Distributing coupling constraints

Let ${\mathbf{F}}^R$ and ${\mathbf{M}}^R$ be the load and moment applied to the reference node. The statically admissible force distribution ${\mathbf{F}}^n$ among the coupling nodes satisfies

where ${\mathbf{x}}^R$ and ${\mathbf{x}}^n$ are the positions of the reference and coupling nodes, respectively. For an arbitrary number of coupling nodes there is no unique solution to Equation 1.

The force distribution adopted in Abaqus has the property that the linearized motion of the reference node is compatible with the coupling node group motion in an average sense. This compatibility can be described by considering the momentum of a moving coupling node group in a case where weight factors are considered as masses. In this case the reference node motion is identical to that of a point on a rigid body occupying the position of the reference node, where the center of mass of the rigid body is the center of mass of the coupling nodes and the rigid body moves with the same linear and angular momentum as the coupling node group. Since the element mass is distributed this way, the dynamic behavior of the element also has this property.

where

and the coupling node arrangement inertia tensor is

where $\mathbf{I}$ is the second-order identity tensor. This force distribution is recognized to be equivalent to the classic bolt-pattern force distribution when the weight factors are interpreted as bolt cross-section areas.

The load distribution results in the following linearized constraint on node motions:

where ${\mathbf{r}}^R={\mathbf{x}}^R-\overline{\mathbf{x}}$.

Finite displacement and rotation terms take the form of a constraint on the motion of the reference node as a function of the coupling-node finite incremental motions. A measure of the finite rotation of the coupling node arrangement is developed first and is based on the mid-increment position of the coupling nodes, defined as

and the mid-increment inertia tensor is

The mid-increment “spin” is then

The finite incremental rotation tensor $\mathrm{\Delta }\mathbf{R}$ is deduced from the above expression according to the Hughes and Winget (1980) formula,

This orthogonal tensor yields an incremental finite-rotation vector $\mathrm{\Delta }{\acute{\mathit{\varphi }}}^R$. From this rotation description comes the constraint expressions for finite displacement and rotation:

The virtual work expression for the attached structure is augmented with the contribution of the constraint

where $\delta {\mathrm{\Pi }}^{*}$ is the augmented virtual work expression, $\delta \mathrm{\Pi }$ is the virtual work expression for the attached structure, and ${\mathit{\lambda }}_f$ and ${\mathit{\lambda }}_m$ are the respective Lagrange multiplier variables for force and moment.

The initial stress stiffness terms are derived from a suitable approximation of the exact virtual work expression shown in Equation 2. This approximation is based on an assumption of infinitesimal incremental motions, $\mathrm{\Delta }{\mathbf{r}}^n$ and $\mathrm{\Delta }{\mathit{\varphi }}^R$, that implies

An approximate virtual work expression is obtained:

This expression yields the following initial stress stiffness terms: