Introduction
Finite element analysis (FEA) is the primary computational tool for evaluating stress, deformation, ratchetting and fatigue in pressure vessels where closed-form equations not feasible. ASME Boiler & Pressure Vessel Code Section VIII Division 2 Part 5 — the Design by Analysis (DBA) path within Division 2 — requires detailed stress analysis for vessels with complex geometry, combined loading, or operating conditions that exceed the assumptions underlying the simpler design-by-rule formulas of Division 1. This article explains what FEA is, how it is applied to pressure vessel analysis, and where it fits within the ASME Section VIII DBA framework.
Part 5 calls for stress analysis rather than FEA by name, but in practice FEA is how that analysis is done. The same Part 5 methods also apply to Division 1 vessels through U-2(g) and Mandatory Appendix 46. This article explains what FEA is, how it is applied to pressure vessel analysis, and where it fits within the Section VIII DBA framework.
What Is Finite Element Analysis?
FEA is a numerical method for solving structural and thermal problems. The method divides a complex geometry into a large number of small, discrete elements, called the finite element mesh. Each element is governed by simplified equations relating stress, strain and displacement. The solver assembles these element equations into a global system and solves simultaneously for displacements at every node in the mesh. Stresses and strains are then calculated from the displacement results.
For pressure vessel work, the relevant analysis types are:
- Linear elastic static analysis: The most common type. It assumes material behavior following Hooke’s law. Stresses are linearized through the wall thickness, classified, and checked against Code limits under design by analysis.
- Elastic-plastic analysis: Accounts for material yielding. Part 5 permits it as an alternative to elastic analysis for plastic collapse, local failure, buckling and ratcheting evaluations. It is often used where elastic methods are overly conservative or where stress classification is ambiguous.
- Thermal analysis: Calculates temperature distributions through the vessel wall under steady-state and transient conditions. The results feed into thermal stress analysis.
- Fatigue analysis: Uses FEA stress or strain results as input to the fatigue evaluation in Part 5. It is critical for vessels subject to cyclic pressure, temperature or mechanical loading. The 2025 edition organizes the fatigue methods as Method A (elastic), Method B (elastic-plastic) and Method C (elastic-plastic with welded structural stress curves).
- Buckling analysis: Evaluates stability under compressive loading, vacuum or external pressure. It can be done as a linear (eigenvalue) analysis with code design factors, or as a nonlinear collapse analysis that includes geometric imperfections.
Why FEA Is Needed for Pressure Vessels
Division 1 design by rule covers standard vessel configurations, such as cylindrical shells, formed heads and nozzle reinforcement, using closed-form equations based largely on membrane shell theory. The rules don’t calculate local stresses at geometric discontinuities. Instead, the design margins and the empirically based rules, such as area-replacement nozzle reinforcement, account for them implicitly under static pressure loading.
That approach works for the configurations and loads the rules were written for. It breaks down when a vessel has features or loads outside that scope: external nozzle loads from piping, saddle supports and local attachments such as lugs and clips, non-standard openings and transitions, thermal gradients, and combined mechanical and pressure loading. At these locations, peak stresses can be several times the nominal membrane stress. Those peaks don’t govern plastic collapse, but they do govern fatigue life and local failure, and the design-by-rule formulas don’t evaluate them.
FEA resolves the actual stress state strain at these locations. When the Division 2, Part 4 design-by-rule methods don’t cover a configuration or load, or when a fatigue evaluation is required, Part 5 requires stress analysis to demonstrate protection against each applicable failure mode.
FEA within the ASME Section VIII Division 2 design-by-analysis framework
ASME Section VIII, Division 2, Part 5 requires protection against each applicable failure mode, each addressed by specific analysis methods:
- Protection against plastic collapse: Evaluated by elastic stress analysis, which linearizes and classifies stresses (Pm, PL, Pb) and compares them to allowable limits, or by elastic-plastic analysis using the limit-load or elastic-plastic collapse method.
- Protection against local failure: Evaluated by elastic analysis, which limits the sum of the three principal stresses, or by elastic-plastic analysis, which confirms that local strain limits are not exceeded under triaxial stress at concentrations.
- Protection against collapse from buckling: Evaluated by linear (eigenvalue) buckling analysis or by nonlinear elastic-plastic collapse analysis that includes geometric imperfections, each with the code’s design factors.
- Protection against fatigue failure: Evaluated by Method A (elastic equivalent stress range), Method B (elastic-plastic equivalent strain range) or Method C, new in the 2025 edition, which follows Method B but uses the welded structural stress fatigue curves.Â
- Protection against ratcheting: Required whenever thermal or mechanical cycling acts on top of a sustained primary load. The elastic method limits the range of primary-plus-secondary stress (PL + Pb + Q). The elastic-plastic method simulates repeated load cycles to determine whether the component shakes down.
- Protection against creep damage (new in 2025): Applies to vessels operating in the creep range, where allowable stresses are governed by time-dependent properties in addition to yield and tensile strength. The rules address creep rupture, creep buckling, creep ratcheting and creep-fatigue. For creep-fatigue, the 2025 edition provides screening methods only.
FEA is the tool that produces the stress and strain results these methods require. The analyst builds the model, applies loads and boundary conditions, runs the analysis, linearizes stresses along stress classification lines where elastic methods are used, and then evaluates the results against the Part 5 acceptance criteria.
ASME Section VIII Division 2 — Part 5
Stress classification in design by analysis
Uniform stress through the full wall thickness from pressure or mechanical loads — governs minimum wall thickness.
Elevated membrane stress at a structural discontinuity such as a nozzle or support that decays within a local zone.
Linearly varying stress from applied bending moments — does not self-limit and must be controlled to prevent collapse.
Self-limiting stress from structural restraint or thermal gradients — evaluated for ratcheting, not plastic collapse.
Highest local stress at a notch or surface discontinuity.
Modeling considerations for ASME-compliant FEA
The accuracy of an FEA result depends heavily on modeling decisions made before the analysis runs. Key considerations for pressure vessel analysis include:
- Element type. Axisymmetric elements are appropriate for vessels with axisymmetric geometry and loading. They significantly reduce model size and computation time without losing accuracy. Three-dimensional solid or shell elements are required when geometry or loading breaks axial symmetry, such as at nozzles under piping loads and saddle supports or non-radial penetrations. Shell elements report membrane and bending stress directly but cannot resolve through-thickness variations at welds, thick sections or nozzle-to-shell junctions. Those regions generally need solid elements.
- Mesh density. Stress results at discontinuities are sensitive to mesh refinement. Areas of geometric complexity, such as nozzle-to-shell junctions, head knuckle regions and fillet welds, require finer meshes than the general shell wall, with enough elements through the thickness to support stress linearization. Mesh convergence studies should confirm that results are not mesh-dependent, particularly where they will be used for fatigue evaluation. At sharp re-entrant corners, the peak stress is a singularity that never converges. Linearized membrane and bending stresses do converge, which is one reason the code evaluates linearized values.
- Boundary conditions. Improper boundary conditions introduce artificial constraints that distort results. Boundary conditions must reflect actual support conditions: free thermal expansion where it occurs in service, and support stiffness representative of the actual saddles, skirts or legs. Model cut boundaries and symmetry planes should be placed far enough from the region of interest that local discontinuity stresses have decayed before reaching them.
- Load cases. Part 5 specifies the load case combinations to evaluate, built from the loads in the user’s design specification: internal and external pressure, dead weight, wind, seismic, thermal and other applied loads. The combinations and load factors differ by failure mode. Elastic-plastic collapse checks use factored loads. Thermal loads enter the ratcheting and fatigue evaluations rather than the plastic collapse check, because thermal stress is secondary. Load cases for fatigue must reflect the actual cyclic operating history, not just design extremes, and test conditions such as the hydrostatic test should be evaluated as well.
- Material properties. Linear elastic analysis uses the elastic modulus and Poisson’s ratio at the relevant temperature. Elastic-plastic analysis uses either an elastic-perfectly plastic model, for limit-load analysis, or the Code’s true stress-strain curve at temperature, for elastic-plastic collapse and strain-limit checks. Thermal analysis needs temperature-dependent thermal conductivity, plus specific heat and density for transient cases, along with realistic film coefficients. Thermal stress analysis then needs the temperature-dependent coefficient of thermal expansion. Properties should come from Section II, Part D.Â
When to use design by analysis vs. design by rule
Two separate decisions are often confused here: which division of Section VIII to build to, and which design method to use. Design by rule is the normal choice for standard vessel configurations with well-understood stress distributions, modest cyclic loading, and geometries covered by the Code’s design formulas. It is simpler, faster, and sufficient for the majority of industrial pressure vessels in either division. Design by analysis, using FEA under Division 2, Part 5, is the appropriate and often necessary path when:
- Vessel geometry is complex or non-standard and falls outside the scope of the design-by-rule formulas.
- The vessel is subject to significant cyclic loading and fracture that requires a formal fatigue evaluation. Division 1 has no explicit fatigue rules, so the Part 5 methods are applied through Mandatory Appendix 46.
- Thermal gradients produce significant secondary stresses that must be quantified.
- External loads from piping, wind or seismic events must be evaluated in combination with pressure beyond what the design-by-rule methods cover.
Design by analysis applies to Division 1 vessels through U-2(g) and Mandatory Appendix 46 as well as to Division 2 vessels, so choosing FEA does not mean choosing Division 2. The choice of division is a separate decision. Division 2 permits higher allowable stresses because it uses a lower design margin on tensile strength, in exchange for more rigorous requirements for materials, fabrication, examination and documentation, including a user’s design specification.
FEA is also used outside new design. Fitness-for-service assessments of in-service equipment under API 579-1/ASME FFS-1 often require similar stress analysis methods.
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· Author: Bill O’Donnell   · Last reviewed by Bill O’Donnell, Sr., PE October 2026
This article is based on the 2025 Edition of the ASME Boiler & Pressure Vessel Code. Always consult the relevant ASME Boiler & Pressure Vessel Code edition for complete design requirements and analysis procedures.
FEA at O’Donnell Consulting Engineers
O’Donnell Consulting Engineers performs finite element analysis on pressure vessels, heat exchangers, piping systems and related components for clients in various industries. The firm’s FEA work spans linear elastic stress classification, elastic-plastic collapse analysis, thermal stress analysis, fatigue evaluation and buckling assessment. Design-by-analysis work follows ASME Section VIII, Division 2, Part 5 — for Division 2 vessels directly, and for Division 1 vessels through U-2(g) and Mandatory Appendix 46. Dr. William J. O’Donnell co-developed fatigue design procedures now codified in the ASME B&PV Code. Since 1972, our engineers have been performing thermal, stress, vibration, fatigue and failure analysis – as well as ratcheting and cumulative strain evaluations of equipment.- Service: ASME Design and Analysis for Pressure Vessels and Piping Systems Design and analysis services to ASME Section VIII and the B31 piping codes.
- Article: Design by Analysis vs. Design by Rule
How the two design methods compare and when each is appropriate. - Article: Overview of ASME Design By Analysis
The Part 5 failure modes and analysis methods in more detail, updated for the 2025 edition. - Article: Introduction to ASME Design Approval Process
How code compliance is documented and certified.
