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.
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 of box 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 longitudinal warping stresses and are in addition to longitudinal bending stresses.
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 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.
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.
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.
Coming to the vertical reactions, let equal and opposite vertical forces be applied at A and B. In studying the longitudinal and transverse effects, it should be noted that finally all longitudinal effects have to be superimposed separately on the one hand, and transverse effects on the other. The vertical forces are denoted by P1 and P2 in fig 6. As shown, (a) = (b) +(c). Since (c) = (d) + (e), it is evident that (a) = (b) + (d) + (e). Now (b) and (d) are symmetrical loads and, as in the case of superimposed dead loads and self weight, do not create any torsional effects. Let the sum of all these symmetrical loads be denoted by Q, Q, acting at A and B fig. The loads Q, Q cause simple longitudinal flexure only and the structural effects caused are illustrated in fig 4(a). The loads P, P cause torsional effects in the box girder, and they are shown in fig b, c. The internal forces generated to counteract P, P are shown in 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 their original 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/2A, 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:
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, a box 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.
This restraint of warping gives rise to longitudinal warping stresses and associated shear stresses in the same manner as bending effects in each wall of the box. The shear stresses effectively modify slightly the uniformity of the shear stress calculated by pure torsion theory, usually reducing the stress near corners and increasing it in mid-panel. Because maximum combined effects usually occur at the corners, it is conservative to ignore the warping shear stresses and use the simple uniform distribution. The longitudinal effects are, on the other hand greatest at the corners. They need to be taken into account when considering the occurrence of yield stresses in service and the stress range under fatigue loading. But since the longitudinal stresses do not actually participate in the carrying of the torsion, the occurrence of yield at the corners and the consequent relief of some or all of these warping stresses would not reduce the torsional resistance
Fig 8 Warping of rectangular box subjected to pure torsion.
If torsional loading is applied, there are forces acting on the plate of elements, which tend to deform the cross section. The movements of the plate elements of the cross section cause distortion stresses in transverse direction and warping stresses in longitudinal direction.
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 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.
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 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 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.
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 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.
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, the more 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 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
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.
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