In a multi-body finite element analysis (FEA) simulation, various components of different shapes, sizes, and materials interact with one another. Some of these components may be idealized as rigid, meaning they are non-deformable under the action of applied forces and moments. This assumption simplifies the model of complex mechanical and physical systems, thereby increasing the computational efficiency. The focus can be shifted to overall motion dynamics of the multibody systems instead of non-crucial complex material deformations. This rigid idealization can find its applications in robotics, structural analysis and collision analysis.

Rigid bodies can be categorized into two types: discrete rigid bodies, which are composed of nodes and elements, and analytical surfaces that do not require meshing and they are defined mathematically. In this article, we will focus on the various aspects of discrete rigid bodies.

## Analytical Rigid Surfaces

These are simple 2D or 3D geometric surfaces defined using equations rather than the meshes. They are defined using straight or curved lines. They do not undergo any deformation during the analysis; hence they are assumed to be rigid. They interact with deformable components which are discretized in multibody simulations. Compared to discrete rigid bodies which require meshing, analytical rigid surfaces reduce computational costs for the simulation. A reference node with translational and rotational degrees of freedom is used to determine the motion of the entire analytical surface.

## Discrete Rigid Bodies

A discrete rigid body has two components: reference node and collection of elements defining the shape of the body. The reference node has both translational and rotational degrees of freedom. The motion and behavior of the rigid body entirely depends on the reference node and all the elements of the rigid body are connected to the reference node. It is usually placed at a relevant point on the rigid body, such as the center of mass, a pivot point, or a critical axis. In general, the placement of the rigid body reference node is relevant only when rotations are applied or reaction moments about a specific axis are needed. In these cases, the reference node should be positioned along the desired axis going through the body.

## Properties Of Rigid Bodies

- There is no relative deformation between the nodes.
- As they are not deformable, there are no numerical integration points.
- No optimum element formulations are required
- The output variables like stresses, strains and pressure are not available for the elements.
- The displacement and rotation output variables are available for the rigid elements, but they strictly follow the reference node.
- Only reaction forces and moments are available at the reference point from the analysis output.

## Element Types

The rigid bodies can be modeled as axisymmetric, planar or 3D. The rigid elements can only be quadrilateral (R3D4), triangular (R3D3) or beam (RB3D3). There are no solid/cubic rigid elements. Hence, the whole solid body cannot be modelled using rigid elements. Only outer surfaces of the solid body can be represented using planar quadrilateral rigid elements.

The independent degrees of freedom are available only for the reference node of the rigid body. All the rigid nodes, which are connected to the reference points have secondary degrees of freedom. The motion of these nodes is fully determined by the movement of the rigid body reference node.

## Material Requirements

Material definitions like elastic modulus, poison’s ratio and plastic properties are not required for these as elements as the strain and stress values will not be calculated for these rigid elements. Mass and moment of inertia can be applied to these rigid components to apply correct forces and moments in the simulation. Thermal properties like conductivity can also be assigned to the elements in heat transfer problems. In contact simulations, interaction properties like frictional coefficient and damping can be defined.

## Mesh Size Requirements

As the rigid bodies are perfectly stiff, mesh convergence is not required. In large models, we can get away with using coarse mesh for rigid bodies to increase computational efficiency. However, care must be taken to make sure that the rigid body is sufficiently refined whenever there are contact definitions associated with it to ensure proper load transfer. The mesh on the rigid body should be generally of similar size to the mesh size of the interacting deformable part in the contact areas.

## Examples

Below are some of the examples demonstrating the use of rigid elements in multibody FEA models.

### To Define A Rigid Floor

In this case, the deformable ball is dropped onto the floor. In this interaction, when the contact occurs between the ball and the floor, the floor is not subjected to any noticeable deformation. So, it is modelled using rigid elements. Planar quadrilateral rigid elements (R3D4) are used to model the floor. The reference node is constrained in all DOF to specify that the floor is fixed.

### To Define A Punch

In this example, the punch is used to deform the workpiece. The punch is generally made of much harder material compared to the workpiece. It undergoes much less deformation than the workpiece. So, the punch is modelled using rigid elements in this model and the work piece is modelled as deformable. Beam rigid elements (2 node, 3D RB3D2) are used to discretize the punch here. This will allow the focus to be on the material deformation of the workpiece while reducing the computational costs. The reference node is fixed in all DOF except for the vertical displacement.

### To Define Components Of A Mechanism

In this example, the components of the mechanism are modeled using the rigid elements. Planar quadrilateral rigid elements (R3D4) are used to model the components. Making the components rigid will help the simulation to be focused on the motion of the individual components instead of the internal deformation thereby increasing the computational efficiency.

## Final Thoughts

Hopefully this article has provided insights into using rigid elements in your FEA model to increase computational efficiency. Of course, care must be taken to avoid improper use of rigid elements in a FEA model which could lead to error in the simulation results.

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