Dynamic Analysis of Three-Dimensional Trusses

Abstract

This chapter describes the dynamic behavior of nonlinear three-dimensional pinjointed trusses. In doing so, the chapter aims to introduce in the simpler context of trusses a number of concepts that will be used in the more complex context of solids. The chapter follows closely the presentation of pin-jointed trusses given in Chapter 3 of NL-Statics. However, a number of formulas and derivations will be repeated for completeness, although the reader will occasionally be referred to the derivations in the NL-Statics volume. The chapter starts with a kinematic and dynamic description of an individual pinjointed axial truss member. In particular, the assumption of concentrated mass at the ends of the rod will be introduced at this stage. This is followed by the assembly of individual rod equations into the global dynamic equilibrium equations of a complete truss. The global dynamic equilibrium equations will be recast in Section 2.3 in the form of a variational principle, first in the form of the principle of virtual work and then as the more advanced Hamilton principle, in which concepts such as the Lagrangian, the action integral, phase space, and the simplectic product are introduced. These concepts will be generalized in Chapter 3 for three-dimensional solids. The set of dynamic equilibrium equations require the use of a discrete timestepping scheme in order to advance the solution in time, a process also known as the numerical time integration of the dynamic equilibrium equations. The same leap-frog and mid-point time integration presented in Chapter 1 can be used in the context of pin-jointed trusses. In the case of the leap-frog scheme, this leads to an explicit solution procedure whereby the next positions at each time step can be evaluated from the previous positions and current external forces without the solution of a nonlinear set of equations involving the evaluation of a tangent operator (called the tangent matrix). The explicit nature of this time-stepping scheme makes