Free · Ideal mechanics
Thin-Shell Pressure Stress and Small Elastic Change
Estimate thin cylindrical or spherical shell membrane stresses and small elastic changes under positive internal gauge pressure.
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
For a sphere of diameter 2 m, thickness 0.01 m and pressure 1000 Pa, nominal hoop stress is 50,000 Pa. At E=1 GPa and ν=0.3 the diameter change is 0.00007 m.
Cylinder σh=pr/t, σz=pr/(2t); sphere σ=pD/(4t); small ΔV=(dV/dD)ΔD.
Worked example
Enter these known values and leave the other values blank.
- Shape
- Sphere
- Shell diameter (d)
- 2 m
- Thickness of the shell (t)
- .01 m
- Internal pressure (P)
- 1000 Pa
- Young's modulus (E)
- 1000000000 Pa
- Poisson's ratio (μ)
- .3
- Efficiency of the circumferential joint (η)
- 1
- Shell radius (r)
- 1 m
- Thickness to diameter ratio (t/d)
- 0.005
- Hoop stress (σ==h==)
- 50000 Pa
- Radial stress (σ==r==)
- -500 Pa
- Maximum out of plane shear stress (σ==s2==)
- 25000 Pa
- Change in diameter (𝛿d)
- 0.00007 m
- Change in volume (𝛿V)
- 0.0004398 m³
Assumptions and limitations
- This is a thin-shell, small-strain, linear isotropic elasticity model under uniform internal gauge pressure. The wall-to-diameter ratio must be below 0.05, and strains must remain small for the approximation to apply.
- A cylindrical vessel has closed ends and axial pressure loading. End effects, external loads, weld stress concentrations, fatigue, buckling, corrosion and plasticity are omitted. This does not establish compliance, allowable pressure or a safe vessel design.
- Positive Young’s modulus and 0<ν<0.5 are the supported material domain. Joint efficiencies lie above zero and at most one; they scale the displayed joint stress values.
- Shear and dimensional changes use nominal unscaled membrane stresses, independently of the joint-efficiency factors. They are not a combined effective-stress tensor for an inefficient joint.
- The spherical volume change includes the Poisson factor 1−ν, obtained from the already-calculated diameter strain and derivative of πD³/6.
Common questions
Why is this limited to thin walls?
Membrane formulas approximate stress as uniform through wall thickness. A thick wall has a radial stress distribution and needs a different model.
References
- MIT: thin pressure vessels and elasticity
- NIST Handbook 44: exact unit definitions
- Calculation definition and unit reference
Replace explanatory external images with original accessible descriptions preserving each scenario, rope or machine arrangement and every variant condition. Include the missing Poisson factor in spherical volume strain, repair cylindrical diameter-output visibility, and start metric controls in SI selections while retaining coherent captured millimetre, megapascal and cubic-millimetre formula bases. Restrict the ideal thin-shell approximation to thickness/diameter below 0.05 and positive Poisson ratios below 0.5. Pounds-force per square inch use the exact international pound and inch with standard gravity, replacing the rounded captured factor.