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A method is presented for stability analysis and finding mechanisms of frames with partially rigid connections. Equilibrium equations are formulated with respect to the reduced vector of member-end forces, for which linear constraints are assigned. The modes of self-equilibrium forces and unstable mechanisms are found using singular value decomposition of the matrix representing equilibrium equations and constraints. An infinitesimal mechanism with small degree of kinematic indeterminacy can be found using limit analysis for an artificial proportional loading condition. The effectiveness of proposed method is demonstrated through examples of frames with rectangular and hexagonal grids.

An optimization approach is presented for design of a tetrahedral tuned mass damper called TD-TMD for three-directional seismic response reduction of structures. The mass damper consists of a viscous damper and a mass connected by springs and a rigid bar. By utilizing flexibility of the springs, movement of the mass in three-directions and elongation of the viscous damper are amplified, and vibration energy of the structure is effectively dissipated by the viscous damper. The objective function of the parameter optimization problem is the mean norm of the response displacements of the structure. The bounds of parameters are determined by solving an auxiliary nonlinear programming problem to maximize the minimum deformation of the damper against unit static loads in various directions. Approximate optimal solutions are found using a heuristic approach called simulated annealing combined with pure random search that generates efficient intial solutions. The TD-TMD is attached to a simple three-degree-of-freedom structure, and the seismic responses are compared with those with conventional single-directional tuned mass dampers.

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