The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method

Abstract

For the optimal design and accurate prediction of structural behavior, the nonlinear analysis of large deformation of elastic beams has broad applications in various engineering fields. In this study, the nonlinear equation of flexure of an elastic beam, also known as an elastica, was solved by the Galerkin method for a highly accurate solution. The numerical results showed that the third-order solution of the rotation angle at the free end of the beam is more accurate and efficient in comparison with results of other approximate methods, and is perfectly consistent with the exact solution in elliptic functions. A general procedure with the Galerkin method is demonstrated for efficient solutions of nonlinear differential equations with the potential for adoption and implementation in more applications.

The Approximate Solution of the Nonlinear Exact Equation of Deflection of an Elastic Beam with the Galerkin Method

“…Focusing on the evaluation of the coefficients from nonlinear algebraic equations and the combination of series expansions, the approximate solutions can be obtained from the nonlinear equation to enhance the accuracy with fewer terms. This has been shown in our recent studies of nonlinear problems [36]. This path for better approximation will be taken in this study to demonstrate the new strategy and procedure of solving nonlinear differential equations by solving the nonlinear algebraic equations, in strong contrast to traditional approximate techniques aimed at obtaining much simpler linear equations from continuing linearization process.…”

“…This implies that the computational cost can be neglected because fewer terms can be evaluated if proper nonlinear solutions are tried. This is clearly shown in our recent papers dealing with the elastic beam problem with the new method [22,36] by selecting the trial solution satisfying the boundary conditions and the numerical solutions with the undetermined constants like coefficients of a polynomial or trigonometric function. Focusing on the evaluation of the coefficients from nonlinear algebraic equations and the combination of series expansions, the approximate solutions can be obtained from the nonlinear equation to enhance the accuracy with fewer terms.…”

“…As demonstrated in an earlier paper, the approximate solution is obtained with the determination of coefficients A 1 in Equation (10) [36].…”

“…Specifically, it is found that the Galerkin method, which is one of the essential and popular methods in the late development of numerical methods, including the finite element method for problems of solid mechanics and differential equations in general, can be used for nonlinear problems by just solving the resulting nonlinear algebraic equations for a good approximation. The procedure and effectiveness have recently been shown in a few papers [36,37]. Of course, for nonlinear vibrations, the time factor is considered through the weighted integration over one period with the periodic solutions by introducing the extended Galerkin method [35].…”

Genetic Algorithm Optimization of Rainfall Impact Force Piezoelectric Sensing Device, Analytical and Finite Element Investigation

“…At this stage, Equations ( 14) and ( 15) are employed to generate the time-domain response of the measured voltage signal of the proposed self-sensing device. The nonlinear Equation ( 14) can be solved analytically with several methods, including the extended Galerkin method, where the nonlinear equation of flexure of an elastic beam can be solved for a highly accurate solution and accurate prediction on nonlinear structural behavior [73,74]. In this paper, we use MATLAB to numerically solve Equations ( 14) and (15) and to generate the time-domain response of the output voltage signals of the three types of rainfall, as indicated in Figure 5.…”