# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 2 The Beam Model
### Geometry and Kinematics  
With reference to Figure 1, the axis line of the beam is defined in correspondence of the cross-sections centroids. It is a straight line given as:  
\[\bar{\mathbf{x}}_{0}(x)=\bar{\mathbf{0}}+x\mathbf{e}_{x} \tag{1}\]  
where \(\bar{\mathbf{0}}\) is the position of the origin, which is the centre of the left tip cross-section, and \(x\) is an abscissa which runs in the interval \([0,l]\), where \(l\) is the length of the beam; the unit vectors \((\mathbf{e}_{x},\mathbf{e}_{y},\mathbf{e}_{z})\) are orthogonal to each other and define the canonical (global) basis. The initial position \(\bar{\mathbf{x}}\) of a generic point \(\mathbf{P}\) located at the mid-surface of the wall is:  
\[\bar{\mathbf{x}}(x,s)=\bar{\mathbf{x}}_{0}(x)+\bar{\mathbf{r}}(s) \tag{2}\]  
where  
\[\bar{\mathbf{r}}(s)=y(s)\mathbf{e}_{y}+z(s)\mathbf{e}_{z} \tag{3}\]  
is the vector which defines the mid-curve \(\bar{\gamma}\) of the cross-section, having curvilinear coordinate \(s\) as parameter. Specifically about the mid-curve \(\bar{\gamma}\) of the cross-section, the parameter \(s\) is chosen so that \(\left(\frac{dy}{ds}\right)^{2}+\left(\frac{dz}{ds}\right)^{2}=1\). Furthermore, it is convenient to set-up a local coordinate system along the curve and given by the tangent, normal and binormal unit vectors \((\mathbf{e}_{s},\mathbf{e}_{n},\mathbf{e}_{b})\) where--with respect to global basis--the relations follow:\[ \begin{split}\mathbf{e}_{s}(s)&=\frac{dy}{ds}\mathbf{e}_{y} +\frac{dz}{ds}\mathbf{e}_{z}\\ \mathbf{e}_{n}(s)&=-\frac{dz}{ds}\mathbf{e}_{y}+ \frac{dy}{ds}\mathbf{e}_{z}\\ \mathbf{e}_{b}(s)&=\mathbf{e}_{x}\end{split}  \tag{4}\]  
and the curvature around \(\mathbf{e}_{b}\) is defined as:  
\[\bar{k}(s)=\frac{dy}{ds}\frac{d^{2}z}{ds^{2}}-\frac{dz}{ds}\frac{d^{2}y}{ds^{2}} \tag{5}\]  
Through the beam deformation, the point \(\mathbf{P}\) undergoes a displacement \(\mathbf{u}\), which can be expressed in terms of components on both the global and local bases, namely \(u_{x}\mathbf{e}_{x}+u_{y}\mathbf{e}_{y}+u_{z}\mathbf{e}_{z}\) and \(u\mathbf{e}_{s}+v\mathbf{e}_{n}+w\mathbf{e}_{b}\). According to Equation (4) the two sets of displacements are related as:  
\[ \begin{split} u&=\frac{dy}{ds}u_{y}+\frac{dz}{ds}u_ {z}\\ v&=-\frac{dz}{ds}u_{y}+\frac{dy}{ds}u_{z}\\ w&=u_{x}\end{split}  \tag{6}\]  
In order to take into account the thickness \(h\) of the walls, the webs and flanges are considered as singly curved thin shells, whose mid-surface coincides with \(\bar{\gamma}\) (Figure 2). The displacements of any generic point located off the mid-surface (\(n\
eq 0\)) are obtained from the mid-surface counterpart ones using [PERSON]'s thin plate assumption. Figure 1: Thin-walled beam in initial configuration (**a**) and details of the mid-curve \(\bar{\gamma}\) of the cross-section (**b**). Thus, every line normal to the mid-surface at \(\bar{\mathbf{x}}\) before deformation remains normal after the deformation and relevant rotations about \(\mathbf{e}_{s}\) and \(\mathbf{e}_{b}\) directions are:  
\[\theta_{s} =-\frac{\partial v}{\partial x} \tag{7}\] \[\theta_{b} =\frac{\partial v}{\partial s}+ku \tag{8}\]  
respectively. It is worth noting how the second term in the right hand side of Equation (8) results from cross-section curvature about binormal axis [33]. As a consequence, any point out of the mid-surface performs displacement \(\mathbf{d}\) which has the following components on \((\mathbf{e}_{s},\mathbf{e}_{n},\mathbf{e}_{b})\):  
\[d_{s} =u-n\theta_{b} \tag{9}\] \[d_{n} =v\] \[d_{b} =w+n\theta_{s}\]  
respectively.