Analysis Of Results

The initial phase of validation has shown a satisfactory behaviour of the model with regards to the following aspects:

- The method seems to be numerically stable.

- Modelling normal contact force with the penalty method has given good qualitative results.

- The frictional behaviour of the numerical model is the one anticipated from the elasto-plastic shear-slip relationship used in the implementation of the code.

- Contact updating is working properly, perceiving changes in the contact configuration.

- Damping cause significance effects on the oscillation response of the block. Contact damping seems to be the more suitable and manageable for structural analysis, at least when the elastic deformations of the blocks are neglectable.

Despite the overall good behaviour of the model, a problem has been encountered in the rocking motion example. The block reaches the horizontal position for the first time, for an instant t, lower than the one obtained with the analytical solution and the difference decreases with the refinement of the discretization of the block.

The error introduced by the numerical method is significative when the parallelepiped is modelled using only 1 hexahedron and becomes tolerable when the mesh consisted in 4 identical hexahedrons.

This behaviour is due to the undervaluation of the moment of inertia and a resulting overvaluation of the rotation of a block when it has not been modelled with an adequate number of finite elements. That depends on the fact that the algorithm has been developed considering the mass concentrated in the barycentre of the element.

Obviously quadrupling the number of the elements would critically increase the computational cost for complex structures.

Figure 13. Rocking: (a) geometrical model; (b) forces and constraints; (c) node numbering of block A.

The geometrical configuration, forces and constraints and node numbering are summarize in Figure 13.

At first no damping is considered. This configuration should lead to a rocking behaviour of A. The oscillations should not change amplitude during the analysis. Using the same geometrical configuration, contact damping is included. Depending on the magnitude of damping considered, the system should evolve with different velocities towards the equilibrium.

A comparison between the analysis of an undamped, a 0.2 and a 0.4 contact damping model is shown in Figure 14.

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