# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 4 Solution Methodology  
Based on the theories developed above, a FORTRAN computer program was written to compute the elastica. The input beam parameters are the span length \(l\), mechanical properties of \(E\) and \(n\), beam geometries of \(B_{c}\), \(H_{c}\), \(B_{f}\), and \(H_{f}\), and individual loads of \(W\), \(P\), and \(C\). These parameters are given in dimensional units but are transformed into nondimensional parameters (i.e., \(b_{c}\), \(h_{c}\), \(r_{B}\), and \(r_{H}\), and \(p\), \(w\), and \(c\)). To integrate the differential equations, the Runge-Kutta method Runge andKutta (1941) was adopted. To determine the unknown \(d\theta/d\lambda\) at the clamped end, the Regula-Falsi method (1941) was adopted. The proposed solution methodology can be summarized as follows:Set the input parameters of \(n\), \(b_{c}\), \(h_{c}\), \(r_{B}\), \(r_{H}\), \(w\), \(p\), and \(c\).
2. Consider a trial \(\left(d\theta/d\lambda\right)_{t}\) at the clamped end \(\left(\lambda=0\right)\). The first trial is numbered as zero.
3. Integrate Equations (21) and (22) using the boundary conditions in Equation (24) and a trial \(\left(d\theta/d\lambda\right)_{t}\). After completing integration, the trial solution includes \(\left(\xi,\eta,\theta,d\theta/d\lambda\right)\) in \(0\leq\lambda\leq 1\).
4. Calculate \(D\) (i.e., the trial boundary condition at the tip \(\left(\lambda=1\right)\)) for Equation (25) as follows: \[D=\frac{d\theta}{d\lambda}-\left(\frac{c}{i_{c}r_{B}r_{H}{}^{2+1/n}}\right)^{n}.\] (26) If \(D=0\), the trial solution computed in step three is the characteristic solution. If \(\left|D\right|\leq ER_{1}\), the trial solution is the numerical solution. Here, the first convergence criterion is set to \(ER_{1}=1\times 10^{-8}\).
5. If the criterion in step four is not satisfied, increment \(\Delta\) to the previous trial \(\left(d\theta/d\lambda\right)_{1}\) to calculate the next trial as \(\left(d\theta/d\lambda\right)_{2}=\left(d\theta/d\lambda\right)_{1}+\Delta\), and repeat steps 2-5.
6. Note the sign of \(D_{1}\times D_{2}\), where \(D_{1}\) and \(D_{2}\) are the values of \(D\) in the previous and current steps, respectively. If the sign changes, the characteristic solution of \(d\theta/d\lambda\) lies between \(\left(d\theta/d\lambda\right)_{1}\) and \(\left(d\theta/d\lambda\right)_{2}\). The next trial \(\left(d\theta/d\lambda\right)_{3}\) is defined using the Regula-Falsi method, and a solution for the nonlinear equations is defined as \[\left(\frac{d\theta}{d\lambda}\right)_{3}=\frac{\left(d\theta/d\lambda\right) _{1}\left|D_{2}\right|+\left(d\theta/d\lambda\right)_{2}\left|D_{1}\right|}{ \left|D_{1}\right|+\left|D_{2}\right|}.\] (27)
7. Once the numerical process is entered into the Regula-Falsi scheme, repeat step six until the following criterion is satisfied: \[\left|\frac{\left(d\theta/d\lambda\right)_{1}-\left(d\theta/d\lambda\right)_{ 3}}{\left(d\theta/d\lambda\right)_{3}}\right|\leq ER_{2},\] (28) where \(D_{1}\times D_{3}<0\) and the second convergence criterion is set to \(ER_{2}=1\times 10^{-5}\).  
If the two convergence criteria are satisfied, output the characteristic solution \(\left(\xi,\eta,\theta\right)\) with the tip responses \(\left(\delta_{v},\delta_{h},\varphi\right)\) and end calculation.