Applies To Product(s): RAM Elements, RAM Structural System, Ram Concept, STAAD.pro Version(s): Any Environment: N/A Area: N/A Subarea: N/A Original Author: Bentley Technical Support Group

# Instability in Finite Element Analysis

A typical 3-dimensional Finite Element analysis of a structure requires that every node must be stable in all 6 degrees of freedom (TX, TY, TX, RX, RY,RZ). This is achieved by specifying fixity conditions for the columns, beams and braces spanning to a given node or through nodal restraint. While many programs can analyze a structure using fewer degrees of freedom, for this discussion all 6 are assumed to be active.

There are many discussions related to FEA online and whole courses devoted to the topic, but the purpose of this article is merely to show by example a few of the most common causes of instabilities in structural models. The rules apply to RAM Elements (aka RAM Advanse), RAM Frame, RAM Concrete or STAAD.pro as well as other FEA applications. The images and examples below are taken from RAM Elements where a light blue circle indicates a hinge, or member release, at the end of a member. A translational restraint is depicted as a triangle on rollers and a rotational restraint is a "T".

## Pinning the free end of a cantilever.

Take the case of a single member fixed at the base for all 6 DOF similar to a flagpole. This structure is stable, except that the free end of member away from the support is hinged or released for major axis bending. As a result, node 2 can spin about the global z axis.

For some applications, this type of "nodal instability" will terminate the analysis. For other applications, a small stiffness may be automatically assigned to the z axis rotational stiffness of the node and you may only get a warning, so long as a moment about the z axis is not applied directly to node 2. This would cause infinite rotation of the node and should terminate any analysis.

The same situation often occurs for a beam with a cantilever, where the cantilever beam is the only member connected to the node at the tip. In short, the free end of any member, where that member is the only member in the model connected to a particular node should never be released.

## Beam, column and brace intersections.

When multiple members frame to a single node, it is acceptable to release some, but not all of those members. If the beam, column and brace are all released at the same node, then the problem is the same as case 1 above. At least one of those members should be fixed ended. In most situations, it is the column top that should remain fixed to the node.

## Releasing the tops of columns.

In this case we have a fixed ended beam setting on two columns, both of which are released at the top node. This case differs slightly from Case 2 because the nodes are fixed to the beam and not themselves instable. The problem is that the beam along with both top nodes can spin as a group on top of the columns similar to a log on water. This is an example of why it is usually better to keep the tops of the columns fixed and release the beams.

## Torsional releases

It's hard to envision a realistic connection that allows a member to spin or swivel, but most FEA application do allow member torsional releases. A general rule is to leave the member torsion fixed except in a situation where member rotation really is free. The most common problem occurs in a chevron brace configuration where the beam is two finite elements. If each beam half is released in torsion, then the node at the top of the braces is instable.

## 2-dimensional frame in a 3-dimensional analysis

Often it is desirable to analyze a 2-dimensional frame using a 3-dimensional analysis. In some applications there is plane frame option that can be used to ignore the deflection out of plane (e.g. z axis) or rotation about the other axes, but if not, the frame can generally be stabilized one of two ways.

• An out of plane, z axis restraint can be applied to some or all of the nodes to effectively keep the frame from falling over, or
• Rotation about the in-plane axes can be restrained at the base nodes (e.g. rotation about the X axis).

The same situation often occurs in RAM frame when no rigid diaphragm is used. This can leave the model with several, isolated, 2D frames in space with no connection between them. If the frames are pinned at the base then they can fall over and an instability results. Fixing the base of the frames against out-of-plane rotation is generally the solution to this problem, though connecting the frames together with lateral members or some other simulation of a diaphragm is also possible.

## Other Global stability issues

A certain number of nodal restraints are always required to keep the structure as a whole from moving. Another common case is one where a shell or mat foundation is supported by a series of vertical springs. While that is stable in relation to vertical loads, some mechanism must be provided to keep the mat as a whole from sliding around like a skateboard. This is generally achieved through the use of horizontal springs in addition to the vertical springs, or by restraining the translation of a node (or line of nodes) along the edge of the structure.

This is a common problem in Ram Concept if a vertical resistance area spring is the only support for the structure. When there are no lateral loads, you might get away with providing an area spring with only vertical stiffness, but when there is any external load applied in the plan directions, some resistance to sliding must be incorporated into the model.

## Diaphragm stability

In most building type structures there is a horizontal diaphragm that ties the frames together and prevents in-plane deformation of the plan. This is typically modeled using a rigid floor diaphragm. The diaphragm constraint forces the nodes of the floor to move together preventing the plan from racking for example.
In space frame models where no rigid diaphragm is modeled (perhaps because the roof is sloped), there must be some other mechanism to keep the plan from racking. This is generally achieved by providing diagonal members in that plane. Fixing the minor axis of the beams in the plan is another approach. Think of this like creating a Vierendeel truss in plan. Using shell elements is another option, though the interaction between the shells and the members is not always desirable.

## Using tension-only members or compression-only springs

When a model utilizes non-linear members or springs most FEA applications iteratively solve for each load case and load combination. On each iteration, if a tension-only member is found to go into compression, that member is thrown out of the analysis and a new iteration is started. If too many of the members go into compression, the frame or structure as a whole can become instable.

There are a couple of ways to effectively deal with such a situation

• Apply a pre-tensioning force to the braces. By putting the member into an initial tension state it is less likely to go into compression and fall out of the analysis.
• Assign some of the members to be tension and compression members. For X braced frames or other symmetric structures it is typically acceptable to analyze the structure with a single tension + compression brace rather than a pair of tension-only braces. This does affect the load path through the columns somewhat, however, and may require two versions of the model to capture the worst condition. A similar option is to leave both braces in the model, but then check the braces or twice the determined force.
• Applying self weight to tension only braces will cause bending moments in the members which usually is not the intent for tension-only braces. in those cases, a zero density material is suggested.
• Additionally, for X braced frames in Ram Elements, the program may be introducing a node at the intersection of the braces. This can be prevented using Process - analyze - FE Model tab - by turning off the option to "Add intermediate nodes at member intersections".

## P-Delta effects and model instabilities

There are cases where a structure might be perfectly stable under a first-order analysis, but as the analysis incorporates P-Delta effects the deflection is amplified and instability can result. Different applications handle P-Delta analysis in different ways, but there are usually controls for the tolerance required for P-Delta convergence. Increasing the tolerance often leads to a solution, but some structures may have to be stiffened in order to complete a P-Delta analysis on all load cases.

# See Also

RAMSS Eigenvalue Error

Structural Product TechNotes And FAQs

STAAD.Pro Instability And Zero Stiffness

# External Links

Bentley Technical Support KnowledgeBase

Bentley LEARN Server

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