Combined Plastic Bending and Compression or Tension

So far we have assumed that a plastic hinge forms if the bending moment in a critical cross section reaches the plastic limit value, M0. We have tacitly neglected the effect of the other internal forces on the formation of the yield hinge. This effect, however, can be appreciable, for example in multi-storey frames or frames with a large horizontal thrust where the axial forces are large. So, we will now refine our analysis and consider the interaction of bending moment M and normal force N . The resulting fully plasticized cross section will be referred to as a generalized plastic hinge. The effect of the shear force is usually less important, and is also more difficult to incorporate. We will postpone this problem to Part III, because it is beyond the scope of analysis under uniaxial stress. A typical evolution of the strain and stress profiles during the formation of a generalized plastic hinge is depicted in Figure 13.1. As usual, it may be assumed that the cross sections remain planar at all stages of the loading process. Consequently, the variation of normal strains over the cross section is linear. The corresponding stress distribution is linear only in the elastic range (Figure 13.1(a)). As the loading process continues, yielding starts at the top or bottom fibers, and the plastic zone propagates into the interior of the cross section (Figure 13.1(b)). During the elastoplastic stage, the cross section has an elastic core with linear stress variation, and one or two plastic zones with constant stress equal to the positive or negative yield stress. For very large curvatures, the elastic core becomes negligibly small, and the stress distribution approaches a piecewise constant distribution, with one part of the cross section yielding in tension and the remaining part yielding in compression. The plasticization process of course takes place in the neighboring cross sections as well, and the plastic zone occupies a certain volume (Figure 13.2(b)). However, for the purpose of modeling, we can lump the plastic hinge into one single cross section, the same as we did while analyzing the hinge under pure bending. The total plastic deformation in the idealized hinge is replaced by a rotation, θ, and longitudinal displacement, ep (Figure 13.2(c)). From kinematic considerations, it follows that the plastic extension at an arbitrary point of the plasticized cross section can be expressed by a linear function ēp(z) = ep + θz (13.1)