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The analysis of pyramidal trusses has an immediate practical interest since these structures are currently used in many present-day civil constructions, either as main parts or a constitutive element. They can be used to represent tripod-like structures, cap of masts, tower cranes, big span roofs, and even a portion of a single-layer geodesic dome or of a generic-shaped reticulated shell. This paper examines the nonlinear dynamic response and stability for a simple class of space trusses in the shape of a regular pyramid. Joints located at the vertices of the base polygon are fixed while the joint at the apex is subjected to static and/or dynamic loads acting in either the vertical direction, in the horizontal plane, or along a generic oblique direction. Despite their apparent simplicity, these structural systems exhibit a wide variety of post-critical responses, not exhausted by the classical snapping and bifurcation phenomena. In addition to regular primary and secondary branches, the equilibrium paths may include neutral branches, namely branches entirely composed of bifurcation or limit points. The analysis is conducted using the Finite Element Method together with a corotational formulation for the bars. Initially, the numerical results are validated in the elastic domain using the closed-form solutions found in literature. Then, bifurcation diagrams are traced showing the possible dynamic instabilities of the model. Finally, the numerical model is exploited for the consideration of inelastic deformations, break of symmetry and sudden bar removal in the system response.