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Title-Stress concentration in Plate with a hole, surface cracks

What is Stress Concentration?

The concentration of high stress due to discontinuity or abrupt change in the cross section is

known as stress concentration.

Calculating Stress Concentration Factor:

The Stress Concentration factor determined here is theoretical one-

Consider a plate with a centrally located hole and the plate is subjected to uniform tensile load

at the ends. Stress distribution at a section A-A passing through the hole and another section

BB away from the hole are shown in figure. Stress distribution away from the hole is uniform

but at AA there is a sharp rise in stress in the vicinity of the hole. Stress concentration factor kt

or kc is defined here as,

kt=max/nominal,where maximum stress at the discontinuity, at section AA is simply ,P/t(w-2*b)

and nominal =P/w*t.

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This is the theoretical or geometric stress concentration factor and the factor is not affected by

the material properties.

In general case it is possible to calculate the kt by using elliptical discontinuity on the plate .In

that case kt is given by -

kt = [1 + 2*b/a],

Where b is the semi-axis of the hole perpendicular to the direction of the force and a is the

semi axis of the hole in the direction of the force.

Stress concentration factors may also be obtained using any one of the following experimental

techniques:

1. Strain gage method

2. Photoelasticity method

3. Brittle coating technique

4. Grid method

For more accurate estimation numerical methods like Finite element analysis may be

employed. Theoretical stress concentration factors for different configurations are available in

handbooks.

In design under fatigue loading, stress concentration factor is used in modifying the values of

endurance limit while in design under static loading it simply acts as stress modifier. This means

Actual stress= kt calculated stress.

For ductile materials under static loading effect of stress concentration is not very serious but

for brittle materials even for static loading it is important.

It is found that some materials are not very sensitive to the existence of notches or

discontinuity. In such cases it is not necessary to use the full value of kt and instead a reducedvalue is needed. This is given by a factor known as fatigue strength reduction factor Kf and

this is defined as:

Kf= Endurance limit of notch free specimens/ Endurance limit of notched specimens

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Another term called Notch sensitivity factor, q is often used in design and this is defined as

following

q=(kf-1)/(kt -1)

The value of q usually lies between 0 and 1. If q=0, Kf=1 and this indicates no notch sensitivity.If however q=1, then , Kf= Kt and this indicates full notch sensitivity.

Methods of reducing stress concentration

A number of methods are available to reduce stress concentration in machine parts. Some of

them are as follows:

1. Provide a fillet radius so that the cross-section may change gradually.

2. Sometimes an elliptical fillet is also used.

3. If a notch is unavoidable it is better to provide a number of small notches rather than a long

one. This reduces the stress concentration to a large extent.

4. If a projection is unavoidable from design considerations it is preferable to provide a narrow

notch than a wide notch.

5. Stress relieving groove are sometimes provided.

Methodology to determine Stress concentration in the plate with a hole:

Analysis Platform used is -ANSYS 13.0

Modeling Software - Creo Pro/ENGINEER 5.0

Here in this project aluminium a ductile material with following specification has been taken-

For this analysis, we will assume the specimen is made of aluminum (E = 71 GPa and = 0.33)

with a yield strength of Sy = 290 MPa and is 8.2 mm thick.

We will use ANSYS to analyze the stresses and deflections in the specimen under a static load

of P = 3500 N. A linear-elastic, static analysis will be performed using 8 node, 2D isoparametric

plane stress elements (PLANE 82). Specifically, we wish to determine the magnitude of the

stress concentration factor due to the hole in the center of the specimen.

Here we can calculate the nominal stress on the specimen by using the following relation-

avg =P/A ,

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where P is applied load in Newton , and A is the cross sectional area of the specimen given as

A=t*w, where t is the thickness of the plate and w is the width.

Once the nominal stress is obtained we will use ANSYS to analyze the rectangular plate with

hole for static load condition and as results we will obtain the maximum stresses at the

discontinuity.

Then we can use the basic relationship given as

Kf=(Actual maximum stress in the notched specimen)/(Nominal stress in the notch free

specimen)

, to obtain the actual stress concentration factor Kf.

Validation of results:

There are standard charts available for a rectangular plate with a hole which provides the value

of stress concentration factor for a provided diameter to width ratio.

Surface crack:

This requires further study into the subject fracture mechanics. Which will be taken in the

forthcoming sessions.

References and Bibliography:

1- Pilkey, Walter D, Peterson's Stress Concentration Factors, Wiley, 2nd Ed (1999).2- ESDU64001: Guide to stress concentration data.3- Google book-Peterson's stress concentration factors By Walter D. Pilkey, Deborah F.

Pilkey, Rudolph Earl Peterson

4- The Journal of Strain Analysis for Engineering Design.