Geometrical Interpretation of the Arc-length Method

The arc-length method is a powerful solution technique becoming increasingly popular among researchers and engineers. This method is presented here as a particular case of a more general formulation which includes all other solution strategies. The arc-length method is derived in its continuous and discrete formulations. Two versions of the arc-length method (Crisfield and Ramm) are presented and compared using a geometrical interpretation. Advantages and disadvantages of each method are pointed out. A new method, called the modified Crisfield-Ramm method, is proposed. This improved arc-length method combines the advantages of the two parent methods. INTRODUCTION The nature of special problems encountered in structural analysis often requires the use of sophisticated analytical techniques. Particularly, optimization constraints combined with safety requirements enhance the need for better understanding of structural behaviour. For these reasons, nonlinear finite element analysis of structures increases constantly. This type of analysis is used either as a powerful research tool, or in the design process of complex or unique structures. The objective of such an analysis is to obtain the equilibrium states of a structure at various load levels. All these equilibrium states trace the load-displacement response of the structure in which the applied load varies proportionally as a function of a unique load parameter. In such a case, for an n degree-of-freedom (DOF) system, n + 1 unknowns define completely the problem. The finite element method, based on continuum mechanics, provides n relationships describing the equilibrium states of a structure. This is expressed in a single equation {R({u), n)} = ltF> IRint<{u))} = O. (1) The parameters of this equation are {R({u}, A)}, the residual out-of-balance nodal forces; {II}, the n unknown DOF; 1, a scalar load parameter; {F}, the reference load vector; {& ), the internal nodal forces. To complete the definition of the problem with n + 1 unknowns, an additional equation is introduced, depending on the solution strategy adopted