Hand Calculations, FEA & Hierarchy of Evidence

Technical provenance

Applicable standards / specifications

• EASA CS-25 — Certification Specifications and Acceptable Means of Compliance for Large Aeroplanes — Example airworthiness framework only; use the applicable certification basis for the product.

References

• EASA Easy Access Rules for Large Aeroplanes (CS-25) — Example aerospace certification framework. The actual certification basis must be established for each product and project.
• MMPDS — Metallic Materials Properties Development and Standardization — Common aerospace source for statistically based metallic material allowables where contractually applicable.

What Is It?

Structural substantiation draws on multiple types of evidence: hand calculations, finite element analysis and physical test. Each type has strengths and limitations. No single type is sufficient for all structures and all requirements. The hierarchy of evidence describes how these types complement one another — how a hand calculation cross-checks an FEA result, how a test validates an analysis method, how a full-scale test provides the strongest evidence for primary structure. Understanding the hierarchy and the complementary roles of each evidence type is essential for building a defensible substantiation package. This article addresses the relationship between hand calculations, FEA and test, and how they combine to form a hierarchy of evidence.

FEA SHOULD EXTEND ENGINEERING UNDERSTANDING — NOT REPLACE IT.

Why It Matters

A substantiation package that relies on a single type of evidence is fragile. An FEA-only package may contain model errors that no one has caught. A hand-calc-only package may miss local effects that the hand calc cannot capture. A test-only package may not address all load cases or all failure modes. A robust package combines multiple types of evidence, each cross-checking the others. The hand calc verifies the FEA; the FEA extends the hand calc to complex geometry; the test validates the analysis method for the real structure. The hierarchy of evidence is the principle that guides how these types are combined to produce a defensible argument.

• A single type of evidence is fragile — it can contain undetected errors.
• Multiple types of evidence cross-check each other, increasing confidence.
• Hand calc verifies FEA; FEA extends hand calc; test validates the method.
• The hierarchy guides the combination of evidence types into a defensible argument.

Hand Calculations: The Foundation

Hand calculations are the foundation of the evidence hierarchy. They are based on classical mechanics, are transparent and can be followed by any engineer. They provide conservative results for simple geometry and are the primary cross-check for all numerical methods. A hand calculation that produces a positive margin for a simple feature may be sufficient evidence on its own. For complex features, the hand calculation provides a baseline against which the FEA result is checked — if the FEA result is an order of magnitude different from the hand calc, something is wrong. The hand calculation is also the tool for preliminary sizing, for checking reaction forces and for verifying that the FEA loads are in the right ballpark.

• Hand calculations are transparent, based on classical mechanics and verifiable by any engineer.
• They provide conservative results for simple geometry.
• They are the primary cross-check for FEA — if the FEA disagrees by an order of magnitude, investigate.
• They are the tool for preliminary sizing and for checking reaction forces.
• For simple features, a hand calc with a positive margin may be sufficient evidence alone.

FEA: Extending Analysis Capability

Finite element analysis extends analysis capability to complex geometry, stress concentrations and realistic load distributions that hand calculations cannot capture. FEA is not a replacement for engineering understanding — it is a tool that extends it. The engineer must understand the physics of the problem before building the model: what is the load path, what is the expected stress distribution, where are the stress concentrations, what is the governing failure mode. Without this understanding, the FEA model is a black box and its results are not interpretable. The engineer must also verify the model — mesh convergence, boundary conditions, material properties, equilibrium — and cross-check the results against hand calculations. FEA that is not understood and not verified is not evidence.

FEA SHOULD EXTEND ENGINEERING UNDERSTANDING — NOT REPLACE IT. Build the model with a clear understanding of the physics: load path, stress distribution, stress concentrations, governing failure mode. Verify the model. Cross-check the results. An FEA model that is not understood is a black box — its results are not interpretable and not evidence.

The Hand Calculation–FEA Cross-Check

The cross-check between hand calculations and FEA is the most important verification step in structural analysis. The hand calculation provides an independent estimate of the result; the FEA provides a detailed result. The two should be in the same ballpark — within a factor of perhaps 1.5 to 2 for a first check, closer if the hand calc is refined. If the FEA result is an order of magnitude different from the hand calc, the model is likely wrong — a boundary condition, a material property, a load introduction or a mesh issue. The cross-check is not a formality — it is the primary defence against model errors. An FEA result that has not been cross-checked against a hand calc is not verified.

Hand calc – FEA cross-check:

  Hand calculation (independent estimate)
        │                        │
        │   same ballpark?        │
        ▼                        ▼
  ┌────────────────────────────────────┐
  │  Compare: FEA result vs hand calc  │
  └────────────────────────────────────┘
        │
        ├── Within ~1.5–2× ──► Plausible. Investigate remaining difference.
        │
        ├── Order of magnitude off ──► Model likely WRONG.
        │   Check: BCs, material, loads, mesh, units.
        │
        └── Wrong sign ──► Model definitely WRONG.
            Check: load direction, convention, BCs.

The cross-check is the primary defence against model errors.

Test: The Strongest Evidence

Physical test provides the strongest evidence in the hierarchy. A test on the actual structure, or on a representative sub-element, demonstrates that the structure carries the load — not that the model predicts it, but that the structure actually does it. Test evidence is essential for primary structure, for novel configurations and for failure modes that analysis cannot confidently predict. But test evidence is not a substitute for analysis — the test must be designed to represent the load, the boundary condition and the failure mode, and the test result must be correlated with the analysis. A test that does not represent the actual condition is not valid evidence. A test that is not correlated with analysis is a data point without context.

The Hierarchy of Evidence

The hierarchy of evidence describes the relative strength of different evidence types. At the bottom is precedent — similarity to a previously certified design. Above that is hand calculation — conservative, simple, verifiable. Above that is verified FEA — detailed, capturing geometry. Above that is coupon test — material data. Above that is sub-element test — feature-level validation. Above that is component test — assembly-level validation. At the top is full-scale test — the strongest, most representative evidence. The hierarchy is not a prescription — not every structure requires full-scale test. The requirement, the criticality and the novelty determine how high up the hierarchy the evidence must go.

Hierarchy of evidence (strongest at top):

  ┌─────────────────────────┐
  │ Full-scale test          │  ◄── strongest; most representative
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ Component test           │  ◄── major assembly, realistic loads
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ Sub-element test         │  ◄── detail design feature
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ Coupon test              │  ◄── material allowables
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ FEA — verified + cross-checked│  ◄── complex geometry
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ Hand calculation         │  ◄── conservative, simple, verifiable
  └─────────────────────────┘
  ┌─────────────────────────┐
  │ Precedent / similarity   │  ◄── weakest; requires documented similarity
  └─────────────────────────┘

Not every structure needs the top. The requirement and criticality govern how high to go.

Combining Evidence Types

A defensible substantiation package combines multiple evidence types. For primary structure, the package might include: a hand calculation for the global load distribution and for each simple feature; a verified FEA model for the stress concentrations and complex geometry; coupon test data for the material allowables; a sub-element test for the critical joint; and a full-scale test for the complete structure. Each type of evidence supports the others: the hand calc cross-checks the FEA; the FEA extends the hand calc; the coupon test provides the allowables used in both; the sub-element test validates the joint analysis; the full-scale test validates the complete analysis method. The combination is stronger than any single type alone.

• A defensible package combines multiple evidence types, each cross-checking the others.
• Hand calc + FEA: the hand calc cross-checks the FEA; the FEA extends the hand calc.
• Analysis + test: the test validates the analysis method for the real structure.
• Coupon + sub-element + full-scale: each level of test supports the next.
• The combination is stronger than any single type alone.

When Analysis Alone Is Sufficient

Not every structure requires test evidence. For secondary structure with well-established analysis methods, conservative assumptions and a comfortable margin, analysis alone may be sufficient. The governing requirement determines whether test is needed — some requirements mandate test for certain structure types; others accept analysis. The engineer must check the requirement and must not assume that test is or is not required based on previous programmes. When analysis alone is used, the analysis must be verified — hand calc cross-checks, equilibrium, mesh convergence — and the margins must be comfortable. Analysis alone with marginal margins is not defensible; analysis alone with comfortable, verified margins may be.

CONSIDERATION: Analysis alone may be sufficient for secondary structure with well-established methods, conservative assumptions and comfortable margins. But the governing requirement determines whether test is needed — check the requirement, do not assume. And analysis alone with marginal margins is not defensible.

When Test Is Required

Test is required when the analysis method is not confident — for novel configurations, new materials, complex failure modes or when the authority mandates it. Test is also required when the margins are marginal — a small positive margin from an analysis that has not been validated by test is a fragile basis for acceptance. For primary structure, full-scale test is often required by the certification basis. The engineer must identify early whether test is required and must plan the test programme to provide the evidence needed. A late realisation that test is required — after the analysis is complete and the design is frozen — is a programme risk.

• Test is required when the analysis method is not confident (novel configuration, new material, complex failure mode).
• Test is required when the margins are marginal and unvalidated.
• Test is often required for primary structure by the certification basis.
• Identify the test requirement early and plan the test programme accordingly.
• A late realisation that test is required is a programme risk.

The Complete Evidence Package

The complete evidence package for a primary structure typically includes: the compliance matrix showing every requirement and its status; the analysis reports (hand calcs and FEA) with verification evidence; the material allowables with traceability; the test reports (coupon, sub-element, full-scale) with correlation to analysis; the configuration control records linking evidence to design; and the substantiation report presenting the argument from requirement to conclusion. The package is not a collection of documents — it is a structured, traceable set of records that an independent reviewer can follow from requirement to conclusion. The strength of the package lies not in the volume of evidence but in the completeness, traceability and cross-checking of the evidence.

VERIFICATION: The complete evidence package is not measured by volume but by completeness, traceability and cross-checking. A thin package with a clear argument, verified analysis and correlated test is stronger than a thick package of disconnected reports. The reviewer must be able to follow the chain from requirement to conclusion.

Key Takeaways

• No single evidence type is sufficient for all structures — combine hand calc, FEA and test.
• Hand calculations are the foundation: transparent, verifiable, the primary cross-check for FEA.
• FEA extends analysis capability but must be verified and cross-checked — it does not replace understanding.
• Test provides the strongest evidence but must be designed to represent the actual condition and correlated with analysis.
• The hierarchy of evidence guides how high up the chain the evidence must go — the requirement and criticality govern.
• A defensible package combines multiple types, each cross-checking the others, with complete traceability.