Thick-Walled Cylinder Tangential Stress Calculator

Example data

Formula used

For a thick-walled cylinder, the Lame stress field is:

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

Boundary conditions (compression negative):

• σ_r(ri) = −Pi
• σ_r(ro) = −Po

Solving gives:

• A = (Pi·ri² − Po·ro²) / (ro² − ri²)
• B = (ri²·ro²·(Pi − Po)) / (ro² − ri²)

If ends are closed, axial stress is σ_z = A. This calculator also reports von Mises stress for quick screening.

How to use this calculator

1. Enter inner radius, outer radius, and select your length unit.
2. Enter internal and external pressures, then pick a stress unit.
3. Choose an evaluation radius r inside the wall thickness.
4. Select end condition: closed ends add axial stress.
5. Optionally set yield strength and safety factor for checks.
6. Press Calculate to see results above the form.
7. Use the CSV or PDF buttons to export your report.

Practical notes

• This model assumes linear-elastic, isotropic behavior and steady pressures.
• Stresses are valid away from local discontinuities and end effects.
• If Po exceeds Pi, hoop stress may reduce or change sign.
• Always confirm design codes and allowable stress definitions.

1) Why tangential stress matters in thick cylinders

In thick-walled pressure vessels, the tangential (hoop) stress is usually the dominant driver of yielding. Unlike thin-wall formulas, hoop stress varies strongly with radius and is highest near the inner surface when internal pressure exceeds external pressure. This calculator evaluates σ_θ(r) anywhere through the wall, supporting safer sizing and inspection planning.

2) When thick-wall theory is the right choice

Use thick-wall analysis when the wall is not “thin” compared with radius. A practical rule is t/ri ≥ 0.1, where thickness t = ro − ri. Under that range, assuming uniform hoop stress can underpredict the peak inner-wall stress. Thick-wall theory also becomes important for high pressures, autofrettage studies, and cases with significant external pressure.

3) Lame stress distribution and boundary conditions

The Lame solution models an axisymmetric cylinder with steady pressure loading. The radial stress matches the applied pressures at the boundaries: σ_r(ri) = −Pi and σ_r(ro) = −Po. Inside the wall, σ_r and σ_θ change with 1/r², creating a steep gradient near ri when ro/ri is large.

4) Interpreting signs and what “compression” means here

This tool uses a common sign convention: compressive stresses are negative. Therefore, radial stress at the inner wall is typically negative (equal to −Pi), while hoop stress is often positive (tensile) for internal pressure loading. If Po approaches or exceeds Pi, hoop stress can reduce, reverse sign, or shift the maximum location.

5) Closed ends vs open ends: axial stress impact

End conditions influence the axial stress σ_z. For closed ends, pressure acting on the end caps induces an axial membrane stress that, in the classic Lame formulation, is uniform and equal to A. For open ends, σ_z is taken as zero. Because combined stress affects yielding, the calculator also reports von Mises stress.

6) Design screening with yield strength and safety factor

Engineering checks often compare an equivalent stress to an allowable value. If you enter yield strength and a safety factor, the calculator computes allowable = YS/SF and shows utilization = σ_vm/allowable at the evaluation radius. A utilization below 1.0 is a quick “screen,” but code compliance may require additional criteria such as fatigue, fracture, or creep.

7) Example numbers that reveal the inner-wall peak

Consider ri = 50 mm, ro = 100 mm, Pi = 10 MPa, Po = 0 MPa. The hoop stress at the inner wall is about 13.33 MPa, while at the outer wall it drops to about 3.33 MPa. That large variation is exactly why thick-wall methods matter: the inner wall controls yielding and crack initiation risk.

8) Using the stress profile table effectively

The stress profile table samples r from ri to ro. Increase “Profile points” to better visualize gradients, and adjust decimals for reporting quality. For documentation, export CSV for spreadsheets and PDF for project files. When comparing options, keep units consistent and record whether ends are closed or open, since σ_z changes the combined stress state.

1) What is “tangential stress” in a cylinder?

Tangential stress, also called hoop stress, acts circumferentially around the cylinder wall. It resists the tendency of internal pressure to split the cylinder along its length.

2) Why does hoop stress change with radius in thick walls?

In thick walls, equilibrium and elasticity produce a 1/r² term in the Lame solution. That term makes stresses higher near the inner radius and lower toward the outer radius.

3) What should I enter for evaluation radius r?

Choose any radius between ri and ro where you want stresses reported. Use r = ri for the inner-wall peak or use mid-wall values for comparison with strain gauges or instrumentation locations.

4) How do closed ends affect the results?

Closed ends add an axial stress component from end-cap loading. This increases the von Mises stress compared with open ends, so utilization can rise even if hoop stress is unchanged.

5) Can external pressure reduce hoop stress?

Yes. External pressure increases compressive radial loading at ro and can decrease tensile hoop stress. If Po becomes large relative to Pi, hoop stress may become compressive in parts of the wall.

6) Is von Mises stress always the right failure check?

Von Mises is widely used for ductile metals under static loading, but real designs may require additional checks: buckling under external pressure, fatigue, brittle fracture, or code-specific allowable stress rules.

7) Do unit choices change the physics?

No. Units only scale the numbers. Keep radii in a consistent length unit and pressures in a consistent stress unit. The calculator converts internally, then reports back in your selected units.