Stress Concentration⚠ unverified

Understanding the Parameters

• Concentration factor $K_t$ — a purely geometric multiplier read from Peterson/Roark charts. Sharp features give high $K_t$ (a small fillet radius can reach $3$+; a circular hole in a wide plate gives exactly $3$). It is defined relative to a stated nominal stress — always check whether the chart uses the net or gross section.

• Nominal stress $\sigma_{nom}$ — the stress computed ignoring the discontinuity, e.g. $P/A_{net}$ at a hole. $K_t$ multiplies it to give the true peak.

• Peak stress $\sigma_{max}$ — the local maximum at the notch root. For brittle materials or static loading near ultimate, this peak governs directly. For ductile static loading it is largely relieved by local yielding (so $K_t$ is often ignored for static ductile design), but for fatigue it always matters.

• Geometry, not material — $K_t$ is elastic and material-independent; the material's response to the peak (yield redistribution, fatigue sensitivity) is captured separately by $q$ and $K_f$ (Fatigue Notch Factor, Notch Sensitivity).

Derivation (Approaching a Proof)

$K_t$ is defined as the ratio of peak elastic stress to nominal stress:

$$K_t \equiv \frac{\sigma_{max}}{\sigma_{nom}} \;\Longrightarrow\; \sigma_{max} = K_t\,\sigma_{nom}.$$

The values come from elasticity theory or finite-element/photoelastic measurement. The classic closed-form result is Kirsch's solution (1898) for a small circular hole of radius $a$ in a wide plate under remote uniaxial tension $\sigma$: the tangential stress at the hole edge is

$$\sigma_\theta = \sigma\,(1 - 2\cos 2\theta),$$

which peaks at the sides of the hole ($\theta = \pm 90^\circ$) at $\sigma_\theta = 3\sigma$ — hence $K_t = 3$ for a circular hole. Other geometries are tabulated as $K_t$ versus dimension ratios (radius/width, etc.).

Dimensional check. $\sigma_{max} = K_t\,\sigma_{nom} = (\text{–})\cdot\text{Pa} = \text{Pa}$ — a stress, as required ($K_t$ is dimensionless).

History and Development

Ernst Kirsch's 1898 circular-hole solution gave the first analytic stress-concentration factor and the famous $K_t = 3$. Charles Inglis (1913) extended it to elliptical holes, showing that stress rises sharply as the tip radius shrinks — the insight that led directly to Griffith's fracture theory. Modern design relies on Peterson's comprehensive $K_t$ charts. The distinction between the geometric $K_t$ and the fatigue-effective $K_f$ (Fatigue Notch Factor) is the bridge from elastic theory to real notched-part design.

Related Concepts: Fatigue Notch Factor, Notch Sensitivity, Fatigue Stress Concentration, Stress Concentration and Notch Effects, Max Principal Stress, Factor Of Safety Ultimate

Notes: $K_t$ is geometric (material-independent), from Peterson/Roark charts — check net vs gross nominal. Governs brittle/static-ultimate and fatigue; largely relieved for ductile static loading. Fatigue uses $K_f = 1+q(K_t-1)$ (Fatigue Notch Factor). Circular hole → $K_t = 3$.