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

: 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-P á de 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.


Introduction
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. High-rise building is the product of the comprehensive application of modern sciences and engineering technologies. Smooth construction and sustainability of this new structural form depends on more accurate calculation theory and method than those for non-high-rise structures in light of that the extrusion of high-rise building into space could be as high as 800 m plus, and slight artefacts could be exaggerated into intractable unpredictability. Since the 21st century, beam-like structures have achieved overwhelming popularity, such as in high-rise buildings [1,2], long-span bridges [3,4], robots and aerospace; however, analysis methods for this form of structure are still those conventional Euler-Bernoulli laws for large deflection beams, which contain implicit integral to hinder general engineering application. An explicit higher-resolution analysis method is expected to tackle the high nonlinearity in high-rise buildings or long bridges.
The general theory of elastic deflection is called Elastica. To solve a general problem of elastic large deflection, the equivalent load, deflection and longitudinal displacement can be expressed by an analytic expression of elliptic integrals with free-end rotation angle as parameters [5]; indeed, the results have been reported extensively in the classical works of elastic mechanics [6][7][8]. For a cantilever beam subject to point load, the solution to this elastic problem is still an implicit solution with an elliptic integral.
In this study, the high-rise building is simplified as a large deformation of a cantilever beam subject to point load at the free end, for example, of a loading extremity, as shown in Figure 1. 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 θ b , vertical displacement δ v and horizontal displacement δ h to minimize the potential unpredictability in the large deformation of the long cantilever-like building. Unlike the perturbation technique, the homotopy analysis method (HAM) is independent of any small/large physical parameters. HAM provides us great freedom to choose equation-type and solution expression of the high order approximation equations. A convergence-control parameter h is introduced into the series solutions to guarantee the convergence. HAM distinguishes itself from other analytic approaches to be applied to complicated problems with strong nonlinearity [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The Föppl-Von Kármán's plate equations were solved by HAM [28,29]. Especially, the steady-state resonant waves were firstly predicted by the HAM in theory [21,22,30,31], which were then confirmed experimentally [30].
Ji Wang et al. derived an explicit solution to the large deformation of a cantilever beam under point load at the free end with HAM [32]. However, the improficiency of this method is identified to be a low convergence rate due to the step-by-step iteration of solutions of linear differential equations to approximate the exact solutions of complex non-linear differential equations; moreover, it fails to serve explicit solutions of vertical and horizontal displacements for large deformation of the beam.
In our previous paper, "Explicit Solution to Large Deformation Cantilever Beam by Improved Homotopy Analysis Method I: ROTATION ANGLE", an improved homotopy analysis method (IHAM) was proposed. In IHAM, an auxiliary nonlinear operator ε is introduced to improve the convergence rate and extend the convergence range. 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.
For small deflection, the horizontal displacement of the cantilever beam is neglected and only the deflection and rotation angle are calculated. For large deflection, horizontal displacement is signified not to be neglected. 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.

Problem Description
The bending equations of a uniform cross-section beam with large deformation are [5][6][7][8] dθ(s) subject to boundary conditions where s is the arc-coordinate of the neutral axis of the beam; x denotes the horizontal coordinate from the fixed end; L represents the length of the beam; F stands for the point load at the free end; EI is the bending stiffness of the beam; θ specifies the rotation angle of cross-section of the beam; and l is the horizontal distance of two ends and is unknown. Differentiating the equation with respect to s and then using the dimensionless vari- subject to boundary conditions where the prime denotes the differentiation with respect to ξ. For infinitesimal deformation, the below linear equations are sufficiently accurate subject to boundary conditions Accordingly, the solutions are The rotation angle, vertical and horizontal displacement of cross-section plane at end as denoted by In our previous paper, "Explicit Solution to Large Deformation Cantilever Beam by Improved Homotopy Analysis Method I: ROTATION ANGLE", the improved homotopy analysis method was proposed to solve Equations (2.3a) and (2.4a), and a series of solutions to the rotation angle are derived as [27]: 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).

Fundamentals of Improved Homotopy Analysis Method
For quick reference, the new improved homotopy analysis method (IHAM) is briefed. The IHAM is essentially different from the HAM. Table 1 compares the formulations of IHAM and HAM. Table 1. Comparison of improved homotopy analysis method against homotopy analysis method.

Homotopy Analysis Method (HAM) Improved Homotopy Analysis Method (IHAM)
Original N(ξ)= 0 and Auxiliary L(ξ)= 0 ⇒ Higher order H(ξ; q)= 0 1 Original N(ξ)= 0 and Auxiliary N 0 (ξ)= 0 ⇒ New N(ξ; q)= 0 2 New N(ξ; q)= 0 and Auxiliary L(ξ)= 0 ⇒ Higher order H(ξ; q)= 0 Notes: N-nonlinear operator, L-linear operator, N 0 -auxiliary nonlinear operator, N-new nonlinear operator, H-homotopy operator, ξ-independent variable, q-embedding parameter. Substantial difference lies in the different construction methods. HAM constructs a higher order nonlinear iterative homotopy differential equation by the selected linear differential equation and the original non-linear differential equation. However, IHAM builds up a higher-order nonlinear iterative homotopy differential equation using the selected linear differential equation, the selected simple non-linear differential equation and the original non-linear differential equation (See Table 1 where in f (w) is a derivable function in compliance with the boundary conditions of f w (0) = c 0 , f w (0) = c 1 , 1 2! f w (0) = c 2 , 1 3! f w (0) = c 3 ; w(ξ) is the selected polynomial function of an independent variable in Equations (2.9) and (2.10). The boundary condition in Equation (2.4b) or Equation (2.4c) can be expressed as v(0) = 0 (2.12) Note that it is unnecessary to assume the existence of any small or large parameters in Equation (2.11). Thus, the proposed approach is general.
Linearizing original Equation (2.11) yields which complies to the boundary condition of v(0) = 0 (2.14) We choose the simplest auxiliary nonlinear equations of herein, c 0 , c 1 , c 2 and c 3 are undetermined coefficients; and Detailed formulation for the improved homotopy analysis method could be found in our previous paper titled "Explicit Solution to Large Deformation Cantilever Beam by Improved Homotopy Analysis Method I: ROTATION ANGLE".
With reference to Equation (3.17), by expanding ν(ξ; q) into Taylor series of the embedding parameter q, we have It is worth noting that the zero order deformation Equation (3.15) contains auxiliary parameters ε 1 , h 1 . Auxiliary function H 1 (ξ; q, ε 1 ) has the form of Assuming that h 1 and ε 1 are properly chosen, so that Equation (3.18) is convergent at q = 1. Thus, we have The governing equations of V n (ξ) can be deduced from the 0th-order deformation Equations (3.15) and (3.16).

High Order Deformation Equations
Substituting Equation (3.19) into Equation (3.15) and differentiating Equation (3.15) n times with respect to the embedding parameter q, then dividing by n! and setting q = 0, we have the n th-order deformation equation as below, and Combining Equations (3.8) and (3.23), we have These equations can be solved by symbolic calculation software such as Maple, Matlab, etc. Thereafter, the linear high-order deformation Equations (3.21) and (3.22) can be solved explicitly.
The solutions given by the IHAM contains auxiliary parameters ε 1 and h 1 , which control and adjust the convergence region and rate of IHAM solution series.

Accurate Solution of Vertical Displacement
From Equation (2.3), the transcendental equation used to solve V b is as follows herein E(µ) is complete elliptic integral of the second kind; E(φ, µ) is the elliptic integral of the second kind; and From Equation (2.8b), the linear solution of the vertical displacement at the free end is When the relative error ∆ ≤ 1%, the convergent region of the linear solution is Accordingly, the optimal convergence parameter for the full coverage is derived as When n = 2, h = −1, ε = 8/π 2 , the 2nd-order approximation of V b by IHAM is For example, the 2nd-order approximation of When the relative error ∆ ≤ 1%, the convergent region of the 2nd-order approximation by IHAM is Table 2 shows 2nd order numerical solutions V by IHAM with different values ε 1 compared with exact solutions V b . The relative error in Table 2 is calculated by Exact Solution ε 1 = 0 (HAM) ε 1 = 24(π−2)/π 3 (IHAM)   After two iterations, when ε 1 = 0, the convergence region of V b is 0 <α ≤ 0.5. When ε 1 = 24(π − 2)/π 3 , the convergence region of V b is 0 < α ≤ 1.5, whence the convergence region is expanded by 200% compared to the case with ε 1 = 0.
When 0 < α ≤ 0.5, ε 1 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 ε 1 is identified to be Thus, the optimal control parameters for the convergence region of the improved homotopy analysis method is identified to be ε 1 = 24(π − 2)/π 3 .

Control Parameters
When the relative error ∆ ≤ 1%, the convergent regions of the 10th-order approximation by IHAM are Table 3 compares the 10th-order numerical solutions with the exact solutions. The relative error in Table 3 is calculated by It is seen that when h 1 = −0.5, the convergence region is 128% larger than that with h 1 = −0.3, while it is 6.7% larger than that with h 1 = −1 (See Table 3).
Accordingly, the optimal control parameters h 1 of convergence rate for the 10th-order approximation solution to the vertical displacement is identified to be h 1 = −0.5.
From the above comparison, it can be seen that the expansion rate of the convergence region for vertical displacement by IHAM against that by HAM increases with the increasing iterations.
With reference to Equation (4.10), by expanding µ(ξ; q) into Taylor series of the embedding parameter q, we have It is worth noting that the zero order deformation Equation (4.16) contains auxiliary parameters ε 2 , h 2 . And the auxiliary function H 2 (ξ; q, ε 2 ) has the form of Assuming that h 2 and ε 2 are properly chosen, so that Equation (4.19) is convergent at q = 1, thus, we have The governing equations X n (ξ) can be deduced from 0th-order deformation Equations (4.16) and (4.17).

Equation (4.24) is subject to boundary condition of
and Combining Equations (4.8) and (4.26), we have The right-hand side term R u n of the above equations can be derived by symbolic calculation software such as Maple, Matlab, etc. Thereafter, explicit form of solution to the linear high-order deformation Equations (4.24) and (4.25) have been derived.
With reference to Equation (2.8c), the linear solution of the horizontal displacement at the free end is U b = 0 (4.30)

Explicit Solution to Horizontal Displacement by IHAM
The solutions by IHAM contains two auxiliary parameters ε 2 and h 2 , which control and adjust the convergence region and rate of IHAM solution series.

Error Analysis of Vertical Displacement
The influence of the control parameters ε 1 and h 1 on the error of vertical displacement after n= 5 iterations with α= 2.0 is then discussed.
After five iterations, the improved homotopy analysis method is adopted to solve the Equations (3.21) and (3.22) to yield the solution to the original Equations (2.3b) and (2.4b) as below When α= 2.0, Equation (5.1) becomes When α= 2.0, the exact solution is V b = 0.49346. When the iteration is five times, the relative error between the result by IHAM and the exact solution can be expressed as According to Equations (5.3) and (5.4), the relative error curve of control parameter ∆ V − ε 1 is a broken line. When the relative error is less than ∆ V ≤ 1%, the convergence interval is R ε 1 = [0. 4,1], which is the effective region of ε 1 . Figure 4 shows the relative errors curve ∆ V − ε 1 of Equation (2.3b) when α= 2.0, h 1 = −0.68. From Figure 4, when ε 1 = 24(π − 2)/π 3 , the relative error is the minimum. According to Equations (5.3) and (5.5), the relative error curve of control parameter ∆ V − h 1 is a quintic power function. When the relative error is less than ∆ V ≤ 1%, the convergence interval is R h 1 = [−1, − 0.58], which is the effective region of h 1 . Figure 5 gives the relative errors curve ∆ V − h 1 of Equation (2.3b) when α= 2.0, ε 1 = 24(π − 2)/π 3 . From Figure 5, when h 1 = −0.68, the relative error is the minimum.

Error Analysis of Horizontal Displacement
The influence of the control parameter ε 2 and h 2 on the error of horizontal displacement after 5 iterations when α= 2.0 is then discussed.
After five iterations, the improved homotopy analysis method is adopted to solve the Equations (4.24) and (4.25) to derive the solution to the original Equations (2.3c) and (2.4c). Thus, When α= 2.0, Equation (5.6) becomes When α= 2.0, the exact solution is U b = 0.16064. After 5 time iterations, the relative error between the results by IHAM method and the exact solution can be written as When h 2 = −0.465, Equation (5.7) becomes U b = 0.16603 − 0.0065196ε 2 (5.9) According to Equations (5.8) and (5.9), the relative error curve of control parameter ∆ U − ε 2 is a broken line. When the relative error is less than ∆ U ≤ 1%, the convergence region is R ε 2 = [0.58, 1], which is the effective region of ε 2 . Figure 6 shows the relative errors curve ∆ U − ε 2 of Equation (2.3c) when α= 2.0, h 2 = −0.465. From Figure 6, when ε 2 = 8/π 2 , the relative error is the minimum. When ε 2 = 8/π 2 , Equation (5.7) becomes According to Equations (5.8) and (5.10), the relative error curve of control parameter ∆ U − h 2 is a quintic polynomial curve. When the relative error is less than ∆ U ≤ 1%, the convergence interval is R h 2 = [−0.75, −0.42], which is the effective region of h 2 .

Conclusions
In this paper, an improved homotopy analysis method (IHAM) is applied to derive complete expressions of vertical and horizontal displacements of a cantilever beam under point load at the free end. In fact, the proposed method is a semi-analytical method since many numerical algorithms are used.
The linear solution is only valid for small α. For large α, the nonlinear effect of the equation cannot be neglected. A family of the explicit solutions of the rotation angle, the vertical and horizontal displacements of a cantilever beam under a point load at the free end are obtained by IHAM, which are summarized in Table A1 for 30th-order approximation. The current explicit solutions with different parameters and solution schemes match well with the exact solutions from elliptical integrals; however, our solutions for the current problem by the proposed IHAM, especially Equations (3.36) and (4.31), are explicit and simple compared with the exact solution from the elliptical integrals which requires solving a transcendental equation. The current solution is easy to calculate with the explicit polynomial expressions, which should be, thus, prioritized for practical engineering applications with minimal requirements on the calculation and computation.
By solving the exact bending equation of a cantilever beam, we demonstrate the superior proficiency of IHAM over a class of existing equations in solid mechanics. The improved homotopy analysis method has great superiority compared over the traditional homotopy analysis method in view of the very much increased rate and precision of convergence as well as the substantially enlarged convergence range.
The convergence ranges of the rotation angle and vertical displacement are extended by the homotopy-Páde approximation, whereas it does not avail horizontal displacement.
In summary, the 30th-order improved homotopy analysis solutions and exact solutions of the angular, vertical and horizontal displacements of large deflection cantilever beams by Equations (3.42), (3.47) and (4.35) are compared in the following Appendix A table. It can be concluded from this study that the proposed IHAM can be applied to solve a general strong nonlinear problem in structural analysis.

Conflicts of Interest:
The authors declare no conflict of interest. Table A1. 30th-order numerical solutions Θ