# 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
\begin{array}{ll}f(\alpha)=&1+3.98575\times 10^{-2}\alpha^{2}-5.41174\times 10^{- 2}\alpha^{4}\\ &+5.72575\times 10^{-3}\alpha^{6}+3.79533\times 10^{-4}\alpha^{8}\\ &-1.51429\times 10^{-11}\alpha^{14}-2.29142\times 10^{-15}\alpha^{16}\\ &-8.87896\times 10^{-6}\alpha^{10}+~{}2.63041\times 10^{-8}\alpha^{12}\\ &-3.45006\times 10^{-21}\alpha^{18}~{}-~{}7.00678\times 10^{-28}\alpha^{20},\\ &+~{}0.131524\alpha^{2}~{}-~{}5.99231~{}\times 10^{-2}\alpha^{4}\\ &+~{}2.34466\times 10^{-3}\alpha^{6}~{}+~{}9.90299\times 10^{-4}\alpha^{8}\\ &-~{}1.37001\times 10^{-5}\alpha^{10}+~{}3.44172\times 10^{-8}\alpha^{12}\\ &-~{}1.45098\times 10^{-11}\alpha^{14}~{}-~{}2.26721\times 10^{-15}\alpha^{16} \\ &-~{}3.50731\times 10^{-21}\alpha^{18}~{}-~{}7.00678\times 10^{-28}\alpha^{20}. \end{array}  \tag{29}\]  
To visualize the solutions and make comparisons, different approximate solutions and the exact solution are plotted in Figure 2 with parameter \(\alpha\). It is clear from Figure 2 that the approximations are very good with just two terms. With three terms, it is almost the exact solution in the range of parameter \(\alpha\), as shown in the plot. The excellent accuracy is a little surprising given the fact the procedure and technique are relatively simple, as shown in this paper. As for a comparison with the rigorous and sophisticated HAM [15], it also shows that for this particular example, this method provides a relatively accurate approximation with a much simpler procedure. We can plot the rotation angle at the free end \(\theta_{B}\) vs. \(\alpha\) with solutions of different orders in Figure 3.  
From Figure 3, it can be found that there is still a noticeable gap between the first-order and the exact solutions. On the other hand, the second-order solution gradually approaches to the exact solution. The third-order solution is almost exact in a large interval in comparison with the closed-form solution with elliptic functions. It is evident now that the Galerkin method is extremely efficient in obtaining an accurate solution to the nonlinear differential equation of the elastica. The excellent choice of the trigonometric sine function as the basis of trial solutions makes the procedure ideal, and the accuracy is also surprisingly good with just the third-order solution. It is evident that the Galerkin method can be equally powerful for nonlinear problems arising from applications in many fields. The elastica problem has a long and rich history, but it is still a hot topic related to the flexure and buckling of beams and rods; accurate results are essentially obtained from the elliptic functions [38; 39]. The procedure and solution from this study show that with the [PERSON] method, a new technique to solve similar nonlinear differential equations is  
Figure 2: The rotation angle at the free end \(\theta_{B}\) vs. \(\alpha\) with different solutions. available, and it can be much simpler to obtain reasonably accurate solutions. This was the objective of this study: to demonstrate the usefulness and efficiency of the Galerkin method with nonlinear differential equations. It is a neglected technique worth being promoted, as demonstrated by the popular example of the elastica problem. The choice of trial functions is of great importance in the simplification of calculation and determination of accuracy, but the choice should be made from experience in dealing with approximate solutions for the Galerkin and Rayleigh-Ritz methods, which are equivalent in most cases. Furthermore, the solution of coupled nonlinear equations of amplitudes can be hard to acquire, but the utilization of symbolic computing tools such as Mathematica(r) and MATLAB(r) can provide much better solutions with explicit expressions that can only be given and meaningful for the lower orders. For practical problems such as the elastica problem in this study, the combination of ideal trial functions, optimal procedure, iteration, and symbolic calculation produces sufficiently accurate results to meet application requirements.