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.
Highlights
Given that the traditional analytical methods for solving nonlinear equations such as perturbation [9], methods introduced in [10,11], Adomian decomposition method [12], and δ-expansion method [13], in addition to the elliptical integrals mentioned above, cannot provide accurate, convergent and simple solutions, we aim to derive an explicit and accurate expression for rotation angle θb, vertical displacement δv and horizontal displacement δh to minimize the potential unpredictability in the large deformation of the long cantilever-like building
The improved homotopy analysis method was used to obtain the explicit solution to the rotation angle for the large deformation of a cantilever beam under point load at the free end
When 0 < α ≤ 0.5, ε1 increases from 0 to 1, the series of solutions of vertical displacement converge to the exact solution with relative error to the exact solution is less than 1%, so that the effective region of ε1 is identified to be Rε1 = [0, 1]
Summary
In view of the shortage of land worldwide, especially in metropolitan areas, high-rise building is a prioritized land-efficient architectural form developed more than 100 years ago. Given that the traditional analytical methods for solving nonlinear equations such as perturbation [9], methods introduced in [10,11], Adomian decomposition method [12], and δ-expansion method [13], in addition to the elliptical integrals mentioned above, cannot provide accurate, convergent and simple solutions, we aim to derive an explicit and accurate expression for rotation angle θb, vertical displacement δv and horizontal displacement δh to minimize the potential unpredictability in the large deformation of the long cantilever-like building. The improved homotopy analysis method was used to obtain the explicit solution to the rotation angle for the large deformation of a cantilever beam under point load at the free end. Based on the explicit solution to the rotation angle obtained in our previous paper, explicit expressions for the deflection and horizontal displacement are derived using the improved homotopy analysis method in this paper. We apply the improved homotopy analysis method to derive the explicit expressions of deflection and horizontal displacement using Equations (2.3b), (2.4b), (2.3c), and (2.4c) on the basis of the derived explicit expressions of rotation angle in Equations (2.9) and (2.10)
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