Lamé equations for a thick-walled cylinder

The thin-wall relation assumes wall thickness is small relative to radius and treats stress as nearly uniform. In thick walls, radial stress varies substantially through the section, and Lamé also describes the hoop-stress gradient. They answer different model questions and are not interchangeable. Lamé assumes a long, axisymmetric, linear-elastic cylinder. Tension is positive and compression negative; radial boundary stresses must match the prescribed pressures. Application: Estimate stresses in thick pressure pipes, cylinders and sleeves. The numerical example With pi=10 MPa, po=0, a=50 mm, b=100 mm, A=3.333 MPa and B=33,333 MPa·mm². At r=a, σr=−10 MPa and σθ=16.667 MPa; at r=b, σr=0 and σθ=6.667 MPa. Boundary radial stresses match prescribed pressure anchors the idea in a concrete measurement case.

Apply σr(a)=−pi at the inner surface and σr(b)=−po at the outer surface to σr=A−B/r². The two boundary conditions give two linear equations for A and B. The negative signs follow from tension-positive convention: pressure is compressive. Substitute the constants back to check both boundaries. Lamé assumes a long, axisymmetric, linear-elastic cylinder. Tension is positive and compression negative; radial boundary stresses must match the prescribed pressures. The corresponding units are Stress and pressure in Pa; radii in m; A in Pa; B in Pa·m². With pi=10 MPa, po=0, a=50 mm, b=100 mm, A=3.333 MPa and B=33,333 MPa·mm². At r=a, σr=−10 MPa and σθ=16.667 MPa; at r=b, σr=0 and σθ=6.667 MPa. Boundary radial stresses match prescribed pressure. This checks that the reference quantities are compatible.

With internal pressure and a pressure-free outside, B is positive and B/r² decreases as r increases. Thus σθ=A+B/r² is greatest at the inner radius r=a. In the example it is 16.667 MPa. This conclusion uses the stated geometry and pressure boundary conditions, not added thermal or axial loads. Lamé assumes a long, axisymmetric, linear-elastic cylinder. Tension is positive and compression negative; radial boundary stresses must match the prescribed pressures. Scope: For a long axisymmetric linear-elastic cylinder, σr=A−B/r² and σθ=A+B/r². Boundary conditions σr(a)=−pi and σr(b)=−po give A=(pi a²−po b²)/(b²−a²), B=(pi−po)a²b²/(b²−a²). Closed ends add axial stress. End effects, notches, plasticity and code design are outside this solution. A practical application is Estimate stresses in thick pressure pipes, cylinders and sleeves.

Lamé’s solution contains a B/r² term. Its magnitude grows as r decreases, so under internal pressure hoop stress typically peaks at the inner wall. Radial stress there also satisfies the boundary condition −pi. Check both stresses at r=a and r=b rather than reading only the largest value; this confirms that pressure signs and boundary values were applied consistently. Lamé assumes a long, axisymmetric, linear-elastic cylinder. Tension is positive and compression negative; radial boundary stresses must match the prescribed pressures. Check the calculation with this case: With pi=10 MPa, po=0, a=50 mm, b=100 mm, A=3.333 MPa and B=33,333 MPa·mm². At r=a, σr=−10 MPa and σθ=16.667 MPa; at r=b, σr=0 and σθ=6.667 MPa. Boundary radial stresses match prescribed pressure. Keep the model limits and units in view.

For a thin wall, radial stress variation is small relative to hoop stress, so membrane approximations may suffice. In a thick wall, variation with radius matters and Lamé accounts for it explicitly. There is no universal thickness cutoff independent of code and geometry. Compare the approximation with boundary stresses and use the applicable pressure-equipment design rules for approval. Lamé assumes a long, axisymmetric, linear-elastic cylinder. Tension is positive and compression negative; radial boundary stresses must match the prescribed pressures. The relation describes Lamé’s elastic solution gives radial and hoop stress through a thick cylinder under internal and external pressure. This helps distinguish its target quantity from neighbouring quantities.