Inelastic Finite Deformation of Planar Frames

A theoretical analysis and a numerical procedure are presented for the determination of the inealstic load-deformation relationship, as well as buckling or collapse load for a planar frame made of general work-hardening materials subjected to general in-plane loadings and loading histories. Lagrangian descriptions are used to describe the variables involved in the finite deformations. A variational principle involving Kirchhoff stress tensor, green strain tensor, and their rates is established and employed in a numerical procedure to determine, in a quasi-static sense, the velocities caused by given loadings and their rates. The derivatives in the functional are expressed in terms of finite differences of the velocities at chosen locations and the integration is replaced by a finite summation. The variation of the functional with resepct to each velocity leads to a set of linear algebraic equations for the solution of the discrete velocities and the subsequent displacements. The interactions between a number of members meeting at a joint are governed by a set of equations also obtained by the variational principle. The test results of two aluminum frames are presented.