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For a thin-walled cylindrical pressure vessel, hoop stress is approximately pressure times radius divided by wall thickness.
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Formula
σθ and p in Pa; r and t in m
| Symbol | Meaning | Unit |
|---|---|---|
| Circumferential or hoop stress in the wall | Pa | |
| Internal pressure relative to external pressure | Pa | |
| Mean cylinder radius | m | |
| Wall thickness | m |
\sigma_\theta=\frac{p r}{t}The membrane approximation assumes a wall thin relative to radius, a long cylindrical region and uniform internal pressure. Radial stress and local effects at end caps, welds and openings are neglected. For closed ends, ideal axial membrane stress is pr/(2t), half the hoop stress. This is a teaching equation, not a pressure-equipment design approval.
The membrane approximation assumes a wall thin relative to radius, a long cylindrical region and uniform internal pressure. Radial stress and local effects at end caps, welds and openings are neglected. For closed ends, ideal axial membrane stress is pr/(2t), half the hoop stress. This is a teaching equation, not a pressure-equipment design approval.
For a thin-walled cylindrical pressure vessel, hoop stress is approximately pressure times radius divided by wall thickness.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Hoop stress in a thin-walled cylinder” and check its unit and magnitude.
For p=2 MPa, r=0.25 m and t=0.005 m, hoop stress is 100 MPa.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: σθ and p in Pa; r and t in m.
Formula cluster
For a thin-walled cylindrical pressure vessel, hoop stress is approximately pressure times radius divided by wall thickness.
For p=2 MPa, r=0.25 m and t=0.005 m, hoop stress is 100 MPa.
Understand first-order stress levels in thin-walled cylindrical vessels.
Here is how to work through a typical Hoop stress in a thin-walled cylinder (σθ = pr/t) task step by step:
Task
Work through the numerical example for “Hoop stress in a thin-walled cylinder” and check its unit and magnitude.
Solution path
For p=2 MPa, r=0.25 m and t=0.005 m, hoop stress is 100 MPa.
Wall thickness must be small relative to vessel radius so stress can be approximated as a membrane stress through the thickness. The exact limit depends on model and purpose. Thick walls have significant radial stress variation; pr/t is insufficient and a thick-cylinder solution is needed. The membrane relation is a simplified estimate for a long thin cylindrical wall under uniform internal pressure. It excludes end discontinuities, welds, openings, residual stress and code factors. It is not pressure-vessel approval; real equipment requires applicable design rules and qualified review.
For an ideal closed thin cylinder, hoop stress follows from equilibrium of a longitudinal cut, while axial stress balances pressure on the end cap. With the same dimensions, σθ=pr/t and σax=pr/(2t). Open or differently supported vessels can have another axial state. The membrane relation is a simplified estimate for a long thin cylindrical wall under uniform internal pressure. It excludes end discontinuities, welds, openings, residual stress and code factors. It is not pressure-vessel approval; real equipment requires applicable design rules and qualified review.
No. The teaching equation gives ideal membrane stress, not a complete pressure-equipment design. Material condition, welds, openings, temperature, cycling, tolerances and inspection requirements are absent. Applicable rules, material verification and qualified approval govern real design and operation. Use this equation for understanding, not approval. The membrane relation is a simplified estimate for a long thin cylindrical wall under uniform internal pressure. It excludes end discontinuities, welds, openings, residual stress and code factors. It is not pressure-vessel approval; real equipment requires applicable design rules and qualified review.
Write the quantity and unit from the variable list next to every value: σθ and p in Pa; r and t in m. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For p=2 MPa, r=0.25 m and t=0.005 m, hoop stress is 100 MPa. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The membrane relation is a simplified estimate for a long thin cylindrical wall under uniform internal pressure. It excludes end discontinuities, welds, openings, residual stress and code factors. It is not pressure-vessel approval; real equipment requires applicable design rules and qualified review.
The background states this starting condition: The membrane approximation assumes a wall thin relative to radius, a long cylindrical region and uniform internal pressure.. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The membrane relation is a simplified estimate for a long thin cylindrical wall under uniform internal pressure. It excludes end discontinuities, welds, openings, residual stress and code factors. It is not pressure-vessel approval; real equipment requires applicable design rules and qualified review.
Quanta exam set
Curated exam set for "Hoop stress in a thin-walled cylinder":
Question (front)
Which formula describes Hoop stress in a thin-walled cylinder?
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Question (front)
How do you rearrange Hoop stress in a thin-walled cylinder for Base equation for the target quantity?
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Question (front)
Which common mistake happens with Hoop stress in a thin-walled cylinder?
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Create a curated FSRS exam set for σθ = pr/t: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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