Just as the conventional matrix stiffness method of analysis (Art. 3.39) may be modified to directly include the influences of second-order effects (Art. 3.48), it also may be modified to incorporate nonlinear behavior of structural materials. Loads may be applied in increments to a structure and the stiffness and geometry of the frame changed to reflect its current deformed and possibly yielded state. The tangent stiffness matrix Kt in Eq. (3.180) is generated from nonlinear member force-displacement relationships. To incorporate material nonlinear behavior, these relationships may be represented by the nonlinear member stiffness matrix
account for the destabilizing effects of material nonlinearities.
In general, there are two basic inelastic stiffness methods for investigating frames: the plastic-zone or spread of plasticity method and the plastic-hinge or concentrated plasticity method. In the plastic-zone method, yielding is modeled throughout a members volume, and residual stresses and material strain-hardening effects can be included directly in the analysis. In a plastic-hinge analysis, material nonlinear behavior is modeled by the formation of plastic hinges at member ends. Hinge formation and any corresponding plastic deformations are controlled by a yield surface, which may incorporate the interaction of axial force and biaxial bending.
Dynamic loads are one of the types of loads to which structures may be subjected (Art. 3.26). When dynamic effects are insignificant, they usually are taken into account in design by application of an impact factor or an increased factor of safety. In many cases, however, an accurate analysis based on the principles of dynamics is necessary. Such an analysis is paticularly desirable when a structure is acted on by unusually strong wind gusts, earthquake shocks, or impulsive loads, such as blasts.