# Global Analysis of Adjustable Pallet Rack Down-Aisle Frames Using the Generalised Beam Theory
## 2 GBT Formulation  
Two main steps are required to carry out a GBT calculation: (i) cross-sectional and (ii) the member analyses. The GBT cross-sectional deformation modes, as well as the modal matrices, are determined in the first step. On the other hand, member equilibrium is solved during the second step. The member displacement field is described, according to each sectional wall local coordinate system (Figure 2), through the use of Equations (1) and (2), in which \(u_{k}(s)\), \(v_{k}(s)\), \(w_{k}(s)\), and \(\phi_{k}(s)\) are the cross-sectional shape functions and the modal amplitude function, and \(k\) corresponds to each deformation mode. The deformation modes can be classified into different groups according to their main kinematic characteristics: (i) conventional modes (warping and in-plane displacements are considered and Vlasov's hypothesis are assumed), (ii) natural shear modes (only involve in-plane displacements and shear membrane strain is allowed), (iii) transverse extension modes (only involve in-plane displacements and the extension of sectional walls is allowed), and (iv) local shear modes (only involve warping displacements). \[ \begin{bmatrix}U_{x}\\ U_{s}\\ U_{z}\end{bmatrix} = \begin{bmatrix}u(x,s)-z\cdot w_{x}(x,s)\\ v(x,s)-z\cdot w_{s}(x,s)\\ w(x,s)\end{bmatrix}  \tag{1}\]  
\[u(x,s)=\sum_{k=1}^{N}u_{k}(s)\Phi_{k,x}(x)v(x,s)=\sum_{k=1}^{N}v_{k}(s)\Phi_{ k}(x)w(x,s)=\sum_{k=1}^{N}w_{k}(s)\Phi_{k}(x) \tag{2}\]A full set of Green-Lagrange membrane strains, small-strain bending effects, and a total Lagrangian kinematic description were used. Thus, the strains were calculated through the use of Equations (3)-(5). \[\varepsilon_{xx}=u_{,x}-zw_{,xx}+\frac{1}{2}\Big{(}{u_{,x}}^{2}+{v_{,x}}^{2}+{w_{,x}}^{2}\Big{)} \tag{3}\]  
\[\varepsilon_{ss}=v_{,s}-zw_{,ss}+\frac{1}{2}\Big{(}{u_{,s}}^{2}+{v_{,s}}^{2}+{w _{,s}}^{2}\Big{)} \tag{4}\]  
\[\varepsilon_{xs}=u_{,s}+v_{,x}-2 zw_{,xs}+u_{,x}u_{,s}+v_{,x}v_{,s}+w_{,x}w_{,s} \tag{5}\]  
The member strain energy was determined by means of Equation (6), where \(U\) and \(UP\) are the strain energy terms of the total gross member domain (\(V\)) and the perforation domain (\(V_{p}\)), respectively. Conversely, \(\overline{U}\) and \(\overline{UP}\) correspond to the strain energy terms of the initial geometrical imperfection. Notice that \(UP\) and \(\overline{UP}\) only have to be computed to take the influence of perforations into account. \[ \begin{array}{c}U=\frac{1}{2}\int_{V}\sigma_{ij}\varepsilon_{ij}dV-\frac{1}{ 2}\int_{V_{p}}\sigma_{ij}\varepsilon_{ij}dV\\ U=U_{1}+U_{2}+U_{3}-UP_{1}-UP_{2}-UP_{3}-\big{(}\overline{U_{1}}+\overline{ U_{2}}+\overline{U_{3}}-\overline{UP_{1}}-\overline{UP_{2}}-\overline{UP_{3}} \big{)}\end{array}  \tag{6}\]  
An isotropic elastic material behaviour and a 2-node GBT beam finite element were applied to solve the member equilibrium equation. The degrees of freedom of the GBT beam element are the values of the amplitude function and/or the amplitude function derivative. Hermite cubic polynomials were used to describe conventional, natural shear, and transverse extension modes. On the other hand, Lagrange cubic polynomials were used for the axial extension and local shear modes (deformation modes that only involve warping displacements). The finite element internal force vector (\(f_{h}^{(e)}\)) and tangent stiffness matrix (\(T_{hi}^{(e)}\)) were calculated by differentiating the strain energy and the internal force vector, respectively (Equations (7) and (8)). Finally, an incremental-iterative process based on the Newton-Raphson algorithm was adopted to solve the geometrically nonlinear analysis. The full procedure and finite element implementation are completely described in [19].