summary

This module calculates deflections and bending stresses in thin plates of uniform thickness subjected to transverse loads.

A plate is a two-dimensional sheet of elastic material that lies in a plane. The Uniform Plate Analysis Module is based upon classical plate theory, which makes the following assumptions:

1. The plate is flat and has uniform thickness.
2. The plate is thin. The thickness is not more than about one-quarter of the smallest length dimension of the plate.
3. The plate material is uniform, isotropic and elastic.
4. All loads are normal to the plate surface. The in-plane load is zero.
5. Deflections of the middle surface are small in comparison with the thickness of the plate. In general, deflections are assumed to not exceed about one-half of the plate thickness.
6. As the plate deflects, the middle surface of the plane has zero bending strain.
7. Normals to the midsurface of the undeformed plate remain straight and normal to the midplane during deformation.
8. The component of stress normal to the midsurface, sz, is negligible.

Background Theory

A plate behaves in large as the two-dimensional equivalent of a beams. When loaded by forces transverse to its surface as shown in Figure 1, a plate experiences bending about the X-axis and about the Y-axis. These deformations cause bending stresses that vary linearly through the plate thickness. Like the neutral axis of a beam, the middle surface of the plate is a plane of zero bending strain.

Figure 1. Uniform plate subject to transverse load

Stress and Strain Equations

The assumptions associated with classical plate theory imply that there is no transverse normal strain (assumption 6) and no shear strain (assumption 7), which results in a condition of plane strain:

The in-plane strains are functions of x and y only, and are proportional to the distance from the middle surface. The normal strains ex and ey and shear strain gxy in the plane of the plate are given by:

where z is the distance perpendicular to the midsurface and w is the transverse deflection of the midsurface.

Assumption 8 states that that sz is negligible and the only non-zero stress components are in the x-y plane. This corresponds to a two-dimensional state of stress in which the stresses in an isotropic material are related to the strains by Hooke's Law for plane stress:

It is appropriate to emphasize that classical plate theory leads to a simultaneous assumption of plane strain and plane stress conditions. This implies that the transverse elastic modulus and shear modulus are infinite, and the transverse Poisson's ratio is zero. This is consistent with the assumption that normals are inextensible and remain normal during deformation.

Plate Deflection

When developing the governing equations for plate deflection, it is convenient to express equilibrium in terms of stress resultants. Stress resultants are forces and moments per unit length. They are computed by integrating through the thickness of the plate, resulting in functions of x and y only.

The bending moments per unit length Mx and My and the twisting moment per unit length Mxy are related to deflections of the plate by:

where the quantity D is defined as

and is referred to as the flexural rigidity of the plate. The flexural rigidity term D in plate theory is similar to the bending stiffness property EI used in beam theory.

The transverse shearing forces per unit length Qx and Qy are defined as:

Although the effects of out-of-plane shear strains gxz and gyz on bending are neglected in thin plate theory, the vertical forces Qx and Qy resulting from txz and tyz are not negligible and must be considered to show equilibrium. The out-of-plane shear stresses txz and tyz will have a parabolic distribution over the plate thickness, maximum at the midsurface and zero at the upper and lower surfaces.

Figure 2. Equilibrium of moments and shear forces (per unit length)

Expressions for Qx and Qy in terms of plate deformation can be obtained from equations of equilibrium for an element on the plate midsurface subject to a uniform pressure q as shown in Figure 2. The equilibrium of the vertical forces is given by:

The equilibrium of moments about the x and y axes is given by:

Simplifying these expressions produces the following equilibrium equations:

By substituting the previous expressions for Mx, My, and Mxy into the moment equilibrium equations, the transverse shearing forces Qx and Qy can be related to the deformation of the plate by:

Substituting Qx and Qy into the force equilibrium equation produces a single expression for the deflection w as a function of the applied uniform pressure load q:

An exact, closed-form analytic solution of this fourth-order partial differential equation is generally not available. Trigonometric series can used to develop approximate solutions for simple plate geometries (such as rectangular or circular plates) and a range of common boundary conditions. The formulations included in the Uniform Plates Analysis module are based on a number of current and classical references. Those as well as other additional sources of information are given in the References section below.

Module Input

The Uniform Plate Analysis module input form is shown in Figure 3 . The geometry, loading, and support conditions are defined by selecting the proper solution type from the tree list. ETBX will automatically update the inputs fields to match the required parameters for the selected solution.

A number of important geometry, loading, and support conditions are included:

1. Rectangular Plates with Simply Supported or Fixed edge conditions subject to Concentrated, Uniformly Distributed, and Hydrostatic Loadings.
2. Circular Plates with Simply Supported or Fixed edge conditions subject to Concentrated, Uniformly Distributed, and Annular Loadings.
3. Annular Plates with Simply Supported or Fixed edge conditions subject to Uniformly Distributed and Annular Loadings.

After selecting a solution type, the user enters geometry and loading parameters as well as material properties. Material properties may be selected from the ETBX Materials Database by clicking the Materials button.

A poisson’s ratio of 0.3 is assumed for stress calculations in most cases, however in all cases, the user specified poisson ratio is used in the deflection calculation. The effects of stress concentration due to supports or geometries is not taken into account.

Figure 3: The Uniform Plate Analysis module input form

Module Results

Results are displayed using the standard ETBX output window shown in Figure 4 .  The following data is tabulated in the output window:

• All problem inputs, including material properties and load spectrum
• Plate deflections and corresponding location
• Plate bending stress and corresponding location

Figure 4. Module tabulated results

References

1. Panc, Vladimir (1975), Theories of Elastic Plates, Noordhoff.
2. Pilkey, Walter D. (2005), Formulas for Stress, Strain, and Structural Matrices, 2nd Ed, Wiley.
3. Szilard, Rudolph (1974), Theory and Analysis of Plates, Prentice-Hall.
4. Timoshenko, S. P., Goodier, J. N., (1959), Theory of Plates and Shells, McGraw-Hill.
5. Young, W. C., Budynas, R. G., (2002), Roark's Formulas for Stress and Strain, 7th Ed, McGraw-Hill.