Detection of rigid body motions

J. Wackerfuß

When designing complex planar or spatial beam structures for the transfer of external loads, it may happen that parts of the structure exhibit kinematics. These kinematics are characterized by the fact that the structure (independent of the stiffness values of the supporting members) has no internal resistance for certain external actions. The resulting rigid body movements do not lead to any constraints or stresses in the overall system or in subsystems. The aim of the structural engineer is to avoid a kinematic load-bearing system and to design usable structural systems, which is not always easy, especially with complex structural topologies. While the recognition of external kinematics (due to missing bearings) is generally uncritical, the recognition of kinematics inside the structure is often more complicated, especially if the connections between the individual members are not rigid for structural or static reasons. It should be remembered that even a formally (!) statically indeterminate system can exhibit kinematics and thus - in the static sense - be unusable.

Numerical investigations of a structure with one or more kinematics with the aid of commercial calculation programs generally lead to a program abort. When solving the equation system, the following (error) message is displayed in most cases: "singular stiffness matrix" or "structure is kinematic" or "structure is not sufficiently supported". However, a program abort can also be caused by the occurrence of very large stiffness differences within the system stiffness matrix, which leads to a poorly conditioned system of equations and thus to numerical problems when solving the resulting system of equations.

As part of this project, a calculation method has been developed that can be used to detect kinematics within arbitrary spatial beam structures and visualize them by means of an animation. The individual rod elements can be rigidly, elastically or articulatedly connected to each other. In addition to the classic spherical joint, any other joint types (e.g. normal force joints, shear force joints,..., or oblique joints) can be taken into account, or any joint situations can be modeled by coupling them. An elastic connection of beam elements is realized by means of translation and rotation springs, whose direction of action (or plane of action) can be entered individually. Any situation can be taken into account for the support of the structure. The possibility of an animated display of the results should make it easier for the structural engineer to recognize the kinematics occurring in the supporting structure and to correct them in a targeted manner. If, on the other hand, the structure does not exhibit any kinematics, the calculation method explicitly indicates the degree of static indeterminacy. Compared to a computationally intensive eigenvalue analysis to determine the zero eigenvalues of the system stiffness matrix (e.g. as part of a finite element calculation), the developed method is characterized in particular by a very low computational effort, which is particularly advantageous for complex load-bearing structures.

The following 3 figures show the results of a calculation to determine the kinematics of a dome support structure. The load-bearing structure is made up of straight individual members that are rigidly connected to each other. Only the meridian rods between the 3rd and 4th parallel are articulated on both sides (spherical joint), which makes the supporting structure 3-fold kinematic (see figures).

Wackerfuß, J.: Algorithmus zur Beschreibung von Starrkörperbewegungen in Mehrkörpersystemen, Diploma thesis, Technische Hochschule Darmstadt, Institute of Statics, 1997