Dis Torsion

8/8/2019 Dis Torsion

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Dis Torsion

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 the cross 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.

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 an elastic 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

longitudinal warping 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 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 also gives rise to


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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.

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 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 torsional loading,

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.

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 slabis 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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It may be concluded that the appearance of the negative shear lag in cantilever box girder 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, themore severe are the effects of positive and negative shear lag.

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 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 alsoappreciable. 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.

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:

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 region influenced by

diaphragms.


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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 longitudinal bending 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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