Pressure Vessels: Thin-Walled Cylinders

Understanding how cylindrical structures like tanks and pipes handle internal pressure is fundamental to engineering design, ensuring safety and efficiency in industries from energy to manufacturing. This analysis focuses on thin-walled cylindrical pressure vessels, where simplified stress calculations prevent overdesign and catastrophic failure. You will learn to derive and apply the key stress equations that govern everything from household water heaters to industrial reactors.

What Makes a Vessel "Thin-Walled"?

A pressure vessel is any closed container designed to hold gases or liquids at a pressure substantially different from ambient pressure. For analysis, we classify cylindrical vessels as thin-walled when the wall thickness $t$ is small compared to the inner radius $r$. Specifically, the thin-wall assumption applies when the radius-to-thickness ratio $r/t$ exceeds ten. This assumption allows you to neglect the variation of stress through the wall thickness, simplifying the analysis to a state of plane stress. Think of a soda can: its metal skin is incredibly thin relative to its diameter, so the stresses are essentially uniform across the thickness. This approximation is valid for most pipes, storage tanks, and boilers, enabling straightforward hand calculations that form the basis of initial design.

Deriving Hoop Stress: The Circumferential Force

Hoop stress (or circumferential stress) is the tensile stress acting tangentially around the circumference of the cylinder. It arises because internal pressure pushes outward against the cylindrical wall, trying to "split" the vessel along its length. To derive it, consider a free-body diagram of a half-cylinder segment of length $L$. The internal pressure $P$ acts on the projected area $2rL$, producing a bursting force. This force is resisted by the tensile force in the two walls, each with area $tL$. Balancing these forces gives the hoop stress formula.

The force due to pressure is $P\times (2rL)$. The resisting force from the wall material is $2\times ({\sigma }_h\times tL)$, where ${\sigma }_h$ is the hoop stress. Setting them equal: $$P\cdot (2rL)=2({\sigma }_h\cdot tL)$$ Solving for ${\sigma }_h$: $${\sigma }_h=\frac{Pr}{t}$$ This is the primary equation for hoop stress. For example, in a pipe with radius 0.5 meters, thickness 0.01 meters, and internal pressure 1 MPa (about 10 atmospheres), the hoop stress is ${\sigma }_h=(1\times 10^6\text{Pa})\times (0.5\text{m})/(0.01\text{m})=50\text{MPa}$. This stress must be less than the material's yield strength for safe operation.

Determining Axial Stress: The Longitudinal Force

Axial stress (or longitudinal stress) acts parallel to the cylinder's axis. It is caused by the pressure acting on the vessel's end caps, which tends to stretch the cylinder lengthwise. Imagine capping both ends of a tube and pressurizing it—the tube wants to elongate. To find this stress, consider a free-body diagram cut perpendicular to the axis. The pressure force on the end cap area $\pi r^2$ is balanced by the axial stress distributed over the circular cross-sectional area of the wall, which is approximately $2\pi rt$ for thin walls.

The force from pressure is $P\times (\pi r^2)$. The resisting force from the wall is ${\sigma }_a\times (2\pi rt)$, where ${\sigma }_a$ is the axial stress. Force balance gives: $$P\cdot (\pi r^2)={\sigma }_a\cdot (2\pi rt)$$ Solving for ${\sigma }_a$: $${\sigma }_a=\frac{Pr}{2t}$$ Notice that axial stress is exactly half the hoop stress: ${\sigma }_a={\sigma }_h/2$. In our previous example, the axial stress would be $50\text{MPa}/2=25\text{MPa}$. This relationship is crucial because it immediately tells you which stress is more critical.

Stress Comparison and Design Implications

The derivation shows that hoop stress is twice as large as axial stress in a thin-walled cylinder under internal pressure. Therefore, hoop stress is the larger principal stress and governs the design. When selecting materials or determining wall thickness, you must ensure that the hoop stress does not exceed the allowable stress, which incorporates factors of safety. Design codes often use the hoop stress equation in a rearranged form to solve for required thickness: $t=Pr/{\sigma }_{\text{allow}}$.

In a state of plane stress, the radial stress through the wall is negligible for thin-walled vessels, as it varies from $-P$ at the inner surface to zero at the outer surface. Since $P$ is typically much smaller than ${\sigma }_h$ and ${\sigma }_a$, we ignore it. This simplifies the Mohr's circle analysis for the cylinder, where hoop and axial stresses represent two principal stresses. For ductile materials, failure theories like Tresca or von Mises are applied using these stresses. Remember, these formulas assume uniform internal pressure and a cylindrical shape without discontinuities like joints or nozzles, which introduce stress concentrations.

Worked Example and Application Limits

Let's walk through a complete design check for a propane storage tank. Suppose the tank has an inner diameter of 1.2 m, a wall thickness of 6 mm, and must withstand an internal pressure of 2 MPa. The material's yield strength is 250 MPa, with a safety factor of 2.5 required.

1. Calculate radius and check thin-wall assumption: Radius $r=1.2/2=0.6\text{m}$. Thickness $t=0.006\text{m}$. Ratio $r/t=0.6/0.006=100>10$, so thin-wall analysis is valid.
2. Compute hoop stress: ${\sigma }_h=Pr/t=(2\times 10^6\text{Pa})\times (0.6\text{m})/(0.006\text{m})=200\text{MPa}$.
3. Compute axial stress: ${\sigma }_a=Pr/(2t)=200\text{MPa}/2=100\text{MPa}$.
4. Check against allowable stress: Allowable stress ${\sigma }_{\text{allow}}=250\text{MPa}/2.5=100\text{MPa}$. Comparing, ${\sigma }_h=200\text{MPa}>100\text{MPa}$, so the hoop stress exceeds the allowable. The design fails; you would need to increase thickness or use a stronger material.

This example highlights practical application. The thin-wall formulas are essential for preliminary sizing, but their limits must be respected. They become inaccurate for thick-walled vessels ($r/t<10$), where stress varies nonlinearly through the wall, requiring Lame's equations. They also don't account for dynamic loads, corrosion, or temperature effects, which require additional analysis.

Common Pitfalls

1. Misapplying the thin-wall assumption: Using these stress equations for vessels with $r/t\le 10$ leads to significant underestimation of true stresses. Always calculate the ratio first. For instance, a high-pressure hydraulic cylinder might have a thick wall, necessitating a thick-walled analysis.
2. Confusing stress directions: Mixing up hoop and axial stresses can lead to incorrect failure predictions. Remember, hoop stress acts circumferentially and is larger. A visual cue: if a cylindrical vessel bursts, it typically splits along a longitudinal line, indicating hoop stress failure.
3. Ignoring unit consistency: Pressure, radius, and thickness must be in consistent units (e.g., Pascals, meters) to yield stress in Pascals. A common error is using pressure in psi, radius in inches, and thickness in inches without proper conversion, resulting in meaningless numbers.
4. Overlooking end conditions: The axial stress formula assumes pressure acts on closed ends. For vessels with open ends or different end caps, the axial stress may be zero or differently calculated. For example, a pipe capped at one end and connected to a system at the other might have a net axial force from pressure only on one cap.

Summary

• Hoop stress ${\sigma }_h=Pr/t$ is the circumferential tensile stress in a thin-walled cylinder and is the dominant stress governing design.
• Axial stress ${\sigma }_a=Pr/(2t)$ is the longitudinal stress, exactly half the hoop stress under internal pressure.
• The thin-wall assumption is valid when the radius-to-thickness ratio $r/t>10$, simplifying stress analysis to a uniform state.
• In design, always check that the calculated hoop stress is below the material's allowable stress, incorporating safety factors.
• These formulas are foundational for analyzing cylindrical tanks and pipes but require careful attention to units, geometry, and assumption limits.