A General Loading on a Box Girder 1

8/10/2019 A General Loading on a Box Girder 1

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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.

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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.

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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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A General Loading on a Box Girder 1

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