Issue 51

S. Merdaci et alii, Frattura ed Integrità Strutturale, 51 (2020) 199-214; DOI: 10.3221/IGF-ESIS.51.16 205 The external force according to Navier’s solution can be expressed as 1 1 ( , ) sin( )sin( ) mn m n q x y q x y         (17) where / m a    and / n b    , « m » and « n »are mode numbers. For the case of a sinusoidally distributed load, we have 1 m n   and 11 0 q q  (18) where q 0 represents the intensity of the load at the plate center. Following the Navier solution procedure, we assume the following form of solution for ( u,v,w b ,w s ) that satisfies the boundary conditions cos( )sin( ) sin( )cos( ) , sin( )sin( ) sin( )sin( ) mn mn b bmn s smn u U x y v V x y w W x y w W x y                                      (19) where U mn , V mn ,W bmn , and W smn are arbitrary parameters. Eqn.(15) in combination with Eqn. (16) can be combined into a system of first order equations as:       , K F   (20) where    and   F denotes the columns     , , , , T mn mn bmn smn U V W W   and     0, 0, , T mn mn F q q    (21) and   11 12 13 14 12 22 23 24 13 23 33 34 14 24 34 44 a a a a a a a a K a a a a a a a a              (22) The elements a ij = a ji of the coefficient matrix [K]. The elements of the symmetric matrix [K] presented in Eqn. (23) are given by   2 2 11 11 66 a A A        12 12 66 a A A      2 2 13 11 12 66 [ ( 2 ) ] a B B B       2 2 14 11 12 66 [ ( 2 ) ] s s s a B B B       (23)