Thin-wall and Lamé thick-wall analysis for cylindrical and spherical vessels. Get hoop stress, axial stress, von Mises equivalent, ASME safety factor, and minimum wall thickness instantly.
The fundamental assumption for thin-wall cylinders is that stress is constant through the wall thickness. The primary stresses are the circumferential "hoop" stress and the longitudinal "axial" stress.
$$\sigma_\theta = \frac{P R}{t}, \quad \sigma_a = \frac{P R}{2t}$$Where $P$ is internal pressure, $R$ is the mean radius, and $t$ is wall thickness. Note: $\sigma_\theta$ is twice as large as $\sigma_a$, making it the governing stress for design.
For thick-wall cylinders (t/R ≥ 0.1), Lamé's equations describe how stress varies with radius $r$. The hoop stress is highest at the inner surface.
$$\sigma_\theta(r) = \frac{R_i^2 P}{R_o^2 - R_i^2}\left(1 + \frac{R_o^2}{r^2}\right)$$Where $R_i$ is inner radius, $R_o$ is outer radius, and $r$ is the radial coordinate ($R_i \le r \le R_o$). This gradient is critical for predicting where failure will initiate.
Industrial Boilers & Heat Exchangers: These are classic thin-wall pressure vessels. Engineers use hoop stress calculations to determine the minimum wall thickness required to safely contain high-pressure steam, ensuring routine inspections occur before stress levels approach the material's limits.
High-Pressure Gas Storage (CNG/LNG Tanks): Tanks for compressed natural gas or hydrogen operate at extremely high pressures (250+ bar). They often fall into the thick-wall regime, requiring Lamé analysis to accurately model the high stress concentration at the inner wall and to optimize material use.
Chemical Reactors: Vessels in chemical plants must withstand not only pressure but also corrosive environments and cyclic loading. Von Mises stress and safety factor calculations are combined with material selection to prevent catastrophic failure over the vessel's decades-long lifespan.
Aerospace Propellant Tanks: Rocket fuel tanks are lightweight, thin-wall structures where minimizing mass is critical. Engineers push safety factors as low as reasonably possible, making precise stress analysis non-negotiable. The axial stress formula is key for designing the tank's end caps and attachment points.
When you start using this tool, there are several pitfalls that beginners in CAE often fall into. First and foremost is the misconception that "if the calculation result is safe, it will absolutely not fail." This calculation is only a primary stress evaluation under the ideal condition of "static and uniform internal pressure." In practice, there are numerous factors you must consider, such as stress concentrations at pipe connections, fatigue from repeated pressure, thermal stress from temperature changes, and wall thickness reduction due to corrosion. For example, even with a safety factor of 3.0, stress can locally spike several times higher at a sharp nozzle fillet.
Secondly, the handling of the "thin-wall" and "thick-wall" boundary. The tool switches calculation formulas at a thickness ratio of t/R=0.1, but this is only a guideline. Actual design codes (like the ASME Boiler and Pressure Vessel Code) make judgments based on more detailed conditions. Since results can change significantly near this boundary (e.g., t/R=0.09 vs. 0.11), the key for designs in this region is to perform both calculations and evaluate them carefully.
Finally, material data input errors. Yield strength changes significantly with temperature. A steel with 450 MPa at room temperature might drop to 350 MPa at 300°C. When entering "Yield Strength" in the tool, be sure to use the value at the anticipated maximum operating temperature. Judging something as "safe!" based on a room-temperature value is a very dangerous misunderstanding.
Stress calculations for pressure vessels are connected to more engineering fields than you might think. The first that come to mind are "Strength of Materials" and "Elasticity Theory." The thin-wall theory and Lamé's equations that form the backbone of this tool are precisely applications of these disciplines. In particular, the stress distribution in a thick-walled cylinder is a classic example of an analytically solved problem in two-dimensional polar coordinates within elasticity theory.
Next is the connection to "Fracture Mechanics." Especially the inner wall of a thick-walled vessel, where maximum stress occurs, is prone to containing tiny defects from manufacturing. Fracture mechanics evaluates how such defects grow under repeated internal pressure loading, leading ultimately to failure. It's deeply linked not only to yield evaluations like von Mises stress but also to the concept of calculating the "stress intensity factor at a crack tip" to prevent brittle fracture.
Furthermore, in applied fields, did you know it's even connected to "Biomechanics"? Blood vessel walls, like those of aneurysms or the aorta, are biological "pressure vessels" that expand under internal blood pressure. When analyzing their wall stress distribution, models modified from Lamé's equations are sometimes used. It's fascinating how industrial design and life sciences are connected by the same mechanical principles.
If you're curious about the theory behind this tool, try taking the next step. First, I recommend trying to derive for yourself "why the hoop stress is twice the axial stress" by drawing a free-body diagram. By considering the force balance on a virtual cross-section cut through half the cylinder, the formula $$\sigma_\theta = \frac{P R}{t}$$ displayed by the tool can be naturally derived. This marks the big difference between "understanding" and "just using."
Next, challenge yourself with deriving Lamé's equations. This is a bit more advanced, but it's a standard exercise in elasticity theory that involves solving by combining the equilibrium equations for stress in polar coordinates, compatibility conditions, and Hooke's law. Getting a textbook (for example, an introductory book on "Elasticity") and working through this problem alone can dramatically improve your ability to interpret mechanics.
As a practical next topic, consider "Detailed Evaluation via Finite Element Analysis (FEA)." This tool only shows a one-dimensional (radial) distribution, but using FEA allows you to visualize the three-dimensional stress state in complex geometries like nozzles or heads. The modern standard design flow is to determine the overall wall thickness and approximate stress with this tool, and then check local stress concentrations with FEA. When you can view the tool's results not as "the final answer" but as "a tool to determine appropriate input conditions for detailed analysis," you're well on your way to becoming a solid mid-level engineer.