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

tH-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

0

^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

0 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.

0 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^~^

»/0*

\

*p

T

I I 1 1

A

BCD

b/0.o.i

E

0 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

65

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 the1
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 EdI^ and Ea1g 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

67

S. 0.30

•s

2 026

i 0.24

o

I 0.22
O

S

Loads ot Midspon

1P (P tP lP

T~

I I I I

A

B C 0 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 0 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> 0 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,
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Part III — Small-Scale Tests of Shear Con-
nectors and Composite T-Beams," Univ. of
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