Thick-Walled Cylinder Stress Calculator (Lamé Equations)
Hoop, radial and axial stress in a thick tube, pressure vessel or gun barrel under internal and external pressure, with the stress through the wall drawn as a chart.
Results
The dashed line marks the value you entered.
What this tool does
The familiar thin-wall formula assumes the stress is the same all through the wall. When the wall is thick — roughly a diameter more than about twenty times the thickness is where it starts to hold — the Lamé equations show the truth: the hoop stress is highest at the bore, falls towards the outside, and the radial stress runs from −p at the bore to zero (or minus the outside pressure) on the outside. With internal pressure only, the bore value is always above the thin-wall figure, which is shown for comparison. Elastic, uniform material is assumed; the axial stress is for closed ends.
Formula
σθ = A + B ÷ r² · σr = A − B ÷ r² · A = (pi·ri² − po·ro²) ÷ (ro² − ri²) · B = (pi − po)·ri²·ro² ÷ (ro² − ri²) (Lamé)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
pn | Internal pressure | MPa |
px | External pressure | MPa |
di | Inner diameter | mm |
dou | Outer diameter | mm |
r | Radius where you want the stress | mm |
HI | Hoop stress at the bore (the highest) | MPa |
HO | Hoop stress at the outside | MPa |
HR | Hoop stress at that radius | MPa |
RR | Radial stress at that radius | MPa |
AX | Axial stress (closed ends) | MPa |
VM | Von Mises stress at the bore | MPa |
TH | Thin-wall formula for comparison (mean diameter) | MPa |
Worked example
- Internal pressure60 MPa
- External pressure0 MPa
- Inner diameter100 mm
- Outer diameter160 mm
- Radius where you want the stress60 mm
- Hoop stress at the bore (the highest)136.92 MPa
- Hoop stress at the outside76.92 MPa
- Hoop stress at that radius106.84 MPa
- Radial stress at that radius-29.91 MPa
- Axial stress (closed ends)38.46 MPa
- Von Mises stress at the bore170.54 MPa
- Thin-wall formula for comparison (mean diameter)130.00 MPa
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.