AFEM.Ch26

. 26 Kirchhoff Plates: Field Equations 26–1 Chapter 26: KIRCHHOFF PLATES: FIELD EQUATIONS 26–2 TABLE OF CONTENTS Page §26.1. Introduction 26–3 §26.2. Plates: Basic Concepts 26–3 §26.2.1. Function . . . . . . . . . . . . . . . . . . . 26–3 §26.2.2. Terminology . . . . . . . . . . . . . . . . . 26–4 §26.2.3. Mathematical Models . . . . . . . . . . . . . . . 26–5 §26.3. The Kirchhoff Plate 26–6 §26.3.1. Kinematic Equations . . . . . . . . . . . . . . 26–6 §26.3.2. Moment-Curvature Relations . . . . . . . . . . . . 26–8 §26.3.3. Transverse Shear Forces and Stresses . . . . . . . . . 26–10 §26.3.4. Equilibrium Equations . . . . . . . . . . . . . . 26–11 §26.3.5. Indicial and Matrix Forms . . . . . . . . . . . . . 26–12 §26.3.6. Skew Cuts . . . . . . . . . . . . . . . . . . 26–12 §26.3.7. The Strong Form Diagram . . . . . . . . . . . . 26–12 §26.4. The Biharmonic Equation 26–13 §26. Notes and Bibliography. . . . . . . . . . . . . . . . . . . . . . 26–13 §26. Exercises . . . . . . . . . . . . . . . . . . . . . . 26–16 26–2 26–3 §26.2 PLATES: BASIC CONCEPTS §26.1. Introduction Multifield variational principles were primarily motivated by “difficult” structural models, such as plate bending, shells and near incompressible behavior. In this Chapter we begin the study of one of those difficult problems: plate bending. Following a review of the wide spectrum of plate models, attention is focused on the Kirchhoff model for bending of thin (but not too thin) plates. The field equations for isotropic and anisotropic plates are then discussed. The Chapter closes with an annotated bibliography. §26.2. Plates: Basic Concepts In the IFEM course [87] a plate was defined as a three-dimensional body endowed with special geometric features. Prominent among them are Thinness: One of the plate dimensions, called its thickness, is much smaller than the other two. Flatness: The midsurface of the plate, which is the locus of the points located half-way between the two plate surfaces, is a plane. In Chapter 14 of that course we studied plates in a plane stress state, also called membrane or lamina state in the literature. This state occurs if the external loads act on the plate midsurface as sketched in Figure 26.1(a). Under these conditions the distribution of stresses and strains across the thickness may be viewed as uniform, and the three dimensional problem can be easily reduced to two dimensions. 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. x y z (a) x y z (b) Figure 26.1. A flat plate structure in: (a) plane stress or membrane state, (b) bending state. §26.2.1. Function In this Chapter we study plates subjected to transverse loads, that is, loads normal to its midsurface as sketched in Figure 26.1(b). 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. Finding those displacements, strains and stresses is the problem of plate bending. Plate bending components occur when plates function as shelters or roadbeds: flat roofs, bridge and ship decks. Their primary function is to carry out lateral loads to the support by a combination of moment and shear forces. This process is often supported by integrating beams and plates. The beams act as stiffeners and edge members. 26–3 Chapter 26: KIRCHHOFF PLATES: FIELD EQUATIONS 26–4 Ω Γ x y Mathematical Idealization Midsurface Plate Thickness h Material normal, also called material filament (a) (b) (c) Figure 26.2. Idealization of plate as two-dimensional mathematical problem. If the applied loads contain both loads and in-plane components, plates work simultaneously in membrane and bending. An example where that happens is in folding plate structures common in some industrial buildings. Those structures are composed of repeating plates that transmit roof loads to the edge beams through a combination of bending and “arch” actions. If so all plates experience both types of action. Such a combination is treated in finite element methods by flat shell models: a superposition of flat membrane and bending elements. Plates designed to resist both membrane and bending actions are sometimes called slabs, as in pavements. §26.2.2. Terminology This subsection defines a plate structure in a more precise form and introduces terminology. Consider first a flat surface, called the plate reference surface or simply its midsurface or midplane. See Figure 26.2. We place the axes x and y on that surface to locate its points. The third axis, z is taken normal to the reference surface forming a right-handed Cartesian system. Axis x and y are placed in the midplane, forming a right-handed Rectangular Cartesian Coordinate (RCC) system. If the plate is shown with an horizontal midsurface, as in Figure 26.2, we shall orient z upwards. Next, imagine material normals, also called material filaments, directed along the normal to the reference surface (that is, in the z direction) and extending h/2 above and h/2 below it. The magnitude h is the plate thickness. We will generally allow the thickness to be a function of x, y, that is h = h(x, y), although most plates used in practice are of uniform thickness because of fabrication considerations. The end points of these filaments describe two bounding surfaces, called the plate surfaces. The one in the +z direction is by convention called the top surface whereas the one in the −z direction is the bottom surface. Such a three dimensional body is called a plate if the thickness dimension h is everywhere small, but not too small, compared to a characteristic length Lc of the plate midsurface. The term “small” is to be interpreted in the engineering sense and not in the mathematical sense. For example, h/Lc is typically 1/5 to 1/100 for most plate structures. A paradox is that an extremely thin plate, such as the fabric of a parachute or a hot air balloon, ceases to structurally function as a thin plate! 26–4 26–5 §26.2 PLATES: BASIC CONCEPTS A plate is bent if it carries loads normal to its midsurface as pctured in Figure 26.1(b). The resulting problems of structural mechanics are called: Inextensional bending: if the plate does not experience appreciable stretching or contractions of its midsurface. Also called simply plate bending. Extensional bending: if the midsurface experiences significant stretching or contraction. Also called combined bending-stretching, coupled membrane-bending, or shell-like behavior. The bent plate problem is reduced to two dimensions as sketched in Figure 26.2(c). The reduction is done through a variety of mathematical models discussed next. §26.2.3. Mathematical Models The behavior of plates in the membrane state of Figure 26.1(a) is adequately covered by two- dimensional continuum mechanics models. On the other hand, bent plates give rise to a wider range of physical behavior because of possible coupling of membrane and bending actions. As a result, several mathematical models have been developed to cover that spectrum. The more important models are listed next. Membrane shell model: for extremely thin plates dominated by membrane effects, such as inflatable structures and fabrics (parachutes, sails, balloon walls, tents, inflatable masts, etc). Von-Karman model: for very thin bent plates in which membrane and bending effects interact strongly on account of finite lateral deflections.1 Important model for post-buckling analysis. Kirchhoff model: for thin bent plates with small deflections, negligible shear energy and uncoupled membrane-bending action.2 Reissner-Mindlin model: for thin and moderately thick bent plates in which first-order transverse shear effects are considered.3 Particularly important in dynamics as well as honeycomb and com- posite wall constructions. High order composite models: for detailed (local) analysis of layered composites including inter- lamina shear effects.4 Exact models: for the analysis of additional effects using three dimensional elasticity.5 The first two models are geometrically nonlinear and thus fall outside the scope of this course. The last four models are geometrically linear in the sense that all governing equations are set up in the 1 Th. v. Ka´rma´n, Festigkeistprobleme im Maschinenbau., Encyklopa¨die der Mathematischen Wissenschaften, 4/4, 311– 385, 1910. 2 G. Kirchhoff, ¨Uber das Gleichgewicht und die Bewegung einer elastichen Scheibe, Crelles J., 40, 51-88, 1850. Also his Vorlesungen u¨ber Mathematischen Physik, Mechanik, 1877., translated to French by Clebsch. 3 E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 12, 69–77, 1945; also E. Reissner, On bending of elastic plates, Quart. Appl. Math., 5, 55–68, 1947. Mindlin’s version, intended for dynamics, was published in R. D. Mindlin, Influence of rotary inertia and shear on flexural vibrations of isotropic, elastic plates, J. Appl. Mech., 18, 31–38, 1951. Timoshenko and Woinowsky-Krieger, cited in §26.5, follow A. E. Green, On Reissner’s theory of bending of elastic plates, Quart. Appl. Math., 7, 223–228, 1949. 4 See for example the book by J. N. Reddy cited in §26.5. 5 The book of Timoshenko and Woinowsky-Krieger cited in §26.5 contains a brief treatment of the exact analysis in Ch 4. 26–5 Chapter 26: KIRCHHOFF PLATES: FIELD EQUATIONS 26–6 reference or initially-flat configuration. The last two models are primarily used in detailed or local stress analysis near edges, point loads or openings. All models may incorporate other types of nonlinearities due to, for example, material behavior, composite fracture, cracking or delamination, as well as certain forms of boundary conditions. In this course, however, we shall look only at the linear elastic versions. Furthermore, we shall focus attention only on the Kirchhoff and Reissner-Mindlin plate models because these are the most commonly used in statics and vibrations, respectively. §26.3. The Kirchhoff Plate Historically the first model of thin plate bending was developed by Lagrange, Poisson and Kirchhoff. It is known as the Kirchhoff plate model, of simply Kirchhoff plate, in honor of the German scientist who “closed” the mathematical formulation through the variational treatment of boundary conditions. In the finite element literature Kirchhoff plate elements are often called C1 plate elements because that is the continuity order nominally required for the transverse displacement shape functions. The Kirchhoff model is applicable to elastic plates that satisfy the following conditions. • The plate is thin in the sense that the thickness h is small compared to the characteristic length(s), but not so thin that the lateral deflection w becomes comparable to h. • The plate thickness is either uniform or varies slowly so that three-dimensional stress effects are ignored. • The plate is symmetric in fabrication about the midsurface. • Applied transverse loads are distributed over plate surface areas of dimension h or greater.6 • The support conditions are such that no significant extension of the midsurface develops. We now describe the field equations for the Kirchhoff plate model. §26.3.1. Kinematic Equations The kinematics of a Bernoulli-Euler beam, as studied in Chapter 12 of [87], is based on the assumption that plane sections remain plane and normal to the deformed longitudinal axis. 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.” This assumption is illustrated in Figure 26.3. Upon bending, particles that were on the midsurface z = 0 undergo a deflection w(x, y) along z. The slopes of the midsurface in the x and y directions are ∂w/∂x and ∂w/∂y. The rotations of the material normal about x and y are denoted by θx and 6 The Kirchhoff model can accept point or line loads and still give reasonably good deflection and bending stress predictions for homogeneous wall constructions. A detailed stress analysis is generally required, however, near the point of application of the loads using more refined models; for example with solid elements. 26–6 26–7 §26.3 THE KIRCHHOFF PLATE Ω Γ x x y Original midsurface Deformed midsurface w(x,y) θ (positive as shown if looking toward −y) θ y Section y = 0 x y θ Figure 26.3. Kinematics of Kirchhoff plate. Lateral deflection w greatly exaggerated for visibility. In practice w << h for the Kirchhoff model to be valid. θy , respectively. These are positive as per the usual rule; see Figure 26.3. For small deflections and rotations the foregoing kinematic assumption relates these rotations to the slopes: θx = ∂w ∂y , θy = −∂w ∂x . (26.1) The displacements { ux , uy, uz } of a plate particle P(x, y, z) not necessarily located on the mid- surface are given by ux = −z ∂w ∂x = zθy, uy = −z ∂w ∂y = −zθx , uz = w. (26.2) The strains associated with these displacements are obtained from the elasticity equations: exx = ∂ux ∂x = −z ∂ 2w ∂x2 = −z κxx , eyy = ∂uy ∂y = −z ∂ 2w ∂y2 = −z κyy, ezz = ∂uz ∂z = −z ∂ 2w ∂z2 = 0, 2exy = ∂ux ∂y + ∂uy ∂x = −2z ∂ 2w ∂x∂y = −2z κxy, 2exz = ∂ux ∂z + ∂uz ∂x = −∂w ∂x + ∂w ∂x = 0, 2eyz = ∂uy ∂z + ∂uz ∂y = −∂w ∂y + ∂w ∂y = 0. (26.3) Here κxy = ∂ 2w ∂x2 , κyy = ∂ 2w ∂y2 , κxy = ∂ 2w ∂x∂y . (26.4) 26–7 Chapter 26: KIRCHHOFF PLATES: FIELD EQUATIONS 26–8 x y Bending moments (+ as shown) 2D view Bending stresses (+ as shown) Mxx Mxx Mxx M = Mxy σyy Myy Myy Myyyx Mxy Mxy Myx M yx x x y y h σxx σ = σxy yx x y z Top surface Bottom surface Normal stresses Inplane shear stresses x y dy dy dy dx dx dx Figure 26.4. Bending stresses and moments in a Kirchhoff plate, illustrating sign conventions. are the curvatures of the deflected midsurface. It is seen that the entire displacement and strain field are fully determined if w(x, y) is given. Remark 26.1. Many entry-level textbooks on plates introduce the foregoing relations gently through geometric arguments, because students may not be familiar with 3D elasticity theory. The geometric approach has the advantage that the kinematic limitations of the plate bending model are more easily visualized. On the other hand the direct approach followed here is more compact. Remark 26.2. Some inconsistencies of the Kirchhoff model emerge on taking a closer look at (26.3). For example, the transverse shear strains are zero. If the plate is isotropic and follows Hooke’s law, this implies σxz = σyz = 0 and consequently there are no transverse shear forces. But these forces appear necessarily from the equilibrium equations as discussed in §26.3.4. Similarly, ezz = 0 says that the plate is in plane strain whereas plane stress: σzz = 0, is a closer approximation to the physics. For a homogeneous isotropic plate, plane strain and plane stress coalesce if and only if Poisson’s ratio is zero. Both inconsistencies are similar to those encountered in the Bernoulli-Euler beam model, and have been the topic of hundreds of learned papers. §26.3.2. Moment-Curvature Relations The nonzero bending strains exx , eyy and exy produce bending stresses σxx , σyy and σxy as depicted in Figure 26.4. The stress σxy = σyx is sometimes referred to as the in-plane shear stress or the bending shear stress, to distinguish it from the transverse shear stresses σxz and σyz studied later. To establish the plate constitutive equations in moment-curvature form, it is necessary to make several assumptions as to plate material and fabrication. We shall assume here that • The plate is homogeneous, that is, fabricated of the same material through the thickness. • Each plate lamina z = constant is in plane stress. 26–8 26–9 §26.3 THE KIRCHHOFF PLATE • The plate material obeys Hooke’s law for plane stress, which in matrix form is [ σxx σyy σxy ] = [ E11 E12 E13 E12 E22 E23 E13 E23 E33 ] [ exx eyy 2exy ] = −z [ E11 E12 E13 E12 E22 E23 E13 E23 E33 ] [ κxx κyy 2κxy ] . (26.5) The bending moments Mxx , Myy and Mxy are stress resultants with dimension of moment per unit length, that is, force. For example, kips-in/in = kips. The positive sign conventions are indicated in Figure 26.4. The moments are calculated by integrating the elementary stress couples through the thickness: Mxx dy = ∫ h/2 −h/2 −σxx z dy dz ⇒ Mxx = − ∫ h/2 −h/2 σxx z dz, Myy dx = ∫ h/2 −h/2 −σyy z dx dz ⇒ Myy = − ∫ h/2 −h/2 σyy z dz, Mxy dy = ∫ h/2 −h/2 −σxy z dy dz ⇒ Mxy = − ∫ h/2 −h/2 σxy z dz, Myx dx = ∫ h/2 −h/2 −σyx z dx dz ⇒ Myx = − ∫ h/2 −h/2 σyx z dz. (26.6) It will be shown later that rotational moment equilibrium implies Mxy = Myx . Consequently only Mxx , Myy and Mxy need to be calculated. Inserting (26.5) into (26.6) and integrating, one obtains the moment-curvature relation [ Mxx Myy Mxy ] = h 3 12 [ E11 E12 E13 E12 E22 E23 E13 E23 E33 ] [ κxx κyy 2κxy ] = [ D11 D12 D13 D12 D22 D23 D13 D23 D33 ] [ κxx κyy 2κxy ] . (26.7) The Di j = Ei j h3/12 for i, j = 1, 2, 3 are called the plate rigidity coefficients. They have dimension of force × length. For an isotropic material of elastic modulus E and Poisson’s ratio ν, (26.7) specializes to [ Mxx Myy Mxy ] = D [ 1 ν 0 ν 1 0 0 0 12 (1 + ν) ] [ κxx κyy 2κxy ] . (26.8) Here D = 112 Eh3/(1 − ν2) is called the isotropic plate rigidity. If the bending moments Mxx , Myy and Mxy are given, the maximum values of the corresponding in-plane stress components can be recovered from σ max,minxx = ± 6Mxx h2 , σ max,minyy = ± 6Myy h2 , σ max,minxy = ± 6Mxy h2 = σ max,minyx . (26.9) These max/min values occur on the plate surfaces as illustrated in Figure 26.4. Formulas (26.9) are useful for stress design. 26–9 Chapter 26: KIRCHHOFF PLATES: FIELD EQUATIONS 26–10 x y Transverse shear forces (+ as shown) 2D view Qy QyQx Qx x x y y h σxz σyz x y z Top surface Bottom surface Transverse shear stresses dy dy dx dx Parabolic distribution across thickness Figure 26.5. Transverse shear forces and stresses in a Kirchhoff plate, illustrating sign conventions. Remark 26.3. For non-homogeneous plates, which are fabricated with different materials, the essential steps are the same but the integration over the plate thickness may be significantly more laborious. (For common structures such as reinforced concrete slabs or laminated composites, the integration process is explained in specialized books.) If the plate fabrication is symmetric about the midsurface, inextensional bending is possible, and the end result is a moment-curvature relation such as (26.8). If the wall fabrication is not symmetric, however, coupling occurs between membrane and bending effects even if the plate is only laterally loaded. The Kirchhoff and the plane stress model need to be linked to account for those effects. This coupling is examined further in chapters dealing with shell models. §26.3.3. Transverse Shear Forces and Stresses The equilibrium equations derived in §26.3.4 require the presence of transverse shear forces. Their components in the { x, y } system are called Qx and Qy . These are defined as shown in Figure 26.5. These are forces per unit of length, with physical dimensions such as N/cm or kips/in. Associated with these shear forces are transverse shear stresses σxz and σyz . For a homogeneous plate and using an equilibrium argument similar to Euler-Bernoulli beams, the stresses may be shown to vary parabolically over the thickness, as illustrated in Figure 26.5: σxz = σ maxxz ( 1 − 4z 2 h2 ) , σyz = σ maxyz ( 1 − 4z 2 h2 ) . (26.10) in which the peak values σ maxxz and σ maxyz , which occur on the midsurface z = 0, are only function of x and y. Integrating over the thickness gives Qx