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Detailed description of the FEMSL calculation code

The FEMSL calculation code has been written in MATLAB programming language and includes the following calculation stages:

· Definition of the elements that make up the structure;

· Automatic determination of the nodes on which the applied external loads act;

· Construction of the local stiffness matrix of each element considering the semi-rigid connections at two ends by means of using the correction matrix C_i.

· Construction of the global stiffness matrix, starting from the local stiffness matrix;

· Definition of the nodal forces in the zone in which the applied external loads act;

· Iterative calculation that foresees:

- A progressive reduction (up to the annulment) of the normal stiffness of the springs which work under traction (because of the displacement of the support towards the tunnel) and the continuous modification of the normal stiffness of the springs which work under compression (because of the displacement of the support towards the ground);

- A continuous modification of the tangential stiffness of the springs;

- A continuous modification of rotational stiffness and therefore the fixity ratios at each end of elements.

in function of the displacement value reached in the previous calculation step;

· Resolution of the system of linear equations and obtaining the nodal displacement vectors, for each step of the iterative calculation;

· Once the iterative process has finished, the nodal displacements under the global coordinates system are first converted considering the local coordinates system of each element and then multiplied by the local stiffness matrix (expressed under the local coordinates system) to obtain the nodal forces and therefore the stress states of each element;

· Plotting of the results.

The data of the problem that are necessary for the calculation are inserted directly into the programme, attributing the respective value to each variable: the name of the variable in the calculation programme is here given together with each datum:

· Type of tunnel lining (i.e. continuous lining or segmental lining) (“segmental_lining”): this can be equal to 0 or 1, which corresponds to the cases of a continuous lining and a segmental lining;

· Type of hypothesis of 3D effect between successive rings (Figure 10) (“hypothesis”): this can be equal to 1, 2 or 3, which corresponds to the hypotheses presented in section 4;

· Number of iterative steps of the calculation (“num_steps”);

· Number of elements in a quarter of the tunnel (“n_el”);

· Tunnel radius (“R”);

· Elastic modulus of the tunnel lining (“E”);

· Width of the segmental tunnel lining (“B”);

· Thickness of the segmental tunnel lining (“thick”);

· Thickness of the narrow part in segmental joint (“a”);

· Depth of the tunnel axis (“H”);

· Lateral earth pressure coefficient of the ground (“K0”);

· Cohesion of the ground (“c”);

· Friction angle of the ground (“fi”)

· Young’s modulus of the ground (“Eground”):

· Poisson’s ratio of the ground (“muy”);

· Unit weight of the ground (“gama_ground”);

· Vertical loads (“pv”);

· Number of segmental joints in a ring (“joint_number”);

· Positions of the joints in two successive rings measured counter clockwise from the tunnel bottom;

The measurement units that have been used are those of the international system (SI): m for the lengths, MPa for the pressures and radian for the angles.