BEGGS DEFORMETER

A WATER

RESOURCES TECHNICAL

PUBLICATION

ENGINEERING MONOGRAPH

No.1 4

Beggs Deformeter Stress Analysis of Single-Barrel Conduits UNITED STATES DEPARTMENT OF THE INTERIOR BUREAU OF RECLAMATION

Mission

of the Bureau

of Reclamation

The Bureau of Reclamation of the U.S. Department responsible for the development and conservation water resources in the Western United States.

of the Interior is of the Nation’s

The Bureau’s original purpose “*to provide for the reclamation of arid and semiarid lands in the West” today covers a wide range of interrelated functions. These include providing municipaland industrial water supplies; hydroelectric power generation; irrigation water for agriculture; water quality improvement; flood control; river navigation; river regulation and control; fish and wildlife enhancement; outdoor recreation; and research on water-related design, construe tion, materials, atmospheric management, and wind and solar power. Bureau programs most frequently are the result of close cooperation with the U.S. Congress, other Federal agencies, States, local governmen ts, academic institutions, water-user organizations, and other concerned groups.

A WATER

RESOURCES

Engineering

Monograph

TECHNICAL

No.

PUBLICATION

14

Beaus Deformeter Stress Analvsis vu

I

of Single-Barrel Conduits By H. B. Phillips and I. E. Allen Experimental Design Analysis Section, Technical Engineering Analysis Branch, Office of Chief Engineer, Denver, Colorado

United

States

Department

OF the Interior

BUREAU OF RECLAMATION

As the Nation’s principal conservation agency, the Department of the Interior has responsibility for most of our nationally owned public lands and natural resources. This includes fostering the wisest use of our land and water resources, protecting our fish and wildlife, preserving the environmental and cultural values of our national parks and historical places, and providing for the enjoyment of life through outdoor recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Department also has a major responsibility for American Indian reservation communities and for people who live in Island Territories under U.S. Administration.

ENGINEERING MONOGRAPHS are prepared and used by the technical staff of the Bureau of Reclamation. In the interest of dissemination of research experience and knowledge, they a,re made available to other interested technical circles in Government and private agencies and to the general public by sale through the Superintendent of Documents, Government Printing Office, Washington, D.C.

First Printing: 1952 First Revised Edition: 1965 Second Revised Edition: 1968 Reprinted: 1986

U.S. GOVERNMENT WASHINGTON

PRINTING

OFFICE

: 1968

For saleby the Superintendentof Documents, U.S. Government Printing Office, Washington, D.C. 20402, or the Bureauof Reclamation,Attention 822A, DenverFederalCenter, Denver, Colorado 80225.

CONTENTS

PW

INTRODUCTION.

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

APPLICATION................................................ DETERMINATION APPENDIX:

OF NORMAL

1 3

STRESS DISTRIBUTION.

THE BEGGS DEFORMETER

..

5

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

57

FIGURES No.

1. Dimensions of conduits and location of points studied. Shapes A,

2. 3. 4.

5. 6.

7. 8. 9. 10. 11.

12. 13.

14. 15. 16.

B,andC............................................,... Dimensions of conduits and location of points studied. Shapes D, E,andF............................................... Dimensions of conduits and location of points studied. Shapes circular, square, and G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients for moment, thrust, and shear for uniform vertical load and uniform foundation reaction. Shapes A, B, and C . . . . . . . . Coefficients for moment, thrust, and shear for uniform vertical load and triangular foundation reaction. Shapes A, B, and C . . . . . . Coefficients for moment, thrust, and shear for concentrated vertical load and uniform foundation reaction. Shapes A, B, and C. . . . Coefficiepts for moment, thrust, and shear for concentrated vertical load and triangular foundation reaction. Shapes A, B, and C . . Coefficients for moment, thrust, and shear for triangular vertical load and uniform foundation reaction. Shapes A, B, and C. . . . Coefficients for moment, thrust, and shear for triangular vertical load and triangular foundation reaction. Shapes A, B, and C . . Coefficients for moment, thrust, and shear for vertical arch load and uniform foundation reaction. Shapes A, B, and C . . . . . . . . . . . Coefficients for moment, thrust, and shear for VerticaLarch load and triangular foundation reaction. Shapes A, B, and C . ; . . . . . . . . Coefficients for moment, thrust, and shear for dead weight of conduit.ShapesA,B, andC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients for moment, thrust, and shear for uniform horizontal load on bothsides. Shapes’A, B, and C.. .*. . . . . . . . . . . . . . . . . . Coafticients for moment, thrust, and shear for triangular horizontal loadonbothsides.ShapesA,B,andC . . . . . . . . . . . . . . . . . . .. . . CJoefficients for moment, thrust, and shear for uniform internal radialload.ShapesA,B, andC.. . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients for moment, thrust, and shear for triangular internal radial load and uniform foundation reaction. Shapes A, B, and c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . * . . * . ., . . . .

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

22 ... ill

NO.

17. Coefficients for moment, thrust, and shear for triangular internal radial load and triangular foundation reaction. Shapes A, B, andc~............................,..................... 18. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load. Shapes A, B, and C. . . . . 19. Coefficients for moment, thrust, and shear for uniform vertical load and uniform foundation reaction. Shapes D, E, and F. . , . . . . . 20. Coefficients for moment, thrust, and shear for uniform vertical load and triangular foundation reaction. Shapes D, E, and F. . . . . . 21. Coefficients for moment, thrust, and shear for concentrated vertical load and uniform foundation reaction. Shapes D, E, and F. . . . . . 22. Coefhcients for moment, thrust, and shear for concentrated vertical load and triangular foundation reaction. Shapes D, E, and F. . . . 23. Coefficients for moment, thrust, and shear for triangular vertical load and uniform foundation reaction. Shapes D, E, and F. . . . . . . . . . 24. Coefficients for moment, thrust, and shear for triangular vertical load and triangular foundation reaction. Shapes D, E, and F . . . . 25. Coefficients for moment, thrust, and shear for vertical arch load and uniform foundation reaction. Shapes D, E, and F. . . . . . . . 26. Coefficients for moment, thrust, and shear for vertical arch load and triangular foundation reaction. Shapes D, E, and F. . . . . . 27. Coefficients for moment, thrust, and shear for dead weight of conduit. Shapes D, E, and F. . . . . , . . . . . . . . . . . . . . . . . . . . . . . . 28. Coefficients for moment, thrust, and shear for uniform horizontal load on both sides. Shapes D, E, and F. . . . . . . . . . . . . . . . . . 29. Coefficients for moment, thrust, and shear for triangular horixontal loadonbothsides.ShapesD,E,andF.. . . .................. 30. Coefficients for moment, thrust, and shear for uniform internal radial load, Shapes D, E, and F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Coefficients for moment, thrust, and shear for triangular internal radial load and uniform foundation reaction. Shapes D, E, and F...................................................... 32. Coefficients for moment, thrust, and shear for triangular internal radial load and triangular foundation reaction. Shapes D, E, and F....,................................................. 33. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load. Shapes D, E, and F . . . . . . . 34. Coefficients for moment, thrust, and shear for uniform vertical load and uniform foundation reaction. Shapes circular, square, andG.................................................. 35. Coefficients for moment, thrust, and shear for uniform vertical load and triangular foundation reaction. Shapes circular, square, andG.................................................. 36. Coefficients for moment, thrust, and shear for concentrated vertical load and uniform foundation reaction. Shapes circular, square, and G.................................................. 37. Coefficients for moment, thrust, and shear for concentrated vertical load and triangular foundation reaction. Shapes circular, square, andG.................................................. iv

23 24 25 26 27 28 29 30 31 32 33 34 35 36

37

38 39

40

41

42

43

NO.

38. Coefficients for moment, thrust, and shear for triangular vertical load and uniform foundation reaction. Shapes circular, square, andG.................................................. 39. Coefficients for moment, thrust, and shear for triangular vertical load and triangular foundation reaction. Shapes circular, square, and G...................................................... 40. Coefficients for moment, thrust, and shear for vertical arch load and uniform foundation reaction. Shapes circular, square, and G . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. Coefficients for moment, thrust, and shear for vertical arch load and triangular foundation reaction. Shapes circular, square, andG.................................................. 42. Coefficients for moment, thrust, and shear for dead weight of conduit. Shapes circular, square, and G.. . . . . . . . . . . . . . . . . . . . . 43. Coefficients for moment, thrust, and shear for uniform horizontal load on both sides. Shapes circular, square, and G. . . . . . . . . . . . . 44. Coefficients for moment, thrust, and shear for triangular horizontal load on both sides. Shapes circular, square, and G. . . . . . . . . . . . . 45. Coefficients for moment, thrust, and shear for uniform internal radial load. Shapes circular, square, and G. . . . . . . . . . . . . . . . . . . . . . . . . 46. Coefficients for moment, thrust, and shear for triangular internal radial load and uniform foundation reaction. Shapes circular, squ~e,andG............................................ 47. Coefficients for moment, thrust, and shear for triangular internal radial load and triangular foundation reaction. Shapes circular, square, and G.......................................... 48. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load. Shapes circular, square, andG.................................................. 49. Coefficients for moment, thrust, and shear for triangular external hydrostatic load including dead load with conduits assumed to float.AUshapes......................................... 50. Coefficients for moment, thrust, and shear for horizontal passive pressure. Circular shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. Beggs Deformeter apparatus and shape B conduit model. . . . . . . . .

44

45

46

47 48 49 50 51

52

53

54

55 56 58

TABLE NO.

1. Correction factors for difTerent radii of curvature.

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

5

INTRODUCTION

This monograph presents the results of the stress analysis, by means of the Beggs Deformeter apparatus,’ of nine shapes of single-barrel conduits. A partial analytical check was made using the least work method to determine the redundant reactions for all shapes due to a uniform vertical load and a uniform horizontal load. All personnel of the Experimental Design Analysis Section, including several rotation engineers who had training assignments in the section, assisted in the experimental work and computations. In particular, the assistance of W. T. Moody in computing the analytical solutions, and the work of H. E. Willmann, who prepared the drawings and also assisted in the experimental work and computations, is gratefully acknowledged. The nine shapes of conduits studied are those most widely used in Bureau of Reclamation structures. All except shape D and the square shape have semicircular top portions of uniform thickness. They can be further described as follows : 1. Shape A: horseshoe-shaped interior with a horizontal exterior base. 2. Shape B: circular-shaped interior with a horizontal exterior base. 3. Shape C: circular-shaped interior with a curved exterior base. 4. Shape D: circular-shaped interior with a square-shaped exterior. 5. Shape E: uniform thickness with a horizontal base. 6. Shape F: uniform thickness of horseshoe shape. 7. Shape G: transition between shape B and shape E with fillets of W r radius in lower interior corners. 8. Circular shape of uniform thickness. 9. Square shape of uniform thickness. Reaction coefficients for bending moment, thrust, and shear at selected locations along the centroidal axis of the conduits have been determined for 15 different loading conditions. 1 See Appendix

for description

of this instrument.

The 15 loading follows : 1. &I33

conditions

top with

considered

f3YJ3 foundation.

2. I1113 top with ‘m

foundation.

3.

1

top with

tflf3

foundation.

4.

I

top with

P-U

foundation.

5. hEr.

top with

Efm

foundation.

6. Q

top with

P-U

foundation.

7. P-Y

top with

Efa3

foundation.

8. 1/

top with

P-U

foundation.

9. Dead load with

PXI

foundation.

10. Uniform 11. B

horizontal

horizontal

12. Uniform

are as

both sides.

both sides.

internal

radial.

13. b

internal radial with

t3333 foundation.

14. b

internal

YV

15. R load.

external

radial with hydrostatic

foundation.

including

dead

Figures 1, 2, and 3 show cross sections of each shape, giving the dimensions and the location of points at which the reaction coefficients have been determined. Each shape was analyzed for three values of crown thickness, t, expressed in terms of the internal crown radius, r. These three values were t==rJ2, t=r/3, and t=r/6. A conduit of unit length was considered in the analysis. Bending moment, thrust, and shear coefficients were determined at the various locations shown, and are expressed in terms of unit intensity of loading and unit internal crown radius. Multiplying the reaction coefficient by the proper load factor gives the total bending moment, thrust, or shear at the centroid of the section under consideration.

1

APPLICATION

The reaction coefficients determined in the study are tabulated in figures 4 through 50 for the various shapes and loading conditions. The reaction coefficients are given for points on the right side of the conduits only, since the conduits and loadings are symmetrical about the vertical centerline. The shear reactions on the left side of the vertical centerline will have an opposite sign from those given for the points on the right side. Consistent units should be employed when using these data. Thus, if loads are expressed in pounds per square inch, all dimensions of the conduit must be expressed in inches. The bending moment will then be in inch-pounds per inch of conduit length and the thrust and shear in pounds per inch of conduit length. If the load is expressed in terms of pounds per square foot, the dimensions of the conduit must be expressed in feet, and the bending moment will be in foot-pounds per foot of conduit length and the thrust and shear in pounds per foot of conduit length. It will be noted that the bending moment in inch-pounds per inch is numerically equal to the bending moment in foot-pounds per foot. One should bear in mind that this analysis assumes no restraint to the deformation of the conduit. In some cases this restraint, or passive pressure, may be important. Some work on passive pressures on tunnel linings through rock has been done by R. S. Sandhu.2 By using his method for determining the intensity of the passive pressure, and using the moment, thrust, and shear coefficients 2 Sandhu, R. S., “Design of Concrete Linings for Large Underground Conduits,” Journal of the American Concrete Institute, December 1961, pp. 737-750.

for a circular conduit given by figure 50, the effect of restraint may be approximated. The foundation load distribution due to a vertical load on the conduit must be assumed, and is influenced by the modulus of elasticity of the foundation material. As the foundation modulus increases, the foundation load distribution approaches a concentration at the outside corners of the conduit, and as it decreases the load approaches a uniform distribution. For all vertical loading conditions except three, two distributions were assumed, viz., uniform, and triangular with zero at the center and maximum at the outside corners. For the dead load the assumed foundation reaction is minimum at the center varying linearly to a maximum at the outside corners, with the intensity at the center equal to the intensity of the weight of the conduit at the center of the base. For the triangular internal radial load the assumed foundation reactions were uniform, and triangular with zero at the outsides and maximum at the center. For the triangular external hydrostatic load; including dead load, the unit weight of the conduit material and the unit weight of water were assumed to be 150 and 62.4 pounds per cubic foot, respectively. With these assumptions the weight of the conduit for the t=r/6 case, except shape D, is less than the uplift, causing the conduit to float. The reaction is assumed to be uniformly distributed across the top. The coefficients for this assumption (conduit floating) are given in figure 49. In the other figures of this loading condition, tension is assumed to develop uniformly along the foundation.

8

DETERMINATION OF NORMAL STRESS DISTRIBUTION

In a curved beam the neutral axis will not be coincident with the centroidal axis, and the normal stress distribution on radial lines, due to moment, will not be linear. However, the radius to the neutral axis and the normal stress distribution may be determined by the following equations, derived from the Winkler-Bach theory for curved beams: 3

T y,, e

is the thrust at the centroidal axis is the distance from the neutral axis to the point of interest (positive outward) is the distance from the centroidal axis to the neutral axis.

As t decreases e approaches zero, and the ug distribution approaches linearity. a#, as computed by equation (2), is only for a constant thickness section. Where the section thickness is not constant, the distribution of stresses must be determined by some other method, such as photoelasticity. The extreme fiber stress in a constant thickness curved beam due to bending moment may be determined by the equation: Mt CT)=Krr

t rn=Gqqq

(1)

where is T is r, is t is In is r,

the the the the the

radius to the neutral axis internal radius external radius wall thickness (rO-r> log to the base e, T ue=-+ t

MY, (r,+y&

where (Tb is M is t is I is K is

the extreme fiber Stress the bending moment at the section the width of the section the moment of inertia of the section the factor by which the extreme fiber stress, assuming linear distribution, is modified to correct for curvature. The following equation for K was obtained by equating equations (2) and (3) : (4) The values of K and e for the t/r ratios used in this study are tabulated below:

(2)

TABLE l.-Correction

factors for different

where a0

M

(3)

radii

of curvature

K is the normal stress in the tangential direction is the bending moment at the centroidal axis

3 Murphy, McGraw-Hill 219.

Glenn, Advanced Mechanics Book Co., Inc., New York,

of Materials, 1946, pp. 217-

e

t Inside fiber 42 r/3 r/6

1.153 1.105 1. 054

o;EP 0.880 0.912 0.951

. 0. 0168r 0.008Or 0. 0021r 6

03lOlllS

SlNlOd

JO

3 ONW ‘8

‘V

NOllWOl

ONV

S3dVHS

SllfMN03

SISAlVNV SS3US U313WYOd30 llnQN03 13clWle 319NIS

A0

SNOISN3WlCl

SO038

OfbE. 66Of JlLZf

'I ‘0

JbSOZ’O

LZLS'Z

ILIZ’b

JSS6b‘O JZbLf

‘0

iad

I

OIJV

Jf699’0

01 ‘6

JblbS’O

II

‘0 I

3Nl-l JOLL61’2

90

HION

JfbZSf.2

3

IlNlOd JCZCOS’I

10

HI

8

3dVHS

Z

3dWS

+nOqO

3NI-l

(O~!J(blUUI~S

I

1 OOEO’E

1 OLIL’b

I 1

(zJ)oW’

1

Symmetrical about vertical centerline-m

A ---------SHAPE

D --_--_-_ Lea

D

SHAPE

I

+j+AJ+&

“Centraidal

about

LA-+SHAPE

It=flt=5lt=fl 0.33333r

0.33333~

0.33333 r

D

I. 50000r

I.33333r

I. l6667r

4.4635

2.7773

I. Z895

7

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS DIMENSIONS

OF

CONDUITS

AND

SHAPES

L,_

oxis

E

A

Area(#) For length of lines for Points 6 thru ond 9 thru 14, se6 Sha’pc 8.

Svmmctrical

I+----..._

LOCATION

0, E, AND

OF F

POINTS

STUDIED

F

’

Symmetrical obout vertical centerline--za

SQUARE

CIRCULAR

t=+ Arto(r*)

3.9t70

t-i

t=+

2.4433

1.1345

Arco(r*)

BEGGS DIMENSIONS

OF

SHAPE

t-f

t=f

t=f

5.0000

3.llII

1.4444

SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS CONDUITS

SHAPES

AND

CIRCULAR,

LOCATION SOUARE,

OF AN!

POINTS G

STUDIED

-%.

A+-----

G

FIGURE

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS +

Sign

convantion

UNIFORM

VERTICAL.

FOR MOMENT, LOADSHAPES

THRUST,

UNIFORM A,

8,

AND

FOUNDATION

AND

SHEAR REACTION

C

X- PEL-372

4

FlGUPE

a L 9 m

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM

VERTICAL

FOR MOMENT,

THRUST,

LOAD-TRIANGULAR SHAPES

A,

AND

FOUNDATION B,

AND

SHEAR REACTION

C

X-PEL-373

5

FIGURE

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATE6i&CAL

FOR MOMENT, LOAD SHAPES

THRUST,

-UNIFORM A,

6,

AND

FOUNDATION AND

SHEAR REACTION

C

X-PEL-374

6

~~~, to.536

to.716

t1.361

to.444

-to.632

tl.21

I

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED

FOR MOMENT,

VERTICAL

SHAPES

5EP

8.lDlO

THRUST,

LOAD -TRIANOVLAR A,

6,

AND

AND

FOUNDATION

SHEAR REACTION

C

X-PEL

- 375

BEGGS

SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS

COEFFICIENTS TRIANGULAR

FOR MOMENT,

VERTICAL SHAPES

LOAD

A,

THRUST,

AN0

UNIFORM

FOUNDATION

8,

C

AND

SHEAR REACTION

X-PEL-

14

1034

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANSULAR

FOR MOMENT,

VERTICAL

LOAD - TRIANGULAR

SHAPES

SEP.

20,

Is*4

THRUST,

A,

8,

AND

AND

FOUNDATION

SHEAR REACTION

C

X-PEL-IO35

15

NOTE’g represents the vea*ht per unit volume of soil cover the conduit sect,on with those of the

on the arch of I” un,+s consistent rod,us r

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS

:03

COEFFICIENTS

U-F

VERTICAL

ARCH


SEP.

16

28.

1964

-

UNIFORM A,

B,

THRUST, FOUNDATION AND

AND

SHEAR

REACTION

C

X-PEL-1036

FIQURE

t‘.-a(r+t) ..i !10 Ia NOTE: g represents the weight per unit volume of soil cover on the arch of the conduit sectton in units consjstent with those of the radius r

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS VERTICAL

ARCH

FOR MOMENT,

THRUST,

LOAD-TRIANGULAR SHAPES

FOUNDATION A,

0,

AND

C

AND

SHEAR REACTION

I I

NOTES c represents the weight per unit volume concrete or other motertal m units consistent wth Ihose of the radius r. See Figure I for net area of shapes

of

BEGGS


COEFFICIENTS

FOR MOMENT, DEAD

+ Sign

convmtion

THRUST,

AND

SHEAR

WEIGNT OF CONDUIT

SHAPES

A, B, AND C

X-PEL-1037

18

FIGURE

SINGLE BARREL CONDUIT BEGGS OEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM

FOR MOMENT,

THRUST,

HORIZONTAL

LOAD

SHAPES

8,

A,

-

AND

BOTH C

AND SIDES

SHEAR

13

t - r6

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR

FOR MOMENT, HORIZONTAL SHAPES

THRUST, LOAD

A,

E,

AND

-

BOTH

AND

SHEAR

SIDES

C

X-PEL-379

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS

FOR MOMENT,

UNIFORM

THRUST,

INTERNAL

SHAPES

A,

RADIAL 8,

AN0

AND

SHEAR

LOAD C

21

Wessun vertical

Wessure vertical

ditiribution along C of conduit -:

dlstnbution along e ot conduit. --,

NOTE: w represents the wright par unit volume of woier in units consistent with those of the radius P.

SINGLE BARREL CbNDUlT BWGS DEFORMETER STRESS ANALYSIS + Sign

convention

COEFFICIENTS TRIANGULAR

INTERNAL

FOR MOMENT, RADIAL SHAPES

LOAD

THRUST, - UNIFORM

A, 8, AND C

AND

SHEAR

FOUNDATION

REACTION

POINT

Pressure vwticol

N

T

s

H

7

T

s

M

T

S

F

distribution obrq C of conduit-..

NOTE : I represents the weight par volume of water In units consiat*nt with those of the radius r.

unit

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS + Sqn

convention

TRIANGULAR

INTERNAL

FOR MOMENT, RADIAL

LOAD

SHAPES

THRUST,

- TRIANGULAR A, 0,

AND

AND

SHEAR

FOUNDATION

REACTION

C

23

m

Deod one-half

Dead

weight of of conduit

WeiQht

of

0

NOTES: Y rWWS~“tr the WlQht per ““I+ volume of water in units consistent with those of the radws r The assumed WeiQht per unit volume of the conduit I?) IsOw/62.4.

l

Tenston is assumption

assumed that

to develop the conduits

at

the foundotion. For the float see Figure 49.


FOR MOMENT,

THRUST,

TRIANGULAR EXTERNAL HYDROSTATIC INCLUDINQ DEAD LOAD SHAPES

AND

SHEAR

LOAD

A, 0, AND C

x-

24

PEL-

1039 -.I

FISURE

I

19

.-1

ftttttfftttftttt?f?ful--I”

BEGGS


COEFFICIENTS UNIFORM

VERTICAL


SEP

ea.

1964

-

THRUST,

UNIFORM D,

E,

AND

SHEAR

FOUNDATION AND

REACTION

F

x-

PEL-

1040

25

BEGGS



VERTICAL

FOR MOMENT, LOAD

-

SHAPES

THRUST,

AND

TRIANGULAR

FOUNDATION

D,

F

E,

AND

SHEAR REACTION

x- PEL-

26

104 I

FIGURE

el

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS CONCENTRATED +

Sign

FOR MOMENT,

VERTICAL

LOAD

-

THRUST,

UNIFORM

AND

FOUNDATION

SHEAR REACTION

convention

SHAPES

D,

E,

AND

F

X-PEL-1042

27

FIQURE

t=j

I +t)

22

t = 5

2vcr

BEGGS


COEFFICIENTS CONCENTRATED +

Sign

VERTICAL

FOR MOMENT, LOAD

conventlo” SHAPES

-

THRUST,

TRIANGULAR D,

E,

AND

AND

FOUNDATION

SHEAR REACTION

F

X-PEL-

1043

FIGURE

BEGGS


COEFFICIENTS TRIANGULAR +

Sign

convention

23

FOR MOMENT,

VERTICAL

LOAD SHAPES

D,

THRUST,

AND

UNIFORM

FOUNDATION

E,

F

AND

SHEAR REACTION

X-PEL-1044

29


FOR MOMENT,

VERTICAL

THRUST,

LOAD - TRIANGULAR

SHAPES

D.

E,

AND

AN0

SHEAR

FOUNDATION

REACTION

F

X-PEL-1045

30

Note:

No

vertical

POINT

1101E: g rmprwentr the weight per unit VO~U~C of soil cover on the arch of the conduit section in units consister with those of the radius r.

7

arch

Y

lood

on

T

S

Shape

D.

M

T

S nrl

-a

H

T

s

11 SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS FOR MOMENT, THRUST, AND SHEAR VERTICAL

+

Sign

convention

ARCH

LOAD

- UNIFORM

SHAPES

0.

E,

FOUNDATION AN0

REACTION

F

s

X-PEL-IO46

31

FIGURE

Note:

No

vertlcol

arch

load

on

Shape

0.

NOTE: g represents the weight per unit volume of soil cover on the arch of the conduit section I” “nits c:onsis+en, with +hose of the rod,“* r

BEGGS iD

COEFFICIENTS

i

VERTICAL

c I

ARCII

FOR MOMENT, LOAD-TRIANGULAR SHAPES

+

Sign

SEP. es. ,964

32

SINGLE BARREL CONDUIT DEFORMETER STRESS ANALYSIS THRUST, FOUNDATION D,

E,

AND

AND

SHEAR REACTION

F

convention

X-PEL-1047

26

NOTES’ c represents the ueiqht per unit volume concrete or other moterod in units consistent wth those of the radius r. See Figure 2 for net orea ot shows.

of

BEGGS


COEFFICIENTS +

SEP.

2%

Slqn

1964

convantion

FOR MOMENT,

THRUST,

DEAD

OF

CONDUIT

E,

AND

WEIGHT

SHAPES

D,

AND

SHEAR

F

X -PEL-1048

33

-y

h “-

I

BEGGS



FOR MOMENT, HORIZONTAL SHAPES

THRUST,

LOAD D,

E,

AND

BOTH

AND

SHEAR

SIDES

F

X-PEL-

34

1049


FOR MOMENT, IfORltONTAL SHAPES

THRUST, LOAD

0, E,

AND

-

BOTH

AND

SHEAR

SIOES

F

X-PEL-

1010

35

FIGURE

BEGGS


COEFFICIENTS

FOR MOMENT,

UNIFORM t

Sign

INTERNAL

THRUST, RADIAL

AND

SHEAR

LOAD

Convention SHAPES

D,

E,

AND

F

X-PEL-

36

30

1051

FIGURE

Pressure vertical

Pressure vertical

31

distribution along k of conduit-.:

dirtributlon along C of conduit--%.

NOTE: u represents the weight par unit volumr of voter in units consistent with those of the radius r.

BEGGS


COEFFICIENTS TRIANGULAR +

Sign

convention

INTERNAL

FOR MOMENT, RADIAL SHAPES

LOAD D,

THRUST, -

E.

UNIFORM AND

AND

SHEAR

FOUNDATION

REACTION

F

X-PEL-IO52

37

FIGURE

t = i Pressure vertical

Pressure vertical

Pressure VWtiCOI

t = f

t=f

distribution olonp t of oonduit..

dishlbution olonp C of Conduit-.,

distribution along C of conduit-.,

NOTE: Y represents the weight per unit wotunw of rater in units consistent with thow of the radius r.

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANOULAR

INmNAL

R)R MOMENT, RADIAL

THRUST,

LOAD - TRIANGULAR

SHAPES

D, E, AND

AND

SHEAR

FOUNDATION

REACTION

F

x-PEL-IO63

38

39

FIQIJRE

Dead on5-half

33

Y ‘ei ht of o P conduit

t-f, t-i, t-i.

D5ad one-halt

5.SeIwr*

t-:* .t

wight af of mndult

-f* t = f.

D5od one-hall

5.55.

wr’

I.55OW’

r5Wt of of conduit

w;155: w reprewnb the wlght p.r unit VOhmw of ratw In unite con5lrtent with tho55 of th. rodiu5 r. Th5 ouumod roight p51 unit voIum5 of t)u conduit I5 l5Ow/55.4.

l

Tension is assumed to olsumption that the

develop

conduits

at the foundation. For the float see Figure 49.

SINGLE ARREL CONDUIT BEGGS DEFORM1 TER STRESS ANALYSIS COEFFICIENTS

FOR MOMENT,

THRUST,

TRIANGULAR EXTERNAL HYDROSTATIC INCLUDINQ DEAD LOAD SHAPES

AND

SHEAR

LOAD

D, E, AND F

39

utttttrtltrltttttttt11 --fL SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS CQEFFICIENTS UNIFORM +

Sign

VERTICAL

FOR MOMENT, LOAD

-

THRUST,

UNIFORM

AND

FOUNDATION

SHEAR REACTION

convention SHAPES

CIRCULAR,

SGUARE,

AND

G

X-PEL-I055

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM

VERTICAL SHAPES

FOR MOMENT, LOAD

-

CIRCULAR,

THRUST,

TRIANGULAR SQUARE,

AN0

SHEAR

FOUNDATION AND

REACTION G

X-PEL-1056

41

FIGURE


FOR MOMENT,

VERTICAL SHAPES

LOAD CIRCULAR,

-

THRUST,

UNIFORM SPUARE,

AND

FOUNDATION AND

SHEAR REACTION

0

X-PEL-

42

1057

36

: a

--r----

7 -

___

2

f LlLzziJ

(I w a. a is

I

i

(( _ ,

, -‘-

t

i ‘I:

-


FOR MOMENT,

THRUST,

TRIANOULAR

VERTICAL

LOAD -

SHAPES

CIRCULAR,

SQUARE,

AND

SHEAR

FOUNDATION AND

REACTION

0

X-PLL-

1058

43

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR +

Sign

VERTICAL

ee,

LOAD

-

THRUST,

UNIFORM

AND

SHEAR

FOUNDATION

REACTION

convention SHAPES

SEP.

FOR MOMENT,

1964

CIRCULAR,

SQUARE,

AND

0

X-PEL-IO59

FIGURE

39


VERTICAL SHAPES

FOR MOMENT, LOAD

-

THRUST,

TRIANGULAR

CIRCULAR,

SQUARE,

AND

SHEAR

FOUNDATION AND

REACTION

G

X-PEL-1060

45

FIQURE

t-f

t-L

t-S-

3

I

6

utttltffttflnmltms.: o.*l446(r

l t1..-a’

Note:

No

vertical

arch

1006

on

square

shopc.

NOTE: g represents ttm wight par unit volume of wil cover on the arch of the conduit section in units conrirteni with those of the rodiur r.


SHAPES

SEP.

46

Le.

1964

FOR MOMENT,

THRUST,

ARCH LOAQ - UNIFORM CIRCULAR,

AN0

FOUNDATION

SQUARE.

AND

SHEAR

REACTION 6

X-PEL-IO61

40

Note :

No

vertical

arch

load

on

square

shape.

-f 0 1(r+t)

i

nom:

g represents the n@ht PM unit volum. of ~11 COYW on the arch of +he conduit aaction in units consistent with those of the radius

r


ARCH SHAPES

FOR MOMENT, LOAD

THRUST,

- TRIANDULAR

CIRCULAR,

AND

FOUNDATION

SQUARE,

AND

SHEAR REACTION

B

X-PEL-106R

47

FIQURE

NOTES: c represents VOlumC of concrete in units consirtant radius r. See FIgwe 3 for

the weight per unit or other motwial with those ot the net

or.x

of

sh.,,xs.


FOR MOMENT, DEAD WElSHf

+

SEP. PI.

4-a

Sign

I,d.

conwntion

SHAPES

CIRCULAR,

THRUST,

AND

SHEAR

OF CONDUIT SQUARE,

AND

0

X-PEL-1063

42

-1

h p-

-04

h b-

*

-1

h p-

-4

h )r-

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS UNIFORM SHAPES

FOR MOMENT, HORIZONTAL CIRCULAR,

THRUST,

LOAO

-

SQUARE,

AN0

BOTH

SHEAR

SIDES

AND

0

x-

PEL-

IO64

49

FIQURE


50

FOR MOMENT,

TRIANQULAR

HORIZONTAL

SHAPES

CIRCULAR,

THRUST, LOAD

-

SQUARE,

AND

BOTH AND

SHEAR

SIDES 0

44

FIQURL -- 45

I

I)

D

0

, -

1

0

I-t

,.. I-I

SINGLE BARREL CONDUIT BEGGS DEFDRMETER STRESS ANALYSIS COEFFICIENTS

FOR MOMENT,

UNIFORM +

Sign

INTERNAL

THRUST, RADIAL

AND

SHEAR

LOAD

convention

SHAPES

CIRCULAR.

SQUARE,

AN0

Q

X-PLL-

1055

51

Prrrrum vertical

Pressure vertical

distribution along e of conduit-.,

distribution along E of conduit.,

NOTE’ w reweeentr the weight per unit volume of voter in units consistent wlh those of the radius r.

)I+ SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS

I

COEFFlClENfS

m-

TRIANGULAR L q

+

Sqn

convention

INTERNAL SliAPES

FOR MOMENT, RADIAL

LOAD

CIRCULAR,

THRUST, -

UNIFORM SOUARE,

AND

SHEAR

FOUNDATION AND

REACTION

G

> r SEP.

52

er,

1964

X-PEL-1067

FIQURE

Pressure

distribution

Pressure vertical

distribution along C of conduit-.,

Pressure vertlcol

w repmsante volume of rater with those of

NOTE:

47

along

distribution along c of conduit-..

the weight per unit in units conristrnt

the rodtur

r.

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANGULAR + Sinp convmtion

SEP.

er,

1004

INTERNAL SHAPES

FOR MOMENT, RADIAL

LOAD -

CIRCULAR,

THRUST,

TRIANGULAR SQUARE,

AND

SHEAR

FOUNDATION AND

REACTION

G

X - PEL-IO68

53

hod wai ht of one-holf f COIldUll

D*Od wqht of one-half of conduit 6.010 wr’ t-f, t - 5, 5.7.0 wr’ 1.75* wr’ t-f,

MOTES:w r5pr555nts the w5ipht p5r unit volum* of water In units conatmt with those of th5 rodlur r. Ths owumed weight p5r unit voIum5 of th5 conduit I5 150~/6~.4. ,+ D

l

Tension is assumed ta develop at the foundation. Far the assumption that the conduits float rep Figure 49.

SINGLE BARREL CONDUIT BEGGS DEFORMETER STRESS ANALYSIS COEFFICIENTS TRIANWLAR SHAPES

FOR MOMENT,

THRUST,

AND

EXTERNAL HYDROSTATIC INCLUDING DEAD LOAD CIRCULAR,

SOUARE,

AND

SHEAR

LOAD Q X-PEL-1069

54

SHAPE POINT

7

M

A

SHAPE

T wr’

0

SHAPE

C

SHAPE

G

S

-1

SHAPE

SUAPE

E

F

Top rtoction of uniform

SOUARE

CIRCULAR

is ossumed to be intensity, v.

For loading diagram, t sign convention, ond the assumption that the conduits do not floot see Figures 16, 33, ond 48. Note:

Shape

D

does

not

floot


FOR MOMENT,

THRUST,

TAIANQULAR EXTERNAL HYDAOSTAtlO INCLUDINS DEAD LOAD CONDUITS

ASSUMED ALL

em,

SHEAR

LOAD

TO FLOAT

SHAPES t-

,EP.

AN0

i x- PEL-

1944

.

1070

1-1

r -0.218 ~~

II +0.5301

L

2

-0.196

0

1

to.512

-0.137 -0.265

II II

-0.

IB3

1 -0.164 11 -0.109

I to.471 I

II

1 to.455

1 -0.122

1 to.4061

6 -0.236

~l--o.t I, --

50 I z.412 --I -...-I

-0.135

I +0.39q

r- a -

-0.107

ll -0.090

-0.238

8

1+0.2l

9

~t0.103~t0.242~t0.420~i+0.065~t0.215~+0.373

1

IO

I-0.024itO.3751

I

II

I-0.130

I2

-0.196

to.512

I3

I-O.218

I +0.530

Sign

convention

t

I

to.082

l+o.459

to.306

t 0. I79

to.073

to.272

to.149 II- +0.069

H

I

+ 0.357 + 0.398 to.412


FOR MOMENT,

HORIZONTAL

PASSIVE

CIRCULAR

56

THRUST, PRESSURE SHAPE

AN0

SHEAR

1

APPENDIX: THE BEGGS DEFORMETER

This study has been made, using the Beggs Deformeter apparatus 4 6 e (figure 51). The basis of the method is a direct application of Maxwell’s Theorem of Reciprocal Deflections, which states that. for any two points on a structure, the ratio of the displacement at the fist point to the load causing it, applied at the second point, is equal to the ratio of the displacement at the second point to the load causing it, applied at the fist point. Displacements are measured in the load directions. In the general application of this method of stress analysis, an elastic scale model of the structure under consideration is deformed at a cut in the model by use of a special set of gage blocks and plugs. Three sets of plugs are used to apply a rotational, a normal, and a shearing displacement at the gage block. Microscopes equipped with filar eyepieces are used to measure the model deflections at points corresponding to the load points of the actual structure. Deflections are measured in the direction of the prototype loads. No loads are applied to the model. Deflections of the model are read at prototype load points for displacements applied at the gage block. The difference in microscope readings is a measure of the model deflection induced by the change at the gage block from the first position of the plugs to the second position of the plugs. From Maxwell’s Theorem the following equations may be written for the redundant reactions at the cut section:

-.

For a concentrated load

M,=PFn .

M

For a distributed load

MI=:

pe,dl M s

4 Beggs, G. E., “An Accurate Solution of Statically Indeterminate Structures by Paper Models and Special Gages,” Proceedings ACI, vol. XVIII, 1922, pp. 58-78. 5 McCullough, C. B., and Thayer, E. S., Elastic Arch iii&es, John Wiley and Sons, New York, 1931, pp. 2826 ‘Phillips, H. B., and Allen, formeter Theory and Technique,” Denver, Colo., July 1965.

I. E., “The Beggs DeBureau of Reclamation,

SFPF

Sl=$ S

TI==PF T

S

s

pes dl

Tl= $spe, T

dl

where

d, is the angular rotation ds is dT is eM is es is eT is I

is

MI is n p

is is is

$

is

P

T1 is The

applied at the cut by the moment plugs the displacement applied at the cut by the shear plugs the displacement applied at the cut by the thrust plugs the measured deflection at a load point, in the direction of the load, due to d, the measured deflection at a load point, in the direction of the load, due to ds the measured deflection at a load point, in the direction of the load, due to dT the load length the redundant moment reaction at the cut the scale factor (prototype to model) a load acting at a point on the prototype the load intensity on the prototype at the deflection point the redundant shear reaction at the cut the redundant thrust reaction at the cut.

only

unknowns

in

these

equations

are

MI, T,, and S,. In the actual operation of the Beggs Deformeter the arithmetic is simplified by the use of calibration factors based on the plug dimensions and the eyepiece scales. An influence line through points obtained by multiplying the deflection ordinates by the proper calibration factor gives directly the magnitude of the moment, thrust, or shear at the gage block position for a unit traveling load. It should be pointed out that the Beggs Deformeter method automatically takes into account the strain energy in a structure due to moment, thrust, and shear as well as haunch effects and other shape changes.

57

FIGURE

51.

-Beggs

Deformeter

apparatus

and

shape

B conduit

model.

58 GPO

850-512

PARTIAL LIST OF WATER RESQURCES TECHNICAL PUBLICATIONS Engineering

Monographs

No. 31

Ground-Water

32

Stress Analysis

33

Hydraulic

34

Control

35

Effect

36

Guide

37

Hydraulic

Research

Movement of Wye

Branches

Design

of Transitions for Small Canals in Mass Concrete Structures of Cracking

of Snow

Compaction

for Preliminary Model

Studies

Aquatic

Weed

on Snow

Dams

for Morrow

Methods Control

Soils Tests Computer Hydraulic Park

Point

Dam

Effects

Forces

Atmospheric

Report

Synthetic

of Progress

Resin Report

Technical

Report

Buttes

Morrow

for

Wildlife-A

Reservoir

Evaporation

From

Division

Structural

Behavior

Results

Research-l

for

Flaming

967

Construction

Features

of the Central

Valley

Project

Dam

Point

Sale.”

obtained

Lining

on Engineering

and

966

Aquifers

and

of Progress

Research-l

Deposits

Canal

Water

of Design

Enamel

on Engineering

of Analytical

River

Twin

I 965

Dam

in Lake

Membrane

Records

Trinity

Research-I

Fish and

for Coal-Tar

of Progress

of Saline

Comparison Dam Annual

Insects,

of Monticello

Structures

Asphalt

Removal

Program-Phase

on Engineering

on

Primer

Studies

Pile Supported Buried

Gates

Resources

Study

Vibration Annual

Materials

on Large

Water

of Monolayers

Reduction

and

Studies

Programs

Downpull

Range

Annual

15

From Rain

of Arch

Reports

Research-Engineering

8 9 10 11 12 13 14

on Runoff

Design

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and

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It describes upon

request

some from

Foundation

of the

technical

the Bureau

Investigations

publications

of Reclamation,

Attn

currently D-822A,

Gorge

BEGGS DEFORMETER

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