Learning topic
Kinematic Indeterminacy and Primary System
Learn how to identify independent joint rotations and translations, determine kinematic indeterminacy, and construct the primary system for displacement-method analysis.
Kinematic indeterminacy is the number of independent joint displacements required to describe the deformed configuration within the adopted structural model. In the classical displacement method these displacements are the primary unknowns.
Rotational unknowns
Each rigid joint that can rotate independently may contribute a rotational coordinate $Z_i$. Member ends rigidly connected at one joint share the same joint rotation.
Translational unknowns
Translations occur when a joint or group of joints can move without violating the adopted kinematic constraints. A common example in frames is a horizontal storey or sway displacement.
Primary system
To form the primary system, all independent unknown displacements are temporarily restrained by artificial constraints. A rotational restraint suppresses a joint rotation; a translational restraint suppresses the corresponding linear displacement.
Unit displacements
To obtain equation coefficients, impose $Z_j=1$ at one restraint while all other primary displacements remain zero. The resulting reactions in the artificial restraints form the stiffness coefficients $r_{ij}$.
Dependence on modeling assumptions
The degree of kinematic indeterminacy depends on the model. For example, neglecting axial deformation may constrain translations of several joints to be equal and reduce the number of independent coordinates.
About this topic
This topic explains the degree of kinematic indeterminacy, selection of independent joint rotations and translations, and introduction of restraints in the displacement-method primary system.