Problem description #
Static systems are considered “unstable” when no unique equilibrium is possible. An unstable system contains components that deform, move or twist without resistance. The linear static analysis of unstable models is possible and returns results that can be helpful in order to find instability. However, the reliability of these results is questionable. Other types of analysis are generally not possible on unstable models.
Instabilities can only be partially localized automatically. In cases where it is not possible to find the problem automatically, a message appears after the calculation: “The model contains singularities”
Reason – Possible causes of instabilities #
Support #
If the support lacks in one direction, the model can move without resistance.
It is important to note, that spatial models always need to be supported in all three directions. This principle applies in planar systems analogously in two directions.
Number of hinges #
If all beams have a hinged connection, this node can rotate freely around every axis (assuming the use of bending joints, the transmission of all forces/exclusion of moments)
Analogically, a line can rotate freely around itself in case all connected domains embody edge hinges.
Hinge chains #
More hinges in a row can build a hinge chain. The “middle” nodes can move without resistance perpendicularly to the chain.
Hinges on free bar ends or domain edges #
Hinges on a free bar end or a free domain edge generate an unstable node or an unstable line.
Hinge arrangements #
An unfavorable arrangement of hinges can generate instabilities.
- Global instabilities (e.g.: unstable frames)
- Local instabilities (e.g.: node rotation without resistance by changing edge hinges)
Finding instabilities #
Automatic avoidance of instabilities
Many of the above-explained instabilities are automatically recognized by AxisVM and, when possible, avoided. However, some types of instabilities cannot be recognized automatically. They are only indicated with the message “The model contains singularities”
- Hinge chains
- Global instabilities
Visual control #
In many cases, the instabilities can be found with a visual control of the inputs. As a first step in locating instabilities, control of the model with the appropriate considerations must be done.
Small stiffnesses #
In certain circumstances, instability has no further importance since there is no force acting in the direction of the possible resistance-free movement. In this case, a small value in the hinge stiffness can stabilize the model without any relevant influence on the results.
Principle forces and moments can indeed be transferred with small stiffnesses, however, they cause large deformations even at low intensities. With the stiffness-oriented calculation (linear elastic correlation between internal forces and deformations), the transferred forces and moments in the corresponding direction are negligibly small.
Step-by-step approach #
If an assumption of which part of the model can contain the instability already exists, it is suitable to use the step-by-step approach.
- Save a copy of the model (File menu → Save as…)
- Remove all hinges (delete edge hinges, block beam releases, high values for all rigidities in connected elements and supports) in one part of the model (e.g.: per stories)
- Start linear analysis
- If the singularities message does not appear (the model is stable), it can be assumed that the instability is generated by a hinge in the region of the removed hinges.
- If the singularities message still appears, repeat the procedure for other parts of the model
The analysis results have no particular meaning in this approach, to minimize the calculation time, a larger mesh can be applied.
Systematic approach #
- Creation of a new loadcase
- Definition of a “Node load” for each node in the model
- Display the entire model (turn off all parts in the model)
- Activate the function “Nodal loads”
- Selection of all nodes
- Start linear static analysis
- Control of the deformation (resultant translation er, resultant rotation fr)
- The unstable part of the model shows a large displacement
Finding large displacements #
- Color scale: Result values that lie outside the color scale limits are shown as hatched surfaces or lines
- Min, Max values: The function “Min/Max values” determines the lowest and highest occurring values of the chosen result component in the visible elements
Related articles #
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