NASA-CR-1457_MANUAL FOR STRUCTURAL STABILITY ANALYSIS OF SANDWICH PLATES AND SHELLS

CONTRACTOR _N A S A

_-

i_

NASA

CR-1457

REPORT

B I

!z MANUAL

FOR

STRUCTURAL

STABILITY

ANALYSIS

I SANDWICH

PLATES

I by R.

T. Sulli,is,

i _Prepared E7

Fi LE

CASE

G.

IFC Smith,

OF AND a,id

E. E.

Spier

b),

:

_ GENERAL

DYNAMICS

_:San Diego,

Calif.

_r

Spacecraf

Manned

CORPORATION

Ce,zter

E NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

E_

SHELLS

•

WASHINGTON,

D. C.

•

DECEMBER-1969

k

_k

|

|

_____ ".

!

_

-........

=.....

--

•!

|

E

i

!

NASA

MANUAL

FOR OF

By

SANDWICH

R. T.

Distribution

Sullins,

of this

information resides

STRUCTURAL PLATES

G. W.

report

the

AND

Smith,

or

San

NATIONAL For

sale

by the

E.

Springfield,

for

by

Center

Scientific 22151

-

and Price

Technical

$3.00

of

contents

AND SPACE ADMINISTRATION

Federal

Virginia

the

prepared

Calif.

Spacecraft

AERONAUTICS Clearinghouse

for that

Spier

interest

No. NAS 9-8244 CORPORATION

Diego,

Manned

E.

in the

organization

Prepared under Contract GENERAL DYNAMICS

for

SHELLS

Responsibility

author

ANALYSIS

and

is provided

exchange. in

STABILITY

Information

it.

CR-1457

ACE

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RIBBON DIRECTION

HONEYCOMB

SANDWICH iii

CONSTRUCTION

l--

ABSTRACT

The basic objective of this study was to develop and compile would

include practical and up-to-date methods

a manual

which

for analyzing the structural

stabilityof sandwich plates and she]Is for typical loading conditions which might be encountered

in aerospace

applications.

use would include known analytical approaches with applicable test data. The for

data

presented

a wide

range

ing combined cluded

loads.

on the

those

here

items

covers

of structural

design where

recommended

curves little

is presented along with data and should be used

or

design

configurations

In a number

of cases,

to substantiate no test

data

the notation with some

The methods

proposed for

as modified for correlation

equations

and

loading

actual

test

the exists

and

data

basic

that this represented caution and judgment

includ-

points

recommendations the

are made.

analytical the until

following Local

subjects

are

among

those

Instability

of Flat

General

Instability

of Circular

General

Instability

of Truncated

General

Instability

of Dome-Shaped

Instability

Inelastic

in the

manual:

Instability

General

Effects

covered

of Sandwich of Cutouts Behavior

Panels

Shell

on the

Cylinders Cones

Shells

Segments

General

of Sandwich

Circular

Instability Plates

and

of Sandwich Shells

inFor

approach

'best available" substantiated

by test. The

curves

conditions,

Shells

TABLE

OF

CONTENTS

Page

Section

INTRODUCTION i.

1

GE 2:ERA_"

I. 2

FAILURE

LOCAL 2.1

i-i

................. MODES

INSTABILITY

1-4

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

2-1

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

INTRACELLULAR

BUCKLING

(Face

2-1

Dimpling)

Sandwich

with

Honeycomb

Core

.........

2-1

2.1.2

Sandwich

with

Corrugated

Core

.........

2-8

2.2

FACE

2.2.1

Sandwich

2. i.

1

WRINKLING with

Wrinkling) 2.2.2

Sandwich

2.3

SHEAR

2.3.1

Basic

2.3.2

Design

GENERAL 3.1

with

Core

(Antisymmetric 2-21

Core

(Symmetric 2-28

CRIMPING Principles Equations

INSTABILITY

OF

RECTANGULAR

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

2-37

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

2-37

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

2 -40

FLAT

PANELS

PLATES

General

3.1

2

ITniaxial

Edgewise

3.1

3

Edgewise

Shear

3.1

4

Edgewise

Bending

3.1

5

Other

3.1

6

Combined

3-1

.......

3-1

..........

3-1

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

Single

Compression

Loading

CIRCULAR Available

Single

3.2.2

Available

Combined

3.3

PLATES

3.3.1

Framed

3.3.2

Unframed

GENERAL AXIAL

4.2.1

Basic

Conditions

3-73

CUTOUTS

3-73

.......

Conditions

3-73

......

3-74

..........

3-74

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

3-75

............. OF

CIRCULAR

CYLINDERS

...........

Curves

vii

4-1

4-5 4-5

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

....

4-1

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

Equations

3-47

.........

Loading

COMPRESSION Principles

3-46

........

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

Loading

Cutouts

INSTABILITY

4.2

Conditions

PLATES

Cutouts

3-37

..........

Conditions

3.2.1

4.1

3-25

Moment

Loading

WITH

3-5

........

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

3.2

Design

Foam

Honeycomb

1

4.2.2

or

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

3.1

GENERAL

2-21

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

Solid

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

Wrinkling)

4

I-i

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

.........

4-17

TABLE

OF CONTENTS,

Cont'd.

Section

Page 4.3

PURE

BENDING

4.3.1

Basic

Principles

4.3.2 4.4

Design Equations and EXTERNAL LATERAL

4.4.1

Basic

4.4.2 4.5

Design Equations and TORSION ................

4.5.1

Basic

4.5.2 4.6

Design Equations and TRANSVERSE SHEAR

4.6.1

Basic

4.6.2 4.7

Design Equations and COMBINED LOADING

4.7.1

General

4,7.2

Axial

Compression

Plus

Bending

........

4.7.3

Axial

Compression

Plus

External

Lateral

4.7.4

Axial

Compression

Plus

Torsion

........

4.7.5

Other

Loading

GENERAL CONES

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

4-25 4-25

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

Principles

Curves ......... PRESSURE

4-29 4-32

.......

4-32

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

Principles

Curves

4-42

.........

4-47 4-47

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

Principles

Curves ............

4-54

.........

4-62 4-62

............. Curves ......... CONDITIONS

4-64 .......

4-65

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

INSTABILITY

4-65

Combinations

4-67 Pressure.

4-89 4-95

..........

OF TRUNCATED

4-71

CIRCULAR

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

5-1

5.1

AXIAL

5.1.1

Basic

COMPRESSION

5.1.2 5.2

Design PURE

5.2.1

Basic

5.2.2 5.3

Design Equations and EXTERNAL LATERAL

5.3.1

Basic

5,3.2

Design

5.4

TORSION

5.4.1

Basic

5.4.2 5.5

Design Equations and TRANSVERSE SHEAR

5.5.1

Basic

5.5.2 5.6

Design Equations and COMBINED LOADING

5.6.1

General

5.6.2

Axial

Principles Equations BENDING

...........

5-1 5-1

............. and Curves ..............

Principles

Principles Equations

5-4

.........

5-6 5-6

............. Curves ......... PRESSURE

5-7 .......

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

Curves

5-9 5-12

.........

5-14

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

Principles

Principles

5-14

............. Curves ............

5-16

.........

5-18

............. Curves ......... CONDITIONS

5-18 5-20 .......

................ Compression

5-9

5-21 5-21

Plus viii

Bending

........

5-23

TABLE

OF

CONTENTS,

Cont'd.

Section

Page

5.6.3

Uniform

5.6.4

Axial

Compression

5.6.5

Other

Loading

GENERAL

External

GENERAL

6.2

EXTERNAL

6.2.1

Basic

6.2.2

Design

6.3

OTHER

INSTABILITY

OF

5-39 SHELLS

....

Axial

Compression

Other

Loading

7.2

OTHER

...........

and

6-3

Curves

SHELL

CURVED

6-12

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

6-15

SEGMENTS

PANELS

.....

........

7-1 7-11

...........

CONFIGURATIONS

CUTOUTS

ON THE

SHELLS

.......

GENERAL

OF

7-11

INSTABILITY

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

BEHAVIOR

7-1 7-1

............. Conditions

PANEL

6-1

6-3

CONDITIONS

SANDWICH

7.1.2

SHE

5-34

6-1

LOADING

7. i. 1

INELASTIC

........

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

Equations

CYLINDRICAL

OF

5-27

..........

DOME-SHAPED

PRESSURE

Principles

OF

SANDWICH

Torsion

......

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

7.1

OF

Plus

Pressure

Combinations

INSTABILITY

6.1

EFFECTS

Hydrostatic

8-1

SANDWICH

PLATES

AND

L LS

9-1

9.1

SINGLE

9.1.1

Basic

9.1.2

Design

9.2

COMBINED

9.2.1

Basic

9.2.2

Suggested

LOADING

CONDITIONS

........

9-1

Principles

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

9-1

Equations

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

9-3

LOADING

Principles Method

CONDITIONS

.......

9-i0

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

9-10

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

9-13

ix

LIST

OF FIGURES P age

Figure 1.1-1

Typical

1.2-1

Localized

1.2-2

Ultimate

1.2-3

Non-Localized

2.1-1

Sandwich

Construction

Instability

Modes

Failures

Critical

Stresses

for

by

Modes

of Dimensions

Chart

for

Determination

Intracellular

Buckling

Not

2.1-5

Buckling

2.1-6

Local Buckling Coefficient Sandwich ...................

for

Local Buckling Coefficient Sandwich ...................

for

Local Buckling Coefficient Sandwich ..................

for

Local Buckling Coefficient Sandwich ...................

for

Local Buckling Coefficient Sandwich ...................

for

Local

for

2.1-9

2.1-10

2.1-11

Configurations

Buckling

Sandwich 2.1-12

2.2-3

Buckling

Under 2-3 2-5

Cell

Size

Occur

Such

That 2-6

.........

2-8

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

2-10

................. Single-Truss-Core

2-14 Single-Truss-Core 2-15 Single-Truss-Core 2-16 Double-Truss-Core 2-17 Double-Truss-Core 2-18

Coefficient

Double-Truss-Core 2-19

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

Local Buckling Coefficient San&rich ...................

for

Typical

q .............

Variation

Comparison

of Theory Constructions

Parameters

for

vs Test

2-24

Results

Having

Determination

Constructions

Truss-Core 2-20

of Q vs.

in Sanchvich

Sandwich

1-6

...........

of Core Will

Corrugation

2.1-8

1-5

.....

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

2.1-4

2.1-7

Wrinkling

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

Definition

Modes

Face

Intracellular

Unia.xi al Compression

1-5

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

Precipitated Instability

1-2

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

Solid

of Face

Having

Solid xi

for

Face

or

Foam

Wrinkling or Foam

Wrinkling 2-25

Cores. in Cores

....

2-27

LIST OF FIGURES,Cont'd. Figure 2.2-4 2.2-5 2.2-6

Page

T_3_ical DesignCurvesfor FaceWrinklingin Sanchvich ConstructionsHavingHoneycomb Cores.........

2-30

Comparisonof Theoryvs Test Resultsfor FaceWrinkling in SandwichConstructionsllaving HoneycombCores .....

2-31

Relationship and

2.2-7

of K6 to Honeycomb

Facing

Graphs Facings Cores

Waviness

of Equation in San_vieh ....................

2.3-1

Uniaxial

2.3-2

Pure

3.1-1

Elastic

Properties

Typical

Sandwich

3.1-2

Shear

3.1-4

3.1-5

3.1-6

3.1-7

of

2 -40 2-41

and Dimensional Panel

Notations

for

a 3-4

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

Panel with Facings,

Ends and Sides and Orthotropie

Simply Core, 3-10

Ends and Sides and Isotropic

Simply Core, 3-11

................... Ends and Sides Simply and Orthotropic Core,

3-12

................... Panel with Ends Isotropic Facings,

Simply Supported and Orthotropic

and Core, 3-13

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

K M for Sandwich Sides Clamped,

Panel with Ends Isotropic Facings,

Simply Supported and and Isotropic Core, 3-14

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

K M for Sandwich Sides Clamped, (R = 2.50)

2 -34

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

K M for Sandwich Sides Clamped,

(R = 1.00)

(Fc/Ee)

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

K M for Sandwich Panel with Supported, Isotropic Facings,

(R = 0.40)

..........

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

K M for Sanchvich Panel with Supported, Isotropic Facings,

(R = 2.50)

Properties

2-35

K M for a Sandwich Supported, Isotropie

(R = 1.00)

Core (6/te)

(2.2-12) for the Wrinkling Stress Constructions ttaving ttoncycomb

Compression

(R = 0.40)

3.1-3

Parameter

Panel with Ends Isotropic Facings,

Simply Supported and Orthotropic

and Core, 3-15

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

xii

OF FIGURES,

LIST

Cont'd.

Figure 3.1-8

Page K M for Sandwich Supported, (R = 0.40)

3.1-9

K M for

Sandwich

Supported, (R = 1.00)

3.1-10

K M for

K M for

3.1-12

K M for

3.1-13

K M for

3.1-14

K M for

Corrugated Perpendicular 3.1-15

3.1-16

3.1-17

3.1-19

3.1-20

Panel and

Supported

to the

Load

KM o for Edgewise

Sandwich Panel Compression a Sandwich

an Isotropic

K M for a Sandwich and an Orthotropic

and

Isotropic

Ends

and

Orthotropic

with

Ends

and

with

with

Ends

Simply

Sides

Simply

Ends

Core,

Sandwich Corrugation

Panel

Panel Core,

.....

(R = 2.50)

Panel

Having

Flutes ..........

with

All Edges

Facings

3-22 a

Simply

Supported,

...........

3-29

All Edges

Simply

Core. ............

Core

xiii

in 3-24

K M for

with

3-21

3-23

with All Edges Simply (R = 2.50) ..........

and Corrugated to Side a

....

a

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

(R = 1.00)

3-20

are

Panel Having Flutes are

With Isotropic ..............

3-19

Clamped,

Simply ........

Panel

....

Clamped,

(R = 1.00)

and Sides

Orthotropic Sandwich

Clamped, (R = 0.40)

and Sides Core,

Direction

Core,

Sides

Core,

All Edges (R = 0.40)

a Sandwich

Sides

Core,

K M for a Sandwich Panel with and with an Orthotropic Core,

Isotropic Facings Flutes are Parallel

Simply

Core,

Clamped

and

Core. Core Corrugation to the Load Direction

Parallel

K M for

Sides Core,

3-18

K M for Simply Supported Corrugated Core. Core

and 3.1-18

with

and Isotropic

Facings, Simply

Clamped

and

Facings,

Panel

Sandwich

Isotropic

Ends

and Orthotropic

Facings,

and

Orthotropic

3-17

Panel

Sandwich

Isotropic

with

Panel

Facings,

Clamped

and

Facings,

Isotropic ...................

Sandwich

Ends

3-16

Panel

Sandwich

Isotropic

with Facings,

Isotropic ...................

Supported, (R = 2.50) 3.1-11

Panel

Isotropic ...................

Supported, 3-30 Supported, 3-31 Supported, Corrugation 3-32

LIST

OF

FIGURES,

Cont'd.

Page

Figure

3.1-21

K M for

a Sandwich

Isotropic Flutes 3.1-2'2

are

K M for

K M for

K M for

Core, 3.1-26

K M

3

1-27

K M

3

1-28

1-29

a Simply

Core.

Core

3

1-30

Interaction Subjected

3

1-31

Interaction Subjected

3

1-32

,,'). 1-33

Interaction

3

1-34

Buckling in

3.1-35

Curve

Coefficients

Biaxial

Buckling in

Biaxial

.........

Edges

All

Isotropic 3-34

Clamped,

Isotropic 3-35

........

Edges

Clamped,

(R = 0.40)

Isotropie 3-36

........

Panel

with

an

Isotropic 3-42

Panel

with

an Orthotropic 3-43

San(kvich

Panel

with

an Orthotropie 3 -44

Sanchvieh Flutes

for

a Honeycomb

for

Core

for

Bending

a Itoneyeomb

for

Compression Coefficients

for

Compression

for

= 1.0)

Corrugated (a/b

Sandwich

Panel 3-60

Sand_vich

Panel 3-61

.......... Core

= 1/2)

= 2.0)

xiv

3-45

Panel

Sandwich

Panel 3-62

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

Corrugated (a/b

.....

3-59

Core

Shear

Corrugated (a/b

Compression Coefficients

Shear

a

.........

a Honeycomb and

Corrugated

Side

Sanchvich

Core

Compression

and

with In

...........

a tloneyeomb and

Panel

Parallel

Compression

Compression

Buckling in Biaxial

for

Bending

to

Clamped,

1.00)

Sandwich

Supported

Bimxial

Subjected

Edges

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

Curve to

3-33

Sanchvich

Supported

Curve to

Supported, Corrugation

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

Curve to

Core

(R = 2.50)

with

Supporled

Corrugation

Interacti,m

Simt)ly

Core.

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

(R = 0.40)

K M for

All

Core,

Supported

(H = 2.50)

Subjected

with

Panel

(R - 1.00)

[or a Simply

Core, 3

a Simply

All (It =

Core,

Orthotropic

for a Simply

Core,

Panel

Edges

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

with

Orthotropie

and

All

b

Core,

a Sanchvich

K M for

Side

Panel

Isotropic

and

with

Corrugated

to

a Sandwich

Facings 3.1-25

Parallel

and

Facings 3.1-24

and

a Sanchvich

Facings "5.1-23

Panel

Facings

Core

Sandwich

Panels 3-63

.......... Core

Sandwich

Panels 3-64

.......... Core ..........

Sandwich

Panels 3-65

LISTOF FIGURES,Cont'd. Figure 3.1-36

Page

Buckling

Coefficients

Under

Combined

Longitudinal 3.1-37

Core

Buckling

Combined

Combined

Buckling

Combined

Transverse Buckling

Combined

Transverse Buckling Under

Combined

Transverse Buckling

Combined

and

Equilibrium

Paths

Cylinders

Equilibrium

Schematic for

Panels

Shear

with

Sandwich and

Panels

Shear

with 3-70

Core

Sandwich and

Panels

Shear

with 3-71

.......... Core

Sandwich

Compresmon, with

Axially

Panels

Transverse

Longitudinal

Core

Compressed

Circular

3-72

......

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

Typical

V

for

and

Core

Corrugated

Shear

Sandwich

Compression

Longitudinal

Compression,

with

3-69

Corrugated

for

Shear

..........

= 2.0)

Coefficients

Under

4.1-1

(a/b

Core

Compressloh

Longitudinal

Core

and

Panels 3-68

Corrugated

for

with

..........

= 1.0)

Coefficients

Panels

Shear

Sandwich

Compresmon

for

(a/b

Core

Corrugated

Longitudinal

Core

and

..........

= 1/2)

Coefficients

Under

3.1-42

(a/b

Sandwich

Compresmon

Longitudinal

Core

with

3-67

Corrugated

for

Panels

Shear

..........

= 2.0)

Coefficients

Under

3.1-41

(a/b

Core

Compression

for

Core

and

3-66

Corrugated

Longitudinal

Longitudinal

Sandwich

..........

= 1.0)

Coefficients

Under

3.1-40

(a/b

Core

Compresmon

= 1/2) for

Core

Buckling

Corrugated

Longitudinal

Longitudinal

3.1-39

(a/b

Coefficients

Under

3.1-38

for Longitudinal

Paths

for

Representation 0 _ 1

of

Circular

Cylinders

Relationship

.....

Between

K c and 4-7

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

C

4.2-2

Semi-Logarithmic Sandwich)

4.2-3

Knock-Down Subjected

4.2-4

Factor to

Comparison Data jected

Plot

Cylinders

for to

Axial of

Axial

Axial

3Zc for

Circular

R/t'

Compression

Isotropic

(Non......

Sandwich

Criterion Sandwich

............. XV

4-9

Cylinders 4-10

...........

Design Circular

for

Compression

Compression

Proposed

Weak-Core

of _/c vs

Under

Against Cylinders

Test Sub4-12

LIST

OF

Cont'd.

FIGURES,

Page

Figure

4.2-5

of Proposed

Comparison Result

Subjected

4.2-8

4.2-9

4.3-1

to Axial Involved

Buckling

Coefficient

Sandwich

Cylinders

Subjected

Buckling

Coefficient

Cylinders

Subjected

Knock-Down

4.3-2

Design

to

4.4-1

4.4-2

L/R

4.4-3

for

Short

to

Axial

_o

for

to

Pure

Simply-Supported

Sandwich (0 = 1) ......

Sandwich

Cylinders

Circular

Sandwich 4-30

.......... to

External 4-32

of Log-Log Cylinders

Pressure

Coefficients

Cp

Subjected

to

Lateral

Shear

Plot

of

Cp

Subjected

Versus to 4-36

............. for

Properties

Circular

Sandwich

Pressure; of

Core

Cylinders

Isotropic Isotropie

Facings; or

Ortho4-44

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

Coefficients to External Shear

4-24

4-28

Subjected

Sandwich

External

4-15

4-20

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

Buckling

Transverse

........

Bending

Cylinder

Lateral

Buckling Subjected

Sandwich

Mo for

Representation

= 0

Circular

...........

Factor

Circular

Vp

......

Circular

Compression

Circular

External

tropic;

Data

Compressed

Compression

Bending

Sandwich

Transverse

4.4-4

Pure

for

Axial

for

Pressure

Schematic

of Test

_-19

Factor to

Subjected

Circular Lateral

4-13

...........

for Axially

Knock-Down

Cylinders

a Test

Cylinder

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

Factor

Subjected

Against

Sandwich

in Interpretation

Knock-Down

Cylinders

Criterion

Circular

Compression

Stresses

Design

Design

for a Weak-Core

Cp for Lateral Properties

Circular Pressure; of

Core

Sandwich Isotropic

Cylinders Facings;

Isotropic

or

Ortho4-45

tropic; 4.4-5

Vp = 0.05

Buckling

Coefficients

Subjected

to

Transverse

External

................ Cp

for

Lateral

Shear

Properties

= 0.10

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

Circular

Sandwich

Pressure; of

Core

Cylinders

Isotropic Isotropic

Facings; or

Ortho4-46

tropic; 4.5-1

Circular

Vp

Sandwich

Cylinder

Subjected

xvi

to

Torsion

......

4-47

LIST OF

FIGURES,

Cont'd. Page

Figure 4.5-2

4.5-3

4.5-4

4.5-5

4.5-6

4.5-7

4.5-8

Typical Circular

Log-Log Plot of the Sandwich Cylinders

Buckling

Coefficients

Subjected

to Torsion

Buckling

Coefficients

Subjected

to Torsion

Buckling

Coefficients

Subjected

to Torsion

Buckling

Coefficients

Subjected

to Torsion

Buckling

Coefficients

Subjected

to Torsion

Buckling

Coefficients

Subjected

to Torsion

Sample

Interaction

Design

Interaction

Subjected 4.7-3

Design

to Axial Interaction

Subjected 4.7-4

Circular Plus

4.7-5

Typical Subjected Pressure

4.7-6

Interaction to Axial

4.7-7

Interaction to Axial

4.7-8

Interaction to Axial

4.7-9

Interaction to Axial

Circular

Sandwich

for

Circular

for

Compression

Cylinders 4-57

Sandwich

Cylinders 4-58

Circular

Sandwich

Cylinders 4-59

............... for

Circular

Sandwich

Cylinders 4-60

............... for

Circular

Sandwich

Cylinders 4-61

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

4-66

Curve

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

Curve

for

Circular

Compression

Plus

for

Circular

Compression

Plus

Cylinder

Curves

for

for

for

for

for

Cylinders 4-70

....... Compression

4-71 Sandwich

External

Sandwich

External

Cylinders

Lateral

Cylinders

Lateral

Sandwich

External

External

xvii

Pressure

Cylinders

Lateral

Sandwich

External

Pressure Cylinders

Lateral

Sandwich

Circular Plus

to Axial

4-68

4-74

Circular Plus

Sandwich Bending

Circular

Circular Plus

Cylinders .......

..........

Plus

Circular Plus

Sandwich Bending

Subjected

Pressure

Compression Curves

Sandwich

Circular

for

Compression Curve

Cylinders

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

Compression Curves

4-50

4-56

to Axial Compression ................... Curves

Sandwich

K s for .....

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

Lateral

Interaction

Coefficient to Torsion

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

Curve

to Axial

External

for

Buckling Subjected

Lateral

Pressure Cylinders Pressure

Subjected ....

4-79

Subjected ....

4-80

Subjected ....

4-81

Subjected ....

4-82

LIST

OF FIGURES,

Cont'd.

Figure 4.7-10 4.7-11 4.7-12 4.7-13

4.7-14

4.7-15

4.7-16

4.7-17

Page Interaction Curves to Axial Compresmon

for

Circular Sandwich Cylinders Plus External Lateral Pressure

Subjected ....

4-83

Interaction Curves to Axial Compress:on

for

Circular Sandwich Cylinders Plus External Lateral Pressure

Subjected ....

4-84

Interaction Curves to Axial Con:press:on

for

Circular Sandwich Cylinders Plus External Lateral Pressure

Subjected ....

4-85

Interaction Curves for Circular Sandwich Cylinders to Axial Con:press:on Plus External Lateral Pressure

Subjected ....

4-86

Interaction Curves to Axial Compresmon

for

Circular Sandwich Cylinders Plus External Lateral Pressure

Subjected ....

4-87

Interaction C_rves to Axial Compresswn

for

Circular Sandwich Cylinders Plus External Lateral Pressure

Subjected ....

4-88

Circular Sandwich Cylinder Plus Torsion .................. Conditional Subjected

4.7-18

Interaction to Axial

Conservative Subjected

to Axial

for

Circular

Compression

Plus

Curve

Torsion

for

Compression

Sandwich

Knock-Down

5.1-2

Truncated

Sandwich

Cone

Subjected

to Axial

5.2-1

Truncated

Sandwich

Cone

Subjected

to Pure

5.3-1

Truncated Pressure

Cone

Subjected

•

•

.

Factors

•

.

,

5.3-2

Truncated

Sandwich

Cone ..............

5.4-1

Truncated

Sandwich

Cone

5.5-1

Truncated

Cone

5.6-1

Sample

Interaction

5.6-2

Design

Interaction

Subjected

to Axial

Subjected

Cylinders

.......

4-94

...........

to Uniform o

4-93

Sandwich

Torsion

Empirical

Cylinders

.......

Circular

Plus

5.1-1

,

Compression 4-89

Curve

Interaction to Axial

Subjected

.

Curve

for

,

5-7

.

.

.

o

.

•

.

5-9

5-17

......

5-18

Shear

......

5-21

Plus

XVIII

.

.....

Lateral

to Torsion

Truncated

Compression

Bending

5-4

5-12

to Transverse ..............

Compression.

External

.

Subjected

Curve

5-2

Sandwich Bending

.......

Cones 5-26

LIST

OF FIGURES,

Cont'd. P age

Figure 5.6-3

5.6-6

Truncated

Cone

Pressure

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

Truncated

Sandwich

Truncated Torsion

Cone Subjected ...................

Conditional Cones

5.6-7

Conservative

Bending

Plus

Curve

Sanchvich

6.2-2

Schematic

6.2-3

Knock-Down Factor 7d for Uniform External Pressure

7.1-3

Schematic

Graphical (7,1-8)

5-39

.....

Plus 5--40 6-1

to External

Logarithmic

Pressure

of Relationship Sandwich .............

and Associated

(Non-Sandwich)

Compression

Compression

Plot Skin

Domes

Subjected

Subjected

6-7

to

to

Flat-Plate

Configuration

Criterion

for

of Test Panels

7-2

Non7-3

........... Data

for

Under

Cylindrical Axial 7-6

................ Representation

V c.

6-13

of Schapitz

Skin

Kc and

6-9

Panels

Plot

6-3

......

Between

Domes

for Sandwich ...............

Cylindrical

Isotropic

7.1--4

Subjected

Logarithmic

Sandwich

Sandwich Torsion

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

Representation

Schematic

Plus

5-38

.....

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

Shapes

Panel

Torsion

Truncated

to Axial

Coefficient Pressure

Sandwich

Plus

for

Subjected

6.2-1

Cylindrical

Truncated

Compression

Structural

Buckling External

for

Compression

6.1-1

6.2-4

Plus 5-34

Curve

Torsion

Dome

Compression

to Axial

Dome

Hydrostatic

5-31

to Axial

Interaction

Cone

External

Cone ..............

to Axial

Subjected

Truncated

to Uniform

5-27

Interaction

Subjected

Cones 5.6-8

Subjected

of Equations

(7.1-5)

through 7-10

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

xix

LIST OF TABLES Table 2-1 3'1 4-1 5-1 6-1 7-1 9-1 9-2 9-3

Page Summaryof DesignEquationsfor Local Instability Modes of Failure ...................

2-42

Summaryof DesignEquationsfor GeneralInstability of Flat SandwichPanels ...............

3-76

Summaryof DesignEquationsfor Instability of Circular Cylinders ...................

4-98

Summaryof DesignEquationsfor Instability of Truncated Circular Cones.................

5-42

Summaryof DesignEquationsfor Instability of DomeShapedShells..................

6-16

Summaryof DesignEquationsfor Instability of Cylindrical, CurvedPanels .................

7-12

Recommended Plasticity ReductionFactorsfor Local Instability Modes ................

9-7

Recommended Plasticity ReductionFactorsfor the General Instability of Flat Sanchvich Plates...........

9-8

Reeommended Plasticity ReductionFactorsfor the General Instability of Circular SandwichCylinders, Truncated Circular SandwichCones,andAxisymmetricSandwich Domes....................

9-9

XX

LIST

a

R

a

P b

Panel

length,

Axial

length

Length Panel

b

R

inches.

of the width,

fiat

Circumferential

b

Width

of the

C O

C P

D

D

Length

flat

panel

in Figure

shown

panel,

by Equation

by Equations

of sandwich

cylinders

wall

wall

E C

Ef E S

Et

inches.

(4.7-25),

(4.2-21)

of sandwich

or panel

or panel

dimensionless.

and

(6.2-19),

thickness

of sandwich

Young's

modulus,

psi.

Young's psi.

modulus

of the

Young's

modulus

of facing,

Secant Tangent

modulus modulus

wall

core

of facing, of facing,

xxi

or panel

in the

psi. psi. psi.

direction

dimensionless.

subjected

to external

_(Elt

1) (E2t2)h2

)'(Eltl

+ E2t2)

=

= h2(G

)/t XZ

Total

inches.

inches.

7. i-i,

q d

inches.

of an ellipse,

in Figure

Bending inch-lbs.

stiffness

inches.

inches.

for sandwich dimensionless.

Shear

7. I-i,

semi-axis

Buckling coefficient lateral pressure,

stiffness

of an ellipse,

inches.

of a cylindrical

defined

defined

panel,

shown

core,

parameter

Parameter

semi-axis

Minor

width

of corrugated

L

panel

inches.

Pitch

C

Major

of a cylindrical

bf

P

OF SYMBOLS

, lb/inch. C

(d = t 1 + t2 + tc),

normal

to the

inches.

facings,

E 1, E 2

Young's

moduli

for facings

Strain intensity defined

e

i F

Transverse

shear

I and 2 respectively,

by Equation

force,

(9.2-2),

psi.

in./in.

Ibs.

V

(Fv)

Critical transverse

shear

force,

ibs.

cr

F C

Flatwise flatwise

sandwich core

strength

tensile,

and

(the 10wcr flatwise

of flatwise

core-to-facing

core

compressive,

bond

strengths),

psi.

G

Transverse

shear

modulus

Core

modulus

of core,

psi.

e

G

shear

associated

with

the plane

perpendicular

to the

perpendicular

to the

perpendicular

to the

ca

facings

Gcb

Core

and parallel

shear

modulus

associated

facings

and parallel

Gf

Elastic

shear

G..

Core

shear

to side a of panel,

with

the plane

to side b of panel,

modulus

of facing,

modulus

associated

psi.

psi.

psi.

with the plane

lj facings G

Secant

and parallel

shear

to the direction

modulus

of facing,

of loading,

psi.

psi.

S

G

Core

shear

modulus

associated

with the plane

perpendicular

to the

XZ

facings

G yz

h

Core

and parallel

shear

modulus

of a cylinder,

Distance

middle

between

Buckling

coefficient

under

edgewise K F

psi.

perpendicular

to the

psi.

surfaces

of the two

for an isotropic

Buckling edgewise

bending

Theoretical

with the plane

of a cylinder,

facings

of a sandwich

inches.

dimensionless. pane]

of revolution

associated

axis of revolution

construction,

K

to the axis

flat panel

facing stiffness

coefficient

compression

(Eb).

K

= KF

buckling

and panel

(non-sandwich)

xxii

for fiat rectangular

(Kc) , edgewise

shear

sandwich (Ks) , or

+ KM.

coefficient

aspect

flat plate,

which

is dependent

ratio, dimensionless.

on

Theoretical

K M

sandwich applied

loading,

Buckling

K C

K

and

buckling

and

shear

coefficient which

rigidities,

panel

is dependent

aspect

on

ralio, and

dimensionless.

coefficient

sandwich

Buckling

I

fiat panel

bending

for sandwich

domes

under

coefficient

cylinders

external

for short

under

pressure,

sandwich

axial compression

dimenionlcss.

cylinders

under

axial com-

c

pression,

dimensionless.

Parameter

K

defined

by Equation

(4.4-2),

dimensionless.

P Buckling

K S

coefficient

for sandwich

cylinder

subjected

to torsion,

dimensionless.

K

Parameter

defined

by Equation

Buckling

coefficient,

Buckling

coefficient

(2.2-4),

dimensionless.

5

k X

the x direction,

Loading X

Buckling Y

k

coefficient

Loading

Y

coefficient

Over-all

stress

acting in

for applied

compressive

stress

which

is acting

associated

with

compressive

stress

acting

for applied

compressive

stress

which

dimensionless.

length,

Effective

L

compressive

inches.

length,

inches.

e

Applied

M

bending

moment,

Critical bending

M

in the

dimensionless.

the y direction,

L

with

dimensionless.

coefficient

y direction,

I

associated

dimensionless.

in the x direction,

k

dimensionless.

in-lbs.

moment,

in-lbs.

cr

M. S.

Margin

of safety,

dimensionless.

N

Critical

compressive

running

Number

of circumferential

load,

lbs/inch.

cr

full-waves

dimensionless. xxiii

in the

buckle

pattern,

is acting

in

P

Axial

load,

pl

Equivalent

p

Critical

lbs. axial

axial

load

load,

defined

by Equation

(5.6-32),

lbs.

axial

when

lbs.

cr

(Per)

Empirical

Empirical lower-bound alone, ibs. External

P P

Critical

cr

(Pc r ) Te st

(P×tCL

Py

pressure, value

Experimental Classical

value

for

critical

load

acting

psi.

for

external

value

for

theoretical

pressure, critical

critical

psi.

external

pressure,

pressure

psi.

for a cylinder

external

pressure

acting

only

on the

end closures,

External

pressure

acting

only

on the

lateral

subjected

to

psi.

surface

of a cylinder,

psi.

(PY)cL

Classical external

theoretical

The relative

Q

critical

pressure

acting

only

minimum,

with

pressure

for

on the

lateral

respect

a cylinder surface,

to {,

subjected

to

psi.

of expression

(2.2-2),

dimensionless.

q

Quantity

defined

R

Degree of core dimensionless.

by

Equation

(2.2-3),

dimensionless.

shear modulus orthotropicity Radius to middle surface,

= G /G _, inehee, a CD

Stress

ratio

defined

by Equation

(4.7-9),

dimensionless.

Stress

ratio

defined

by Equation

(4.7-5),

dimensionless.

Load,

stress,

of this

handbook,

(Rb)cL R C

(Re)

R

CL

Stress ratios dimensionless. Effective

or pressure

as

radius,

ratios

as defined

in appropriate

sections

dimensionless. defined

in appropriate

inches.

e

xxiv

sections

of this

handbook,

Stress or load ratio for the particular type of loading associated with

R.

1

the subscript i, dimensionless. Stress or load ratio for the particular type of loading associated with

R

J

the subscript j, dimensionless. Radius to middle surface at the large end of a truncated conical shell,

Rlarge

measured Maxium

R Max

perpendicular to the axis of revolution, inches. radius of curvature for middle

surface of a dome-shaped

shell, inches. Pressure

R P

ratios as defined in appropriate sections of this handbook,

dimensionless.

Pressure

ratio defined by Equation (4.7-15), dimensionless.

CL Load or stress ratios as defined in appropriate sections of this

R s

handbook,

dimensionless.

Stress ratio defined by Equation (4.7-29), dimensionless. (Rs)cL Radius to middle surface at the small end of a truncated conical shell,

R small

R

measured

perpendicular to the axis of revolution, inches.

Stress or load ratio corresponding

to the x direction, dimensionless.

stress or load ratio corresponding

to the y direction, dimensionless.

x

R Y

Middle-surface

R 2

meridian, Parameter

r

radius of curvature in the plane perpendicular to the

inches. defined by Equation (4.2-37), dimensionless.

a

Cell size of honeycomb T T

External

torque,

in-lbs.

Critical

external

torque,

core, inches.

in-lbs.

er

(¥cr) Empirical

t

Empirical

lower-bound

value

in-lbs. Thickness,

inches.

xxv

for

critical

torque

when

acting

alone,

Total

t

thickness

of the

cylindrical

panel

shown

in Figure

7.1-1,

inches.

R

t e

t

f

Thickness inches.

of core

Thickness

of a single

Total

t

thickness

(measured

in the

facing,

of the

fiat

direction

narmal

to the

facings),

inches.

panel

shown

in Figure

7.1-1,

inches.

P Thickness

t

of material

from

which

corrugated

core

is formed,

inches.

0

t 1, t 2

Thicknesses (there

U

of the

respective

is no preference

1 or 2),

inches.

Sandwich lbs. per

transverse inch.

facings

as to which

shear

of a sandwich facing

stiffness,

construction

is denoted

defined

as

by the

subscript

h2 U = T-- Ge _ hGe' e 2 lr D

Bending and dimensionless,

_7

shear

rigidity

parameter

which

is defined

as

V b2U

Parameter

defined

in Sections

4.2

and

6.2,

dimensionless.

Parameter

defined

by Equation

(4.4-4),

dimensionless.

Parameter

defined

by

(4.5-4),

dimensionless.

Parameter

defined

by Equation

(4.7-13),

dimensionless.

Parameter

defined

by Equation

(4.7-14),

dimensionless.

C

_T

P _7

Equation

S

V XZ

V

yz W

Bending

and

shear

rigidity

parameter

for

fiat

sandwich

panels

?T2tc (Eltl)(E2t2) corrugated

core

which

is defined

as

dimensionless. Running

W

compression

T?

W Xb 2 Gcb(Eltl+E2t2

load,

lbs/inch.

C

z z

Length

parameter

defined

by Equation

(4.2-33),

Length

parameter

defined

by Equation

(4.5-3),

S

xxvi

dimensionless. dimensionless.

)

with

Angle

of rotation

construction conical

at appropriate

(sec

shell,

Knock-down

in corrugated-core

degrees.

at appropriate Figure 2.1-5),

factor,

Knock-down bending,

joint

2.1-5),

Vertex

sanchvich half-angle

of

degrees.

Angle of rotation construction (see

7

Figure

joint in corrugated-core degrees.

dimensionless.

/actor

Ratio

= _y/a

sandwich

x,

dimensionless.

associated

with

general

instability

under

pure

associated

with

general

instability

under

axial

with

the

dimensionless.

Knock-down

factor

"Ye

compression,

Td

(Ti)Test

dimensionless.

Knock-down

factor

associated

shaped

under

external

shell

Knock-down loading

factor

condition

pressure,

determined

from

corresponding

a test

to the

Knock-down

with

associated

specimen

subscript

with general pressure,

factor

instability

of a dome-

dimensionless.

Knock-down factor associated under uniform external lateral

torsion,

i,

subjected

instability dimensionless.

general

to the

dimensionless.

instability

of a cylinder

under

pure

dimensionless.

Amplitude E

general

of initial

waviness

in facing,

Normal

strain

in the

x direction,

in/in.

Normal

strain

in the

y direction,

in/in.

inches.

x E

Y Shear

E

strain

in the

xy plane,

in/in.

xy Parameter buckle

_Test

involving wavelength,

the

core

Plasticity

reduction

factor,

Plasticity

reduction

factor

stress

value,

moduli,

core

thickness,

and

dimensionless. corresponding

to an experimental

critical

dimensionless.

(1 - PaPb ) = (1 - p_) for r/_×,

elastic

dimensionless.

isotropic

dimensionless. xxvii

facings,

dimensionless.

Ratio

=

Ratio of transverse dimensionless. Actual v e

shear

Poisson's

ratio

Poisson's

ratio

Radius

of gyration

for

shell

Peak psi.

((_b)c L

Classical theoretical value for under a bending moment acting

_CL

Classical

c

Uniform psi.

compressive

value

stress

compressive

(%)

Uniform axial load, psi.

Critical

cr

and

non-sandwich whose

stress,

to an applied

the critical alone, psi.

two

solely

defined

stress

peak

bending

moment,

compressive

stress

psi.

due

stress

compressive

Classical theoretical stress under an axial

solely

stress,

stress

compressive

Peak axial compressive moment, psi.

cr

due

of critical

((_c)b

(Y

of sandwich

psi.

Effective

(_c)CL

(4.2-1)],

dimensionless.

wall

_b

c

Equation

(p _ h/2 for sandwich constructions of equal thickness), inches.

Stress,

e

Esee

dimensionless.

of facing,

(l

(_!

of core

of facing,

Elastic

constructions facings are

moduli

due

stress

to an applied

by Equation solely

due

axial

(4.7-38),

to an applied

solely

psi.

bending

to an applied

value for the critical uniform load acting alone, psi.

load,

axial

compressive

psi.

test

Experimental men, psi.

critical

stress

test

Ex_perimental critical stress the test specimen remained

obtained

from

a particular

test

speci-

!

(I

cr

xxviii

which elastic,

would psi.

have

been

attained

had

ff

Critical

cr x

value

for

Compressive

compressive

stresses

ence

of the

ence

as

wich

critical

in

stress

facing

acting

1 and

for

is

compressive

2,

general

denoted

stress

constructions,

Stress

facings

loading

to which

Uniaxial crimp

the

in

the

x direction,

psi.

respectively,

instability

by

the

at which

(there

subscript

shear

1 or

crimping

in

the

pres-

is

no

prefer-

2),

psi.

occurs

in

sand-

psi.

intensity

defined

membrane

stress,

by

Equation

(9.2-1),

psi.

1

a

ttoop

H

Meridional

M

{YMax

O'MI N

psi.

membrane

stress,

Maximum

possible

material,

psi.

Minimum

value

of

Predicted

value

for

psi.

critical

stress

stress

for

critical

corresponding

the

post-buckling

stress,

psi.

to

a particular

equilibrium

path,

predicted o" P

a R (y

wr

(y

Critical

buckling

stress

for

a flat

Critical

buckling

stress

for

a complete

Facing

wrinkling

Stress

acting

stress

due

stress,

in the

plate,

psi.

cylinder,

psi.

psi.

x direction,

psi.

Uniform

axial

compressive

X

Effective

0 -I

to

an

applied

axial

compressive

load,

stress

psi.

defined

by

Equation

(4.7-37),

psi.

X

Peak

(ax) b

axial

moment,

Classical CL

stress

theoretical when

Uniform

(z x) C

load,

Stress

compressive

stress

due

solely

to

an

applied

bending

psi.

axial

acting

value alone,

for

critical

compressive

stress

due

psi.

acting

in the

uniform

axial

compressive

psi.

y direction,

Y

xxix

psi.

solely

to

an

applied

axial

psi.

Shear

T

T

I

stress,

psi.

Effective

shear

stress

Classical

theoretical

defined

value

by

for

Equation

critical

(4.7-39),

psi.

uniform

shear

stress

cylinder

subjected

when

(T)C L acting

alone,

Critical

T

psi.

shear

stress,

Critical

shear

stress

applied

torque,

psi.

psi.

cr

T

I

cr

T

Pure crimp

TT

T V

shear

crimping

Peak

stress, occurs

Uniform

shear

shear

for

an

acting in

equivalent

coplanar

sandwich

stress

stress

due

with

the

constructions,

solely

due

solely

of

corrugated

to

to

an

facings,

to

at which

an

shear

psi.

an

applied

applied

torque,

transverse

psi.

shear

force,

psi.

Angular

dimension

Quantity

defined

Angle

of

construction Equation

rotation (see (4.4-3),

by

Equation

core

at appropriate Figure

2.1-5),

dimensionless.

XXX

(see

(4.2-10),

joint

Figure

2.1-4),

degrees.

dimensionless.

in corrugated-core

degrees.

Parameter

sandwich defined

by

CONVERSION OF U.S. INTERNATIONAL

CUSTOMARY UNITS TO THE SYSTEM OF UNITS 1

(Reference: U.S.

MIL-HDBK-23)

Customary Unit

Quantity

Conversion Factor 2

lbm/in.

Density

lbm/ff

27.68 3

SI Unit

× 103

16.02

kilograms/meter3 kilograms/meter

(kg/m33) ;_ (kg/m)

meters

(m)

Length

ft in.

0.304854 0.02

meters

(m)

Stress

psi

6. 895 x 103

newtons/meter

2 (N/m 2)

lb/in.

6. 895

newtons/meter

2 (N/m 2)

newtons/meter

2 (N/m 2)

newtons/meter

2 (N/m 2)

Pressure

Tempe

r atu re

The rmal

lb/ft 2

47.88

psi

6. 895

(° F + 460)

5/9

degrees

0. 1240

kg cal/hr

3

IElastieity _ Rigidity

Moduli

x 103

Btu in./hr

conductivity

Prefixes

ft 2 ° F

to indicate

multiples

Prefix giga

kilo

of units

are

as

Kelvin

(°K)

m °C follows:

Multiple

(G)

mcga

x 10

109

(M)

10 6

(k)

103 -3

milli

(m)

micro

10

_)

10 -6

1 The

2

International

System

Eleventh Resolution

General Conference No. 12.

Multiply

value

equivalent

value

given

of Units

in U.S.

[Syste'me

on Weights

Customary

International and

Unit

in SI unit.

xxxi

Measures,

by conversion

(S1)] was Paris,

factor

adopted

October

by the 1960,

to obtain

in

1 INTRODUCTION

]. 1 GENERAL

This handbook sandwich

presents practical methods

for the structural stabilityanalysis of

plates and shells. The configurations and loading conditions covered here

are those which are like]y to be encountered tions, design curves,

For

the

purposes

construction depicted and

verse

shear the

beam. while

the

Numbers section

The

facings

primary

in brackets (1;

core

a structural

sandwich

thin

to a comparatively

the

and

concept

perform

difference

[ ] in the

a function between

text

these

denote

2; etc.).

1-1

core

to that

is analogous very two

references

much types

listed

as bending

to position all

against

the

of the local

transbuckling.

of a conventional

to that like

core

of the over-all

virtually

the facings

similar

as a layered thick

serves

provides

stabilizes

which

all

The

axis,

is quite

a role

is defined

practically

sandwich.

neutral

sandwich,

plays

facings provide

to the

from

sandwich

sandwich

sandwich

rigidity

of the

structural

two

applications. Basic equa-

of theory against test data are included.

The facings

removed

rigidity

The

flanges.

1.1-1.

extensional

at locations

handbook,

by bonding

in Figure

faces

Thus

of this

formed

in-plane

and comparisons

in aerospace

that

of the

I beam

of the

I beam

of construction

at end

I

of each

lies

web

in the

major

fact

that

the

behavior;

case

of

transverse

shear

whereas,

for

relatively

deflections

I beams,

short,

deep

these

The

of

can

sandwich

materials

be

sandwich

conducive

The

is

use

innovation.

an

and

achieved.

is

attractive

Since

particularly

deflections

Typical

are

Sandwich

structural

design

constructions

rigidity

well

significant

only

to the

important

sandwich

for

the

special

J

1.1-1.

geometry,

usually

beams.

FACING"

Figure

are

is

suited

having

required

to

Construction

concept

high

to prevent

applications

since,

ratios

by

of

the

proper

choice

stiffness-to-weight

structural

where

the

instability,

loading

the

conditions

to buckling.

of sandwich

The

construction

British

de

in aerospace

Havilland

Mosquito

1-2

vehicles

is

certainly

bomber

of World

not

War

a recent

II employed

are

structural form

sandwich

of birch

ing the

face

B-58,

weight

the

Centaur

and other

information that

need.

problems ferred

which

for

it should

be kept

fully

to as a "best-available"

numerical in this

computations handbook

as

sections

(SLA) tank

latest

and

This

in mind

that,

in many

and

one

In these

with

cases

Such the

have

it has

analysis

become

analyst

all

need

the

this

to fulfill

practical

what

might

it is advisable

appropriate

desirfor

is meant

areas, employ

areas

on the

criteria

document

can only

strength-to-

fairings

stress

use.

high

bulkheads.

sandwich,

and

in the includ-

applications

Adapter

design

was

airplanes,

of the

vehicle

designer

sandwich other

advantage

propellant

testing.

dealing

Many

rapid

resolved

suitable

the

of structural

approach.

with

in the

taken

practicing easy,

core.

LM

presents

The

case,

Space

as well

application

suitable

not yet been

have

construction.

construction.

However, have

etc.,

vehicles,

increasing

in a form

wood

the Spacecraft

a handbook

of such

In this

to a balsa

spacecraft,

ever

airframe.

C-5A,

launch

to assemble

stability

bonded

by sandwich

Apollo

of the

the

F-Ill,

enjoyed

included

able

sheets

B-70,

ratio

In view

throughout

be re-

to supplement

of uncertainty

are

configurations

identified and loading

conditions.

In the

sections

equations

along

is followed facilitate the range

list

to follow with

by the their

of references of loading

conclusions

design

use,

a discussion

derived

equations

a table

and

from

along

of these

in Sections

conditions

is given

2,

with

of the

basic

an analysis any

limitations

equations

and

restrictions

3,

5 since

4,

and

considerations. 1-3

these

principles

behind

of available

test

on their

use.

immediately sections

cover

the

design

data. Also, precedes a wide

This to

1.2 FAILURE MODES Structuralinstability of a sandwichconstructioncanmanifestitself in a numberof different modes. Thevariouspossibilities are as describedbelowandas shownin Figures1.2-1 through1.2-3. Intracellular Buckling which the

occurs

regions

facings

only

buckle

Wrinkling

form

of short

of wrinkles

final

surface failure

tensile

rupture

if proper ably

care

assume

pressive

is a localized

involves

may

core,

is exercised that

strengths

the

in Figure

original

wrinkling

of the

the

be either

bond core

must

or

result rupture

selection strength

will

proper.

1-4

itself cells

to facings)

straining

the

As shown from

of the

core-to-facing

adhesive both

The mode

manifests

antisymmetrical

exceed

supports.

to individual

consider

with

the

respect

core,

bond.

tensile

to the

1.2-2,

of the

one

of the occurrence

in Figure

system,

in the

of

possible

crushing

in

the

the buckling

either

of the

1.2-1,

core),

as edge

which

(normal

one

in Figure

precipitate

sandwich.

usually

in the

tensile

of the

1.2-1,

tensile

acting

is not confined

symmetrical

will

walls

transverse

of instability

of a honeycomb

of instability

undeformed

or

cell

mode

mode

As depicted

eventually

in the facings,

and

is a localized

as those

the

can

- This

of the from

buckles

wrinkling.

As shown which

(such with

as face

cores,

material.

cells

fashion

of these

- This

is not continuous.

core

wavelengths

cellular-type

core

in plate-like

below

Face

Dimpling)

the

above

growth

identified

middle

when

directly

progressive

core

(Face

However, can

and

reason-

com-

A - Intraceilular Buckling (Face Dimpling)

SYMMETRIC

Figure

ANTISYMMETRIC B - Face

Wrinkling

C - Shear

Crimping

1°2-1,

Localized

A - Core

Modes

Crushing

B - TensileRupture of Bond Figure 1.2-2.

Instability

C - Tensile Rupture of Core Proper

Ultimate Failures Precipitated by Face Wrinkling

1-5

i ,ut|

,.,_|

"_Sl

I .mm_' ..--m_ D"--'4

L_

C

!

o=

Z

4_

1-6

Shear

Crimping

-

Shear

is actually

a special

very

due

short

occurs also

quite cause

occur

the

suddenly

the

appear the

mechanism

1.2-3A. with

-

Except the

larger

than

configurations the

term

having general

to an additional

the

general

will first

is

it may

sometimes

develops. shear

general

but

phenomenon

however,

In such

stresses

buckle

to an erroneous

may

at dis-

conclusion

instability bending

shear

deformations.

a significant

role

localized

phenomena,

special

identified

in this

case

mode of the

under

takes as

on new panel

1-7

composite

as to

significance

instability.

is of a more

are

at locations

For

"Shear normally

and face

other and

exten-

intracellular

identification

buckling

as

coupled

transverse

Whereas

instability

(such

in Figure wall

instability the

in intracellular

stiffening

is depicted

behavior.

general

stiffening

Usually,

general

cited

with

encountered

instability

no supplementary

over-all

supplementary

mode

This

transverse the

of failure

wavelength

in shear;

instability

lead

having

associated

those

core.

Crimping

local

then

the buckle

the

bond.

mode

failure.

the

wavelengths

core

develops,

would

to facings)

for

to fail

crimp

involves

are

the

general

configurations

do not play

nature.

siderably

For

(normal

and wrinkling

Crimping",

made

initiated

phenomenon

strains

gross

aries,

which

transverse

buckling

examination

which for

of severe

As the

at the boundaries,

The

sional

For

patterns.

for

core-to-facing

because

to as a local

modulus

long-wave

appears

Instability except

instability

causes

in the

relatively

crimp

referred

shear

usually

failure

and a post-test

rings)

of general

and

where

of buckle

General

is often

to a low transverse

in cases

ends

form

a shear

instances

crimping

than

reference this

case,

conwrinkling.

the boundis also general

instability is as definedabovebut with the addedprovisionthat thebucklepattern involvessimultaneousradial displacementof boththe sandwichwall andthe intermediatestiffeners. As shownin Figure 1.2-3B, the appropriatehalf-wavelength of thebucklepatternmustthereforeexceedthe spacingbetweenintermediatestiffeners. Theexampleusedin Figure 1.2-3B is thatof a sandwichcylinder stiffenedby a seriesof rings whichhaveinsufficientstiffnessto enforcenodalpointsat their respectivelocations.

Panel

Instability

supplementary depicts series

this

stiffening mode

of rings.

nodal

points

tion.

Therefore,

between

rings.

bending

of the

again,

- This

using

However,

composite

case wall

extensional

than

example

case

the

locations.

half-wavelength

As in the

applies

other

the

in this

respective

the

of instability

at locations

by again

at their

transverse

mode

of general

the boundaries.

rings

have rings

instability,

this

transverse

strains

do not play

cylinder

no

cannot mode

shear

a significant

radial

exceed involves

deformations. role

which

in the

have

1.2-3C

stiffened

stiffness

experience pattern

with

Figure

sufficient

buckle

coupled

1-8

to configurations

of a sandwich

The of the

only

by a

to enforce deformathe

spacing

over-all Here behavior.

2 LOCAL

2.1

INTRACELLULAR

2.1.1

Sandwich

2.1.1.1 From

the

viewpoint,

Even

honeycomb

assumption. expressed

Honeycomb

(Face

Dimpling)

Core

Principles

a practical

behavior.

BUCKLING

with

Basic

INSTABILITY

where core

intracellular

curvature

cell

As noted

size from

buckling

is present,

will

as

normally

Reference

be

2-1,

can in the

be regarded cases

sufficiently

the

critical

as flat-plate

of cylinders small

stress

and

to justify for

flat

such

plates

spheres, an can be

in the form

Crcr -

12(1-v0)

(2.1-1)

_s

whe re

%r k

= Critical

compressive

= Coefficient conditions,

reduction

= Young's

modulus

cellular ve

= Elastic

psi.

which depends on the plate geometry, and type of loading, dimensionless.

= Plasticity Ef

stress,

buckling), Poisson's

intracellular

factor,

boundary

dimensionless.

(for facing

material

in the

case

of intra-

psi. ratio

buckling),

(for

facing

dimensionless.

2-1

material

in the

case

of

tf

= Thicknessof plate (Facingthicknessin the caseof intracellular buckling), inches.

s

- A selectedcharacteristic dimensionof the plate, inches.

It is convenienthereto combineseveralof the constantsin Equation(2.1-1) to obtain r/Ef _tf_ _ (1- Vea )

(2.1-2

\s/

O_r

K

%r

(l-re)

)

or

= K

(2.1-3)

_TEf

To apply define

these

Norris

this

is taken cell.

case

s

equal

to the

test

the

choice

results

fall

formula.

This

honeycomb

sandwich

sufficiently

large

Figure

2.1-1,

considerably design

curve.

diameter

of the

This which

of

K = 2.0

will

could

reasonably

Under

the

buckling these

since not

remain

conditions,

then

from

than some

would

bound

for

fit to the

test

2-2.

the results

It should

to the data.

failure

Six of

cells

in a

so long by the

of unbuckled by the

of stress

be

recommended

of several

As indicated

redistribution

within

K = 2.0

by the

be indicated

to

In Reference

be inscribed

Reference

majority

necessary

By convention,

good

to catastrophic

the

K.

chose

predicted

unbuckled.

2-2

can

the dimpling

lead

expect

strengths

that

a lower

values

for size.

a reasonably directly

can be tolerated

of cells

Norris

it is only

cell

circle

not provide

below

construction

greater

taken

does

significantly

data,

value core

largest

provides

was

number

one

honeycomb

of test

2.1-1

buckling,

a corresponding

to the

analysis

situation

of intracellular

and establish

compression.

in Figure

that

case

to be equal

on the

of uniaxial

noted

s

took

Based

as shown

the

to the

the dimension

2-2,

the

equations

as a

cells

scatter

to possess

proposed would

in

occur

0.1

Theoryfor Simply _Suppor_d SquarePla_

0.01

Legend: Dimplingin Elastic Range • Test Datafor SpruceCorewith SingleCircular Hole • Test Dimpling

Data for Beyond

Honeycomb Core Elastic Range

O

Test Data for Single Circular

Spruce Hole

[2

Test

Honeycomb

Data

for

Core

with Core

0.001

0.0005 0.01

1.0

Figure

2.1-1.

Critical Under

Stresses Uniaxial

for

Intracellular

Compression

2-3

Buckling

but the structure couldcontinueto supportthe appliedload. In addition, it is pointed out that thedimpledregionsretain significantpost-bucklingload-carryingcapability sincetheybehaveessentiallyas flat plates. This doesnot mean,however,that one canpermit the dimplesto growwithoutbound. The pointcanbe reachedwherethese deformationsprecipitatewrinkling andthis cannotbe tolerated. It is alsoof importanceto noteherethat mostof the test datashownin Figure 2.1-1 wereobtainedfrom sandwichplateshavinga solid sprucecore throughwhicha singlecircular holewasdrilled to representa core cell. It is questionable that suchspecimenstruly simulatethe cell edgesupportlikely to be encounteredin practical honeycomb configurations. Onlythree datapointswereobtainedfor specimensactuallyhavinghoneycomb coresand, as shownin Figure 2.1-1, thesepoints lie in the lower regionof the total bandof scatter. In viewof theforegoingdiscussion,it is evidentthat the useof Equation2.1-3 togetherwith the selectionof K = analysis this

area,

approximate

of intracellular

buckling.

it is recommended design

that

2.0

is certainly

However, this

criterion

tool.

2-4

not a rigorous

until

further

be employed

work as

approach

to the

is accomplished a 'best-available",

in

2.1.1.2 DesignEquationsandCurves Thefacingstress atwhich intracellular bucklingwill occur underuniaxialcompressionis givenby thefollowing semi-empiricalformula:

_cr The the

dimension cell

s

shape.

measured

is the

For

diameter

example,

as shown

= 2.0

of the in the

(l_Pe_) _Ef

(_)_

largest

circle

cases

(2.1-4)

that

of hexagonal

can be inscribed

and

square

cells,

within s

is

below.

@ Figure

Solving

Equation

(2.1-4)

for

2.1-2.

s

Definition

gives

the

of Dimension

s

result 1

s = tf

This

equation

sponding

may

to particular

family

of plots

0.001

to

For

elastic

Section

be used

tf

facing

of Equation

[_cr

,,_

to determine

the

materials (2.1-5)

for

and

(2.1-5)

_(1- Ve_)]"j -g

maximum

permissible

thicknesses.

selected

values

Figure of

tf

cell 2.1-3

ranging

size

corre-

presents from

tf

a =

= 0.100.

cases,

9 must

use

_? = 1.

Whenever

the

be employed.

2-5

behavior

is inelastic,

the

methods

of

2.00

1.00 0.80

0.60

0.40 0.30

0.04 0.03

0.02

1

2

"0Ef

Figure

2.1-'3.

Chart

for Determination

Intracellular

Buckling

2-6

of Core Will

Not

Cell Size Occur

Such

That

When use

the the

facings

are

interaction

subjected

to biaxial

compression,

it is recommended

that

one

formula

R x + Ry

= 1

(2.1-6)

whe re

Applied Compressive Loading] in Subscript Direction J Ri

This

straight-line

Reference are

2-1

coplanar

computed ever should

for with

and

one

that

= [Critical [acting

Compressive Loading (when] alone) in Subscript DirectionJ

interaction

relationship

square

plates.

flat

the facings, these

values

of the

principal

stresses

be based

on the

assumption

is based For

cases

be used is tensile that

the

alone.

2-7

that

in the and

information

involving

it is recommended then

on the

shearing

the principal

above

interaction

the behavior

compressive

(2.1-7)

provided stresses

which

stresses

first

equation.

is elastic,

principal

in

stress

the

be

Whenanalysis

is acting

2.1.2 Sandwich With CorrugatedCore 2.1.2.1 BasicPrinciples This sectiondealswith corrugated-coresandwichconstructionswhosecross sections maybe idealizedas shownin Figure 2.1-4. here For

is that flat

where

plates,

or transverse

the axis however,

of the the

For

corrugations

corrugations

cylinders,

is parallel can

case

to the

axis

of revolution.

in either

the

Doub le-T

russ

of the following

2.1-4.

loading

Corrugation

conditions

is considered:

Uniaxial

compression

acting

parallel

to the

axis

b.

Uniaxial

compression

acting

parallel

to the

facings

axis

The design entirely

longitudinal

Configurations

a.

c.

treated

directions.

Figure

based

only

be oriented

Single-Truss

Each

the

of the

Biaxial curves

of the

corrugations.

but normal

to the

corrugations.

compression presented

on theoretical

resulting here

are

from

taken

considerations.

combinations

directly

from

No comparisons

2-8

of

a

Reference are

and 2-3 made

b

above. and

are

against

test

datato confirm the validity of thesesolutions. Until suchsubstantiationis obtained, the recommended designcurvescanonlybe consideredas a 'best-available"criterion. It is pointedout, however,thatthere doesnot appearto be anyreasonto suspectthat test datawoulddisagreewith the curves. AlthoughReference2-3 is devotedsolelyto flat plates, the results are consideredto be applicableto thecylindrical configurationsshownin Figure 2.1-4 sincethe dimensions bf will usuallybe smallwith respectto the radius. Undersuchconditions, curvatureinfluenceswill be negligible. Thetheoreticaldevelopmentincludesconsiderationof eachof the bucklingmodes shownin Figure 2.1-5. Both of thefollowing possibilities are covered: a.

Theface sheetsare the unstableelementsandare restrainedby the core.

b.

Thecore is the unstableelementandis restrainedby theface sheets.

Bucklingis assumedto be accompanied by rotationof the joints but with nodeflection of the joints. Theanglesbetweenthe various elementsat anyonejoint are takento remain unchanged duringbuckling. It is alsoassumedthatthe over-all sandwich dimensionsare sufficientlylarge suchthatendeffectsare negligible.

2-9

cl

-oL

Q

ci

cl

ol

01

Clomped

oL

-cl

o_

ci

-_

rz

Cl

_1

ol

-cl

i1

ol

- rx

rl

o.

-el.

i::i

cl

-cl

ol

-cI,

-_

_1

cl

__ply

Single-T (a,

fl,

and

t

russ-Core

Double-Truss-Core

q_ denote

-oI

cl -_

-el

,gl cl

angles

Figure

of rotations

2.1-5.

Buckling

2-10

at the

appropriate

Modes

joints)

2.1.2.2 DesignEquationsandCurves Thetheoreticalstress at whichintracellular bucklingof thefacingsor bucklingof the corrugatedcore will occuris givenby the followingformula:

°_cr -

kl zru 77E

/tf_

12(1-V0)

\bf]

_

(2.1-8)

where = Critical

%r ki

The

only

and the

= Coefficient conditions,

_7

= Plasticity

E

= Young's

stress,

which depends dimensionless. reduction modulus

Poisson's

ratio

tf

= Facing

thickness,

inches.

bf

= Pitch

of corrugated

considered

here

sandwich

2.1-6

through

is that

construction 2.1-12

give

upon

and core,

of facings

core where (facings

values

for

(see

The

kx

when

k_

= 0

b.

kx

when

k_

= 0.5

c.

kx

when

k_

=

1.0

d.

ky

when

kx

=

0

k_

and

coefficients

k_

are

defined

the

two

and

k i for

core,

2.1-4),

facings

and core)

as follows:

2-11

and

loading

psi.

Figure

combinations:

a.

the geometry

dimensionless.

of facings

= Elastic

case

psi.

factor,

re

entire

Figures

compressive

are

is made each

psi.

inches. of the

same

of a single

of the following

thickness material. loading

12(1-Ve kx -

_ ) (bf._ _

rr _rIE

_-f_

re2 ) ky - 12(1_ 17E

The subscript

x

(for k

and

(Applied

Compressive

¢rx)

(2. I-9)

(Applied

Compressive

O'y)

(2.1-10)

k') is used to identify cases where the loading is

directed along the axis of the corrugations and

k') is used to identify cases where the loading is acting in the y

is parallel

to the facings

a

c,

through

core

plots

configurations.

The

regions.

Above by the

by the face

sheets.

to

the

the

core.

are

shown sample

furnished

lines, the

given

face

family

applied

load

through

2.1-11

sheets

dashed

lines,

in Figures

corrugations.

are the

the

y

and of curves

double-truss-

is transferred

unstable

combinations

covers

divide

is unstable

2.1-6

through

2.1-12,

the

corrugations

both

through

the

charts

elements

core

(for k

direction which For

single-truss-core

a single

2.1-6

the

of the

for

d,

in Figures

charts

axis

corresponding

Below

= Thickness (see Figure

the

of the

dashed

definitions

are

to the

combination

lines

design

_b = Angle In addition,

all

dashed

restrained

additional

For

since

facings.

clarify

but normal

separate

arrangements

To

(x direction). The subscript

and

the

the into

and

in Figure problem

from inches.

which

2.1-4, given

are

formed

following

to the

user

degrees.

below

handbook.

2-12

should

be helpful

are

is restrained

provided:

of material 2.1-4),

two

of this

Given: SampleProblem Data for Single-TrussCore Type SandwichPanel E = 30

x 106p si

v e = .30

Proportional Required:

Limit

Find

k_

acr x ;

= .016"

tf

= .020"

90,000

psi

Assuming

12ay(1-re2) 7r2_E

=

%=.o16

_k_2 \tf /

¢

(ry = 16,300

_? = 1,

=

bf = .700"

one

linear

interpolation

k x = 2.68.

Hence,

the

obtains

12 x 16,300 x ,910 9.87 x 1 x 30 x I0'

critical

between

stress

values

in the

given

x direction

on

_.70012 \.020/

aCrx assuming

_Crx

=

stress

_ = 1,

one

-

Figures

(parallel

kx _2_ E

The

(Compression)

= 0.736

.020

obtains

and,

psi

= 65 °

= .800

tf Using

a=

to

12(1-Ve2

to the

ai

and

2.1-8

corrugation

one

axis)

is

/tf_ 2 ) _bf/

obtains

2,68 x 9,87 x 1 x 30 x 106f.O20h 2 12 x . 910 \. 700/ = 59,300 intensity

2.1-7

(See

Section

9)

can

now

be

psi

computed

(Compression) as

follows:

o"i = _/O-x2 + O'y2 - ax(_y + 31"2

=

Since In

this

cases

9 must

103_

value where

be

f

(59.3) 2 + is below the

qi

(16.3) the

value

2 -

(59.3

x

proportional exceeds

the

16.3)

limit, proportional

employed.

2-13

+ 0 = 53,100 the

assumption limit,

psi _ = 1 is

the

methods

valid. of Section

o I

E_ I

o

_d !

_4

©

2-14

0

¢

hL

.E

!

2-15

r..

i 0

?

<.i.-.i H

2

r,D

i

I.-i .,.-I

0

,,,.,

2-16

0

?

O

=

2

O

I

_9

_D

t_

re)

2-17

_xj

0

=1

,g 11 eo

".>

g

!

d

.,,-i

©

2-18

[

ll[i

,.

I

;I

1ii 1_]

:iiiiiii

iiiitiii :_!!!!!!! :::::::::

iiiiiiiii .'__!!!!!!!

!N!!iiJi iiNiiiii H Mili_iii J[l{I {1ill '_ILI ] I IAI I%J I I ]/I_l

I I I I

ll_lllll [ I I_I I Ii I t I i_i_ll I [nlL[ i [l I kl l_l t'_ll 1 IN ]l I_ I I |k] i I I_1 II Ill

U

i 1 i ll_i 1 111 I I I • II III ! [_1 t i lil.l

i

] /!/ i i

i

!11]_1|1 l_Jl

1 I Illi

1

_ i i 1 IRl.Jll i flllll_ I ] 'WL] I ] I II{ J {

iiiiiiiii

iiiiiii;.: ,,!!144_

2-19

I

_:..i .......l.... I .... H

__._-_-....

-+

-: .......

ZZ<01 _....

C_ I r_

0

i::::b:::

....

....

"X

U

I

.....

+ ...............

_.

_

0 i_

f_J

c
2-20

2.2

FACE

2.2.1

WRINKLING

Sandwich

2.2.1.1 The

With

Basic

problem

far

as

2-4

through

The

initial

thick,

the

same

critical

wrinkle

The

isotropic

the

been

Wrinkling)

pattern solid

governing subjected

handbook,

or foam

of the for

the

other

the

facing.

or foam

cores

equation

was

The

development

modes out that,

and strains these

to predict

back

that

the only

there

results to sandincludes

along

with

the

the

core

is sufficiently

when

of each

influences

other

and

modes.

However,

introduced

by one facing

influ-

conditions, to wrinkle this

form

it was

found

that

antisymmetrically. of wrinkling

for

1

{rwr

:

_L

Ec Gc_ _

(1-Pea)_

whe re, Crwr

= Facing

17 = Plasticity Ef

= Young's

wrinkling reduction modulus

stress, factor, of facing,

psi. dimensionless. psi.

2-21

the

antisymmetric

compression:

c_ _r]Ef

as

as References

applies

be independent

can be expected

to uniaxial

decided

cores.

Under

derived

listed

latter

will

core

are

The

symmetric

cores,

subject

dating

useful.

It is pointed two facings

investigators

it was

and antisymmetric

facings.

in the

by many on this

be the most

symmetric

thinner

treated

of this

solid

is obtained

having

facings

have

patterns

having

following

(Antisymmetric

publications

would

of the

load

the wave

sandwiches

Core

purposes

2-9

which

waviness

sandwiches

the

and

of both

has

important

For

configurations

from

ence

most

2-7

consideration

for

wrinkling

2-14.

in References wich

or Foam

Principles of face

1940.

Solid

(2.2-1)

= Young's

E c

Core

Gc

_e The

modulus shear

parallel

to the direction

Elastic

Poisson's

is the

in the

associated

and

Q

core

modulus

facings

quantity

of the

ratio

relative

with

_ 30V ÷ 16q

perpendicular

applied

respect

cosh

sinh_-

( 11cosh_sinh

KS_

plane

to the facings,

load,

psi.

to the psi.

dimensionless.

with

(

the

normal

of the

of facings,

minimum,

1 4 6.4

direction

to

1

_ , of the

expression

5)

(2.2-2)

_4 1 5 )

whe re

tc q = _f Gc

K8

I (1-Ve_)13 LrlEf Ec Gc]

(2.2-3)

6Ec - tc Fc

(2.2-4)

and -

Parameter buckle

tc

involving

Thickness

of core,

tf

:

Thickness

of facing,

6

:

Amplitude

of initial

Fc

the

wavelength,

: Flatwise

core

elastic

moduli,

core

thickness,

and

inches. inches. waviness

sandwich

flatwise

core

dimensionless.

strength

tensile,

in facing, (the

inches.

lower

and flatwise

of flatwise

core-to-facing

core

compressive,

bond

strengths),

psi. The

initial

waviness

causes

transverse

small.

As the

lead

to transverse

plays facing

load

an important deflections

increases, tensile

these

role

in the

to develop deflections

or compressive

failure

2-22

wrinkling

even grow

when

phenomenon the

applied

at steadily

of the

core

or

since loading

increasing tensile

rupture

it is very

rates

and

of the

core-to-facingbond. Thesefailures occur, of course, at loadvaluesbelowthe predictionsfrom classical theoryin whichit is assumedthat thefacingsare initially perfect (K6 = 0).

The

results

from

accompanied is given The

by plots

crimping antisymmetric

practice

data.

This

are

since

the

design

here

but

existing

configurations can only

be

of the

by the

straight

Section

conservative

2.2-2

which

the

same

made

by

data

likely regarded

value

are

for

Plantema for

were

data

test

value

given

in Reference

from

to be encountered

design

2-15

in realistic

as a "best-available"

that

2-6 This

that

to be

rarely of face

become

on available from

been

which

the

selected are

value

In view

since

2-9 as a

with has

much

of the

representative

of

the selection of the

test

not shown

Q = 0.50

not very

com-

Reference

is in conformance

However, were

are

it has

has

al"e for

prediction

selected

= 0.50

2.2-1.

do not prove

the

curves

shear

curves

based

structures,

approach.

2-23

Q

purposes.

specimens

type

for

in Reference Q.

to the

on the

cores,

data

Q

a lower-bound

practical

obtained

means or foam

of such

structures

lower-bound

in Figure

to the

of this

solid

Elastic

Additional

curves

(2.2-1)

in Figure

correspond points

a practical

here.

value.

shown

to particular

having

from

0A

of Equation

A family

All other

appropriate

constructions

form

shape

line

practice,

to provide

a single

general

2.3).

In actual

in order

recommended

test

are

in the

as a parameter.

is followed

lead

been

K8

approach

observation

often

(see

be summarized

with

K 8 values

to select

plotted

safe

the

of failure

in sandwich

mon

q

established

Therefore,

wrinkling

vs

can

and they

wrinkling.

helpful

known.

Q

2-9

values

mode

2-9

of

in Reference

limiting

very

Reference

of Q = o. 50

uncertainties

K6 = 0

Q

K 8 = Constant K 6 = Constant 0

q

Figure

involved,

it is recommended

be performed The

method

an approximate

on specimens presented

here

2.2-1.

Typical

that

for

which for

the

are

Variation

the verification truly

prediction

guideline.

2-24

of

Q

vs.

of final

representative of wrinkling

q

designs, of the

should

wrinkling

actual only

tests

configuration. be

regarded

as

o

.i 0 O

O

0

°,-_

O

o 0

Qo

¢.0

5'q t

2-25

¢.j

2.2.1.2 The

Design

following

stress

Equations equation

may

face

wrinkling

at which

foam

cores

and

Culwes

be used

to compute

will

occur

where

Figure

2.2-3

mended

that

For

When the

the

amplitude

to establish the value

elastic

Section

approximate

in sandwich

uniaxial

constructions

compressive

having

solid

or

:

awr In cases

the

cases,

9 must

waviness

Whenever

Q = 0.50

}7= 1.

such

be used

(2.2-5)

is known,

one

information

to obtain

can

the behavior

to biaxial

compression,

the

is unavailable,

a lower-bound

Whenever

use

curves

of

it is recom-

prediction.

is inelastic,

the

methods

of

be employed.

the facings

interaction

of initial Q.

use

= Q L (1-pe 2

are

subjected

it is recommended

that

one

use

formula Rxa + Ry

= 1

(2.2-6)

whe re

Ri= and

the

action

y

flat

plates

are

coplanar

ever should

iCritieal

Compressive Loading (when Subscript Direction

corresponds is based

having

very

with and that

one

Compressive

direction

relationship

computed

[Applied

aspect

the facings, these

of the principal

be based

to the

on the

large

on the

Loading

values stresses assumption

in Subscript

direction

information ratios.

be used is tensile that

that

in the

the principal interaction

behavior

(2.2-7)

compression.

involving

above

and the

in ?

in Reference

cases

the compressive

2-26

alone)

of maximum

it is recommended then

acting

provided For

Direction]

2-1

for

shearing

stress

which

first

equation. the

inter-

rectangular

stresses

stresses

is elastic,

principal

This

be

Whenanalysis

is acting

alone.

I

0

_o

/

2_

//

\

d

O0

L"-

(y

2-27

2.2.2

Sandwich

With

2.2.2.1

Basic

As noted

in Section

figurations

which

is capable

Honeycomb

2.2.1.1, have

2-7.

nize

honeycomb

very

small

Full modes

in this

case

symmetric

facings

The

2-7

extension core

along

with

it was

found

wrinkling

Wrinkling)

resulted in sandwich

of Reference

cores.

will

core

given

influences

that,

except

to both

for

at stress

occur. following

constructions

moduli the

the levels

Based equation having

for

in the

are

cores

which

facings.

lower

than

to the

crimping

subjected

(low q),

at which

the development

and

wrinHowever,

those

of wrinkling

recogare

normal

by shear

prediction

honeycomb

is accomplished

to the facings

of the

conreport

and antisymmetric

observation, the

this

direction

controlled

which

on this

parallel

waviness

region

and

of that

conditions

symmetric

initial

to sandwich

theory

cores

incorporating

elastic

only

the basic

in the plane

from

apply

honeycomb by

moduli

the

was

the

in the

is achieved

with

2-9

However,

having

elastic

develops

mode

results

or foam

consideration

antisymmetric ence

solid

in comparison

facings. kling

the

to constructions

in Reference the

(Symmetric

Principles

of extension

that

Core

of Referfor

isotropic

to uniaxial

compression: 1

082(Ec ( Ef, awr

= .

\r/Ef te/ 1 + 0.64

K8

(2.2-8)

where 8 Ec K8

-

tc Fc

2-28

the

(2.2-9)

and awr

= Facingwrinkling stress, psi.

Ec

= Young'smodulusof the core in thedirection normal to the facings, psi.

tf 77

cal

=

inches.

of initial

waviness

psi.

can be used

to plot

is based

curve obtained

agrees

be noted on the very

that

and

awr Comparison gives

critical

Bartelds

Numbers section

of Equations stresses

and

Mayers

in brackets (1;

2;

(2.2-8) which

of design

the

curve

for

that

the

Mayers

= 0.86

are

[2-14]

of the form

facings

compresbond

shown

is an upper-bound are

initially

symmetrical

classi-

perfect. wrinkling

(r/Ef)

[ ] in the

text

denote

shows

that,

5 percent

references

2-29

This equation

(2.2-10) when less

K8

= 0,

the former

than

those

obtained

[2-14].

etc.)

in

:

L_TEf tc j

approximately

curves K8 = 0

the following

(2.2-10)

inches.

(the lower of flatwise core and flatwise core-to-facing

a family

with

and

in facing,

strength tensile,

assumption

closely

by Bartelds

psi.

of core,

strengths),

It should

dimensionless.

of facing,

= Flatwise sandwich sive, flatwise core

which

recently

Thickness

inches. factor,

modulus

= Amplitude

2.2-4.

particular

reduction

tc

(2.2-8)

value

= Plasticity

= YoungTs

Fc

Figure

of facing,

Ef

8

Equation

= Thickness

listed

at end of each

major

by

K(_ = 0

= Constant

K 6 = Constant

i

_TEf t c ]

Figure 2.2-4.

In actual helpful

practice, since

Typical Design Curves Constructions Having

curves

the

K5

values

Therefore,

in order

in sandwich

constructions

in this are

handbook.

plotted

elastic these

data

were

data

points

are

One

point

mum, a similar wrinkled

and

discarded. plotted

under

in Figure

means

test

remaining

2.2-5

for

by

group

maximum

test

value

for

for

the

average.

The

data

from

inelastic

added

restrictions.

conditions.

Since 2-30

of face

wrinkling

group,

A number crude

is taken

2-7

of shear

Reference

rather

known.

2-7

failed

and

one point 2-10 of these plasticity

three

specimens.

for

were

the

and 2-7,

identical

2-10

within

crimping

in Reference

of nominally

the

rarely

References

reported

to be very

approach

Reference

means

tests

each

are

prediction

from

from

the

several

the

selected

occurred

do not prove

a lower-bound

specimens

failures the

for

cores, data

in Sandwich

structures

for

with highly

For

2.2-4

to particular

All of the

of these

Wrinkling" Cores

in Figure

honeycomb

purpose,

2.2-5.

Several

one point manner

this

shown

a practical

having

For

is plotted

type

appropriate

to provide

in Figure

range.

of the

for Face Honeycomb

the selected

specimens reduction

miniin

°1°l o

o

0

0 o

_-'

0

o

o

_._

_=

e_ o

r/3

o

•C_'_

_._ o

d

00 o 0

2-31

factors (77 Et/Ef) wereusedin thedatareduction, it wasdecidedto plot dataonly for thosespecimenswhichwrinkledat stress levels where (E t/Ef tion,

as

many

of the

test

by

flatwise

measured

for

those

specimens

specimens

compressive

proper

tensile

whose

one

can

plot

of

tensile

Adhesive

usually

2-10

strengths.

flatwise

strengths.

care,

of Reference

an

very

It was

therefor(,

strengths

were

technology

select

had

has

adhesive

poor

)

/> 0.85.

eore-to-faeingbonds

decided

at

now

advanced

system

which

In addi-

least

to

to plot

equal

data

to

the

.the point

satisfies

only

flatwise

where,

such

with

a require-

ment.

Based

on

the

Figure

2.2-5,

the

relationship 1

/Eetf O'w r has been

factor

selected

here

to provide

of approximately

this is not a rigorous

design

equation

representative

only be

0.4

regarded

licate the selected

regarded

selection

practical

wrinkling

sandwich

as an approximate

(_TSf)

values.

This

to this wrinkling

to the problem

as a '_est-aw_ilable"

fication of final designs,

_ f te /

safe design

wider

of contemporary

0.33\r/E

is applicable

approach

on a much

=

and

it would

implies

that a knock-down

phenomenon.

be advisable

of test data of specimens

designs.

approach

Therefore,

and

The

guideline.

2-32

Equation

method

on specimens

presented

Obviously,

to base

which

it is recommended

tests be performed

configuration.

(2.2-11)

were

the

truly

(2.2-11)

can

that, for veri-

that actually

here

should

dup-

only be

2.2.2.2

Design

The following stress

Equations

and

equation

may

face

wrinkling

at which

Curves

be used will

to compute occur

the

approximate

in sandwich

uniaxial

constructions

compressive

having

honeycomb

cores:

z/Ec 0.8 tc/ (,El) °'wr-

1 + 0.64

K(5

(2.2-12)

whe re (]Ec K6 In cases

where

equations

the

or the

stress.

Both

ever

the

used

to obtain

amplitude

curves

given

of these

initial

of initial

waviness

are

a lower-bound

elastic

Section

When the

cases,

9 must

the

facings

interaction

use

and

directly

2.2-7

from

one

can

either

to establish

that

the

these

the wrinkling

MIL-HDBK-23

it is recommended

use

[2-16].

following

Whenequation

be

prediction:

awr

For

is known,

2.2-6

taken

is unknown,

(2.2-13)

waviness

in Figures

figures

tc Fc

77 = 1.

= 0.33

Whenever

\_--_f tc/

(UEf)

the behavior

(2.2-14)

is inelastic,

the

methods

of

it is recommended

that

one

be employed.

are

subjected

to biaxial

compression,

use

formula 3

R x + Ry

= 1

(2.2-15)

where

Ri=

[Applied Critical

Compressive

Loading

in Subscript

Compressive Loading (when Subscript Direction

2-33

acting

Direction] alone)

(2.2-16) in]

CD

i

0

0

0

0

0

0

0

0

O0

4hl/,/l ll " / _o CD

LO

//

, /// z.//, / jjz// f

/

_I _

CO 0 0

o I>

0 C_

!

&

2-34

0 ?]

II

11 ¢£: 0

.....

!

oi

0

b b _J

0

and the

y

direction

interaction

relationship

rectangular

flat

stresses

which

stresses

first

equation. the

analysis

is acting

corresponds is based

plates are

having

coplanar

be computed

Whenever should

to the

one

on the

very with

large

be based

these

principal

on the

of maximum

information aspect

the facings,

and that of the

direction

values

provided ratios.

alone.

2-36

in Reference

For

cases

it is recommended then

stresses

assumption

compression.

be used

is tensile that

the

in the

compressive

2-1

involving that

and

This

the above

for shearing

principal interaction

the behavior principal

is elastic, stress

2.3 SHEARCRIMPING 2.3.1 BasicPrinciples To understandthe phenomenon of shearcrimping, onemustkeepin mindthatthis modeof failure is simplya limiting caseof generalinstability. Theequationsfor predictingshearcrimpingemergefrom generalinstability theorywhenthe analytical treatmentextendsinto the regionof lowshearmodulifor the core. For example,the theoreticalderivationof Reference2-17, as reformulatedin Section4.2. I. 1 of this handbook,yields the result that, whenthetwo facingsare of the samematerial, shear crimpingwill occurin axially compressedsandwichcylinderswhenever Vc _ 2

(2.3-1)

where _o V c

(2.3-2)

=

Crerimp h (ro = *)Ef_

2 xJtz t2 l___pe2(tl

+ re)

(2.3-3)

Gxz

(2.3-4)

h_ °'crimp

= Plasticity

t_ and

reduction

-

(tl + t2) tc

factor,

Ef

=

Young's

modulus

h

=

Distance

between

middle

surfaces

R

=

Radius

to middle

surface

of cylindrical

t_ =

Pe

=

Thicknesses

of facings,

dimensionless.

of the facings

psi.

(There

of facings,

sandwich,

is no preference

facing

is denoted

by the

subscript

1 or

Elastic

Poisson's

ratio

of facings,

dimensionless.

2-37

inches.

2.),

inches.

inches. as to which

'l'hickn_,ss

t C

Gxz

The

critical

stress

of

core,

inches.

,_. _r-< shear

modulus

associate,I

facings

crienled

in the

c:m

and

t)(_ dt, termined

from

when

the

Inequality

(2. :{-1)

holds

axial

the

% r'

and,

with

Kc

true,

the

plane

direction,

perpendicular

to the

psi.

eq_mtion

(2.3-5)

%

Kc: can

be

computed

as

follows:

1 (2.3-6)

Hence, '{; rimp _'_-7-

"i?, .......

Therefore,

can

be

under

when

written

axial

the

for

the

two

facings

c_'itk_'t!

",re

stres_

_r°

made

for

of

:

the

_h¢,ar

'rcri'nP

(2.3-7)

same

material,

crimpir_g

the

in a circular

following

equation

sandwich

cylinder

compression: h _

_r_r

An

I ' equlvalen_

je,"led

of

t}l(_

4

rest)It

[(_ _mif,__,'m

.'a{!til('

Illa[eFi:'J,

can

be

e>:t,",_i

:

obtained

!g_!,,ra!

_}l i,

(r

( ;_n

',vl'Jte

rrcr

::

. crimp

7t:

from

Referen¢'e

pressure.

Tim!

(r crimp .

-

(t t

_ixz

! t>_) t c

.)

:_-t

,_,

• .

_ for

,.:i_,,c_,

+ re)t

e

iated

with

(2.3-8)

sandwich

the

two

cylinders

facings

sub-

are

made

(2

O.vz

3-9)

x\' h e 9"e

Gyz

;'-,_',,

sl_( :z_' ul _d_!t

'l'
s us,;¢)( psi.

the

tqane

perpendicular

to the

In addition, circular same

the

development

sandwich material

cylinders

It should

be noted

sandwich

cylinders,

equations

under

2-19

pure

leads

one

and

having

torsion

to the both

following

formula

facings

made

for

of the

:

-rcr

these

of Reference

that,

have

all

= "rcrim

p

-

although

Equations

of these

final

a general

h2 (t 1 + t2) tc (2.3-8)

expressions

applicability

which

figuration.

2-39

JGxz

through are

(2.3-10)

GY z (2.3-10)

independent

is not limited

were

derived

of curvature. to the

cylindrical

for Thus, con-

2.3.2

The

Design

Equations

following

equations

crimping

mate

ao

will occur

may

be used

in sandwich

to compute

constructions

the facing

having

stresses

both facings

at which

made

shear

of the same

rial:

For

uniaxial

compression

acting

coplanar

with

the

facings

(see

Figure

2.3-1),

use h _

_crimp

=

(t_ + t_)

tc

Gij

(2.3-11)

where Gij

=

Core

shear

the facings

modulus

associated

and parallel

with the plane

to the direction

perpendicular

of loading,

O', psi

o-,

ps

o-,

.i___

psi

_

Figure

2.3-1.

Unia×ial

2-40

Compression

psi.

to

For

bo

pure

shear

acting

coplanar

Tcrimp

with

-

the

facings

(see

Figure

2.3-2),

use

h_ (t_+ t2) tc _GxzGyz

(2 3-12)

X

/

Figure 2.3-2.

Pure Shear

The foregoing equations are valid regardless of the overall dimensions m

addition, no knock-down

initialimperfections.

factors are required since shear crimping

of the structure. is insensitive to

The predictions from these equations will be somewhat

conserva-

tive since their derivations neglect bending of the facings about their own middle surfaces.

Although such bending is of negligible importance

phenomena,

Further on general

to most

sandwich buckling

in the case of shear crimping thls influence can be considerable.

mention

of the

instability

shear included

crimping in this

mode

of failure

handbook.

2-41

is made

in the

various

sections

_4

=

i-

ECJ

o

2_ •

,_ ©

O

_

<

_

g

_D

• m

_ b_ _

.Z

b£.

o c-i

,"

b

_

r,9

o_

._ ¢-/

_

o

e

° m

._

b

._

c'd

la.,

el

Z

i_

iII

i ¢',1

g .-A

i

i ©

i,-ii )-'( I _,,._

e_

_'i

,--i

2 -42

;

i#_

_

_-)

r2_ L:I

=

4o

7 ¢o

o

°_,

°}

_'_

_

_ S_

•

I

_

m

a

_

"0

0

_ ._ ._f

•.,_,_-_ o _, ._

i "_

b _

_

_

_

i ,° _

._

._

:i

.

_

.

0

be

%

0

g

o

1ii c,|

2-43

g

g

8

RE FERENC

2-1

Gerard, Part

2-2

2-3

2-4

George I - Buckling

Norris,

C.

Forest

Service

M.

Sandwich

Plate,"

2-6

2-7

D.,

C.

Leggett, End

June

1941.

Hoff,

N.

and

Mautner,

Journal

of the

Norris,

C.

1810-A,

Yusuff,

S.,

K.

B.,

H.,

Edgewise

C,

Boller,

"Wz2nkling

1957.

Construc-

Materials,"

U.

S.

R-30,

of a Truss-Core 1959.

N.

A.,

Medium,"

"The

Journal

Stabilization of the

of a

Royal

Aero-

and Hopkins,

Loads,"

Sandwich Report

Establishment

'_rhe

Buckling

of Sandwich

Vol.

and Voss,

Subjected

"Flat

Aircraft

Sciences,

H.,

G.,

Royal

S. E.,

K.

H.

12,

A.

No.

W.,

3,

to Edgewise

July

Type

Panels No.

Panels,"

1945.

'WVrinkling

of the

Facings

FPL

Report

Compression,"

1953.

"Theory

B.,

July

1964.

Elements

deBruyne,

3781,

1940.

Construction June

and

Supporting

Aeronautical

Aeronautical

Norris,

F.,

Report

Stability,

of Sandwich

Core

January

of the

D.M.A.,

3174,

J.

FPL-026,

Technical

Note

Strength

of Honeycomb

Instability

January

Compressive

Royal

2-9

Elam,

Society,

of Sandwich

2-8

NASA

of Structural

Technical

Compressive

of Cells

"Local

"Handbook

NACA

Note,

A.D.

No.

Plates,"

By a Continuous

Williams, Under

S.,

S.,

Sheet

nautical

by Size Research

Anderson,

C.

Herbert,

"Short-Column

as Affected

Gough,

of Flat

B.,

tions

Thin

2-5

and Becker,

ES

of Wrinkling Society,

Ericksen, of the

Compression,"

Vol.

W.

S.,

Facings FPL

in Sandwich

Construction,"

59,

1955.

January

March,

H. W.,

of Sandwich Report

2-45

No.

Smith,

Constructions 1810,

March

Journal

C.

B.,

and Boller,

Subjected 1956.

of the

to

2-10

Jenklnson, P. M.

and Kuenzi,

E. W,,

"Wrinkling of the Facings of Aluminum

and Stainless Steel Sandwich Subjected to Edgewise

2-11

2-12

Report No.

2171, December

1959.

Yusuff,

S.,

"Face

and Core

Journal

of the

Royal

Harris,

B. J.

and Crisman,

Sandwich

Panels,"

Journal

2-13

of the

Benson,

A.

Sandwich

2-16

S.

C., of the

J.,

- Unified

New York,

No. 287, November

Plantema,

F.

J.,

Copyright

1966.

Department

Zatm,

J.

J.

E.

Sandwich S.

and

and Kuenzi,

Structural

E.

W.,

AIAA

Wrinkling

Paper

No.

of 66-138

for the Bending and Buckling of Circular Cylindrical

1966.

John

Wiley

Sandwich

"Classical

& Sons,

Inc.,

Composites,

B.,

of Finite Research

Buckling

- Orthotropic

November

Bohannan,

Service

W.,

Compression

FPL-018,

Cylinders

Forest

Face

New York,

MIL-HDBK-23,

1968.

Note

Kuenzi,

Engineers,

and Astronautics, Stanford University

Construction,

of Defense,

in Axial

Research

U.

Sandwich

of Civil

of

1966.

J., "Unified Theory

Report No. SUDAAR

of Buckling

1965.

Instability

January

1960.

Mode

and Applications,"

of Aeronautics

S.

March

June

" FPL

Construction,"

Society

Division,

Shells," Department

U.

64,

American

"General

Theory

Vol.

"Face-Wrinkling

Mechanics

m_d Mayers,

in Sandwich

Shells - Application to Axially Compressed

Construction

2-18

W.

Bartelds, G. and Mayers,

30 December

2-17

Strength

Society,

Proceedings

in New York,

Sandwich

2-15

Aeronautical

Engineering

Plates

Presented

2-14

Wrinkling

Compression,

Note

2-46

Cores,"

U.

of Sandwich

S.

Forest

Service

1963.

and Stevens,

Length

of Cylinders

Under

G.

H.,

Uniform

FPL-0104,

"Buckling External

September

Coefficients Lateral 1965.

Pressure,

for

2-19

March, Torsion,"

H.

W. FPL

and Kuenzi, Report

No.

E.

W.,

1840,

"Buckling January

2-47

of Sandwich 1958.

Cylinders

in

3 GENERAL

3.1

RECTANGULAR

3. i.

As

1

which

cause

a failure

fiat,

due

for

problems

token,

to

flat

has

fidence.

for

PANELS

and

data

for

test

that

and

as

far

been

and

fiat

This

inclusive,

an

panel

loads.

may

or

failure

for

sandwich

becomes

elastically

Further,

it

may

as

represents

has

this

applications

panels,

of

not

be

of

panels

unstable

should

such

is

be

under

noted,

magnitude

that

the

as

to

materials.

panel

solutions

the

of in-plane

of

costs

when

sandwich

fact

modes

instability

basic

the

been

this

type

been

appreciable

that

configuration

accumulated

configuration

and

have

that

was

it

the

is

for

the

adapted

represented

developed

amount

over

best

of construction

for

past

to

the

minimum

range

for

correlation

By

of

the

This

structural

fabrication

the

loading

with

vast

decade.

in

concerned.

a wide

of testing

which

same

applications

these

solu-

accomplished.

a consequence

tions

occurs

a number

analytical

tions

As

critical

of the

potential

types

of fabrication

needs

3-7,

are

probably

for

This

rectangular

majority

of the

of certain

loads

is

one

instability.

application

The

FLAT

PLATES

noted,

general

the

OF

General

previously

of

INSTABILITY

of this

past

as

given

panels,

view

for

is

basic

supported

panel

work,

it

is

now

possible

in MIL-HDBK-23,

by

design.

recommendations

Therefore,

3-1

to

[3-1],

employ

with

given

with

this

the

a high

in

analytical

degree

References

background

solu-

of

3-2

con-

through

in mind,

the

buckling

coefficients,

conditions "knock

will

The

down"

development

face

following

as a function use

of plate

use

or

for

the

various

of Reference

of (a/b),

V,

type

sandwich

are:

1) the

of dissimilar

for these

plate 3-1,

loading

with

no

core

The

possibilities;

materials

and

since

degree

for

this

the

general the

support

face

plates

curves

typical

and,

given

in

showing

conditions

is largely

the of the

equations

however, edge

requires

of orthotropicitsr

materials

material.

of loading

construction

will

K

assume

of aerospace

practices.

the

basic

design

principles,

final

The

design

The

core

shear

shall

failure core

that

wrinkling

If the

of either

core

is a cellular

dimpling

of the

spacing core

high

enough

shall spaces

be will

not

under

enough flatwise facing

comply

with

enough

of the

ultimate

have

sufficient

the

following

shear

the

loads.

design

of elasticity, and compressive

not occur or

the

under

constructed dimpling design

the

four

rigidity

and the the

and

cell

of either loads.

and

sandwich

strength design

size

such

loads.

of corrugated the

chosen

loads.

deflection,

moduli

so that

3-2

design

tensile will

under

to withstand

excessive

is not permissible,

enough occur

thick

buckling,

honeycomb

facings

small

and

sandwich

not occur

have

great

must

application

enough

over-all

will

panel

be at least

the

be thick

shall

have

shall

under

so that

shall

and

sandwich 3-1;

facings

stresses

strength

The

of the

Reference

sandwich

design

d.

same

faceplate

section

sections

for

some

account

cases,

c.

of factors,

the

applicable

coefficients

sections

In all

b.

buckling

of the

in this

to them.

of the

design

a.

the

of orthotropicity

of isotropic

vehicle

be given

from

to be applied

2) the

degree

will

taken

of a ntmlber

plates,

3) the

the

be those factor

consideration

the

K, which

or

facing

material corrugation into

the

Otherrequirementsincludethe useof moduli of elasticity andstressvaluesrepresentativeof thosevalueswhichprevail underthe conditionsof use. Also, wherethe stressesare beyondtheproportionallimit, the appropriatereducedmodulusof elasticity shouldbeused. The followingsectionson specifictypesof panelloadsdefinethe appropriateequations for eachparticular situationanddiscussusefullimits andother considerations,as applicable. A summarytable, (Table3-1), listing the panelinstability equationsgiven in the variousparts of this section, alongwith a definitionof terms, equationlimitations if any, andreferencesfor the appropriatebucklingcurvesimmediatelyprecedesthe list of referencesto facilitate useof the manualfor specificproblemsolution. Figure3.1-1 showselasticproperties anddimensionsfor the typical sandwichpanel underconsiderationin this section.

3-3

--_

b__..t

ttf

_,_ b[ __

_

•/

W-bf --q \¢

SINGLE

_tf

t¢

(TYPICAL)

_-_-'t

f DOUBLE

-TRuss

CORRUGATED

CORE

-TRUSS

CONFIGURATIONS

_h-_

N v

CORE

E b,

--,

Gcb tl-_

_--t

4-t4 C

2

.-t-o

ItONEYCOMB

-e_

b

---.--]_

Figure

y

3.1-1.

Elastic for

Properties

a Typical

and Sandwich

3-4

Dimensional Panel

Notations

CORE

3.1.2

Uniaxial

Edgewise

Basic

Principles

3.1.2.1

The buckling

coefficient

pression

those

arc

cluded

in the

in the

use

for

and

loads for

panels

various

3.1.2.2

panel

As previously Ericksen Supporting along

that shear

the

with

basic

face

support

With

to the

edgewise

_3-8],

The

basic

are

in-

principles

instability where

and

com-

and

equations

required

are

to limit

their

sandwich

section.

Curves

and both follow

the

panel for

edgewise

panel

orthotropic

com-

buckling

and

coeffi-

isotropic

cores

equations.

Curves

equations

presented

in this

in MIL-HDBK-23, assumptions

and

section

arc

as well definition

those

developed

as in other

of terms

by

documents.

are

also

in-

equations.

Honeycomb

assumptions

plates

then.

except

allowable

conditions

as pertinent

the

March,

general

here

facepk-ttes

and presented

such

Panels

One of the

the

since

uniaxial

imposed.

following

and

for

and

of these

of the

isotropic

Equations

March,

data

Sandwich

in the

here

Ericksen

issued

restrictions

support

given

not repeated

given

noted,

and

are

calculation

edge

by

development

for

having

Design

cluded

original

are

curves

documents,

references of the

and

developed

in the

equations

pression cients

originally

employed

because

The basic

equations

MIL-HDBK-23

assumptions noted

Compression

carry face

Cores

used the plates

inplane

in the loads

required

design

and

applied for them

3-5

analysis and that to act

as

of sandwich the

core

a unit

panels

provides in preventing

is that early

individual

given

buckling.

by

the

From

following

this,

the

equations,

edgewise

which

N

compression

are

=

taken

capability

from

Section

of the

5.3,

panel

Reference

is

3-1:

(_2/b_)(K)(D)

(3.1-1)

cr

where

ses

D is

gives

the

the

sandwich

bending

equal

Solving

this

equation

for

the

facing

sires-

following:

l'c_,

For

stiffness.

=

=

(E_t_)(E_t_) (E_t_ _ E't2)2

_7_K

(h_) (b _)

(_

(3.1-2) k

facings:

_K F

-

e

E

(h)_

4

I

f (3.1-3)

(b) 2 )_

w he re

K

=

buckling

coefficient

=

K

F

+ K

(see

M

definitions

in following

work).

E

= (,'a

J

1

=

effective

modulus

()f elasticity

for

orthotropic

facings.

X

Since

of

(1 - _ta_b)

_a'_b

=

Poisson's

f, 1, 2

=

subscripts

denoting

h,b

=

see

3.1-1.

the

buckling

isotropic

face

sanchvich

this

=

Figure

coefficient

plates,

applications,

ratio

curves

which

the

is

affected

as

measured

parallel

to

the

subscript

direction.

facings.

to be

presented

representative

equations

of the

given

situation.

3-6

here

large

previously

are

being

majority

arc

limited

of

revised

to the

case

structural

below

for

For

isotropic

facings:

Eai' where

= E' bi

77i = plasticity

As noted given

above

by the

=

E.'1 =

77iEi;

correction

the

buckling

and

factor

(see

coefficient

=

_ai

_bi

Section

for

the

=

_i

9.0).

panel

under

this

loading

condition

is

equation

K

=

KF+K

M

where

KF

-

12 E_t a (E/t2)

h_

(3.1-4)

KM 0

KM

= KM for

the

case

where

V=0

Esee

Figure

(3.1-16)1

(3.1-5)

O

Values

of K F are

is to assume = K M may

generally

it is equal be used

rigidities

of this core.

for

calculation

does

of this

not propose

initial

the

coefficient

to repeat

3.1-2

through

3.1-15

aspect

ratio,

and the

give

face

these values

bending-shear

panel

plate

Other

edge

support

considerations are

to KM,

given

thicknesses

for

sandwich

panel

on the factors

V

which

conditions

parameter,

3-7

y_D b2U

influence

the

of edge

and

K

bending

and

magnitude

orthotropicity

3-8.

curves support

V which

basis,

the panel.

the

development 3-1

approximation

On this

and the

with

however,

of K M as a function rigidity

first

and core

along

here;

a safe

is made.

in References

equations

thus

check

is dependent

ratio.

panel

of these

relative

a final

which

aspect

include

A discussion

small until

coefficient and panel

coefficient

the

to zero

to develop

K M is a theoretical shear

quite

of

of the

equations

This

manual

shown

in Figures

condition,

is defined

as

panel follows

(3.1-6)

which further canbewritten as: -It C kb; Ge (E_t,+E;t:)

V =

(3.1-7)

_2 t t c Eftf V

-

(for

2). b 2 G

equal

facings)

(:3.1-7a)

C

where axes

U is sandwich parallel

to the

An indication

inspection

later.

Holding

all

of V to be used an increased

cores;

of the

terms

with

value

Panels

equations

above

With

of the

a.

For

the

tion

of the

the

side

For

the

where Geb,

case

loading, poYtion

the

curves

with

the

a) and per-

curves,

modulus

giving

G , an increase e

in its

this

values

value

reduced

may

be obtained

of K M given

reduces

value

then

the calls

value for

Core given

are

to cover

the

for

the

corrugation

application, is very

where

the high

the

sandwich

case

flutes shear

with

corrugation

corrugations

to their

is replaced

of length

shear

panels

of panels

are

with

with

oriented

modulus

respect

in the

to the

the direction 2f andl°ading'R Ge :}; thus,= 0.the previous letting Gcb = = Gca/Gcb b.

associated

honeycomb

corrugated

cores

modifications:

load

flutes,

core

V m_d the

coefficient

be adapted

following case

for

previously

may

by means

modulus

to panel

of the

except

Corrugated

they

shear

M"

and fornmlas

however,

equations

buckling

of K

parallel

and importance

constant

the

(also

core

panel.

influence

of the

Gc is the

of loading

plane

of the

from

The

stiffness;

to direction

pendicular

Sandwich

shear

area

may

be assumed

and elastic

by the parameter

flutes modulus.

W, which 3-8

are

normal direction

shear curves

parallel

to carry The is defined

load

parameter as

modulus

to the

direc-

parallel

to

parallel

may

be used

to the

direction

in a direct V for this

to by

of procase

e

= W

Or,

for equal

edge

of K M as a function support

conditions

representing

the

Figure

3.1-16

values

conditions obtained

for use for

specific

The

curves

tion

to checking

process. approach new

design

of (b/a),

case

2 Gab

R = (Gea/Gcb), 3.1-2

of KMo

through having

and

V,

3.1-15,

are

given

for

Figures

3.1-14

ratio

and edge

various and

cores.

of panel

of K F in order

or W,

with

corrugated

as a function

values

(3.1-8a)

aspect that

final

values

for

K

support may

be

designs.

just

adequacy

As a consequence, described

cE'[t/2xb

of panels

in determining

and equations the

= _t

in Figures

3.1-15

gives

(3.1-8)

facings,

W

Values

i (E_t_ + E_t,)

).b2Gcb

in Reference

given

may

be used

of an existing this

design;

manual 3-1

in developing however,

recommends

since

process.

3-9

it was

the

specifically

a panel this

use

design

is a slow

of the developed

in addiiterative

design-procedures to expedite

the

]

10 2.5 Gc END

V 05 0.15 0.30 1 FOR

o 0

Figure

V>0.40

.__ 0.2

3.1-2.

0.4

0.6

0.8

1.0

0.8

tC M

_k .... 1 0.6

0.4

a

b

b

a

K M for Isotropic

a Sandwich Facings,

Panel with Ends and Orthotropic

3-10

=--

V

1 0.2

and Sides Simply Core, (R = 0.40)

0

Supported,

14

12

10

G

.20 .40

2 FORV 0

i

0

0.2

Figure 3. i-3.

0.4

0.6

0.8

1.0

0.8

> 1.00 _.'= M

i

0.6

.

0.4

a

b

b

a

0.60 1.00

1 -0.2

0

K M for Sandwich Panel with Ends and Sides Simply Supported, Isotropic Facings, and Isotropic Core, (R = 1.00)

3-11

14 ....

[

..........

l-i

I I

12 I 10

V -

2

II

1_

rr D

_

!

b2U

0.4G 8

END

I_ KM

I

'r

_1

I-

X

i

°

b

-I I

3 0.10

FOR

o 0

0.2

0.4

0.6

0.8

1.0

3.1-4.

K M for Isotropic

I( M

t

i

I

0.8

0.6

0.4

a b

Figure

1 V >2.50

= 1 V 0.2

b a

Sandwich Facings,

Panel with Ends and Orthotropic

3-12

and Sides Simply Supported, Core, (R = 2.50)

0.20 0.30 O.5O 2.50 0

16 14 12

V

m

2 ?r D b2 U

10 V 0 _0.05 .10 .15 0.20 0.25 0.30

0 0

0.2

0.4

0.6

0.8

1.0

0.8

0.6

0.4

0.2

b a a

b

Figure

3.1-5.

K M for Sandwich Panel with Clamped, Isotropic Facings,

3-13

Ends Simply Supported and Orthotropic Core,

and Sides (R=0.40)

0

16

T

14"I

I

....

1

12

10

2 ?r D V

m

b2 U

G c

END

_,

\

S

v 0

0.I0 0.20 0.30 0.40 0.60 0.75

1 0 0

0.2

0.4

0.6

0.8

1.0

FOR 0.8

V a>0.75 0.6

KM=__ V 0.4 0.2

a

.-_

b a

Figure

3.1-6.

K M for Sandwich Clamped,

Isotropic

Panel

with

Facings,

3-14

Ends and

Simply

Supported

and

Isotropic

Core,

(R=I.00)

Sides

16

i

T

i I

14

i

l L_o

2

12

V

D

m

b2U

0.4G C

END

10

L_

-]

-- -

I V

8 6

\

0

\ 0.10 0.20 0.30 1

0 0

0.2

0.4

0.6

0.8

1.0

FOR J 0.8

b

a

a

b

Figure

3.1-7.

lK_" .l, =_77:1-0.4 0.2

V > i1.875 0.6

K M for Sandwich Panel with Clamped, Isotropic Facings,

3-15

Ends and

Simply Supported Orthotropic Core,

and Sides (R= 2.50)

0.60 1.00 1. 875 0

16 14 12

8 6 4 .30 O. 40 1 FOR

V >0.40

0.8

0.6

K

0 o

Figure

0.2

3.1-8.

0.4

0.6

0.8

1.0

0.4

a

b

b

a

K M for Sandwich Panel with Supported, Isotropic Facings,

3-16

Ends Clamped and Orthotropic

-

V 0.2

0

and Sides Simply Core, (R=0.40)

16 14 2

12

V

D

b2U G C

END

\

1 0 0

Figure

0.2

3.1-9.

0.4

0.6

0.8

1.0

FORVi 0.8

> i.?0 0.6

K M=_ 0.4

a

b

b

a

K M for Sandwich Panel with Supported, Isotropic Facings,

3-17

Ends and

Clamped Isotropic

V 0.2

and Sides Simply Core, (R = 1.00)

I

16-14

......

l,

\

2

\

12

D

b2 U 0.4G C

\

END

K

t

M 6

4

2

0 0

0.2

0.4 a

b

Figure 3.1-10.

K M for Sandwich Panel with Ends Clamped and Sides Simply Supported, Isotropic Facings, and Orthotropic Core, (R= 2.50)

3-18

;_iA 14

2 V

12

\

0°

= __ rr D

1'

b2U

\,

2.5G e

END

2

?

o 0

Figure

1

FoR v > 0.30 % - V 1 0.2

3.1-11.

0.4

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for Facings,

Sandwich Panel and Orthotroptc

3-19

with Ends and Sides Core, (R = 0.40)

0.2

Clamped,

0

Isotropic

.... _.....]

16

1

i i

14

b2U 12 END

I

_'_

10

8

6 0.i0 O. 20

I FOR

V >0.75

K M

0

i

/

0.2

0

Figure

0.4

3.1-12.

0.6

0.8

1.0

0.8

0

0.4

a

b a

Sandwich

Facings,

and

Panel Isotropic

with Core,

3-20

Ends

1 =-\7

•

6

b

K M for

0.30 0.40 0.50 0.75

and

(R = 1.00)

Sides

0.2

Clamped,

0

Isotropic

16 L

14

2 V

D

=

b2 U 12 0.4G e

END

_'

V

8

0

O. 10 0.20 0.30

FORV 1 0.8

0 0

Figure

0.2

3.1-13.

0.4

0.6

0.8

1.0

> 1.875 J 0.6

M

0.4

a

b

b

a

KM for Facings,

Sandwich Panel and Orthotropic

with Ends and Sides Core, (R = 2.50}

3-21

0.60 1.00 1.875

1 -- --_ v

K.o

Clamped,

0.2

0

Isotropic

14 I

V =0

12

lp I

+I

10 2 Tr Dt

=0.1

t

c V

Gb=_

_t:

b2h2 G ca

_L

I t?

_,

\ °v0

\ [

l

V_0.4

0 0

Figure

0.2

3.1-14.

0.4

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for Simply Supported Core. Core Corrugation Load Direction 3-22

Sandwich Panel Having Flutes are Perpendicular

0.2

a Corrugated to the

0

14

12

2

Dt C

W

]0

b 2 h 2 Gob

W

0 0.1 _'--0o2

_- o.4

0.4

Figure

3.1-15.

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for Simply Supported Core. Core Corrugation Direction 3-23

0.2

0

Sandwich Panel Having a Corrugated Flutes are Parallel to the Load

END

\ \ K Mo

\

I

0.i

0.2 a

b

Figure

3.1-16.

KMo for Sandwich Panel in Edgewise Compression 3-24

with

Isotropie

Facings

t

3.1.3

Edgewise

3.1.3.1

Shear

Basic

As noted wich

earlier

panels

in Section

has

approaches curves

Principles

been

in the

MIL-ttDBK-23 and

described

in Section

equations

will

The

equations

basic

shear and

loads

faceplates

3.1.3.2 The

ing data

The

given

shear

and

presented constraints

load

shear

of the

those

were

general

and

taken

from

originally

assumptions

or restrictions

sand-

analytical

equations

are

equations

same

of the section

and buckling

the

adequacy

of flat

the

developed as those

on the

use

of these

consideration.

and isotropic

follow

Equations

equations

require

following

orthotropic

These

and testing

coefficient

edgewise

limitations

these

curves

the

the

in calculation

in the

conditions

and design

edgewise

Specific

use

for

employ

design,

buckling

in use.

and

where

Design

Design design

E3-13]

for

and both edge

presently

be noted

are

the panel

paragraphs

3.1.1.

analysis,

to demonstrate Thus,

following

Ericksen

assumptions.

clamped

in use.

documents

by Kuenzi

sufficient

accom,lished

presently

given

3.1.1,

allowable

along

with

sandwich applicable

coefficients cores

panel

background

for panels

for both

edgewise

simply

having supported

data isotropic and

equations.

Curves here are

carrying

are also

taken noted

capability

from

Reference

and discussed

of a sandwich

3-1

and

3-13.

Support-

as required.

panel

is given

by the

follow-

ing equation:

Nser

= (Tr_/b_) (Ks) (D)

3-25

(3.1-9)

where Nscr

= critical D

Solving

this

for

the

=

edgewise

shear

load,

lb per

inch

sandwich bending stiffness

facing

st_'esses

gives

the

following

/

equation:

I

f

(E_tI)(E_t_)(he)E z, Fsz,_

Or,

for

equal

=

_K

, + i (Elt 1 E_t_)

s

(3.1-10) (be) x

facings y2K s (h e) Ef' F

-

s

(3.1-10a)

4 (b_)k

where E'

is the

effective

r_ = plasticity X = 1-D = _ =

b

= panel

of elasticity

correction

factor

of facing

(Section

at stress

F s = 7/E

9.0)

2

= p_ = Poisson's

h

Ks

modulus

distance

ratio

between width

(
of facings

facing

centroids

(Figure

3.1-1)

= KF + KM (Note:

These

terms

(Figure

differ

from

3.1-1)

those

of Section

3.1.2)

whe re i

3

t

(E_t_3 + E_t2 ) (E'_t_+ E_t_)KMo KF Or,

for

equal

=

12(E_t_)(E,2t_ ) h2

(3.1-11)

facings

(tf_KMo KF

KMo

= value

ofK M for

=

3h 2

V=W=0

3-26

(3.l-lla)

The equation defining the value of K M

is quite complex

on panel aspect ratio, (a/b), the number

and involved, being dependent

of half-waves,

(n), for the minimum

buckle pattern, and the panel bending and shear rigidityparameter, manual

proposes

energy

(V, or W).

This

to follow general practice in the literature and provide curves only

for the definition of this buckling coefficient. Those and its development

will find this in Reference

interested in the basic equation

3-13.

Values of K M are given in Figures 3.1-17 through 3.1-24 as a function of the panel aspect ratio and the parameter These

V, or W,

for various panel edge support conditions.

figures cover panels with isotropic faceplates and both isotropic and orthotropie

core, including panels using corrugated flutes for cores.

cient, KMo,

may

also be obtained from the same

The equations defining the parameters

V and W

previous section for edgewise compression; facilitate

their

use.

The equation numbers

Values of the buckling coeffi-

set of figures.

are the same

however,

as those given in the

they are repeated below to

previously assigned to them

are retained

below

V

V

For

a sandwich

to the fined

edge

panel

of length

(E_tl)(E'_t2) (Y% tc = k(E[t_ +E_t 2) (b 2) G

= _r_tcE'ft/2),b2

with a,

the

a corrugated parameter

Gca

core

(equal

in which

V is replaced

as follows:

3-27

(3.1-7) ca

(3.1-7a)

facings)

the corrugation by the

parameter

flutes

are

W which

parallel is de-

W

)_b2 %b

Or,

for equal

a particular

through

3.1-21

Curves

of K M for

3.1-22

values

It should value,

the

value value

cal panel actual

The

of the the

stress.

of F scr

Fscr,

Figures

having

all edges

simply

supported.

edges

clamped

are

given

in order

orthotropicity,

and curves

of new

Thus,

3.1-17

in Figures to obtain

the

(R = Gea/Gcb),

iterative approach

designs

case

may

of V,

may or

and

be.

be used

design;

in the

however,

process. described

is required. 3-28

This

Thus,

inter-

(3.1-7)

same

manual 3-1

or (3.1-7a)

effective

value the

criti-

to establish

the

limit.

development

this

and this

calculating

proportional

the

limit

level,

be required

as was

in Reference

stress

when

will the

proportional

Equation

(3.1-10a)

it exceeds

the

on that

interations

where

given

is above

based

value

(3.1-10), several

cases

just

the

as the

in Equation

set

value

in computing

an existing

is a lengthy

of F

be an effective

in those

design-procedures

initiation

stress,

be interpolated

value

or (3.1-8a),

buckling

this

may

of core

resulting

be used

(3.1-8)

as in checking

pression,

values

all

(3.1-8a)

buckling

panels

having

curves

other

of E' shall

be used

equations

panels

if the

also

value

as well

that

shall

Equation

for E I shall

critical

for those

These

for

the

e Gcb

of V or W°

be noted

effective or W,

3.1-24.

= rr_t c E_t/2kb

for

be used

sandwich

coefficients

mediate

design

should

through

buckling

(3.1-8)

facings

W In checking

{EIlt_ + E_ta)

of panel

case

for

uniaxial

recommends for those

cases

designs comthe

use

where

10

l

0

/

2 7r 1) b2U

/

0.05

0.10

KM

----

[

0.2

0

0.4

0.6

0.8

0.40

1.0

b a

Figure

3.1-17.

K M for a Sandwich Panel with All Edges and an Isotropie Core, (R = 1.00)

3-29

Simply

Supported,

10 V

0.4

G

8

4

0.40

b a

Figure

3.1-18. K M for a Sandwich and an Orthotropic

Panel Core,

3-30

with All Edges (R = 2.50)

Simply

Supported,

J ff

V 10

/

2 D

t_

b2 U

t_L

/

/

/

0.05

_J"

J

O. 20

0.40

C 0

0.2

0.4

b

0.6

0.8

1.0

a

Figure

3.1-19.

K M for a Sandwich Panel with and with an Orthotropic Core,

3-31

All Edges Simply (R = 0.40)

Supported,

10

K M

5 i

0 0

0.4 b a

Figure3.1-20. K M for a Sandwich Panel with All Edges Isotropic Facings and Corrugated Core. Corrugation Flutes are Parallel to Side 3-32

Simply Core a

Supported,

10 V

0

-r I ' _L

9-

2 _r t

D c

b2 h2 G ca

_b_ / 7

1

i

I

KM

i

0.05

/ " / /

/ 0.20

J

_J i

i

O. 50 I

jl

4

0 0

1

-

f

// J

/

V

0.2

0.4

b

11

0.6

0.8

1.0

a

Figure 3.1-2 i.

K M for Isotropic

a Sandwich Facings

Corrugation

Flutes

Panel with All Edges and Corrugated Core. are 3-33

Parallel

to Side

Simply Core b

Supported,

16 V 0 14

l

Gc

12

10 0.05

K M O. i0

O. 20

0 0

0.2

0.4

0.6

0.8

1.0

b a

Figure

3.1-22.

K M for Isotropic

a Sandwich Facings

Panel and

3-34

with

Isotropic

All

Edges

Core,

Clamped, (R = 1.00)

16 V 2 77 D

14 i

V I

c

12

_-b

10

/

----"--

b2 U

If

0.4G

0

/

/

---4

|

i

r_

I jj J

8

5_ _..I

6 _J

J

0.05

J

0.10

7

J

v

I

I

I

0.20

o 0

I 0.2

0.4

0.6

0.8

1.0

All Edges

Clamped,

Core,

(R=2.50)

b a

Figure

3.1-23.

K M for

a Sandwich

Isotropic

Facings

3-35

Panel and

with

Orthotropic

16 V

2

0

_r D

14 Il 2.s "_-----

C

_

/

II

12

10

0.05

0.i0 K

M

.2O

0

0.2

0.6

0.4

0.8

1.0

b a

Figure

3.1-24.

K M for Isotropic

a Sandwich Facings

3-36

Panel with All Edges and Orthotropic Core,

Clamped, (R= 0.40)

3.1.4

Edgewise

3.1.4.1

The

Bending

Basic

Principles

application

produces

of

an

edgewise

condition

such

as

different

situation

from

the

on

one

half

of the

panel

on

the

other

half

of the

mum

value,

panel

buckling

ation

N,

been

as

panel

edge

support

along

been

on

given

The

equations

panels

The

bility

loading

with

assumptions

coefficient

in

3-1

for

behavior

condition

for

were

core

employed

this

the

this

in

loading

line

since

The

zero

at

edge

the

panel

represents

the

a

tension

loading

compression

neutral

loading

axis

which

compression

case;

of maximum

of loading

evaluation

load

to

can

a maxi-

produce

however,

the

loading

forces

consider-

been

covered

sandwich

for

this

of

of flat

flat,

by

development

are

3-37

to

enable

condition

with

rectangular

Kimel

Harris

of

the

the

basic

testing

has

use

of the

buckling

whose

Auelman,

same

confidence.

sandwich

equation

the

3-17).

and

complete

while

and

in the

(Reference

honeycomb

E3-15]

by

generally

plates

development

panels

loading

developed

condition

have

analytical

developed

were

the

type

the

rectangular

the

uniaxial

This

covered,

compression

sufficient

Reference

a corrugated

this

for

noted,

flat,

3.1-25.

effect.

from

the

for

techniques

previously

Figure

sandwich

mode.

considerations

accomplished

this

failure

analytical

coefficients

under

It is

rectangular

previously

linearly

edge.

a flat,

in

a stabilizing

varies

panel

to

shown

ones

fashion

of

general

that

same

mode

has

panel

complex

failure

as

represents

moment

in the

of a more

development

hlso,

at the

of the

These

bending

a loading

somewhat

presence

Moment

as

applicable

E3-14]

for

panels

the

those

and

panel

described

to

E3-16].

sta-

in Section

3.1.1,

critical the

design

This

loading

is exeeede.d beyond modulus extrapolation

equations

buckling

The

modulus

basic

loading

are

given

for panels

having

on simply

supported

3.1.4.2

Design

The

design

edge

a linear is given

edge

within

by the

in the

following

presented

variation

and

width

given

the

the

elastic

are

with

thus

the the

strictly

on a stress stress

an effective Since

consider

for

buckling

to a buckling

formulas.

the

stress

is based

by using

associated

that

buckling

analysis

Once

must

here

elastic

the

elastic

proper

variation stress

of variation,

applicable

only to

range.

calculation

of the

Design

and both follow

elastic

be done

range

section.

conditions

that

buckling

elastic

faceplates

any applicable

stress

the

the

requires

and extrapolation

in the

coefficier_ts

fact

cannot

the panel

Equations

and

stresses

across

exception

panel.

linear,

the

isotropic

equations

assumptions

Using

in the

of the

beyond

to be used

the

edge

modulus,

stresses

equations

the

This

not exct,ed

stresses

of facing

and buckling

at facing

shall

is no longer

tangent

to stresses elastic

Fcr,

across

range

as the

exception.

requirement

variation

elastic

such

effective

stress,

variation this

the

one particulal'

faeeplate

faceplates.

linear

the

with

allowable

curves

isolropic

these

sandwich

and buckling and

panel

edge

coefficients

orthotropic

cores

based

equations.

Curves

here

arc

design

those

taken

constraints

as previously

from are

discussed,

Reference

also

the

3-1.

Background

covered.

value

of N at

the

panel

equation:

N

=

6M/b _

3-38

(3.1-12)

where N

_-_

M

edgewise

equation,

= panel

bending

taken

per

unit

= edgewise

b

The

load

load

from

width

bending

width

moment

(Figure

capability

Reference

of edge

3.1-25)

of a sandwich

panel

is given

by the

following

3-1: N

=

er

(_,/be)

(%)

(3.1-13)

(D)

where N

er

D

The

critical

are

as

= critical =

faceplate

edgewise

loading,

sandwiehbending

stresses

are

lb per

stiffness

obtained

by solution

of the

previous

equation

and

follows:

(E1t_)(E2t_) Fe_,_ = Or,

inch

for

equal

Y2Kb

(h_) (El,s)

(Eli a +E_t_) 2 (be)

1

(3.1-14)

facings,

F

c

=

4

(3.1-14a)

(b _) ),

whe re E

= modulus

I

= (1 - _)

= Poisson's h

=

distance length

of elasticity

of facing

ratio

of facings:

between

facing

of loaded

edge

/_a = _b assumed centroids

of panel

K F + K M (Note: The values from those given in Sections 3-39

above

for these buckling 3.1.2 and 3.1.3)

terms

differ

(E_tl_+ E_t2 3) (Elt1+ Eat) (3.1-15) KF

Or,

for

equal

=

12 (E1t_)(E_t 2) (he)

KMo

facings (tf 2) KM o KF

=

(3.1-15a)

2 h2

where KM

Values tion

of K M of the

isotropic

value

for panel

buckling

using

flute-type

section

facilitate

their

for use.

however,

both

are

V=W=0

given

and the

in Figures

panel

isotropic

aspect

and

3.1-25

through These

ratio.

orthotropic

cores,

3.1-28

cover

as

panels

including

a funchaving

those

using

cores.

defining

previous

ofK M for

\T or W,

faceplates

equations

below;

=

parameter

corrugated

The

o

the parameters edgewise The

values

V and W are

compression;

equation of E _ are

however,

numbers replaced

by those

same they

previously

(Elt _ ) (E2t_) (z) V

the

as those are

assigned of E for this

repeated to them

given

in the

below are

to retained

case.

tc

=

(3.1-7) k (Eit_ + E_t_)(b2 ) G ca

V

For

a sandwich

to the fined

edge

panel

of length

as follows

=

1r_t e Eftf/2k

b 2 Gca

with

a corrugated

core

a, the

parameter

V is

(equal

in which replaced

:

3-4O

facings)

the by the

corrugation parameter

(3.1-7a)

flutes W,

are

parallel

which

is de-

C

w

=

(3.1-8) I b_Gcb

(Elh

+E2t _)

Or, for equalfacings, W

A particular 3.1-28

may

to determine

(3.1-15), which

or involves

new panel approach such

design

design

the

(3.1-15a) trial

designs; described

_r_tc Eft/2X

be checked

by using

appropriate

value

to compute and

=

error

however,

the

this

in Reference

in Figures

coefficient

buckling

be considered

3-41

given

buckling

recommends

caleulations.

(3.1-8a)

graphs

by iteration,

manual 3-1

the

of the

critical

solutions

b 2 Geb

stress, may that

since

also the it was

Fer.

3.1-25

to use This

in Equation approach,

be employed

to develop

design-procedures set

through

up to facilitate

36

°°°I

......

C

_-

_--

_|_T

2 ?r D

32 V

V--0

--

b2 U

28 ....... I r

/

"_

n--2/

n--3

24

16

i

V=0.215

8 FOR

v => 0.21_ KM = 1.ssG/v

4

0 o

Figure

0.2

3.1-25.

0.6

0.4

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for Isotropic

a Simply Core,

Supported (R = 1.00)

3-42

Sandwich

Panel

0.2

with

0

an

4O

a -------_ G

_ e

c,

,,

|

3_

y2 D V

32

b2 U

!V=0

A

_4 !

,o "X

16 \

",

/ f S

n=l

n=2vxn=3

v=o.osl

\_.

\ \

12

-'_..L-o.2 8

V--0 .4 I

FOR

I----

V _> 0.4 K M

-- 1. ss6/v

{ I

0 0

Figure

0.2

3.1-26.

0.4

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for a Simply Supported Sandwich Orthotropic Core, (R = 2.50)

3-43

Panel

0.2

with

0

an

4O

36

2 D

32

V=0

V

--

b

2

U

28

K

M

0.15 FOR

V >0

15

KM=

1.886/V

0 0

Figure

0.2

3.1-27.

0.4

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

KM for a Simply Supported Sandwich Orthotropic Core, (R = 0.40)

3-44

Panel

0.2

with

0

an

4O v

a-----_

cr rl

36

G

--_

y2t

'i

±

D e

32 b2h 2 Gcb

28

Ill/ \ "_..

24---

n=21

n=3

W=O.05

2O

W=0.20

16 _W=0.50

12

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

L .......................

4

C 0

Figure

3.1-28.

0.2

0.4

0.6

0.8

1.0

0.8

0.6

0.4

a

b

b

a

K M for a Simply Core Corrugation

Supported Sandwich Panel with Flutes Parallel to Side a

3-45

0.2

Corrugated

0

Core.

3.1.5

OtherSingleLoadingConditions

A searchof the literature, aswell as contactswith a numberofpeoplewhohavebeen activein the analyticalmethodsfield for this type of construction,revealednoother singleloadingconditionswhichmight leadto panelinstability problems. Consequently, the previously'describedloadingconditionsrepresentthe extentof the fiat panelstability datawhichwill begivenherefor individualloadingcases.

3-46

3.1.6

Combined

3.1.6.1

Basic

A study

Loading

Conditions

Principles

of the effects of combined

requires

the consideration

of a number

a.

The

mode

b.

The

interaction

The

on the buckling

of factors.

of failure of the panel

buckling

c.

loadings

between

under

different

Some

each

modes

of fiat sandwich

panels

are:

of the applied

for precipitation

loads.

of panel

or failure.

influence

interaction

of variations

equations

in the core

for panel

shear

rigidity values

instability failure under

on the

combined

loadings.

Since

little specific testing

fiat sandwich

panels,

this part

equations

for combination

cations.

Additionally,

along with

for biaxial instability modes

appropriate

of the manual

of the stress

some

discussion

references

has been

will provide

ratios which

are

analytically

conservative

of the considerations

in case

more

specific

accomplished

solutions

developed

for most

involved

for

appli-

is included

or background

is

needed.

The

equations

the stress

given

ratios,

failure by overall

stress

on the following

(R i = Ni/Nicr) panel

ratio relationships

to Section

2.

These

pages

, for each

instability under

which

cover

produce

latter equations

the interaction

of the separate

the action

panel

loadings

of the combined

which loads.

discussion

between

produce For

failure by local instability only,

and pertinent

3--47

relationships

are not repeated

the

refer

here

although local

the

specific

instability

Face

Wrinkling

(Asymmetric):

c.

Face

Wrinkling

{Symmetric):

be noted

might

problem

to the

first

ratios;

the

as

the

loading

one.

action

tests

some

question

effects

there

herewith

made

an effective field.

calculated

listed

the

below

for

each

of the

one edge

loading

applied

most

of these

would

also

of the

combined

to the

discussion

equations

plasticity Once

the

by means

are

define

an effective

value

factor

and

design

of (y. is known,

of Equation

1

(9.2-3).

3-48

panel

sandwich

accounts the

edge very

in later

cases

insta-

perpendicular high

aspect

instability

failure

been

where

stress

previously there

ratios,

paragraphs.

is

stress,

cr., 1

for the

for

effects

reduction

R.,i

Reference

in Section

to Equation

plasticity

situation

as has

the

given

uniaxial

failures

component.

in calculating given

panel

a general

local

in all

in particular

which

2-35.

however,

and recommendations report

Page

with

a potential

of the

2-26.

instability

having

cases,

for

in this

reduction

panels

a final

which

the

Page

potential

a local

along

In all

be accounted

for

in conjunction

for

adequacy

equations

(2.2-15),

indicate

to substantiate

must

interaction

loads.

2-7.

(2.2-6),

available

along

structural

Page

between

occur

CONDITIONS,

of these

of interaction

might

as to the

in the

data

situation

be run

(2.1-6),

Equation

no known

applied from

of plasticity

LOADING

are

Equation

This

should

be used

Equation

are

a result

arising

noted,

be

that

however,

under

Buckling:

occur

from

bility

page

modes:

b.

arising

stress

and report

Intracellular

which

Either

number

a.

It should

The

equation

9.2, (9.2-1) use

factor,

is

COMBINED or

(9.2-1a).

in determining

of the

to

biaxial _7, may

3.1.6.2

The

Design

design

equations

those which

be uscd

each

should

with

type

Equations

The

be used

having

and loading

Sandwich

Panels

interaction

buckling.

with

Honeycomb

covering

In view

edge

any

having

are separated

honeycomb

cores

and

Supporting

references

limitations

or restrictions

the stress

core

each

support

the core

relationships

for panel

typical range

of aerospace

combined

are

shear

presents

propose

into

those

given

to

for

on the use

buckling.

etc.,

application

definition

somewhat

and have

[3-I].

3 -49

been

One

of

for minimum

V,

energy

with

the panel

plate

the other

aspect

of general

equations

problem.

of combined

of the following

give

complex

is the determination

the establishment

a formidable

are

of general

briefly here.

of not only its relationship

in an exact

These

the onset

loadings

equation

rigidity parameter,

the use

define

will be covered

of the interaction

considerations,

involved

of this manual

under

of these

is a function

influences

of the complexity

ratios which

in both the x and y directions

of these

upon

panels

Some

for the development

all of these

the writers

with

between

of factors.

but is also dependent

ratio, panel

cores.

conditions

Cores

of honeycomb

of half-waves

Since

along

loading

panels

corrugated

condition

of a number

of the number

for sandwich

relationships

the prerequisites

for combined

equation.

instability buckling

functions

Curves

and curves

panels

of the interaction

and

load

simplified

conservative

recommended

interactions,

stress

ratio

results

over

for general

the

use,

A°

Biaxial

The

Compression.

buckling

of

a panel

following

subjected

to

formula

biaxial

R

is

recommended

for

estimating

compression:

+R

=1

cx

(3.1-16)

cy

where

R

=

N/N

e

cr

N

N

=

Loading

along

panel

=

Critical

loading

edge,

along

lbs/inch.

panel

edge,

lbs/inch.

(See

Equation

cr 3.1-1,)

x, y

A plot

design

noted

Bo

References

regime

have

reasonably

Bending

method

and

for

direction

given

and

panels

for

aspect

ratio,

(V _> 0.3).

stiff

prediction

and

for

is

3-1

of large

core

method

(3.1-16)

sandwich

panels

denoting

in

of loading.

Figure

3.1-29

(See

to

Figure

facilitiate

3.1-1.)

its

use

in

checks.

in

isotropic

for

Subscripts

of Equation

making

As

=

For

compression

estimation

cores,

of panel

and

aspect

for

correct

for

appreciably

panels

of

2.0

or

square,

conservative

bordering

(3.1-16)

on

less,

the

and

weak

which

provides

a satisfactory

a sufficiently

reliable

buckling.

(3.1-17)

buckling

is

ratios

Equation

of panel

Equation

equation

It becomes

_ 3.0)

with

onset

above

V _ 0.

(a/b

honeycomb

of the

the

which

panels

Compression.

the

3-23,

provides

under

the

action

of combined

bending

loads.

R

CX

+ (RBx)3/2

3-50

=

1

(3.1-17)

where Rc RB

=

N

= Load

C.

unit

(3.1-30)

plots

enable

its

use.

ready

3-19

loading

the

are

for

Equation

3.1-16.)

loading

due to edgewise on panel

bending,

due to bending

lbs/in.

moment,

3.1-13.)

relationship

recommended,

given

in case

by Equation

more

(3.1-17)

accurate

to

analysis

is desired.

Shear.

for the

of edge

interaction

and 3-23

and

method

of Terms

(See Equation

combination

Compression

width

edgewise

lbs/in.

Figure

able

per

Critical cr

References

definition

(N/Ncr)bending

N

of this

(See

= N/Ncr

The

prediction

following

interaction

of panel

buckling

formula under

this

furnishes particular

a dependcombination

of loads:

Rc + (Rs)2

=

1

(3.1-18)

whe re Re

= N/Ncr

RS

=

N

= Shear

(See definition

of terms

for

Equation

3.1-16.)

(Ns/Nsc r) loading

per

unit

width

of panel

edge,

lbs/in.

s

N

= Critical

edgewise

shear

loading,

lbs/in.

(See

Equation

3.1-9.)

ser

Equation in the

(3.1-18) solution

interaction

is plotted of specific

relationship

in Figure problems.

in greater

3.1-31

to enable

References depth

3-51

for those

3-21

it to be more and 3-23

needing

this

easily

develop information.

this

used

Bending

O_

and

Shear.

approximation

bending

The

of

and

tile

shear

following

buckling

interaction

behavior

equation

of panels

represents

under

a close

combined

edgewise

loads.

(RB)2+

(Rs)2

=

(3.1-19)

1

w he r c R

=

B

These

(N/Ner)bending

terms

Equation

Rs

=

As

(Ns/Nscr)

Again,

as for the previous

Figure

3.1-32

vides

additional

to make

combined

it more

easily and

information

Corrugated

Cores

defined

defined

Equation

readily

as

before

for

7).

previously

loadings,

background

are

(3.1-1

for

Equation

(3.1-19)

usable.

is plotted

Reference

on the development

(3.1-18).

3-19

in

pro-

of this interaction

equation.

Sandwich

The

Panels

with

interaction

sandwich

plex

equations

panels

with

relationships

corrugated

also,

for fluted corrugations

modulus

shear

normal

modulus

to carry

cates

cores

as before,

measured

the problem

involve

with

core

the distribution

and

case.

the additional

along

3-52

The

of a number

same

consideration

is negligible

Also,

of com-

influences

prevail

that the core

shear

in comparison

to the

the ability of the corrugations

the axis of the flutes, further

of this loading

material

instability failure for

the consideration

to the flutes.

it is applied

on the geometry

of general

of flute orientation

parallel

when

since

the onset

as for the honeycomb

to the direction

axial loading

flutes depends

for predicting

between

thicknesses.

the faceplates

compli-

and the

In view the

of the

magnitude

interaction

in this

performed

curves

aspect direction

the

and

respect

3.1-33

W,

and for

the

1.0,

and

with

respect

A discussion

Ao

this

through

carried

and

Buckling

D = bending

be obtained

The

following

load

load

for

are

other

by the

of sandwich

curves

curves given

panel

shear

core

reference

buckling

in the of panel

between

by the

flutes

are

repeated

rigidity ratio,

(D

c

load with here

parameter, a/b

corrugations

= 0.

= 1/2, is negligible

= bending

stiffness

panel.)

follows.

relating

in Figures aspeo*

latter

_!ation

aspect

De/D

studies

as a function

curves

of the

i.e.,

The

the

for

extensive

carried

1) Panel

of interaction

condition

other

loading

equations

of panel

interaction

carried

stiffness

sets

3-16].

W,

values

faceplates,

Interaction

coefficients

may

several

of axial

Compression.

combined

These

of the

onset

to each

relationships:

by the

of the

of the

of the

faceplates. for

and

parameter,

ratio

specific

advantage

[3-14

rigidity

additional

to that

take

coefficients

3.1-42

2) Amount

will

prediction

and the

and

of each

Biaxial

buckling

orientation,

following

of corrugations,

for the

by the

in developing

and Auelman,

bending-shear

carried

in Figures

2.0,

core

flute

to that

the

involved

manual

Harris

equations

relating

ratio,

this

by

interaction

form'of

problem

relationships,

area

presents

of the

the buckling 3.1-33

ratios

coefficients

through

and

different

to demonstrate

how

for

3.1-35. values

of W

by interpolation.

example

be used

to predict

Given:

Panel

the

withN

D = 3.0

problem onset

x

=2000

× 10 _ lbs/in

is offered

of panel

lbs/in, e, use

these

curves

buckling.

N

y

=400

lbs/in,

W = 0 for example 3-53

a=30 problem.

in,

b--

60 in,

may

Figure3.1-33 is usedfor this casesince(a/b) = applies

since

W = 0.

The

interaction R

equation + R

cx

1/2.

takes

The the

top line

following

of this

figure

general

<_1

form: (3.1-20)

cy

or

N

N x

+

__

N

y

1

N xcr

(3.1-21)

ycr

where = Stress

Rcx'Rcy N

=

xcr

ratios

Critical

for

panel

loading

(rr_/b _) (Kx) (D), N

=

ycr

K,K x

Critical lbs per

y

Buckling direction.

D

= Sandwich

loads

panel inch.

in subscript for

lbs per

loading

loading

for

applied

dimensionless.

in the

x direction,

=

inch.

for

coefficients

directions,

the

y direction,

loading

parallel

= (rr_/b _) (%)

to the

(D),

subscript

E; (tf)(t e + tf) 2 bending

stiffness

for

=

equal

facings.

2 (1 _Nfe) W

= _r_ (t c) (EO (tf)/2

Ef ' = Effective Gcb t

--

Core

shear

(1 -_Zfe) (b 2) Gcb

Young's

modulus

modulus

for

in the

= Thickness

of core,

=

of faceplates,

Thickness

_Zf = Poisson's a,b

=

ratio

inches.

inches.

of faceplates.

Panel dimensions,

inches.

3-54

faces,

direction

C

tf

for equal

facings.

psi.

parallel

to the

flutes,

psi.

Substitutingin Equation(3.1-21): N

N Y

X +

(K x) (_) and,

D/b _)

(%)

1.0

(3.1-22)

(_'_D/b2)

letting r

=

[

1

N /N y x

(3.1-23)

then

Nxb_

Since may

D,

N , and r will x

be obtained

checking same

b,

from

the panel for each

the

stability

loading

be known

for the

appropriate on the

direction.

design

curve, basis

in question,

Equation

that

the

panel

(3.1-24) margin

and

K

can

be used

of safety

x

and

K

y

in

is the

Thus,

(M.S.)x

From

(3.1-24)

r]

=

(M.S.) Y

(3.1-25)

which

(_) 1o:(_yc4 lO .Nx.

\Ny/

or

= r Nxer/

(3.1-26)

=

the n

(Ky) (y2D/b 2)

KY -----r

-

K (Kx) (_'_D/b _) 3-55

x

(3.1-27)

Returningto the datagivenfor the exampleproblemto demonstratethe method for checkingpanelstability: r = N /N

=

y K /K y x

=

(400/2000)

=

0.20

x 0.20,

from

Equation

(3.1-27)

Using Figure 3.1-33, erect a line passing through the origin and having a slope of K /K = 0.20 y x dinates and

of this

K

and

extend

intersection

it until point,

it intersects as taken

the

from

the

line

for W = 0.

figure,

are:

The K

X

coor-

= 6.0,

= 1.2. Y

Then, n

N

= K xcr

N

N

= ycr

Solving

Since margin

Equation

the

total of safety

(_2D/bZ)

=

(6.0)

(_r_ × 3.0

× 102/602

)

X

=

xer

K

(6.0)(822.0)

(_r_D/b 2)

=

= 4930

(1.2)(822.0)

lbs/inch

=

986 lbs/inch

y

(3.1-21)

for

2000

400

4930

986

is less

than

for panel

a panel

1.0,

-

stability

0.406

the panel

buckling

is:

3-56

check:

+ 0.406

is stable M.S.

=: 0.812

under

= (1.0/0.812)

the

applied - 1.0

loads.

= +0.232.

The

B°

Combined 3.1-38

Compression give

eurves

coefficients ratios

for panels

and values

rigidity

given.

condition

are

made

in the

given

on page

Combined for the

obtain

values

ships

for the

Biaxial

These the

small

manner.

stability

the

as

for

may

other

buckling

panel

aspect

be developed

by inter-

for this

for the

biaxial

compression

for

t2

done

for

the

combined

the I_

cy

s

loading

term

term

through

the

checks

calculations

as was

3.1-36

between

Curves

parameter

to Core

covering through

design

fashion

to those

in item

compression

change

and are

for

in the

and

ease. in the

in the

inter-

example

may

and the

biaxial

checks

Interaction

combination curves

study

for the

of loads

are

be interpolated

stability

compression

on panels

curves

loaded

to

checks

may

case.

The

in this

manner

above.

Shear.

shear

Figure

buckling

a square values

Shear.

particular

under

(B)

and

These

design

Compression

curves

this

specific

noted

Flutes

3.1-41.

in performing

as that

for the

dition. from

3.1-39

to be used

Combined

Normal

in a similar

same

relationships

manner

way

coefficients

in Figures

is the

same

Compression

given

method

same

Figures

3-53.

buckling

be made

Panel

by handling

in the

and Shear.

interaction

shear

those

equation

the

Flutes

in this

of the

is accomplished

Core

loaded

from

action

D°

showing

polation

This

C,

Along

panel of K

3.1-42

coefficients only,

between

shows for this

however, the

the

loading

as may

various

relationcon-

be noted

values

of the

Y shear

rigidity

parameter,

basis

of ratios

obtained

W, from

approximate the

curves

3-57

interpolations of Figures

3.1-36

may

be made

through

on the

3.1-38.

Panelstability checksare madein basicallythe samemanneras for the example problemgivenonpage3-53, exceptthatthe stress ratio, Rcy, is handleddifferently. Thebasic interactionequationfor this conditiontakesthe following generalform: R +R +R _1 cx cy s where R andR are as definedonpage3-54. cx cy Rs

=

(N xy/Nscr

K

=

buckling

) =

[Nxy /(_/b

coefficient

for

2) (Ks) (D)]

shear

s

Since,

and

as may

be seen

is independent

lated

imm

of the

:diately

Rcx The

design

problem way noted, which the

+ Rs may

on page

3-53,

R

cy

term

however, is less

example. of safety

would

now become:

(1.0-

if the and

1.0

Thus,

margin

of Kx and

interaction

=

K

value

equation

put

in the

: Or,

Rcx

Rcy )

s

the term and this

on the

right

side

value

should

buckling

for

= 0.10,

then

as calculated

(0.90/0.812)-

3-58

1.0

may

case

be calcu-

form:

as for the

are

handled

for the

be used

for this

= C way

of the

Rcy

following

and calculations handled

cy

of W only

+Rs

same

were

assumingR

=

Ks,

in the

term

calculations

for panel

M.S.

R

is a function

Y

the

now be performed

that than

3.1-42,

values

and the

check

as the

in Figure

in place

on page

= +0.109

in the

example.

equation,

C = 1.0

example

-0.1 3-57

same

It is to be

C,

has

a value

of the

1.0

used

=0.9, for

and the the

example

in

R cx

0

0.2

0.4

0.6

R

0.8

cy

Figure

3.1-29.

Interaction Panel

Curve

Subjected

for

a Honeycomb

to Biaxial

3-59

Compression

Core

Sandwich

1.0_::: : r!','_T

!_

....

!':

Itt

_:

.

_!i+ i I

i!

:+'i:

i_

+-

i_

::i

1! ?}

:: +

t;+ '++

,

:i!: if!! _'_

_it_ +P+_

|

: ;:{!:..

....

: ,:--!:

,,_i,],,i

!:i

:':i ;:.

::+_lli-'ii_

i:+

:::: til + +_ :,-::: " "+ :

[ T++[ ,_!_

:iiii:i: il!i ::-t:l

![;:

+ --.+v'":=;!:

Ittt]t::iit 0.6

_t

::

;:,- tit,

+;++I+

_'.|

',t:

.,+.

CX

;4

_++ ]

,+|

•

, 111

+ i':,:

'

!i" +:; '+

+,+

......

ii

"

, ,+_

++:_ +. +P_

i!i _;

1+

:::

+4,)

?f

I.

:'I!

!!

'

I

::"'

Irt++'P_+l?':

.la

":r

|

_'

!:++

+_+

N_:

: 1.0):_,+. lii: JJl.

+ RBx 3/2

ii["

,_

0.4

::--i

N_!! (Rcx

H.

+_I :....... 144

;i,:' ::: : ::"

!]ii _ !_!il!i[i :+++: _i_

...... lll.+'.l;

+i::

+!t+l+++: :'.'.i +:': +!+ ;+++t!G:_ ',',+

.rg_!!i!!:Fi:i i;I !!! _i:

It

:::-::::,:ii _!:iii!L

t[

+r +

'+li_li:i: :_

++'

r

il "I _ _ !_ilii!:

,,,

'

P

i.

: ,:

:fill+:

II

....

'::;::.t

.i..

i

....

+_+ ....... t_+:

::::I:::: i,+!t

_i i!

:ii

I +

'

;I fl

i+

.... ....

ft+. +++.

+i

!! !'

o.8i!.!!!...ii::i _+fl

'

hi

_t_'l][t

fff+l_+'f

....

I+++++N +: ....... : : _bJ: :!+`:!ti:ifill+

+::"

++:I++++I++++:N +++:+_,,, + : : _,++If+1+ ',_tt++, ttTT

0.2

tit

tf:

T[

r!

!t

F_F!i!!!i:_

TTC_

iii

tr _

+ !+I++++ ....... + +!+_ +4++t++++t+!i+ +1++ FiHtN .

......

t!t:Ir _ +|:tr _t_;t:'+:: itt_

;Ut

[[i[ "_]_

It l_tt

'

I

' i!::

!-F i:

U4ttt!iiilii!t 0

Hf+H_ii_t+xl

_ ]PI

o

tt

0.2

11

ii _t!,_ii!l I _++_ !d+:]l+_fi..

Pt

0°4 R

Figure

3.1-30.

Interaction Panel

Curve

Subjected

0. 6

for

a Honeycomb

to Bending

3-60

o. 8

1. o

Bx

and

Core

Compression

Sandwich

i!

I!+t:: : +!:iii _!< - [![ ' !1

_

![![i[I

+':t!tlii!

.... ,,

!i'. t!'.: ;;::

!;;

ii:

[b!:

_iiil _;t ii :_i ilti !!!: tr_r,i;

,,_+.:

tt _i:: :_ i_ tR + R = 1 O) i[ _:i _ c s " I[ :t_l liT! T.... :: ,lii:*+

ii!

.

_ 2

R

.:

.-

: ;+ ,

"TrY gt

t_; !_

]q[

_I_

,

+

ii

1I.... tti_

Jill C

!! iTii

_':_it:_l}_ l;ii _.XiitiliaS:.-!i

+÷,.

';'

t;t:

+,+l

++ _!!1

I ¸

!_i

tell :::?='¢tii _!" i

i: !i!! !111

'

::;ilf!+

::',. :::;+,+u m;:

:I1iI2:;

::;;

+.1!!Ill:

+:i....

'_+

1tit 4;;;

4

r!t !!! r i}_+ _iii!!iT:!!!+! +

,

I

i

,

,

+ i

'

_ '

4

=1

"

:

l

r

t!

f1

t

_ ]

L

i

,

i]4!ti!i::;;i:_i;I +t_;liilli) t_ !+'.,.J,i ;++_ ++++i;+++++l++ 11 ........ !+,+:t;;_p.+!:_i-]-Ll ++ L+

+:_i

'

Lilt

t;:i

'.1+

L

;i

!![

1%I:

,+++i;+11++1+Pl_+_ + H

0.6

0.8

R S

Figure

3.1-31.

Interaction Curve for a Honeycomb Core Sandwich Panel Subjected to Compression and Shear

3-61

_ t_

1,0

Figure

3.1-32.

Interaction Panel

Curve

Subjected

for to

3-62

a Honeycomb

Bending

and

Core Shear

Sandwich

20.0

Y

N

x

___

+.

b

16.0 W=0

L...×

-tttt _ N

\

Y 2

i

12.0

Nb X

W = 0.05

K

2 _r D

X

K Y . Z y

8.0

N b2 Y = -2 D

I W = 0.204.0 (a/b

= 1/2)

I W

0.50-

\ 0

2.0

4.0

6.0

K X

Figure 3.1-33.

Buckling Coefficients for Corrugated Core Sandwich Panels in Biaxial Compression (a/b = 1/2) 3-63

8.0

5.0

Figure

3.1-34.

Buckling Panels

Coefficients in Biaxial

for Compression

3-64

Corrugated (a/b

Core = 1.0)

Sandwich

2.0

Y l L I-

a

_1 -I

N

b

X

1.6 N

_'_

N b2

Y K X

D 2

1.2 K

\

K Y

-

2

I w

0.8

\

"-,g_,.

"_......

I

-.., \'_

O. 50 --_

0.4 (a/b

= 2. O)

_

_

\_

o 0

1.0

2.0

3.0

4.0

K X

Figure

3.1-35.

Buckling Panels

Coefficients in Biaxial

for

Corrugated

Compression 3-65

(a/b

Core = 2.0)

Sandwich

10.0

y_ N X

8.0

"k-

4

'-_" b

4.._

b

-"_---4

(a/b = 1/2) xy

I 6.o

x

K X

2D

2

__

K

2

Nb

N _

×

\

4.0

xy

K

\

b

S

"_,

\

2D

\ \\ \

W = 0

2.0 W=0.20

-

..W = O. 05

0 0

8.0

24.0

16.0

32.0

K S

Figure

3.1-36.

Buckling Coefficients for Corrugated Panels Under Combined Longitudinal and

Shear

with

Longitudinal 3-66

Core

Core Sandwich Compression (a/b

= 1/2)

4.0 Y

a

_1

N

_

b

3.0 x

'

"K

_

= x

x 2D2

K X

2.0

_

o_ g1>w\_--io. w_-o._o

\

\ jA% 1.0

w=o. o_l \ ,w:o (a/b

0

= 1.0)

2.0

4.0

6.0

8.0

10. O

K S

Figure 3.1-37.

Buckling Coefficients for Corrugated Core Sandwich Panels Under Combined Longitudinal Compression and Shear

with

Longitudinal 3-67

Core

(a/b

= 1.0)

5.0

Y,I a

_1 vI

Nx_'_

(a/b 4.0

= 2. o)

:

*-b

......

_ "-

_

xy

_-_'_

_

!

Nxb2

_'_'\ _'

ff _

Kx

-

D 2

3.0 Ks

_rxy2D

K X

__

N

2.0

1.0

,

0 0

2.0

..... o,,

4.0

8.0

6.0

K S

Figure

3.1-38.

Buckling Panels

Coefficients Under

and Shear

with

for

Combined Longitudinal 3-68

b2

Corrugated

Core

Longitudinal Core

Sandwich

Compression (a/b

= 2.0)

10.0

•

(a/b

= 1/2)

N

_

x.... 8.0

F I

t q

_t

t_

b

I

""

¢-N

_" "-32"_" x xy

i 6.0

K

"_

_

KX

_

x

_,

N xb 2 2

N

\

Ks

b

2

xy

4.0

2.0 PW

I _, w_o._o0

\

8.0

= 0.05

i

'

16. 0

32.0

24.0

K S

Figure

3.1-39.

Buckling Panels and

C(_fficients Under

Shear

with

for

Combined Transverse 3-69

Corrugated

Core

Longitudinal Core

(a/b

Sandwich

Compression = 1/2)

--N

f I

I

I /o__

' b

2

Kx Nbx2

3.0

W=

0.

N

Ir D b2

= 0.05

2.0

K X

_-W=O

1.0 W

(a/b

0

= 1.0)

2.0

\

4,0

6.0

8.0

10.0

K S

Figure

3.1-40.

Buckling Coefficients for Corrugated Panels Under Combined Longitudinal

Core Sandwich Compression

and Shear

= 1.0)

with

Transverse 3-70

Core

(a/b

5.0

Y ii a

l

N x -_ (a/b

-- 2.0)

""t.,_

l

I

l

_-

b

, i i i t"-AL.. ,,_

,,,_

4.0

I

_1

X

\N xy

I

L

K

x

-

_x_ _

2

D

3.0 N

\

K

b2 xy 2 17 D

S

\

X

2.0

05

1.0

w=o._\

0

_

4.0

2.0

8.0

6.0

K S

Figure

3.1-41.

Buckling Panels and

Coefficients Under

Shear

with

for

Combined Transverse 3-71

Corrugated

Core

Longitudinal Core

Sandwich

Compression (a/b

= 2.0)

2.5

Y 1

(a/b

: 1. O)

Nx

._e_q_ -----

=

b

2.0

Y

_N_

_

1.5

N b2 rr D

Nxy

rr D

N b2

K X

N

N_ 2_

__ '

Ky = 1.

=-

2

KY -

\

1.0 K

Ks

=1.7.

Y 7r 2D

W=0

-W = 0.05 0.5

_

0 50 \i

o

" ()'20

_

,_,o

__

4o

=_ K 1"9y

oo

=o

Ks

Figure

3.1-42.

Buckling Coefficients for Corrugated Core Under Combined Longitudinal Compression,

Sandwich Panels Transverse

Compression,

Core

and

Shear 3-72

with

Longitudinal

3.2 CIRCULARPLATES 3.2.1 AvailableSingleLoadingConditions A searchof the availableliterature as well as contactswith otherswhoare familiar with sandwichpanelstability referencesandstudiesin progressuncoveredno stability solutionsfor anysingleloadingcondition. This result might havebeenanticipated sincethe flat, circular sandwichplate hasvery few applicationsin aerospacevehicle structures in whichit mustbestableunderthe appliedloads. Consequently, this manualmakesno recommendations for techniquesto be usedin design, andstrongly suggeststhatall final configurationsbetestedas requiredto demonstratetheir adequacystructurally. 3.2.2 AvailableCombinedLoadingConditions Nopanelstability solutionswere foundfor anycombinedloadingconditionsapplicable to flat, circular platesin the courseof the literature searchnotedin Section3.2.1. Consequently, this manualmakesno recommendations for possibleanalyticalapproacheswhichwoulddescribeanystability limits for circular, flat sandwichplates.

3-73

3.3 PLATESWITH CUTOUTS 3.3.1 FramedCutouts Whileit is highly desirableto avoidcutoutsin aerospacestructuresbecauseof the attendantweightproblemsas well as uncertaintiesaboutloadpile-up andredistribution, theseare a practical necessitybecauseof accessandother requirementsand every effort shouldbe madeto derive reliable designapproaches which minimizethese drawbacks. Mostgeneralizedsolutionsfor plateswith cutoutsemployframing membersandbase the analysisonthe assumptionof buckledskinpanelswhichcarry only shearloads. Obviously,the solutionbecomesmuchmore complexwhenskin bucklingdoesnot occur, as wouldbe the casefor a framedcutoutin a sandwichpanel. Despitethe increasedcomplexity,however,solutionsfor the load distributionaroundthe cutout canbe obtainedfor variousload applicationsawayfrom the opening. Knowingthe load distributionadjacentto the cutoutdoesnot necessarilyprovideananswerto all questionsregardingthe adequacyof the design, however,particularly in the caseof sandwichconstruction. In the caseof monocoque or semi-monocoque panels, the lateral momentsof inertia of the framingmembersare generallysufficientlygreaterthanthoseof the skin such thatthey maybe consideredto providelateral supportfor the paneledge. This is not necessarilythe casefor sandwichpanels,thus settingupthe caseof a free, or nearly free, edgefor thepanelandfor whichconditionnogeneralstability solutionsor data were foundin the courseof this study. 3-74

It maybepossiblefor specificdesignsto be assessed,onthe basisof goodengineering judgment,to be critical in local instability rather thanfor generalinstability. This beingthe case, designchecksmaybe madeonthe basisof the equationsgivenin Section2. This manualmakesno recommendations for thosecaseswherethe general instability modeappearsto controlbeyondthe exerciseof goodjudgmentin the developmentof the design, andsufficient testingas neededto insure its integrity. 3.3.2 UnframedCutouts Unframedcutoutsin sandwichpanelshaveall of the disadvantages notedfor framed cutoutsandrepresenta muchmore seriousdesignproblemlocally, insofar as the free edgeis concerned. Thewriters of this manualencounteredno instancesin which sucha designapproachwasusedin primary or secondarystructure and, in general, recommendavoidanceof this practice. This recommendation is basednot only onthe lack of anyanalyticalor test databut alsoonpotentialproblemsof faceplate-corebond separationalongthe free edgedueto damagewhile in use, adhesivedeterioration, load cycling, etc.

3-75

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REFERENCES

3-1

U.

S. Department

HDBK-23,

3-2

3-3

30 December

General

Dynamics

of Analysis,

Konishi,

D.,

North May

3-5

Manual

American

General

3-9

" MIL-

Structures

Manual

- Volume

1,

Sandwich Angeles

11.00:

Inc.

Structures Division,

Report

Honeycomb

Space

and

Manual,

NA58-899.

Sandwich

Information

North

Structures,

Systems

Division,

Tests

A Division

for C-5A

Aircraft

GELAC

Report

Corp.,

No.

ER

8976,

Forth

F-111

Worth

Division,

Airplane,"

Report

"Design

No.

Allowables

FZS-12-141,

1965.

Report

No.

Ericksen,

"Design

TSB

W.

123,

S. and

Handbook October

March,

of a Flat Rectangular

Sandwich

Panels

Products

Laboratory

C.

B.,

Isotropic

Products

Airplane,"

Corporation

of Analysis,

Corp.,

Norris,

of Lockheed

1967.

Hexcel

with

Composites,

1959.

Los

Company,

Dynamics

1 August

Core

Corp.

Aviation,

Panel

and Methods

3-8

November

- Section

Lockheed-Georgia

22 February

3-7

San_vich

1964.

"Sandwich

3-6

Wo,__th Division,

et a], Honeycomb

Aviation

Structures

"Structural

1968.

Fort

Methods

American

3-4

of Defense,

Having

Laboratory

H.

W.,

Dissimilar

Report

Sandwich

Structures,"

1967.

Sandwich

"Compressive Facings

for Honeycomb

1583-B,

"Fffects Panel

- Compressive

Facings

of Unequal

Revised

Buckling

and

Isotropic

Report

1854,

3-85

of Shear

Buckling Thickness,"

in the of Forest

1958.

Curves

for Flat Sandwich

or Orthotropic Revised

Deformation

1958.

Cores,

" Forest

Panels

3-10

Norris, with

C. B., Dissimilar

September 3-11

Kuenzi,

E.

W.,

cients

for

Under

Edgewise

P.

E.

M.

Harris,

W.,

L.

A.

Flat

Sandwich

Products

Laboratory

and Jenkinson,

P.

Clamped

Rectangular

and

Flat,

" Forest

E.

Panels

W.,

with

Panels

Report

M.,

1875,

"Buckling

Coeffi-

Sandwich

Products

Laboratory

"Buckling

Coefficients

Corrugated

Products

Cores

Laboratory

Ericksen,

W.

Panels

Research

for

Under

Research

L.

Journal

Note

Flat,

Edgewise

Paper

of Aerospace S.

FPL

Company,

Gerard,

G.,

and

Buckling

of Flat

Technical

Note

Inc.,

25,

Becker,

3781,

R.,

J.

H.,

p.

1957. 3-86

Supported

and

North

Shear,"

STR

67,

1959.

Rectangular

Bending

and

Compression,

of Flat,

Simply

No. Theory

Under 7,

p.

Combined 525-534,

of Elastic

Supported Loadings,"

1960.

Stability,

McGraw-

373-379.

Handbook

National

1951.

1956.

Plates

M.,

Simply

Supported

"Stability

27,

1560,

of

Compression

Report

1857A,

Panels

Longitudinal

Bending,

Edgewise

Vol.

1961,

of Flat,

of a Simply

Rectangular

and Gere,

Plates,

and

of Flat Report

Combined

Division,

R.

Sciences,

P.,

Laboratory

Under

Report

Sandwich

Stability

"Stability

to Combined

and Auelman,

Core

R.,

Buckling

Laboratory

Timoshenko, Book

R.

Missile

Subjected

A.,

"Shear

Products

Plate

Inc.,

Products

S.,

Compression

"Elastic

Panel

Harris,

Auelman,

Sandwich

R.,

W.

" Forest

Transverse

Corrugated

3-18

and

Aviation,

Sandwich Forest

and

Core

Bending,

Kimel,

Hill

B.

and Kuenzi,

Construction,

American

3-17

for

1964.

" Forest

Corrugated

3-16

Supported

Sandwich

Sandwich

3-15

C.

Curves

1965.

Kuenzi,

and

Norris,

Compression,

Compression,

3-14

" Forest

December

Rectangular

3-13

Facings,

Simply

Jenkinson,

May

Buckling

1960.

FPL-070, 3-12

"Compressive

Advisory

of Structural Committee

Stability, for

Part

Aeronautics

I-

"

3-19

Kimel, W. R., "Elastic Bucklingof a SimplySupportedRectangularSandwich PanelSubjectedto CombinedEdgewiseBending,Compression,and Shear," ForestProductsLaboratoryReport1859,1956.

3-20

Noel, R. G., "Elastic Stabilityof SimplySupportedFlat RectangularPlates UnderCritical Combinationsof LongitudinalBending,LongitudinalCompression, andLateral Compression,"Journalof the AeronauticalSciences, Vol.

3-21

19,

Norris, Flat

C.

1838,

3-24

3-25

and

W.

J.,

to Two

Laboratory

Kommers,

Panel

Direct,

Direct

Loads,

of a Rectangular,

Combined

1833,

within

a Shear

a Rectangular,

Distributed

" Forest

with

1952.

"Stresses

to a Uniformly

Shear

Loads

Loads

Report

W. J.,

Subjected and

"Critical

Normal

Products

Load

Laboratory

and

Report

1953.

Plantema,

F.

York,

1966.

U. S.

Department

forced

Plastics,

1959,

Available

J.,

Sandwich

Construction,

of Defense,

"Plastics

" MIL-HDBK-17, from

U. S. Department space

Vehicle

U.

Government

S.

1952.

Subjected

Products

B.,

Sandwich

Edgewise,

829-834,

Panel

Forest

Norris,

p.

and Kommers,

Sandwich

Flat

3-23

12,

C. B.,

Load,"

3-22

No.

U.

Armed

S.

Government

of Defense,

"Metallic

Structures," Printing

John

for

3-87

Flight

and

Supply

Printing

Office,

Materials

Washington,

Sons,

Vehicles:

Forces

MIL-HDBK-5A, Office,

Wiley

and

February D.

Part

Support

Elements

New

I, ReinCenter,

Washington,

1966, C.

Inc.,

for

Available

D.

C.

Aerofrom

4 GENERAL

4.1

OF

CIRCULAR

CYLINDERS

GENERAL

In the has

INSTABILITY

case

long

of axially

been

classical

compressed,

recognized

that

small-deflection

primarily

thin-walled, test

results

theory

isotropic

usually These

[4-1].

fall

(non-sandwich) far

below

discrepancies

the are

cylinders,

predictions usually

it

from

attributed

to the

a.

shape

of the

of initial

post-buckling

equilibrium

path

coupled

with the

presence

imperfections

and the

b.

fact

that

buckling Neglecting

the

pressed 4.1-1.

a stress

[

°'eL

the

level

behavior rrtio ent

equal

- R _/3(1-Ve _) cylinder

is initially will

for

cylinder.

will

the

until

to the

general

by curve

be dependent However,

shape

from

isotropic and

since

this

path

shown and

classical

the

buckling magnitude

information

4-1

of the for

an

by the solid general

occur of the

is not

However,

at the

normally

occurs

at

theory

are

initial

comin Figure

instability

distortions

pre-

boundaries.

curve

cylinders

will

for

axially

small-deflection

(non-sandwich)

0B and

not account

neighborhood

the discontinuity

upon

does

equilibrium

A is reached

result

imperfect

theory in the

the

point

elastic,

be as shown

(Crcr/erCL) in the

is of

is linear

crCL

distortions

distortions,

cylinder

path

small-deflection

discontinuity

discontinuity

perfect This

classical

if

considered, stress

the

O'cr.

The

imperfections available,

presone

usuallyfinds it necessaryto resort to either of the followingpracticesto obtainpractical designvalues: a.

Setthe allowablecompressivestress equalto the valuecrMi N Figu_e

b.

shown

in

4. i-I.

Use the classical small-deflection value o-CL in conjunction with a suitable knock-down

factor )'e which is based on the results from a large array of

test data.

The allowable compressive

(rcr =

stress is then obtained from

)'c_CL

(4.1-I)

Perfect Cylinder Imperfect Cylinder

A

crCL

Axial Compressive Stress O'cr

C

O'MiN

0 End

Figure

For these the

isotropic

4.1-1.

(non-sandwich)

approaches radius-to-thickness

and,

for

Equilibrium

Paths

Compressed

Circular

cylinders such

ratio

Shortening

cylinders,

it is common the

(R/t).

4-2

test

for

Axially Cylinders

practice data

shows

to follow that

the

second

_'c is a function

of of

In the caseof sandwichcylindershavingrelatively rigid cores, the behavioris similar to that of the isotropic (non-sandwich) cylinder andonecanexpectimperfectionsand boundarydisturbancesto precipitategeneralinstability at compressivestressesbelow the predictionsfrom classical small-deflectionsandwichtheory, ttowever, in most practical applications,the sandwichwall will provide aneffectiverelatively thick shell so thatthe discrepancieswill not be as large as thosenormally encountered in thinwalledisotropic (non-sandwich) cylinders. In addition,as the coretransverseshear rigidity decreases,the differencesbetweentest results andclassicalpredictionswill diminish. In the extremecasewhereshearcrimping occurs, initial imperfectionsdo not appearto haveanyinfluence. Oneof the mostprominentof the early designcriteria developedfor axially compressed circular sandwichcylindersis that of Reference4-2. This solutionemployedlargedeflectiontheorytogetherwith approach(a) cited above(Crcr= now

rather

generally

agreed

too conservative. that

In addition,

CrMi N can be decreased

terms has

in the become

common

and 4-5].

with

an empirical

In the

treatment

characteristics torsion,

the

which

knock-down

of the

equilibrium

or external

zero

radial

are

pressure.

not For

4-3

values

Therefore,

followed

by method

purposes

for

cases

which

are

indicates of

years,

(b) cited

theory in this

it is

number

in recent

it

above

in conjunction

handbook.

it is important

identical

4-3

a sufficient

small-deflection

loading,

However,

of Reference

cylinders

is likewise

of external

design

by including

functions.

employs

paths

provides

development

sandwich

factor,

types

often

theoretical

to design

approach,

of various

criterion

displacement

practice

This

this

to essentially

large-deflection

[4-4

sion,

that

°'MIN).

to note of axial

of comparison,

that

the

compresFigure

4.1-2

depictsthe generalshapesof thesepathsfor eachloadingcondition[4-6] assuming that the cylindersare initially perfect andthatnodiscontinuitydistortions are present.

Deflection Axial Compression

Deflection Torsion

Deflection ExternalPressure

Figure 4.1-2. Typical Equilibrium Pathsfor Circular Cylinders Basedonthe relative shapesof thesecurves, onewouldexpectthat, undertorsion or externalpressure, the cylinderswouldbemuchless sensitiveto initial imperfections thanin the caseof axial compression. This has beenborneout by the availabletest datafrom thin-walledisotropic (non-sandwich) cylinders.

4-4

4.2

AXIAL

4.2.1

Basic

4.2.1.1

The

COMPRESSION

Principles

Theoretical

theoretical

Kuenzi

[4-7]

a.

Considerations

basis

which

The

used

includes

facings

shear

here

is the classical

small-deflection

solution

of Zahn

and

the follox_ing assumptions:

are

isotropic

but the core

may

have

orthotropic

transverse

properties.

b.

Bending

c.

The

core

of the facings

about

their own

has infinite extensional

middle

surfaces

can be neglected.

stiffness in the direction

normal

to the

facings.

d.

The

core

parallel

e.

The

In

this

equivalent

where

nific

ant

the

core

shear

rigidities are negligible

in directions

short

(a quantitative

limit is specified

[4-8]

can

without

in

4.2.2).

approximations

formulations

the

is not extremely

The

handbook,

shear

to the facings.

cylinder

Section

f.

e×_tensional and

of Donnell

be

applied

introducing

sig-

error.

final

equations

which

moduli

should

satisfy

of Reference

be

the

more

4-7

meaningful

have

been

to

the

transformed

user.

For

into

those

cases

condition Gxz _;1

O-

(4.2-1)

Gyz the

following

expression

is

obtained:

_cr

=

Kc(ro

4-5

(4.2-2)

where h _o

2 _r}-_l t 2

= _Ef--_

l_:-_ee

(tl

(4.2-3)

+ t_)

and

When

Vc _ 2

When

Vc > 2

1 1 - -_- V c

Kc =

(4.2-4)

1 Kc

(4.2-5)

= V--c

where o-o Vc -

(4.2-6)

°'crimp

5 2

_rcrim p -

z

t

Plasticity

E f=

Young'

h=

Distance

R=

Radius

andt_

=

ue =

reduction

--

The

relationship

important different

types

that

psi.

surfaces

of facings,

to middle

surface

of cylindrical

Thicknesses of the facings (There facing is denoted by the subscript Elastic

Core

Poisson'

s ratio

of core,

shear

modulus and

the

of behavior.

Kc

and

value The

associated in the

associated psi.

can

V c = 2.0 region

sandwich,

inches.

is no preference 1 or 2.), inches.

of facings,

oriented

Vc

inches.

as to which

dimensionless.

inches.

Core shear modulus the axis of revolution,

between

to note

of facings, middle

the facings Gy z =

dimensionless.

between

tc = Thickness Gxz

factor,

s modulus

(4.2-7)

(t 1 + t2)t c Gxz

be plotted

with axial with

as

establishes

where

4-6

V c _ 2.0

the

plane

perpendicular

direction, the plane

perpendicular

shown

in Figure

a dividing

line

covers

to

psi.

the

to

4.2-1. between

so-called

It is two stiff-core

1.0

K c

I I

I 2.0 Vc Figure

4.2-1.

Schematic

Representation

Between

and

moderately-stiff-core

zero,

the

core

shear

initial

imperfections.

factors

applicable

from

zero

less.

The

crimping enced

stiffness

to such

to a value domain

of

to such

knock-down is beyond

2.0,

relationship the

scope

any given

the

is the

constructions

of initial

of the

present

0_< 1

When

Ve

is in the

exhibits

of maximum

severity.

As

beeomes

so-called

weak-core

region

which

within

category

fall and

the handbook.

4-7

this

a knock-down

to develop variable

neighborhood

of

sensitivity

to

maximum ratio,

to Imperfections

be possible

recognizes

for

sandwich

imperfections,

It should which

of Relationship

Vc

radius-to-thickness

are

sensitivity

V c > 2.0

Sandwich

structures.

and the

constructions

where

occurs.

for

and

constructions

is high

Hence,

by the presence

applied

this

sandwich

Kc

factor

a continuous

influence

of the

the

knock-down

Ve

increases

progressively where are

shear not

of unity

influean

be

transitional core

rigidity

but

4.2.1.2 Empirical Knock-DownFactor Asnotedin Section4.1, the allowablestress intensitiesfor axially compressed,thinwalled, isotropic (non-sandwich) cylinders areusually computedusingthefollo_ving equation: _cr =

The quantity

(4.2-8)

_c °-cL

Yc is referred to as the knock-down

factor and this value is generally

recognized to be a function of the radius-to-thickness ratio (R/t). Various investigators have proposed

different relationships in this regard.

The differences arise out

of the chosen statisticalcriteria and/or out of the particular test data selected as the empirical basis. the lower-bound

One of the most widely used of the relationships proposed criterion of Seide, et al. [4-9] which can be expressed

}c =

1 - 0.901(1

to date is

as follows:

- e -¢)

(4.2-9)

where 1

R (4.2-10)

This

gives

a knock-down

curve

purposes

of this

handbook,

provided

for

design

achievement

of this

empirieal the

data

ness points.

the

concept

isotropie

of sandwieh

objeetivo,

this

that

lack

with

(non-sandwich)

end,

shape

factors it is usually

One of the

of sufficient

this

deficiency,

cylinders which

depicted

an empirical

cylinders.

is the Faced

and eorrection

Toward

general

it is desired

determination. from

of the

are

assumed

4-8

in Figure means

of this

major

obstacles

sandwich

test

one finds

in conjunction based that,

on the when

4.2-2.

data

for

be

a thorough to employ

an effective

available

V c _< 2.0,

also

the

to the

it expedient with

few

type

For

sandwich equal

thicktest

sensitivity

1.0

Yc

Log Scale

Figure

4.2-2.

to imperfections

Semi-Logarithmic Sandwich) Cylinders

results

from

h (_ _

for

sandwich

constructions

fore,

the

approach

taken

of P.

The

revised

4.2-3.

Also

axially

compressed

the test

weak-core points

[4-10]

which

included

in

also

because

taken

equal

likewise Figure the

that

to the

the

give figure

Rz

are

axially

compressed

did

not

in the

weak-core

of

scarcity

in view were

(finite

curve

analyzed principal

on this of

limit_i

Figul"e

as

data

the as

radius

conical

4-9

of

curve

which

Figure

obtained

from

not

In addition,

sandwich

constructions

available

test

design

fall

at the data, purposes.

into two

data

results

whose

values

of sandwich

conical

test

cylinders

for

in terms in

did

The

p There-

shown.

of curvature)

be used

gyration

(4.2-10)

points

region.

equivalent

amount 4.2-3

are

(Non-

thickness). and

4-28]

points

of

a dashed test

4-10,

for

lie

of equal

appropriate [4-2,

radii

(4.2-9)

shown

such

Based solid

plot

R/t for Isotropie Compression

shell-wall

facings

the

cylinders

the

Equations

the

are

YC vs Axial

Eleven

4.2-3 cones

two

is to rewrite

sandwich

shown

of the specimens. mended

in this

of

whose

here

category. are

equivalence

formulations

sho_

Plot of Under

are

and

radii

were

small

end

it is recomThis

1.0

.8 .7 .6

.3 .2 .1 10

103

Figure

4.2-3.

Knock-Down Factor Cylinders Subjected

4-10

Yc for to Axial

Circular Sandwich Compression

gives Ycvaluesthat are 75percentof thoseobtainedfrom the dashedcurve whichwas basedonthe empirical formulaof ,':_eide, eta].

[4-9].

In additionto the test results describedabove,a considerablenumberof test points are availablefrom cylindrical sandwichconstructionswhichfall into the weak-core classification. Asnotedin Section4.2.1.1, the methodsrecommended in this handbookare suchthat, in the weak-coreregion, no empirical reductionwill be applied to the theoreticalresults of Reference4-7. In order to explorethe validity of this approach,plots are furnishedin Figures4.2-4 and4.2-5 whichcomparethe weakcore test results of References4-2 and4-11againstpredictionsfrom the recommendeddesigncriterion. It canbe seenthat all but oneof the test results exceed the predictedstrengths,andthatthe singleexceptionfailed at 86percentof the predictedvalue. In manyof the caseswhere (O-CrTest/O-Predicted) > 1.0, althoughthe discrepanciesmeasuredin units of psi werenot very great, thepercentagedifferenceswerequitelarge. This behaviorcanbe explainedby the fact that the theoretical basis [4-7] proposedin this handbookassumesthat bendingof the facingsabout their ownmiddle surfacescanbe neglected. As shownin Reference4-12, this assumptioncanbe very conservativein the weak-coreregion. However,in the interest of simplicity, the methodsof this handbookretain this assumptionespecially sinceit is a conservativepractice andmostpractical sandwichconstructionswill not bedesignedas weak-corestructures.

In view stiff-core

of the

meager

sandwich

compressive cylinders,

test the

method

data

available

proposed 4-11

from here

stiff-core

is not very

and reliable

moderatelywhen

Vc < 2.0. Therefore, in suchcasesthe methodcanonlybe consideredas a "bestavailable"approach. On (V c _>2.0),

the

method

the

other

is quite

hand,

reliable

where

and

the

will,

failure

in fact,

is by shear

usually

give

crimping

conservative

predictions.

2O

15

9

•

_,_b.

8

10 _D

%

Test

Data

from

I 5

(Neglecting

Figure

4.2-4.

bending

Comparison of Proposed Core Circular Sandwich

O'cr, stiffness

4-2

I

10 Predicted

Reference

15

20

ksi of individual

facings)

Design Criterion Against Test Data for Cylinders Subjected to Axial Compression

4-12

Weak-

2OO

Test Point Reference

from 4-11

lOO

100 Predicted (Neglecting

200

Crcr , ksi bending

of individual Figure

4.2-5.

Comparison of Proposed Design a Test Result for a Weak-Core Cylinder

4.2.1.2.1 As

Interpretation

indicated

to arrive misled lie

in this

in the

for

when

region. is limited

to the

ders

for

V c = 0.

Then

be expressed

Against Sandwich

Compression

appropriate

knock-down data

case

the

this

point

the

of axially

recommended

test

factor.

and/or

To demonstrate

discussion

can

the

the test

sent

which

to Axial

Criterion Circular

Data

paragraphs,

values

endeavor

inelastic

Subjected

of Test

in the preceding at practical

stiffness facings)

data

However, classical

be used one

in order be easily

predictions

as possible,

circular value

can

theoretical

as simply

compressed design

must

for

sandwich the

critical

the

pre-

cylinstress

as follows:

°'cr

=

Yc _ Ef

h R

4-13

2 _ t_ I_Z-Y-Ye2 (tl + t2)

(4.2-11)

For anyparticular test specimen,the relatedvaluefor the knock-downfactor shouldbe computedfrom the followingexpressionwhichis obtainedby a simpletranspositionof Equation

(4.2-11):

C(rCrTest _ \ "_Test /

= (YC)Test

The

plasticity

reduction

stress.

By inspection

conclude

that

this

factor

DTest

of the

formula

[E fRh

and

wEXperimental

(Yc)Test

_lassical

figure,

the

rial.

illustrated solid

Suppose

indicated

that

in the

so that

NTest

assume

that

critical

stress

represents

this

figure, will

_?Test

the

-

0.80.

would

have

meaningful

critical"

should

specimen

one

can

form:

(4.2-13)

the

help

curve

that

For

for

stress

purposes

had

the

than

lies

equal in the

In this

specimen

to

the .

This

mate-

O-CrTest

inelastic

discussion,

elastic, OcrTest

concept.

test

of this

remained

higher

this

at a stress

this

the

material

somewhat

to clarify

buckled

here

unity.

If the been

(4.2-12),

value]

assuming

stress-strain

it is assumed than

stress

more

buckling

to be elastic

4.2-6

particular

be less

experimental

of Equation

following

theoretical value

behavior

in Figure

line

actual

would have been attained had] material remained elastic ]

L

example

at the

in the

critical

| stress

The

t_) ]

denominator

be rewritten

hich the

2vf_-lt_(t_

is evaluated

numerator

may

_

(4.2-12}

.

As

region further

experimental greater

value

I

will

be denoted

as

CrcrTest

.

Then

it follows

°'CrTest CrcrTest

-

_)Test

that (4.2-14)

°-CrTest 0.80 4-14

-

1.25

(rCrTest

¢rCL

(r_rTest O'Ma x _rcr

Test

Strain Figure

4.2-6.

Now let it also

be assumed

retical

stress

formula

critical would

then

Stresses

that,

equals give

the

the

Involved

using

elastic

value

o-CL

proper

value

for

O'CrTest ( ¥C)Test

in Interpretation

material indicated the

properties, in Figure

experimental

1.25

of Test

Data

the 4.2-6. knock-down

classical The

thee-

following

factor:

¢rcrTe st

-

(4.2-15) oCL

¢rCL where h (rCL

= Ef _

2v/t _ tz _pe

4-15

z {tl +t2)

(4.2-16)

The abovediscussionis givenhere sincesomeof the results presentedin the literature canbe quitemisleading. That is, comparisonsare oftenshownbetweenthe actualtest value (rcrTest (withoutregard as to whetherelastic or not) andthe inelastic classical theoreticalprediction. For the caseshownin Figure4.2-6, the latter valuecannot exceed _Max andthis type of comparisonmight lead oneto believethat the appropriate knock-downfactor is very closeto unity. However,use of the correct approachas expressedby Equations(4.2-13)and(4.2-15)givesa muchlower given O-Ma x a flat

geometry, simply post-yield

one

could

by choosing stress-strain

always

show

a material

very

with

close

agreement

a sufficiently

curve.

4-16

low

Yc

value.

between yield

strength

For

O'crTest

any and

and having

4.2.2 DesignEquationsandCurves For simply supportedcircular sandwichcylinders subjectedto axial compression,the critical stressesmaybe computedfrom the relationshipsgivenonpage4-18wherethe subscripts1 and 2 refer to the separatefacings. facing

is denoted

tension

of the

where The

the

formulas

behavior

extended

thickness where

by either

versions

two

facings

only

when

the

behavior

only

be made

when

El and

buckling

not made

are

of course,

coefficients

Kc

4-7

handbook

were

of the of the

which

same

of the

material,

Application

made

of the

obtained

only are

derived

moduli

is no preference

were

of elasticity

(7 = 1).

facings

E_ will,

moduli

ratios

is elastic

both

equations

in Reference

in this

on the

are

These

and the

given

based

the

The

developed

is elastic

concepts

tions,

subscript.

"I2aere

by a simple

considered

the

identical

through two

for the

these

equations

material.

For

facings.

of equivalent-

For

cases

are

cases

ex-

case

both

use

facings.

to inelastic

same

as to which

valid

(_? _ 1) can

such

configura-

be equal.

can be obtained

from

Figure

4.2-7.

Curves

are

given

Gxz there

for both

0 _< i

and

0 = 5

where

0 -

Since

these

two

plots

are

not very

Gyz different mates

of

Whenever When

For

from

each

Kc

when

1<

V c < 2.0,

V c _> 2.0,

elastic

Section

other,

cases,

9 must

use

use

Figure

4.2-7

to obtain

rather

accurate

esti-

0 <5.

the

use

one may

knock-down

factor

Yc can be obtained

from

Figure

4.2-8.

Yc = 1.0.

_) = 1.

Whenever

the

be employed.

4-17

behavior

is inelastic,

the

methods

of

!

I

I cq

I L'_

I

N

O

C +

IJ

j O

17

oJ

O

A

v ¢--..J

v-4 I

I

I O,1

I

I o,1

-.>

O

O

¢,J

¢7

+

o L) g-

°,.._

4-18

¢7 _>

JiJ,

i¸

........

il

:!:i!

_9

0

_9

<

i

}.

1 _9

4-19

1.o _i:il ..... :::i_/_'!!I ..... :i!_

!_....' e ...............

ii_i ::i i:,

.9

•8

x/(EI tl) (E2 t2) h

'ki_ i_L[ii_4.!i_ i

T i_

............. '!t i i,I

_'C

T:t:T!_-T, ,,

:

.2

J'_' [71_i!:7 _ _ti3

jklt.... ,

:

dii_t _:ii11

i!ii

lit,

" 3!7!t7['_i

10 _

I0

Figure

F3

4.2-8•

Design Knock-Down Cylinders Subjected

4-20

Factor for Circular to Axial Compression

10 a

Sandwich

The

critical

axial

load

(in units

Pcr

In the

special

Equations

case

where

(4.2-17)

of pounds)

= 27rR

[_cr_

t_ = t2 --- tf

through

(4.2-26)

can

and

can be

(rcr

be computed

tl

+ _cre

both

facings

as

follows:

t_]

(4.2-26)

are

simplified

made

to the

of the

same

following:

= Yc Kc ¢ro

(_Ef)

material,

(4.2-27)

h

(4.2-28)

Gxz

(4.2-29)

5 2

°'crimp

2 tf t c

o-o Vc -

(4.2-30) arcrimp

Pcr

Equations of the

(4.2-17)

cylinder

through

(4.2-31)

is greater

than

the

= 41rRtf

and

Figure

length

(4.2-31)

o-cr

4.2-7

of a single

are

valid

axial

only

half-wave

when

the

length

in the

buckle

L

pat-

Gxz tern

for

one can

the apply

corresponding

infinite-length

the

following

When

V

test

<

cylinder.

to determine

if the

For cylinder

2

and

Figure

(4.2-17) 4.2-7

through are

valid

case

(4.2-31)

Equations

only

and

where

value

4-21

(4.2-17)

Figure of

4.2-7 L.

where

length

When

el

Equations

the

_ - Gy z -

is sufficiently

Vcl

> 2

through are

large:

valid

(4.2-31) for

any

1,

For constructionswhere e _ 1, nocorrespondingnumericalcriterion is presently available. In suchcases, onecanonly usethe abovetest in conjunctionwith engineeringjudgement. It is helpfulto pointout, however,thatmostpractical sandwichcylinders for aerospaceapplicationswill besufficiently longfor Equations(4.2-17)through (4.2-31) use

and

of these

Figure

which

as short

cylinders.

under

to be valid.

relationships

Cylinders

ders

4.2-7

fail

for

to meet The

axial

shorter

the

only

compression

In addition, cylinders

foregoing

it is comforting results

length

means

available

is the

solution

the

of Stein

that

the

in conservatism.

requirement for

to note

are

analysis and

usually

of such

Mayers

referred

to

sandwich

cylin-

which

is only

[4-13]

valid a.

when

0 = 1

b.

when

both

ends

c.

when

both

facings

d,

the

and of the

cylinder

are

simply

supported

and are

made

of the same

material

and thickness

other

For curves

short

sandwich

of Figure

of one

facing

is not

more

than

twice

the

thickness

one

can

of the

facing.

cylinders 4.2-9

which

which

satisfy

involves

Z -

these

the

following

2L_ Rh 1-_/_e2

4-22

conditions,

use

the

design

parameters:

(4.2-32)

rr r a

2

D

(4.2-33)

-

L 2 Dq

•

_cr

(t_ + t_) Le

Kc =

(4.2-34)

ycrr 2 D

where (_ h)(E2 ts)h2

(4.2-35)

D = r](l_Pe_) [(F__tl) + (F__t_i]

h _

Dq

= _-c Gxz

(4.2-36)

and L

During

the

pressed However, long

for

= Over-all

preparation

sandwich in most such

fixity

length

of this

of cylinder,

handbook,

cylinders

having

practical

aerospace

to have

negligible

inches.

no solutions

any degree

of rotational

applications, effects

were

on the

4-23

the

uncovered

for

restraint cylinders

buckling

loads.

axially

at the will

com-

boundaries.

be sufficiently

...... ]

70

----

4o-

-

-_

1

+ .... •_

30-:

_

_

I

I

'

t

20-='

i ::ii4'

:! 1

10_

_,

7!

--'_-,

4-

K

:

:

.....

i

C •

.5-

_

.4-

i:

:: ,!

I

• 9

1

Z

Figure

4.2-9.

Buckling Subjected

Coefficient to

Axial

for

Short

Compression

4-24

Simply-Supl_orted (0

::

1)

Sandwich

Cylinders

4.3

PUIIE

4.3.1

BEN1)ING

Basic

Principles

4.3.1.1

Theoretical

Based

on small-deflection

and

4-16

jeeted

of elastic

to axial

theory,

instability

results

of these

the

circumferential

stresses

references, stress

compressive

investigations

it can

satisfies

vary

classical

solution

be recalled for

theory

been

erences erence

the

4-17

deals

the peak pirical

infinitely

value

of the

knock-dow_l

which applied factors

this

are

might only

obtained

the

peak

that

of

the

peak

4.2

(axial

to pure stress. for 4-25

the

The two

stress

need

compression. the

case.

It Ref-

cylinders

while

in the

stiff-core

and moder-

of this

handbook,

it is assumed

if the

only

cases.

axial

small-deflection

is indeed

compression)

bending

to uniform

and axial

fall

of Section

small-deflection

that

sandwich

which

the

compressive

this

weak-core

purposes

subjected

expect

axial

the

recommended

From

nature

when

subjected

bending

and 4-19 for

from

reasonably

for

compressive

of the

is reached

cylinders

cylinders

Therefore,

considerations

sub-

(4.3-1)

is also

4-18,

long

4-15,

direction.

regardless

instability

or combined

4-17,

categories.

cylinders

that

demonstrate

with

theoretical

one

bending

in References

circumferential

that,

(non-sandwich)

indicate

cylinders

Et R

. 6 Et/R

result,

of pure

and 4-18

4-19

to sandwich

also

in cases

ately-stiff-core that

of this

would

shown

value

isotropic

In view

be considered has

the

thin-walled,

compression. sandwich

that

4-14,

condition

o" _.6

It should

in References

(non-sandwich)

in the

be concluded

the

made

isotropic

which

distribution,

stress

were

in thin-walled,

compressive

the

axial

Considerations

apply

analysis

differences

equally considers

lie

in the

Ref-

well only em-

4.3.1.2 Empirical Knock-I)ownFactor In the caseof ence

pure

experiences

consequent

stress

reduced

imperfection, be as

only levels

probability

a relatively

which

as

(non-sandwich)

the

cylinders

lower-bound

that

corresponding pure

the

portion buckling

stresses the

factors

under

small

initiate

for peak

it is to be expected

severe

following

bending,

axial

bending,

with

factors load.

Seide,

cylinder's

process.

to coincide

knock-down for

of the

et al.

Because the

for

For

circumferof the

location

pure

bending

thin-walled,

[4-9]

have

of an will

not

isotropic

proposed

the

relationships:

Yb = 1 - 0.731

(1 - e -¢)

(4.3-2)

where

= _

Comparison

against Equations (4.2-9) and (4.2-10) shows that this bending criterion

does indeed give Yb compressed

(4.3-3)

values of lesser severity than those which apply to the axially

cylinders.

Following the same

approach

as that taken in Section 4.2, the h

above equations are rewritten in terms

of the shell-wall radius of gyration p(_

sandwich constructions whose two facings are of equal thickness). lations then give the plot shown as a dashed curve in Figure 4.3-1.

The Also

revised shown

for formuin this

figure are the appropriate test points from stiff-core sandwich cylinders subjected to pure bending [4-201. Since only three such data points are available, it was thought to be helpful to include the axial compression

Figure

4.2-3.

portant for

the

sandwich data points previously shown

To fully understand the information given in Figure reader

to be aware

of the

data

4-26

reduction

techniques

in

4.3-1, it is imused

here.

For

an

explanation related

of the

discussion

Based

on the

solid

curve

shown

to pure

from

the dashed

This

is consistent

which

obtained

in such the

other

reliable

the

design from

meager

proposed

cases,

4.3-1

curve

the value

method

in Figure

the

hand, and will,

the

test

gives

when

the

in fact,

reference

practice

followed factor

available 4.3.2

can only failure usually

for

on the

corresponding

data

data,

T b values

is based

in Section method

test

be used

knock-down the

handbook,

of available

This

where

of the

amount

with

in this

should

be made

to the

4.2.1.2.1.

bending.

pression

In view

used

in Section

limited

jected

the

procedures

the

design

that

are

in Section

4.2

sandwich

be regarded

give

reliable as

crimping

conservative

4-27

likewise derived

is not very

is by shear

75 percent formula

was

here

of sandwich

empirical

curve

from

it is recommended

for

from

cylinders of those

of Seide, the

taken

Reference

under

when

V c < 2.0.

predictions.

com-

percent

pure

of

method

bending,

Therefore,

approach. the

[4-9].

4-9.

cylinders

a "best-available"

obtained

of axial 75

the sub-

et al.

case

to be

(V c >_ 2.0),

that

On is quite

•

(_)

Data

from

Ref.

4-20

(Cylindrical;

Pure

+

Data

from

Ref.

4-2

(Cylindrical;

Axial

Compr.)

•

Data

from

Ref.

4-10

(Cylindrical;

Axial

Compr.)_

Bend.)

10 a 10

Figure

4.3-1.

Knock-Down Cylinders

Factor Subjected

4-28

_

for

to

Pure

Circular Bending

Sandwich

!

t

4.3.2 For same

Design simply

Equations supported

design

except

and

cylinders

curves

as

subjected

are

given

to pure

in Section

bending, 4.2.2

one may

(for

axial

use

the

compression)

following:

For

the

ease

factor

For

bo

Curves

sandwich

equations

for the a.

and

of pure

bending,

Yb whenever

the

case

equations

V e < 2.0

of pure

which

furthest

removed

peak

values

when

the

lies

from

within

computed

from

variable

is elastic, the

4.2.2

axis.

4.3.2

to obtain

Vc >_2.0,

use

critical

stresses

side Hence

critical

knock-down

= 1.0).

from

axis

and

stresses

are

distribution. bending

the

circumferential

neutral

computed

circumferential the

Yb

to the

of the

the

the

obtained

correspond

compressive

that

the

behavior

the

of Section on the

Figure

(When

bending,

and curves

location

use

moment

is the

Therefore, Mcr

can be

following:

Mcr

= rrR _ [¢rcr 1 t_ + _cr2

t_]

(4.3-4)

where

Mcr

= Critical

R = Radius

(rcr:

and

O-cra

Note:

and

ts

moment,

to middle

surface

= Critical tively,

tl

bending

compressive which

= Thicknesses

There subscripts

of the

is no preference 1 and

result

in.-lbs. of sandwich

4-29

inches.

stresses

in facings

1 and

2,

in general

instability

of the

cylinder,

facings

as to which

2.

cylinder,

1 and

facing

2,

respectively,

is denoted

respec-

inches.

by the

psi.

1.0 .9

6 _b

5

.2 .I io_

I0

(s) P

Figure

4.3-2.

Design Cylinders

Knock-Down Subjected

Factor

"¢b for

to Pure

Bending

4-30

Circular

Sandwich

To computel_cr whenthe behavioris inelastic, onemust resort to numericalintegration techniques. Sincethe procedurerecommended heremakesuseof the methodsof Section4.2.2, all of the limitations of that sectionare equallyapplicableto the presentcase. Thatis, onlysimply supportedboundariesare consideredandthe primary solutionis excessively conservativefor the so--calledshort-cylinderconstructions. In addition,only very limited meansare availableto facilitate a quantitativeassessmentof whetheror not a particular constructionfalls within the short-cylinderclassification. Furthermore, the computationof critical stressesfor short-cylinder constructionscanonly be accomplished for rather specialcasesas cited in Section4.2.2. Asnotedin Section4.2.2, duringthe preparationof this handbook,no solutionswere uncoveredfor axially compressedsandwichcylindershavinganydegreeof rotational restraint at the boundaries. However,it was alsonotedthat, in mostpractical aerospaceapplications,the cylinders will be sufficiently longfor suchfixity to havenegligible effectsonthe critical stresses. The samesituationexists for the caseof pure bending.

4-31

4.4

EXTERNAL

4.4.1

Basic

4.4.1.1

This

section

Considerations

deals

with

cylinder

axial

is

loading

ported.

PRESSURE

Principles

Theoretical

sandwich

No

LATERAL

is

That

placements

is,

the

loading

subjected

to

applied.

In

during

condition

external

in Figure

pressure

addition,

buckling,

and no bending

depicted

it is

both

ends

only

specified

of

the

4.4-1.

over

that

the

the

cylinder

Note

cylindrical

ends

are

experience

no

radial

s uppo rt_

Figure

theoretical

4.4-1.

basis

which

Circular External

used

here

includes

the following

facings

are

b.

The

facings

may

be of equal

c.

The

facings

may

be

d.

Poisson's

ratio

e.

Bending

of the

f.

The

has

Cylinder Pressure

is the classical

The

core

psi

Sandwich Lateral

a.

Subjected

small-deflection

to

solution

of Kuenzi,

assumptions:

isotropic.

is

of the

facings infinite

the

or unequal

same

or

same

for

about

their

different

both

materials.

facings.

own

extensional

thicknesses.

stiffness

middle

surfaces m the

direction

can

be

neglected.

normal

facings. g.

dis-

moments.

p,

et al. [4-21]

sup-

p, psi

Both

The

the

surface.

simply

ends s imply

that

The parallel

core

extensional to the

and

shear

rigidities

facings.

4-32

are

negligible

in directions

to

the

The

h.

transverse

shear

properties

of the

core

may

be either

isotropic

or

o rthotropic.

The

2R _- >> 1

i.

The

j.

Several

additional

below

in connection

solution

inequality

of Kuenzi,

order-of-magnitude with

et al.

in References

4-22,

4-23,

develop

design

curves

which

are

Kuenzi,

et al.

[4-21]

constitutes

most

up-to-date

has

been

slightly

tion

required

the

need

in practical

for

still

required

The

final

modified

each

theoretical

published

of the

of the

relationships

Pcr

Zahn

as noted

for

in this

Vp

handbook

[(Eltl)

the is used

since

[see

reports

as

The

of their scope

to

work and

of is

results

of interpola-

here.

separate

Equation

are

these

by

of reports

the format

eliminated

values

series

format

laid

and 4-26.

to reduce

revised

entirely

Cp = R (l-re _)

to this

in order

used

4-25

However,

4-5

used

valid,

groundwork

and

revision

The

selected

earlier

in References

subject.

not been

the

Norris

latest

in Reference

has

upon

4-24.

applications.

interpolation for

and

are

(4.4-2).

draws

the

treatment

assumptions

Equation

[4-21]

Raville

the

is satisfied.

However, families

are

(4.4-4)].

follows:

(4.4-1)

+ (E_)]

where Cp

= Minimum

value

(with

respect

to

n) of

Kp

, dimensionless.

and

(4.4-2) _2 (n 2 _ 1) (3 + --_--_/L\_-'_R n2 L2 _[/n2 L22

1)(nL

+ 4-33

7v:'R2 1+ _-)-2]+

98 [1+

(n 2 + _]Vp]

(Ez tl)

(E2 t_)

h2

L_/2

[(Eltl)

(Eltl) Vp

(4.4-3)

* (E_t_)laW

(Ezt2)

h

= [(El

tl)

+ (Ez

t2)]

(1;,seg)R_-Gi,

z

(4.4-4)

whe re

Per

= Critical

It

e

value

= Radius

to middle

surface

-

Poisson's

ratio

Elastic

= Plasticity Et

and

tx

E2

and

Over-all

length

h

-

Distance

between

-

Core shear to the axis

Note:

formulas cases For

are

factor,

1 and 1 and

the

two

only

( E _ 1) can

only

configurations,

facings when

the

be made E_

and

not

behavior

psi.

respectively,

surfaces

of the

is elastic

inches.

in the

buckle

of facings, with

the

facing

pattern,

same

both

facings

are

E_

will,

of course,

made

inches.

plane

perpendicular

is denoted

material,

(_ = 1).

when

4-34

respectively,

inches.

as to which

made

inches.

dimensionless.

2,

2,

modulus associated of revolution, psi.

are

sandwich,

full-waves

middle

psi.

dimensionless.

of cylinder,

There is no preference subscript 1 or 2.

valid

such

of facings,

of facings

-

pressure,

of cylindrical

of facings

L

where

lateral

= Number of circumferential dimensionless.

Gyz

cases

modtdi

= Thicknesses

n

For

reduction

= Young's

te

of external

the foregoing

Application of the

be equal.

by the

same

to inelastic material.

Equation(4.4-2) constitutesanapproximateexpressionfor Kp sinceit embodiesthe assumptionscited earlier in this sectionin additionto the following: a.

Terms containingKp and 4 R2 wereneglected.

b.

It wasassumedthat (1 +

By using The

Equation

design

erence

curves

4-5.

cient

Cp

and

[4-8] than

critical

are

Such

the

related

predictions number

Donnell's 4-21

cylinders.

That

is equal

to the value

scribing

the

associated

leads

is,

behavior with

the

that

ring

when

Gy z -* o0

obtained where

from n = 2.

crimping

The mode

of the

This

end

are in the

obtained

are

to accurately

ring

theory

upper

limit

of failure 4-35

the

predict is large,

which

is capable

to the which

curve involves

for

the

Refcoeffiwith

constraints

and to

neighborhoods from

such

rings.

33 percent

higher

to the

that

fact high

theory

the

L/R

from

approximations

is due

that

4.4-2.

subjected

(n = 2} is not sufficiently to observe

number.

is associated

Donnell

which

formulations.

{Vp -_ 0) and

that

which

as are

pressures

It is important of terms

limit

do not lie

same

application

exist

by the

rings

that

directly

limits lower

whole

in Figure

taken

unaffected

be the

full-waves

number

shear

will

to critical

accurate

upper

for

shown

were

The

are

cylinders

be noted

and

is a small

form

and

4.4-2.

of the

patterns

assumptions.

a sufficient

lower

to those

m

of the type

equal

of circumferential

[4-8]

retains

from

that

in Figure

portions

rings

of this

configurations

the buckle

to non-sandwich

here

are

it should

be generated are

identified

For

connection,

can

to note

pressures

boundaries,

In this

handbook

these

pressure.

of the

of this

behavior.

related

external

plots

It is helpful

long-cylinder the

(4.4-2),

= 1, where

m2_)

to justify

of Reference

buckling the

the

of long

critical

pressure

of properly of Figure extremely

de-

4.4-2 short

is

circumferential

(4.4-4)

load

wavelengths

to this case

Ncr

gives

(n--oo).

the following

in units

measured

Specialization

of

formula

of Equations

(4.4-i) through

for the critical compressive

running

lbs/inch:

Ncr

= h Gy z

(4.4-5)

Ncr

= Per

(4.4-6)

where

By

using

equivalent

the approximation

h _ tc , it can

to the crimping

formula

It

easily be shown

presented

•

earlier

that Equation

as Equation

"

(4.4-5)

is

(2.3-9).

Crimping)

Cp

Lower Note:

Figure

Vp

= Constant

0 a

:- Constant

4.4-2.

(n = 2)

Schematic Versus

Representation L/R

Subjected

Another

[Equation

important

(4.4-2)]

perpendicular

has

very

point

which

does

not

to the

little

influence

facings

on

Limit

for

to

contain

and

be

the

core

is

4-36

that

shear

in the

longer

Sandwich

Lateral

noted

oriented

cylinders

Circular

External

should

of Log-Log

than

Cylinders

Pressure

the

approximate

modulus

axial

Plot of Cp

direction

approximately

associated

(Gxz).

one

formula

for

with

the

This

modulus

diameter

Kp

plane

and

has

therefore ence

disappeared

4-21.

Thus

through the theory

the and

of the

handbook

can be considered

either

isotropic

or orthotropic

approximations design

curves

applicable transverse

made presented to sandwich

shear

4-37

moduli.

in the development in this cylinders

section having

of Refer(Section

4.4)

cores

with

4.4.1.2 Empirical Knock-DownFactor In Section4.1 it is pointedoutthat, for circular cylinderssubjectedto external lateral pressure,the shapeol 0m post-bucklingequilibriumpathis suchthatone wouldnot expectstrongsensitivity to the presenceof initial imlxerfections. This has indeedbeenshownto be the casefor isotropic (non-sandwich) cylinderswherethe availabletestdatashowrather goodagreementwith the predictionsfrom classical small-deflectiontheory. In viewof this, it hasbecomewidespreadpracticeto either acceptuncorrectedsmall-deflectiontheoreticalresults asdesignvaluesor to applya uniformknock-downfactor _p of 0.90 regardlessof the radius-to-thicknessratio. In Reference4-4 the latter practiceis also recommended for sandwichcylindersand this approachhaslikewisebeenselectedas the criterion for this handbook. Theonly availabletest datafor sandwichcylinderssubjectedto externallateral pressltre are thosegivenin References4-27and 4-28. In the first of thesedocuments, Kazimi reports the results from two specimenswhichwereidenticalexceptlor the use of normal-expanded core in onecylinder while the other incorporatedover-expanded core. Thefollowingresults wereobtained: Comparison

of Theoretical

Test

@

Results

Predictions

of Kazimi

Versus

_4-27_

@

@

@

Theoretical Core

Type

Based

Test Pcr (psi)

Pcr on

Ref.

Over-

Expanded

!

(Test

Pcr)

m

4-5

and

(Theo.

"/p : 1.0

(psi) Normal-Expanded

('(P)Test

:

Pcr)

®-®

17

30.5

.56

27

30.5

.88

4-38

Kazimi the

[4-27]

attributes

over-expanded

condition

from

normal-expanded

rests

on the

fact

deformation

4-28

facings.

individual which

in his

gives

more

the

core

Jenkinson

Each

results

to the

circumstance

uniform

core

properties

than

argument core

to the

identical facing

of the

report

construction. was

put forth

exhibits

shape

and Kuenzi

controlled

essentially

test

The

the over-expanded

orientations

were

cylinders

that

of nominally

plastic

scatter

honeycomb.

in forming

In Reference cylinders

the

the

less

results

These

to provide

a laminate

following

on behalf

®

(saddle-type)

obtained

layers

all

from had

of glass

five

test

glass-reinforced

fabric

with

having

in-plane

properties

were

obtained

from

results

of Theoretical Predictions Results of Reference 4-28

their

these

®

Versus

®

®

Theoretical Cylinder No.

Based

Test Pcr (psi)

4-5

specimens pressures

2 through which

(YP)Test

on Ref.

(Test

55.2

1.09

45.2

1.16

52.5

52.6

1.00

52.5

45.2 47.6

1.16

from

that

50 to

55 psi.

4-39

initial

buckling

Therefore,

Pcr)

®+®

52.5

reported

Pcr)

(Theo.

60

5 it was ranged

Per

and 7p = 1.0 (psi)

52.5

lateral

viewpoint

: Comparison Test

For

obtained

of this

anticlastic

cylinders

of three

The

are

cylinder.

composed

isotropic.

whereby

1.10

occurred in the

at external above

tabulation

it wasassumedthateachof thesefour cylindersbuckledat 52.5 psi. In general, the test valuesare somewhathigherthanthe theoreticalpredictions. This is probably due to a.

the

absence

b.

inaccuracies

C.

the

of precise due

data

on the

material

to interpolation

between

properties the

theoretical

curves

and fact

that

sandwich

The

foregoing

justification it would mens

results

for the

use

having

the

fact

shear

perfections. safely the

bending

spection occur

the

for

simplicity of a uniform

failures

of the

low

and the

moderate

knock-down

L/R

these

would

with

the

4-28

data

use

this

with

factor,

additional

of a unifornl

mode

since

about

added ltowever,

tests

on speci-

of configurations

values.

This

fact,

value

factor.

4-40

middle that

"_p - 0.90,

Fp

= 0.90

of initial

prevails,

one

basis surfaces.

this

coupled

of

presence

theoretical

own shows

value

to the of failure

the

their

4.4.2

of the

to provide

representative

be insensitive

where

seem

cylinders.

of Section

nature

in comparison

knock-down

be truly

the

will

facings

and

a lower-bound

especially

cuz_ves

extremely

as

concerning

= 1.0,

stiffnesses

4-27

sandwich

region

thick

.25).

which

full-size

,(p

relatively

to supplement

ratios

in the

of the design

_

'{1) = 0.90

crimping

value

were

References

of interest

Hence,

use

of

tf/h

point

that

from

in realistic

An additional

(_

be desirable

small

found

facings

thickness

test

certainly

usually

the

type

with

used

is im-

could here

neglects

However, of failure

considerations

led to the

selection

in-

will

only

of

here

In viewof the meagertest dataavailablefrom sandwichcylinderssubjectedto external lateral pressure, the methodrecommended herecanpresentlybe regardedas onlya "best-available"approach. However,there appearsto be little reasonto doubtthat f_'ther testingwouldshowtheseproceduresto bequitereliable.

4-41

4.4.2 DesignEquationsandCurves For simply supportedcircular sandwichcylinderssubjectedto externallateral pressure, the critical pressuremaybe computedfrom the equation 7p T, Cp - R(l__e _) [(Eltz)

Pcr

(4.4-7)

_ (Eet2)]

where "_p = 0.90

and

Cp

is obtained

one

must

from

compute

the

1,'igures

following

4.4-3

through

In order

4.4-5.

to use

these

curves,

values:

(E _t_ )(Ea te) he

,2

(4.4-8)

uJ

:

[(Eltl)

+ (E2ta)]

(E 1 tl )(Ea

2 112

ta ) h (4.4-9)

Vp

For

elastic

Section

For

9 must

cases

formuIas cases For

Since values

cases,

use

the

valid

two

only

(_ _ 1) can only such

configurations,

separate of

extrapolation

Vp,

_] = 1.

+(Eata)]

(l-re

the

behavior

Whenever

_) He Gyz

is inelastic,

the

methods

of

be employed.

where are

= TI [(Eltt)

facings when

are

the

be made E_

not made

behavior when

and

families

of design

one

will

usually

to establish

Cp

find for

the

facings

will,

curves

same

is elastic

both

Ee

of the

(,Z - 1). are

of course,

(Cp

vs

it necessary configuration

4-42

material,

L/R) to use

the

Application

made

of the

same

foregoing to inelastic material.

be equal.

are

provided

graphical

of interest.

for

only

interpolation Where

desired,

three or

improved in order

The cores

accuracy to obtain

results with

given either

can

be obtained

by minimizing

Equation

(4.4-2)

with

respect

to n

Cp.

by the isotropic

procedures

specified

or orthotropic

here

transverse

4-43

apply shear

to sandwich moduli.

cylinders

having

0.01 Yp rtC Per

'2

I_:]

- v

[(I

lt])

+

(E2t,2)]

)

\5 9O

\

P

",v I

\.

defined

by Eq. (4.,t-8)

•] Vp defh]ed

' CIo00]

2

by

Eq.

(,i.4-'J)

----*,-

_x NN

N

<

2 = O. 0001

N-<

/oll,oJo!ol i _-ll 0.000]

C

-

......

p

w..:

---:oTioii I

\

O. 00001

\

---

X_

0.000001--.--

--1

Figure

4.4-3.

10

Buckling Subjected Transverse tropic;

Coefficients Cp for to External Lateral Shear Properties

Vp : 0 4-44

Circular Pressure; of Core

100

Sandwich Cylinders Isotropic Facings; Isotropic or Ortho-

0.0

:,.-

......

_p

,7c 2

-.....

Per

-

R(1 -v eP )

_2

[(Eltl)

+ (E2t2)]

defined

by Eq.

(4.4-8)

V P defined

by,Eq.

(4.4-9)

0.00]

O. O00l

C P

0.00001

0,000001 10

Figure

4.4-4.

Buckling Coefficients Cp for Subjected to External Lateral Transverse Shear Properties tropic; V = 0.05 P 4-45

Circular Pressure; of Core

Sandwich Cylinders Isotropic Facings; Isotropic or Ortho-

0.01

¢' yz 1

0.001

0.0001

C P

0.0000]

0.000001

Figure

4.4-5.

-1

10

Buckling Subjected Transverse tropic;V

Coefficients Cp for to External Lateral Shear Properties P

= 0. i0 4-46

100

Circular Sandwich Cylinders Pressure; Isotropic Facings; of Core Isotropic or Ortho-

4.5 TORSION 4.5.1 BasicPrinciples 4.5. I. 1 TheoreticalConsiderations This sectiondealswith the loadingconditiondepictedin Figure 4.5-1. Notethat the onlyconsiderationgivento boundaryconditionsis that, during buckling, it is assumed that no radial displacementsoccurat either end. Further conditionsat theseboundaries are completelydisregarded. This approachshouldbesufficiently accuratefor all simply supportedcylinders exceptthosewhichare very short.

T,

In.-Lbs.

Torque

__

T,

In.-Lbs.

Torque

It is assumed that, during buckling, no radial displacements occur at either end.

Figure

The

buckling

treated

of isotropic

by Donnell

concerning theory. ling

4.5-1.

of long

consideration

the

circular

Sandwich

(non-sandwich),

in Reference

reasonable Using

Circular

4-8

Donnell

approximations,

sandwich

whatsoever

cylinders

to the

circular

which

approximations

boundary

Cylinder

has

which

Subjected

cylinders

become

be employed

Gerard

[4-29]

conditions.

4-47

subjected

a standard

can

subjected

to Torsion

to torsion. Such

to torsion

source

of information

in practical has

thin-shell

investigated This

the

solution

an approach

was

buck-

gives

is valid

no in

view

of the

4-30,

qssumed

March

both

finite

cylinders ence

and and

are

4-30

Kuenzi

infinite as

configuration.

The

to provide

the

given

in Section

the

following

a.

The

facings

are

isotropic.

b.

The

facings

are

of equal

c.

Young's

d.

Poissonts

e.

The

sandwich

of one

most

the

taken

purposes

up-to-date 4.5.2

taken

in Reference

sandwich lot

cylinders

the

of this

treatment

were

has

However, having

is not more

is the

ratio

thickness. cylinders

facing

modulus

core

For

for

hand,

finite-length handbook,

of the

directly

of

Refer-

subject.

from

The

that

report

assumptions:

for

thickness

conditions

4.5-1.

embody

accurate

solutions

boundary

in Figure

curves

On the other

small-deflection

lengths.

indicated

design

long

develop

is considered

theoretical

and

extremely

is the

same

for

same

infinite

for

unequal

than

both both

extensional

twice

the

curves

facings, the

are

reasonably

provided

that

the

other.

facings. facings.

stiffness

in the direction

normal

to the

facings. f.

The

core

extensional

parallel g.

The

to the

and

shearing

stiffncsses

are

negligible

may

be either

in directions

facings.

transverse

shear

properties

of the

core

isotropie

or

o rthotropic. h.

The

The

design

bending situations,

approximations

curves

of the

include

facings

it is assumed

of Donnell

separate

about

their

that

the

[4-81

families own

which

middle

facings

are

4-48

can be applied.

respectively

surfaces. thin.

However,

neglect

and

for

both

include of these

The theoreticalbucklingrelationshipusedhereis d rcr

which

is based

rial.

The

on the

notation

rcr Ks

Ef d

further

used

= K s ]] ErR

assumption

here

is as

=

Critical

value

:

Torsional

buckling

=

Plasticity

reduction

-

Young's

=

Total

that

thickness

shear

t l and

te

=

Thickness

:

Thicknesses

factor,

R

The

-

buckling

Radius

pression

given

it should

be noted

plotted

in the

in Reference that

general

t c +tz

facings

form

shown

mate-

(4.5-2)

is no preference

1 or

surface

2.),

at by the

This

in Figure

is not leads

4.5-2

15 (tl

4-49

to

of a complicated reproduced Ks

values

here. which

exHowever, can

be

where

L_ - dR

+t2)

facing

inches.

minimization

(4.5-3)

16 t c tlt_ -

cylinder,

formulation

minimization

as to which

inches.

of sandwich

Zs

Vs

same

+ te

(There

subscript

4-30. indicated

of the

inches.

is arrived

the

made

dimensionless.

wall.

by the

are

dimensionless.

of sandwich

of the

Ks

psi.

psi.

to middle

coefficient

stress,

of facings,

of core,

is denoted

facings

coefficient,

d

t c

both

follows:

of facing

modulus

(4.5-1)

Ef Rd

Gxz

(4.5-4)

Gxz (4.5-5) Gyz

and L Gxz

-

Over-all

=

Core

shear

facings Gy z

=

Core

of cylinder,

modulus

Number d imens

m the

modulus

of revolution,

inches.

associated

and oriented shear

axis n

length

axial

associated

with

the

direction, with

the

perpendicular

to the

perpendicular

to the

psi. plane

psi.

of circumferential

full-waves

in the

buckle

pattern,

ionl es s.

t c/d K s

plane

Upper

Limit

- Constant

I

_ Constant

'/

(Shear Crimping) : Constant ] Long

Cylinder (,1

: 2)

ZS

Figure

4.5-2.

Typical for

Log-Log

Circular

Plot

Sandwich

of the

Cylinders

The curves given in Section 4.5.2 are of this type. buckling coefficient K s correspends involves extremely

Buckling

Coefficient

Subjected

K_

to Torsion

Note that the upper limit for the

to the shear crimping

short circumferential wave-lengths

mode of failure which

(n -. _o). Specialization of

the buckling equations to this case leads to the following result when it is assumed that tc/d

_ I :

4-50

5

Tcr

= rcrimp

- (tz

2

+ t_)

tc

v/Gxz

GY z

(4.5-6)

where h

=

Distance

In connection (long for

with

cylinders),

which

point,

Donnell

attention

obtains

sandwich

the

out that

to the

result

[4-8]

the

In Reference solution

4-32,

which

Timoshenko

does

more

the

Donnekl

nal

lateral

Since tions,

the they

exact

result

torsional must

Donnell

:

to the

situation

with

of Section

caution

in the

4-51

illustrate

(n = 2) this

circular

Gerard

[4-31]

cylinders:

result

from

a more

rigorous

(.___.) 3/_

is similar

curves

shape

approximations:

This

design

To

Zs

(4.5-7)

which

where

an oval

(non-sandwich),

of such

following

E 3 _]2 (1-Ve2)a/a

4.4)

into

parameter

E

stress

Section

buckle valid.

the

approximations,

stress

a critical

(see

be used

the

for

isotropic

Donnell

shear

inches.

values

no longer

gives

approach, pressure

the

will

for

- (l_Ve_)a/_

the

rcr

The

obtained

presents

not invoke

are

critical

rcr

large

cylinder

By using

for

of facings,

having

results

to torsion.

following

surfaces

the

approximations

is drawn

subjected

middle

constructions

it is pointed

the

cylinders

between

the 4.5.2 case

is only

87 percent

encountered

difference

is even

incorporate of long

(4.5-8)

the

cylinders

of that

in the more

case

= 2).

by

of exter-

pronounced.

Donnell (n

given

approxima-

4.5.1.2

In

Empirical

Section

4.1

shape

of

tile

tivity

to

initial

Knock-Down

it

is

more

pressure,

4-8

the

of

indicates

60

of

cal

theoretical

To

date,

consider_t

exists

Based

on

[4-51

recommend

takes

a more

lor

the

the

cautious

cylinders

should

no

is

lead

circular

range

classical

small-deflection

approximated

the

case

be

using

axial

expected

to

external

lateral

loaded

be

in torsion

and

obtained

80

sensi-

of

proportions,

can

by

the

expect

cylinders

of sizes,

data

be

under

circtdar

torsion,

in tile

cylinders

test

are

of

of

for

the

by

taking

theory

{_

-

0.60).

percent

of

the

classi-

post-buclding

made

value

to expect

moderate

that

did

the

( "_s :: 1.0).

the

use

largely

not

on

furnish

usually

reductions

4-52

lactors

are

path,

the

_s

.80

basis

of

a lower-bound

greater

than

apply

the

types

cases.

some

sources

Reference

for

the

the

4-4

sand-

isotropic

to

thicknesses

to

the

empirical

such

ttowever,

of

of

no

in

equilibrium

employed

was

which

Therefore,

knock-down

recommending

this

cylinders

to torsion.

reliable

be

in

sandwich

subjected

selection

more

not

would

available

ptd)lished

approach

to

iil torsion

the

be

would

encountered

enormous

reduction

reasonable

one

to

= 0.80)

and

Although

by

that

that

sensitivity

from

drop-off

This

data.

it

to

determination

that

as

subjected

an

can

been

hmld!yook

configuration.

points,

data

such

(non-sandwich),

over

(3's

has

moderate

(non-sandwich)

test

data

in this

basis

wich

test

cylinders

is

strong

exhibited

obtained

of the

as

tile

curve

values

test

be

isotropic

predictions

no

is

that,

a lower-bound

values

circtdar

path

h:md,

than

materials,

Average

to

other

case

the

for

equilibrium

severe

hi the

percent

that,

imperfections

On

I{eference

out

post-buckling

compression.

somewhat

pointed

Factor

the

of

isotropic

isotropic

sandwich

(non-

sandwich) will the

conligurations.

continue

to supt_)rt

torsional

instability value

buckling

be regarded

test

data

compressed

has

it should

considerable mechanism

of axially

"is - O. 80

sandwich

In addition,

been

selected

to substantiate

as a "best-available"

torque

be noted

well

into

should

not be nearly

cylinders.

With

for this

use

in this

selection,

approach.

4-53

that

the

postbuckled

so catastropic

these

several

handbook. the

cylinders

methods

factors

under region, as

the

torsion ttence general

in mind,

In view

of the

proposed

here

lack

the of

can only

4.5.2

For

Design

simply

shear

Equations

and

supported

stress

may

Curves

circular

be

sandwich

computed

from

cylinders

the

subjected

to torsion,

the

critical

equation

d

rcr

=

Xs Ks

_

Ef

_

(4.5-9)

where "¢s

d

and

Ks

one

must

is

obtained

first

from

compute

Figures

each

= 0.80

= tc + t_

4.5-3

of

the

(4.5-10)

through

following

(4.5-11)

+ t_

border

4.5-8.

to use

these

cu_,es,

values: L2

Zs

V

= s

-

(4.5-12)

dR

16 tc t_ t_ TiEr 15 (tl + t_) R d Gxz

(4.5-13)

Gxz

@ -

(4.5-14) Gyz

It is

required

For

elastic

Section

The

here

cases,

9 must

critical

following

be

torque

for

both

that

both

facings

use

r1 = 1.

be

made

Whenever

the

of the

same

behavior

material.

is

inelastic,

the

methods

computed

from

of

employed.

Tcr

elastic

, measured

and

inelastic

Tcr

in

units

of

in.-lbs,

can

be

the

cases:

= 2rrR

2 (tl

4 -54

+ ts)

rcr

(4.5-15)

Culwesfor for

Ks

other

The

the

latter

noted

may

Ks

of the

@ -0.4;

1.0;

and

2.5.

Estimates

of

Ks

by interpolation.

given

for

bending

to obtain

4.5.1.1,

(long

curves

in this

speaking,

Figures

curves

provided

of

values

of the

numerical

of

facings estimates

t c/d

= 1.0

about

their

and own

0.7.

The

middle

of the

conservatism

somewhat

inaccurate

former

surfaces. introduced

stiffnesses.

is large

the

values

are

from

be used

in Section

cation

facings,

for

these

Zs

However,

be obtained

@ can

contribution

where

Strictly

of

curves

by neglecting

As

for

values

In addition, neglect

aregiven

that

are the

the

design

cylinders).

curves

Some

are

caution

should

in the

be exercised

in the

region

appli-

region.

4.5-3 reasonably thickness

through

4.5-8

accurate of one

apply for

facing

4-55

only

when

sandwich is not more

the

cylinders than

facings

are

having twice

the

equal.

unequal other.

e_

©

r_

.! m

El

4-56

0

¢)

II

II

_J

¢)

0

0 C)

iZ/

4-57

0

0

b or,,i

°,,-I

r..)

II

4-58

S

0

£

E)

4-59

.2 0

_D

"0

.2 e_

.2

0

I t_

O(3 o

4-60

S

©

©

@

0

m

,.,,-.i

% ©

@

©

ca

od I

u

4-61

4.6

TRANSVERSE

4.6.1

Basic

Principles

In Reference

4-33,

(non-sandwich),

The

same

Both

sets

of data

of these

type

stresses

pare

stress

theory

for

Becker the

following

of data

were

obtained to the

values

the subjected

from

ratios

where

in Reference

transversc

shear

loading.

theoretical

results

circular

for nominally the

elliptical

permits

It has

from

loaded

are

such

obtained

bend-

cylinders. lengths.

Extrapo-

useful

to com-

small-deflection

in torsion.

specimens,

and

a determination

proven

obtained

predictions

on isotropic shear

of varied

stress

cylinder,_

identical

theoretical

for

cylinders bending

of tests

transverse

,I-34

of zero

the

a series

to combined

condition

against

that,

from

cantilevered

(non-sand_vich),

report

results

is published

for pure

isotropic

[4-35j

reports

cylinders

results

of critical these

Lundquist circular

ing.

lation

SI[EAR

Gerard

comparisons by using

and yield

Reference

4-36:

Average Transverse

of "rer Values Shear Test Loading

f¢)r t

Small-Deflection Theoretical [ Values for Torsional Loading

Ter i

Lower-BoundTcr Values Transverse Shear Test Loading Small-Deflection Values

To properly

interpret

shear

stress

hand,

under

Tcr

then

Tcr

is uniformly

transverse

corresponds

these

shear to the

for

Theoretical

Torsional

ratios,

distributed loading, peak

intensity

around the

Vc r ]

(4.6-1)

_

1.25

(4.6-2)

j

out that, the

1.(;

for]

Loading

it is pointed

a,

for torsional

circumference.

shear

stress

is non-uniform

which

occurs

at the

4-62

neutral

loading, On the

the other

and the axis.

value

For the lack of a better approach,it is recommonded that Equation(4._;-2)beused[or the designandanalysisof circular sandwichcylindersthat are subjectedto transverse shearforces. In suchcases,the requiredsmall-deflectiontheoreticalrcr valuesfor torsional loadingshouldbeobtainedas specifiedin Section4.5 of this handbook with the exceptionthat 7s = 1.0 shouldbeusedhere. Notest dataare availableto substantiatethe reliability of this practice. Until suchdatadobecomeavailable, onecan onlyregardthis procedureas a "best-available"approach.

4-63

4.6.2

DesignEquationsandCurves

For simply supportedcircular sandwichcylinderssubjectedto a transverseshear force andhavingboth facingsmadeofthe samematerial, the critical shearstress maybe computedfrom the equation d rcr = 1.25KsrlEf -_

(4.6-3)

where thebucklingcoefficientKs is obtainedfrom Figures4.5-3 through4.5-8 and the notationis the sameas thatemployedthroughoutSeetion4.5. As notedin Section 4.5.1.1, thesefigures are somewhatinaccuratein the regionwhere Zs is large (long cylinders)andoneshouldexercisesomecautionwhendealingwith suchconfigurations. Strictly speaking,Figures4.5-3 through4.5-8 applyonlywhenthe facingsare of equal thickness. Itowever,the curvesare reasonablyaccuratefor san_viehcylinders having unequalfacings,providedthatthe thicknessof onefacingis not more thantwice the other. For elastic eases,use r? = Section

For

9 must

elastic

pounds,

1.

Whenever

eylinders

the

critical from

transverse the

(Fv)er

gration

behavi'n"

is inelastic,

the

methods

of

be employed.

can be computed

To compute

the

(Fv)cr

when

the behavior

shear

force

(Fv)cr,

measured

in units

following:

= rrR (t 1 + ta) rer

is inelastic,

techniques.

4-64

one

(4.6-4)

must

resort

to numerical

inte-

of

4.7 COMBINED LOADINGCONDITIONS 4.7.1 General For structural memberssubjectedto combinedloads, it is critical

loading

shows

the

graphic

of an applied alone. this

The form

usually

or stress

quantity

give

by means

format

load

figurations. curve

conditions

a very

clear

points

coordinate

axes

outside

of the

interaction

curve

shown

in Figure

assume

that

for

computed

the

actual

to point

proportional from

the

loading

of the point

structure

increases

type

of loading.

margin

the

area

will

curve

and

to the

when

bounded

Curves

by the

All points

interaction lying

Furthermore,

is given

by the

origin.

combined

loading

on or as

ratio

For

of

con-

occur.

to the

ratio

acting

of particular

structures.

of safety

is subjected

B of Figure

a second

integrity

4.7-1

R. is the 1

of loading

buckling

to the

quantity

type

within

that

Figure

that

to stable

indicate

The

structural

fall

correspond

a measure

a particular

corresponding

Then,

4.7-1,

for

as to the which

for

to represent

curves.

purpose.

value

defined

picture

interaction

for this

critical

R. is similarly J

and the

from

used

to the

All computed

distances

of so-called

customary

of

example,

condition

4.7-1.

in R. and 1

R., j

the

margin

of safety

(M.S.)

can be

following:

(Rj)D M.S.

-

1

(4.7-I)

(Rj)B As an alternative

procedure,

which

is based

on the

Point

M is located

one might

assumption

in such

that

a position

choose loading

that

to compute beyond

BM is the

4-65

point shortest

a minimum B follows line

that

margin the path

of safety BM.

can be drawn

between

calculated

point

B and

as

the

interaction

curve.

The

minimum

margin

can

then

follows:

OB Minimum

M.S.

_ BM OB

1

1.0 (Rj) D _M

(Rj)B

R

of safety

J iiiiii

0

R

1.0 i

Figure

4.7-1.

Smnple

4-66

Interaction

Curve

(4.7-2)

be

4.7.2

Axial

i.7.2.1

Compression

Basic

In References cylinders

Bending

Principles 4-17,

4-18,

subjected

interaction

Plus

and

to axial

curve

may

4-19

it has

compression

be accurately

been plus

described

shown

that,

for

bending,

the

by the

equation

circular

classical

sandwich theoretical

(4.7-3)

(Re)c L + (Rb) C L = 1 where U e

(Rc)cL

-

(4.7-4)

(5c)CZ

_b

(.i. 7-_) {Rb)cL-

((_b) CL

(Sb)CL

=

(4. v-6)

((_e) C L

and =

(Y

Uniform

compressive

stress

due

solely

to applied

axial

load,

e

psi. =

_b

Peak

compressive

ment, =

(_c)CL

Classical stress

=

((_b) C L

which

fail

4-17 in the

Classical

with

infinitely

core

categories.

and 4-18 shear

long

due

solely

load

acting

value

for

a bending

the mode.

which

Equation

for

an axial

under

crimping

value

theoretical

develop

cylinders Since

theoretical under

stress

References

stress

to applied

bending

mo-

psi.

fall (4.7-3)

foregoing On the in the

moment

result other

4-f;7

uniform

alone,

compressive

psi.

critical

peak

acting

alone,

compressive psi.

for weak-core hand,

stiff-core

is written

critical

Reference

and the

in terms

constructions 4-19

deals

moderately-stiff-

of classical

theoretical

allowables,it doesnot includeanyconsiderationof the detrimentalinfluencesfrom initial imperfections. For the pullmsesof this handbook,theseinfluencesare treated by'introducingtheknock-dt_vnfactors ye spectively)

and Tb (see

Figures

4.2-8

and

4.3-2,

re-

to obtain

Rc +Rb

=

1

(4.7-7)

where

e

R

-

(4.7-8)

c

'Yc ((_c) C L

(_b %

Therefore,

the

no test

data

bending,

tors and

general

curve

for

sandwich

validity

of this

can

curve

correlation

is inherent

3/c and 7b were

established,

in part,

4.3).

only

ttc_vever,

even

these

substantiation be considered

data

were

is obtained, a "best-available"

lo0

(4.7-9)

be drawn

cylinders

of empirical

experimental can

interaction

is available

the

degree

design

Yb ((_c) CL

as

subjected

has

not been

in the

sandwich

few

in number.

in Figure

to combined experimentally

approach

from

the

shown

since

test

axial

load

verified. the

data

(see

Therefore,

recommended

4.7-2.

interaction

Sections until

Design Subjected

relationship

-

Interaction to Axial

Curve

1.0 for

Circular

Compression

Plus

4-68

Sandwich Bending

Some fae-

4.2

further

%

Figure 4.7-2.

and

knock-down

method.

Rc

Since

Cylinders

4.7.2.2

Design

For

simply

plus

bending,

Equations

suppol_ed,

the

and

Curves

circular,

following

sand_vich

interaction

cylinders

equation

Re

+ Rb

subjected

may

=

be

to

axial

compression

employed:

1

(4.7-10)

w h e re (y R

c

of

Equation

In Equations

tained

The

(4.7-11)

from

is

and

Figures

quantity

Section

(4.7-10)

is

in

(4.7-12),

4.2-8

(ffc)CL

given

and

simply

(4.7-12)

_b (_c)CL

Figure

the

4.3-2,

the

(4.7-11)

Tc ((7c) C L

au A plot

c

4.7-3.

knock-down

factors

Tc

and

_b

are

those

ob-

respectively.

result

obtained

by

using

Tc = 1.0

in the

method

of

4.2.2.

Plasticity

considerations

in this

case,

one

may

should

be

handled

as

specified

in Section

9.2

except

that,

use

IlVel Et

ao

r7 =

[ 1---2-_1

_ff

for

short

cylinders,

and

1

t b.

_

tl-u_] Equation

(4.7-10)

considerations

into

the

s

for

--

may

should

short-cylinder

moderate-length

through

long

cylinders.

Ef be

be

applied

included

range

(see

to

sandwich

in the

Section

cylinders

computation

4.2.2). 4-69

of

of

any

length.

((_c)CL

when

However,

the

structure

length

falls

0.8

!i!ii _' j

0.6

i!il Rb

0

0. '2

0.4

0.6

0.8

1.0

R C

Figure

4.7-3.

I)esigm Subjected

Interaction to Axial

for

Circular

Compression

Curve

Plus

4-70

Sandwich tk, nding

Cylinders

4.7.3

Axial

4.7.

:5.1

This

Compression

Basic

section

cylinder

deals

including

ures.

In addition,

or

with

is,

the

loading

to uniform

is imposed

source

This

External

Lateral

Pressure

Principles

is subjected

loading

Plus

external

as indicated external

condition

by the

pressures

during

it is specified buckling,

circumferential

the

[orces

the

ends

and

These uniformly of the

constrained they

are

in Figure over

P. are

are

displacements

pressure

which that

ends

depicted

forces

that

of bending

originate

they

over simply

the

Axial from

any

end

elos-

supported.

experience

no radial

moments. , psi Y I

lbs

lbs__

/

lllttIlIIlt'( Both

Ends

Simply

Figure

theoretical

[4-37]. and

are

p

Y

The

can

sandwich

surface.

distributed

p , psi

P,

The

cylindrical

eylinder

such free

the

4.7-4.

The

embody

Supported

4.7--4.

basis

design

Circular Sandwich Compression Plus

used

here

curves

is the

given

the

following

ao

The

facings

b.

Both

facings

are

e.

Both

facings

have

Cylinder External

classical

in this

handbook

small--deflection

were

assumptions: are

isotropic. of the

same

identical

thickness. material

4-71

Subjected to Axial Lateral Pressure

properties.

taken

solution

directly

of Maki

from

that

source

d.

Poisson's

ratio

e°

Bending

of the

f.

The

core

has

for

the

facings

facings infinite

about

is equal their

to 0.33.

own middle

extensional

stiffness

surfaces

in the

can

direction

be neglected. normal

to the

facings. The

g*

core

parallel The

ho

to the

The

theoretical

sixth

order

when had

determinant

of terms the

been

number based

not be applicable

The

interaction

in Figure

4.7-5

in this were

radius

of the

and

arc

no significant

retained

= G

xz

negligible

in directions

yz

full-waves

is in the would

derivation

buckle

equals

in this

4-37

are

in the

two

with

form

circum-

sandwich

by reproducing

to note

that

a sufficient

accurate

(n = 2). [4-81,

the

of a complicated

be gained

to obtain

approximations

in Reference

same ).

it is important the

Donnell

the

in comparison

advantage

throughout

which

are

(G

[4-37_

However,

well-known

core

is large

by Maki

handbook.

given

of the

cylinder

derived

to structures

curves

rigidities

directions

of circumferential on the

shear

moduli

longitudinal

relationship

formulation

number

shear

and

The mean thickness.

io

and

facings.

transverse

ferential

that

extensional

If the the

results derivation

results

would

manner.

of the

two

different

types

depicted

where Ef tf h V

xz

-

2 (1-.33e)

(4.7-13)

RaG XZ

Eftfh Vyz

= 2 (1-

.33_)

(4.7-14)

RaG yz

_ (Rp) CL

P

Y

_y) CL 4-72

(4.7-15)

cr x

(Rc)CL

(4.7-16) (C_x)C L

and Ef

G

: Young's

tf

=

Thickness

h

=

I)istancc

R

=

Radius

=

XZ

Core

G

= yz

Y

(1-iy)CL

facing, middle

surfaces

to middle

surface

of cylindrical

modulus

Core

shear

to the

axis

and modulus

associated

in the

the

lateral

pressure,

(C_x)C L

=

Classical theoretical value for critical sive stress when acting alone, psi.

L

=

Over-all

axial

value psi.

compressive

length

of Reference

they

do not include

ratios

psi.

plane

perpendicular

for

critical

stress

of cylinder,

external

lateral

due to applied tmiform

pressure

axial axial

load,

psi.

compres-

inches.

in Equations value for the

(4.7-13) elastic

and (4.7-14) is an Poissonts ratio of

facings.

curves

This

perpendicular

direction,

psi.

Uniform

imperfections.

used

with

inches.

psi.

=

the

approach,

the plane

axial

Classical theoretical when acting alone,

X

inches.

sandwich,

with

in the

of revolution,

external

of facings,

associated oriented

Note : The value .33 appearing assumed representative

the

inches.

=

g

Since

psi.

between

facings

= Applied

of facings,

of single

shear

to the

p

modulus

is evident

(Rp)cL

and

4-37

were

developed

any consideration from

(Rc)CL.

the

fact

For

4-73

from of the

that

the

detrimental

classical

purposes

a classical,

effects

theoretical

of this

small-deflection from allowables

handbook,

the

initial are

effects

1.0

1.0

:

4>

-._I

% .. o (Rc)CL

(Rc)CL

Jill/ = \XZ

-

- \VZ

[-_ :

O/

Constant

\

_

Constant]

_

. 1.0

1.0

(tlp)CL

Figure

from

4.7-5.

initial

(Re)CL

Typical

Interaction

to

Compression

Axial

imperfections

by

the

(Rp)CL

are

ratios

Rp

and

Curves

for

Plus

introduced R

which

c

R

Circular

External

through

the

are

defined

as

=

Py -

P

Sanchvieh Lateral

Cylinders

Subjected

of

and

Pressure

replacement

(%)CL

follows:

(4.7-17)

Tp (Py) CL

(y R

=

x

e

The

quantities

yp

respectively.

equal

No

for

yc

the

knock-down

factors

can

be

from

obtained

discussed

Figure

in

4.2-8

Sections

while

yp

4.4

may

and

be

4.2,

taken

0.90.

data

here

and

the

general

verified.

Yc are

Values

to

test

and

(4.7-18)

Yc ((_x) C L

are

are

available

subjected

validity

Some

for

to

of the

degree

o[

sandwich

axial

cylinders

compression

design

empirical

curves

which

plus

external

recommended

correlation

4-74

are

is

of the

lateral

here

inherent

has

in the

types

.ressure.

not

considered

Therefore,

been

experimentally

approach

since

the

knock-down Sections until curves

factors 4.2

further can

and

Te and 4.4).

Tp were

ttowever,

eN)erimental only be considered

established, even

substantiation as

these

in part, data

were

is obtained,

"best-available"

4-75

from few

the criteria.

sandwich

in nmnber.

recommended

test

data

Therefore, interaction

(see

4.7.3.2

DesignEquationsandCurves

For simply supported,circular, sandwichcylinderssubjectedto axial compression plus externallateral pressure, onemayemploythe interactioncurvesof Figures 4.7-6 through4.7-15where Eftf h 2(1-. 33z')R_ Gxz

xz

Eftf

(4.7-19)

h (4.7-20)

Vyz

2(1-.332)R P R

= P

R

Figure

The

(4.7-21)

4.2-8

while

quantity

and

Tp may

(4.7-22),

=

the

be taken

yz

Y

(4.7-21)

Yp (Py)CL

x Tc (fix) C L

c

In Equations

_G

knock-down

equal

(4.7-22)

factor

Tc is that

obtained

from

to 0.90.

(15y)CL

is simply

the

result

obtained

by using

7p = 1.0

in the

methods

of

((_x)CL

is simply

the

result

obtained

by using

Tc = 1.0

in the

methods

of

Section 4.4.

The

quantity

Section

4.2.

Plasticity

Figures V

considerations

4.7-6

= 0 (G yz

values

for

through = G

should

4.7-12

be handled

give

as specified

interaction

curves

in Section

only

for

9.2.

cases

where

Vxz =

- co). Separate families are provided for each of three selected

xz

yz

the

parameter

-_

= 50;

160

and 500 4-76

.

Graphical

interpolation

may

be

used

to obtain results

separate

curves

view

of the

for

for intermediate

ten

restrietions

different on

V

values

values and

behavior

it

is

of

stiff-core

proposed

of the

V

xz

that

ratio

, these

--L--

curves

--

can

Each

=

0.1;

only

be

family

0.'2; used

---

includes

1.

.

In

to describe

the

yz

eonstruetions.

here

of this parameter.

For

Figures

4.7-6

the

purposes

through

4.7-12

of praetieal

be

design

considered

and

applicable

analysis,

only

when

Rt C

-h_

V xz

± 0.05

(4.7-23)

_ 0.05

(4.7-24)

Rt C

V h*

yz

\vhcre

t

=

Thickness

of

core,

inches.

C

It is

ex_pected

Fi_ires

4.7-13

in V

(= V XZ

cases

to

that many

through

) will have

realistic sandwich

4.7-15

present

configurations

will satisfy these

a partial picture

on the interaction

requirements.

of the effects which

relationships.

These

figures

variations

only treat

yZ for

which

conjecture

--rrR _= 0.1. I, that

the

}tc_vever,

curves

given

for

the

trends

V

= V xz

dictions

if they

equalities

were

(4.7-23)

and

sweeping

application

shown

Figures

in

applied

to

(4.7-24).

sandwich

4.7-13

through

furnish

= 0 would

result

one

in view

4.7-15.

4-77

some

in

basis

for

conservative

one

pre-

yz

configurations

However,

of this observation

displayed

should

which

be

do

not

cautioned

of the limited

scope

satisfy

the

In-

againstmaking

of the information

It shouldbe kept in mindthatthe interactioncurvesgivenin Figures4.7-6 through 4.7-12

include

C L values

ranging

only

from

C

0.1

L

it

follows

that

these

curves

only

embrace

through

_R

1.0.

Since

(4.7-25)

L the

range

where

L 3.14

_ -_- <- 31.4

4-78

(4.7-2(;)

h= O.

8

_.

50

Vxz = Vyz : 0,0

__C

rrI1

0.6

c

O. ,1

0,2

0

0.2

0.4

0.6

0°8

1,0

R P

Figure

4.7-6.

Interaction Subjected Lateral

Curves to Axial

for

Circular

Compression

Pressure 4-79

Sandwich Plus

Cylinders

External

1 I1 11 V

50

= V xz

C

=

0.0

yz 'gR

I_

L

0.6 R C

0.4

0.2

0

0

0.2

CL

0.7"

CL C

0.8O. 9_

0.6

0.4

0.8

1.0

R P

Figure

4.7-7.

Interaction Curves Subjected to Axial I,ateral Pressure

for Circular Compression

4-80

Sandwich Cylinders Plus External

1.0 R N h V

0.8

= 50

= V xz

=

0.0

yz _R

C

L

=

L

0,6 '0

R

\

c

0.4

0.2

0

0.2

0.4

0.6

R

0.8

1.0

P

Figure

4.7-8.

Interaction

Curve

for

Subjected to Axial Lateral Pressure 4-81

Circular

Compression

Sandwich Plus

Cylinders

External

I H -h

V -

160

:-: V xz

::

0.0

yz

--

=0.]

C

0

0.2

0.4

0.6

0.8

1.0

R P

Figure

4.7-9.

Interaction Subjected Lateral

Curves

for

to Axial Pressure 4-82

Circular

Compression

Sandwich Plus

Cylinders

External

I R h 0.8

-- 160

= V

C LC_ 0_ 8(_ ) _

0.6

KC

L = 0.6 I

Vxz CL

0.0

?rRYZ _[[

e

C

0.4

L

0.2

0 0

0.2

0.4

0.6

0.8

R P

Figure

4.7-10.

Interaction

Curves

for

Circular

Sandwich

Cylinders Subjected to Axial Compression Plus External Lateral Pressure

4-83

1.0

1.0

I R

CLI0. 4 _

jt/ICL

h

= 0.

V

0.8

xz

= V

yz

=

0.0

-

_R

C L

C L = 0.5'__

5O0

L

0.6 R

c 0.4

C

L

C L = 0.3

=0.1

0.2

0 0

0.2

0.4

0.6

0.8

R P

Figure

4.7-11.

Interaction Curves Subjected to Axial Lateral Pressure

for Circular Compression

4-84

Sandwich Cylinders Plus External

1.0

= 0o0

R C

0.4 =0.8 CL=0.

0.2

0

0

0.2

0.4

0.6

0.8

R P

Figure

4.7-12.

Interaction Curves Subjected to Axial Lateral Pressure 4-85

for Circular Compression

Sandwich Cylinders Plus External

1.0

l°01

R C

R P

Figure

4.7-13.

Interaction Subjected Lateral

Curves to

Axial

Pressure

4-86

for

Circular

Compression

Sandwich Plus

Cylinders

External

R C

0.2

0 o

0.2

0.4

0.6

0.8

1.0

R P

Figure

4.7-14.

Interaction Curves Subjected to Axial Lateral Pressure

4-87

for Circular Compression

Sandwich Cylinders Plus External

1.0

0.8

t 5

0.6

%

kk

--

%

R C

\

0.4

h 0.2

=

500

--

CL

0

= _R I--_- =0.1

0.2

0.4

0,6

0.8

1.0

R P

Figure

4.7-15.

Interaction

Curves

Subjected to Axial Lateral Pressure

for

Circular

Compression

4-88

Sandwich Plus

Cylinders

External

4.7.4 Axial CompressionPlus Torsion 4.7.4.1

BasicPrinciples

This sectiondealswith the loadingconditiondepictedin Figure4.7-1(;. Thesanchvich cylinder is subjectedto endtorqueT plus axial loadingindicatedby the forces P.

T, in-lbs Torque

T, in-lbs Torque

BothEnds SimplySupported

Figure

4.7-16.

In Reference

4-18

of weak-core

configurations

assume

that

small-deflection the

following

the

Wang,

cylinder analysis

interaction

Circular

et al.

Compression

treat

this

which is long makes

Sandwich

Axial

fail

in the

so that use

type

Cylinder Plus

of problem shear

but

crimping

the boundary

of the

Subjected

Donnell

to

Torsion

only mode.

conditions approximations

consider

the

In addition

case they

can be ignored. [4-8]

This

to arrive

at

relationship:

(Rc)CL

+ (Rs) CL 2

=

1

(4.7-2

where

C

(Rc)CL

-

4-89

_

(_c)CL

(4.7-28)

7)

I(Rs)CL-

(4.7-29)

(_)CL

and (I c

=

Uniform

=

Classical

axial

compressive

theoretical

stress

value

for

due

to

crilical

applied

axial

uniform

load,

axial

psi.

eompres-

((_e)CL sire

_-

(T)CL

Because

it

Equation

does

not

tions.

Classical theoretical value duc to torque acting alone,

and

initial imperfections

the

ratios

R

and

was

stress

developed

consideration

from

due

from

of the

(Rs)CL.

For

are

defined

as

respectively.

equal

to

gives

the

Values

0.80.

following

are

for

Incorporation

interaction

effects

Yc

theoretical

initial

imperfee-

allc_vables

are used

of this handbook,

in

the effects from

of (Rc)cL

and

(Rs)CL

c

by

(4.7-30)

_

17s

(4.7-31)

(_)CL

the

knock-down

factors

can

be

from

of the

from

approach,

"/c(ffc)C L

s

_s

stress

-

R

and

shear

follows:

c

_c

psi.

small-deflection

the replacement

(y

quantities

torque,

for critical uniform psi.

detrimental

through

R

The

applied

a classical,

the purposes

are introduced

psi.

to

the fact that classical

1R which s

c

alone,

=

is evident

shear

acting

Uniform

any

the ratios (Rc)cL

when

=

(4.7-27)

include

That

stress

obtained

foregoing

relationship

c

Figure

substitutions

for

R

discussed

+ R 2 s

4-90

weak-core =

1

in Sections

4.2-8

into

while

Equation

4.2

:Ys may

(4.7-27)

and

be

4.5,

taken

then

constructions: (4.7-32)

In

Reference

walled,

4-38,

isotropic

transverse

Batdorf,

as

thai

el.

Equation

(4.7-32).

only

be

the

observed

extremes

although

probably

be

the

in

actual

cylinders

which

are

compression

plus

of this,

since

data

(see

Section

until

further

ttowever,

here

even

Equation

general

these

only

validity

loading

In

repre-

available

and

are

for

sandwich

subjected

to

(4.7-32)

in part,

from

as

that

a reasonable

established,

are

fact

to estab-

are

in

caution

made

of Equation

few

as

should

is

were

(4.7-32)

condition

were

is

above

considered.

correlation

data

given

of the

been

into

results,

some

of empirical

considered

4-91

data

handbook

investigations

be

have

(4.7-32)

test

that

fall

test

Equation

investigations

no

_'e was

experimental

can

subject

no

as

by

because

the

one

which

ttowever,

core

for

degree

factor

cylinders

of the

in this

the

Some

knock-down

cited

partially

considered

Therefore

and

viewpoint,

which

importance,

to view

cylinders,

Furthermore,

types

choose

thin-

constructions,

modified

sandwich

stiffness

over

negligible

sandwich

for

such

expression

might

parameter,

lengths

verified.

4.2).

to

relationship

behavior.

theoretical

relationship

this

of

interaction

one

condition

for

considerations

same

shear

loading

Since,

are

applied

for

a length

torsion.

the

the

formula

of the

experimentally

approach

action

In view

cylinder

of the

been

at

subject

wall

theoretical

arrived

upon

sentation

be

implementing

dependent

sandwich

could

interaction

the

cylinders.

shell

on

of transverse

the

with

of the

work

interaction

addition,

lish

deal

circular

Based

c4-38n:

a comprehensive

should

this

category.

et

el.

deformations

conjecture

stiff-core

et

(non-smldwich),

shear

might

the

Batdorf,

number.

accomplished,

a "best-available"

axial

has

inherent

not

in the

sandwich

test

Therefore,

the

inter-

criterion.

4.7.4.2

Design

For

simply

plus

torsion,

Equations

supported,

and

Curves

circular,

one might

sandwich

choose

cylinders

to employ

the

R

+R

is plotted

in Figure

4.7-17

interaction _ =

to axial

compression

formula

1

(4.7-33)

c _'c (_c) CL

(4.7-34)

c

which

subjected

s

and where a

R

c

-

R s

In Equations

Figure

The

4.2-8

quantity

Section

The

(4.7-34)

((_c)CL

quantity

Attention

often

may

the

be taken

is simply

the

knock-down

equal

result

factor

_c is that

obtained

from

to 0.80.

obtained

by using

)'c = 1.0

in the

methods

of

(_)

CL

is

simply

the

result

obtained

by using

_s = 1.0

in the

methods

of

4.5.

Plasticity

use

_s

(4.7-35),

(4.7-35)

4.2.

Section

shed

while

and

_" Ys (_) C L

considerations

is drawn

considerable of Equation choose

to the doubt (4.7-33)

to employ

should

fact

that,

upon

the

and the

be handled

as specified

in Section reliability

Figure

4.7.4.1,

several

factors

of results

obtained

from

4.7-17.

straight-line

In view

interaction R

+ R c

in Section

= s

4-92

of these

9.2.

are the

uncertainties,

cited

which

indiscriminate one might

formula 1

(4.7-36)

which

any

is

plotted

length

Figure

of cylinder

experience

shell

in

has

stability

shown

problems,

4.7-18.

This

and

for

any

that

the

linear

ttowever,

relationship

region

of transverse

interaction

in

can

many

be

shear

formula

cases

used

is

it will,

with

rigidity

never

of

confidence

for

of the

since

core

unconservative

course,

for

introduce

execs-

sive conservatism.

(}. (!

CO

Figure

4.7-17.

0.2

0.4

Conditional Subjected

0. (;

Interaction to

Axial

Curve Compression 4-93

for

0. _

Circular Plus

Torsion

Sandwich

I . 0

Cylinders

0.6 R s

0

0.2

0.4

0.6

R

0.8

1.0

c

Figure

4.7-18.

Conservative Subjected

Interaction to

Axial

Curve

Compression

4-94

for Plus

Circular Torsion

Sandwich

Cylinders

4.7.5

Other

4.7.5.1

Basic

In Sections

The

Loading

Principles

4.7.3

and

a.

Axial

Compression

plus

External

b.

Axial

Compression

plus

Torsion.

corresponding

binations

by

a.

recognizing

the

b.

the

the

in

PRESSURE

loading

Lateral

can

conditions

are

treated:

certain

additional

Pressure.

be

used

for

com-

that

equivalent

peak

shear

an

equivalent

tmiform

the

equations

one

combined

relationships

an

o[

if

following

axial

mind,

combination

the

peak

into

this

4.7.4

interaction

inio

With

Combinations

stress

to

uniform

stress

design

AXIAL

simply

due

an

applied

a.xial

due

to

stress,

and

COMPRESSION

the

shear

shear

curves

PLUS

moment

,

=

of

force

Section

BENDLNG

quantity

((:rx)c

be

converted

g_

+()'c_

can

be

converted

stress.

for

4.7.3.2

PLUS

whe

(_

X

%

can

and

a transverse

torsional

substitutes

bending

can

be

used

EXTERNAL

LATERAL

re

X

((rx)b

\rb/

for

(4.7-37)

and

= ((_X)

Uniform

axial

compressive

stress

due

solely

to

applied

axial

C

(Crx) b

=

load,

psi.

Peak

axial

moment,

%

=

Knock-down given

=

compressive

stress

in

factor Figure

Knock-down

associated

4.2-8,

solely

to

applied

bending

[actor

4. ;/-2,

with

axial

compression

pure

bending

and

as

dimensionless.

associated

_zb

Figure

due

psi.

dimensionless. 4-95

with

and

as

given

in

This

formula

In addition,

the

combination VERSE

is based

on the

design

equations

of AXIAL

SHEAR

respectively,

findings

reported

and

curves

COMPRESSION

FORCE

in Section

of Section

4.7.4.2

BENDING

PLUS

PLUS

if one simply

4.3.

substitutes

the

can be used TORSION

quantities

_'

C

for

PLUS

the TRANS-

and T _ for (_

C

and 1",

where

(r'e

, T

=

(_c)c

+

(4.7-38)

(_c)b

O. 80 = T T + 1.2----55rV

= 7"T + 0"641"V

(4.7-39)

and

(gc) c

=

(gc) b

= Peak axial compressive moment, psi.

_'T

=

Uniform

TV

=

Peak psi.

7c and :_b Equation

(4.7-38)

(4.7-39)

stems

Since can

from

regarded

shear

on the

factors findings

data as

are

stress

stress

stress

stress

a comparison

test

compressive

shear

= Knock-down

is based

no sandwich only be

Uniform axial load, psi.

due due

"best-available"

solely

4-9G

to applied

to applied

to applied

torque,

transverse

axial

bending

psi. shear

force,

above. in Section

(4.5-9)

to substantiate criteria.

solely

solely

to applied

specified

of Equations

available

due

solely

reported

due

and

the

4.3

while

Equation

(4.6-3).

foregoing

procedures,

they

4.7.5.2

DesignEquationsandCmwes

For the combinationof AXIAL COMPRESSION PLUSBENDINGPLUSEXTERNAL LATERALPRESSURE, substitute(_X for(_X andusethe designequationsandcurves given

in Section

4.7.3.2.

The

0"X

Itowever,

the

quantity

(CTx)CL

(i X is defined

quantity

=

used

(ax) c +

in Section

follows:

as

(4.7-4O)

((Ix) b

4.7.3.2

remains

as defined

in thai

sectior_.

For

the

combination

TRANSVERSE

of AXIAL

SIIEAR

FORCE,

COMPRESSION substitute

PLUS

cr I for a e

and curves

given

in Section

4.7.4.2.

The

e = (<:'c)e\ 'bl

However, in that

The

the

((Tc)CL

and

(_)CL

and T _ for

"r in the

TORSION design

PLUS equations

=

used

a t and e

_-_ are

defined

as

u

(4.7-41)

TT + 0.64"r V

in Section

follows:

(4.7-42)

4.7.4.2

remain

as

defined

section.

foregoing

applied

quantities

PLUS

c

quantities

0.80 7" = T T + 1.2-----_o "rV

BENDING

loads

criteria equal

will

still

apply,

of course,

zero.

4-97

where

one

or more

of the

specified

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4-107

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4-1

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Development

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4-10

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of

Technical

5 GENERAL

5.1

AXIAL

5.1.1

no significant

pressed

sandwich

cones.

cylinder

concept

of Seide,

on a large cones,

Seide,

taken

equal

to the

a.

The wall

theoretical Therefore,

array

cated

eta!.

of test

et al. values

thickness

In the

ease

dition nesses

is that found

for [5-11

data

that

circular of the

of sandwich

been

the

CONES

equivalent

cylinder

c.

The length e one.

of the

cylinder

the have

for

handbook,

satisfy

for the

logical

expediency.

such

following

is equal cone.

to the

finite

is equal

to the

slant

of the of this

and

cone. con-

core

principal

length

5-2, Baker presents test data from two axially compressed,

cones having vertex half-angles equal to 15 degrees.

can be

conditions:

to that facing

trun-

cones

extension

same

com-

equivalent-

(non-sandwich),

is equal the

axially

the

as a practical

stresses

which

cylinder

published

isotropie

cylinders

of the equivalent cylinder at the small end of the

sandwich

adopted

critical

The radius of curvature

equivalent

been of this

thin-walled,

constructions,

the equivalent in the cone.

have

the purposes

has

from

concluded for

solutions

b.

In Reference

CIRCULAR

Principles

that

Based

OF TRUNCATED

COMPRESSION

Basic

It appears

INSTABILITY

thick-

radius

of the

truncated

These data were

used

in conjunction with the foregoing equivalent-cylinder concept to arrive at knock-down

factors _c"

The results are shown in Figure 5.1-i, along with data obtained from

5-1

!!

i

i

+ t

_t

t

!

2_ I

0

t

4

J

, !

-ill

i!:

I

I

5-2

axially compressedsandwichcylinders. This figure alsoincludesthe designcurve recommended in Section4.2.2 for suchcylinders. It canbe seenthat the datafrom the conesare in favorableagreementwith the results obtainedfrom cylinders. This providesat least a small degreeof experimentalsubstantiationfor application(>_ the equivalent-cylinderapproachto sand_vich cones. However,in view of the scarcity of test points from conicalspecimens,this methodcanpresentlybe consideredas only a "best-available"criterion.

5-3

5.1.2

For

Design Equations and Curves

simply

supported,

compression, may

the

be computed

following

critical

right-circular,

stresses

a

and a

crl

sandwich

cG

(for

cones

facings

subjected

1 and 2

to axial

respectively)

from the equations and curves of Section 4.2.2, provided that the

substitutions

So

truncated,

are

made:

The values t_, t_, tC , and h are measured •

as shown in Figure 5.1-2.

(There is no preference as to which facing is denoted by the subscripts 1 or2.)

6

The

radius

R is replaced

by the

effective

radius

R

shown

in Figure

5.1-2.

e C°

The

length

L is replaced

by the

Both

lbs

length

L

shown

e

in Figure

5.1-2.

Ends

Simply

P,

effective

P,

Supported

Axis of Revolution

lbs

R small

View

A

N()F}: t 1, [ c

12 art

it1( hes

Figure

5.1-2.

Truncated

Sandwich

Cone

5-4

Subjected

to Axial

t

,

all

h,

I_"lll a] r '

o_t a,l_rttl

w]li}t

¸ _

ul

let ' IllrJlq

I_ r41ca_url'd

Compression

a_ld (i I irl

The

applied

indicated

axial in

end

of the

and

inelastic

load

Figure

cone,

P

and

5.1-2.

the

cases,

the

computed

stresses

In addition,

critical

one

values

can

Pcr

since

are

2_Re

the

associated

maximum

associated

therefore

=

are

with

with

stresses

this

the

occur

location.

For

directions at the

both

small

elastic

write

((rcr

It_

+_cr_t_)

cos2_

(5.1-I)

where Rsmall R e

It is

recommended

_ 30

Plasticity

the

cone

that

the

approach

(5. i-2)

cos

specified

o_

here

be

applied

based

on

the

only

to

cases

where

degrees.

reduction

(see

Section

factors

should

always

be

9).

5-5

stress

at the

small

end

of

5.2

PURE

5.2.1

BENDING

Basic

It appears cones

Principles

that

no significant

subjected

to pure

equivalent-cylinder Based

cones,

be taken

array

Seide,

equal

following

bending.

concept

on a large

cated

theoretical

et al.

to the

Therefore,

of Seide,

of test

solutions

for the

et al.

data

from

concluded

has

the

values

published of this

adopted

circular

sandwich

handbook,

as a practical

isotropic peak

for

the

expediency.

(non-sandwich),

stresses

cylinders

for which

trun-

such

cones

satisfy

can

the

conditions:

a°

The

wall

In the

thickness

case

of the

of sandwich

equivalent

cylinder

constructions,

the

tion is that the equivalent cylinder nesses as are found in the cone.

have

b.

The radius of curvature

of the equivalent cylinder at the small end of the

c.

The

of the

No test

been

critical

for

been

purposes

thin-walled,

that

corresponding

have

data

handbook

and

recommended a "best-available"

are

length

available

are here

for

subjected has

equivalent

sandwich

to pure

not been

cylinder

cones

bending.

experimentally

approach.

5-6

is equal logical

the

same

facing

is equal cone.

to the

is equal

to the

which

are

of the

Therefore,

the

verified

and

to that

extension

finite

slant

types validity

can

of the of this

and core

cone. condithick-

principal

length

radius

of the

considered of the

cone.

in this

method

only be considered

as

5.2.2

For

Design

simply

bending,

Equations

and

supported,

the

Cu_wes

truncated,

critical

peak

right-circular,

stresses

_

and

san_vich

ff

e r 1

may

be

computed

folh_ving

from

substitutions

a.

The

the

equations

are

made:

values

(There

is

t_, no

t.z,

re,

preference

and

and

facings

subjected

1 and

2,

to pure

respectively)

c r>

curves

h are as

(for

cones

of

Section

measured

as

shown

is

denoted

to which

facing

1 or

2.)

b.

The

radius

R is

replaced

by

the

effective

radius

c.

The

length

L is

replaced

by

the

effeetive

length

4.:'.2,

provided

in

that

Figure by

5.2-1.

the

R e shown

subseripts

in

L e shown

the

Figure

5.2-1.

in Figure

5.2-1.

[ 1

"r C a_nraJll

- Axis

\

r
A

Rsmall

L Note:

tl,

t2,

t c,

are

all

measured

while

Figure

5.2-1.

Truncated

Sandwich

Cone

5-7

Subjected

Olis

to

h,

R small' in

measured

Pure

Re, units in

Bending

and

of degrees.

inches

1, e

The

applied

bending

directions at the When

moment

indicated

small the

end

in Figure

of the

behavior

M and the

cone,

5.2-1. the

is elastic,

Mcr

computed

stresses

In addition,

critical

since

values

are

are the

associated

maximum

associated

one can therefore

write

= _r R 2e ((Ycr 1 tl

+Crcr_t2)

with

with

the

stresses this

occur

location.

cos 3(_

(5.2-1)

whe re Rsmall R

To compute gration

M cr

when

the

behavior

=

e

cos

is inelastic,

(5.2-2)

(_

one must

resort

to numerical

inte-

techniques.

It is recommended

that

the

approach

specified

here

be applied

only

to cases

where

o_ _ 30 degrees.

Plasticity the

small

reduction end

of the

factors cone

should (see

always

Section

be based

9).

5-8

on the

peak

compressive

stress

at

5.3

EXTERNAL

5.3.1

Basic

The

is

LATERAL

loading

Principles

condition

subjected

PRESSURE

to

considered

a uniform

here

external

is

depicted

lateral

in

Figure

pressure.

The

5.3-1.

axial

As

shown,

component

of

the

this

cone

loading

i o_

R R

F

-IR smal 1

!

is

Figure

5.3-1.

reacted

by

results

core

a uniform

in principal

has

Truncated

a relatively

% Cone

high

stresses

extensional

(t_ + tp)

to

running

Uniform

load

which

may

stiffness

(YH

(TM

w c, lbs/in

Subjected

compressive

membrane

R large

External

at the

be

in the

large

computed

direction

P Ram

-

Lateral

end

Pressure

of the

as

normal

cone.

follows,

to

This

when

the

the

facings:

(5.3-!)

(t_ + t_)

2c-_s

5-9

_

R

(5.3-2)

where R cos (_

R_ -

(5.3-3)

and

(YH =

Hoop

(rM

Meridional

=

p

membrane

external

to which = Radius

1 =

Radius

e

=

Since

the

uniform occur

radii over

at the

It appears

large

end

measured

method,

of the

sandwich for

in Reference the

critical

surIace,

to the

to the

{There

inches.

is no preference

subscripts

as

1 or 2.)

perpendicular

with

the

at small

to the

axis

The

end

of revolution,

measured

per-

of cone,

measured

per-

inches.

degrees.

location,

maximum

of cone,

inches.

at large

of cone,

axial

end

of revolution,

axis

half-angle

the values

stresses

¢YH and a M are

for each

of these

non-

quantities

cone.

which

the purposes 5-11

axis

surface,

theoretical

cones

surface,

inches.

surface.

no significant

Therefore,

suggested this

conical

that

of truncated sure.

the

of middle

inches.

surface

pendicular

R and I_ vary

facings,

of middle

of middle

= Vertex

of curvature

by the

of middle

Radius

psi.

is denoted

pendicular

Rlarg

pressure,

radius of the

psi.

facing

revolution,

Rsmal

stress,

lateral

principal

= Thicknesses

R

psi.

membrane

= Uniform = Finite

stress,

has

lateral

are

solutions subjected

of this been pressure

have

to uniform

handbook,

adopted for

5-10

been

the

as a practical the

truncated

published external

for the

stability

hydrostatic

pres-

equivalent-cylinder expediency. cone

may

approach Based

be taken

on equal

of

to that for anequivalentcircular sandwichcylinderwhichsatisfies the following conditions: a.

Thefacingandcorethicknessesof the equivalentcylinder are the same as thosefoundin the cone.

b.

Thelengthof the equivalentcylinder is equalto the slant lengthof thc conc.

c.

The radius ofthe equivalentcylinder is equalto the averagefinite principal radius of curvatureof the cone. Thatis, Rsmal

R

The

critical

equations

Since lateral verified

lateral and

no test

pressure

curves

data

pressure, and can

for

e

=

the

of Section

4.4.2.

are

available

from

the

reliability

only

be considered

1 + Rlarg

e

(5.3-4)

2 cos

equivalent

cylinder

can be obtained

truncated

sandwich

cones

subjected

to external

has

not been

experimentally

of the as

foregoing

approach

a "best-available"

5-11

technique.

by using

the

5.3.2

For

Design

a simply

form,

Equations

supported,

external,

lateral

lateral

pressure

critical

and

Curves

truncated,

right-circular,

pressure,

for

the

an

critical

equivalent

sandwich

pressure

sandwich

cone

may

be

cylinder

subjected

taken

which

to uni-

equal

to

satisfies

the

the

following:

a,

The

values

t_,

b.

The

length

is taken

c.

The

radius

is

t:.,

tc,

and equal

denoted

R

e

h are

measured

to

slant

the

and

is

Rsmal

1,

Rlarg

1 + Rlarg

shown L

from

e

in Figure

5.3-2.

.

the

formula

e

= e

where

length

computed

Rsmal R

as

e,

(5.3-5)

2 cos_

and

_

are

as

shown

in Figure

5.3-2.

t k_\ ,..--- BOTH ENDS SIMPLY SUPPORTED

VIEW A

tl,

t 2, Ic, h, R, R2, Rsmall,,

a_d Lu aR: all measured

it]

in{'hcs while _ is measured NOTE:

Figure

5.3-2.

Truncated

5-12

Sandwich

Cone

units

Rlarge' of

in degrees,

The critical lateral pressurefor the equivalentsandwichcylinder canbe obtainedby usingthe equation,_ andcurvesof Section4.4.2. Plastieily c_)nsiderations shouldbehandledas roduclion

lat-ge

end

factor

77 should

of the

cone

always

be

based

on

specified

in Section

theprincit)al

9.2.

nlembrane

Tile

stresses

plasticity

at the

where

P R1 arge =

_I!

(L

_ t:.) cos

P Re _M

It is

ruconm_(,nded

c_ _ 30

thai

the

-

at)l)roach

(L

.

t_) ' _

specified

degrees.

5-13

in. 3-6)

q

(

R_m_ll /

\1

-Rlarg

here

be

(5.3-7)

e /

applied

only

to

cases

where

5.4 TORSION 5.i. 1 BasicPrinciples It appearsthat no significantlheoreticalsolutionshavebeen1)ublished for sandwich conessubjectedto torsion. Therefore, for thepurl)(,s(,s,)1this handbook,the equivalent-cylinderconcepl()t'Seide[5-3i hasbeenadopledas a practical expediency. Basedonthe analysisof his numericalcomputationsfor lhin-walled, isotropie {nonsand_vich), truncatedcones,Scideconcludedthat the critical l¢)rquesfor suchshells canbe takenequallo the valuesfor circular cylinderswhichsalisfy the following conditions: The

a.

wall

In the tion

thickness

case is

that

nesses

as

of the

of sandwich the arc

equivalent

equivalent found

cylinder

constructions,

the

cylinder

in the

have

is

the

same

The

leng-th

of the

equivalent

cylinder

is

equal

c,

The

radius

of the

equivalent

cylinder

is

coml)uted

::

(Rsmall

cosc_)1

I[( R _. _

to

that

of the

extension facing

cone.

of this and

core

condithick-

cone.

[?.

'lc

equal

logical

1 _ R---_---_maH/j-

to

1

the

axial

from

length

the

of the

relationshit)

>]1

1 )I/larg_eRsmall

,,5.4-1)

where

R

=

Radius

of equivalent

cylinder,

inches.

C

It

=

small

Radius to

Rlarg

the

c

::

Radius to the

cy

:_

Vertex

at small axis

end

of

cone,

inches

(measured

perpendicular

cone,

inch(_s

(measured

perpendicular

of revolution).

at large end of axis of revolution).

half-angle

of cone,

5-14

degrees.

cone.

In R(_ference truncated

cones

having

vertex

results

with

parisons

5-1,

which

were

data

to use

4.5

for sandwich

No test

data

are are

available subjected

ex-perimentally

test

results

30 and

isotropic

cylinders

for

For

similar

conical

knock-down under

sandwich

to torsion. verified

was

(non-sandwich)

solutions.

same

These

(;0 degrees.

predictions

the

from

to torsion.

(c_ of both

from

Section

not been

subjected

theoretical

decided

and

present

equivalent-cylinder

small-deflection

handbook

et al.

half-angles

of test

therefore

Seide,

and can

ten

isotropic

(non-sandwich),

tests

included

specim¢,ns

The

agreement

of these,

to that

obtained

cylinders

against

sandwich

constructions

factor

(7 s = 0.80)

the

as was

from

com-

corresponding it was

selected

in

torsion.

cones

which

Therefore only

approach.

5-15

are the

of the method

be considered

t33)es

considered

recommended as a "best-available"

in this here

has

5.,1.2

For

the

Design

simply

Equations

and

supported,

critical

Curves

truncated,

torque

may

be

right-circular

computed

T

from

=

er

sanck_'ich

the

2vR

e

cones

subjected

to torsion,

equation

z (t t + t._) y_ • cr

(5.4-2)

where

T

Critical

torque

for

sandwich

cone

subjected

to

torsion,

in-lbs.

cr

R

Radius

=:

of equivalent

sandwic:h

cylinder,

inches

r see

Equation

o

(5.,,_-3)]. t 1 and

t2

Ttticknesses

=

as /

T

of the

which

Critieal

=

cr

to

shear

to torsion,

not

to

the

radius

R

is

coml)uted

inches.

denoted

stress

construction.

The

is

subjected equal

facings,

faeing

for

(There

the

equivalent

psi.

critical

by

(it

should

sh,-ar

stress

is

no

preference

subscript

1 or

sandwich

cylinder

be

noted

of the

that conical

2.)

when

this

value

is

sandwich

)

from

e

1

Re

where

The

=

Rsmal

stress

(Rsmal

1,

7_

1 c_)scs)

1 + [2

Rlarg

e,

and

may

be

computed

(1

c_ are

as

1

+ Rlarge_]'Rsmall/]

shown

from

in

the

Figure

equations

[1(I

+ Rlar_e_] Rsmall/l

-_

5.4-1.

and

cur_'es

of

Section

4.5.2

in

Figure

5.4-1.

or

vided

that

a.

The

values

bg

The radius

t_,

t_,

te,

and

R is replaced

d

are

measured

by the effective

as

radius

shown

R . e

co

The

length

L is

taken

equal

to

the

5.4-1).

5-16

axial

(,_. 4-:_')

length

of the

cone

(see

Figure

pro-

Both

Ends

Simply

Supported

7

/

(

d

.! i

R

small

Axis l,

of

Revolution

Hl-/bs

T,

View A

itl-lbs

N

_Tt-

i1,

12,

it,

l/,

tl,

I,

aud Rlarg c art al_ ill II[lllS O[ il_C lies d i_ (t('_rt

rneastlrt

Figure In a truncated

cone

at the

small

end.

value

is associated

5.4-1.

Truncated

which

is subjected

Hence,

for

with

that

Sandwich

sandwich same

Subjected

to Torsion

the

maximum

shear

constructions

location.

cs,

Cone

to torsion,

of this

One can

type,

therefore

P'srl_al[,

illtasclr_ tl wllilt' t+_ iS

stress

the

will

critical

occur

stress

write

T cr

rcr

=

2rr R _ small

stress

for

(t_+t 2)

(5.4-4)

where r

cr

= Critical

It is recommended

to torsion, that

the

shear

truncated

sandwich

cone

when

subjected

psi. approach

specified

here

be applied

only

to cases

where

_ 30 degrees.

Plasticity

the

cone

reduction

(see

Section

factors

should

always

be

9).

5-17

based

on

the

stress

at the

small

end

of

:).;_

TRANSVERSE

5.5.1

Basic

Th(,

case

Principles

considered

to transverse tions,

SIIEAR

here

shear

such

as

is that

forces

A-A,

are

of a truncated

a,_ shown subjected

sandwich

in Fig_,re to the

cone

5.5-1.

same

which

No_c that

magnitude

all

<)f shear

is subjected

only

transverse

see-

load.

--,.-- A

J

l.i

J

____

--

_.--A

Figxlre This,

of course,

over-all

static

the

the

shear

effects

meridional stress

cones

subjected

that

stress

no significant to transverse

condition

an external

bending

loading shear

equation. under

loading

to Transverse

To obtain

of transverse

an interaction

Subjected

structure.

unbalanced

corresponding

that

Cone

h3_pothetical of the

hypothetical

combined

It appears

is a pur_qy

it is required

zed by using peak

Truncated

equilibrium

and moments, theless,

5.5-1.

Such

a bending to the

shear.

does

associated

artificial

solutions Therefore, 5-18

acting

for

also

of forces

be present.

Never-

are both

and the

since

usually

analy-

the

critical

critical

peak

condition.

been the

in

to be of interest

invulves alone

not result

balance

bending

loading

have

it does

necessary

prove

a relationship

moment

subject

theoretical

its

the moment

system

and

since

Shear

published

purposes

for sandwich of this

handbook,

the conceptusedfor sandwichcylinders (see a practical [5-4

expediency.

and

5-5]

conclusion

As noted

on isotropic

[5-6]

in Section

rcr To properly

understand

thin-walled

circular

For

the

(5.5-1)

lack

(5.5-1)

quired as

No sanchvich Until

such

neutral

available"

and

also

be adopted

from

elliptic

a series

here

as

of tests

cylinders

led

to the

the

other

hand,

forces.

also

be used

data

value

design

shear

test

Torsional

1.25

that

for

(5.5-1)

]

be observed

shear

stress

under

_'cr then

_

Loading]

it should

approach,

for the

in Section

for 1

Tcr

torsional

is uniformly

transverse

shear

corresponds

to the

loading

distributed

loading,

peak

the

of a around

shear

intensity

which

axis.

of a better

data

will

results

circular

TestValues Loading

section

and the

it was

the

same

for truncated theoretical

5.4, are

do become

reconm_ended

and analysis

For

small-deflection

specified

the

Theoretical for

ratio,

On the

be used

transverse tion

this

is nonuniform at the

Values

cross

circumference.

occurs

4.6)

that

Small-Deflection

stress

4.6,

(non-sanchvieh),

Lower-Bonndrcr Transverse Shear

the

Section

with

reason,

cones.

values

exception

In the

Equation

are

subjected

here

latter

ease,

loading

that

now be taken

7s must

to substantiate

the

available,

one can

regard

5-19

that

that

for torsional

available

approach.

cylinders

4.6

it is recommended

sandwich Tcr

the

of sandwich

in Section

only

reliability this

should

of this proeedm_e

that

to

Equa-

the

re-

be obtained equal

to unity.

practice. as a "best-

5.5.2 DesignEquationsandCurves For simply supported,right-circular, truncatedsandwichconessubjectedto transverse shearforces, the critical peakshearstress maybe comtmtedfrom theequation rcr

= 1.25

(T) cr

(5.5-2)

Torsion Ys

1.0

where

(_r)

= Torsion

The critical torsional shear stress obtained by substituting Ys = 1.0 throughout the meth_gls cited in Section 5.4, psi.

Ys = 1.0 In a truncated will

occur

cone at the

which

small

is subjected

end.

ttence,

to transverse the

critical

shear, stress

the

value

maximum

shear

is associated

stress

with

that

1ocation.

Plasticity

reduction

cone

(see

Section

When

the

behavior

puted

from

the

factors

should

always

gration

on the

stress

at the

small

end

of the

9).

is elastic,

the

critical

transverse

shear

force

.._Fv)cr can

be corn-

following:

(F v) cr

To compute

be based

(Fv)cr

when

the

behavior

= _ Rsmall

is inelastic,

techniques.

5-20

(t_ +%)r

(5.5-3)

cr

one must

resort

to numerical

inte-

5.6

COMBINED

5.6.1

CONDITIONS

General

For

structural

critical

an

members

loading

shows

of

LOADING

the

conditions

graphic

applied

alone.

load

The

this form

format

or

distances

of

used

to the

clear

All computed

for

axes

of the interaction

in Figure

from

5.6-1,

the actual loading

that a particular

corresponding

to point

structure

for

that

for a second

fall within

point to the curve

is subjected

quantity

of loading

type

of loading.

is given

to the combined

M

1/111

].0 Ri 5.6-I.

Sample 5-21

Interaction

Curve

of

con-

on

by the ratio of

loading

1.0 (Rj) B --__

Figure

Curves

Furthermore,

and to the origin.

0

acting

by the interaction

will occur.

I

Rj

when

ratio

All points lying

5.6-1.

-- -- --

the

of particular

bounded

of safety

5.6-1

R i is

type

integrity

represent

Figure

D (Rj) D

to

curves.

the area

that buckling

of the margin

B of Figure

customary

to stable structures.

indicate

a measure

is

The

as to the structural

correspond

curve

purpose.

value

defined

it

interaction

this

critical

picture

loads,

so-called

points which

and the coordinate

as shown

combined

means

usually

stress

give a very

or outside

assume

by

to

quantity" Rj is similarly

figurations.

curve

subjected

For

example,

condition

Then, puted

for proportional from

the

increases

in R. and R., 1 j

the margin

of safety

(MS)

can be com-

following:

(Rj) D MS

-

i (Rj) B

As

an alternative

procedure,

which

is based

Point

M is located

between

point

be calculated

on the

one might

assumption

in such

B and the

that

a position

interaction

choose loading

that

(5.6-1)

to compute beyond

BM is the

curve.

The

a minimum

point

B follows

shoo-test

minimum

line margin

that

margin the

path

of safety BM.

can be drawn

of safety

can then

as follows:

Minimum

MS

-

5-22

OB + BM OB

1 (5.6-2)

5.6.2

Axial

5.6.2.1

Compression

Basic

In Section

Plus

Bending

Principles

4.7.2

this

loading

cylinders.

For

such

interaction

relationship:

condition

is treated

configurations,

it was

for

the

concluded

case that

of circular

one

may

sandwich

use

the

following

(5.6-3) R c

+%=1

where c

R

(5.6-4)

= c

"/c ((_c) C L

ab

%=

(5.6-5)

Tb ((_c) CL

and (Y

= Uniform

compressive

stress

due

solely

to applied

axial

load,

c

psi. = Pcak

compressive

stress

due

solely

to applied

bending

moment,

_b psi. = (_c)C L

In this

handbook

bending

and

cylinder of the the

equivalent

small

stress

end

theoretical under

value

for

load

acting

an axial

critical

uniform

alone,

compressive

psi.

Tc

= Knock-down

factor

given

by

Figure

4.2-8,

dimensionless.

Tb

= Knock-down

factor

given

by Figure

4.3-2,

dimensionless.

it is proposed

of axial

concept

Classical

(see

load

of the

acting

Sections

cylinder cone.

that

for truncated

alone 5.1

is taken It should

both

and equal

sandwich

be treated

5.2). to the

be noted 5-23

For

by means both

finite that

cones

the

types

principal maximum

the

cases

of pure

of an equivalentof loading, radius stresses

the

radius

of curvature from

at both

bendingandaxial compressionoccurat this samelocation. In viewof theseseveral considerations,it is assumedherethatEquations(5.6-2,)through(5.6-5) canbe appliedto truncatedsanchvich conesif a,

o"c and gb are bcCh computed end of the cone, and

for

the

meridional

direction

and at the

small

b.

the

values

for

sandwich

keel) in mind ((_c) C L • )

Since

no test

compression verified

data

have

plus

bending,

and can

only

;¢e' and

cylinder that

been

Yb'

and

described 7c must

published the

be regarded

((Te)r_T are

in Sect_i_ns be taken

equal

for truncated,

recommended

approach

as a "best-available,,

5-24

those 5.1

which

and

to 1.0

when

sandwich has

apply

5.2.

computing

cones

not been

method.

to the

(It is important the

subjected experimentally

equivalent to value

to axial

5.6.2.2 DesignEquationsandCurves For simply supported,truncated, right-circular sandwichconessubjectedto axial compressionplusbending,the followinginteractionequationmaybeemployed: Re+ Rb = 1

(5.6-6)

where R

c

-

(5.6-7)

7e (6c)CL

c

_b

ab Equation

(5.6-6)

may

be used

for

7t) (5c)

cones

of any

(,_.(i-s)

CL

length.

A plot

of this

equation

is given

in Figure 5.6-2.

The quantity gc is the uniform

meridional compressive

stress, at the small end of the

cone, due to the axial force acting alone.

The

quantity

cone,

The

due to the

quantities

cylinder

from

Figures

quantity

Section

peak

bending

Ye'

described

In Equations

The

Crb is the

_b'

meridional

moment

and

acting

in Sections

and

(5.6-8),

4.2-8

and

4.3-2,

is simply

5.1

those

and

the

stress,

at the

small

end

of the

alone.

(Cre)CL- are

(5.6-7)

(_c)CL

compressive

which

apply

to the

equivalent

sandwich

5.2.

knock-down

factors

)'e

and_b

are

those

obtained

respectively.

the

result

obtained

4.2.2.

5-25

by using

Yc = 1.0

in the

method

of

Plasticity considerationsshouldbehandledas specificdin Section9.2 except,that in this case, one may

use

(a) r? -:

[

l

[ _-]--_ 1

1

EZ

for

short

cones, and

l

(b)

The plasticity at the

reduction

small

end

of the

Ii- VEtEs o l

r7 =

1---_ l

factor

for

Ef

77 should

always

moderate-length

be based

on the

through

peak

long

compressive

stress

cone.

1.0t I

'iii

0.8

\

........

T

0.6

It b O. 4 ...................

4-..........

'

' '

_

'

'

.....

0.2

1

! _

i

I

_

r

-

t

'

I

0 0

i

.

0.2

0.4

0.6

0.8

.0

R e

Figure

5.6-2.

Design Subjected

Interaction to Axial

Curve

for Truncated

Compression 5-2 6

Plus

Sandwich Bending

Cones

cones.

5.6.3

Uniform

5.6.3.1

The

is

External

Basic

loading

Pressure

Principles

condition

subjected

Itydrostatic

to

considered

a uniform

here

external

is

depicted

pressure

over

in

the

Figure

lateral

5.6-3.

As

surface

and

shcavn,

both

the

cnd

cone

closures.

Both Ends Simply

p,

c_ p,

SuppoEtLd

psi

R Rz

-I

psi

..q-_

_t_

Rlarg c

Figure

5.6-3.

Truncated

Cone

External

This

the

results

core

has

in principal

membrane

a relatively

high

Subjected

Itydrostatic

stresses

which

extensional

stiffness

to

Uniform

Pressure

may

in the

be

computed

direction

as

normal

follows

to

when

the

facings:

p R_ CrH

-

(t_ + t_)

cr M

-

pR._ 2 (t_ + t_)

(5.6-9)

(5.6-10)

where

R R

2

(5.6-11) cos

5-27

and =: Itoop

-

crM

membrane

Meridional

p -

--

external

R

to

Radius

:

of

(_

Since

the

uniform

occur

radii over

the

at the

large

It appears

of

sure.

of

Seide,

end

no

Scide,

et

be

taken

following

with

al.

equal

of

inches.

denoted

surface

psi.

middle

(There

1)3, the

surface,

is

no preference

subscripts

measured

inches.

1 or

perpendicular

2.)

to

the

axis

of cone,

degrees.

location,

the

maximum

values

stresses

(Ytt

for

of

each

and

r;

are

non-

M these

quantities

cone.

theoretical

which

purposes

has

been

solutions

are

concluded

adopted

as

values

for

the

lo

been

uniform

published

_he

apractical

expediency.

critical

equivalent

the

hydrostatic

equivalent-cylinder

cylinders

hydrostatic

pressures

cylinders

stability

pres-

at)preach

Based

(non-sandwich),

circular

for

external

handbook,

isotropic

that

have

subjected

of this

thin-walled,

to the

facings,

axial

The

lh(,

from

pressure,

inches.

the

surface.

c(mes

[5-11

data

vary

psi.

of curvature

is

half-angle

significant

i'or

of test

revolution,

of the

sandwich

et al.

cones,

R

the

facing

of middle

Vertex

conical

Therefore,

array

can

that

truncated

=

[/ and

radius

of

which

stress,

hydrostatic

principal

Thicknesses as

psi.

membrane

Uniform

Finite

tI andt

stress,

on

and

for

which

such

satisfy"

alarge

truncated

cones

the

conditions:

a.

The

wall

In the tion nesses

thickness

case is

thai as

of the are

of the

sanokvich

equivalent constructions,

equivalent found

cylinder

in the

cylinder cone.

5-28

the have

the

is logical same

equal

to that

extension facing

of the of this

and

core

cone. condithick-

b°

C.

The

length

of the

equivalent

cylinder

is equal

to the

slant

The

radius

of the

equivalent

cylinder

is equal

to the

average

of eurvature

of the

radius

R

cone.

of the

finite

cone.

principal

is,

Rsmall + Rlarge 2cos

-:

e

That

length

(5.6-12)

where =

R

Radius

of middle

surface

for equivalent

Radius

of middle

surface

at small

cylinder,

inches.

e

R

= small

perpendicular =

Radius

Rlargc

The

critical

the

equations

The

only

external

assist

eN)erimental

hydrostatic

pressure

in the

_5-7

and

preparation 5-7.

the plastic

The

other

region.

stresses

in this

instance

approach

of the

present

Sect{on

9,

the

design

faclot3'

agreement

section

with

the

for

at large

cylinder

(measured

inches.

end

of cone

taxis of revolution),

conical

data

from

in conjunction

(measured

inches.

can be obtained

low

was

the

of Reference to permit

was

with

5 ')9

under

eondueted

by using

Navajo

missile

made

since

of the

it was

computations.

of 43.6

to bc 36.4

psi.

result

stressed

reliable

plasticity

inelastic

To published too

This

deeply

but the Using

reduction psi.

American

program.

also

the

uniform

by North

5-7 was

computed

value

shells

was

not studied

in conjunction

experimental

tests

an analysis

enough

pressure

san&vieh

two

with

handbook,

specimen

were

critieal

the

specimen

The

of cone

4.7.3.

are

of this

end

of revolution),

surface

equivalent

results

5-8_

axis

to the

for the

of Section

available

in Reference into

pressure

and curves

Corp.

of middle

perpendicular

hydrostatic

Rock, veil,

to the

criteria

the of

is in satis-

The foregoingsubstantiates,to a very small degree,the reliability of the equivalentcylinder conceptrecommended here. However,in viewof the lack of a sufficient numberof test results, this approachcanpresentlybe consideredas only a "bestavailable"method.

5-30

5.6.3.2

I,'or

Design

a simply

external,

Equations

supported,

the

cylinder

a.

The

values

t_,

b.

The

length

is

Figx_re

The

right-circular

pressure,

sandwich

C0

Curves

truncated,

hydrostatic

equivalent

and

for

t:.,

pressure

cone

may

be

subjected

taken

equal

in

Figure

1o uniform,

to that

for

tin

which

t c,

taken

critical

sandwich

and

equal

h are

measured

as

to the

slant

and

computed

length

shown of the

Le

cone

5.6-4. as

shown

in

5.6-<1.

radius

is

denoted

R

is

from

the

fornmla

e

Rsmal R

where

Rsmal

1,

e

=

Rlarg

e,

,

Rlargc

1 + Rlarg

and

2cos

a

are

e

(5.6-13)

(_

as

shown

in

Figure

R

L_L\I _-T

5.6-4.

R2

VIEW A P'smail BOTH ENDS SIMPLY SUPPORTED

NOTE:

t 1, l,.

i c , h, R, R,2

Rslnall'

Rlarge'

and Le are all measured in units of inches while o/is measured in degrees.

Figure

The

critical

from

the

as

follows

hydrostatic

equations

pressure

and

curves

5.6-4.

Truncated

for

of Section

the

Sandwich

equivalent

4.7.3

: 5-31

Cone

sandwich

if the

ratios

cylinder

R

c

and

R

p

can

be

are

now

obtained

defined

R

=

P

c

ye

(l_x)C L

(5.6-14)

P

R

=

(5. (;-15)

p

Wp (_y)CL

w he re

external,

Uniform,

p

surfaces

and

hydrostatic

end

closures

pressure

of the

applied

equivalent

to

lateral

sandwich

cylinder,

psi.

In

Equations

Figure

(5.6-14)

4.2-8,

It should

be

and

while

noted

v 'p

(5.6-15),

may

be

the taken

knoek-d(r_cn

equal

to

factor

_/e is

that

obtained

from

0.90.

that

(Px)CL Re ((rx) CL

-

2 (t_ * t:_)

(5.6-16)

or

2 (gx)CL(t (ISx)CL

=

! _ t_)

R

(5.6-17) e

whe

re

(C_x)CL

The

value

Section

(tSy)c

L can

=

Classical

theoretieal stress

cylinder. equations

This value and curves

be

obtained

by

when

using

for

acting

critical

alone

can be obtained of Section 4.2.

Tp = 1.0

uniform

on

in the

the by

axial

equivalent using

equations

comsandwich

Yc = 1.0

and

in the

curves

of

The

plasticity

4.4.

Plasticity

considerations

should

be

handled

reduction

factor

r7 should

always

be

based

of the

cone

large

value

pressive

end

as

on

where,

5-32

specified

the

principal

in

Section

membrane

9.2.

stresses

at the

pR large (YH=

P Rlarg _M

It is

recommended

_ 30

that

the

approach

=

(5.G-18)

(t1 _ t2) (cos _) e

2 (t 1 + t_)

(cos

specified

here

degrees.

5-33

(5. (_)

be

applied

only

to

cases

where

(;-19)

5.6.4 Axial CompressionPlus Torsion 5.6.4.1 Basic Principles Theloadingconditionconsideredhereis depictedin Figure 5.t;-5. Theaxial load P canoriginatefrom anysourceincludingexternalpressureswhichare distributed uniformly over the endclosures.

T,

Hl-lbs

],

torque

(orqtle_

f_ P.

m-lbs

/

\\

I bs _.__________

____

_

_

_

___

_

P,

Bol:h

l bs

Ends

Simply

Figure

5.6-5.

Truncated

Cone

Compression

It appears

that

of truncated

Matthiesen

such

loading

expedient

on

a large

significant

sanchvich

and

an

no

cones

_5-9]

and,

have

for

engineering

array

of

the

following

cluded

that

tropic

(non-sanchvich),

theoretical

test

the

under

arrived

purposes

approach

data

from

interaction truncated

Subjected

Plus

solutions

this

at

of this

have

been

handbook,

case

Mylar

specimens,

these

results

sandwich

be

the

applied

provide

shells

the

constructions.

and

stability

MacCalden

non-sandwich

MacCalden

could

for

ltowevcr,

for

of conical

relationship

published

of loads,

conclusions

to the

Axial

Torsion

combination

certain

to

Supported

Matthiesen

to thin-walled,

under

basis

for

Based

con-

iso-

cones:

R

+R c

_ = S

5-34

1

(5.6-20)

where R

P

:2

c

(5.6-21)

m

(Pcr)Empirical

T

R

(5.6-22)

s

(Ter) Empirical and (Pcr)

= Empirical

Empirical

when

result

is identical

sandwich)

cylinders

conjecture cated

that

cones

sented

subjected

under

in Section

recommended

acting to that

in the

case

the

alone,

given

loading

for

arc

based

on this

for

the

critical

axial

load

value

for

the

critical

torque

when

in-lbs. 5-10

compression

for

plus

constructions condition

circular

value

lbs.

in Reference

of sandwich

subject

alone, lower-bound

to axial

4.7.4.2

here

acting

= Empirical

(Tcr) Empirical

This

lcnver-bound

of the

cylinders.

The

premise.

torsion.

the

are

That

thin-walled,

One might,

interaction same

design is,

isotropic

therefore,

curves

shape

one might

for trun-

as those

equations

(non-

pre-

and curves

choose

to view

the

formula, R

+ R2 s

c as

a comprehensive

sanchvich)

and sanchvich

MacCalden made

the

interaction

construction.

and Matthicsen torsionally-loaded

was

the

case

that

whenever

when

equation

no axial

R c is non-zero,

for

that

conical

shell

was the

1

(5.6-23)

truncated

However,

observed

load

=

the

applied same

more

at all.

knock-down

5-35

of both

it is important

presence

much

cones

of even sensitive They, factor

isotropic

to note a very

here

small

(nonthat axial

to imperfections

therefore, be employed

load than

recommended in computing

m

(Tcr)Empirica fied the

that

1 as is used

this

case

single

of axial

knock-down

should

because

only the

extremes

Section

4.7.4.1).

(see

subject

be exercised

loading

condition

investigations

were

is a reasonable been

obtained

fore,

the

and

can

general only

acting

validity

be regarded

should

be taken

alone.

o[ transverse

should

of the cones

foregoing

the

be dependent the

sandwich

actual

subjected

of Equation

the

although

probably

to establish

same

rigidity

(5.6-23)

as a "best-available"

5-36

has

to that

practice

of the

which

core

have

over

not been

partially been

considfor the

parameter,

which

Furthermore, compression

for

here.

relationship

a length

speci-

applies

is adopted

interaction

lengths

approach.

further

recommendations,

upon

behavior. to axial

It was

equal

The

shear

In addition,

made

sandwich

of (Pcr)Empirieal"

in implementing

representation for

calculation factor

compression

Caution

ered

in the

Equation no test

plus

experimentally

no

torsion.

(5.6-23) data

have

Thereverified

5.6.4.2 DesignEquationsandCurves For simply supported,truncated,right-circular sandwichconessubjectedto axial compressionplustorsion, onemight chooseto employthe interactionformLda, (5.6-24)

R +R e =1 C

which

is plotted

in Figure

5.6-6

and

S

where, P (5.6-25)

c

(Per)Empirical

T -

(5.6-26)

3,c

p

= Applied

axial

load,

T

= Applied

torque,

in-lbs.

Lower-bound (Pcr)Empirical

acting

value

alone.

equations

1=

alone.

The

and

knock-down

critical can

for the

axial

5.1.2,

critical

factor

obtained

by using

when

by using

5.4.2,

when the

lbs.

torque

can be obtained of Section

load

be obtained

of Section

value value

curves

the

value

curves

Tkis

tions

for

This

and

Lower-bound (Tcr)Empirica

lbs.

the

acting equa-

in-lbs. from

Figure

4.2-8

_C

(dimensionless). case, the quantity equal

to the

For the purposes of the present R (see Figure 4.2-8) must be set

equivalent

radius

R e which

is computed

as follows: R Re R

= Radius small

pendicular = Vertex

=

small cosc_

at small to the half-angle 5-37

(5.6-27)

end axis

of cone,

inches

of revolution).

of cone,

degrees.

(measured

per-

0

The

Figure

5.6-6.

factor

_

shed use often

For

be taken

Attention

equal

is drawn

considerable

0.4

Interaction Curve to Axial Compression

be introduced

the special

into

case

to the

(5.6-24)

fact

that

any

length

upon

the

the

in Figure

of cone

and

no axial

in Section reliability

mid Figure

to employ

is plotted

the demoninator

where

Sandwich

of the

load

straight-line

for

5.6-7. any

5.6.4.1,

ratio

is present

This region

In view interaction

c

several

of results

5.6-6.

R

which

for Truncated :Plus Torsion

Cones

R s only (R c

0),

when Rs

to T + (Ter)Empirical"

doubt

of Equation choose

Conditional Subjected

should

R c is non-zero. should

O. 2

+It

s

obtained of these

from

are the

cited

of transverse

which

indiscriminate

uncertainties,

one

might

formula,

=1

relationship

5-3 8

factors

(5.6-28) can

be used

shear

rigidity

with

confidence

of the

core,

for since

experiencehas shownthat the linear interactionformulais neveruneonservative for shell stability problems, ttowever, in manyeasesit will, of course, introduce excessiveconservatism. Plasticity considerationsshouldbehandledas specifiedin Section9.2. Theplasticity reductionfactor r{ should

always

be based

on the

stresses

at the

small

end

of the

cone.

0o

0°

0.4

li (Ii

0.2

0.4

0.6

0.8

1.0

R e

Figure

5.6.5

Other

5.6.5.1 In Section treated.

5.6-7.

Loading

Basic 5.6.4, The

Conservative Interaction Curve Subjected to Axial Compression

for Truncated Plus Torsion

Sandwich

Cones

Combinations

Principles the interaction

combined

loading

relationships

condition

of axial

presented

there

5-39

compression can be used

plus for

torsion

an additional

is

loadingcombination1)5, recognizing peak

meridional

equivalent era'yes bending

stress

uniform of Section plus

To accomplish

due

at any given

to an applied

meridional 5.6.4.2

that

bending

stress.

can

torsion

which

Fig_lre

5, 6-8.

this

it is simply

With

be used

for

is depicted

this

the

axial

can be converted

in mind,

the

tlmt

design

of axial

cone

the

into

equations compression

an and plus

5.6-S.

Truncated Cone Subjected Plus Bending Plus Torsion

required

on the

moment

combination

in Figure

location

the

to Axial

quantity

Compression

Rc be redefined

as

follows:

p#

R c

(o. 6-29)

(tScr) .... r_mplmcal

whcre

(5, 6-30) and P

Applied

axial

M

Applied

bending

Yc

Note:

load,

lbs.

moment,

= Axial comt)ression dimensionless.

Vor

the purposes

tmock-down

of the

Figure 4.2-8) must be set is comt)uted as follows:

in_-lbs.

present

equal 5-40

to the

factor

ease,

from

the quantity

equivalent

radius

Figure

4.2-8

R (see Re which

Rsmall Re -

Tb Note:

Bending For

the

4.3-2) must as follows:

lo]ock-down

pro'poses

be set

:: Radius

small

axis

at small

The

foregoing

formula

Since

no sandwich

here,

they

5.6.5.2 For

test

can only

Design simply

for

data

are

Equations

supported,

end

and

as

dimensionless.

quantity

radius

R (see

R e which

Fignre

is computed

(5.6-32)

of cone

(measured

perpendicular

to the

inches. of cone, on the

available

be regarded

the

4.3-2,

Rsmall coset

-

tmlf-angle

P_ is based

Figxlre

case,

equivalent

of revolution),

=: Vertex

from

present

to the

Re

R

factor

of the

equal

(5.6-31)

cosa

degrees.

principles

cited

to substantiate

a "best-available"

the

in Section

5.2

recommendations

made

criterion.

Curves

truncated,

right-circular

loading

condition

depicted

in Figure

curves

of Section

5.6.4.2,

except

5.6-8, that

the

sandwich one

may

quantity

use

cones the

R c must

design

subjected equations

now be defined

to the and as follows:

p' Rc =

(Pcr)Empirical

(5.6-33)

where

(5.6-34) _,Tb/

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5-54

REFERENCES 5-1

Seide,

P.,

opment

Weingarten,

of Design

V.

Criteria

TR-60-0000-19425, 5-2

Baker,

E.

Subjected

I. , and Morgan, for

Elastic

31 December

If.,

"Experimental

to Axial

E.

Stability

J.,

"Final

of Thin

Report

Shell

on the

Structures",

DevelSTL-

1960. Investigatim

Compression",

of Sandwich

AIAA

Journal,

Cylinders

Volume

6,

and Cones

No.

9,

September

1968. 5-3

Seide,

P.,

of Applied 5-4

Lundquist,

Gerard,

P.

Batdorf,

Note

and 1345,

B.

TN 527,

NACA

Corp.

of Conical

Axial

TN 3783, Fuel

Report Booster

Corp.

M.,

in Torsion",

Shells",

Fuel

for

No. B.,

Journal

Thin-Walled

5-55

Tests

Cylinders

in Com-

1935. of Thin-Walled

Duralumin

1935. Stability,

August

Conical

AL-2622,

No.

15 September

AL-2623,

M.,

HI - Buckling

Bulkhead

Conical

Journal,

Part

1957.

No.

16 September Torsion February

3,"

1957.

Bulkhead

"Combination

AIAA

1947.

523,

Tank

and Schildcrout,

Stress

TN

Tank

No.

Report R.

Duralumin

of Structural

Booster

and Matthiesen,

Stein,

Direct June

NACA

of XSM-64.A

Tests S. B.,

"Strength

"Handbook

Shells",

Rockwell,

NACA

F.,

of XSM-64A

"Test

MacCalden,

W.

tt.,

Rockwell,

Shells

of Thin-Walled

Bending",

Section",

and

American

American

Tests

and

Becker,

Conical

1962.

and Burke,

"Test

Anonymous,

Torsion

E.

Plates

Compression 5-10

Shear

and

of Truncated

"Strength

of Elliptic

Anonymous,

North 5-9

E.,

E.

G.

North 5-8

June

Transverse

of Curved 5-7

Mechanics, E.

Cylinders 5-6

Buckling

Lundquist, bined

5-5

"On the

4,"

1957. and Axial

1967.

"Critical

Combinations

Cylinders,"

NACA

of

Technical

5-11 Baker, R.

M.

for

the

Center,

E.

tt.,

, Shell

Cappelli, Analysis

National [[ouston,

A.

Kovalevsky,

Mmmal,

Aeronautics Texas,

P.,

Prepared and

June

Space

1966.

5-56

L., by

North

Administration,

Rish,

F.

American Manned

L.,

andVerette,

Rockwell Spacecraft

Corp.

6 GENERAL

6.1

GENERAL

This

section

Figure

6. i--i

tor.

Note

blends

cover

deals

with

shows

that

the

the

large

,)

shapes

One

consists

It is expected

dome

observe

all

surfaces

are

for

likely

each

to

toroidal

be

of revolution.

truncated

at the

segment

that the configurations

2

,?

are

of which

of a lower

structures

that

SHELLS

contours

here,

shape

of the

should

whose

considered

cap.

DOME-SHAPED

shells

torispherical

majority

applications.

OF

dome-shaped

into a spherical

the

INSTABILITY

shown

encountered

which

here

will

in aerospace

of

2

x

y

_2

b2

Y _...._Spheri

,

(a)

-

Boundary

or

(b)

ttemispherical

cal

domes

the

engineering

maximum

expediency,

6.1-1.

radius

analysis

t-Boundary

Simply-

Supported Clamped

Figure these

ca 1

I.

_---x

Supported Clamped

equa-

Supported

or

Simplyor

Clamped (c)

Ellipsoidal

Structural

of curvature

of all

Dome RMa

the

on this radius.

6-1

Shapes

x occurs

illustrated

Torispherical

at

the

configurations

apex.

As

will

a practi-

be

based

In the has

case

long

from

of externally

been

recognized

classical

spheres.

The

base theory Section

the

thin-walled, test

discrepancies

are

b.

the fact that large-deflection cal stresses approximately symmetric values, and

c.

the

that

case

analyses

modified 4.1

to the

of circular

stability

for

classical

discontinuity

For

design

procedures

knock-down

cylinders

and

6-2

does

the

it

predictions

of complete

earlier

in this

latter,

it has use This

adopted

with

(see common

of classical

for

of

critiaxi-

for pre-

of the boundary.

become

here

presence

behavior yield small-deflection

handbook

approach

the

not account

neighborhood

on the

is also

coupled

theory

factors.

below

of asymmetric lower than the

in the

the

far

buckling

path

small-deflection

cylinders.

by empirical sandwich

analyses 20 percent

described

fall

domes,

to,

equilibrium

distortions

(non-sandwich)

axisymmelric

attributed

situation

and

isotropic

normally

for the

usually

the shape of the postbuckling initial imperfections,

fact

results

theory

a.

is analogous

for the

that

small-deflection

buckling This

pressurized,

Section

4.1)

practice

small-deflection was

sandwich

selected domes.

in

to

6.2

EXTERNAL

6.2.1

PRESSURE

Basic

6.2.1.1

Principles

Theoretical

This

section

form

external

Considerations

deals

with

the

pressure

loading

acts

condition

over

the

depicted

entire

in Figure

surface

of the

6.2-1.

That

sandwich

dome.

is,

a uni-

The

net

p, psi

l

+ lbs

l

w w

-iI1

_'

Figure

vertical

Figure

edges.

That

placements

the

as

of

6.1-1,

is,

boundary

this

during

that

by

the

reacted

domes

the

clamped

by

can

here

Plantema

is

ao

The

are

b.

Both

facings

are

C.

Both

facings

have

such

either

to

classical,

This

result

same

identical

edge

6-3

are

wall

on

the

embodies

properties.

or

that

is

no

free

completely

rotation

small--deflection

thickness.

material

in

load

such

shell

isotropie.

of the

c

simply-supported

constrained

the

j --

Pressure

rtmning

rotations

restraints

is the

External

a uniform

case,

edges

[6-2].

to

have

boundary

intermediate

used

facings

is

Subjected

simply-supported

for

basis

reformulated

that

In the

of course,

Dome

loading

buckling

whereas

theoretical

Sandwich

note

occur.

follows,

The

6.2-1.

component

From

l

lbs

clamped

radial

to

dis-

rotate

suppressed.

are

also

solution

the

boundary.

following

along

It

acceptable.

by

Yao

assumptions:

[6-1]

d.

Bending

of the

facings

e.

The core facings.

f.

The eore extensional and pacallel to the facings.

g.

The

transverse

11.

The

inequality

has

about

infinite

their

own

extensional

shear

middle

stiffness

shear

in the

stiffnesses

properties

of the

surfaces

are

core

can be neglected.

direction

normal

negligible

are

to the

in directions

isotropie.

R 7-- >> 1 is satisfied, C

W hc

re

It t

i.

The

speaking, small

this

buckles

development

could

then

undertook

to ex-press

verse

in tkis shear

handbook through factor,

solution that

stiffness,

for axially

Y'd'

derived

are

axisymmetric

one

such

interpretation

regard,

(4.2-30)

was

and Equations

for

buckle

able

sphere,

results

body

involved. which

is identical

to the

equations

(4.2-4)

and

(4.2-5)].

core

cylinders TEat

sphere.

Plantema

foster

has

isotropie

given

when

theory

is not conducive

would

rsee is,

which

shallow-shell

Therefore,

the

sandwich

of the

which

when

circular

spheres

so that

in a form

in a manner

can be applied.

to a radius

to sh(rvv that,

solution

inches.

sandwich

respect

as a free his

TG-a]

complete

with

relationships

tie was

compressed

of I)omlell

of the phenomena

final

Yao's

of sandwich

inches.

to those

Yao presented

the

surface

of core,

equivalent

be employed. physical

to middle

Thickness

isolated

to a ready

insight

=

C

Approximations

Strictly exhibit

_: Radius

earlier

Equations the

some transin this (4.2-27)

knock-down

is included,

(_er

= 3/d Kc°'o

6--4

(6.2-1)

where _TEf

h (6.2-2)

R

O" o

and When

V

---2 C

K

When

V

1 = !---V 4

c

(6.2-3)

e

>-2 C

1 K

-

c

(6.2-4)

V C

where

0

V

(6.2-5)

C

crimp

5 2

(7

crimp

(I

=

Critical

=

Plasticity

compressive

(6.2-6)

G 2 tft c

stress

c

for

sandwich

sphere,

psi.

cr

Ef h

v

reduction

= Young's

modulus

= Distance

between'

factor,

dimensionless.

of facings, middle

psi. surfaces

of facings,

inches.

= Elastic Poisson's ratio of facings, dimensionless. e

R

=

Radius

to middle

surface

tf

=

Thickness

of a single

t

=

Thickness

of core,

=

Transverse

shear

of sandwich

facing,

sphere,

inches.

inches.

C

G

modulus

C

6-5

of core,

psi.

inches.

Theequivalence betweenanaxially compressedsandwichcylinderandanexternally pressurizedsandwichspherehasbeenanalyticallydemonstratedonlyfor the case wherethe two facingshaveidenticalmaterial propertiesandare of the samethickness. If oneassumesthatthis equivalencestill holdstrue whenthe facingsare of different thicknesses,Equations(4.2-2) through(4.2-7) canthenbe usedhereif Gxzis replaced by Gc so that, whenthe knock-downfactor Ydis included, o"cr

=

yd K c o o

(6.2-7)

where

_7o

h R

= rtEf

2_/tl_ (6.2-8) e

and When

V

22 C

K

When

V

=

c

1 1---V 4

(6.2-9)

c

'2 C

1 K

c

-

(6.2-10)

V C

whe re

(7 0

V

-

e

(6.2-11)

(7

crimp

5 2

crimp t

1

and

t

2

=

Thicknesses which facing

G (t 1 + t2) t c

(6.2-12) c

of the facings (There is no preference is denoted by the subscript 1 or 2.),

6-6

as to inches.

The relationshipbetweenKe andVe canbeplottedas shownin Figure6.2-2. It is importantto notethat the valueVc = different

types

core

and

moderately-stiff-core

hood

of zero,

maximum

of behavior.

the

core

The

region

establishes where

sandwich

transverse

sensitivity

2.0

Vc < 2.0

stiffness

imperfections.

V

between

the

so-called

stiff-

is in the

neighbor-

When

is high As

line

covers

constructions.

shear

to initial

a dividing

e

V

C

and the

increases

two

sandwich

exhibits

from

1.0

K e

I L

Figure

zero

to a value

where

V

c

Sandwich ence

Schematic

of 2.0,

_ 2.0

is the

this

of initial

the

sensitivity

which

imperfections It should

relationship

Representation

so-called

constructions

structures.

beyond

6.2-2.

I 2.0 Vc

which scope

of the

fall

be possible

present

becomes

weak-core

and

recognizes

of Relationship

the

within

progressively

region

where

this

category

a knock-down

factor

to develop

a continuous

variable

influence

handbook.

6-7

Between

less.

K

The

shear

crimping

not

influenced

are

of unity

of the

--e

rigidity

V

domain occurs. by the pres-

can be applied

transitional core

and

e

to such

knock-down but this

is

6.2.1.2 Empirical Kmock-D(_vn Factor As notedin Section6.1, for the purposesof this handbook,the allowablestressesfor externallypressurizedsandwichdomesare establishedby applyinganempirical knockdownfactor (_d)to the results from classical small-deflectiontheory, ttowever, since the availabletest datafrom sandwichdomeconstructionsare very scarce, onecannot yet determineYdvalueswith a highdegreeof reliability. Theonlyusefuldatauncoveredduringthepreparationof this handbook are thosewhichwere obtainedby NorthAmericanRockwelli 6-4_in conjunctionwith the SaturnS-II developmentprogram. Theseresults givethe yd

values

points

data

from

hemispheres

ellipsoidal. ranged felt

Reference all

that

faulty

the way

these

reduction appear

edge

subjected

these

domes,

Still

in the

the

inner

+10 °F.

to arrive 6.2-3

having

these

the

at the

another

test.

to a thernml

apt)roximately

mens

6.2-/t.

to the

too high

Therefore,

facing This related

points

in Figure

from

specimens

the elastic were

condition

were

in Figure

from

factors.

six

(;-4 includes

stresses

in Figure

formed

and

shown

the

particular

data

experimental

gradient

along

was

at roughly

gradient

was

Yd values.

they

fall within

same

basic

contour.

the

as with

were

zones.

computation were point

noted

external

+280°F

while

Nevertheless, band

inehides

stresses

cases

pressure.

neglected these displayed

they

do not

because

6.2-3,

outer

it was

plasticity

and

discarded

the

data

at failure

In three

discarded was

two

approximately

of reliable

in Figure

the

scatter

which

membrane

completely

6-8

that

plastic

to permit

In addition,

since

domes

whose

deeply

6.2-3

two

specimens

For

each

facing

in the results by the

of a

analysis

was

of at

per-

are

retained

other

speci-

i

1°0

L .....

Lt

•

I

io,

LEGEND • ,,

--[]

0,

ttEMISPHERES AT ROOM TEMPERATURE APPROXIMATE EI_LIPSOIDS AT ROOM TEMPE RATURE APPROXIMATE EI, LIPSOIDS WITH TItERMAI_ GRADIENT

p

= SHE]_I,-WAI,I, (p _ h/2 FOR WtIOSE

TWO

RADIUS OF GYRATION SANDWICtt CONSTRUCTIONS FACINGS

ARr._EOF EQUAL

i [ ' ' THICKNESS)

I ....

0o

• ' '

•

_d

' ........ , . '' _

.....

'

t

.... '•:

i .

i i RECOMMENDED DESIGN. _ IVAI,UE FOR EI,I_IPSOIDS] AND TORISt tlERICAI, _.. 'liiDOMES

' i +

'

0o

0._

'

. •

: ......

_'

'. i

,

0

I

'.. '.

RECOMMENDED i• ' _[

_"i VALUE , r

'

•

FOR

DESIGN HEMISPttERES " " •

i

I

100

Figure

i ...... _' _

6.2-3.

200

300

(->)

Knock-Down Factor Subjected to Uniform

6-9

400

_/d for Sandwich Domes External Pressure

aO0

To fully understandthe informationgivenin Figure6.2-3, it is importantfor the readerto beawareofthe datareductiontechniquesemployedhere. For anexplanation of theseprocedures,relbrencemaybemadeto the discussionin Section4.2.1.2.1. Althoughthat sectionis concernedwith sandwichcylinders, the samebasicapproach wasusedin analyzingthe domes.

Basedon Figure6.2-3, it is recommended that, exceptwhereshearcrimping occurs, the followingvaluesma3be usedfor Td: )'d "Yd =

Insufficient the

data

ratio

RMax/P.

lead

to the

for

isotropic

is lower

in the

fact

occur

at the

hand,

the

there than

This

that,

for the

apex

which

membrane

Discontinuity theory

but,

possible

that

domes

of the

even

be consistent

justification

ellipsoids

latter

two

(6.2-14)

knock-down

a large

array

with

the practice

for the

hse

of a 7d value

and torispherical

configurations,

is well-removed

from

in a hemisphere

at the in reality,

and precipitate

torispherical

any dependence

would

for

s_resses

distortions

(6.2-13)

factor of data

usually

on

would accepted

domes.

is physical

that

and

to discern it is quite

(non-san&rich)

which

perfections

available

conclusion.

that

for hemispheres

for ellipsoids

ttowever

same

It is thought

stability

are

0.35

= 0.20

boundaries these

buckling.

are

deformations This

fact,

6-10

domes.

the

maximum

the

boundary

are

uniform

ignored can coupled

This

lies

stresses

disturbances.

On the

over

surface.

the

the

entire

other

small-deflection

somewhat

with

hemispheres

justification

membrane

in classical act

for

like

uniform

initial

im-

membrane

stress in the hemisphere,canlead to earlier failure thanwouldbe encounteredfor shapeswherethe peakmembranestressesdonot extendinto the boundaryregions. Sincethe recommended valuesfor Tdare basedonmeagertest results, the method proposedhereis not very reliable andcanonlybe regardedas a "best-available" technique. It shouldonlybeusedas a roughguidelineandfinal designsmustbe substantiated by test.

6-11

6.2.2 DesignEquationsandCurves For sandwichdomesof the typesshownin Figure6.1-1 andsubjectedto uniform externalpressure, the given

in the

facings.

equations

There

the

the

material,

the

same

The

buckling

The

knock-down

facings

where

For

such

coefficients

factor

the

obtained

arc

where are

(7)/1)

wtlid can

be chosen

as

occurs

the

when

E_ and

E_, will,

I"ig_rc

both

pre-

to cover

of the

not made

behavior

be made

KC can be obtained from

Yd may

are

formulas

This

ratios

1 or 2.

was

of course,

achieved

moduli

of the

is elastie facings

some

for

same (r_= 1).

are

made

of

be equal.

(;.2-4.

follows: When

V

_ 2.0 e

?/d

quantity

measured

only when

only

configxlrations,

facings

of the

material.

on the

separate

subscript

in order

same

relationships

to the

by the

extension

based

the

2 refer

is denoted

of the

the two

from

1 and

accomplished

concepts

2.0

USe_d

The

facing

was

C

Use

subscripts

not made

cases

be computed

by a simple

extension

equations cases

may

as to which

were

For

resulting

material.

V

stresses

of equivalent-thiekness

to inelastic

the

(;-14

The

two

facings.

Application

When

the

use

respective

6-14

6.2.1.1.

where

through

on page

on page

in Section

situations

apex

is no preference

The equations sented

critical

:: 0.20 _

for

,

Use

)Jd

=

in units

maximum

of inches.

For

principal all

1.0

for hemispheres,

ellipsoids, spherical

0.35 forellips_dds and torispherical domes.

RMa x is the

at the

hemisl)heres

of the

apex. 6-12

radius shapes

of curvature shown

in Figure

for the 6.1-1,

and torldomes.

dome this

and

is

value

t_

[

....

•

.

i':T

i

0

©

.!

ttL! T T

!!

6-13

The

formulations

stiffness

this

given

of the

stiffness

will

structures,

one

modc

is

In

other

all

core

critical

here

is

are

based

isotropic.

vary

with

on the

assumption

However,

direction.

In

in

most

order

to

apply

select

a single

effective

G c value.

(V c

: 2.0),

G c must

be

equal

cases

one

must

rely

on

engineering

the

practical

must

taken

that

transverse

sandwich

the

given

judgment

the

minimum

in

constructions,

criteria

Whenever

to the

shear

to

such

shear

value

crimping

for

the

making

an

appropriate

stress

at

the

core.

selection.

The

plasticity

dome.

reduction

For

of Section

elastic

factor

cases,

9 must

be

a

er 1

o"

=

Yd Ke !

=

01

r?E 1 C

always

r?= 1.

Whenever

be

based

the

on the

behavior

is

inelastic,

1

(i

Facing {6.2-15)

oi

(6

0

h

2

_

o

"

2-17)

cr_

=: Yd Kcr_o_

cr

:r/E

<

_

(Eztl)

(6.2-16)

(6.2-18)

(Estp) (6.2-19)

2

=

G

h _

(6.2-20)

G C

Its+

E(_(.)t_]

tc

c

ta+t

(7 _

(7

O 1

(6.2-22)

(7

methods

.:

o

Max

crimp_

of the

2

C

R

5

the

Apex

%

C

apex

employed.

Facing

Apex

use

should

V

crimp_

=V C 1

C_

t

o (6.2-23)

(7 crimp_

6-14

(6.2-21)

Thecritical pressurePer (in units of psi) maybe computedas follows: 2 t2] Pcr - RMax ECrcrl tl +acr2

(6.2-24)

In the specialcasewheret1= t2_tf andbothfacingsare madeofthe samematerial, Equations(6.2-15)through(6.2-24)canbe simplifiedto the following: Apex

=

_cr

(6.2-25)

_d Kc(Y o

(fiEf)

h

(6.2-26)

(7 O

ffe_ RMax

5 2

-

G

Crcrimp

2 tft c

(6.2-27) c

(7

o V

-

(6.2-28)

---

e

(7

c rimp

4 Pcr

-

(6.2-29)

R

((_crtf) Max

6.3

OTHER

No

information

shells

covered

under

LOADING

is

loading

in Section

CONDITIONS

available

conditions

concerning

other

the

than

general

that

6.2.

6-15

instability

of uniform

of dome-shaped

external

pressure

sandwich

which

is

r

r_

i

c_

_J

C,

L_

" '_ "b

b-,

o

_'-go

o c,i

'_@

H

AI

p d-'

l

/I,

H b

c,_

3

o

_D

m

-_ _:o

o Ph

d_

0 "_ b

,_ P'-

._

o

_._ o U

+

g._ M 0

;,.I,

<:

cq

lI

II

'-'

H

,¢

g

-,,11--

--0

_q b

6-15

L; 0

2, o '-c

II

,-,o b _

o

d_ el

o

0

_._

_

b

o

II

o

:I li

h rd''

o

p-

b

6-17

o o

@

,_

It

REFERENCES 6-1

Yao, J. C., "Bucklingof Sandwich SphereUnderNormalPressure", Journal of the AerospaceSciences,March 1962.

6-2

Plantema,F. J., SandwichConstruction, Copyright

6-3

6-4

Donnell,

Wiley

& Sons,

Inc.,

New

York,

1966.

L.

H.,

"Stability

of Thin-Walled

Technical

Report

No.

479,

Gonzalez,

H.

and

Patton,

Diameter

Scale

Corp.

John

Report

M.

Model No.

SDL

S-II 468,

Tubes

Under

Torsion",

NACA

1934.

R. Common May

J.,

"Development Bulkheads",

1964.

(_-19

and North

Fabrication American

of

55-Inch

Rockwell,

7 INSTA

7.1

CYLINDRICAL

7.1.1

Axial

7.1.1.1

cases,

putation

here skin

the

as

SE GME N TS

PANELS

to first panels

Schapitz

of critical

geometry

CURVED

CH S HE LL

Principles

be helpful

(non-sandwich) such

OF SANDWI

Compression

Basic

It will

BILITY

ticular,

Schapitz

verified

by the

for

to narrow proposed rederivation

case

four

furnishes

from

which

of axially

boundaries

criterion

is made

that

all _7-1]

This

panels

the

which

criterion

stresses.

the transition

flfll cylinders,

consider

the

of Reference

simply

which

the

behavior

following

means

for the

panels,

approach

one use

are

a practical

accounts

wide

compressed,

effects behave

relationships

of fiat

isotropic supported. for the

In corn-

of skin-panel essentially plates. which

as In par-

have

been

7-2:

When

(7.1-1)

then 2

c_R (l

=

U

er

4_

p

(7.1-2) P

when (YR

>2_

(7.1-3) P

then U cr

= crR

7-1

(7.1-4)

where, (Y

P

=

(rfl

Critical stress for buckling figuration shown in Figure

Critical cylinder

of a simply 7.1-1, psi.

supported

flat

plate

stress (in units of psi) for buckling of a simply of radius R, length aR, and thickness t R (see

quantities R, aiR, and t R are all measured knock-down factor should be incorporated mental effects from initial imperfections.

of the

supported complete Figure 7.1-1). The

in units of inches. An empirical here to account for the detri-

(Y, psi

/

a,

a p

R

g,

psi t P a

=a

R

R b

=b

t

7.1-1.

Cylindrical

Panel

::b p

=t R

Figure

_a

p

R

t p

and Associated 7-2

con-

Flat-Plate

Configuration

psi

b.

For sandwichpanelswhichfall in the moderately-stiffor weak-corecategories, gcr shouldbetakenequalto thehigher of the two valuesut)and(_R"

In the courseof preparingthis handbook,noanalysiswasmadeof test datafrom sandwich panels. Therefore,the reliability of this approachhasnot beenestablished,and, until ex_perimental substantiationis obtained,onecanonlyregardthe methodas a "bestavailable"teebxlique. In view of the lack of sandwichdatacomparisons,it is informativeto notethata large collectionof test results from isotropic (non-sandwich) specimensis evaluatedin Reference7-3 andit is shownthere thatthe Sehapitzcriterion is a reliable approach for suchpanels. Thetest configurationsembraceda wide rangeof ratios.

Narrow,

between

those

wide, for the

where

all

itative

presentation

and

relative

logarithmic

four

and ease

boundaries

intermediate where are

of Figure

positioning

format

all

fully

panels four

This

of the

The

theoretical

buckling

b.

The

classical,

small-deflection,

complete

d.

are

The

following

simply

results

figure

shows when

The

are the

-_

, and

K values

supported

and

summarized

general

displayed

fell the

in the

characteristics

in a nondimensional

:

a.

c.

included.

boundaries

clamped.

7.1-3.

for each

were

,

relationship

for

fiat

theoretical

plates. buckling

A lower-bound

buckling

relationship

for

complete

obtained by multiplying the values from knock-down factor of Reference 7-4.

b; above,

The

criterion.

design

relationship

for

cylinders.

curve

based

on the

Sehapitz

7-5

cylinders. by the

This empirical

is

case qual-

\

\

\ ,

* _

O" er

•

t

.\\

%

\

'\ De sign on

Figure

7.1-3.

Cum'e

Schapitz

Schematic (Non-Sanchvich)

Based Criterion

Logarithmic Skin

Plot Panels

7-6

of

Test

Under

Data Axial

for

Cylindrical

Compression

Isotropic

Althoughderivedspecificallyfor the easeof simple support,this criterion hasbeen successfullyemployed[7-3_ wheretheboundariesprovidevariousdegreesof rotational restraint alongwith the conditionof no radial displacement. This wasaccomplishedby simply adjustingthe val_,efor ap to correspondwith the appropriateedgerestraints. For the case

under

criterion

be

can

curves

of this

(7.1-2)

becomes

greater

than

For

curve

all

other

immediate

graphically

type

are

that

(R/t)

(non-sandwich

represented

given

tangent

asymptotically

discussion

to

of the

values,

in

the

Reference

shown

7-3.

full-cylinder

tangency

the

as

point,

the

the

line

The

skin

when

panel

Crp.

gR

= 2gp.

as

applies.

(b)

For

= C°nstat_

Schematic for

Logarithmic

Non-Sandwich

Plot Cylindrical

7-3

of

Schapitz Skin

Panels

Criterion

Equation

cylinder.

that

K denoted

design

values

a complete

!

7.1-2.

by

(R/t)

:

Figure

of

defined

Note

quantity

Schapitz

A series

curve

behaves

The

the

7.1-2.

transition

relationship

for

panels),

in Figure

curve

transitional

approaches

skin

the

in

transition

Figure

7.1-2

is the conventionallist-plate bucldingcoefficientwhichis oh,pendent ratio

(a/b),

seen

that,

1)oundary

if the

c(mditions,

critical

and

stress

t33)e

weco

of loading.

taken

equal

Fr,.m_

to the

this

higl_er

upon

the

fig-ere,

of the

it

two

aspect

can

1)e

values

cr P

and,7

R,

one

would

cross-hatched

only

be

region.

neglecting

When

(_-_R / \

design

value

ratio the

{_f{ /, \Crp/

the

lhe

nmde

by

computing

vided

here

that

in Sections

the

core

of Equations

(7.1-1)

panel

buckling

strength,

stiff

will

Consequently

a.

it

the

sandwich

lail

tfowever,

For

it should

the

restllt

for

in a

here

other

hand,

between

will

makt.

the

inust

t P,

small.

speculation

panels

merely

are

pro-

recognize

For

direct

stiff-

application

weak-core

region,

not

contribute

(7.1-4)

would

foregoing

oi(;tl \

which

having

of the

quite

stiffness.

constructions

the

be

cr R,

one

in the

through

ranges

would

curves

(:ore

to

curvature

values

sandwich

and

the

possible

sandwich

most

extension,

upon

(7.1-1)

somewhere

an

()/her

(71) and

cylindrical

be

and

Equations

situation

sandwich

the

crimping,

cases,

fall

dependent

On

area

equations such

all

for

values

to

design

4.2),

shear

lhe

application

is

recommended

stiff-core

with

would

For

Indeed,

on

in making

{7.1-.t).

The

ccmtribution

cross-hatched

solely

panel

So.ellen

by

this

associated

prediction.

the

cJt{ from

lhr_ugh

of

sig_,ificanl.

neglecting

its

of course,

is

less

extend

in such

would,

be

might

predictions.

cores

hy

.I.

strenglh

neglect

Schapitz

on(,

(see

sandwich

servative

would

of a sandwich

constructions

of the

is: dependent

:; al?d

behavior

=: 1,

criterion

crp and

tnulsitional

P/

sl) pc, rccnt

introduced

Schapitz

is

is

differences

conservatism

Since

that

which

the

to

the

the

yield

uneon-

moderately-

limiting

eases.

that,

panels,

Equations

applied.

7-4

(7.1-1)

through

(7.1-4)

can

be

Also

sh(_vn

in Figure

non-sandwich of the

reported

Based

7.1-3

many

design

as a flat plate.

curve

Except

a.

The

recommended

b.

The

values

7.1-3

ditions

include

displacement. buckling

the

Schapitz

the

of the

between

have

been

verifies

the

of the

skin

panels,

partly

usually

criterion

from plate

utilizes

that

exhibited

7-7

fact

by wide

This

panels

full theoretical

to the

to support

region.

for wide

even

on the

can continue

postbuckling

observed

basic

test

curves

points

which

essentially regions

of

t_vo bounds:

reliability

is based

the

appropriate in Figure

for the

following

of these

not incorporate

factor.

conclusion

into

data

course

on the

beltaves

if _R did

This

fiat

of the

the

for each

several

predicted

in addition

The

the

inserted

panel

test

the

restraint

well

in load

where

rotational

different

to the

but four

some

is quite

made

were

all

all

During

fr()m

curve.

(non-san&rich)

cylinders.

loading

drop-off

Figure

region

fall

design would

that

data

located

points

relative

points,

behavior

which

of isotropic

the

four

that

case

shows

were

accurately test

positions

figure

knock-down

for the

plane

the

an empirical It is concluded

complete

plots,

test

7-8.

plots

different actual

of the

through

quantitative were

for those

full-cylinder

7-5

points

lie within

and

locations

test

This

transitional

plate

7-3,

to their

of behavior. the

approximate

of References

corresponding

in approximation

fall below

the

panels

on these

regions

are

in Reference

and the

graph.

and

cylindrical

study

specimens

7.1-3

and

predictions

Schapitz

where

the

criterion boundary

requirement that

the

of no radial character

cylindrical steadily

is in contrast full cylinders. as the

con-

of flat-

panels increasing to the

and in-

sudden

Consequently limiting

case

of a

flat

plate

will

display

bucause st ructur,,tl

is

approached.

some

of tile

small

physical

One

degFee

behavior

might,

therefore,

of scatter

cited

on

above,

de fieieneies.

7-8

expecl

both

this

sides

generally

that

within

of th(,

this

design

will

not

region

curve,

lead

test

data

ttowever,

to

any

seFious

7.1.1.2 DesignEquationsandCurves For cylindrical sandwichpanelssubjectedto axial compression,the critical stress maybe computedfrom the following: Weak-Core

and Core

Stiff-Core Constructions When

(y

(7. i-5)

The cr

=

Moderately-Stiff-

Constructions

higher

|values [

_

of the P

and

(7.1-9)

two]

gR

! J

then (_R (3" cr

=

(7.1-6)

+ 4--_-- ' and P

p

when

(7.1-7)

then U

whe

=

cr

U

re, =

u P

Critical

axial

flat

sandwich

cal

panel

and,

(see

puting

this

value.

Critical

axial

complete is

down

factor

effects

a rule-of-thumb,

the

has

for

Figure

identical

from

initial

one

may

curved

same

An here

buckling as

geometry factor

of psi) for

the

the

for

the

as

the

required

cylinin

buckling

com-

of a dimen-

empirical for

a

cylindri-

circumferential

appropriate

to account

is

of

the

knock-

the

detrimental

are

those

imperfections.

assume

that

stiff-core

constructions

which

inequality V

(7.

<0.25 C

where

units

except

panel.

incorporated

of the

for

conditions

knock-down

(in

which,

of psi)

boundary is

No

stress

the be

units

same

curvature,

cylinder to

should

(in

the

7.1-1).

compressive

sandwich

sion,

stress

which

except

panel

=

satisfy

compressive plate

drical

(;R

As

(7.1-8) R

V c is

computed

as

specified

in Section 7-9

4.2.

i-i0)

The

quantity

Section

The

be computed

by using

the

design

equations

and

curves

given

in

(_R should

be computed

by using

the

design

equations

and

curves

given

in

3.

quantity

Section

_p should

4.

A graphical

representation

of Equations

(7.1-5)

through

(7.1-8)

is provided

in Figure

7.1-4.

The method

given

against

radial

the

boundaries

four

from

a hinged

here

applies

only where

displacement.

Therefore,

may

condition

include to fully

all

four

no free

rotational

boundaries edges

restraint

are

of any

are

completely

permitted. degree

restrained

Any ranging

or all

all

clamped.

4.0

+ 4_-

-T.H }--p_

3.0

4.0

(ffR/(_p) Figure

7.1-4. Graphical

Representation

of Equations 7-10

(7.1-5)

through

(7.1-8)

the

of way

7.1.2 OtherLoadingConditions 7.].2.1

BasicPrinciples

In the preparationof this handbook,almostno considerationwasgivento the buckling of cylindrical sandwichpanelssubjectedto loadingsotherthanaxial compression. Therefore, nofirm recommendations canbe madehere concerningdesignequations andcurves, llowever, the suggestionis offeredthat, for suchcases, onemight consider an extensionof the conceptspresentedin Section7.1.1. Inparticular, for all regionsof core stiffness, it mightbepossibleto applythe equation [The higherof the two] Crcr if one

simply

computes

In conformance only

when

all

with four

the

Therefore,

may

rotational

dition

7.1.2.2

to fully

No information shapes

other

Crp and

_p and o R for the

of Section

of the panel

no free

(_R

edges of any

are

are

loading

7.1.1,

degree

the

completely

permitted.

Any

ranging

(7.1-11)

J

all

condition

of interest.

foregoing

suggestion

restrained

against

or all the

way

of the from

four

applies radial

dis-

boundaries

a hinged

con-

segments

of

clamped.

Design

OTHER

[values

restrictions

restraint

Equations

No rcconm]endations

7.2

values

boundaries

placement. include

the

=

are

PANEL

the

made

Curves here.

CONFIGURATIONS

is available than

and

concerning

cylindrical

the

instability

configurations

7-11

of sandwich

considered

shell

in Section

7.1.

©

b

b II

t,

-< 0 Z

7-12

b

_L _

_

°

c_

q_

_Q

m_

r!

b

_o

7-13

REFERENCES

7-1

Schapitz,

E.,

Festigkeitslehre

D_isseldorf, 7-2

Spier,

Copyright

E.

E.

Simply

ln'ession",

Contract

Smith,

G. W.,

cally Skin

Convair 7-4

Axial

division

Seide,

P.,

opment

and

J.

Cox,

It.

Thin

Sheet

Crate,

Jackson, Research

I.,

R.

GmbtI

Elaslic

Convair

to Axial division

L.

S.,

and

"The

Stability

Buckling Com-

Memo

AS-

of Eccentri-

Volume

II - Buckling

of Curved

Contract

NAS8-11181,

General

20 June

Morgan,

E.

Stability

J.,

Dynamics

1967.

"Final

of Thin

Isotropic

Report

Shell

on the

Devel-

Structures",

STL-

1960.

O.,

and

of Circular

Deaton, Cylinders

J.

W.,

with

"Structural

Behavior

Longitudinal

Stiffening",

1962. Clenshaw,

W.

J.,

British

"Compression

A.R.C.

Technical

Tests Report

on Curved R&M

Piates No.

of

1894,

1941. It.

and

Levin,

Compression", 7-8

the

Subjected

Dynamic

Fossum,

for Elastic

Strength

and

for

Panels

GDC-DDG-67-006,

V.

Duralumin",

November

7-7

No.

Whitley,

May L.

and

31 December

Compressive

TN-D-1251, 7-6

E.,

Criteria

P.,

VDI-Verlag

Criterion

Skin

General

Cylinders,

Weingarten,

Petcrson,

Sehapitz

Cylindrical

Compression",

TR-60-0000-19425,

7-5

E.

Report

of Design

2 Aufl.,

1967.

Circular

Panels;

"The

NAS8-11181,

Spier,

Stiffened

G. W.,

Supported

31 January

Leichtbau,

1963.

Smith,

of Isotropie

D-1029, 7-3

and

f_r den

L.

NACA

K.

B.

Council

and

R.,

"Data

WR L-557,

Hall,

of Canada

A.

H.,

on Buckling

Strength

of Curved

Sheet

in

1943. "Curved

Aeronautical

7-15

Plates Report

in Compression", AR-1,

1947.

National

8 EFFECTS

OF CUTOUTS

ON TttE

GENERAL

SAND_\rICtt

In many for

practical

purposes

of access,

experimental having

such

rares,

this

data

vide

in this

made

in the

received

little

handbook

problem,

and,

stress

design

criterion.

exists

for

in view

further

is not

theoretical

one

struchave

in isotropic

work

design

been

shells with

comprehensive

experimental

no related

shell

dealing

or

shells

solutions

(8-1)

sufficiently

and

situation,

cutouts paper

solutions

of sandwich

theoretical

around

be incorporated

no theoretical

(non-sandwich)

Some

of only

paper

OF

cutouts

instability

of isotropic

attention.

aware

this

of this

case

that

However,

general

distributions

are

and

etc.

[or the

Even

of this

need

area,

tins

it is required

venting,

published

the

a practical

An obvious

been

concerning

instability

structures,

lightening,

have

problem

the authors

general

shell

penetrations.

accomplished but

aerospace

INSTABILITY

SttELLS

the

to pro

to be accomplished

recommendations

can be

at the present tirne.

REFERENCE

8-1

Snyder,

R.

on the

Buckling

pression," Institute ics,

E. , "An Experimental

May

Strength

Thesis in candidacy

of Circular

submitted for

to the the

degTee

Investigation Cylindrical Graduate of blaster

1965. 8-i

of the

Effect

Shells Faculty

Loaded

of Circular in Axial

of the Virginia

of Science

Cutouts Com-

Polyteclmic

in Engineering

Mechan-

9 INELASTIC

9.1

SINGLE

9.1.1 For

Basic

LOADING

many

members

on plasticity

times

resort

which

are

provided

In this

theory

AND SItE_LLS

but,

recommended

of these

stresses

such for

when

in this

modes

or

the proportional

this

reduction

9-5

factors

9.1.2

of so-called

by the

it is plasticity

symbol

r7 •

derivations

impractical,

one must

the

sandwich are

for isotropic the

use

material,

by theoretical

gives

equations

involves

of the

denoted

proves

for various

These

through

the

established

Section

of instability. 9-1

are

approach

handbook

limit

through

factors

_7 are

expressions.

in References

Application

loads

formulas

to empirical

and

beyond

handbook,

appropriate

based

of loading,

stressed critical

factors.

cases,

PLATES

CONDITIONS

to compute

reduction

OF SANDWICH

Principles

structural

customary

BEHAVIOR

formulations

on the

(non-sandwich)

types

information

plates

trial-and-error

some-

for

configurations,

based

In

and

procedure

shells.

outlined

below:

a.

First,

assume

7) = 1 and

configuration,

b.

c.

loading

If the

critical

limit

of the

ever,

if the

computed

must

continue

as

Assume portional

stress facing

condition, computed

material,

for

critical

stress

and

mode

of failure.

in a, above,

critical

but less

the

no further

specified

a new value limit

compute

stress

is less

for

than

computations exceeds

the

the

the are

appropriate

proportional

required,

ttow-

proportional

limit,

one

is in excess

of the

pro-

below. the than

critical the 9-1

value

stress

which

computed

in a,

above.

dp

_°

f.

Based

stress

for the

facing

duction 1ol- this

factor. purpose.

Using If the with ever, ration until lated

A numerical

on the

critical

assumed

material, The

lh(. r_ value

the

level

compute formulas

a value of Tables

computed

stress

value

in c,

in d above,

calculated

assumed

in e,

above,

and the

for the 9. i-1

appropriate

through

recalculate

in e,

above,

no further

stress-strain

the

plasticity

9.1-3

are

re-

can be used

critical

is in reasonable

computations

curve

stress. agreement

required.

How-

if such agreement is not aehieved, one must then repeat the eompucycle starting with e. This iterative procedure must be continued acceptable agr¢_ement critical stlvsses.

example

of the

fc,regoing

is attained

procedure

9-2

between

is provided

the

assumed

in Section

and the

9.1.2.

ealeu-

9.1.2 DesignEquations Recommended formulasfor plasticity reductionfactors are givenin Tables9.1-1 through9.1-3 where Ef = CompressiveYoung'smodulusof facings,psi. E = Compressivesecantmodulusof facings, psi. s E = Compressivetangentmodulusof facings, psi. t Gf = Elastic shearmodulusof facings,psi. G = Secantshearmodulusof facings, psi. s v = Elastic Poisson'sratio of facings,dimensionless. e

v Values

Actual

Poisson's

for v can be obtained

ratio

of facings,

dimensionless.

by using

v

=

0.50-

(0_. 50 - re)

(9.1-1)

or

(9.1-2)

= o. o -\ El/ The

technique

means

for

applying

of a numerical

the

plasticity

example

assumed

to be of sufficient

assumed

that

for

length

reduction

an axially to fall

factors

compressed

outside

the

al

both

facings

are

of the

same

thickness,

b.

both

facings

are

made

of the

same

e.

is demonstrated sandwich

short-cylinder

material,

cylinder range.

below which

(Gxz/Gyz)

= 1. 9-3

is

It is further

and

the transverse shear properties of the core are isotropic so that 0=

by

For

such

stabilit},

cylinders, may

Section

be

4.2.2

computed

specifies

that

the

critical

stress

for

general

in-

[rom

(Tcr

=

Yc Kc(Yo

(9.1-3)

where

(71E f)

h

cr

(9.1-4)

o

_/c is

obtained

from

Figure

4.2-8.

K e is

obtained

from

Figure

4.2-7

where

Gr o V

-

(9.1-5) (Ycrimp

c and

5

_crimp

For

the purposes

of the present

Ef

=

i0 ×106

v

=

0.30

R

=

:_2.0"

h

=

.320"

tf =

.020"

=

.300"

=

20,000

-

sample

2

2tft

problem,

c

Gxz

assume

psi

e

t C

G

psi

XZ

h f) -

2

R

32.0" -

p

Facing

. 160" -

200

.160"

Proportional

Limit

=

25,000

9-4

psi

(9.1-6)

that

By using

these

values

and

assuming

that

7c

=

0.49

cr

=

104,900

=

170,800

=

104,900/170,800

_ = 1,

it is found

that

0

crimp V

=

.614

C

K

= 0.85 C

The re fore,

Note

that

(25,000 proceed stress

the psi)

7cKcCro

computed

critical

of the

facings.

value

which

Hence

for

the

be computed

that

facing from

the

the

(43,600 the

use That

psi)

is,

following

limit.

than

select

For

the

is taken

from

and

purposes

reduction Table

limit

one must

an assumed

By using

plasticity

which

the proportional

be valid

is selected.

corresponding

formula

is higher

one must

(_cr = 30,000 the

43,600

of _ = 1 cannot

proportional

value

material, the

49 x.85x104,900

basis.

exceeds

suppose

=

stress

on a trial-and-error

problem, curve

gcr

critical

of this

the

now

sample

stress-strain

factor

can

then

9.1-3:

1

1--=71 Suppose

that

this

gives

the

result

that

= 0. 900 so that

one now

obtains

7c

= 0.49

(remains

=

×104,900

.900

unchanged) =

94,400

O

9-5

Ef

(9.1-7)

g

erimt) V

=

170,800

=

9,t,

=

0.8G

(remains

unchanged)

!t00/170,800

=

.553

C

K e

There

fore,

(rcr

Note

the

that

the

assumed

=YcKc

eon_puted

value

selecting

a new

Equation

(9.1-7)

critical

1:_0,000

assumed

the

that

one

now

=

.49

stress

psi).

×.86

×94,400

(39,800

psi)

Therefore,

critical

stress,

corresponding

r? = so

cro

=

:_9,800

does

not

another

iteration

35,000

psi.

say

plasticity

reduction

agree

very

closely

be

performed

will

Suppose

factor

is

that

found

by

to

with

using

be

0. 790

obtains

:/c

=

0.,i9

(remains

(Yo

=

"790

×104,900

=

170,800

=

82,900/170,800

=

0.87

=

,/cKcffo

cr crimp V

unchanged)

=

82,900

(remains

unchanged)

=

.486

C

K C

"rile re fore,

cr

Note

that

with

the

and

the

the

computed

assumed

design

value

value

for

=

.49

×.87

×82,900

psi)

critical

stress

(35,400

(35,000

psi).

Therefore,

the

critical

stress

is

9-6

is

no

35,000

=

35,400

now

in

further

psi.

reasonable

iterations

agreement

are

required

by

=o._ "_

_

0

_2J L,-_

_

j

M r_

_--

0

_

o

.,..4 [

o

i

0

0 0

O_ ..-i

+

0

_o

¢)

_

0

o

,--_I cq + ,-_

0

I c"q

I

|

U9

0

r_

0 0 I.-..4

0 0

I

0

I

I

J II

0

0

r

1

_'_ r_

I_

_L

_

+

0

•r_ 3 'tJ

+

c_ o r..)

0

_ _

0

I

0

0 0 0

r--¢

b_ I

.a

l li

I _b

N r..) t:_

b_

r_

,-_ 0

0

9-7

[-, © 2;

r

_I_

c'o I '._-t._ _1'_

+

IJ

_1_1

L

J

_1_ f

] i I

1 L

J

L""*

J ][

*3

@

©

© r_

p

q3

l

I

0

i[

Ir ir

,....,

I

c_ E_

9-8

Table

Recommended

9-3.

plasticity

of Circular and

Reduction

Sandwich

Axisymmctric

Cylinders, Sandwich

Factors

for the

Truncated

General

Circular

Instability

Sandwich

Cones,

Domes

PlasiicJly

Heducliol_

i.'zLclol's

Mo(h:ralc (:ylinders

Short _tnd

Through

l,englh

l,ong

Cones

ail(I

('ylinders

(:(rues

I

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

1 - Lit_

]

External

Laieral

T ors

I)rcssure

I -u(:' _

Et

,:: L,-:V ]

_

rl-

, : L,_-77]

ll_--_J

Ef

Torispherical cated External

formula

for

r l is

essentially

not

as will

it is informative

at

the

and

(All

trun-

equator)

be

to

valid

a long,

when

the

flat

encountered

note

that,

cylinder

plate.

in aerospace for

such

or

However,

cone

is

so

it is unlikely applications.

constructions,

the

short

that

that

it

such

Furthermore, given

formula

for

conservative.

**This

formula

behaves

for

rl is

essentially

configurations it is

Ellipsoidal, Domes

Pressure

configuralions

rl is

Es)

,I

Shear

ile, mispherical,

behaves

+

ion

Transverse

*This

_4

Ef

informative

approximately

not

as will

he to

when flat

encountered

note

13-1)ercent

valid

a long,

that,

the

cylinder

plale,

ttowever,

in aerospace for

such

constructions,

unconscrvative.

9-9

or

cone

is so

it is unlikely applications. the

short

that that

it

such

Furthermore, given

formul.a

is

9.2

COMBINED

9.2.1

Basic

As

noted

l)ility

of

shell

for

fields

due

structures

to the

criteria

one

for

one

e i.

intensity

linear

as

stress

subjected

not

is

been

note

to

a purely

conditions,

the

comt)ined

this

type

of the

a fundamental

the

stress

(unloading

elastic

case.

and

single-valued

Based

strain

on

the

intensities

phtsticity

reduc-

necessary

Toward

which

the

shear

be

to

this

specifies

that,

condition),

strain

between

c i can

these

engineering

of

oetahedral

and

done

becomes

relationship

gi

in

(loading

ftmction

the

been

has

theory

increasing

sta-

situation

conditions.

of plasticity

condition),

it

inelastic

A similar

practical

and

loading

is

the

related

many

problem

intensity

defined,

work

general,

critical

on

conditions.

in

of

hypothesis

a tmiquely

stress

in

t[owever,

with

available

theoretical

and,

esklblished,

when

is

loading

little

problem

estimate

decreasing

in

the

co_ffronted

and

ilfformation

Very

of

a rough

cri is

cri is

limited

constructions.

material

When

[ 9-]]

least

should

a given

stress

is

at

only

complexity

have

applications,

end,

9-(;,

flat-plate

determine

CONDITIONS

Principles

in llefe_'cnce

exists

tion

LOADING

law

intensity,

cr i and

for

defined

the

e i

plane

as

follows

:

,.

ei

It sJ_o'_ld

be

noted

facilitate

its

use:

that

1

,/,_

Eq'_tation

_<,

a:-,

_

3'

¢2 x

_ ¢2 y

(9.2-1)

is

+3r 2

xy

+ ¢

x

¢

sometimes

y

(9.2-1)

+'¢:_ /4 xy

written

(9.2-2)

in

ti,c

following

form

to

[ cri

(Crx)y

I - 7 + )Z 9-10

+ 3),::

(9.2-1a)

where

cr

-

Normal

stress

in the

x direction,

psi.

Normal

stress

in the

y dh'ection,

psi.

x

Y _-

:

Shear

stress

in the

xy

plane,

psi.

-

Normal

strain

in

the

x direction,

in/in.

=

Normal

strain

in the

y direction,

in/in.

=

Shear

x £ Y E

strain

in the

xy

plane,

in/in.

it

be

concluded

xy

y

x

x

From

cy

i

the

foregoing

(loading

tional

be

condition),

the

stress-strain

evident

tional

lies

discussion

that

limit

above

portant

be

of

the

to

must

curve

although

the

a rigorous

that

foregoing

conjunction

from

individual

material,

this

between

obtained

each

the

limit

phenomenon

that,

(yi and

a uniaxial

stress

in

the

mind

behavior

when

deciding

the

of

stresses

be

whether

conven-

It should

therefore

less

the

can

actually

increasing

to the

test.

may

is

case

identical

loading

of these

that

e i is

component

combination

so

for

than

propor-

give

a ffi value

inelastic.

It is

or

not

plasticity

which

im-

effects

considered.

Lacking

the

relationship

proportional

keep

can

with

approach

generalization

the

plasticity

to

the

of

subject

the

reduction

stability

stress-strain

problem,

relationship

it

is

conjectured

might

be

here

used

in

factor Et (9.2-3)

9-11

to obtain

conservative

quantities

E t and

predictions

Ef are

Et

of inelastic

modulus

of facing

(_i vs e i at a prescribed

The above overall

procedure

thought

that

The

details

to keep has

as

modulus

for _ was suggested

tttis

conservatism

of the

suggested

in mind

not been

regarded

Young's

formula

that

evaluated

this

under

combined

loadings.

The

as h)llows:

Tangent

Ef

instabilit%_

of facing

selected here

is based

are

the

curve

of

psi.

of its conservative

purely

outlined

not give

by comparisons

a "best-available"

from

psi.

nature.

on an engineering

Since

estimation,

the it is

justified.

approach does

obtained

of gi'

material,

in view

is well

method

material value

technique

against and

9-12

in Section

a rigorous test one

data. should

9.2.2.

solution,

It is important and its

reliability

Therefore,

this

can only

be cautious

in its

application.

be

9.2.2

Suggested

The

method

and

shells

terial

Method

suggested first

have

here

requires

the

that

stress

as e. 1

By completely

proceed

to establish

handbook.

are

for the

in the

That

is,

nents then

same during

the

is maintained. be inserted

considerations

into

for

the

to determine

the

associated

proportio_ml

limit

of the

the

tional must

critical

limit then

from

outlined

in Section a.

Assume for

the

the

for

strain

the

elastic

should

first-estimate

be

components

loading

individual

This

in earlier

stress

applied

several

then

condition.

assumption

individual

actual

should

provided

the

ma-

relabeled

one

stress

relationships

for the

facing

coordinates

(_ = 1),

the

the

plates

condition.

stress

compo-

computation

must

equation

stress

cYi versus

of the cYi versus to the

the

curve

computation,

between

of sandwich

combined

condition

as exist

stresses

combination.

resort

tiffs

stress

other

critical

interaction

_ cy a + (y2 y xy

cYi = 4a;

fact

the

appropriate

proportionality The

stress-strain

plasticity

combined

loading,

stability

as cYi and

all

to each

inelastic

relabeled

In performing

critical ratios

of the

conventional

a first-estimate

sections that

the

ignoring

by using

made

analysis

coordinates

can be achieved of tiffs

for

intensity

value.

e i curve,

the

However, e. curve,

following

if the the

+ 3-r 2

(9.2-4)

If this

first-estimate related

procedure

does

stress

a i value

first-estimate

trial-and-error

value

results

exceeds are

which

not exceed

the

values

are

in

the

propor-

not valid is similar

and

to that

9.1: a new (Yi versus

value

for

cri which

e i curve.

9-13

is in excess

of the

proportional

one

limit

b

°

For the cri value factor

assumed

in a, above,

compute

the plasticity reduction

(9.2-5)

who re

co

Using

Et

Tangent

Ef

Elastic

the

7?, value

_.1 This by _.

do

If the with

is

new the

factor

value

rl is

If the with not

vMue the

valid.

This attained

from

for

Then of the

One

itcratlve between

simply

procedure the

critical

in a,

then

above,

must

the be and

is

related

computed

until

9 -14

agreement

is by

in reasonable

cycle acceptable or. values.

this

obtained

by

rl value.

agreement

reduction

1

a i value

reduction

stresses

plasticity

computation

intensity

reasonable plasticity

of

not

stress

first-estimate

components

continued the

in

related

combination

the

repeat

assumed

is

psi.

critical the

stress

in c,

assumed

the

above, the

first-estimate

cyi computed

must

c,

above,

psi.

e i curve,

multiplying

in

in a, the

e i curve,

recalculate

cri computed

assumed

cri versus

gi versus

above, by

each

of

the

of the

b,

valid.

cri value

of

modulus

accomplished

c_i value

multiplying

modulus

starting

factor with

agreement

is a. is

REFERENCES

9-1

Gerard,

G.,

Sciences, 9-2

9-3

"Plastic

April

Gerard,

G.

of Flat

Plates,"

Gerard,

G.

Gerard,

and

Yield 9-5

9-6

Baker,

E.

--

NACA

1962.

H.,

Cappelli,

the

and

Technical

Analysis

National

CR-!45_

Texas,

Note

3781,

Journal

NACA

of the Aeronautical

A.

P.,

Manual," and June

Kovalevsky, Prepared Space

August

Stability

L.,

Administration,

9-15

I - Buckling

Stability,

Part

III - Buckl-

3783,

August

of Thin-Wall

1957.

Cylinders

in

1956.

Theory,

by North

1966.

Part

Note

Buelding

3726,

Stability, 1957.

Technical

Torsional Note

July

of Structural

to Strueturai

Aeronautics

Itouston,

32

Shells,"

of Structural

"Handbook

and Shells,"

Inc.,

for

1969

H.,

Introduction

M., "Shell

of Thin

"Handbook

Technical

"Compressive

R.

Center,

NASA-I.aIt!:I"v,

PIates

G.,

Company,

tt.,

and Becket,

Region,"

Gerard,

Becker, NACA

G.,

Theory

1957.

ing of Curved 9-4

Stability

Rish,

McGraw-Hill

F.

L.,

Ameriean Manned

and

Book

Verette,

Roe_vell, Spaceeraft

Corp.

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