Lamé Equation Stress Calculator

Calculator Inputs

Enter geometry and pressures, then choose an evaluation radius within the wall thickness.

Inner radius (r_i)

Outer radius (r_o)

Evaluation radius (r)

Must satisfy r_i ≤ r ≤ r_o.

Internal pressure (p_i)

External pressure (p_o)

Length unit

Pressure unit

End condition

This affects σ_z, Von Mises, and shear.

Formula Used

For a thick-walled cylinder with internal pressure p_i and external pressure p_o, Lamé theory gives:

σ_r(r) = A − B / r²
σ_θ(r) = A + B / r²

Constants are found from boundary conditions σ_r(r_i) = −p_i and σ_r(r_o) = −p_o:

A = (p_i r_i² − p_o r_o²) / (r_o² − r_i²)
B = (p_i − p_o) r_i² r_o² / (r_o² − r_i²)

How to Use This Calculator

Enter inner and outer radii using your chosen length unit.
Enter internal and external pressures in the selected pressure unit.
Set the evaluation radius where you need stresses reported.
Select the end condition to include or omit axial stress.
Press Calculate Stress to see results above the form.
Use Download CSV or Download PDF to export.

Example Data Table

Sample values for quick verification and learning.

r_i	r_o	r	p_i	p_o	σ_θ(r) approx.	σ_r(r) approx.
50 mm	100 mm	50 mm	10 MPa	0 MPa	~16.667 MPa	~−10.000 MPa
30 mm	60 mm	45 mm	8 MPa	1 MPa	~6.111 MPa	~−2.111 MPa
1.5 in	3.0 in	2.0 in	1200 psi	0 psi	~960 psi	~−240 psi

Lamé Stress Analysis Guide

1) Purpose of Lamé stress checks

Thick-walled pressure parts can develop high local stresses even when overall deformation looks small. Lamé relations estimate radial and hoop stress through the wall and help identify where the peak hoop stress occurs. This calculator reports stresses at a chosen radius and at both surfaces for fast sanity checks.

2) When the model is appropriate

Use the method for straight, axisymmetric cylinders under uniform internal and/or external pressure. The assumptions are linear-elastic behavior, small strain, and no plastic yielding. As a rule of thumb, if wall thickness exceeds about 10% of the inner radius, thick-wall analysis is preferred over thin-wall estimates.

3) Meaning of the inputs

Enter inner radius r_i, outer radius r_o, and an evaluation radius r within the wall. Internal pressure p_i loads the bore, while external pressure p_o loads the outside surface. The tool enforces r_i ≤ r ≤ r_o to avoid extrapolated results.

4) What the stress distribution tells you

Lamé solutions combine a constant term and a 1/r² term, so stresses are not uniform across thickness. For typical cases where p_i > p_o, hoop stress is largest at the inner surface and decreases outward. Radial stress matches the boundary values: σ_r(r_i) = −p_i and σ_r(r_o) = −p_o.

5) End condition and axial stress

Closed ends develop axial stress because pressure acts over the end area. Open ends vent that load, so axial stress is near zero. Because von Mises stress uses all three principal stresses, selecting the correct end condition can change the reported equivalent stress and maximum shear stress.

6) Units and helpful reference values

Radii can be entered in mm, cm, m, inches, or feet, and pressure can be entered in Pa, kPa, MPa, bar, or psi. Common conversions: 1 bar = 100 kPa, 1 MPa = 10 bar, and 1 psi ≈ 6.895 kPa. Keeping consistent units avoids scaling mistakes.

7) Using outputs for engineering decisions

Compare hoop stress and von Mises stress against allowable limits with appropriate factors. If hoop stress dominates, it usually drives the check; if axial stress is included, equivalent stress may rise. Increasing outer radius, lowering pressure, or reducing bore radius all reduce peak hoop stress. External pressure can reduce tensile hoop stress but may introduce buckling concerns beyond this model.

FAQs

1) Why is radial stress negative in many cases?

Radial stress is compressive where pressure acts on the wall. With internal pressure, σ_r equals −p_i at the inner surface and rises toward −p_o at the outer surface, following the sign convention used in elasticity.

2) What does the evaluation radius r mean?

It is the location inside the wall where you want stresses reported. Choose any value between r_i and r_o. The calculator also prints stresses at r_i and r_o for reference.

3) Closed ends vs open ends: what changes?

Closed ends develop axial stress due to pressure acting on the end cap area, while open ends do not. Axial stress affects von Mises and maximum shear results, so select the end condition that matches your physical component.

4) Can I set external pressure greater than internal pressure?

Yes. The formulas support any nonnegative p_i and p_o. If p_o exceeds p_i, hoop stress can become less tensile or even compressive, depending on geometry and the chosen radius.

5) When should I avoid thin-wall approximations?

If the wall thickness is more than about 10% of the inner radius, stress varies significantly through the thickness and thin-wall assumptions can underpredict peak hoop stress. Thick-wall analysis is the safer default in that range.

6) Does this handle tapered or noncylindrical parts?

No. The method assumes a straight, axisymmetric cylinder with uniform pressure and no geometric taper. For cones, elbows, threaded regions, or sharp transitions, use finite element analysis or appropriate code formulas.

7) Are these results enough for final design approval?

They are a solid preliminary check. Final design should consider material strength limits, fatigue, temperature effects, corrosion allowances, manufacturing tolerances, buckling under external pressure, and the relevant pressure vessel or piping code.