A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry

• A necessary stability condition for pin-jointed structures is established from the viewpoint of symmetry.
• A prestressed structure contains at least one fully symmetric prestress mode.
• Positive definiteness of stiffness matrices can be efficiently identified.
• All blocks of symmetry-adapted geometric stiffness matrix along the main diagonal are empty if the necessary condition is not satisfied.

Stability conditions are the key to transform kinematically indeterminate structures into prestressed structures or deployable structures. From the viewpoint of symmetry, a necessary condition is presented for the stability of symmetric pin-jointed structures with kinematic indeterminacy. The condition is derived from the positive definiteness of the quadratic form of the tangent stiffness matrix. Numerical examples verify that the proposed necessary stability condition is in accord with the conventional theory of structural rigidity, and is considered to be more comprehensible. It is robust and easy to implement. Results show that a symmetric prestressed structure is guaranteed to possess integral prestress modes, if the necessary condition is satisfied. Further, a pin-jointed structure with fully symmetric mechanism modes is proved to be unstable, if it does not satisfy the condition.