Bending of plates
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load. Bending of Kirchhoff-Love plates. Definitions. For a thin rectangular plate of thickness formula_1, Young's modulus formula_2, and Poisson 's ratio formula_3, we can define parameters in terms of the plate deflection, formula_4. The flexural rigidity is given by Moments. The bending moments per unit length are given by The twisting moment per unit length is given by Forces. The shear forces per unit length are given by Stresses. The bending stresses are given by The shear stress is given by Strains. The bending strains for small-deflection theory are given by The shear strain for small-deflection theory is given by For large-deflection plate theory, we consider the inclusion of membrane strains Deflections. The deflections are given  by Derivation. In the Kirchhoff–Love plate theory for plates the governing equations are and In expanded form, and where formula_26 is an applied transverse load per unit area, the thickness of the plate is formula_27, the stresses are formula_28, and The quantity formula_30 has units of force per unit length. The quantity formula_31 has units of moment per unit length. For isotropic, homogeneous, plates with Young's modulus formula_2 and Poisson's ratio formula_3 these equations reduce to where formula_35 is the deflection of the mid-surface of the plate. Small deflection of thin rectangular plates. This is governed  by the Germain-Lagrange plate equation This equation was first derived by Lagrange in December 1811 in correcting the work of Germain who provided the basis of the theory. Large deflection of thin rectangular plates. This is governed by the Föppl–von Kármán plate equations where formula_39 is the stress function. Circular Kirchhoff-Love plates. The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829. Cylindrical coordinates are convenient for such problems. Here formula_40 is the distance of a point from the midplane of  the plate. The governing equation in coordinate-free form is In cylindrical coordinates formula_42, For symmetrically loaded circular plates, formula_44, and we have Therefore, the governing equation is If formula_47 and formula_48 are constant, direct integration of the governing equation gives us where formula_50 are constants. The slope of the deflection surface is For a circular plate, the requirement that the deflection and the slope of the deflection are finite at formula_52 implies that formula_53. However, formula_54 need not equal 0, as the limit of formula_55 exists as you approach formula_52 from the right. Clamped edges. For a circular plate  with clamped edges, we have formula_57 and formula_58 at the edge of the plate (radius formula_59). Using these boundary conditions we get The in-plane displacements in the plate are The in-plane strains in the plate are The in-plane stresses in the plate are For a plate of thickness formula_64, the bending stiffness is formula_65 and we have The moment resultants (bending moments) are The maximum radial stress is at formula_68 and formula_69: where formula_71. The bending moments at the boundary and the center of the plate are Rectangular Kirchhoff-Love plates. For rectangular plates, Navier in 1820  introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load. Sinusoidal load. Let us assume that the load is of the form Here formula_74 is the amplitude, formula_59 is the width of the plate in the formula_76-direction, and formula_77 is the width of the plate in the formula_78-direction. Since the plate is simply supported , the displacement formula_79 along the edges of the plate is zero, the bending moment formula_80 is zero at formula_81 and formula_82, and formula_83 is zero at formula_84 and formula_85. If we apply these boundary conditions and solve the plate equation, we get the solution Where D is the flexural rigidity Analogous to flexural stiffness EI. We can calculate the stresses and strains in the plate once we know the displacement. For a more general load of the form where formula_89 and formula_90 are integers, we get the solution Navier solution. Double trigonometric series equation. We define a general load formula_92  of the following form where formula_94 is a Fourier coefficient given by The classical rectangular plate equation for small deflections thus becomes: Simply-supported plate with general load. We assume a solution formula_79 of the following form The partial differentials of this function are given by Substituting these expressions in the plate equation, we have Equating the two expressions, we have which can be rearranged to give The deflection of a simply-supported plate (of corner-origin) with general load is given by Simply-supported plate with uniformly-distributed load. For a uniformly-distributed load, we have The corresponding Fourier  coefficient is thus given by Evaluating the double integral, we have or alternatively in a piecewise format, we have The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by The bending moments per unit length in the plate are given by Lévy solution. Another approach was proposed by Lévy in 1899. In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied. The goal is to find formula_113 such that it satisfies the boundary conditions  at formula_114 and formula_115 and, of course, the governing equation formula_116. Let us assume that For a plate that is simply-supported along formula_81 and formula_82, the boundary conditions are formula_120 and formula_121. Note that there is no variation in displacement along these edges meaning that formula_122 and formula_123, thus reducing the moment boundary condition to an equivalent expression formula_124. Moments along edges. Consider the case of pure moment loading. In that case formula_125 and formula_79 has to satisfy formula_127. Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as Plugging the expression for formula_79  in the governing equation gives us or This is an ordinary differential equation which has the general solution where formula_133 are constants that can be determined from the boundary conditions. Therefore, the displacement solution has the form Let us choose the coordinate system such that the boundaries of the plate are at formula_135 and formula_136 (same as before) and at formula_137 (and not formula_84 and formula_85). Then the moment boundary conditions at the formula_137 boundaries are where formula_142 are known functions. The solution can be found by applying these boundary conditions. We can show that for the "symmetrical" case where  and we have where Similarly, for the "antisymmetrical" case where we have We can superpose the symmetric and antisymmetric solutions to get more general solutions. Simply-supported plate with uniformly-distributed load. For a uniformly-distributed load, we have The deflection of a simply-supported plate with centre formula_150 with uniformly-distributed load is given by The bending moments per unit length in the plate are given by Uniform and symmetric moment load. For the special case where the loading is symmetric and the moment is uniform, we have at formula_154, The resulting displacement is where The bending moments and  shear forces corresponding to the displacement formula_4 are The stresses are Cylindrical plate bending. Cylindrical bending occurs when a rectangular plate that has dimensions formula_161, where formula_162 and the thickness formula_163 is small, is subjected to a uniform distributed load perpendicular to the plane of the plate. Such a plate takes the shape of the surface of a cylinder. Simply supported plate with axially fixed ends. For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed formula_164. Cylindrical bending solutions can be found using the Navier and Levy techniques. Bending of  thick Mindlin plates. For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations. Governing equations. The canonical governing equation for isotropic thick plates can be expressed as where formula_47 is the applied transverse load, formula_167 is the shear modulus, formula_168 is the bending rigidity, formula_163 is the plate thickness, formula_170 , formula_171 is the shear correction factor, formula_2 is the Young's modulus, formula_3 is the Poisson's ratio, and In Mindlin's theory, formula_4 is the transverse displacement of the mid-surface of the plate and the quantities formula_176 and formula_177 are the rotations of the mid-surface normal about the formula_178 and formula_164-axes, respectively. The canonical parameters for this theory are formula_180 and formula_181. The shear correction factor formula_171 usually has the value formula_183. The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations where formula_185 is  the displacement predicted for a Kirchhoff-Love plate, formula_186 is a biharmonic function such that formula_187, formula_188 is a function that satisfies the Laplace equation, formula_189, and Simply supported rectangular plates. For simply supported plates, the "Marcus moment" sum vanishes, i.e., In that case the functions formula_186, formula_188, formula_194 vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by Bending of Reissner-Stein cantilever plates. Reissner-Stein theory for cantilever plates leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load formula_196 at formula_82. and the boundary conditions at formula_82 are  Solution of this system of two ODEs gives where formula_202. The bending moments and shear forces corresponding to the displacement formula_203 are The stresses are If the applied load at the edge is constant, we recover the solutions for a beam under a concentrated end load. If the applied load is a linear function of formula_78, then