# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 3 Mathematical Modelling  
The symbols and loads for the cantilever beam are defined in Figure 2. The beam was subjected to three types of loading, either individually or in combination: a uniform load \(W\), tip point load \(P\), and tip couple load \(C\). The straight line \(cf\) and solid curve \(cf\) are the axes of undeflected and deflected (elastica) beams, respectively. The material point of the elastica at \(s\) is defined in Cartesian coordinates \((x,y)\), where the beam's rotation is \(\theta\) and the shear force and bending moment are \(Q\) and \(M\), respectively. At the free tip \((s=l)\), the vertical displacement is \(\Delta_{v}\), horizontal displacement is \(\Delta_{h}\), and rotation angle is \(\theta_{f}\). In this study, the beam material was nonlinearly elastic with a stress-strain relationship defined by [PERSON]'s constitutive law as follows:  
\[\sigma=E|\epsilon|^{1/n}, \tag{4}\]  
where \(\sigma\) and \(\epsilon\) represent the stress and strain, respectively, and \(E\) and \(n\) represent the Young's modulus and exponential constant of the material, respectively. Note that the sign of \(\sigma\) follows that of \(\epsilon\). Figure 1: Geometry of the non-prismatic cantilever beam with a rectangular cross-section. Figure 2: Elastica of cantilever beam under combined loading. The Bernoulli-Euler bending moment-curvature relationship for the rectangular beam with \(B\) and \(H\) fabricated from a Ludwick-type material can be written as follows [8]:  
\[M=E\frac{BH^{2+1/n}}{(2+1/n)2^{1+1/n}}\kappa^{1/n}=E\frac{BH^{2+1/n}}{(2+1/n)2^{1 +1/n}}\bigg{(}\frac{d\theta}{ds}\bigg{)}^{1/n}, \tag{5}\]  
where \(\kappa\) is the curvature \(d\theta/ds\) of the elastica. The shear force \(Q\) is expressed using the free body shown in Figure 2 as follows:  
\[Q=-[W(l-s)+P]\cos\theta. \tag{6}\]  
Substituting Equations (3) and (4) into Equation (6) yields  
\[M=EI_{c}\Big{(}\alpha_{B}\frac{s}{l}+1\Big{)}\Big{(}\alpha_{H}\frac{s}{l}+1 \Big{)}^{2+1/n}\bigg{(}\frac{d\theta}{ds}\bigg{)}^{1/n}, \tag{7}\]  
where  
\[I_{c}=\frac{B_{c}H_{c}{}^{2+1/n}}{(2+1/n)2^{1+1/n}}. \tag{8}\]  
The first derivative of Equation (7) with respect to \(s\) is obtained as follows:  
\[ \begin{array}{c}\frac{1}{EI_{c}}\frac{dM}{ds}=\frac{\alpha_{B}}{l}\big{(} \alpha_{H}\frac{s}{l}+1\big{)}^{2+1/n}\Big{(}\frac{d\theta}{ds}\Big{)}^{1/n}+ \Big{(}2+\frac{1}{n}\Big{)}\frac{\alpha_{H}}{l}\big{(}\alpha_{B}\frac{s}{l} \big{)}\big{(}\alpha_{H}\frac{s}{l}+1\big{)}^{1+1/n}\Big{(}\frac{d\theta}{ds} \Big{)}^{1/n}\\ +\frac{1}{n}\big{(}\alpha_{B}\frac{s}{l}+1\big{)}\big{(}\alpha_{H}\frac{s}{l} +1\big{)}^{2+1/n}\Big{(}\frac{d\theta}{ds}\Big{)}^{1/n-1}\frac{d\theta}{ds^{2 }}.\end{array}  \tag{9}\]  
Considering \(dM/ds=Q\), \(dM/ds\) is obtained from Equation (8) as  
\[\frac{dM}{ds}=-[W(l-s)+P]\cos\theta. \tag{10}\]  
Substituting Equation (10) into Equation (9) yields the following ordinary differential equation:  
\[ \begin{array}{c}\frac{d^{2}\theta}{ds^{2}}=-\frac{n[W(l-s)+P]}{EI_{c}} \big{(}\alpha_{B}\frac{s}{l}+1\big{)}^{-1}\big{(}\alpha_{H}\frac{s}{l}+1\big{)} ^{-(2+1/n)}\cos\theta\Big{(}\frac{d\theta}{ds}\Big{)}^{1-1/n}\\ -\Big{[}\frac{n\alpha_{B}}{l}\big{(}\alpha_{B}\frac{s}{l}+1\big{)}^{-1}+(2n+1) \frac{\alpha_{H}}{l}\big{(}\alpha_{H}\frac{s}{l}+1\big{)}^{-1}\Big{]}\frac{d \theta}{ds}.\end{array}  \tag{11}\]  
The geometric relationships of the Bernoulli-Euler nonlinear beam are defined as follows:  
\[\frac{dx}{ds}=\cos\theta,\ \frac{dy}{ds}=\sin\theta\ \text{for}\ 0\leq s\leq l. \tag{12}\]  
Now, we consider the boundary conditions. At the clamped end \((s=0)\), no deflection occurs, resulting in  
\[x=0,\ y=0,\frac{dy}{ds}=0. \tag{13}\]  
Note that the boundary condition of \(d\theta/ds\) at the clamped end is unknown, meaning Equations (11) and (12) cannot be solved as initial value problems. To calculate the unknown \(d\theta/ds\) as a boundary value problem, the boundary condition at the free tip \((s=l)\) is considered.