Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels

$Learning Objectives$

Check whether the thin-wall approximation is appropriate for a vessel.
Derive and apply hoop and longitudinal membrane stresses in cylindrical vessels.
Apply the membrane-stress relation for spherical vessels.
Explain why cylinder hoop stress governs over longitudinal stress for equal geometry and pressure.
Account conceptually for joint efficiency without confusing it with wall-thickness geometry.

$Thin-Walled Pressure Vessel$

A pressure vessel is treated as thin-walled when the wall thickness is small enough relative to radius that membrane stress can be approximated as uniform through the thickness. A common introductory criterion is $t/r\le0.1$ .

$Thin-Wall Applicability$

When $t/r>0.1$ , through-thickness stress variation is no longer negligible and thin-wall membrane equations are not appropriate. Use a thick-cylinder formulation suitable for the problem instead.

Thin-Wall Vessel Stress Route

Check the introductory thin-wall limit, choose the vessel-specific membrane relation, and handle joint efficiency only when the adopted design method includes it.

Use t/r ≤ 0.1 as the introductory applicability gate. If the gate fails, leave the thin-wall equations and use an appropriate thick-wall method; for a thick cylinder, use a suitable thick-cylinder formulation such as Lamé analysis. For a thin cylinder, hoop stress applies for either end condition, while cap-induced longitudinal membrane stress applies only to closed ends. For a thin sphere, the tangential membrane stress is equal in every direction. Joint efficiency is a design-method assumption, not a geometry correction: apply it consistently to capacity or required thickness only when the adopted method specifies it.

Start → Record pressure, size, thickness, and vessel details; Record pressure, size, thickness, and vessel details → Does t/r ≤ 0.1?; Does t/r ≤ 0.1? — No (> 0.1) → Use an appropriate thick-wall method; Does t/r ≤ 0.1? — Yes (≤ 0.1) → Cylinder or sphere?; Use an appropriate thick-wall method → End; Cylinder or sphere? — Cylinder → Are the cylinder ends closed?; Cylinder or sphere? — Sphere → Calculate equal tangential stress; Are the cylinder ends closed? — Yes → Calculate hoop and longitudinal stress; Are the cylinder ends closed? — No → Calculate hoop stress only; Calculate hoop and longitudinal stress → Does the adopted design method use joint efficiency?; Calculate hoop stress only → Does the adopted design method use joint efficiency?; Calculate equal tangential stress → Does the adopted design method use joint efficiency?; Does the adopted design method use joint efficiency? — Yes → Apply η consistently to capacity or thickness; Does the adopted design method use joint efficiency? — No → Keep the method's unadjusted basis; Apply η consistently to capacity or thickness → End; Keep the method's unadjusted basis → End

Start: terminator
Record pressure, size, thickness, and vessel details: input output. For the applicability ratio, use the vessel radius r with wall thickness t. Record the geometry and, for a cylinder, whether the pressure-loaded ends are open or closed.
Does t/r ≤ 0.1?: decision. This is the stated introductory thin-wall criterion, not a design-code acceptance check.
Use an appropriate thick-wall method: process. Do not use the thin-wall membrane equations when t/r exceeds 0.1. For a thick cylinder, use an appropriate thick-cylinder formulation such as Lamé analysis.
Cylinder or sphere?: decision
Are the cylinder ends closed?: decision. The longitudinal membrane stress in this lesson arises from pressure acting on closed end caps.
Calculate hoop and longitudinal stress: process. For a closed-end thin cylinder, use σ_h = pD/(2t) and σ_l = pD/(4t). The longitudinal relation accounts for pressure on the closed end caps.
Calculate hoop stress only: process. Use σ_h = pD/(2t). Do not assign the closed-cap-induced σ_l = pD/(4t) to an open-ended vessel; any other axial stress requires its own stated source and model.
Calculate equal tangential stress: process. For a thin sphere, use σ_s = pD/(4t); the membrane stress is equal in every tangential direction.
Does the adopted design method use joint efficiency?: decision. Joint efficiency η is included only when the adopted design method calls for it; it is not a correction to vessel geometry.
Apply η consistently to capacity or thickness: process. Use the efficiency consistently with the adopted design model and allowable-capacity or required-thickness calculation.
Keep the method's unadjusted basis: process. Do not add a joint-efficiency adjustment when the adopted design method does not specify one, and never use η as a geometry correction.
End: terminator

Text-free formed cylindrical shell on simple supports with a clean cutaway exposing its curved wall. — Formed Thin Shell in Context
The formed shell shows the physical curved wall that is idealized as a membrane in the following equations; it carries no thickness or manufacturing claim.

$From Formed Shell to Membrane Model$

Observe the continuous curved wall and exposed interior surface before treating the vessel as a thin membrane. The raster is a qualitative physical-context aid: it supplies no thickness dimension, fabrication sequence, or design conclusion.

Cylinder Hoop Stress

Tangential membrane stress in a thin cylindrical pressure vessel.

\sigma_h=\frac{pD}{2t}

Cylinder Longitudinal Stress

Axial membrane stress in a closed-end thin cylindrical pressure vessel.

\sigma_l=\frac{pD}{4t}

$Cylinder Free-Body Interpretation$

Hoop stress resists the tendency of the cylinder to split along a longitudinal plane. Longitudinal stress resists the pressure force on the closed end caps. For the same $p$ , $D$ , and $t$ , the hoop stress is twice the longitudinal stress, so hoop stress usually controls a simple membrane-stress thickness check.

Text-free longitudinal cutaway of a cylindrical shell flowing continuously into a rounded end cap. — Closed End-Cap Junction
The sectional close-up shows the cylindrical shell flowing into a continuous rounded end cap, giving physical context for the closed-end load path behind longitudinal stress.

$Reading the End-Cap Closure$

Treat the cap-to-shell junction as the continuous boundary that closes the vessel. The raster shows geometry only—not pressure magnitude, free-body arrows, stress values, or a design detail.

Text-free cutaway of a closed cylindrical vessel showing a thin shell and an intact end cap. — Cylindrical Shell and End Cap
The cutaway makes the thin shell and closed-end geometry visible before the exact circumferential and longitudinal membrane-stress equations are applied.

$Reading the Vessel Cutaway$

Use the shell's circumferential direction and the closed end cap as the two physical free-body orientations behind the hoop- and longitudinal-stress formulas. The raster is a qualitative geometry aid: it supplies no pressure, dimension, stress value, or code threshold, so use the stated thin-wall check and the interactive model for exact interpretation.

Spherical Vessel Stress

Uniform tangential membrane stress in a thin spherical pressure vessel.

\sigma_s=\frac{pD}{4t}

$Why Spheres Are Efficient$

A sphere carries internal pressure through equal membrane stress in every tangential direction. For the same diameter, wall thickness, and pressure, its membrane stress equals the cylinder's longitudinal stress and is half the cylinder's hoop stress.

Text-free cutaway of a closed spherical metallic shell with a continuous curved wall. — Closed Spherical Vessel
The cutaway makes the sphere's continuous closed-shell geometry visible beside the membrane-stress relation; it supplies no stress value or sizing detail.

$Reading the Spherical Shell Geometry$

Observe how the shell curves continuously around the vessel rather than changing into a cylindrical wall. The raster supports the geometric idea of equal tangential directions, but it supplies no membrane-stress value, thickness criterion, or design detail.

$Interactive Exploration$

Vary pressure, diameter, wall thickness, and vessel type. Confirm the linear effect of pressure and diameter, the inverse effect of thickness, and the two-to-one cylinder hoop-to-longitudinal stress ratio.

Controls

2.0 MPa

1000 mm

10 mm

t/r

0.020

Within the lesson's t/r ≤ 0.1 criterion.

Hoop stress

100.0 MPa

σh = pD/(2t)

Longitudinal stress

50.0 MPa

σl = pD/(4t)

Hoop / longitudinal

2.00

For the ideal closed thin cylinder.

$Joint Efficiency$

Joint efficiency $\eta$ represents the reduction in effective strength associated with a seam or joint relative to the base material.

$Using Joint Efficiency$

When a design method includes joint efficiency, apply it consistently to the allowable capacity or required thickness according to the adopted design model. Do not silently mix an efficiency-adjusted formula with a separate unadjusted allowable-stress assumption.

Text-free close-up of a thin cylindrical shell with a continuous longitudinal seam and joined wall edges. — Continuous Shell Seam
The local section reveals a continuous longitudinal seam in the thin shell; joint efficiency remains an authored concept rather than a raster claim.

$Reading the Seam Detail$

Use the seam as a physical reminder that a shell is assembled from joined material surfaces. The raster shows qualitative joint continuity only; it gives no weld size, efficiency, leak condition, strength, or design conclusion.

Key Takeaways

Verify thin-wall applicability before using membrane equations.
Cylinder hoop stress is $pD/(2t)$ and is twice the longitudinal stress $pD/(4t)$ .
A thin sphere carries uniform tangential membrane stress $pD/(4t)$ .
Pressure and diameter increase membrane stress; greater wall thickness reduces it.
Joint efficiency modifies capacity assumptions and must be stated explicitly.