Tools › Structures › Stress Concentrations › Round Hole Uniaxial Stress › Example

Kirsch (1898): round hole in an infinite plate under uniaxial tension, K_t = 3 and −1

Tool	Round Hole Uniaxial Stress
Source	Kirsch, G. (1898). "Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre." Zeitschrift des Vereines deutscher Ingenieure, Vol. 42, No. 28, pp. 797–807. Equations on p. 799.
Level	Undergraduate

Problem

An infinite plate carries a uniaxial tensile stress of 150 ksi. It contains a round hole of 0.25 in diameter. Find the radial, tangential and shear stresses on the hole edge at the point 90° from the load axis, the stress concentration factor there, and the tangential stress at the point on the load axis.

Inputs to Enter

σ_x = 150 ksi (uniaxial stress, drawn vertical on the page)
D = 0.25 in (hole diameter)
r = 0.125 in (= D/2, the hole edge)
θ = 90° from the load axis

Open the Round Hole Uniaxial Stress tool (loaded with this example)

Answers

σ_θθ = 450 ksi = 3 σ_x, σ_rr = 0, σ_rθ = 0; K_t = 3 (Kirsch)
σ_max = 450 ksi, σ_min = 0, τ_max = 225 ksi
At θ = 0 (change the angle): σ_θθ = −σ_x = −150 ksi, K_t = −1
At r = 2a = 0.25 in, θ = 90°: σ_θθ / σ_x = 1.219 (Kirsch field, off the edge)
At r = 2a, θ = 45°: σ_rr = 56.25 ksi, σ_θθ = 93.75 ksi, σ_rθ = −98.44 ksi; principal 175.2 / −25.2 ksi, τ_max 100.2 ksi

Notes

The tool's default inputs are this case; press Submit. θ is measured from the load axis, so θ = 90° is the point of maximum hoop stress; the page prints the angle-0 and angle-90 edge values in a separate section whatever the probe angle.