# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 2 The Beam Model
### Geometry and Kinematics
Substituting Equations (7) and (8) into the above definitions, the local basis displacement components are as follows:  
\[d_{s} =u-n\Big{(}\frac{\partial v}{\partial s}+ku\Big{)} \tag{10}\] \[d_{n} =v\] \[d_{b} =w-n\frac{\partial v}{\partial x}\]  
The infinitesimal strain components within the curved thin wall are defined as [33]:  
\[\varepsilon_{x} =\frac{\partial d_{b}}{\partial x} \tag{11}\] \[\theta_{s} =-\frac{\partial v}{\partial x} \tag{7}\]0 \[\theta_{s} =-\frac{\partial v}{\partial x} \tag{7}\]1  
where \(\varepsilon_{x},\varepsilon_{s}\) are longitudinal strain along \(x,s\), respectively, and \(\gamma_{xs}\) is the shear strain. After substituting Equation (10) in Equation (11), the strain components attain the following expressions:  
\[\theta_{s} =-\frac{\partial v}{\partial x} \tag{7}\]2 Local sketch of the singly curved thin shell. \[ \begin{split}\varepsilon_{x}&=\varepsilon_{x}^{M}+\varepsilon_{x }^{F}\\ \varepsilon_{s}&=\varepsilon_{s}^{M}+\varepsilon_{s}^{F} \\ \gamma_{xs}&=\gamma_{xs}^{M}+\gamma_{xs}^{F}\end{split}  \tag{12}\]  
where the pure membrane strain components, i.e., relevant to the mid-surface (\(n=0\)) and indicated with superscript \(M\), and the flexural strain components, i.e., proportional to \(n\) and indicated with superscript \(F\), are:  
\[ \begin{split}\varepsilon_{x}^{M}&=\frac{\partial w}{ \partial x}\\ \varepsilon_{s}^{M}&=\frac{\partial u}{\partial s}-kv \\ \gamma_{xs}^{M}&=\frac{\partial u}{\partial x}+ \frac{\partial w}{\partial s}\end{split}  \tag{13}\]  
and  
\[ \begin{split}\varepsilon_{x}^{F}&=-n\frac{\partial^{2 }v}{\partial x^{2}}\\ \varepsilon_{s}^{F}&=-n\Big{(}\frac{\partial^{2}v}{ \partial s^{2}}+k\frac{\partial u}{\partial s}+\frac{\partial\bar{k}}{ \partial s}u\Big{)}\\ \gamma_{xs}^{F}&=-n\Big{(}2\frac{\partial^{2}v}{ \partial x\partial s}+\bar{k}\frac{\partial u}{\partial x}\Big{)}\end{split}  \tag{14}\]  
Following the idea of the Kantorovitch semi-variational method the individual components of the mid-curve \(\tilde{\gamma}\) points displacement are expressed as a linear combination of \(K\) triplets of trial functions \((U_{j}(s),V_{j}(s),W_{j}(s))\), \(j=1,\ldots,K\), and unknown amplitude parameters \(a_{j}(x)\):  
\[ \begin{split} u(s,x)&=\sum_{j=1}^{K}U_{j}(s)a_{j}(x )\\ v(s,x)&=\sum_{j=1}^{K}V_{j}(s)a_{j}(x)\\ w(s,x)&=\sum_{j=1}^{K}W_{j}(s)a_{j}^{\prime}(x) \end{split}  \tag{15}\]  
where the prime indicates derivative with respect to \(x\). The functions \((U_{j}(s),V_{j}(s),W_{j}(s))\) represent the three components of the assumed \(j-th\) deformation field, defined on the section profile. The use of Equation (15) in Equations (13) and (14) allows one to express the strain measures in terms of the kinematic descriptors \(a_{j}(x)\), for \(j=1,\ldots,K\):  
\[ \begin{split}\varepsilon_{x}^{M}&=\sum_{j=1}^{K}W_{ j}(s)a_{j}^{\prime\prime}(x)\\ \varepsilon_{s}^{M}&=\sum_{j=1}^{K}\Big{[}\frac{dU_{j }(s)}{ds}-\bar{k}V_{j}(s)\Big{]}a_{j}(x)\\ \gamma_{xs}^{M}&=\sum_{j=1}^{K}\Big{[}U_{j}(s)+ \frac{dW_{j}(s)}{ds}\Big{]}a_{j}^{\prime}(x)\end{split}  \tag{16}\]and  
\[ \begin{split}\varepsilon_{x}^{F}&=-n\sum_{j=1}^{K}V_{j}(s )a_{j}^{\prime\prime}(x)\\ \varepsilon_{s}^{F}&=-n\sum_{j=1}^{K}\Big{[}\frac{d^{2 }V_{j}(s)}{ds^{2}}+k\frac{dU_{j}(s)}{ds}+\frac{d\bar{k}}{ds}U_{j}(s)\Big{]}a_{j }(x)\\ \gamma_{xs}^{F}&=-n\sum_{j=1}^{K}\Big{[}2\frac{dV_{j} (s)}{ds}+\bar{k}U_{j}(s)\Big{]}a_{j}^{\prime}(x)\end{split}  \tag{17}\]