Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24 290 where W K is the modulus of subgrade reaction (elastic coefficient of the foundation) and 1 S K and 2 S K are the shear moduli of the subgrade (shear layer foundation stiffness). If foundation is homogeneous and isotropic, we will get 1 2 S S S K K K   . If the shear layer foundation stiffness is neglected, Pasternak foundation becomes a Winkler foundation. The variation of kinetic energy of the plate can be expressed as: (13) where dot-superscript convention indicates the differentiation with respect to the time variable t ; ( ) z  is the mass density given by Eq. (1b); and ( i I , i J , i K ) are mass inertias expressed by     /2 2 0 1 2 /2 , , 1, , ( ) h h I I I z z z dz     (14a)     /2 2 1 2 2 /2 , , , , ( ) h h J J K f z f f z dz     (14b) Substituting Eqs. (9), (11), and (13) into Eq. (8), integrating by parts, and collecting the coefficients of 0 u  , 0 v  , 0 w  and   ; the following equations of motion are obtained: 0 0 0 0 1 1 0 0 0 0 1 1 2 2 2 2 2 0 0 0 0 0 1 2 0 2 2 2 2 2 2 : : : 2 : 2 xy x xy y b b b xy y x e s x N N w u I u I J x y x x N N w v I v I J x y y y M M M u v w f I w I I w J x y x y x y M M x                                                                           2 0 0 1 2 2 2 2 0 2 s s s s xy y xz yz M S S u v J x y x y x y y J w K                                 (15) where 2 2 2 2 2 / / x y        is the Laplacian operator in two-dimensional Cartesian coordinate system.      0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 ( ) V A K u u v v w w z dV I u u v v w w w w w w I u u v v x x y y J u u v v x x y y w w w w I x x y y                                                                                                             2 0 0 0 0 2 K x x y y w w w w J dA x x x x y y y y                                                                            