Stress concentrations occur when stress in a material is increased due to the presence of geometrical irregularities like holes, notches, or sharp corners.

These irregularities cause localised “hot spots” where stress is much higher than in the surrounding material, which can lead to failure. Stress concentrations are especially critical in engineering design since they can significantly affect the structural integrity of components.

1. Understanding Stress Concentration Factors (SCF)

The Stress Concentration Factor (often denoted as ) quantifies the increase in stress around an irregularity relative to the nominal or average stress in the material. The general formula for the stress concentration factor is:

where:

• is the maximum stress at the irregularity,
• is the nominal stress in the absence of irregularities.

The SCF is typically determined experimentally or through Finite Element Analysis (FEA), though approximate values are available for standard shapes and irregularities from resources like stress concentration factor charts.

2. Nominal Stress Calculation

The nominal stress is calculated based on simple load assumptions. For example, for a bar in tension:

where:

• is the applied force,
• is the cross-sectional area.

In bending applications, nominal stress can be determined by the bending stress formula:

where:

• is the bending moment,
• is the distance from the neutral axis to the outer fibre,
• is the second moment of area of the cross-section.

3. Key Causes of Stress Concentrations

Common causes include:

• Holes: A circular hole in a plate under tension creates a stress concentration. The SCF depends on the hole diameter and the plate width.
• Notches: A notch, such as a U- or V-shaped groove, introduces sharp edges that lead to high localised stresses.
• Sharp Corners: Square or rectangular sections with sharp internal corners have higher stress concentrations than those with rounded edges.

4. Example of a Stress Concentration Factor: Hole in a Plate under Tension

For a circular hole in a plate under uniform tension, the SCF can be calculated using the equation:

where:

• is the diameter of the hole,
• is the width of the plate.

This equation shows that as the hole size increases relative to the plate width, the stress concentration increases.

5. Avoiding or Reducing Stress Concentrations

Stress concentrations can often be reduced by modifying design features. Common methods include:

• Filleting sharp corners: Rounding corners reduces the SCF, as it softens the stress transition.
• Adding relief holes: Relief holes near sharp changes in geometry help to distribute stress more evenly.
• Tapered or gradual changes: For parts with a sudden change in cross-sectional area, using tapered transitions can smooth the stress flow.

For example, the SCF for a round fillet with radius in a rectangular bar under tension may be approximated by:

where is the height of the bar.

6. Example Problem

Consider a rectangular bar 50 mm wide with a circular hole of diameter 10 mm at its centre, subject to an axial tensile load. If the nominal stress is calculated to be 100 MPa, the SCF can be estimated as follows:

Thus, the maximum stress is:

This increase from 100 MPa to 140 MPa is significant and demonstrates why these concentrations are critical in design.

7. Practical Applications of Stress Concentration

In real-world applications, avoiding or reducing stress concentrations is essential in components such as:

• Aircraft fuselage frames where windows or rivet holes are common,
• Machine shafts with keyways or grooves,
• Structural beams with bolt holes or notches.

Summary

Stress concentrations are critical considerations in design, as they can drastically increase localised stress, potentially leading to material failure. By understanding and calculating SCFs, engineers can make informed design choices to either reinforce areas of high stress or adjust geometries to minimise these concentrations.