Copyrighted Materials 0CpUyPress o CHAPTER FOUR Thin Plates In practice we frequently encounter "thin"plates whose thickness is small com- pared with all other dimensions.Such a plate,undergoing small displacements, may be analyzed with the approximations that the strains vary linearly across the plate,(out-of-plane)shear deformations are negligible,and the out-of-plane normal stress o:and shear stresses tx,tyz are small compared with the in-plane normal ox,oy,and shear try stresses. Under certain conditions,solutions may be obtained for thin plates either by the solution of the differential equations representing equilibrium or by energy methods.Here we demonstrate the use of the first method via the example of long plates and the second method via examples of rectangular plates either with sym- metrical layup or with orthotropic and symmetrical layup.(For orthotropic plates the directions of orthotropy are parallel to the edges of the plate.)We chose these three types of problems because (i)they illustrate the analytical approaches and the use of the relevant equations,(ii)solutions can be obtained without extensive numerical algorithms,and last,but not least,(iii)they are of practical interest. Additionally,and importantly,these problems provide insights that are useful when analyzing plates by numerical methods. Although the specification of orthotropy may seem to be overly restrictive,in fact it does not unduly limit the applicability of the analyses.The reason for this is that plates are often made according to the 10-percent rule,and such plates behave similarly to orthotropic plates.2 Therefore,solutions for orthotropic plates provide good approximations of the deflections,maximum bending moments,buckling loads,and natural frequencies of nonorthotropic plates that have symmetrical layup and are constructed according to the 10-percent rule.The 10-percent rule 1 J.M.Whitney,Structural Analysis of Laminated Anisotropic Plates.Technomic,Lancaster, Pennsylvania,1987. 2 I.Veres and L.P.Kollar,Approximate Analysis of Mid-plane Symmetric Rectangular Composite Plates.Journal of Composite Materials,Vol.36,673-684,2002. 89CHAPTER FOUR Thin Plates In practice we frequently encounter “thin” plates whose thickness is small com￾pared with all other dimensions. Such a plate, undergoing small displacements, may be analyzed with the approximations that the strains vary linearly across the plate, (out-of-plane) shear deformations are negligible, and the out-of-plane normal stress σz and shear stresses τxz, τyz are small compared with the in-plane normal σx, σy, and shear τxy stresses. Under certain conditions, solutions may be obtained for thin plates either by the solution of the differential equations representing equilibrium or by energy methods.1 Here we demonstrate the use of the first method via the example of long plates and the second method via examples of rectangular plates either with sym￾metrical layup or with orthotropic and symmetrical layup. (For orthotropic plates the directions of orthotropy are parallel to the edges of the plate.) We chose these three types of problems because (i) they illustrate the analytical approaches and the use of the relevant equations, (ii) solutions can be obtained without extensive numerical algorithms, and last, but not least, (iii) they are of practical interest. Additionally, and importantly, these problems provide insights that are useful when analyzing plates by numerical methods. Although the specification of orthotropy may seem to be overly restrictive, in fact it does not unduly limit the applicability of the analyses. The reason for this is that plates are often made according to the 10-percent rule, and such plates behave similarly to orthotropic plates.2 Therefore, solutions for orthotropic plates provide good approximations of the deflections, maximum bending moments, buckling loads, and natural frequencies of nonorthotropic plates that have symmetrical layup and are constructed according to the 10-percent rule. The 10-percent rule 1 J. M. Whitney, Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, Pennsylvania, 1987. 2 I. Veres and L. P. Koll´ar, Approximate Analysis of Mid-plane Symmetric Rectangular Composite Plates. Journal of Composite Materials, Vol. 36, 673–684, 2002. 89