Stress concentration is the buildup of stress at a specific point in a material, caused by an abrupt change in shape such as a hole, notch, corner, or sudden narrowing of a cross-section. Instead of spreading evenly across the entire surface, the internal forces crowd together around these geometric irregularities, and the local stress at that spot can reach two to three times the average stress in the rest of the material. This phenomenon explains why cracks tend to start at corners, why bolted joints fail at the bolt holes, and why aircraft windows are rounded instead of square.
How Stress Concentration Works
When a force is applied to a uniform bar or plate, the internal stress distributes fairly evenly across the cross-section. You can picture the force traveling through the material in smooth, parallel lines. But when those lines of force encounter a disruption, like a hole or a sharp corner, they have to reroute around the obstacle. The lines bunch together at the edges of the disruption, and that crowding translates directly into higher stress at those points.
The resulting stress is not uniform. It spikes sharply right at the edge of the irregularity and then drops off quickly as you move away from it. A plate with a circular hole under tension, for example, experiences its peak stress right at the top and bottom edges of the hole, while the material a short distance away remains at or near the average stress level. This localized spike is what engineers mean by “stress concentration.”
The Stress Concentration Factor
Engineers quantify this effect using a number called the stress concentration factor, written as Kt. The relationship is straightforward: the maximum stress at the trouble spot equals Kt multiplied by the nominal (average) stress in the part. If Kt is 2.5 and the average stress across the cross-section is 100 MPa, the peak stress at the discontinuity is 250 MPa.
For most common geometric features, Kt falls between 1 and about 3. A smooth, shallow groove might have a Kt near 1.5, meaning the local stress is only 50% higher than the average. A deep, sharp notch or a hole in a narrow plate can push Kt above 2.5. The sharper the corner and the more abrupt the change in cross-section, the higher the factor climbs. A perfectly uniform bar with no shape changes has a Kt of 1, meaning no concentration at all.
Common Geometric Causes
Several shapes are well-known stress raisers:
- Holes: Bolt holes, rivet holes, and access openings all force stress to reroute around them. A circular hole in a wide plate under tension has a Kt of roughly 3.
- Sharp internal corners: A 90-degree inside corner in a bracket or frame concentrates stress far more than a gently rounded one.
- Sudden changes in width or thickness: A shaft that steps down abruptly from a large diameter to a small one creates a stress spike at the shoulder. Adding a smooth, curved transition (called a fillet) at that step reduces Kt significantly.
- Notches and grooves: Keyways in shafts, threads on bolts, and machining marks all act as small notches that elevate local stress.
The critical variables are always the same: the radius of the curve at the transition (sharper means worse) and the ratio of the feature’s size to the overall cross-section (a hole that takes up a large fraction of the width creates more concentration than a tiny one in a wide plate).
Why It Matters for Fatigue and Failure
Stress concentration is the starting point for most structural failures, especially under repeated loading. A part that experiences millions of load cycles, like a rotating shaft or an aircraft wing panel, doesn’t usually fail because the average stress exceeds the material’s strength. It fails because the peak stress at a stress concentration point is high enough to initiate a tiny crack, and that crack grows a little with every load cycle until the part breaks.
This process, called fatigue failure, is extremely sensitive to stress concentrations. Even a small scratch or a poorly drilled hole can cut a component’s fatigue life dramatically. Classic engineering disasters illustrate the point: the square windows on early de Havilland Comet airliners concentrated stress at their corners, leading to catastrophic fuselage cracks during pressurization cycles. Modern aircraft use oval or rounded-rectangle windows specifically to keep Kt low.
Corrosion makes the problem worse over time. Pitting on a metal surface creates small, irregular notches that act as new stress concentrators. Research has shown that the fatigue life of corroded metal plates depends heavily on the resulting thickness variations, because uneven thinning produces stress concentrations that would not exist on a smooth surface.
Stress Concentration in the Human Body
The same physics applies to bone. Orthopedic surgeons deal with stress concentration regularly, and they refer to these vulnerable spots as “stress risers.” Screw holes drilled into bone during surgery, the tips of metal rods or plates, sharp edges left by incomplete healing, and even the gap where two different implants meet all create localized stress spikes in the surrounding bone.
Periprosthetic fractures, where bone breaks near a joint replacement or metal implant, are a direct consequence. The implant changes the stiffness of the structure at that point, and the bone at the boundary between stiff metal and flexible bone absorbs a disproportionate share of the load. After implant removal, the empty screw holes left behind are classic stress risers, and surgeons often advise patients to limit high-impact activity for weeks or months while the bone fills in those defects. Defects in the outer layer of the thighbone from screw holes or misdirected drilling during surgery are known to predispose patients to fractures at those exact locations.
How Engineers Reduce Stress Concentration
The most effective strategy is to eliminate sharp transitions in geometry. Replacing a sharp inside corner with a generous, rounded fillet can cut the stress concentration factor in half or more. The larger the radius of the curve, the more gently the stress redistributes, and the lower the peak value.
Other common techniques include making holes oval rather than circular when the loading direction is known (elongating the hole along the load direction reduces the peak stress at its edges), adding reinforcement around necessary holes, and distributing loads over a wider area rather than channeling them through a single bolt or pin. Surface finish also matters: polishing away machining marks and tool scratches removes tiny notches that would otherwise serve as fatigue crack initiation sites.
In situations where a geometric discontinuity cannot be avoided, engineers design the part so the nominal stress is low enough that even after multiplication by Kt, the peak stress stays well below the material’s fatigue limit. This approach is why bolted joints in bridges and aircraft use thicker plates and more fasteners than a simple static-strength calculation would require. The extra material keeps the average stress low so the stress concentration at each bolt hole never reaches a dangerous level.