Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Abstract

Explicit solutions to vertical and horizontal displacements are derived for large deformation of a cantilever beam under point load at the free end by an improved homotopy analysis method (IHAM). Quadratic and cubic nonlinear differential equations are adopted to construct more proficient nonlinear equations for vertical and horizontal displacements respectively combined with their currently available nonlinear displacement equations. Higher-order nonlinear iterative homotopy equations are established to solve the vertical and horizontal displacements by combining simultaneous equations of the constructed nonlinear equations and the auxiliary linear equations. The convergence range of vertical displacement is extended by the homotopy-Páde approximation. The explicit solutions to the vertical and horizontal displacements are in favorable agreements with the respective exact solutions. The convergence ranges for a relative error of 1% by the improved homotopy analysis method for vertical and horizontal displacements increases by 60% and 7%, respectively. These explicit formulas are helpful in practical engineering design for very slender structures, such as high-rise buildings and long bridges.

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

“…With the procedure and algorithm outlined above, a typical nonlinear equation is needed to test the procedure and accuracy. Among the many typical nonlinear differential equations, the large deformation of a beam under a concentrated force at the free end, also known as the elastica, is [14,15,[31][32][33]…”

“…Wang et al used the homotopy analysis method (HAM) for this problem with approximate solutions in power series with a large radius of convergence [15]. An elastic beam with relatively large deformation under a concentrated load is a common structural problem in both civil and mechanical engineering [10,11,14,15,[32][33][34]. The analytical methods and exact solution to this problem are very important for the design and optimization of these structures.…”

“…With classical methods of solutions failing in these contexts, in recent decades, some special methods with gradual sophistication have been developed, such as the multiscale method [10], homotopy analysis method [12][13][14][15], homotopy perturbation method [16], and others dealing with emerging differential equations in many fields [17,18]. Generally, most of these methods are asymptotic in nature, and can find solutions in series form with an improved converging rate and numerical stability over a relatively large interval.…”

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.…”

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

“…In addition to many approximate methods, the problem was eventually solved with the definition of elliptic functions by Bisshopp and Drucker [15]. Since then, the exact solutions have been used to compare and validate many approximate solutions [16][17][18][19][20]. The search for reasonable approximate solutions has never stopped, and there are many attempts for simple and accurate solutions as part of the study of nonlinear differential equations [21][22][23][24][25][26].…”