# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## 2 The Beam Model
### Trial Functions and Vlasov's Constraints  
The regular Vlasov's internal constraints specific for any open cross-section require that the profile mid-curve \(\tilde{\gamma}\) is inextensible and the middle surface of the thin-walled beam is shear indeformable. Thereby two relations follow:  
\[\begin{split}\varepsilon_{s}^{M}&=0\\ \gamma_{xs}^{M}&=0\end{split} \tag{22}\]As a consequence, regarding Equation (16), every \(j\)-th set of assumed trial functions must satisfy the following conditions:  
\[\frac{dU_{j}(s)}{ds}-\tilde{k}(s)V_{j}(s) =0 \tag{23}\] \[U_{j}(s)+\frac{dW_{j}(s)}{ds} =0 \tag{24}\]  
In this study, the trial functions \((U_{j}(s),V_{j}(s),W_{j}(s))\), \(j=1,\dots,K\) are evaluated in two stages as proposed by [PERSON] and [PERSON] in [21]. At first, the in-plane deformation field components \((U_{j}(s),V_{j}(s))\) are evaluated as the dynamical normal modes of the planar unconstrained inextensional (i.e., satisfying the first [PERSON]'s condition) frame having the shape similar to profile mid-curve \(\tilde{\gamma}\). Thereby the considered frame is constituted by a monodimensional curved beam element or assembly of them in case of piece-wise regularity of \(\tilde{\gamma}\). Having found the in-plane deformation modes, at the next stage the third trial function \(W_{j}(s)\) is determined. This represents the component relevant to the out-of-plane warping of the cross-section and it is consistently evaluated by integration of Equation (24). The emerging unknown integration constant is obtained by imposing the orthogonality condition of \(W_{j}(s)\) to the uniform axial extension, i.e., \(\int_{\tilde{\gamma}}W_{j}(s)ds=0\). Therefore, the triple of thus deduced deformation fields \((U_{j}(s),V_{j}(s),W_{j}(s))\) satisfy both Vlasov's conditions given by Equation (22).  
In the above analysis a uniform mass per unit length is assigned to the referenced frame in order to evaluate the in-plane deformation components; the latter must also be consistent with Equation (23), i.e., the curved beam constituting \(\tilde{\gamma}\) is axially indeformable. Furthermore, a suitable normalization condition on the normal modes is imposed.  
It is worth noting that, due to the lack of external constraints, among the set of determined in-plane modes, three independent rigid motions of the cross-section in its plane are present as well. These are two translations in directions \(\mathbf{e}_{y}\) and \(\mathbf{e}_{z}\), respectively, and rotation about \(\mathbf{e}_{x}\). Moreover, consistently with the internal constraint in Equation (24), the out-of-plane components relevant to the two translations in directions \(\mathbf{e}_{y}\) and \(\mathbf{e}_{z}\) turn out to describe rotations of the cross-section about axes \(\mathbf{e}_{z}\) and \(\mathbf{e}_{y}\), respectively, as in an Euler-Bernoulli beam bending. Thus, it should be pointed out that, with the present approach, it is precluded the possibility of introducing independent bending rotations as it would be done in Timoshenko beam models.  
An additional trial function, which describes the uniform extension in direction \(\mathbf{e}_{x}\) is defined as well, and it is assigned to ordering \(j=1\). Therefore, the triplets \((U_{j}(s),V_{j}(s),W_{j}(s))\) for \(j=1,\dots,4\) describe rigid motions of the profile and are given as follows: \(j=1\): \((0,0,1)\); \(j=2\): \((\frac{dy}{ds},-\frac{dz}{ds},-y)\); \(j=3\): \((\frac{dz}{ds},\frac{dy}{ds},-z)\); \(j=4\): \((y\frac{dz}{ds}-z\frac{dy}{ds},y\frac{dy}{ds}+z\frac{dz}{ds},-\int(y\frac{dz}{ ds}-z\frac{dy}{ds})ds+c_{4})\). The other triplets (\(j>4\)) describe in-plane deformation of the cross-section and its corresponding kinematically consistent out-of-plane warping.  
It is worth mentioning that the Vlasov's conditions (22) can be removed in the framework of GBT, as well: This is the case, for instance, of structures where shear-lag might occur, as shown in [22]; there, further trial functions are introduced, which account for both shear and extensional deformation.