# The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method
## 3 Equation of Flexure of an Elastic Beam and Its Solution  
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]  
\[\theta^{\prime\prime}+\alpha\cos\theta=0,\ \theta(0)=0,\ \ \ \theta^{\prime}(1)=0, \tag{9}\]  
where \(\theta=\theta(x)\) is the rotation of the cross-section of beam, and \(\alpha=Pl^{2}/EI\) is a constant of the property of the beam problem, with \(P\), \(l\), and \(EI\) as the point load at the free tip, beam length, and bending stiffness of the beam, respectively. This is widely known as the elastica problem, which has been extensively studied [32; 33; 34]. The elastica of nonprismatic cantilever beams with rectangular cross-sections that are subjected to combined loading was studied by [PERSON] [35]. [PERSON] 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. Therefore, a new solution technique with improved results is important for validating the proposed approach with the Galerkin method. In previous studies, the traditional Galerkin method was first used for analysis of wave propagation in finite solids with nonlinear complications [36]. By assuming the spatial variation in displacements in the exact form of the known linear mode shape, [PERSON] et al. used the orthogonal properties of the linear mode shapes to obtain the equations to approximately determine the temporal behavior [37]. [PERSON] further expanded the application scope of the Galerkin method by integrating the weighted equation of motion over the time of one period of vibrations to eliminate the harmonics from the deformation function [31]. This is an extension of [PERSON] method that has broad applications in approximate solutions, particularly for solid mechanics problems [29]. This extended [PERSON] method can also be utilized for the analysis of nonlinear differential equations of free and forced nonlinear vibrations of structures as a new technique with many advantages [28; 29; 30; 31]. The procedure outlined in this paper for the determination of displacement coefficients from nonlinear algebraic equations is new, but it is a natural extension of the popular [PERSON] method meriting wide publicization and promotion. By examining Equation (9), it is clear that the trial solution with trigonometric functions satisfying the boundary conditions is in the form of  
\[\theta(x)=\sum_{n=0}^{\infty}A_{n}\sin\frac{2n+1}{2}\pi x. \tag{10}\]  
The amplitudes \(A_{n}\) are determined with the procedure outlined above. One must substitute Equation (10) into Equation (9) with a standard Galerkin formulation  
\[\int_{0}^{1}(\theta^{\prime\prime}+\alpha\cos\theta)\delta\theta\mathrm{d}x=0, \tag{11}\]  
where  
\[\delta\theta=\sum_{n=0}^{\infty}\delta A_{n}\sin\frac{2n+1}{2}\pi x, \tag{12}\]  
with arbitrary variation in amplitudes \(\delta A_{n}\). For the evaluation of the integral, there are several choices of analytical expressions and power series expressions. It is more favorable to use  
\[\cos\theta=\sum_{n=0}^{\infty}(-1)^{n}\frac{\theta^{2n}}{(2n)!}=1-\frac{ \theta^{2}}{2}+\frac{\theta^{4}}{24}-\frac{\theta^{6}}{720}+\ldots. \tag{13}\]Then, the problem will be the evaluation of individual equations without coupling as  
\[\int_{0}^{1}(\theta^{\prime\prime}+\alpha\cos\theta)\sin\frac{2n+1}{2}\pi x\; \mathrm{d}x=0,\;n=1,\;2,\;3,\;\cdots,M. \tag{14}\]  
With Equation (4), Equation (6) can be written as  
\[