Summary: The paper deals with a stochastic analysis of random displacements governed by isotropic elasticity equations. The elasticity constants are assumed to be random fields with Gaussian distribution. Under the assumption of small fluctuations of elasticity constants but not ignoring their distribution and correlation structure, we derive the spectral tensor of the random displacement field. A randomized spectral representation is then used to simulate this random field numerically. The same approach was applied to the case of random loading. In this case, the intensity of fluctuations may be arbitrarily large. A series of test calculations confirm the high accuracy and computational efficiency of the method.