For the Shell interface, the displacement is taken on the plate side defined by a connected location. This side can be determined automatically or specified by the user. In either case, the key assumption is that there is no offset in the direction of the shell normal between the measurement points. In other words:

The stiffness in the normal direction, kn, is a given by the axial stiffness of a bar

Here, Ef is the fastener Young’s modulus and df is the fastener diameter. The total length is the sum of the thickness of the plate on the destination side, tdst, and the thickness on the source side, tsrc.

In the Solid Mechanics interface, the Fastener node can be used to model a pretensioned bolt connection. To account for this, the effective length of the fastener includes an estimated bolt head thickness. The normal stiffness then becomes

The stiffness in the transverse direction, ks, is more complicated. There are several contributions to the deformation of the rivet-plate system. Four compliance terms connected in series are considered,

Here, Cb is the fastener bending flexibility, Cs is the fastener shear flexibility, Cbr is the fastener bearing flexibility, and Cpbr is the plate bearing flexibility.

In case of a pretension load, the clamped plates transfer a large part of the applied load. This partitioning of the load is a function of the relative stiffness of the bolt and the joint members. With an increment in the external load, ΔFext, then

Here, ΔFn is the change in load carried by the bolt, and ΔFg is the change in the load carried by the plates. The joint constant, Cj, is defined as

where kg is the called grip stiffness, explained below.

In the following, un is the displacement in the normal direction between the plates, measured positive when the fastener is extended, so that the increment in the fastener force is

Now, assume a preload, Fpre. After pretensioning the bolt, there is no resulting external force. The preload in the fastener is balanced by the stress in the surrounding plates. That is

The connection will be compressed a distance un0, which is negative. The force in the joint members is compressive,

which provides a value for Δun0. It is now possible to write the expression for the total fastener force under an arbitrary extension as

To approximate the grip stiffness, the following approach is used. The pressure distribution through the thickness of the grip is approximated by a conical shape, also called frustum. The total grip stiffness is a combination of the frustum stiffnesses as described below

The grip stiffness is approximated by giving a value for the frustum angle αsh, which can also depend on the material, the thickness of the plates, and the applied external load.

Using the normal to the shell on the destination side, n, the displacement difference can be separated into normal and tangential parts: