The stiffness of prestressed frameworks: a unifying approach

This paper gives a simple derivation of the tangent stiffness matrix for a prestressed pinjointed structure: the stiffness is found by differentiating equilibrium expressions at nodes of the structure with respect to the position of the nodes. It uses this derivation to compare the diverse formulations that are applied to understanding the mechanics of prestressed structures by different academic communities. Two basic approaches to understanding the mechanics of pin-jointed structures are common. In the computational mechanics approach, the results of computations are used to gain insight into structural response. In this context, it is sensible to use an exact tangent stiffness matrix, as described by e.g. Argyris and Scharpf (1972). Another approach is to gain understanding through the basic formulation of the problem: an exact formulation is less important, and it may prove sensible to use a simplified set of equations. This paper shows the links between various such formulations by describing the exact tangent stiffness matrix using equilibrium and stress matrices, each of which has been used individually to gain understanding of structural response in different circumstances. Using the equilibrium matrix to understand structural response is described in e.g. Pellegrino and Calladine (1986), or Pellegrino (1993). A basic assumption is that the key structural action comes through the deformation of members — a common assumption in structural engineering. Study of the equilibrium matrix (or equivalently its transpose, the compatibility matrix) enables small movements of the structure to be decomposed