Plane Stress

The Plane Stress Problem

1. Introduction

The plane stress analysis refers to the problems where the thickness of the structure is very small compared to other dimensions of the structure in the XY plane. The plane stress problem is a 2D problem. The stresses in the z direction are considered to be negligible i.e ;

The stress-strain compliance relationship for an isotropic material becomes,

The three zero stress entries in the stress vector indicate that we can ignore their associated columns in the compliance matrix (i.e. columns 3, 4, and 5). If we also ignore the rows associated with the strain components with z-subscripts, the compliance matrix reduces to a simple 3x3 matrix,

The stiffness matrix for plane stress is found by inverting the plane stress compliance matrix, and is given by,

2. Plate in plane stress

In structural mechanics, a flat thin sheet of material is called a plate. The distance between the plate faces is called the thickness and denoted by h. The midplane lies halfway between the two faces. The direction normal to the mid-plane is the transverse direction. Directions parallel to the midplane are called in-plane directions.

A class of common engineering problems involving stresses in thin plates is thin-walled pressure vessels under external or internal pressure, the free surfaces of shafts in torsion and beams under transverse load etc. By assuming the plane stress condition, the three-dimensional stress state can be reduced to two dimensions. These simplified 2D problems are called plane stress problems.

Assuming that the negligible principal stress is oriented in the z-direction. On reducing the 3D stress matrix to the 2D plane stress matrix, by removing all components with z subscripts we get,

Where T_xy=T_yx.. . ...for static equilibrium.

3. Problem Unknowns

The unknown fields are displacements, strains and stresses. As the stress along the thickness of the plate is negligible the dependence on z disappears and all such components become functions of x and y only. Displacements: The in-plane displacement field is defined by two components:

The transverse displacement component u_z(x, y, z) component is generally nonzero because of Poisson’s ratio effects, and depends on z. However, this displacement does not appear in the governing equations.

Strains: The in-plane strain field forms a tensor defined by three independent components: e_xx , e_yy and e_xy.

Stresses: The in-plane stress field forms a tensor defined by three independent components:

4. Finite Element Formulation of the problem

The plate is divided into ‘N’ number of finite elements having ‘n’ nodes each. The element has 2n degrees of freedom. These are collected in the element node displacement vector in a node by node arrangement as follows:

The displacement field u_e(x, y) over the element is interpolated from the node displacements.

Where N_i^e. (x, y) are the element shape functions. In matrix form:

This N (with superscript e omitted to reduce clutter) is called the shape function matrix. It has dimensions 2 × 2n.

Differentiating the finite element displacement field yields the strain-displacement relations:

This B = DN is called the strain-displacement matrix. It is dimensioned 3 × 2n.

Thus the element stiffness matrix is given by:

And the nodal force vector is given by:

Where .first integration is for the force on the interior of plate and . the second integration is for for the boundary surface of the plate.

5. Example Problem for Plane Stress Analysis

A code is written in MATLAB to do the Plane stress analysis of a plate. A plane stress problem i.e a plate under uniform tension at its edges is solved. Plate is dicretized using isoparametric Q4 elements. The values obtained with present code are compared with standard FEM software and are in good agreement.

Here an example problem is considered. A thin plate under uniform traction forces at extremes is considered. The following are the data used in the code

Young's Modulus, E = 2.1*1011 Nm/kg2

Plate thickness, h = 0.02504 m

Poisson’s Ration, . = 0.3

Length of the plate, a = 1m

Breadth of the plate, b = 1m

Load = 1*105 N

Four noded Isoparametric elements are used. 100 elements are used. The problem case and the meshing of the plate is shown in figure 2. The static analysis is done and the results obtained are checked with standard FEM software. To show the accuracy of the code, the present code values and FEM software

Figure 2: Plate under traction

Table 1: Comparison of results with FEM software

The following figures show the deformed plate under the traction, displacement field distribution using the present code and standard FEM software. It shows that both the present code is in perfect match with standard FEM software package.

Figure 3: Plate deformation under tension at extremes

Figure 4: Displacement UX field profile on the plate using present CODE

Figure 5: Displacement field UX profile on the plate from FEM software

Figure 6: Displacement UY field profile on the plate using present CODE

Figure 7: Displacement field UY profile on the plate from FEM software

Code can be obtained from the following link:

The Plane Stress Problem:

http://www.mathworks.ch/matlabcentral/fileexchange/31788-the-plane-stress-problem

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