Plates and Shells Victor Birman Engineering Education Center, Missouri University of Science and Technology, St. Louis, MO, USA 1. Kinematic equations determine a displacement field 1 Introduction 1 throughout the structure experiencing deformations. A 2 Classical Theory of Plates and Shells 3 particular theory is based on assumptions adopted for the kinematic equations. For example, plates and shells 3 Bending and Buckling of Thin Isotropic Plates 4 are often analyzed by assumption that the thickness of the 4 Plates and Shells with Stiffeners and Cut-Outs 6 structure remains constant under loading, implying that 5 Composite and Sandwich Plates and Shells 7 deflections along a normal to the middle surface of the 6 Summary 8 structure (the surface that is equidistant from the outer Related Chapters 8 surfaces) are constant. 2. Strain-displacement relationships reflect the magnitude References 9 of deformations relative to characteristic dimensions of the structure. These relationships are affected by kine- matic equations, enabling us to introduce such effects as transverse shear deformability into the analysis. If 1 INTRODUCTION displacements of the plate or shell are relatively small, the strains can be assumed linear functions of displace- Plates and shells represent principal elements of aerospace ments. However, in case of large displacements, strains structures, including fuselages of planes and missiles, con- are nonlinear functions of deformations. Linear strain- trol surfaces, bulkheads, helicopter blades, and others. The displacement relationships always represent a particular multiple applications, shapes, and materials found in plate case of a more general nonlinear formulation. and shell structures dictate the necessity of a comprehensive 3. The constitutive relations account for the physical prop- approach to their analysis reflected in relevant theories and erties of the material defining the stress tensor in terms methodologies. of the strain tensor. Thus, elastic, elastic–plastic, viscoe- Various aspects of the theory and analysis of these leastic, viscoplasctic, shape memory, piezoelectric and structures are found in the books by Timoshenko and other materials can be characterized by the appropri- Woinowsky-Krieger (1959), Novozhilov (1964), Dym ate theory. With the exception of sandwich structures (1974), Libai and Simmonds (1998), Ugural (1999), Ventsel with a soft core, the thickness of the plate or shell can and Krauthammer (2001), and Reddy (2007). usually be assumed unaffected by deformations, that is, All theories of plates and shells rely on the following the normal strain εz = 0 (the Cartesian coordinate sys- relationships: tem adopted in the analysis of rectangular plates and the cylindrical coordinate system used for cylindrical shells are shown in Figures 1 and 2, respectively). In nearly Encyclopedia of Aerospace Engineering. all plates and shells found in applications, the applied Edited by Richard Blockley and Wei Shyy c 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-68665-2 pressure and normal stresses through the thickness are 2 Structural Mechanics Figure 1. Cartesian coordinate system and stresses acting on an infinitesimal element (for simplicity, stresses on the top and bottom surfaces z = const are not shown). of equilibrium of an infinitesimal element detached from the structure where the effect of adjacent parts is repre- sented by the stresses applied to the element (Figure 1). The increments in the stresses on the opposite faces of the element shown in Figure 1 indicate that they are con- tinuous functions of the coordinates. The other method is the energy approach using the Hamilton principle. An alternative to using equilibrium equations for the analy- sis is the application of one of the energy methods, such as the Rayleigh–Ritz method. The use of energy meth- ods is conditioned on specifying the expressions for the potential energy of the structure (kinetic energy is also necessary in dynamic problems). There are five equations of equilibrium including those Figure 2. Cylindrical coordinate system and stress notation. enforcing the equilibrium of forces acting on an infinitesimal element in the x, y, and z-directions and the equilibrium of moments about the x and y coordinate axes. The sixth equa- at least an order of magnitude higher than the maximum tion of equilibrium of moments about the z-axis is identically in plane stresses, so that σz ≈ 0. The reciprocity law of satisfied since τxy = τyx. shear stresses dictating the symmetry of the stress tensor Equilibrium equations for plates and shells involve func- implies thatτmn = τnm (m, n = x, y, z). tions of stresses, that is, the forces and moments per unit 4. Equilibrium equations (equations of motion in dynamic width of the cross section (they are called stress resultants problems) and the boundary conditions can be derived by and stress couples, respectively). These stress resultants and two methods. One of these methods employs the analysis stress couples are defined by integrals of the stresses and Plates and Shells 3 their moments with respect to the middle surface through the thickness. For example, in the cylindrical coordinate system (Figure 2) the in plane stress resultants Ni,Nij , transverse stress resultants Qi and stress couples Mi,Mij are h h Nx 2 z 1 Ny 2 1 = σx 1 − dz, = σy dz, Mx − h R z My − h z 2 2 h h Nxy 2 z 1 Nyx 2 1 = τxy 1 − dz, = τyx dz, Mxy − h R z Myx − h z 2 2 h h Figure 3. Stress resultants and stress couples in the Cartesian coor- 2 z 2 Qx = τxz 1 − dz, Qy = τyzdz dinate system (for simplicity, stress resultants and couples are shown − h R − h (1) only on two edges of an infinitesimal element). 2 2 transverse pressure become = = where σi,τin (i x, y, n x, y, z) are normal and shear M M N + N = 0,N+ N + xy,x + y,y = 0, stresses oriented similarly to those in Figure 1, R is the radius x,x xy,y xy,x y,y R R of the middle surface, and h is the thickness of the shell. In Ny the case of a flat plate, R = 0 andNxy = Nyx,Mxy = Myx. Mx,xx + 2Mxy,xy + My,yy − + q = 0 (3) While these stress resultants and couples are not equal to R z each other in cylindrical shells, in thin shells the ratio R << 1 where q is pressure applied in the perpendicular direction to since− h ≤ z ≤ h , so that the differences between N and 2 2 xy the middle surface and (...), i ≡ ∂(...)/∂i, i = x, y. Three Nyx and between Mxy and Myx are negligible. The simplified equations of equilibrium (3) are not sufficient to determine version of equations (1) becomes six stress resultants and stress couples they contain, that is, the problem is statically indeterminate. This obstacle is elim- inated if the equations of equilibrium are expressed in terms h h 2 2 of displacements or by the use of the stress function illus- Nx 1 Ny 1 = σx dz, = σy dz, trated below for a flat plate. Equations of equilibrium of flat Mx z My z − h − h plates in the Cartesian coordinate system are obtained from 2 2 equation (3) if R =∞. h h 2 2 Nxy Nyx 1 = = τxy dz, Qx = τxzdz, Mxy Myx z − h − h 2 CLASSICAL THEORY OF PLATES AND 2 2 h SHELLS 2 Qy = τyzdz (2) The classical theory of thin plates and shells is based on − h the Kirchhoff–Love hypothesis. Two assumptions involved 2 in this hypothesis are: The stress resultants and stress couples defined by equations 1. A cross-section perpendicular to the middle surface prior (2) are depicted in Figure 3 in a Cartesian coordinate system. to deformation remains plane and perpendicular to the The difference between results obtained by various shell deformed middle surface (Figure 4). This assumption can theories is usually small, with the exception of the Donnell also be formulated in the sense that a normal to the mid- theory that is less reliable for long cylindrical shells (e.g., Bert dle surface remains straight and normal to this surface. and Kim, 1995). Here we consider cylindrical shells using the Accordingly, transverse shear strains in the planes xz and Love shell theory. Eliminating transverse shear stress resul- yz are equal to zero. Moreover, the length of the normal tants Qx and Qy from the moment equilibrium equations, the remains constant(εz = 0). remaining three equations for the cylindrical shell subject to 2. The transverse normal stress σz = 0. 4 Structural Mechanics and v v κ =−w ,κ=−w + ,y ,κ=−2w + ,x x ,xx y ,yy R xy ,xy R (7) respectively. Upon the substitution of equations (5) into the plane-stress Hookean constitutive relations and the subsequent integration of the stresses according to (2), the stress resultants and stress couples become Eh Eh N = ε0 + νε0 ,N= ε0 + νε0 , x 1 − ν2 x y y 1 − ν2 y x = 0 Nxy Ghγxy, M = D κ + νκ ,M= D κ + νκ , Figure 4. Deformation of the cross section in the xz plane according x x y y y x to the classical plate theory. 1 − ν M = D κ (8) xy 2 xy The kinematic equations reflecting this hypothesis define the following vector of displacements where E and ν are the modulus of elasticity and the Poisson ratio of the material, respectively, and the so-called Eh3 u (x, y, z) u0 (x, y) − zw(x, y) cylindrical stiffness is D = . Transverse shear stress ,x 12(1−ν2) = = − u v (x, y, z) v0 (x, y) zw(x, y),y (4) resultants Qi are equal to zero according to the assumption w (x, y) w (x, y) of plane stress. where u0, v0, and w are in plane displacements of the mid- dle surface in the x- and y-directions and the deflection in 3 BENDING AND BUCKLING OF THIN the z-direction, respectively, while u and v are in plane dis- ISOTROPIC PLATES placements of a point located at a distance z from the middle surface.