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

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 



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ie 
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tit 



ite 
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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.