Stress Analysis of Infinite Plate with Circular hole Cutout

Abstract:

In this paper the elastostatic problem for homogeneous isotropic infinite plate with the loading at infinity (in X and Y ) for circular hole is solved .For this we have used the Schawarz’s alternating technique given by Sokolnicoff and Ukadgaonkar. After that the analytical solution is used for generation of code in C++ to find out the stress components at a given particular point and also the code for generating the stress distribution around the hole.

Keywords: Circular Hole, stresses around hole,

Introduction:

The presence of circular hole in plates and machined components which are subjected to loading introduces stress concentration in the vicinity of holes. Sudden changes in cross section lead to stress concentration at a localized point. It is impossible to estimate the rise of stresses in neighborhood of hole using

elementary theory of strength of materials. The stress concentration depends on geometry of components, size of hole, and loading condition. A classical example is plate loaded with tension which contains centrally located hole is considered here. The solutions in form of simple analytical equations from the method of complex variable approach can give us the check against the effect of change in the parameter in design process.

Theoretical Solution:

Problem Definition:-

-Infinite Plate (neglecting edge effect)

-Hole of radius=R

-Boundary Condition:

Sol:

The above problem can be solved by various methods. The elasto plastic problem for homogeneous isotropic plate with loading at infinity for circular, elliptical, triangular, rectangular holes and cracks are solved using a novel method called Schwarz’s Alternating Method given by Sokolnikoff and Ukadgaonkar with successive approximations. Initially a plate without any hole is considered. The boundary condition on the hole boundary is found out. To nullify this, the negative of this boundary condition is applied on the hole .The solution of this problem is superimposed on the first solution which give the required closed form solution.

By the method we can superposition the two problems to get the final solution.

First Problem Second Problem

To obtain the boundary condition for first problem can be found out as given by the formula

For second problem boundary condition can be imposed as = f(t)2 – f(t)1

First problem:

Putting the given boundary

condition and integrating we get

the

;

Now,

We get ,

;

(t is boundary point lie on hole)

……. (t.t =R2)

Second Problem:

f(t)2 = –

f(t)1

Solving cauchy’s integral,

Finally we get

Finding stresses:

Separating real and imaginary part and converting in polar form we get

…….(1)

(1)

Separating real and imaginary part

Solving equation 1 and 2 we get

2) Problem Definition:-

-Infinite Plate (neglecting edge effect)

-Hole of radius=R

-Boundary Condition:

Sol:

By the method we can superposition the two problems to get the final solution.

+

First Problem Second Problem

To obtain the boundary condition for first problem can be found out as given by the formula

For second problem boundary condition can be imposed as = f(t)2 – f(t)1

First problem:

Putting the given boundary

condition and integrating we get

the

;

Now,

We get ,

;

=Pt/4 + tP/4 + Pt/2

(t is boundary point lie on hole)

=Pt/2+ PR2/(2Z) (t.t

=R2)

Second Problem:

f(t)2 = –

f(t)1

Solving cauchy’s integral,

= -PR2/(2z)

Finally we get

Finding stresses:

Separating real and imaginary part and converting in polar form we get

(1)

Separating real and imaginary part

(3)

Solving equation 1 and 2 we get

Computer Code for the

Analytical solution:

The above analytical

expressions are used to find out

stress filed around the hole. Also it

generates the text result file of all

the nodes (in the region which is

four times the hole radius)

information (all stress

components).

Results:

By adopting the computer code we can find out the stress at any given particular point in the neighborhood of circular hole. The result file so as generated can be used for visual display of result so obtain such as meshing and then contour plot for the stress components. Consider Plate of 200 x 200 x 20 and circular hole at the centre of plate of radius 10 mm. The model is constrained on opposite sides of plate and load is applied so as to get boundary conditions σ x =200 for first case and σ y = 200 for second case at edge of plate (we have assumed the boundary conditions at infinity)

Table 6.3Comparison of

Stresses by analytical & FEM

Loading in X direction

Point Stress Analytical FEM

1(10,0)

σx 0

σy -200

2(0,10)

σx 600 484

σy 0 1.65

Loading in Y direction

Point Stress Analytical FEM

1(10,0)

σx 0 1.65

σy 600 484

2(0,10)

σx -200 -3

σy 0 32

Stress Plots with exact solution

Normal Stress σx plot

References:

Muskhelishvili N. I., “Some Basic

Problems of the Mathematical

Theory of Elasticity”, P. Noordoff

Ltd., (1963).

Jess Comer, Julie Bannantine;

“Fundamentals of metal fatigue

analysis”.

A.E.H Love I. S., “Mathematical

Theory of Elasticity”, Cambridge

university press.

Ukadgaonkar V. G. and P. J.

Awasare; “A Novel Method of Stress

Analysis of an Infinite Plate with

Elliptical Hole with Uniform Tensile

Stress”, IE (I) Journal-ME, Vol. 73,

pp. 309-311.

Complex Analysis, Complex

Variables, Advance Mathematics

Mohammed Ameen “Computational

Elasticity “ Narosa Pub.