Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09 118 where  is the top surface, and e f is the density of reaction force of foundation. For the Pasternak foundation model: 2 2 ( , ) ( , ) e w p w x y f k w x y k x     (10) The equilibrium equations can be acquired using the Hamilton principle. 2 3 3 '2 0 0 0 1 2 3 1 2 2 2 2 2 2 3 4 4 0 0 0 0 0 0 1 2 4 5 2 2 2 2 2 2 2 2 2 2 4 ' '2 '2 ' 0 1 1 3 1 5 1 2 2 2 : 0 : ( , ) 0 : x b x e s xz x N u w u I I I k A x t t x t x M w w u w w q x t N f I I I I x x t t x t x t x Q M u k A k A I k A I k A x x t x t                                                         4 2 '4 6 1 2 2 2 0 I k A x t x        (11) where   x N denote the resulting force in-plane,     , b s x x M M denote the total moment resultants and   xz Q are transverse shear stress resultants and they are defined as /2 /2 /2 /2 /2 /2 /2 /2 , , ( ) , ( ) h h b x x x x h h h h s x x xz xz h h N dz M zdz M f z dz Q g z dz                 (12) Following the Navier solution process, we assume the following solution form for   0 0 , , u w  and that check the boundary conditions, 0 0 1 cos( ) sin( ) e sin( ) i t m u U x w W x x                               (13) where , , U W and  are arbitrary parameters to be determined,  is the natural frequency, and m L    . The transverse load ( ) q x is also expanded in Fourier series as:   1 ( ) sin m m q x Q x      (14) where 0 2 ( )sin( ) L m Q q x x dx L    (15) In the case where a sinusoidally distributed load, we have 1 0 1 , m Q q   (16) In the case where uniform distributed the load, we have