Stress concentration occurs when a part has a sudden change in shape, such as a hole, notch, groove, shoulder, keyway, or sharp corner. These geometric discontinuities disturb the smooth flow of internal stress through the material. The result is a local peak stress that can be much higher than the average stress calculated from force divided by area.
This matters because cracks and fatigue failures often start at these high-stress locations even when the overall load seems safe.
Engineers describe the severity of a stress raiser using the theoretical stress concentration factor Kt. It compares the maximum local elastic stress near the feature to the nominal stress in the part. A larger Kt means the geometry creates a stronger stress amplification, especially near sharp radii or small holes in highly loaded regions.
Good design reduces peak stress by using larger fillet radii, smoother transitions, better hole placement, lower nominal stress, and surface finishing that removes crack-like defects.
Understanding Engineering: Stress Concentration Factors
A loaded part carries force through countless tiny regions of material. Near a free edge, the material cannot pull across the empty space, so the internal force has to curve around that boundary. This creates a strong pulling or shearing effect at particular points.
In a plate pulled from left to right, a circular hole has its highest tensile stress at the top and bottom edges, not at the left and right edges. The exact location changes when the load changes from tension to bending, torsion, or a combination of loads. Engineers must therefore consider both the shape and the direction of loading.
The theoretical concentration factor comes from an elastic model. It assumes the material returns to its original shape after unloading and has no major flaws. Handbooks give factors for common shapes, while computer simulation can estimate them for complex parts.
A value of three for a small circular hole in a very wide plate is useful, but real plates have finite width. If the hole takes up a large part of the width, the remaining material is more heavily loaded and the result differs.
Nearby holes, edges, threads, welds, and contact forces can change the local pattern too. A published factor is only valid when the actual geometry and loading are close to the stated case.
Materials do not all react to a peak stress in the same way. A ductile metal may yield in a tiny zone at a notch tip. This local yielding can reduce the elastic peak, although it does not make the feature harmless.
A brittle material has much less ability to redistribute stress and may crack suddenly. Under repeated loading, even a small local plastic region can be important. A crack can start at a rough machining mark, a thread root, or the edge of a drilled hole.
The crack then grows a little during many load cycles. For fatigue work, engineers often use a fatigue notch factor rather than relying only on the theoretical factor. Its value depends on the material, surface condition, part size, and the sharpness of the feature.
Students meet these ideas in bicycle frames, aircraft windows, bolted joints, shafts with keyways, phone casings, and plastic clips. Notice how well-made parts often use rounded inside corners and gradual changes in thickness. When solving a problem, first identify the nominal stress region and state which area is being used.
Gross area and net area can give different nominal values near a hole. Then identify the relevant geometry chart or model, apply the factor only within its limits, and compare the local result with the material strength or fatigue data.
Pay close attention to units, load type, surface quality, and whether the load is steady or repeated. These details often decide whether a small feature is safe or becomes the first failure point.
Key Facts
- Stress concentration factor: Kt = sigma_max / sigma_nom
- Nominal tensile stress in a simple plate: sigma_nom = F / A
- For an infinite plate with a small circular hole in uniaxial tension, Kt = 3 at the hole edge
- Sharper notches usually produce larger Kt because stress lines are forced to turn over a smaller radius
- Increasing fillet radius, blending shoulders, and avoiding sharp internal corners reduce peak stress
- Fatigue cracks often begin at stress concentrations, so local stress range matters: Delta sigma_max = Kt Delta sigma_nom
Vocabulary
- Stress concentration
- A localized increase in stress caused by a change in geometry or material condition.
- Stress concentration factor
- The ratio of maximum local stress to nominal stress, written as Kt = sigma_max / sigma_nom.
- Nominal stress
- The average stress calculated from the applied load and a simple cross-sectional area, without including local geometric effects.
- Fillet radius
- The rounded radius used at a corner or shoulder to make a smoother transition between surfaces.
- Fatigue failure
- Failure caused by repeated loading that grows cracks over many cycles, often starting at a stress concentration.
Common Mistakes to Avoid
- Using sigma_nom as the true maximum stress. This is wrong because holes, notches, and sharp corners can make the local stress much larger than the average value.
- Assuming a small hole has no effect because little material was removed. This is wrong because even a small circular hole can triple the local tensile stress in an ideal wide plate.
- Making corners sharp to save space. This is wrong because a small radius forces stress paths to bend abruptly and raises Kt.
- Applying Kt blindly in every situation. This is wrong because Kt depends on geometry, loading direction, part dimensions, and whether the material is behaving elastically or plastically.
Practice Questions
- 1 A flat bar has a nominal tensile stress of 80 MPa and a circular hole with Kt = 2.6. What is the maximum local elastic stress at the hole edge?
- 2 A stepped shaft has a nominal bending stress of 120 MPa at the shoulder. If the fillet geometry gives Kt = 1.8, calculate the peak elastic stress at the shoulder.
- 3 Two plates carry the same tensile load and have the same width and thickness. Plate A has a sharp rectangular notch, while Plate B has a large rounded notch. Explain which plate is more likely to have a higher peak stress and why.