A Elements of the nodal displacement correlation matrix for a clamped-free beam

B Elements of the nodal displacement correlation matrix for a simply supported beam

C Elements of the nodal displacement correlation matrix for a clamped simply supported beam

D Correlation matrix of elements of the nodal displacement correlation matrix for a clamped clamped beam

E Stochastic clamped free beam under stochastic loading

If the beam is clamped at x = 0 and free at x = L and is subjected to a load q( x), then the governing equations for the bending moment are given by eqns (4.2) and (4.6). The boundary conditions are

For an arbitrary bending moment m( y), we obtain:

Similarly,

The boundary conditions (E2) and (E3) in explicit form become, after substituting into eqn (4.8):

We first determine f ₂( x) from the fourth equation; the third equation then yields f ₁( x), after substituting f ₂( x) into it; the second equation yields g ₂( y), whereas the first equation yields g ₁( y).

The solution of covariance function becomes:

where

To obtain the boundary conditions for the covariance function we first note that the boundary conditions for displacement for the beam are:

For an arbitrary displacement w( y), we have:

Similarly,

In explicit form, the boundary conditions (E9) and (E10) are:

By solving for F ₁( x), F ₂( x), G ₁(