Bending and Extension of Thin Plates, W.E. Haisler
Bending and Extension of Thin Plates
1. IntroductionEquation Section 1
A theory for the bending and extension of slender beams was developed in Chapter 4 with particular application to non-homogeneous and semimonocoque type beams typical to aerospace construction. While a typical beam is clearly a three-dimensional structure, it is idealized by assuming that the axial and transverse deformations are functions of only one variable, x, which is the neutral axis along the length of the beam (as shown in Fig. 4.9). Consequently, slender beams are considered to be one-dimensional problems. This chapter considers the extension of Euler-Bernoulli beam theory to Kirchhoff plate theory. Both the beam and plate theories are referred to as classical or strength of materials theories in that the following assumptions are made: a straight line perpendicular to the neutral axis of the beam or plate is inextensible, remains straight and only rotates about the undeformed axis. In classical plate theory, the same general assumptions of beam theory are extended to thin planar bodies (see Fig. 1) wherein the geometry is now slender in only one direction. This will result in a two-dimensional problem wherein deformations are now functions of the two in-plane coordinates (x and y). In beam theory, bending and extension is considered in only one direction; in plate theory, bending and extension is considered in two directions (x and y).
While beam theory may be quite adequate in many situations (as in the examples illustrated in Chapter 4 of Allen and Haisler), many structural configurations and stress analysis requirements require a two dimensional analysis. Referring to structures such as shown in Figs. 4.22a or 4.29a of Allen and Haisler, the web and skin of the beam are in reality twodimensional flat or curved plates. In beam theory, the shear stress (shear flow) in the webs and skin of the multicell beam are assumed to be constant between longitudinal stringers and for a given length of beam span. In Chapter 4, it was assumed that the skin and web carried only shear and that the stringers carried all bending loads (as axial stresses). In many cases, the webs will actually carry some bending loads. Other platelike structural components, such as floor panels, equipment support panels, etc. will experience both bending and inplane stresses that may not be adequately analyzed by the beam theory of Chapter 4 (A&H).
In the present chapter, the development of plate and membrane theory will be restricted to small deformations and strains. It is possible for thin plates subjected to large transverse loads to experience large transverse deformations and large strains. In those cases where the thickness is very small and/or the bending stiffness is very small (referred to as a membrane), the bending stresses will be small in comparison to the in-plane stresses and the transverse deformations and strains will quite often be large. The development of plate theory which accounts for large strains requires the inclusion of nonlinear straindisplacement terms such as that shown in Eq. 241 and results in a nonlinear set of partial differential equations which are beyond the scope of this text. The treatment of large deformation plate theory is contained in Rivello, "Theory and Analysis of Flight Structures" and many journal publications.
2. Geometry and Deformation
Thin plates are characterized by a structure that is bounded by upper and lower surface planes that are separated by a distance h as shown in Fig. 1. The xy coordinate axes are located on the neutral plane of the plate (the "in-plane" directions) and the z-axis is normal to the xy plane. In the present development of classical plate theory, it will be assumed that h is a constant and that material properties are homogeneous through the thickness. Consequently, the location of the xy axes will lie at the mid-surface plane (z=0) with the upper and lower surfaces corresponding to z=h/2 and z=h/2, respectively. The treatment of non-homogeneous plates utilizing modulus weighted section properties is discussed in Rivello.
Fig. 1. Plate Geometry
In most plate applications, the external loading includes distributed load normal to the plate (z direction), concentrated loads normal to the plate, or in-plane tensile, bending or shear loads applied to the edge of the plate. Such loading will produce deformations of the plate in the x,y,z coordinate directions which in general can be characterized by displacements u(x,y,z), v(x,y,z) and w(x,y,z) in the x, y and z directions, respectively.
As in beam theory, classical plate theory makes two major assumptions: 1) a line normal to the mid-surface of the plate is inextensible (does not stretch), and 2) a straight line originally normal to the undeformed mid-surface remains straight and rotates so as to remain straight and normal to the deformed mid-surface plane. These assumptions imply that there is no transverse normal strain (assumption 1) or shear strain (assumption 2), i.e.,
(1.1)
Given that the only non-zero strains lie in the x-y plane, we have what was referred to as plane strain in Chapter 2 of A&H. It should be noted that since , this implies that (or that the transverse shear modulus is infinity). Since the transverse stress can be no larger than the normal pressure and in general will be much smaller than the in-plane stresses, one can assume that. This implies that the only non-zero stress components are in the x-y plane and we have a plane stress condition. The stress-strain relations that will be utilized later will make the plane stress assumption.
Consistent with the assumptions made in Euler-Bernoulli beam theory, the plate deformations will be restricted such that the normal displacement w is a function of x and y, and only an assumed linear function of z (analogous to the assumption of "plane sections remain plane" in beam theory). Likewise, the in-plane displacements u and v are assumed to be functions of x and y only. As a result of these assumptions, the deformations can be described entirely in terms of the deformation of the mid-surface plane; hence, the plate is reduced to the study of a two-dimensional problem consisting of the plate mid-surface. Figure 2 shows the u and v displacement assumptions in the x-z and y-z planes respectively.
Fig. 2. Mid-plane Displacements
From Figure 2, the following displacement patterns may be assumed:
(1.2)
where and are rotations of a normal the mid-plane about the x and negative y-axes, respectively.
3. Stress Resultants
In beam theory, the stress distribution that acts over a cross section may be integrated over the cross section to define equivalent forces and moments acting at the neutral axis. For two-dimensional plate analysis, it is more convenient to define stress resultants per unit length by integrating only through the thickness. Thus, stress resultants are defined to be forces and moments per unit length. The general stress components acting on an infinitesimal element are shown in Fig. 3.
Fig. 3. Free Body of Stress Components
The differential equations of equilibrium for an infinitesimal element in terms of stresses were previously derived in Chapter 2 of A&H (see Equations 2.15).
(1.3)
where X, Y and Z are body forces per unit volume.
In order to make the analysis easier, we define 8 stress resultants (forces and moments per unit length) by integrating through the thickness of the plate:
(1.4)
These force and moment resultants are shown in Fig. 4.
Fig. 4a. Force Stress ResultantsFig. 4b. Moment Stress Resultants
where and are in plane membrane forces per unit length (due to stretching of the plate mid surface), and are bending moments per unit length about the y and x axes, respectively, and are transverse shear forces per unit length, is an in-plane shear force per unit length (same as shear flow in a shear panel), and and are twisting moments per unit length (similar to torsion in a beam).
Note that you have to be careful about the sign convention for the forces and moments; some follow "usual" directions and notation, but some don't. For instance, the bending moment notation is opposite to what you normally think of ( is a bending moment about the y-axis!) and the shear forces and are opposite to the shear stresses. This notation and sign convention for force and moment resultants has been used for a long time in plate theory and is still used.
4. Equilibrium Equations in terms of Stress Resultants
Equilibrium of the plate mid-surface in terms of stress resultants is derived in exactly the same manner as was done for stresses in Chapter 2 of A&H, i.e., summation of forces and moments in the x, y and z directions. Figures 4a and 4b show separately the force and moment stress resultants. It is assumed that the plate mid-surface is subjected to applied distributed loads , and (force per unit area). In reality, these loads are applied to the upper or lower surface but in terms of the free body diagram of the plate mid-surface and assumptions made here, they can be considered to act at the mid-surface. To illustrate, consider equilibrium in the x direction for a plate differential region by :
We note that and can be written as Taylor series in x and y, respectively. Then,
Divide by and we obtain:
Similarly, we can do force equilibrium in y and z directions, and moment equilibrium about the x and y-axes. Hence, we have 5 equilibrium equations in terms of force and moment stress resultants:
(1.5)
5. StrainDisplacement Relations
As in beam theory, we will assume that all displacements and strains are small (infinitesimal). Similar to the assumptions made in Euler-Bernoulli beam theory (refer to Chapter 4.3 in A&H, in particular equations 4.3.1 and 4.3.3), we assume displacement patterns of the mid-surface as discussed in Section 2 above. Substituting these displacement assumptions (equations (1.2))
into the infinitesimal strain-displacement equations developed in Chapter 2 (Equations 2.42) results in
In the above expressions, all displacements are at the mid-surface and are functions of x and y only. To simply the notation, the functional notation of (x,y,0) will be dropped and the above expressions will be written as
(1.6)
6. StressStrain Relations
As noted in Section 2, classical plate theory leads to a simultaneous assumption of plane strain and plane stress conditions. While this requires that the transverse (z direction) Young's modulus and shear modulus be infinity and the transverse Poisson's ratio be zero, this requirement is consistent with the assumption that normals are inextensible and remainnormal during deformation (i.e., the material appears to have an infinite modulus in the transverse direction). Following the notion that the dominant stresses lie in the xy plane, classical plate theory utilizes the plane stress assumption for an isotropic material. From Chapter 3 of A&H, Equations 3.99 give
(1.7)
where E, and are in-plane values of the isotropic material properties defined in Chapter 3 of A&H (Young's modulus, Poisson's ratio and coefficient of thermal expansion, respectively, and is the temperature change above a reference zero stress state. As noted before, we will assume that the material properties E, and are homogeneous through the thickness; however the temperature change may vary through the thickness.
7. StressDisplacement Relations
Substituting the strain-displacement relations in Section 5 into the stress-strain relations in
Section 6 gives the following stress-displacement relations:
(1.8)
In Chapter 4 of A&H, the stresses for beam theory were also expressed in terms of equivalent forces, moments and thermal loads acting at the neutral surface (see Equation 4.55). Substituting equations (1.8) into the definitions for stress resultants (1.4) given in Section 3 and integrating through the thickness (remember that u, v and 2 do not depend upon z), yields the following equations:
(1.9)
and are obtained by using the last two equilibrium equations (1.5). In the above, the following "stiffness" properties (per unit length) have been defined
(1.10)
and the following thermal force and moment resultants have been defined:
(1.11)
The terms K and D in plate theory are similar to the axial and bending stiffness terms EA and EI defined in beam theory.
Similar to beam theory (see Equation 4.61 in A&H), we can utilize the definitions for stress resultants in terms of displacements and rewrite the equations relating stresses to displacements to obtain stress equations in terms of stress resultants. For example, we note that the term is equivalent to , and the term is equivalent to where . Thus, the first stress equation (see (1.8)) can be written in terms of stress resultants as
(1.12)
and in a similar fashion we obtain
(1.13)
where I = moment of inertia (about the x or y axis) for a unit width of plate = .
8. Formulation and Solution of the Plate Equations
We can now develop the differential equations of equilibrium in terms of the displacements u, v and w and the applied loads. Consider the in-plane force equilibrium equations in the x and y directions given in Section 4. Substituting the stress resultant-displacement relations derived in Section 7 into these two force equilibrium equations results in
(1.14)
Substituting the two moment equilibrium equations into the transverse force equilibrium equation [see (1.5)] gives
(1.15)
Noting that , then the above equation may be written as
(1.16)
The last equation can be written in terms of the transverse displacement by substituting for the moment resultants in terms of w from Section 7 [see (1.9)]. Using the notation (the Laplacian operator)
we obtain the following equation
(1.17)
If we assume constant material properties and thickness for the plate, then D is a constant and the terms in the bracket are zero; and the above equation reduces to
(1.18)
where
Solution of the three partial differential equations given by (1.14) and (1.18) constitutes the solution of the plate bending problems. It should be noted that all five equilibrium equations (two in-plane and one transverse force equilibrium equations and the moment equilibrium equation) have been utilized to obtain the three partial differential equations above. The partial differential equations defining the in-plane displacements u and v are coupled. Consistent with small strain theory, the partial differential equation for the transverse displacement w is uncoupled from u and v.
9. Boundary Conditions
The solution of the differential equations defining u, v and w require careful attention to boundary equations. For the present discussion, we only consider rectangular plates whose edges are parallel to the x and y coordinates axes. In the x and y directions, the u and v displacements each must be specified along at least one of the plate boundaries (similar to a simply supported beam). The boundary conditions for the transverse displacement w are somewhat more complicated but are analogous to those in beam theory. Letting n and s be directions normal and parallel, respectively, to a boundary, then we can write the following boundary conditions:
Clamped edge: and
Simply supported edge: and
Free edge:
The clamped boundary condition is equivalent to saying that the displacement and slope are zero. For the case of no thermal edge loads (), the simply supported boundary condition requires that the displacement and curvature (i.e., the moment normal to the edge) be zero. The free edge boundary conditions require that the moment and shear be zero on the free edge (for the case of ). When the thermal moment is not zero, the above equations require that curvatures at the plate edge satisfy the above relations (i.e., the internal moment at the edge must equal the thermal moment (or its gradient for the shear boundary condition on a free edge).
10. Some Simple Plate Solutions
The exact solution of the fourth-order, partial differential defining the transverse deflection w is generally quite difficult. For certain special cases, approximate analytical solutions can be obtained by assuming a displacement w(x,y) and obtaining a particular and complimentary solution in the traditional manner of solving differential equations. Other approximate solutions may be obtained by using finite difference or finite element methods. In practice, numerical finite difference methods (which replace derivatives by algebraic approximations) tend to cumbersome and difficult to apply for general plate geometries. The finite element method, which uses a combination of assumed displacement solutions and energy principles to solve the differential equations, is better suited for arbitrary plate geometries. The finite element solution of plate problems will be considered in a later section.
Consider the analytical solution of a thin, rectangular plate that has uniform material properties, is loaded with a uniform normal pressure , and is simply supported along all edges. Assume the plate has dimensions a and b in the x and y directions, respectively, and that the coordinate system is located at one corner of the plate as shown below.
Since the thermal loading is assumed to be zero, we are faced with solving the following fourth-order differential equation
As with the usual solution of differential equations, we assume a solution that satisfies the boundary equations. Trigonometric series can generally be used to satisfy many type of boundary conditions. For the simply supported plate of dimensions a and b we can utilize the Navier series:
(1.19)
where the constants . have to be determined and will depend on the loading. It should be noted that the above solution satisfies the simply supported boundary conditions, i.e., w=0 at x=0 and x=a and y=0 and y=b, at x=0 and x=a, and at y=0 and y=b. Since the solution for w(x,y) has been assumed in the form of a double trigonometric series, it is customary and convenient to also expand the loading in a similar double Fourier series as follows: