Stress Concentration

Introduction

Basic stress analysis calculations assume that the components are smooth, have a uniform section and no irregularities.

In practice virtually all engineering components have to have changes in section and / or shape. Common examples are shoulders on shafts, oil holes, key ways and screw threads. Any discontinuity changes the stress distribution in the vicinity of the discontinuity, so that the basic stress analysis equations no longer apply. Such 'discontinuities' or 'stress raisers' cause local increase of stress referred to as 'stress concentration'.

The 'theoretical' or 'geometric' stress concentration factor Kt or Kts is used to relate the actual maximum stress at the discontinuity to the nominal stress.

Kt = max direct stress / nominal direct stress and Kts = max shear stress / nominal shear stress.

In published information relating to stress concentration values the nominal stress may be defined on either the original 'gross cross section' or on the 'reduced net cross section' and care needs to be taken that the correct nominal stress is used.

The subscript 't' indicates that the stress concentration value is a theoretical calculation based only on the geometry of the component and discontinuity.

Notch Sensitivity

Some materials are not as sensitive to notches as implied by the theoretical stress concentration factor. For these materials a reduced value of Kt is used: Kf. In these materials the maximum stress is:

max. stress = Kf x nominal stress

The notch sensitivity, q, is defined as: q = (Kf - 1) / (Kt - 1) where q is between 0 and 1.

This equation shows that if q = 0, then Kf = 1 as the material has no sensitivity to notches. If q = 1, then Kf = Kt and the material is fully notch sensitive.

When designing, a frequent procedure is to first find Kt from the geometry of the component, then specify the material and look up the notch sensitivity, q, for the notch radius from a chart. Then by rearranging the above equation, determine Kf.

Kf = 1 + q(Kt - 1).

Curves for q values are normally plotted up to notch radii of 4mm. For larger notch radii, the q value at 4 mm can be used.

Most cast irons have a very low q value. This is because their microstructures contain many notches, so additional machined ones make little difference. A value of q = 0.2 will be on the safe side for all grades of cast iron.

When to Use Stress Concentration Values

To apply stress concentration calculations, the part and notch geometry must be known. However where a part is known to contain cracks, the geometry of these may not be known and in any case as the notch radius tends to zero, as it does in a crack, then the stress concentration value tends to infinity and the stress concentration is no longer a helpful design tool. In these cases 'Fracture Mechanics' techniques are used.

Where the geometry is known, then for brittle materials, stress concentration values should be used.

In the case of ductile materials that are subject only to one load cycle during their lifetime (fairly unusual in Mechanical Engineering) it is not necessary to use stress concentration factors as local plastic flow and work hardening will prevent failure provided the average stress is below the yield stress.

NB Not all ductile materials are ductile under all conditions, many become brittle under some circumstances. The most common cause of brittle behavior in materials normally considered to be ductile is being exposed to low temperatures.

For ductile materials subjected to cyclic loading the stress concentration factor has to be included in the factors that reduce the fatigue strength of a component.

Java Applet to calculate the theoretical stress concentration factor in a rectangular member with a transverse hole subject to tension

Java Applet to calculate the theoretical stress concentration factors in a stepped shaft with a shoulder fillet subject to torsion or tension or bending.

David Grieve, 8th April 2008.

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