New and powerful methods for stress and displacement analysis of rigid plane frames subjected to material nonlinearities are proposed on the basis of the principle of minimum complementary energy, the Crotti-Engesser theorem and mathematical programming techniques. The stress analysis problem is formulated as a minimization of the total complementary energy subject to the equilibrium equations at the free nodes. The unknown member end forces are determined by solving the energy minimization problem using a modified sequential quadratic programming and linear programming algorithms. The displacement at an arbitrary point i is determined as the first partial derivative of the minimized total complementary energy with respect to the load P i acting at the point i. The partial derivatives are numerically calculated using a finite difference formula which is embedded in the stress analysis algorithm. The problem formulation and analysis algorithms of the proposed methods are applicable to rigid frame structures with any type of material nonlinearity. The accuracy, reliability and efficiency of the proposed methods are demonstrated by comparing their results with those of the displacement method of analysis for several statically indeterminate rigid frame structures with three types of material nonlinearities.