Thin-Walled Pressure Vessels - Cheat Sheet
Your last-minute reference for cylindrical and spherical pressure vessel stress formulas, design equations, and exam-critical relationships. Covers thin-wall criterion, hoop vs. longitudinal stress, joint efficiency, and design for allowable stress.
Thin-Wall Criterion & Validity
Formulas
t/r ≤ 1/10 OR r/t ≥ 10
Meaning
t = wall thickness (mm); r = inner radius (mm). Condition for thin-wall formulas to apply.
Watch out
If t/r > 1/10, use thick-cylinder (Lamé) theory, NOT these formulas. Always use INNER radius/diameter, not outer.
When to use
ALWAYS verify this first before using any thin-wall stress formula.
Important facts
- Thin-wall assumption: stress is constant through the wall thickness → simple equilibrium formulas work.
- Derived from cutting the vessel and applying force balance on half or ring element.
- t/r > 1/10 → thick-wall (Lamé) formulas required; thin-wall formulas are INVALID.
- Always use gauge pressure (not absolute) and INNER diameter/radius in formulas.
- Radial stress through thin wall is negligible (approximately zero) compared to membrane stresses.
Key definitions
Thin-Walled Pressure Vessel
Example
Water pipe D = 600 mm, t = 8 mm: t/r = 8/300 = 0.027 ✓ thin-wall.
Closed container with wall thickness ≤ 1/10 of inner radius; stress is essentially uniform across thickness; allows equilibrium-based (membrane stress) formulas.
Gauge Pressure
Example
Boiler at 1.5 MPa gauge = 1.6 MPa absolute (if 0.1 MPa atm).
Internal pressure p measured relative to atmospheric; absolute pressure = p_gauge + p_atm. Always use gauge pressure in these formulas.
Hoop (Circumferential) Stress
Example
Why pressure pipes fail along a longitudinal seam: hoop stress tears the seam.
Stress acting perpendicular to a longitudinal section (trying to split the vessel along its length); largest stress in cylinders.
Longitudinal (Axial) Stress
Example
σ_l = pD/4t governs circumferential seams and end attachment.
Stress along the cylinder axis; caused by internal pressure pushing on end caps; equals half the hoop stress in cylinders.
Diagrams to know
- Free-body diagram of half-cylinder showing internal pressure and hoop stress on the seam.
- Transverse section showing longitudinal stress on end caps and axial force.
- Mohr's circle for cylinder stresses (σ_h and σ_l as principal stresses; radial stress ≈ 0).
CYLINDRICAL VESSELS — STRESS FORMULAS
Formulas
σ_h = pD/(2t) = pr/t
Meaning
Hoop (circumferential) stress; p = gauge pressure (Pa or MPa); D = inner diameter (mm); r = inner radius (mm); t = wall thickness (mm).
Watch out
Denominator is 2t for D or t for r — don't mix. This is TWICE the longitudinal stress. Forget the factor of 2 and fail the design.
When to use
Always for cylinders under internal pressure; HOOP GOVERNS DESIGN (larger stress).
σ_l = pD/(4t) = pr/(2t)
Meaning
Longitudinal (axial) stress along cylinder axis.
Watch out
Denominator is 4t for D or 2t for r. Exactly HALF of hoop stress. Do not use hoop formula for longitudinal.
When to use
When checking circumferential seams or end attachment; lesser of the two cylinder stresses.
σ_h = 2σ_l
Meaning
Relationship: hoop stress is always twice the longitudinal in a cylinder.
Watch out
This is the KEY characteristic of cylinders — memorize it. Sphere is DIFFERENT.
When to use
Check your calculated stresses; if ratio is not 2:1, you made an error.
Important facts
- Hoop stress σ_h = pD/(2t) is THE CRITICAL stress in cylinder design — governs wall thickness.
- Longitudinal stress σ_l = pD/(4t) acts on the end caps and circumferential seams.
- Both stresses are TENSILE (positive) for internal gauge pressure; no compression.
- Hoop stress is twice longitudinal — cylinder fails first along a LONGITUDINAL seam.
- Shear stress: τ_max (in-plane) = (σ_h − σ_l)/2 = pr/(4t); absolute max = σ_h/2 = pr/(2t).
Key definitions
Membrane Stress
Example
In a cylinder, σ_h and σ_l are membrane stresses; radial stress through wall is ~0.
Average stress in thin-walled vessel, assumed constant through thickness; acts in the plane of the wall (hoop or longitudinal in cylinders).
Principal Stresses
Example
1.5 MPa pressure → σ_h ≈ 60 MPa, σ_l ≈ 30 MPa, σ_r ≈ 0.
In a thin cylinder: σ₁ = σ_h, σ₂ = σ_l, σ₃ ≈ 0 (radial). All tensile for internal pressure.
Diagrams to know
- Half-cylinder FBD with pressure load and hoop stress on the seam.
- Transverse section FBD showing end-cap pressure and longitudinal stress on the wall.
- Mohr's circle: σ_h and σ_l as principal stresses; diameter = σ_h (max shear = σ_h/2).
Reactions or equations
Hoop stress derivation: p·D·L/2 = σ_h·t·L → σ_h = pD/(2t)
Note
This equilibrium approach (not complex theory) is why thin-wall formulas are so simple and powerful.
Conditions
Free-body: half-cylinder length L; pressure p acts on projected area D·L; stress acts on seam area t·L.
Longitudinal stress derivation: p·π·r² = σ_l·2π·r·t → σ_l = pr/(2t) = pD/(4t)
Note
End caps push the cylinder apart; stress is half hoop (factor of 2 difference in areas).
Conditions
Free-body: transverse slice; pressure acts on end cap (πr²); stress acts on wall annulus (2πrt).
SPHERICAL VESSELS — STRESS FORMULA
Formulas
σ = pD/(4t) = pr/(2t)
Meaning
Membrane stress in ALL directions (sphere); p = gauge pressure; D, r, t as before.
Watch out
Denominator is 4t (for D) or 2t (for r) — SAME as cylinder's LONGITUDINAL. Do not confuse with cylinder's hoop. A sphere is more efficient.
When to use
All spherical pressure vessels (LPG tanks, gas spheres, high-pressure storage). Same pressure → sphere is stressed HALF of cylinder.
Important facts
- Sphere stress = pD/(4t) = pr/(2t) — equals cylinder LONGITUDINAL stress, NOT hoop.
- For equal p, D, t: sphere stress is HALF cylinder hoop stress → sphere is STRONGER.
- No directional weakness in a sphere (unlike cylinder's longitudinal seam); best for high pressure.
- Sphere always fails at a lower internal pressure than an equal-size cylinder.
- Used in practice: LPG tanks, compressed air storage, nuclear reactor vessels.
Key definitions
Spherical Symmetry
Example
A sphere has no hoop-vs-longitudinal distinction; all directions see the same tensile stress.
Every great-circle cut of a sphere is identical by geometry; only ONE membrane stress exists, the same in all directions.
Diagrams to know
- Sphere sectioned by a diametral plane; pressure force on hemisphere equals stress on the edge.
- Comparison sketch: cylinder hoop vs. sphere stress for same p, D, t.
Reactions or equations
Sphere equilibrium: p·π·r² = σ·2π·r·t → σ = pr/(2t)
Note
Same form as cylinder longitudinal; geometry gives the 2:1 advantage over cylinder hoop.
Conditions
Any diametral cut; pressure on hemisphere (πr²) balanced by stress on semicircular edge (2πrt).
DESIGN FOR ALLOWABLE STRESS
Formulas
t = pD/(2·η·σ_allow) [CYLINDER, hoop governs]
Meaning
Minimum wall thickness; η = joint efficiency (≤ 1); σ_allow = allowable stress (MPa); p, D in consistent units.
Watch out
Hoop governs (not longitudinal) because σ_h = 2σ_l. If problem is ambiguous, size for hoop (larger). Always verify t/r ≤ 1/10 afterward.
When to use
Design any cylindrical pressure vessel when p, D, material, and safety factor (implicit in σ_allow) are given.
t = pD/(4·η·σ_allow) [SPHERE]
Meaning
Minimum wall thickness for spherical vessel; same parameters as cylinder formula but denominator is 4, not 2.
Watch out
Denominator is 4t (half the cylinder's 2t denominator). Sphere is more efficient. Do not confuse with cylinder hoop formula.
When to use
Design spherical tanks (LPG, gas). For SAME p, D, σ_allow: sphere t is HALF cylinder t.
p_max = 2t·η·σ_allow / D [CYLINDER]
Meaning
Maximum allowable internal pressure given wall t, efficiency η, and material allowable stress.
Watch out
This is just the design formula rearranged. Check thin-wall validity: if given t/r approaches or exceeds 1/10, the result is unreliable.
When to use
When a vessel already exists and you need to find safe pressure limit.
p_max = 4t·η·σ_allow / D [SPHERE]
Meaning
Maximum pressure for spherical vessel; denominator is D (not 2D as in cylinder).
Watch out
Coefficient is 4, not 2. Sphere can hold twice the pressure of equal-size cylinder under same wall and material.
When to use
Assess existing spherical tank safety or rated pressure.
Important facts
- Hoop stress GOVERNS cylinder design (factor of 2 in numerator vs. 4 for longitudinal).
- Always account for joint efficiency η when seams are present; reduces allowable stress.
- After calculating t, VERIFY thin-wall criterion t/r ≤ 1/10; if violated, formulas are invalid.
- Design formula is rearranged equilibrium + allowable stress limit; no complex theory.
- Sphere requires half the wall thickness of a cylinder for identical p, D, material.
Key definitions
Joint Efficiency η
Example
η = 0.85 for a specific weld → apply 0.85 × σ_allow across the seam.
Fraction (≤ 1) representing strength of welded/riveted seam relative to parent plate; reduces effective allowable stress at seams.
Allowable Stress σ_allow
Example
Steel at room temp, η = 1: σ_allow ≈ 80–150 MPa depending on grade and safety factor.
Maximum working stress; typically material yield/ultimate divided by a safety factor (2–4 for vessels); includes material type, temp, code (ASME, etc.).
Diagrams to know
- Design flowchart: given p, D, material → find σ_allow → apply η → calculate t → verify t/r ≤ 1/10.
- Graph of wall thickness t vs. pressure p for fixed D and material (linear relationship).
Reactions or equations
Design equilibrium: σ_working = pD/(2t) ≤ η·σ_allow → t ≥ pD/(2·η·σ_allow)
Note
Rearrange to solve for maximum p or required t depending on the problem.
Conditions
Apply factor of safety through σ_allow; joint efficiency reduces seam strength.
SHEAR STRESS & STRESS COMPONENTS
Formulas
τ_max (in-plane) = (σ_h − σ_l) / 2 = pr / (4t) [CYLINDER]
Meaning
Maximum in-plane shear stress in the hoop-longitudinal plane; derived from Mohr's circle of two principal stresses.
Watch out
This is NOT the absolute maximum shear. Absolute max includes radial stress ≈ 0, giving τ_abs = σ_h/2.
When to use
When checking failure by shear (von Mises, Tresca) or needing complete stress state.
τ_abs = σ_h / 2 = pr / (2t) [CYLINDER]
Meaning
Absolute maximum shear stress, accounting for radial stress σ_r ≈ 0; equal to half the hoop stress.
Watch out
Often the exam uses Tresca (max shear) criterion; this τ_abs is the relevant quantity.
When to use
Failure theories (max shear / Tresca criterion); more conservative than in-plane shear.
Radial stress σ_r ≈ 0 (thin-wall assumption)
Meaning
Stress through the wall thickness is negligible in thin-walled vessels; only hoop and longitudinal are significant.
Watch out
Thick walls (t/r > 1/10) have significant radial stress; thin-wall formulas break down.
When to use
Justifies treating the vessel as a 2D stress state (hoop and longitudinal principal stresses).
Important facts
- Thin-wall cylinder is a 2D stress state: only hoop and longitudinal are significant; radial is negligible.
- Mohr's circle has diameter = σ_h (the larger principal stress); max shear = σ_h/2.
- In-plane shear (hoop-long plane) is (σ_h − σ_l)/2; absolute max is σ_h/2 (includes radial dimension).
- Sphere has only ONE non-zero principal stress → max shear = σ/2.
Key definitions
Principal Stresses
Example
1.5 MPa pressure, D = 600 mm, t = 8 mm: σ₁ ≈ 56 MPa, σ₂ ≈ 28 MPa, σ₃ ≈ 0.
Normal stresses on planes where shear = 0; in thin cylinder: σ₁ = σ_h (hoop), σ₂ = σ_l (long.), σ₃ ≈ 0 (radial).
Diagrams to know
- Mohr's circle for cylinder stress state: σ_h and σ_l on horizontal axis, circle diameter = σ_h.
- 3D stress element showing hoop, longitudinal, and radial components (last one ≈ 0).
Reactions or equations
Mohr's circle for cylinder: center = (σ_h + σ_l)/2, radius = (σ_h − σ_l)/2 = pr/(4t)
Note
Absolute max shear is radius to the σ_r axis, which equals σ_h/2.
Conditions
σ_r ≈ 0; both hoop and longitudinal tensile (positive).
COMPARISON: CYLINDER vs. SPHERE
Important facts
- For identical p, D, t: sphere stress = pr/(2t); cylinder hoop = pr/t → SPHERE IS HALF-STRESSED.
- Design: for equal p, D, σ_allow: t_sphere = t_cylinder / 2 → SPHERE NEEDS HALF THE WALL.
- Cylinder has two stresses (hoop 2×, long. 1×); sphere has one (isotropic).
- Cylinder weak along longitudinal seam (hoop pulls it apart); sphere has no weak seam direction.
- Practical outcome: high-pressure storage (LPG, gas) uses SPHERES; low-to-medium pressure uses cylinders (cheaper fabrication).
Diagrams to know
- Side-by-side stress distribution diagrams: cylinder (hoop vs. long.) vs. sphere (uniform).
- Wall thickness comparison chart: t vs. pressure p for cylinder and sphere.
SEAMS & JOINT EFFICIENCY
Formulas
Effective allowable stress = η · σ_allow
Meaning
Joint efficiency η reduces the strength of a welded or riveted seam; apply this reduced value in design formulas.
Watch out
If no η is given, assume η = 1 (parent plate strength). If η is given but not used in design, the vessel is over-stressed at the seam.
When to use
Always when the problem specifies a joint efficiency (e.g., η = 0.85 for longitudinal seam).
Common values
Value
η = 1.0
Symbol
η
Quantity
Joint Efficiency (fully radiographed weld)
Value
η = 0.85
Symbol
η
Quantity
Joint Efficiency (spot radiographed)
Value
η = 0.60–0.70
Symbol
η
Quantity
Joint Efficiency (no radiography, visual only)
Important facts
- Hoop stress σ_h = pD/(2t) acts PERPENDICULAR to longitudinal seam → tries to split the seam.
- Longitudinal stress σ_l = pD/(4t) acts ACROSS circumferential seam; smaller stress, less likely to fail.
- Joint efficiency η typically 0.50–1.00 depending on inspection and weld type (radiography increases η).
- If η is given for one seam, problem often requires separate check for the other seam.
- Pressure vessels in code practice (ASME, NSCP 2015) specify η by seam type and inspection method.
Key definitions
Longitudinal Seam
Example
Cylinder wall thickness is set by hoop stress and the efficiency of the longitudinal seam.
Weld or rivet line running along the cylinder length; stressed by HOOP stress (the larger one); governs cylinder thickness design.
Circumferential (Girth) Seam
Example
Typically requires less inspection/testing than longitudinal seam because stress is lower.
Weld running around the cylinder circumference; stressed by LONGITUDINAL stress; less critical (σ_l is smaller).
Diagrams to know
- Cylinder with visible longitudinal and circumferential seams; annotate which stress acts on which seam.
- Cross-section showing seam orientation and direction of hoop and longitudinal stresses.
Reactions or equations
Hoop-seam relationship: σ_h = pD/(2t) acts perpendicular (across) longitudinal seam.
Note
Why cylinders fail along the length in a spectacular burst — hoop stress is the largest.
Conditions
Longitudinal seam runs along vessel axis; hoop stress pulls it open.
WORKED PROBLEM EXAMPLES — CRITICAL PATTERNS
Important facts
- PATTERN 1: Given p, D, t → FIND stresses: σ_h = pD/(2t), σ_l = pD/(4t); verify t/r ≤ 1/10.
- PATTERN 2: Given p, D, σ_allow, η → FIND t: t = pD/(2·η·σ_allow); round up; verify thin-wall.
- PATTERN 3: Given D, t, σ_allow, η → FIND p_max: p = 2·η·σ_allow·t / D.
- PATTERN 4: Compare cylinder vs. sphere: sphere t ≈ (1/2) × cylinder t for equal p, D, material.
- PATTERN 5: Seam efficiency: if η is given, include in all stress/thickness calculations. If not given, η = 1.
Must remember
- THIN-WALL CRITERION: t/r ≤ 1/10 MUST BE VERIFIED FIRST. If false, these formulas do not apply.
- CYLINDER HOOP STRESS σ_h = pD/(2t) is ALWAYS the governing stress for cylinder design (larger than σ_l).
- CYLINDER LONGITUDINAL σ_l = pD/(4t) = σ_h/2. This relationship (2:1) is unique to cylinders and is exam-critical.
- SPHERE SINGLE STRESS σ = pD/(4t) = pr/(2t) equals cylinder LONGITUDINAL, NOT hoop. Sphere requires half the wall of a cylinder for the same p, D, material.
- DESIGN FORMULA for cylinder: t = pD/(2·η·σ_allow), where η is joint efficiency. For sphere: t = pD/(4·η·σ_allow).
- HOOP STRESS acts perpendicular to LONGITUDINAL SEAM; longitudinal stress acts across CIRCUMFERENTIAL seam. Hoop is larger → longitudinal seam governs and often has lower η.
- JOINT EFFICIENCY η reduces allowable stress at seams (e.g., η = 0.85 for spot-radiographed welds). Always include in thickness calculation if given; if not given, assume η = 1.
- GAUGE PRESSURE is used in all formulas (internal pressure relative to atmosphere, not absolute). Do not add atmospheric pressure.
- USE INNER DIAMETER/RADIUS in all formulas. If outer dimensions are given, subtract 2t to convert to inner.
- MAXIMUM SHEAR in a cylinder: τ_abs = σ_h/2 = pr/(2t) (accounts for zero radial stress). In-plane shear is (σ_h − σ_l)/2 = pr/(4t) (smaller).
Last minute tips
- On any exam problem, DRAW A SMALL CIRCLE OR CYLINDER and LABEL p, D, t, r. Then identify which formula to use. Visual orientation prevents swapping hoop/longitudinal.
- If the problem says 'compare a cylinder and a sphere of the same size and pressure,' answer IMMEDIATELY: sphere wall = (1/2) cylinder wall. Sphere stress = (1/2) cylinder hoop stress. This relationship is tested often.
- ALWAYS calculate t/r at the end of a design problem to verify thin-wall. If it creeps above 0.1, state 'thin-wall assumption invalid; thick-wall theory or iteration required.' Showing this check gains partial credit even if the initial design is slightly off.
- If joint efficiency η is mentioned ANYWHERE in the problem, it MUST appear in your thickness formula. Forgetting it is an instant point loss. Write: t = pD / (2·η·σ_allow) and circle the η.
- When comparing stresses or pressures, always verify the RATIO FIRST before computing. For cylinders: σ_h/σ_l must equal 2.0. For sphere vs. cylinder: stresses must differ by factor of 2. If ratios are wrong, stop and recheck the formula.
Comparison tables
Cylinder vs. Sphere — Formulas & Behavior
| Property | CYLINDER | SPHERE |
|---|---|---|
| Hoop/Radial Stress | σ_h = pD/(2t) = pr/t | N/A — single stress only |
| Longitudinal/Single Stress | σ_l = pD/(4t) = pr/(2t) | σ = pD/(4t) = pr/(2t) [SAME as cylinder longitudinal] |
| Stress Ratio | σ_h = 2σ_l (hoop is twice longitudinal) | Single stress, isotropic (no ratio) |
| Governing Design (largest stress) | HOOP σ_h (use in thickness formula) | Single stress σ (no choice) |
| Design Thickness Formula | t = pD / (2·η·σ_allow) | t = pD / (4·η·σ_allow) → HALF of cylinder |
| For equal p, D, σ_allow | Reference thickness | Requires t/2 ← SPHERE IS MORE EFFICIENT |
| Weak Seam Direction | Longitudinal seam (hoop pulls it open) | No directional weakness (isotropic) |
| Maximum Shear (absolute) | τ_abs = σ_h/2 = pr/(2t) | τ = σ/2 = pr/(4t) → HALF of cylinder |
| Common Applications | Water mains, penstocks, low–medium pressure; cheaper to fabricate (rolled plates) | LPG, compressed gas, high-pressure storage; more robust but costlier to fabricate (hemispherical shells) |
Thin-Wall vs. Thick-Wall Decision Tree
| Check t/r | Classification | Formulas to Use | Example |
|---|---|---|---|
| t/r ≤ 0.1 | THIN-WALLED | σ_h = pD/(2t), σ_l = pD/(4t), σ = pD/(4t) for sphere | D = 600 mm, t = 8 mm, t/r = 0.027 ✓ |
| t/r > 0.1 | THICK-WALLED | Use Lamé equations (radial stress varies through wall, complex) | D = 100 mm, t = 15 mm, t/r = 0.3 ✗ (do NOT use simple formulas) |
Common Exam Mistakes — What Goes Wrong
| Mistake | Consequence | How to Avoid |
|---|---|---|
| Use outer radius/diameter instead of INNER | Stress calculated is too low (unsafe design); thickness is undersized | ALWAYS read 'inner diameter'; if only 'outer' is given, subtract 2t to get inner D. |
| Forget to check t/r ≤ 1/10 | Apply thin-wall formula to a thick vessel; results are meaningless | Every problem: calculate t/r first. If > 0.1, stop and state 'thick-wall theory required.' |
| Swap hoop and longitudinal stress formulas | Wrong stress for design; vessel over- or under-sized | MEMORIZE: hoop has '2' in denom., longitudinal has '4'. Hoop is twice longitudinal. Hoop GOVERNS cylinder. |
| Use absolute (gauge + atm) pressure instead of gauge | Over-design (too thick wall, wasted material) or under-design (unsafe) | Problem gives 'internal pressure 1.5 MPa' → use 1.5 MPa as p in formulas (it IS gauge pressure). |
| Forget joint efficiency η when it's given | Design thickness is too small; vessel fails at seam at lower-than-expected pressure | If η is mentioned, include in formula: t = pD / (2·η·σ_allow). |
| Apply cylinder hoop formula to a sphere | Sphere stress is calculated 2× too high; over-design by factor of 2 | Sphere: σ = pD/(4t). Cylinder hoop: σ_h = pD/(2t). Sphere is the DENOMINATOR with 4, not 2. |
| Mix up D and r in the formula (e.g., use D in formula written for r) | Factor of 2 error (doubled or halved result) | Read the formula carefully: σ_h = pD/(2t) uses DIAMETER D. σ_h = pr/t uses RADIUS r. Convert if needed: r = D/2. |
| Not rounding wall thickness UP after calculation | Calculated t = 14.1 mm used as-is; actual wall is 14 mm and is slightly under-designed | Always round UP to the next practical size (10 mm, 12 mm, 15 mm, etc.) and RE-CHECK thin-wall criterion. |
Diagrams
Thin-Walled Pressure Vessels — Formula Map
Complete roadmap of thin-walled vessel formulas, showing the relationship between hoop and longitudinal stresses in cylinders, sphere efficiency, and design thickness. Use this to orient yourself before solving any exam problem.
Thin-Walled Vessel Problem-Solving Flowchart
Step-by-step decision tree for solving any thin-walled pressure vessel problem. Start by identifying vessel type and problem type, then apply the correct formula, verify thin-wall criterion, and handle joint efficiency. Use this to stay organized under exam pressure.