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.

Tangential (Hoop) 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).	-

Longitudinal Stress ( $\sigma_l$ )

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.

Pressure Vessel Stress Calculator

Internal Pressure (p)

2 MPa

Diameter (D)

2000 mm

Thickness (t)

10 mm

Hoop Stress (Tangential)

200.0 MPa

Governing Design Stress (Max)

Longitudinal Stress

100.0 MPa

Axial stress along the length

Hoop stress tries to split the cylinder (burst). Longitudinal stress tries to blow the ends off.

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.