ABSTRACT

Elements of variational calculus were discussed in Chapter 3, and the Principle of Virtual Work was introduced in Chapter 5. Under static conditions, the principle is repeated here as

∫δEijSijdV+∫δuiρüidV=∫δuiτidS. (10.1)

As before, δ represents the variational operator. We assume for our purposes that the displacement, the strain, and the stress satisfy representations of the form

u(x, t)−φT(x)Φγ(t), E=βT(x)Φγ(t), S=DE, (10.2)

in which E and S are written as one-dimensional arrays in accordance with traditional finite-element notation. For use in the Principle of Virtual Work, we need D′, which introduces the factor 2 into the entries corresponding to shear. We suppose that the boundary is decomposed into four segments: S=SI+SII+SIII+SIV. On SI, u is prescribed, in which event δu vanishes. On SII, the traction τ is prescribed as τ0. On SIII, there is an elastic foundation described by τ=τ0−A(x)u, in which A(x) is a known matrix function of x. On SIV, there are inertial boundary conditions, by virtue of which τ=τ0−Bü. The term on the right now becomes

(10.3)

The term on the left in Equation 10.1 becomes ∫δEijSijdV=δγTKγ(t), K=∫ΦTβ(x)D′βT(x)ΦdV ∫δuiρüidV=δγTMÿ(t) M=∫ρΦTφ(x)φT(x)ΦdV, (10.4)

in which K is called the stiffness matrix and M is called the mass matrix. Canceling the arbitrary variation and bringing terms with unknowns to the left side furnishes the equation as follows:

(10.5)

Clearly, elastic supports on SIII furnish a boundary contribution to the stiffness matrix, while mass on the boundary segment SIV furnishes a contribution to the mass matrix.