The Plane Stress Problem

8/3/2019 The Plane Stress Problem

http://slidepdf.com/reader/full/the-plane-stress-problem 1/7

The Plane Stress Problem

Siva Srinivas Kolukula

Structural Mechanics Laboratory

IGCAR, Kalpakkam

INDIA Email – alwayzitzme(at)gmail.com

_____________________________________________________________________________________

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, , 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 thecompliance 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 isgiven 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 midplane 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 for static equilibrium.

Figure 1 : Internal forces of thin plate in plane stress

3. Problem Unknowns

The unknown fields are displacements, strains and stresses. As the stress along the thickness of the plateis 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 uz(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: exx , eyy and

exy.





And the nodal force vector is given by:

Where e stands for the force on the interior of plate and гe for the boundary surface of the plate.

5. Example Problem:

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

Youngs Modulus, E = 2.1*1011

Nm/kg2

Plate thickness, h = 0.02504 m

Poisson’s Ration, υ = 0.3Length of the plate, a = 1m

Breadth of the plate, b = 1m

Load = 1*105

N

Four nodes 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

values are compared in table 1.

Unknown MATLAB FEM software

Maximum UX 3.4744e-007 m 3.47441E-07 m

Minimum UX -3.4744e-007 -3.47441E-07 m

Maximum UY 1.2783e-007 m 1.27826E-07 m

Minimum UY -8.5757e-008 m -8.57570E-08

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 2: Plate under Traction



Figure 3 : Deformation of plate

Figure 4: Plate in Tension, displacement UX from FEM software



Figure 5: Displacement field UX from present Code

Figure 6: Plate in Tension, displacement UY from FEM software



Figure 7: Displacement field UY from present Code

References:

1. Introduction to Finite Element Methods – University of Colorado2. Concepts and Applications of Finite Element Analysis – Robert D. Cook

Documents

The Plane Stress Problem