Finite Deflections of Sandwich Plates

This paper gives the basic differential equations for finite transverse deflections of sandwich plates under the following assumptions. The plate consists of a core layer and of two face layers of such construction that the face-parallel stresses in the core and the variation of the face stresses over the thickness of the face layers are negligible. The resultant equations permit the analysis of the effect of transverse shear stress deformation and transverse normal stress deformation in the core on the overall behavior of the plate. I t is shown that, in general, the effect of the transverse normal stresses in the core is negligibly small compared with the effect of the transverse shear stresses. The equations, when simplified by the omission of the transverse normal stress terms, are brought into a form suitable for the solution of rectangular-plate problems. It is further shown that the range of deflections for which the linear ("small deflection") theory is adequate decreases in accordance with a simple explicit formula as the core is made softer relative to the faces. The finite deflection equations are used to obtain plate-buckling equations that include the effect of transverse shear stress deformation on buckling loads. A typical buckling problem is solved and discussed. • (1) I N T R O D U C T I O N I THIS NOTE we derive a system of differential equations for the small finite deflections of sandwich plates. This system of equations is a generalization of the now well-known results for homogeneous plates, where the problem can be reduced to two simultaneous equations for the transverse deflection w of the middle surface and for an Airy stress function F. We consider a sandwich plate consisting of a core layer of thickness h — t and two face layers of thickness t each. We assume tha t / is small compared with h and tha t the values of the elastic constants Ef, Gf for the face layers are large compared with the values of the elastic constants ECf Gc for the core layer. We further assume tha t the products tEf, tGf are large compared with the values of hEC) and hGc. On the basis of t h e assumption tha t t <C h, we may assume tha t the stresses in the faces parallel to their planes are distributed uniformly over the thickness of the face layers.f On the basis of the assumption tha t hEc <C tEf) we may neglect the face-parallel stresses in the core layer and their effect on the deformation of the composite plate. Thus, we t reat the sandwich plate as a combination of two plates without bending stiffness (the face layers), and of a third plate (the core layer) Received January 16, 1948. * Department of Mathematics. f By this assumption we exclude the possibility of investigating local (skin) instability. offering resistance only to transverse shear stresses and transverse normal stresses. A question of primary interest for such a composite plate is whether nonlinear effects have to be taken into account as soon as the transverse deflections are of the order of the face layer thickness t or whether this is only the case when the transverse deflections become of the order of the core thickness h. We shall decide this question on the basis of the system of equations obtained in this note. We somewhat limit the general applicability of the present results by the assumptions made concerning order of magnitude of thickness ratio and ratio of elastic constants. These assumptions should, however, be adequate in those cases where metal face layers are separated by a balsawood or synthetic material core layer. More general results can be obtained along the lines of the following work with the help of somewhat lengthier algebra. We restrict at tention to face layer materials t h a t are isotropic in the plane of the layers. This restriction also may be removed should the eventual usefulness of the present considerations warrant further developments. (2) EQUILIBRIUM EQUATIONS AND STRESS-STRAIN RELATIONS FOR F A C E AND C O R E LAYERS The face layers are treated as thin plates with negligible bending stiffness normal to their planes. In the equilibrium equations for these plates (membranes), the effect of small finite displacements is taken into account. The loads applied to the membranes consist in external loads p and in the stresses a t the junction of face layers and core layer. We indicate quantities referring to the upper membrane (z = -\-h/2) by a subscript + and quantities referring to the lower membrane (z = —h/2) by a subscript —. With the notation of Fig. 1 the equilibrium differential equations for the membranes are the following: (Wx±/bx) + $S±/by) =F rxsb = 0 (1) ( 5 5 ± / b x ) + (bNy+/by). =F TV± = 0 (2) d / dw±\ c) / 5 w ± \ dx\ z ± dx / dy \ * bx J d x \ * dy / dy\ ydB dy /