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# Plastic ‚ÄúHinges‚Äù

Once the cross-section is fully plasticized, it can rotate with no additional moment applied.

Plastic ‚ÄúHinges‚Äù

Once the cross-section is fully plasticized, it can rotate with no additional moment applied. Thus for a beam subjected to pure moment (with a constant moment diagram), the entire beam plasticizes simultaneously and there is unrestrained plastic flow. In the normal situation where the moment diagram varies, this is not the case.

Consider for example the EPS cantilever beams of rectangular cross-section with a concentrated load, P, at its free end shown in Figure 11.3a. Assume that the cross-section at the wall is on the verge of full plastification the stress distribution will be as shown in Figure 11.3d where the elastic zone extends ye from the neutral axis. Therefore . Since MA is a known function of x or x* for a given loading, the elastic-plastic interface can be determined from Equation (11.3). In this case, As long as the elastic core remains, plastic flow is constrained. However, when P increases such that PL = MP , the cross-section becomes fully plastic and the curvature can increase without limit but contained within the plas-tic zone. The beam rotates like a rigid body about a ‚Äúplastic hinge‚Äù holding the ultimate plastic moment MP . If, as in this case, the structure is determi-nate, it collapses and PL MP/L = KsMY/L. It should be noted that no pro-vision is made in this simple strength-of-materials approach for the effect of any shear stresses, œÑxy , or vertical stress, œÉy , which may be of some signifi-cance in short beams. The chapter problems explore these questions in a preliminary way.

On the other hand, the effect of axial load N combined with bending moment M in the plastic range is straightforward. Suppose N and M combine to give the stress state at yield shown in Figure 11.4b. The maximum stress in the bottom fiber is as the elastic interaction curve for yield shown in Figure 11.4d. When N1 is zero,  M1 = MY and similarly when M1 = 0, N1 = NY. Between these extremes there are any number of combinations of moment and normal force to cause yield.

If we call Nu and Mu the combination of normal force and moment to fully plasticize the cross-section, it is clear from Figure 11.4c that This interaction relationship plotted in Figure 11.4d gives all combinations of N and M for full plastification.

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Civil : Principles of Solid Mechanics : One Dimensional Plasticity for Design : Plastic ‚ÄúHinges‚Äù |