5.2 DEFLECTION OF RECTANGULAR SANDWICH PLATES 179 A 众 余 L Figure 5.8:The different types of supports along the long edges of a long sandwich plate. deflection of the plate wo and the rotation xr do not vary along y: aw° =0 aXx0 (5.40) ay ay We neglect the shear deformation in the y-z plane (yy=0).Consequently,the rotation of the normal is zero(Eq.5.3): Xyz=0. (5.41) The equilibrium equations are (Egs.4.22 and 4.23) :+p=0 (5.42) dx dM:-V.=0. (5.43) dx When the sandwich plate is symmetrical with respect to the midplane([B]=0) from Eqs..(5.12),(5.15),(5.40),and(5.41),we have M=-n2 V SuYzz. (5.44) Equations (5.42),(5.43),and(5.44),together with Eq.(5.2),give sandwich plate,symmetrical layup: xz -Du dx +p=0 (5.45) hx琴+Sx-c Du- =0. (5.46) For a transversely loaded isotropic sandwich beam the corresponding equa- tions are (Eqs.7.83 and 7.84) isotropic sandwich beam: (5.47) +s(-x=0 (5.48)5.2 DEFLECTION OF RECTANGULAR SANDWICH PLATES 179 y x Ly Lx p z Lx Figure 5.8: The different types of supports along the long edges of a long sandwich plate. deflection of the plate wo and the rotation χxz do not vary along y: ∂wo ∂y = 0 ∂χxz ∂y = 0 . (5.40) We neglect the shear deformation in the y–z plane (γyz = 0). Consequently, the rotation of the normal is zero (Eq. 5.3): χyz = 0. (5.41) The equilibrium equations are (Eqs. 4.22 and 4.23) dVx dx + p = 0 (5.42) dMx dx − Vx = 0. (5.43) When the sandwich plate is symmetrical with respect to the midplane ([B] = 0) from Eqs. (5.12), (5.15), (5.40), and (5.41), we have Mx = −D11 ∂χxz ∂x Vx = S 11γxz . (5.44) Equations (5.42), (5.43), and (5.44), together with Eq. (5.2), give sandwich plate, symmetrical layup: −D11 d3χxz dx3 + p = 0 (5.45) D11 d2χxz dx2 + S 11 dwo dx − χxz = 0. (5.46) For a transversely loaded isotropic sandwich beam the corresponding equa￾tions are (Eqs. 7.83 and 7.84) isotropic sandwich beam: −EI d3χ dx3 + p = 0 (5.47) EI d2χ dx2 + S
dw dx − χ  = 0, (5.48)