# Behaviour of Square and Rectangular Tunnels Using an Improved Finite Element Method
## 2 Methods Calculation  
### Improved (Hypersatic Reaction Method) HRM Method  
In the HRM method, the tunnel lining is segmented into 1D beam elements. These beam elements are linked to each other by nodes assigned at two ends of the beam element. The tunnel lining element is linked to the surrounding ground by normal and tangent springs at the defined element nodes. Loads of the surrounding ground are applied to the tunnel lining through the springs mentioned above. In the HRM method, when the displacements of the nodes on the tunnel lining elements are determined, internal forces induced in the tunnel lining can be calculated through the global stiffness matrix of the tunnel lining elements. The detail of the HRM method applied to circular tunnels was introduced by [PERSON] [18] and [PERSON] et al. [4]. In this paper, some important improvements of the HRM method applied in square and rectangular tunnels are proposed. In the case of square and rectangular tunnels, the length of tunnel lining elements was recalculated according to the corresponding coordinates of these elements. The tunnel lining is divided into 360 elements. The lengths of elements are fixed while the angle made by two consecutive nodes and the centre of the tunnel, i.e., point O in Figure 1, is changed depending on the node's position on the tunnel boundary. On the cross-section of the square and rectangular tunnel lining, tunnel lining was divided into 6 regions specified by points \(A_{j}\) (\(j\) counted from 0 to 6) (see Figure 1). Figure 1: Diagram of the square tunnel in the HRM method. The tunnel lining beam element \"\(i\)\" has two nodes, with \(L_{i}\) is the length of the beam element. Bending (\(E_{i}J\)) and normal stiffness of tunnel lining (\(E_{i}A\)) (see Figure 2). _The Ground-Support Interaction Impact_  
In the HRM method, the interaction between the tunnel lining and the surrounding soil is considered through the active vertical load and the horizontal load applied at all sides of the tunnel. Springs at the element nodes are specified by their normal stiffnesses (\(k_{n}\)) and shear stiffnesses (\(k_{s}\)), (Figure 1) [4,18]. The global stiffness matrix of the tunnel lining element is defined as follows:  
\[ \begin{array}{l}K_{3i-2,3i-2}^{*}=K_{3i-2,3i-2}+k_{n,i}\times\cos^{2}(\frac{ i_{i+1}}{2}+\frac{\alpha_{i}}{2}-\frac{\pi}{2})+k_{s,i}\times\sin^{2}(\frac{ \alpha_{i+1}}{2}+\frac{\alpha_{i}}{2}-\frac{\pi}{2})\\ K_{3i-1,3i-1}^{*}=K_{3i-1,3i-1}+k_{n,i}\times\sin^{2}(\frac{\alpha_{i+1}}{2}+ \frac{\alpha_{i}}{2}-\frac{\pi}{2})+k_{s,i}\times\cos^{2}(\frac{\alpha_{i+1}}{ 2}+\frac{\alpha_{i}}{2}-\frac{\pi}{2})\\ K_{3i-1,3i-2}^{*}=K_{3i-1,3i-2}+(k_{n,i}-k_{s,i})\times\sin^{2}(\frac{\alpha_{ i+1}}{2}+\frac{\alpha_{i}}{2}-\frac{\pi}{2})\times\cos(\frac{\alpha_{i+1}}{2}+ \frac{\alpha_{i}}{2}-\frac{\pi}{2})\\ K_{3i-2,3i-1}^{*}=K_{3i-2,3i-2}+(k_{n,i}-k_{s,i})\times\sin(\frac{\alpha_{i+1} }{2}+\frac{\alpha_{i}}{2}-\frac{\pi}{2})\times\cos(\frac{\alpha_{i+1}}{2}+ \frac{\alpha_{i}}{2}-\frac{\pi}{2})\end{array}  \tag{1}\]  
where \(i\) is the number of the generic node; \(k_{n,i}\) is the stiffness of the normal spring linked to node \(i\); \(k_{s,i}\) is the stiffness of the tangential spring linked to node \(i\); \(\alpha_{i}\) and \(\alpha_{i+1}\) is the angle between the local and global reference systems, of element \(i\) and element (\(i+1\)). The reaction pressure \(p\) of the surrounding ground is represented by a nonlinear relationship (hyperbolic) with the tunnel lining deformation \(\delta\) (Figure 3). The parameters representing the soil properties surrounding the tunnel are presented through the following equations [4,18]:  
\[\eta^{*}=\frac{p_{\rm lim}}{\delta}\times(1-\frac{p_{\rm lim}}{p_{\rm lim}+ \eta_{0}\delta}) \tag{2}\]  
\[\eta_{n,0}=\beta\times\left(\frac{1}{1+\
u}\right)\times\left(\frac{E}{R_{i}}\right) \tag{3}\]  
\[\eta_{s}=\frac{1}{3}\times\eta_{n} \tag{4}\]  
Figure 3: Relationship between the reaction pressure \(p\) caused by the surrounding soil and the tunnel lining deformation \(\delta\).