Arc‐Length Method for Passing Limit Points in Structural Calculation

Abstract

A new solution procedure is presented, based on the arc‐length method, for passing limit points (load or displacement peaks) in nonlinear finite element analysis of structures. In addition to the usual equilibrium equations, a quadratic arc‐length constraint equation is specified so that the nonlinear solution is sought on a small ellipsoidal surface in load‐deflection space. The main new feature of the proposed procedure is the resolution of the out‐of‐balance loads into a parallel and an orthogonal component with respect to the vector of applied external loads. Methods for avoiding complex roots to the quadratic constraint equation, and for selecting the appropriate root, are presented. Applications of the proposed procedure to two reinforced concrete problems give satisfactory results. The method is effective and versatile in handling both snap‐through and snap‐back problems.