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Non Linear Static Analysis

The nonlinear static solver works using the dynamic relaxation method. This is an iterative method which simulates a process of damped vibration in small time increments (cycles). This is a specialisation of the explicit time-history solution method. Fictitious masses and inertias are computed for each free node.

At each cycle the forces and moments which elements exert on each node are summed for the current displacements. The linear and angular accelerations of each node are computed from its fictitious mass and inertia, damping is applied to the node’s current linear and angular velocities and the node’s shifts and rotations are calculated for the cycle.

This process is repeated until it is terminated by the user or the solution has converged (the out-of-balance forces and moments (residuals) at every free node are less than target values).

If the damping is too high or the fictitious masses and inertias of the nodes are too large, their shifts and rotations at each cycle will be very small and many cycles will be needed to achieve a result. If on the other hand the damping is too low or the masses and inertias are too small, the simulated damped vibration becomes unstable.

The two cases of an unstable structure and of unstable simulated damped vibration can be distinguished by inspecting the results. When the structure is unstable, the element forces change little from cycle to cycle and the shifts of the nodes at each cycle may be very large but do not vary significantly from cycle to cycle. If the simulated damped vibration is unstable, the forces and nodal displacements oscillate wildly between cycles and usually increase to enormous values. The third case of stable simulated damped vibration converging to a stable solution can be recognised because the residuals and the shifts of the nodes decrease overall from cycle to cycle.

It should be noted that very few structures are so unstable that they do not eventually converge to a solution. Generally secondary effects become operative with large deflections and allow the structure to reach some kind of equilibrium.