A comparative study on application of Chebyshev and spline methods for geometrically non-linear analysis of truss structures

• An efficient iterative method is developed for non-linear analysis of truss structures.
• The competency of four popular iterative methods was comparatively evaluated.
• The very rapid convergence scheme capable of non-linear analysis is developed.
• Geometrical non-linearity of large-scaled truss structures is investigated.
• A comprehensive algorithm capable of computer programing is improved.

In this paper, the effectiveness of the modified Chebyshev and cubic spline׳s iterative methods is comparatively evaluated on geometrically non-linear analysis of truss structures. For the purpose of a comprehensive comparison, we have also proposed an iterative method free from second derivative originated from modified Chebyshev and cubic spline׳s schemes. The method involves a set of predictor–corrector schemes constructed by Chebyshev as the predictor for spline correctors to improve the approximation of the tangential stiffness matrix. The numerical assessment of the proposed method lies on three-step algorithm with satisfactory convergence of results. The analysis of convergence is carried out and is shown that the proposed method is at least third-order convergent. A simple step-by-step algorithm is developed capable of tracing the non-linear equilibrium curve until the first limit point through an incremental approach. The robustness and efficiency of the proposed schemes are comparatively investigated against classical Newton–Raphson׳s method for solving practical non-linear problems. It is concluded that for the large structural systems, where a large-scaled stiffness matrix is being iteratively updated, the best computational time, thus the optimum cost of analysis is accomplished by the proposed algorithm using reasonably less number of incremental loads. Finally, it is demonstrated that the proposed procedure and spline׳s method require considerably less number of iterations to reach the sufficient accuracy.