Tools › Structures › Stress Concentrations › Pressurized Hole › Example

Lamé pressurized hole superposed on Kirsch: 3σ + p at the edge

Tool	Pressurized Hole
Source	Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 4, thick cylinder under internal pressure (Lamé) with b → ∞, and the Kirsch hole solution.
Level	Graduate

Problem

A 0.1 in plate (E = 10 Msi, ν = 0.33) carries 5,000 lb/in in y and has a 0.25 in hole under 20 ksi internal pressure. Find the hoop stress from the pressure alone, the extreme hoop stresses on the edge with their positions, the stress concentration factor referred to the far-field stress and the radial growth of the hole.

Inputs to Enter

N_xx = 0, N_yy = 5000 lb/in, N_xy = 0, t = 0.1 in (σ_yy∞ = 50 ksi)
D = 0.25 in, p = 20 ksi, E = 10 Msi, ν = 0.33
r = 0.125 in (edge), θ = 90°

Open the Pressurized Hole tool (loaded with this example)

Answers

Pressure alone: σ_θθ = p a²/r² = 20 ksi all round the edge, σ_rr = −p
σ_edge,max = 3S_1 − S_2 + p = 170 ksi at θ = 90°; σ_edge,min = 3S_2 − S_1 + p = −30 ksi at θ = 0
K_t = σ_edge,max / S_1 = 3.40; at the probe σ_max = 170 ksi, τ_max = 95 ksi
Hole-edge displacement u_a = (1+ν) p a / E = 0.0003325 in

Notes

The tool's default inputs are this case. Entering a rigid-pin interference δ = 0.0005 in instead of p gives p = Eδ/((1+ν)a) = 30.08 ksi (hand 30.075) and u_a = δ; with p alone the tool reports K = 1 referred to p.