued)

in a Plate Containing

Vol.

48

Around

Two

International

E(lual Reinforced

Journal

Circular

Openings

of Solids and Structures,

Vol. 4,

1968.

12.

Howard, Plane

13.

I.

Wittrick,

Hicks,

H.:

R.:

Analysis

Eccentrically

The

W.

Plate. It.:

On

D.:

Stress

The

Stresses

Quarterly,

Plate-Reinforced

1965.

Hole

Reinforced

Stress ina

Plate.

in

Holes

1960. llole

May

Axisymmetrical Circular

at

Circular

Holes

1967.

AugusL

TheAeronautica[')uartcriy, the

Around

November

Concentration

Aeronautical

Reinforced

February

S.

Ouarterly, of

Asymmetrical

Flat

L.

Aeronautical

Sheets.

Wittrick,

Ouarterly,

Morley,

The

W.

End-Load 15.

and

Sheets.

in Infinite i4.

C.,

ina

1959.

Concentration The

Uniformly

at

Aeronautical

an

Section

BI_)

15 May Page

1979_

4q

REFERENCES (Concluded) 1(;. tlicks, R.:

Variably

Reinforced

The Aeronautical 17.

I8.

Hicks,

R.:

the

Royal

Lo,

C.

Aeta

19.

20.

Quarterly,

Reinforced

Elliptical

and

A.

Bending

of

Leissa, Vol.

F.:

Elastic

Jot_rnal

of

the

Structural

of

Engineers,

U.S.

in Stressed

1957.

J.

state

Iloles

Oc ix)bee

C.,

E.

Offset

Government

F.:

4,

No.

Joumm

Plates.

Plates

With

Circular

l of

Holes.

Around

tloles

inWide-F[ange

Proceedings

of

the

Beams.

American

Society

Structures.

Tri-

1966. and

Designof

Cincinnati,

Office:

Plates.

19(;7.

Division,

Analysis

Printing

1,

Stresses

April

Company,

W.:

in Stressed

1958.

Society,

Bower,

Bruhn,

August

Iloles

Aeronautical

Meehaniea,

Civil

Circular

1972

Flight

Ohio,

--

745-386/4112

Vehicle

1965.

Region

No.

4

SECTION C STABILITYANALYSIS

SECTION C1 COLUMNS

TABLE

OF

CONTENTS Page

CI.O.O

Columns

.............................................

i.i.0

Introduction

....................................

1.2.0 1.3.0

Long Columns Short Columns

.................................... ...................................

1.3.1 1.3.2

Crippling Stress ............................ Column Curve for Torsionally Stable Sections .................................. 1.3.3 Sheet Stiffener Combinations ................ 1.4.0 Columns with Variable Cross Section ............. 1.5.0

Torsional

Instability

Cl-iii

of Columns

................

18 21 25 29

Section

C

I

25 May 1961 Page 1

C

1.0.0

COLUMNS

C

I.I.0

Introduction

In

general,

column

(I)

Primary

(2)

Secondary

failure

may

failure

be

classed

under

(general

instability)

(local

instability)

failure

two

headings:

Primary or general instability failure is any type of column failure, whether elastic or inelastic, in which the cross-sections are translated and/or rotated but not distorted in their own planes. Secondary or local instability failure of a column is defined as any type of failure in which cross-sections are distorted in their own planes but not translated or rotated. However, the distinction between primary and secondary failure is largely theoretical because most column failures are a combination of the two types. Fig. C I.I.0-i failure.

column

m

illustrates

the

curves

for

several

types

of

r

_Tangent

_

Euler

O

Fcc

,-4

s

m o -,4

L--M°difled

-,4

abola

_D

Johnson

Par ! ! !

!

0

a Slenderness Fig.

upon mum

Ratio, C

b

L'/p

i.I.0-I

L' represents _he effective length of the column and is dependent the manner in which the column is constrained, and _ is the miniradius

of

gyration

For a value in the classical

of

the

cross-sectional

of L'/p in the range Euler manner. If the

area

"a" to "b", slenderness

of

the

column.

the column buckles ratio, L'/P , is in

C

I.I.0

the

range

Introduction of"O"to

"a",

Section

C

25 May Page 2

1961

I

_Cont'd) a column

may

fail

in one

of

the

three

following

ways :

(i)

Inelastic bendin_ failure. This is a primary failure described by the Tangent Modulus equation, curve nm. This type of failure depends only on the mechanical properties of the material.

(2)

Combined inelastic bendin_ and local Instability. The elements of a column section may buckle, but the column can continue to carry load until complete failure occurs. This failure is predicted by a modified Johnson Parabola, "pq", a curve defined by the crippling strength of the section. At low values of L'/D the tendency to cripple predominates; while at L'/ P approaching the point "q", the failure is primarily inelastic bending. Geometry of the section, as well as material properties, influences this combined type of failure.

(3)

Torsional instability. This failure is characterized by twisting of the column and depends on both material and section properties. The curve "rs" is superimposed on Fig. C i.I.0-I for illustration. Torsional instability is presented in Section C 1.5.0.

These curves are discussed in detail in _ections C 1.3.0, CI.3.2, and C 1.5.0. For a given value of L'/D between (0) and (a), the critical column stress is the minimum stress predicted by these three failure curves.

L'/D

Each of values

the Tangent Modulus (point "_'). This

FO. 2 for ductile E t is the Tangent

material, Modulus

curves cutoff

has a stress

cutoff stress at low has been chosen as

and is the stress for which Et/E = 0.2. and E is the Modulus of Elasticity.

Section 25 Page C

1.2.0

Lon$

A

column

critical long

with

(elastic a

column.

ratio

This

of

a

type

lack

The

critical

ended

column

of

of

l

1961

3

buckling)

slenderness

slenderness

instead

pin

Columns

C

May

ratio

(L'/

(point

"a"

column

fails

P)

Fig.

C

greater

than

I.I.0-i)

through

is

lack

of

the

called

a

stiffness

strength.

column (L'

load,

= L)

of

Pc,aS

given

constant

by

cross

the

Euler

section,

formula

for

a

is

_2EI 2 c

.......................................

(i)

(L)

where E i

= Young's Modulus = least moment of

L

= Length

End

conditions

end

conditions;

A

column

in

column,

the

The shown

end

nor

otherwise

of

in

of

which

(a)

C

P c

a

ends

of

so

slope is

column

is

in

degree

of

end-fixity

that at

fixed

which

are

loading

and is and

surfaces,

bear

against

load

for

1.2.0-1

flat

there

either

constrained,

rigid

critical Fig.

the

which

of

of

words,the

change

one

against

ends

strength

at

nor

end-surfaces

evenly

column

other

supported

supported both

the

- The

displacement A

of

inertia

columns

dependent

be is

or neither

called

other

end

on lateral

fixed-ended.

neither

laterally

free-ended.

A

normal

to

and

is

called pins,

with

the

axis,

flat-ended. is

called

various

end

column, bear

A

column,

pin-ended. conditions

are:

_2EI = -4L12

(2) ....................................

2.05_2EI (b)

e

=

(3)

C

2

.....o..o.

....

°......°°°.°.°o.°

L 2 4_2EI

(c) P

C

=-

2

L 3

,,.,.°,°,°.°°°°,°

.,.,..°,.°°,o°,

the

constraint.

called

transverse

long

can end

the

part

(4)

as

P

P

f

y

C

25 May Page 4

1961

1

4

If, T ¥

Section

3

L3

w

L'

= 2L 1 P

(a)

I

(b) Fig.

C

(c)

1.2.0-1

The effective column lengths L' for Fig. C 1.2.0-1 are2Ll, 0.7 L2, and 0.5 L 3 respectively. For the general case, L' = L/Ve_, where c is a constant dependent on end restraints. Fixity columns are

coefficients (c) given in Figs. C

Limitations

of

the

Euler

for several types 1.2.0-2 through C

Formulas.

The

of elastically 1.2.0-4.

elastic

modulus

restrained

(E) was

used

in the derivation of the Euler formulas. Therefore, all the reasoning is applicable while the material behaves elastically. To bring out this significant limitation, Eq. I will be written in a different form. By definition, I = AO 2 , where A is the cross-sectional area and @ is its gives

Pc

radius

of

2El = -(e') _

P Fc

The over

_

Substitution

_ 2 EAP

of

this

relation

into

............................

=

Eq.

i

(5)

(e')2

_2 c

_

critical the

gyration.

E

stress

.................................

(Fc)

cross-sectional

for area

a column of

is

a column

defined at

the

as

an

(6)

average

critical

load

stress (Pc)"

C 1.2.0

Long

Columns

Section

C1

July Page

9, 5

1964

I

I

(Cont'd)

I

4.0

For

I

_ = oo , C = 4.0

I I

3.8 3oth 3.6

Ends

Equally

I

Restrained

///i

////

/

3.4

3.2

_e = . _f_ cot E = Modulus

E1

of Elasticity

"-

2

3.0 CI = = Moment Fixity

Coefficient of Inertia

L

of

._

%)

-2.8

/

.,4

02.6 u.w

82.4

Column

= Bending Restraint Coefficient - Spring Constant (inmlb/Rad)

I

I

/

1.1

"_22 X

= Length

"

J

llJJF°r _

2.0

f

One

=

!

_

End

, C

I

= !1"05

Restrained

1.8

1.6

/

1.4

.L

/

I

EI

1.2

I I 1

1.0

0

I0

20

30

40

50

60

70

80

90

I00

Ii0

E1 Fig.

CI.2.0-2

Fixity Having

Coefficient for a Known Bending

a Column Restraint

with

End

Supports

II

Section 25 Page

C 1.2.0

L__on_Columns

C

May

1

1961 6

(Cont'd) NOTE: A Center Rigidly

Support If

q

Behaves _

16

_2

L rL-4 ._.

4.0

(l/) i 3.8

"fc

Mu'st

be

Determined

t

!

i

= I

E"

i C

(_)2

I

I

l

of !

3.4

-q

E I

=" Modulus

3.6 I

=

i (p

l

from

Exper'iment

i

._ ×,'/ /971/, q " 14o

/L) 2 I

i

i

Elasticity

.1

E

i

1

"

1

I"

3.2

_.o

i,

!

oo

_2.6

2.2

,_j"

/

i

!-6o

'

11 .I:_/',," I'Z

/

i_

"_

"° / :, I II._'/'/

i ].0

..

';'" _'"_.,_

"-

_-'_

q

.I

O

.2

,3

.4

=

.5

r Fig.

CI.2.0-3

Fixity Ends

Coefficient and

an

for

Intermediate

a

Column Support

with of

Simply Spring

Supported Constant,

0

Sect 25

ion May

Page

C 1.2.0

Long

Columns

7

(Cont'd)

P

q

=

8El

Where

(_)

Constant C _ 2El =

L 2

Is

Equal

C J[2E To

Pc

= Spring Which

or

fc

-

(L/

The

Number

of

Pounds

p)2 Necessary

The

To

Spring

Deflect

One

Extrapolated

to

Inch Zero

Deflection.

7

0

.i

.2

"i

.3

.4

.5

.6

.7

Two

Elastic,

.8

.9

1.0

b Fig.

CI.2.0-4

Fixity

of

Placed

Supports

a Column

with

Having

Spring

Constants,

Symmetrically p

C 1961

I

Section

C

1

25 May 1961 Page 8 C

1.3.0 Most

Short

Columns

columns

fall

into

the

range

generally

described

as

the

short

column range. With reference to Fig. C I.i.0-i of Section C I.I.0, this may be described by 0 < L'/p < a. This distinction is made on the basis that column behavior departs from that described by the classical Euler equation, Eq. (6). The average stress on the crosssection at buckling exceeds the stress defined by the proportional limit of the material. The slenderness ratio corresponding to the stress at the proportional limit defines the transition. In the short column range a torsionally stable by crippling or inelastic bending, or a combination described in Sections C 1.3.1 and C 1.3.2.

column may fall of both, as

Section

C

i April, Page CI.3.1

Crippling

When

the

restrained tinue

be

When

the

tion

loses

average

corners

loaded

stress

in

its

crippling

on

stress

tortion

Fig.

section

C

after

the

corners to

before

exceeds

at C

one

any

shows

the

occurred

its

critical

in

in

stress

a

the

can

the

and

is

called

con-

section.

stress, load

load

shows

are

material

the

sec-

fails.

The

the

cross-sectional

typical

dis-

thin-walled

distribution

over

sec-

the

cross-

crippling.

vl

/

Ib-"

1

P

Fcrit'

n j

l

has

failure

length

the

compression

corner

additional

1.3.1-1a

wave

in

the

buckling

support

Fig.

over

section

movement,

section

Fcc.

I

thin-walled

even

1.3.1-1b

just

a

lateral

the

occurring

tion.

of

any

ability

stress

9

Stress

against

to

i

J

__I

(b)

C

Fig. p_

i j

1.3.1-1

Ca) The extruded crippllng

empirical

method

and

metal

column

the

The

not

is

the

equal

A etc.

common

attached less

stress

stress

than

of for

load and

the

to the

a

thin

crippling

to short

a

column

stress,

component skin.

is

The

stress

short of

the

to

member.

is

composed

buckling the

stress

this

It thin

of stress

stiffener.

This

lengths

and

constitutes

the

elements.

the

of

of

section.

column

product

member;

summation

the

of

in

equal

of

the of

crippling

failure.

composed

area

area

the presented

extremely

member

actual

actual

structural

is

sections

of

the

crippling

to

predicting

Fcc , applies

crippling

calculating

for

elements

beginning

cut-off

crippling

is

sheet

stress,

indicates

] 1972

of

however, the

an of

the

in

element

angle, the Taking

areas

tee, skin a

zee, panel

thin

Section I

Page C

1.3.1

Crippling

Stress

C 1

April,

1972 I0

(Cont'd)

panel plus angle stiffeners at spacing, b, as shown on Fig. C 1.3.1-2, apply a compressive load. Up to the critical buckling load for the skin, the direct compressive stress is uniformly distributed. After the skin buckles, the central portion of the plate can carry little or no additional load; however, the edges of the plate, being restrained by the stiffeners, can and do carry an increasing amount of load. The stress distribution is shown in Fig. C 1.3.1-2.

e-- rWe Fcc

._"

I

I

!

I I

!

_'crl I 1

J _

b

_ OL stiffeners

Fig.

C

1.3.1-2

For the purpose of analysis, the true stress distribution shown by the solid llne in Fig. C 1.3.1-2 is replaced by a uniform distribution as shown by the dotted lines. Essentially, an averaging process is used is held Notat

to determine constant.

the

effective

width,

We,

in which

the

stress,

Fcc ,

ton F¢c fccn bn bfn tn R

= = = = = =

the crippling stress of a section. the crippling stress of an element. effective width of an element. flat portion of effective width of an element. thickness of an element. bend radius of formed stiffeners measured to the

Rb We E

= extruded bulb radius. = effective width of skin. = Modulus of Elasticity.

centerllne.

,'_ect 25

ion Max,

]'ago CI.3.1

Crippling

Use

of

the

the

I. The followin_

Stress

crippling

C

[

19(,1

I 1

(Cont'd)

curves

crippling expression

stress,

Fcc

, at

a

stiffener

is

computed

by

Xbntnfccn

F

=

The

method

cc

II. into

elements

is

2; bnt

. .................................

n

for

shown

dividing

in

Fig.

formed C

(t)

sheet

and

extruded

stiffeners;

1.3.1-3.

bfl

Formed

Extruded

Fig.

III. in

the

Angle

stiffeners

one-edge-free

leg.

The

shown

in

have

condition

method Fig.

of C

dividing

C

1.3.1-3

low and

crippling offers

such

stresses little

stiffeners

as

support into

each to

effective

1.3.1-4.

b I b b 2 I == bf2 bfl

__

+ +

1.57 1.57

R R

L Extruded

Formed Fig.

C

1.3.1-4

the

leg

is

other

elements

is

C

1.3.1

Crippling

Stress

Section

CI

July Page

1964

9, 12

_Cont'd)

IV. Certain types of formed stiffeners, as shown in Figurel C 1.3.1-5, C 1.3.1-6, and C 1.3.1-7, whose radii are equal and whose centers are on the same side of the sheet, require special consideration. Table C 1.3.1-1 explains the handling of these cases.

Fig.

C

1.3.i-5

Fig.

C

1.3.1-6

Fig.

C

1.3.1-7

when

And

As

b f2

bfl

in Fig.

=0 R R

=0 =0 =0
CI.3.1-5 CI.3.1-6 CI.3.1-6 CI.3.1-7 CI.3.1-7

b I = 2.10R b 2 = 2.10R

(one (one

b 2 = bf2 + b 2 = 2.10R b 2 _ bf2 +

>R

CI.3.1-7

b I = bfl b 2 = bf2

1.07R (Avg. one & no edge free) (one edge free, neglect bl) 1.07R (Avg. one& no edge free, neglect bl) 1.07R (one edge free) 1.07R (no edge free)

shown

D

>0

Use

Table

+ +

C

edge edge

free) free)

1.3.1-i

Vt

Special

conditions

for

extrusions

bl

_'k_Y\\\\\'_

T

"_

bl

(a)

(b)

I Rb

Fig.

C

1.3.1-8

The crippling stress of an outstanding leg with bulb is 0.7 of the value for the no-edge-free condition if R b is greater than or equal to the thickness of the adjacent leg (t I Fig. C 1.3.1-8). When Rb < free.

tI,

the

outstanding

leg

shall

be

considered

as

having

one

edge

S_ction 25 Page C

1.3.1

Cripplin_

Stress

_,-b 2

__

t2

b I

>

neglect

VI.

The

C

compression,

(Fig.

C

Note

13

Avg.

of

one

edge

no

and and

edge

<_

Bt 2

Fcc 2

=

free

and

free.

> 3t 2 method.

1.3.1-9

effective

under

t2 bl;

for b I Regular

Fig.

is

width

of

determined

sheet, from

in

the

a

sheet-stiffener

plot

of

2We/t

combination versus

fstiff

1.3.1-12).

the

following

special

cases

one

edge

m

_

free

t

chart

(no

edge

free)

no One-Edge-Free Fig.

C

edge

free

Sheet 1.3.1-I0 =

J

(a)

(b) J

l

I 4--

Sheet

Hat

Fig.

If

chart

effective

reduce

Effective

with

Stiffener

C

1.3.1-11

i

1961

(Cont'd)

for

-D-

C

May

2WeJ

Large

Calculate free

widths

overlap,

accordingly.

as

as

one

necessary.

or

no

edge

Section 25

Page

C 1.3.1

Crippling

Effective

width

The sides

Stress of

the

stress

level

weld,

or

For

for

inter

skin

rivet

the

is

the

following

2W e --{--=

, is at

the

the

width

of

stiffener

obtainable

skin

on

stress

only

if

of

sheet

either

level.

there

is

no

or

both

This inter

spot-

buckling.

calculating

stiffener

(We)

acting

the

14

sheet.

width,

stiffener

1961

_qont'd)

stiffened

effective

of

C

May

effective

width

equation

is

K (E s) skin V(-E_s)stiff

graphed

1 stiff

on

acting

Fig.

C

with

the

1.3.1-12

.....................

(2)

Where We t

= =

effective thickness

Es

=

secant

modulus

f

=

stress

(KSI)

K

=

1.7

for

simply

K

=

1.3

for

one

For

a

width of skin of skin (ins at

(ins

stress

level

supported

edge

of

case

(no

stiffener

edge

(ksi)

free)

free.

sheet-stiffener

combination

of

the

same

material,

Eq.

2

becomes 2We -_--

The stiffener

(i)

I =

K _

procedure

for

Determine

(3)

determining

Load

P Astif

is

spot-weld crippling

by not

stress

for

a

sheet-

using

or

Eq.

inter

I.

level

of

stiffener.

f

fstiff

applicable

stress by

crippling

stress

= Area

(We)

calculated

the

(3)

is

approximate

procedure

The

. .................................

panel

Determine

inter

'

fstiff --

compression

fstiff

(2)

Es

for

if rivet the

and

Fig.

the

sheet

C

1.3.1-12. is

This

subjected

buckling. composite

section

is

then

to

1

Sect 25 Pn_,e

C 1.3.1

Crippling

Stress

(Cont'd)

[

0

O0

0

0

0

0

0

0

,-4

21o Fig.

CI.3.1-12a

Effective

Width

of

Stiffened

Sheet

ion May

C 1961

l 5

]

C

1.3.1

Cripplin_

Stress

Section

C1

July Page

1964

9, 16

(C0nt'd)

0

m

0 0

Fig.

0 0

C 1.3.1-12b

O0 0 _

0 o_

Effective

0 _

0 _0

0 _

Width

0 -_

of

0 _

0 ¢'_

Stiffened

0 ,-4

Sheet

Section

C 1

25 M_y 196i Page 17 C

1.3.1

Cripplin$

Stress

(Cont'd)

O

O

00

u_ °

Fig.

C

1.3.1-13

°

°

°

Nondimensional

°

Crippling

Curves

Section

C1

1 April, Page 18 C

1.3.2

Column

Curve

for Torsionally

Stable

1972

Sections

The column curves in Fig. C 1.3.2-3 are presented for the determination of the critical column stress for torsionally stable sections. The modes of failure are discussed in sections C 1.2.0 and C 1.3.0.

These curves are Euler's long column curve and Johnson's modified 2.0 parabolas. They are to be used to determine the critical stress, Fc, for columns at both room and elevated temperatures. It is noted that the modulus of elasticity, E, corresponds to the temperature at which the critical stress is deslred. The Fig.

C

following

1.3.2-3

Illustrative

to

sample

problem

determine

the

problem

is used critical

to

illustrate

stress,

the

use

F c.

P 6061-T6

Square

tubing

3.00

x

.065

L = 60 inches (pin ended) Temperature, T = 500 ° F (exposed E =

7.9 x

106 psi

Fcy = 26.6 ksi O = i. 730 in.

Fig.

C

of

@

@

_ hr)

500 o F

500 ° F

Determine column.

the

critical

stress

the

section

stress

for

the

1.3.2-i

Solution Determine the outlined in Section

crippling C 1.3.1. b = 3

of

- .065

= 2.935

(Center here)

\.O--_-J

7.9

. 065

t =

by

the method

line

values

.065

= 2.62

From

Fig.

C

1.3.2-2

Fig.

C

1.3.1-13

fccn -fcy

=

.64

=

x

106

used

S_ctJon March Page C

1.3.2

Use

Colum

Eq.

I Section

Fee

Fcy

Fee

Z bnt

-

.64(26.6)

critical

stress

Fee -_-=

For

C

Then

m L'

Torsionally

Stable

Sections

= n

4(.64) (2.935) (.065) 4(2.935) (.065)

-

17,030

for

the

=

60

L__i'= D

60 1.73

=

34.7

from

Fig.

C

1.3.2-3

x

is

obtained

x

10 -3

Giving

a Fc

critical

10 -3

in.

" Z.0Z

stress x

10 -3

(7.9

of x

.64

from

F c

= Z. OZ

(Cont'd)

psi column

2.16

=

column =

-E

19

1.3.1

17_030, 7.9 x ]0 6

pin-ended L

for

E fccnbntn

Fcy

The

Curve

(, i l,

106 )

= 15,960

psi

Fig.

C

1.3.2-3.

1965

C

1.3.2

Column

Curves

for

Torsionally

CI

0 •,q

=

Columns

C

July

9,

1964

1

Page

20

_Cont'd)

O_

> II

-,-4

"0 OJ 4o

Stable

Section

II_

_

0 ,.0 c_

-,-4

:=_

/

0

il

/

0

/I > % _ U

_ 0 m I=

m

0

0 Z

,I/I //// _Y/ !

/.4;.//?,I,'1 / ! I //,/1"1 ! ! l ! I //III If' ! / If'' i i

Fig.

C

1.3.2-3

Critical

Stress

for

Torsionally

LL_

Stable

Columns

.._4

Section

C

1.3.3

Sheet

Flat

Stiffener

Sheet-stiffener as of

combinations

columns.

sheet

Each

acting

at

of

flat

stiffener the

of

stiffener

compressi,_n the

panel

stress

restraint

stiffener

failure

an

the

itself

against

even

though

The

stress

buckled is

sheet

distribution

is

shown

by

the

assumed

widths.

The

effective

sections

are

given

over

the

panel

curve

(We)

Section

C

axis

buckled

1964

Fig.

for

an

sheet.

after

the

the

sheet

The

concept

torsionally

to

stiffeners.

1.3.3-I.

the

be

individual

perpendicular

C

using

may

effective

the panel considerable

between section

in

distribution

widths

in

the

solid

stress

about has

an

constitutes

parallel to and offers

curve

21

panels

plus

about an axis is continuous

has

Pa_e Combinations

column that is free to bend The sheet between stiffeners

sheet

9,

panel

analyzed width

Cl

July

stable

of

dotted effective

and

unstable

1.3.1.

i ,-L

i' r--

-7

I

!

I

Fig.

The on

a

procedure

for

sheet-stiffener

(I)

p

section

(2)

From the Fc/E

the

the is

about

L'/p

J

i

of

curve

0

in

the

Fig.

I

crltical

L'/

gyration

(Fig.

of the

parallel

C The

C

0 of

axis

cross-section. =

fstiff

stress

and

load

is

ratio of

centroldal

crippling

stiffener at

radius a

I

panel

slenderness

the

I

1.3.3-I

determination

compression

Determine where

C

y

r--"

1.3.1-13) value

1.3.2-3.

the

stiffener

stiffener to

the

sheet.

determine Fcc/E

is

alone

cross-

given

Fcc by

of

Section July Page C

1.3.3

Sheet

(3)

Stiffener

Using the determined

column curves in steps (i)

(Interpolate (4)

Determine Fig.

between

C 1.3.1-12

curves

Use Fig. effective

C

(6)

Re-enter

the

where

1.3.3-3 sheet.

L'/p

and

Repeat

steps

as

widths

column

(4),

fstiff

curve

The Pc

Where

Ast

critical

value

(5),

and

= Fe

is the

Curved

[

load, Ast

+

Pc'

ts Z

cross-sectional

22

and Fcc of F c.

by

using

= Fc" P

(Fig.

the

with L'/P the value

sheet

of

C

the

stiffener

1.3.2-3)

with

plus

new

of F c. (6)

until

to a final stress, Fc, is obtained. occurs after two trials. Fc is the stiffened sheet.

(8)

l

1964

required) of

to compute

record

C

_Cont'd_

(Fig. 1.3.2-3), and (2), record

the effective

(s)

(7)

Combinations

9,

satisfactory

convergence

Convergence generally critical stress of the

is We

J

area

........................ of the

(i)

stiffener.

panels

Analysis of curved stiffened panels requires but a slight extension in procedure beyond that described for flat panels. Fig. C 1.3.3-2 shows a curved panel with the effective widths of sheet that act with the stiffeners. W

We

'

b'2We

Fig.

C

The load-carrying capacity of flat panel plus an additional load curvature of the sheet between the

Pc

= Pflat

+ Pcurved

We

1.3.3-2 such a panel attributable stiffeners.

is equal to that of a to the effect of the The critical load is

Section

C

1.3.3

or

for

Sheet Fig.

Stiffener

C

Pc

Combinations

CI

July

9,

Page

23

]964

(Cont'd)

1.3.3-2

=

(Fc)

column

(Ast

+

4tsWe)

+

(For)

curved

(b

- 2W,_)

ts

panel

The

critical

equations entire b

-

curved

of width

2We,

is panel.

stress section "b"

used

(Fcr)Of C

3.0.0.

of

the

in

calculating

curved

the Note

curw-d that

panel in

panel

is

used.

the

load

that

is

calculated

computina

this

(_nly is

the

contributed

by stress

reduced

the the

width, by

the

Sec tion

C

1

25 May 1961 Page 24 C

1.3.3

Sheet

Fig.

C

Stiffener

1.3.3-3

Combinations

Variation Stiffener

(Cont'd_

of Radius of Gyration Combinations

for

Sheet

Sect

ion

Page CI.4.0

Columns

The the

with

modified

critical

Euler

load

subjected

to

of

This

t

section

The a

to

determine

column

not

is

. .....................................

gives

following

of

used

stable

(i)

(L,)2 appropriate

and formulas for com_uting columns. Where m = ± in c

load

modulus)

torsionally

failure

_

Sections

(tangent

prismatic,

i9_,1

I

= c

Cross

equation

a

crippling c_2E

p

Variable

('

Mny

25

example

stepped

column

the Euler Eq. I.

is

buckling

loads

typical

for

for

coefficients

varying

calculating

(m)

cross-section

the

critical

column.

Example

P IIII

III

Given:

E1 = I I

i0 x 1_ 6 .30 in

psi

(aluminum)

E2

=

30

psi

(steel)

12

=

.50

A I

=

1.94

x

106 in 4

IIIZll

in 2

ll l

Determine

critical

buckling

load,

Pc Fig.

--12

P

C

1.4.0-1

Solution: a .... L

12 36

From

Fig.

C

1.4.0-2,

2(Etl)l Pc

Stress

=

level

of

the

stress the

10

x

106

(.30)

E212

30

x

106

(.50)

= AI

If

fl

is

Pc

is

the

level

critical

at

=

=

.20

.545

2

(I,0) 1 06

.545

aluminum

=

(.30)

:

41.,800

lb

(36)2

section

(max.

of

column)

41800

fl

stress

m

(3.14)

mL 2

Pc

then

ElI1 .33

=

I. 9------4 = 21,600

below

the

critical the

psi

proportional load

of

proportional

must

be

used.

load

of

the

limit

the limit,

This column.

of

column.

leads

the to

the

material

However,

if

in fl

question,

is

tangent

modulus

(Et)

a

process

to

trial

above at

the

determine

1

Sect ion

C

i

25 May 1961 Page 26 C

1.4.O

Coh,mns

with

Variable

Cross

Sections

(Con t'd}

P

2 Pc-

(Etl)l

_

m

L_" a/L

= 0

1.0

jO.8

07 0

.I

a/L

.2

I I = 0.9, 1.0

J

i

.3

.4

.5

.6

.7

(Etl)l (EtI) 2 Fig.

CI.4.0-2

Buckling

Coefficient

.8

.9

1.0

Sect ion

C

[

25 May 1961 Pa_e 27

CI.4.0

Columns

With

P

Variable

._

Cross

I I --

=

(Cont'd)

12 --7

_ 2 Pc

a/L

Sections

I

__

(EtI)l

m

L2

= 0

I. (;

0.3 0.9

I-2.

11---0.8

/_

04

0.7

0.5

/

s

/

o.(//; /.

o.s

/ o._-_- _I_: o._. _.o

/GY 0,3

0.2

0.1

/

,.

- y
0

0. i

0.2

0.3

0.4

0.5

(Etl) I/(Etl) Fig.

CI.4.0-3

0.6

0.7

0.8

2

Buckling

Coefficient

0.9

1.0

Section

C

i

25 May 1961 Page 28

CI.4.0

Columns

with

Variable

k_--

Cross

Sections

(Cont'd)

e

--

L-a

__

F

2 I2

Pc

f (Etl) I

m

L "_

=

= I

-_

2 I2 P

_2

a/L

1.0

a I1

L-a

I

--_0_.8 -0.7----"---

-0.6 _-

0.9

_

0.5

I-

1

0.8

_.-._

_

o.4

L

E _J

_If

o.3

.,4 .,4

_

.//

/_

// /

_

I

0.6

..0.2

0.5

// _"

/" /t

J 0 C.)

./-/f,,,

_'_

I

0.7

--

._/

y

/

0.4

I

//

/ /

/

I /

0.1

.-4

0°3

L

/

/ J

0.2

/

[ a/L

0.1

= 0

-

/

/ 0

0.1

0.2

0.3

0.4

0.5

(Etl)l/(Etl) Fig.

CI.4.0-4

0.6

0.7

2

Buckling

Coefficient

0.8

0.9

1.0

Section 25 Page C

1.5.0

Torsional

The mined

critical

by

Manual

is

I.

2.

.

Instability

use

of

of

torsional the

29

or

load

references

for

until

a

column

this

is

to

section

be

of

deter-

the

completed.

Argyrls,

John

Aircraft

Engineering,

H.,

Kappus,

Robert,

Section

Columns

Niles, Vol.

Alfred II

June,

Twisting in

S.

Third

Flexure-Torsion

the

and

Failure Elastic

J.

Edition,

Failure

of

Panels,

1954.

of

Centrally

Range,

S.

Newell,

John

Wiley

Loaded

T.M.851,

Airplane & Sons,

Open-

N.A.C.A.

Structures, Inc.,

New

York,

1943.

.

1961

Columns

stress

following

C

May

Sechler,

Ernest

Analysis 1942.

and

E.

Deslgn,

and

L. John

G.

Dunn,

Wiley

Airplane &

Sons,

Inc.,

Structural New

York,

1938.

1

Section

C

1

25 May 1961 Page 30 BI_MEOGIAFIPf Text

books

I.

Timoshenko, Edition, D.

S. Van

2.

Seely, Second

3.

Popov, E. New York,

4.

Sechler, Ernest E. and L. G. Dunn, Airplane Structural and Design, John Wiley & Sons, Inc., New York, 1942.

5.

Steinbacher, Franz R. and G. Pitman Publishing Corporation,

6.

Niles, Third

Fred B., Edition,

StrenKth Nostrand

of

Materials,

Convair

Div.

S. and J. S. Newall, John Wiley & Sons,

2.

Chrysler

3.

North

4.

Martin

5.

Grumman

6.

McDonnell

of General

Corp.

American

Missile Aviation

Aircraft Aircraft

Dynamics Div.

Prentlce-Hall,

Gerard, Aircraft New York, 1952. Airplane Inc., New

Manuals I.

Part I and II, Third New York, 1957.

and J. O. Smith, Advanced Mechanics of Materials, John Wiley & Sons, Inc., New York, 1957.

P., Mechanics 1954.

Alfred Edition,

of Mat.erials, Company, Inc.,

Corp.

Inc.,

Structural

Structures, York, 1943.

Analysis

Mechanics,

Vol.

II,

SECTION C1.5 TORSIONAL INSTABILITY OF COLUMNS

SccLion Ci. ;,._, 15 May

TABLE

OF

CONTENTS

C1.5.0

Torsional

Instability o_ Coltuuns

C1.5.1

Centrally

Loaded

Two

of Symmetry

I

II

Axes

General

III

One

C1.5.2

Cross

Axis

Special

Columns

Section

of Symmetry

Cases

.......

1 '_(P.J

.....................

........................

i

2

...........................

2

...........................

3

...........................

5

'..........................

I_

I

Continuous

Elastic Supports

.......................

1

II

Prescribed

Axis

.......................

20

HI

Prescribed

Plane

C1.5.3

Eccentrically I

General

II

One

III

Two

C1.5.4

Axes

Example of Columns

of Deflection

Loaded

Cross

Axis

of Rotation

Columns

Section

of Symmetry

of Symmetry Problems

.....................

21

.....................

24

...........................

2,1

...........................

26

...........................

27

for Torsional-Flexural

Instability

...................................

2,_

I

Example

Problem

I

.............................

2_

II

Example

Problem

2

............

31

III

Example

Problem

3

.............................

CI.

5-iii

'.................

3(;

-,,_s

Section

C 1.5.0

15 Mn)

195!}

Page TORSIONAL

CI.5.0

In the stable;

P

cross

there

occur

if the

walled

open

section the

torsional

the

Euler

load.

open

column

warp,

failure,

can

but

is

only

primary equations

the

their

assumed

of the

t)ecause,

buckling. developed

curves,

Section

C1.3.2).

a theory

which

would

include

therefore,

an

analysis

would

and

which However,

the

cube at

the not

cross local

torsionally include

this

wall well

and

sections

during

below in thin-

In such

opposed

sections, flexure

of the

buckling;

i.e.

to secondary

sections. buckling the're stable coupling has

been

of torsion

and

flexure

be extremely

cross

important

cross

change as

its

torsion

plane

or

of thin-

open

of

symmetry.

n(_ attempt

coupling

a bar

loads

buckling

in gen('rnl,

For

for

bending.

twisting failures

of an

therefore,

does

of primary

as

of double

of columns

distortion

as

by twisting

torsional

that

shape

low,

roughly

and

by

buckling

in behavior

and,

torsionally

,_t symnletry

either

torsional

buckle

lack

was

of crippling

is very

coincide

failure

results

secondary been

be

column

in a plane buckle

difference

makes

geometric by

approximate

have

not

primary

investigation

and

that

frequent do

it will

characterized

the

factor

center

consider

Separate give

varies

section,

theories

section

sections

Another

will Such

rigidity

shear

In this

of the

the

bending

aeon_bination

twisting.

torsional

sections

and

by

Since

open

that

by

a colmnn

section.

thin-walled

centroid interact.

the

rigidity

fail or

and

COI,UMNS

assmned

either

in which

of bending

is that

walled

cases

cross

thickness,

it was

would by crippling,

are

a combination

()F

sections,

column

section,

However, by

previous

i. e., the

of the

INSTABII,ITY

1

complicated.

can will

necessarily i)e coupling

sections,

()f

approximate

(,I()hnson-Euler made and

to formulate local

buckling,

,

Section

C 1.5.1

15 Ma 3 1969 Page '2 CI.5.

CENTRALLY

i

Centrally (1)

They

can

LOADEI)

loaded bend

COLUMNS

columns

in the

can

plane

buckle

of one

in one of three

of the

principal

possible

axes;

modes:

(2)

they

can

twist about the shear center axis; or (3) they can bend and twist simultaneously. For any given member, depending on its length and the geometry its cross section, one of these three modes will be critical. Mode (1) has been discussed cussed below. I

Two Axes

the

in the

previous

load

P0

Modes

(2)

and

(3)

-will be dis-

of Symmetry

When the cross section shear center and centroid

buckling

sections.

of

about

the

shear

has two axes will coincide. center

axis

of symmetry or is point symmetric, In this case, the purely torsional is given

by Reference

8.

l,:l" 7r2 1 GJ + _f2

: lr _O

where: r

o

G

:

:

of gyration

modulus

torsion

of the

constant

I See

Section

= Young's

modulus

of elasticity

F

-- warping

constant

of the

length

of member

effective

Thus, for a cross section values of the axial load.

principal

Depending

axes, on the

P shape

x

section

about

its

shear

center

of elasticity

E

:

the

radius

= shear

J

critical

polar

and

P

of cross

B8.4.1-IV

section

(See

Section

B8.4.

1-IV

E}

with two axes of symmetry there are They are the flexural buckling loads y

, and the section

purely and

torsional

length

loads will have the lowest value and will determine In this case there is no interaction, and the column bending or in pure twisting. Z-sections, and cruciform

A )

Shapes sections.

in this

category

buckling

of member,

three about

load,

one

of these

the mode of buckling. fails either in pure include

I-sections,

P

_"

_S('cLJon C J..;. i 13 .".[:_y 19(;fl Page ;/ II

General

Cross

Section

In the general buckling occurs purely torsional consider x and Xo

case

of a column

I)y a combination bucMing cannot

of thin-walled

Yo

are

cross

section

by the

deflections

the will

coordinates undergo u

the shear center moves to o' and

of the

v

and

in the

x

o. Thus, during point c to c'.

solution

culating

of these

the

critical

ro2(Pcr-Py _

p

equations value

2x 2(p

-P

ero

er

fh'xurcql or of buckling,

y

The

buckling,

translation

directions,

The and the is defined

respectively,

of tim cross of the cross

section, section

of point about

o the

d_, and the final position of the centroid element of a column deformed in this differential equations (l{eference 8).

yields

the

following

cubic

equation

for cal-

load: - P

(Pcz,-P4))

)

During

rotation.

and

of I)uekling

) (Pcr-Px)

center.

translation The rotation

shear center is denoted by the angle is c". Equilibrium of a longitudinal manner leads to three simultaneous The

section,

shown in Vi_urc CI. 5-1. axes of the cross section

shear

translation

and

cross

of torsion an(I I)emlin_. I)urely occur. To investigate this type

the unsymmetrical ct',Jss section y axes are the principal centroidal

and

open

°ri2Yo2 p•

CI"

PX

)

: 0

y

where P

7r2EI x : _

x

P

'

_r2El y 2 _

y

,

and

G

ErTr2 _

O

Solution of the cubic equation then gives P of which the smallest will be used

three values of the critical in practical applications.

load, The

or'

lowest three

value

of

P

er

parameters,

it represents effective solution

can Px'

always Py'

an interaction

length, above.

_,

various

be shown and

P4)"

of the three end

conditions

to be less This

than

the

lowest

is to be expected,

individual can

modes. be incorporated

of the

noting

By use

that

of the

in the

Section

C1.5.

15 May

1969

Page

xo

1

4

u

|_

t

% %%%

! C

s....,

c

Centro,d

o

Shear Center

FIGURE

C1.

5-1.

DISPLACEMENT FLEXURA

OF

SECTION

I_ EUCKI,ING

j

o

I)UItING

TORSIONAL

!°

--

Section

C I.,;.

1

15 7xla3 19t;!) P:l:_,c 5 III

for

One Axis

of Symmetry

K tile a general

x-axis section

(Pcr-Py)

There

again

purely

three

of the quadratic

give

two torsional-flexural

will

always

Which

a singly in either

about

inside

and

the

P

.

syn_metrieal of two modes,

two actually

0

one

buckling Px

evaluation

tedious.

efficient equation their load are

0 and the

Yo

of which the

square

It may,

however,

depends

is

(1)

P

= P and er y Tile other two are

brackets The

section (such by bendhlg,

occurs

.

y-axis.

loads.

equation

equated

lowest

to zero,

the and

torsional-flexural be above

load

or below

as an angle, channel, or in torsional-flexural

(m th,, _limcnsions

and

Py.

or hat) lmcMing. shape

()f the,

section. The

and

buclding

term

be below

of these

given

solutions,

flexural

roots

Therefore, can I)ucMe

then

Iroa(Pcr-Px)(Per-P0-Pcr_Xo:_l=

are

represents

is an axis of symn_etry, reduces to

of the

Chajes

and

procedure for (1) for singly

approach, are clearly singly

load

from

(Reference

evaluating symmetrical

equation 7) have

(1)

is often

devised

the torsional-flexural sections shown

lengthy

a simple

bucMing load in Vi_ure C1.5-2.

and from In

the essential parameters and their effect on tile critical evident. Since most shapes used for compression members

symmetric, Critical

A.

bucMing

Winter

their Mode

method

is quite

useful

as described

below.

of Failure

Failure of singly symmetrical sections can occur either in pure bending or in simultaneous bending and twisting. Because the evaluation of the torsional-flexural buckling load, r(,gar_llcss ol the .mcttl(_(l used, c_n ncv('r be made as simple as the determination of the Euler load, it would be convenient to know if there are certain combinations of dimensions for which torsionalflexural

buetding

a method modes

of delineating of failure

boundary be limited offer

need

equal

has

conditions. to members restraint

not be considered the been

regions

governed

developed.

For the purpose with compatible to bending

at all.

about

The

To obtain

by each method

this

of the

the

principal

two possible

is applicable

of this investigation, end conditions; i. e., axes

and

information, to any

however, supports to warping.

set

of

it will that

Section C1.5.1 15 May 1969 Page6

--

p.

O

]{--

iL {]

1 '

O

i

--

O

i j-¢ I=

b

FIGURE For sections is given by equation member equation,

buckling

I

IT -i

C1.5-2

b SINGLY

SYMMETRICAL

£

SECTIONS

symmetrical about the x axis, the critical (1). According to this equation, the load

actually buckles is either whichever is smaller.

The These defined

_,

domain

P

or the

root

load the

of the quadratic

Y

can be visualized

are shown schematically by the width ratio, b/a.

smaller

buckling at which

as

being

in Figure C1.5-3 Region 1 contains

composed

for a section all sections

of three

regions.

whose shape for which I

y

is >I

x"

Scuti_,n

Page hi this 1

re_ion,

I

x

fall

3

into

the

parameter

tile

two

the

buckling

in the

') or

l_/a 2.

mode

flexural

Figure

with

C1..5-,t In this

torsional-flexural

pure

of the

boundaries

as_mlptote,

which

correslxmding

b/a

axis.

th(,s(,

Th(, the

is

the

plot

curves

in

various

asymptote,

bet_veen I

which

the

transition

buckling, wllue

regardless for

either

the

by

region,

Region the

b/a are

B.

Interaction

The

critical

-Px)

to

2.

to

(71.7,

axis.

I . y

K

the

right and

of the

z,

fail

and

b/a

curw, buelding

which These el

(l,)('s n_)t occurs,

curves

located h/a

is

smaller

tsvo

at

where l)/a

is

the less

sections

int(.rs(,c,.t does not

I)/n

than

will

For

the

can

tile

b(,

in this

possil/le

channel

2.

The

larger

2 ;inti l"lilurc

of the

t_/a

l()_':_ti_)n

figure.

mode.

occurs

for

the

of the

is

curve fail

in torsional-flexural

lipped

3.

of the

in the

with

i)) I{egi(m

which

value

fail

abo\e

ai\('

Each

1 and

ahvays

transition

Sections

ahva_,s

left

line

Sections

plain

in Region

_vhicl_

;m
t als,,

(Iomains.

determine

This

at

The

3 \viii

the

a dashed

falls

of the

regardless

modes

section,

there

(t_/a

2)

than

the

nhvays lipped

lhll value fail antic

the b/a axis. exist for these.

Equation

buclding

axis of symmetry) lowest root of

r ca(Per

will

case

the (ff/a2jlim only flexural

and

those

will

sccti(m

I)c|\\ccn

of the

in thetorsional-flexural

located

mode,

and hat sections Region 3, where sections.

the the

t_/a 2,

below

dilnensions.

thc

In the

boundary

in theflexural

other

hueklingor

par,'uneter,

intersection

asymptote

, then

flexurnl

intersects

at this

the

of their

is critical.

is a lower

at

as3ml}t()Ic

pure

the

of failure

curve

value

e

('h:lnn('ls,

l{egions

x than

angles,

as -

3


ff/a 2.

1)uckling

for

o1 tf,:l

intersection

l"iKur('

indicated

I)oundary

to sections

the

of

whereas

value

in Hegion

for

that

depends

I)()untl;IF

the

\_hich

to torsional-flcxural. at

value

for

l)ucklin_

J

T

Sections

of

()ltlle

Sections

cu)'v('s

hers

mode

flexural

of the

nccut'.

l'epL'osen[N

located

mode,

between

a vertical

_), the

II ix a plot 3 is

men,

can

CUI'VC

purely

rood(,.

approaches

lllill

from

the

figure,

t,cnding

h(.glon

2 and

d('fin(,s

sections. in the

In

regardless

fail

in the

3.

of failure.

chnn._es

mode

I_uckling

(L(:/L!)

Regions

curve

m in

The

modes

between

2)

torsion:lt-flcxurnl

Hegions

possible

1ooundary (tI/a

only

('!.5.

load

that

(Per

-Pc)

for

bucMe

-

singly in the

Per 2x20o

symmetrical torsional-flexurnl

=

sections mode

(x-axis is

given

is by

(2)

Section

CI.

15 May Page Flexurol

Suckling

mode depends

buckling

t{

o_ly

o

1969 8

/ | Torsional/

o. vol_, of _

5. i

[

flexuraJ

/ I /

b,_ck_i,_g

ii

only

=1% o o

1

n.,

,

1

J_-

1, Ratio

b O

C1.5-3

FIGURE

BUCKLING

REGIONS

ol

=1%

I

!

f/I,'_l

!l

O

_1

I

Y

I

I

I

I

Ifg/

I o'o "_°

____!____L___._M__.___

I 0

0

0.2

0.4

0.6

0.81.0.0.20.40.6

08

1.0

Ratio

1.2 1,40

E J"

0.2

0.4

0.6

I

I

0.8

1.0

od

_

IVo_J-o'{r

1.2

1.4 1.6 1.8

b o

FIGUI{E

CI.5-4

BUCKLING

Dividing this equation by

MOI)E

OF

SINGI_Y

SYMM/
SECTIONS

PxPoro 2, and rcarranzing results in the following;

interaction equation:

P <

P +

-'P-x

K -

= Px

t

(a)

St'clion

_'1.._.

P;I ,_t'

]

!)

m which K

is a shape

factor that depends

FignJre method

C1.5-5

for

flexural

checking

is

a plot

the

safety

on geometrical

of

equation

properties

(?,).

of a column

This

of the cross

I)l_t

against

pr_)vidcs

failure

section.

n sil_ll)l(,

by torsional-

buckling.

To determine

if a given nmml)er

is only necessary

to compute

P

can safely carry

and

P

X

knowing

K,

use the correct

tile argtmlents

P/P

pertinent curve.

and

x

instead of ascertaining

=--

er

which

is

+

another

The

0

form

P

it can

-

x

P

of equation

interaction

whether

P,

it

and then,

tile point determined

(safe}

to determine

whether

2K

to check

falls below

If it is desired

load,

in question

(b

curve

P/P

a certain

for the section

or above

by

the

tilecritical load of a member

safely carry

+

d)

(unsafe)

a given

-

4

load, use

(5)

,

(3).

equation

(eq.

3)

indicates

that

P

depends

on three

CF

factors: the

the

two

factors

interact. and

loads,

The

tile

which do

therefore

To

and

x

bendin_ not

O'

and

the

while and

coincide.

causes

evaluate

P

interact,

reason

eentroid

points,

P

detern_ines

twistin_

interact

A decrease

for determining

these two

to know

parameters

in •

in the

the torsional-flexural it is necessary

factor,

K

a decrease

action equation,

shape

the is x

*

mteractmn.

bucMing P

K. extent

tn;tt

, the

P

the

P

to which ,shear

are

¢

they

center

between

these

0

K.

is therefore

and

(list_mee

load by means

and

x

A

of the inter-

convenient

an essential

method

part of the pro-

cedure,

C.

For the shape

Evaluation

of K

any given section, of the section.

K

is a function of certain

Starting with equation

(4)

parameters

and substituting

that define for

Sccti_m

C 1.5.

15 May

t969

Page

10

I.Q

0.9

0.8

I

0.7

...... -_--_,-X

0.6

' O I-

!

_ i..... Ni A

\'_

_,

I

_I

0.5

i

0.4

P

0.3

P_

x _.,,;

i

X

\\_

0.2_

0.1

0

J

0

x

and

,

r

o

K

0,2

can

0.3

0.4

o.s

RATIO

P/Px

FIGIIIiK

('1.5-5

be reduced

to an

0.6

0.7

IN'I'I':ICAt'_'I'II)N

expression

0.8

0.9

1.0

('UI{VI.;H

in terms

ol' one

or

more

of

o

these will

0.1

parameters. be

of the

components (4) can be

If the form

of the reduced

b/a,

thickness in which

section. to

In the

of a case

the anti of

member b

are

a tee

is the section,

unilt)rm, widths

the of two

for

example,

parameters of

the

flat

equation

4

K

-- t

-

[1

_- b/a]

L'r'b/a) a +"1 _J

(¢_)

1

_('('tton

C1. 5. 1

I3 .kla5 l'.)_;t) I):_gc I1 f

in which

the

b/a

is the ratio

o[ tilt' fl:mgc

of the

relation

lips have the same and lipped angles, channels K (Fig.

and hat C1.5-2).

Curves

for

K.

Because

t Fig.

sections

for

require

determination

of curves D.

stituting

with

the equation

ro,

K

J,

and

a general

relation

sectional member;

area; t = the a = the width

C 1 and of the

Equation their

effect

directly with the St. Venant

for

for

obtained

for

c/a,

to define

angles,

channels,

and C1.5-7. The value of

= 0. 625, is given by the point For hats an(I channels ( Figure C 1.5-7)

F

the

same

I)

_iv('n

scheme in

as that

used

(:1.5.

l)arabWal)h

to determine 1-1,

and

Pr_'

(7)

in which,

E

= Young's

modulus,

A

:

cross-

thickness of the section; £ : effective length of one of the elements of the section; and of

sub-

yields

b/a

and

c/a,

in which

b

and

c

are

of the

the widths

elements. (4)

indicates

on the buckling

the

important load.

E and A. The term torsional resistance t/a, wall

a/l

in warping

the

K

follows

of these, the parameter, with decreasing relative shows

been

and

2 + eclair)el

C 2 = functions remaining

have

b/a

shown in l"i_tr(.'s C1.5-_; lipped angle sections.

angles, C1.5-(;).

P

EA[C_(t/a)

:

without

d)

of

for Pd)

of P

evaluation

Starting

angles

is gdven.

Evaluation

The

is composed and determine the

equal-legged

two parameters,

of

K for all plain equal-legged b/a =- 0 on this curve ( Fig. a series

all

C1.5-").

shape, K is a constant for this section. For channels K is a function of a single varial)le, b/a, while lipped

and hat sections. These curves are A single curve covers all equal-legged

and

le R"width

In general, the number of elements of which a section number of width ratios required to define its shape will

complexity

K.

t,) the

decrease

parameters

Similar

in torsional

to Euler

bueMing,

inside the bracket consists and the warping resistance.

indicates thickness;

the decrease in torsional whereas, in the second

resistance

with

increasing

bucMing P

varies

of two parts, In the first resistance the parameter slenderness.

,

Section 15

C 1.5.

May

Page

19(;9 12

0.7

0.6

,,,/ . 0,5 u

0,4 0

0.3

0.2 0.2

0.4

0.6

FIGURE

CI.

5-6

0.8

1.0

b _

Ratio St[APE

F\CTOI(

K

FIGURE

C1.5-7

SllAPE

F()I(

I':QUAI,-LI.:(iGED

FACTOR

K

ANGLES

1

Seeti_m 15

C 1. :;. 1

May

PaRe The

coefficients,

ftmctions

of

effect

that

lga

tile

at

Because reduces

C a is

section

point

on

P

elcl_lcnts

negligible

to

F,

tile

:lad

tel'IllS

warping

terms

(hcrel'ort,

are

indicate

the

@"

rt'(:t:lll.t2,]u:ll"

have

proportional

Velmnt

'l'hcsc

has

()|" thin

a common

ill tilL' St.

warping

torsional

\vh()s(,

iilid_tlc

stiffness-

i. c.,

bucMing

load

lines 1" -

of these

0.

sections

to

P_

the

: EACl(t/a)

plain

further

in which the legs.

G

is the

For

(s_

angle,

which

falls

into

this

P

category,

can

be

modulus

C l

in Figures

cross

center

(9_

however,

given

other

2

shear

in general, values are and channels.

.

to

= AG(t/a)

¢

2

equal-legged

reduced

P

of shear

C2,

respectiw'ly.

of the

COlllposed

intersect

nnd

e/a,

shqpe

Sections

For

CI and

191;!1 13

are

of

elasticity,

and

C,

C1.5-8,

sections given

and

must

C1.5-9,

values

in Table

of

he and

the

a

is

the

length

evaluated. el.

warping

5-10

for

of one

Curves

[or

angles,

hats,

constant,

1", and

of

these

location

I.

0.1

(a) o

0

o

0.2

0.4

0.6

0.8,0

Ratio

0.4

0.6

0.8

b_ o

FIGURE

C1.5--8

TOIISIONAL FOIl

BUCKLING EQUA

LOADCOEFIICIENIS, • , t'1 ANGLES

L--LEGGED

,

r,

(_, I

AND

Ca,

Section

C 1. ,_. 1

15 Nay

1969

Page

o 0,2

0,4

06

0

I0

Ratio

FIGURE

C1.5-9

TOItSIONAL

I;IICKLINt; F()It HAT

14

12

b

LO\I) SECTIONS

COI"I;'I"[CII+:NTS,

C i

ANI)

C2,

1.0

0.$

0.6

I "3 0.4

0.2

o 0.6

0.4 o

-;

0.2

o 02

FIGURE

(.'1.5-10

TORSIONA FOR

Ol

06

L l/IlC CHANNE

08 Ratio

10

1.2

14

b

K LING L()AI) L SECTIONS

COEI,'

I,'ICII';NTS,

C t

AND

CI,

St'¢[io_1

('i..,

1,_ .\1:_\

]'.16,.i

Pagc TABLI.;

I.

SIiE.\I_

CENI'I.]i_ FOR

I"

[XX'.\'I'IONS

VAIlI()Us

CIIOSS

.\NI) SI,R'

\\.\Ill'IN(;

• i

15

Ct)NSI'.\N

i'_

s = shear

center

l'l()NS

t

!

!

t

tf

--

t-

tfh2b3

c = centroid

24

I' = war_ina

constant

tf

,

f

I.t

I

b

t t tf

|

"1

1--

h

._---t w

h

0

tf t

I-

e:h

bl3

b2 b

w

t

"I

tf o

3b2tf e -

6btf •.- t w

tfb3h 2

3btf + 2ht w

I-' =

+ ht w

12

6btf

+ ht w

tf

t t

o

b3h2 12 (2b + h) 2 I

2tf(b 2+bh

+h 2) +3twbh

Section

C 1. S. 1

1S .May

196g

Page

TA gL_E: i

16

Continued/

•

:

Sill

a

a

- Sin

- a COS

cJ

20 a COS

•

o

!-" 2ta$ l_

6 (s'n a - o cos "P-I / cl - sin

!

c

a

cos

a..J

t O is

I' "_

located

A3

at

tl_e

intersection

A : cross-sectional

of

the

two

legs

areo of the engJe

144

o is

Iocoted

o!

the

intersection

of

the

o is

located

at

t_e

interlect_on

of

flange

t

t1 [-

b

f O

e = b(bl)2(3b'2bl),

[2o-i . ,132

two

legs

ond

web

Section 15

.M:t}

Page 1",\ I;I,E

!b_4__

b!i

C i. 7). I l:t(i',,

17

I

:i

,I

; bl _,, t_+6(b*b

.
1)

tt

-F-

Volu,4ms o! • 1.0 0

o

t (small)

e

.I J

b

O8

"h

06

04

02

0430

0 330

0236

0141

0 055

O.I

0 477"

0 3tBO

0280

0

0 087

02

0.530

0425

0.325

0.222

0

0.3

0 575

0 470

O365

0 258

0 138

04

0610

0 5O3

0.394

0.280

0 155

0.5

0.621

0 517

0.405

0 290

0 161

183

I1.5

*t

Values

at e _,

b h |0

04

o2

0.430

0 330

0 2_6

0 141

o O55

0 464

0.367

0 270

0

0 O8O

02

0 474

0 377

0.2gO

0 102

0 090

0.3

0 451

0 358

0.265

0 172

0.085

0 4

0 410

0320

0 215

0 150

o _ss ..... o 36o......

0 _75 o 22s

0,196

0.123

0 O5,6

o tss

o oes

0 O40

o.s ---%_ .... J

06

OI

0

_

0.8

'

173

Section

C 1.5.2

15 May

1969

Page C1.5.2

I

SPECIAL

Continuous

CASES

Elastic

Consider elastically

18

Supports

the

stability

throughout

its

of a centrally length

and

compressed

defined

by

bar

which

coordinates

h

is

supported

and

h

x ( Fig.

y

5-11).

CI.

v

x

YO

FIGURE For

C1.5-11

this

(Reference

case,

8).

SECTION three

They

WlTII

simultaneous

d4u dz "-r"

+

l"_dz-_

+ Yo ,17/

EI x

_

+

V_dz--_

-

- ( GJ

+

k x[u

+

(Yo-

+

koO

=

0

I':I,ASTIC

differential

equations

SUl_polfrs can

be

are:

El y

d4_b EFdz-_-

CON'I'INIJ()US

Xo_rz

+

] +

Io P]dT--_, d2_ - _--

by) 4)l

(Yo-

kx

ky

+

o

-

d2v p (x odz--_

hy)-

(Yo

-

ky[V-

-

=

hy),

-

=

0

0

d2u_ Yodz-_.]

(x °

-

hx)_X

°

obtained

Section C Page I1" the

ends

about

the

rotate take

the

u

x

solution

=

of same

AaP a

art,

simply

y

,axes

but

equations

v

expressions

a determinant manner

as

+

+

A2t¢

supported, with

:

and

the

hene{,

described

AlP

in the

A 2 sin

into

in

+

A0

that

11o rotation

above

nrrz

of these

evaluation

bar

and

of the

A/sin

Substitution

in the is

of the

is,

about

free

19

to \_arp

the

z

1.5.2

nnd

axis,

we

tc_ can

form

nTrz

c5

differential

to a cubic

equations equation

Paragraph

C 1.5.

leads

for

:',-I.

nTr.____z

= A a sin

the

to the

critical

The

cubic

loa_ls, _'qu:_ti_m

= 0

where

Aa

=

-

A

7 A2

=

E

Ix

A +

2

+

k

(n_ _

I)y

+

h yx

IyYo2

2

+

+

k@

Ix x o 2

+

(kx

+

I'J(nrr_ \_-/

+

k)( y

_ +

Ix

GJk_

+ I)l A

/

(n_r_4 ',7-/

III A I

1.22

-

-

E

xY°A

Iykx(Y

+

°

(I x

-

hy)2

+x.e +i_ A

xx

_

A

I Y k Y(X °

+(I_

-

hx_2]

+_0(_x+_y)I(V-)' d

h 2y

+

I xyk h 2x

_)_0+(i +;)'t (v-)':

I( I +I

_o_(_+k)(_)'- %

_- I xxk

A

}r

+

h2 x

+

h2y

)

Section CI. 5.g 15 May

1969

Pa_e _0

Ao

x y

x )

y

xlykx\Yo

+,++,+o+,++o +:i++ ial,l (_ +

+ KlxGJ

ky

+ kkGJ xy

It can on

be seen

n.

The

equation

that

value

must

+ kkk xy_

the values n

which

be found.

The

use

of a computer.

II

Prescribed

of

Axis

Using

the

_2 /x5 o - h y?

+ E

of the

coefficients

to the

minimizes

the

lowest

comph_xity

of this

cubic

positive

solution

equation root

may

depend

of the

cubic

necessitate

the

of Rotation

same

differential

equations

given

in the

previous

paragraph,

we can investigate buckling of a bar for which the axis is prescribed about which the cross sections rotate during buckling. To obtain a rigid axis of rotation, we have only to assume that k : k = _. Then the n axis x y (Fig, C1.5-11) will remain straight during buckling and the _'voss s(,(-tions will rotate with respect to this axis. The resulting differentialequation m:

I EF

[ GJ

-

Taking

the

=

P cr

+ EIyCy °

oA -_ Ip

solution

_F

+ EI

-hy)Z

+

p (x02

of this

Y

-

+ Yo2 ) -

p (hx 2 + h 2Y)I _d2_

equation

I

o

_'-

-

hx

in the form

- h ) 2 + EIx Y

Yo

d'°

+ EIx(X °

x

+ Yo

o

:

_

= A 3 sin

+k0_

nTrx

2 nTr z - hx ..... _- + GJ

y

=0

+

k

_n_

Section

C 1.5.

15 May

19(;!)

Page we

can

calculate If the

bar

P

the

critical

has

two

EF

=

+

buckling

planes

EI

Y

h2

load

in each

of symmetry,

the

EIx h2xj _

+

+

Y

particular

+

21

case.

solution

GJ

2

is:

k0

cr

p

h2 x

In each be found,

must

If the

particular

+

case,

h

the

y

2

+(Io)\--_/

value

of

which

makes

t)

a minimum cr

fixed

axis

of rotation

is the

fn2_'2_ P

n

shear-center

axis,

the

solution

becomes

/ _\

= cr

I O

A

This

expression

III

is

Prescribed

of the

valid

Plane

design

bar

in

is welded

to a thin

with

the

sheet

the

contact

plane

sheet, must

In

are no longer

of this

type,

only

y, parallel and perpendicular

it is

principal axes 1

arises

in which

buclding.

C1.5-12,

plane

deflect

situation

during

in Figure in the

sheet.

axes

shapes.

the

direction

x

and

problems

as

deflect

nn

cross-sectional

of eolunms,

a known

cannot

all

of Deflection

In practical deflect

for

the

of the in the

fib¢'rs

sheet.

advantageous

certain

example, of the

Instead,

direction

to the sheet.

of the cross

For

bar the

fibers if a bar

in contact

fibers

perpendicular to take

the

Usually

this means

centroidal

axes, that the

section.

C 1(

]

FIGURE

C1.5-12

SECTION

WITH

PItESCRIBEI)

fl

PLANE

()I"

along

to the

Di,:I,'I_I':CTI()N

Sectlon

C i._. 2

15 .May 1_{_9 Pa_c, 22 For this case, ( Reference

They

two simultaneous

differential equations

can be obtained

8).

are d4v

_d2v

EI x _

F

+

+

o

E

yo

Elxy

/

Pdz--_

-

-

_d4d)

EI xy \_/v ° -

-

-v-

hy

h Y) d4v _zz4

j_o

d2v Px odz--_

-

d2cb

hy) dz--_-

=

Px o

-

A

P

+

2

PYo

0

-

:= 0

These equations can be used to find the critical buckling loads for a given case. As before, taking simple supports and a solution in the form

ffZ

v

The

7TZ

- A 2 sin-_-

following

_}-

determinant

can

be

A:_sin (,

obtained:

_

rr2

_

1_ol

=

-Elxy(Y

O

-

hy)_-

+

PX O

EI'_-

+ G,J

From which

this determinant

the critical load can

A2_

+ AlP

+

A 0

a quadratic be calculated

=

0

-

+

EIy(y

o

p

equation in each

O

+

for

-

hy)Z}-,y

pyo2

P

0

Ph

Y

_

is obtained

particular

case.

from

Section

(' 1.5.

,)

15 ,1lay f,)o,)' ' Page 23 where I o A

A2

2 -

2

Yo

-x

2 eh

o

y

I

A1

I Op

= - A-

+

+

x

py2

x o

2XoPxy(Y

°

ph

-

-

x y

0

2

--p A

l)y (Yo

-

hy) 2

hy)

I 0

p

A0

= _--zP_

P

=

EI

:

EI

+

p

x

x Py(Yo

-hy)

2

-

-

P2xy(Yo

hy) 2

7r2

x

x-_

7r2

P

y

y_-_ rr2

Pxy

P$

EIxy

-_

=i--

J

+ Er_

O

Io

If the

=

bar

channel, the

is

Ix

substitution

plane

+

A(Xo2

and

the

y

symmetrical

the

independent. the

+I

x axis of

I

The

with

xy

first

=

0

to the become

x

o

EI Y (Yo

-

torsional

Yo

buckling

y the

0,

the

axis,

as

principal two

gives

equation

(:o) the

=

equations

second

-

represents

respect and

The

+

y_

which

yo 2)

y axis

of these

of symmetry.

P

+

the

+

Euler

GJ

y

load

for

case Then,

equations

gives

h l)_-_Z]

in the axes.

this

case.

of a with

become load

for

buckling

in

Section

C1.5.3

15 May

1969

Page

ECCENTRICALLY

CI.5.3 I

General

Cross

In the subjected force,

previous is

COLU/vlNS

LOADED

Section

sections

to centrally P,

24

applied

applied

eccentrically

we

have

considered

compressive

the

loads

(Fig.

buckling

only.

C1.5-13)

will

of columns

The

case

not

be

when

the

considered.

Y

i Ip

I/ P

FIGURE

In considering calculation

investigating the of

the

case the

C1.5-13

stability

of simply

critical

loads

ECCENTRICALLY

of the supported is obtained

APPLIED

deflected ends,

form the

( Reference

LOAD

of equilibrium,

following 8).

determinant

and for

Soclion

C 1.5.

15 Ma 5 Page

1.969

25

P - Py) 0

P

P(Yo

-

ey)

-

P(ex

Px)

_

Xo)

Pey[31

+

Pc;,/_2

I

I O

The lating

solution

of this

P

determinant

gives

the

following

cubic

o

equation

for

calcu-

: cr

AaP a

+

A21_

+

AlP

+

Ao

--- 0

where

Aa

--T

+°/,-

0

A2

=T£A

AI

-

+

I

y x 0

A o

=

_p

P xy

P

=

E1

rr2 x_2

x

P

q)

-(°x

Y

ey )z

[ px(Yo-

-ey[Jl(Px

° +

Py)]-

p y(X °

('}x

x

y Y

_

"

ex)2

py

_

_

exB 2 (Px

.;

py)

P(b)

y

y

c/)

x

(

:l

St,ction

C 1.5.

:'

15 May. 19(;9 Page 26 -

J

_ Ere--, *

o

Io

:

Ix

fll

= Ix

+ Iy

+ A(x o 2 + yo 2)

- 2y°

/3 2 : _y

xadA

+

In the general case, torsion. In each particular evaluated nmnerically for

( xyZdA A

and

:

x

the

X

,

o

buckling

e

:

y

loads

very simple ifthe thrust, P, acts along the shear-

y

o

become

independent

buckling in the two principal planes ently. Thus, the critical load will P

Y

, and

the

load

o

buckling of the bar occurs by combined bending and case, the three roots of the cubic equation can be the lowest value of the critical load.

The solution becomes center axis. We then have

e

- 2x

corresponding

of each

and torsional be the lowest

to purely

other.

In this

case,

lateral

buckling may occur independof the two Euler loads, P , x

torsional

buckling,

which

is:

I ___Op A p P

I e fll Y

II

One

Axis

Another

+ exfl2 +

"

o _-

of Symmetry special

case

occurs

when

the

bar

has

one

(which is true for many common sections). Assuming tile plant: of symmetry (Fig. C1.5-13) , the x 1_2 (, the critical

buckling

loads

is _)btain,'d

in the san,c

plane

of symmetry

that the yz plane is 0. The solution for

manner

as in Paragraph

1.

Section 15

C 1.5.

May

Page A case

of common

()f symmetry;

then

e

interest O.

x

When

metry

takes

place

independently

as

Euler

load.

However,

the

buckling the

are

coupled,

following

/Py

III

Two

-

Axes

If the center simplifies

and

and

quadratic

P/

occurs this and lateral

tile

when

the

happ(,ns,

the

load,

P,

buckling

corresponding

buckling

xz

critical

loads

plane

and

are

plane of s_m-

load

plane

196!)

27

in the

in the critical

in the

corresponding

acts

is

th('

torsional

obtained

from

equation:

P_-

P(e/

_

=

p2 ly °

0

of Symmetry

cross

section

of the

the

centroid

coincide.

the

solutions

of

bar

Paragraphs

has Then I

two

axes

Yo

:

and

Xo II

of symmetry, =

fil

somewhat.

-

the f12

:

:_

shear

0 , which

same

Section

C 1.5.4

15 May Page EXAMPLE

C1.5.4

OF

FOR

TORSION.\L-FI.EXUR.\

L INST;\BILIT_"

COI_UMNS

Example

I

PROBLEMS

1969

"2S

Problem

1 Given: /

t:

14"

=

60

in.

A = 3.5 in.2

S

6

Ix : 22.S in. 4 L

Xo

_-

i-

ly : 6.05 in. 4 |

4

[_

I-

E = 10.5 x 106 psi

"1 G:

c - centroid

4.0x

106pli

J = .073 in. 4

s - shear center Find

critical

general

load

method A.

Method

From

Pcr

applied

at

centroid,

also

use

n_t,thod

and

c.

and

of S('cti()n

the

mode

C1.5.1-IIi.

1

Section

CI. 5. l-Ill,equation

(5),

P "x) [( J(.,,

_ 1-2K

\ro/ From

Table

I,

3b2tf e

3(4)

1.6 6btf

+

ht W

fi(4)('l)

and

tfb3h 2 F

2(,)

-=

3btf

+

2ht w

._

12

6btf

+

hi w

+

6(11

in.

of buckling.

Use

Section

C1.5.4

15 May 1969 Page '29 ('_) l_

(4)3(6)

2

+

3(4)([) 6_4)(])

12

F

38.4

=

I

=

I

I x

+ .2(6)(',)] 6(I)

in._

+

I y

+

AX

2 o

=

22.5

+

6.05

=

54.75

in. 4

+

3.5(2.74)

-

15.(;5

2

O

I O

I 2

o A

o

54.75 3.5

Tr2EIx D

_

7r210.5

x

in. 2

i0 c' x

22.5

.¢

x

p y

__

((;0) 2

_-ZEj = ______ _z

P_

-: r--,z

=

7r210.5

J

x 106 (60) 2

x

6.05

647,691

lbs

174,000

lbs

+

0

10.5 Pcb

-

15.65

4

x

Pq5

=

89,200

Ibs

x

+

106 (. 073)

10G(38.4}_ (60) z

(2.74)2 K

:=

=

I

1

-

15.65

:

0.55

\ro/

' E,

P

2(0.55)

cr

q

/

(89,200

89,200

+

647,691)

+

647, 691)

2

-

-

4(0.55)(89,200)(647,691)]

2]

Section

C1.5.4

15 May Pagt, P

-

82

727

1969

:Ill

lbs

cr

therefore,

critical

load

is _2,727

l_mnds

and the

mode

is torsional-flexural

buckling. B.

Method

Check

Figure

b/a

Since

From

C1.5-4(b)

= 4/6

the

buckling

2

point will

for

= 0.66

plots

are

,

critical

c/a

below

the curve

C1.5-7(a1,

From

Figure

C1.5-10,

From

Section

C 1.5.1-III,

10.5

x

P_b

= 93,500

t_ a-,Z

for

c/a

: 0,

C2

- 0.20

with

= 0.416

the

critical

mode

buckling.

K

-

C 1 -:

0.53 0.31

equation

and

( 7),

-,-

: EA[c, Pd)

of buckling

,

= 0

be torsional-flexural

Figure

mode

10s(3.5)

[ 0.31x-_(0.25]' /

({i + 0.2\601

_21 j

lbs

IrZEI P

p

X

x

cr

f

-

v--

1 2( 0. 53)

-V(93,500

= 647,691

F / k

(93,500

lbs

_- 647,(;91)

+ 647,6917

-

4(0.53)(93,500)(647,691)1 J

of

Sc'ction

C1.5.

15

1:)(;_*

._lay

l)agL ' 31 P

:

S(;,668

lbs

Problem

2

cr

II

Example

c

3.o "I j_

t

1.56

s

Given:

1,89

" '

I y

locate

applied

center

at

Yo

=

4

-2.

and

=

546

+

bl _

+

0.3(3.3)

F

evaluate

To

=

1.68

calculate

Io

Ix

the

+

warping

3 (2.0)3

-- -2.54¢;

-0._;5:;

b2 _ (3)3(2) (3) _

+

/?1,

Iy

f12 refer

A(Yo)_

A=

constant,

3

_- (2) _

in. 6

Io ,

::

(3.0)3

1.89

2

12

F

of

to

load

application

2.1 in 2

L=

b13b23

12

paint

50in.

Q.

3.3(3.0)

b23

t hz f

F

Q =

ly = .88 in, 4

4,0 x 106 psi

point

• hbl 3 = bl 3

center

Ix = 4.43 in4

E = 10.5 x 106 psi

'0.3

load

shear

shear

J = ,053 in, 4

G=

Critical

s =

o.3 x

To

centroid

I_

' "" '-0.2

Find:

=

P_ragraph

C1.5.3-I:

refer

to

Table

I.

t

Section C i.5.4 15 May

1969

Page 32 4.43

I

+ 0.88

+ (2.1)(-0.656) 2

O

= 6.21 in.4

I O

fil _--

.

•

-

2Yo

X

+ 2. 0( 0. 3) (1. 89} _] 1 [(3.0)(0.3)(_1.41)3 /)I - 4.43

+ 3.0(0.2)(0.24) 3

-2( -0.656)

fll

-

fi2

1.66

in.

: 0

_2EI x

7r210.5

pX = _

=

x

106(4.43)

183,633

Ibs

(50)2

v2EI

¢p

36,477

_2

Y

EFIr 2] - 2.1 + _'-FrJ _.21 [i

I

x

+ 10.5

10(;(0. 053}

X (50)2 106(1.

68)7r 2]

O

95,239

p

lbs

|bs

¢o

Now calculate

A3

the

A° Io

coefficients

Iexfi 2

eye,

to th(: cubic

(ey

equation

yo) 2

in Parai_raph

iex

-

C 1.5.3-I:

Yo) 2] - (0.2-0)

A3

A3

2.1 6.21

_0.2)(0)

= 0. 6789

+ (0.89)(1.66)-

(0.89

+0.655)

2

2] + !

Section

C 1.5.

15 May

1_)6!)

l)ag

AE r°

A2

A2

o

y

2. 1 [183,633(-0. 6.2--"1

:

-0.2(0)

A2

-- -276,596

A,

-

h i

-

A'i

( 183,633

[PxPyexfl2+P

o

6.21

(;55-0.89)

2 + 36,

+ 36,477)

- (0.89)

x P y e y fl _

-

31.

Ao

--

A 0

=

Ao

= -

A3I_

Dividing

042

x

-

pp xy¢

P

-

(183,

633)

+

by

477(

x

0-0.

2) 2

(1.66)

(183,633

Py

_

+

A3

(36,

_-36,477(95,394)

x

1012

AlP

+

l_ ,

477)

Ao

+ A2 P Aa

+PxP_

+ 36,477)]

o)

+ 183,633(3(i,477)(0.89)(1.66}

+ _183,633)(95,3._4)J

10 'q

638.985

A2p 2

\

(P x P y +P

83,633(36,477)(0.2)(0)

+ [183,633(36,4771

A 1

×

.,.,

( .95, 394)

.

0

+

A1 P Aa

+ A-AAa

0

,4

Section

C 1.3.4

15

1969

.May

Page

let k = A2 A3

For

k

=

-4.

q

=

4.5725

r

:

0742

-

x

+

aX

x

+

=_

ho A3

10 l° 1014

of cubic,

X3

r

A3

l0 s

x

-9.4123

solution

A_a -

q

let

b

:

P

X

-

k/3

, then

the

cubic

0

where

a

=

b

l_ = --:-(2k 27

Since

:

a3

4

27

<

X k

x

k 2)

-

9kq

+

we

2

27r)

I0 i°

1015

-0.01605

0,

-

-

x

0.259

b2

Q

3

: -0.9605

a

b

i/3(3q

have

x

three

cos

1030

real,

_

+

unequal

120K

roots

,

K

given

0,

where

¢b

=

arc

co

34

.'--

-

d)

:

45"39'

1,

by

2

reduces

to

.%t,ctionC 1.5.4 15 Muy 1969 Page35 X : -109,200 o X_ : 80,324 £-

X2

Substitute

::

28,876

these

roots

into

P

:

x

-

k/3

x

10 '_)

for

critical

F :/

Pl

=

X1

-

1/3

Pl

= -109,200

Pl

=

26,606

Ibs

P2

=

80,324

+

P2

=

216,130

Ibs

P3

: 28,876

+

P3

:

lbs

{-4.

Therefore,

1{14,682

the

0742

-

critical

3

135,806

135,806

load

is

Pl

-

2{;,606

pounds.

load

values.

Sccti_n C1.5. t 13 ._[:lx1!tl;!t Pa_c III

Example

Problem

26

3 c - centroid s - shear center I "= 38,4 in 6 A = 2.1 in, 2 I XX

G = 4 x 106 psi J =

073

ly = 22.5 ino4

L = 60 in.

Find

the

normal the x

critical to plane axis the

load

applied

nt point

n-n critical

(refer load

to Section is

c

= 6 05 in. 4

E = 10 5 x 106

(centr()id)

with l:,,r

(',1.5.2-1II).

prescribed ]':ulcr

psi

deflection I,uckling

:_bout

?r2EI x P

The

-

x

7r210.5

torsional

buckling

load

\ f -/ I o --A

=

10c'(6.05) 174,157

( 60)2

(,

-

Yo

+

2

e

_1 0. 5xl 0_;(22.5)

y_b

P

54.75 2.1

=

146,672

GJ

h 2 Y

10.5x10G(38.4)

P

lbs

is

y

_r2

Py_

x

_ 2

_

(2.74)

( '2. 7,1- J. Z) _ 60

2

,

(1.2)

_ lx 101;( 0. 073)

2

lbs

yo

Therefore,

the

critical

load

is

P

( 146, y(1)

torsional

buckling,

assuming

no

local

failures.

(;72

II)s)

and

tiw

m()de

of

failure

is

Section

C1.5.

o

15 May 1_5_o Pa_c. 3,

Timoshenko,

lo

S. : Strength

D. Van

Nostrand

Seely,

F.

Second

Edition,

Popov, 1954.

E. P. :

o[ Materials.

Company,

Inc.,

Part

New York,

I

and

II, Third

Edition,

1957.

F o

1

J

e

Sechler, Design.

Niles,

1

J

Smith,

John

F.

A.

S., John

J.

Wiley

Mechanics

R.,

Publishing

Edition, o

and

E. E., and John Wiley

Steinbacher, Pitman

e

B.,

O. : Advanced

Mechanics

& Sons,

New

of Materials.

and

Gerard,

and Newall,

S.

P.,

McGraw-Hill

Book

1957.

Prentice-Hall,

G. : Aircraft

Inc.,

J.

& Sons,

New S.:

York,

Airplane

Inc.,

New

and Gere, Company,

J. Inc.,

Bleich, F. : Buckling Strength of Metal Company, Inc., New York, 1952.

and

Mechanics.

1952. Structures.

York,

M. : Theory New

New York,

Analysis

Structural

Chajes, A., and Winter, G. : Torsional-Flexural Numbers. Journal of Structural Division, ASCE 1965, p. 103. Thnoshenko,

of Materials.

York,

Dunn, L. G. : Airplane Structural & Sons, Inc., New York, 1942.

Corporation,

Wiley

Inc.,

York, Structures.

Vol.

II,

Third

1943. Buckling of Thin-Walled Proceedings, August

of Elastic

Stability.

1961. McGraw-Hill

Book

SECTION (2 STABILITY OF PLATES

-,.,...J

TABLE

OF

CONTENTS Page

C2.0 2.1

STABILITY

OF PLATES

BUCKLING

OF FLAT

2.1.1

Unstiffened I. II. 2.1.1.1

•

•

•

Plates

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

1

....................

1

......................

Plasticity Cladding

Compressive Shear

III. IV.

Bending Buckling

V.

Special

1

Reduction Reduction

Factor Factors

Plates

.............

Rectangular

II.

I. H.

•

PLATES

I.

2.1.1.2

2.1.2

•

Buckling

Buckling

....... .......

2 4 4

...........

4

................

7

Buckling .............. Under Combined Cases

Loads

................

Parallelogram

Plates

....

8 8 15

...........

16

Compression ................. Shear ......................

16 17

2.1.1.3

Triangular

18

2.1.1.4

Trapezoidal

2.1.1.5

Circular

Stiffened 2.1.2.1

Plates

Plates

..............

Plates Plates

............. ...............

Conventionally

21

Stiffened

Plates

...............

I.

Stiffeners

Parallel

H.

Stiffeners

Transverse

22

to Load

Conventionally Stiffened in Shear ....................

C2-iii

20

.......................

in Compression

2. I. 2.2

19

........

to Load

......

22 26

Plates 26

TABLE

OF

CONTENTS

(Continued) Page

2.1.2.3

Conventionally Stiffened in Bending ...................

2.1.2.4

Plates Stiffened in Compression

2.1.2.5

2. 1.2.

6

2. 1.2.7 2.2

CURVED 2.2.

2.2.

1

2

Sandwich

31

Plates

2. 2. 1.1

Compression

2. 2. 1.2

Shear Curved

Curved

2. 2. 3.1

Curved Stiffeners Plates

Under

C2-iv

................

with

32 .......

with

in Shear

Plates

with

33 Circumferential 35 ............

37

Axial

.................... Plates

with

38 Circumferential

.................... Combined

33

Axial

....................

Plates

Stiffeners

32

.................... Plates

Curved

...........

in Compression

Plates

Curved

31

32

Buckling

Plates

Stiffeners

Curved

......

.................

Buckling

Curved

2. 2. 2.2

2.2.4

Material

32

Plates

Stiffeners

2. 2. 3.2

................

of Composite

Curved

Stiffened

27

.............................

2. 2. 2. I

2. 2.3

with Corrugations ...............

29"

Unstiffened

Stiffened

27

Plates Stiffened with Corrugations in Shear .....................

Plates

PLATES

Plates

39 Loading

.......

39

TABLE

OF

CONTENTS(Concluded) Page

2. 2. 4. 1

REFERENCES

Longitudinal Plus Normal

2. 2. 4. 2

Shear

2. 2. 4.3

Longitudinal Plus Shear

.......................................

Plus

Compression Pressure Normal

............

Pressure

Compression ....................

40 ........

41

41 125

C2-v

Section

C2

1 May Page SECTION

2. 1

BUCKLING This

of flat

plates.

common

may

plates

subjected

should

be made

2. 1. 1

contains

Various

geometrical

be used

following

design

information

shapes are

in either

to thermal

the

elastic which

D4. 0.2,

for predicting

under

several

considered.

gradients

to Section

OF PLATES

PLATES.

structures

UNSTIFFENED With

STABILITY

FLAT

section

to aerospace

presented

"Thermal

types

In most

or plastic may

stress

cause

Buckling

the buckling of loading

cases

the

range.

buckling, of Plates.

methods For

reference "

PLATES.

few exceptions,

plate

critical

stress

equations

take

the

form:

F

= _?_ cr

where

OF

C2.

1971 1

the

terms

F cr

kTr2E 12 (1 - _ 2) e

are buckling

defined

plasticity

which

reduction reduction

k

buckling

coefficient

E

Young's

Modulus

elastic

includes

the

(psi)

cladding

e

(1)

as follows:

stress

and cladding

2

Poissonts

factor factor

(elasticity) ratio

(psi)

effects

of plasticity

Section

C2

1 May 1971 Page 2 t

plate

b

dimension

The excluding

its

the

material,

v e,

_7, and

thickness

for

must

only

the

establishing

curves

are

plate

critical

to advantage

sibly

the first

Whatever the tion

material

the

carry

values

used

more

inelastic range

plate

values

buckling

boundary of

For of

E,

k

coefficient,

conditions.

By

be read

directly.

can

is an important

A plate

the natural

edges.

above. of the

surface

load.

at the

the proper

values

dimensions,

will buckle

This the

into

principle

in

a "natural"

has

efficiency

wavelength

factor

been

of the flat of buckling

sheet. can

be

has

been

load.

Factor of theoretical so-called

by design for

on the

equation

to increase

of the

expressi.on

in the plastic

a/b,

in. )

of support

and various

stress.

amount

value

only

to find values

means

Reduction

to the

in the

used

dimension,

used,

the

to a minimum

A tremendous relative

level

into

buckling

will

Plasticity

done

stress

ratio,

structural

the plate I.

condition

in structures

if by any

prevented,

the

of the buckled

corresponding

is,

depends

conditions aspect

short

k,

be substituted

wavelength

the

wavelength

That

upon and

loading

The

applied

and

temperature, _

(usually

constant,

thickness,

numerous

knowing

of plate

buckling

Buckling k,

(in.)

is nonlinear,

involve

range;

experimental

work

correction

factor.

plasticity

engineers

r/, it must stress

and

and,

a resort

were

7? = Et/E

a measure since must

the

or

Pos-

q = E sec /E.

of the stiffness stress-strain

be made

to the

of rela-

Section C2 I May Page

stress-strain curve to obtain a plasticity correction factor. tion

is greatly

simplified

stress-strain

curve

by using

which

the

involves

Ramberg

three

and

simple

1971 3

This complica-

Osgood

equations

for

the

parameters:

n

F0.7

where

F0.?

n = 1 + 10ge(17/7)/log

e (F0.?/F0.85),

and

the

terms

are

defined

as

follows: F0.7

secant and

n

yield

stress

a slope

a parameter curve

F0. 85

stress

materials;

some

drawn

which

describes

on the yield at the

There considered differs

sented

unsafe with

established

is usally to stress

given

for

each

design.

in Table

C2-2.

for

A check

the

shape

curve

F0. 7 , F0.85,

in Table or

the material.

of loading,

origin

of the

a maximum,

type

curve

of the

stress-strain

by a line

of slope

of

origin

values

the

of the

region

the

are

the intersection

from

intersection

1 gives

of these

as

0.7E

0. 85E through Reference

taken

Suggested should

many

flight

vehicle

C2-1. "cutoff" The

and may

and

stress,

value

vary

according

values

of the

be made

above

of this

cutoff

which stress

to the design cutoff

to ensure

stress that

it i_

are

criteria pre-

the buckling

stress is equal to, or less than, the cutoffstress. With the use of the Ramberg-Osgood

parameters,

plasticityreduc-

tion factors will be given for various types of loading in the paragraphs which follows.

Section C2 1 May Page II.

cally

Cladding

alloy

sheets

aluminum

and

is widely

is referred

to as alclad

properties

of this

the clad

located

where

Figure

C2-I

stress-strain correction clad

must

covering

factors

stitute

one

Rectangular

plates

vehicles.

webs,

panels,

and

of the

sheet

alelad

and

sheets

Such

than

strength

sheet, takes

Fig.

material. it is

place.

C2-2

combinations.

shows

Thus,

of the

simplified

material

the core

alclad

and

of practi-

mechanical

buckling

because

I gives

covering

lower

cladding

the

a further

strength reduction

C2-3.

elements

to loads

encountered

plate

simulation

which

cause

instability

in the

structural

occurs

in such

con-

design areas

of as beam

flanges.

Compressive

boundary

lower

of an alclad

subjected

Rectangular

Figure

restrained.

major

The

fibers

core,

a thin

structures.

alloy.

when

in Table Plates.

in aircraft

value

Reference

Rectangular

space

I.

for

material.

of the

their

cladding,

be made

as summarized

2. 1.1.1

as the

attain

for

with

is considerably

the makeup

curves

used

at the extreme

strains

shows

available

aluminum

material

is located the

are

or clad

clad

4

Factors

Aluminum pure

Since

Reduction

1971

Buckling

C2-3

shows

conditions In Fig.

C2-3(a)

are

the

change

changed the

sides

in buckled on the are

shape

unloaded

free;

thus,

of rectangular

edges the

plate

from acts

plates

free

to

as a

Section C2 1 May 1971 Page 5 column. In Fig. C2-3(b) oneside is restrained and the other side is free; such a restrained plate is referred to as a flange. In Fig. C2-3(c) both sides are restrained; this restrained element is referred to as a plate. Critical compressive stress for buckling of plate columns (free at two unloadededges) can be obtainedfrom Fig. C2-4. As can be seenfrom this figure, a transition occurs with changing b/L_ the varying value of ¢ betweenthe limits load-carrying bending the

capacity

effects

equation

in the

ratio

to the C2-5

in equation of the

rotation.

effect

letter

of the

opposite

The

buckling

forces

located

in Reference

P

cr

2.

-

For

column, For

for

boundary,

a free

curves edge

edge

condition

of a rectangular

simply

plate

midpoint supported

2b

The

increased

to antielastic

columns,

_

the buckling

¢ = 1 - v 2, e

means and

clamped SS means

that

-

coefficient,

conditions

for

long

k,

and

or fixed

a/b against

simply

supported

plates,

(a/b)

> 4,

is negligible. compressed

of its

long

sides,

the

side

by two equal (Fig.

following

___fl 2

7rfl tanh

narrow

or edge,

it can be seen support

at the

< _ < 1.

¢ = 1, is due

finding

C on an edge

means

Ve2)

equation.

curves

letter F

these loaded

Euler

for various

The

From

plate

at buckling.

gives

(1)

plate. The

or hinged. the

plate

reduces Figure

to use

of a wide

(1 -

values as evidencedby

c°sh 2 -7-Trfl

and

C2-6)

is given

equation

is true:

(3)

Section

C2

1 May 1971 Page 6 For

long

plates,

P

If the

P

sides

(#)

rotational flange

that

to

(4)

are

clamped,

the

solution

reduces

to

8_TD

(5)

b

Figure restraint

reduces

"

of the plate

=

cr

> 2, this

4_D b

=

cr

long

(a/b)

C2-7

along

rigidity has

shows

the

sides

of the

one

curves

for

of the

sheet

plate.

edge

Figure

k

for

C

panel, C2-8

free

and

the

other

gives

the

k

factor

various where

shows with

degrees #

is the

curves

various

of

for

ratio k

degrees

of

for

C

a

of edge

re straint.

extremes weak

Figure

C2-9

of edge

stiffener,

stiffener

a relatively

and

a hat

strong

the

plasticity

section

for

reduction

Paragraph

2. 1. 1 along

compression

buckling Cladding

namely

stiffener

To account

c

for a long

a zee-stiffener

which

is a closed

sheet

which

panel

with

two

is a torsionally

section

and,

range,

one

therefore,

torsionally. buckling

factor.

inelastic

By using

with

Figs.

stress

for

reduction

in the

C2-i0 flat

factors

the

Ramberg-Osgood

and

C2-i

plates should

with

i,

one

various

be obtained

must

obtain

parameters can find

boundary from

of

the conditions.

Paragraph

2. 1.1.

Section

C2

1 May 1971 Page 7

II.

Shear The

various

Buckling

critical

boundary

elastic

shear

conditions

_2k s cr

buckling

is given

stress

for flat

by the following

plates

with

equation:

E ((_)

\ol 12(l-v

2) e

where

b

shear. plate

is always

The aspect

clamped

the

shorter

shear

buckling

ratio

a/b

in Fig.

the

produces

plate

to buckle the

buckle

has

tabulated

Thus,

in patterns patterns

been

have

can shear

is plotted

simply

a long

these

edges

carry

as a function

supported

rectangular

stresses

compressive

of the

edges

by Cook

and

Rockey

subjected

to

at 45 degrees

cause

the

as illustrated

of 1.25

plates and

on planes

edges

length

plate

stresses

to the plate

of rectangular

investigated

long panel

in Fig.

C2-13;

b.

with

mixed

[ 3].

The

boundary results

condiare

C2-4. occurs

correction

be taken secant

that

a half-wave

buckling

in Table

a plasticity

the

for

compressive

at an angle

If buckling

factor

C2-12

to note

internal

edges.

Shear tions

k s,

as all

edges.

shear

with

of the plate,

coefficient,

It is interesting pure

dimension

factor

as

modulus

at a stress should

_?s = Gs/G as obtained

above

be included

where from

G

the

proportional

in equation

is the

a shear

shear

limit (6).

modulus

stress-strain

stress,

This

and diagram

Gs

Section C2 I May 1971 Page 8 for the material.

Also, Fig. C2-14 can be used for panels with edge rota-

tional restraint if the values of (70. 7 and n are known. III.

Bending The

Fb

Buckling

critical

-

elastic

12(1-u

buckles

equal

(Fig.

C2-15).

cient

kb

for

Thus,

to be larger

in the

C2-17

gives

supported

and

tong smaller

than

kc

gives

relatively

with

buckle

flat

compression

plates

is

a/b

wavelength

supported cause

edges

the

buckling

coeffi-

k.s

with

stress

all

edges

the case side

of

short

simply

patterns

the critical

for

as a function

plates

or

of the plate

coefficients

the

it involves

the

C2-16

plane

the

coefficients

for

(7)

in bending,

to (2/3)b

Figure bending

stress

e

a plate

buckles

buckling

2}

cr When

bending

for

coefficients simply

when

Figure

various

degrees

a plate

supported.

the plate

is fixed.

for

tension C2-18

Figure side

gives

of edge

in

is simply

the

rotational

restraint. The using

simply IV.

plasticity supported

Buckling Practical

The

buckling

use

of interaction

reduction

can

be obtained

from

Fig.

C2-10

edges. Under design

strength

factor

Combined of plates

of plates

equations.

under

Loads usually combined

involves loads

a combined will

load

be determined

system. by

Section

C2

1 May Page A.

Combined

The and

longitudinal

Rbl.

+

This for

various

interaction

RC

Rb2

the

+

expression

is

plotted

Bending

:

1.0

for

margin

ltb2

also

combined

bending

C2-19.

Also

shown

are

curves

and

Shear

equation

for

combined

bending

and

shear

is

,

(9)

of

safety

-

+

C2-20

are

for

S. ) values.

1

Figure values

in Fig.

(M.

= J

accepted

(8)

interaction

Rs2

is

1.0

Combined

M.S.

M.S.

=

margin-of-safety

The

that

Compression

is

equation

B.

and

equation

compression

75

and Longitudinal

Bending

1971 9

R

is

S

is

1

(10)

2

a plot

shown.

of equation

R

is

the

(9).

stress

Curves ratio

showing

due

various

to torsional

shear

s

stress

and

R

is

the

stress

ratio

for

transverse

or

flexural

shear

stress.

S

C.

Combined

Shear

and

Longitudinal

Direct

Stress

(Tension

or

Compression)

RL

and

the

+

The

interaction

R2S

=

expression

M.S.

for

equation

for

combination

margin

of safety

is

is

2

+

of loads

(11)

1.0

= (RL

this

JRL

-1 2+4R

)s2

.

(12)

Section

C2

1 May 1971 Page 10 Figure

C2-21

tension,

it is included

pression

allowable. D.

is a plot of equation on the figure

Combined

bending, This

and shear

figure

margin R

than

the

are

whether

of safety.

Given

defined

given The

determined

of

margin

from

Rc = 0. 161,

for

specific

value

origin

0 through

point

yields

the allowable

safety

calculations,

under

combined

plate

will buckle

the

Rc,

Rs,

ratios value

C2-23.

the

panel

iNote:

and will

and and

the

and bending When

E.

in other

cases

lines

buckling

of simply

investigation

supported

fiat

the

of the

numerically

plates

indicate

dashed

than

line curve

determined

from

margin

use

the

at point

the desired R

be

application

1 is the first

Rc/Rs for

may

a typical

diagonal

related

is less

the left

C2-22.

not give

If the value

flat

Point

the

right

2, of

half

S

half.

)

Combined Longitudinal Bending, and Transverse Compression A theoretical

the

use

com-

of Fig.

is greater

C

of the figure;

the

buckle.

stresses

R

Rb:

buckled

The

is

compression,

but will

RS

R b = 0. 38. R b.

using

curves

and

of elastically

1, intersecting shear

Rb

The dashed

Rs. "= 0.23, Rs

of

stress

and Shear

by the interaction

R c, then

of

compression

or not the

of safety

Fig.

where the

for buckling

by the given

value

If the direct

Bending,

represented

tells

curve

C

as negative

Compression,

The conditions

(11).

Longitudinal

by Noel rectangular

[4] plates

has

Compression,

been under

performed critical

on

Section

C2

1 May Page combinations

of longitudinal

compression.

Interaction

for

various

case

of the the

reduction

compression

is greatly

longitudinal

compressive

infinitely support

long along

Bending,

bending) The

plates.

edge

resulting

Batdort

and

It was

found

be applied

sive

stress

simply

The

clamping

curves

are

case

that

and

shown

is shown

C2-24

limiting

to zero.

curves

addition

lateral in Fig.

the

equal

to the

The

indicate

additon

of only

of lateral

a small

required in Fig.

simple the

aspect

modification

C2-27.

considered

support

C2-25

simple (due

to bending)

shear

edges

in the

Batdorf

for

[ 5] for

tension

acting

to

edge.

and

found

finite

alone,

elastically

critical'stress

and

trans-

C2-26.

with

ratio

the

(due

and

reduction

and

were

along

and

of the

buckling.

shear,

and Buckert

compression

any

Compression

bending,

of support

longplates

without

of finite

Transverse

by Johnston

fraction

to produce

plates

condition

for

these

due

compression

[ 6] examined

plates

by)

combined

in Fig.

of transverse

necessary

long

for

along

to the plate

infinitely

and

two types

an appreciable

supported

Shear,

established

long edges,

Houbolt

may

been

with

In the

ratio

stress further

presented

be used

stress

by the

surfaces

have

both

can

verified

bending

are

and

load.

Combined

compression

loadings

one

up to (and

magnified

compression,

curves

by setting

allowable

Interaction verse

these

These

leading

in the

longitudinal

for

ratios.

conditions

studies

F.

curves

aspect

of two loading

results that

plate

bending,

1971 11

restrained.

in pure

transverse Stein that

aspect

the

shear

compres[7]

examined

curve

ratios.

for This

Section

C2

I May 1971 Page 12 Combined and Shear

Go

Johnson compression,

plates.

C2-28

procedure

line

to theyk/k

Compression,

examined

data

To make

1.

Calculate

2.

On the curve

from

the

the

origin

ratios

of k/k

supported

graphically curves,

the

flat

in following

At the

intersection

s ratio

of step

1, read

Determine

{f ) x appl.

x-k/ks

= Nx/N s and to the

ky/ks

plate

a/b,

= Ny/Ns" lay

off a

s. of this

ky

required

k x

simply

presented

of these

corresponding

3.

=

use

for

of longitudinal

be observed:

4.

N

are

Compression,

combinations

and shear

calculated

C2-32.

Transverse

critical

compression,

The

through must

straight

[ 8] has

transverse

rectangular Figs.

Longitudinal

and/or plate

_2E

x

line

and

the

curve

corresponding

ks . thickness

from

t3

(t) = 12(1

- v 2) e

b2

or

t3 = req'd,

(N) B x appl.

(N) B Y appl.

_-

k

=

k y

x

(N) B s apph k s

where 12(1 - v 2) b 2 e

B

_.2E If desired,

the

value

of

k

y

may

be determined

from

k

s

and

k /k . y s

Section C2 1 May 1971 Page 13 5. Determine the margin of safety. Assuming that increase other

at the

same

at all load

M,

levels,

=

S,

rate

and are the

therefore

margin

of safety

td

given

cr

is the

-1

=

on load

is given

by

based

on stress

-1

thickness.

Margin

of safety

is

:

t = 0.051in., N

S

a simply

E_ = 0.30, e

= 80 lb/in.

supported

the

stress

ratios

kx/k s = Nx/N s = i00/80

the

loads.

origin

k/k

of slope

s = 0.4

From

k

y

the k /k x

C2-33,

, = 0. 8.

with

a = 10 in.,

= 1001b/in.,

x

N

b = 5 in. ,

y

= 321b/in.,

of safety.

and

load

ratios:

0.4

= 1.25

interaction

S

determines

Fig.

kx = 2.5 therefore,

C2-33,

N

the margin

ky/k s = Ny/N s = 32/80=

On Fig.

plate

E = l0 Tlb/in.2,

Determine

Calculate

curve

ttd) tr ,d 21

\fappl 1 Consider

_-

to each

by

EXAMPLE

from

based

proportion

treq'd.

design

M.S.

and

same

loads

(N) (t )3 Nappl.

where

in the

the

the

ks = 2.0

= 1.25.

the

The

critical

following

,

curves

and

for

a/b

intersection

buckling values

are

= 2, lay of this

coefficients obtained:

ky/k s = 0.4

;

off a line

line

for

with

three

the

SectionC2 1 May 1971 Page To determine

any one of the three the

following

the

plate

buckling

value

thickness

required

coefficients

determined

to sustain

may

these

be used.

14 loads,

Using

kx,

is obtained: 1/3 X

treq'd.

Ve 2)

=

=

k 7r2E

0.048

in.

X

Since based

the actual

on stress

M,

design

thickness

t°l

- 1 =

tr_q,d. of safety

M.S.

= (

td treq'd.

Bieich to combined Fig.

based

Combined

H,

shear

C2-33.

The

o_>- 1:

the margin

of safety

is

Si

The margin

is 0. 051 in.,

k

on load

- 1 = +0. 1289.

0. 048 /

is

and Nonuniform

[ 9] presents

and nonuniform

=

0. 051_2

_3 _ I = +0.1995 ]

Shear

critical

(

buckling

a solution longitudinal coefficient

- 1.0+

3.855'2fl

Longitudinal for buckling compression

Compression of a plate as shown

subjected in

is for

1+

4)

(13)

where 4 . _?=

5.34+ 7.7

;

-j

Section

C2

1May

1971

Page

1-
1: k

j

= 3.85T2fl

fl2+3

15

( J -1+

1+--

\

(14)

72fl 2

where 5.34

4-

_2

T

= 7.7+33

V.

(i-_)

Special Cases A.

EfficientlyTapered When

brium,

3

Plate

a tapered plate has attained the state of unstable equili-

instabilityis characterized by deflections out of the plane of the plate

in one region only.. The other portions of the plate remain essentially free of such

deflections.

design,

since

This

the

same

condition

loading

distribution

by a lighter

plate

loading

be characterized

by deflections

Pines

[ 10] have

this

will

reason

simply Fig.

and

supported C2-34.

by shear

stresses

aspect

ratio

subjected

load

variation

small of the

in such

Gerard

plate

The

characteristics plate

tapered

of instability

plate.

is plotted

presumably

a manner

that

to have

The in Fig.

instability

examined

the

C2-34

was

under

the

entire

plate.

influence

various

specified For

tapered

as shown

assumed

buckling for

be sustained

an exponentially

negligible

resulting

the

loads

plate

an inefficient

could

throughout

to compressive along

enough

constitutes

in

to be produced upon

coefficient amounts

the

buckling

versus

the

of plate

taper.

Section C2 1 May 1971 Page 16 B.

compressed Reusch axial

Compressed

Plate

The

of determining

fiat

rectangular

[ 11 ] for load

(Long

problem

a simply

gradient.

plates

will C.

buckle

buckling

elastic and

to a plate

with

2. 1.1.2

pattern. well

plate

the buckling

plate

with

appearing

stress

of an axially

by Libove,

constant

in Fig.

where

long,

It is shown

Ferdman,

thickness

and

depict

their

C2-35

the maximum

drastically

attached

foundation. Plates.

Parallelogram

plates

technology

the the

may

some

effect

load

and a linear results.

is applied.

basic

load

respect

loading

problem

)

of the

plates

are

of the

webs

plate

as compared

in an oblique

plates

available

conditions

on an

of the plate

or

to such

com-

resting

of nonattachment

in beam

solutions

the

supported

buckling

with

several

for

simply

exist

of analysis

However, for

a solution flat,

that

reduces

coefficients I.

Loading

investigated

obtained

of infintely

developed.

was

end

Parallelogram

The

buckling

at the

[ 12] has

foundation.

foundation

curves

Variable

Foundation

Seide pressive

plate supported

The

Elastic

with

panel

is not very

which

present

and boundary

conditions.

Compression

with

Wittrick

[ 13] has

clamped

edges

Results

in the

of these

curves

form with

examined

for

of buckling those

for

the

case curves

rectangular

the

buckling

of uniform are

stress

of a parallelogram

compression

shown

in Fig.

plates

shows

in one direction.

C2-36. that

for

Comparison compressive

Section C2 1 May 1971 Page 17 loads, parallelogram plates are move efficient than equivalent area rectangular plates of the same length. References 14 and 15 contain solutions for simply supported parallelogram plates subjected to longitudinal compression. A stability analysis of a continuous flat sheet divided by nondeflecting supports into parallelogram-shaped areas (Fig. C2-37} under compressive loads has been performed by Anderson [ 16]. The results show that, over a wide range of panel aspect ratios, such panels are decidedly more stable than equivalent rectangular panels of the same area. Buckling coefficients are plotted in Fig. C2-37 for both transverse compression and longitudinal compression. An interaction curve for equal-sided skew panels is shown in Fig. C2-38. Listed in Table C2-5 is a completion of critical plate buckling parameters obtained by Durvasula [17]. II. Shear The buckling stress of a parallelogram with clamped edges subjected to shear loads has also been investigated by Wittrick [ 18]. It is worth noting that the shear loads are applied in such a manner that every infinitesimal rectangular element is in a state of pure shear. For such a condition to exist, the plate must be loaded as shownin Fig. C2-39. To signify this condition, the shear stresses are drawn along the y-axis in Fig. C2-40. As might be expected, unlike a rectangular plate it was found that a reversal of the

Section C2 1 May Page direction of the shear load causes a change in its criticalvalue. shear stress value occurs when

1971 18

The lower

the shear is tending to increase the obliquity

of the plate. The smaller criticalshear stress values are plotted in Fig. C2-40. Table C2-5 presents criticalshear stress parameters

for both directions of

shear for several plate geometries. 2. i. I. 3

TrianBular Plates. Several investigations have been performed

Cox and Klein [19] analyzed buckling for normal triangles of any vertex angle.

on triangular plates.

stress alone in isosceles

The results are shown in Fig. C2-41.

The

buckling of a right-angled isosceles triangular plate subjected to shear along the two perpendicular edges together with uniform tions has been considered by Wittrick [20-23]. boundary

compression

in all direc-

Four combinations

conditions were considered, and the buckle is assumed

symmetrical

about the bisector for the right angle.

interaction curve in terms of shear and compressive

of

to be

Figure C2-42 depicts the stresses.

ing cases, these results agree with those of Cox and Klein.

In the limit-

In Wittrick's

study it was shown that for a plate subjected to shear only, the criticalstress Is changed considerably upon reversal of the shear. interaction curve is unsymmetrical

Because

of this, the

'and the criticalcompressive

be appreciably increased by the application of a suitable amount

stress can of shear.

Section

C2

1 May 1971 Page 19 2. 1. 1.4

Trapezoidal Klein

ported

flat

Plates.

[24]

plates

has

of isosceles

along

the parallel

edges.

edges

so that

ratio

given tion

plate. function

along

the

where

any

obtained

loads

of axial

loads

assumed

not satisfy

sides are

Shear

does

edges.

comprise

shown

Pope

a large

in Fig.

[25]

has

C2-43

tapered

symmetrically

in plaaform

loading

on the parallel

ends.

1. brium

being

Different

the

act along

the parallel

edges

to obtain

results

of the

and

of a plate

subjected are

stresses flows

for long

Buckling

plates curves

C2-44.

the buckling

by shear

for

edges.)

deflec-

moment

enough

incorrect

of the

The

for

accurate

plate

sloping

results.

condition

are

sup-

in compression

his

to be more

Two cases

uniform

maintained

and

analyzed

to act along

the boundary

percent

of simply

assumed

used

appear

loads

loaded

may

the

results

buckling

planform are

was

However,

{His

elastic

trapezoidal

method

purposes. the

the

A collocation

sloping

practical

determined

of constant

to uniform

thickness

compressive

considered:

applied along

normal

the

sides

to the {Figs.

ends,

equili-

C2-45

through

C2-56). 2. of the

sides

Equal prevented

uniform normal

stresses

applied

to the

to the direction

ends,

of taper

with

{Figs.

displacement C2-57

through

either

simply

C2-60). Boundary

conditions

supported

or clamped.

are

such Pope

that has

opposite used

pairs

a more

of edges rigorous

are analysis

than

Klein;

Section C2 I May 1971 Page 20 and for comparable are more

plates, Pope's results (which represent an upper bound)

correct and will give buckling values lower than Klein's. However,

the range of applicabilityof Pope's curves is limited to taper angles, 0, of less than 15 degrees. 2. i. I. 5

Circular Plates. The buckling values of circular plates subjected to radial com-

pressive loads (Fig. C2-61) have been investigated [2]. It has been shown that the criticalbuckling stress for a circular plate with clamped

f

r

edges as shown

is

14. 68 D

-

(15)

a2t

cr

Similarly, critical

in Fig. C2-61

stress

for the

case

of a plate

with

a simply

supported

edge,

the

is 4.20D

f

-

r

The with

clamped

N

case

=

has

investigated

subjected [ 26]

to unidirectional and

found

compression

to be

(17)

a 2

plates

compressive

stress

been

plate

32 D

Circular

ling

of a circular

edges

O

to radial

(16)

a2t

cr

for

these

with

forces plates

a cutout have

is

also

center been

hole

of radius,

investigated.

b, The

subjected

critical

buck-

Section

C2

1 May

1971

Page f

where

kD

r

(18)

cr

the

a2t

values

2.1.2

of

Critical rigidity

increasing to the

the

stability tudinal load

used.

by introducing

and

local

small,

stiffened

"Local

Buckling

plates

are

stability

of the

plate

analysis

of flat,

smaller plate

of the along

therefore,

of Stiffen.ed

rectangular

carry

a portion

The

stiffeners nearly

by

with

respect

is obtained

by

increasing

the

plates

longi-

of the

with

compressive

considerably

stiffened "

plates

should

general

mode

increasing

account

for

local

along)

the

stiffener-skin

The

local

plates

exists,

for

both

of instability

while,

two modes

neglected.

Plates.

and

For

stiffened

these,

integrally

solution

thus

the

will buckle.

(or

between

upon

can be increased

as possible

panels,

of instability.

nodes

economical

not only

dependent

will not be economical

ribs.

the

and

buckling

as small

at which

coupling and,

plate

into

modes

plate

a design

the plate

with

Some

C2-62.

A more

stiffeners

by deflection occurs

such

reinforcing

the

stress

juncture. usually

of the

for

The

of material

characterized buckling

plate. but

but subdivide

general

of load

thickness,

stiffeners,

in Fig.

PLATES.

thickness

critical

given

values

Stability

ally

are

of the

its

weight

keeping

the

k

STIFFENED

flexural

21

is

instability,

but this

instability is presented

effect

is

of conventionin Section

C4,

•

Section C2

°

1 May

1971

Page This plate.

It should

ultimate the

section

load

is concerned

be emphasized,

is distinctly

two must

be calculated

different

2. 1.2.

1

using

Buckling stiffened

plate

In this plate

presented parallel

be determined

k

of the

of finding

the

the buckling

of sheet-stringer

load,

t

is the

for to the

the

combinations

in Compression.

genera[ from

instability the

general

of a conventionally equation

A.

t,

which

(Fig.

C2-63).

of several of the

skin.

k for

both

and

case

where

the

the

Parallel

The is

Plate

a rectangular

area

with plate

by a longitudinal of the cross

I, taken

the

of the

tables

stiffened

and charts

case

where

stiffeners

are

the

will

be

stiffeners

are

perpendicular

to

to Load

Supported

is reinforced

parameters

Design

of

Simply

of inertia

of the

evaluation

Consider ness

(19)

thickness

load

and

2

is a function

Stiffehers

moment

load

12(1 -u 2)

case, and

problem

of finding

loads

Plates

from

c

cr

the

buckling

C1.

Stiffened

k 7r2E F

that

Ultimate

resulting

may

that

from

Section

Conventionally

the critical

however,

not be confused.

should

with

22

with

section respect

One

Stiffener

of length

a,

stiffener of the to the

or Centerline width on the

stiffener

axis

coinciding

b,

and

thick-

centerline is

A,

and

with

the

its

Section

C2

1 May

1971

Page outer

surface

as small

of the

and

flange.

12(1

Db

plate

the

below with

notation

is regarded

is used:

- v 2) I

(2O)

b t3

(21)

7

is the

of width

to the

shown

when

following

stiffener

A

area

If the as

the

of the

bt

coefficient

stiffener

Also,

rigidity

- --

5

of the

torsional

will be neglected.

EI

The

The

23

ratio

b,

and

bt

of the

stiffener C2-63(c).

rigidity

ratio

7

will

rigidity

ratio

of the

of the

stiffener

to that

cross-sectional

area

of the

plate. straight

the

buckling

antisymmetric

is larger

occur.

flexural

is the

This

the symmetric

the plate

6

remains

in Fig.

7 o,

of the

than

form

At the

ratio

T°

the

following

is antisymmetric

displacement

a certain

displacement

mode

value

in which both

form

7o. the

For

will

values

stiffener

configurations

are

occur

of

deflects equally

possible. Bleich

To where

11.4_

a = a/b

inertia,

[ 9] has

and

I , to keep O

IO

= 0.092bt

+

derived

(1.25+165)_2

0 -< 6 < 0.20. the

stiffener

3 7°

formula

for

"YO:

(22)

5.4_aUsing

straight

this,

the

required

moment

of

is

(23)

5'

Section

C2

1 May Page Timoshenko various The

[ 2] gives

parameters,

values

of

proportions when

c_, 5, and

k

above

of the

the plate

case

of bi/b

stiffener

Also, is equal

located = 1/3,

this

results

are

which

in equation

given

in Table

a value

undeflected

3+0.4At

for

as shown

he determines

(or

Having

solutions

stiffener

to remain

(19)

in Table

C2-6

indicate

for

C2-6. those

the

stiffener

remains

One

Stiffener

Eccentrically

a rectangular

in Fig. for

during

2

greater)

Simply For

equal

panels,

given

in Table

the

to be used

straight

the

C2-64.

of I, the

For

moment

critical

stiffened the

of inertia

buckling.

with

particular of the

It is

(c_ -< 1)

value

plate

.

(24)

buckling

coefficient,

k,

to 10. 42. C.

Bleich

for

Plates

[ 9] obtains

= 1.85bt

with

k

lines

and plate

Simply Supported Located

required I

These

the horizontal

stiffener

Bleich eccentrically

%

of

buckles.

B.

one

values

1971 24

has

stiffener

_o

Supported

the

case

Timoshenko C2-7

obtained

Plates

Having

of two stiffeners obtained

values

for various

values

of the

for

to remain

undeflected

= 96 + 6105

+ 97552

values during

Equidistant

subdividing

has

formulas

Two

for

the

the plate coefficient

parameters,

of stiffener buckling.

Stiffeners

rigidity They

_,

into

three

k; these 5, and

necessary

% for

are

(25)

are

SectionC2 1 May 1971 Page25 for 0 < 6 < 0.20 and Io with

the

= 0. 092 bt 3 To

critical

stress

,

for

(26)

the plate

given

by

2

Fcr

=

D.

32.5

E /b )

Plates

Having

When the

stiffened

following

plate equation

the

(27}

More

number

Than

Two

Stiffeners

of stiffeners

can

be treated

for

the compression

is equal

as an orthotropic

to or greater

plate.

buckling

This

than

three,

results

in the

coefficient: i/2 Az I

2 k

Ii

=

N-I + -- N

+ 1 s -bD-

1+

0.88A bt

(28)

where

the

terms

are

defined

N

number

A

area

I

bending

as follows:

of bays of stiffener

cross

moment

section

of inertia

of stiffener

cross

section

taken

S

about

the

stiffener

centroidal

z

distance

from

D

flexural

rigidity

b

spacing

of stiffeners

midsurface of skin

axis of skin

per

inch

to stiffener of width,

centroidal E t3/12(

axis

1 - v z)

Section C2 1 May 1971 Page 26 II.

Stiffeners

Transverse

Timoshenko applied

load.

stiffener given

in Table

verse

stiffeners.

plate

obtained

straight C2-8

For the

[ 2] has

He has

remains

the

F

studied

plates

several

limiting

during

buckling

for various

case

values

of a large

is considered

perpendicular

to Load

to have

directions.

of

number

critical

stiffeners

values

of the

plate.

a

one,

for

of equal

two different

The

with

T at which These

two,

and

flexural

stress

of

transverse the

values

and three

equidistant

rigidities

is given

to the

are trans-

stiffeners, in the

two

as

2 7r2 b2 t

er

(

_]D1D 2 + D 3 )

(29)

where D1 =

(EI)

D 2 --

2.1.2.2

1/2(v

xy

two stiffeners

C2-10 and

two

x

give

the

stiffeners,

- v p ), x y

flexural

rigidity

in longitudinal

- Px py )'

flexural

rigidity

in transverse

D 2+

is the

Conventionally The

and

/(1

(EI)y/(1

D3 = 2(GI)

x

simple

p D1) y

average

have limiting

torsional

Stiffened cases been values

respectively.

+ 2(GI}

; and

xy

in Shear.

supported

investigated of the

direction;

rigidity.

Plates

of simply

direction;

rectangular

by Timoshenko. ratio

T in the case

plates Tables of one

with C2-9

stiffener

one and

Section C2 1 May 1971 Page 27 Additional analysis of stiffened plates in shear is given in Section C4. 4. 0 and in Section B4. 8. 1. 2. 1.2. 3

Conventionally The

case

Stiffened

of a rectangular

stiffener

under

bending

For

case,

reference

this

2. 1.2.4

Plates

and

local

results

in complete to develop

for

these

simply

applied

of failure

can usually

two modes

edges

for

shear

beams.

of corrugated

to the are

since

the

All

corrugations

strength. stress

ultimate.

in such edges

a manner are

are

instability,

buckling

be considered

edges.

instability

post-buckling

compression

supported

Both

General

of local

some

plates

corrugations.

treated.

In the case

lower

are

these

analysis

develop

the

of webs B4. 8. 1.1.

plate,

of failure

along

the

parallel

strength.

that

design

a longitudinal

in Compression.

for

applied

with

to Section

of a corrugated

the corrugated

is uniformly

below

modes

post-buckling

calculated

load

Corrugations

failure

it is recommended

in the

With

instability

However,

reinforced

be made

corrugations

that

is common

load

the

assumed

plate

is presented

however,

in Bending.

should

to a compressive

general

unable

load

Stiffened

A method subjected

Plates

assumed

It is that

the

to be

supported.

of failure

The

compression

may

be found

k F

cr

= 77

buckling

stress

by orthotropic

7r2E

c 12 o( 1 - _2)

plate

for the analysis

general

instability

mode

to be

2 It)a

(30)

Section

C2

1 May 1971 Page 28

where k ,1 12I =

-t3L

o

other

terms

are

E

b

plate

d

centerline

to centerline

L

developed

length

of the loaded dimension

moment the

plate

approximately

equation

applies:

per

aspect

above

behaves

as

_2 E

edge

in the

of inertia

computations

+

(b)21

}

and

in compression

length

When the

of elasticity

a

I

2v+2

as follows:

modulus

C

1/3,

defined

+

direction spacing width

a wide

plate parallel

to the

load

of corrugations

d

of width ratio

may

of the

d about

a/b

neutral

is greater

axis

than

approximately

be simplified

since

the corrugated

column.

these

cases,

For

the

plate following

I C

Fc

= _?

The failure

may

composed

Lt b 2

compression

be found of flat

from

e

= V cr

buckling the

following

stress

for

equation

the local when

the

instability corrugation

mode is

elements:

k F

(31)

_r2E

o c

12 ( 1 - u 2)

(32)

of

Section

C2

1 May Page where

k

Fig. flat

represents

c

C2-5, plate

supported

t is the thickness element

be described Fig.

simply

of the

of the

plate,

corrugation

by presenting

edge

form.

some

typical

conditions and

b

The

latter

and

is the

examples

is taken

width

from

of the widest

dimension

such

1971 29

as those

may

best

shown

in

C2-65. In the

local

instability

case

of Fig.

should

C2-65(d),

be based

R (see

2. 1. 2.5

Plates

Stiffened

With

Corrugations

Plates

stiffened

with

corrugations

modes

for

light

of failure

assumed

that

conditidn force

are

in these

is such

bending

produce

treated

in the

only

or torsion).

brazing,

a spar General

plate

because

mode

for

the

cap

in the

following

that

support

form

of the

shearing

on the

for

unbuckled

stress

load

for

cylinder

in Shear. may

state

this

inner

corrugations

are

local

methods the

of post-buckling

and

edges

cannot

an externally

of the plate

corrugated

of a wing

skin.

(i. e.,

by welding plate

failure

of the

stresses

In contrast,

This

shearing

plate

be met

to redistribute

It is

applied

of the

stress.

instability

be distorted.

corrugated may

weight

general

of analysis.

in the complete unable

a structural

corrugated

condition

surface

results

Both

in the

joining

instability

provide

corrugation

stresses

mechanical

the development

of an axially

conditions.

In practice,

or by rigorous

instance,

loading

methods

that

buckling

C3. 1).

the unbuckled

means

will

shear

compression

on the buckling

of radius

advantage

Section

the

to,

no or for

corrugated in this

local

SectionC2 1 May 1971 Page 30 instablility

of the corrugations

post-buckling however, modes

strength that

the

of failure

lower

shear

mean

failure,

that case.

stress

since

some

It is recommended,

calculated

here

for these

two

ultimate.

buckling

Reference

for

buckling

be considered

is from

not necessily

can be developed

The shear failure

does

stress

for

the general

instability

mode

of

1:

4 Fs

= T/ cr

F

where

and or

_/Dl(D2)

= _/-- 4k b2t

s er

D 1 and

directions,

4.._k b2t

D 2 are

_r D2 D3

the

respectively;

H is equal

to

3

flexural D3

when

H>

1

(33)

when

H < 1

(34)

stiffnesses

is a function

_]-D1D2/Da.

of the of the

The values

for

analysis,

the

plate

in the

torsional k

are

rigidity

taken

from

x

and

y

of the Fig.

plate,

C2-66

C2-67. For

corrugations plate.

general for

For

this

instability

a reversible orientation,

shear the plate

flow

optimum

is parallel

flexural

to the

stiffnesses

orientation short may

of the side

of the

be expressed

as follows: E

t3d C

D1

=

12L E

--

D2

(35)

I C

d

(36)

Section C2 1 May 1971 Page 31 E t3L C

D3

= uDl+

The may

be found

posed

of flat

12d

shear

buckling

from

= _

k

Fig. C2-5.

of Fig. based

on the

2. 1.2.6

select

(d},

the

torsional

corrugation

of failure

form

is com-

(38)

p2)

best be described by referring

such as those presented in Fig. C2-65.

shear

buckling

buckling

stress

of a cylinder

for

local

instability

of radius

R (see

In the case should

Section

be C3. 1).

Plates.

in Reference

for

the design

27 which

technology. and

the

mode

2

The latterdimension may

Sandwich

sandwich

when

instability

represents simply supported edge conditions and is taken

s

Procedures found

local

s c (b)

typical examples C2-G5

the

equation

?r2E

12 (1

Here

to some

for

elements:

s cr

from

stress

the following

k F

(37)

contains

It contains

check

designs

and

Plates

of Composite

and analysis the latest

many

its use

information

formulas

is quite

of sandwich

and

can be

in structural

charts

widespread

plates

necessary

in the

to

aerospace

industry. 2.1.2.7

The buckling sented

in Section

F and

of plates

Material. constructed

in Reference

28.

of composite

materials

is pre-

Section C2 1 May 1971 Page 32 2. 2

CURVED Design

of buckling

PLATES. information

in plates

• is presented

of single

curvature

which

are

section

for

the prediction

both

stiffened

and

2. 2. 1

UNSTIFFENED

2.2.

Compression

Buckling.

The

bermvior

of curved

plates

uniformly

compressed

is similar

in many

respects

to that

of a circular

cylinder

considerably

below

1.1

curved axial

edges

compression

predictions

CURVED

in this

(e. g.,

of small

semi-empirical

both

methods

axially

compressed

buckling 2. 2. 1.2

Shear

following

shear

where

k

methods

cylinders

the

for

(Section

buckling

stresses

for

available

predicting C3. 1.1)

C2-68,

and

curved

formula:

rl 7r2E S

s cr

the

with

to resort test

their under the to

results).

buckling be used

of to predict

Buckling.

k F

it is necessary

along

plates.

Critical by the

and

agreement

that

monocoque

of curved

at stresses

theory,

to show

It is recommended

PLATES.

buckle

deflection

unstiffened.

12(1

is determined

v 2) e

from

Fig.

T/=

plates

are

calculated

Section C2 1 May 1971 Page 33 2.2.2

STIFFENED CURVED PLATES IN COMPRESSION. Information is presented in the following paragraphs for stiffened

plates of single curvature in compression where the stiffening members are either axial or circumferential.

In these considerations, both the local and

general modes of instability must be considered. 2. 2. 2. 1

Curved

Plates

A method with

a single

method plates with

specified

coefficients,

when

has

presented in that

for chart.

which

may

the

the

been

same

developed

However,

in the

Figures

C2-69a

general through

the following

curved

in Reference

2. 1.2. equation

and the

with

supported

basic

local

be used

of simply

in Paragraph

coefficients.

same

: cr

equation

may

k

7r2E

1 for is used

present

modes C2-69d equation

plates 29.

stiffened

This flat

in conjunction case,

of instability present to predict

the are these buckling

2

c

12(1

also

c

cr

= 12

j

(40)

- v 2) e be written

k F

buckling

stiffener

to that

buckling

Stiffeners.

-_ O. 25:

Zb

c

This

axial

coefficients on the

Axial

predicting

in compression,

buckling shown

for

central

is similar

With

as

7r2E c

(41)

l-u

2 e

Section

C2

1 May

1971

Page

where

Zb

radius

is the plate

of curvature;

curvature

and

b

parameter,

R-'_-

is the half-width

of the

34

1 - v e 2 .' R is the plate loaded

(curved)

edge

of

plate. Figures function

of

4/3,

2, 3,

•

I

Zb,

C2-69a A/bt,

and

4,

and

about D

stiffness

yield

buckling

values

terms

of inertia

stiffener

flexural

for

where

moment the

C2-69d

ELs/bD

respectively,

bending

s

through

of the are

of the

coefficients ratio

defined

axis

of the plate

per

inch

equal

to

as follows:

stiffener

centroidal

a/b

as a

cross

section

of width,

taken

Et3/12(

1- v 2) e

a

length The

of Figs.

lines

inertia

through

right

design,

instability

portions

and in the

to the

left

designs

Local

instability

is represented

chart.

less

a lowering stiffener

curves

represent

in each since

of the

C2-69d

is critical.

to the

efficient

sloping

C2-69a

of instability

of plate

intersection

moment-of-inertia of the

has

The

buckling

no effect

in each

wherein

the

stress,

while

on the buckling

more

stress

mode

horizontal

curves

stiffener

charts

general

by the

of these in the

of the

represents

induces

general

moment-ofof the

stiffened

plate. Although the

increase

central

axial

in the

not specifically curved-plate

stiffener,

shown buckling

is negligible

when

in Fig. stress,

C2-69a

through

due to the addition

Zb > 2.5.

Thus,

plates

C2-69d, of a

with

a

SectionC2 1 May 1971 Page35 large degree of

curvature

member.

In this

in Section

C3. 1.1.

case,

Also, axial

the

when for

the

stiffeners,

satisfy

2.2.

2.2

stiffener

(where

is small.

With

With

Curved

plates

stiffened

been

that

the

curved

addition

stress

This

plate

stiffener,

geometric

a/b

restrictions

must

with be 0.6

as a function

given

value

of

value

read

from

from

the

chart,

a/b, the

the

for

chart.

a/b

the If

design Zb

for

in the buckling plate.

to

described

Schildcrout

[ 30].

central but

only

within

of both width

For

the

buckling central

parameter

Zb

shown

in Fig.

be equal

design

stress

the

from

stiffener restricted

ratio

a/b

curved,

stress

loaded

of the

circumferential imposes

further

C2-70.

For

to or smaller

is larger

results

deter-

a rather

of the

of a single

must

They

circumferential

is the

are

the

methods

circumferential

b

The

which

two or more

be used.

is a function

addition

techniques

of stiffeners

central

plate

Z b.

number

axial

Stiffeners.

plate,

or less.

of

Zb

no gain

to the curved

of the parameter

to increase

should

of a single

range

needed

the

and

of a curved

requirements

stiffeners,

a single

by Batdorf

geometries.

the

with

by the

with

to the

C3. 1.2

a central

not be applied

sensitive

Circumferential

considered

is the half-length

and

multiple

in Section

Plates

a

edge)

stiffener

are

with

be determined

should

of instability

Curved

of plate

above

mode

the buckling

range

should

geometrical

analytically

increases

cited

the

cylinders

have

mined

methods

stress

by stiffening

stiffener

general

orthotropic

not benefited

buckling

since

the number

are

than the

than

the value addition

a

that read of the

Section C2 1 May 1971 Page 36 Small deflection theory was used in Reference 30 to predict the buckling the

stress

results

are

in buckling where both

of curved

plates

presented

in terms

stress

the

gain

for

configurations. by multiplying

curved

plate

of the

curved

on theoretical The

same

gain

stiffeners.

factor

plate

which

over

here,

dimensions

factors

are

the

stress

by methods

plate,

stress

for

may

for

given

gain

curved

therefore,

by the buckling

overall

indicates

of the buckling

presented

factor

Consequently,

an unstiffened

predictions

information

the

circumferential

of a gain

a stiffened

is based

applied

with

be

an unstiffened in Paragraph

2.2.1.1.

Maximum

gain

in Fig.

C2-71.

The

bending

rigidity

to enforce

The node

at the

with

a maximum

required

stiffener

to enforce limitations the

gain

of Fig.

stiffener

line

along

C2-72

may

the

is labeled

in Fig. from

that

at the

the

rigidity

needed

C2-72

when

Fig.

to determine

more

or less

bending

C2-70

here to take

may this

apply

must

not be maximum; possibility

and

has

Zb

sufficient

line. to enforce the figure

gain

rigidity

In this and

a/b

a buckle is entered

C2-71.

be used

line.

of

stiffener

stiffener

also

stiffener

in Fig.

obtained

bending

obtained

as a function

implies node

is defined

either

stipulated factors

a buckle

factor

C2-72 has

a node

"maximum"

stiffener

gain

Figure existing

term

presented

case,

factors

than the

when

that

same

required geometrical

be observed. therefore,

into account.

(Note the

)

an

that

ordinate

Section C2 1 May J971 Page 37 After first referring to Fig. C2-70 to ascertain whether or not a gain is indeed possible, find the gain factor (from Fig. C2-72) based on the properties of the existing stiffener.

Now plot this gain factor on Fig. C2-71.

If the point is below and to the left of the a/b curve to which it relates, then the gain factor is less than the maximum permissible and the bending rigidity of the stiffener is less than the minimum required.

In this case, general

instability of the curved plate represents the critical mode, andbuckling may be predicted using the gain factor obtainedfrom Fig. C2-70. Whenthe point is above andto the right of the a/b curve in Fig. C2-71 to which it relates, the contrary is true, and local instability of the curved plate represents the critical mode. In this case, buckling may be predicted using the maximum gain factor obtained from the a/b,

Zb intersection in Fig. C2-71.

The methodsof this section should not be applied to curved plates with two or more circumferential stiffeners.

The general instability stresses

predicted by the design charts are sensitive to the number of stiffeners when their total number is small.

In this case, recourse should be had to

Section C3. 1.2. 2.2.3

STIFFENED CURVED PLATES IN SHEAR. Methodsare presented in the following paragraphs for predicting

the buckling stress of plates of single curvature in shear having a single stiffener in either the axial or circumferential direction.

The methods

account for both the local and general modes of instability, and charts are

SectionC2 1 May 1971 Page 38 given that present the buckling coefficient ks versus EI/bD, where at low values of EI/bD the general mode of instability is critical.

As EI/bD

increases, the local mode of instability becomes critical and is signified by a constant value must

have

Note

that

line

an the

yields

critical

the

of

kS .

to enforce

EI/bD

which

falls

EI/bD

value

representing

most

efficient

design;

a node

at the

on the horizontal

portion

the extreme local

and

stiffener of the

left

general

the design

point

design

curve.

of the horizontal

instability

are

both

here.

2. 2. 3. 1

Curved The

stiffener

Plates

With

buckling

may

the

plates

with

a single,

central

equation:

_2E ,

2)

\_

(42)

e

k

curved

plate,

and

applies

when

axial

is taken

Fig.

C2-73(b)

Note

that

from t

Curves

are

as well

as of the

cases,

presented

are

plate based

Fig.

is the

length

applies

in both

C2-73

Stiffeners. curved

from

12(1

where

S

for

be determined

c cr

Axial

stress

k

Fig.

Thus,

C2-73,

b

thickness

of the

is greater

when b

axial

than length

is denoted

as a function curvature on small

is the

overall

curved

plate.

than

the

short

overall

of the

aspect

ratio

deflection

Figure

circumferential is less

parameter,

dimension

Z b. theory

width,

C2-73(a) and

circumferential dimension of the

Note and

of the

also

agree

width. of the

plate, that

plate.

a/b, the

satisfactorily

data

of

Section C2 1 May 1971 Page 39 with experimental results except in the case of cylinders for which a 16 percent reduction is recommended. The preceding method should not be extended to apply to curved plates with multiple axial stiffeners.

The bending rigidity required of each

stiffener to support general instability is sensitive to the total number of stiffeners when this number is small. 2.2. 3.2

Curved The

with

a single

buckling

buckling central

The

local data

circumferential Paragraph 2.2.

4

for

pressure,

plates Section

are

of the horizontal

instability)

from

C2-74.

not be applied

the

reasons

for

combined with

curved

may

UNDER

relations shear

presently

C3. 1.2

Fig.

circumferentially equation

(42)

with

As in Fig.

C2-73

for

portions

of the curves

the a (the

is recommended.

should for

stiffened

be determined

reduction

PLATES

combined

unstiffened,

may

plates

to curved

noted

plates

previously

with

multiple

in

1.

CURVED

compression

curved

from

above

Interaction normal

for

Stiffeners.

k , taken s

stiffeners 2. 2.3.

Circumferential

stiffener

a 16 percent signifying

With stress

coefficients,

cylinder, portion

Plates

shear

plates.

compression

with

normal

pressure,

presented

in the

Interaction however, The

LOADING.

longitudinal

are

unavailable; be used.

COMBINED

normal

relations techniques pressure

combined and

longitudinal

following for

in the

paragraphs

stiffened,

discussed first

with

curved in two cases

is

SectionC2 1 May 1971 Page40 applied to the concave face of the curved plate. The interaction relations apply only to elastic stress conditions, since verification of their application to plastic stress conditions is lacking at present. 2.2. 4. 1

Longitudinal The

pressure

interaction

applied

=

R c = Fc/F

c

F F

concave

for face

Normal

Pressure.

longitudinal

compression

of an unstiffened

plus

curved

plate

normal

is

1

(43)

and cr

applied

C

Plus

equation

to the

R 2_ R c p where

Compression

Rp

= P/Pcr'

longitudinal

buckling

stress

where

the

compression of the

curved

following

definitions

apply:

stress plate

where

subjected

to simple

C cr

axial

Pcr

compression,

absolute

value

of the applied

absolute

value

of the external

the

cylinder

methods

Note stituted

into

accounted the

concave

sion

load

that

the

for

absolute

the

normal

equation

since

2. 2. 1. 1

pressure

pressure

which

is a section,

quantities their

It can be seen

unstiffened,

of Section

would

buckle

determined

by the

C3. 1.1.5

of the

be carried

by the methods

plate

values

equation.

of the may

of which

of Section

interaction

in the

face which

determined

curved by the

plate

prior

and

difference

that plate

p

Pcr

are

in sign

normal

pressure

increases

the

to buckling.

axial

sub-

is already applied compres-

to

Section C2 1 May 1971 Page 41 2. 2.4. 2

Shear

Plus

When with

normal

2-R

the

R = Fs/F

curved

4.3

S

cr when

of Section

is applied

2. 2. 1.2),

face

+ R

C

represents

2

=

and

R

of the

for shear

and

plate,

FS

in-plane

the

combined following

is buckling

stress

cr

shear,

is as previously

P

Plus

equation and

stress

to simple

Compression

compression

C

shear

subjected

interaction

to longitudinal

R

concave

to shear

(44)

(F

S

plate

The

where

is subjected

i

Longitudinal

R

on the

plate

p

methods

2.2.

curved

applies:

=

s

of the

acting

equation

R

Pressure.

an unstiffened

pressure

interaction

where

Normal

determined

by

defined.

Shear.

an unstiffened

curved

plate

subjected

is

(45}

1

S

and

R

S

are

as defined

approximately

results

while

the lower

linear

relation

between

in previous

an average bound R

of the and

e

curve test

R . s

paragraphs. through

results

may

the

This available

be represented

relationship experimental by a

Section

C2

1 May

1971

Page

42 u_

tt_ ¢q

,v

¢,i

¢g.-_

t"oO

t.O L'.-

_ _

og_

O0 t"-

tt_

u_ t_

o,1

_.-

¢xl

co

o,1

¢q

¢q

°,-I

O0 ¢Xl

¢xl

O,,1

O0

tO

Oq

_

Oq

o_o_

.Jog

Oq

-.-_

¢_

Oq

O_

O0

_

O0

0 o 0

t_

v

O0

°,-.I

O0

t_

tt_ 0

tm

oo

¢Xl

oq

o_

tt_

oq

o,1

o o,1

¢m

! eq

0 0

oq

¢Xl

o,1 oq

o

0

_

0

_

co

_ 0

oo

_w _

[--4

Section

C2

1 May

1971

Page ¢q

cg

¢.o

t--

tt_

_4

_t_

¢o

¢D

tt_

t_

4cq_; ¢q

tr_ e_.

.,-4

o

00

_d

_

d,g

tr_

Ue_

L_

4

or-:

L_

Lea 0b

c_

_4

_tC

ea_

_4c;

cO

v

o_

00

¢D

D-

io i,,-4

Lea

tr_

L_

L_

C_ 12x1

¢D _:1

t'-

tD

¢q _e)

¢q _l_

_ i---

tra t_

t--.

@,1 oo

¢D ",D

o tt_

o e_ o

_9 v

o_

¢q

I

[o _D c_

c_ _o

¢q ¢,1

¢,]

¢q

¢q

¢q


,-_ _l

V

r_ _

_.

md o

LQ D

0

.ha

0

_O0

_ o

_

V

_ o

Vt

_ o

Vl

_ o

Vl

L'--

_

II

< Vl

¢q

43

Section

C2

1 May 1971 Page 44 Table

C 2- 2.

Cutoff

Stresses

for

Buckling

Cutoff

of Flat

Unstiffened

Plates

Stress Shear

Material

Compression

Buckling

Bending

Buckling

cy

2024-T

0.61

2014-T

Fcy

200Fcy000 )

1 +

Fcy

0.61

Fcy000 ) 1 + 200

6061-T

0.61

7075-T 18-8(1/2

H) a

1.075

F

0.835

F

cy cy

(3/4

H)

0. 875 F

(FH)

0.866

1.075

F

0.835

F

All Other

0.875

cy

F

F

0.866

with

Table

Summary

C2-3.

grain,

F

Short

Plate

Plate

Compression Shear Panels

0.61 cy

based

on MIL-HDBK-5

of Simplified

Cladding

<_ cr

<_ pl

properties.

Reduction

cr

Columns l+3f

Long

0.53

F

acl

Loading

0.53

cy

cy

Cold-rolled,

0.51

cy

F

Materials

0.61

cy cy

cy

a.

Buckling F

Columns

and

l+3f

l+3f

l+3f

l+3flf l+3f

l+3f

Factors

pl

Section 1 May Page Table

C2-4.

Shear

Buckling

with

Mixed

Coefficients

for

Boundary

Rectangular

Plates

Conditions

f

I_

a

fcr _----

NOTE:

Two

Short

Edges

Aspect

Ratio

Two

Long Simply

/r 2E -12 (1 - Ve2)

blSSMALLER

Clamped, Edge Conditions

ks

Edges Supported k

DIMENSION

One

Short

Clamped,

2

ALWAYS

Edge Three

Edges

Simply

Supported k

s

/t\

One

Long

Clamped,

Edge Three

Edges

Simply

Supported k

s

s

7.07

b/a 0

5.35

5.35

0.2

5.58

5. 58

0. 333

6.13

0.5

6.72

6.72

8.43

0.667

7.83

7.59

9.31

0.80

9.34

8.57

9.85

0.90

10.83

9.66

10.38

1.00

12. 60

10.98

10.98

7.96

C 2 1971 45

Section

C2

1 May 1971 Page 46 Table

C2-5.

Critical

Plate Plates,

a

k _b

x

0

19.35 21.63

0.5

15 30

0.75

1o0

13o. 5

0 15

14.92 16.49

30

22.55

45

39.73

60

90. 50

1.50

12. 76

30 45

16,71 27, 06

6O

60.59

Parallelogram

Clamped

fy

k y

k s

32.13

- 42.28

+

42. 28

34.09 39.72

- 34.58 - 31.58

+ +

55.36 76. 90

53,22 86.20

- 40.54 - 85.0

+ 128.3 531.5

0

10. 08

10.0_

-

14.83

14.83

15

10. 87

10.43

30

13. 58

11.76

- 14. 39 - 16.66

17.24 23.64

45 60

20.44

15. 26 25.78

-

24.08 46.58

32.56 69.86

- 11.56 - 12.01

11.56 12.73

42. 14 9.25

15

9.92

30

12. 32

45 60

18.50 38.01

o 15

8.33 8.91

30

11.16

14.05

15.19

45

17.10

10. 10

- 20.21

22.37

60

36.84

18. 54

- 40.24

45.83

0 2.0

Edges

for

11.70

15

0 1.25

30.38 55.26

45 60

0

all

Parameters

x

a/b

0.6

Buckling

78 6. 151 7. 271 5.

-

8.033

4.

838

-

10.57

t0.57

15

8.70

5. 132

-

10.84

ll.

30 45

10.53 15.74

6. 208 8. 938

- 13,34 - 19.24

J3.73 20.35

60

39.35

- 39.38

44.40

17. 08

lO

Section

C2

i May

1971

Page

II ,o

_9 CD

I ,0

n=l

o

0

I ,0

©

o

n_ e_ q) q_ II

ir

'-_ I

o

0 g_ _D

,o

I CXl

_9 _9

k_

_

,,o

_

_

_

e0

_

c_

_

e_

c_

_

_

_.

e_

47

Section

C2

1 May

1971

Page

¢q

_4 O_

oO

¢D

¢q

¢q

*O II u_ O

Oq

¢q

o6 II

cO

¢q

¢q

@0 _J aJ

¢/ II

_5

¢q

oo

o_

o_

¢4

u_

t_

t_

c/

_4

¢q

t_ ¢D

O0

cd

¢q

II u_ oo

¢/ II

O

_d ¢q

¢Xl

¢q

t'--

t_

¢q

c4

¢/

o_

Oq

¢O

¢D

¢q

O ¢q

o_ ° II O

oO

_D

t_ oO

O t--

¢D

o_ O-

tO

¢D

u_

u_ O

c/

t_

¢4 ¢q

I

L) O II

¢x]

¢/

O0

u'3

¢D o_

og

v:

Yo

II

II

¢D

u/

¢q

od

tZ

¢q

_O

oO

c/

¢q

¢D

oO

O

:4

48

Section i May

C2

Page Table

C2-8.

0.5

One

Rib

12. 8

Two

Ribs

65.

Three

Ribs

Table

5

Values Transverse

0.6

0.7

0.8

0.9

1.0

1.2

4. 42

2. 82

1.84

1.19

0.435

0

7.94

4.43

2. 53

37.8

7

15.8

102

64.

4

43.

Limiting

Values

Stiffener

Table

of the

Under

_/ For One, Stiffeners

Two,

11.0

23.

177

C2-9.

Limiting Three

7.25

of

30.

1

2

Ratio

Shearing

T

For

Plate

1.25

1.5

2

T = EI/Da

15

6.3

2.9

0.83

Limiting

a/b 7 = EI/Da

Values

Stiffeners

Under

1.2

1.5

22.6

10.7

of the

7.44

With

One

Stress

1

Two

and

12.6

21.9

a/b

C2-10.

1971 49

Ratio

Shearing

2 3.53

_/ For

Plate

With

Stress

2.5 1. 37

3 0.64

Section

C 2

1 May

1971

Page

50

ft/2 1

f

CLADDING

IL.. ft/2 FIGURE

C2-1.

ALUMINUM

°pl OF CORE

CLADDING

"ALC

a

= l-f,,sf

(;core Ocore

°'cl

FIGURE

C2-2.

CLADDING

STRESS-STRAIN AND

"ALCAD"

CURVES

FOR

COMBINATIONS

CLADDING,

CORE,

Section 1 May Page

C- 2 1971 51

Z

r COLUMN BUCKLED

FORM

ORIGINAL (a) COLUMN FLAT STRIP Y

(b) FLANGE

(c) PLATE

FIGURE

C2-3.

TRANSITION

ADDED

ALONG

FROM

COLUMN

UNLOADED buckle

EDGES

configurations.

TO

PLATE (Note )

AS changes

SUPPORTS in

ARE

Section

C 2

1 May

1971

Page

52

Ve = 0.3

1.00

|,,

rt I_III FIXED ENDS,,_ II (c - 4) _.. IlY

0.96

V/" 1171rl fl IIFIII _

II

I i'

1

IIII

I I

II_°''' :

0.92

jr J L[IJJfffll !1 [111 IJll!

0.88 0.1

0.2

0.4

1

2

4

10

20

40

100

b

L/V'_-

fcr

/r2E

fcr

2

for

t

_,"N

-lL

I "--

t

I I

m

! ENDS SIMPLY SUPPORTED (c " 1) FIGURE

C2-4.

CRITICAL

,2

l t

ENDS FIXED (c = 4) COMPRESSIVE

STRESS

FOR

PLATE

COLUMNS

Section I May Page

C2 1971

53

16 F

C B SS

C D FR 12

10 TYPE OF SUPPORT ALONG UNLOADED EDGES

kc

8

B

C

i

I_'-_

--

ellll

*_III

_

"fIll

m

_Inl

mill

Wilm

E 0 -]lllllltJ 0

lltJlltll

I,,,11il1

1

2

Illil,ll,

Ill,lllil

3

4

alb

FIGURE

C2-5. FOR

COMPRESSIVE-BUCKLING FLAT

RECTANGULAR

COEFFICIENTS PLATES

5

Section 1 May Page

C 2 1971 54

o9 o9

0

Z

a_

o9 o9

o9 o9

_

Q.

0 Z 0 0

!

0

0

o9 o9

Section 1 May Page

15

C2 1971 55

:

14

13 .//////////,/////.

5"--

12

2

•--*

SS

_/

11

#

SS

///I//,"///I//

=

g _%%!_ --

10

LOADED

EDGES CLAMPED

k c

2

=_

r-

3

:

,,t,_l,,,

0

n,la*,

0.4

,,,I,,

0.8

,,,I,,J

1.2

|ilJlll

1.6

iltiall

2.0

Itli*ll

2.4

ill

2.8

I

III

l|

I_lll

3.2

II

t lit

3.6

4.0

a/b

FIGURE

C2-7.

COMPRESSIVE-BUCKLING-STRESS

PLATES

AS A FUNCTION

OF

a/b

ROTATIONA

FOR

COEFFICIENT VARIOUS

L RESTRAINT

AMOUNTS

OF

OF EDGE

Section C2 1 May 1971 Page 56

1.9 g.

I

I

1.8

FR

_SS

1,7.

SS "___.

r///.//.///////,,,,J 1.6 _,..._._---

LOADED EDGES CLAMPED

1.5 :

\\i 1.4

1.3 =

1.2 rk c

1.1

1.0

0.9 _---

-------"

2

0.8 % 0.7 0.5 0.2

0.6

0.1 0.5 0 0.4 [,,I,,,, 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

a/b

FIGURE

C2-8,

FLANGES

COMPRESSIVE-BUCKLING-STRESS AS A FUNCTION OF a/b EDGE ROTATIONAL

FOR VARIOUS RESTRAINT

COEFFICIENT AMOUNTS

OF OF

r

Section

C2

1 May

1971

Page

57 c_ Z <

r

Z

N

N

a

Z_

W O.

,< .,.I

MZ NO

¢j ¢,n w

t_4

a IJJ

(3 Z 0 .J

©M _Z

>.er .J z

!

Z_ ©©

M M _

rr

r..) m

_

_Z M< ©

2: M _Z ,
z

I

M (.9 m _

ll_lllllJllJJ

IJf,l,lll

II,lllJ

llJl

o

M< Nm ©

(.1

!

©

Section

C2

1 May 1971 Page 58

Z_ OZ o; _,8

o.

_\\\\\\\\\\\

%

|

u

_m M m_

z< t-

tj

_@

\\\\\_\\\\\

*

M

04w

r_

M_ _m

qF-

c9

Na_

I__

w

Q_

tu ÷

Z_ o ÷

Z_

N_

w N

I,,,,-4 CO

d

w

Zm

II

°

°


,,,il,,,,,,,,I,,,

,,,I,,,,] 0 f.,. 0 LL

0

0

M_

M

N

Section

C2

i May

1971

Page

59

©

\fill \

c_ c_

o

Z

c[: IJ.

¢D s.=.

rJ)

k

c_

@d s-*

Z

A

r_

%

"-I'-

..x

v-

_Z II

Z_ © ©

JLai],*,i

llliilll

*¢i,li,a o.

IN

r-.

s.-

I

Section

C2

1 May Page

1971 60

15 i

i I m

13

I

m I m m I.

m

m

11 n

SYM

CLAMPED

EDGES

k$ B

9 m

i m

ANTISYMMETRIC

SYMMETRIC

m

i m

7 D

INGED

5o_,,, I, ,,,

,,,,I,,,,

,,,,I,,,, 2

EDGES

,,,,I,,,,

,,,,I,,,,

3

4

5

a/b

FIGURE

SHEAR-BUCKLING-STRESS

C2-12.

AS

A

FUNCTION

OF

a/b

FOR

,

,,÷

C2-13.

AND

1.25 b

HINGED

OF

PLATES

EDGES

"I

,, , ,...-.. I /

,--:.',,/( f @/X,

/ /--, FIGURE

COEFFICIENT CLAMPED

SHEAR

t

"--BUCKLING

__ ." // PATTERN

%',----_Jvo FOR

RECTANGULAR

/

PLATE

1

1

Section

C2

1 May

1971

Page

17 = (Es/E)

(1

-Ve2)l(1

61

-v 2)

0.7

n 2

0.6

n 0.5

J

0.4

Fs/Fo.7 0.3

2

I '_

"_'_'-"_

*" r

0,2

.,, ___J_ 1_\_\_\_, t' x_ 0.1

0

0.1

02

0.3

0.4

0.5

k s _'2E 12 (1

FIGURE

C2-14.

CHART

FOR

PANELS

OF

- _p2)Fo.

0.6

0.7

EDGE

0.9

1.0

(t/b)2 7

NONDIMENSIONAL

WITH

0.8

SHEAR

ROTATIONAL

BUCKLING

STRESS

RESTRAINT

.,J-.-2/3 b_._

l

bA

I .Iw-j

i

/;:;-, i /'w._

i

i

"%..

I

i

%...

_J

w=0

I FIGURE

C2-15.

BUCKLE RECTANGULAR

PATTERN PLATE

FOR

r

',k

BENDING

OF

Section i May Page

t°U.k 32

b

lr2S

(b)

2

C2 1971 62

to

P-

/

N.A.

28

0 I0 I-

.

4

_4

fb"

24

'1 __

;

¸

•

m-2

I

fo I1

m-3

I

m-4 -

23.9

20,

kb

16'

12 _

__._1.3

i

_: 10.00

__ 8

_-- _

_ "-----

4

0.5

1

1"!

-

lrO

0.5O 9 .75 0.25

_

0.0

1.5

8.ss --

2

!

--

'

7.80

--

5.3O 6.35 4.55

,,

4.0

_

2.5

a_

FIGURE

C2-16.

CRITICAL BENDING ALL

STRESS

COEFFICIENTS

IN THE PLANE EDGES SIMPLY

FOR A FLAT

OF THE PLATE, SUPPORTED

PLATE

IN

Section

C2

I May Page

fOcr=k__ 'q

_2 E 12 [1 - Ve 2]

1971 63

(____)2

r

[_

56

SS

= N.A.

'_._

;I

FIXED SS J_ I-

SS

1'

0 I-

_

"-----I

48

m=2

m=3

m=5

m=4

i

,1.8 40

I''"_L"_'_--'--_

-

32

:A

_

_.8

_

31.2

_"

27.4

24

23.9

20.7

1.3 16

--

__

1.25 1,2

18.0 16.8 15.6 13.8

1.0 __

12.3

0.8

10.0

8 _

7.7 _...._

__-_

--.---

0.5 0 5.4

0 0

0.4

0.8

1.2

1.6

2.0

2.4

a/b

FIGURE

C2-17.

BENDING

CRITICAL

STRESS

COEFFICIENTS

IN THE PLANE OF THE PLATE, SUPPORTED AND COMPRESSION

FOR

A PLATE

TENSION SIDE SIDE FIXED

SIMPLY

IN

Section 1 May Page

C 2 1971 64

6O m

56 _ I/////////////.

I

- /-////////////// 48

P 100 50

3e

_

-_ 24

2O 10 5 3 1 0

,,---

2O

m

8

o ..I ....... .h,, ,,.I., 3 5 7

,.,h,,,i.,,h,,, .... h,,, ,,,,h,,, ,,,,I,,,, ,,,,h,,,,,,,I,,,,l 9 11 13 15 17 19 21 23 a/b

FIGURE

C2-18.

AS A FUNCTION

BENDING-BUCKLING OF

a/b

FOR

ROTATIONAL

COEFFICIENT VARIOUS RESTRAINT

AMOUNTS

OF

PLATES

OF

EDGE

Section

C2

1 May 1971 Page 65

0 L, O< O=

Z_

Z 2;

d_ I

L) M

om _o oo o<

< Z 0

I 11//7/ r I, /lJ7

"-;z

°

MZ _Z

{b

_m

Section

C 2

1 May

1971

Page

66

!

!

0.6

0.8

1.8 1.6 1.4 1.2 n,.

1.0

n,.

o 0.8

COMBINED SHEAR & LONG. DIRECT STRESS

0.6 m

R L + R$2 = 1.0 0.4 0.2 0 -2.0 -1.8

-1.6

-1.4

-1.2

-1.0

R L LONGITUDINAL FIGURE

C2-21.

INTERACTION

-0.8

-0.6

-0.4

-0.2

TENSION OF DIRECT

0

0.2

0.4

R L LONGITUDINAL

COMBINED

SHEAR

AND

C2-22.

LONGITUDINAL

STRESS

INTERACTION

OF

BENDING,

AND

COMBINED SHEAR

1.2

COMPRESSION

R$

FIGURE

1.0

COMPRESSION,

Section

C2

1 May 1971 Page 67

-'T-T'TRIR

s-

(.0" _

(.2"

_(,.s-

_,,.7 .._ j.q_.e"

_/.o-

II!

1.0

...........

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1

0.2

0.3

0.4

0.5

Rc

FIGURE

C2-23.

MARGIN OF SAFETY DETERMINATION COMPRESSION BENDING AND SHEAR

0.6

0.7

0.8

0.9

R$

FOR

COMBINED

1.0

Section

C2

1 May Page

(a)

1971 68

(b)

1.0

1.o :_,_.

a/b = 0.8

a/b-

1.0

0.8

0.6 Rb

N\\_o.,"

:.\,_\X _' \\\\ \

\

'

0.4

0.2

\ \

;.o\ \ \\\\"

._ o_\\ o_\\ \',

\\\\_\

o,\\\ %

\\

\ ,\

-\\

\

,,,

0 0.2

0.4

0.8

0.6

%

1.0

0.2

....

!

,,,a

0.4

.,,.

0.6

Ry

0.8

1.0

Ry

(c) 1.0 a/b = 1.20

0.8 fy

llllllllllllllll. Rb

o._,,,\\ o.,\i\\ \_ \\

\\ \'o, \\,_\\_

Illlllllllllllll

-

\

°_. \\\\\\\ !.............. :: ....l............ t 0.2

0.4

0.6

0.8

1.0

Ry

FIGURE

C2-24.

RECTANGULAR

INTERACTION PLATES

UNDER

LONGITUDINAL

CURVES

FOR

COMBINED BENDING

SIMPLY BIAXIAL LOADINGS

SUPPORTED COMPRESSION

FLAT AND

Section C2 1 May 1971 Page 69

(d)

[ m

1.0

(e)

a/b = 1.60

[

a/b = 2.0

l

,,,i

i,,1

0

m

0.8 0,

"" 0., 0_ 0

0.6

Rb

Rb 0 ,.80

0.2

'

"0.9)

0

\

0.8 1.0

_

""--

Rb

0

"Villi

Ilia

I

II

Ill

l

I ,Ib=3.0 I

IlJl

I_il

Ill

q,)

I ,/b-_

I

._......,._.._._...

!-

0.6

, i

,,I,

If)

\o.,

z

_0.50

_

_

-""-_

__-'1"-"

llll

___

_.

_

0.4

_

.._,o_

IIII o, _..._o \ \ IIII \ \ IIII o., _\o.,o \

\ 0.2

_

III1

_0.9

;= 0

")ala

0

"

IILI

":

_

_

%%

iJ.,

0.2

0.4

0.6

0.8

1.0

0

Ry

FIGURE

0.2

0.4

0.6 Ry

C2-24.

Concluded

0.8

1.0

Section 1 May Page

C2 1971 70

fs

IJ

2

f --T--T--T-- 1-1.0

0.8

0.6 Rb 0.4

0.2

0

''''l'

0

'''

,,_l,,,_

0.2

*_,,I,

,_

0.4

_lJt

0.6

|l

J,

,j

,fjjj,

0.8

1.0

Rs FIGURE FLAT

C2-25. PLATES

INTERACTION

CURVES

FOR

SIMPLY

UNDER VARIOUS COMBINATIONS OF AND TRANSVERSE COMPRESSION

SUPPORTED, SHEAR,

LONG

BENDING,

Section

C2

1 May

1971

Page 1.0 _

,_.,_

I ._

71

'r

0.8

j_i-

0.6 Rb 0.4

0.2

t ,11111

1.0

a I

J

allllln

I

I

0.8

aillql

I

0.6

Jpl

1.0}J

pare

allJ

I1

I

IJ"

0.4

0.2

0

0,6

0,8

1.0

Ry

R$

0

. 0.4

0.2

0.4

........ ,....... .j..... /....... jy

7//

!

Ry

: 0.6-

0.8

/

Rb _

_...__

1.0

• FIGURE

C2-25.

Concluded

Section

C2

I May Page

1971 72

Rs

0

O.2

0 ;,,llW111

0.4

0.6

0.8

1.0

1.2

,11,|,,11

rllll|l|l

0.2

I

|1

III/

II/

/

y,,;, ///

0.4 [ 10

./

Ry

o.,/¢',::

0.6

0.8

1.0

1.0

0 0.5 0.8

0.8

i(

/ (

(

i

Rs

.o.s :

Rb 0.4

i,,,l,*,, 1.0

h,,Jl 0.8

_.,,Jl,,,, 0.4

0.6

,,,,,,,,,_ 0.2

0 0

Ry

FIGURE

C2-26.

SUPPORTED, COMBINATIONS

INTERACTION

LOWER OF

EDGES SHEAR,

CURVES CLAMPED, BENDING,

FOR LONG

AND

UPPER PLATES

TRANSVERSE

EDGES UNDER

SIMPLY VARIOUS

COMPRESSION

Section

C2

1 May

1971

Page

fy

IIIIII

I I I f$

CLAMPED

tltttt

t t t

1.0

0.8

\

0.6 Rb

/J

0.4

0.2

J '/

:,,,,1,,, 0

IliIIIll

0.8 R$

FIGURE

C2-26.

Concluded

II

1.0

1.2

73

Section 1 May Page

C2 1971 74

f_

_

ff

v e_ eL

J

0 ¢J

f

N_z

o_

_m 0

/

-1

_N

,n_

t"qz o

/

-

j

- I

Z I,U

Z 0

I-

t_

Z M o _',

m _

Section

C2

1 May

1971

Page

4.0

75

!

lllllll 3.5

tlt1111 fv 3.0

2.5

kx

2.0

1.5

1.0

0.5

8 o o

2

4

5

8

10

12

ks

FIGURE SUPPORTED

C2-28. FLAT

CRITICAL PLATES

TRANSVERSE

STRESS UNDER

COEFFICIENTS LONGITUDINAL

COMPRESSION,

AND

FOR

SIMPLY

COMPRESSION, SHEAR

Section

C2

1 May

1971

Page

76

kx

0

1

2

3

4

5

6

7

8

ks

FIGURE SUPPORTED

C2-29.

CRITICAL

FLAT PLATES TRANSVERSE

STRESS

COEFFICIENTS

FOR

SIMPLY

UNDER LONGITUDINAL COMPRESSION, COMPRESSION, AND SHEAR

9

Section

C2

1 May Page

4.8

!

1

1971 77

!

F"

,lllllllllllL m-2 4.0

r1TTTi1TTTTIT fy

3.0

k x

2.0

/i °-: 1.0

I

2

3

4

5

6

7

8

k S

FIGURE SUPPORTED

C2-30. FLAT

CRITICAL PLATES

TRANSVERSE

STRESS UNDER

COEFFICIENTS LONGITUDINAL

COMPRESSION,

AND

FOR

SIMPLY

COMPRESSION, SHEAR

9

Section

C2

1 May Page

4.8

I

fx "-" --"

,.o ""- _.

l

I

1971 78

i

l".b-,.o] b |

i a

vTtiTvTt(tvit fy

3.0

Im'2

"IX '

2.0

1.0

0.8 m-1 2

0 0

1

2

3

4

5

6

7

8

ks

FIGURE SUPPORTED

C2-31. FLAT

CRITICAL PLATES

TRANSVERSE

STRESS UNDER

COEFFICIENTS

FOR SIMPLY

LONGITUDINAL

COMPRESSION,

COMPRESSION,

AND SHEAR

Section

C2

1 May

1971

Page

79

4.8

I

I

I [

I ,,/b-a.o

I I

lllllllllllllllllllll

kx

).8

I1 2 5

1

2

3

4

5

6

7

8

ks

FIGURE SUPPORTED

C2-32.

CRITICAL

FLAT PLATES TRANSVERSE

STRESS

COEFFICIENTS

FOR

SIMPLY

UNDER LONGITUDINAL COMPRESSION, COMPRESSION, AND SHEAR

9

Section 1 May Page

o1

C 2 1971 80

o1

i

1

'rxy

"rxy

°2

I_

'

I_

a

a = a/b

FIGURE

C2-33. AND

RECTANGULAR NONUNIFORM

I °2 i1

I

_ = 01 / "rxy

PLATE LONGITUDINAL

UNDER

COMBINED

COMPRESSION

SHEAR

Section C2 1 May 1971 Page 81

0 I

I

I

Z

lillllii

--I _LA'3

t t t t t t t

X

II

<[ x

+.J

C_ Z
OU

t t

¢,_

If)

_' t_T_t_

a_

2

ml

o A

l

t

tl m

x

,

o

Od

r._

M

•

.,Q v

II

L_

Z

marc Z< M

ff

X

9' GA II

//

Z x

Z

/ i

d

f

N

I!

i J i

4 © "--" m_

o_

Z

/

A_

J

Z

M

II

o

i I

ial_iJl_L ILllJllll

llllJllll O0

LLIIILIII r',,,,

llllJlll,

JJllll,

l'

nnilJ,lll

anJtJ]aJJ 0

Section

C2

I May

1971

Page

82

+T2 E

(t/b)2

kcav Fay=

12(1 -re2)

IlllJ

I I I.I

7

o ,t,l,Jt, 0

I I Ill

[IJ

I

1

2

I It

JI I III

3

III

I J I]

4

I J

I I1 I1 I I I I 5

a/b FIGURE

C2-35.

COEFFICIENT THICKNESS

AVERAGE FOR

COMPRESSIVE-BUCKLING-STRESS

RECTANGULAR

WITH

LINEARLY

FLAT VARYING

PLATE AXIAL

OF

CONSTANT LOAD

m

Section 1 May Page

Z 0

X

0

z 04 ¢'--Q--"1

(Y 0

0 Z

t,-

I r_ 0 m

0 e-

04

M r..)

00 I

C 2 1971 83

Section

C2

1 May

1971

Page

(a) LOADING

84

IN x-DIRECTION

25 Fy 20

15

" I \

k% 10

I::y

[

B

5

0 _ 0

nnllJlnl

_1

IJllJllll

'1

2

3

11 |

4

]

5

a/b (b) LOADING

FIGURE

C2-37.

IN y-DIRECTION

COMPRESSIVE-BUCKLING

SHEET INTO

ON

NONDEFLECTING PARALLELOGRAM-SHAPED (All panels

COEFFICIENTS SUPPORTS

DIVIDED PANELS

sides are equal. )

FOR

FLAT

Section

C2

1May Page

1971 85

10 a/$ = 1 ky

___11111_1111.__ kx _

6

_ttttttlttt

ky a/b

¢ 4

2

1.155

0 0

2

4

6

10

8

k x

FIGURE

C2-38. AND

INTERACTION TRANSVERSE

CURVES LOADING

FOR OF

SKEW

COMBINED PANELS

AXIAL

12

Section C2 1 May Page

1971 86

f$

b

a fs

FIGURE

C2-39.

SHEAR

LOADING

OF

PARALLELOGRAM

PLATE

Y

f

3O

\

2O

X

_| _

=0°

k s

10 m

m

alb =w

0

for

I 0

1.0

2.0

1 cos2a

?r 2E 12 (l-ue

Ft'1 2 2)

3.0

a/b

FIGURE

C2-40,

BUCKLING FLAT

COEFFICIENTS PLATES

OF

IN SHEAR

CLAMPED

OBLIQUE

Section

C2

1 May

1971

Page

87

0

tJ.I I'C£

<

,r,.

o Q,.

J

a. --

er

< _t_iA

Z <

J _J

I I i i

r,J

I i 1 i

I i I I

c5

I I i i

IIII

L_

Itll

0 N

0

0

0

0

© klJ v f_J

tl

i

/ <

I nl

I

ill

I

I ii

Z 0

2_ o •

y I

i I I 1

I I I I

O O

u

m

a.

@

Y l I I

©

W

°)

i

J /:

0

U

0

I ¢q

Section 1 May Page

C 2 1971 88

f$ .

fc fcr --

/r 2 E = k

r/

t2/a 2 12(1 -v:)

>450

,-V-Y-Yfc

15 ALL

EDGES

CLAMPED

kc COMPRESSIVE STRESS COEFFICIENT _

PERPENDICULAR CLAMPED,

10

SIMPLY

EDGES

HYPOTENUSE SUPPORTED

PERPENDICULAR SIMPLY

HYPOTENUSE /ALL

EDGES

SUPPORTED

EDGES

CLAMPED SIMPLY

SUPPORTED

/ Sp 5

I

w

v

v

,

-

-25

,

0 ks

FIGURE

C2-42.

RIGHT-ANGLED

SHEAR

INTERACTION ISOSCELES

_

i

b

,I +25

STRESS

b

b

i +50

COEFFICIENT

CURVES

FOR

TRIANGULAR

BUCKLING PLATE

OF

Section

C2

1 May Page

12

12

I

1971 89

I

N 2 = 1/2 N 1 11

11

/i

10

b2/a 71 10

9

Nl(b)2

/7'

9

8

Nllb)2

DTr 2

/

8

D_ 2

7

7

6

6

5

__j

,/ f

5

4

4 0o

10 °

20 °

30 °

40 °

0°

10°

20 °

0

30 °

40 °

0

!

I

N 2 = 2N 1

N1

b2/a = 1

5

1/2 N11_12

4

1/4

_b2/a = N1 (_}2 2

1.0

D_ 2

0o

10 °

20 °

30 °

40 °

% 0.5 \ 0.7

b 1 + b2

\ 0.25 \0.1 0_(TRIANG

=

2

N1 = fl t

LE)

0

/ 0 0o

20 °

40 °

60 °

0

D=

Et 3

P'---- "

12(1 -_J:)

FIGURE

C2-43.

BUCKLING

CURVES

FOR

TRAPEZOIDAL

"1

PLATES

_L

Section

C2

1 May

1971

Page

90

10 b2/l

" 1/4

\

• b2/I

- 1/2

,,I I ' !'

Nl(b)2 D _2

01 45o400

35°o 25o _

2

15°200

0° 2 0

1 N2/N 1

b2 iI , 14 ! 11 10

b2/a - 3/4

,o\ \'\

_¸

1

9

D 72

'o _.

,

_,_..o _.

b 1 +b 2

0

/N

. ©_,k

1

_%

D

_.---.

FIGURE

_

C2-44.

_.-_...L--

---_

BUCKLING

Et3 12il

CURVES

FOR

40 °

N2/N1

• _ 2)

TRAPEZOIDAL

PLATES

_

Section

C2

1 May 1971 Page 91

b2

L !

_1 -I

Nx 2

Ve

=0.3

0

<

15 °

12

It

t I ll-_x b1

"t Nx 1

10

fcr 1 -

t = 0.8

Nx2/Nx

I

and

ends

\

T

b2

0.4 0.6 0.8 1.0

0 0

3

4

a

b1

FIGURE

C2-45,

BUCKLING

simply-supported.

STRESS No stress

DIAGRAM normal

(Sides to sides.

)

1

Section

C2

1 May

1971

Page ve = 0.3

92

0 < 15 °

10

6

-10. 6 0.8 1.0

4

0 0

1

2

3

4

a/b I h

b2

:I

--__

Nx2

Nx2/Nx 1 = 1

-_

fcr I = Nxl/t

Nx 1 k r FIGURE

C2-46.

BUCKLING

simply-supported.

STRESS No

stress

DIAGRAM normal

bl

(Sides to sides.

J -i and )

ends

Section C2 1 May 1971 Page 93

N x

1 _,--

fcr 1 =

v e = 0.3

b2 _

_Nx

0 < 15 °

2

10

"-_'a

l _Ol

\

I i l_Nxl

b2

fcr 1 ml

--

E

b1

t2 0.4 0.6 0.8

__---

1.0

Nx2/Nxl

= 1.2

0 0

3

4

a

bI

FIGURE

C2-47.

BUCKLING

simply-supported.

STRESS No

stress

DIAGRAM normal

(Sides to sides.

and )

ends

Section

C2

1 May

1971

Page

94

_.-- b2-- _ ve " 0.3

# < 15 °

14

i.!

12

10

Nx 1 =

fcrl

fcrl.

b21

E

t2

t

b2 b1 0.4 0.6 0.8 1.0

Nx2/Nxl

= 0.8

0 0

2

3

4

a/b 1

FIGURE

C2-48.

BUCKLING Ends

clamped.

STRESS

DIAGRAM

No stress

normal

(Sides

simply-supported.

to sides.

)

Section C2 1 May 1971 Page 95

,f

J_- b2---t Ue = 0.3

fNx

0 < 15°

2

14

a

1

12

10

\

_'Nxl

I'---

bl --'t

Nx 1 f©r1 --e

b21

fcr I =

--

E

t2 b2 b1 0.6

0.8 -_

1.0 Nx2/Nxl

= 1.0

2 a

bI

FIGURE

C2-49.

BUCKLING Ends clamped.

STRESS DIAGRAM No stress normal

(Sides simply-supported. to sides.)

Sectioa C2 1 May 1971 Page 96

}.....t,2......J

V e = 0.3

0<15

°

a

12

-l.]_

\ 10

,,,,

"'

_._.. bl.___..( N'_' 8

6

4 1.0

2

Nx2/Nxl

= 1.2

0 0

1

2

3

4

I!

b1

FIGURE

C2-50.

BUCKLING Ends clamped.

STRESS DIAGRAM No stress normal

(Sides simply-supported. to sides. )

Section

C 2

1 May

1971

Page

re=0.3

0<

97

15 °

14

12

10

b2/b 1 0.4 fcr I

bl 2

E

t2

0.6 0.8 1.0

!_

Nx2

a

a/b I

T J_ Nx 1

_--- _,---t Nxl

Nx2/Nxl

fcr 1 _

FIGURE

C2-51.

BUCKLING

simply-supported.

STRESS No

stress

DIAGRAM normal

(Sides to sides.

clamped. )

Ends

=

0.8

Section

C2

1 May

1971

Page

L ,-

0 < 15 °

_-0.3

I

14

b2

98

_l 7

I

12

10

for 1 E

2 b1 t2

0.8 --

--

1.0 J

0 0

2

3

4

bI

FIGURE

C2-52.

BUCKLING

simply-supported.

STRESS

DIAGRAM

No stress

normal

(Sides to sides.

clamped. )

Ends

Section

C 2

1 May

1971

Page

v e'0.3

<15

99

° Nx 2

14

12

b_

bl

'1

Nx 1 b2

A

8 for 1

t"

\

10

fcr I =

b1 0.4

Nx2/Nxl

=

1.2

0.6 E

0.8

t2 6

1.0

0 0

a

3

b1

FIGURE

C2-53.

BUCKLING

simply-supported.

STRESS No

DIAGRAM stress

normal

(Sides

clamped.

to sides. )

Ends

"1

Section

C2

1 May 1971 Page 100

2O

18

16

14

fcr 1 E

by

12

Nx2/Nxl

= 0.8

t2 Nx I

10

b2-"[

2

2

x1

0

-,

0

4 a/b 1

FIGURE

C2-54.

BUCKLING STRESS DIAGRAM (Sides No stress normal to sides, )

and

ends

clamped.

Section C2 1 May 1971 Page 101

2O

18 v e = 0.3 0 < 15 ° 16

14

12

b2 fcr 1

bl2

E

t2

Nx 1

b1

fcr I =

10

T

Nx2/Nxl

= 1

0.6 0.8

1.0

}-,- b2- _ Nx 2

I,,,,,,,L Nix1

0 0

2 8 b1

FIGURE

C2-55.

BUCKLING STRESS DIAGRAM (Sides No stress normal to sides. )

and

ends

clamped.

Section C2 I May 1971 Page 102

_e = 0.3

O< 15°

18 Nx 2

16

14

I12

"1 fcr I = Nxllt

Nx2/Nxl

10

" 1.2

0.4 E

t2

_"

6

0.6

__--

0.8

__

1.0

2

3

4

a/b I

FIGURE

C2-56.

BUCKLING STRESS DIAGRAM (Sides No stress normal to sides. )

and ends

clamped.

Section C2 1 May 1971 Page 103

=0.3

u e

0 <15 °

/Nx

t

_FFITI-_

2

T ± a

fcr I E

Nx 1

b_

--o--

d

fcr I =

t2

b2/b 1 0.4 0.6 0.8 1.0

Nx2/Nxl

2

3

= 1

4

a

bI

FIGURE supported.

C2-57. BUCKLING No displacement

STRESS DIAGRAM of the sides normal

(Sides and ends simplyto the direction of taper.

)

Section C2 1 May 1971 Page

104

I...--b:,....._ N x ue = 0.3

0<15

°

10

T 1 8

2_iNx

f

1

Nx 1 for 1" fa

1

b21 Nx2

z

J_ i

/ Nxl

o.4

b..2

0.6 0.8

b1

" 1

1.0

0 0

3

2

4

| b1

FIGURE

C2-58.

clamped.

BUCKLING

No displacement

STRESS

DIAGRAM

(Sides simply-supported.

Ends

of the sides normal to the direction of taper. )

Section 1 May Page

C 2 1971 105

.,,,o--b2

,ve = 0.3

0 < 15°

['--[--_"-_

14

Nx2

12 Nx I

10

8 fcr 1

b21

E

t2

Nx2/Nxl

--

= 1

0.4 0.6 0.8 1.0 b2

3

4

a/b 1

FIGURE supported.

C2-59.

BUCKLING No

displacement

STRESS of the

DIAGRAM sides

normal

(Sides to the

clamped. direction

Ends

simply-

of taper.

)

Section

C2

1 May 1971 Page 106

re=0.3

0<15

°

16

T l

14

It

\

12

1111111 11_1

I-- b,--H

10

t

Nxl fcr 1 "

Nx2/Nxl

b2_!1 E

t2

.-_

- 1

0.4 0.4 --

0.8 b2 1.0

_ b1

2

"'

0 1

0

2

3

8

b1

FIGURE

C2-60.

BUCKLING

displacement

STRESS

DIAGRAM

(Sides and ends clamped.

of the sides normal to the direction of taper.)

No

Section

C 2

1 May

1971

Page

N

Nr

FIGURE

C2-61.

CIRCULAII

f rcr -77

PLATE

= k

fr -

t2

12 (1 -v 21

a2

r

t

SUBJECTED

E

107

TO

RADIAL

LOAD

k 25

CLAMPED EDGES AT

U e = 0.3

1 20

I SIMPLY

r=8 i

15 v s = 0.3

3 2 1 0

_/_

14 _ 13

J

120

_=

SUPPORTED EDGES AT

"* _. _ -

r=a 0

0.5

1.0

b

0.5

a

b a

--

FIGURE

C2-62.

BUCKLING

COEFFICIENTS

FOR

ANNULAR

PLATE

Section 1 May Page

a

C2 1971 108

i,i

b/2 STIFFENER

b12

(a) FIGURE

C2-63.

(b)

RECTANGULAR STIFFENER

PLATE UNDER

WITH

(c)

CENTRAL

COMPRESSIVE

LONGITUDINAL

LOAD

e.m.m.--

b STIFFENER b1 -I

(a} FIGURE

C2-64.

(b)

RECTANGULAR

ECCENTRICALLY

PLATE

LOCATED

WITH

UNDER

(c) ONE

COMPRESSION

(a)

(b)

(c)

(d)

FIGURE

C2-65.

TYPICAL

STIFFENER

CORRUGATIONS

Section C2 1 May 1971 Page 109

24 22

T

20 w

i

b

18

.//'j

a

16 at

14

12

10

f

8 6 0

0.2

04

0.6

bla _"D

FIGURE

C2-66.

SHEAR

SUPPORTED

BUCKLING CORRUGATED

0.8

1.0

11D2

COI_FFICIENTS PLATESWIIEN

FOR

FLAT II>I

SIMPLY

Section

C2

i May Page

1971 110

20

18

J¢

16

14 N,. m

12 L

___-_'_

r

10 0.1

I

I

!

0.2

SIMPLY SUPPORTED

I lll 0.5

I 1

I 2

EDGES

J

I III 5

I 10

! 20

H

FIGURE

C2-67.

SHEAR

INFINITELY

LONG

SUPPORTED

BUCKLING

COEFFICIENTS

CORRUGATED

EDGES,

H
AND

PLATES: CLAMPED

FOR SIMPLY EDGES

FLAT

I 50

Section C2 1 May 1971 Page 111

ksTT2E

(t/b) 2 : Z b=b2/rt

(1-_2)

F,_ 12(1-v_)

½

_

103 -----

10 2

a/b 10 • \ 2.0_ 3.0--

k$

1.5 J

10

5

r a

t 1

, 1

I 2

, I,l,l,l,hhl, 5

,

I

, I,l,l,l,l,l,h

10

,

t

, I,l,l,l,l,hh

10 2

10 3

Zb (a) FIGURE

C2-68.

LONG

SHEAR

SIMPLY

BUCKLING CURVED

SUPPORTED COEFFICIENTS PLATES

PLATES FOR

VARIOUS

Section 1 May Page

C 2 1971 112

10 3

10 2

ks

10

10

10 2 Zb

(b)

FIGURE

LONG

C2-68.

CLAMPED

PLATES

(Continued)

103

Section

C2

1 May

1971

Page

113

10 3

lO 2

a/b

ks

10

_._

1.5 _

,

2.0 ----,\

-

1.0

\

5

tT 1 2

it

i..__/t

_

b I

B

1

,

1

I , I, I,l,l,l,l,h 2

5

_

I ] I,

I

I,l,l,l,hh

10

10 2 Zb

(c)

WIDE.

SIMPLY

FIGURE

C2-68.

SUPPORTED

(Continued)

PLATES

,

I , I,l,10hhh 10 3

SectionC2 I May 1971 Page

114

10 3 ,

b

1 ..,._._..,jS.L I

I

)

I

I

]

IIII

1

I

!

I

)111

]

lo2

10 Zb

(d) WIDE CLAMPED PLATES

FIGURE

C2-68.

(Concluded)

I

[

]

] ]11

Section 1 May Page

f_

6

C 2 1971 115

i

- Z b ,, 2.50 1.25

_-- Zb = 2.50 _\I_

1.25 I

4 A/bt = 0 kc

3 A/bt ,, 0.2

0

6

Zb

5m

4

8

=.

i

2.50 1.25 0 --

12

0

16

4

8

12

16

Z b - 2.50

I i i

0

_

1.25

kc

3

/,¢//

_

2

A/bt = O.6

Y/

,//

f

0 0

_

4

8

12

0

16

4

8

12

16

EI/bD

(a) FIGURE

C2-69.

SUPPORTED

a/b=

COMPRESSIVE-BUCKLING CURVED

PLATES

4[3 COEFFICIENTS

WITH

CENTER

AXIAL

FOR STIFFENER

SIMPLY

Section 1 May Page

i '

I

r

A/bt ,, 0 5

A/lit - 0.2 I

i

I l

l

._

4

i_;

3

_

-1.25

_zb. I

.o2.so

_7_

i

1.25 0 i

I

,// . ' -__ 1_. . - H//II/IIAOV

0 0

8

I 16

I

I

A/bt" l

.u

|

,"

-

-

t_

1 24

32

0

0.4

8

16

I

I

24

32

A/bt - 0.6 I

i

I

3

/

2.50 1.25

1.25 0

//

_

f

f 0 0

8

16

24

32

0

8

EI/bD

(b)

FIGURED

a/b=2

C2-69.

(Continued)

16

24

32

C2 1971 116

Section

C 2

1 May

1971

Page

I

A/b! = 0

_

_'- "b = 2.50

\

kc

A/bt = 0.2

_ Z b = 2.50 ,--- 1.25

r--1.25

3

'/

#/(////////,z

Y

'___

i-,,,:,,#,,7 ,. l, 0

16

--

32

48

64

0

16

32

48

64

48

64

A/bt = 0.6

A/bt = 0.4

_- Z b = 2.50 /

1.25

_

,

kc 3

/// 0

16

Y

32

48

0

64

16

EI/bD

(c) FIGURE

C2-69.

a/b

= 3 (Continued)

32

117

SectionC2 1 May

1971

Page

118

48

64

80

48

64

80

6

A/bt

= 0

Albt

= 0.2

4

3

2

2

- ///////////0 0

16

'

I

I

J

!

32

48

64

80

I

1

I

96

0

16

32

96

'

. 0 0

16

32

48

64

80

96

0

16

EI/bD

(d) FIGURE

a/b=4

C2-69,

(Concluded)

32

96

Section

C2

1 May 1971 Page 119

P

0.6

J

0.5

f

_ \\\\_.x\\\\\\\\\

]

0.4

\...,.,\.\.\\..\\

/

0.3

0.2

I=I

=II

GAIN

IN BUCKLING

STRESS

NO GAIN

IN BUCKLING

STRESS

0.1

0 0

FIGURE

20

C2-70.

DEFINITION

OF

a/b

VS

Zb

RELATIONSHIP

BUCKLING STRESS OF AXIALLY COMPRESSED CURVED ADDITION OF SINGLE CENTRAL CIRCUMFERENTIAL

24

FOR

28

GAIN

PLATES DUE STIFFENER

IN TO

Section 1 May Page

0 -----.__.

Q

j

M

N_ N

a_ N

Z_

e4_

0

2:

O

_

"

II

N

,,

_0 !

0

o

q

H!_sUl'l :ooi/

o

= HOlOV:i

o.

NIV9

"XVIN

r..)

<

C 2 1971 120

Section i May

r_

Page

C2 1971

121

F_

--OLu
:E .. _. < 0.,_ _

i

" (J c_ _-z.-

-J <_

>Z < J_,-

k.r a=,.,i-

xi-_

w
.j'_-_ 0

(,_

II

UJ

--

-J

m_

o _

I-

II

B

(3

*'

_o

U

"Q N

_II

_Z m<

e-

Na_ m<

ZO

ii,

ON O<

z_ Z

_m _M

0

I

I

I

LII

(: e-

e,-

('5

"

oZ s_

M _r,.) 0

Section

C2

1 May 1971 Page 122

<

a;

a; )1

n

II

u

N

I-..:Jl

..=I m .

.<

= =

N

°

•_1

o

r_ r_

_

I.M

M.I

O " Z • =.I >. •

i Z'

z

.3

--I

_z

0

>. tJ

>.

<

ll/

-

M Z

. o

o ° o O

r_

ell

N

N .< M

"

,.D °

L.--.

8

!

o

I

o

I

M

o

_N

(.) Z M M r_

Section C2 1 May 1971 Page

/-

123

Section 1 May Page

C 2 1971 124

Section

C2

1 May 1971 Page 125 REFERENCES

Bruhn,

.

E.

F. : Analysis

Tri-State o

Offset

Timoshenko, McGraw-Hill

and

Printing

Design

of Flight

Company,

Cincinnati,

1

Noel, Plates nal no.

Q

R. G. : Elastic Under Critical

Johnston,

A.

Bending, of Infinitely Batdorf,

,

,

E.,

Shear,

Lateral

Jr.,

and

and

Long

Batdorf,

and

Stein,

J.

H. , Jr.

Bleich, Book

10.

F. :

Pines,

S.,

Libove, a Simply Linearly

Inc.,

Tapered 14, no. C.

L.,

Critical

M. : Critical Supported

Aeron.

Sci.,

vol.

Combinations

Dec.

for

of

Buckling

of Shear

Long Flat Plate NACA Report

Combinations

with 847,

of Shear

Fiat

Stresses

19,

1951.

Combinations

Rectangular

Buckling

Flat Rectangular Bending, Longitudi-

Stresses

TN 2536,

Plates

and

Plates.

NACA

of Simply

Supported

Under Combined Longitudinal Compression, and Shear. J. Aeron. Sci., June 1954. Strength New York,

and Gerard,

Efficiently Sci., vol. 11.

Buckling

Company,

C. :

Stability.

of Rectangular Nov. 1963.

P. : Critical

for an Infinitely Against Rotation.

: Critical

Fiat Rectangular Plates Transverse Compression, .

J.

Direct Stress Restrained

Johnson,

NACA

Houbolt,

and Transverse Edges Elastically 1946.

K.

J.

Compressive

Plates.

Direct Stress for Simply TN 1223, Mar. 1947. .

Buckert,

Transverse

Flat

S. B. , and

S. B.,

Buckling Quart.,

Compression.

1965.

of Elastic 1961.

Stability of Simply Supported Combinations of Longitudinal

Compression, and 12, Dec. 1952.

Structures.

Ohio,

S. P., and Gere, J. M. : Theory Book Company, Inc. , New York,

Cook, I. T. , and Rockey, K. C. : Shear with Mixed Boundary Conditions. Aeron.

.

Vehicle

of Metal

Structures.

G. : Instability

Analysis

Plate Under Compressive 10, Oct. 1947.

Ferdman,

Supported Plate in the Direction

S., Under of the

McGraw-Hill

1952.

and Reusch,

and

Design

Loading.

J.

J. :

Elastic

J.

of an Aeron.

Buckling

a Compressive Stress that Varies Loading. NACA TN 1891, 1949.

of

Section C2 I May

1971

Page 126 RE FERENCES

(Continued)

12.

Seide, P. : Compression Buckling of a Long Simply Supported Plate on an Elastic Foundation. J. Aeron. Sci. , June 1958.

13.

Wittrick, Under

14.

W.

H. : Buckling

of Oblique

Uniform

Compression.

Aeron.

Guest, J. : The Compressive

Plates

Guest

J.,

and

Supported Aero. 16.

J.

Parallelogram

Res.

Labs.,

P.

O. : A Note

Plates,

Structures (Melbourne),

Anderson,

R. A. : Charts

Giving

Critical

Continuous

Flat

TN 2392,

Durvasula, vol.

8,

S. : no.

Divided

July

1951.

Buckling

1, Jan.

into

on the and

Supply

Sheet

Compressive

Skew

Plates.

204,

of

Journal,

1970.

Wittrick, W. H. : Buckling of a Right-Angled Plate in Combined Compression and Shear

Isosceles (Perpendicular

Clamped,

Rep.

Plates vol.

with Clamped V, May 1954.

L., and Klein, B. : The Buckling of Isosceles J. Aeron. Sci., vol. 22, no. 5, May 1955.

Hypotenuse

Simply

Supply

Supported).

(Melbourne),

June

Isosceles (Perpendicular

Simply

Supported,

Hypotenuse

Rep.

Labs.,

Dept.

Supply

(Melbourne),

Wittrick, W. H. : Buckling in Combined Compression (Melbourne),

Nov.

of a Simply and Shear. July

1952.

Triangular Edges Aero.

Res.

1953.

Buckling of a Right-Angled Compression and Shear Clamped}.

Edges

Triangular

SM 211,

Wittrick, W. H. : Plate in Combined

Supply

Note

AIAA

20.

Dept.

of Simply

Panels.

Cox, H. Plates.

22.

Plate Simply

Stress

19.

21.

1953.

1953.

Parallelogram-Shaped

of Clamped

Feb.

Res. Labs.,

Materials. May

Wittrick, W. H. : Buckling of Oblique Under Uniform Shear. Aeron. Quart.,

, Dept.

II,

Buckling

18.

Labs.

Edges

4, pt.

199, Aero.

Dept.

NACA 17.

Silberstein,

Clamped

vol.

Buckling of a Parallelogram

Supported Along all Four Edges. Rep. SM Dept. Supply (Melbourne), Sept. 1952. 15.

with

Quart.,

Triangular Edges

SM 220,

Aero.

ires.

1953. Supported Triangular Rep. SM 197, Aero.

Plate Res. Labs.,

Section

C2

1 May i971 Page 127

L

"REFERENCES

23. f

(Concluded)

Wittrick,

W.

Triangular

H. : Symmetrical

Plates.

Buckling

Aeron.

Quart.,

24.

Klein, B. : Buckling of Simply J. Appl. Mech., June 1956.

25.

26.

of Right-Angled

vol.

V, Aug.

in Planform.

Pope, G. G. : The Buckling of Plates Tapered Aircraft Establishment, Report No. Structures

in Planform. 274, Apr.

Royal 1962.

Tang,

Under

S. :

Elastic

MIL-HDBK-23, Structural pared

of a Circular

Spacecraft,

Structural

Design

by the

Corporation

Stability

J.

Los

Guide Angeles

Plates

1954.

Tapered

Compression.

Supported

Isosceles

vol. Sandwich

for

for Wright-Patterson

1, Jan.

Air

July

Composite

of the

North

Force

Unidirectional

1969.

Composites,

Advanced

Division

Plate

6, no.

1968.

Application. American

Base,

Ohio,

29.

Schildcrout, M. , and Stein, M. : Critical Axial-Compressive of a Curved Rectangular Panel With a Central Longitudinal NACA TN 1879, 1949.

30.

Batdorf,

S. B.,

and

Schildcrout,

M. :

Stress of a Curved Rectangular Panel Stiffener. NACA TN 1661, 1948.

Critical With

PreRockwell Aug.

Stress Stiffener.

Axial-Compressive

a Central

1969.

Chordwise

--__j

SECTION C3 STABILITY OF SHELLS

v-"

TABLE OF CONTENTS Page C3.0 3.1

STABILITY

OF

CYLINDERS

3. 1. 1

SHELLS

......................

1

..............................

Isotrooie 3.1.1.1

5

Unstiffened Axial

Cylinders

Compression

Unpressurized 3.1.1.2

Axial

-...............

Compression

Pressurized

8

1.3

Bending

--

Unpressurized

3.1.

1.4

B(mding

--

Pressurized

1. 1.5

External

Pressure

Shear,)r

Torsion

3.1.1.(;

Unt)r(.'ssurized 3.1.1.7

Shear

or

3. 1.1.8

Combined

-14

Torsion-................

Loading

C()mt)r(;ssion

16

............

Axial

and

Bending

Axial Compression Pressure ..................

and

External

III.

Axial

nnd

Torsion

IV.

Bending

Compression and

Torsion

3.1.2.2

Bending

16

...........

. . .

17 17

18

Compression

C3-iii

. . .

17

Cylinders Axial

12

............

lI.

3.1.2.1

11

.........

I.

Orthotropie

10

.......

...............

Pr_ssurizcd

3. 1.2

--

................

3.1.

3.

6

...........

...................

............

19 22

TABLEOF CONTENTS(Continued) Page 3.1.2.3

External

Pressure

3.1.2.4

Torsion

...................

25

3.1.2.5

Combined Bending and Axial Compression ................

26

Elastic

26

3.1.2.6

Constants Multilayered

II.

Orthotropic Isotropic

Cylinders .......... Cylinders with

Ill.

Stiffeners and R ing-Stiffened Cylinders

IV. V.

3.1.4

.............

Stiffened

I,

3.1.3

............

Isotropic

Rings ........... Corrugated

3.1.3.1

Axial

3.1.3.2

Bending

3.1.3.3

Lateral

3.1.3.4

Torsion

Cylinders

with

3.1.4.1

Axial

3.1.4.2

Cylinders

Compression

32 32 32

............

34 38

.............

.......

an Elastic

30

............

................... Pressure

27

31

..................

Waffle-Stiffened Cylinders ....... Special Cons iderations ......... Sandwich

23

............ Core

...........

39 40 42

Compression

............

43

External

Pressure

............

44

3.1.4.3

Torsion

...................

47

3.1.4.4

Combined Axial Compression and Lateral Pressure ..........

48

C3-iv

TABLE OF CONTENTS(Continued) Page

3.2

3. i. 5

Design

of Rings

3. I. 6

Plasticity Correction

CONICAL

SHELLS

3.2.1

Isotropic

49

.....................

Factor

50

............

61

..........................

Conical

3.2. I. I

Axial Compression

3.2. I. 2

Bending

...................

3.2. I. 3

Uniform

Hydrostatic

3.2. I. 4

Torsion

...................

3.2. i. 5

Combined

I.

Pressurized

III.

Combined

63

Pressure

Conical

Axial Compression

Bending

6l

............

....

Shells

72

in 72

............

Conical

Shells

in

................. Axial

Compression

••

Combined

V.

Axial Compression Combined Torsion Pressure

3.2.2

Orthotropic

3.2.2. I.

I

Conical

Uniform

76

. ..

Pressure

76 77

Shells ..............

Hydrostatic

74

and

or Axial Compression

....

77

Orthotropic

...................

Stiffened Conical

C3-v

Pressure

............ and External

Cons rant-Thickness Material

If.

External

73

and

Bending for Unpressurized and Pressurized Conical Shells ...... IV.

67 70

Loads ..............

Pressurized

II.

61

Shells ................

Shells .........

77 78

TABLE OF CONTENTS (Continued) Page 3.2.2.2

Torsion

I.

Constant-Thickness Material ....................

H.

Ring-Stiffened Shells

3.2.3

3.3

DOUBLY 3.3.1

....................

San_ich

Isotropic

3.2.3.2

Orthotropic

3.2.3.3

Local SHELLS

Isotropic

Doubly

3.3.

3.3.1.2

l. 1

3.3.1.3

3.3.1.4

3.3.1.5

3.3.1.6

...........

Sheets

8O

.........

82

................

83

..................... Curved

85

Shells

............

86

Caps Under Uniform Pressure ............. Caps

Under

at the Apex

Spherical External

86

Concentrated

..............

88

Caps Under Uniform Pressure and Concentrated

at the Apex

Complete Uniform

..............

9O

Ellipsoidal Shells External Pressure

Complete Under

8O

Sheets

Face

Failure

Spherical

Load

79

.................

Face

Spherical External

Load

78

Conical

Shells

3.2.3.1

CURVED

Orthotropic

......................

Conical

78

Oblate Uniform

Spheroidal Internal

91

Shells Pressure...

Ellipsoidal and Torispherical heads Under Internal Pressure

C3-vi

Under .......

Bulk.....

94

94

TABLE OF CONTENTS (Concluded) Page 3.3.1.7

Complete

Circular

Shells Under Pressure 3.3.1.8

Bowed-Out

Segments

Toroidal"

Under

3.3.3

Isotropic 3.3.3.

3.3.3.2 3.4

COMPUTER ANA LYSIS

3.5

External

Doubly

Sandwich I

Gener-d

Local

Segments Pressure

........

Shells

.........

Doubly

Curved

Shells

Failure

IN SItELL

................................

98

Curved

Failure

PROGRAMS

Axial

...................

Shallow

Orthotropic

96

Toroidal

Under

Loading

3.3.2

External

..................

Shallow

3.3.1.9

Toroidal

Uniform

..............

...............

102 105 ....

109

110 111

STABILITY 113

R FF ER ENC ES .............................

C3-vii

119

DEFINITION OF SYMBOLS Definition

Symbol A

,A s

Stiffener area, and ring area, respectively r

A,B

Lengths of semiaxes

a

Radius of curvature of circular toroidal-sheIl cross

of ellipsoidalshells

section

B!

Extensional

b

Stiffener

spacing;

circular

cross

section

to axis

Effective

b

stiffness

of isotropic also,

distance

section

sandwich from

of circular

wall

center

toroidal

of shell

cross

of revolution

width

of skin

between

stiffeners

e

Coefficients

C,o

ij

Coupling

of constitutive constants

Coefficient

c

for

of fixity

equations

orthotropic

in Euler

cylinders

column

formula

3

D

Wall

D

flexural

Transverse

q

sandwich

stiffness

per

shear-stiffness

unit width,

12(1

parameter

for

- p2) isotropic

wall,

52

G xz

h-

1 _ (11 + t2)

m

D

x

, D

y

Bending y-directions,

stiffness

per

unit

respectively;

cylinder

C3-viii

width D

x

of wall =D

y

in x- and

= D for

isotropic

DEFINITION OF SYMBOLS (Continued) Definition

Symbol

w

d

D

Modified

twisting stiffness of wall;

D

= 2D

for

xy

xy isotropic cylinder

E

th 2 S

m_

Flexural

d

Ring

E

Young's

modulus

Reduced

modulus

Young's

moduli:

face

Young's

modulus

of elastic core

Young's

moduli:

rings; stiffeners,

Young's

moduli

gR

E S , Ef E

stiffness of isotropic

sandwich

wail,

2(1 - _2)

spacing

sheet; sandwich,

respectively

C

E

,

E

respectively

r

F,s , E 0

0-directions,

Secant

E

of orthotropic

material

in the s- and

respectively

modulus

for uniaxial stress-strain

curve

sec

E tan ,

E

E

x

Tangent

modulus

for uniaxial stress-strain

Young's

modulus

of orthotropic

material

curve

in x-

and

y y-directions,

E

Young's

respectively modulus

of sandwich

core

in direction

perpen-

Z

dicular

Et

,

E2

to face

Young's wich

moduli

sheet

of

sandwich

of the face sheets

shell

C3-ix

for isotropie

sand-

DEFINITIONOF SYMBOLS{Continued) Definition

Symbol E

Equivalent

Young's

modulus

Young's

moduli

for

isotropic

sandwich

shell Equivalent

of orthotropic

material

in

the s- and 0-directions, respectively

Ex'

Ey'

Exy

Extensional stiffnessof wall in x- and y-directions, Et respectively;

e r

f

G G

s

,G

r

G Sz

G G

xz

isotropie

cylinder

Distance

of the centroid

from

the

middle

Ratio

of minimum

,G

yz

Shear

modulus

Shear

moduli:

stiffeners;

Shear

modulus

of core

Shear

moduli

planes,

of the

to maximum

in face

shear

_

,

Et 1_-_

-Exy=

for

ring-shell

combination

principal

compressive

surface

stress

Inplane

xy

Ex=Ey-1_-

sheets

modulus of core

rings,

respectively

of sandwich

wall

of orthotropic of sandwich

in s-z

plane

material wall

in x-z

and

y-z

respectively

w

G G

Equivalent

xy

Shear plane;

shear

stiffness G

xy

modulus of orthotropic

= Gt

C3-x

for

lsotropic

or sandwich cylinder

wall

in x-y

DEFINITION OF SYMBOLS (Continued) Definition

Symbol h

Depth

of sandwich

wall measured

between

centroids

of

two face sheets

Moment

I

Moments

,I r

of inertia per unit width

of corrugated

of inertia of rings and stiffeners,

cylinder

respectively,

S

about their centroid

J

Beam

,J r

torsion

constants

of rings and stiffeners,

respec-

S

tively

k

Buckling

coefficient of cylinder

subject

to hydrostatic

with

elastic

P pressure,

pr f2/Tr21)

l_uckling

k

coefficient

of cylinder

an

core

pc subject

k

to lateral

Buckling

pressure,

coefficient

pr3/D

of cylinder

subject

to axial

compres-

x

sion,

N f2/Tr2D

or

N f2/_l)

x

]_uckling

k

I

x

coefficient

of cylinder

subjected

to torsion,

xy N

_2/Tr2D

or

N

xy Buckling

k

_2/Tr2D t xy

coefficient

of cylinder

subject

to lateral

pres-

Y sure, L

N/2/Tr2I)

or

Nyf2/Tr2Dl

Slant length of cone Ring

L

spacing

me:Isurcd

along cone

generator

0

l_ength of cylinder, toroidal-shcll

axial length of cone,

scl.m_ en t

C3-xi

or length of

DEFINITIONOFSYMBOLS(Continued) Definition

Symbol M M cr

M

press

Bending

moment

on cylinder

or cone

Critical

bending

moment

Bending

moment

at collapse

of a pressurized

moment

at collapse

of a nonpressurized

on cone

or cylinder cylinder

or cone

Mp=0

Bending cylinder

Mt

Twisting

moment

Mi , M2 , Mi2

Moment

resultants

m

Number

of buckle

N

Axial

tension

a toroidal

on cylinder per

unit of middle

half-waves

force

per

surface

in the axial

unit circumference

length

direction applied

to

segment

N O

N x

Axial load per unit width of circumference

for cylinder

subjected to axial compression N

Shear load per unit width of circumference

for cylinder

xy subjected to torsion N

Circumferential

load per unit width of circumference

Y for cylinder subjected to lateral pressure N1 , N2 , N12

Force resuL_nts Number

per unit of middle surface length

of buckle waves

in the circumferential direc-

tion

C3-xii

___,

DEFINITION OF SYMBOLS (Continued) Definition

Symbol P

Axial

load on cylinder

apex

/

P

of spherical

or cone; concentrated

load at

cap

Critical axial load on cone; critical concentrated

load

cr

at apex

P

of spherical

cap

Axial

load on nonpressurized

Axial

load on pressurized

cylinder

at buckling

p=0 P

cylinder

at buckling

press Applied

P

uniform

internal

or

external

hydrostatic

I)res-

sure

Pc_

Classical

uniform

spherical

shell

Critical

Per

buckling

hydrostatic

pressure

(uniform)

for

a complete

pressure 7r2 D

R

Shear

flexibility

coefficient,

_2 D q

RA Rb

Effective Ratio

radius of bending

to more ing only

than

moment

moment

on

type

of loading

one for

oblate

the

cylinder

cylinder

or

or

to the cone

I3 _-

spheroid, c(me

subjected

allowable

when

bend-

subjected

to bending

Ratio

R

of a thin-walled

of axial

load

in cylinder

or

cone

subjected

to more

C

than

one

the

cylinder

type

of or

pression

C3-xiii

loading cone

when

to the

allowable

subjected

axial only

load

to axial

for com-

DEFINITION OF SYMBOLS (Continued) Definition

Symbol R m

R P

Maximum Ratio

radius

of external

to more

R s

Rt

pressure

than one type

nal pressure only

of torispherical

for

Radius

of torsional

to more

than

al moment

subjected

to the allowable

or cone

when

exter-

subjected

shell moment

one type

for

of loading

or cone

pressure

of spherical

Ratio

on cylinder

the cylinder

to external

shell

on cylinder

of loading

the cylinder

or cone

subjected

to the allowable

or cone

when

torsion-

subjected

only

to torsion

Rtr r

Toroidal Radius

radius

of torispherical

of cylinder,

equator

equivalent

of toroidal

shell

Radius

of small

end of the cone

r2

Radius

of large

end of the cone

S

Cell

Distance

of honeycomb

cylindrical

shell

or

segment

r 1

size

shell

core

along

cone

generator

measured

from

vertex

of

along

cone

generator

measured

from

vertex

of

cone

Sl

Distance cone

T

to small

Torsional

end of cone

moment

C3-xiv

on cone

_-_

°!

DEFINITION OF SYMBOLS (Continued)

P

Definition

Symbol Critical

T

torsional

moment

on cone

isotropic

cylinder

er

Skin

thickness

of

of corrugated

T

Effective width

sandwich thickness

equal

thickness thickness

having

x,

y,

z

der

and

tropic

z

in tile

s

,

Z

of iso-

layer

having

of layered

for

faces

of

cylinder

sandwich

construction

thickness

axial,

circumferential,

and

radial

_2 _2 _-_ ",/-1 --

parameter:

tor,_idal-shell

segTnent:

for

P 27- _

isotropic

_!{-:__2

cylin-

[oriso-

cylinder

of center

of

k th

layer

of

layeredcylin(fer

reference

l)istance

surface

ofcentroid

(positive of stiffeners

outward) and

rings,

respectively,

r from are

.

unit

k from

Z

per

respectively

sandwich

Distance

k th

of unequal

Coordinates

area

thickness

cylinder

thicknesses

faces

Curwtture

thickness

cone

of

directions,

Z

cylinder; skin

of sandwich

Facing-sheet

t I , t_

of corrugated

of circumference;-effective

Face

Skin

tk

cone:

cylinder thickness

tropic

tf

or

reference

surface

on _utside)

C 3- xv

(positive

when

stiffeners

or

rings

DEFINITION OF SYMBOLS (Continued) Definition

Symbol o/

Semivertex Buckle

aspect

Correlation

Y

A

AT

factor

Distance

of reference

Ratio

El2

of core

from

correlation

loads

inner

factor

of honeycomb

sheet

of sandwich

of centroid

inner

surface

of

resulting

from

of

k th

layer

sandwich

plate

to

plate of layered

cylinder

surface

Reference-surface

strains

reduction

_o

Ring-geometry

0

Coordinate

factor

parameter in the

Spherical-cap

circumferential

geometry

Poisson's

ratio

Poisson's

ratio

Poisson's

ratios

tropic

surface

density

of face

Plasticity

/_c

instability

between

pressure

Distance

E2,

and predicted

in buckling

density

from

for difference

wall

internal

5k

to account

theory

Increase

5

n_ (_-_m)

ratio

classical

layered

El j

angle of cone

material

C3-xvi

parameter

of core

material

associated in the

direction

s-

with

stretching

and 0-directions,

of an orthorespectively

____-

DEFINITION OF SYMBOLS (Continued) Definition

Symbol Poisson's tropic

ratios

associated

material

Equivalent

in x-

with

and

Poisson's

stretching

y-directions,

ratios

in the

of an ortho-

respectively s- and

0-directions,

respectively r1 + r2 2 cos a,

P

Average

Pl

Radius of curvature at small end of cone,

02

Radius

radius

of curvature

Normal

_N

of cone,

at large

end of cone,

r__D__ cos

O/

stress

Mmximum

(7

of curvature

compressive

membrane

stress

max

Critical

fr

,axial

stress

for

a cylinder

with

an elastic

P (7

Local

failure

stress

Axial

stress,

critical

S O" X

cr Circumferential

(y

stress

Y Shear

stress

Torsional

T

bucMing

stress

of an unfilled

cr

critical Shear

T

shear stress

stress in the x-y

plane,

critical

XYer

4)

Ilalf

the

included

C3-xvii

angle

of spherical

cap

cylinder;

core

DEFINITION OF SYMBOLS (Concluded) Symbol _2

Definition Half the included torispherical

angle

closure

Reference-surface [See

eq.

of spherical

(17).

C3-xviii

curvature ]

changes

cap portion

of

Section

C3.0

Decemberl5, Page C3.0

STABILITY

1970

1

OF SHELLS.

F

The

v

load

at which

in the

buckling

configuration

shell

or may

from

the

conditions,

decrease the will

stress

be discussed

collapse

load

load

nearly

the

in this

are

often

The depends and

on its

components, shell

of the

imporkmt

_3ffects

that

affect

buckling,

with

time

are

method

predicts load

less

load

of analysis

or may the

not

geometry the

buckling

available load

of

and

load

on collapse before

load

manner

of a structure in which

stiffnesses

at which

it is stiffened

of its wtrious

characteristics.

characteristics, unloaded

buckling

For

such

shape,

gener:dly

may

will

as small

also

have

occur.

Other

and

variation

stiffnesses,

thindevi:_tions

quite factors of loading

here. and flat

plates,

lo:ld quite

as the

cannot

loading

tile deformations

static

well-defined

as nonuniform

the buckling is used

of the

identified.

Only

buckling

different

the

certain

nominal

such

columns

the

critical

reasonably

Qn the

etc.

shape

and

may

information

and extensional

not considered

For

buckling

of the

its

of shells

shell,

large.

bending

from

the

basic

of loading,

shell,

are

proportions,

structures

structure

theory

very

or other

type

In gener_d,

magnitude

the

because

of the

in the

shell

change

in equilibrium

and easily

of the

on the

but if they

geometric

supported,

wailed

same,

change

types

pronounced

of the buctded

limited.

the

most

capability

section

The

applied

in a large

in the deflections

For

depending

as the

results

by a change

is quite

levels

is very

collapse

load

buckling

the

are

shape.

load-carrying

after

shell,

increase

is defined load

shell.

not be accompanied

buckling

The

in that

of the

a large

prebuckled

the

structure

increase

configuration

is usually

may

of a shell

an infinitesimal

equilibrium

which

load

dcsi_m

be used

the well;

allow-tble generally

classical in _enernl, buckling; for

shell

small

deflection

the

theoretical

load. structures.

ltowever, The

this

Section

C3.0

December Page 2 buckling

load

the load

predicted

by classical

of the test

data may

scatter

for some

discrepancies

are

a statistical

determining

a design most One

reduction

of the primary

obtain

the design

curves

may

which

the design

curves

are

used

Whenever design

allowable

available factors Such

this

or "knockdown"

factors

to reduce

raising

the design Most with

supported

simply edges

the effects

computations

analysis

is obtained,

a statistical

until analysis

curves. to obtain

a statistical are

recommending

the theoretical

be too conservative

made

on

correction buckling

in some

investigations

procedures

supported should

are

of the actual has been

allowable

long

presented

edges.

be assumed

and graphs. quite

to

cases;

loads. nevertheless,

necessary

to justify

curves.

An attempt so that the design

used

structure

However,

involves

and experimental

conditions

recommendations

Usually

theoretical

sometimes

design

of obtaining

particular

design

used

section.

to analyze.

to obtain

information.

further

indicate

used

in

has been

of this method

data do not exist

load,

may

in this

of the

the

these

be useful

For

most

unless

test

boundary made

buckling

results

condition

loads

may

which

cylinders,

section

applications

to simplify

The analyses

(orthotropic

in this

obtained

which

of the design. the analysis

be obtained have

for

are for

simply

are

been

instance)

procedures from

1970

than

When sufficient

method

and boundary

stability

sufficient

a recommendation

shells

presented

not be typical

possible

buckling

This

less

in addition,

for

5.

data may

load.

specimens

on shell

and,

i through

curves

be much

Explanations

of the test

being

may

theory

shortcomings

the test

whenever

large.

buckling

of the design

information

deflection

in References

curves

has been

small

allowable

is that

and loadings

be quite

design

additional

of shells

discussed

data exist,

to determine

types

15,

hand

presented

are

but,

in general,

Section

C3.0

December Page results

can

cases,

computer

The are

be obtained

applicable noted.

Computer

to keep theoretical

programs programs

They

can

Utilization As

revisions

quickly

to this abreast and

more

with

are are

generally

a few

available

simple for

described,

computations.

a more

and their

be obtained

from

15,

3/4 In many

sophisticated limitations

COSMIC

analysis. and

or from

availability the

Manual. information

section

will

of the changes experimental

on shell be made. in current investigations.

stability However, technology

becomes the

analyst because

available, should of recent

attempt

1970

Section

C3.0

December

r

Page 3.1

cylindrical

shells

understanding has

been

computers.

This

formulations

of the theory

cylindrical

with

axisymmetric

loading

distributions,

generally

and

or stiffness

longitudinally various

have

Problems

for

of this

be discussed

shells

experiment.

are

analysis type

application

been

complicated

For

shear

or

generally

shells

to establish

generally

understood.

of cylindrical

cylindrical specimen

shape usually

to the

by apparent

subjected

surface

to

loadings but nonuniform

of the

available

are

nonuniform

effects

information

computer

of is

circumferentially. programs

but still

of actual

and

will

The

For the

to require

causes

of such

deviations Because

stringently

large.

enough

is the

the

the

shells

in

discrcp:,ncics

discrepancies

of the

nominal

unloaded

controlled,

and

experimental

dependence

from

theory

of the cylinder

predominates,

of error

cylindrical

between

compression

can be quite

large

data.

on small

not been

design

discrepancies

compression

of the structure. has

thickness

longitudinal

Also,

longitudinal

source

shells

properties,

axisymmetric

by digital

of theory

design

primary

of uniform

3.4.

in which

severe

programs

load

be treated

of circular

investigations

that

rigorous investigation.

simple

are

digital

by more

involving

made.

the discrepancies

less

and

certain

that

been

both

shells

but detailed

circumferential

The

have

of loadings

can

The

predominates,

which

not

of electronic

on buckling

Problems

solved,

in Subsection

has

which

of circular

on experimental

stiffness

variations

have

aided

information

uniform.

been

the

been

by reliance

uniform

of buckling

by use

to unstiffened

states

parameters

inadequate

has

and

theory

possible

is restricted

shells

thickness

made

of the available

shells

to stiffened

of the

understanding

Most

wall

1970

CYLINDERS. A better

or

15,

5

most

shape test

are

i)uclding

circular of a test results

for

Section

C3.0

December

nominally

identical

theoretical

values.

specimens

Another loads

of cylindrical

displacements usually test

source

results

design

shells

precisely

Also,

tend to treat Results

without

regard

to specimen

to yield

to simplified

lower

versions

satisfactory

to date Within

procedures shells

are

3.1.1

conditions 3.1.1.1

for

in this

to axial

methods

are

considered

correction

factors

This

of

of establishing

conditions

of testing

not

as

together and

are

to be applied

technique

has proved

by the state loads

of the art,

on circular

acceptable

cylindrical

section.

isotropic

CYLINDERS.

circular

considered

Compression design

of the scatter

and edge

results.

of critical

Unstiffened

Axial

Current

or methods

imposed

UNSTIFFENED

are

some

have

purposes.

ISOTROPIC

of loading

tests,

tests

or statistical

the limitations

the estimation

edge

1970

of buckling

conditions

all available

for design

the

tangential

imperfections

construction

bound

below

and circumferential

source.

of the theoretical

described

The subjected

from

6

of longitudinal

in buckling

to this

and fall

Page

is the dependence

because

both initial

effects.

scatter

values

controlled

random

analyzed

on edge

can be attributed

data

larger

of discrepancy

or forces.

been

have

15,

cylinders

subjected

to various

below.

-- Unpressurized.

allowable

compression

buckling is given

stress

for

a circular

cylinder

by

OK

cr

yE

71

_

t/r

(1)

3 (1-p 2)

0 K

cr

-

0.6_/

Et -r

(for

_ =0.3)

Section C3.0 December 15, 1970 Page7 where the factor 3/ should be taken as q/

= i.0 - 0. 901

(1-e -0)

(2)

where

-

1

Equation bound

(2) for

used

with

five

since

Very

long

is shown

graphically

test

[6].

most caution

data

for

cylinders

The

has

should

<

1500)

in Figure

cylinders

the correlation

(tr

for

information

with not

3. 1-1

be checked

verified for

provides

in Figure

length-radius

been

and

3.1-1

ratios

greater

lower

should

be

than

by experiment

Euler-column

a good

in this

about range.

buckling.

0.9

0.8

0.7

o

I,,,,'-

0.6

u u,. z o

0.5

V-- 0.4 ..J uJ

¢

nO 0

0.3

0.2

0.1

2 10

4

6

8

2

4

6

102

8

1.5 103

r/t

FIGURE

3.1-1. CORRELATION FACTORS FOR ISOTROPlC CYLINDEItS SUBJI_3CTED TO AXIAL COMPI/IgSSION

CII1CUI.,A//

Section

C3.0

December Page When geometric

and

material

properties

are

such

15,

1970

8

that the

ff X cr

computed stress

buckling a

stress

should

X

is in the plastic

be calculated

by applying

range,

the actual

the plasticity

buckling

coefficient,

_ .

cr

This For

calculation

is facilitated

moderately

long

by the use

cylinders

of the curves

the critical

stress

_

of Paragraph should

X

3.1.6.

be determined

cr

by using

curve

E 1 in Paragraph

(Z -_0)

curve

G should

be used.

For

a cylinder

having

curves

E 1 and

G apply,

a plasticity

a linear

interpolation

would

should

be a function

stress-strain

Axial

cylinders

unpressurized

short

those

lengths

is not available.

satisfactory

results.

as well

cylinders

for which Presumably,

Such a factor

as of the usual

material

in axial

coincide

compression,

for internally just

Pressurization

pressurized

as in the case

increases

of the

the buckling

load

in

ways: 1.

pressurization

The total load

2.

The

destabilizing tensile

buckling

buckles buckling

compressive

p _ r 2 before

The circumferential the diamond

-- Pressurized

and collapse

cylinder.

the following

classical

extremely

between

factor

geometry

Compression

Buckling

cylinder

a length

provide

of cylinder

For

curve.

3.1.1.2

circular

3.1.6.

in the classical stress.

buckling effect

stress

pattern,

load must

and,

imperfections

by the pressurization

at sufficiently

axisymmetric

than the tensile

can occur.

of initial

induced

be greater

is reduced. inhibits

high pressurization, mode

at approximately

the the

Section

C3.0

December Page It is recommended

that the total load for buckling,

stantiated by testing, be obtained p _ r2 ,

the buckling

and an increase

9 unless

by the addition of the pressurization

load for the unpressurized

in the buckling

15, 1970

load caused

cylinder

[equation

by pressurization;

subload

(i)],

that is:

3 (1-_ 2)

where

A7

is

obtained

For

from

p=0.3

P

Figure

3.1-2.

,

= 2 7r Et 2 (0.67

+AT)

+P_r

2

(4)

press

1 8 6

,_

f

7

10 1

_v

B

f

6

j f

10 .2 2

4

6

10-2

8

2

4

6

B

10 "1

FIGURE

3.1-2.

BUCKLING-STRESS RESULTING

2 1

INCREASE

IN AXIAL-COMPRESSIVE

COEFFICIENT FROM

INTERNAL

OF

CYLINDERS

PRESSURE

6

8 10

Section C3.0 January

15, 1972

Page I0 3.1.1.3

Bending

-- Unpressurized.

Buckling and collapse coincide for Isotropic, unpressurized circular cylinders in bending. in axial compression

may

The procedure given for isotropic cylinders

be used also to obtain the criticalmaximum

stress for isotropic cylinders in bending, except that a correlation factor based on bending tests should be used in place of the factor given by equation (2) for cylinders in axial compression.

The correlation factor

for the cylinder in bending is taken as

"y = 1.0

- O. 731

(1-e -_b)

(5)

where

Equation

(5)

equation

should

data

not available

are

stress

for

axial

correlation

buckling

with in this

compression

factor

is primarily triggered

be used

is presented

for

because by any

caution

and

r/t

[7]. that

buckling

for

> 1500

bending than

of a cylinder

on the shell

initiated

in the

in Figure

the

theoretical the

for

compression.

that

of the

greatest

critical

same,

in compression whereas

This

experimental

are

surface,

region

3.1-3.

because

Although

is greater

imperfection

is generally

for

range

bending the

graphically

the This can be

in bending, compressive

stress. For using

curves

allowable

inelastic

buckling

E 1 in Paragraph

moment

M

3.1.6.

critical If the

stress stresses

may are

be found elastic

by the

is

=rr2_ cr

the

t x or

(6)

Section C3.0 February 15, Page l 1

1976

f0.9

0.8

0.7

E 0

0.6

u 0.5 z 0 m

nnO 0

0.4

0.3

0.2

0.1

0 4

6

8

2

4

8

6

1.5 10 3

10 2

101

r/t

FIGURE

3.1-3.

CORRELATION

CIRCULAR 3.1.1.4

Bending For

pressure, adding the

the

thin-walled

moment

Ay

is obtained

that

subjected the buckling

capability

for

in the

Mpress

where

cylinders

moment-carrying

an increase

SUBJECTED

FOR

ISOTROPIC

TO BENDING

-- Pressurized.

it is recommended

buckling

and

CYLINDER

FACTORS

= 7frEt2

from

moment

(*J

Figure

moment

of a pressurized

the unpressurized

critical

to bending

cylinder caused

3 _/(l_p 2)

3.1-2.

and

internal

be obtained membrane [equations

cylinder (1)

by pressurization.

+ AT)

+ 0.5prr

by

and

(5)],

Then

3

,

(7)

Section C3, 0 February 15, Page 12 For

_ = 0.3

Mpres

3.1.1.5

s

= lrr

External

load

acts

only

The

term

on both loads

the

by

N

=at

for

x

Y

+0.5

pTr r3

(8)

designates

an external

of the cylinder

pressure

and not on the ends.

The

by

designates

the ends

an external

of the

cylinder.

pressure The

which

cylinder

acts

wall

pr 2

short

(10)

cylinders,

not significantly

pressure

(9)

=pr

the critical

pressures

for

the

two types

different.

An approximate by lateral

walls

is given

and

=at= x

unusually

are

pressure"

pressure" walls

given

Y

+AT)

= o t = pr Y

Y

curved

N

of loads

wall

"hydrostatic

are

Except

"lateral

on the curved

in the cylinder

N

Et 2 (0.6T

Pressure.

The term which

1976

buckling

is given

equation

for

supported

cylinders

loaded

as

_D Nycr

The

buckling

obtained

= ky

equation

by replacing

_

for

.

cylinders

ky by kp in the

loaded formula

(II)

by hydrostatic above.

pressure

is

Section C3.0 February 15, Page 13

1976

As sho_q3in Figure 3.1-4, except for unusually short cylinders, the critical pressures for the two types of loads are not sigmificantly different.

For

short cylinders equations suitable for design solutions to the hydrostatic (Z < 30) andlateral (Z < 70) cases respectively are: k

=

1.853

+

.141714

=

3.98o

+

.02150

Z "83666

(12a)

Y and

k

Z 1"125

(12b)

Y The verge

solutions

for

to an equation

k

intermediate

given

length

cylinders

(100

_/ Z---

4000)

con-

by

= .78o

(13a)

Y or the

critical

p

prcsstu'e

. O4125

-

The

family

radius-thickness into

is given

an oval

E

of curves ratio

shape,

for long

of the

y

_

( i:_lD

cylinders

cylinder

as given

(Z > 3000)

an(I corresponds

is dependent

to buckling

of the

Z r t

upon

the

cylinder

by

2.7(}

k

by

(14a)

J1

- p2

or

P -

and

applies

4(1

for

-it 2) rx/1 20 < _-

(lqt,) _

p

=

< 100.

_l;t U.S

GOVERNMENT

PRINTING

OFFICE

1976-641-255/391

REGION

NO

4

Section

C3.0

February Page 13A

15,

L) © ©

_4 M H rJ3

©

m N

M

Z _ M M _

m

_ m M r_ 0 _ M _

Z

-40 k-,,t

• (.) M

d)l'A)l IN3110hl=1300 ONll)13FIg

1976

Section

C3.0

December Page For obtained curves the

from

curve 3.1.1.6

should

For

given

For

short

moderate

be used.

For

cylinders

length long

correction

(TZ < 5)

cylinders

cylinders

factor

1970

should the

be

C

(5 < TZ < 4000)

(TZ > 4000)

the

E

be used. Shear

or Torsion

-- Unpressurized.

The theoretical be obtained

the plasticity

3.1.6.

be used.

curve

should

stresses

Paragraph

should

El

inelastic

15,

14

from

Figure

buckling 3.1-5.

coefficient

for

The straight-line

cylinders portion

in torsion of the curve

can is

by the equation

N

and applies

kxy

-

for

50 < yZ

fz

_xL'--D --

0.85

< 78

(t)2

(15)

(yZ) 3/4

(l-gZ).

Equation

(15)

can

be written

as

45]

0. 747 T°/4 E

T

For circumferential

TZ > 78 (t)z_ (l-p 2) , waves.

k

xy

(16)

The

buckling

the cylinder coefficient

2 q-YTz

buckles is then

given

with

two

by

(17)

r 1/2

1/4

Section

C 3.0 15, 1970

December Page

15

10 3

,

8

r

J

6

, /

4

2

J

x

z UJ

I

1o_

i

/ /

6 uu w. w

/

/

/

4

o (J

I

I

2 z ,.J v U

./<

! 10 8 6

i

I

4

,

i

2 i

1 1

2

4

6

810

2

4

6 8 102

2

4

6

8103

2

4

6 8 104

2

4

6 8105

_z FIGURE

3. 1-5.

ISOTROPIC

BUCKLING

COEFFICIENTS

CIRCULAR

CYLINDEI{S

FOR

SIMPLY

SUBJECTED

TO

SUPP()I_TED TOHSION

or

yi,:

: XYcr

To

approximate

3 _

the

T3/4

is recommended

_

lower

0.

limit

of most

data,

the

value

(19)

for moderately

3. I. 6.

(is)

r-

(;7

Plasticity' should Paragraph

3/4 (t___3/2

(l-p2)

long cylinders.

bc accounted

for by using curves

A

in

Section

C3.0

December Page 3.1.1.7

Shear The

be calculated 3.1.1.8

increase

Combined

is frequently

empirical) the ratio

and

and

Axial

allowable in axial

equations

general

stated,

may

etc.

the

relationship the

term

caused

by a particular

the exponents

(usually

of the quantities

"stress-ratio"

combined

equation,

stress

), and

under

for

is used

failure

of

to denote

the

stress. and Bending. interaction

equation

for

combined

compressive

is

Rc

+ Rb = 1

The

quantities

load loads

or stress

(20)

Rc

and

cylinders or

(20)

and

and

The

given

is also

(7)

(8).

respectively,

denominators

(1)

and

recommended

axial or

are,

by equations

by equations

in combined (4)

Rb

ratios.

or stresses

compression

(3)

pressure

of a member

of a stress-ratio

denote

shear,

recommended

Equation circular

by internal

1.

failure

in terms

Compression

bending

bending

structural

subscripts

to allowable

The load

The

the

Simply

of applied

caused

in Reference

expressed

express

I.

for

(compression,

member.

stress

Loading.

criterion

of loading

1970

Pressurized.

the curves

R1 x + R2 y + Ra z = 1 . kind

--

in buckling

by using

The loading

or Torsion

15,

16

compression

of the

(1)

and

(5)

for

for

the

and

ratios

(2) for cylinders

internally bending

compressive are

the

cylinders in bending. pressurized by using

Section C3.0 December 15, 1970 Page 17 II.

Axial Compression andExternal Pressure. The recommended interaction equationfor combinedcompressive

load andhydrostatic or lateral pressure is R + R =1 c P

(21)

The quantities R

and R are, respectively, the compressive andhydroc p static or lateral pressure load or stress ratios. The denominators of the ratios are the allowable stresses given by equations (1) and (2) for cylinders in axial compression andby equations (11) or (12) for cylinders subjected to external pressure. III. Axial Compression andTorsion. For cylindrical shells under torsion and axial compression, the anaiytical interaction curve is a function of Z andvaries from a parabolic shape at low-Z values to a straight line at high-Z values. The scatter of experimental data suggests the use of a straight-line interaction formula. Therefore, the recommended interaction equation is (22)

Rc +R t =1 The quantities load

or stress

stresses and

RC

Rt

ratios.

given

The

(16)

Bending

or and

bending

and

(18)

and

for

(2)

the

of the for

cylinders

compressive

ratios

cylinders

are

and

torsion

the allowable

in axial

compression

in torsion.

Torsion.

torsion

R b + Rt2 = i

respectively,

denominators (1)

A conservative combined

are,

by equations

by equations IV.

and

estimate

of the

interaction

for

cylinders

under

is (23)

Section C3.0 December 15, Page The torsion

load

allowable and

quantities

or

stress

stresses

ratios.

given

by equations

3.1.2

R b and

(16)

term

orthotropic

material

cylinder

to be approximated

averaged axes

are

the

a single

orthotropic

material,

coupling

between

force

resultants

and

extensional

serious significant having vanishes

errors.

stiffeners when

resultants

the

in bending

of the

types

and

enough

of for

Generally, the

the

bending

stiffening

elements

the

directions

longitudinal

of orthotropic

of which

and

neglect the

in theoretical surface

of the inclusion results or the

equations

strains.

strains

terms

of coupling for

stiffened

outer

surface.

of

the

between

or for cylinders coupling

cylinders

similar

be

and

to neglect and

is then

may

of

force For

permissible

theory

cylinders

are

cylinders

between

bending

bending

The

stiffened

is omitted.

types

orthotropic

individual

with

of

of a single

denotes

whose

relationships

and

example,

variety

made

is small

sheet

elements

strains.

on the inner coupling

spacing

it is generally

however,

difference

and

cylinder.

extensional

For

For

are

a wide

or areas.

structure,

and

layers,

ratios

cylinders

It'also

to coincide

the

buckled

cylinders.'

layers.

those

taken

th¢ory,

resultants,

orthotropic

cylinders

of the various

moment

isotropic

it denotes

widths

of the

behavior

for

covers

stiffener

representative

by a single

equilibrium

the

include

directions The

for

bending

in torsion.

by a fictitious

of orthotropy

circumferential

described

which

properties

out over

(5)

cylinders"

or of orthotropic for

the

of the

and

cylinders

sense,

cylinders

extensional

for

(1)

18

CYLINDERS.

stiffened

of the

(18)

respectively,

denominators

"orthotropic

In its strictest

are,

by equations

or

cylinders.

and

The

ORTHOTROPIC The

Rt

1970

moment

to that

for

having can

terms

lead yields

cylinder The

to a

configurations difference

Section

C3.0

December Page

P

Theoretical generally

and experimental

in better agreement

possibility of local buckling should

results

between

19

for stiffened shells are

than those for unstiffened of the cylinder

15, 1970

shells.

The

stiffening elements

be checked. In general,

cylinders

necessitates

solutions

are discussed

3. I. 2. i

of the analysis

the use of a computer in Subsection

for orthotropic

solution.

Applicable

computer

3.4.

Axial Compression.

A compression

the complexity

buckling

equation

for stiffened orthotropic

cylinders

in

[8] is given by:

A11

A12

A13

A21

A22

A23

A._.

A._._

for

(n -> 4)

,

(24)

Ai2

A21

A22

in which

A11

-

= Ex

_

-

+ Gxy

(25)

(26)

Section

C3o 0

December Page

15,

1970

20

A33=\(_)'+_y(_--)'(n)' +_(n), +r_+-_(r AI2

= A21

=

(

+--_r --

+%)

Ly

m_

(275

(28)

rn

(29) m

E

, x,7_+(%+_Kx,)_-(_)'

xy A31 = A13 -

m_

r

s

+ _- _m_"_

--

.

(30) Values are given

in Paragraph

into account assumed

to be supported

specialized

with

for

eccentricity with

corrugated 9 and

J0. n

For

by rings

unstfffened

neglected

to be used

effects

are

cylinder those

are

and

edges

rigid

in their

own plane

plane.

The

have

stiffened

identical

minimize

been

theory

dimensions, N

. X

For

not taken are but offer

equation treated

orthotropic

or stiffened

into account.

and stiffener which

The cylinder

or stiffened

but not

of construction are

which

taken

types

deformations

out of their

example,

a similar

given

that

of cylinders

eccentricity

cylinders,

of the equation.

types

effects

for various

Prebuclding

or bending

various

literature;

cylinders

and

to rotation for

to be used

3.1.2.6.

in the derivation

no resistance

in the

of stiffeners

can

be

separately cylinders

orthotropic ring-stiffened

is given the values

in References of

m

Section

C3.0

December f

Page The does

not permit

programs

any

axial

that

be more

effective

buckling

load

values

of the

buckling

[9-17] theoretical

buckling spaced,

stiffener

for

On the

basis

large

coefficients cylinders Although

the

with theory

from

covering

and

11.

can

and conservative cylinders.

load

with

limited good

An extensive

various

stiffener

experimental

agreement

data

with

the

that

the

investigated.

it in recommended (24)]

[ 16]

edge rotation and longitudinalmovement

or both,

values

of cylinders

having

by a factor from

stiffeners

experiment

rings

calculations

unconservative

transition

spaced

shells,

that

yield

be multiplied the

shown

Generally,

The

data,

of the

compression.

of parameters

stiffeners

been

computer

axial

stiffened

equation

It has

cylinders

buckling

of available

closely

for

in reasonably

range

[calculated

moderately

stiffened

are

solution

or

1970

(24)

The

for

of stiffened

stringers

externally

in Reference

for

design.

stiffened

of the

shells

be used

eccentricity

internally load

results

loads

correlation

gated.

stiffening

for

stiffened

a given

internal

is reported for

for

should

in equation

to be shown.

representative

of the variation

parameters

3.4

whether

neglecting

investigation

load

results

stiffening,

than

of the

of parameters

numerical

of parameters external

calculations

number

in Subsection

compressive

combinations

indicate

large

definitive

discussed

critical for

unusually

15,

21

have

indicate

of 0.75.

unstiffened not been that

closely

restraint

The

cylinders fully

to

investiagainst

sigmificantlyincreases the buckling

load, not enough is known about the edge restraint of actual cylinders to warrant

taking advantage of these effects unless such effects are substantiated

by tests. For layered or unstiffened orthotropic cylindrical shells, the available test data are quite meager

[18, 19].

For configurations where

the coupling coefficients _ x , C y , C-xy ' and Kxy

can be neglected,

Section

C3.0

December Page it is recommended

that the buckling

load

be calculated

from

15,

1970

22

the equation

X

m

72 14

_m2"_

m

u

ExEy - E 2 xy

X

r_

('_'_

_g2 Gxy

xy)

fl2+'_

Y (31)

The correlation replaced

correlation

factor by the

circumferential

for

factor

isotropic

geometric

T = 1.0

cylinders

mean

directions.

T is taken

to be of the same

[equation

of the radii

(2)]

of gyration

with for

form

as the

the thickness

the axial

and

Thus

- O. 901

(1-e -_b)

(32)

where

,/2 (33)

3.1.2.2

Bending. Theoretical

the

critical

maximum

and load

experimental per

unit

results

circumference

[10,

20-23],

of a stiffened

indicate cylinder

that

Section

C3.0

December Page in bending absence

can of an

maximum

load

stiffeners

be

is

exceed

calculated

extensive per

calculated

unit

load

in axial

of a cylinder

from

equation

to the (24)

or

critical

load

multiplied

by

unstiffened

it is recommended

equation

(31)

5/ = 1.0-

0.731

with

(1-e

5/

in axial a factor

orthotropic that replaced

the

closely

critical spaced

compression, of

cylinders the

In the

that with

maximum

1970

23

compression.

circumference equal

layered

unit

it is recommended

as

coefficients, by

critical investigation,

taken

For coupling

the

15,

which

0.75. with unit

negligible load

be

by

-(p)

(34)

where

1/2

1

r

(35)

2.). 8

E x l,:y

3. 1.2.3

External The

cylinders

Pressure.

counterpart

under

later:ll

r -

of pressure

equation is

_iven

(24)

for

by

All

A12

A13 ]

A21

A22

A23

A._ 1

A,_2

A_._

n 2

I All A2!

AI2 A22

I

stiffened

orthotropic

Section

C3.0

1 April

,1972

Page For

hydrostatic

pressure,

n2

the quantity

shown

in equation

24

(36)

is replaced

by

n _ + 1/2

(

In the case

of lateral

n must

be varied

to yield

not less

than 2.

In the case

be varied

as well

as

)2

m _ r l

n .

pressure,

a minimum

m

value

of hydrostatic

For

long

is equal

to unity,

of the critical pressure,

cylinders,

pressure

the value

equation

whereas

(36)

of

but m

is replaced

should by

3 _y-E --YY p =

If the neglected,

the

r3

coupling

critical

(37)

coefficients

buckling

5. 513 P'_

Cx'

pressure

_ y' Cxy' and can be approximated

y

KLy

can

be

by [24]:

(38)

_

Ey

for

x _Dx/

y

12E y

j.._2> 500 x

/

r

(39)

Section

C3.0

January Page

1 5, 1972 25

F

Equation cylinders

with

References depends of

less

whereas

stiffeners

25 and

26 show

than for

of the

pressures

warrant

its use. The

in Reference results

multiplied

unstiffened

If coupling cylinders

relative

the

good

However,

buckling of 0.75

for

pressure for

use

cylinders

rings

with

effective

than

inside

rings,

the

reverse

shell

differences

for

or outside shells

of outside

results

stiffening,

for

500,

to the

ring

of inside

is true.

stiffening

complex are

not so significant

with

cylinders

of all

calculated

of moderate

the

to increase lower

accurate

described

theoretical

types,

from as has

the

as to

cylinders

agreement

values

Somewhat but more

ring-stiffened

in dcsigm,

As

tends

increases.

by the extremely

but the

spaced

isotropic

Generally, more

than

for

it is

equation been

(36)

be

recommended

length.

Torsion.

treated

coupling

greater

rings

isotropic

The been

are

in reasonably

by a factor

3.1.2.4

rings

given

closely

effectiveness

effectiveness

{36).

that

the

primarily

For

geometries.

experimental

of equation

recommended

ring

28,

29 are

investigated

the

are

of Reference

that

Z

varies,

theory

for

of

been

[25-27].

outside

values

stiffness

buckling

and

100,

geometry

as the

has

ring

on the shell

Z

ring

(36)

problem

of torsional

in References

between effects can

Mt

}_nding are

24 and

21.75D

30,

and extension,

negligible,

be estimated

buckling

from

5/8 Y

the

which and critical

the relationship

x

of orthotropic do not

take

cylinders into account

in Reference torque

has

31,

which

of moderately

does. long

[24]:

y xy _E2 13/8 l,_y

r5/4 _I/'T

(40)

Section

C3.0

December Page

15,

1970

26

for

x

-

_2

r 12EyD

Reference important internal shells

for cylinders rings

are

the entire

equations

range

of Reference

theoretical theory.

3.1.2.5

Combined

[9-10],

orthotropic

a straight-line cylinders

The critical

combinations

Rc

3.1.2.6

buckling construction.

in good

short

the

case

considered. with

an adequate

long

critical

test

of

torques

cylinders.

Compression. 21]

curve

to combined are

shells

for

agreement

to provide

20,

interaction

long

or graphs

specific

that theoretical

[10,

quite

be considered,

each

for moderately

of loading

rings;

are

and limited

experimental

is recommended

for

bending

thus given

and axial

compression.

by

+ Rb = 1

Elastic The

32 are

and Axial

subjected

for

For

formulas

that should

insufficient

of theory

rings.

of general

recommended

Bending

(41)

effects

thdn outside

be solved

of 0.67

On the basis

spaced

effective

of Reference

but are

by a factor

that coupling

In the absence

It is therefore

be multiplied

data

more

of parameters

data

predictions

shows by closely

31 should

The test

the

stiffened

is true.

.

/

however,

generally

the reverse

to cover

31,

500

Constants.

values

of orthotropie

(42)

of the various

elastic

cylinders

different

are

constants for

used

different

in the theory types

of

of

Section C3.0 December t5, 1970 Page 27 I.

Stiffened Multilayered Orthotropic Cylinders. Somewidely usedexpressions for this type of cylinder are: --

S

_' E = _ x k=l

x 1 - #x_y

-Ey

y 1 - PxPy

=

-Exy

}_ k=l

=

_ k=l

S

(43)

tk + b

k

tk+

r

d

(44)

r

k

1 \

tk - #xPY]

k

i k=l

pyEx 1 - PxPy

/

tk k (45)

N (4_;) k:=l

EA

E I S

x

_

x

k:l

- PxPY

1

3 + tk

+

S

b

s +

z 2 s

S

b

k (47)

EI -Dy

\, = kLJl

r

y 1 -PxPy

1 (-i-2

tk3 4_tkz_ "

/ +

b

r

E A r r d

/k

(48)

Section

C3.0

December Page

% _

(4Gxy

GJ ss b

N k=l

x

C-

#x E

+

2 = k=lN

(

_

Ex / #xPY

Y

k=l

xy

1

k=l

_yEx 1-.x.y

)

ek + tkz

1 k

(49)

S

PxPy

tkzk

+ zr

k

- _x_Y]

(50)

EA b

k

Y

-

28

GJ rr d

N(E)

:

1970

+

1 - PxUy

+

1

_

15,

k

EA

(51)

rd r

k--1

x y

k

(52) N -Kxy

where

the

A

make is taken

_ k=l

subscript

of an N-layered can

=

at least as

k

shell one

(Gxy)

refers

k

tk_ k

to the

,

material

(Fig.

3.1-6).

A proper

of the

coupling

coefficients

and

(53)

geometry

choice

of the

of the

vanish.

k th

reference

For

example,

layer surface ff

Section

C3.0

December Page

N(_yEx_

k=l

15,

1970

29

tk6 k

- Px/_Y/k

(54) E xy

the

coefficient

C

vanishes,

and

ff

xy

N

V k_l

(Gxy)

k

tkSk (55)

z

G xy

the

coefficient

K

vanishes. xy

Z

LAYER k th

1

FIGURE

3.1-6.

MULTILAYEREI)

CYLINDRICA

L SHF

LL

OIi'FttO'FROPIC GEOMETRY

Section

C3.0

December Page II.

Isotropic For

for

a cylinder

a reference

reduce

Cylinders

with

Stiffeners

surface

consisting at the

center

1970

and Rings.

of a stiffened of the

15,

30

single

Layer,

equations

isotropic (43)

layer

and

to (53)

to

_

Et 1 - p_

x

--_ E y -

+

Et 1 -p2

+

EA s s b

(56)

EA r r d

(57)

Exy

(58)

-Gxy

D

-

x

D --

Et 2(1 +p)

12(1

Et 2 - t_2)

_2(1

Et_ -p2)

y

_

(59)

+

+

Et3 -

Dxy

EISS b

+ T 2s

EAS

EIrr d

+ _'2 r

EA r r d

GJ

6(1 +p)

+

sb s +

S

b

(6O)

(61)

GJ rd r

(62)

EA C

=

x

z

S

s

b

S

(63)

Section C:I.0 December

15, 1970

Page 31 EA -C

_" = z

y

r

r

C

=K

of formula

((i5)

xy

Ring-Stiffened The

required

(6,t)

=0

xy

III.

r

d

following

stiffnesses

formulas

of ring-stiffened

depending

E

Corrugated

on the different

= ET,

E A r r d

E

x

y

Gxy

are

commonly

used

to calculate

corrugated

cylinders,

assumptions

which

with may

the the

choice

be made:

(66)

(67)

= Gt

E1

I)

Cylinders.

((;_)

x

D

y

EI rr d

+ z2 r

GJ r_____r d

D xy

EA r r d

(69)

(70)

E A -C

_ = z

y

r

r

d

r

(Tt)

Section

C3.0

December Page E

= C xy

Slightly

different IV.

= C x

stiffnesses

References

Special

of the

given

in Reference

22.

cylinders

with

waffle-like

walls

are

described

in

Considerations. designs

cylinder.

decreased

3.1.3

stiffness

can 13,

ISOTROPIC The by bonding

facings

provide

separates

axis.

23,

and

all the

The

and core

stiff

skin

than

by methods

may

buckle

unbuckled similar

before

sheet.

to those

pre-

CYLINDERS.

sandwich"

designates

to a thick

bending

transmits

provides

the

36.

two thin facings

the facings

is less

be calculated

"isotropic

nearly

cylinders,

sheet

SANDWICH

term

formed

of stiffened

Buckled

in References

neutral

(72)

Cylinders. for

In some

sented

= 0

33 to 35.

V.

The

32

xy

are

Waffle-Stiffened Stiffnesses

failure

= K xy

1970

15,

the

core.

rigidity shear shear

a layered Generally,

of the

so that

construction

construction.

the

rigidity

the thin

facings

of the

bend

sandwich

The

core

about

a

construc-

tion. Sandwich of instability core

and

dimpling

failure:

facings of the

bending

little stiffness

(1)

acting faces

If the relatively

construction general

together, or wrinkling

isotropic

bending is given

sandwich stiffness, by

should

be checked

instability and

(2)

failure local

of the faces shell then

for

has

for two where

instability (Fig.

the taking

shell

modes fails

with

the form

of

3.1-7).

thin facings

unequal

possible

thickness

and

the

facings

core the

has

Section

C3.0

December Page

,,, f

15,

1970

33

1111

I

FACING _ HONEYCOMB f

CORE _,ISEPARATION

COR!

FACING

FROM CORE

7

I

J

ttT

tttt GENERAL

BUCKLING

FIGURE

3.1-7.

D1 -

and

for

equal

DIMPLING FACINGS

TYPES

Etl -p2)

(1

thiclmess

-

The

2(1

and

for

equal

(j

OF

SANDWICH

SHELl,S

t2 h2 (h +t2)

(73)

(74)

- /32 )

extensional

-

FAILURE

OF FACINGS

h2

E ]31

OF

tttt

WRINKLING

facings,

Etf D1

tttt OF

_%.,NG

stiffness

(h

+ t2)

for

unequal

thickness

facings

is

given

by

(75)

_ /_2)

thickness,

2 Etf B1

( 1 -/_2)

(76)

Section

C3.0

December Page 34 The

transverse

shear

stiffness

for

an isotropic

core

15,

1970

is given

by

h2

D

= G q

and

for

equal

stiffness 38,

(78) xz

h-tf

of other

and

3.1.3.1

_+h 2

52

=G q

37,

h

thickness,

D

The

(77) xz

types

Axial

circular

cylinders

Design

information

is applicable

be used dict

in axial

3.1-9

from

buckling

are

loads.

these

_b -

procedure

with Figure For

in References

references

on the

convenient

present

application

the

3.1-9.

to use

and

Figures

theory

and

3. 1-10

to pre-

(32),

given

parameter

is consistent when

shearing

to the

deformations,

41.

and

+ R) ].

of Figure

on equation

40 and

3.1-8

two figures

buckling

factor

is based

in Figures

[_/Z < _/(1

small-deflection

3.1-10

sandwich

in References

of the

cylinders

the correlation

the

reported is given

short

of isotropic

should

for _

becomes

29.8

of construction buckling

are

is the more

based

cylinders.

behavior

compression

3.1-9

in conjunction

buckling

This

of the

to all but unusually

orthotropic

is given

Compression.

Figure

and

construction

39.

Investigations

3.1-8

of sandwich

(79)

with the of the that

is,

procedures core when

does

given

earlier

not contribute

No/D q of Figure

for

other

types

significantly 3.1-9

is small.

Section

C3.0

December Page

102

11

15, 1970

35

III

!

[[

Io.o,

i/" ,,x

2

i-"

0.1

Z w

•

10

f

II

6

ii. k_

If

I,,,"r

[I I I

8

1.0

II

..

II

4

I

[ 10"1

2

4

6

I

O

4

6

2

4

102

10

2

68

4

8m

103

10 4

-_z

FIGURE

3.1-8.

SUPPOI/TE

BUCKLING

D ISOTI/OPIC

SUBJECTED

TO

COEFFICIFNTS

SANDWICH

AXIAl,

CIRC

COMPRESSION

FOIl

SIMPLY

U LA I/ CY LINI)FIlS Gxz/Gy

z

1.0

1.0

0.9

\\

0.8

0.7

m

0.6

--

G

Gx z

z°

yz

0.5

Zx 0.4

0.3

0.2

0,1

4

Nolo

FIGUIIE

3. i29.

SUPPORTED,

BUCKLING

ISOTI/OPIC SUBJECTEI)

OF TO

5,

q

MODERATELY

SANI)WICIt CIRCULAR AXIA I_ COMPIIESSION

LONG,

SIMPI_Y

CYLINI)EIIS

Section C3.0 December Page

15,

1970

36

0.9

0.8

0.7

0.6 I,-

I--

0.4

uu _ M,

O.3

8

11.2

0.1

0 2

4

6

8

2

10

4

6

8

10 2

10 3

r_

FIGURE

3.1-10.

CORRELATION

CIRCULAR

FACTORS

CYLINDERS

FOR

ISOTROPIC

SUBJECTED

TO

SANDWICH

AXIAL

COMPRESSION

As

shearing

from

deformations

application

becomes

of the

zero

[(No/Dq)

become

in the

factor

_/ ,

limiting

be been

for

data limited found

additional

pronounced, prescribed

the above,

of buckling

from

correction

resulting

decreases

and

a weak

core

> 2].

the

lightweight test

as

condition

A weight-strength values

more

shear

cores are

study

stiffness are

obtained

more

of honeycomb desirable

with

that

plates

modes

than

to substantiate

to sandwiches sandwich

based

of deformation,

rather with and

on

Figure cores

heavier

this

indication,

heavy

cores

light failure

honeycomb may

3.1-9 [42]

and

indicate

cores.

Until

however, (5

result

are from

that

unusually

adequate designs

>-- 0.03). cores

published

Also, susceptible intracell

should it has to

Section C3.0 December 15, 1970 Page 37 buckling,

face

wrinkling,

a cylinder-buckling occurred

the

mode.

in actual

reached

(see,

and

be

directly

relatively

these

capable

may

cause

The

only

heavy

Honeycomb

failures. be

small

before

Ref.

43,

buckling

known

or

method

which

Large

with

shear

failure

can

rather

penalty.

Moreover,

adequate

for

predicting

failures

in the

modes.

The

following

equations

may

heavy

these

the

for

this

moments

failures

is

in crushing be

and

adequate

little

buclding

used

in addition

core.

approximate

intracell

that

the

in intracell with

used

shears

core

should

cores

are

be

of

was

requires

and

with have

load

cylinders,

failure

heavy

cores

behavior

strength

against

wrinkling

when

sandwich

6 : 0.03

margins

obtained

of

modes

deformations

moments

considerable ratio

of these

buckling

This

of preventing

have

a density

217).

case

both

theoretical

internal

In the

or

buckle-like

the

p.

resisting

core

of one

addition,

loads.

with

interaction

long

of

cores

cores

core

In

example,

applied

shears

an

structures

for

structure

to the

or

and

no weight

equations and

shear.

to prevent buckling-

or

to use

are

face-wrinlding

purpose.

For

intracell

buc kl ing:

"x _ 2.51_ R• (t/s) 2

where

S

inscribed

is the circle

core

cell

size

(do)

expressed

as

the

diameter

of

the

largest

and

4 E

Etan (8i)

where materi:/l.

E

and

l':tn n

If initial

are

the

dimpling

elastic is to be

and

tangent

checked,

moduli the

of the

equation

face-sheet

Section

C3.0

December Page Ox = 2.2

should

be used.

Critical

where and

E G

is the

xz

shear

stresses

the

must

equations

strength

applied

The

plasticity

compression

factor what

44,

3.1.3.2

that

the

Figure orthotropic

by

perpendicular

in the

x-z

initial

plane.

then

waviness

coefficients

of

where

stress stress

which

If biaxial

the

(1 + f_)-t/s

compressive compressive

to the core,

in facings in facings

consider

of the

facings

given

for

(84)

strength are

of bond,

given

in

correction may

as the

factor

be applied

shear

to isotropic

cylinders stiffness

isotropic

sandwich

cylinders

in

cylinders.

The

becomes

some-

with

stiff

cores

and

of the

core

is decreased

[46].

Bending. The

cylinders

core

equations

to sandwich

conservative

occurs.

and 45.

also

is applicable

dimpling

(83)

by the factor

intracell-buckling

39,

predicted

to the sandwich,

principal principal

and

if initial

in a direction

of the

be reduced

of foundation,

References

axial

and

load

Ez Gxz )1/3

modulus are

carry are

of the core

f _ minimum maximum

Wrinkling

still

stresses

(Esec

modulus

compressive

(82)

will

wrinkling

= 0.50

is the

z

1970

E R (t/S) z

The sandwich

ax

15,

38

buckling

in axial

compression

correlation 3.1-10.

equations

factor Figure

cylinders

given

may

be used

_/ is taken 3.1-11

in bending.

in Paragraph

is based

from

for

3.1.3.1

cylinders

Figure

on equation

3.1-11 (34),

for

circular

in bending, instead given

provided

of from earlier

for

Section C2.0 15, 1970

December Page 3,_)

0.9

0.8

0.7

d O pu

0.6

0.5 z O <

0.4

..J w

0.3 O ¢J 0.2

0.1

0 2

4

6

8

10

2

4

6

8

10 2

10 3 r/h

FIGURE

3.1-11.

SANI)WICH 3. I.3.3

CORRELATION

CIRCULAR

FACTORS

CYLINDERS

FOIl

SUBJECTED

ISO'I'I{OPIC TO I;ENI)ING

Lateral Pressure. A plot of k

against TZ , constructed from the dat't()fllclerence Y 47, is given in Figure 3. 1-12. The straight-line portion of tilecurve ()[ Figure 3. 1-12 for a sandwich cylinder with a rigid core

(6=0)

is given by

the equation

N k

--_ 7ri)t

y

There ence

are with

to 0.56

_2 -

no experimental isotropic

be use.d

with

0.56

data

cylinders, this

figure.

(ss)

_v yZ

to substantiate however,

l"il4_re

it is suggested

3.1-12. that

From a l'aet()r

experi3, equal

Section C3.0 December 15, 1970 Page 40

10 2 8 6

. o/i

4

2

.......

10 8

6

!" __o O,S

u

2 1,0

6 4

2

•

10 "1 2

4

68

2

1

I 4

68

2

10

4

68

2

10 2

4

68

10 3

2

4

68

10 4

108

-),z FIGURE 3.1-12. BUCKLING COEFFICIENTS FOR SIMPLY SUPPORTED ISOTROPIC SANDWICH CIRCULAR CYLINDERS SUBJECTED Here, designs

should

is 0.03

or greater, For

obtained

TO LATERAL

as with

be limited

sandwich

unless inelastic

the

stresses

should

be used.

For

moderate-length

should

be used.

For

long

which

short

correction

cylinders

the

< 4000 E

curve

bending, ratio

tests.

factor

(_Z < 5)

5 < _Z ,

density

by adequate

cylinders

_/Z > 4000

the

or

should the the

C

be curves

E 1 curve

should

be used.

Torsion. Isotropic

criteria

plasticity

For

cylinders

for

z = 1.0

compression

is substantiated

the

3.1.6.

Gxz/Gy

in axial

cylinders

design

Paragraph

attention

cylinders

to sandwich

from

3.1.3.4

PRESSURE

as cylinders are

reasonably

sandwich

cylinders

in compression, well

defined.

in torsion although Whereas

both

have

not received

rigid-

the transition

the

same

and weak-core region

between

Section

C3.0

December Page

r

rigid

and

ably

weak

cores

sufficient

given

for

is not design

in References

Figure

3. 1-13,

isotropic

which

shear

continuous equal

37

at

to

1/R

,

well

and

47;

7Z

indicating

latter

was

to sandwich

Gxz/Gy of

the

methods

Information

the

applies

value

defined,

purposes.

behavior the

as

the

a change

used

The

are

transition

with

curves

buckling

of mode

presented

the

of

the

figure k

at

plot

is of

exhibiting

coefficient

of buckling

prob-

region

cores this

1970

41

to construct

cylinders

z = t . where

on

15,

that

xy

are

dis-

becomes

point.

102 q

i I x

I

10 Z w u. u. Ill 0 U Z ,,.I

i

D

i i i

i

I i L

10 "1 2

4

1

6

I

J 8

2

4

68

10

2

4

68

2

102

68

4

2 104

103 "yZ

FIGURE SUPPOI1TEI)

3. 1-13.

BUCKLING

ISOTROPIC SUBJECTED

COEFFICIENTS

SANDWICII TO

TORSION

FOIl

CIRCULAR G

/G xz

= 1.0 yz

SIMPLY

CY I_INDEIIS

4

68 105

Section

C3.0

December Page Reference a sufficiently tion

of the

scatter

wide

given

In addition,

where

be made,

discrepancies

of the

=

sandwich

is indicated

that

sandwich

the

same

tests. graph

_

xy

the

data

there

charts

was

3.1-13

for

some

of that

of References

were

construc-

reference.

37 and

47

noted.

The

straight-line

a rigid

core

(R=0)

is

(vZ) 3/4

0. 586 is the for

experience

factor which

may

be taken

(86)

not available

From

Plasticity

_ the

into

Figure

with

cylinders,

to be used density

unless

to substantiate

the

with

ratio

design

account

isotropic the

of

5

figure. is 0.03

is substantiated

by using

the

A

3.1-13 it

Here,

as

or greater, by adequate

curves

of Para-

3.1.6. CYLINDERS

structure.

strain

are

be used

center.

tension

data

should

enclosing

fore

0.34

cylinders.

cylinders

The

its

the

of Figure

-

D1

factor

3.1.4

shell

curve

in the

12

Experimental

with

to construct

rather

small

to be used

not cover

equation

kxy

most

used

it does

that

1970

42

but

37 indicates

between

only

behavior,

proportions

Reference

results

N

for

this

comparisons

(_/Z > 170) by the

not support

of geometric

calculated

In the ranges

portion

range

figure.

in the

could

37 does

15,

term

This

type

or compression

with

material

of shell

propellant

is strain-rate

AN ELASTIC

"cylinder

an elastic

The

rate.

WITH

an elastic that

closely

can

tests

The of the

core"

be either

approximates

is generally

sensitive.

CORE. defines solid

core

modulus material

or have

a hole

a propellant-filled

of a viscoelastic

core

a thin cylindrical

should

material be obtained

simulating

in

missile and

there-

from

its expected

Section

C3.0

December Page

/

Although design lem

curves

are

of a core

Reference curves

or

cushion

on

this

problem.

Axial The

this

only

Not

3.1.4.1

enough

some for

the data

analytical

isotropic

shells

outside are

dam

of the

available,

for

and

cylindrical

4.7

orthotropic

cores.

shells

The shell

however,

1970

[48],

inverse

prob-

is analyzed

to recommend

in design

Compression. buckling

comprc:_sion

reference

are

given

49. for

in axial

there

15,

is

are

behavior given

shown

of cylindrical

in Iteference

graphically

shells with a solid elastic core

50.

in Figure

Analytical 3. 1-14.

results For

oblained

small

values

from of

_j

(_7)

\

p

"/11(1

_r

iS

t}](.

x

c!I'iti_

: ';i

CF

_:illl('

{_I

c'(_:'_l_;:s:-i_*_

:lxi::l

f:_l" aw J:_,_tra,pi¢'

('ir(,lll:ll"

2_

(.[-

(,Vilrl

"1

,,.

1_'_'

.

:Is

li_:m

1._*',!_,

_m,,-l;:JIl.

i!l

[}:ll':t_

Vor

l':l[I}l

I;_r:_er

,}.

1.

'I'hi,;

I.i.

?,

va.a,?,,

<

_)f '/'1 ,

glI]l)l'l_iX_::.;Iti_)tl

1<.:" ,,x:_m|)lc,

i:-,

:iCCIlI';I|('

,:,! greater

l',_l

th:lll

:{,

Section

C3.0

December Page

15,

1970

44

10 8 6 ::::: :::::

2

%/o.)- 1,. 1

||l|l ||l

:,;!!

8 6

::'."." :::::

4

A

:::::

fjf v" :::::

2

10 "1 8 6

rill--

4

/

:::::

/

2 10 .2

:::::

2

J

.....

4

68

10 .2

2 10 "1

I_ 1 "

FIGURE

3.1-14.

VARIATION WITH

3.1.4.2

External

is

shown

50.

A plot

graphically

k pc

These

curves

are

COMPRESSIVE

STIFFNESS

BUCKLING

STRESS

PARAMETER

Pressure.

Analytical Reference

OF

CORE

curves

for

of

against

kpc

in Figure

the

3.1-15.

lateral _r -_-

pressure for

The

r t

core _

parameter

are

100,

presented

200, k

pc

500

or

is expressed

= pr_ D

to be

in 1000 by

(90)

used

for

finite

cylinders

loaded

by lateral

pressures.

Section C3.0 Deceml)er 15, Pa!_,e45

1048:

i

_ 6

= Ec/E

4

-"10

1970

i

3

2 1038

-104 6 4

-10-5

-10-6

1 ln-1

(TT r} / I'

10

61- c :

-t

5

_

:

'

-=

4 _-_--:

-H1_t

lIlitl 4

68

"1

I I

IIlIil 4

3. 1-15.

-

I _

-

°

,+

" '

',_

I-_---I--_

_t_f

2

4

10"1

;.I 1

I a.!;; _,

6

U

7

4

MODULUS

BUCKLING RATIO

PRESSURE (p

=0.3

COEFHCIENT , Pc

68 10

('fir)/t'

AND

:_

_ ;_I_,,=

1

r)/_

OF

'

t !

- _ :_" _:_

,,,

'

I L__L_.-L

10

VARIATION

LENGTH

-

""

_...

68

1

WITH

rI

41---- .....

11 Illtll

2

('/r

FIGURE

i J::,"J_:l

iltill

2

....

_'"'"

z 10

r

_

1o -_

1

_-::----,'T_

- _ _, •

8

/F-

....

1°'8 -_----_:-I

-10

E

_ 0.5)

Section

C3.0

December Page Some independent The

cylinders

of length;

straight-line

are

the

portion

long

single

enough

curve

of the

for the

shown

curve

can

critical

in Figure

15,

46

pressure

3.1-16

be approximated

1970

to be

can

by the

then

be used.

equation

k

- 3 (¢2)s/_

E r

(91)

C

1+

Et (1 - #c )_ where

E

¢2 - s (1__2)

3

(92)

E

C

103 8

J jv

6

/

J

4

2 f

1o2 /

8 M

f

0

/

4

I= u MS ÷ P

/ 10 8 0

J

1 2

4

6

8

4

68

10 "1

2

_2"

68

3.1-16. LONG

BUCKLING CYLINDER

2

4

102

68

2 103

3_1-_) Eo c÷)3 1 -_©

FIGURE

4

10

2

E

PRESSURE WITH

COEFFICIENTS

A SOLID

CORE

FOR

4

68 104

Section

C3.0

December Page The between

analysis

Hence, the

experimental

data

experiment,

but

and

a correlation

curves

warranted by

few

in Figures as

using

factor

more

curves

3.1.4.3

of 0.90

3. 1-15 data

A

and

become

points one

available

test

fails

is recommended 3.1-16.

for

Plasticity

good

4 percent use

below

of the should

1970

agreement

in conjunction

A reinvestigation

available.

in Paragraph

indicate

point

15,

47

factor

theory. with may

bc accounted

for

3.1-6.

Torsion.

The analytically

buckling

ch_scribed

behavior

of cylindrical

in Reference

51

shells

and

is shown

2

4

with

an

elastic

g_raphically

core

in Figure

3. 1-17.

102

10

I

10 "1

lq -2

in -3 2 10-?.

4

6

_

2

4

6

R

_.¢_-1

_3

:;. 1-17.

=

'I'()I{,<]_)N.\I,

('YI.IN!)!'II',,_

2

4

68

10

E¢ F

I'IGUI{I';

tK R

1

V,I Ftl

_

(r_

2

r

I',ITCI,[].IN(; AN

102

C()I,:J,'IqCJJ:NTN

HI...'_S'I'IC

COI{_,_

I,'(_,{

is

be

Section

C3.0

December Page For approximated

small

values

of

Cs (¢s < 7),

the

analytical

15,

1970

48

\

results

can

be

by

T

-

T

1 +0.16

_)s

(93)

XYcr i

where

E C

(94)

and

r

is the

torsional

bucking

stress

given

by equation

(16),

with

7

XYcr equal

to unity.

When

@3 is greater

than

10,

the analytical

results

follow

the

curve

T

- I + 0.25

T

(_b3)3/4

(95)

XYcr

Experimental The

experimental

compression theory Hence,

points and

than

data

the

are

obtained

external

not available from

pressure,

corresponding

cylinders however,

experimental

conservative-design

for

curves

can

this

with show

results

condition.

an elastic better

for

be obtained

loading

core

for

correlation

the

unfilled

axial

with cylinder.

by calculating

r

in XYcr

equations the

(93)

plasticity

3.1.4.4

and factor

Combined Interaction

combined

axial

(95)with given

the correlation by curves

Axial curves

compression

factor

A

in Paragraph

Compression for and

and

cylinders lateral

given

pressure

(19)

and

3.1-6.

Lateral with

by equation

Pressure.

an elastic are

shown

core

subjected

in Figure

to

3.1-18.

Section

C3.0

December Page

f

15,

1970

49

1.o

0.8

0.6

EclE = 0 {//r

_

= QO

n. u

_/r=

1

0.4

-%

\ \ 0.2

-'1 \

0

02

0.4

0.6

0.8

1.0

R P

FIGUIlt,,

3.1-18.

INTERACTION

(r/t These

curves

sufficienlly h/feral

stiff

obtained core,

pressure,

axial of

were

compression.

I. 5

.52

cylinders is

pression.

frequently

to

()1,'

critical

cited Unforlu_mt_'ly,

axial

more

and

indicate

stress

lateral data

pressure become

rec,,mmended

is

for

that

for

insensitive is

insensitiw_'

aw_iiable,

to

the

conservative

is

available instability

applicable this

on

to criteri_m

which

to

failures. cylinders Js

empirical

base The

subjected

the

use

design.

to and

design

criterion

based

of of

bending on

or data

a

to

I{INGS.

lzeneral as

is

50

compressive

critical

CYLINI)I,:IIS

COIIE

Ilefcrence

experimental curve

FOIt

ELASTIC

in

the

information exclude

AN

analytically

inLeraction

DESIGN

WITH

similarly, Until

l.ittle for

the

:m(l

a straiaht-linc

3.

_ 3o0)

CURVES

rings

llcference comfrom

Section

C3.0

December Page

15,

1970

50 v

test

cylinders

few checks

with made

proportions

on cylinders

is conservative,

but this A less

sists

simply

failing

calculations

are

against

weight,

ring

made

can

be ascertained.

ties

in the calculations

action

between

by the

calculations

the

and

designer

mine

ring

ring

can

be judged.

modes;

thus,

of wall for

the

of References

of ring-stiffened

isotropic

theory

dividing

as general

the

ratio

cylinder

PLASTICITY The

the

rings

in the

there

may

rings

as cal-

rings.

calculations

in weight

heavier

is,

of course, It also

influence

53 and

result

by the

as well

Both

are

desired from

plotted mode

uncertain-

be some

inter-

those

indicated

than

of the

applicable

has

the

various

to all

advantage

factors

types

of giving

which

deter-

weight.

the

for

somewhat

construction.

indicates

3.1.6

of error

It con-

so-called

between

failure

Presumably,

of designing

in compression,

ring

to force

be used.

rings,

If such

usually

53].

in the

failure

A

be used.

cylinders same

of the

for wall

criterion

may

cylinder

failure

design.

[10,

rings

of the

amount

the

cases

weights.

the

feeling

A study analyses

several

In addition,

all types some

involves

necessary

method

that

designing

load

the weight

should

indicated

of the cylinder

for

failing

This

for

failing

which

load

in contemporary

not be so in certain

the

mode,

interest

have

procedure

of calculating

of the

of loading

in use

may

direct

general-instability culation

of little

use

effect of the

¢Y

-

cr e

into

cylinders that for

CORRECTION

plasticity

which

present

in torsion

the

linear

except

procedure those

with

gives a single

bays. FACTOR.

on the buckling

coefficient,

general

and of orthotropic

recommended

all cylinders

two equal

of plasticity

54,

77 .

of shells This

can

coefficient

be accounted is defined

by

Section C3.0 December 15, 1970 Page 51 where

a

= the actual

buckling

stress.

cr

= the

(7 e

elastic

would level).

buckling

stress

if the

material

occur

(the

stress

at which

remained

elastic

buckling

at any

stress

The elastic buckling stress, therefore, is given by the equation (I O"

the

loading,

cr

-

e

q

The

definition

the type

struction.

For

cylindrical

shells

of

_? depends

of shell,

example, with

the

the

_

simply

on

°cr/(re

boundary

which

conditions,

recommended

supported

'

for

edges

and

is a function the

type

homog(mc,)us

subjected

of

of con-

isotropic

to mxial compression

is

,/2

r

JEt's E where

Et , ES

ratio,

resoeetively,

Poisson's

and

p

are

L'.o 1 - it 2 the

tangent

at the actual

modulus,

buckling

stress,

pared

for

(_cr/__

,

o the (_

cr

/7

versus

actual This

buckling method

cr

otherwise

and

modulus Pc

and

is the

Poisson's

elastic

ratio. For a given material, temperature,

versus

secant

be necessary.

(_ er

By first

stress eliminates

(_cr

and _ , a chart may

calculating

the

c'm

from

bc read

an iterative

elastic

procedure

the

be pre-

buckling chart which

of

stress, ,,cr/71_

would

Section C3.0 December 15, 1970 Page 52 Figures a

cr

for

some

space

industry.

drawn

as one

3.1-19

materials

through and

In many

3.1-25

temperatures

cases,

the

present

curves

commonly

curves

are

of

acr/_?

encountered

so close

together

versus

in the that

they

curve.

The

_/ used

to determine

each

curve

is defined

as follows:

Curve A

Es/E

E s

B

0.330

C

1/2

+ 0.670

+ 1/2

J

=t] S

p2 + (1__2) F

J

S

Et D

352 + 0.648

J

p2 + (1__2) S

E

F

G

p2 Es/E

0.046

+ (l_p2)

Es/E

Et/E

+ 0.954

Et/E

Et/E

112 E1

aero-

(U = 0.33)

are

Section

C3.0

December f

Page

!

MINIMUM

GUARANTEED

!

E,F,G

GUARANTEED

I

3()o

-423°F E

26o

F

c

i

E1 ._.._._j___

1970

i

I

I

MINIMUM

15, 53

E,F,G

1

I "

.30O°F

- 12.4 x 106 psi

E --

" 74 ksi

" 11.9x 106 psi ¢

F " 67 ksi cy FO.2 ,, 67 ksi

cy FO.2 - 74 ksi

220 E1

-

/

180

i i

140

B,D

c

I

_ I B'D

//r/ /,///,,

100 r

¢<"

6O

20 55

66

75

85

45

@cr (ksi)

FIGURE

3.1-19. -T651

55

65

_cr {ksi)

PLASTICITY ALUMINUM

CORRECTION ALLOY SHEER'

(-423" F, -300 °F)

CURVES FOI{ AND PLATE

2014-T6,

75

Section

C3.0

December Page

! MINIMUM

I

1

28O ROOM

I

I

E,F,G

- 10.7x

I I06

F

- (10 ksl

I

MINIMUM

m60ksi

I

I

200°F. Ixi

1970

54

GUARANTEED

I

TEMPERATURE

E© cy F..u,

I

GUARANTEED

15,

E,F,G

1

1/2 hr

Ec

-

10.4 x 106 psi

FLy

- 55 ksi

F0. 2

- 56 ksi

24O

20O

J*

_"

leo B,C,D,E 1

12o

i

/

/ I1

.o

//

B,C,D

/,/j

X/.,,,'"

,,o..__

_j"

0 40

50

60

70

38

45

Ocr (ksi)

FIGURE

3.1-20. -T651

PLASTICITY ALUMINUM {room

55 Ocr (ksi}

CORRECTION ALLOY SHEET temp.

; 200°F,

CURVES FOR AND PLATE

1/2 hr)

2014-T6,

65

Section

C3.0

December

F

Page

"

90% PROBABILITY

LONGITUDINAL OOUO

15,

1970

55

GRAIN

o

r._

"

©

¢$

1 <

_

,"

o.

r,:

t,_

U..

(_"

UJ

Z

Ilrd

< o

_ I

<

> ,_

r.) Z (3

© r..)

<

=

..J I

Section December Page

PROIIABI

C3.0 15, 1970

56

LITY

0 .< 0 r_

_'_. mo

z_

O----

OZ

_4 I

_4

r_

Section

C3.0

December Page

f

15,

57

!

MINIMUM

GUARANTEED

500

A

3OO

2OO

100

0 10o

12o

140

160

180

200

_r (ksi) FIGURE

3.1-23.

STAINLESS

PLASTICITY STEEL

CORRECTION

SHEET

AND

(room

CURVES

PLATE temp.

)

FOR

-- RH 1050,

Ftt

PH 15-7 1075

Mo

1970

Section

C3.0

December Page

MINIMUM

6OO

15,

58

GUARANTEED

G

5OO ROOM

F

//

TEMP.

" 15.8 x 10 6 psi I" 125 ksi

cy FO. 2"

///

120 ksi

,///

_- 3¢X}

5 f

E1

_.¢/. .¢"//.°

2@@

100

/

B

0 100

106

110

115

120

_r {k,i)

FIGURE

3.1-24. ALLOY

PLASTICITY SHEET

< 0.25

CORRECTION 6AL-4V

CURVES

ANNEALED

FOR

LB0170-113

TITANIUM

1970

Section

C3.0 15, t970

December Page

MINIMUM

59

GUARANTEED

I/)

Q I.l._

O_

_Z zZ

a.

=. o .; .; O

I._ t,¢3

Nz >.H U 0

II

© m_

o u3

,rr

¢,3

4.-

Z ©

<

(.9

O cL

O io rr

_

"> "N

n W

u u.

u.

v 0

g

0 0

0 t_

_

Section

C3.0

December Page Although in excess ing the of

of the

and

value

and,

It can be seen most tion

that

buckling

_

is small noting

curves

A

is a function the

/_ = I/3

.

for curves,

and

G

whereas

curve

load

to plasticity.

due

plasticity

except

that

on the remaining

conservative, in the

of limit,

of

# =1/2

It is "worth 77 = Et/E

value

proportional

conservative

_ =I/3

the

A

The

stresses

curves

were

obtained

assum-

between

using

the value

difference E

A , 77 =E

and s

77 is a function bound results

1970

of the

for curves

curve

15,

60

the range

of

/E

for

F . ; for

of both T/ .

in the smallest

stresses

curve Et

Curve possible

G, and G

Es is the

reduc-

r

Section

C3.0

December Page 3.2

CONICAL This

,f

form

stiffened

static

loading

loads

which

are

suggests

various

shells, the

however,

behavior

differences are

In addition,

ditinns

remain

compu_rs.

two types

such

torsion,

3.2.1.

I

along Axial For

disagreement These

are

axial with

under

buckling

estimates

of uni-

various

types

of

of static

buckling

of these

Frequently,

and

problems

there

geometric

the

effects

ean

in References

param-

of edge

be treated

con-

by digit_a[

! ,and 2.

L SHELLS.

the

recommended

compression,

those

Whereas

significant

of conical-shell cases

shells

of conical

shells.

to be similar,

range

is given

stability

unexplained.

loading

Some

elastic

of cylindrical

remain

of conical

for

design

l_nding,

combined

procedures

unifnrm

for

hydrost:_tic

isotropic pressure,

loads.

Compression. conical

between

discrepancies

shells

have

axial

as well

conditions

as to shortcomings

A theoretical can

analysis be expressed

compression,

loads

tx, en attributed

analysis,

shells

under

experimental

and of edge-support

conical

shells

of the

appears

the wide

CONICA

structure

long

of shells

pro_ram

following under

predicting

of the buckling

as that

important

ISOTROPIC The

for

that yield

Knowledge

to In studio, d.

shells

conical

conducted

results

some

One

3.2.1

been

conditions.

_tata to cover

eters.

and

have

in experimental

insufficicnt

conical

circular

is not as extensive

of the

61

to be conservative.

studies

loading

practices

procedures

considered

Many under

recommends

unstiffened

and

1970

SHELLS.

section

and

15,

and

to the different

of the [3]

loads

effects from

that

is considerable

predicted

by theory.

of imperfections

small-deflection

indicates as

the

there

the

those

assumed

of the

theory

in the

critical

nxial

used. load

for

Section C3.0 December 15, 1970 Page 62 2_ Et 2 cos 2

=T

P

(1)

cr

with

the

that

theoretical

within

apparent fore,

_] 3 (l_p2)

the

range

effect 7

be taken

can

value

of 3/ equal

of the geometries

of conical-shell be taken

as the

for

gives

cone

as a constant.

constant

data

coefficient ness

a lower

can

as the

radius

are

bound

shell

shells varies

to differ effects for

from

of the

moduli

along

should

cone,

yield

correlation

there factor.

is no There-

it is recommended

that

7

region.

are

used

correspond

Because

shells.

75 deg

must

For shell

equal

to the

(2)

Buckling-load

coefficients

be verified a < 10 deg having slant

by experiment the buckling

the

same

height

on the compressive the nominal

load

wall

and

thick-

average

of plasticity

A conservative

(Paragraph

3.1.1.1).

The

are

compressive

likely

of plasticity

reduction secant

of

in a conical

shells

estimate

if the

membrane

level

in conical

however,

to the maximum

buckling

stress

be obtained,

P 2_r Plt

radius

the effects

could

data.

range.

published

in cylindrical

shells

than

,

respectively.

been

length,

shells

max

specimens,

75deg)

of a cylindrical and

have

its

those

in conical

cylindrical

as that

<

experimental

in this

and a length

in the

on the

< _

greater

not available

No studies conical

to the

angles

of curvature

tested

[4, 5] indicate

value.

be taken

cone

Experiments

At present,

(10deg

semivertex

because

of the

geometry

7 = 0.33

which

to unity.

factors and

tangent

stress

(3) cos z

Section

C3.0

December Page Figure axialfi)rce

(P

semivertex

an_le

range

for

stress

from

equation

)

cr

(cz)

are

alignment

chart

(1)

known.

where

This

devised the

to determine

shell

nomograph

thickness is

1970

the (t)

applicable

critical

and

the

in the

elastic

alloy.

From

eauations

(1)

c r

t = 2/ I,: -r 1

and

(3)

the

maximum

membrane

compressive

is

3.2-2a

stress

wh,_n

The

is lhc

f()ll,:win:4

thiel_m,ss

(t)

axial

force.

0.06

on

point

draw

t

the

scale

a line

psi,

which

3.2.1.2

compressive

radius

in.,

the

the

of the

of I:i?ur(_

and

the the

critical

3.2-21)

is

line

nomo:_,r:lph. (r 1)

3.2-'2 until

scale.

u<.cful

in.,

are shell

_'_ ru, sultiniz

line

r 1

QR

inl_,rs(.ets

km)wn. hz_s a

a semiverlox

.lfl (m the

itmcels

,'Lxial

an_tv

and

slrc.

line

critie:,l

A conical

of 40

join

This

t)uckliJ,;Z

the

._(,mivurtex

compre:_,ive

nomograph

the

a_(!

radius

criliea]

extend

to determine

radius,

use

a small

(.1)

from

scale

an

with

.

l'r[,v_

this

th(,

,J ec

sc:,le

stress.

if th(,

:_tress

lalls

inl,_

the

t,lasti(:

r:un_z('

I',)r

shown.

Ben(ling. For

pressive

small

shrews

is

materials

(4)

thickness,

to (;0 (leg_n

Figxlre three

_

of equation

[kq(:rmin(; ()n

the

2600

_hvll

of 0.0[;

of 6flde_4.

cos

a n()mograph

cx:lpap](_

anglo

the

is an

aluminum

Figure

at

3.2-t

15,

63

conical

stress stress

at

shells the

of a cylinder

of curvature.

M

The

: cr

in bending,

small

end

having buckling

bucklin

of the the mom,,nt

T 7r Et 2 r 1 cos 2 f_' [ 3 (1_1_2) ]1/2

cone

same

,,"(_ccur_; is

wall is given

eaual thickness

when t() the

tim critica!

an(l

the

m:_>dm:lm c,)msame

l,)cal

I)y

(5)

Section

C3.0

December Page

t (In.) .03

15,

1970

64

Per (IW|

--

1000

70

100O0

,10

--

_,,,,

--eO

qm

100 000

.2O

• SO

,30

40 10ooo00 3O 2O --10

_0-

FIGURE

3.2-1.

CRITICAL

(ALUMINUM with

the

theoretical

it is recommended

_/ = 0.41

ALLOY

value that

AXIAL

the

of

MATERIAL),

-y equal coefficient

(10

deg

LOAD

to unity.

FOR

E = 10.4 Based

_/ be taken

<

_

LONG

< 60 deg)

CONICAL

SHELL

x 106 psi

on experimental

as the constant

data

[6 ]

value,

(6)

Section

C3.0

December Page

f

15,

65

0.1 °)

r I (in.)

10-175

--

10 2

70

.01

- 10 3

f

.10

6O

_ 1114 100 -

5O

1.0

"

4O

20_--

m

3O

- 10 5 2O 30t)_ _0

FIGURE

3.2-2a. SHELL

CI¢ITICAL (A],UMINUM

COMPRESSIVE STRESS FOR LONG CONICAL ALLOY MATERIAL), E = 10.4 × 106 psi

1970

Section

C3.0

December Page 66

15,

1970

80

70

00

50

40

30

20

10

0 10

50

lO0

50O

r1 tcos (1

FIGURE

3.2-2b.

BUCKLING OF ISOTROPIC UNDER AXIAL COMPRESSION

CONICAL

SHELL

1000

Section

C3.0

December Page

f

Buckling 60 deg

must

shells

in bending

conical

be verified

shells

shells

can be used

3.2-3,

when angle

proper

sequence

design

purposes

3.2.1.3

(_ are

when

mate

form

known.

cr

The

the

theoretical

less

greater

equivalent than

correction

may

be obtained

t ,

the

lines

small

1 and

alignment.

Hydrostatic

This

is given

than

cylindrical

10 deg.

For

suggested

for

conical

and

from

the nomograph

radius

rI ,

2 on Figure

illustration

a required

and

3.2-3 can

the

show

also

thickness

pressure

eircumferentialwaves

the

be used

(n>

of a conical

2)

can

shell

be expressed

which [7]

buckles

in the

approxi-

E T

of

_, but indicate

that

3/ :

should

provide

solution cable

[8, the

(7)

9] show

constant

a relatively

wide

scatter

band

for

the

(8)

a lower

in the elastic

value

value

0.75

to equation

for

is needed.

Pcr

Experiments

of

Pressure.

buckling

0.92

angles

67

angles for

1970

be used. M

a moment

semivertex

coefficients

stresses,

thickness

of parameter

The several

shell

cone

semivertex

moment

the

Uniform

into

with

may

buckling

for

Buckling

to plastic

compression

The

semivertex

coefficients by test.

subjected

in axial

Figure

load

15,

(7) range

bound for

for the values

only.

of

available p/t

and

Figure

data. L/p

.

The

3.2-4 curves

gives are

the

appli-

Section

C3.0

December Page

in/////// I M

(in-lb)

_r

15,

68

a(O)

.03

r 1 (in)

-r_5

10

104 .O4

-

105

20

.00 .07

J

J

J 30

.00 .09

-40 108 +60

-

50 6O 7O 8O

107

_so

_

90 -100

.20-

lo"

.30-

I"

-2OO

0

10

R

FIGURE SHELL

3.2-3.

BUCKLING

(ALUMINUM

MOMENT

ALLOY

FOR

MATERIAL),

CONICAL E = 10.4

TRUNCATED x 10 s psi

_

-3OO

1970

Section

C3.0

December

F

Page

15,

1970

69

O

J

O2

O Z ©

_O2 0 _ 0

m

5O

r_

|

o2_ O.

_Z

g×M "t

0 Z

r..)M _Z !

t

d

Section

C3.0

December Page For correction

for

the conical curves

conical

in the

3.1.6.

The

The

moduli

stress

plastic

shells

considered.

compressive

max

buckle

cylindrical

geometries

in Paragraph

circumferential

end

1970

plasticity

for

the

range

is to use

the

E1

correspond

large

the

be used

procedure

should

at the

may

range,

to the

of the

conical

of

maximum shell:

= Pcr (_/t)

(9)

To rs ion. An approximate

[10]

which

moderate-length

shell

3.2.1.4

shells

15,

70

equation

for

the critical

torque

of a conical

shell

is

Tcr

(t)1/3 -_

= 52.8_/D

(10)

( _r)S/,

where

r = r_ cos

1 +

1/2(1

rl ) r__

1/3 -

1/2_1

+ rl

-t/2

r2 (11)

The

variation

purposes (10)

of

r/r 2 cos

_

it is recommended

be taken

with that

as tim constant

rl/r 2 is shown the

in Figure

torsional-moment

3.2-5.

coefficient

For

in equation

value

_/ = 0.67

torque

design

(12)

Figure

3.2-6

of a conical

shell

is a nomograph (equation

devised

10) when

t,

to determine t/_,

and

r/t

the

buckling

are

known.

Section

C3.0

December Page

rJ"_

15,

1970

71

1.0

s

/

/

r

_

rcos(L

r

{.1+

[ 1(1

+r-_

r2 )]1/2

[.__(l+r

I

-1/2

,

)]

rl ___r._

I

0 0

.5

1 .On

r 1 r2

FIGURE

3.2-5.

No data torsion.

The

however,

give

VARIATION

are available

plasticity

the maximum

factor

conservative shear

stress

()F

for the used

plastic

The small

end

c_ WITH

buckling

for cylindrical

results. at the

r/r 2 cos

secant of the

shells modulus cone,

rj/r

2

of conical

shtqls

in torsi
in

._h(,uld,

corresp,md

to

by

T cr

_'cr

-

2r rt 2 t

(13)

Section

C3.0

December Page

t |_.)

TORQUE

r/t

.O3

15,

1970

72

10

U

-

.0003

10

.08-

n

.001

.10 -

1000_

.18 -

1ooooo:

.201000000-,r

.2S

10000--m.006 NOTE:

SEE FIGURE 3.2-5 TO DETERMINE

FIGURE

3.2.1.5

3.2-6.

Combined I.

"'r".

ALLOWABLE TORQUE FOR (E = I0.4× 108)

CONICAL

SHELL

Loads.

Pressurized

Conical Shells in Axial Compression.

The theory for predicting buckling of internally pressurized conical shells under axial compression two respects.

[II ] differs from that for cylindrical shells in

First, the axial load-carrying capacity is a function of internal

Section

C3.0

December Page

J

pressure

and

exceeds

shell

the

pressure

and

analyses

for

significant dent

shells

conditions are

surized at the

small

shell.

indicate

edge

conditions

that

cone

long

the

to the

design

axial

small

of

end

results

use

are

the

of the

conical

compressive

compressive

(2_

the

results

have

indepen-

load

in-

shells.

load

pressurization buckling

entire

for

a pres-

load at the

It

Tr rl 2 p conical

Then

Pc r ::[

The

The

pressurized

by adding

unpressurized

cones.

to warrant

critical

of the

at the

1970

73

Second,

capacity.

of internally

be determined

of the

of the cone.

end for data

capacity

end

load-carrying

capacity

shell

end

that

insufficient

recommended

conical

load-carrying

small

at the large

in load-carrying

is therefore

of the

at the

on the axial

There crease

sum

load

conical effect

of edge

the

15,

unprcssurized increase

is given value

in the buckling

given

by equation

bending. surized

in the

(14),

no theory For conical

has

yet

conservative shell

Mpress

A_/

critical

Shells

is written

_1'3

is equal

equivalent

may

increase

to 0.33,

cylindrical

be increased

above

is justified

by test.

and sholl the

in Bending.

developed

design,

y

the

load

if the

of unpressurized been

for

axial

however,

Conical case

coefficient

cocfficient The

Pressurized As

bending,

3.2-7.

(14)

Et 2 cos 2 (_) +Tr rl 2 p

+ Ay]

compressive-buckling

in Figure

II.

d 3 (l-p2) Y

therefore,

conical for

shells

pressurized the

subjcct('(l

t(, pure

conical

shells

desi_,m moment

()f the

under pres-

us

1-_ 2

2 (_5)

a

Section C3.0 December Page

1.0

7 s

III III

III III

111 111 III

111 Ill J[[

I!1

II1

I1[

JlJ

Ill

2

__

Ill

ill

] [_'_ J_rl 7

._

5

]_lll

/

/ •

2

/

'

I11 IIJ

] i l Ill Ill

'

I IT I]]

1[I

ill

111

I[[

IIJ

Itl

Ill

IlJ

111

o.o, 2

3

5

7

2

0.01

3

5

3.2-7.

STRESS

INCREASE

IN AXIAL

compressive-buckling

the increase in buckling coefficient Ay can be obtained from

5

7 I0

2

COMPRESSIVE

coefficient y

BUCKLINGRESULTING

is equal to 0.41, and

for the equivalent cylindrical shell

Figure 3.2-7.

III. Combined Pressurized Some

3 1.0

COEFFICIENTS OF CONICAL SHELLS FROM INTERNAL PRESSURE

The unpressurized

Axial Compression Conical Shells.

experimental

pressurized

and

pressurized

and

[6].

These

bending

7

0,10 rI

FIGURE

1970

111

IlJ

0.10

15,

74

and Bending for Unpressurized

interaction conioal

investigations

curves

shells

under

indicate

have

been

combined that

the

obtained axial

following

for

and

un-

compression straight-line

Section

C3.0

December Page interaction

curve

Rc

for

conical

shells

is adequate

for

design

15,

1970

75

purposes:

(16)

+Rb=l

where

P R

__

c

(17)

P CF

and

M M

Rb

cr

For

equations

(17)

P

P

and

(!_),

np!)liod

e()mpressive

hind.

critic31

compressive

load

for

cone

not

sut)je('ted

Io b('n,lin_,

cr

oblain(,(l

M

M

from

shells,

:ln(l

applied

bendin_

critical

cr

equntions groin

as

for

obtained

test

values

of

P

and cr

curve

may

substantiated

no

longer by

test.

be

conservative,

(14)

cone

from

surized,shells, shells.

If actual

cqu:/tion

and

(2)

for

for

unpressurized

pressurizl,(l

sh_Hls.

mon_('nt.

moment

sion,

(1)

and

M

m.,t

from

are

subjected

eqtmtions equation

used,

the

(5)

to a.xi;ll and

(15)

compr(,_-

(C,) for for

straight-line

unpres-

pressurized

int(,r_ction

cr

and

the

entire

interaction

curve

must

be

Section

C 3.0

December Page IV.

Combined For

axial

External

a conical

compression,

R

c

the

+ R

is recommended

Compression.

subjected

to combined

external

pressure

and

relationship

(19)

design

purposes.

Where

P

p

and Axial

= i

p

for

R

shell

Pressure

1970

15,

76

(20)

P cr

P

R

is given by equation (7) and (8), and cr

V.

Combined For

Torsion

conical

shells

pressure the following

and

External

under

is given by equation (17). c

Pressure

combined

or Axial

torsion

and

Compression.

external

interaction formula is recommended

hydrostatic

for design purpose:

Rt + RP = i

(2i)

with T T

Rt-

(22)

cr

where

T

is given

by equations

(10),

(11),

and

(12),

and

R

cr

equation

following

conical

Rt

shells

interaction

R t + Rc where

by

(20). For

the

is given p

is given

formula

under

combined

is recommended

torsion for

and

axial

design

purposes:

= i

by equation

compression,

(23)

(22)

and

R

c

by equation

(17).

Section

C3.0

December Page 3.2.2

ORTHOTROPIC The

determining

theory

because

as well

as for

shells

two directions. to conical

13.

limited are

also

are

1

Uniform

I.

properties

isotropic

conical

'File computer

programs

Few

in Reference parameters,

cxpcrimcnls

reeommen(lations discussed

theory

12, whereas

of the many

generalizations.

in the

shell

is considered

to only a few values

the design

stiffening, differ

in Reference

orthotropy

arc

Rased

in Subsection

on tim 3.,1

theoretical

slant

orthotropic

the

conical

L,

conical

theoretical

hydrostatic

orthotropic

conical

of the

shells

_

conical

buckling

_,[1

[15,

indicates

that

and

shell; shell,

pressures

_0"86T PsPO _]_'/a

ES

,/,

the for

as one

a/4

shell

between

equal

(I_,)(t)

lxctwccn

ami of the the

so-

buckling

a length

to the avcragu

thickness.

supported

m_;hip

isotropic

having

same

be expressed

E0

rc!uti,

equivalent

a radius and

16] can

to that

of the

is defined

the

conical

is similar

shell

cylinder

_ ,

Per

cylinder

of the

Material.

of an orthotropic

equivalent

length,

of curvature,

[14]

pressures

of an isotropic eases

Orthotropic

investigation

bucMing

equivalent

In both

Pressure.

Constant-Thickness

pressures

the

whose

in

geometrically

or circumferential

is given

tentative

II_'drostatic

A limited

called

are

is valuable

recommended.

3.2.2.

the

shells

which

meridional

geometric

Following

available.

shells

orthotropy

for

conical

Donnell-type

limited

the basis

conducted.

data

of the

with

results

for

of a material

material

shells

provklc

been

spaced

with

Numerical

have

of closely constructed

of conical

but these

criteria

1970

SHELLS.

of orthotropic

buckling

An extension

shells

buckling

of buclding

adequate

orthotropic

CONICAL

15,

77

Thus,

cylinder. ,'qual

to

radius the

moderate-lengl.h

as

5/2

,

(24)

Section

C3.0

December Page 78 which

reduces

E

to the corresponding

=E

8

expression

for the isotropic

15,

cone

1970

when

=E

e

(25) l_s

= _ 0 = I_

Only for test

an orthotropic results,

be taken

material

as 0.75

hydrostatic all

but could

for

both

Stiffened The

tions,

experimental [17].

pressure rings

has

were

have

that

and

shells

also

investigated

been

to have

spacing.

The

are

not recommended

of substantiating

test

data

of the

constructed

extensive

correlation

range

of

coefficient

7

cones.

becomes

stiffened

the

[ 13,

same

use

under

18].

In these

cross-sectional

approximate for

by rings

buckling in design

uniform investiga-

shape formulas

until

and

area

given

in

a larger

amount

available.

Torsion. I.

Constant-Thickness The

theoretical

buckling

of the conical Refer

Orthotropic

investigation

that of an equivalent

(11).

of a more

isotropic

of conical

references

3.2.2.2

shells

Shells.

assumed

variable

for conical

absence

the value

orthotropic

Conical

stability

data exist In the

it is recommended

II.

these

limited

shell

torque

reported

to Figure

having 3.2-5

in Reference

of an orthotropic

orthotropic and

Material.

cylinder the for

same

conical having

shell

a length

thickness

the variation

19 indicates

is approximated equal

and radius of

that the by

to the height, given

r with r z cos o_

in equation r-t rz

I,

Section

C3.0

December Pa_e The may

then

be

critical

torque

approximated

by

the

expression

E

5/a

E

A shells)

reduction

conical

band

factor

is recommended.

epoxy

for

[17]

isotropic

II.

sh_.'ll

Although

rings

equally

procedure adequate thus the

spaced

The

be approximated leng{h,

ring-stiffened

and

cone

T

with

the

where

(Fig.

the

1/2

value

given

points

available

for

value

of

"/

but

for

isotr,,pi_'

conical

fib_.'r_lass-reirfforced

fallwitl,in

calculations n few

the

same

torque critical

()1' such

the

indicate

s_:att-r

c'(mical

rings

The is

nnd

sh(_ll c'_)nicaI

wh,m

giv(m

5/_

ih(_

:_rc.a, vii'

a

3 i.,,ld

sbeql

cvlind(,r criticql

then

: _,_._ie fo_-

that

shnt)(,

a rinb_-stiffon(.'d

above.

spaced

b(,.(,n

of a ring-stiffcm,,!

described

: 4.57y

[17]

orthotropic

torque

uniformly

t(:sts

have

cross-secti(m:,l

mqt(_rially

thickness

cr

shell

Shells.

have

for

conical

thickness.

in torsi,)n,

critical by

(the

theoretic'_i

and

to that

results.

radius,

larger

no accurate

similar

data

of constant

shells

5/4

1970

79

orthotropic

r2t

"/ - 0. _;7

(7onic'd

c_)nical

are

a/a

few

yielda

Ring-Stiffened

ring-stiffened

of

The

shells

the

of a moderate-leng-tb

15,

may

h,qvin!:

t()rrl_.e

o1:_

by

1 +rl0

(l,-PspO)

3.2-8)

77o = 12(1-P2)

-E-I_:[r

ITor_

+

I_ Ar t

(Zr-

t er)

21

(28)

+ 12 (__)2

o

and

the

test

results

scatter

factor

band

T also for

is

recommended

indicate the

to be

a larger

isotropic

value

conical

taken of

shell

T ,

equal but

of constant

to 0.67. these

The again

thickness.

fall

few

available

within

the

Section

C3.0

December Page

15,

1970

80 .,.,..r

t

/ /

Zr

_

t

' \"

/

i

/

L

FIGURE

3.2-8.

3.2.3

NOTATION

SANDWICH

FOR

CONICAL

RING-STIFFENED

CONICAL

SHELLS

SHELLS.

If the sandwich core is resistant to transverse shear so that its shear stiffness can be assumed

to be infinite,the previous results for isotropic

and orthotropic conical shells may

readily be adapted to the analysis of sand-

wich conical shells by the following method. 3.2.3. I

Isotropic Face Sheets. If the core

and no load-carrying the analysis isotropic

for

capacity isotropic

sandwich

and thickness

is assumed

must

conical

infinite

in the meridional

conical shells

be defined

to have

shells

sandwich

shear

or circumferential

of constant

of constant

for the

transverse

thickness

thickness. shell.

stiffness directions,

may

be used

An equivalent The face

sheets

for

modulus may

be

Section C3. 0 April 15, 1573 Page 81 of different

thicknesses

that

the

Poisson's

and

bending

ratios

and

having

F

t

an isotropic

stiffnesses

same

neutral

= E1h

+ E2_

materials,

two materials

of such

bending

the

of different

of the

stiffnesses

stretching shell

and

subject

be identical. sandwich

of an equivalent surface

shell

to the

restriction

If the

stretching

are

equated

constant-thickness

dimensions,

to the isotropic

then

{29a)

Q E(_) 12

=

h2 ! Elh

Then pic

the shell

modulus

and

the

E_t2

thickness

¢.f '._c equivalent

conatant-thmkncss

is_tro-

are

E2h

E

= E_.____+__+ E_.__

L The as

the

listed

buckling

bucl:lia_ loads

load:_

of the equiv

Comnression

_ler, t isotrop_c

sandwich _;he[l

sh_,]l

may

of constant

now

l,_: 'a_.e _

thick::e_

a_

.rl

P:, ra_-a__b

P,eferer_t.e

3.2.1.1 3.2.1.2

Bending Uniform

i_tr._9ic

be.low.

L_:3

Axial

of the

hydrostatic

pressure

3."

1 3

Section

C3.0

December Page 82 Load

Paragraph

Torsion conical

Pressurized

conical shells in bending

Combined

shells

in axial

axial,compression

unpressurized

3.2.1.5-II

and bending for

and pressurized conical shells

external pressure and axial compression

Combined

torsion and external pressure or axial 3.2.1.5-V

compression

In the factors

absence

for

isotropic

sandwich

shells.

3.2.3.2

of experimental

shells

Orthotropic If the

Face

core

capacity

the

available

for

may

as

be used

sheets long as

procedure following tropic

may their as

for

for

conical

shells conical

shells

principal

are

shells

j_-

or correlation

recommended

axes shells

oriented having

material

properties

of constant

thickness:

+ j-_-tl

transverse

for

shear

or circumferential

of constant-thickness

thicknesses

and

are

infinite

meridional

be of different

thickness conical

thickness

to have

in the

sandwich

sandwich

the reduction

isotro-

Sheets.

is assumed

no load-carrying results

data,

of constant

and

face

Reference

3.2.1.5-I

compression

Combined

rial

1970

3.2.1.4

Pressurized

pic

15,

orthotropic

but of the

same

isotropic of the

same face

directions,

orthotropic

having

in the

stiffness

faces.

sheets

equivalent

The

orthotropic

direction.

mate-

material The

leads materially

same

to the ortho-

(31a)

Section

C3.0

December Page E

E

s

c

_o -

-

#s

buckling

load

thickness

tropic

sheets. Local Thus

of failure. core

buckling, usually

data

become

however,

can

shell

for

core

heavy

honeycomb-core

under

uniaxial

may The

crushing.

stress

that

loading.

are

conical

of orthotropic factors

use

for

has

been

the

stress

For

because

use

of relatively

shell

is then

material for

having

isotropic

sandwich

shells

shells with

ortho-

intracell

For

of core

crushing,

heavy may

honeycomb-

prove

(6 > 0.03) to be justified

that

are

a function

of position.

varies

only developed

shells

buckling with

slightly for and

intracell

cores

conducted

intracell

conical

as a criterion

however.

cores

equations from

considered

been

that state

approxlmate

sandwich

occur

have

states

failure

possible,

Lighter

No studies

under

to predict

buckling

of failure

wrinkling.

the following

be used

sandwich

or correlation

overall

failure

available.

assume,

conical

recommended

only

modes

prevent

failures

region,

far,

shells,

buckling

orthotropic

reduction

are

and face

will

of the

Failure.

Other

sandwich

load

The

of constant

3.2.3.3

(31c)

of the equivalent

thickness.

face

1

#0

The

constant

(31b)

7

0

is

buckling

83

t,+t_

G

the

1970

--

o

s

15,

predict

over

localized If we

the

buckled

cylindrical face

equal-thickness

as

shells

wrinkling face

of sheets

buckling

(32)

Section

C3.0

December Page where

S is the

scribed

circle

core

cell

size

expressed

as the

diameter

15,

1970

84

of the

largest

in-

and

4 Ef Eta n ER

=

(33) _f'Ef

where

Ef

and

material.

+ _-'_tan

Eta n are

If initial

the elastic

dimpling

(_s = 2"2ER

should For

be used.

where

E

plane

of the

biaxial

tangent

moduli

is to be checked,

(_)

The

sandwich

= 0.50

(Esec

is the

z

modulus

core,

and

compressive

equations

must

of minimum

of the face-sheet

the equation

2

(34)

will

still

carry

Loads

if initial

dimpling

occurs.

of bond, given

strength

20, plasticity may

as the

in a direction

applied

intracell-buckling

21,

and

sandwich,

stress equations

initial

waviness

,

to the of the

core.

If

the coefficients where

f

is the ratio

in the face which of the

sheets.

consider face

of

sheets

strength are

22.

correction

factor

be applied

also

to sandwich shear

modulus

(1 + i¢)-1/3

compressive

and

perpendicular

shear to the

by the factor

of foundation,

is applicable

conservative

are

(35)

transverse

principal and

compression

factor

is the

sz

be reduced

in References

in axial

of the core

G

to maximum

The

Ez Gsz )l/_

stresses

Wrinkling

what

and

wrinkling

Os

The

2

cones

stiffness

given

for

to isotropic with

stiff

of the core

isotropic

conical

shells

sandwich

conical

shells.

cores

and becomes

is decreased

[23].

some-

Section

C3.0

December Page

i

3.3

DOUBLY Doubly

external

curved

closures

heads.

When

reaction

doubly

buckling

its

the

properties

supported,

and

geometric

deviations

adverse

effect scatter

This doubly

design

of complete

Most data

of the are

curved

data

also

The small

dynamic

accounted

shells.

are

for for

shells point

reduction

a lower-bound

ficient

data

available

shells plasticity lations factor.

design.

of arbitrary effects, in applied

from

loads

forces

in

is often

depends

stiffening,

ideal

when

in which Initial,

shape

upon

can

curved

present),

its edges

although have

its

are

small,

a significant

shells

and can

cause

practices arc

and

for

practices

and toroids,

as well

to uniform

for

the

as bulkheads.

pressure

buckling

loads

loads,

conditions,

caused and

buckling

conservative

estimate.

and nonuniform

of compressively

recommended

the theoretical

for

design

although

on spheres.

correlation

Experimental

loading

manner

of doubly

subjected

to obtain

shape

shell

the

loading. its

boundary

to obtain

the

the

of critical

factor

to verify

curved

(including

ellipsoids,

by multiplying

are

of a doubly

Included

oscillations,

for

capacity

recommends

spheres,

given

load-carrying

bulk-

results.

paragraph

loaded

common

their

strength

of experimental

or an internal

as

or buckling.

shell

on the buckling

vehicles

membrane

of the applied

of the

in space

compressive

proportions

nature

used

vehicles

of its materials,

the

85

develop

loads,

strength

geometric

frequently

or entry

instability,

curvature, elastic

are

shells

applied

by structural The

tanks

curved

1970

SHELLS.

shells

of fuel

to externally

limited

large

CURVED

15,

verification shells shell

is considered

like

loads

is usually

by a correlation

However,

factors,

testing

is also

of revolution stiffness.

the

by imperfections,

to be accounted

cutouts,

effect for

insuf-

is recommended

recommended

having The

when

of small

for joints, oscil-

by the correlation

Section C3.0 December

15, 1970

Page 86 For doubly curved shells, considerable capability for theoretical analysis is available although experimental

investigations of the stabilityof

doubly curved shells lag far behind analytical capabilities;the shallow spherical cap under external pressure

is the only problem which has been investigated

extensively. The growing use of digitalcomputers has greatly improved example,

the available analyses which can be performed.

a comprehensive

computer

stabilityanalysis of segmented, gram

for analysis of shell structures

program,

BOSOR

For

3, [I] performs

a

ring-stiffened shells of revolution. The pro-

is quite general with respect to types of loading, geometry,

conditions, and wall stiffness variation. All the programs

boundary

for doubly curved

shells, including both finite-differenceand finite-element, treat only those cases in which the shell does not become

plastic before buckling.

Although the capability for stabilityanalysis has increased, parametric optimization studies for problems be because of the relative newness computer

programs

for comparisons

of interest are lacking. This may well

of most computer

programs.

To date, most

have been used for spot checks of approximate

with experimental

solutions and

data.

The designer is advised to be alert to new developments

in shell-

stabilityanalysis. 3.3.1

ditions

ISOTROPIC

DOUBLY

CURVED

Unstiffened

isotropic

doubly

of loadings

spherical,

3.3-1)

and

buckling

has

in References

been

toroidal

2 and

paragraph.

Uniform cap

extensively.

3 for

shells

subjected

to various

Solutions

are

con-

limited

to

shells.

of a spherical treated

curved

in this

Spherical Caps Under The

sented

considered

ellipsoidal,

3.3. I. 1

(Fig.

are

SHELLS.

axisymmetric

External Pressure. under

uniform

The

theoretical snap-through

external results

pressure are

of shallow

prespherical

Section

C3.0

December Page f---

shells

with

edges

that

15,

1970

87

are

restrained

P

against rotate

or are

metric

f-

translation

5 for

are

the

The

results

are

presented

classical

for

3.3-1.

GEOMETRY

SPHERICAL CAP EXTERNAL

ratio

the

spherical

Pcr

UNDER UNIFORM PRESSURE

Pc_

-

asym-

references

of the buckling

spherical

cap

pressure

Pc/

shell

parameter

OF

for

to

conditions.

in these

buckling

of a geometry

free

in References

boundary

as the

a complete

FIGURE

given

same

Pcr

either

Results

reported

pressure the

clamped.

buckling

4 and

but are

and for

as a function X :

(1)

f(_)

with

Pci

= [3(1

k = [12(1

where

_

function

is half f(k)

values

are

discrepancy

lower

from

the

and

actual

the

than

ideal

test

spherical

(3)

spherical

conditions data

apply

predicted

experiment [3, [8,

6, 9].

cap

(Fig.

imposed to spherical

Most

The

shell.

shells, pressures.

be attributed 7] and

3.3-1).

on the

buckling

can

shape

conditions

(2)

2 sin -_ 2

boundary

and

edge

,

of the

theoretically

theory

assumed

angle

available the

E

(R/t)t/2

on the

of the

the

_2)]1/2

included

between

deviations

2

-p2)]I14

depends Most

-

and

the

The

largely

to initial

to differences

between

of the

available

data

are

Section C3. 0 April 15, 1973 P_ge summarized 6 and

11.

in Reference A lower

i0;

bound

some

other

to the data for

test

results

clamped

are

shells

given

88

in References

is given

by

Per

-0.14

(k > 2)

+._2

.

(4)

Pc/

This

curve

shallow-shell to shallow

is plotted

in Fi._are

analysis,

Figure

3.3-2. 3.3-2

Whereas may

the

k

be applied

parameter

to deep

is used

shells

as well

in as

shells.

1.0

i

i

L

t

,.

0._

iI

i

'

I

I

!

I

,

t

!

!

,

t

'

i

a

Per

-_

3._

- o,,+--_

_

0.4

p,

_

I

i

''

0.2

t,

0

]FIGURE

3.3-2.

will

buckle

are

free

Spherical

Caps

Under

Snherical

caps

under

under to rotate

certain and

[

RECOMMENDED

PREF_SUR,T, 3.3.1.2

!

Conc:ntrated concentrated

in the

The

PUCKLI,_:G

CAPS Lo,_'] at the Apex. load

at the

theoretical

direction

I

!

_E$iGN

C_" SPXERICAL

conditions.

to expand

i

'normal

apex

results

(Fig. for

to the axis

3.3-3)

edges

that

of revolution,

Section

C3.0

January

1 5, 1 972

Page

89

f and

P

for

clamped

edges,

are

given

in

Cf

Reference

12 for

through

and

axisymmetric

in References

asymmetric for

centrated

loads

For

// FIGURE

3.3-3.

edges,

shell

GEOMETRY

SPHERICAL

CAP

CONCENTRATED

OF

UNDER

LOAD

AT

results

agree this

when

loads

15,

and

have

that

collapse

been

16 for

P

and

shells

1

E t3

For will

24

spherical

not

8 and

9,

axisymmetric

carry

an

increasing

occur

if

occurs.

For

greater

than

this

if

with

is

of

will

in-

until

plasticity

shells

)_

range

load

from

3.8,

asymmetric

buckling

coleffects

values

of

apparently

not

the

geometry

should

and

edges

first. and

relationship

experi-

but

occur

synonymous,

parameter

unrestrained

theorctical snap-through

A lower-bound

between

for

the

dnta

is given

by

of

with

clamped

)_ is

less

than

For

larger

In

the

References

(5)

caps

will

dis-

only

x2 (4 =-__. _ is)

snap-through load.

resulting

In

increasing

axisymmetric

are

with

r

cr

lapse

occur

deformation

for

measured.

parameter and

buckling

agreement

indicates

buckling

collapse-load 13,

in good

theory

case,

collapse

are

with

not

unrestrained

THE

APEX mental

crease

with

3.8.

geometry,

in

19.

will

about

con-

described

shells

buckling than

approximate

are

15 to

14 for

Experimental

which

loading

References

less

13 and

buckling.

results

snap-

edges,

about

8.

occur,

values

of

theory For

indicates

values

with

the

shell

)_ ,

asymmetrical

of

that )_ between

continuing buckling

to

Section C3.0 January 1 5, 1 972 Page 90 will occur first, but the shell will continue to carry load. Although imperfections influence the initiationof symmetric surements

have been made

of the load at which symmetric

deformations first occur. of clamped mated

or asymmetric

Experimental

or asymmetric

results indicate that the collapse loads

spherical caps loaded over a small area are conservatively esti-

by the loads calculated in Reference

When

buckling, few mea-

the area of loading becomes

13 and shown in Figure 3.3-4.

large, large buckling may

occur at a lower

level.

18

\ a_ v

10

A

4

6

8

FIGURE

3.3-4.

10

12

14

THEORETICAL

BUCKLING

CLAMPED SPHERICAL CONCENTRATED 3.3.1.3

Spherical Caps Under Load at the Apex. Clamped

nal The

pressure experimental

and

spherical

concentrated and

theoretical

load

External

subjected at the

data

18

LOADS

20

25

FOR

CAP UNDER LOAD

Uniform

caps

16

to combinations

apex

given

Pressure

are

there

discussed are

and Concentrated

of uniform

exter-

in Reference

20.

insufficient,

however,

r

Section

C3.0

December Page

f to yield conclusive

P P

where Per

P

+-2--=

cr

external

Complete

as shown

4 > A/B

> 1.5

of Reference

3.3-5,

spheroids

A/B-

1.5,

AXIS

3.3. I. I.

agreement

Uniform

subjected

External

are shown

in Figures

22 for prolate

a lower

to the data. shell

(A/B

= 3)

J B

spheroid

FIGUHI':

spheroid

B>A

A>B

3.3-5.

GEOMETRY

OF

ELIAPSOIDAL

SHELI,S

3.3-6b.

results

be multiresults

closed

shell.

REVOLUTION AXIS OF

theo-

shells with

The

OF

b. Oblate

a. Prolate

and

should

REVOLUTION

/

pres-

Calculated

spherical

pressure

the

external

3.3-6a

with those for the complete

l

Pcr

with the theoretical

bound

spheroidal

21.

pressure,

Pressure.

to uniform

in Reference

the theoretical

23 for half ()fa prolate

uniform

in Paragraph

close agreement

0.75 to provide

in good

is the applied

3.3. i. 2, and

are treated

given in Reference

For

plied by tilefactor

given

p

Shells Under

are in reasonably

end plate are

recommended:

in Paragraph

shells of revolution

for prolate

21.

in Reference

pressure

in Figure

results

load,

load given

Ellipsoidal

Ellipsoidal

Experimental

is

(6)

concentrated

the critical concentrated

retical results

curve

I

is the applied

3.3. I. 4

interaction

1970

Per

critical uniform

sure,

A straight-line

results.

15,

91

given

by an

Section

C3.0

December Page

15,

1970

92

1.0

l I'----_

,,---

--_

BUCKLING

PRESSURES

OF LONG

CYLINDERS]

0.1

B

0.001

'_

__

1

il &

0.00021

5

10

16

20

25

A

T FIGURE 3.3-6a. THEORETICAL PRESSURES OF PROLATE

EXTERNAL SPHEROIDS

BUCKLING (# = 0.3)

30

Section

C3.0

December Page

15,

1970

93

1.0 ,,

0.1

\\\\ u g_

B 0.05 ._/_

- 20

t

0.01

1

2

3

4

5

6

BUCKLING

PRESSURES

A B

FIGURE OF

3.3-6b. PROLATE

THEORETICAL SPHEROIDS

EXTERNAL (p

=:0.3)

FOR

A/B

RATIO

OF

1:6

Section

C3.0

December Page The analysis thin,

oblate

of Reference

spheroidal

shells

are

21 indicates

similar

for a sphere

1970

94

that theoretical

to those

15,

results

for

of radius

B_

RA-

The similar

A

data

as well.

spheroid

may

is the

3.3.1.5

internal allows

Thus,

limit

Oblate

When

ratio

the

instability

to occur.

given

results

of Reference

without a certain

the ratio

buckling critical

h

that

with

value.

This

pressure, problem

(8)

Under

Uniform

spheroid

Internal

is less

than

in the shell,

values

_f2"/2

of the critical

internal

showri

in Figure

but the study

of the

imperfection

0.5

should

be good

< A/B

< 0.7.

Bulkheads

and major

provided is investigated

sensibetween

Internal

Pressure.

bulkheads axes

3.3-7.

agreement

Under

(ellipsoidal)

,

and hence

21 are

there

of minor

internal

oblate

large.

stresses

of Reference

spheroidal

of length

under

becomes

of an oblate

and Torispherical oblate

a thin,

are

'

Shells

available,

for shells

E!lipsoidal

as

The theoretical

are

for

= 0.14

compressive

21 indicates

and experiment

have

A/B

pressure

results

relationship

Spheroidal

by the analysis

Clamped may

(4)

that the experimental

buckling

by the

of equation

produces

3.3.1.6

external

P E

Complete Pressure.

(7)

24 show

(1 -_t 2) 2

No experimental

theory

the

pressure

pressures

tivity

of Reference

be approximated

_J3

which

"

(A/B)

(Fig. less

than

that the thickness in Reference

3.3-8) _-2/2 exceeds 25.

Section

C 3.0

December Page

AXIS

OF

r_EVOLUT'OI_

15,

1970

95

I 1

t

Rt

0 0

0.1

02

0.3

0.4

0.5

0.6

0.7

A B

FIGURE

3.3-7.

THEORETICAL

SPHEROIDS

UNDER

BUCKLING INTERNAL A

AXIS

PRESSURES

PI:_ESSURE nonlinear

determine

OF

OF (p

bending the

OBLATE

-- 0.3) theory

is used

prebuekling

to

stress

distri-

RFVOLUTION

A

L ////?///

B---"

i

////////

bution.

The

shown

in Figure

of buckling

thickness

is

3.3-S.

ELLIPSOIDAL INTERNAL

CLAMPED

BULKIIEAD PRESSURE

UNDER

theory

shown has

experimental sb(_uld

be

of stability

3.3-9;

variation

The FIGURE

regions

not

the

calculated

pressure

with

in Figure

3.3-10.

been

results, used

are

cautiously.

verified h_wev(;r,

by :tnd

Section

C3.0

December Page

15,

1970

96

20 x 10 "3

16

UNSTABLE

STABLE

12

\ \ 0 o

0,1

0.2

0.3

0.4

0.5

0.6

A B

FIGURE

3.3-9.

CLOSURES

REGION

Torispherical

end

Reference

25.

Calculations

in these

bulkheads

for

pressures

for

buckling.

The

of Reference should

results

Complete Pressure,

{Fig.

the

theoretical

results

made

shown

has

(Fig. for

with

in Figure

the

the

prebuckling

circular

toroidal

been

investigated

obtained

are

shown

shells

3.3-12.

The

predicted

Shells

shell and

also

and

to 0.7.

Under

Uniform

uniform

is described

in Figure

for

buckling after

experimental

equal

3.3-14.

in

distribution

conditions

buckling

under

investigated

stress

edge

T

(# = 0.3)

are

clamped

factor

Toroidal

ELLIPSOIDAL

3.3-11)

theoretically

by a correlation Circular

FOR

PRESSURE

by cylindrical

bulkheads

that

complete

3.3-13)

closures

restrained

are

26 indicate

The sure

ends

STABILITY

TO INTERNAL

are

torispherical

be multiplied

3.3.1.7

OF

SUBJECTED

results

pressures

External

external

in Reference

pres27;

Section

C3.0

December Page 97

15,

1970

20 x 106

/ 10

_-

/

0.40

/ /

/

/

/

/

J

0,7

/

O.5

/

0,4

/

/

0,3

0.2

0.1 5

6

7

8

9

10

11

12

13

t A

FIGURE

3.3-10.

ELLIPSOIDAL

THEORETICAL BULKHEADS

INTERNAL

RESULTS SUBJECTED

PRESSURE

FOR

CLAMPED

TO UNIFORM

(p =0.3)

14

16 x 10 .3

Section

C3.0

December Page

15,

1970

98

AXIS OF REVOLUTION

FIGURE

3.3-11.

The b/a equal

of 6.3

8 and

greater

multiplied been

experimental

and

to or

than

recommended

should

of _.

be verified Shallow

For

3.3-15)

be susceptible is given

will

in Reference with

theoretical

values.

less

For

This

than

correction

the

which buckling

of

of b/a

should

shells

6.3,

values

values

pressure

long cylindrical

of b/a

27 for

theory.

buckling

design

28 for

values

given

agreement

to yield

Bowed-Out

Toroidal

equatorial

undergo

to buckling.

in Reference

are

CLOSURE

be factor

has

correspond pressure

bytest.

A bowed-out (Fig,

the

in Reference

of b/a

3.3.1.8

good

6.3,

of 0.9

OF TORISPHERICAL

results

indicate

by a factor

to a value

GEOMETRY

29 and

toroidal

compressive An analysis yields

Segments

Under

segment circumferential

for

simply

the relationship

Axial

under

Loading.

axial stress

supported

tension and will

shallow

thus

segments

Section

C3.0

December Page

f 30x

99

10 -6

_9

/,/<

2O

10

7

5

1111_

4

! X i'll

3

o.

/

'

I

t

LEGEND:

/,liT1

2

/ /,7 /

0.7

0.5 0.4

(

/, ///' ! ,/ / , /

0.2

7o:19° _< :o._,6

I--]

_0

O

Rt _)0 = 20"80 '_

"

350

Rt ' -_c

= 0.346

= 0.242

C

70

Rt " 27.5 ° ._--

= 0.194

c

R_j.t .,1 <_o-35°,R

U

0.3

R t

0

0.167

¢

il_

_0

= 14"1°'-_c

= 0.146

0.1 1.5

2

3

4

5

7

10 x 10 .3

t

Rt

FIGURE

3.3-12.

CLOSURES

THEORETICAL

SUBJECTED

TO

RESULTS UNIFORM (/_ = 0.3)

FOR INTERNAL

TORISPHERICAL PRESSURE

15,

1970

Section

C3.0

December Page

15,

1970

100

P

b -

FIGURE

3.3-13.

=-

GEOMETRY OF A TOROIDAL UNIFORM EXTERNAL PRESSURE

SHELL

UNDER

0.1

0.01

0.002 0

0.1

0.2

0.3

0.4

0.5

e+b

FIGURE

3.3-14.

TOROIDAL

THEORETICAL SHELLS

UNDER

BUCKLING UNIFORM

COEFFICIENTS

EXTERNAL

FOR

PRESSURE

Section C3.0 December 15, 1970 Page 101 1000

r

m

0.25

100--

..,If

_f

O.5O

/ 10 N

1.0 (SPHERICAL

SEGMENTS)

._.

1

1 10

1

100

1000

'Tz FIGURE

3.3-t5.

BUCKLING

SEGMENTS

N£

2

1

:_3D

-

,)

correlation

coefficient

where

the

ancies

between

equation portion

theory

(9) with of the

N_

respect

curves

2

n2D -

and

[

_2

(t

+f12)

7

AXIAL

2

has

experiment. to

/3 are

is represented

4_f-3

OF BOWED-OUT

UNDER

vz

+ 12

been The

shown by the

TOROIDAL

TENSION

_4

inserted values

in Figure

2a

(9)

1 +/3 2

to account obtained 3.3-15.

for

discrep-

by minimizing The

straight-line

relationship

(_o)

Section

C3.0

December Page A similar

analytical

truncated

hemispheres

those

the curve

for

investigation in axial of Figure

equals

30 indicate

tension 3.3-15

The experimental Reference

described

in Reference

yields for

results

results

r/a

for

t970

102

30 for clamped

in close

agreement

with

= I .

the truncated

that the correlation

15,

hemisphere

coefficient

given

for the curve

for

in r/a

1 is

= o.35

The same

value

(11)

of the correlation

coefficient

may

be used

for other

toroidal

segments

values

of

r/a. Some axial

results

compression

segment

loaded

3.3.1.9

are

given

Toroidal

The term which

acts

static

pressure"

walls

and the ends

static

only

toroidal

pressure

segments

ri

for

lateral

P or

that

acts

for simply

to uniform

3.3-17)

are

pressure

and not on the ends;

pressure

given

"hydro-

on both the curved

supported

external

shell

lateral

in Reference

shallow or hydro33 as

1+-1

(1 +#2)

- #2

pressure,

r_ 2

_2 D

and

an external

2

cr

_D

Pressure.

Expressions subjected

3.3-16

External

of the shell

an external

of the shell.

(Figs.

P

walls

32.

designates

under

spherical

in Reference

Under

pressure"

curved

designates

31; the equatorial

is treated

Segments

"lateral

on the

equatorial

in Reference

by its own weight

Shallow

equatorial

for bowed-out

and

2

12

+-_

_/2Z 2

.

a

1+#5rfl2)

(12)

as

i

(13)

#2(1_

1/2r)+

1/2

Section

C3.0

December Page

15,

1970

103

1000

r=

1.0

a

f,,_j

0.75

0.25 IO0

0 CYLINDER

10

jj_ --

yl

I

_ 0,50 "0.75

1 1

10

100

lOOO

"Tz FIGURE

3.3-16.

BUCKLING

UNDER

for to

hydrostatic

pressure.

segments

of type

segqnents been The

The 3.3-18

of type

introduced results

ferential

UNIFORM

In equations

(a) (b)

of

straight-line

buckling

parameter

represented

of by

p cr r/

ri 2

the

the

r

2

a

8 ,_"

v2D

for

lower

correlation

pressure

with

shells

sigm

sign

theory

and

respect

type

,/

to has

experin_ent.

to the

circum-

and

3.3-17.

3.3-16 of

refers

refers

coefficient

in Figures

the

upper

(a)

of

l.'igure

3' Z

(lateral

3/Z

(hydrostatic

pressure)

(14a)

r r

2--

curve

tile

between

shown

the

relationships

4 x]'-3 -

_I)

[_ are

(13),

whereas The

SEGMENTS

PlIESSUllE

and

discrep'meies

the

portions

Per

3.3-18.

for

TOHOIDAI,

(12)

3.3-18,

of Figure

of minimizing

arc

Figure

to account

wavelenigth

OF

LATERAL

--

a

a

pressure)

(14b)

Section

C3.0

December Page

1970

15,

104

_-1.0 1000

100

0 CYLINDER

J

lo

.

_

0_0

oJs 1,0

"

1

10

FIGURE

100

3.3-17.

BUCKLING

OF TOROIDAL

UNIFORM

EXTERNAL

HYDROSTATIC

No experimental for which

1000

a correlation

data

factor

are

available

SEGMENTS

UNDER

PRESSURE

except

for

the

cylindrical

shell,

of

2/ -- 0.56

was for

recommended shells

type

(b)

r/a

near

buckling

with with

in Reference r/a

near

values

unity, pressure

(15)

of

the

shell

should

28.

zero r/a

but should

near can

The

unity.

same

be used For

be conservatively

be verified

correlation with

shells

factor caution

of type

treated

can

for (a)

shells

with

as a sphere,

be used of

values or

of

the

by test.

--.._j

Section

C3.0 15, 1970

December Page

r

105

la}

,

\±

Ib}

FIGURE

3.3-18.

GEOMETRY NEAR

3.3.2

ORTHOTROPIC

The of

shells.

made

In

for

shells

in which by

representative

widths The

of equilibrium

those

behavior

a single for

the

curved single-

section,

shell. spacing

sheet

shells"

whose

individual

or the

assumed

stiffener

of the or

are of the

a fictitious

include

by

revolution

the

SEGMENTS

SHELLS.

it denotes In this

directions

approximated

described

sense,

of

CURVED

doubly

materials.

shells

properties

DOUBLY

strictest

circumferential

stiffened be

its

TOROIDAL

EQUATORS

"orthotropic

of orthotropic

orthotropy and

term

OF

covers

directions

The

term

is

small

variety

multiple-layered

to coincide

shells

of the with

also

axes

the

for

bending elements

types the

and

of

meridional

denotes

enough

orthotropic stiffening

a wide

of

shell

to

extensional

averaged

out

over

areas. of the

various

theory,

the

buckled

structure,

types

governing

of orthotropic equations

and

relationships

shells

of which

are

between

may

be

equations force

and

Section

C3.0

December Page moment

resultants

relating

the

curvatures and

inplane for

shells

circumferential

and extensional forces

and

and bending bending

of revolution directions

moments

with

can

strains.

axes

be written

The

to the

matrix

inplane

of orthotropy following

1970

equation

strains

in the

in the

15,

106

and

meridional

form:

C1_

Ci2

0

Ci4

Cls

0

Ct2

C22

0

C24

C2s

0

0

0

Ca3

O-

0

0

Ct4

C24

0

C44

C45

0

C15

C25

0

C45

C55

0

0

0

0

0

0

C66 J

(16) Zero entries in the preceding matrix generally refer to coupling terms for layers whose

individual principal axes of stiffnesses are not aligned

in meridional and circumferential directions. The values-of the various elastic constants used in determining buckling loads of orthotropic shells are different for different types of construction. Some in References

widely used expressions are given

I and 34.

The theory for single-layered shells of orthotropic material is similar to that for ihotropic shells since the coupling terms and

C25 may

be set equal to zero.

C14 , Ct5 , C24 ,

For stiffeneddoubly curved shells or for

shells having multiple orthotropic layers, this is not generally possible, and it is shown in References lead to serious errors. a significant dome face.

difference

configurations The

difference

35 and 36 that the neglect of coupling terms For example,

in theoretical having vanishes

stiffeners when

can

the inclusion of coupling terms yields results

for

on the inner coupling

stiffened surface

is neglected.

shallow

spherical-

or on the outer

sur-

Section

C3.0

December Page Very

little theoretical

tropic and stiffened doubly pressurized

shallow

in Reference

shells.

domes

caps.

38.

in Reference

design

This formula

Buckling

39.

The

107

data are available for ortho-

general

with meridional

37, and a semiempirical

in Reference

domes

curved

spherical

for stiffened spherical given

or experimental

15, 1970

formula

closely

instability loads of stiffeners are determined is given in I_eference

approximates

the test data

loads are given for grid-stiffened

References

37 and 39 do not include

38

spherical

the effect of

stiffener eccentricity. Stiffener-eccentricity grid-stiffened

spherical

toroidal shells under Reference coupling

40.

The

effects are investigated

domes.

Eccentrically

axial load and uniform development

as well as nonlinear

prebuekling

of revolution

is discussed

this program

is given in Subsection

this program

are available from

Numerical with

results obtained

selected

determine

in References

experimental the

buckling

The

of the

computer

stringer

stiffening.

with

ring

2.

Shells

with

skew

3.

Fiber-reinforced

4.

Layered

5.

Corrugated

6.

Shells with one corrugated

in

that includes

description

and a computer of Georgia,

[34] were

orthotropic

Shells

shells

and

cards

following

1.

program

A further

University

this program

results. load

(The

equatorial

effects for orthotropie

i and 34.

3.4.

35 for

are investigated

computer

bending

COSMIC,

from

stiffened shallow

pressure

of a buckling

in Reference

shells of

listing for

Athens,

Ga. )

in good agreement

program

can

be

used

to

shells:

stiffeners.

(layered)

shells.

(isotropic or orthotropic).

ring-stiffened

shells.

and one smooth

skin (with rings).

Section

C3.0

January 1 5, 1972 Page 108 Boundary free, ary

fixed,

or elastically

or as discrete This

mentally shells

conditions

computer

program

correlation

also

The

should

buckling

also

or both ends

are

permitted

in conjunction buckling

of the program

recommendations

of local

rings

to obtain

in Subsection

the recommendations

possibility

factors

discussed

at one

can be used

The limitations

The design domes;

Edge

in the shell.

of revolution.

be closed

restrained.

rings

determined

and 34 and are

may

be

on the bound-

with

loads

are

or may

experi-

for

orthotropic

given

in References

limited

to spherical

3.4. that follow

are

be verified

by test,

of the shell

between

where

stiffening

feasible.

elements

should

be checked. The investigation pressure

of a grid-stiffened

of Reference

This

analysis

assumes

many

buckle

wavelengths.

effect

on the buckling

results

given

geometrically

dome

that the spherical In thiscase,

load.

analysis

24 tend

orthotropic

shell,

the

the theoretical uniform

is "deep"

the boundary effects

to support

of Reference

under

dome

Eccentricity

in Reference If the

expressed

spherical

39 gives

the

are

hydrostatic

external

pressure.

and that it contains

conditions

have

neglected.

assumptions

39 is extended

buckling

to the

buckling

little

Experimental of the

analysis.

materially

pressure

can

or be

as

Pr_ I/2

= 4_

C_A CaA 1 +2 CL_ +Cs_ + C__ 1/2 1 + C-22 + 2 C--2z )

C44 _1

(17)

where

¢_ = C2_r2 C44

( 1-

C222 C11 C22 )

,

(18a)

1

Section

C._. 0

December Page

15,

1970

109

2 Ca._ _2

=

C22

I-

CIt

The defined materials. since

Cll

34 for

Equation the

C22

constants

in Reference

coupling

not ,

neglected.

Only

orthotropic

spherical

domes

subjected

the

of more

extensive

test

absence

spherical tropic

cap

reduction

spherical

shell.

y

Refer to be

to Figure used

C24 ,

experimental

factor

for

in obtaining

( 1 8b)

and effect

and

C25

in equation

exist

for

geometrically

results,

it is

factor

is

also

given

are

eccentricity (16)

have or

381.

that

the

be used

for

been

materially

[24,

recommended (4)

C66 orthotropic

of stiffener

pressure

in equation

correlation

geometrically

the

to hydrostatic

and

In isotropic

the

ortho-

by

3.2 k2

+

3.3-2

data

shown

The

= 0.14

materially

include

, Cls

"

, C22 , C33 , C44 , C45 , C55 ,

various

does Cf4

limited

, Ct2

the

(17) terms

C12 C_._ "" C11 C22

-2

(19)

the

plot

of this

equation.

;_ is recommended

The

effective

shell

thickness

as

4 (20) x/

3.3.3

C22

ISOTROPIC

The formed

by isotropic

The

core

"isotropic

two

facings

separates

a common

SANI)WICIt

term

bonding

thin

about

Ctj

thin

neutral

facings axis.

sandwich"

isotropic

provide the

DOUBLY

nearly and

SHELI.S.

desi_,mates

facings all

CURVED

the

transmits

to a thick

a layered

bending shear

core. rigidity so that

construction Generally,

of the the

the

conslruction.

facings

bend

Section

C3.0

December Page Sandwich instability

failure:

core

and

form

of dimpling

(1)

facings

3.3.3.1

acting

General

stiffness

isotropic

instability

together,

and

sandwich

local

for

two possible

where

the

instability

of the

faces

shell

(Fig.

modes

fails

failure

material

shell.

thickness

to transverse

to be infinite, For

the

unequal

and modulus

of

with

taking

the

so that

its

3.1-7).

shear

sandwich

thickness

shell

facings,

of elasticity

are

_f_h

then

can

be treated

the

equivalent

given

by

,

E_t_h + E 2 t2

for

(2)

is resistant

be assumed

=

and

failure

or wrinkling

core

isotropic

=

be checked

1970

Failure.

can

as an equivalent

should

general

of the faces

If the shear

construction

15,

110

(21a)

E 1 t_

Et t_ + E_ t_

equal-thickness

(21b)

facings

with the

= _,f3"h

same

modulus

of elasticity,

by

,

(22a)

_ 2Etf E = ----r----

(22b)

4- h These recommended Reference

equivalent

practices 34 to analyze

properties

in Paragraph isotropic

can be used 3.3.1

sandwich

and

in conjunction

with

doubly

the

curved

computer shells.

with

the

program

of

Section C3.0 December 15, f

Page Only is available. a core

mental

In Reference

layer

material

one theoretical

data,

3.3.3.2

Local

intracell (5 > 0.03) justified cell

will as data

buckling

and

there

of a sandwich two equal

are

recommendations

of failure sandwich

buckling,

and

includes

face

insufficient can

shear

sphere

comprised

layers

of high-modulus

theoretical

be given

flexibility

for

this

and

of

experi-

case.

Failure.

Modes honeycomb-core

material

Because

no design

which

41 the bucMing

of low-modulus

is discussed.

investigation

1970

111/112

and usually become

other shells,

face

than failure

wrinkling.

prevent

core

available.

face-wrinkling

overall

buckling

may The

use

crushing. Procedures

loads

are

given

occur

are

because

of relatively Lighter for

cores

possible. of core

crushing,

heavy

cores

may

prove

the determination

in Reference

For

42.

to be

of intra-

Section C3.0 December 15, Page 113 3.4

COMPUTER The

Table

3.4-1,

classes and

names

which

of various

indicates

of problems

general

PROGRAMS

are

IN SHELL

digital

their

programs

a shell

cylindrical

3.4-1.

Computer Stability

Symmetric Nonsymmetric

of Shells

are

stability

shells,

listed

in

analysis.

shells

Three

of revolution,

Programs

for

Shell

Analysis

System Displacements

CORCY L

Cylindrical Shells

Nonsymmetric

a

STAGS

IV a

BOSOR

of

System

INTACT

DBS TA B a SCAR MARK

Shells

for

ANA LYSIS.

shells.

Table

Types

computer

scope

specified:

STABILITY

1970

BOSO It 3

R evolution

SABOR3-F General

BERK3

Shell

NASTRAN I/EXBA

a.

Programs

available

Often closed

form.

Those

fied

by separation

are

assumed

linear

stability

shells

of variables.

of shells

variation

at Marshall

analysis

concerning

to be periodic,

analysis

meridional

the

for use

and

Flight

of cylindrical of revolution

Center.

shells can

can

in the circumferential

the method

of superposition

subjected by series

in

be simplidirection

is used

to nonsymmetric expansion,

be solved

frequently

Variations

of revolution

is determined

Space

T

for

loads.

the method

the The

of finite

SectionC3.0 December 15, 1970 Page 114 differences,

numerical

analysis

of general

required,

since

and

the

shells,

time

or terms

References

3.4-2,

1 through

numerical

per

case series

very

and

3.4-4

they

are

and briefly

elements.

For

numerical The

rapidly

expansion

3.4-3,

used,

of finite

be separated.

increase

12 in which

analysis

method

a two-dimensional

cannot

in a double

Tables

or the

however,

the variables

computer

points

integration,

core

the

analysis

storage

is

required

as the number

of mesh

increases. list

the

programs

documented,

describe

by name,

specify

the major

cite

the method

features

of

of the

analysis. In general, BOSOR, dence

BOSOR3, index"

than

CORCYL,

and SABOR3-F the other

programs

but because

of the

obtained.

In STAGS,

BERK3,

too small within

to ascertain

a percent

expensive

a two-dimensional

lems)

is given

DBSTAB,

1.

Structural NASA/MSFC

and

Analysis

that

input

and

internal

are

more

input

1 for

data

has

those

with

programs

is often to

very

programs

must

been

accurate

generally

than

(along the

in the

storage are

computer

explanation

document

core

are

to use

"con/i-

convergence

checks

harder

since

a higher

obtained

of the

INTACT,

of defects

that

stresses

Most

output

to have

AND REXBAT, the

time.

IV,

is not because

convergence

analysis

MARK

of proving

NASTRAN,

analysis

in an MSFC

BOSOR,

ease

In addition,

numerical A simplified

This

relative

of computer

SCAR,

be expected

programs.

numerical

one-dimensional

might

conclusively

or so.

in terms

DBSTAB,

requiring requiring

a

be specified. example

prob-

CORCYL,

NASTRAN.

Computer

(to be published).

Utilization

Manual,

Astronautics

Laboratory,

Section C3.0 December Page

Table 3.4-2.

Computer

Programs

of Cylindrical

for

Stability

15,

1970

115

Analysis

Shells

Method Reference

Program Name

No.

of a

Comments

Analysis Linear

CORCYL

small-deflection

R ing-stiffened under axial are

distributed

along

Eccentricity to corrugation sidered.

cylinder Rings

the

cylinder.

of rings with respect centerline is con-

Small-deflection

DBS TA B

theory.

corrugated compression.

theory.

Ortho-

gonally stiffened cylindrical shell under axial compression and lateral pressure. or heavily

Restricted stiffened

to moderately cylinders.

Rings and stringers are considered eccentric with respect to the skin's middle surface. Local buckling of the

skin

between

adjacent

before general instability and the resulting reduction stiffness is determined. Membrane

SCAR

prebuckled

stringers is allowed, in skin

theory

and

simply supported edges. Various types of wall construction permitted, as well as combined pressure and axial and MARK

IV

4

compression. longitudinal

Ring stringers

stiffeners permitted.

SCAR-type analysis for optimization of integrally stiffened cylinders with

respect

to weight.

Section C3.0 December 15, 1970 Page 116 Table 3.4-2.

(Concluded)

Method

Program Name

of

Reference No.

Analysis

INTAC T

2

a

Comments Buckling

of cylinders

under

bend-

ing, axial compression, and pressure. Interaction curves calculated. Otherwise, same as SCAR. S TA GS

Nonlinear tions

and

analysis. elastic-plastic

Large

deflec-

behavior

permitted. Discrete rings and stringers included. Maximum number of unknowns is 4300.

a.

Method

of Analysis:

1

= Closed

form expansion

2

= Series

3 4

= Numerical integration = Finite difference

5

= Finite

element

Section C3.0 December Page Table 3.4-3.

15, 1970

117

Computer Programs for StabilityAnalysis of Shells of Revolution

Method Reference

Program Name

of

BOSOR

Comments

Analysis

No.

7

4

Nonlinear General meridian,

prebuckling

conditions, and metric loading. 8

BOSO R 3

4

effects.

with respect shell wall

to geometry design, edge

loading.

Rings elastic

can be treated structures.

linear

prebuckling

Axisym-

as discrete Option of noneffects

or linear

bending analysis.

theory for prebuckling Scgqnented shells can

analyzed pendent

with each segment of other segments.

wise,

same

of

be

indeOther-

as I_OSOR.

a

SABOR3-F

a.

Also, report,

9

W.

A.

5

Loden:

Lockheed

Calculation of vibration frequencies of stacked and branched shells.

SABOR3-F/EIGSYS

Missiles

and

Space

Instructions, Company,

August

unpublished 1967.

Section

C3.0

December Page

Table

3.4-4.

Computer Programs for Stability of General Shells

15,

i970

118

Analysis

Method Program Name BERK3

Beference No. 10

of Comments

Analysis Flat

triangular

elements

sional and bending for the calculation

with exten-

stiffness are used of stresses and

vibration frequencies of general shells or shells of revolution with cutouts. Up to 6000 be handled. Discrete

unknowns stiffeners

can

permitted. NASTRAN

ll

General-purpose tic structural stricted

program analysis.

to shells.

Contains

of elements including shear panels, plates, REXBAT

12

General-purpose

library

rods, beams, and shells.

program

structural analysis static stresses and quencies. be handled.

for elasNot re-

for

linear

with respect to vibration fre-

Up to 6000

unknowns

can

Section

C3.0

December Page

f

15,

1970

119

3.5 REFERENCES REFERENCES

Baker,

1.

E.

FOR

H.,

et

al.

CYLINDERS

:

Shell

(SUBSECTION

Analysis

Manual.

3.1)

NASA

CR-912,

April

1968.

o

Almroth, the at

B. ; Holmes,

Buckling SESA

of Cylinders

Spring

Koiter,

.

W.

Buckling

Meeting,

T.:

B.

Axially

O. :

D. : An

under

Axial

Compression.

Lake

City,

Utah,

of Axisymmetric

Shells Missiles

Influence

Compressed

January

Brush,

Effect

Lockheed

Almroth,

Study

Paper May

6-8,

Axial

Compression.

and

Space

Company,

Conditions

1964.

Shells.

AIAA

on

August

J.,

the

Rpt.

on the

4,

6-901963.

Stability

vol.

of

presented

Imperfections

under

of Edge

Cylindrical

Experimental

of no.

1,

1966.

Buckling

.

The

; and

Salt

of Cylindrical

6-90-63-86,

.

A.

of Thin-Walled

Circular

NASA

Cylinders.

SP-8007,

August

1968.

_°

Weingarten,

V.

Thin-Walled

Cylindrical

AIAA

.

J.,

Seide,

vol.

I.;Morgan,

3,

and

no.

3,

P. ; Weingarten,

of Desii,m

for

.

Jones, Orthotropic December

M.:

(now Buckling

Layers 1968,

I. ; and

pp.

and pp.

TRW

Systems),

of Circular Eccentric

2301-2305.

Axial

Stability

of

Compression.

500-505.

Morgan,

Stability

Elastic

Under

E.

of Thin

(AFBMD/TR-61-7),

Inc. R.

P.:

Shells

1965,

Elastic

STL/TR-60-0000-19425

J.;andSeide,

Conical

March

V.

Criteria

Laboratory,

E.

J.:

The

Shell Space

Dec. Cylindrical

Stiffeners.

31,

Development

Structures. Technology

1960. Shells

AIAA

with J.,

vol.

Multiple 6,

no.

12,

Section

C3.0

December Page 120

15,

1970

Instability

of Ring-

REFERENCES (Continued)

o

Dickson,

J.

Stiffened

Corrugated

3089, i0.

January

Meyer,

Aircraft

Singer,

J. ; Barueh,

Solids 12.

J.

Shells

vol.

P. ; and

Harari,

3,

no.

Dow,

1967,

pp.

by Closely

Strength

Stiffening.

pressive

General

3639,

for

Milligan, lity AIAA

R. ; Gerard,

vol.

J.

Space

on Circular

Z Section

Stringers.

of Circular

TN D-1251,

R.

1963, M.:

pp.

Structural

Cylinders

Experiments

with

on Axial

Ring-Stiffened

Com-

Cylinders.

1614-1618.

Experimental

of Eccentrically

W.:

1962.

R. :

of Monolithic July

J.

Stiffened

and

Theoretical

Cylinders.

NASA

TN D-

1966.

of Orthotropically J.,

7,

Jones,

Buckling

October

NASA

Instability

F. ;and

Tests

Deaton,

G. ; and Winter,

1, no.

Int.

1959.

Compressive

H. ; Gerard,

of Eccentric-

445-470.

B. : Compression

Longitudinally

Becker,

Sub-

DAC-60698,

July

4,

M.

Report

Compression.

and

M.

TN D-

Cylinders

Axial

Behavior

Card,

NASA

Corrugated

under

R. O. ;and

vol.

Compression.

O. : On the Stability

P. ;Whitley,

J.,

Axial

1967.

J.

Results

16.

July

Peterson,

AIAA

General

and Bending.

M. ; and

2-12-59L,

Longitudinal

15.

under

Load

Company,

Stiffened

Memo

H. : The

of Ring-Stiffened

Cylindrical

Peterson,

NASA

14.

Axial

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Cylinders

13.

Cylinders

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Douglas

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jected

lt.

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C. : General Axial

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Section

C3.0

December Page

f-

1970

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REFERENCES (Continued)

17.

Singer,

19.

20.

22.

Influence

of Axially

liminary

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Compressed

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1962,

Card,

F. ; and

Block,

D.

under

Pure

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J.

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no.

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Orthotropic

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on the Theory

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May

Instability Hydrostatic

1957,

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C 3.0

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1970

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26.

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General

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C3.0

December Page

1970

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33.

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N.

F.

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; Libove,

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37.

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E.:

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the

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NACA

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1954.

Crawford,

R.

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C. ; and

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J.

Aeron.

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Zahn,

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E.

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2289,

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C3.0

December Page

1970

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Armed 43.

45.

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v

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C3.0

De('.

15,

Pa_e

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REFERENCES(Continued) 51.

Weingarten, _4th

52.

an

V. Elastic

Shanley,

F.

Block,

R. :

L. :

Cylinders 54.

55.

Stein,

,

Battier/',

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CONICAL

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With

Various

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Orthotropic

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Stress

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Stability

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(SUBSECTION

Sobel,

L.

II. :

3.2)

Bucklin_

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II,

of

Complex

o1 Shells and

III.

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NASA

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Circular

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Shells Lockheed

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3,

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s Manual

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J.,

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Constructions,

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Axisymmetric

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AIAA

Wall

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SHELLS

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Structures.

Instability

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1968.

Bushnell,

SAMSO

on TN

Crate,

April

Shells

1952.

Stiffeners

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B.

4,

Cylinder

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York,

NACA

Shells.

Almroth,

no.

NASA

Cylindrical

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°

Inc.,

J.

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A Simplified

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°

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Torsion

Compression.

M. ; Sanders,

Revolution

.

Co.,

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Thin

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Stiffened

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Weight-Strength

Book

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McGraw-tti]l 53.

I. :

March

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no.

J. ; and

4,

Conical 1965.

Scide,

Shells pp.

Under

December

P. : Under

500-505.

195G,

Elastic Axial

Axial pp.

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Com625-628.

of

Section

C3.0

December Page

1970

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126

REFERENCES (Continued)

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Hausrath,

A.

Levels

for

Collected

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Structures.

Rpt.

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Under

Strength

Axial

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E.

Final

Com-

NASA

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Delft,

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on

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Uniform on the

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Netherlands,

1960,

363-388.

Axial

V.

I. ; and Seide,

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Singer,

J. ; and

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Seide, Under

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Congress Berkeley,

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Israel

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1965,

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Pressure

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Shells AIAA

Shells

Conference,

11.

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Cones

Morgan,

TR-60-0000-19425

North-Holland

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J.

Criteria

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Cylindrical

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1959,

Development

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.

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December Page

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1970

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12.

Singer,

J. : Donnell-Type

tropic pp. 13.

Conical

Baruch,

Israel

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17.

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Fersht-Scher,

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Israel

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Technion-

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Singer,

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M. ; Singer,

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1965,

TAE

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Feb.

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Gerard,

J. : On the

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NACA

Singer,

Baruch,

Pressure.

Mayers,

Sci.,

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Pres-

Section

C3.0

December Page

15,

1970

128

REFERENCES (Continued)

20.

Plantema,

F.

Sandwich

Beams,

York, 21.

22.

Yusuff,

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J.

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FOR

CURVED

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and Vibration

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Math.

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3.3)

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Lockheed

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H. : On the Stability

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1960,

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ProShells,

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1964,

pp.

447-

Section

C3.0

December Page

15, 1970

129

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e

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f_

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o

So

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December Page

1970

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14.

Bushnell,

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Inward

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Shells

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Under

Con-

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Leo,

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1962,

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an

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The

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CR-265,

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U. S.

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Under

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T.

C. ; and

Concentrated Mech., 21.

vol.

Danielson, Spheroidal pp.

936-944.

Evan-Iwanowski,

Loads 33,

Acting

no.

Shells

on Shallow

3, September

D. A. : Buckling Under

R.

and

Pressure.

M. : Interaction Spherical

1966, Initial

pp.

of Critical

Shells.

J.,

Appl.

612-616.

Postbuckling

AIAA

J.

vol.

Behavior 7,

no.

of

5, May

1969,

Section

C3.0

December Page

15,

1970

131

REFERENCES(Continued)

22.

23.

Hyman,

B.

Shells

Under

1967,

pp.

Nickell,

Shells

Subjected

June

30,

of Prolate

AIAA

J.,

vol.

Spheroidal 5,

no.

8, August

vol.

IV,

Duckling

Tests

to External Lockheed

of Magnesium

Hydrostatic

Missiles

Monocoque

Pressure.

and Space

Tech.

Company,

1961. R.

R. ; and

Bellinfante,

of Common

SM-47742,

26.

Experimental

3-42-61-2,

Douglas

Thurston, Shell

J. : Buckling

Pressure.

It. :

Rpt.

Evaluation

25.

J.

Hydrostatic

E.

Meyer,

Healey,

1469-1477.

Ellipsoidal

24.

I. ; and

G.

End

Domes

Aircraft

A. ; and

Closures

Adachi,

J. ; and

Internal

Pressure.

It. Having

Exp.

Inc.

A.E.,

by Internal

Fabrication

and

Waffle-like

Company,

Holston,

Benieek,

J. :

Buckling

Pressure.

Mech.,

vol.

W.:

Stability

Stiffening.

NASA

of Cylindrical

CR-540,

1966.

of Torispherical 4,

I_pt.

, 1964.

Jr.:

M. : Duckling

Experimental

no.

Shells

8, August

1964,

Under pp.

217-222. 27.

Sobel,

I,.

Uniform pp. 28.

External

Anon:

Buckling Desigm

Itutchinson, Segments. pp.

Flugge, Pressure.

AIAA

J.,

of Toroidal vol.

5,

Shells

no.

3,

Under

March

1.9(;7,

425-431.

Vehicle 29.

It. ; and

97-115.

J. Int.

of Thin-Walled Criteria W. : Initial J.

Solidg

Circular

(Structures),

Cylinders. NASA SP-8007,

Post-Buckling Structures,

NASA

Behavior vol.

3, no.

i,

Space

revised

1968.

of Toroidal

Shell

January

1967,

Section

C3.0

December Page 132

15,

1970

REFERENCES (Continued)

30.

31.

Yao,

J.

C. : Buckling

AIAA

J.,

Babcock,

C.

on the

Blum,

R.

1968.

Stein,

M. ; and

34.

J.,

Almroth,

35.

Crawford,

R. AIAA

F. :

Effect

Shells. TN D-1510,

of Initial

Papers

1962,

135-142.

: Buckling

pp.

by its Own Weight.

Buckling

pp.

D. ; and Sobel,

L.

Constructions.

on Insta-

of an Equatorial NASA

of Segments

1965,

Imperfections

Collected

Loaded

A.:

Tension.

2316-2320.

Jr.

Wall

Effects

Paper

TN D-

of Toroidal

Shells.

1704-1709. H. : Buckling Vols.

8, August

of Asymmetric

No.

Eccentrically

versteifter

The

Axial

I-III,

of Shells NASA

CR-

1968.

Deformed

Ebner,

pp.

Under

H. G.,

J.

Various

D. : Symmetric

no.

E. :

9, September

Bushnell,

5, 37.

Shell

3, no.

to CR-1051,

Shells. 36.

McComb,

McElman,

with

E.

NASA

B. O. ; Bushnell,

Revolution 1049

Structures.

vol.

1963,

of Cylindrical

of a Spherical

4921,

AIAA

Sechler,

Stress

E. ; and

Hemisphere

10, October

D. ; and

of Shell

Segment

33.

1, no.

Buckling

bility 32.

vol.

of a Truncated

1967,

65-371,

1965.

and

Nonsymmetrie

Stiffened

Shells

pp.

H. : Angencherte Kugelschalen

IUTAM

Symposium

Holland

Publishing

on the Co.,

Stiffening

on Buckling

Buckling

of Revolution.

of

of Finitely AIAA

J.,

vol.

1455-1462.

Bestimmung unter

der

Tragfahigkeit

Druckbelastung.

Theory Amsterdam,

of Thin

Proceedings

Elastic

1960,

radial

pp.

Shells. 95-121.

of the North-

of

Section

C3.0

December Page

15,

1970

133

REFERENCES (Continued)

38.

Kloppel,

K. ; and

dunnwandiger 130.

39.

1966.

40.

available

R.

Optimum

Design

3,

J.

J. 42.

of

A. :

C. :

David

zum

Stahlbau,

TayLor

Schwartz, Stiffened

Durchschlagproblem

Vol.

Model

pp.

511-515.

TN

Stiffened

D-3826,

VI,

Basin

29,

Sandwich

General

1953,

pp.

121-

Translation

308.,

3,

Domes.

and

AIAA

Shells

J.,

vol.

3,

of Double

1967.

Sphere

no.

Instability

Shallow

February

of Sandwich vol.

B. :

Spherical

Eccentrically

Sci.,

Structural

D.

Grid

Buckling

Aerospace

Anon:

; and

NASA

J.

Der

as

1965,

Curvature. Yao,

F.

March

McElman,

41.

Beitrag

)

Crawford,

no.

O. :

Kugelschalen.

(Also

May

Jungbluth,

Under

March

Composites.

Normal

1962,

pp.

Pressure.

264-268.

MIL-HDBK

23,

December

30,

1968.

REFERENCES

I,

FOR

Dickson,

J.

Stiffened

Corrugated

3089,

N. ; and

January

Dickson,

2.

J.

,

Burns,

Company,

Brolliar,

Stiffened Pressure,

A.

R.

Cylinders

N. ; and

B. :

of Ring-Stringer Lateral

Brolliar,

PROGRAMS

H. :

(SUBSECTION

The

Under

General

Axial

3.4)

Instability

Compression,

of RingNASA

TN

D-

1966.

Eccentrically Lateral

COMPUTER

Pressure. March

R.

Cylindrical

NASA Computer

CR-1280, Program

Stiffened

Cylinders

Rpt.

4-11-(;(;-i,

1966.

H. :

The

Shells

General Under

January for

the

Subjected Lockheed

Instability Axial

of

Compression

and

1969. General

Instability

to Axial Missiles

Analysis

Compression and

Space

and

Section

C3.0

December Page 134

15,

1970

REFERENCES (Continued)

1

Burns,

A.

pression vol. .

and Internal

5,

Mah,

1968,

G.

tropic a

B. : Optimum

Almroth,

with

TR-69-375,

Percy,

J.

Program

Revolution

Under

Method,

and

Com-

Rockets,

Buckling

of Ortho-

598-602.

of Cylinders

Aerospace

Sciences

Element

Arbitrary

Thin

L.

H.:

with

Cutouts.

Meeting,

Buckling Vols.

the

New

I,

of Shells H,

and III,

Linear

Program

Loading

BOSOR3,

Klein,

S.:

Shells

Lockheed

by Using

of Rpt.

Linear

December

Rpt.

of Thin

Institute

Laboratory,

Static

May G.

and

E.,

Dynamic

5-59-69-1,

1968.

A Shells

the Matrix

G. ; and Striekland, for

SABORIII:

Analysis

Massachusetts

with Stiffeners.

Company,

for

Elastic

Research F.

of Complex

1969.

TR 121-6,

Structures

Vibration

Manual

D. R. ;and for

Shells

and

User's

Asymmetric

Space

Sobel,

Constructions.

Stability,

A. ; Morton,

and

V. :

pp.

O. : Buckling

Wall

and

ASRL

A Finite

Missiles

Spacecraft

E.

1968,

D. ;and

H.;Navaratna,

W.

6,

Seventh

September

FORTRAN

Loden,

J.

Axial

1968.

Analysis

Aeroelastic 10.

B.

Various

D. : Stress,

SAMSO

ment

Combined

1969.

CR 1049-1051,

Bushnell,

Pittner,

vol.

B. O. ; Buslmell,

Revolution.

o

O. ; and

J.,

of the AIAA

of Revolution

.

AIAA

January

NASA

B.

F. ; and Almroth,

York,

Pressure.

for

690-699.

Cylinders.

Brogan,

Cylinders

or External

B. ; Almroth,

Proceedings

o

pp.

Stiffened

of

Displace-

of Technology 1965. Jr.

:

BERKIII:

Analysis Lockheed

of

Section

C3.0

December Page

15,

1970

135

REFERENCES (Concluded)

11.

McCormick, October

12.

Loden, Space

C. W.,

ed.:

The

NASTRAN

User's

Manual,

NASA

SP-222,

1969. W.

A.:

Company,

User's January

Manual 1969.

in Preparation.

Lockheed

Missiles

and

_.B4

SECTION C4 LOCAL INSTAB ILITY

Section C4 1 December

TABLE

1969

OF CONTENTS Page

P

C4.0.

0 4. 1.0

Local Instability............................ Introduction...............................

4. 2.0 4.2.1

Conventionally Stiffened Flat Panels in Compression Local Skin Buckling .........................

I 1 ...

6 7

4.2.2

Local Stiffener Buckling

4.2.3

Inter-Fastener

4.2.4

Panel Wrinkling (Forced Crippling) .............. Torsional Instability.........................

23 28

4.4.0

Integrally Stiffened Fiat Panels in Compression ....... Stiffened Fiat Panels in Shear ...................

34 40

4.5. O 4.6.0

Fiat Panel Stiffened with Corrugations* . ........... Stiffened Curved Panels"' . ....................

4.2.5 4.3.0

References Bibliography

......................

Buckling (Interrivet Buckling) ........

.........................................

9 17

45

........................................

45

* To be supplied

C4-iii

Section

C4

1 December Page C4.0.0

LOCAL

C4.1.0

Introduction

1969

1

INSTABILITY

f

curved plates

This

section

panels.

The

and

longitudinal

for

are panels

and are

panel

refers

term

"plate"

refers

importance

and

failures

to a composite

{e. g.,

information

analyses

and

section.

them

should

account

and

consisting

or skin

stiffeners

in this

of flat

structure

to sheet

concerning

occurs

juncture

(Fig.

of panels

The

is characterized

general

by deflection

with

modes

C4.1.0-1).

is usually

General FIGURE

instabilities

members

of primary

of instability.

effect

local

bounded

frames).

of

by

Panels

Although

panels

is not as extensive

in in

as that

in compression.

buckling

this

The

discussed,

with "panel"

tranverse

Stability modes

term

stiffeners.

compression shear

deals

along Some

small

and

mode

of instability

of the (or

is generally

STIFFENED

TYPICAL PLATES

BUCKLE UNDER

general

the

between

and

for

localinstability,

stiffener-plate

these

modes

Local

Instability

exists,

neglected.

MODES

IN LONGITUDINALIJY

LONGITUDINAL

local

a compression-loaded

whereas,

along)

Instability

C4.1.0-1.

both for

stiffeners;

nearly

coupling

for

LOAD.

but

Section

C4

1 December Page Definition

1969

2

of Symbols Definition

Symbol dimension

of a plate,

J

a

long-side

in.

b

short-side

dimension

of a plate,

b

short-side

dimension

of a rectangular

bf

width

b

geometric

in.

!

O

b

of stiffener

stiffener

S

flange,

fastener

width

of hat top for

b

depth

of stiffener

d

fastener

df

frame

hat-section

web,

diameter, spacing,

e

end-fixity

E

Young's

in.

coefficient modulus

of elasticity, moduli,

f

actual

stress,

also

F

allowable

F0.

TD F0.85

stress

D

flexural

g

spacing

in.

in.

and tangent

Et

stiffeners,

in.

secant

E S'

in.

in.

bT

W

in.

in.

offset,

spacing,

tube,

psi;

or buckling at secant

stiffness between

psi psi

effective

stress,

modulus,

of skin staggered

fastener

offset,

in.

psi 0.7

per

E or 0.85

inch

columns

of width,

E of skin

material'

Ets3/12(1-v2

of fasteners,

in.

)

_J

Section

C4

1 December

1969

Page 3 Definition of Symbols

Definition

Symbol G

(Continued)

elastic shear modulus,

psi

!

h

long-side dimension

of rectangular tube, in.

spacing between staggered rows of fasteners, in. bending moment of inertia of stiffener cross section taken about the stiffener centroidal axis, in.4 I P

polar

moment

torsion (GJ k

constant

spring

kh, kt, kw

compressed

stiffener

k

shear

k

buckling

compressed

wr

k

local

integral

panel

wrinkling

stiffeners. of safety

N

number

of stiffeners

n

shape par,'Lmeter

r

radius, In.

8

fastener pitch, in.

S°

in. 4

coefficient

buckling

coefficients.

coefficient

margin

M°

of rotation,

length)

buckling local

compressive-local-buckling SC

unit

center

constant

skin

S

about

stiffener,

per

compressed

C

of section

of the

= torque/twist

rotational

k

of inertia

coefficient coefficient

for

panels

with

in. 4

Section

C4

1 December

1969

+ J

Page Definition

of Symbols

4

(Continued) Definition

Symbol t

thickness,

11

plasticity

in. reduction

cladding

reduction

v

Poisson's

ratio

F

torsional-bending

factor factor

in elastic constant,

range in.6

Subscript C

com press

ion

e

effective

f

flange

t

tension

S

skin,

W

web

av

ave rage

clr

local

co

cutoff

CRI

local

CS

compressive

or top web

of hat-section

stiffeners

compressive

inter-fastener

buckling

compressive

integral

shear

skin

stiffener

panel

buckling

buckling ..

csr

shear

skin buckling

Section C4 1 December Page Definition

of Symbols

(Concluded)

Definition

Subscripts csk

compressive

local

skin

cst

compressive

local

stiffener

ct

compressive

panel

torsional

cw

compressive

panel

wrinkling

cy

compressive

yield

tr

tensile

pl

proportional

fastener

stress limit

buckling buckling buckling

5

1969

Section

C4

1 December Page C4.2.0

Conventionally Buckling

conventionally in either

Stiffened

resulting

stiffened

the stiffener

from

(Fig.

Flat local

Panels

elements

"S"

6

in Compression

compression

C4.2.0-1)

1969

instability

in the direction

in a panel

of the

load may

occur

or the plate.

""

J

"/

"/

""

..... FIGURE

C4.2.0-1.

TYPICAL

STIFFENED

CONVENTIONALLY PANEL.

The forms of local instabilitiesand failures which will be discussed in this section are: I. Local skin buckling 2.

Local stiffener buckling

3. Inter-fastener buckling 4.

Panel wrinkling {Forced crippling)

5.

Torsional instability

Although these are distinctii_stability modes, usually a combination of two or more

modes.

ultimate buckling failure is

Section

C4

1 December Page Other phenomena f

local

are

discussed

section.

in "Shell

this

is not discussed 3.

unit

Monolithic

of skin

when

subjected

in the

documents

since

this

Three

C4.2.

cited

I_ocal

This

skin

general

mode

Although

failure The

the

instability

c_Jmpression

p_nel.

precipitates

2 and

manual

modes

are

reader

since

time.

that

crippling,

they

Analytical

3 are

section, technique

to the

of stiffener

together levels.

in this

stiffener

analytical

present

an extension

stress

of the The

at the

stiffener

act

at the

presented

to the

present

a direct

as a monolithic

methods

recommended

is

reader

time.

function

of the

fastener

SkinBucklin/_

buckling.

stiffened

local

not

C4.2.0-2).

A lo(:a[ skin

manual

in References

are

manual.

is recommended

so fastened

is not discussed

cited

(Fig. 1

stiffener

section.

[1]

failure,

to crippling

mode

of the

spacing

and

of this

of a twisting

Manual"

panel

but

of a composite

C1.3.1

cross

in this

post-buckling

paragraphs

collapse

consists

Analysis

7

manual.

is a local

of the

as column

three

in Section

by a distortion

mode

of the

instability

presented

a failure

section

is covered

Lateral

classified

following

crippling

Crippling

accompanied

in the

in this

Stiffener

2.

of failure

described

further 1.

modes

1969

instability

m_)de

can

occur

this

mode

carl

mode

if the

in another

in skins

normal

analytical

procedure

as a simply

support(_d

flat

equation

for

skin

_ 77_ F0.7

buckling

be a failure load for

is often

between

observed

or under in itself,

this

of infinite

stress

is written

type

stiffen_.rs

of a

increased.

instability

length.

is local

it usu:tlly

is appreciably

pl_te

c 12(1-v

that

The

is to treat following

in nondimensional

form:

(1) 2) F0.7

Section

C4

1 December Page

1969

8

FORCED CRIPPLING

• IVET FIGURE

This

equation

table,

C4.2.0-2. FAILURE MODES RIVETED PANELS.

can be solved

by the following

1.

Determine

F0. 7 from

2.

Determine

k c

3.

Determine

n from

it may

4.

SPAC/NO

be obtained

from

from

procedure:

appropriate Table

Table

OF SHORT

stress-strain

curve.

C2.1.5.5. C2.1.5.6.

data given

If

in Section

n is not given

in the

C4.2.3.

Calculate 12 (1-v

2) Fo. 7 J

Fcs"/F°'a

5.

Enter

Figure

C2.1.5-4

T,

using

appropriate

at value

n curve.

calculated

in step

four

to obtain

Section

C4

1 December Page

6. Fcs

k

should

Tables

Calculate

Fcs

include

plasticity

C2.1.5. The

l and margin

-

9

k and

cladding

C2.1.5.2

respectively,

of safety,

M.S.,

Fcs M.S.

1969

reduction

factors,

as

given

in

if applicable. can

be

calculated

as

follows:

k

f

1

,

(2)

C

where

f

is the compressive

stress

in the skin.

C

C4.2.2

from

Stiffener BucklinK

Local

instability or local buckling

crippling

in Section occur

Local

of a section.

C I. 3. il, while

at much

lower

Crippling

is an ultimate

local buckling

stresses.

of a section

each

in which

side buckles

the meets

the same

angles

the order

of the cross-section Two

buckling

of the simplest

of the shapes a more

taking into account

to use the following

(Fig.

flange-web

Information

However, stress,

dimensions

as compound

C4.2.2-2. stresses

parallel

shapes

plate elements found

direct method the interaction

equation:

as that mode

stationary.

Thus,

the half-wavelength

is of

C4.2.2-I). are

angles

and zees.

and can be broken

in Section

described

mode.

to the load rotate through

as those of the adjoining sides and

can be considered in Figure

local instability is defined

edges

may

failure in another

of adjoining sides remain

as a plate whose

which

(discussed

this type of instability will not

constitute failure in itself but will usually precipitate

of distortion

type of failure

is an elastic condition

Generally,

In stiffeners with flat sides,

is to be distinguished

C2

can

be used

These

up as shown to determine

above. for predicting of the sides,

the local buckling in the elastic range

is

Section C4 i December Page

Fcet

where

{k i)

Figures

for

-3 through

H-sections,

determined

C4.2.2-6

local

show

(k l)

(3)

buckling values

rectangular-tube-sections,

stiffened-plate

10

,

12 (i-v z)

is an experimentally

C4.2.2.

Z-sections, used

= _

1969

v

coefficient.

for various

and hat-sections

channels, often

construction.

F ¢st

s

!-I

FIGURE

C4.2.2-I. TYPICAL SHOWING ONLY

2-2

STIFFENER LOCAL TWO HALF-WAVES.

3-3

4-4

BUCKLING

A discussion of cladding and pLasticity-reductlon factors can be found in Section C2. i.I.

Figure C4.2.2-7

gives experimentally determined

plastlcity-reduction factors for stiffener shapes presented in Figures C4.2.2-3 through C4.2.2-6.

The and

extruded

parameters

reader

should

stiffeners, when

using

also

observe

as shown the buckling

dimensioning

in Figure coefficient

C4.2.2-8, curves

differences prior cited

for formed

to calculating above.

---J

.

Section C4 1 December Page

1969

11

_t

I I _ _ $$

•

$

$$

i

FIGURE

C4.2.2-2. BREAKDOWN INTO COMPONENT

OF ANGLE AND Z-STIFFENERS PLATE ELEMENTS.

Section C4 1 December Page

Fcs t

FIGURE COEFFICIENTS

= rl_

C4.2.2-3. FOR

12 (l-v

COMPRESSION

CHANNEL

AND

12

2)

LOCAL Z-SECTION

BUCKLING STIFFENERS.

1969

Section C4 1 December Page

7

13

WEB BUCKLES FtRST"_ I

fLANGe"

8_ICLLr5

F'N_T__

5

4

k. J

2

1 0

1

I

1111

0

.g

,6

4

B

10

% Fcst

FIGURE C4.2.2-4. COEFFICIENT

= _ _

12 (1-u

z)

COMPRESSION FOR H-SECTION

LOCAL BUCKLING STIFFENERS.

1969

Section C4 1 December Page

1969

14

J

\

2

I

Fcs t

FIGURE COEFFICIENT

= r/_

C4.2.2-5.

12 (1-v 2)

COMPRESSION

\h'/

LOCAL

FOR RECTANGULAR-TUBE-SECTION

BUCKLING STIFFENERS.

Section C4 I December Page

1969

15

,.

,r,j

_

2

•

.8

£

kt _2 E Fcst

FIGURE COEFFICIENT

C4.2.2-6. FOR

= _ _

/D

/

/_

•

/4

_

12 (1-zJ 2)

COMPRESSION HAT-SECTION

LOCAL

STIFFENERS,

BUCKLING t = tf = tw = t t

Section

C4

1 December Page Section

Fig.

Buckling coefficient

16

P ias ticity-reduc tion factor

i

(Es/E) ( l-VeZ)/(1-v 2) k

C4.2.2-3

is about

W

5 percent

conservative

C4.2.2-4

k

None

reported

None

reported

None

reported

W

C4.2.2-5

C4.2.2-6

kt

FIGURE

1969

C4.2.2-7.

EXPERIMENTALLY

PLASTICITY-REDUCTION COMPOSITE SHAPES C4.2.2-3 THROUGH

DETERMINED

FACTORS IN FIGURES C4.2.2-6.

FOR

0 J

Section

C4

! December Page

1969

i7

i__. bf

T b W

b

(a)

Formed

FIGUI_E

C4.2.3

C4.2.2-8.

Inter-Fastener This

fasteners

causing

a separation

action

than the

Any

by the skin

skin;

in load

deformation

from

failure

the

skin

as shown

the

gage

inter-fastener

redistribution

with

buckling

skin

Section

FORMED

in Figure

stiffened

column

Inter-fastener

above

FOR

and an essentially

of a wide

the

Extruded

AND

Buckling)

of longitudinally

where

therefore,

A criterion result

skin

spacing.

K (Interrivet

instability,

that

designs

increase

DIMENSION STIFFENERS.

between

approximates

stiffened-panel

TYPICAL EXTRUDED

of local in the

fastener

(b)

Bucklin

mode

between

The

Section

panels

equal

is usually

is less

than

to the

the load

occurs

irl compression,

undistorted

a width

buckling

of load

C4.2.3-1,

stiffener. to or leqs

found

in

stiffener cannot

stiffeners

ga_e. be supported

and excessive

occurs. for

fastener

spacing

in the inter-fastener

is determined buckling

mode

from rather

test from

data

which

panel

Section C4 1 December Page wrinkling

(Section

C4.2.4

1969

18 ):

t

1/1

b

S/bs >- 1.27/(kwr)

q

i

.J

(4)

I where

the wrinkling

coefficient

(k

) wr

FIGURE

C4.2.3-1.

INTER-FASTENER determined

rivet

for formed When

following

v ,

ts ,

curve

.

in Section

C4.2.4.

is a function

Dimensioning

stiffener

is analyzed

coef-

.j

of the experimentally

rules

sections

This

given

should

in Figure

be observed.

as a Wide column,

the

applies:

s ,

e,

of

as indicated

(5)

e_Ec 12 (l-v 2) /_- /

presents

The values

Values

(f)

buckling

C4.2.3-2

above.

ficient

and extruded

: T}_

Figure equation

offset

inter-fastener

equation

Fcir

Ec ,

BUCKLING.

effective

C4.2.2-8

is given

TYPICAL

and

a graphical

needed

to enter

nondimensional this

chart

form

are:

F0. To,

of the F0. I

,

n.

F0. To and in Figure

F0. s5 may C4.2.3-3

be obtained

(a).

Values

from for

a stress-strain

E

and

v

can be

,

will

C

obtained

from

MIL-HDBK-5A

If cladding include

the

cladding

the pattern spacing C4.2.3-4

is used material.

(a)

be the actual

well-qualified

on the sheet,

of the fasteners.

will

or other

a single

distance

and C4.2.3-4

the sheet

The fastener For

(b).

between For

sources. thickness,

spacing row

to be used

or double

fasteners,

staggered

rows

as shown

rows,

t

s

will

not

depend

on

the fastener in Figures

an effective

fastener i

spacing

must

be used.

This

effective

spacing,

s,

may be calculated:

Section

C4

1 December Page

1969

19

1.3 1.2 I.I I.II

r,-

o7

U.

:"*J

.6

L

.$ .4 J .2 .!

.2

.3

.4 .5.6

.8

I

2

3

4

5 6

0

20

10

30 40 5060 80100

4 FO. 7• (lEe-v 2) siT/2

FIGURE

C4.2.3-2. INTERRIVET

g 2 where use

g 2h

and

+ h

h

are

as the value

CHART

BUCKLING

(0 <- g<-

shown of s.

OF NONDIMENSIONAL

in Figure

STRESS.

2s)

C4. 2. 3-4

(6)

(c).

If

g

is greater

than

2h,

SectionC4 1 December 1969 Page 20

_J

Section

C4

1 December Page

O

O

O

O

O-

O

O-

O

O

O

(a)

Single

row

FIGURE

(b)

Double

FASTENER

C4.2.3-4.

FASTENER

rows

(c)

SPACINGS

FOR

1969

21

Staggered

rows

TYPICAL

PATTERNS.

The value of e is dependent on the type of fastener. Values of e to

Table C4.2.3. i. Values of EndFixity Coefficient "e" for Several Types of Fasteners

be used are listed in Table C4.2.3. i for

several

types

Values

of fasteners.

of the shape

Type

of Fastener

parameter Flathead

n

for

several

Table

C2.1.5.6.

given,

the

obtained If

n

in that from

materials For

shape

from

is out of the figure, the

are

range

it may

following

given

materials

parameter

Fi_,mre

e

may

C4.2.3-3

4

rivet

in not

Spotweld

be

Brazierhead

3.5 rivet

3

(b). Machine

csk.

Dimpled

rivet

rivet

1

of the curve be calculated

equation:

n = 1 * log e (17/7)/1og

e (F0. 70/F0. 85)

(7)

Section C4 1 December Page For performed These

temperatures using

values

the

values

of

can be obtained

It should calculations. shown

other

than room F

cy

from

,

temperature,

F0.70,

F0. u

the appropriate

n for this

stress-strain

be noted

that a cutoff

stress

is used

The values

of the cutoff

stress

recommended

in Table

22

the analysis and

1969

may

be

temperature.

curve.

in the interrivet for use

buckling here

are

C4.2.3.2.

Table

C4.2.3.2.

Recommended

Values

Material

for

Cutoff

Cutoff

Stress

Stress

(Fco)

2024-T 2014-T

Fcy

Fcy 200,_00

1 +

]

6061-T 7075-T

1. 075 F cy

18-8

(i/2

H)*

0. 835 F

(3/4

H)

0.875

cy F cy 0. 866 F

(FH)

cy All other

materials

F cy

* Cold-rolled, A general margin

of safety

with grain, procedure is listed

based

for

on MIL-HDBK-5A

calculating

inter-fastener

properties. buckling

stress

and

below:

1.

Determine

F0. 70 and

2.

Determine

n from

F0. 8s from

Table

C2.1.5.6.

appropriate If

n

stress-strain is not given

curve. in the table,

.w,

Section

C4

I December Page it may

be

obtained

from

3.

Obtain

4.

Calculate

e

Enter

.

from

(b)

C4.2.3.1

or

equation

23

(7).

.

4Fn7,1 2 eEc the

C4.2.3-2

at

appropriate

Calculate

6.

C4.2.3-3

Table

Figure

7 using

Fcir/Fo.

Figure

1969

F

.

n

value

calculated

in step

four

to obtain

curve.

as

elr

F

.

:

CIF

Obtain

.

(Fo.

7)

the

(Value

cutoff

determined

stress,

in step

F

five)

from

]'able

F

and

C4.2.3.2.

CO'

Calculate

8.

the

M.S.

as

F cr

M.S.

f

-

-

1

C

where

F

is

cr

compressive C.4.2.4

the

sheet Panel

designing

skin

This existence acts

is

mode

as

a column The

to a degree

instability

failure

is

equal

panel

wrinkling.

to or

greater

shown

attachment

supported

at the foundation

dependent

on

. clr

F

co

, and

f

c

is

the

fasteners. Crippling)

of failure,

elastic

values

(Forced

panels

gage

two

between

of local

of a flexible

foundation. flange

stress

stiffened the

of the

Wrinkling

A mode

where

lower

its

sometimes This than

in Fiffure

the

encountered generally

the

skin

fastener

attachment

occurs

stiffener

and

results stiffener. points

is provided

by

the

geometry:

the

offset

in designs

gage.

C4.2.4-1,

between

when

on

stiffener distance

an

from

the

The

skin

elastic

attachment of the

Section C4 1 December Page

1969

24

J

!

J

(a) General

Wrinkling

C4.2.4-1.

from

the stiffener

In the wrinkling contour

mode,

and causes

precipitating

The most buckling

stresses

web,

PANEL

the

fastener

the attachment

flange

plate

elements

of the panel commonly

A Cross

Wrinkle Showing of Stiffener

TYPICAL

other

failure

(b)

Failure

FIGURE fastener

of a

Appearance

WRINKLING spacing,

Distortions and Skin

diameter,

stiffener

at

FAILURE.

of the stiffener

of the

Section

and strength.

follows

to distort,

the

skin

thereby

as a whole.

used

analytical

is sere[empirical

method

in nature.

for determining

The general

equation

wrinklingfor

wrinkling is

F

where

kwr

cw

_

wr c k12 (1-u _E 2)

=V_

the wrinkling

coefficient

is a function

(f)

is obtained

which

Figure Cladding agreement

C4.2.2-8 (_)

for

coefficient

Figure

formed

and plasticity

with Section

is given

of the experimentally from

C2.1.1

'

I___Ss 1 2

C4.2.4-3.

and extruded (_)

correction

and should

(8)

in Figure determined

C4.2.4-2. effective

Dimensioning stiffener factors be used

sections should accordingly.

rules should

This rivet

offset

given

in

be observed.

be determined

in

Section

C4

1 December Page

25

IO

0

$

0

.2

A

.8

•

I.O

I.Z

s,$

".6

l*

gO

,_ g

?.4

g$

#w / ?_ ba / t.

FIGURE

C4.2.4-2.

EXPERIMENTAI,I,Y

COEFFICIENT

FOIl

DETERMINED

FAILURE

IN

THE

BUCKLING

WRINKLING

MODE.

._..----410 9

jj_

2

4

$

$

6

7

$

9

/0

•

-

l/

12

_

t$

9

14

-'

_"

7 FIGURE

C4.2.4-3.

EXPERIMENTALLY OF

EFFECTIVE

RIVET

DETERMINED OFFSET.

VALUES

1969

Section

C4

1 December Page A criterion result

from

required

a wrinkling

l'27/(kwr)_/2

Wrinkling

imposes

to make

The area,

ds

head

For

a high

tensile

attachment for the tensile

sd

load

from

test

on the fasteners

flange

data which

of the fastener

are

to the wrinkled

sheet.

is

"

(Ftr)

with either through

that are being alloy,

which

of the fastener

( Fcw )

be associated

aluminum

conform

strength

of the fastener

the fasteners

should

is determined

(9)

strength

than 2117-T4

expressions

ff

Est

and it may

When

criteria

b

tensile

countersunk

other

0.7

spacing

26

failure:

the stiffener expression

ftr >

the

mode

S/bs <

An approximate

shank

for fastener

1969

(10)

is defined

shank

failure

in terms

of the

or pulling

of J

the sheet. analyzed

the following

are

rivets

of materials

experimentally

proven

be used.

2117-T4

rivets

whose

tensile

strength

is

F t = 57 ksi,

the

are:

F t =57ksi

,

de/tav

;

_-< 1.67

(11)

or

190 Ft -

de/tav

160 (de/tav)2

,

(12)

Section

C4

1 December Page if

d /t e

where

>

av

t

1.67

(in

av

effective

1969

27

;

inches)

diameter

d

is the

average

of sheet

is the

diameter

for

and

a rivet

stiffener made

thickness.

from

The

2117-T4

material.

e

The

effective

diameter

of a rivet

of another

material

is,

1

d e /d where load

Ftr

=

is the

divided The

,

(Ftr/Ft)2 tensile

by shank

strength area

following

of a rivet,

defined

as maximum

tensile

in ksi units.

procedure

is recommended

when

analyzing

a panel

for

wrinkling:

f/tw,

1.

Calculate

2.

Enter

using

the

s/d. Figure

C4.2.4-3

appropriate

b e /t

3.

Calculate 1.

4.

Calculate

b

using

the 6.

return

Enter

7.

Figaare

f/b

calculated

in step

4 to obtain

W

S

C4.2.4-2

W

equation

at value

b.

If equation

is not satisfied,

continue

CW

using

equation

to step

wrinkling

:3. F

wr

(9). is satisfied,

C4.2.

k

curve.

If equation

Calculate

1 to obtain

curve.

a.

to Section

in step

b /t

approprinte Solve

calculated

/t W

S

.

w

at value

(8).

7.

is not the

critical

mode;

Section C4 1 December 1969 Page 28 8.

Checkfastener tensile stresses using equation (10), and equations

(11), (12), and (13) if necessary. 9.

Calculate the panel M.S. as F M.S.

CW

-

-

f

1

e

where f

is the compressive

e

stress

in the

skin,

and the

fastener

M.S.

frames

occurs

is

Ftr M.S.

=

1 ftr

where

Ftr

C4.2.5

is the

tensile

Torsional

cross

instability

section

its own plane. instability The sheets, plates

of the Typical

are

of the

fastener.

Instability

Torsional the

allowable

shown

analysis

of a stiffened

stiffener antisymmetric

in Figure methods

as suggested or cylinders

rotates

panel but

and

between

does

not

symmetric

distort

when

or translate

torsional

modes

in of

C4.2.5-1. of torsional

in Reference with typical

instability

4, will frame

of stiffeners

be described.

For

attached the case

to

of flat

spacing,

1

df the

> 7r (E_GI/k)_

allowable

Figure

torsional

C4.2.5-1

Fct

=

, instability

stress,

(14) Fct , for

the mode

shown

in

is

G_A

. +

2

_

E_?Gk

(15)

Section

C4

1 De_'ember Page

(al

Symmetric

C4.2.5-1.

MODES ] NSTA

and

I

an

iteratiw_

MIL

TORSIONAL

for

Z an(I

J type

stiffeners

may

be

obtained

P

Figures The

values

I

P from

OF

BI I,ITY.

,J F

J

where

29

Antisymmetric

(b) H(;URE

1969

C4.2.5-2

:mdC4.2.5-3,

plasticity

correction

procedure

tlDBb:-SA

for

uMng most

respectively. factors

_A

stress-strain

materi;ds)

for

the

and

rl G

curves

(which

given

malerial

may

be c_,leulated

can :rod

be fl)und by

the

by in

following

expressions:

_A

=

_,IG =

Curves

for

71A

Es/g

(I(;)

1,2,1,/Iq

( 17 )

m_,l

7?G

for

several

mnterial_

at

various

temperature

levels

Section

C4

I December Page

1969

30

0.16

o.2

0.14

I ,/:/,.--

0.2 0.,3

o.,/, /

p

o/

0.12 0.2

0.3

0.4

0.6

0.5

0.7

0.8

bf b W

t'_i

0.8

_>,\'\\

0.7

_\i\< Ip "1 [i'i_]

"\

\ \\

1 0.6

\

\ \ '\\, " ,,, \v ,_

0.5

\.

\

\

o

0.4

-_ 0.3

0.2

0.3

0.4

0.5

0.6

bt b W

FIGURE FOR

C4.2.5-2. LIPPED

TORSIONAl.

SECTION

Z-STIFFENER--SHEET

PROPERTIES PANELS.

_,

o.2 "-'-" 0.3 0.7

0.8

Section

C4

I December Page

.01

1969

31

__. r

2

.04

.02 I

3

7

5

9

!1

13

b w r

bf I

r

.04

2 4 b Wq

w

bt_

'\\ P

_t

6 _

.°2

_.__

.01

!

3

5

7

9

13

II

b w f

FIGURE

C4.2. 5-3.

TORSIONAL

J STIFFENER-SHEET

SECTION

PROPERTIES

PANELS.

FOR

Section

C4

1 December Page may

be found

Section

in References

C2.1.0

1 and 5.

of this manual

The rotational

Similar

at a later

spring

constant

curves

will

1969

32

be provided

in

date. (k)

may

be found

using

the following

expression:

1

1

-

k

1

+

.+ (18)

keb

kshee t

where Et

3 W

keb

(19)

4bw + 6bf

ksheet

-

(20)

b S

for the

k

=

1 for the symmetric

k

-

3

_A '

>

Fct

4.33

and

_G

(22)

---Fcrs

mode.

<

F crs

(21)


1+0.6

antisymmetric If

Fct

I

mode

4.33 '

depend

and-error

procedure.

calculate

Fct

F crs

the antisymmetric

'

the symmetric on

Fct,

Starting

and correct

mode

of failure

the solution with

mode

for

Fct

the assumption

for plasticity

if required.

is critical.

is critical. is, that

Since

in general, _, = 1,

Correct

If

a trial°

r_A = _G = 1, _

if required

Section

C4

1 December Page

and repeat procedure until desired convergence is obtained.

Then

1969

33 check

to

z/4 see

whether

or not df >

If

df

r

"

-

the torsional

( E_Gr' ) k

< _

instability

ktdf/x) Fct

= G_ A

+

_p

f

I

stress

is

2 (23)

P

where

F

=

(I-_p

)

The formulas with values

sections

other

of

J

Ip,

Z Ip2

which

than those and

F

are

have

•

been

shown known.

presented

in Figures

(24)

may be used C4.2.5-2

for stiffeners

and C4.2.5-3

if the

SectionC4 1 Decembex 1969 Page 34 C4.3.0

Integrally

Stiffened

The allowable certain may

integrally

be found

C4.3.0-1

Flat

buclding

stiffened

C4.3.0-5

The

integral

tw/t f

=

shapes

and

solving

stress

for

local

which

have

been

C4.3.0-1

shown

local

compression

parallel

instability

to the integral

coefficient

k

from

$

of

stiffeners Figures

the equation:

C

presented

(_)

include

webs,

(25)

•

S

zees,

and

tees

for various

values. Also,

that

loaded

12 k (1-v z_E 2)

_

for

the buckling

S

FCRI

in Compression

stress

plates

by determining

through

Panels

the

these

may

compression idealized

through

rules

C4.2.2-8,

be used

instability into

C4.3.0-5.

dimensioning in Figure

charts

geometries When

this

pertaining are

to determine

the

of conventionalty similar is done, to formed

allowable stiffened

to those care and

shown

should extruded

buckling plates in Figures

be exercised shapes,

as

observed.

_J

Section

C4

1 December Page

•

1969

35

! OUCKLIN6 0l r SKIN .rRA/NED BY .r//r._N£R.

6

_-'_'*--'----'_

//X_m-a:._.k'.LIN$OIrSTIFYENER

--

3

O ,,,t,,,,[ 0 _'

1 4

.8

£

_

b

0.5 FIGURE

C4.3.0=1. FOR

<

tw/t s <

2.0

COMPRESSIVE-LOCAL-BUCKLING

INFINITELY TYPE

WIDE

FLAT

INTEGRAL

PLATES STIFFENERS.

COEFFICIENTS HAVING

WEB-

,J /2

Section

C4

1 December Page

1969

36

? J

6

5

4

3

2_

I

0

_il_llltzl 0

I 2

.4

£

.8

1.0

/.t

b$ tw/t s = 0.50and0.79 FIGURE

C4.3.0-2.

COMPRESSIVE-LOCAL-BUCKLING

FOR INFINITELY SECTION

WIDE FLAT INTEGRAL

PLATES

COEFFICIENTS HAVING

Z-

STIFFENERS.

.r

Section C4 1 December Page

i

1969

37

i

6

/.0

b,

5

¥

.63_

0.3 .4

4

3

.3

2

.4 .5

0

2

0

.4

tw/t s = 0.63 FIGURE

C4.3.0-3. FOR

.6

and

.8

1.0

COMPRESSIVE-LOCAL-BUCKLING

INFINITELY SECTION

I_

COEFFICIENT

WIDE FLAT PLATES HAVING INTEGRAL STIFFENERS.

Z-

12

Section

C4

1 December Page

1969

-J

38

?

6

5

4

k, 3 J

2

tw/t f = FIGURE

1.0;

bf/tf

>

10;

bw/b s >

C4.3.0-4. COMPRESSIVE-LOCAL-BUCKLING FOR INFINITELY WIDE FLAT PLATES SECTION

INTEGRAL

0.25

HAVING

STIFFENERS.

COEFFICIENT T-

Section

C4

1 December Page

1969

39

7

bf

6

5

4

3

g

RESTRdlNED 87" SKIN _ SUCKLING OFSTIFFENER

t

\

..4 ,6

\

_

\.

D B

0_,, 0

i_IRl|

4

.2

.6

.8

LO

b/b w

tw/t FIGURE

C4.3.0-5. FOR

f

= 0.7;

bf/tf

>

$

10;

bw/b

s >

0.25

COMPRESSION-LOCAL-BUCKLING INFINITELY SECTION

WIDE

FLAT

INTEGRAL

PLATES STIFFENERS.

COEFFICIENT HAVING

T-

/2'

Section C4 1 December Page C4.4.0

Stiffened Local

shear For

FLat Panels shear

are

presented

shear

buckling

distinguished •the panel short

methods

for panels

stiffened

either

these

two types

calculations,

by considering

side

panels

square

panels

panels

is recommended,

of analysis

and panels

as transversely

are encountered,

longitudinally

parallel with

advantage

panels

stiffeners

for

may

be

parallel where

extensive

of

to the

J

stiffened

transversely

of the more

in

to the Long side

In instances

of the analysis

panels

or transversely.

of stiffened

stiffened.

use

to take

for stiffened

with stiffeners

stiffened,

of the panel

40

in Shear

instability

as longitudinallY

1969

stiffened test

data

(no deflection

of

available. The stiffeners)

analysis

that follows

instability

mode

necessary

for determining

low values

of

EI/b

s

value EI/b

D

increases,

of the

s

D.

EI/bsD

Thus,

stiffener

moment

buckling

stress

area

instability

instability

of

and less

stiffening

depends

of the panel

becomes

does

s

s

D which

nothing induces

point

arrangements,

the

k

critical

general

is At As

a constant

required since

D

is critical.

of further

is important

S

mode.

and yields

stiffness

that for similar equal

mode

regardless

to increase

EI/b

instability

deflect)

the fie×ural

EI/b

is a function

The parameter

(stiffeners

mode

inertia

at this

upon

only.

coefficient

of inertia

It is noted

Local

the general

in determining

transition

for the local

the criticality

buckling

it is this

different

of failure

the local

shear

accounts

increases

of

in the stiffeners, additional

allowable

local

plate

instability. bay geometries

local

instability

stresses

of only

the local

geometry;

whereas,

orientation

of the

stiffeners

with

for the

two

will

result.

respect

general

j

to the

panel's long dimension.

J

Section

C4

1 December Page The shear

equation

for

local

instability

of the

skin

of

1969

41

a stiffened

panel

in

is

Fsc

where

k

r

,

=

the

7) _

12

shear

(1-v

(26)

2)

buckling

coefficient,

can

be

found

by

referring

to

S

Figure

C2.1.5-14. For

longitudinally

stiffened

panels,

k

using

determined

above,

S

enter

Figure

skin

buckling.

as

C4.4.0-1

and

The

ratio

solve

for

between

a stiffener

local

I

buckling

required

and

to

general

prevent

local

instability

is

follows:

k

/Keneral) S

ks

where

N

general

(local)

is the

=

IN

number

instability

+1)2

,

of stiffeners.

can

be

(27)

A stiffener

calculated

in a similar

I manner

required using

to prevent the

ratio

cited. For above

can

buckling;

transversely

be

used

this

stiffened

to calculate

is done

by

an

entering

panels, I

for

Figure

the

a similar

procedure

stiffeners

required

C4.4.0-2

with

a known

to that

given

to resist k

.

local The

S

relationship

between

k

local

(local)

buckling

k

S

and

general

instability

is

(general) S

b

2

{28)

=

a 2

s

Using calculated

this

relationship

and

to prevent

general

the

same instability.

figure,

a required

stiffener

I

can

be

Section C4 1 December Page

1969

42

j'

Qm

r..>

4_

m Fa.l

+

o

o ,o

F.r.l

o

-+j

Section

C4

1 December Page

1969

43

:/b

= I i

I0

32

/-

II

120 •

F-

24

a/b

.

= 2

a/b ! = 5

7, .Ig t_

.

80

w .D Z tU -r

z uJ x ._

40'

16

//

t_

8

O.

,>.b_,

/

M .lg

0 0

4

8

12 16 O WHEN a/'b = I

I

6

40

80

I

_

I

200

400

600

EI/b S i

,

0

$ I

120 DWHEN

EI/b

I

in

0

EI/b

FIGURE

C4.4.0-2.

LONG

SIMPLY

SHEAR

TRANSVERSE

,i

I

800

D WHEN a/b

BUCKLING

SUPPORTED

200

$

$

I OOO

= 5 $

COEFFICIENTS

FLAT STIFFENERS.

PLATES

WITH

d

I

f60 : 2

a/b

| |

OO

20

FOH

,

d O0

Section C4 I December Page The preceding instability.

It is then quite

structural

integrity When

factors

relationships

of the

Fsc r

associate

simple

to discern

stiffened

it should

which

mode

modes

is critical

of

to the J

including

not exveed

plutlclty

F CO

that

and general

44

panel.

is calculated,

if necessary,

local

1969

and cladding

given

in Table

C4.4.0.

correction i,

is, F scr

Table

_

F co

C4.4.0.1.

(29)

Recommended

Values

Material

for Shear

Cutoff

Cutoff

Stress

Stress

( Fco )

2024-T 2014-T 0.61

F cy

6061-T 7075-T 18-8

(1/2

H)*

0.51

F

(3/4

H)

0.53

F

0.53

F

cy cy (FH)

cy All other

materials

0.61

F cy

* Cold-rolled,

with grain,

C4.5.0

Flat

C4.6.0

Stiffened

t tt

Panel

to be supplied to be supplied

based

Stiffened Curved

on MIL-HDBK-5A

with Corrugations

Panels

tt

properties. t

Section

C4

1 December Page

1969

45

References

.

o

o

NASA-Manned Manual. Gerard,

G. : Handbook

Strength

of Flat

Bruhn, Offset

.

5.

Spacecraft Center, Houston, Texas:

of Structural

Stiffened

E. : Analysis Company,

Panels.

and Design

Cincinnati,

American

Aviation,

TN 3785,

of Flight

Vehicle

V -- Compressive 1957. Structures.

Tri-State

1965.

Failure

Inc.

Part

NACA

Ohio,

Argyris, J. : Flexure-Torsion June - July 1954. North

Stability,

Shell Analysis

of Panels.

: Structures

Aircraft

Engineering,

Manual.

Bibliography

Argyris, J. and Dunne, P. : Structural Analysis. ttandbook of Aero., No. 1, Ltd. London, 1952. Beeker,

H. :

Elements.

Handbook

NACA

of Structu_:a!

TN 3782,

G.

and

Becker,

H. :

Buckling

of Flat

Plates.

NACA

Gerard,

G. : Handbook Elements.

Lockheed

-- Missiles

Semonian,

J.

and

Handbook

1955.

of Composite

Stability,

Part

I --

Part

IV -- Failure

of Plates

and

1957.

Division:

Structural

J. : An Analysis

Compressive Strength of Short Sheet-Stringer to the Influence of Riveted Connection Between 3431,

II -- Buckling

1957.

Stability,

TN 3784,

and Space

Peterson,

Part

of Structural

TN 3781,

of Structural NACA

Stability,

1957.

Gerard,

Composite

Principles and Data, Part 2, Structural Fourth Edition, Sir Isaac Pitman & Co.,

Methods

of the Stability Panels Sheet

Handbook. and

with Special and Stringer.

Ultimate Reference NACA TN

g

..t