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which relates all the element displacements {.v} to the nodal displacements {p\ for the whole structure.

Internal element compatibility

Again, for simplicity consider an axial force element. For the displacement within such an element not to introduce any voids or overlaps the displacement along the element, u, needs to be a continuous function of position, x. The compatibility condition is satisfied by du/dx = sx

9.5.3. Stress!strain law

Assuming for simplicity the material behaviour to be homogeneous, isotropic and linearly elastic, then Hooke's law applies giving, for a one-dimensional stress system in the absence of thermal strain,

in which E is the empirical modulus of elasticity.

9.5.4. Force!displacement relation

Combining eqns. (9.2), (9.4) and (9.5) and taking u to be a function of x only, gives

Integrating, and taking m(0) = m, and u(L) = Uj, corresponding to displacements at nodes i and j of an axial force element of length L, gives the force/displacement relationship

in which (uj — ut) denotes the deformation of the element. Thus the force/displacement relationship for an axial force element has been derived from equilibrium, compatibility and stress/strain arguments.

9.6. The principle of virtual work

In the previous section the three basic arguments of equilibrium, compatibility and constitutive relations were invoked and, in the subsequent sections, it will be seen how these arguments can be used to derive rod and simple beam element equations. However, some situations, for example, may require elements of non-uniform cross-section or representation of complex geometry, and are not amenable to solution by this approach. In such situations, alternative approaches using energy principles are used, which allow the field variables to be represented by approximating functions whilst still satisfying the three fundamental arguments. Amongst the number of energy principles which can be used, the one known as the principle of virtual work will be considered here.

The equation of the principle of virtual work

Virtual work is produced by perturbing a system slightly from an equilibrium state. This can be achieved by allowing small, kinematically possible displacements, which are not necessarily real, and hence are virtual displacements. In the following brief treatment the corresponding equation of virtual work is derived by considering the linearly elastic, uniform cross-section, axial force element in Fig. 9.21. For a more rigorous treatment the reader is referred to Ref. 8 (p. 350). In Fig. 9.21 the nodes are shown detached to distinguish between the nodal and element quantities.

' i j Fig. 9.21. Axial force element, shown with detached nodes.

' i j Fig. 9.21. Axial force element, shown with detached nodes.

Giving the nodal points virtual displacements w, and uj, the virtual work for the two nodal points is

This virtual work must be zero since the two nodal points are rigid bodies. It follows, since the virtual displacements are arbitrary and independent, that

which, for the single element, are the equations of external equilibrium.

Applying Newton's third law, the forces between the nodes and element are related as

The first quantity (PiUi+PjUj), to first order approximation assuming linearly elastic behaviour represents the virtual work done by the applied external forces, denoted as We. The second quantity, (5,m, + SjUj), again to first order approximation represents the virtual work done by element internal forces, denoted as W, . Hence, eqn. (9.9) can be abbreviated to

which is the equation of the principle of virtual work for a deformable body.

The external virtual work will be found from the product of external loads and corresponding virtual displacements, recognising that no work is done by reactions since they are associated with suppressed dof. The internal virtual work will be given by the strain energy, expressed using real stress and virtual strain (arising from virtual displacements), as

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