# Full text of "Distribution of loads to girders in slab-and-girder bridges : theoretical analyses and their relation to field tests"

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ENGINEERING EXPERIMENT STATION STRUCTURAL RESEARCH SERIES NO. S-13 6 29s ENGINEERING LIBRARY I pif^Ty OF ILLINOIS DISTRIBUTION OFLOADS TO GIRDERS IN I SLAB-AND-GIRDER BRIDGES: THEORETICAL I ANALYSES AND THEIR RELATION I TO FIELD TESTS I I I an. coil By C. P. SIESS and A. S. VELETSOS A Report of the Concrete Slab Investigation Sponsored by THE ILLINOIS DIVISION OF HIGHWAYS and THE U. S. BUREAU OF PUBLIC ROADS UNIVERSITY OF ILLINOIS URBANA, ILLINOIS Distribution of Loads to Girders in Slat>and-Girder Bridges: Theoretical Analyses and Their Relation to Field Tests C. P. Siess, Research Associate Professor and A. S. Veletsos, Research Associate Department of Civil Engineering, University of Illinois Reprinted from Research Report 14-B (1953) Highway Research Board, Washington 25. D. C. SYNOPSIS THE object of this paper is to present a picture, based on theoretical analyses, of the manner in which loads on slab-and-girder highway bridges are distributed to the supporting girders. The discussion is restricted to simple-span, right bridges consisting of a slab of constant thick- ness supported on five girders, spaced equidistantly, and having equal flexural stiffnesses but no torsional stiffness. The numerous variables influencing the behavior of this type of structure are listed, and the effects of the following are considered in detail: (i) the relative stiffness of girders and slab, H\ (2) the ratio of girder spacing to span of bridge, b/a; (7/) the number and arrange- ment of the loads on the bridge; and (4) the effect of diaphragms, their stiffness, number, and location on the structure. Particular emphasis is placed on the relative magnitudes of the maximum moments in interior and exterior girders. It is shown that when the slab is fairly flexible in comparison to the girders, the maximum moment in an interior girder will usually be larger than the corresponding maximum moment in an exterior girder, if the loads in each case are arranged so as to produce maximum effects in the girder considered. This condition of maximum moment in an interior girder is found to be typical for reinforced-concrete T-beam brides having no diaphragms. However, if the transverse stiffness of the structure is fairly large in comparison with the stiffness of the gir- ders, then the maximum moment in the exterior girder will generally be the greatest. Such conditions will usually be encountered for typical I-beam bridges and for concrete-girder bridges having adequate transverse diaphragms. For those arrangements of loads which are critical in design, an increase in relative stiff- ness of the slab and the girders (decrease in H) will general'y reduce the maximum moment in the interior girders. For exterior girders, a corresponding decrease in H may either in- crease or decrease the maximum moment. A change in the ratio b/a affects the distribution of loads to the girders in much the same way as a change in H, since both of these quantities are measures of the relative stiffness of the slab and girders. Thus, a decrease in b/a improves the load distribution in about the same manner as a decrease in H. The behavior of a slab-and-girder bridge under a single wheel load is found to be dif- ferent from the behavior of the same structure under multiple wheel loads. Unless the per- formance of the structure and the effects of the numerous variables affecting its behavior are investigated for all possible conditions of loading to which the bridge may be subjected, cer- tain aspects of the action of the structure may be overlooked. The addition of diaphragms in slab-and-girder bridges supplements the capacity of the roadway slab to distribute loads to the supporting girders. The manner and extent to which diaphragms modify the distribution of load depends on such factors as the stiffness of the diaphragm, the number employed, their longitudinal location, and also on all those param- eters influencing the behavior of slab-and-girder bridges without diaphragms. Diaphragms will almost always reduce the maximum moment in an interior girder but they will usually increase the maximum moment in an exterior girder. These effects, which are a function of the many variables referred to above, may be beneficial or harmful depending on whether the moment controlling design occurs in an interior or exterior girder. The conditions under 58 U *'<■ SIESS AND VELETSOS: THEORETICAL ANALYSES which diaphragms will increase or decrease the controlling design moments are described in the body of the report. rThe simplifying assumptions involved in the analyses and the limitations imposed by these assumptions are discussed in detail, and consideration is given to the probable effects of the neglected variables. The relationship between thoretical analyses and the behavior of actual structures is also considered, and the paper concludes with a discussion of the manner in which theoretical an- alyses can best be used in planning field tests on slab-and-girder bridges, and in interpreting the results obtained. The slab-and-girder highway bridge is a structure for which neither theoretical analyses nor laboratory or field tests alone can be expected to yield a complete and trustworthy descrip- tion of its action. Only by considering together the results of both analyses and tests can we hope to understand a type of structure whose behavior depends on so many variables. 59 • THE slab-and-girder highway bridge as con- sidered in this paper consists essentially of a rein- forced-concrete slab supported by a number of paral- lel steel or concrete girders extending in the direction of traffic. The wide use of such bridges, together with an increasing awareness of their inherent com- plexity, has emphasized the need for a better under- standing of the way in which they function. Of par- ticular interest has been the manner in which wheel loads from vehicles are distributed to the supporting teams. Studies of slab-and-girder bridges were begun in 1936 at the University of Illinois in cooperation with he Illinois Division of Highways and the U. S. Bureau of Public Roads. The results of these studies ve been presented in several publications (/, 2, _j, !, 5, 6). Included in this program were extensive heoretical analyses in which the effects of several im- lortant variables were studied, and a rather complete licture of the behavior of such structures was ob- jined. In addition, numerous laboratory tests on :ale-model I-beam bridges were made to determine le accuracy of certain assumptions in the analyses ad to study the behavior of the bridges at ultimate '.ids. The object of this paper is to present a picture, ised on theoretical analyses, of the manner in which •ids arc distributed to the girders in slab-and-girder idges. The scope of these analyses, and thus also le scope of this paper, has been limited to the be- l.vior of the bridge under working loads. This is an Important limitation, since both the ultimate strength I the structure and its behavior at loads producing ■ elding are factors which should be given greal •ight in the selection of design methods. ]A second purpose of this paper is to consider the 'ationship between the results obtained from the Irtical analyses and those obtained from tests ot actual structures. This is a two-way relationship: neither approach to the problem can be considered alone and each can benefit from a study of the other. The theoretical approach cannot be accepted with entire confidence until its predictions have been veri- fied by comparison with the behavior of real bridges. On the other hand, no field test can give the full pic- ture, since the number of variables that can be con- sidered is necessarily quite limited. Only by con- sidering the two together can we obtain a complete and generally applicable solution to the problem. Analyses of Slab-and-Girder Bridges Variables The slab-and-girder bridge is a complex structure, and an exact analysis can be made only by relatively complex means. In essence, this structure consists of a slab continuous in one direction over .1 sun flexible girders. The presence of the slab as a major element of the structure is, ol course, one complicating factor. However, the complexity ol the structure is further increased by the continuity ol the slab and by the deflections of the supporting girders. The problem of studing analytically the slab-and- girder bridge is further complicated by the largei number of variables that may conceivably affect its behavior. The more significant variables ma) bi listed as follows: Variables relating to the geometry of the structun (1) Whether girders are simpl] d, continu- ous, ot cantilevered; (2) whethei the bridge is right or skewed; (3) the number ol girders; (4) the span length oi th (5) the spacing ol thi girders, and whether or not it is uniform; .\w\ ('•) the number and locations of diaphragms. Variables relating to the stillness ol the bridgi ments: (7) The flexural stillness ol the girders (this 62 LOAD STRESS IN BRIDGES T 0.15 b /o = 0.1 N ^H-20 0.10 t H-5\ A//^H-2\\\ £ •S 005 --' C -H.0.5 * E c (a) Moment in Girder C o Z o V \s H ■ zo ~ 0.15 • V \V-H=5 a m o o 0.10 « o c ■ 3 C 0^)5 H-2V\ H-O.JMJ ^r^ " u *^— d (b) Moment in Girder A Figure 2. Influence lines for moment in girders at mid-pan for load moving transversely across bridge at midspan. The effects of these variables are discussed in the following sections of this paper. Effect of Relative Stiffness H The relative stiffness of the girders and the slab, as expressed by the ratio H, is one of the most im- portant variables affecting the load distribution to the girders. The effectiveness of the slab in distributing loads will increase as its stiffness increases. More- over, a slab of a given stiffness will be more effective when the potential relative deflections of the girders are large; that is, when the girder stiffness is small. Thus the distribution of load will generally become greater as the value of H decreases, whether the change is due to a decrease in girder stiffness or to an increase in slab stiffness. The effects of variations in H can best be illustrated by means of examples taken from the analyses of five- girder bridges. Typical influence lines for moment at midspan of the girders are shown in Figure 2 for a structure with b/a=o.i and for various values of H. Figure 2(a) shows the influence lines for the cen- ter girder. For small values of H, corresponding to a relatively stiff slab, the curves are rather flat, indi- cating that the slab is quite effective in distributing the moment among the girders. As the value of H increases, the moment becomes more and more con- centrated in the loaded girder, and for f/=infinity, would theoretically be carried entirely by that girder. Figure 2(b) shows influence lines for an edge girder. Although the shape of these curves is quite different, owing to the location of the girder, the trends with changes in H are similar to those for Figure 2(a). It may also be seen from the influence lines in Figure 2 that the effects of a concentrated load on the more distant girders is relatively small. Thus, the addition of more girders on either side in Figure 2(a), or on the side opposite the load in Figure 2(b), would obviously have little effect on the character or magnitudes of the influence lines. Although this 5 10 15 20 2! Relative Stiffness of Girders ond Slab , H Figure 3. Variation of moment in loaded girder as a function of H for concentrated load at midspan. SIESS AND VELETSOS: THEORETICAL ANALYSES 63 conclusion does not apply without reservation for all possible values of H and b/a, it is reasonably valid for practically all structures having the proportions considered in the analyses. This observation then provides justification for extending the results of the analyses to bridges having more than five girders, and possibly also in some cases to bridges having only four girders. The effects of changes in the relative stiffness H may be shown more directly by the curves of Figure 3 for a bridge having b/a=o.i. Relative moments at midspan of girders A, B, and C for a single, con- contrated load directly over the girder at midspan are shown as a function of H. The moments are given in percent of the total moment in all the gir- ders; that is, neglecting the portion of the static mo- ment carried directly by the slab. 1 The close agreement between the curves for Girders B and C suggests that the behavior of all interior girders is much the same regardless of their location. It also provides further justification for extending the results of these analyses to bridges having more than five girders or to ' dges having only four girders. It can also be seen from Figure 3 that relatively much less distribution of moment occurs for a con- centrated load over an edge beam than for a load over an interior beam. When a load is applied over Beam A, the slab, no matter how stiff, cannot trans- fer the load effectively to the more distant girders, which are relatively farther away for this loading than for a load over Beam C. Such a reduction in the degree of distribution is evident also from Fig- ure 2(b). A further illustration of the way in which the moments resulting from a single, concentrated load are distributed among the beams is provided by Fig- ure 4 for a bridge having five girders and b/a=o.i. Relative moments in all girders for a load over Gir- der B are plotted as a function of H in this figure. The curve for moment in Girder B is the same as that on Figure 3. For this girder the moment in- creases continuously as the value of H increases. For an infinitely stiff slab, corresponding to f/=o, all girders participate equally in carrying the load, while for W=infinity all of the moment is carried by the loaded girder. A study of the variation of moment in the remaining girders as H decreases from near in- finity to zero in Figure 4 gives further insight into the behavior of this type of structure. Consider first the moments in Girder A. At H equals infinity this 1 The portion of the longitudinal moment tarried by the slab is usually quite small. An approximate expression for determining this moment is given on pp. 24-25 of Reference 2. 5 10 15 20 Relative Stiffness of Girders and Slab , H Figure 4. Variation of moment in girders as a func- tion of H for a concentrated load over Girder B at midspan. moment is zero. As the slab becomes stiffer and H decreases, this moment gradually increases until a value of H=2 or 3 is reached. At this point, the moment in Girder A begins to decrease with fur- ther decrease in H and finally reaches a value of 20 percent at H=o. This rather interesting behavior can be explained in terms of the increasing ability of the slab to distribute moment to the more distant girders as its stiffness increases. Note first that the moment in Girder C changes very little for the range of H on the figure. For values ol // greater than about 5, the moments in Girders D and E are rela- tively small and do not change rapidly with //. 111 dicating that in this range the stiffness of the slab is not sufficient to transfer an appreciable portion ol the load to these more distant girders. Consequently, most of the decrease in moment in Girder H as // decreases is accomplished by transfer of moment to Girder A. However, for values til // less than s in Figure 4 the stiffness of the slab becomes great enough to increase appreciably the | on ol girders O and E, and the moment in these girders begin to increase more rapidly as // decreases. In this stage the load applied over Girder B is more widely distributed and the adjacent Girdei \ is no >4 LOAD STRESS IN BRIDGES *70 c50 « e E40 n.i «/0^~^ »/ * \ * p T I I 1 1 A BCD b/ .o.i E S 10 15 20 25 Relative Stiffness of Girders and Slob , H ? igure 5. Effect of b/a on midspan moment in loaded Girder C for concentrated load at midspan. onger required to resist as much moment as before, rhus the moment in Girder A ceases to increase and ictually decreases to its final value of 20 percent at f/=o. The nature of the curve for Girder A in this lgure is generally typical of those for this loading con- dition and for other values of b/a. However, as b/a ncreases, the maximum moment in girder A occurs or smaller values of H than that shown in Figure 4 or b/a— 0.1. Effect of Ratio b/a The second major variable included in the analyses s the ratio of girder spacing to span, b/a. A change n the relative span lengths of the slab and the gir- ders, as represented by a change in b/a, causes a cor- esponding change in the relative stiffnesses of these :wo elements; that is, an increase in b/a corresponds :o a decrease in the transverse stiffness of the bridge. Thus, in general, the effect of increasing b/a is simi- ar to that of increasing H. This is illustrated in Figure 5 which contains curves of relative moments it midspan of Girder C for a concentrated load over Sirder C. The variation of moment with H is shown For structures having b/a=o.i, 0.2, and 0.3. The relative effects of changing b/a and H are easily seen from this figure. For example, an increase of b/a from 0.1 to 0.2 produces an increase in moment in Girder C approximately equal to that resulting from about a sixfold increase in H. That is, a change from b/a=o.i, H=4 to £/a=o.2, H—\ is equivalent to a change from b/a=o.i, H=4 to b/a=o.i, H=2$. Similar relations hold for an increase in b/a from 0.2 to 0.3 but the equivalent change in H in this case is less than threefold. Although an increase in b/a will always result in less distribution of load, the effect for an actual slab- and-girder bridge will usually be less than indicated in Figure 5 because of changes in H that occur as a result of changes in b/a. For example, if b/a is increased by shortening the span a, the change in span results in smaller and less stiff girders and thus causes a decrease in H which partially offsets the effects of increasing b/a. Similarly, if b/a is in- creased by making the girder spacing b larger, changes in H are again produced, chiefly because of increase in slab thickness which usually results from the changed span of the slab. Although the girder stiffness may also be increased as a result of the wider spacing, the net result is usually a decrease in H. since the slab stiffness varies as the cube of the thickness and may be increased a fairly large amount. Effect of Loading The preceding discussions of the manner in which load distribution depends on H and b/a have been confined to the case of a single, concentrated load on the structure. This loading condition was chosen partly for its simplicity but also because all of the effects discussed are greater for a single, concentrated load than for multiple loads. For this reason it is necessary to discuss also the behavior of the structure for the case of more than one load applied at a given section, since highway bridges are always subjected to multiple loads. In some cases, two loads cor- responding to a single truck may be considered, but more commonly the loading will consist of four loads representative of two trucks. The curves in Figure 6 show the variation with H of the maximum moments in Girders A and C of a five-girder bridge having £/<7=o.i. In each case the loads are placed transversely in the position to produce maximum moment in the girder considered. The spacing of the loads corresponds to the spacing of truck wheels on a bridge having a girder spacing of 6 ft. Consider first the curve for Girder C in Figure 6. This curve is very similar to that for the same girder in Figure 3, except that the decrease in moment with a decrease in H is much less. For a concen- SIESS AND VELETSOS: THEORETICAL ANALYSES 6 5 trated load (Fig. 3), the moment decreases from 54 xrcent of the total moment at f/=25 to only 20 per- cent at H=o. However, for four loads (Fig. 6), the moment in Girder C for f/^25 is only about 30.3 percent of the total, since the application of four loads provides in itself a better distribution of total moment among the girders. Since this girder must esist 20 percent of the moment at H=o, it is evident that a decrease in H can produce much less reduc- tion in moment for multiple loads than for a single load. The curve for Girder A in Figure 6 is quite dif- ferent from that for Girder C, in that there is a range of H in which the moment increases as H decreases. This phenomenon was observed also in the curve for moment in Girder A for a single load over Girder B (Fig. 4). The similarity between these two curves is to be expected since the center of gravity of the four loads in Figure 6 is very close to Girder B. Thus, the explanation for the peculiarities of this curve are the same as those given in the discussion of Figure 4. It can be seen from Figure 6 that for H less than about 10 the moment in the edge girder is the greater while for H greater than 10 the opposite is true. This condition is fairly typical for other structures with a load over the edge girder as shown in Figure 6, but the value of H at which the two curves cross will de- pend on the values of other variables, such as b/a and the spacing of the wheel loads relative to the spacing of the girders. Obviously, the magnitude of the moment in an edge girder will be decreased it the loads are shifted away from it. If conditions are such that the outer wheel load cannot be placed directly over the edge girder or sufficiently close to it, the moment in the edge girder may be less than that in an interior girder for all values of H. Another difference in the behavior of edge and in- terior girders is the way in which the moments vary with H. For an interior girder, the maximum mo- ment always decreases as H becomes smaller and this trend is independent ot the type or number of loads. However, the moment in an edge girder first increases and then decreases as II is made smaller. The value of H at which this change takes place depends some- what on the other variables not shown in Figure 6. Another characteristic ot the structure loaded with several loads is worthy of mention although it is not illustrated in Figure 6. As the number of loads in- creases, the distribution of load along the girders be- comes more nearly alike for the several girders. Con- sequently, the differences between relative loads, mo- ments, and deflections become less. For example, con- sider a structure having />/a=<).i and f/*=5. For a concentrated load over Girder C the moment in that girder is 2.05 times the average moment for all the girders, while the deflection of Girder C is only 1.55 times the average. However, for four loads placed as in Figure 6, the corresponding ratios of maximum to average are 1.28 for moment and r.23 for deflec- tion. This relatively close agreement between the distribution of moment and deflection for a practical case of loading is quite convenient in that it makes it possible to use the same assumptions for the com- putation of moments and deflections in the design ol slab-and-girder bridges. Action of Diaphragms in Distributing Loads Diaphragms or other kinds of transverse bracing between the girders are often used in slab-and-girder bridges, in an attempt to improve the distribution of loads among the girders. The results ol analyses show, however, that the addition of diaphragms does not always accomplish this aim since in certain cases it may actually increase the maximum moment in a girder. The conditions which determine whether diaphragms will decrease or increase the moment in a particuler girder can best be described by con- sidering two typical examples. First, consider a five-girder bridge with lour loads 09 10 is to to Relative Stiffness ot Girders ond Slob , H Figure S. Variation "ith " <>f maximum moment in exterior and interior girders for four wheel loads ai midspan. 66 LOAD STRESS IN BRIDGES placed to produce maximum moment in the center girder. The moments in this girder as a function of H are shown in Figure 6. Note that the loads are located symmetrically about the longitudinal center- line of the structure, and that it is the moment in Girder C that is being considered. If no diaphragms are present, the effect of increasing the transverse stiffness by increasing the stiffness of the slab causes a continuous decrease in moment as illustrated by the curve in Figure 6 for decreasing values of H. When the slab becomes infinitely stiff (H=o), the load and moment is distributed equally to all of the girders, and the maximum distribution is thus obtained. Now consider the same structure, having a slab with a stiffness corresponding to say H==20, but having a diaphragm added at midspan. If the diaphragm is assumed to be infinitely stiff, the load and moment will be distributed uniformly among the girders, since the applied loads are placed symmetrically about the longitudinal centerline of the bridge. The effect of providing infinite transverse stiffness is therefore the same whether the added stiffness is provided in the slab or by means of a diaphragm. It is reasonable to assume, therefore, that this equivalence in effect of slab and diaphragm will hold also for intermediate diaphragm stiffnesses, and analysis has shown this to be true. Thus, for a symmetrically loaded bridge, the addition of transverse stiffness by means of dia- phragms produces a reduction in the maximum girder moments in much the same manner as would an in- crease in slab stiffness (decrease in H). Consider next the other loading condition illus- trated in Figure 6 with loads placed eccentrically in the transverse direction so as to produce maximum moments in an exterior girder. In the structure with- out diaphragms, the effect of increasing the slab stiff- ness is shown by the curve in Figure 6 as H decreases. At first, the moment in the edge girder increases. Then, as the stiffness becomes very great (H small), the moment begins to decrease. And finally, for infinite slab stiffness (H=-o), the load and moment is again distributed uniformly to all of the girders just as it was for symmetrically placed loads. This ability of an infinitely stiff slab to provide uniform distribution of load for any arrangement of the loads results from the torsional stiffness of the slab which, in theory, becomes infinite when the transverse stiff- ness does. This property of the slab is not possessed by a diaphragm. Thus, if the transverse stiffness is increased by the addition of a diaphragm at midspan the behavior of the bridge is quite different from that produced by an increase in slab stiffness. Consider the limiting case of an infinitely stiff diaphragm. For this condition, the deflection of the girders, anc thus the distribution of load to equally stiff girders becomes linear, but not uniform. In other words the structure tilts because of the eccentricity of the 1 loading, and the moment in Girder A becomes something greater than 20 percent. Actually, for the loading arrangement shown in Figure 6, the mo inent in Girder A for an infinitely stiff diaphragm is theoretically equal to 33.3 percent. Thus, if the load is eccentrically located on the bridge, the addi- tion of diaphragms may result in an appreciable in crease in the edge-girder moment. Magnitude of Effects The foregoing discussion has shown clearly chat beneficial effects are not always produced by the addi- tion of diaphragms. It is important, therefore, to know under which conditions a diaphragm is able to exert its greatest effects and to have some idea of how great these effects might be. Since a diaphragm, like the slab, derives its effectiveness in transferring load from its ability to resist relative deflections of the girders, any condition leading to large relative de- flections, or to more nonuniform distribution of load or moment, will provide the diaphragm with a better opportunity to transfer loads. Thus, the following conditions should lead to the greatest effects of dia- phragms: large values of H; large values of b/a; or a decrease in the number of loads. The effects of these variables, as well as others, are discussed in the sections following. Effect of H and Diaphragm Stiffness The relative stiffnesses of the slab, the diaphragms, and the girders are all related in their effect on the load distribution. It is convenient to combine these three stiffnesses in two dimensionless ratios. One of these is, of course, H, which relates the stiffness of the girders to the stiffness of the slab. The other is defined as EJt where E d I^ and E a 1 g are the moduli of elasticity and moments of inertia of a diaphragm and a girder, re- spectively. It is obvious that the effectiveness of the diaphragm is a function of its stiffness, and that it increases with an increase in ^. However, the change in moment produced by the addition of a diaphragm of given stiffness depends on the stiffness of the slab already present. This can best be illustrated by reference to the moment curve for Girder C in Figure 6. The structure considered in this figure is representative SIESS AND VELETSOS: THEORETICAL ANALYSES 6 7 S. 0.30 •s 2 026 i 0.24 o I 0.22 O S Loads ot Midspon 1 P ( P t P l P T~ I I I I A B C E <>/ a .0.10 >f' H 20 H.5-^*^~ 0.20 k . Figure 7. Effect of adding diaphragm at midspan of bridge on moments at midspan. of a bridge having a girder spacing of 6 ft. and a span of 60 ft. A concrete-girder bridge of these di- mensions would have a value of H in the neighbor- hood of 20 to 50, while a noncomposite I-beam bridge would have an H of about 5. Since results of an- alyses are available for values of H=5 and 20, these will be used for comparisons; they can be considered roughly typical of the two types of bridges men- tioned. First consider the larger value of H. The moment in Girder C for no diaphragm is found to be 0.298 Pa. If a diaphragm is now added at mid- span with a stiffness corresponding to ^=0.40, a fairly large value, the moment in Girder G at mid- span is reduced to 0.217. The reduction in this case is 27 percent. Now consider a bridge having //=5, and add the same diaphragm. For no diaphragm the moment in C is 0.256 Pa, and with a diaphragm having ^=0.40 it becomes 0.215. The reduction in this case is only 16 percent, or a little more than halt as much as for the other bridge. The reason for this becomes evident if it is noted thai the moment after the diaphragm was added was approximately the same in both structures, 0.217 alH ' 0-215. This means that the action of a diaphragm ol this stiffness dominates the action of the slab and leads to about the same result in the two cases. However, since the Dridge with //=5 initially has a somewhat smaller Tioment than the bridge with H=2o. the chai Produced by the diaphragm is correspondingl) less. Hie relations just discussed are illustrated better in Figure 7 which gives moments for the same struc- ture and loading as in Figure 6. The moment in Girder C for symmetrical loading is shown as a func- tion of \ for the two values of //. It is easily seen from this figure that a given diaphragm stiffness provides a much greater reduction of moment if H=20 than if H=$. Figure 8 is similar to Figure 7, except that the moment given is that in Girder A for the eccentric load arrangement shown. Again, the bridge and loading are the same as in Figure 6. In Figure 8, the maximum moment in an edge girder increases as the diaphragm stiffness increases, for the reasons given previously. Comparisons can be made as be- fore for structures having values of W=5 and 20. For f/=20, the addition of a diaphragm with ^=0.4 in- creases the moment from 0.268 Pa to 0.319 Pa, an in- crease of 19 percent. For #=5, the corresponding increase is from 0.283 t0 0.302, or only 7 percent. Thus in this case also, the effect of adding a dia- phragm is greater for the larger value of H. Figures 7 and 8 show also that the diaphragm has a diminishing effect as its stiffness increases; that is the moment curves tend to flatten out as { increases. For example, for Girder C and H=2o in F'igure 7. an increase in ^ from to 0.40 reduces the moment 27 percent, while a further increase in l( from 0.40 to infinity would produce an additional decrease of only about 6 percent in terms of the moment for ^=0. S 0.22 Figure 8. Effect of adding diaphragm at midspan of bridge on momenta at midspan. 68 LOAD STRESS IN BRIDGES The comparisons in the preceding paragraphs have been presented only to give a picture of the relative effects of adding diaphragms to structures having dif- ferent values of H. The numerical values are ap- plicable only to the particular structures considered and no general conclusions regarding the absolute ef- fects of diaphragms can be drawn from them, since there are several other variables whose effects have not yet been considered. It is also important to note that the theoretical analyses on which the foregoing discussions are based involve the assumption that the longitudinal girders have no torsional stiffness. If such stiffness is pres- ent, the action of a diaphragm for eccentric loading approaches more nearly that of the slab. However, a relatively high degree of torsional stiffness and a fairly- stiff connection between diaphragms and girders is required before this effect becomes appreciable. These conditions are more likely to be present in bridges with concrete girders and diaphragms than in the I-beam type of bridge. Effect of b/a The relative deflections of the girders in a bridge without diaphragms become greater as the value of b/a increases. Therefore, the effects of the dia- phragms, which are dependent on the relative deflec- tions, will tend to be greater for larger values of b/a. The actual effects will be similar to those discussed in the preceding sections; that is, the moment in an interior girder for symmetrical loading will be de- creased, while the moment in an exterior girder will be increased if the loads are placed eccentrically with respect to the longitudinal centerline of the bridge. In either case, the changes in moment will be greater for larger values of b/a. Effect of Number of Loads The effects produced by adding diaphragms will depend on the number of loads considered to act on the structure at a given transverse section. The choices in either analyses or test programs are normally three: (1) a single concentrated load; (2) two loads, repre- senting a single truck; or (3) four loads, represent- ing two trucks. Data have been presented previously to show that the distribution of load and the deflec- tions of the girders tend to become more uniform as the number of loads is increased. Obviously then, added diaphragms will be more effective for a single load than for two or four loads. Effect of Transverse Location of Loads If the loads are placed symmetrically with respect to the longitudinal centerline of the bridge, the ad- dition of diaphragms will 'always produce a mor< uniform distribution of load, and the largest girdei moment, occurring for this case in an interior girder will be decreased. However, if the loads are shiftec transversely toward one side of the bridge, the largesi moment may occur in the edge girder, and will be increased by the addition of diaphragms. The practical significance of an increase in edge- girder moment depends on the relative magnitudes of the moments in edge and interior girders, the loads being placed in each case to produce maximum mo- ments in the girder being considered. If truck loads can be placed on the bridge with one wheel load directly over or very close to an edge girder and if the value of H is relatively small, the moment in an edge girder will usually be greater than that in an interior girder when each is loaded for maximum effect (see Fig. 6). In this case, the addition of dia- phragms will increase the moment in the edge girder, while decreasing the moment in the interior girder. The governing moment is thus increased and the effect of adding diaphragms may be considered to be harmful for these conditions. On the other hand, if the layout of the bridge and the locations of the curbs are such that a large transverse eccentricity of load is not possible, or if H is large, the governing moment will usually be that in an interior girder. The addition of diaphragms will again cause a de- crease in moment in the interior girder and an in- crease in moment in the exterior girder. If the final , result is equal moments in the two girders, each for its own loading condition, the effect of diaphragms is beneficial, since the governing moment has been re- duced. However, the diaphragms may change the moments so much that the edge-girder moment is the greater, and may even produce the condition in which the edge-girder moment with diaphragms is greater than the interior-girder moment without them. In this case, the effect of the diaphragms is again harm- ful. It is evident from the foregoing discussion that the transverse location of the loads has an important bear- ing on whether the effect of adding diaphragms is to increase or decrease the governing moment in the girders. However, the effects of the other variables affecting the behavior of the structure should not be ignored. Whether the governing moments in a given bridge will be increased or decreased, and to what degree, will depend also on the values of H, b/a, \, and on the longitudinal location of the dia- phragms as discussed in the following sections. This phase of the action of bridges with diaphragms is quite complex and the theoretical studies are still too SIESS AND VELETSOS: THEORETICAL ANALYSES 69 limited in scope to state, in terms of all the variables, the conditions under which added diaphragms will be beneficial or harmful. Effect of Longitudinal Location of Diaphragms Relative to Load It is almost obvious that a diaphragm will be most effective when it is located in the structure at the same longitudinal location as the loads being con- sidered. However, in a highway bridge the loads may be applied at any point along the girders, while diaphragms can be placed at only a few loca- tions. Since maximum moments in a bridge will usually be produced by loads applied in the neigh- borhood of midspan, a diaphragm or diaphragms located at or near midspan should be most effective. Consider the examples given previously for the struc- tures and loadings shown in Figures 6, 7, and 8. In this case, the loads and moments are at midspan, and the effects of adding a single diaphragm at midspan have been discussed. If, instead, two diaphragms had been added at the third points, each having a stiffness corresponding to ^=0.40, the results would have been somewhat different. For example, for the interior girder, the addition of two diaphragms at the third points would decrease the moment by 9 and 23 percent, respectively, for H=5 and 20, as com- pared to reductions of 16 and 27 percent for a single diaphragm at midspan. Similarly, the moment in Girder A would be increased 3 and 13 percent, re- spectively, for W=5 and 20, by the addition of dia- phragms at the third points, as compared to increases of 7 and 19 percent for a diaphragm at midspan. It should be noted that although the total diaphragm stillness is twice as great in one case as in the other, the effect is still reduced significantly because of the less advantageous location with respect to the load. Of course, if loads were applied at a third point of the span the diaphragm at this location would be quite effective, but the gin lei moments produced for this location ol the load would not he significant in de- sign. Analyses have shown also that il .1 diaphragm las been added at midspan. the addition of other liaphragms, say at the quarter points, will have little Sect for loads at or near midspan. This can be 'xplained by the fact that the relative deflections of he girders at the quarter points have been decreased iy the addition ol .1 diaphragm at midspan. It has been shown that il the loads are applied at lidspan, the effectiveness ol diaphragms will ' le more distant they are from the loads. < a diaphragm is located .it midspan, its effectiveness will decrease as the loads move away from midspan. Analyses have shown that the maximum girder mo- ments in a bridge with a diaphragm at midspan will be obtained for loads placed a short distance from midspan. The exact location of the loads lor maxi- mum moment will depend on the values of //, \, b/a, and the number of loads on the structure. For the bridges and loading of Figures 6, 7 and 8, and for a single diaphragm at midspan having ^=0.40, the maximum moments in Girder C for loads off mid- span are 2 and 6 percent greater, respectively for II g and 20, than the moments for loads at midspan. The magnitude of this increase depends on a number of factors and the above values should be considered only illustrative. Since the moment in Girder A is in- creased by the addition of a diaphragm, it will be a maximum for loads applied at the location of the diaphragm. The foregoing remarks may be summarized as follows: Diaphragms, unlike the slab (which acts at all points along the girders), can be added only at discrete points; their effectiveness is therefore not equal at all locations but extends only for some dis- tance either side of the diaphragm. Consequently, for greatest effectiveness, diaphragms should be placed near the locations at which loads will be placed for maximum moments, usually near midspan. Fur- thermore, since maximum moments do not decrease greatly as the loads are moved away from midspan, analyses have shown that in many cases the optimum arrangement will consist of two diaphragms placed a short distance either side of midspan. Flexibility of Diaphragm Connections All of the analyses used as a basis for the foregoing discussions of the effects of diaphragms involve the assumption that the diaphragms are continuous mem- bers extending across the full width of the bridge. However diaphragms in [-beam bridges comnv consist of short sections of rolled beams or ol trans verse Iraines spanning between adjacent girders. In such cases, the continuity of the diaphragm is derived solely from the rigidity of its connections to the girders. If these connections are not sufficiently rigid to provide llexural stiffness equal to that of tin phragms proper, the effective stiffness of the ilia phragm, and thus its listribute load, will be decreased. It seems reasonable to assume that the condition of a fully continuous diaphragm is approached closely where rcinforccd-concrct> lot diaphragms, as is the case in concrctc-girdcr bl and in some I-beam bridges. 70 LOAD STRESS IN BRIDGES The problem of determining the effective rigidity of a diaphragm, taking into account the flexibility of the connections, and the problem of evaluating the stiffness of framed bracing are outside the scope of this paper. Nevertheless, it is one of the most im- portant problems confronting the designer who wishes to use diaphragms as an aid to load distri- bution. Another problem of similar nature is represented by the skew bridge in which the diaphragms are frequently staggered longitudinally and thus depend on the torsional rigidity of the girders as well as on the rigidity of the connection to provide continuity across the bridge. This problem is also outside the scope of this paper. Limitations of Analyses The applicability of the analyses described in this paper is necessarily limited by the simplifying as- sumptions that have been made and by the fact that not all of the variables affecting the behavior of slab- and-girder bridges have been considered. Conse- quently, close agreement between the predictions of the analyses and the real behavior of actual bridges should not be expected unless the properties and characteristics of the structure are reasonably simi- lar to those assumed in the analyses. It becomes de- sirable, therefore, to consider the assumptions of the analyses and the limitations imposed by those assumptions, and to consider so far as possible the effects of the neglected variables. Properties of Materials A basic assumption in the analyses is that the slab is homogeneous, elastic, and isotropic. Although a reinforced-concrete slab satisfies none of these con- ditions, especially after cracking has occurred, the results of tests on scale-model I-beam bridges have shown that the distribution of load to the girders is predicted very closely by an elastic analysis. This conclusion, of course, does not apply after extensive yielding of the slab reinforcement has occurred. Ultimate Strength Another basic assumption is that the entire struc- ture — slab, girders, and diaphragms — behaves elas- tically; that is, deflections, moments, and shears are linear functions of load, and thus, superposition of effects is possible. Obviously, this condition is not satisfied after significant yielding has taken place in any element of the bridge, and these analyses are there- fore not suitable for predicting ultimate capacities which are attained usually only after considerable in- elastic acion. Values of b/a Of the several variables relating to the geometry of the structure, only the ratio of girder spacing to span, b/a, has been considered in the analysis, and this only for values of o.i, 0.2, and 0.3. This range of values includes a majority of actual structures, and some extrapolation is possible, especially to lower values of b/a since the load distribution for b/a=o is theoret- ically uniform. Number of Girders Although only bridges having five girders have been considered, it has been pointed out in a previ- ous section that the influence lines for moments in the girders (Fig. 2) may be used for bridges with more than five girders and even, in some cases, for bridges with only four girders. Analyses have also been made for a three-girder structure; some of these have been published (S), while the others have not (9). Continuous Bridges A further limitation of the analyses is that only simple-span bridges have been considered. However, some analyses, and fairly extensive tests on scale models (not yet published), have shown that the distribution of moment to the girders in a continuous bridge is approximately the same as that in a simple- span structure having values of H and b/a correspond- ing to those for the continuous bridge using for a the span between points of contraflexure. This similarity extends also to the distribution of girder moments over an interior support. Sfyetv Bridges Only right bridges have been considered, and no analyses for skew bridges are available. However, tests on scale models (5) have indicated that for angles of skew up to about 30 deg. the distribution of load is very similar to that for a right bridge. For larger angles of skew, the distribution of load is af- fected adversely; however, at the same time, the total moment in the girder is decreased in such a manner that the maximum girder moment is also decreased in spite of the changed distribution (5, 6). The effects of diaphragms in skew bridges have not been studied. Nonuniform Girder Spacing It has been assumed in all of the analyses that the girder spacing b is uniform. If this spacing varies slightly it is probable that the use of an average value when computing b/a will be satisfactory. However, this approximation may not be valid if the variation in b is great; fortunately this condition is not com- mon in slab-and-girder bridges. Stiffness of Slab Some uncertainty always exists regarding the abso- lute stiffness of a reinforced-concrete slab, since it is affected by the degree and extent of cracking. How- ever, the tests of scale-model bridges (4) showed an excellent correlation between the results of analyses and tests when H was based on a slab stiffness com- puted for the gross concrete section, neglecting the reinforcement, and taking Poisson's ratio equal to zero. Whether a similar approximation will also be satisfactory when applied to actual structures can be determined only by studying the results of field tests. Stiffness of Girders The other quantity entering into the expression for H is the stiffness of the girders, and this too is sub- ject to some uncertainty. For I-beam bridges the major problem is estimating the degree of composite action which exists between the slab and the girders of the bridge in question. If no composite action ; exists, the girder stiffness is easily determined. If composite action is provided by means of positive anchorage between the slab and girder, the stiffness of the composite T-beam may be computed easily by including a width of slab extending half the distance to the adjacent girder on each side. Tests in the '• laboratory as well as in the field have shown that I some degree of interaction probably exists in most ac- tual bridges, even if positive shear connection is not provided. The source of shear transfer in these struc- tures is either bond or friction between the slab and I-beam, or perhaps both. Since the stiffness of an I-beam is increased markedly by the existence of even 1 small amount of interaction, the value of girder itiffness, and thus of H, may be quite indeterminate n a real bridge. For this reason, it is desirable that ests on such structures include strain measurements in both top and bottom flanges of the I-beams, so hat the position of the neutral axis can be deter- lined and the degree of interaction estimated. The absolute stiffness of reinforced-concrete girders I aKo uncertain because of the indeterminate effects I tracking. It is customary in reinforced-concrete rames to compute relative stiffnesses on the basis f the gross concrete sections of the various membi rs, 'his procedure may be used also for computing II 'hen both the girder and the slab are reinforced con- ■«e. However, the possibility should not be ovcr- loked that the absolute stiffnesses oi these two mem- :rs may be affected differently by cracking mil that their relative stiffnesses may be changed. Thus, again there may be some uncertainty regarding the real value of H lor a particular bridge. However, the value of H will usually be fairly large for concrete- girder bridges and the moments in the girdei not especially sensitive to variations in H when H is large (Figs. 3 to 6). Unequal Girder Stiffnesses Only bridges in which all girders have the same stiffness have been considered in this paper. This condition, however, is frequently not satisfied in actual structures. In concrete-girder or composite I-beam bridges, the edge girders may have an in- creased stiffness because of the greater cross section of the curbs or sidewalks as compared to the slab prop- er. Also, some I-beam bridges have been designed with the edge beams smaller than the interior beams. The effects of unequal girder stiffnesses have been studied analytically for one bridge having edge girders 20 percent stiffer than the interior girders (2, 9). These effects have also been observed in tests of scale- model I-beam bridges in which the edge beams were less stiff than the interior beams. In both cases the bridges had five girders. Although these data arc not sufficient to permit precise statements regarding the behavior of bridges with girders of unequal still ness, some idea can be given of how such a bi will behave. Consider a structure in which the edge girders are stiffer than the interior girder, since this is a fairly common condition in actual highway bridges. In this case, the stiffer girders attract addi- tional load, the amount of which depends on how much sutler these girders are in comparison to tin- others, as well as mi the transverse silliness oi the slab or diaphragms, through which loads reach the girders. '["he limited data available indicate lh.it the increase in load is not as great as the increase in stillness. Thus, the deflections of the stiller girder will not be increased. An increase in load produces also an in crease in moment in about the same prop hi >u ever, this does not necessarily lead to an ini in stress, since the section modulus is usually in creased by the same factors which cause the ini in stillness. Whether or not the stresses will be in creased in any given t.ise will depend on th( live magnitudes of the increases in moment and section modulus. Torsional Stiffness of Girders The torsional si the analyses di scribed herein. I on the side ol safety, since such stiffness alwaj w LOAD STRESS IN BRIDGES tributes to a more-uniform distribution of load. The torsional stiffness of noncomposite I-beams is negli- gible compared to the flexural stiffness of the slab, and even for composite I-beams the effect may still be small. However, the torsional stiffness of concrete girders may be appreciable and may produce notice- able improvements in the load distribution, especially as it reduces the harmful effects of stiff diaphragms. If H is large and the diaphragm is relatively stiff, the contribution of the slab will be relatively small and the structure may be analyzed relatively easily, but with fairly good accuracy, by means of a crossing- beam or grid analysis, including the effects of torsion but neglecting the presence of the slab. Stiffness of Diaphragms A major uncertainty will always exist regarding the stiffness of the diaphragms. If rolled sections or framed bracing are used, the rigidity of the connec- tions at the girders is the major problem. If rein- forced-concrete diaphragms are used, the effect of cracking must be evaluated. This latter is particu- larly important where concrete diaphragms are used in a bridge with steel stringers, since the relative stiff- ness of diaphragms and girders, ^, becomes quite un- certain, because of the two different materials in- volved. However, for these conditions the value of ^ is likely to be relatively large, and variations in l{ will consequently be less important ( see Figs. 7 and 8). Use of Analyses in Planning and Interpreting Field Tests An important use of the results of analyses is in the planning of field tests to yield significant results, and in the interpretation of field tests to provide the greatest amount of useful information. Load, Moment, and Deflection Frequent reference has been made in this paper to the distribution of load. However, since the girders are designed for moment and shear, not load itself, a knowledge of the distribution of total load to the girders is of little value to the designer unless he knows also how the load is distributed along the length of each girder. For this reason, the meas- urement of load itself, for example, by measuring reactions, may provide little useful information ex- cept as a check on other measured quantities. Since moments are of primary interest to the de- signer, it is certainly desirable that they be determined in field tests, if at all possible. Although moment cannot be measured directly, it can usually be com- puted from measured strains. In reintorced-concrete ' : girders, the determination of moments from measure strains is usually a difficult problem because of th effects of cracking on the moment-strain relation. Th calculation of moments from measured strains ma be somewhat easier in the case of steel stringers, bu even here the effective section modulus may not b known exactly, because of the existence of a partia | interaction between the slab and girders in bridge without mechanical shear connectors. However strains are measured on both the top and bottorr flange of the beam so as to locate the position of th< neutral axis, the degree of interaction can be deter, L mined approximately and the effective section modu lus and moment of inertia for the composite beam can be estimated from the theory of partial interac tion presented in Reference 10. Measurements of deflection in tests of slab-and- girder bridges are always of value since the deflec tions are of interest in themselves. However, the as- sumption should not be made that the distribution of load or moment among the girders is the same as the distribution of deflection. Although these dis tributions may be nearly the same under certain conditions, they may be greatly different under others. Obviously, if the girders are of different stiffnesses, the distribution of deflection will depend on the rel- ative stiffnesses of the girders as well as on the loads that they carry. Moreover, even if the girders are of equal stiffnesses, the distribution of deflection may not be the same as the distribution of moment, or even of total load, since the longitudinal distri- bution of load along the various girders may be quite different (Fig. 1). This difference will be es- t pecially pronounced if only a single concentrated load is used in the test, and comparisons of moments and deflections for this case have been given else- where in this paper. If several loads are applied to the bridge, the distribution of deflection and moment will become more nearly alike, and in many tests ad- vantage may be taken of this relation if it is not possible or convenient to determine moments from measurements of strain. Loading The analyses have shown that the effects of varia- tions in H, b/a, diaphragm stiffness, or diaphragm location will depend to a considerable extent on both the number and locations of the loads used in a test. The loading considered in the design of a bridge usually consists of not less than two trucks for a two-lane bridge, the most common type, and it is the behavior of the bridge under this loading that is ot greatest interest. Frequently, however, it is SIESS AND VELETSOS: THEORETICAL ANALYSES 73 not possible to make field tests with two trucks, and only a single-truck loading is used. For this case, the maximum moments, the distribution of moment or deflection, and the effect of adding diaphragms will be different than for a two-truck loading. More- over, the distribution of moment will be different from the distribution of deflection. These differ- ences present certain difficulties in interpreting the results but they can be overcome partially by ob- taining data for various transverse positions of the single truck and combining the results to simulate the effects of two trucks on the bridge. Such super- position of effects is valid only if all of the observed a phenomena are linear functions of load; this con- dition will usually be satisfied, however, except pos- sibly for concrete-girder bridges in which the de- cree and extent of cracking may increase as suc- ;essive tests are made. In such bridges, it is usually desirable to load the structure at all of the test loca- ions at least once before any measurements are made. \ similar problem may be encountered in I-beam nridges in which the degree of composite action may thange during the tests. In some cases it may be more convenient to test ' ihe bridge under a single, concentrated load. The rarious phenomena observed for this loading will ie greatly different from those corresponding to a jad consisting of two trucks, and the results can be ntcrpreted correctly only by obtaining influence ines, or an influence surface, for the desired quan- ity by placing the single load at several different ransverse and longitudinal locations on the bridge. 'he problem of superposition is even more acute in his case than for single-truck loading, and special are should be taken to determine if the relation etween load and moment or deflection is truly linear ver the range necessary to permit addition oi elicits. The transverse location of the loads at any see- on has been shown to have an appreciable effect n the maximum moments in the girder, especially diaphragms are present. Consequently, an effort tould be made in any field test to place the loads i eccentrically as permitted by the spicing and nee requirements of the specifications. Il this is ot done, an erroneous concept of the action of dia- hragms may be obtained. The longitiHlinal location ol the test loads will sually be that producing maximum moments in the, ridge. If the bridge does not have diaphragm., the laximum moment in a simple span will occur under ie rear axle of the trikk or trucks when thai axle is :•■ cated a short distance from midspan. However, !, ill MK le| ie W '- | tit ite tin j since the moment at midspan lor the rear axle at midspan is only slightly less than the maximum, it is frequently more convenient to measure strain or deflection at midspan with the rear-axle loads it midspan. This procedure should prove entirely sal islactory if no diaphragms are present. Howi if a diaphragm is present at midspan, the moments and deflections at midspan tor load at midspan may be significantly less than those which may be found under a load placed a short distance away from the diaphragm. Obviously, such shifting of the loca- tions at which the load is placed and measurements are made adds much to the complexity of the test. How ever, it is important to recognize that the effect of diaphragms depends on the longitudinal location of the load, and this variable should either be included in the test program or its effect should be evaluated theoretically. Other factors influencing the results of tests ire H and b a. Although these quantities arc not likelv to vary in a single test structure, it is necessary to recognize that a concrete-girder bridge havil value of H will not behave the same as an I beam bridge having a small value ot //. The same is tru of bridges having different values of b a. Obviously, then, tests made on a single bridge cannot be general- ized to apply to all slab-and-girder bridges. Even tests on a number of bridges arc not capable of giv- ing a complete or general picture of the behavioi ol such bridges, since such a complex structure does not lend itself readily to a purely empirical study. The importance and usefulness ol theory becomes evident at this point. II field tests cm be planned and car- ried out so as to yield significant comparisons with the predictions of the analyses, and it these compari- sons show reasonable agreement, the theory then be- comes a tool which cm lie used with confidence to understand and predict the behavior of slab .on! girder bridges. Willi. ml verification from field tests, the theory is of limited value; and without the aid ot the theory, field tests, unless very great in number, cannot give a general picture applicable to the lull range of the variables. Conclusion The numerous variables affecting the distribution of load to girders in slab and girder budges have been discussed solely on the basis ol tin icsulis ol theo 1 analyses. The following majoi variables have been (i) Rela girders ind slab. //; (;) ratio oi girdei spacing to span, b .. number and arrangement ol loads; and (4) dia phragms, including effect ol diaphragm stiffness ind 74 LOAD STRESS IN BRIDGES longitudinal location. The discussion has been limited throughout to simple-span, right bridges having five girders spaced equidistantly and all having the same stiffness. Torsional stiffness of the girders has been neglected. The slab-and-girder bridge is a complex structure. Nevertheless, its behavior can be predicted and un- derstood with the aid of theoretical analyses involving a number of the more important variables. The ad- dition of diaphragms still further complicates the ac- tion of this type of bridge, but even here some in- sight into the effect of diaphragms can be obtained from analyses. This phase of the problem, however, has not yet been studied as fully as the action of the slab and girders alone. Of course, an understanding of the theoretical be- havior of this type of bridge is not enough. What we really desire is the ability to understand and predict the behavior of actual slab-and-girder bridges. To this end, the predictions of the analysis must be com- pared with the results of field tests; only in this way can we hope to understand a type of structure whose behavior depends on so many variables. Acknowledgment The studies of slab-and-girder highway bridges described in this paper were made as part of the Con- crete Slab Investigation, a research project under- taken by the University of Illinois Engineering Ex- periment Station in cooperation with the Illinois Di- vision of Highways and the U. S. Bureau of Public Roads. The analyses for bridges without diaphragms were made chiefly by the senior author, and the anal- yses for bridges with diaphragms were made by B. C. F. Wei, A. D. Kalivopoulos, and the junior author. However, considerable credit must go also to the many others who performed the detailed and frequently tedious numerical calculations required by the analyses. All of the analyses were made under the direction of N. M. Newmark, research professor of structural engineering, who planned and guided the work at all stages. References i. Newmark, N. M., "A Distribution Procedure for the Analysis of Slabs Continuous over Flex- ible Beams," Univ. of 111. Eng. Exp. Sta. Bulletin 304, 1938. 2. Newmark, N. M. and C. P. Siess, "Moments in I-Beam Bridges," Univ. of 111. Eng. Exp. Sta. Bulletin 336, 1942. 3. Newmark, N. M. and C. P. Siess, "Design of Slab and Stringer Highway Bridges," Pub- lic Roads, Vol. 23, No. 7, pp. 157-165, Jan.- Feb.-Mar. 1943. 4. Newmark, N. M., C. P. Siess, and R. R. Pen- man, "Studies of Slab and Beam Highway" Bridges: Part I — Tests of Simple-Span Right I-Beam Bridges," Univ. of 111. Eng. Exp. Sta. Bulletin 363, 1946. 5. Newmark, N. M., C. P. Siess, and W. M. Peck- ham, "Studies of Slab and Beam Highway Bridges: Part II — Tests of Simple-Span Skew I-Beam Bridges," Univ. of 111. Eng. Exp. Sta. Bulletin 375, 1948. 6. Richart, F. E., N. M. Newmark, and C. P. Siess, "Highway Bridge Floors," Transac- tions, American Society of Civil Engineers, Vol. 114, pp. 979-1072, 1949. (Also Univ. of 111. Eng. Exp. Sta. Reprint 4$). 7. Wei, B. C. F., "Effects of Diaphragms in I-Beam Bridges," Ph.D. Thesis, University of Illi- nois, Urbana, 1951. 8. Jensen, V. P., "Solutions for Certain Rectangu- lar Slabs Continuous over Flexible Supports," Univ. of 111. Eng. Exp. Sta. Bulletin 303, 1938. 9. Siess, C. P., "Moments in the Simple-Span Slab and Girder Bridge," M.S. Thesis, University of Illinois, Urbana, 1939. 10. Siess, C. P., I. M. Viest, and N. M. Newmark, "Studies of Slab and Beam Highway Bridges: Part III — Small-Scale Tests of Shear Con- nectors and Composite T-Beams," Univ. of 111. Eng. Exp. Sta. Bulletin 396, 1952.