Slide 1

Stress Concentration Localization of high stresses due to the the geometrical discontinuities present in machine components or abrupt changes in the cross-section of the machine components is called Stress Concentration .

In design of machine elements, following three elementary equations are used :

These equations are based on the assumptions that there are no geometrical discontinuities in the cross-section of the component and also the members cross-section is uniform throughout.

However in practice, Discontinuities and abrupt changes in the cross-section are unavoidable due to various features of the machine component such as oil holes and grooves, keyways, splines, screw threads and shoulders.

Under these circumstances the stresses calculated near discontinuities have been found much higher than the stresses calculated by elementary equations.

Experimental Evidence of Stress ConcentrationUsing Photo-elasticity technique Figure shows the stress concentration introduced by notches and fillets in a flat bar subjected to bending moment. The stress effects were measured using photo-elastic techniques, and the resulting fringes indicate the stress distribution in the part when loaded.

At the right end of the part where the cross-section is uniform, the fringe lines are straight, of uniform width, and equispaced. This indicates a linear distribution across this portion of the bar.

But at the fillet where the width of the part is reduced from D to d, the fringe lines indicate a disruption and concentration of stresses at this sudden change in geometry.

The same effect is seen at the left hand end around the two notches.

This is the experimental evidence of the existence of stress concentrations at any change in geometry.

Such geometric changes in a part are called stress-raisers and should be minimized.

Theoretical Stress Concentration Factor: Stress concentration factor is defined as:

Kt = Highest value of actual stress near discontinuity Nominal stress obtained by elementary equations for minimum cross-section

where and are the nominal stresses calculated for the particular applied loading and net cross-section, assuming a stress distribution across the section that would obtain for a uniform geometry.

For example in the Figure below

The stress at the notches would be then

The nominal stresses are calculated using the net cross-section which is reduced by the notch geometry ,i.e. using d instead of D as the width at the notches.

Stress Concentration under Static loading:Ductile Material:

Ductile materials yield locally at the stress-raiser while the lower stressed material far from the geometrical discontinuity remains below the yield point.

When the material yields locally , its stress-strain curve becomes nonlinear and of low slope, which prevents further significant increase in the stress at that point.As the load is increased , more material is yielded, bringing more of the cross-section to that stress.Only when the entire cross-section has been brought to the yield point will the part continue up to the fracture point( according to - curve).Thus, it is common to ignore the effects of stress concentration in ductile materials under static loading.

Brittle Material:

Brittle materials do not yield locally , since they do not have a plastic range. Thus Stress concentration do not have an effect on their behavior under static loads.Once the stress at the stress raiser exceeds the fracture strength , a crack begins to form.

This reduces the material available to resist the load and also increases the stress concentration in the narrow crack. The part then quickly goes to failure.

So, for brittle materials under static loads , the stress concentration factor should be applied to increase the apparent maximum stress according to equation (1).

Stress Concentration under Dynamic loading:Ductile Material:

Ductile materials under dynamic loading behave and fail as if they were brittle.

So, regardless of the ductility or brittleness of the brittle material, the stress concentration factor should be applied when dynamics load (fatigue or impact) are present.

While all materials are affected by stress concentration under dynamic loads ,some materials are more sensitive than others.

For such materials, a parameter called notch sensitivity q is defined for various materials and used to modify the geometric factors Kt and Kts for a given material under dynamic loads.

Determination of Stress Concentration Factor:Experimental Methods:

Photo elasticity Brittle Coating Method Electric Strain Gages

Numerical Techniques:

Theory of Elasticity Finite Element Analysis and Boundary Element Analysis

Figure shows an elliptical hole in a semi-infinite plate subjected to axial tension. The nominal stress in this case is given by P/A. The theoretical stress concentration at the edge of the hole was developed by Inglis in 1913 using the theory of elasticity and is given by:

The charts of stress concentration factors for different geometric shapes and conditions ofloading were originally developed by RE. Peterson. Some of these charts are as follows:

Force-Flow Analogy:Design to avoid Stress ConcentrationA useful way to visualize the difference in stress states in contoured parts is to use the force flow analogy.Force flow analogy considers the forces ( and thus the stresses ) to flow around contours in a way similar to the flow of an ideal incompressible fluid inside pipe or duct of changing contour. There is a similarity between velocity distribution in a fluid flow in a channel and the stress distribution in an axially loaded plate .The equation of flow potential in fluid mechanics and stress potential in solid mechanics are the same.Therefore, it is perfectly logical to use fluid analogy to understand the phenomena of stress concentration..When the cross-section of channel has uniform dimensions throughout, the velocities are uniform and the streamlines are equally spaced.When the cross-section of channel has uniform dimensions throughout, the velocities are uniform and the stream lines are equally spaced. The flow at any cross-section within the channel is given by Q = v da.In a similar way, when the cross-section of the plate has same dimensions throughout, the stresses uniform and stress lines are equally spaced. The stress at any cross-section is given by P = da.When the cross-section of the channel is suddenly reduced , the velocity increases in order to maintain the same flow and the streamlines become narrower and narrower and crowd together. Similar phenomena is observed in stressed plate. In order to transmit the same force, the stress lines come closer and closer as the cross-section is reduced. At the change of cross-section, the streamlines as well as stress lines bend.When there is sudden change in cross-section, bending of stress lines (in a similar way as stream lines in fluid flow ) is very sharp and severe resulting in stress concentrationStreamlined shapes (i.e. rounding the corners) are used in channels like aircrafts and boats to reduce the turbulence and resistance to flow.Streamlining or rounding the contours of mechanical components has similar beneficial effects in reducing stress concentration.

Reduction of Stress Concentration:Additional notches and holes in tension member.

Fillet radius, Undercutting and Notch for member in bending.

Drilling additional holes for shaft.

Undercutting and reduction of shank diameter in Threaded members.Additional notches and holes in tension member:

A flat plate with a V-notch subjected to tensile stresses is subjected to a high degree of stress concentration.

The severity of the stress concentration is reduced by the following methods:

Use of multiple notchesDrilling additional holesRemoval of undesired material. ( Principle of minimization of the material)

Fillet radius, Undercutting and Notch for member in bending:A bar of circular cross-section with a shoulder and subjected to bending moment is shown in figure. Ball bearings, gears and pulleys are seated against this shoulderThe shoulder creates changes in cross-section of the shaft resulting in stress concentration.There are three ways to reduce stress concentrationUse of fillet radius UndercuttingAdditional notch

Drilling additional holes for shaft:A transmission shaft with a keyway is shown in figureThe four corners of the keyway viz. m1, m2,n1,n2 are shown in figure. It has been observed that torsional shear stresses at two points viz. m1 and m2 are negligibly small in practice and theoretically equal to zero.On the other hand it has been observed that the torsional shear stresses at two points viz. n1 and n2 are excessive and theoretically infinite, that means even a small torque will produce a permanent set at these points.There are two methods to reduce the stress concentration at these points:Rounding corners at points n1 and n2 by means of fillet radiusDrilling two symmetrical holes on the sides of the keyway. These holes press the force flow lines and minimize their bending in the vicinity of the keyway.

Undercutting and reduction of shank diameter in Threaded members:A threaded component is shown in Figure:There are two ways to reduce stress concentration which results while passing from shank portion to threaded portion of the component:

A small undercut is taken and fillet radius is provided for this undercutThe shank diameter is reduced and made equal to the core diameter of the thread.

ShankThreaded portion