Plates on Random Elastic Foundation

Consider a simply supported rectangular plate with unit stiffness sitting on a linear elastic founda­tion with stiffness K(x, y) and subjected to a spatially distributed load Q(x, y). The displacement W(x, y) of the plate is the solution of the partial differential equation

AAW{x, y) + K(x, y)W(x, y) = Q(x, y), (x, y) є D = (0,a) x (0,b), (14)

where A = d2/dx2 + d2/dy2 denotes the Laplace operator. The plate displacement satisfies the conditions W = 0 on the boundary dD of D, d2W/dx2 = 0 on {0} x (0, b) and {a} x (0, b), and d2W/dy2 = 0 on (0, a) x {0} and (0, a) x {b}. It is assumed that the foundation stiffness K is random and the applied load Q is deterministic.

The foundation stiffness is modeled by the homogeneous translation field

K(x, y) = a1 + (a2 – a1) Ф^О^, y) = a1 + (a2 – a^ Ф(0(x, y) Z), (15)

where 0 < a1 < a2 < ж are some constants,

n/2 ,

G(x, y) = ^ otiAk cos(vk ■ (x, y)) + Bk sin(vк ■ (x, y)) k=1 ^

= в(x, y) Z, (x, y) є D = (0,a)x (0,b), (16)

is a homogeneous Gaussian field, n/2 > 1 is an integer,

в (x, y) = [01 cos(v1 ■ (x, y))…On cos(Vn ■ (x, y)) 01 sin (v 1 ■ (x, y))…On sin(Vn ■ (x, y))|, ZT = [A1 …An/2 B1 …Bn/2], (17)

vk = (vk, x, vk, y) are wave frequencies, vk, x, vk, y > 0 are some constants, vk ■ (x, y) = vk, x x + vk, y y, ok > 0 are some constants such that ^=1 ot = 1, and (Ak, Bk) are independent N(0, 1) variables. We note that K depends on n independent N(0, 1) random variables, the entries of Z, and its marginal distribution is uniform in (a1, a2). The Gaussian field О has mean 0, variance 1, and covariance function

n

E[G(x, y) G(x’, y) = ^2ok2 cos (vt ■ (x – x’,y – y’)) (18)

k=1

so that it is homogeneous. The translation field K in Equation (15) with О in Equation (16) is homogeneous and its covariance and correlation functions can be calculated from the probability

law of G. In applications the covariance function of K, rather than that of G, is given, and we need to find a Gaussian field G such that K in Equation (15) has the required properties. The existence of G and the determination of the second-moment properties of G, if it exists, are discussed in (Grigoriu, 1995: section 3.1).