Developing and Investigation the Numerical Efficiency of Modified Energy Method in Solid Mechanics With Geometric Nonlinearity and Bifurcation Points

Geometric nonlinear analyses are used in many structural problems, such as the determination of failure load, as well as the study of buckling mechanism. Nevertheless, due to the complex nature of this type of problems and the absence of a comprehensive analytical solution for them, numerical methods are utilized in practice to approximate the exact response of these systems. In the application of numerical methods, there are also some difficulties such as divergence or finding the correct path of equilibrium, especially in the case of bifurcation points. Hence, the main purpose of this research is to apply the modified energy method (introduced in the dynamics of structures) in quasi-static problems with geometric nonlinearity and bifurcation points so that the efficiency of this method can be compared to others, such as Newtonian numerical techniques and force-displacement-constraint approaches. To achieve the objectives of this research, after briefly reviewing the current force-based computational methods in practice, the energy method is described for such problems, and then the step-by-step process of its computer implementation will be presented. Afterward, by coding in MATLAB software and applying the method to numerical examples employed by other researchers such as truss and frame structures, the numerical results are verified by analytical solution as well as those obtained by other methods, such as Newton-Raphson and Arc Length techniques. Generally, the interpretation of the results obtained from performed simulations has shown that the presented numerical method in analyzing nonlinear geometric problems has better accuracy compared to the Arc Length method; moreover, it can well pass through bifurcation points in the force-displacement curve without divergence in comparison with the Newton-Raphson method.