Thick-Walled Pressure Vessel Analysis

When designing containers for high internal pressure, engineers cannot rely on simplified formulas forever. Thick-walled pressure vessels, where the wall thickness is a significant fraction of the vessel's radius, require a more sophisticated analysis to ensure safety and reliability. This is critical for applications like hydraulic actuators, industrial reactors, and firearm barrels, where failure can be catastrophic. Understanding the precise stress distribution through the wall is the key to preventing such failures and optimizing material use.

The Limits of Thin-Wall Theory and the Thick-Wall Threshold

In introductory mechanics, you often use the thin-wall pressure vessel equations. These assume the wall stress is uniformly distributed because the wall is very thin compared to the vessel's radius. This simplification leads to the familiar formula for hoop (circumferential) stress: ${\sigma }_{hoop}=Pr/t$, where $P$ is internal pressure, $r$ is the mean radius, and $t$ is the wall thickness.

However, this assumption breaks down as the wall gets thicker. The standard rule of thumb states that thin-wall theory becomes inadequate and inaccurate when the radius-to-thickness ratio ($r/t$) is less than 10. For a cylinder, this is often evaluated using the ratio of the inner radius to the wall thickness. When this ratio falls below 10, the stress can no longer be considered constant across the wall thickness. Instead, it varies significantly, with the highest stress concentration occurring at the innermost surface. This is the domain of thick-walled vessel analysis, governed by the Lamé equations.

Stress Distribution: The Lamé Equations

For a long, cylindrical thick-walled vessel with closed ends, subject to internal pressure $P_i$ and external pressure $P_o$, the stress state is biaxial. The two principal stresses of interest are the hoop stress (${\sigma }_{\theta }$) and the radial stress (${\sigma }_r$). The axial stress (${\sigma }_z$) also exists but is often constant for a closed cylinder. The brilliant contribution of Gabriel Lamé was to provide exact equations for these stresses as functions of the radial coordinate $r$.

The general Lamé equations for radial and hoop stress at any radius $r$ are:

$${\sigma }_r=\frac{P_i{r}_i^2-P_o{r}_o^2}{r_o^2-r_i^2}-\frac{(P_i-P_o)r_i^2{r}_o^2}{(r_o^2-r_i^2)r^2}$$

$${\sigma }_{\theta }=\frac{P_i{r}_i^2-P_o{r}_o^2}{r_o^2-r_i^2}+\frac{(P_i-P_o)r_i^2{r}_o^2}{(r_o^2-r_i^2)r^2}$$

Where $r_i$ is the inner radius and $r_o$ is the outer radius. These equations reveal that both stresses vary hyperbolically with $1/r^2$. For the common case of internal pressure only ($P_o=0$), the equations simplify to:

$${\sigma }_r=\frac{P_i{r}_i^2}{r_o^2-r_i^2}\left(1-\frac{r_o^2}{r^2}\right)$$

$${\sigma }_{\theta }=\frac{P_i{r}_i^2}{r_o^2-r_i^2}\left(1+\frac{r_o^2}{r^2}\right)$$

Key Insights from the Stress Profiles

Plotting these simplified equations reveals crucial design insights. The radial stress (${\sigma }_r$) is always compressive. It starts at a value equal to $-P_i$ at the inner surface (balancing the applied internal pressure), and decreases to zero at the free outer surface. The hoop stress (${\sigma }_{\theta }$), which is tensile, is not uniform. It has its maximum value at the inner surface and its minimum value at the outer surface.

You can calculate these exact values by substituting $r=r_i$ and $r=r_o$ into the equations. For internal pressure only:

• Inner Surface ($r=r_i$):

${\sigma }_{\theta ,max}=P_i\frac{r_o^2+r_i^2}{r_o^2-r_i^2}$ and ${\sigma }_r=-P_i$

• Outer Surface ($r=r_o$):

${\sigma }_{\theta ,min}=P_i\frac{2r_i^2}{r_o^2-r_i^2}$ and ${\sigma }_r=0$

The fact that the maximum hoop stress is at the inner surface explains why failures (cracks) in thick-walled cylinders under internal pressure typically initiate from the inside and propagate outward. This stress gradient means the material at the outer wall is under-utilized, a key consideration in advanced design techniques like autofrettage, which pre-stresses the inner wall in compression to better distribute the operational load.

Application to Design and Failure Prediction

The Lamé equations are not merely academic; they are the foundation for designing high-pressure equipment. Engineers use them to:

1. Size hydraulic cylinders and accumulators: Ensuring the wall is thick enough to keep the maximum von Mises or Tresca stress below the material's yield strength, with a safety factor.
2. Design gun barrels and launchers: Withstanding the immense, transient internal pressure from propellant gases.
3. Analyze reactor vessels and pipelines: Particularly in chemical, petroleum, and nuclear industries where pressures and temperatures are extreme.

The design process involves selecting materials, inner/outer radii, and verifying that stresses remain safe under all loading conditions. Failure prediction often uses a failure theory (like Maximum Shear Stress-Tresca or Distortion Energy-von Mises) applied to the stress state at the critical inner radius. For a ductile material under internal pressure, the Tresca criterion is commonly used, where the maximum shear stress is half the difference between the maximum and minimum principal stresses at a point: ${\tau }_{max}=({\sigma }_{\theta }-{\sigma }_r)/2$. This is maximum at the inner wall.

Common Pitfalls

1. Misapplying Thin-Wall Equations: The most frequent error is using ${\sigma }_{hoop}=Pr/t$ for a vessel with an $r/t<10$. This will significantly underestimate the true maximum stress at the inner wall, leading to an under-designed and dangerous vessel. Always check the radius-to-thickness ratio first.
2. Ignoring the Stress Gradient: Assuming stress is constant and selecting material based on an average stress value. This overlooks the critical stress concentration at the inner surface, which is the most likely origin of failure.
3. Incorrect Boundary Condition Application: Forgetting that the Lamé equations in their common form assume a state of plane strain (long cylinder) or plane stress (disk). Applying the closed-end cylinder formulas to an open-ended tube (like a sleeve) requires a different treatment of the axial stress. Always match the equations to the physical constraints of your problem.
4. Sign Convention Errors: Radial stress is compressive (negative) under internal pressure. Mixing up signs when calculating principal stresses or applying failure criteria can lead to incorrect safety factors. Maintain a consistent sign convention (tension positive, compression negative) throughout the analysis.

Summary

• Thick-walled pressure vessel analysis is required when the inner radius-to-thickness ratio is less than 10, rendering thin-wall assumptions invalid and unsafe.
• The exact Lamé equations describe how hoop and radial stress distributions vary hyperbolically through the wall thickness, with the maximum hoop stress occurring at the inner surface.
• This non-uniform stress profile means material is used inefficiently; the outer layers carry less load, which informs advanced manufacturing processes.
• These equations are essential for the safe and efficient design of hydraulic cylinders, gun barrels, and reactor vessels, where high internal pressures are present.
• Design and failure analysis must apply an appropriate failure theory (e.g., Tresca) to the stress state at the critical inner radius, not to an average stress value.