Based on a generalized non-homogeneous shear-beam model[1]

for stratified foundations with exponential function shear modulus

closed-form analytical expressions are derived for determining natural frequencies

modal displacements

participation factors and steadystate response function. The orthogonality of the modal displacements is proved for the first time. And then

dynamic response of the stratified foundations to earthquake is studied by the theory of random vibration.Two cases of power spectral intensity function for the earthquake acceleration imput at the base rock are considered: the white noise spectrum and filter white noise spectrum. The numerical calculation results show that 1) under the case that the power spectral density of the earthquake acceleration input at the base rock is the white noise spectrum

the maximum expected response of the soil layer is different from that for the case of the filter white noise spectrum; 2) both the earthquake acceleration input the base rock and the response of the soil layer are stationary random process

which overestimates the random response of the soil layer.