A QUADRATICALLY CONVERGENT METHOD FOR A DIRECT COMPUTATION OF SINGULAR POINTS OF ELASTIC-PLASTIC PLANE FRAMES

This paper presents the implementation and effectivness of the quadrat ically convergent method for the direct computation of singular points , i.e. bifurcation and limit points of an elastic-plastic plane frame. The method was first used in mathematical literature by Moore and Spe nce [1] in computing simple limit points. Its application in the finit e element analysis was first presented in 1988 by Wriggers, Wagner, an d Miehe [4] to determine limit and bifurcation points of non-linear el astic structures, particularly of three-dimensional truss and axisymme tric shell structures. Their formulation was generalized by Wriggers a nd Simo [5] to include elastoplasticity. The basis of the procedure is an extension of the system of discretized non-linear equilibrium equa tions by a set of constraint equations that characterize the presence of the singular point. The resulting extended system of algebraic equa tions is then solved by the quadratically convergent Newton's method. The solution of the extended system also provides the characteristic m ode and information to allow classification between limit and bifurcat ion points. Newton's method shows the quadratical convergence only if (i) the tangent stiffness matrix of the system is regular, and (ii) th e directional derivatives of the tangent stiffness matrix are accurate ly evaluated. In order to by-pass the need for explicit closed-form de rivatives, Wriggers and Simo [5] used the finite difference approximat ion of the directional derivatives. To improve the regularity of the t angent stiffness matrix and the rate of convergence of Newton's method near a bifurcation point, they used a penalty regularization of the m atrix. The present article shows the results of the direct method when implemented in the Saje elastic-plastic planar beam finite elements [ 2]. The characteristics of the Saje finite elements are: (i) only rota tion of the axis needs to be approximated; (ii) rotations of internal nodes and two additional unknowns, Lagrangian multipliers, are elimina ted (condensed) on the element level. Directional derivatives, needed in the direct procedure, are evaluated analytically and numerically, a nd the rates of convergence are compared. It is shown that the rate of convergence not only strongly depends on the finite difference increm ent, but depends also on material parameters. Numerical results indica te that the penalty regularization of the tangent stiffness matrix is not necessary.