Thick-Wall Cylinder Stress (Lame)

The stress in a thick-wall cylinder - a hydraulic-cylinder tube, thick pipe.

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Checked against published worked examples from: McGraw-Hill (Shigley's MED 11th ed. solutions, via Oakland University).

Details, formula, and sources

ScopeThe stress in a thick-wall cylinder - a hydraulic-cylinder tube, thick pipe, or gun barrel - where the thin-wall hoop formula (valid only for D/t >= 20) does not apply. Below that the wall is thick, the stress varies through it, and the peak sits at the BORE. Lame for internal pressure: sigma_hoop(bore) = P(ro^2+ri^2)/(ro^2-ri^2), sigma_hoop(outer) = 2P ri^2/(ro^2-ri^2), sigma_radial(bore) = -P, sigma_long(closed) = P ri^2/(ro^2-ri^2). A 4 in bore, 0.5 in wall tube at 3,000 psi (D/t 8) sees 13,667 psi at the bore vs the thin-wall P D/(2t) = 12,000 psi - 12% low (Lame runs 14% higher), so designing on thin-wall leaves the bore over-stressed. External pressure, press fit, compound/autofrettaged tubes, and end caps are separate; apply a safety factor with a failure theory. A design aid; Shigley / Roark and the engineer of record govern.

Formula or tablero = ri + t; sigma_hoop(bore) = P(ro^2+ri^2)/(ro^2-ri^2); sigma_hoop(outer) = 2P ri^2/(ro^2-ri^2); sigma_radial(bore) = -P; sigma_long(closed) = P ri^2/(ro^2-ri^2); thin-wall P D/(2t) shown for comparison.

Source and editionThe Lame thick-wall cylinder equations for internal pressure (Shigley, Mechanical Engineering Design; Roark's Formulas for Stress and Strain), by name; the companion below D/t = 20 to the hoop-stress-thin-wall tile.

Free accessThe Lame equations are standard published elasticity results; the pressure, inner radius, and wall thickness are the user's inputs.

What governsEstimate. AHJ and licensed professional govern.

API field namespressure_psi, inner_radius_in, wall_thickness_in, hoop_bore_psi, hoop_outer_psi, thin_wall_estimate_psi

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