Thin-Walled Assumption

The walls of many boilers, pipes, and storage tanks are thin relative to their radius. If the thickness $t$ is less than about one-tenth the inner radius $r$, stresses across the wall are nearly uniform. This thin-walled approximation greatly simplifies the stress analysis, allowing designers to use straightforward formulas to ensure safe operation.

Stress in a Cylindrical Vessel

When a cylinder is pressurized, two principal stresses develop. The hoop stress acts circumferentially around the shell, while the longitudinal stress runs along its length. For internal pressure $P$, the hoop stress is given by

${\sigma }_h=\frac{Pr}{t}$

The longitudinal stress is half this value:

${\sigma }_l=\frac{Pr}{2t}$

These expressions reveal why thin walls amplify stress: smaller thickness values place more load on the material for a given pressure and radius. Engineers select wall thicknesses to keep the maximum stress well below the material’s yield strength, often incorporating generous safety factors.

Stress in a Spherical Vessel

A spherical tank experiences the same stress magnitude in all directions. The stress equals the longitudinal expression for a cylinder:

${\sigma }_s=\frac{Pr}{2t}$

Because the stress is uniformly distributed, spheres can hold pressure more efficiently than cylinders. However, they are more difficult and expensive to fabricate, so cylindrical vessels are common for industrial pipelines and storage tanks.

Using the Calculator

Enter the internal pressure in kilopascals, along with the inner radius and wall thickness in meters. Select either cylinder or sphere from the dropdown menu. Press Calculate Stress to obtain the resulting hoop and longitudinal stresses. Only hoop stress is reported for spheres because the stresses are equal in all directions. Use the copy button to place the results on your clipboard for design reports or documentation.

This calculator performs direct arithmetic in your browser. No data leaves your computer, so you can safely experiment with different scenarios or materials to see how the stresses change.

Example Material Table

The following table lists approximate yield strengths for common vessel materials. Designs typically apply a safety factor to ensure operating stress remains well below these values.

Material	Yield Strength (MPa)
Carbon Steel	250
Stainless Steel	300
Aluminum Alloy	200

Worked Example

Suppose a cylindrical vessel has an internal pressure of $400$ kPa, an inner radius of $0.5$ m, and a wall thickness of $0.01$ m. The hoop stress is

${\sigma }_h=\frac{4000.5}{0.01}=\mathrm{20 000}\text{kPa}$

The longitudinal stress is half that value, $\mathrm{10 000}$ kPa. These stresses must be compared with the material’s yield strength to ensure a safe factor of safety.

Cylinder vs Sphere Comparison

The table summarizes the stress relationships for each shape, assuming identical pressure, radius, and thickness.

Shape	Hoop Stress	Longitudinal Stress
Cylinder	$\frac{Pr}{t}$	$\frac{Pr}{2t}$
Sphere	$\frac{Pr}{2t}$

Spherical vessels distribute stress more evenly, allowing thinner walls for the same pressure, but their fabrication complexity often outweighs the material savings.

Importance of Proper Design

Pressure vessel failures can be catastrophic. Understanding the stresses involved is the first step toward safe construction. Engineers must also consider factors such as weld quality, corrosion allowance, material fatigue, and cyclic loading. Codes like the ASME Boiler and Pressure Vessel Code provide detailed guidelines to ensure vessels meet stringent safety standards. This calculator is an educational tool, not a substitute for professional engineering analysis, yet it highlights the fundamental relationships that govern design.

History and Applications

Pressurized containers have existed since steam engines powered the industrial revolution. Over time, improved metallurgy and welding techniques allowed vessels to withstand greater pressures. Today they appear in chemical plants, fuel storage depots, space vehicles, and even medical oxygen tanks. While the shapes and materials vary, the thin-walled stress equations remain a cornerstone of mechanical design.

Summary

By applying simple formulas to wall thickness, radius, and internal pressure, you can estimate the primary stresses in a vessel. Whether you are studying for an engineering exam or sketching out a preliminary design, this tool gives quick insight into how geometry and pressure interact. Always consult relevant codes and experienced engineers before committing to a design.

Limitations and Assumptions

The thin-walled formulas assume the wall thickness is less than one-tenth of the radius and that the pressure is uniformly distributed. They neglect weld efficiencies, stress concentrations at openings, and external loads such as thermal gradients or external pressure. For thicker shells or high-temperature service, more sophisticated analyses like finite element models are required.

Related Calculators

For further design insight, see the Pipeline Pressure Drop Calculator and the Deep Sea Pressure Hull Thickness Calculator.