Pressure vessel design covers the stresses, thickness requirements, material limits, and safety checks used for tanks, cylinders, spheres, and piping components that hold internal or external pressure. Students need this cheat sheet because pressure vessels combine mechanics of materials, failure theory, and engineering codes in one practical design problem. It provides a quick reference for common equations, assumptions, and checks used in introductory mechanical and chemical engineering design courses.
The core ideas are pressure-induced membrane stress, allowable stress, weld or joint efficiency, corrosion allowance, and code-based safety margins. Thin-walled vessels are usually analyzed with simple hoop and longitudinal stress formulas when the radius-to-thickness ratio is large enough. Thick-walled vessels require radial stress variation and Lame equations instead of constant stress assumptions.
Final designs should always be checked against the governing code, such as ASME Section VIII, because real vessels also involve openings, welds, fatigue, temperature, inspection, and testing.
Key Facts
- For a thin cylindrical pressure vessel, the hoop stress is sigma_h = p r / t, where p is internal pressure, r is inner radius, and t is wall thickness.
- For a thin cylindrical pressure vessel with closed ends, the longitudinal stress is sigma_l = p r / (2 t), so hoop stress is twice the longitudinal stress.
- For a thin spherical pressure vessel, the membrane stress is sigma = p r / (2 t), which is the same in every tangential direction.
- The thin-wall approximation is commonly acceptable when r / t >= 10, or equivalently when the wall thickness is small compared with the vessel radius.
- A basic allowable-stress thickness estimate for a thin cylinder is t = p r / (S E), where S is allowable stress and E is joint efficiency, before adding corrosion allowance.
- The required design thickness is t_required = t_pressure + corrosion allowance + manufacturing tolerance allowance.
- Hydrostatic test pressure is often greater than design pressure, and a simplified classroom estimate is p_test = 1.3 to 1.5 times the design pressure depending on the code and material limits.
- For a thick cylinder, stresses vary through the wall, so radial stress, hoop stress, and longitudinal stress must be checked instead of using one constant membrane stress.
Vocabulary
- Design pressure
- The pressure used for sizing and checking a vessel, usually chosen above the maximum expected operating pressure.
- Hoop stress
- The circumferential tensile stress in a cylindrical vessel caused by internal pressure trying to split the cylinder lengthwise.
- Longitudinal stress
- The axial tensile stress in a closed cylindrical vessel caused by pressure acting on the end caps.
- Allowable stress
- The maximum stress permitted for design after applying safety factors, temperature effects, and material limitations.
- Joint efficiency
- A factor less than or equal to 1 that accounts for the strength reduction caused by welds, seams, or inspection quality.
- Corrosion allowance
- Extra wall thickness added so the vessel remains safe after expected material loss during service.
Common Mistakes to Avoid
- Using diameter instead of radius in sigma_h = p r / t is wrong because it doubles the calculated stress or required thickness.
- Applying thin-wall formulas when r / t is too small is wrong because thick vessels have large stress variation across the wall.
- Forgetting joint efficiency is wrong because welded joints may carry less load than seamless base material and require greater thickness.
- Adding corrosion allowance before computing pressure thickness can be wrong in many design procedures because the pressure-resisting thickness is first calculated, then allowances are added.
- Treating design pressure as normal operating pressure is wrong because design pressure must include expected pressure surges, control limits, and safety margin.
Practice Questions
- 1 A thin cylindrical vessel has internal pressure p = 2.0 MPa, inner radius r = 0.50 m, and wall thickness t = 10 mm. Calculate the hoop stress and longitudinal stress.
- 2 Estimate the minimum pressure thickness for a thin cylindrical vessel with p = 1.5 MPa, r = 0.40 m, allowable stress S = 120 MPa, and joint efficiency E = 0.85 using t = p r / (S E).
- 3 A spherical pressure vessel has p = 3.0 MPa, r = 0.75 m, and t = 15 mm. Calculate the membrane stress using sigma = p r / (2 t).
- 4 Explain why a thick-walled high-pressure vessel cannot be safely designed using only the thin-wall hoop stress equation.