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# Limit Load (Collapse) of Beams

With the concept of a plastic hinge it is possible to determine the collapse or limit load of structures by the upper-bound theorem (as was done for thick rings and the bearing capacity problem in previews pages).

With the concept of a plastic hinge it is possible to determine the collapse or limit load of structures by the upper-bound theorem (as was done for thick rings and the bearing capacity problem in previews pages). Any type of structure: beams, frames, arches, plates, even shells, are susceptible to this approach no matter how complicated or irregular the geometry. Various collapse â€śmechanismsâ€ť are assumed and the limit or collapse load computed for each from equilibrium or conservation of energy. That mechanism corresponding to the lowest load is critical. For linear structures (one-dimensional from the standpoint of equilib-rium) this minimization procedure by analytic analysis or by trial and error leads to an â€śexactâ€ť upper-bound solution since only the number and longitudi-nal position of the discrete plastic hinges are unknown for a limited number of possible types of collapse mechanisms. For plates and shells, the exact mecha-nism is much more difficult to determine, but it is often not really necessary for a good approximation since the type of mechanism is seldom in doubt.

Only a few examples will be given in this and the following section to illus-trate the fundamental idea and to show how similar it is to the yield surface concept used for upper-bound analysis for two-dimensional stress fields in the previous chapter.* Take first the simple case of a once-indeterminate beam with a uniform load shown in Figure 11.5. Two plastic hinges, one at the wall and one near midspan, is the only possible collapse mechanism. The second hinge will be, by the upper-bound theorem, located such that the limit load wL is a minimum. The minimization procedure can be done graph-ically by trial and error as shown or formally. Since (L x)Î±=xÎ¸the internal energy dissipation from the rotation of the plastic hinges equated to external work done by wL gives: The calculation of the limit load is seen to be independent of the elastic analysis. The individual pieces of the structure move as rigid bodies after the plastic hinges form and the plastic energy, concentrated at the hinges, can be made arbitrarily large for an EPS material. Thus the elastic contribution can be neglected.

With the formation of the first plastic hinge, the one-degree indeterminate beam becomes determinate and when the second one forms, the structure is unstable (on the verge of becoming dynamic). This gives the extra equation to calculate the limit load. Since the most complicated indeterminate struc-ture is reduced to a simple computation in statics, limit analysis is very pop-ular. Moreover, small support settlements or rotations will not affect the limit load while they have a great influence on the elastic forces and moments in indeterminate structures.

The price paid for this simplification is the loss of any straightforward cal-culation of displacements beyond first yield.* As outlined in Figure 11.5, two interacting nonlinear effects are occurring. The plastification of the first hinge redistributes the moment diagram which, in turn, influences the location of the next plastic hinge. The increase in moment, and therefore load to cause the first plastic hinge is the shape factor, Ks, which, as we have seen, is a geo-metric property of the shape of the cross-section. The added strength intro-duced by the indeterminacy and geometry of the structure can be called the redistribution factor, KR, defined as For the beam in Figure 11.5, KR = 1.46 and Ks for a rectangular cross-section is 1.5 giving a combined plasticity factor KP = Ks KR= wL/wY = 2.19. Thus, if the beam were designed to give a factor of safety of 1.5 against yield, it would have a factor of safety against collapse of (1.5)(2.19) = 3.28. Even in an H-beam with Ks = 1.12 and a working load of two-thirds, the yield load would have a factor of safety against collapse of 2.46 if supported as in Figure 11.5. Such large reserves against flexural failure are one reason indeterminate bridges and other structures built in the first half of this century using the working-stress design code are still functioning well, even though trucks and trains are now so much heavier.

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Civil : Principles of Solid Mechanics : One Dimensional Plasticity for Design : Limit Load (Collapse) of Beams |