-
yENGINEERING MONOGRAPHS No. 15
- - - |-- Cº. ºOº.United States Department of the Interior º º º
ºBUREAU OF RECLAMATIONMAR 9 1954ºº ".
PHOTOELASTIC STRESSANALYSIS OF BRIDGE PIERS
| - By H. B. Phillips and C. N. Zangar
ºlº tººº ºf ºilºlººs 24, Call.
| Denver, ColoradoApril 1952 *
-
United States Department of the InteriorOSCAR L. CHAPMAN,
Secretary
Bureau of Reclamation
MICHAEL W. STRAUS, CommissionerL. N. McCLELLAN, Chief
Engineer
Engineering Monograph
No. 15
PHOTOELASTIC STRESS
ANALYSIS OF BRIDGE PIERS
by H. B. Phillips and C. N. ZangarPhotoelastic UnitDams
Branch
Design and Construction Division
Technical Information OfficeDenver Federal CenterDenver,
Colorado
-
INTRODUCTION . . . . . . . .DETAILS OF THE STUDY . .APPLICATION
OF THE DATAAPPENDIX . . . . . . . . . .
EngineeringLibrary
CONTENTS
3.2. (). 5!.
*~
_{sx * 2-º
17
-
Number
10
11
12
13
14
15
16
17
18
. Interference fringes . . . . . . . . . . . . . . . . . . . . .
-
LIST OF FIGURES
The photoelastic interferometer . . . . . . . . . . . . . .
The photoelastic interferometer, simplified diagram . . . . . .
. . .
Loading frame and model . . . . . . . . . . . . . . . . . . . .
. . .
Dimensions of piers and locations of reference lines . . . . . .
. . . .Stress coefficients for uniform vertical load symmetrical
aboutvertical centerline and applied over 0.90 width of pier . . .
. . . .Stress coefficients for uniform vertical load symmetrical
aboutvertical centerline and applied over 0.70 width of pier . . .
. . . . .Stress coefficients for uniform vertical load symmetrical
aboutvertical centerline and applied over 0.50 width of pier . . .
. . . . .Stress coefficients for two uniform loads symmetrical
aboutvertical centerline of pier with each load 0.05 width of pier
andload centerlines 0.40 width of pier apart . . . . . . . . . . .
. . . . .Stress coefficients for two uniform vertical loads
symmetricalabout vertical centerline of pier with each load 0.10
width of pierand load centerlines 0.40 width of Pier apart. . . . .
. . . . . . . . .Stress coefficients for two uniform vertical loads
symmetrical aboutvertical centerline of pier with each load 0.20
width of pier andload centerlines 0.40 width of pier apart . . . .
. . . . . . . . . . . .Stress coefficients for two uniform vertical
loads symmetrical aboutvertical centerline of pier with each load
0.05 width of pier andload centerlines 0. 60 width of pier apart .
. . . . . . . . . . . . . .Stress coefficients for two uniform
vertical loads symmetrical aboutvertical centerline of pier with
each load 0.10 width of pier andload centerlines 0. 60 width of
pier apart . . . . . . . . . . . . . . .Stress coefficients for two
uniform vertical loads symmetrical aboutvertical centerline of pier
with each load 0.20 width of pier andload centerlines 0. 60 width
of pier apart . . . . . . . . . . . . . . .Stress coefficients for
two uniform vertical loads symmetrical aboutvertical centerline of
pier with each load 0.05 width of pier andload centerlines 0.80
width of pier apart . . . . . . . . . . . . . . .Stress
coefficients for two uniform vertical loads symmetrical
aboutvertical centerline of pier with each load 0.10 width of pier
andload centerlines 0.80 width of pier apart . . . . . . . . . . .
. . . .Stress coefficients for two uniform vertical loads
symmetrical aboutvertical centerline of pier with each load 0.20
width of pier andload centerlines 0.80 width of pier apart . . . .
. . . . . . . . . . .Stress coefficients for uniform vertical load
symmetrical aboutvertical centerline and applied over 0.25 width of
pier . . . . . . . .
10
11
12
13
14
15
18
ii
-
EngineeringLibrary
Number Page19 Stress coefficients for uniform vertical load
symmetrical about
vertical centerline and applied over 0.125 width of pier . . . .
. . . . 19
2O Stress coefficients for triangular vertical load applied over
0.25width of pier . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 20
iii
-
FIGURE l - The photoelastic interferometer.
iv.
-
INTRODUCTION
This monograph presents the results ofa photoelastic analysis of
stress distributionin bridge piers of three different
width-toheight ratios, each subjected to 12 differentloading
conditions. The study considers thepier to have a finite width with
the load acting over a definite part of the pier width.The effect
of the ratio of pier width to heighthas also been considered in the
analysis sothat the bending effect present in shallowpiers is
evaluated.The three ratios of bridge pier width to
height studied were:
1. Width and height of pier equal
2. Width equal to twice the height
3. Width equal to four times theheight
The twelve loading conditions applied toeach of these pier sizes
were further dividedinto single and double loadings.
In the single loadings three uniform loads,symmetrical about the
vertical center lineand differing only in the length of load
appli
cation, were used. The three lengths were0. 9, 0.7, and 0.5
times the width of the pier.
In the double loadings two variables, thespacing between centers
and the length ofload. were introduced. The three center
linespacings were 0.4, 0.6, and 0.8 times thewidth of the pier. The
load lengths were 0.05,0.1, and 0.2 times the width of the pier
foreach load.
For each condition the principal stresses,O1 and O2, and their
direction, oº
, for a load
of unit intensity were determined experimentally at points along
certain lines traversingthe pier, by use of the photoelastic
interferometer. Then these were resolved to givethe stresses normal
to the reference lines,On, the stresses tangential to the
referencelines, C, , and the shear stresses along thereference
lines, Tat.
Acknowledgements
H. J. Kahm, I. E. Allen, W. J. Brown,Norman S. Johnson, N. F.
Forell, and H. E.Willmann assisted in the collection of the
data,the computations, and the preparation of thedrawings.
DETAILS OF THE STUDY
The photoelastic interferometer 2 (Figure
1) can be used to measure changes in the refractive index of
isotropic transparent materials as they undergo stress changes.
In
Figure 2, a highly simplified representation,L1 and L4 are
partially-reflecting, partially-transmitting optical flats; L2 and
L3are fully-reflecting optical flats. Polarizedmonochromatic light
from the point source
S is split at L1, travels along the inside
*Timoshenko, S., and MacCullough, G. H.
Elements of Strength of Materials, D. VanNostrand, New York,
1935, pp. 42-43.*Only a brief description of the basic principle of
operation of the interferometer is
given here. For a more detailed descriptionof the instrument
used in these studies seeAppendix of Engineering Monograph No.
12,Photoelastic Interferometer Determinationof Stresses Around an
Inspection Gallery
and outside paths, and is recombined at L4,where it is observed
in the eyepiece. At acertain very slight angular difference between
the two beams of light as they leaveL4 the eye sees in the combined
beam a
series of alternately bright and dark bandswhere the waves of
the two beams first reinforce, then cancel, each other (Figure
3).These interference fringes shift with a change
in stress within the model, and the change
EYE pieces4.
Point source ofPLANE POLARizedMONOCHROMATICLight,
tº insidepain.” l-s
FIGURE 2 - The photoelastic interferometer, simplified
diagram.
-
(B).
INTENSITY:VARIATION”
@ (;|}
FIGURE 3 - Interference fringes.(Polarized light is represented
hereby alternate full and broken linescorresponding to troughs and
peaks,respectively, as at A. B shows howthe light from the inside
and outside paths combines in different raysto give a resultant
intensity variation. C shows the actual appearanceof these
"interference fringes" inthe eyepiece. Sizes and quantitiesin A and
B are greatly exaggerated.)
in stress is linearly related to the numberof fringes that cross
a fixed point in the fieldof the eyepiece.
Thus, with suitably elaborate apparatus,methods of calibration,
and operating procedures, and with the use of the equationsrelating
changes in refractive index to principal stresses, this basic idea
may be usedto determine stresses in scale models ofvarious
structures.
In this study both pier and foundationmodel material were
Columbia Resin, CR
39. Figure 4 is a photograph of the model,mounted in the loading
frame, for a doubleloading condition where pier width is fourtimes
pier height. It was assumed that nobond existed between pier and
foundation, andthat the modulus of elasticity of the pier wasequal
to that of the foundation.
As often occurs in such studies, difficultywas encountered in
obtaining a uniform loading over the entire loaded area, especially
inapplying the single uniform loading to thepiers of w/h ratios two
and four. In the firstattempt a single solid shoe of the proper
lengthwas used. The bending of the model was ofsufficient magnitude
to cause a wide deviationfrom uniform loading. A more
successfulmethod was the use of multiple shoes on theloaded area;
four shoes of the proper lengthgave satisfactory results. As can be
seenfrom some of the tabulated values, however,particularly in
those cases where the loadedarea was large, the load applied was
not entirely uniform. In most cases the non-uniformity was small
and such non-uniformitymight more closely approach actual
conditions.
Checks made on the results of the studyby use of Boussinesq's
equations and comparison with parts of similar experimentalstudies
showed consistent values betweenstudies 3 Equilibrium checks were
madeacross horizontal lines in all cases and werefound to be
satisfactory.
3 Brahtz, J. H. A., "The Stress Functionand Photoelasticity
Applied to Dams, "Transactions ASCE. Vol. 101, 1936, pp.
1251-1253
FIGURE * - Loading frame and model.
-
APPLICATION OF THE DATA
Results are summarized in the accompanying figures 5 through 17.
Figure 5 givesdimensions of piers and locations of referencelines.
Each of the remaining figures givesstress coefficients for all
three ratios of pierwidth to pier height, loading condition
used,and the locations of the reference lines alongwhich the
stresses are given. The stresscoefficients are expressed in terms
of an intensity of loading p where p represents theaverage uniform
pressure applied over the
loaded area. All dimensions are given interms of the pier width,
w.The appendix gives additional data for apier with width and
height equal for three
different loadings. These were a uniformvertical load applied
over 0.25 width of pier(Figure 18); a uniform vertical load
appliedover 0.125 width of pier (Figure 19); and atriangular
vertical load applied over 0.25width of pier (Figure 20).
-
----------- ------ w ------------------------->:
|---------------- b --------------- :t - r :*----0 ---> F---
--->} | . l ; :, Ti" 1 :{{||||||| || || ||y y 1—-----| | | | {|
-| | | | :| t | | :i ! !
| | | l{ | | t | !| | :1 ––––––––. 1. DOUBLE UNIFORM LOADING...A
1. * h q=005W,OIOwGnd0.20w---- y rt-1--, | b=0.40w,060wGnd0.80wI |
| | | | ! h:0.25w,0.50wGnd 00w9--L-9. t | | i|-}+; - :| | | -
} || | | | ----|--|--|--|--—l—— ———4–– ——— ——l–––––.0.05h | A |
| | | y
*zzzzz 2. zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz…” i ! t i
+Stressesdeterminedclong~~~~1--------- + --- --thesereferencelines
-- w( )."
Foundationondpiermoduliof ... ---eldsticityare equal ---
------------------------- w -------------------------; :---- o
-------------------------1
i------I|| | | | | | | | | | | | | | | | | | | | | }-1---
w_w_w_Y Y Y Y Y Y M_ --y- I | |: |- - | | |! | |! I | | |t | I
|; | | |
SINGLE UNIFORM LOADING : 1– —", |-y—1–W-1–1q=0.50w,070wond0.90%
h T r— º * I *—T-T—h:0.25w,050w0nd1.00w ! | | |! | I
|F-----4-------------- -- | |t 4 |
r | | | | |! ------- ——— —— —————— ——— –––l--1y | | |
TNTT|oosh---~~~777 WZZzz Zzz zºz. T! i - \\ s |------------- --
---------- -------------- + ----X------->
--Stressesdeterminedclong- Yº, thesereferencelinesWS`s,
FoundotonOndper moduliof
`-- elasticityore equalX-Pel-341
FIGURE 5 - Dimensions of piers and locations of reference-
lines.
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