This chapter presents the foundation of stiffness formulations for four well-known hysteresis models, namely, elasto-plastic, bilinear, curvilinear, and Ramberg-Osgood. These models have a wide range of engineering applications for both reinforced concrete and steel structures. The chapter illustrates numerical procedures for response analysis of members' forces and structural displacements. It shows various effects of P-∆ large deflection, and material nonlinearity on structural response behavior. The chapter demonstrates that correctness check of inelastic numerical solutions based on equilibrium and compatibility requirements. The Bauschinger effect can be observed in the moment-curvature relationship of a structural member. This relationship is determined by the material stress-strain relationship and the cross-section shape of that member. The reduced moment-curvature relationship should be used in the hysteresis models for nonlinear analysis. The chapter provides eight numerical examples to illustrate detailed calculations of system response and member forces as well as equilibrium and compatibility checks.