Investigation of critical load of structures using modified energy method in nonlinear-geometry solid mechanics problems

Abstract

Geometrically nonlinear analysis is required for resolving issues such as loading causes failure and structure buckling analysis. Although numerical methods are recommended for estimating the exact solution, they lack the necessary convergence in the presence of bifurcation points, making it challenging to find the equilibrium path using these methods. Thus, the modified energy method is employed instead of the numerical method, frequently used to solve quasi-static problems with nonlinear nature and bifurcation points. The ultimate goal of this study is to determine the critical load of structures through the modified energy method rather than other methods in which the relationship between force, displacement, and constraint is used to solve the problem. This study first describes the energy method for this type of problem and then details its computational steps progressively. This method yields numerical results when applied to numerical examples such as truss and frame structures and coded in MATLAB software. These findings are compared to the analytical results. The energy method is more precise than the alternative methods and superior to the Newton–Raphson method at crossing the load–displacement curve’s bifurcation points.

Applications of common energy variability methods for establishing the equilibrium equation in mechanics

“…The Euler -Bernoulli beam theory transforms the 2D problem to this of 1D with the beam motion is the deformation w(x). The internal force in the beam is the bending moment M determined by the following relationship Equation (5). (5) where q is the distribution of the external force, EJ is stiffness against bending.…”

“…The internal force in the beam is the bending moment M determined by the following relationship Equation (5). (5) where q is the distribution of the external force, EJ is stiffness against bending. Equation (5) provides the relationship between bending moment M and deformation (curvature of the elastic line) According to the Lagrange equation, the kinetic energy and deformation potential of the beam can be found in Equation (5) Let wi be the displacement of point i; qi is the external force of point i, mi is corresponding mass.…”

“…From the 19th century, the available approaches for structural engineers basically followed the energy principles and the equilibrium equation of forces as well as strain. In structural analysis, energy methods have been demonstrated as a significant approach, particularly in indeterminate structures [5,6]. Energy methods can be utilized for various structural types or even can be applied as indirect methods of establishing equilibrium equations.…”