A pressure vessel stress analysis example is most useful when it connects the quick equations engineers use at concept stage with the finite element model required to make a defensible design decision. The membrane stresses in a cylindrical shell are easy to calculate. The engineering judgment begins where geometry changes, loads combine, and code requirements define what must be evaluated.
Consider a horizontal process vessel with a cylindrical shell, welded heads, and a radial nozzle. This is a simplified example, not a substitute for a completed ASME design-by-rule or design-by-analysis calculation. Its purpose is to show how hand calculations and Nastran-based FEA should work together.
Pressure Vessel Stress Analysis Example: Start With Membrane Stress
Assume the vessel has an internal diameter of 48 in., a nominal shell thickness of 0.50 in., and an internal design pressure of 150 psi. The inside radius is therefore 24 in. For a first-pass thin-wall calculation, the radius-to-thickness ratio is 48. Since this is comfortably above the usual thin-wall screening threshold, membrane theory is appropriate for estimating the general shell response away from discontinuities.
For a closed-end cylindrical vessel, the hoop stress is:
`σh = pR / t`
Using the stated dimensions:
`σh = 150 psi × 24 in. / 0.50 in. = 7,200 psi`
The longitudinal stress from pressure acting on the closed ends is:
`σL = pR / 2t`
`σL = 150 psi × 24 in. / (2 × 0.50 in.) = 3,600 psi`
These values establish the baseline. The hoop stress is the governing pressure membrane stress in the cylindrical shell, while the longitudinal stress becomes particularly relevant when it combines with bending from vessel supports, piping loads, or thermal restraint.
If the shell includes a corrosion allowance, the stress calculation should use the effective thickness at the required condition, not the nominal plate thickness. For example, if 0.06 in. is reserved for corrosion, the effective thickness is 0.44 in. The pressure hoop stress rises to approximately 8,180 psi. That change is not academic. It can affect allowable-stress margin, fatigue performance, and the interpretation of local FEA results.
What the Hand Calculation Does Not Capture
The shell equations assume a uniform cylinder with pressure as the only load. They do not represent the localized stress field at a nozzle opening, the shell-to-head transition, a saddle support, or a welded attachment. They also do not address stress concentration, local bending, through-thickness gradients, or nonlinear contact behavior.
A hand calculation remains valuable because it gives the FEA analyst an expected answer. In the cylindrical shell far from openings and supports, an FEA model should recover approximately 7,200 psi hoop stress and 3,600 psi longitudinal stress for the nominal-thickness case. If it does not, the analyst should investigate units, pressure application, material definition, constraints, mesh quality, or result orientation before evaluating peak stresses.
Moving From the Equation to an FEA Model
For this vessel, a global shell-element model is usually the efficient first model. Shell elements represent the primary vessel geometry well when the thickness is small relative to radius and when the objective is to evaluate overall membrane and bending response. The model should include the shell, heads, nozzle necks, reinforcement pads if present, saddles, and meaningful attachment geometry.
Internal pressure must be applied consistently to all wetted surfaces. In a shell model, element normals must be checked carefully because incorrect normal direction can reverse the applied pressure. The model also needs realistic support representation. Fully fixing every node on a saddle may make the model stable, but it can create artificial restraint and nonphysical bending stresses.
A better approach depends on the actual support design. One saddle may restrain axial motion while the other permits axial thermal growth. Contact or distributed bearing behavior may be needed where the vessel rests on saddles. If the support details are still preliminary, analysts should document the assumed degrees of freedom and run sensitivity cases. A result that changes materially with a modest boundary-condition adjustment is not yet a mature design result.
Load Cases Must Reflect the Service Condition
Pressure alone rarely controls every region of a vessel. A practical analysis matrix may include design pressure, hydrotest pressure, operating temperature, dead weight, insulation weight, liquid contents, wind, seismic acceleration, and piping loads at nozzles. Thermal gradients deserve particular attention in thick heads, jacketed vessels, startup conditions, and systems with restrained attachments.
For a nozzle, a pressure-only model may show acceptable membrane stress while the combined pressure-plus-piping case produces a significant local bending increase at the nozzle-to-shell junction. Conversely, a highly refined model may show a sharp stress peak at a weld toe that is not relevant to the same acceptance criterion as a gross plastic-collapse check. The load case and the acceptance method must be aligned.
Use the Right Level of Model Detail
A global model identifies overall load paths and highlights areas requiring scrutiny. It is not always the right model for a nozzle intersection or a local weld detail. At those locations, a submodel or refined local solid model can provide better resolution of stress gradients and through-thickness behavior.
The transition from shell elements to solids should be deliberate. The local model needs displacement boundary conditions transferred from the global model at a sufficient distance from the discontinuity. Applying arbitrary fixed constraints close to the nozzle can distort the very stress field under investigation. The objective is to preserve the global deformation pattern while resolving the local geometry.
Mesh refinement should follow stress gradients rather than visual preference. A coarse mesh may underpredict local bending at a nozzle or knuckle. An excessively fine mesh at a sharp geometric corner may generate a mathematically singular peak that continues to rise as element size decreases. Neither result is automatically meaningful.
A useful verification exercise is to compare membrane stress from the shell model with the hand result in a smooth shell region, then perform a mesh-convergence study at the discontinuity of interest. Convergence should be judged on a defined reporting quantity, such as membrane-plus-bending stress linearized through a section, reaction force, or a code-relevant stress measure. Reporting the single highest nodal stress without context is not validation.
Interpreting Stress Results for Design Decisions
Stress contour plots are diagnostic tools, not acceptance criteria by themselves. The analyst must distinguish general membrane stress, local membrane stress, bending stress, and peak stress. These categories have different physical meanings and may have different allowable limits under the governing code methodology.
For the baseline cylinder, the FEA result should show a nearly uniform hoop membrane stress away from discontinuities. Near a nozzle, the stress field will be multiaxial. Principal stresses can help visualize the load path, but stress linearization is often required when a design-by-analysis assessment calls for membrane and bending components across a defined stress classification line.
The equivalent von Mises stress can be useful for screening ductile material response, yet it should not replace code-specific evaluation. ASME Section VIII requirements, material allowable stresses, weld efficiency, fatigue provisions, and the selected design route govern the actual acceptance process. The applicable code edition, vessel category, and service conditions matter. A pressure vessel intended for cyclic service requires a different level of attention than a vessel with limited pressure and temperature cycles.
Common Failure Modes in Otherwise Good Models
Several modeling errors appear repeatedly in pressure-vessel work. The first is comparing a local singular peak directly with an allowable stress. The second is using nominal thickness in one part of the analysis and corroded thickness in another. The third is representing piping loads as isolated forces without the associated moments or without confirming the load coordinate system.
Another common issue is overconstraint. A vessel can appear unusually stiff, show elevated support stresses, and still pass a superficial solver check because the constraints satisfy equilibrium. Reactions should be reviewed against hand expectations, and deformed-shape plots should make physical sense. For a pressure-loaded closed vessel, pressure thrust, support reactions, and imposed loads must balance.
Material behavior also deserves review. Elastic isotropic properties may be adequate for a linear screening analysis, but temperature-dependent modulus, yield behavior, creep considerations, and nonlinear contact can become necessary depending on service. The appropriate fidelity is driven by the decision at stake, not by a preference for either the simplest or most elaborate model.
A Defensible Analysis Delivers More Than a Color Plot
The strongest pressure-vessel analysis package traces a clear path from requirements to conclusions. It identifies the design condition, material basis, effective thickness, governing code approach, load combinations, assumptions, mesh strategy, validation checks, and acceptance criteria. It also records limitations, especially where nozzle loads, thermal transients, support contact, or fatigue data remain uncertain.
For engineering teams, that traceability reduces more than calculation risk. It supports design reviews, accelerates changes when operating conditions evolve, and makes physical test results easier to reconcile with simulation. Experienced FEA support is most valuable when it challenges assumptions early, before a detailed model gives false confidence. A well-validated vessel model should leave the team with a clear next action: refine a local detail, adjust geometry, confirm a load, or proceed with justified confidence.