SectionNM

$N$-$M$ Interaction Abstract Section

References

1. 10.1061/JSENDH.STENG-12176

2. 10.1007/978-94-007-6573-3_3

3. 10.1002/nme.1620371506

Please note the formulation is based on the first reference, which is a revised version of the one shown in the last two references.

Remarks

Strictly speaking, the SectionNM is not a section, as a conventional section takes sectional elongation and curvature as input and computes sectional force and moment as output. The SectionNM takes elemental deformation and resistance (which are local quantities of two end nodes) and directly compute the force and moment. Loosely speaking, the SectionNM is a special model that accounts for two end sections simultaneously.

According to [1], the whole model can be implemented at the element level. It is chosen to separate the model into the element part (NMB* elements) and the section part (NM2D* and NM3D* sections) for better code organisation.

Apart from the provided sections, other interaction surfaces can be defined.

One can define a derived class based on, for example, SurfaceNM2D, and implement methods.

[[nodiscard]] double compute_sf(const vec&, const vec&) const;
[[nodiscard]] vec compute_dsf(const vec&, const vec&) const;
[[nodiscard]] mat compute_ddsf(const vec&, const vec&) const;

All methods take the normalised shifted resistance $\mathbf{s}=\mathbf{q}-\mathbf{\beta}$ and equivalent plastic strain $\alpha$ as input where $\mathbf{q}$ is the normalised nodal resistance and $\mathbf{\beta}$ is the back resistance, which is similar to the concept of back stress.