Detection of rigid body motions

J. Wackerfuß

In the design of complex planar or spatial beam structures for the transfer of external loads, it may occur that subsections 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 goal of the structural designer is to avoid a kinematic load-bearing system and to design usable static systems, which is not always easy, especially in the case of complex structural topologies. While the detection of external kinematics (due to missing supports) is generally uncritical, the detection 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 is thus - in the static sense - unusable.

Numerical investigations of a structure with one or more kinematics with the help of commercial calculation programs usually lead to a program abort. When solving the system of equations, the following (error) message is output in most cases: "singular stiffness matrix" or "structure is kinematic" or "structure is not sufficiently supported". Causally, however, a program termination can also be related to 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.

In this project, a computational method has been developed to detect kinematics within arbitrary spatial bar structures and visualize them by means of an animation. The individual beam 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, transverse force joints,..., or even 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 translational and rotational springs, whose direction of action (or plane of action) can be entered individually. Any situation can be considered 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 supporting structure does not exhibit any kinematics, the calculation method explicitly indicates the degree of static indeterminacy. Compared to a computationally intensive eigenvalue analysis for the determination of the zero eigenvalues of the system stiffness matrix (e.g. in the context of a finite element calculation), the developed method is characterized in particular by a very low computational effort, which has a favorable effect especially for complex load-bearing structures.

The 3 following figures show the results of a calculation to determine the kinematics of a dome support structure. The supporting structure is composed of straight individual members that are rigidly connected to each other. Only the meridian bars between the 3rd and 4th circle of latitude are articulated on both sides (ball-and-socket joint), which makes the supporting structure 3-fold kinematic (see figures).

Wackerfuß, J.: Algorithmus zur Beschreibung von Starrkörperbewegungen in Mehrkörpersystemen, Diplomarbeit, Technische Hochschule Darmstadt, Institut für Statik, 1997