# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 2 The Beam Model
### Equilibrium Equations  
Equilibrium equations in terms of generalized amplitudes \(a_{j}(x)\) are obtained imposing the stationary condition to the sum of elastic and load potential energy (conservative external loads are assumed). In particular, the elastic potential energy has the following expression:  
\[\mathcal{U}_{e}=\frac{1}{2}\int_{\mathcal{V}}[\sigma_{x}^{M}\epsilon_{x}^{M}+ \sigma_{x}^{F}\epsilon_{x}^{F}+\sigma_{s}^{F}\epsilon_{s}^{F}+\tau_{xs}^{F} \gamma_{xs}^{F}]d\mathcal{V} \tag{25}\]where \(\mathcal{V}\) is the total volume of the beam and the differential volume element being \(d\mathcal{V}=dndsdx\). In the above the membrane components of strain in tangential direction \(s\) and shear mid-surface indeformability are omitted according to [PERSON]'s conditions Equation (22). After expressing the stress and strain components in Equation (25) in terms of the unknown amplitudes \(a_{j}(x)\), by means of the relevant equations from Sections 2.1 and 2.2, the elastic potential energy becomes:  
\[\mathcal{U}_{e}=\int\limits_{0}^{l}[\frac{1}{2}\big{(}\mathbf{a}\cdot\mathbf{ Aa}+\mathbf{a}^{\prime}\cdot\mathbf{Ba}^{\prime}+\mathbf{a}^{\prime\prime} \cdot\mathbf{Ca}^{\prime\prime}\big{)}+\mathbf{a}\cdot\mathbf{Fa}^{\prime \prime}]dx \tag{26}\]  
where \(\mathbf{a}(x):=(a_{1}(x),\ldots,a_{k}(x))^{T}\), the dot stands for scalar product, the prime indicates derivative with respect to \(x\) and the expressions of the elements of the individual matrices are:  
\[ \begin{split} A_{i,j}&=\int\limits_{\gamma}\!\!\int \limits_{-\frac{b}{2}}^{\frac{b}{2}}\frac{n^{2}E}{1-v^{2}}\frac{d}{ds}\Big{[} \frac{dV_{i}}{ds}+kU_{i}\Big{]}\frac{d}{ds}\Big{[}\frac{dV_{j}}{ds}+kU_{j} \Big{]}dnds\\ B_{i,j}&=\int\limits_{\gamma}\!\!\int\limits_{- \frac{b}{2}}^{\frac{b}{2}}\frac{n^{2}E}{2(1+v)}\Big{[}2\frac{dV_{i}}{ds}+kU_{i }\Big{]}\Big{[}2\frac{dV_{j}}{ds}+kU_{j}\Big{]}dnds\\ C_{i,j}&=\int\limits_{\gamma}\!\!\int\limits_{- \frac{b}{2}}^{\frac{b}{2}}\frac{E}{1-v^{2}}[W_{i}W_{j}+n^{2}V_{i}V_{j}]dnds\\ F_{i,j}&=\int\limits_{\gamma}\!\!\int\limits_{- \frac{b}{2}}^{\frac{b}{2}}\frac{n^{2}Ev}{(1-v^{2})}\frac{d}{ds}\Big{[}\frac{ dV_{i}}{ds}+kU_{i}\Big{]}V_{j}dnds\end{split}  \tag{27}\]  
It is worth noting that \(\mathbf{A},\mathbf{B},\mathbf{C}\) are symmetric matrices. The load potential energy is evaluated for a general case of a force per unit area acting in correspondence of the mid-surface. The generally oriented force is defined by the three components vector \(\mathbf{p}(x,s)=p_{x}\mathbf{e}_{x}+p_{y}\mathbf{e}_{y}+p_{z}\mathbf{e}_{z}\) that yields the energy:  
\[\mathcal{U}_{l}=-\int_{0}^{l}\!\!\int_{\gamma}[p_{x}u_{x}+p_{y}u_{y}+p_{z}u_{ z}]dsdx \tag{28}\]  
where the displacement components can be expressed in terms of \(\mathbf{a}(x)\) as well, by means of Equations (6) and (15). The minimum of the total potential energy is attained imposing that \(\delta(\mathcal{U}_{e}+\mathcal{U}_{l})=0\), \(\forall\delta a_{j}(x)\), \(j=1,\ldots,K\), where \(\delta\) is the variation operator, which produces the following set of Euler-Lagrange equations:  
\[\mathbf{Ca}^{\prime\prime\prime\prime}+(\mathbf{F}+\mathbf{F}^{T}-\mathbf{B}) \mathbf{a}^{\prime\prime}+\mathbf{A}\mathbf{a}=\mathbf{f} \tag{29}\]  
and boundary conditions in \(x=0,l\):  
\[ \begin{split}\mathbf{a}&=\mathbf{0}\quad\text{or} \quad-\mathbf{Ca}^{\prime\prime\prime}+(\mathbf{B}-\mathbf{F}^{T})\mathbf{a}^{ \prime}=\mathbf{g}\\ \mathbf{a}^{\prime}&=\mathbf{0}\quad\text{or}\quad \quad\mathbf{Ca}^{\prime\prime}+\mathbf{F}^{T}\mathbf{a}=\mathbf{0}\end{split}  \tag{30}\]  
where the elements of \(\mathbf{f}\) and \(\mathbf{g}\) are:  
\[ \begin{split} f_{j}&=\int_{\gamma}\!\!\Big{[}p_{y} \Big{(}\frac{dy}{ds}U_{j}-\frac{dz}{ds}V_{j}\Big{)}+p_{z}\Big{(}\frac{dz}{ds}U_{ j}+\frac{dy}{ds}V_{j}\Big{)}-p_{x}^{\prime}W_{j}\Big{]}ds\\ g_{j}&=\!\bigg{[}\int_{\gamma}\!\!p_{x}W_{j}ds\Big{]}_ {x=0,l}\end{split}