point in some abstract ‘configuration space' V. Here, Rhas 21 dimensions; this number could be reduced by taking into account a priori information about constraints, like e.g. no displacement at points like A or B. By applying the laws of elasticity, one may compute what the energy of the structure would be in state v for a given load. Let be this energy. The state u which is actually reached under this load is that (if unique) which minimizes J among all possible states. This may be written, in condensed form, as follows: The search for such a u is a mathematical problem (known as an ‘optimization problem', or ‘variational problem') that can be solved, at least approximately, with a computer. This is the cornerstone of structural analysis, at least in the static case. Straightforward generalizations to dynamic problems exist. Formulation (1) is nevertheless too general for the purpose of this paper and we need extra assumptions. First, a ‘superposition principle': if v and w are two possible displacements, v + w also is one. Similarly, A® is an element of V for any real A. Next, the potential energy of the structure in the field of imposed forces is supposed to be linear, as a function of displacements, so if E(v) is this energy, E^v) = vh where the V, are the real parameters which constitute v. We note this as follows: meaning that E(v) is the scalar product of the vector of forces by the vector of displacements. (‘Force' is to be taken in a large sense, for if Vj is a rotation angle,/- is a torque.) Last, the elastic energy of the structure is supposed to be a quadratic function of displacements, which means that reactions are proportional to displacements, according to Hooke's law. Thus there exists a symmetric matrix J (21 X 21, in our example), known as the ‘stiffness matrix', such that the elastic energy is L v, Vj, which again we denote as
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