Lehigh University Lehigh Preserve Fritz Laboratory Reports Civil and Environmental Engineering 1957 Investigation of multi-beam bridges (Progress Report No. 14), Proc. ACI, 29 (6) December 1957, Reprint No. 128 (58-1) R. E. Walther Follow this and additional works at: hp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab- reports is Technical Report is brought to you for free and open access by the Civil and Environmental Engineering at Lehigh Preserve. It has been accepted for inclusion in Fritz Laboratory Reports by an authorized administrator of Lehigh Preserve. For more information, please contact [email protected]. Recommended Citation Walther, R. E., "Investigation of multi-beam bridges (Progress Report No. 14), Proc. ACI, 29 (6) December 1957, Reprint No. 128 (58-1)" (1957). Fritz Laboratory Reports. Paper 49. hp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/49
Transcript
Lehigh UniversityLehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1957
Investigation of multi-beam bridges (ProgressReport No. 14), Proc. ACI, 29 (6) December 1957,Reprint No. 128 (58-1)R. E. Walther
Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports
This Technical Report is brought to you for free and open access by the Civil and Environmental Engineering at Lehigh Preserve. It has been acceptedfor inclusion in Fritz Laboratory Reports by an authorized administrator of Lehigh Preserve. For more information, please [email protected].
Recommended CitationWalther, R. E., "Investigation of multi-beam bridges (Progress Report No. 14), Proc. ACI, 29 (6) December 1957, Reprint No. 128(58-1)" (1957). Fritz Laboratory Reports. Paper 49.http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/49
1. The Assumed Load Carrying System of Multi-Beam, Bridges
An exact physical interpretation of themanner in whiGh a multi-beam bridge transmits an externally applied load to its supports is somewhat complex because, not onlymay the structure be non-homogeneous andanisotropic, but also discontinuous due toslip between adjacent beams. In order totreat the problem mathematically, we mustintroduce simplifying assumptions, which inreality do not always hold true. The mathematical solution of the plate equation, forexample, is made possible only by assumingthat no slip occurs. As we shall show laterthis assumption may be far from valid.
The different possible load-carrying systems, with their respective assumptions, aregiven in tabular form on the following page.For a more extensive treatment of this subject, we refer to Progress Report No. 10 (A.Roesli, "Lateral Load Distribution in MultiBeam Bridges," Lehigh University, July,1955) .
The analysis of a gridwork (with orwithout consideration of the torsion) ismathematically easy to express. However,in order to obtain reasonable accuracy, onemust take into account many redundants,making this method somewhat lengthy. Onthe other hand, simplification of the analysisby means of finite differences produces veryinaccurate results for concentrated loads.
Another possibility would be to considerthe multi-beam bridge acting as a plate. Theisotropic plate (i.e., a plate with the samebending properties in all directions) is onlymentioned for completeness of discussion,because in practice the degree of lateral posttensioning is never great enough to raise
3
the lateral bending stiffness to the level ofthe longitudinal one. For a practical approach we must therefore assume that thebridge acts as an orthotropic plate, whereinthe lateral bending stiffness Elx is smallerthan the longiFudinal bending stiffness Ely.*Finding the solution of an orthotropic plateproblem involves excessive calculations. InProgress Report No. 10, A. Roesli has presented the latter solutions for the most important loading conditions and for bridgesof various sizes; hence, the designer's problem of finding the internal forces is reducedto a simple interpolation of values takenfrom the tables of the mentioned report.
The question arises as to whether or notthe simplifications necessary in the theory oforthotropic plates are of negligible influence.At this point we ,are not concerl1;ed with theformal assumptions of the plate theory. Weare specifically interested in the effects ofslip, which cannot be taken into laccountmathematically. As proved later in thispaper, slip may occur in practice. Furthermore, even when small, slip has a decisiveinfluence on the deflection and thus on theinternal forces - especially the bendingmoments.
Nevertheless, the analysis of our problem based on the assumptions of an orthotropic plate remains the best existing approximation for the analysis of multi-beambridges. As will be seen later, it is possibleto modify this method according to theamount of existing slip in a very simplemanner.
* Since the modulus of elasticity is considered constant, the expression (EI)x can be replaced by EIx.
POSSIBLE ASSUMPTIONS
OFLOAD CARRYING SYSTEMS
(A) (B)
IGRIDWORK I(I) (2)
Consists of a network of beams inwhich there are two systems ofparallel beams spanning orthogonal directions.
(2)
Slip between adiacent beams
Plate with discontinuous lateralbendibg and sheer stiffness.
No slip between
adiacent beams
Plate with different
lateral andlongitudinal
bending stlffnesses.
O(=~(EI)x
Torsionconsidered
Torsionneglected
Notration: (EI) x == Longitudinal bending stiffness per unit width.(EI) y=== Lateral bending stiffness per unit width&
0<. == (EI)y(EI)x
Plate with uniform bendingproperties in all directions.
(EI)x === (EI)y0( == I
Plate with no lateralbenJdi ng stiffness.
(Elh~ === 0ex == 0
Lateral load transmissiononly by shear.
.4
2. The Distribution of Internal Forces and Deflections of Orthotropic Plates
The design of multi-beam bridges is governed principally by the longitudinal bending moments (Mx ). Hence, the most important characteristic of such a structure is thelateral distribution of these moments overthe cross section of the bridge. This momentdistribution is generally known as the "lateral load distribution," probably to avoidconfusion with the terminology used inHardy Cross' method of stress analysis. Thisexpression 'is somewhat misleading, becauseit is not the load which is distributed butonly the internal forces and moments. Theterm "lateral load distribution" should beused only as a collective expression for thecombined action of all the bridge membersunder a concentrated load. The idea of dividing the load into parts, proportional to thelongitudinal bending moments of each beam,may be expedient for design purposes, but itdoes not give the correct picture of the trueplate action. In a plate the- load is not onlytransmitted by the longitudinal bending(Mx ), 'but also by the lateral bending moments (My), the twisting moments (MxyandMyx), and the shear forces (Qx and Qy). AsFig. 2_ illustrates, their distribution over across-section is quite different. (This example is for the particular case where:
Since internal forces cannot be measureddirectly, it .is necessary to relate them theoretically with the deformations, which canbe obtained directly from tests. This was
5
done by using the theory of orthotropicplates.
Of special interest is the relrationship between longitudinal bending moments and deflections. It should be noticed that thesemoments are not proportional to the deflections. Therefore, the distribution curves forthe moment coefficients* and the deflectioncoefficients are not identical. This can bevisualized by considering the following: witha concentrated load at the center point of thebridge, the deflected sha1pe of an edge beam.~s somewhat similar to the shape of a uniformly loaded beam, whereas, the middlebeam acts more or less like a simple beamunder a single concentrated load. The moment-deflection ratio for these two casesdiffers by 25 per cent. Another explanationmay be found in the theory of thin plates,where under a point load, the maximum deflection is finite; yet the maximum momenttheoretically approaches infinity. In thepractical case of a multi-beam bridge, theseeffects lead to a deviation between the coefficients of the longitudina'l bending momentsand the deflection coefficients which may befas high as 50 per cent.
*It is convenient to present distribution coefficients,rather than the actual moments, shear forces or deflections, themselves. The coefficient f~r a particular point is defined as the ratio of the moment(shear force or deflection) at this point to th-e average moment (shear force or deflection) of the entirecross-section. These coefficients are dimensionlessand - in the range of elastic deformation - inde-pendent of the amount of lo~d. '
Beam Cross Section
Longitudinal BendingMoment (M x) +
Shear Force (Q'~.)
Lateral BendingMoment (M J,)
+
+Deflection (w)
-----
l _--------------------------.
FIG. 2
J
~"C' c , .r;J ~ .,
lPtPfP 401'''~
DISTRIBUTION OF MOMENTS. SHEAR FORCES AND DEFLECTIONS
AT THE MIDSPAN CROSS.SECTION (center loading)
Magnitude of the parameters for this particular case:
B/L = O.5~ a = 0.5; aYh = 1.0
6
3. The Parameters of the Theory of Orthotropic Plates
According to" the theory of orthotropicplates the internal forces or their distribution coefficients depend on three parameters:
The first parameter is merely dependenton the geometry of the bridge. The secondone is a function of the cross-section of thebridge members-the beams. It replaces thecoefficient of torsional rigidity, which occursin the differential equration of the orthotropicplate. The relationship of alh and the torsional rigidity was found by theoretical considerations, which are explained in ProgressReport 10.
The derivation of 'the parameter 0(" issomewhat difficult and no theoretical ap·proach has yielded satisfactory results; thereason being that the latera~ bending stiffness is by no means a constant. It variesnot only from point to point in the bridge,but it is also dependent on the' magnitudeand location of the concentrated load. Thiscan be visualized by considering a tr·ansversestrip taken out of the bridge. It .consists ofrectangular -blocks with no mutual cohesion;the lateral b'ending stiffness is provided onlyby the post-tensi.oning (Fig. ~). In practicethis post-tensioning force is never largeenough to prevent small openings at the bottom of the joints. In these joints, the stressdistribution becomes similar to one of acracked cross-section of a ,.prestressed beam,i.e., this cross-section is inhomogeneous. Themoment of inertila of an inhomogeneous section is variable under the combined action ofmoments and normal forces (post-tensioning) , making it impossible to derive octheoretically. This is one of the main reasons why an empirical investigation on alaboratory bridge was nHcessary. It yieldedsimple empirical formulas for an average
7
0< value over the entire bridge, as a function of the load and the total post-tensioningonly.
If slip occlirs between adjacent beams,the problem of ~hat should be called thelateral bending stiffness becomes more complex. Due to this discontinuity (slip), thedeflections and the stress distribution do' notfollow the rules of the plate theory at all.The idea of compensating for the effect ofslip by reducing the lateral bending stiffnesswould lead to a contradiction, because in thecase of an, articulated plate, where a development of slip is very likely, 'the lateral bending stiffness is ze'ro and cannot be reduced.Therefore, it w,as arbitrarily assumed thatslip does not change the lateral bending stiffness, or in other words, that 0( is independent of slip. The effect of slip was taken intoaccount by increasing the longitudina~lbending moments as follows: the ratio of moment coefficient to deflection coefficient in thecase with no slip was calculated for eachpoint of the cross'-section. The actual moment coefficients (considering slip) were obtained by multiplying the coefficients of themeasured deflections by this ratio. This procedure, although far from being theoreticallycorrect, was felt to be justified, since it overestimates the maximum bending momentwhich can occur under such conditions.
prellrelllnvShand's
FIG. 3DISTRIBUTION OF CONCRETE ... STRESSES.
ALONG A BRIDGE CROSS SECTION~
It illustrates the effects of the lateral bending moments on the stress distribution. As the lateralmoment increases, the joint between adjacentbeams opens, causing a !'eduction in the lateralmoment of inertia, I y. Thus ex: = EIy/EIx variesover the entire beam.
c. THE TESTS
1. ,GENERAL
In order to carry out the objectives (PartI, Introduction), the dimensions of the laboratory bridge were chosen large enough togive a reliable comparison with an actualbridge, but were limited to some extent bythe availa.ble facilities. TJhe bridge of ninebeams, with which the majority of testswere performed, had a span of 16 ft. and awidth of 10 ft. 9 in. (Fig. 1). Prestressingwas applied in two directtons, namely: longitudinally by means of pretensioning thetwelve 5/16 in. strands in each beam, andlaterally by post-tensioning with five 1118in. Stressteel bars, lqcated at the center, thequarter points and over the supports of thebridge. This system was convenient becausethe lateral post-tensi!oning could easily bevaried, during the course of the tests, to anydesired degree. Th~ cross-sectional properties of the individual beams forming thebridge are given iIi Fig. 4.
To study the transfer of the shear forces,
large keyways were put along the sides ofthe beams; in the first test series (see Program) the keyways were filled with woodso that only friction participated in thistransfer, whereas real shear transfer wasrealized in the second series by means ofgrouted concrete shear keys. Because of theinitial warping of the beams, additional concrete had to be placed between adjacentbeams (spaced 1Y2 in.) to guarantee a full'bearing area for the lateral post-tensioning.
All tests were performed on the newlycompleted test bed of the Fritz EngineeringLaboratory (Fig. 5). The loads were appliedby the Amsler Equipment (hydraulifl, jacksfed by a pendulum manometer. A detaileddescription of_ this test equipment can befound in a paper by B. Thiirlimann andW. J. Eney.*
* B. Thurlimann and W. J. EneyModern Installation for Testing of Large Assemblies Under Static and Fatigue L'oading. Fritz Laboratory Report~No. 237-7. Lehigh University.
Pretensi-oning (after 20% lOSS>'Strands - 12 - 5/16 in. cablesTensi-oning ForcejStrand- 7 700 lbs.Total Tensioning Force - 92,400 lbs.
Moment of Inertia - 664 in4Concrete Cylinder Strength - 4,540 psi
8
FIG. 5TEST SET.UP OF THE LABORATORY BRIDGE
FIG. 6DEFLECTION GAGES ON FRAMES
\
9
FIG. 7REACTION DYNAMOMETER
FIG. 8EQUIPMENT TO MEASURE SLIP BETWEEN
ADJACENT BEAMS
10
2. THE INSTRUMENTATION
t3.. DeflectionThe most conclusive data in a plate in
vestigation - as was our laboratory bridgetest - are the deflections; if the deflectedshape of the bridge is known in all details,all internal forces, reactions etc., can befound by differentiation. Therefore, thecomparison between the theoretical and theactual behavior of the bridge was essentiallya comparison between the theoretical! andactual deflections.
The deflections were measured at thecenterpoint and at the quarterpoints of eachbridge member by means. of Ames DialGages (sensitivity 1/1000 in.). Fig. 6 showssome of the 29 gages mounted on measuringframes. For the final destruction test, thedial gages were replaced by scales, in orderto permit measurements up to the ultimateload.
b. ReactionsThe reactions at the support of each
beam were measured with rea.ction-dynamometers, each consisting of two aluminumpins and two bearing plates (Fig. 7). Theshortening of the pins was measured by electri~al strain gages (SR4) glued diametrically opposite on the pins. The upper bearingplate was grouted to each beam; the lowerbearing plates, in which the pins were fixed,were supported either by rollers or by rectangular bars, depending on whether thedynamometer was placed at the fixed ormovable end of the bridge.
c. SlipAnswering the important question, "Does
slip occur and if so, how much?" required aprecise measuring device; because, evenwhen very small, the slip has a decisive effect on the distribution of deflections andinternal forces. This high precision was,achieved by measuring directly the relativeslip between two adjacent beams (Fig. 8) :two dial gages (sensitivity 1/10,000 in.)were fixed on a rectangular bar which spanned the two beams from center to center.Small half-round bars,- fixed with sealing wax
to the beams, served as a support for thedevice. The points of the di,als touched twosmall bearing plates on both sides of theshear keys.
3. THE TEST PROGRAM
The 58 different tests, performed in a sequence dictated by the operations they involved, can 'be divided into three major categories, which are:
(1) Tests to determine the stiffness properties of the bridgea. The properties of the individual
beamsb. The longitudinal bending stiffness of
the bridgec. The lateral bending stiffness of the
bridged. The torsional rigidity of the bridge
(2) Tests to investigate the influence ofa. Degree of lateral post-tensioningb. Location of lateral post-tensioningc. Location of the external loadd. Interaction of shear keyse. Slip between adjacent beams
(3) Tests to determine the l,ateral load distribution in the inelastic range up todestructioll of the bridge.
Table I gives a description, the numeration and some important results of the tests.
First, an individual beam was tested todestruction, in order to find the elastic andplastic deflections, the cracking load and theultimate load of the bridge members (Fig.9).
The tests (70 to 82) to determine theeLastic properties of the entire bridge (EIx,
EIn etc.) were all performed with groutedshear keys, varying only the post-tensioning
_force of the bars. Fig. 10 shows the setupof tests 80, 81 .and 82.
The bridge was supported across theends, the jack load being distributed to aline load by a heavy I-beam. These testsyielded values for the longitudinal bendingstiffness, EIx •
The lateral bending stiffness was foundsimilarly (tests 70 to 72, Fig. 11) : with lon-
11
FIG. 9DESTRUCTION TEST OF AN INDIVIDUAL BEAM
FIG. 10DETERMINATION OF THE LONGITUDINAL
BENDING STIFFNESS
13
FIG. IIDETERMINATION OF THE LATERAL
BENDING STIFFNESS
FIG. 12DETERMINATION OF THE TORSIONAL RIGIDITY
14
gitudinal supports along the two edge beamsand a line load along the center beam.
During the tests for the determinationof the torsional rigidity (tests 75 to 77, Fig.12) the bridge was in an indifferent state ofequilibrium: two diagonally related cornerswere loaded, the other two .supported. Theload was transmitt~d to the corners by anI-beam, supported by half round blocks atthe other two corners; the reactions weretaken by spherical bearings. Since thesetests implied a certain risk of local destruction, they were performed {liter the I important tests 3 to 65.
The investigation of different influencingfactors (tests 3 to 65) represents the essenceof this study. The variation of these factorscould not follow a logical sequence as givenin the beginning of this chapter. Since many.operations such as changing jacks, groutingof the shear keys, etc., were involved, thesequence of tests was dictated by practicalconsiderations.
The variation of the shear-transfer leadsto two major groups: Tests without shearkeys (with or without grouting) and tests
WITHOUT SHEAR KEY, JOINTS GROUTED
(Keyway filled with wood)
with shear keys (Fig. 13).
With the exception of tests 2 to 6 (determination of the influence of the ratio B/L)the two groups involved the same variables,namely: the degree rand location of lateralpost-tensioning and the location of the concentrated load. Only the two load conditions,which can yield maximum moments, wereconsidered, Le., a load at the center point ofthe bridge and a load at midspan of the edgebeam. The post-tensioning was varied bychoosing different forces in the bars as wellas by changing the number of bars. The variation of the slip measurements in these testswas big enough, so that no separate tests forthe determination of its influence were necessary.
Finally the 13.teral load distribution wasstudied in the inelrastic range of deflection.This test (No. 90), up to destruction, wasperformed with five kips post-tensioning, oneach of the five bars and with a concentratedload at midspan.
The extensive testing necessary in thisinvestigation covered a period of threemonths.
WITHOUT SHEAR KEY
JOINTS NOT GROUTED
• • ••••grouting
• • •• • •
FIG. 13
WITH GROUTED SHEAR KEY
~routing
• •• • •
15
D. THE TEST RESULTS
Rather than to present all the test results, involving about 30,000 readings, aneffort will be made in this chapter to showthe relationship between the different influ,encing factors. The most important data,however, is given in the Tables 1, 2a-f.
I. THE STIFFNESS PROPERTIES OF THE
BRIDGE
a. The properties of the individual beamsPreliminary to the bridge tests, a single
beam was tested under third-point loadingup to failure (Fig. 9). The load-deflection(respectively moment - deflection) curve isshown in Fig. 14.
The bending stiffness before crackingwas 25 x 108 lb. in.2• In order to compare thisvalue with the bending stiffness of the bridge-in plate problems always given per unitwidth-it had to be divided by the width ofthe beam (13 in.) ; i.e.,
Elx == 196 x 106 1b. in.2/in.
b. The longitudi'nal and lateral bendingstiffness of the bridge
Due to the effect of the lateral contraction (Poisson's Ratio) and the fact that thejoints of the beams were grouted, the longitudinal be.nding stiffness per unit width ofan individual beam and of the entire bridgewas different, furthermore the latter depends on whether or not one includes thegrouting in the total bridge width.
Longitudinal Bending Stiffness(per unit width)
Single Beam Elx===196 x 106 Ib.in.2/in.Entire Bridge
grouting excluded EIx===205 x 106 Ib.in.2/in.grouting included EIx==185 x l06Ib-in.2/in.
FrQm the first two values we can conclude that the effect of Poisson's Ratio increased the bending stiffness about 5 percent. In both cases, single beam and bridge,the longitudinal bending stiffness could be
65 am 30 20 With 75 82 90 100 III 130 152 179 208shear key (0.60) (0.66) (0.72) (0.80) (0.89) (1.04) (1.22) (1.43) (1.67) 0.19
NOTATIONS:• Point of load* Figures without parenthesis are measured deflectibns in thousandths of an inch.
Figures in parenthesis are the coefficients of deflection.
TABLE 2f
22
I IPOST-TENSIONING PER BAR
p (xy)EIx
10,000 lhs.
NGTE: All originally horizontal lines, whichwere parallel -to the X axis or the Y axis,remained straight after deformation.
FIG. 16TORSIONAL RIGIDITY - TEST NO. 75
(deflections in inches)
Besides the ratio of the bending stifflless( \X ), it contains a coefficient 2f3, called thecoefficient of torsional rigidity. It can experimentally be found by loading a plate attwo diagonally opposite corners land supporting the other two corners (Fig. 12).Under these loading conditions no bendillgmoments Mx and My occur; the only existing internal moments acting on an elementdxdy are the twisting moments Mxy andMj,x.
fined to a small area under the load whereas they extend over the who'le length of thebridge in the prescribed tests. The experimental 0( values for concentrated loads(tests 50, 52 and 53) are also given in theabove table.
For this reason, the way of finding CX'" byseparately bending the bridge longitudinallyand laterally was abandoned. T'he methodof calculating oc. used in this study, will begiven in section 2,a of this chapter.
c. The Torsional Rigidity of the BridgeThe differential equation of an ortllo
tropic plate can be written in the followingform:
6542o
\\~~
TEST No. 70
30 tons post-tensioning
30 tons 5 tons I ton
c{ 0.41 0;0174 0
(obtained by line loading)
0.65 I 0.38 I 0.20
(concentra ted center loa ding)
I I
LOAD AT CENTER POINT (Kips)
FIG. 15
In order to derive the factor oC === Elj../EIxfrom these tests, the lateral bending stiffnesswas taken at the point where the curve ofFig. 15 becomes horizontal, yielding the following ratios:
These oC values, however, are much toosmall for the same bridge subjected to concentrated loads, because in this case themaximum lateral bending moments are con-
This did not hold true for the tests toderive the lateral bending stiffness (Fig. 11) .The reason was, again, that the post-tensioning could not prevent openings between adjacent beams and thus the bending stiffnessw,as dependent on the magnitude of the load.Fig. 15 shows the large variation of Ely asthe load increases.
considered as constant up to the crackingload.
120
.~'-.N
.~ 100
~ID0
- 80VlV)WZu..u..i= 60V)
<1>zc
40zLUco
-J
~20LU
I-:s0
23
More adequate is a consideration of thelateral deflection distribution which, basedon the theory of orthotropic plates, canyield the longitudinal bending moments farbetter than the method mentioned above.First, however, it had to be proven thatthis theory applies with sufficient accuracy.
* See introduction.
For this proof on'ly those tests whichfulfilled the basic assumption that no slipoccurs between adjacent bea.ms could be
2. COMPARISON BETWEEN THE THEORY
AND THE TESTS
a. DeflectionsOne of our major problems, the deter
mination of the longitudinal bending moment of each bridge member, can be solvedby using the longitudinal or the lateral deflection distribution.
The first method would make use of theformula
d 2wMx = -Elx -ax 2
(effect of Poisson's Ratio neglected)
Thus the longitudinal bending moment ateach point could be found by differentiatingtwice the measured longitudinal deflectioncurve. This method would require a verylarge number of measuring points andwould not be sensitive enough to indicatepossible moment concentrations.
Fig. 17 gives the theoretical curve and thetwo points which resulted from tests 75and 76.
Th-e correlation between the empiricaland theoretical results indicates that underthe various assumptions proposed for thisrelationship, the one established by A.Roesli comes closest to reality.
Both the discussion of our test resultsand the resume of the analysis are basedon this theoretical reLa.tionship. Since f3 isa function of K and 0( where K depends ona/h, the effect of torsional rigidity can beexpressed by the parameters 0< and a/h.
• Measured Relationship(1) Theoretical curve according to A. Roesli(2) Theoretical curve according to M. T.
Huber
//
/
0.6 IF----+----S---+-----+-----I----.......
0.8 1~--+-~:::"'-+---~---I------1
1 Mxy + Myxw=-EI
x2f3 xy
2f1 = - ~xywEIx
In Progress Report 10* a theoretical relationship between 0( and f3 had been established, which had to be checked by our tests.
1.0
0.4 II---...,-/~__-+---_-+-----+-----I
I ~ a---+lI hlf------j K == f f..!!-)/ T;:-11: 'h
The deflections follow the formula:
This expression is linear in x and y, thusthe deformed bridge has two straight linesas generatrices, the x and y axis. This wasconfirmed in the tests 75 land 76, for allbeams and strips perpendicular to thebeams remained approximately straight.Fig. 16 shows a perspective view of the'shape of the deformed bridge, and the deflections of test 75 at the load of 10,000 lbs.at both load points.
Under these loading conditions, thetwisting moments can be expressed by theload:
Mxy + Myx = PHence the coefficient of torsional rigiditybecomes
TOTAL POST..TENSIONING FORCE YS. MAXIMU-M VALUE OF Sw
2.00
LU::>-oJ
~ 1.20
~::>~
~~
1.00
~cD....
...!!!
"S 1.60.....c(I)
·0~
(I)
oo~ 1.40
V')
co
+:::s
~....lit
:a." 1.80
111
~
26
CROSS SECTION
TEST #90
0.80
0.90
t-Z 1.00~Uu:u..w0 1.10UZ0~U 1.20w-Ju..L1J0
1.30DEFLECTION COEFFICIENT DISTRIBUTION
20 K
40 K
80 K110 K
0
40
80
.~ 120
0d
z 1600i=0W-JLL 200w0
240
20 K
"--- - 40 K
r------- ~80 K
-------- '..-------...............~.....-----~
~ ~ 110 K
~ V~~L,/
VDEFLECTION DISTRIBUTION
FIG. 21
27
considered. This was the case in all testswith grouted shear keys. A certain difficulty developed from the fact that one ofthe three parameters of the problem, i.e.,«, cannot be found by theoretical considerations or by special tests, for reasons explained before. Thus the 0\. values have tobe derived from the s.a.me tests which serveto check the adaptability of tHe theory. Thismeans that only the deflected shape, but notthe magnitude of the deflections, can becompared for this particular purpose. Forthis comparison the distribution of thelateral deflection coefficients was calculatedas a function of c:(; the other two pa.rameters, B/L and alh, are constants for agiven bridge. By varying ~, the theoretical curve with the s,ame maximum 'deflectioncoefficient as the experimental one could befound and the correlation between the distribution of the meas-ured and theoreticalcoefficients could' be checked. Fortunately,the correlation was excellent throughout thetest series and it 0an be con'cluded that thetheory is well applicable as long as no, oronly little, slip occurs. This shall be documented with some examples (Fig. 18).
It may be noticed that for the, two caseswithout any' post-tensioning DC was smallbut not zero, as expected. When the bridgeWras disassembled the reason for the latterwas found; the grout had adhered sostrongly to th(3 beam, that it had provideda considerable lateral bending stiffnessi •
3. THE INFLUENCE OF THE B/L RATIO
The less the width of a bridge as compared to its length, the better is the lateralload distribution, i.e., the smaller are maximum moment and deflection coefficientS'.
The relationship between these coefficients and the B/L ratio could' be determinedfrom tests 3 to 15, where the width of thebridge 'was varied by consecutivelyassembling 5, 7, ,and 9 beams,. In Fig. 19 two typical curves are plotted for post-tensioningforces of 10 and 30 tons per bar, with a loadof 10 kips at the bridge center. The devia-
28
tion of the measured curve from the theoretical is the result of slip, which could notbe prevented in these tests without shearkeys.
4. THE INFLUENCE OF THE DEGREE AND OF
THE LOCATION OF POST-TENSIONING
The maximum moment and deflection aredependent not only upon the degree,* butalso upon the location of the lateral posttensioning. The influence of' the latter wasstudied by post-tensioning four different barcombinations as shown in Fig. 20, where thetotal post- tensioning force is plotted againstthe maximum deflection coefficient. Thesmaller these coefficients a:t;e for a givenpost~tensioning force, the more efficient isthe combination. The best load distributionwas achieved by post-tensioning only at thequarter points. This is somewhat surprisingbecause, analogous to a gridwork, one mightexpect post-tensioning of the center bar togive the optimum load distribution. A plausible explanation for the former can be foundby considering the different shear characteristic for each case.
A smEctller, but still considerable contribution to the lateral load distribution, is obtained by post-tensioning over the supports,which increases the torsional rigidity of thebridge.
From Fig. 20 it can also be concluded'that the method often used in practice, i.e.,post-tensioning at midspan and over the sup-ports, is fully satisfactory.
5. THE INFLUENCE OF THE MAGNITUDE OF
THE LOAD (DESTRUCTION TEST)
Since the lateral bending stiffness of abridge is, dependent upon .the magnitude ofthe applied load, the moment and deflectioncoefficients cannot remain constant. 'Only inthe case of OC = 1.0 (isotropic plate), andeven then only so long as the resulting
*We define the degree of lateral post-tensioning asthe total lateral post-tensioning force of the bardivided by the lateral surface of the bridge.
strains stay in the elastic range, are thesecoefficients independent of the load.
The magnitude and significance of thevariation in these deflection coefficients couldbe observed in the final destruction test(post-tensioning 5 tons at each bar). Themeasured deflections over the midspan crosssection and the corresponding deflection coefficients are shown in Fig. 21. The variationof the maximum deflection coefficient- as afunction of the load is plotted in Fig. 22;the irregular shape of this curve was a result of the consecutive cracking of the individual longitudinal booms. It should benoticed that Sw max. increased only 2.5 % inthe range from zero load up to cracking ofthe first beam. The total change in Sw max.
amounted to about 14%.1.30
z
~0
1.25:i
:.cJ-
~<3 1.20
H:00
Z 1.150i=
~a
~ 1.10::>~
x-<~
1.0520 80 100 120
terest is the fact that the ultimate capacityof the bridge was only 4 percent smallerthan the sum of the ultimate capacities ofall the beams. This means that for an actualbridge, where such load concentrations areimpossible, the ultimate capacity of the entire structure can be obtained by adding thecapacities of all the members.
6. THE LONGITUDINAL BENDING MOMENTS
a. Complete interaction of the shea.r keysand no development of slip
In Section 2 of Chapter B it is explainedwhy the distribution curves of the deflectioncoefficients do not coincide with the distribution curves of the moment coefficients. Theconversion from one to the other needs atheoretical relationship, in our case thetheory of orthotropic plates. The v,alues ofthe coefficients for the measured deflectionsand for the calculated longitudinal bendingmoments are presented in Tables 2a-2f. TheLatter were found by the following procedure: first, the 0<: value wa.s determined bycomparing the measured with the theoreticalmaximum deflection coefficient. Since 0( ,
B/L and alh were then known, the actuallongitudinal bending moments of each beamcould be calculated.
CENTER LOAD (Kips)
"FIG. 22VARIATION OF MAXIMUM DEFLECTION COEFFICIENT
AS A FUNCTION OF THE LOAD
The bridge finally failed at a load of113.4 Kips by punching shear (Fig. 23). Anunderside view of the bridge ,after destruction is presented in Fig. 24. Of special in-
29
b. With slip and incomplete interactionof the shear keys
Under these conditions no direct correlation between the theory and the test resultsexists. In order to estimate the longitudinalbending moments for each beam, necessaryfor the design, a method had to be estab-
FIG. 23DESTRUCTION OF THE BRIDGE BY PUNCHING SHEAR
FIG. 24UNDERSIDE VIEW OF THE BRIDGE AFTER FAILURE
30
8M = ~ . s·wB Sw B
201612
a:O'Iff'
8
0.4
0.3
0
~ 0.2
lLJIIU
0.1
0'°0 4
7. EMPIRICAL FORMULAS
a. The ParameterSince the theoretical prediction of oC
seems to be impossible, an attempt was madeto find a method for its approximation.Hence, empirical formulas were developed interms of the post-tensioning and the load.
SM == moment coefficient with slipSM'B=== moment coefficient without slipSw = deflection coefficient with slip
B (measured)Sw = deflection coefficient without slip
(measured)
It can be proved that this method alwaysover-estimates the maximum bending moment and is, therefore, satisfactory for design purposes.
deflection due to slip. Expressed in terms ofcoefficients, this becomes
0.80
0.6
0
_K0
~ILl 04
II
tt
a=o,23/f
1.0 .----------,.------~-------,---~----.
lished to ,approximate these moments suchthat they ~re always on the safe side. Thismethod is based on the "following assumption: the slip does not change the lateralbending stiffness of the bridge. This meansthat the deflection of a lateral strip is considered to consist of two separate phenomena-on~ is a continuous rotation of the stripelements (i.e., simple bending), the other isa discontinuous vertical displaceme,nt ofsome strip segments (a strip segment = across-section of an individual beam). Theaction of the plate, therefore, oo,n also beimagined to consist of two separate phases:in the first, the bridge deflects as a normalorthotropic plate (plate bending), in thesecond, caused by slip, the longitudinalbeams deflect individually without torsion(additional beam bending). The additionralbeam bending can be p'ositive or negative depending upon the proximity to the load. Thelongitudinal bending moments resulting fromthe plate bending follow the same rules usedso far. The change, percentage-wise, in thesebending moments is considered to be proportional to the change, percentage-wise, in the
F Total post-tensioning forceP e = Applied edge load
31
For these formulas, all details such as thenumber or location of· the post-tensioningbars were neglected; only the total post-tensioning force was considered, excluding thatcase where only bars over the supports areprovided. Center and edge loading yieldeddifferent formulas, namely:
Center load: 0<. = 0.23 .J~c
Edge load: ex: = 0.10 JFPe
The points resulting from our tests andthe curves given by these formulas (chdsento be a lower boundary of the test points)are shown in Fig. 25 and Fig. 26. If a bridgeis subjected to a load resulting from severaltrucks, P (center or edge) is not the totalload, but only the portion concentrated atthe peak of the moment coefficient curve.This means, for most practical cases, thatthe applied center load (Pc) is equal to two
wheel loads, whereas the applied edge load(Pe ) is usually equal to 0l'l:e wheel load.
b. The increase in moment coefficient dueto slip and incomplete interl3.ction ofthe shear keys
It was felt ~hat a suitable way of takinginto account the effect of slip and incomplete interaction of the shear keys was toincrease the value of the maximum momentcoefficient. This increase (in per cent) verslis F/p. 'was plotted as· an upper boundaryto the test points for the two cases - with'and without shea,f keys (Fig., 27). Since theshear keys of our laboratory bridge werestronger. than those found in practice, theresults of a field test* on a bridge with normal shear key~ were also included to find thelower curve of Fig. 27.
*A. Roesli, C. E. Ekberg, Jr. A.Smislova W JEney "Field Tests on a Prestr~ssed Concrete' M~lti~Beam Bridge" (Progress Report 9).
so r---e-~---r-------r-----'------"""----
FIG. 27
IN-CREASE (per cent) IN MOMENT CO
EFFICIENT DUE TO SLIP AND INCOMPLETE INTERACTION OF SHEAR KEY
Maximum moment resultingfrom superimposing theeffect of all wheel loads
I
Center Load
0.4
Max. Moment Coefficient
0.20{
FIG. 28
o90 l-- --l.__---1-__--L-__-....L- ---"
0.8 1.0
170 .n----~----l-----+---_+--_\
140
180
120
130
110
160
150
* See introduction.t The definition and application of these momentcoefficients are given in the Resume of the Analysisof Multi-Beam Bridges.
An error in estimating 0< produces a decisive change in the maximum moment coefficient. Fortunately, the mlfl,ximum longitudinal bending moment resulting from asuperposition of all load influences is lesssensitive to errors in C(. In Fig. ·28 the maximum moment coefficient of a bridge and itsmaximum longitudinal bending momentsunder a four lane loading are plotted as afunction of 0(, taking both moment and coefficient arbitrarily as 100% for 0( = 1.While the maximum moment coefficientvaries about 80% over the entire range of
CONCLUSIONS AND RECOMMENDATIONS(6)
No factor of safety should be requiredfor uncertainties in the lateral load distribution as such for the fo'llowing reason:
E.
(5)Shear can be critical either by 'punching
through of a wheel load or in diagonal tension in an individual beam. The factor ofsafety ag1ainst punching shear maybe ascertained in the usual manner. The diagonaltension stresses may be calculated approximately by subjecting a.n individual beam toa certain fraction of wheel loads. It is safeto take the same fraction as derived forbending.
(3)The th~oretical relationship between 0(
(coefficient of the bending stiffnesses) and f3(coefficient of torsional rigidity) establishedin Progress Report 10* was confirmed byour tests. This means that the influence ofthe torsional rigidity of the bridge can beexpressed in the final solution by the parameter a/h (width of bearn/depth of beam).
(2)When the relative displacement between
adjacent beams becomes decisive, the theoryof orthotropic plates may still be used, however an empirical modification is necessary.
(1)The theory of orthotropic plates, as es
tablished in Progress Report 10*, was foundto be well applicable to the analysis of multibeam bridges. The correlation between theresults of the theory and those of the testswas very close as long as only little displacement developed between adjacent beams.
The design of a multibeam bridge is mainly governed by the longitudinal bending moments. Two load conditions can be critical:either a load concentration at the center ofthe bridge ,or at'midspan of an edge beam.The maximum SM valuet (coefficient of longitudinal bending moment) for a given bridgeis always higher for edge loading than forcenter loading, but since in the latter thepossible load concentration is larger, thecritical bending moment may occur In thecenter beam. Therefore, both load conditionshave to be checked.
33
0( (from zero to one), the maximum moment increases only by 1:2% (for edge load~
ing only 8%).
(7)
The ultimate capacity of a prestressedmulti-beam bridge of normal constructioncan be expected to closely approach the sumof the capacities of all the members, provided that no punching shear failure occurs.
(8)From (6) and (7) it follows that the
safety of a multi-beam bridge can be guaranteed if:
(a) the individual beams under theloads derived from the theory of lateralload distribution fulfill all the requirements of the specifications.
(b) that the punching shear stressesunder a wheel. load do not exceed theallowable limits.
(9)
For the calculation of cJ\. (ratio of lateralto longitudinal bending stiffness) empiricalformulas were derived from our tests as afunction of FjP (total post-tensioning force/applied concentrated load). A distinction between center .and edge loading must be made.
2/FCenter loading: d\. = 0.23 VP:
Edge loading: if = 0:10 V< :eThese c< values are conservative, as can beseen in Fig. 25 and Fig. 26.
(10)
The influence of slip and incomplete interaction of the shear keys can be considered
by increasing the maximum moment coefficient, 8M, as shown in Fig. 27. The pointsindicating test results were found from thisinvestigation as well .as from a field test.*Again the empirical curves are chosen suchthat the obtained bending moments are onthe conservative side.
(11)
A good lateral load distribution of a geometrically given bridge is depende~t uponthe effectiveness of post-tensioning arid sheartransfer between adjacent beams. The predomin,ant influence is that of strong shearkeys. The question of when it is economicalto apply post-tensiop.ing can be judged fromcase to case by making use of the variouscharts given in this report.
(12)If post-tensioning is provided, care has
to be taken to prevent the beams from cracking laterally. The beams are always a bitwarped with respect to a vertical longitudinal plane, due to the unequal pretensioningof the longitudinal tendon~. The lateralpost-tensioning tends to straighten thebeams; this can cause l~,teral cracks acrossthe sides of the beams. In order to preventsuch cracks, grouted beraring areias should beprovided between all beams at the ,locationof the post-tensioning bars.
(13)
Among various combinations of post-tensioning, the greatest efficiency is attained byarranging the bars either at the quarterpoints or by distributing them over the entire bridge. A relatively high efficiency mayalso be obtained by providing bars over thesupports and at midspan, as might be morepractical in the field.
*See 'fntroduction.
4' /'Fo RESUME OF THE ANALYSIS OF MULTI-BEAM BRIDGES
Edge loading:
The purposes of this chapter are to briefly explain the analysis of multi-beam bridgesaccording to the theory of orthotropic platesand to give all the formulas and charts necessary for their design. Only results andrecommendations on the procedure are reported; for all the details or reasoning werefer to the preceeding chapters ffild toProgress Reports 9 and 10.* The numericalsolutions of the partial differential equationof orthotropic plates were :a11 taken from thetheoretical study by A. Roesli.*
Due to all these investigations the designof multi-beam bridges becomes ratherstraightforward. It is essentially reduced tothe problem of how to load ran individualbeam such that the resulting maximum ofthe critical internal forces is equal to theirmaximum in the bridge. The critical internal force is usually the tongitudinal bendin!gmoment, which can be calculated if the .lateral distribution of the longitudinal bendingmoments is known. This shall be explainedmore extensively.
Since the bridge is a plate, simply supported along two opposite edges, the calculation of the total bending moment of alateral bridge cross·-section is a staticallydeterminate problem, identical to that of asingle beam. This total bending moment,divided by the width of the bridge, represents the laverage bending moment of thecross-section (Mx ay). The actual bendingmoment per unit width at each point in thecross-section can be expressed as a proportion of Mx ay. The factor of proportionalityof each pOInt is, by definition, the coefficientof the longitudinal bending moment (81\1).
MxS1\I ===-M
xav
The essential part of the analysis is thedetermination of the distribution over across-section of these dimensionless coefficients. As will be seen later, two distributioncurves may be of concern: one- for a IToadconcentration at bridge center, the other for
35
a load concentration at midspan of an edgebeam.
The distribution of the moment coefficients is dependent on three parameters:
B/L and alh are merely geometricalproperties of the bridge, whereas 0( may h-ecalculated by the empirical formulas:
2/FCenter loading: ex = -O.23V~
,3!F0\ = O.10V~
F == total post-tensioning forcePc == applied concentrated load rat bridge
centerP e == applied concentrated load at mid
span of an edge beam
If a. bridge is subjected to a load resulting from several trucks, Pc and Pe do notrepresent the tota~l load, but only the portionconcentrated at the peak of the moment coefficient curve. This means, for most of thepractical cases, that the applied center load(Pc) is equal to two wheel loads, whereasthe applied edge load (Pe) is usually equalto one wheel load.
In comparin,g the solution8 of the theoryof orthotropic plates, it Wras found that thedistribution curve ean be considered as dependent on S1\1 max only; no mla.tter from whatcombination of B/L, a/h and oc. this S1\1 max
may result. Hence the distribution curve isknown if 8M max is known. The value ofS:u mux, dependent upon a/h, ElL and ()(, isgiven in Figs. 29 to 32. Linear interpolation
Maximum values of SM coefficient of lateral moment distribution} as a function of theparameters 0(, • B/L. a/h and the location of the concentrated load.
wheel load r----I r---- II I 1'1I! I II I i II I I 1
M:::-o
I ; ~II iI I 2BI I I..... I I
( II II r
IIIIIII'
II
FIG. 35EVALUATION OF THE MAXIMUM MOMENT
Distribution curve of the coefficients of the longitudinal bending moments
(5:\1) =::::: influence line for the longituqinal bending moment of the center beam.
In order to design an individual beam. wehave to find ,an equivalent load (Weq ) which,applied to the beam, would produce the samemoment M as is present in the bridge; i.e.:
M =k.Weq"=Meq, a
n4 8:\1
ill
Wn = one wheel loadm = number of beamsn == number of wheel loads over one
cross-section
The lateral moment distribution is leastuniform at midspan; this means that Weq
becomes greatest at the midspan cross-section. If the above ratio Wer/Wn is used as areduction factor for all the loads along onebea.m, the resulting design is conservative.Thus the fina~l step is to design a single beamunder critical wheel load combinations,where each wheel load may be reduced bythe ratio WeqjWn. Although this reductionfactor was derived for bending, it may alsobe used to calculate the diagonal tensionstresses of an individual beam.
To illustrate this method a typical example is presented:
Thus:
a n (F )Weq == ~ •~ ~. W' 2B ..-tf- bn n
2BSince -- represents the number of beamsa(m), the formula becomes:
In most of the pra,ctical cases, all thewheel loads in one cross-section are equaland the area (Fn) under the curve a.pproximates that of a trapezoid. This simplifies theformula to
k-
2B
or extrapolation will be necessary if a/h isnot equal to either 1.0 or '1.7.
If no, or only a small, post-tensioning isprovided, 8M max has to be increased, to compensate for slip or incomplete inter3ction ofthe shear keys. Fig. 27 indicates how largethis percentage increase should be made.Again, P is not the total load but only thelargest possible load concentration.
Once the maximum value for 8M is calculated in this manner, the distri,bution curvecan be taken from Fig. 33 and Fig. 34 forcenter and edge load respectively. These distribution curves can also be considered asinfluence lines for the longitudinal1 bendingmoments.
The in:tluence, of a wheel load (Wn ) ofthe width b on the moment at the center of.a given cross-se'ction (Fig. 35) oon be expressed as:
Mn= -:nn · Mnnv
M n av is the average moment of the crosssection due to Wn alone and can be ,writtenin the form:
Mnnv=k · ~where k is a constant depending only uponthe length of the bridge and the location ofthe cross-section.
If -the bridge is subjected to a number ofn different wheel loads the moment becomes:
41
Length of the bridge = 60 feetLoading = H20-S16-44
Example
GIVEN:Width of the bridge = 4 lanesA. A. S. H. O. Specifications
CHOSEN:Post-tensioning = 3-0/8" bars; working forcejbar = 26,050 lbs.Assumed cross-section = 36" x 36"Shear keys will be usedCenter ITanes - 13 feet; Outside lanes = 12.5 feet
CALCULATIONS:1. Determination of OC :
center loadingww
~IIJPc = 32,000 lbs. .
F 26,050'3 _ 2 44Pc = 32,000 -.
"'= 0.36
edge loading
w
~II~Pe == 16,000 lbs.
F 26,050'3Pe = 16,000 = 4.88
0(= 0.17
F total post-tensioning forceP == maximum concentrated load
oc may be read directly from Figures 25 and 26 :
2. Determination of the maximum coefficient -of lateral moment distribution.
a _ width of the beam = 36" = 1h - depth of the beam 36"
B half the bridge width 51/2L = length of the bridge = 60 = 0.43
The coefficient of later,aJ moment distribution for center loading, independent of slip,may not be read directly from Fig. 29. If alh was between 1.0 and 1.7, it would then havebeen necessary to interpolate between Fig. 29 and Fig. 30. Figures 31 and 32 are used in thesame manner to determine the coefficient for the edge load. From Fig. 27 we may estimatethe effect of the slip and incomplete interaction of the shear keys.
From 29: SM = 1.67
From Fig. 27 - 3.5% increase8M total = 1.67 + O.03~' (1.67) = 1'1.73
42
From 31: 8M = 4.10
From Fig. 27 - 1.3% increaseSM total = 4.10 + 0.013 (4.10) = 4.1L
3. Finral lateral moment distribution curves:From figures 33 an.d 34 the desired curves are interpolated. These curves are also
the influence lines for the longitudi,nal bending moments. To determine the effect of severalloads, we shall load the influence lines. The percentage of wheel load which is to be carriedby one beam is then ::E 8MI'm :
Weq• ~SM-W- == Jil==
Center loading:
Summation of the coefficientsnumber of beams
w w
~w w
~w w
Mw w
~
0 0 0C'V! ": M
14 51- ~
~SM: _ 8.74 -0 514m - 17 -.
Edge loading:
w w w w w w w w
~ ~ ~ ~~sM=8.74
~_.---
tv")00d
o0'd
o.l'Vl
51'
00-00
d
-------,1:%.~9.47_0557.
Irl. - 17-· ·
4. Analysis of a single beam
In this example the edge loading yields the maximum moment. Thus a single beamhas to be loaded with as many wheel loads as the specifications prescribe, but each wheel]load may be reduced by the factor 0.557.
-44
G. NOTATIONS
a Width of sIngle beam
B 1;2 width of bridge
Elx Longitudinal bending stiffness
Ely Lateral bending stiffness
F Total post-tensioning force
h Depth of single beam
Iy Later,al moment of inertia
L Length of bridge
m Number of individual beams
Mx Longitudina'l bending moment
My Lateral bending moment
M yx Twisting moment
Mxy Twisting moment
n Number of wheel loads over one cross-section
P Applied concentrated load (Pe===edge,Pc=center)
Qx Longitudinal shear force
Qy Lateral shear force
s . Denotes coefficient
w Deflection
0< Ely,/EIx
f3 Coefficient of torsional rigidity
45
FIGURES,Page
1 The Test Bridge 22 Distribution of Moments, Shear Forces and Deflections at the
Midspan Cross-Section (center loading) 63 Distribution of Concrete Stresses Along a Bridge Cross-Section__ 74 Beam Cross Section .___________________________________________________ 85 Test Set-Up of Laboratory Brid-ge______________________________________________________ 96 Deflection Gages on Frames________________________________________________________________ 97 Reaction Dynamometer 108 Equipment to Measure Slip Between Adjacent Beams____________________ 109 Destruction Test of an Individufal Beam____________________________________________ 13
10 Determination of the Longitudinal Stiffness------------------------------------' 1311 Determination of the Lateral Bending Stiffness_____________________________ 1412 Determination of the Torsional Rigidity____________________________________________ 1413 Variation of Shear Transfer 1514 Load (Moment)-Deflection Curve of an Individual Beam--________ 1615 Lateral Bending Stiffness vs. Center Point Load______________________________ 2316 Torsiana,} Rigidity ~________________________________________________________________________23
17 Relationship Between 0<.. and f3-------------------~---------------------------------_______ 2418 Theoretical and Measured Deflection Distribution__~_______________________ 2519 The Influence of the BlL Ratio ~ ~_________ 2620 Maximum Value of Sw vs. Total Post-Tensioning Force__________________ 2621 Measured Deflection and Corresponding Coefficients at Midspan
Cross-Secti0 n -------- --______ _ 28
22 Variation of Maximum Deflection Coefficient as a Function of theLoad ---------------------------------------___ 29
23 Destruction of the Bridge by Punching Shea.r ~___________________ 3024 Underside View of the Bridge After Failure__________________________________ 3025 Relationship Between ex and FIP c (center load) ----------_________________ 3126 Relationship Between 0( and F/Pe (edge load) 3127 Increase (Percent) in Moment Coefficient Due to Slip and Incom-
plete Interaction of Shear Key_______________________________________________________ 3228 Maximum Moment Coefficient and Maximum Longitudinal Bend-
ing Moment as a Function of do. .--------- ~________________ 3329 Maximum value of SM fas a Function of B/L and at
(center load, aih :::::::: 1.0) 3630 Maximum va::ue of S1\{ as a Function of B/L ando<.
(center load, aIh :::::::: 1.7) 3731 Maximum value of SM as a Function of B/L and at
(edge load, a/h ==: 1.0) 3632 Maximum value of Si\C as a Function of B/L and at.
(edge load, ,alh === 1.7) 3733 Distribution oj Moment Coefficients for Center Load-__________________ 3834 Distribution of Moment Coefficients for Edge Loa.d--______________________ 3935 Evaluation of the Maximum Moments______________________________________________ 40
TABLES1 Tests Performed and Results ~________________ 122 Deflections and Corresponding Deflection Coefficients 17-22
46
CORRECTIONSPage 6, Fig. 2 top:
Bridge Cross-Section instead of BeamCross-Section.
Tables 1 and 2a-2fSlip (1/1000") should be Slip (1/10000")
Page 29, Paragraph 6a:should read:The values of the coefficients for the measured deflections are presented in Tables2a - 2f. The calculated longitudinal bending moments were found by ...
Page 81:Heading 7a: "the parameter" should read"the parameter 0( ."
Fig. 25: Relationship between ex and ~
Fig. 26: Relationship between oc. and ;.
Page 42 bottom:instead of "may not be read"should be "may be read"
