Nonlinear springs

Ashes gives the possibility to implement nonlinear springs, i.e. springs with a stiffness that depends on the displacement of the node. Nonlinear springs can currently only be applied as translational springs (not as rotational springs).

Nonlinear springs are input in the text file describing the imported support structure (see Support section files) as look-up tables of displacement vs force, so-called P-y curves (see API (2011)). When a node is displaced, it experiences a force corresponding to the value of the projection of the displacement in the spring direction in the look-up table, in the opposite direction of the displacement. If a displacement falls between two values of the look-up table, the force is linearly interpolated between the closest values.

The figure below illustrates an example of P-y curve:

Note: in Ashes, the look-up table is given as Force against Displacement (and not pressure against displacement).

The displacements and forces in the look-up table are given in absolute value, and p-y curves are considered symmetric.

Nonlinear springs can be applied in all three translational directions, which enables the implementation of the PISA model (see Byrne et al. (2019)).

When a nonlinear spring is applied to a model, the Support sensor will show the output Stiffness. If a look-up table contains a force 
and a displacement 
defined as follows
$$F=[F_1,F_2,F_3\cdots F_n]$$

$$d=[d_1,d_2,d_3,\dots d_n]$$

then for a displacement 
such that 
$$d_i\leq d_x < d_{i+1}$$
, the stiffness 
is defined as