An Efficient Strategy for Solving Structural Nonlinear Equations by Combining the Orthogonal Residual Method and Normal Flow Technique

This paper presents a new procedure for solving structural nonlinear problems by combining the orthogonal residual method (ORM) and normal flow technique (NFT). The perpendicularity condition to the Davidenko flow, introduced by the NFT, which must be satisfied during the iterative process, overcome the difficulties, i.e. the poor convergence and inefficiency of the ORM close to the limit points, particularly the displacement limit points (snap-back behavior). Basically, the idea of the proposed strategy is to adjust the load parameter, which is treated as a variable in the nonlinear incremental-iterative solution process, assuming that the unbalanced forces (residual forces) must be orthogonal to the incremental displacements. This constraint is used together with the NFT perpendicularity condition. The proposed procedure is tested, and its efficiency is corroborated through the analyses of slender shallow and nonshallow arches and an L-frame since they exhibit highly nonlinear behaviors under certain loading conditions. It is concluded that the proposed procedure can overcome the numerical instability problems in the neighborhood of critical points when using only the conventional OR process, and the procedure compares favorably with the arc-length method, minimum residual displacement method, and generalized displacement control method.

Strong-Form Formulated Generalized Displacement Control Method for Large Deformation Analysis

The traditional analysis of geometric nonlinearity is mostly based on the weak-formulated Galerkin method such as the finite element method. The element nature has limited its application as a result of numerical integration in the governing equation and quality control of deformed mesh. In the middle of 1990s, the meshfree methods have been developed and become one leading research topic in computational mechanics. Especially, the strong form collocation methods require no additional efforts to process numerical integration and impose Dirichlet boundary condition, thereby making the collocation methods computationally efficient. In the incremental–iterative process, how to accurately reflect the change in the slope of the load–deflection curve of the structure and remain numerically stable are of major concerns. Thus, we propose a strong-form formulated generalized displacement control method to analyze geometric nonlinear problems, where the radial basis collocation method is adopted. The numerical examples demonstrate the ability of the proposed method for large deformation analysis.

GEOMETRIC NONLINEAR ANALYSIS OF CABLE STRUCTURES WITH A TWO-NODE CABLE ELEMENT BY GENERALIZED DISPLACEMENT CONTROL METHOD

This paper presents a two-node catenary cable element for the analysis of three-dimensional cable-supported structures. The stiffness matrix of the catenary cable element was derived as the inverse of the flexibility matrix, with allowances for selfweight and pretension effects. The element was then included, along with the beam and truss elements, in a geometric nonlinear analysis program, for which the procedure for computing the stiffness matrix and for performing iterations was clearly outlined. With the present element, each cable with no internal joints can be modeled by a single element, even for cables with large sags, as encountered in cable nets, suspension bridges and long-span cable-stayed bridges. The solutions obtained for all the examples are in good agreement with the existing ones, which indicates the accuracy and applicability of the element presented.