Thin-Walled Pressure Vessels

Thin-Walled Pressure Vessels

$Learning Objectives$

Define thin-walled and thick-walled pressure vessels and their key assumptions.
Calculate tangential (hoop) stress and longitudinal stress in cylindrical vessels.
Determine the maximum shear stress in thin-walled vessels.
Calculate the stress in spherical pressure vessels.
Understand Lame's Equations for thick-walled cylinders and the concept of autofrettage.
Apply failure theories (Rankine, Tresca, Von Mises) to evaluate pressure vessel safety.

Pressure vessels are enclosed containers designed to hold gases or liquids at a pressure substantially different from the ambient pressure. A vessel is classified as "thin-walled" when its wall thickness ( $t$ ) is small compared to its inner radius ( $r$ ), generally when $t/r < 0.1$ (or $D/t > 20$ ). For such structures, we can reasonably assume that the internal stress distribution across the thickness of the wall is uniform, greatly simplifying the analysis.

Stresses in Thin-Walled Vessels

$Stresses in Cylindrical Vessels$

When a cylindrical tank is subjected to internal fluid pressure $p$ , two types of stresses are developed in the wall: tangential stress and longitudinal stress.

$\sigma_t$

This stress acts along the circumference of the cylinder. It tends to split the cylinder into two troughs.

Tangential (Hoop) Stress

Formula for calculating the tangential or hoop stress in a thin-walled cylindrical vessel.

\sigma_t = \frac{pD}{2t\eta}

Variables

Symbol	Description	Unit
$\sigma_t$	Tangential (hoop) stress.	-
$p$	Internal pressure.	Pa, psi
$D$	Inside diameter.	-
$t$	Wall thickness.	-
$\eta$	Joint Efficiency (a dimensionless factor $\le 1.0$ accounting for the strength reduction at welds or riveted joints. Use $\eta = 1.0$ for seamless pipes).	-

$\sigma_l$

This stress acts along the length of the cylinder. It tends to blow off the ends of the cylinder.

Longitudinal Stress

Formula for calculating the longitudinal stress in a thin-walled cylindrical vessel.

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

Variables

Symbol	Description	Unit
$\sigma_l$	Longitudinal stress.	-
$p$	Internal pressure.	Pa, psi
$D$	Inside diameter.	-
$t$	Wall thickness.	-
$\eta$	Joint Efficiency (a dimensionless factor $\le 1.0$ accounting for the strength reduction at welds or riveted joints. Use $\eta = 1.0$ for seamless pipes).	-

$Design Note$

The tangential stress is twice the longitudinal stress ( $\sigma_t = 2\sigma_l$ ). Therefore, design is typically governed by the tangential stress (Hoop Stress).

$Stresses in Spherical Vessels$

A spherical vessel is the most efficient shape for resisting internal pressure. Due to symmetry, the stress is the same in all tangential directions.

Stress in Spherical Vessels

Formula for calculating the uniform tangential stress in a thin-walled spherical vessel.

\sigma = \frac{pD}{4t\eta}

Variables

Symbol	Description	Unit
$\sigma$	Uniform tangential stress.	-
$p$	Internal pressure.	Pa, psi
$D$	Inside diameter.	-
$t$	Wall thickness.	-
$\eta$	Joint Efficiency (a dimensionless factor $\le 1.0$ accounting for the strength reduction at welds or riveted joints. Use $\eta = 1.0$ for seamless pipes).	-

$Comparison to Cylindrical Vessels$

Note that for the same diameter and pressure, the stress in a sphere is half the hoop stress in a cylinder.

Maximum Shear Stress in Thin-Walled Vessels

$In-Plane vs. Absolute Maximum Shear Stress$

In pressure vessels, determining the maximum shear stress ( $\tau_{\text{max}}$ ) is critical for materials that follow the Tresca failure criterion. Because the stress state at the surface is biaxial (with radial stress approximately zero), we must consider both in-plane and absolute maximum shear stress.

For cylindrical vessels, the absolute maximum shear stress occurs on an out-of-plane plane (rotated 45 degrees relative to the hoop and radial directions).
For spherical vessels, the absolute maximum shear stress is also out-of-plane.

Maximum Shear Stress (Cylinder)

Absolute maximum shear stress for a thin-walled cylindrical vessel.

\tau_{\text{max}} = \frac{\sigma_t}{2} = \frac{pD}{4t}

Variables

Symbol	Description	Unit
$\tau_{\text{max}}$	Absolute maximum shear stress.	-
$\sigma_t$	Tangential (hoop) stress.	-
$p$	Internal pressure.	Pa, psi
$D$	Inside diameter.	-
$t$	Wall thickness.	-

Maximum Shear Stress (Sphere)

Absolute maximum shear stress for a thin-walled spherical vessel.

\tau_{\text{max}} = \frac{\sigma}{2} = \frac{pD}{8t}

Variables

Symbol	Description	Unit
$\tau_{\text{max}}$	Absolute maximum shear stress.	-
$\sigma$	Uniform tangential stress.	-
$p$	Internal pressure.	Pa, psi
$D$	Inside diameter.	-
$t$	Wall thickness.	-

Thick-Walled Vessels and Failure

While thin-walled equations ( $\sigma_t = pr/t$ ) assume a uniform stress distribution across the thickness, thick-walled cylinders (where $t/r > 0.1$ ) cannot use this simplification.

$Lame's Equations (Thick-Walled Cylinders)$

For thick-walled cylinders under internal pressure $p_i$ and external pressure $p_o$ , the radial stress ( $\sigma_r$ ) and tangential stress ( $\sigma_t$ ) at a radius $r$ are given by Lame's Equations:

Lame's Equation: Radial Stress

Radial stress at radius r in a thick-walled cylinder.

\sigma_r = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} - \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2 (r_o^2 - r_i^2)}

Variables

Symbol	Description	Unit
$\sigma_r$	Radial stress at radius $r$ .	-
$p_i$	Internal pressure.	-
$p_o$	External pressure.	-
$r_i$	Inner radius.	-
$r_o$	Outer radius.	-
$r$	Radial position.	-

Lame's Equation: Tangential Stress

Tangential (hoop) stress at radius r in a thick-walled cylinder.

\sigma_t = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} + \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2 (r_o^2 - r_i^2)}

Variables

Symbol	Description	Unit
$\sigma_t$	Tangential stress at radius $r$ .	-
$p_i$	Internal pressure.	-
$p_o$	External pressure.	-
$r_i$	Inner radius.	-
$r_o$	Outer radius.	-
$r$	Radial position.	-

Notice that unlike thin-walled vessels, radial stress ( $\sigma_r$ ) is not assumed to be zero, and the maximum hoop stress occurs at the inner surface ( $r = r_i$ ).

$Error in Thin-Walled Assumption$

The assumption of uniform stress in thin-walled vessels implies a maximum error of roughly 5%. As the wall thickens, this error grows exponentially, necessitating Lame's thick-wall formulation for safety and material efficiency.

$Autofrettage$

Autofrettage (from French for "self-hooping") is a manufacturing technique used to increase the pressure capacity and fatigue life of thick-walled cylinders, particularly in high-pressure applications like cannon barrels, fuel injection systems, and deep-sea vessels.

According to Lame's Equations, the highest hoop stress always occurs at the inner wall ( $r_i$ ). If the internal pressure is increased until this inner layer yields plastically, while the outer layers remain elastic, a permanent deformation is created.

When the massive internal pressure is removed, the outer elastic layers try to shrink back to their original size, but they are resisted by the permanently expanded inner plastic layer. This creates a state of residual compressive stress at the inner wall and residual tensile stress at the outer wall.

In service, when the cylinder is pressurized again, the operational tensile hoop stress must first overcome the residual compressive stress at the inner wall before it can cause yielding or fatigue crack initiation. This dramatically increases the effective elastic limit and fatigue life of the vessel.

$Failure Theories and Yield Criteria$

When designing thin-walled pressure vessels, we analyze a biaxial state of stress (longitudinal and tangential stress). For ductile materials, we must ensure the stress state doesn't exceed yield thresholds using advanced failure criteria.

Maximum Principal Stress Theory (Rankine): Fails if maximum principal stress equals or exceeds yield stress ( $\sigma_{\text{max}} = \sigma_t \ge S_y$ ). Simplest, but not conservative for ductile materials.
Maximum Shear Stress Theory (Tresca): Fails if maximum shear stress equals or exceeds yield shear stress ( $\tau_{\text{max}} \ge S_{sy}$ ). Commonly used in pressure vessel codes (ASME Section VIII Div 1).
Von Mises Yield Criterion (Distortion Energy): Fails if distortion energy exceeds the yield threshold. Uses an effective equivalent stress ( $\sigma_{\text{vm}} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}$ ). Typically the most accurate for ductile metals like steel.

Interactive Tool: Pressure Vessel Calculator

$Interactive Simulation$

Compare the stresses in Cylindrical and Spherical vessels by adjusting the pressure and geometry below.

Controls

2.0 MPa

1000 mm

10 mm

Transverse cylinder section · wall at true scale

Inner 1000 mm · outer 1020 mm · arrows indicate pressure, not membrane stress.

Thickness / inner radius

0.020

Thin-wall hoop stress

100.000 MPa

Thin-wall axial stress

50.000 MPa

Thin-wall in-plane shear

25.000 MPa

Thin-wall absolute shear

50.000 MPa

Lamé inner hoop stress

101.010 MPa

Exact closed-end axial stress

49.505 MPa

Within the teaching t/r ≤ 0.1 approximation; stresses do not establish design adequacy.

$Membrane approximation versus thick-wall response$

Increase wall thickness until the thickness-to-inner-radius ratio exceeds 0.1. The tool withholds thin-wall stresses outside that approximation. For cylinders, compare the membrane estimate with Lamé's inner-wall hoop stress; pressure arrows in the section represent radial pressure, not tangential membrane stress.

Mechanics Visual Synthesis

Use these deterministic figures to connect the lesson equations to the corresponding free-body, section, transformation, or response geometry. Each visual addresses a separate mechanics question and is intentionally static so it can be compared directly with the interactive model above.

Thin cylindrical vessel membrane free bodies

Longitudinal and transverse cuts expose different pressure resultants, producing hoop stress twice the longitudinal membrane stress for a closed cylinder.

Spherical vessel membrane equilibrium

Spherical symmetry produces equal membrane stress in every tangent direction, with magnitude pD/4t under the thin-wall idealization.

Thin-wall validity versus through-thickness gradient

As t/r grows, the membrane approximation becomes less representative because radial and hoop stresses vary appreciably through the wall.

Pressure-vessel membrane stresses create coupled biaxial strain

A closed thin cylindrical vessel develops hoop and longitudinal membrane stresses simultaneously, so each normal strain includes direct elastic strain and Poisson coupling from the orthogonal stress.

Pressure Vessel Model Selection Workflow

Select a pressure-vessel stress model using the lesson's thin-wall approximation criterion and the vessel geometry.

This lesson uses t/r < 0.1 as a teaching screen for the thin-wall membrane approximation. For a thin cylinder, evaluate hoop and longitudinal membrane stresses; for a thin sphere, evaluate equal tangential membrane stresses. For a thick cylinder, use Lamé's radial and hoop stress distribution. A thick sphere requires a thick-wall spherical solution outside this lesson. After selecting the model, evaluate principal or equivalent stress using the applicable material/design criterion, and confirm pressure-vessel code limits and load cases for design.

Key Takeaways

Thin-Walled Assumption ( $t/r < 0.1$ ): Valid for most tanks and pipes. Stress is assumed uniform across the thickness.
Hoop Stress ( $\sigma_t = pD/2t\eta$ ): Always the governing stress in a cylinder (twice the longitudinal stress). It acts to split the pipe along its length.
Longitudinal Stress ( $\sigma_l = pD/4t\eta$ ): Acts to separate the ends.
Spherical Stress ( $\sigma = pD/4t\eta$ ): The most efficient shape; stress is half that of a cylinder's hoop stress.
Joint Efficiency ( $\eta$ ): Accounts for the reduced strength of welded or riveted joints.
Thick-Walled vs. Thin-Walled: Thin-walled assumes uniform stress; thick-walled assumes varying stress (Lame's Equations). In thick-walled vessels, the highest stresses occur at the inner surface.
Autofrettage: A technique where a thick-walled cylinder is intentionally yielded to create beneficial residual compressive stresses at the inner wall, drastically increasing its pressure capacity and fatigue life.
Failure Theories: Use Von Mises or Tresca yield criteria to accurately predict failure under biaxial stress states in ductile materials.

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$Learning Objectives$

Stresses in Thin-Walled Vessels

$Stresses in Cylindrical Vessels$

$\sigma_t$

Tangential (Hoop) Stress

$\sigma_l$

Longitudinal Stress

$Design Note$

$Stresses in Spherical Vessels$

Stress in Spherical Vessels

$Comparison to Cylindrical Vessels$

Maximum Shear Stress in Thin-Walled Vessels

$In-Plane vs. Absolute Maximum Shear Stress$

Maximum Shear Stress (Cylinder)

Maximum Shear Stress (Sphere)

Thick-Walled Vessels and Failure

$Lame's Equations (Thick-Walled Cylinders)$

Lame's Equation: Radial Stress

Lame's Equation: Tangential Stress

$Error in Thin-Walled Assumption$

$Autofrettage$

$Failure Theories and Yield Criteria$

Interactive Tool: Pressure Vessel Calculator

$Interactive Simulation$

Pressure Vessel: Thin-Wall Limit

Controls

Internal gauge pressure

Inner diameter

Wall thickness

Vessel geometry

$Membrane approximation versus thick-wall response$

Mechanics Visual Synthesis

Thin cylindrical vessel membrane free bodies

Spherical vessel membrane equilibrium

Thin-wall validity versus through-thickness gradient

Pressure-vessel membrane stresses create coupled biaxial strain

Thin-Walled Pressure Vessels

Learning Objectives

Stresses in Thin-Walled Vessels

Stresses in Cylindrical Vessels

Tangential (Hoop) Stress (σt\sigma_tσt​)

Tangential (Hoop) Stress

Longitudinal Stress (σl\sigma_lσl​)

Longitudinal Stress

Design Note

Stresses in Spherical Vessels

Stress in Spherical Vessels

Comparison to Cylindrical Vessels

Maximum Shear Stress in Thin-Walled Vessels

In-Plane vs. Absolute Maximum Shear Stress

Maximum Shear Stress (Cylinder)

Maximum Shear Stress (Sphere)

Thick-Walled Vessels and Failure

Lame's Equations (Thick-Walled Cylinders)

Lame's Equation: Radial Stress

Lame's Equation: Tangential Stress

Error in Thin-Walled Assumption

Autofrettage

Failure Theories and Yield Criteria

Interactive Tool: Pressure Vessel Calculator

Interactive Simulation

Controls

Membrane approximation versus thick-wall response

Mechanics Visual Synthesis

Thin cylindrical vessel membrane free bodies

Spherical vessel membrane equilibrium

Thin-wall validity versus through-thickness gradient

Pressure-vessel membrane stresses create coupled biaxial strain

$Learning Objectives$

$Stresses in Cylindrical Vessels$

$\sigma_t$

$\sigma_l$

$Design Note$

$Stresses in Spherical Vessels$

$Comparison to Cylindrical Vessels$

$In-Plane vs. Absolute Maximum Shear Stress$

$Lame's Equations (Thick-Walled Cylinders)$

$Error in Thin-Walled Assumption$

$Autofrettage$

$Failure Theories and Yield Criteria$

$Interactive Simulation$

$Membrane approximation versus thick-wall response$