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Background Structural Theory

This page contains background theory that will be useful for the extended exercises invloving statics experiments.

Equipment for the extended exercises

These extended exercises will allow you to study the structural mechanics of the model building. Our basic assumption (which you may like to question) is that the floors of the building can only move horizontally, resulting in a three-degree of freedom problem. The experimental rig available for the structures extended exercise allows measurement of the displacement at each of the three floors, d₁, d₂ and d₃, and allows forces p₁, p₂ and p₃ to be applied at the floors. We will write these displacements and forces as vectors:

d and p are work-conjugate, that is d.p is the work done when the loads p move by displacements d.

In most practical structural analysis the relationship between displacements and forces is written as a stiffness matrix, K (here a 3 x 3 matrix):

K d = p

but it might sometimes be easier to find the inverse, the flexibility matrix, F:

F p = d,    F = K^-1

It is possible to show that both K and F must be symmetric matrices.

The best way to understand the stiffness matrix is by considering its eigenvectors and eigenvalues. For an eigenvector v_i and its associated eigenvalue λ_i ,

K v_i = λ_i v_i

The eigenvectors are dimensionless mode shapes, and the eigenvalues are the stiffnesses of those modes, and have units of force/displacement. For a 3 x 3 symmetric matrix, there are three eigenvectors, which can be chosen to be orthonormal (orthogonal, and with unit magnitude), and three real eigenvalues (in general, eigenvalues may be complex).

It is often assumed that the stiffness matrix is unchanged by loads (i.e. the stiffness equations are linear), but in fact variation in stiffness with load is often a critical factor in the design of structures, particularly when structures buckle. Buckling can be thought of as when the stiffness of the structure reduces to zero, sometimes in an unexpected mode. Thus, on the experimental rig, it is possible to add vertical loads to each of the floors, w₁, w₂, w₃, and to investigate how the stiffness matrix K depends on these parameters.

© Cambridge University, Engineering Department
Last updated 12/10/2012 by hrs@eng