Stress Concentration Factor for a Plate with a Hole: Formula and Method
A stress concentration factor (Kt) describes how much a local stress rises above the average, or "nominal," stress that elementary strength-of-materials theory would predict for a member, once you account for a geometric discontinuity such as a hole, notch, fillet, or groove. By definition, Kt = σmax / σnom, where σmax is the true peak stress found at the edge of the discontinuity (from theory of elasticity or experiment) and σnom is the simple average stress computed as if the discontinuity were not there. This calculator implements the most common textbook case: a centrally located circular hole in a flat plate of finite width W, loaded in uniaxial (axial) tension.
How the calculation works
First the calculator forms the ratio of hole diameter to plate width, d/W, and evaluates the empirical curve-fit formula compiled in Peterson's/Roark's stress-concentration charts: Kt = 3.000 − 3.140(d/W) + 3.667(d/W)² − 1.527(d/W)³. This cubic is a fit to Howland's 1929 exact elasticity solution and is valid for 0 ≤ d/W ≤ 1, agreeing with the exact solution to within about 2%. It is referenced to the net-section stress, σnet = P / ((W − d)·t), which uses the actual reduced cross-section at the hole. The calculator also reports the gross (nominal) stress, σnom = P / (W·t), where P is the applied axial force and t is the plate thickness; σnet is always higher than σnom. Finally, it multiplies Kt by the net-section stress to get the true peak stress at the hole boundary: σmax = Kt × σnet. As a check, when d/W → 0 the formula reduces to Kt = 3.000, matching Kirsch's classical result for a small hole in an infinite plate; as d/W → 1 it approaches Kt = 2.000.
Common mistakes
- Mixing gross and net stress: the Kt from this formula must be multiplied by the net-section stress (P/((W−d)·t)), not the gross-section stress (P/(W·t)) — combining the wrong pair overstates or understates σmax.
- Reusing the hole formula for other geometries: shoulder fillets, U-notches, grooves, and keyways each have their own Kt curves; a circular-hole factor does not transfer to a fillet radius or a notch root.
- Applying Kt past yield without adjustment: Kt assumes linear-elastic behavior. For ductile metals under static load, local yielding redistributes stress so the true safety margin is better than Kt alone suggests; for fatigue, use the notch-sensitivity-adjusted Kf instead.
Real-world applications
- Sizing rivet, bolt, and pin holes in structural plates and brackets so the peak edge stress stays below the material's allowable stress.
- Predicting fatigue crack initiation sites in machine components, since cracks almost always start where Kt (or its fatigue counterpart Kf) is highest.
- Comparing design alternatives, such as widening a plate or relocating a hole, to reduce peak stress without adding material.
- Cross-checking finite-element analysis results against a known closed-form solution for a simple hole-in-plate geometry.