a general loading on a box girder 1
TRANSCRIPT
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A general loading on a box girder, such as shown in fig 1 for single cell box, has
components which bend, twist, and deform the cross section. Thin walled closed section
girders are so stiff and strong in torsion that the designer might assume, after
computations based on the elemental torsional theory, that the torsional component of
loading in fig 1(c). has negligible influence on box girder response. If the torsional
component of the loading is applied as shears on the plate elements that are in
proportion to St. Venant torsion shear flows, fig 1 (e), the section is twisted without
deformation of the cross section. The resulting longitudinal warping stresses are small,
and no transverse flexural distortion stresses are induced. However, if the torsional
loading is applied as shown in fig 1 (c), there are also forces acting on the plate
elements fig 1 (f), which tend to deform the cross section. As indicated in fig 2 the
movements of the plate elements of the cross section cause distortion stresses in the
transverse direction and warping stresses in the longitudinal direction.
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FLEXURE :
Fig: 2
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A vehicle load, placed on the upper flange of box girder can occupy any position,
transverse as well as longitudinal. This load is transferred transversely by flexure of
deck to the webs of box girder.
For understanding the various stresses generated, initially consider that the webs ofbox girder are not allowed to deflect. The structure resembles a portal frame. The
flexure of deck would induce transverse bending stresses in the webs, and consequently
in the bottom flanges of the girder. Any vehicle load can thus be replaced by the forces
at the intersections of deck and web as shown in fig 3.
Now the supports under the web are allowed to yield. This results in deflection of web
and consequently redistribution of forces among web and flanges.
Distortion of cross section occurs as a result of the fact that m1 and m2 are not equal
resulting in sway of frame, due to eccentrically placed load. The section of box tries to
resist this distortion, resulting in the transverse stresses. These stresses are called
distortional transverse stresses. The distortion of cross section is not uniform along the
span, either due to non uniform loading or due to presence of diaphragms or due to
both. However the compatibility of displacements must be satisfied along the
longitudinal edges of plate forming the box, which implies that these plates must bend
individually in their own plane, thus inducing longitudinal warping displacements. Any
restraint to these displacements causes stresses. These stresses are called longitudinalwarping stresses and are in addition to longitudinal bending stresses.
.
TORSION :
The main reason for box section being more efficient is that for eccentrically placed live
loads on the deck slabs, the distribution of longitudinal flexural stresses across the
section remains more or less identical to that produced by symmetrical transverse
loading. In other words, the high torsional strength of the box section makes it very
suitable for long span bridges.
Investigations have shown that the box girders subjected to torsion undergo
deformation or distortion of the section, giving rise to transverse as well as longitudinal
stresses. These stresses cannot be predicted by the conventional theories of bending
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and torsion. One line of approach to the analysis of box girders subjected to torsion is
based on the study of THIN WALLED BEAM THEORY. The major assumptions are:
a) Plate action by bending in the longitudinal direction for all plates forming the cross
section, namely webs, slabs is negligible.
b) Longitudinal stresses vary linearly between the longitudinal joints, or the meeting
points of the plates forming the cross section.
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Fig: 3
The kerb, footpath, parapet, and wearing coat generally form the superimposed dead
loads acting on the effective section which is responsible for carrying all loads safely
and transmitting them to the substructure. Because of symmetry, the self weight of the
effective section and the superimposed dead loads do not create any torsional effects.
However the non-symmetrical live loads which consist of concentrated wheel loads from
vehicles on any part of carriage way and the equivalent uniformly distributed load on
one of the footpaths can subject the box girder to torsion.
Fig:4
If the deck slab is considered to be resting on non deflecting supports at A and B in fig
3(b) , the vertical reactions and the moments created by the live loads at these points
can be computed. The effects of moments at this stage are treated as separately since
they cause only local transverse flexure fig 5 and can be evaluated by considering a
slice of unit length from the box girder. The effect of superimposed and dead loads
should also be taken into account in such evaluations.
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Fig: 7
In rigid body rotation or pure torsion effects, the section merely twists or rotates
causing St.Venant shear stresses and associated warping stresses which can be
evaluated by the elemental theory of torsion as applied to closed sections of thin walled
members. It may be emphasized that due to very high stiffness in pure torsion, the
box girder will twist very little, and that the webs will remain almost vertical in theiroriginal unloaded position. Also the associated longitudinal stresses due to warping
restraint when present are negligible as compared to those induced by the longitudinal
flexure due to forces Q, Q.
The theoretical behavior of a thin-walled box section subject to pure torsion is well
known. For a single cell box, the torque is resisted by a shear flow which acts around
the walls of the box. This shear flow (force/unit length) is constant around the box and
is given by q = T /2 A , where T is the torque and A is the area enclosed by the box. The
shear flow produces shear stresses and strains in the walls and gives rise to a twist per
unit length, theta, which is given by the general expression:
Or,
Where J is the torsion constant.
However, pure torsion of a thin walled section will also produce a warping of the cross-
section, Of course, for a simple uniform box section subject to pure torsion, warping is
unrestrained and does not give rise to any secondary stresses. But if, for example, abox is supported and torsionally restrained at both ends and then subjected to applied
torque in the middle, warping is fully restrained in the middle by virtue of symmetry
and torsional warping stresses are generated. Similar restraint occurs in continuous box
sections which are torsionally restrained at intermediate supports.
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Fig 9: Distortional effects
When torsion is applied directly around the perimeter of a box section, by forces exactly
equal to the shear flow in each of the sides of the box, there is no tendency for thecross section to change its shape. Torsion can be applied in this manner if, at the
position where the force couple is applied, a diaphragm or stiff frame is provided to
ensure that the section remains square and that torque is in fact fed into the box walls
as a shear flow around the perimeter. Provision of such diaphragms or frames is
practical, and indeed necessary, at supports and at positions where heavy point loads
are introduced. But such restraint can only be provided at discrete positions. When the
load is distributed along the beam, or when point loads can occur anywhere along the
beam such as concentrated axle loads from vehicles, the distortional effects must be
carried by other means.
The distortional forces shown are tending to increase the length of one diagonal and
shorten the other. This tendency is resisted in two ways, by in-plane bending of each of
the wall of the box and by out-of-plane bending, is illustrated in Figure.
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Fig 10 Distortional displacements in box girder.
In general the distortional behavior depends on interaction between the two sorts of
bending. The behavior has been demonstrated to be analogous to that of a beam on anelastic foundation (BEF), and this analogy is frequently used to evaluate the distortional
effects.
If the only resistance to transverse distortional bending is provided by out-of-plane
bending of the flange plates there were no intermediate restraints to distortion, the
distortional deflections in most situations would be significant and would affect the
global behavior. For this reason it is usual to provide intermediate cross-frames or
diaphragms; consideration of distortional displacements and stresses can then be
limited to the lengths between cross-frames.
The distortion of section is not same throughout the span. It may be completely nil or
non-existent at points where diaphragms are provided, simply because distortion at
such points is physically not possible. The warping stresses produced by distortion are
different from those induced by the restraint to warping in pure torsion which is
encountered in elementary theory of torsion. The compatibility of displacements must
be satisfied along the longitudinal edges of the plate forming the box, which implies
that these plates must bend individually in their own plane, thus inducing longitudinalwarping displacements. Any restraint to this displacement causes stresses. These
stresses are called longitudinal warping stresses and are in addition to longitudinal
bending stresses. A general loading on a box girder such as for a single cell box, has
components, which bend twice and deform the cross section. Using the principles of
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super position, the effects of each section could be analyzed independently and results
superimposed.
Distortional stresses also occur under flexural component, due to poisson effect and the
beam reductance of the flange in multi cellular box, the symmetrical component alsogives rise to distortion stresses and it is significant percentage of total stresses. With
increase in number of cells, the proportion of transverse distortional stresses also
increase. How ever for a single cell box the procedure of considering only the
distortional component of loading for evaluation of distortional stresses in adequate for
practical purposes.
The concrete boxes in general have sufficient distortional stiffness to limit the warping
stresses to small fraction of the bending stresses, without internal diaphragms. But for
steel boxes either internal diaphragms or stiffer transverse frames are necessary to
prevent buckling of flanges as well as of webs and in most cases these will be sufficient
to limit the deformation of the cross section.
Sloping of the webs of box girder increase distortional stiffness and hence transverse
load distribution is improved. If section is fully triangulated, the transverse distortional
bending stresses are eliminated. This form could be particularly advantageous for
multicell steel boxes. Therefore distortion of box girder depends on arrangement of load
transversely, shape of the box girder, number of cells and their arrangement, type of
bridge such as concrete or steel, distortional stiffness provided by internal diaphragms
and transverse bracings provided to check buckling of webs and flanges.
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WARPING OF CROSS SECTION :
Warping is an out of plane on the points of cross section, arising due to torsional
loading. Initially considering a box beam whose cross section cannot distort because of
the existence of rigid transverse diaphragms all along the span. These diaphragms are
assumed to restrict longitudinal displacements of cross sections except at midspan
where, by symmetry the cross section remains plane. The longitudinal displacements
are called torsional warping displacements and are associated with shear deformations
in the planes of flanges and webs.
In further stage assume that transverse diaphragms other than those at supports are
removed so that the cross section can distort. (Fig). It results in additional twisting of
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cross section under torsional loading. The additional vertical deflection of each web also
increases the out of plane displacements of the cross sections. These additional warping
displacements are called distortional warping displacements/
Thus concrete box beams with no intermediate diaphragms when subjected to torsionalloading, undergo warping displacements composing of two components viz, torsional
and distortional warping displacements. Both these give rise to longitudinal normal
stresses i.e. warping stresses whenever warping is constrained. Distortion of cross
section is the main source of warping stresses in concrete box girders, when distortion
is mainly resisted by transverse bending strength of the walls and not by diaphragms.
.
SHEAR LAG :
In a box girder a large shear flow is normally transmitted from vertical webs to
horizontal flanges, causes in plane shear deformation of flange plates, the consequence
of which is that the longitudinal displacements in central portion of flange plate lag
behind those behind those near the web, where as the bending theory predicts equal
displacements which thus produces out of plane warping of an initially planar cross
section resulting in the SHEAR LAG". Another form of warping which arises when a box
beam is subjected to bending without torsion, as with symmetrical loading is known as
SHEAR LAG IN BENDING.
Shear lag can also arise in torsion when one end of box beam is restrained against
warping and a torsional load is applied from the other end fig 11. The restraint against
warping induces longitudinal stresses in the region of built-in-end and shear stresses in
this area are redistributed as a result which is an effect of shear deformation sometimes
called as shear lag. Shear distribution is not uniform across the flange being more at
edges and less at the centre fig 13.
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Fig:11
In a box beam with wide, thin flanges shear strains may be sufficient to cause the
central longitudinal displacements to lag behind at the edges of the flange causing a
redistribution of bending stresses shown in fig 12. This phenomenon is termed as STRESS DIFFUSION.
The shear lag that causes increase of bending stresses near the web in a wide flange of
girder is known as positive shear lag. Whereas the shear lag, that results in reduction of
bending stresses near the web and increases away from flange is called negative shear
lag fig 12. When a cantilever box girder is subjected to uniform load, positive as well as
negative shear lag is produced. However it should be pointed out that positive shear lag
is differed from negative shear lag in shear deformations at various points across the
girder.
At a distance away from the fixed end in a cantilever box girder say half of the span;
the fixity of slab is gradually diminished, as is the intensity of shear. From the
compatibility of deformation, the negative shear lag yields. Although positive shear lag
may occur under both point as well as uniform loading, negative shear lag occur only
under uniform load.
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Fig:12
It may be concluded that the appearance of the negative shear lag in cantilever boxgirder is due to the boundary conditions and the type of loading applied. These are
respectively external and internal causes producing negative shear lag effect.
Negative shear lag is also dependent upon ratio of span to width of slab. The smaller
the ratio, the more severe are the effects of positive and negative shear lag.
Fig:13
The more important consideration regarding shear lag is that it increases the
deflections of box girder. The shear lag effect increases with the width of the box and
so it is particularly important for modern bridge designs which often feature wide single
cell box cross sections. The shear lag effect becomes more pronounced with an increase
in the ratio of box width to the span length, which typically occurs in the side spans of
bridge girders. The no uniformity of the longitudinal stress distribution is particularly
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pronounced in the vicinity of large concentrated loads. Aside from its adverse effects on
transverse stress distribution it also alters the longitudinal bending moment and shear
force distributions in redundant structural systems. Finally, the effect of shear lag on
shear stress distribution in the flange of the box, as compared to the prediction of
bending theory is also appreciable. A typical situation in which large stress
redistributions are caused by creep is the development of a negative bending moment
over the support when two adjacent spans are initially erected as separate simply
supported beams and are subsequently made continuous over the support. In the
absence of creep, the bending moment over the support due to own weight remains
zero, and thus the negative bending moment which develops is entirely caused by
creep.
Fig 14 Effect of shear lag on distribution of stresses at the support of a box girder
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DIAPHRAGMS :
Advantage of closed section is realized only when distortion of cross section is
restricted. Distortion could be checked by two ways: First by improving the bending
stiffness of web and flanges by appropriate reinforcement, so as additional stresses
generated due to restraint to distortion are within safe limits. The Second alternative to
check distortion may be to provide diaphragms as shear walls at the section where it is
to be checked. These diaphragms distribute the differential shears of web to flanges
also by bending in plate ad by shear forces in diaphragm.
The introduction of diaphragms into box girders will have two effects on transverse
moments in slabs:
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1) If the diaphragm spacing is approximately equal to transverse spacing of webs,
transverse bending moments may be reduced as a result of two way slab action of
diaphragm support.
2) The moments caused by differential deflection will be eliminated over the regioninfluenced by diaphragms.
By the provision of diaphragms, transverse bending stresses caused by the moments,
resulting from differential deflection of top and bottom slabs are eliminated. Proper
spacing of diaphragms can be determined by the use of beam on elastic foundation
concept to effectively control differential deflection. The use of diaphragms at supports
which are definite locations of concentrated loading significantly diminishes the
differential deflections near the supports and should always be provided.
As far as possible interior diaphragms are avoided as they not only result in additional
load but also disrupt and delay the casting cycle resulting in overall delay
in construction . In general interior diaphragms would be needed for the box section,
which has light webs and supported by relatively stiff slabs. Such a form of cross
section is not appropriate for concrete box girders, although prestressing is done
externally this type of cross section is not justified.
Diaphragms which are stiff out of their planes, when provided at the supports, restrain
warping in continuous spans, resulting in stresses. These stresses add to longitudinalbending stresses. As conditions of maximum torque do not generally coincide with
conditions of maximum bending, and the warping stresses, if they occur, may not
therefore increase bending stresses to unacceptable values
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