Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

Abstract

An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

A Method for Comparison of Large Deflection in Beams

“…In the first approximation of the Euler-Bernoulli beam theory [1,5,8,10,11,13,[15][16][17][18][19][20][21][22][23][24][25][26][27], the following simplifying assumptions are considered in calculating the displacement field:…”

“…Li et al [13] improved the homotopy analysis method to solve the strongly nonlinear differential equation, for example for a cantilever beam subjected to point a load at the free end. The results were validated with the traditional homotopy method.…”

Chaotic vibration control of a composite cantilever beam

“…Repka et al 6 applied the Timoshenko beam model in the analysis of the flexoelectric effect for a cantilever beam under large deformations, and considered the geometric nonlinearity with von Kármán strains. Meanwhile, some methods, such as a homotopy analysis method 7 , a rational elliptic balance method 8 , an enriched multiple scales method 9 , and an improved homotopy analysis method 10 , 11 , etc., have been gradually developed to solve nonlinear differential equations.…”

Chaotic Vibration Control of a Composite Cantilever Beam

“…Repka [6] et al applied the Timoshenko beam model to the analysis of the flexoelectric effect for a cantilever beam under large deformations, and considered the geometric nonlinearity with von Kármán strains. Meanwhile, some methods, such as homotopy analysis method [7], rational elliptic balance method [8], enriched multiple scales method [9], improved homotopy analysis method [10][11], etc, have been gradually developed to solve nonlinear differential equations.…”