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

Abstract

Calculating the large deflection of a cantilever beam is one of the common problems in engineering. The differential equation of a beam under large deformation, or the typical elastica problem, is hard to approximate and solve with the known solutions and techniques in Cartesian coordinates. The exact solutions in elliptic functions are available, but not the explicit expressions in elementary functions in expectation. This paper attempts to solve the nonlinear differential equation of deflection of an elastic beam with the Galerkin method by successfully solving a series of nonlinear algebraic equations as a novel approach. The approximate solution based on the trigonometric function is assumed, and the coefficients of the trigonometric series solution are fitted with Chebyshev polynomials. The numerical results of solving the nonlinear algebraic equations show that the third-order approximate solution is highly consistent with the exact solution of the elliptic function. The effectiveness and advantages of the Galerkin method in solving nonlinear differential equations are further demonstrated.

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

“…The exact solution to Equation ( 2) can be given in elliptic functions known as the analytical solution of an elastica [26,27], as has been shown in earlier solutions [43,45,46]. The vibrations of such a beam has been studied with the EGM also in a similar procedure [4].…”

“…Mathematical Problems in Engineering is the iterative process by the successive determination of coefficients one at each time for the successive approximation. Both approaches have been demonstrated as part of the solution procedures of the EGM [4,5,[44][45][46]. In this study, the approximation starts from the linear solution…”