This study also assumed that [l.sub.a] denotes the length of each block (i.e., the length of the gob), [l.sub.b] denotes the width of the protective coal pillar between the blocks, [h.sub.k] denotes the mining height of the coal seam, [h.sub.1], [h.sub.2], [h.sub.3], ..., [h.sub.j] denote the thicknesses of various strata above the coal seam (total thickness of m layers of strata above the coal rock and thus can be written as [H.sub.m] = [SIGMA] [h.sub.m] = 1, 2, 3, ..., j)), [k.sub.a] denotes the coal's elastic foundation coefficient, kb denotes the backfill body's elastic foundation coefficient, and [k.sub.1], [k.sub.2], ...
where p denotes the pressure on the foundation per unit area, w(x) denotes the subsidence of the foundation, and k denotes the elastic foundation coefficient.
The elastic foundation coefficient of the backfill body and the backfill ratio satisfy the following expression :
The elastic foundation coefficients of the backfill body, the coal seam, and (m - 1) strata above the bedrock can be written as
In addition, the influence of the foundation coefficients as well as the boundary restraint parameters are also analyzed, which can serve as the benchmark data for the future research technique.
The foundation coefficients are [K.sub.W] = 10 and [K.sub.S] = 10.
In this table, first four frequency parameters Q are obtained for various boundary conditions and foundation coefficients according to the present formulation and compared with those given by Zhou et al.
Thus, based on the comparison studies, the first known results of rectangular thick plates rested on Pasternak foundation with various elastic boundary conditions as well as foundation coefficients are presented in Table 3.
Lastly, the vibration analysis of the moderately thick FG sandwich sector plate resting on two-parameter foundations with different boundary conditions is conducted and the effects of the elastic foundation coefficients on the free vibration characteristic of the sector plate are presented.
Thus, the author will present some new results of moderately thick FG sandwich sector plates resting on two-parameter elastic foundation with various boundary conditions, material types, and foundation coefficients using the present method in the next examples.
The effects of elastic boundary restraint parameters, power-law exponents, sector angles, material types, thickness schemes, and foundation coefficients on the frequency values are examined and discussed in detail.
(3) The vibration characteristics of the moderately thick FG sandwich sector plate strongly depend on boundary conditions, power-law exponents, sector angles, material types, thickness schemes, and foundation coefficients.