20Kirchhoff Plates: Field Equations 20–1 3. . . §20. . . . . §20.1. . . . The Biharmonic Equation Notes and Bibliography .Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS TABLE OF CONTENTS Page §20. . §20. . .2.3. . . . Mathematical Models . . . . . . . . . . . . . . . . . . . . Introduction Plates: Basic Concepts §20. . . . . . . . . . . . . .3.3. . . §20. . .4. . . . . . . . . Terminology . . Transverse Shear Forces and Stresses . . The Strong Form Diagram . . Indicial and Matrix Forms . Skew Cuts . . . .3. . . Kinematic Equations . . 20–3 20–3 20–3 20–4 20–5 20–6 20–6 20–8 20–9 20–10 20–11 20–12 20–13 20–13 20–14 20–16 . . . . . . . . . . §20. . §20. §20.3.7. §20. . . .4. . . . . . Exercises . . . . . . . . . . . . . . . . .1. . . . . .3. . .2. . . . .3. . . .3. . . §20. . . . . . . . §20. . .2. . .2. §20. . . . . . 20–2 . . . . . . . . . .2. . . . The Kirchhoff Plate Model §20. Equilibrium Equations .1. . . Structural Function . . §20.2. . . . . Moment-Curvature Constitutive Relations §20. . .3. .5. . . . .6. . . . . . . . The field equations for isotropic and anisotropic plates are then discussed. §20. The Chapter closes with an annotated bibliography.1(a).2. shells and near incompressible behavior. Plate bending components occur when plates function as shelters or roadbeds: flat roofs. Their primary function is to carry out lateral loads to the support by a combination of moment and shear forces. Flatness: The midsurface of the plate. that is. In this Chapter we begin the study of one of those difficult problems: plate bending. attention is focused on the Kirchhoff model for bending of thin (but not too thin) plates. Finding those displacements.1.1. called its thickness. loads normal to its midsurface as sketched in Figure 20.1. which is the locus of the points located half-way between the two plate surfaces.2 PLATES: BASIC CONCEPTS §20. As a result of such actions the plate displacements out of its plane and the distribution of stresses and strains across the thickness is no longer uniform. strains and stresses is the problem of plate bending.1(b). If the plate displays linear elastic behavior under the range of applied loads then we have effectively reduced the problem to one of two-dimensional elasticity. This process is often supported by integrating beams and plates. Prominent among them are Thinness: One of the plate dimensions. Following a review of the wide spectrum of plate models. This state occurs if the external loads act on the plate midsurface as sketched in Figure 20.2. The beams act as stiffeners and edge members. 20–3 . is much smaller than the other two. Plates: Basic Concepts In the IFEM course [255] a plate was defined as a three-dimensional body endowed with special geometric features. is a plane. Under these conditions the distribution of stresses and strains across the thickness may be viewed as uniform. also called membrane or lamina state in the literature. §20. and the three dimensional problem can be easily reduced to two dimensions. such as plate bending. In Chapter 14 of that course we studied plates in a plane stress state. A flat plate structure in: (a) plane stress or membrane state. Structural Function In this Chapter we study plates subjected to transverse loads. bridge and ship decks.§20. (a) (b) z y z y x x Figure 20. Introduction Multifield variational principles were primarily motivated by “difficult” structural models. (b) bending state. Those structures are composed of repeating plates that transmit roof loads to the edge beams through a combination of bending and “arch” actions. Such a three dimensional body is called a plate if the thickness dimension h is everywhere small. Terminology This subsection defines a plate structure in a more precise form and introduces terminology. If so all plates experience both types of action. as in pavements. Idealization of plate as two-dimensional mathematical problem. h/L c is typically 1/5 to 1/100 for most plate structures. also called material filament Figure 20. The end points of these filaments describe two bounding surfaces. If the plate is shown with an horizontal midsurface. An example where that happens is in folding plate structures common in some industrial buildings. §20. directed along the normal to the reference surface (that is. in the z direction) and extending h/2 above and h/2 below it. imagine material normals. but not too small. We will generally allow the thickness to be a function of x. The term “small” is to be interpreted in the engineering sense and not in the mathematical sense. as in Figure 20. Axis x and y are placed in the midplane. The third axis. We place the axes x and y on that surface to locate its points. See Figure 20. Plates designed to resist both membrane and bending actions are sometimes called slabs. ceases to structurally function as a thin plate! 20–4 . called the plate surfaces.2. such as the fabric of a parachute or a hot air balloon. Consider first a flat surface. For example. The one in the +z direction is by convention called the top surface whereas the one in the −z direction is the bottom surface. y).2. although most plates used in practice are of uniform thickness because of fabrication considerations.Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS Midsurface y (b) Mathematical Idealization Plate Γ x (c) Ω (a) Thickness h Material normal. plates work simultaneously in membrane and bending. The magnitude h is the plate thickness. we shall orient z upwards.2. A paradox is that an extremely thin plate. Such a combination is treated in finite element methods by flat shell models: a superposition of flat membrane and bending elements. called the plate reference surface or simply its midsurface or midplane. compared to a characteristic length L c of the plate midsurface. Next. y.2.2. z is taken normal to the reference surface forming a right-handed Cartesian system. If the applied loads contain both loads and in-plane components. that is h = h(x. also called material filaments. forming a right-handed Rectangular Cartesian Coordinate (RCC) system. The more important models are listed next. §20. The last two models are primarily used in detailed or local stress analysis near edges. inflatable masts. such as inflatable structures and fabrics (parachutes. or shell-like behavior. Exact models: for the analysis of additional effects (e. etc).2.§20.g. for example. balloon walls. Also called simply plate bending.. Proposed by von K´arm´an in 1910 [776]. The book of Timoshenko and Woinowsky-Krieger [739] follows Green’s exposition of Reissner’s theory in [316]. as well as certain forms of boundary conditions. Mindlin’s version [493] was intended as a more accurate model for plate vibrations.625]. several mathematical models have been developed to cover that spectrum. The resulting problems of structural mechanics are called: Inextensional bending: if the plate does not experience appreciable stretching or contractions of its midsurface. The bent plate problem is reduced to two dimensions as sketched in Figure 20.1(b). See.2 PLATES: BASIC CONCEPTS A plate is bent if it carries loads normal to its midsurface as pctured in Figure 20. for example. As a result. Kirchhoff model: for thin bent plates with small deflections. material behavior. Also called combined bending-stretching. This model is particularly important in dynamics and vibration. 20–5 . point loads or openings. since shorter wavelengths enhance the importance of transverse shear effects. All models may incorporate other types of nonlinearities due to. Mathematical Models The behavior of plates in the membrane state of Figure 20. negligible shear energy and uncoupled membrane-bending action. von-K´arm´an model: for very thin bent plates in which membrane and bending effects interact strongly on account of finite lateral deflections. Kirchhoff’s seminal paper appeared in 1850 [416]. The book of Timoshenko and Woinowsky-Krieger [739] contains a brief treatment of the exact analysis in Ch 4. bent plates give rise to a wider range of physical behavior because of possible coupling of membrane and bending actions. Extensional bending: if the midsurface experiences significant stretching or contraction. In 1 Kirchhoff’s classic book on Mechanics [417] is available in digital form on the web. Membrane shell model: for extremely thin plates dominated by membrane effects. On the other hand. The reduction is done through a variety of mathematical models discussed next. sails. composite fracture. coupled membrane-bending. The last four models are geometrically linear in the sense that all governing equations are set up in the reference or initially-flat configuration. Important model for post-buckling analysis. the book by Reddy [622] and references therein. non-negligible normal stress σzz ) using three dimensional elasticity. Reissner proposed his model for static analysis of moderately thick plates [624. The first two models are geometrically nonlinear and thus fall outside the scope of this course. tents.3.1 Reissner-Mindlin model: for thin and moderately thick bent plates in which first-order transverse shear effects are considered.2(c). High order composite models: for detailed (local) analysis of layered composites including interlamina shear effects.1(a) is adequately covered by twodimensional continuum mechanics models. cracking or delamination. The slopes of the midsurface in the x and y directions are ∂w/∂ x and ∂w/∂ y. • Applied transverse loads are distributed over plate surface areas of dimension h or greater.3. 20–6 . ∂x (20.2 • The support conditions are such that no significant extension of the midsurface develops. for example with solid elements.Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS this course. as studied in Chapter 12 of [255]. respectively. near the point of application of the loads using more refined models. is based on the assumption that plane sections remain plane and normal to the deformed longitudinal axis.3. respectively.1) The Kirchhoff model can accept point or line loads and still give reasonably good deflection and bending stress predictions for homogeneous wall constructions. however. we shall focus attention only on the Kirchhoff and Reissner-Mindlin plate models because these are the most commonly used in statics and vibrations. • The plate is symmetric in fabrication about the midsurface.3. For small deflections and rotations the foregoing kinematic assumption relates these rotations to the slopes: θx = 2 ∂w . We now describe the field equations for the Kirchhoff plate model. • The plate thickness is either uniform or varies slowly so that three-dimensional stress effects are ignored. see Figure 20. In the finite element literature Kirchhoff plate elements are often called C 1 plate elements because that is the continuity order nominally required for the transverse displacement shape functions. • The plate is thin in the sense that the thickness h is small compared to the characteristic length(s). These are positive as per the usual rule. Kinematic Equations The kinematics of a Bernoulli-Euler beam. Poisson and Kirchhoff. The kinematics of the Kirchhoff plate is based on the extension of this to biaxial bending: “Material normals to the original reference surface remain straight and normal to the deformed reference surface. we shall look only at the linear elastic versions. §20. §20. particles that were on the midsurface z = 0 undergo a deflection w(x. however.” This assumption is illustrated in Figure 20.1. Furthermore. but not so thin that the lateral deflection w becomes comparable to h. It is known as the Kirchhoff plate model. ∂y θy = − ∂w . Upon bending. The Kirchhoff Plate Model Historically the first model of thin plate bending was developed by Lagrange.3. The rotations of the material normal about x and y are denoted by θx and θ y . The Kirchhoff model is applicable to elastic plates that satisfy the following conditions. in honor of the German scientist who “closed” the mathematical formulation through the variational treatment of boundary conditions. A detailed stress analysis is generally required. y) along z. of simply Kirchhoff plate. ∂y ∂y ∂y = −z (20. ∂x2 ∂ 2w = −z 2 = −z κ yy . u z } of a plate particle P(x. ∂x ∂x ∂x ∂u z ∂w ∂w + =− + = 0.3. ∂y ∂ 2w = −z 2 = 0. Plainly the full displacement and strain fields are fully determined if w(x. y) is given. (20. z) not necessarily located on the midsurface are given by u x = −z ∂w = zθ y . ∂ x∂ y (20. ∂x ∂ x∂ y ∂u z ∂w ∂w + =− + = 0. In practice w << h for the Kirchhoff model to be valid. ∂x2 κ yy = ∂ 2w . 20–7 . Lateral deflection w greatly exaggerated for visibility. The displacements { u x . u y . Kinematics of Kirchhoff plate.2) The strains associated with these displacements are obtained from the elasticity equations: ex x = e yy = ezz = 2ex y = 2ex z = 2e yz = ∂u x ∂x ∂u y ∂y ∂u z ∂z ∂u x ∂y ∂u x ∂z ∂u y ∂z ∂ 2w = −z κx x .§20. ∂z ∂u y ∂ 2w + = −2z = −2z κx y .4) are the curvatures of the deflected midsurface. ∂y u z = w. ∂x u y = −z ∂w = −zθx . ∂ y2 κx y = ∂ 2w .y) θx x Ω Section y = 0 Figure 20.3) Here κx y = ∂ 2w .3 THE KIRCHHOFF PLATE MODEL θ y (positive as shown if looking toward −y) θy y Deformed midsurface Γ Original midsurface x w(x. y. σx y E 13 E 23 E 33 2ex y E 13 E 23 E 33 2κx y 20–8 (20. Moment-Curvature Constitutive Relations The nonzero bending strains ex x . The stress σx y = σ yx is sometimes referred to as the in-plane shear stress or the bending shear stress. Similarly.3). • Each plate lamina z = constant is in plane stress.3. to distinguish it from the transverse shear stresses σx z and σ yz studied later. plane strain and plane stress coalesce if and only if Poisson’s ratio is zero. §20.4. illustrating sign conventions. Remark 20. which in matrix form is σx x E 11 E 12 E 13 ex x E 11 E 12 E 13 κx x σ yy = E 12 E 22 E 23 e yy = −z E 12 E 22 E 23 κ yy . But these forces appear necessarily from the equilibrium equations as discussed in §20. fabricated of the same material through the thickness. Bending stresses and moments in a Kirchhoff plate. For example. it is necessary to make several assumptions as to plate material and fabrication. the transverse shear strains are zero. The geometric approach has the advantage that the kinematic limitations of the plate bending model are more easily visualized.2. For a homogeneous isotropic plate. Some inconsistencies of the Kirchhoff model emerge on taking a closer look at (20. We shall assume here that • The plate is homogeneous.3. Remark 20.2.4. is a closer approximation to the physics.1. ezz = 0 says that the plate is in plane strain whereas plane stress: σzz = 0. this implies σx z = σ yz = 0 and consequently there are no transverse shear forces.5) . • The plate material obeys Hooke’s law for plane stress. To establish the plate constitutive equations in moment-curvature form.4. If the plate is isotropic and follows Hooke’s law. On the other hand the direct approach followed here is more compact. Both inconsistencies are similar to those encountered in the Bernoulli-Euler beam model. Many entry-level textbooks on plates introduce the foregoing relations gently through geometric arguments. e yy and ex y produce bending stresses σx x . because students may not be familiar with 3D elasticity theory. that is. and have been the topic of hundreds of learned papers that fill applied mechanics journals but nobody reads. σ yy and σx y as depicted in Figure 20.Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS Normal stresses Inplane shear stresses dy dx dy dx h Top surface z Bending stresses (+ as shown) y y dx dy x σyy σxx σxy = σyx x y y x Myx Bottom surface M yy Bending moments (+ as shown) Mxx M yy Mxx Mxy y M xy x Mxy = M yx x M xx Myy M yx 2D view Figure 20. h2 max. Mx x dy = −h/2 h/2 M yy d x = Mx y dy = M yx d x = −h/2 h/2 −h/2 h/2 −h/2 −σ yy z d x dz ⇒ M yy = − −σx y z dy dz ⇒ Mx y = − −σ yx z d x dz ⇒ M yx = − −h/2 h/2 −h/2 h/2 −h/2 h/2 −h/2 σ yy z dz. The moments are calculated by integrating the elementary stress couples through the thickness: h/2 h/2 −σx x z dy dz ⇒ Mx x = − σx x z dz. however. For example. which are fabricated with different materials. and the end result are moment-curvature relations as in (20. Formulas (20. the integration process is explained in specialized books.4.min σ yy =± 6M yy . inextensional bending is possible. 0 1 Mx y 2κx y 0 0 2 (1 + ν) Here D = 1 Eh 3 /(1 12 − ν 2 ) is called the isotropic plate rigidity. (20. σ yx z dz. M yy and Mx y are given.3 THE KIRCHHOFF PLATE MODEL The bending moments Mx x . the essential steps are the same but the integration over the plate thickness may be significantly more laborious. (20.§20. It will be shown later that rotational moment equilibrium implies Mx y = M yx . j = 1.7) The Di j = E i j h 3 /12 for i.5) into (20. (20. 2. They have dimension of force × length.9) These max/min values occur on the plate surfaces as illustrated in Figure 20. h2 σxmax.7) specializes to κx x 1 ν 0 Mx x (20. The positive sign conventions are indicated in Figure 20. the maximum values of the corresponding in-plane stress components can be recovered from σxmax. coupling occurs between membrane and bending effects even if the plate is only laterally loaded. 3 are called the plate rigidity coefficients. This coupling is examined further in chapters dealing with shell models. (For common structures such as reinforced concrete slabs or laminated composites. If the wall fabrication is not symmetric. 20–9 . that is.4. Remark 20. 2 h (20. one obtains the moment-curvature constitutive relations Mx x M yy Mx y h3 = 12 E 11 E 12 E 13 E 12 E 22 E 23 E 13 E 23 E 33 κx x κ yy 2κx y = D11 D12 D13 D12 D22 D23 D13 D23 D33 κx x κ yy 2κx y .min = σ yx . The Kirchhoff and the plane stress model need to be linked to account for those effects.9) are useful for stress driven design. kips-in/in = kips.6) σx y z dz.min =± y 6Mx y max.3.8) M yy = D ν 1 κ yy . For an isotropic material of elastic modulus E and Poisson’s ratio ν. force. M yy and Mx y are stress resultants with dimension of moment per unit length. For non-homogeneous plates. Consequently only Mx x .min =± x 6Mx x .) If the plate fabrication is symmetric about the midsurface. If the bending moments Mx x .6) and carrying out the integrations. M yy and Mx y need to be calculated. On inserting (20.8). 12) should be significantly smaller than the maximum inplane stresses (20.2. predictions such as (20.12) must come entirely from equilibrium analysis. h/2 −h/2 max σ yz dz = 23 σ yz h. h (20.3.11) If Q x and Q y are given. 20–10 .5. For a homogeneous plate and using an equilibrium argument similar to Euler-Bernoulli beams. Transverse Shear Forces and Stresses The equilibrium equations derived in §20.5. with physical dimensions such as N/cm or kips/in. σx z = σxmax z h2 4z 2 max σ yz = σ yz 1− 2 . Associated with these shear forces are transverse shear stresses σx z and σ yz . h (20. Transverse shear forces and stresses in a Kirchhoff plate. The Kirchhoff plate model ignores the transverse shear energy and in fact predicts σx z = σ yz = 0 from the strain-displacement (kinematic) equations (20.12) As in the case of Bernoulli-Euler beams. y } system are called Q x and Q y . In practical terms this means that stresses (20.3.Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS Top surface z dx dy Parabolic distribution across thickness dy dx h Transverse shear stresses σxz x y σ yz y y x Bottom surface Qy Qy Qx Transverse shear forces (+ as shown) Qx y x 2D view x Figure 20.10) max and σ yz .3. as illustrated in Figure 20. are only function in which the peak values σxmax z of x and y. These are defined as shown in Figure 20. which occur on the midsurface z = 0. Remark 20. These are forces per unit of length. §20.5: 4z 2 1 − . which accounts for transverse shear energy to first order. and one should move to the Reissner-Mindlin model. the stresses may be shown to vary parabolically over the thickness.4 require the presence of transverse shear forces. Integrating over the thickness gives the tranverse shear forces Qx = h/2 −h/2 σx z dz = 2 max σ 3 xz Qy = h. the Kirchhoff model is inappropriate. (20.3).9) . Their components in the { x. Cf. the maximum transverse shear stresses are given by = σxmax z 3 2 Qx . h max = σ yz 3 2 Qy . If they are not. illustrating sign conventions. 3.14).16) For cylindrical bending in the x direction we recover the well known Bernoulli-Euler beam equilibrium equation M = q.15) The four equilibrium equations (20.13) through (20. in which M ≡ Mx x and q are understood as given per unit of y width.15) relate six fields: Mx x . 20–11 . 3 Timoshenko’s book [739] unfortunately defines M yx so that Mx y = −M yx . (20. Consequently plate problems are statically indeterminate.14) ∂x ∂y ∂x ∂y Moment equilibrium about z gives3 Mx y = M yx .6.13) and (20. Mx y . Force equilibrium along the x and y axes is automatically satisfied and does not provide additional equilibrium equations. §20. (20. The present convention is far more common in recent books and agrees with the shear reciprocity σx y = σ yx of elasticity theory. Consideration of force equilibrium along the z direction in Figure 20. + = −Q y .13) in which q is the applied lateral force per unit of area. Consideration of moment equilibrium about y and x in Figure 20. Q x and Q y . Equilibrium Equations To derive the interior equilibrium equations we consider differential midsurface elements d x × dy aligned with the { x. y } axes as illustrated in Figure 20.6. M yy . Eliminating Q x and Q y from (20. and setting M yx = Mx y yields the following moment equilibrium equation in terms of the applied load: ∂ 2 M yy ∂ 2 Mx y ∂ 2 Mx x + + 2 = q.6(b) yields two moment differential equations: ∂ Mx y ∂ M yy ∂ M yx ∂ Mx x + = −Q x .§20. Differential midsurface elements used to derive the equilibrium equations of a Kirchhoff plate.6(a) yields the transverse shear equilibrium equation ∂ Qy ∂ Qx + = −q.4.3 THE KIRCHHOFF PLATE MODEL z z (a) (b) Qx Qy q dy dx Q y+ x Q x+ ∂ Qx dx ∂x Myy ∂ Qy dy ∂y y M yx Mxy Mxx dy y dx Myy+ x Distributed transverse load (force per unit area) Mxx + ∂ M yx ∂ Mx x dx M yx + dx ∂x ∂x ∂ Mx y Mxy + dy ∂y ∂ M yy dy ∂y Figure 20. M yx . ∂x ∂y (20. ∂x2 ∂ x∂ y ∂ y2 (20. Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS Table 20. MT = [ Mx x M yy Mx y ] = [ M11 M22 M12 ]. The exterior normal n to this cut forms an angle φ with x.7.1 Kirchhoff Plate Field Equations in Matrix and Indicial Form Field eqn Matrix form Indicial form Equation name for plate problem KE CE BE κ = Pw M = Dκ PT M = q καβ = w. κT = [ κx x κ yy 2κx y ] = [ κ11 κ22 2κ12 ].5. run over 1. Skew Cuts The preceding relations were obtained by assuming that differential elements were aligned with x and y. Greek indices. such as α.6.1. Skew plane sections in Kirchhoff plate.7. 20–12 .αβ = q Kinematic equation Moment-curvature equation Internal equilibrium equation Here PT = [ ∂ 2 /∂ x 2 ∂ 2 /∂ y 2 2 ∂ 2 /∂ x∂ y ] = [ ∂ 2 /∂ x1 ∂ x1 ∂ 2 /∂ x2 ∂ x2 2 ∂ 2 /∂ x1 ∂ x2 ].3. positive counterclockwise about z. The tangential direction s forms an angle 90◦ + φ with x.3. §20. Parabolic distribution across thickness z Stresses σnn and σns are + as shown Top surface dy dx ds dy dx dx ds s s y σnz y σnn n φ n x φ x y y s s (b) n y z x (a) Mxx φ Qy Qn ds dy Mns x dy s ds y σns φ n n φ x ds M xy z Mnn dy dx dx M yy Qx M yx Figure 20.αβ Mαβ = Dαβγ δ κγ δ Mαβ. §20. which defines notation and sign conventions.2 only. Consider now the case of stress resultants acting on a plane section skewed with respect to the x and y axes. To facilitate reading the existing literature and the working out the finite element formulation. both matrix and indicial forms are displayed in Table 20. This is illustrated in Figure 20. Indicial and Matrix Forms The foregoing field equations have been given in full Cartesian form. These relations can be obtained directly by considering the equilibrium of the midsurface triangles shown in the bottom of Figure 20.4. Strong Form diagram of the field equations of the Kirchhoff plate model.4 Transverse load q Deflection w Kinematic THE BIHARMONIC EQUATION κ=Pw in Ω Equilibrium Γ PT M = q in Ω Ω Constitutive Curvatures κ M=Dκ in Ω Bending moments M Figure 20. (20. Alternatively they can be derived by transforming the wall stresses σx x . 20–13 . If the lateral load q vanishes.7. Elimination of the bending moments and curvatures from the field equations yields the famous equation for thin plates. Thus.§20.18) in which ∇ 4 = ∇ 2∇ 2 ≡ ∂4 ∂4 ∂4 + 2 + . as happens with plates loaded only on their boundaries. found posthumously in his Notes (1813). The boundary conditions are omitted since those are treated in the next Chapter.17) sφ2 ). Mnn = Mx x cφ2 + M yy sφ2 + 2Mx y sφ cφ . §20. The Biharmonic Equation Consider a homogeneous isotropic plate of constant rigidity D.7. Boundary condition links omitted: those are considered in te next Chapter. and then integrating through the thickness. ∂x4 ∂ x 2∂ y2 ∂ y4 (20.19) is the biharmonic operator. Mns = (M yy − Mx x )sφ cφ + Mx y (cφ2 − (20.8 shows the Strong Form diagram for the field equations of the Kirchhoff plate. σx y .8. σx z and σ yz to an oblique cut. under the foregoing constitutive and fabrication assumptions the plate deflection w(x. σx y . 4 Never published by Lagrange. the deflection w satisfies the homogeneous biharmonic equation.3.7 are related to the Cartesian components by Q n = Q x cφ + Q y sφ . in which cφ = cos φ and sφ = sin φ. first derived by Lagrange4 D ∇ 4 w = D ∇ 2 ∇ 2 w = q. §20. The Strong Form Diagram Figure 20. y) satisfies a non-homogeneous biharmonic equation. The bending moments and transverse shear forces defined in Figure 20. largely obsolete. vibration and stability analysis. Others are simply listed. both theory and computational methods treated. Densely written. Love [449]. The title 5 The advent of finite elements has made the distinction between isotropic and anisotropic plates unimportant from a computational viewpoint. Useful mostly for tracing historical references to specific problems. Emphasis is on technology oriented books of interest to engineers. Focus is on analytical solutions of tractable problems of anisotropic plates. Appropriate for supplementary reading in a senior level course. Calladine [119]. See comments for Letnitskii’s below. Morley [498]. Ling (ed. L. Saving grace: contains the famous Donnell shell equations for cylindrical shells. A classical (late XIX century) treatment of elasticity by a renowned applied mathematician contains several sections on thin plates. Lowe [450]. Still reprinted as a Dover book. book is well written. An elementary treatment of plate mechanics that covers a lot of ground. including plastic and optimal design. in 175 pages. Donnell [189]. Dynamics and vibration of composite plates. Quintaessential Timoshenko: a minimum of generalities and lots of examples. which detracts from its usefulness. Devoted to plate vibrations. Entirely analytical. Deals with anisotropic fabrication of plates. A textbook containing a brief treatment of plastic behavior of homogeneous (metal) plates. Letnitskii [440] See comments for next one. Compact and well written. Good coverage of methods for composite wall constructions. still worth consulting for FEM benchmarks.5 Liew et al. The treatment of numerical methods is obsolete. Covers analysis and design of composite plates. Equation (20. Letnitskii [441] Translated from Russian. Reddy [622]. After the advent of finite elements it is largely a mathematical curiosity. 20–14 . Unlike Timoshenko’s book.) [446].Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS The biharmonic equation is the analog of the Bernoulli-Euler beam equation for uniform bending rigidity E I : E I w I V = q. Gorman [314] . A classical reference book. dynamics. both analytical and numerical (the latter by either finite difference or Galerkin methods). Articles on composite plate constructions. One of the earliest monographs devoted to the anisotropic plates. Not a textbook (lacks exercises) but case driven. This tiny monograph was motivated by the increasing importance of skew shapes in high speed aircraft after World War II. Vibration of plate and shell structures. The problem is one of selection.) [445]. Soedel [680]. Ambartsumyan [18].18) was the basis of much of early (pre-1960) work in plates. remembered (a la Proust) by a few old-timers. Don’t expect any physical insight: Love has no interest in engineering applications. with too many pages devoted to finite difference methods in vogue before 1960. Only books examined by the writer are commented upon. Many exercises. in which w I V = d 4 w/d x 4 . no sparks. Szilard [715]. Hussein [389]. Notes and Bibliography There is not shortage of reading material on plates. Material and notation is outdated. H. Although intended as a graduate textbook it contains no exercises. Timoshenko and Woinowsky-Krieger [739]. Selected English textbooks and monographs are listed below. (eds. A massive work (710 pages) similar in level to Timoshenko’s but covering more ground: numerical methods. Notes and Bibliography is a bit misleading: only shells of special geometries are dealt with.6 The reader will find no composite wall constructions. vectors or tensors.com is recommended. Expanded version of a 1970 book by Ashton and Whitney. with references as to source. (Unfortunately the sign conventions are at odds with those presently preferred in the Western literature. [642] contains chapters that collect specific solutions of plate statics and vibration. There are only two chapters treating moderately large deflections. As for plates. The well known handbook by Roark et al. Long portions are devoted to the author’s research and supporting experimental work. The physical insight is unmatched. Despite these shortcomings it is still the reference book to have for solution of specific plate problems. 20–15 . and mechanical engineering.) Exhaustive reference source for all work before 1959. Whitney [794]. civil. There is no treatment of buckling or vibration. The vast compendium of solutions comes handy when checking FEM codes. and for preliminary design work. one with the von Karman model. Concentrates on layered composite and sandwich wall constructions of interest in aircraft structures. Old-fashioned full notation is used: no matrices. Oriented to civil engineers. marine. Wood [809]. For general books on composite materials and their use in aerospace. which can be a source of errors and some confusion. only one chapter on orthotropic plates (called “anisotropic” in the book) and a brief description of the ReissnerMindlin model.addall. 6 The companion volumes Vibrations of Engineering Structures and Theory of Elastic Stability. Strangely. where understanding of the physics was far more important than numbers. in 3 chapters out of 16. which had been published in between the 1st and 2nd editions. this long-windedness fits nicely with the deliberate pace of engineering before computers. Focuses on plastic analysis of reinforced concrete plates by yield line methods.§20. a web search on new and used titles in http://www3. which are covered in other Timoshenko books. the emphasis is on linear static problems using the Kirchhoff model. 7.18) is PT DP w = q. where W is a diagonal weighting matrix and P is defined in Table 20. EXERCISE 20. assuming that D is constant over the plate. EXERCISE 20. 20–16 .2 [A:15] Using the matrix form of the Kirchhoff plate equations given in Table 20.1.Chapter 20: KIRCHHOFF PLATES: FIELD EQUATIONS Homework Exercises for Chapter 20 Kirchhoff Plates: Field Equations EXERCISE 20. θ y and the angle φ defined in Figure 20.1 [A:10] The biharmonic operator (20. EXERCISE 20.3 [A:15] Rewrite PT DP w = q in indicial notation. eliminate the intermediate variables κ and M to show that the generalized form of the biharmonic plate equation (20. in terms of θx . assuming that D is constant over the plate.5 [A:20] Express the rotations θn and θs about the normal and tangential directions.4 [A:20] Rewrite PT DP w = q in glorious full form in Cartesian coordinates.1.19) can be written symbolically as ∇ 4 = PT WP. Find W. This form is valid for anisotropic plates of variable thickness. EXERCISE 20. respectively.