Solutions of linear and nonlinear problems related to torsional rigidity of a beam

Linear and nonlinear boundary value problems related to elastic and elastoplastic torsional rigidity of a beam are considered. As a sample model, pure elastic torsion problem for a square cross section bar is solved by the method of separation variables. The well-known analytical formula for torsional rigidity is obtained. The nonlinear boundary value problem related to quasi-static elastoplastic torsion of a beam, given by monotone increasing values of the angle of twist per unit length, is modeled in view of monotone potential operators. Numerical modeling of the considered problems are demonstrated for power hardening engineering materials. The proposed model permits one to predict some elastoplastic torsional effects arising in the deformed beam, depending on the angle of twist, as well as physical parameters.