ENGINEERING MONOGRAPHS

United States Department of the Interior BUREAU OF RECLAMATION

STRESSES -AROUND A GALLERY­determined by the

photoelastic interferometer

By H. B. Phillips and C. N. Zangar

Den\'er, Colorado

April 1952

No. 12

20 cents

United States Department of the Interior

OSCAR L. CHAPMAN, Secretary

Bureau of Reclamation

MICHAEL W. STRAUS, Commissioner L. N. McCLELLAN, Chief Engineer

Engineering Monograph

No. 12

STRESSES AROUND A GALI .. ERY-

determined by the photoelastic interferometer

by H. B. Phillips and C. N. Zangar Photoelastic Unit

Dams Branch Design and Construction Division

Technical lnformation Office Denver Federal Center

Denver, Colorado

ENGINEERING MONOGRAPHS are published in limited editions for the technical staff of the Bureau of Reclamation and interested technical circles in government and private agencies. Their purpose is to record developments, inno­vations, and progress in the engineering and scientific techniques and practices tl:lat :are em­ployed in the planning,, design, construction, and operation of Reclamation structures and equip­ment. Copies may 'be obtained from the Bureau of Reclamation, Denver Federal Center, Denver, Colorado, and Washington, D. c.

INTRODUCTION • . . . • •

DETAILS OF THE STUDY. Procedure • Conclusions •

NOTATION ••••

APPLICATION OF THE DATA

CONTENTS

APPENDIX: THE PHOTOELASTIC INTERFEROMETER

BIBLIOGRAPHY • • . • • • . . . • • • • . . • • . • • • •

i

. •. . . ·.

Page

1

1 1 2

3

3

19

21

LIST OF FIGURES

Number

1 Dimensions of gallery and location of reference l:i.nes •

2 Photoelastic interferometer .

3 Model and loading mechanism

4 Isochromatic photograph of uniform load applied parallel to vertical axis of gallery • . . . . . . • . . . . • . . . . . .

5 Comparison of analytical solution with experimental results,

6

7

8

9

10

11

12

13

stress distribution around .a circular opening in an infinite elastic plate • . . . . . . . . . . . . . . . . . . . . . .

Uniform load applied parallel to vertical axis of gallery, principal stress at boundary of gallery . . . . . . . . . .

Uniform load applied parallel to vertical axis of gallery, stress normal to reference lines . . . . . . . . . . . .

Uniform load applied parallel to vertical axis of gallery, stress tangential to reference lines . . . . . . . . . .

Uniform load applied parallel to vertical axis of gallery, shear stress along reference lines . . • . . . • . . • •

Uniform load applied normal to vertical axis of gallery, principal stress at boundary of gallery . . . . . . • . .

Uniform load applied. normal to vertical axis of gallery, stress normal to reference lines . . . . . . . . . . • .

Uniform load applied normal to vertical axis of gallery, stress tangential to reference lines • . . • • • . . • .

Uniform load applied normal to vertical axis of gallery, shear stress along reference lines ·. . • . • . • . • • .

for

14 ·Pure shear load applied to gallery, principal stress at boundary

15

16

17

18

of gallery ...•.••..••.•...••....•.••.•.

Pure shear load applied to gallery, stress normal to reference lines . . . . . . . • • . . . . . . . . . •

Pure shear load applied to gallery, stress tangential to reference lines . . . . • . . . . • . • . . • . . •

Pure shear load applied to gallery, shear stress along reference lines • . . • . • . . • . . . • . •

Diagrammatic lay-out of the interferometer . . . • . .

ii

Page

'iv

1

2

2

3

6

7

8

9

10

11

12

13

14

15

16

17

19

Number

I

II

III

LIST OF TABLES

Tabulation for 00 on lit).es A, D, and H on left side of gallery (in Example) . • • , • . • . . • • . . . . . .

T3bulation of O""t on lines A, D. and H ·on left side of gallery (in E)!:ample) . . . • . • • . . • • . . • . • .

Tabulation of ~ on lines A, D, and H on left side of gallery (in Example) • . • . . • . • • . • • . • • . •

iU

.. . . . . . . . . . 4

4

4

F

D

' ---~---

' ---

c

o' ---~ --' '

A ·B. 8

D

E

G ____ ;_ ___________ _

H H

'

y : ' ' ' '

--~- -:---~- -f -------~ I

N Q; ~ ~

' ' ' J -----,.-------------- __ t_ __ t_ __ i_ __ i_ ________ J

:. ,./: ' : ', i ,/" ~ --------- 2 r -----~---------------- ----~ ', 0 ,

. ~~ / I : 1 ', :;_> "":~. ,.// i---------- r ____ ,... ',

• / I ', / )< .. _ i : __ )\

// .... <lso I : .Ar; . ..- ',, K ------~ .------ K

I I I I L

i M

DIMENSION SCALE

1.6r

I I I

L

FIGURE 1 - Dimensions ot gallery and location ot reference lines.

iv

X-PEL-546

INTRODUCTION

The stresses that exist around galleries and other openings are of great importance in the design of concrete dams. The quan­tity and location of steel reinforcement placed around these openings are dependent upon the magnitude and distribution of the stresses. Several analytical solutions1,1,3,4,5* give the stress distribution around various shaped openings in an infinite elastic medium, but none apply exactly to the gallery shape used in concrete dams. For this reason a photo­elastic interferometerS study was under­taken to determine the stresses around the standard five-by seven-foot inspection gal­lery used in Bureau of Reclamation dams.

Stress coefficients were determined in the study at several points on various lines radiating from the gallery (Figure 1). These coefficients are expressed in terms of unit

loads acting on the external boundaries of the model. Dimensions are expressed in terms of the crown radius of the gallery. These procedures permit the data to be used repeatedly on geometrically similar prob­lems. The designer can determine the stress­es around a gallery in a matter of minutes.

It is difficult, if not impossible, to deter­mine the true effect of placing reinforcement steel in mass concrete. This paper is not concerned with that problem; however, a method is given for determining the approxi­mate quantity of steel reinforcement needed.

No attempt has been made here to evaluate the effects of plastic flow, although boundary stresses determined photoelastically may be somewhat higher than actual stresses because of relief of stresses by plastic flow. 7

DETAILS OF THE STUDY

Procedure

The gallery with its center at point 0 (Figure 1) was considered to be situated in an infinite elastic medium and subjected to a uniform stress field composed of the stress­es initially present at point 0. A coordinate system was selected (Figure 1) with the X-axis horizontal, the Y -axis vertical downward, and the Z-axis horizontal and parallel with the axis of the gallery. Since a slice of ma­terial of unit width normal to the Z-axis was used, the study is one of plane stress.

A gallery-shaped opening 0. 714 inch wide and one inch high was milled in a plate of Allite (CR-39) five inches square and one­fourth inch thick. This model was placed in the interferometer** (Figures 2 and 3) and subjected to a uniform vertical load of in­tensity p. Principal stresses and their di­rections were read at several points on vari­ous lines radiating from the gallery, then resolved into normal, tangential, and shear stresses on the radiating lines as shown in Figures 6, 7, 8, and 9. Next, the gallery model was subjected to a uniform horizon­tal load of intensity p and again stresses were determined on the radiating lines as shown in Figures 10, 11, 12, and 13.

*Superscripts refer to numbers in Bibli­ography.

**See Appendix for description of the in­strument used in this study.

1

The gallery was then subjected to a uni­form load of intensity p acting at 45 degrees with the vertical axis. Principal stresses and their directions were read at the same points on the radiating lines. The stresses existing around the gallery due to pure shear, initially acting at the gallery center, were determined by considering the load acting at 45 degrees on each side of the vertical axis of the gallery, with the applied load con­sidered compressive on one side, tensile on the other. The sum of these two effects gave the pure shear situation. The results of this part of the study are shown in Figures 14, 15, 16, and 17.

FIGURE 2 - Photoelaatic interferometer.

FIGURE 3 - Model and loading mechanism.

As a check on boundary stresses around the gallery opening, the model was placed in the photoelastic polariscope and the isochro­matic stress pattern was photographed. Fig­ure 4 is the stress pattern for the gallery with the load applied parallel to its vertical axis. Checks on the accuracy of the inter­ferometer were also made by checking hori­zontal and vertical equilibrium from the measured stresses. Errors were found to be generally less than five percent and in many cases less than one percent.

The accuracy of the interferometer may also be checked, for problems to which both analytic anq interferometer methods are applicable, by comparison of the two methods. This is shown in Figure 5, where analytic 1, Z and interferometer solutions are plot­ted together for the stresses around a cir­cular opening contained in an infinite elastic medium subjected to a uniform stress field.

Conclusions

Stresses around a gallery resemble some­what the stresses around a circular opening' located in an infinite elastic medium subjected to a uniform stress field. They are, however, different in magnitude and distribution.

When the gallery is subjected to a vertical compressive stress field of intensity p, then (Figure 6):

2

(a) The boundary tensions at the top and bottom of the gallery are equal to -1.00 p.

(b) The boundary compressive stress near the spring line of the top arch is equal to 2.75 p, while the vertical boundary com­pressive stress at the center of the gallery side is equal to 2.18 p.

(c) Corner stresses are approximately equal to 3.30 p.

When the gallery is subjected to a uniform horizontal compressive stress field of in­tensity p then (Figure 10):

(a) The boundary tensions at the center point of the sides are equal to -1.10 p.

(b) The boundary compression at the top of the gallery is equal to 4. 00 p, and the boundary compression at the midpoint of the base of the gallery is equal to 2. 72 p.

(c) Corner stresses are approximately equal to 3.30 p.

When the gallery is subjected to a pure shear of intensity p then (Figure 14):

(a) A maximum boundary tension of -6.12 p occurs at a point midway between the crown and the spring line of the gallery and a tension of -7.00 p occurs at the sharp reentrant corner.

(b) Maximum boundary compressions are the same magnitude as the tensions, and occur at corresponding points on the opposite side of the vertical centerline of the gallery.

FIGUBE 4 - Isochromatic photograph of uniform load applied parallel to vertical axis of gallery.

®

®

DISTANCE OUT FROM BOUNDARY OF OPENING IN TERMS OF t0

0

~ : : (+) : : : L-- _L....- - ~--- .l-- _-l

(+) .,

1 1 I I I L.- -- L ...... - -'-- ---1---_j

o (+) I I L ___ _l ___ .J. ___ .l. ___ ..J

(+) I

1 I I I : I I I 1 ~...- __ _j ___ ...L ___ ,L ___ _J

O"n

_.,- I I I

I I I I ....J-- _L __ _j ___ ..J

' .1- -· _ _.

I t t I t t t t I I t I t p

0 4

~;-::.::;:T---:---; I I I I

o ' ' a ' a ' 0 0

~ ---~ -:--- :- --: 0 I I 0 I 0

0 0

~~ =-=..--:----;--- -; I . I I

0 : : 6 I 6 '

NOTES Dimension Scale in

terms·of r0 0 0.5 I 2

I I I ,I 0 0.5 1.5

Stress Scole in terms of 1'

·(~) Compression

(-) Tension

Reference lines ore at 15° increments

--Analytical

Experimental

X-PEL-559

FIGURE 5 - Comparison of analytical so~ution with experimen­tal results 1 for stress distribution around a circular open­ing in an infinite ehstic phte.

3

NOTATION

p magnitude of stress field 0: = horizontal stress ini-initially acting at the cen-

X tially acting at the gal-.

ter of the gallery in pounds lery center per square inch

0:.= vertical stress initially A, B ... M designation of lines ra-

y acting at the gallery

diating from the gallery center face along which have

Txy= been determined normal, shear stress initially tangential, and shear acting at the gallery stresses. center

O"n = stress normal to desig-nated lines

(J1 = maximum principal stress

O"z= minimum principal stress O"t = stress tangential to des-

ignated lines CJ. = direction of principal stress

Tnt= shear stress along desig-nated lines + = compression stress

r = crown radius of gallery tensile stress

APPLICATION OF THE DATA

Since both unit loads and unit dimensions have been used, and all dimenpions of the gallery are expressed in terms of the crown radius r, the study presented here has very general application to similar problems. The data may be used directly to determine the complete state of stress around a gallery if the normal and shear stresses f7x, O"y and Tnt are given at the point in the structure where the gallery is to be located 8 . If the principal stresses and their directions, 01 , 0"2 ,and a: only are given, they must be ro­tated into equivalent normal and shear stress­es. In either case the designer need only multiply the stress coefficients given on the figures by the stresses existing initially at the center of the gallery and sum the separate stress effects. Example:

Assume a standard five-by seven-foot gallery with a crown radius of 2-1/2 feet with initial stresses at the gallery center of--

O"x = + 75 psi

()y = + 200 psi

Txy= + 40 psi

Compute the stresses on Lines A, D; and H on the left side of the gal­lery.

Table I gives the tabulation for O"n.

Table II gives the tabulation for ()t.

Table III gives the tabulation for Tnt. The required reinforcement steel can be

found by determining the tension area on a particular line, then dividing the total ten­sile force so determined by the allowable steel stress.

Acknowledgement

The assistance of Ira E. Allen and Norman S. Johnson in the experimental work and the preparation of the drawings is acknowledged.

4

tl1

TABLE I

Tabulation for O"n on lines A, D, and H on left side of gallery (in Example)

Distance Stress due to Line out on

in feet ox rTy Txy 0 +300 -200 0 +100

0.5 -t212 - 68 0 +144

1.0 +165 - 20 0 +145 A

2.0 +114 0 0 +114

3.0 + 99 + 4 0 +103

4.0 + 92 + 10 0 +102

0 + 91 +174 -245 + 20

0.5 + 82 +158 -106 +134

1.0 + 74 +146 - 87 +133 D

2.0 + 64 +134 - 65 +133

3.0 + 58 +132 - 52 +138

4.0 + 54 +126 - 41 +139

o. - d1 +436 - 18 +337

0.5 - 32 +378 + 4 +350

1.0 - 15 +336 + 10 +331 H

2.0 + 7 +286 + 13 +306

3.0 + 20 +258 + 11 +289

. 4.0 + 26 +234 + 9 +269

TABLE II

Tabulation of Of. on lines A .. D, and H on left side of gallery (in Example,)

Distance Stress due to Line out

Txy ot in feet O"x rFy

0 0 0 0 0

0.5 +29 + 4 0 ·+ 33

1.0 +38 + 26 0 +64 A

2.0 +36 + 76 0 +112

3.0 +24 +118 0 +142

4.0 +17 +142 0 +159

0 0 0 0 0

0.5 +17 + 28 -15 + 30

1.0 +23 + 46 -12 + 57 D

2.0 +28 + 64 - 1 + 91

3.0 +31 + 76 + 6 +113

4.0 +32 + 82 +12 +126

0 0 0 0 0

0.5 - 6 + 18 -25 - 13

1.0 0 + 22 -26 - 4 H

2.0 +12 + 28 -26 + 14

3.0 +25 + 34 -23 + 36

4.0 +35 + 38 -17 + 56

TABLE III

Tabulation of 'Tnt on l.ines A, D, and H on left side of gallery (in Example)

Distance Stress due to Line out

Txy Tnt in feet (J'y_ O"y

0 0 o· 0 0

0.5 0 0 - 8 - 8

1.0 0 0 -15 -15 A

2.0 0 0 -30 -30

3.0 0 0 -42 -42

4.0 0 0 -55 -55

0 0 0 0 0

0.5 -45 + 72 + 2 +29

1.0 -53 +120 - 2 +65 D

2.0 -57 +132 - 6 +69

3.0 -57 +124 - 4 +63

4.0 -57 +116 - 3 +56

0 0 0 0 0

0.5 - 4 0 +12 + 8

1.0 - 5 0 +22 +17 H

2.0 - 5 0 +37 +32

3.0 - 5 0 +48 +43

4.0 - 5 0 +55 +50

(+)

NOTES (+ l. ind icotes compression. 1-l indicates tension

+ / I

···r ntensity and direction of load in structure without gallery.

p-lbs per sq. in compression

(-)

0

0

0.4r

2.0p

STRESS SCALE

(+)

O.Br

4.Qp

FIGUBE 6 - Unif'om load applied parallel to vertical axis of galler,r 1 principal stress at bo'IJllda.r,r of galler,r.

6

X-PEL-547

STRESS COEFFICIENTS IN TERMS OF p

LINES DISTANCE OUT FROM GALLERY 0 0.2r · 04r 0.8r 1:2r 1.6r

-100 -0.34 -0.10 0 +0.02 +0.05 -0.94 -0.11 tOOl t008 +0.11 +0.12

-0.55 +0.34 +0.39 +041 +0.39 +0.36 ,0.87 ,0.79 ,0.73 ,067 .0.66 ,0.63

2.10 tl.62 .1.38 .1.10 .0.97 0.8.9 F +2.75• + 196 .1.56 +127 ,1.10 +1.04

-c;-· +2.50. ,2.01 tl70 +142 +128 .1.20

H +2.18 + 189 +168 +1.43 +129 +1.17

1------:-'-1-::_-::_·~~+:-:~.-:.~:-:~:+-: :"::c~:cl-:,-: :c-2::-4+-:-:-:-:-;c:~+:-:-:-c;::-:-~-t+-:, :-;.:;-:::c-1

Intensity and direction of load in structure without gallery p-1 bs. per sq in compressron ...

M

+3.20 t080 +0.59 +0.53 +0-52 .0.52 +3.20 -0.12 -0.10 -0-04 +0.04 +0.13 -100 -0.53 -0.22 0 +0.15 +0.21

' \ E-...,~' '

F ·' ----:: G ---

H ---

1---::

B A B I I• I I I I

/ f D I I /

I // ,..... ..... E

~---F

·=t·-. .

---

---G

---H

---1

J- -/-; 1-1 ---,-1 ---kl' --J

K" I ~ : ''K L M L

LOCATION OF REFERENCE LINES

NOTES (+) indicates compress1on (-) indicates tension

4.0p

STRESS SCALE

DISTANCE OUT FROM GALLERY

0 0.2r o.4r. o.er A~""~-~.........;----~==

1.2r I.SC

B~

c r-L __ : __ .Jc.I __ L __ ~ _ _; _____ J

·r 1 : ("'") , _' 1 1 0 I ___ L __ .)_ _____ .1 _____ _,_ _____ J

I I

\ 1 (+) : : : E f-- -L--L ______ .L _____ L _____ J

I I I

I I I \+) I : : F L __ .)_ __ _l _____ J. _____ L _____ _j

I I

: I I : : : (+) : : :

G >---L.:__L _____ L ____ L ____ _i

I : I I I I I (+) I I ;

I I I I I I H '-- _L __ L ____ _[ _____ L ____ J

K

I

I I I I

I I I I

I I (+)

I I I 1 I _L __ L _____ .l _____ L ____ .J

I I (+)

I I I I I __ L __ L _____ L-----~-----~

__ L __ .l.. _i~ __ _J _____ _i_ ---- _j

I(+)

Ll-~=-~~---------~~==

X-PEL-548

FIGURE 7 - Uniform load applied parallel to vertical axis of gallery, stress normal to reference linea;

7

STRESS COEFFICIENTS IN TERMS OF p DISTANCE OUT FROM GALLERY

LINES DISTANCE OUT FROM GALLERY 0 0.2r. .0.4r o.ar 1.2r 1.6r

0 0.2r 0.4r O.Sr 1.2r 1.6r

A 0 +002 +Q.I3 +0.38 ~0.59 +Q.71

8 0 +0.14 +0.27 +o.4; +0.59 +0.67

0 +0.10 +0.20 +0.37 +o.so +0.59

D 0 +0.14 +0.23 +0.32 +0.38 +Q.41

E 0 +0.11 +Q.I7 +0.22 +0.24 +o.2s 0 0 +0.01 +0.03 +0.04 +0.06

0 ... 0.14 +0.19 +0.21 +0.18 +0.14

H 0 +0.09 +0.11 +0.14 +0.17 +0.19

0 +0.09 +0.14 +0.18 +0.18 T0.18

0 +0.26 +Q.24 +0.21 +0.12 +0.10

0 +0.92 +0.73 +0.68 +0.64 +0.61

L 0 +0.72 +0.78 +0.88 +0.92 +1.00

M 0 -0.03 0 +0.21 +0.37 +o.47

c ~ t ~ c o, \ \ ~ I I.}>

E, ', \ 1 / ,,/ ..,...E

F--:' t /~--F lntensoly and G- -- ---G d~recflon of load m structure H --~-:: ........... ---H wothaut gallery ............. ·· p-lb~ per sq. on. I--- ---1 compression.

J - - ..,/.~. ---:-~-----!1,-, -- J

// I I I ', K I. I I K

L M L

LOCATION OF REFERENCE LINES

NOTES 1+1 indicates compression 1-1 indicates tension

I 2i0 " •t· STRESS SCALE

A ~-l+.L_l ____ j _____ j

B ~--1+1 __ ~..:. ... ___ -j ______ ~

c GL I .l ' e ..0. _,

D ..---_c l£1 I '--

____ J

E 1±1

F ------------~=

L_ -l

J~._ ,_~+IJ =::

~: (+): I : K __ , __ ..J. _____ , ______ , ______ J

L' I 1 I I L __ ; __ _; __ ~+:_.l_ ____ ~ ____ j

FIGURE 8 - Unif'ora load applied parallel to vertical axis of gallery, stress tangential to reference lines.

8

X- PEL-549

STRESS COEFFICIENTS IN TERMS OF p DISTANCE OUT FROM GALLERY

DISTANCE OUT FROM GALLERY 0 0.2r 0.4r o.er L2r I.Gr LINES

0 0.2r 0.4r 0.8r L2r 1.6r

c D

M

0 0 0 0 0 0

0 +0.38 +0.40 +0.36 .0.32 • 0.29

0 +0.51 +0.55 +0.56 .0.57 +0.58 0 +0.36 +0.60 +0.66 +0.62 +0.58

o · I• o.28 +0.42 +0.56 +0.59 +0.54 0 +0.24 +0.32 • a 37 +0.38 •0.39 0 .. 0.22 +0.18 +007 +002 0 0 0 0 0 0 0

0 0 0 0 0 0

0 +0.05 +0.04 +0.04 +0.04 +0.04 0 -0.72 -0.70 -0.68 -0.66 -0.64 0 -0.15 -0.27 -0.35 -0.27 0 0 0 0 0 0 0

8 I C

I I 0 I / // E

/ /_,..,.,... --F

---G

---H

1--- ---1

J---/·.L,---,----',:,-- _J

K,/: 1 \ ""K M L

LOCATION OF REFERENCE LINES

NOTES S1qn convent1on for left half. (+)-=----(-) --==- ---Shear stresses in right holfore equal in magnitude to those 1n left half, but opposite in sign.

4.0p

STRESS SCALE

A

L_ l. __ !___-=-__ ._l ___ ~ j_ ____ j

o ~-L~ ___ L ___ j ____ j

~~---L _____ L~ __ J

~_:__L=-=----~- __ j ____ _j

L ._..-----~---r----~---

M -----------------------------

X-PEL-550

FIGURE 9 - Uniform load applied parallel to vertical axis of gallery, shear stress along reference lines.

9

-------.. I I I I I I

(-)

v

+ --+--:

: . '--.Intensity and direction

of load in structure without go llery.

1-l

p-lbs per sq. in. compression.

(t)

I I I I I I I I

~ 0 0.4r o.er

NOTES (+l indicates compression. (-J indicates tension

I I I DIMENSION SCALE

0 2.0p 4.0p I 1 I I

STRE$S SCALE

FIGURE 10 - Uniform load applied normal to vertical axis of' gallery, principal stress at bound.ary of gallery.

10

X-PEL-551

STRESS COEFFICIENTS IN TERMS OF p

DISTANCE OUT FROM GALLERY

Intensity and direction of

LINES

A

8

c D

E F

G

H

I

J

K

L

M

0 0.2r 0.4r O.Br L2r

+4.00 +2.83 +2.20 +1.52 +1.32

+3.42 +2.68 +2.13 +1.52 +1.33 +2.33 tf.ss +1.53 +1.22 1.08

+1.2.1 +1.09 +0.99 +0.85 0.77

-0.24 +0.41 +0.40 +0.39 +0.38

-a.s·l +0.06 +0.07 +0.09 +0.11

-1.10 -0.60 -0.40 -0.17 -0.04

-1.08 -0.43 -0.20 +0.09 +0.26

-<).98 -0.64 -0.39 -0.11 0

+3.40 -0.05 -0.22 -0.18 -0.10

+3.40 +0.20 +0.09 +0.05 +0.05

3.40 +2.20 +1.57 +1.33 +1.29

+2.72 +2.09 +I. 86 +1.57 +1.50

B ~ B

' ' I ' \ \ I

E...., ', ' F ', ---G--- -~-- ---G

load in structure H--­without gollery------------­p-lbs. sq. in !--­compression

----~ ---H

---1

J- .-7.!--1 --...,~,---~1 ,- -J

K/ : : : ',K L M L

1.6r

+ 1.23

+1.21 +0.99

+0.72

0.37

+0.12 +0.03

+0.34

+0.04

+0.02

+0.05

+1.23

+1.40

LOCAfiON OF REFERENCE LINES

NOTES (+) indicates compression (-) indicates tension

I I I STRESS SCALE

DISTANCE OUT FROM GALLERY

0 0.2r 0.4r "o.e·r 1.6r

I

: ~.-------~--------1 (t} I I ··I I . I · I I

- _L_- --;"--L---- __ -..t. -- -----.L

' ' I : (+)

I I I I f

~--~----.--L--~---~------~

' : ----~.--------~----~

I 1 I, ·(t) 1. 1 I

c ~ - _.1_ __ l·---- _1_---- _l_---- -~

' : : : {t) 1 1 I

0 L--L--..1..- ____ .,L ______ L. _____ ..J

(- I I (+I I I 1 E __ ..J ___ L------~------~------~

---• :J

G i-- ;--- r-r__:-:t,=::J:-::· -:.-=..r:====~---1

I

I I I·

--- __ _.._:.._..,:. ___ .l

l ,; -- -.·n~ .,m-- ~. I

K ~ - -"'<'""',..,;,------------------~ I I I I I I \_ ___.-1 : : I (+) I 1 I

L L __ _L __ --' _____ --' _____ -L _-- _ -- _.J

I I

I I (+) I I I I I I

ML-- L- _j_-- --L-----L _____ J

X-PEL-552

FIGURE 11 - Uniform load applied normal to vertical axis of gallery, stress normal to reference lines.

11

STRESS GOEFFICIENTS IN TERMS OF p DISTANCE O(JT FROM GALLERY

I.INES 0 0.2r 0.4r 0.8r 1,2r 1.6r

A o ~0.38~0.51 ~0.48 +0.3£+0.23 a o +0.43 +o.sz +0.42 +0.34 o.29 c ·o +0.43 +0.55 +o.so +o~ +ll.34

0 0 +0.23 +6.31 +o.37 +0.41 f+0.43 E o -1-0.23 1"0.32 +0.45 +0.53 +0.59 F o +0.19 +0.28 0.46 +0.61 0.68 G o +0.05 0.12:..0.27 o.41f+0.54

H o -o:oe o o.l6 +o.33 +0.47 r o -o." -o.o 1 +0.23 +0.44 +O. 61

J 0 +i.35 + 1.13 +0.95 +0.89 +0-87

K o +1.00+0.92 +0.79 +0.70+0.66 L o -o.o4 o.o5 +o.1e o.1o +0.07 M o +0.20 0.21 +0.25 +0.14 +o.1o

7 !' / / ,,o I,' ,,.E

... : ___ ... , Intensity and direction G----~r ~ ---·G ·of lood in structure H---- ---- H

without vollery. ---- ----·· p-lba. per sq. ln. __ ••• compression.----· r---- ----J

LOGATION OF REFERENCE LINES

NOTES (+l Indicates compression (-J Indicates tension

f 1 I Zfp

STRESS SCALE

4.0p I

b DISTANCE· OUT FROM GALLERY

d.2r d.4r I o.er

A~ l_ (+) _[ .!.

B ~--l_J~--L----~----~

c ~J--~1 __ L _____ L _____ J

0 ~~--l __ ttl __ l _____ l_ _____ ~

F ~--.1..._:-~,__j ______ L _____ j

I .,

G -'---- ~

H I

...J -"

......, I L I

...L

I

: I (+) I I :

--~--L----~-----~----~

[_ I I . I I I

K __ l __ l_~~--L ____ j _____ j

L ·!I

M ~--j i•1 - z __ - =

X-PEL-553

FIGUBE 12 - Un11'orm load applied normal to vertical axis of galleey, stress tangential to reference lines.

12

STRESS COEFFICIENTS IN TERMS OF p

DISTANCE OUT FROM GALLERY LINES 0 0.2r 0.4r 0. Sr 1.2r 1.6r

~ ___ o_ a' o o.l o -b 8 0 -0.30 ~032 -0.3~-Q.3~ .::~

-c · ,-~_: .. ~o.~1 -o.~ -QJi.ll:fl:§.!i. ~ ~ _.P- -1:),~0- -~0.7Q_ -0}_6•-0.7_6 -=~ ~- _ o_ :oso -057-o6s:_:D,~~ 1- ! ... I- o - 0.4L-.O.'I] -Q.~.L-0.44 -0.43 ___§_ __ _ p_. -0.20 -0?0. -_o,2o__-_Q.20 -0.20

e-li--- 0 -0()5 :-.0·0_6 :0:06 :006 -0.06 I 0 +009- +0.14 +0.19 !0,19 +0.17

0 +_Q42 +(),56_ +9._4_2~_~8 ±Q# 0 _+0.70 ~ ()1_9_ +0_8_()f,: Q16 +_Ql.Q K. :o~8 -od1 -ott' 1.~~T

c a, ~ 0 \ \ : ', \ \ E,,

B I C

/ / ,o

direction of

I / E / ..... ,. .....

· Intensity and 0F_-:-:. _*' -load in structure H---- - -

,---G.

--- H

without gallery .. ------------ -----· p-lbs per sq.'1n. 1 --- --...-I

com pression.

J --/-/~~--"7'-~1,-- J

K / : : ', K

L M L

NOTES Sign convention for left half. (t) --~----

(-) ---------Shear stresses in right half ore

equal in magnitude to those in left half, but opposite in sign.

? I I 210p 410p

STRESS SCALE

DISTANCE OUT FROM GALLERY

0 0.2r 0.4-r Q9r 1.2r 1.6r

A

D ~--,--~T- --~- ----, I I 1 t I I I I

G -...-,-- =

===:::r :X

~· _i__~l_ ____ j_ ___ -;.J

c.l I I I 1 I . I 1 I I

K -..L--L-~--1----.....i ____ ,

M

X-PEL-554

FIGUBE 13 - Uniform load applied. normal. to vertical -e.xis of gallery, shear stress alcuig r~ferenc.e_ .. ;u.nes~~- · ·

13

I I

I I

I I

I I

I I

I I

I

/ /

/

,/

/ /

/ (+)

/ /

/

/

/ /

/

------

-m-

0

Intensity and direction of load in structure without gallery.

0.4r 0.8r

DIMENSION SCALE

2.0P 4.0p

STRESS SCALE

NOTES (+) indicates compression (:..) indicates tension

H

FIGURE 14 · • Pure shear load, 'T'l<y 1 applied· to gallery, · principal stress at bOunda.r,y of galle~.

14

X-PEL-555

STRESS COEFFICIENTS IN TERMS OF p

DISTANCE OUT FROM GALLERY LIN"' 0 Q2r 0.4r 0.8r 1.2r 1.6r

A .o _ o --~ .. o . o_ _Q~ !l -3.27 -1.51 -1.10 -0.78 -0.68 -0.65 c -5.18-2.20 -1.74-1.27 -1.07-0.91'

-6.12 -2.65 -2.17 .1:62 -1.29 -1.03 __i__ ~~t3-18 -2.49-1.61 -1.13 -0.95

F -3.77 -2.17 -1.67 -I. II -0.75 -047 _!.... -~4~r~~~ -0.44 -ol4-o.o2

H -0.45 +0.1 0 +0.26 + 0.33 +0.·28 +0.23 -~-~- +l~o4._(i,4s..g._~o ·~5_2 •04s 1!~

J +_7.00 tl.95 tl.29 +0-!1 +9,~~·~ _K +7.00+2.68 +2.20+1.~_7+1}§l+il§__ L + 7.00 +L~. +1.04 +0.56 +0-26 .o,crr_ MOOOOOO

Pure shear load, "t,y, rn structure. p-lbs. per-,

sq. in. shea~ A \)

J I C I

I I ' / l I I I D I /.~·,.- E

~- =~::': D

G ----

H ----

I ---- ---- I

J--:,,.!-1 --,r----1:'<---J -t(/ : : I ',.,.

L M L

LOCATION OF REFERENCE LINES

NOTES

(+) l_ndicotes compression C-) Indicates tension Stresses in right half ore equal

in magnitude to those in left half but opposite in sign.

0 2.0p 4.0p

STRESS S<;ALE

DISTANCE OUT FROM GALLERY

0 0.2r 04-r O.Sr 1.6r

A

B r---r·- -,-·- ---.,'-·---·-,.--...:~---. I I . I· ·. (-) I ··' . ' I I I I I

c l--1 I I

E

! 1+1 ' , H ~~----.L-----~----~

r--._ - . _j __ l __ (_tJ _ _l ____ l ___ :J

I I I I I

K r I I I I I I

I 1 I (+) _. ----~ LL_J __ L ____ L----~---

M------------------------

X-PEL-556

FIGURE 15 - Pure she!;IX ·load,, Txy, applied. to gallery-, stress normal to r~ference lines.

15.

STRESS COEFFICIENTS IN TERMS OF p

DISTANCE OUT FROM GALL'ERY LINE 0 0.2r 0.4r O.Sr 1.2r /.6r

A 0 0 0 0 0 0 0 -0.33 -0.24 -0.03 +0.15 +0.28 0 -0.34 -0.26 -0.06 +0.17 +0.35

0 -0.37 -0.31 -0.03 +0.15 +0.30

0 -o.31 -0.34 -0.20 -0.02 +0.20 -0.39 -0.49 -0.39 -0.26 -0.19

0 -0.32 -0.49 -0.57 -0.50 -0.35 0· -0.63 -0.65 -0.64 -0.57 -0.43 0 +0.18 t0.26 +0.25 +·0.15

~~ 0 +1.55 +1.46 +1.12 +0. 77 t 0.47

K 0 +0.21 +0.19 -0.01 -0.19 -0.30 L 0 +1.21 +I .42. +I .35 +0.94 +0.56 t.t 0 0 0 0 0 0

Pure shear loaad, 1.,1, in structure. p lbs. per sq. in shear.-----.,

8 A ': 8 I :, C c I •' ,

0' \ J ::,' ,/ ,/D I I ,"'

I i ------ F

E' ',,'

F-->.· G---­

H ----

I -----

l~l ----- G

----- H

----- I

J---,:-;·1-: ---.-: --~:',~--- .. * : : I ",.,_

M L

LOCATION OF REFERENCE LINES

NOTES

(+) Indicates compression (-) Indicates tension Stresses in right holf ore equal

in magnitude to those in left half, but opposite in sign.

A

DISTANCE OUT FROM GALLERY r--.---.------r------.-----,

0. 0.2 r 0.4r O.Br 1.2r I.Gr

B ' --

c "'-.::..i.J::)

H '-J(:.I-;---- --;-------:----- =.J

I -==-=-t' l+L .1... L

I

: (+) r I 1 I I 1 --'---..L _____ J. _____ .L _____ J

K~!.&L.J I

I 1 t (T) 1 I I I I I I t L --.L..--...J. ______ L _____ ..L ______ .J

M----------------------------

FIGURE 16 - Pure shear load, Txy, applied to gallezj, stress tangential to reference lines.

16

X-PEL-557

STRESS COEFFICIENTS 0 !STANCE OUT FROM GALLERY

IN TERMS OF p 0 0.2r 0.4r O.Sr 1.2r 1.6r

INES DISTANCE OVT FROM GALLERY

0 ·o.2r .0.4r 0.8r 1.2r 1.6r

A 0 -0.20 -0.38 -0.74 -1.06 -1.38

B 0 -us -1.20 -1.24 -1.21 -1.14

c 0 -0.74 -0.77 -0.77 -{).70 -0.59

0 0 +0.06 -0.06 -0.14 -0.11 -0.08

E 0 +0.25 +0.37 +0.49 +0.51 0.51

F 0 +0.79 +0.90 +1.00 .f.-1.02 +1.00

G 0 +0.53 -tQ.78 +1.06 +1.26 +1.34

H 0 +0.30 +0.54 +Q.92 + 1.21 +1.37

1 0 +0.45 +0.64 +0.88 +t.os +u7

0 +o.a4 +o.91 •o.92 +0.92 +0.91

K 0 +0.14 +0.15 +0.11 +0.06 0.06

L 0 -0.91 -1.08 -1.15 -us -1.12

M 0 -0.72 -0.82 -o.s8 -0.92 -{).96

Pure shear load,'txy, in structure. p-lbs per sq. ill shear.··--.

'\ A al : / /,c o I I//,//,.,.£

I~ -:::: G---

H---

1--- ---1

J---L-----·- --J /1 I I'

/ I I I '

~ ~ ~ L '+ LOCATION OF REFERENCE LINES

NOTES

::~=-= =-O~!-LI-L1---2~f_P ______ .~jop

STRESS SCALE

A -------r--------;--=..::..::..;-~-------,

o o I o I

E ~j ______ L __ ~ __ !_ _____ j

. L:I: : : : F __ j ___ j _____ .,.j __ ~-~-----~

I : : I 1 I I I I G ___ L ___ J._ _______ L __ ~_J.:, _______ J

' ' ' 1 I I 1 H ___ .L. ___ J ________ l __ ~_..! _______ .J

' ' ' ' ...:. _J.... ___ r ______ _j __ ~-L---·- _J

J L] ___ l_ _____ _l __ ~ __ i _______ j

L'-C1------!--~r-----~

FIGURE 17 - Pure shear load, Txy 1 applied to gallery, shear stress along reference lines.

17

X-PEL- 558

APPENDIX: THE PHOTOELASTIC INTERFEROMETER

Photoelastic stress analysis is ordinarily made by means of the polariscope in con­junction with other experimental or analytical methods. The polariscope alone, however, gives only the boundary principal stresses and the maximum shears at all points in a stressed, transparent plastic medel subjected to po­larized light. But with the photoelastic inter­ferometer it is possible to measure the mag­. nitudes of the separate principal stresses and their directions at a point in a transparent, stressed, two-dimensional structural model.

Henry Favre 9 brought this instrument to the attention of the scientific world in 1929. Basically the instrument measures the abso­lute retardation of a beam of light passing through a transparent model subjected to a change in stress. The Bureau's instrument (Figure 2) was patterned after the one built by Favre. This instrument should not be confused with lateral extensometers employ­ing the interferometer principle in their operation.

Figure 18 is a diagrammatic layout of the interferometer. The light source at S is a high..:intensity, water-cooled, mercury­vapor lamp. A monochromatic point source of polarized light is obtained by filtering, then passing the light through an 'aperture 1/128 of an inch in diameter and through a polaroid. The light next passes through a projection lens so located that it brings the light to focus in the model. After passing through the projection lens the light strikes the optical parallel L1, which has a thin coat­ing of aluminum on one face.

L1 splits the light into two paths of equal intensity. One path, called the inside path, goes to the full reflecting optical flat L3, through the stressed model and then to L6. Ls is an optical parallel like L1 which both transmits and reflects light. A second path, called the outside path, goes from L1 to the fully reflecting optical flat L4 and then to Ls. The interference phenomenon is devel­oped at Ls where the two paths converge, and is visible in the eyepiece as a series of alternate light and dark bands called in­terference fringes. These fringes are en­tirely different from the familiar isochro­matic fringes of the polariscope, particu­larly in that they are visible both before and after loading the model. The interference fringes are merely shifted by a change in stress within the model, and the change in stress is linearly related to the number of fringes that cross a fixed point in the field of the eyepiece.

19

POINT SOURCE OF PLANE POLARIZED MONOCHROMATIC LIGHT-..

s /

FIGURE 18 - Diagrammatic ~-out of the interferometer~

The fringe movement at a point in ~ stressed model is, in general, dependent upon the plane of vibration of the polarized light passing through the model. ~t is therefore essential that the model be penetrated by plane-polarized light vibrating in a princi­pal stress direction. This is accomplished by the use of an auxiliary polarizing disk or analyzer placed in front of the eyepiece. The light from the outside path is cut off at the same time by inserting an opaque disk. The two polaroids are crossed so that the light from the inside path is extinguished. Since the vertically polarized light emitting from the light source will not generally be vibrat­ing in the plane of the principal stresses, the second or auxiliary polaroid set to ex­tinguish vertically polarized light will permit the components of light vibrating in the prin­cipal stress planes to pass through into the eyepiece.

The vertically polarized light is made to vibrate in a principal stress plane by the use of a half-wave plate placed in front of the model. A second half-wave plate placed behind the model rotates the light back to the vertical plane. The direction of the prin­cipal stress is found by rotating the two half­wave plates in unison and noting the angle of rotation necessary to produce extind:ion. With the light thus vibrating in a principal stress direction, the analyzing unit (auxil­iary polarizing disk) is removed from in front of the eyepiece and the opaque disk is

. removed from the outside path. The inter­ference fringes are again visible and by slowly removing the applied load acting on the model the fringe movement in the eyepiece is noted and recorded.

The laws relating changes in refractive indices to principal stresses have been in­vest~ated by such men as Favre9, Max­welll , Nuemannll, and Filon 12. For plane stresses these relations are:

di = CtO"z. + C20"1 (1)

6z = CtO"t + c2o-z (2)

where

6 = represents the change in re­fractive index,

(J = the principal stress, 1 and 2 are subscripts denoting prin­cipal stress directions.

-· the transverse stress-optical · coefficient, and

= is the direct stress-optical coefficient.

. Solving equations (1) and (2) for the principal stresses gives

(Jl Ao1 + B62

0"2, = Bo1 + Ac52

·A - c2

. c22 - c12

cl B.= ·c2._c2

1 2

(3)

(4)

The constants A and B are determined by a simple calibration described later. When these constants are known the principal stresses lrt and lT2 ·can be determined by Equations (3) and (4) in terms of the changes In refractiv·e indices which are measured by the number of interference fringes that pass the cross hairs of the eyepiece while the model is undergoing a stress change.

In actual operation the number of fringes that move past the cross hairs is not counted. Instead the outside path is changed in length by an amount equal to the refractivE:_ change

20

of the inside .. path .. This -is accomplished by using the clear glass optical parallel Ls, which is mounted on a spindle equipped with a tangent arm and tangent motion. Turning . the tangent screw changes the effective length of. glass through whi.ch the outside light path must travel. . Thus the change in length of the outside path is made equ;1l to the change of the inside path. A nlicrometer di!!-1 bears against an agate insert on the tangent arm, and changes in length of the outside path are read directly on the micrometer dial.. The optical parallel L2 is necessary only ·as a compensator for L5.

Stress measurements are made by first determining the direction of principal stress­es and then gradually removing the load act­ing on the model. As the load is removed the interference fringes move across the field of vision in the eyepiece. The fringe dis­placement is nullified by turning the tangent screw, and the d~fference in dial readings befo~e and after ~oaQing is a measure of the refractive change o.

The operation is repeated for the other principal s.tress direction. Magnitudes of the. principal stresses are obtained by sub­stitutingthe difference in dial readings, c5, into Equations (3) and (4).

Calibration

The constants A and B are . determined by a simple calibration experiment. A ten­sile specimen is cut from the same plate .as the experimental model, placed in the inter­ferometer and stressed. The principal stress directions are set in succession on the half­wave plates and the dial readings 61 and 02 obtained in turn by noting the fringe move­ment between the position of load and no load on the tensile specimen. In a. tenspe speci­men 0" 2 = 0 and O"t = pI A. . By inserting the values of o1, c5 2 and o-1 into Equations (3) and (4) the constants A and B may beode­-termined. These constants are functions of the model material, the model thickness, and the wave length of the light.

BIBLIOGRAPHY

1. Kirsch, G., "Die Theorie der Elastizitat und die Bedurfnisse der Festigkeitslehre ", Zeitschrift des Vereines deutscher In­genieure, Vol. 42, No. 29, 1898, pp. 797-807.

2. Timoshenko, S., Theory of Elasticity, McGraw-HUl, New York, 1934.

3. Greenspan, M. , "Effect of a Small Hole on the Stresses in a Uniformly Loaded Plate", Brown University, Quarterly of Applied Mathematics, Vol, 2, No. 1, 1949, pp. 60-71.

4. Inglis, C. E., "Stresses in a Plate Due to the Presence of Cracks anli Sharp Cor­ners", Transactions, Institute of Naval Architects, Vol. 55, Part I, 1913, pp. 219-241.

5. Mindlin, R. D., "Stress Distributioq Around a Hole Near the Edge of a Plate Under Tension", Proceedin~ SESA, Vol. V, No. 2, 1948.

6. Brahtz, J. H. A., and Soehrens, J. E., "Direct Optical Measurement of Individ­ual Principal Stresses", Journal of ARP.lied PhY.§ics, Vol. 10, No. 4, 1939, pp. 242-247.

21

7. Blanks, R. F., and McHenry, D .• "Plas­tic Flow of Concrete Relieves High-Load Stress Concentrations", Civil Engineering, Vol. 19, No. 5, 1949, pp. 40-42.

8. Trial Load Method of Analyzing Arch Dams, Boulder Canyon Project Final Re­ports, Part V--Technical Investigations. Bulletin 1, Bureau of Reclamation, 1938.

9. Favre, H., "Sur une Nouvelle Methode Optique de Determination des Tensions Interieures", Revue d' Optique, Vol. 8, Nos. 5, 6, and7, 1929.

1 0. Maxwell, C., "~ the Equilibrium of Elas­tic Solids", Transactions, Royal Society of Edinburgh, Vol. XX, Part 1, 1853.

11. Neumann, F. E. , ''Uber die Gesetze der Doppelorechung des Lichtes in compri­mierten und ungleichformig erwarmten unkrystallinischen Koerpern", Abh. Akad. Berlin (1841), Part (ii), pp. 1-254. (See also Coker and Filon, Photo-elasticity, page 190.)

12. Coker, E. G., andFilon, L.N,G., Photo­elasticity, Cambridge University Press, London, 1931.

Interior - Reclamation- Denver, Colo.