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Distributing coupling constraints

Distributing coupling constraints provide a means to connect a reference node to a group of coupling nodes in a way that distributes loads according to weight factors that are prescribed individually at each coupling node. The constraints distribute forces and moments at the reference node as a coupling node–force distribution only.

The formulation presented here is used for the distributing coupling elements that are available in Abaqus/Standard, as well as the surface-based distributing coupling and fastener constraints that are available in both Abaqus/Standard and Abaqus/Explicit. For the surface-based distributing coupling and fastener constraints, this section defines the load distribution relationship for the default continuum coupling method.

The following topics are discussed:

Load distribution
Constraint expression
Finite motion
Virtual work contribution
Initial stress stiffness terms

Load distribution

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

$\begin{matrix} (1) & \begin{matrix} \sum_{n} F^{n} & = F^{R} \\ \sum_{n} x^{n} \times F^{n} & = M^{R} + x^{R} \times F^{R}, \end{matrix} \end{matrix}$

where $x^{R}$ and $x^{n}$ 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.

$F^{n} \overset{def}{=} {\hat{w}}^{n} (F^{R} + (T^{- 1} \cdot {\hat{M}}^{R}) \times r^{n}),$

where

$\begin{matrix} {\hat{M}}^{R} & = M^{R} + r^{R} \times F^{R}, \\ r^{n} & = x^{n} - \bar{x}, \\ \bar{x} & = \frac{\sum_{n} w^{n} x^{n}}{\sum_{n} w^{n}} = \sum_{n} {\hat{w}}^{n} x^{n}, \end{matrix}$

and the coupling node arrangement inertia tensor is

$T = \sum_{n} {\hat{w}}^{n} [(r^{n} \cdot r^{n}) I - (r^{n} r^{n})],$

where $I$ 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.

Constraint expression

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

$\begin{matrix} δ x^{R} & = \sum_{n} {\hat{w}}^{n} δ x^{n} + (T^{- 1} \cdot \sum_{n} {\hat{w}}^{n} (r^{n} \times δ x^{n})) \times r^{R} \\ δ ω^{R} & = T^{- 1} \cdot \sum_{n} {\hat{w}}^{n} (r^{n} \times δ x^{n}), \end{matrix}$

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

Finite motion

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

${\overset{ˇ}{r}}^{n} \overset{def}{=} r_{0}^{n} + \frac{1}{2} Δ x^{n},$

and the mid-increment inertia tensor is

$J = \sum_{n} {\hat{w}}^{n} [({\overset{ˇ}{r}}^{n} \cdot {\overset{ˇ}{r}}^{n}) I - ({\overset{ˇ}{r}}^{n} {\overset{ˇ}{r}}^{n})] .$

The mid-increment “spin” is then

$Δ θ^{R} = J^{- 1} \cdot (\sum_{n} {\hat{w}}^{n} {\overset{ˇ}{r}}^{n} \times Δ x^{n}) .$

The finite incremental rotation tensor $Δ R$ is deduced from the above expression according to the Hughes and Winget (1980) formula,

$Δ R = {(I - \frac{1}{2} {\hat{Δ θ}}^{R})}^{- 1} \cdot (I + \frac{1}{2} {\hat{Δ θ}}^{R}) .$

This orthogonal tensor yields an incremental finite-rotation vector $Δ {\overset{´}{ϕ}}^{R}$ . From this rotation description comes the constraint expressions for finite displacement and rotation:

$\begin{matrix} Δ x^{R} & = \sum_{n} {\hat{w}}^{n} Δ x^{n} + Δ R \cdot r_{0}^{R} - r_{0}^{R}, \\ Δ ϕ^{R} & = Δ {\overset{´}{ϕ}}^{R} . \end{matrix}$

Virtual work contribution

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

$\begin{matrix} (2) & \begin{matrix} δ Π^{*} = δ Π + λ_{f} & \cdot [δ x^{R} - \sum_{n} {\hat{w}}^{n} δ x^{n} - δ R^{R} \cdot r_{0}^{R}] + \\ λ_{m} & \cdot [δ ω^{R} - δ {\overset{´}{ϕ}}^{R}] + \\ δ λ_{f} & \cdot [Δ x^{R} - \sum_{n} {\hat{w}}^{n} Δ x^{n} - Δ R \cdot r_{0}^{R} + r_{0}^{R}] + \\ δ λ_{m} & \cdot [Δ ϕ^{R} - Δ {\overset{´}{ϕ}}^{R}], \end{matrix} \end{matrix}$

where $δ Π^{*}$ is the augmented virtual work expression, $δ Π$ is the virtual work expression for the attached structure, and $λ_{f}$ and $λ_{m}$ are the respective Lagrange multiplier variables for force and moment.

Initial stress stiffness terms

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, $Δ r^{n}$ and $Δ ϕ^{R}$ , that implies

$\begin{matrix} δ ϕ^{R} & = δ ω^{R} and \\ δ {\overset{´}{ϕ}}^{R} & = T^{- 1} \cdot \sum_{n} {\hat{w}}^{n} (r^{n} \times δ x^{n}) . \end{matrix}$

An approximate virtual work expression is obtained:

$\begin{matrix} {\tilde{δ Π}}^{*} = δ Π + λ_{f} & \cdot [δ x^{R} - \sum_{n} {\hat{w}}^{n} δ x^{n} - δ ω^{R} \times r^{R}] + \\ λ_{m} & \cdot [δ ω^{R} - T^{- 1} \cdot \sum_{n} {\hat{w}}^{n} (r^{n} \times δ x^{n})] + \end{matrix}$

$\begin{matrix} δ λ_{f} & \cdot [Δ x^{R} - \sum_{n} {\hat{w}}^{n} Δ x^{n} - Δ R \cdot r_{0}^{R} + r_{0}^{R}] + \\ δ λ_{m} & \cdot [Δ ϕ^{R} - Δ {\overset{´}{ϕ}}^{R}] . \end{matrix}$

This expression yields the following initial stress stiffness terms:

$\begin{matrix} λ_{f} \cdot & [(δ ω^{R} \cdot d ω^{R}) r^{R} - \frac{1}{2} (δ ω^{R} \cdot r^{R}) d ω^{R} - \frac{1}{2} (d ω^{R} \cdot r^{R}) δ ω^{R}] + \\ λ_{m} \cdot T^{- 1} \cdot & [δ ω^{R} \sum_{n} {\hat{w}}^{n} r^{n} \cdot d r^{n} - \frac{1}{2} \sum_{n} {\hat{w}}^{n} d r^{n} r^{n} \cdot δ ω^{R} - \frac{1}{2} \sum_{n} {\hat{w}}^{n} r^{n} d r^{n} \cdot δ ω^{R} + \\ d ω^{R} \sum_{n} {\hat{w}}^{n} r^{n} \cdot δ r^{n} - \frac{1}{2} \sum_{n} {\hat{w}}^{n} δ r^{n} r^{n} \cdot d ω^{R} - \frac{1}{2} \sum_{n} {\hat{w}}^{n} r^{n} δ r^{n} \cdot d ω^{R}] . \end{matrix}$