# Linear Static 3D Euler Beam Physics Evaluation Report

## Metadata

- feature_id: `linear-static-3d-euler-beam`
- model_id: `cantilever-beam-b33`
- evaluated_head: `d76d052456ec134a98bcd5aa3b3c18a6b0ad6ba4`
- source_reference_verification_report: `docs/reference-verifications/linear-static-3d-euler-beam-reference-verification.md`
- source_reference_model: `docs/reference-models/linear-static-3d-euler-beam-reference-models.md`
- source_requirement: `docs/requirements/linear-static-3d-euler-beam.md`
- source_formulation: `docs/formulations/3d-isoparametric-euler-beam-formulation.md`
- source_numerical_review: `docs/numerical-reviews/linear-static-3d-euler-beam-review.md`
- source_io_definition: `docs/io-definitions/linear-static-3d-euler-beam-io.md`
- status: `pass-for-release-agent`
- owner_agent: `physics-evaluation-agent`
- date: `2026-08-09`

This verdict means that the documented physical checks pass and the Release Agent may audit
release readiness. It does not approve release readiness and does not re-evaluate the Step 26
reference tolerance decision.

## Input Evidence

The reference-verification prerequisite has status `pass-for-physics-evaluation`. Its checked
artifact identity was reproduced before the physics checks and again after the targeted CTest.

| evidence | exact path or identity | status | notes |
| --- | --- | --- | --- |
| reference verification | `docs/reference-verifications/linear-static-3d-euler-beam-reference-verification.md` | pass-for-physics-evaluation | Required gate satisfied. |
| solver HDF5 | `.harness/build/reference/cantilever-beam-b33/results.h5` | present and readable | 25,336 bytes; post-acceptance-run SHA-256 `13ECCF68262C14BBDE0F63667C0F10896ACD40EFEC56E8C9121C298333FD9B6D`. |
| comparison evidence | `.harness/build/reference/cantilever-beam-b33/comparison.json` | present and passing | 128,118 bytes; SHA-256 `258347AEA791D981AEA9B2BCAD85DE5344D4859ECA3692DC5E7AA01A848F8E0D`; `passed=true`, 176 rows, 16 metrics. |
| reference input | `reference/cantilever beam/cantilever beam.inp` | exact read-only artifact | SHA-256 `E406EA9560321B791DB829E03BD24593B9875E0195D35B86BD931EDA122EF3`; `TYPE=B33`. |
| reference displacement | `reference/cantilever beam/cantilever beam displacements.csv` | exact read-only artifact | SHA-256 `7B3312FBC8848E81D9A0FD4FF2B56BC1954636A2C14B5C1CBB269CB9477D3C31`. |
| reference reaction | `reference/cantilever beam/cantilever beam reactions.csv` | exact read-only artifact | SHA-256 `BF30CDB0CD50106885DE14D63492737736C587426EBD787DE4F7EE6AA86DAA23`. |
| reference elemental force | `reference/cantilever beam/cantilever beam elemental forces.csv` | exact read-only artifact | SHA-256 `E5E77FEC0FA9482AE018DBF296E74D396335C7C711BD2E9AA2315247A34290BA`. |
| solver CSV views | N/A | not used | No derived FESA CSV was generated or used. |
| stress reference CSV | N/A by approved contract | not applicable | `S11` is evaluated from HDF5 schema, formulation, and analytical/unit evidence only. |
| targeted physics tests | `.harness/build`, MSVC x64 Debug | pass | Exact Step 27 CTest passed 17/17: EulerBeam3D 10, ResultRecovery 6, B33ReferenceComparison 1. |

Read-only HDF5 inspection confirmed schema `0`, solver `0.1.0`, feature identity, `Step-1` frame
`0`, formulation `B33-3D-Euler-Bernoulli`, source content identity
`fnv1a64:04543464cc970405`, and coordinate convention
`global-cartesian; beam-local=(t,n1,t-cross-n1)`. The file contains 11 nodes at
`(x,y,z)=(0..10,0,0)` m and 10 consecutive one-metre elements. Every element has identity local
axes, so local `(x,y,z)` equals global `(X,Y,Z)` for this model.

The documented physical model is a 10 m cantilever with node 1 fixed in all six DOFs and a
free-end global/local `FZ=-1.0e6 N` at node 11. The section and material are SI:
`E=2.1e11 Pa`, `Iy=I11=0.0833333 m^4`, and the only expected deformation mode is local-`z`
translation with bending about local `y`.

## Physics Checks

All normalized physics criteria below come from the approved `1e-10` residual/equilibrium,
`1e-12` matrix/transform/end-sign, or `1e-9` analytical contracts. They are independent checks of
physical meaning, not a second application of the Abaqus row tolerance.

### 1. global equilibrium

In global Cartesian coordinates, force equilibrium is

`r_F = sum(F_applied) + sum(R)`,

with criterion

`||r_F|| / max(||sum(F_applied)||, ||sum(R)||) <= 1e-10`.

The comparison physics evidence records `sum(F_applied)=[0,0,-1000000] N` and
`sum(R)=[0,0,999999.9999998808] N`. Therefore
`r_F=[0,0,-1.1920928955078125e-7] N`, its norm is
`1.1920928955078125e-7 N`, and the normalized value is
`1.1920928955078125e-13`. Verdict: **pass**.

Moment equilibrium about the global origin is

`r_M = sum(M_applied) + sum(X cross F_applied) + sum(M_reaction) + sum(X cross R_force)`.

The free-end force gives `[0,1.0e7,0] N*m`; the complete HDF5 reaction field gives
`[0,-9999999.999997258,0] N*m`. Thus `r_M=[0,2.7418136596679688e-6,0] N*m`,
with normalized norm `2.7418136596679688e-13` against `1.0e7 N*m`. Verdict: **pass**.

### 2. reaction consistency and true free residual

The constrained reaction contract is `R_c=(K*d-F)_c`. At fixed node 1 the observed global row is
`[RF1,RF2,RF3,RM1,RM2,RM3] = [0,0,1000000.0000008196,0,-10000000.000005051,0]`
in `[N,N,N,N*m,N*m,N*m]`. It opposes the applied `-Z` force and balances its positive origin
moment. Differences from the physical closed-form reactions are `8.195638656616211e-7 N` and
`5.0514936447143555e-6 N*m`, normalized to `8.195638656616211e-13` and
`5.0514936447143555e-13`. Verdict: **pass**.

The reaction dataset intentionally preserves free residuals. The implementation uses

`rho_f = ||(K*d-F)_f||_2 / max(||K*d||_f, ||F||_f)`

with no artificial unit floor. The serialized true free residual norm is
`9.356339321107032e-7 N-equivalent`; the physical free scale is `1.0e6 N`, so
`rho_f=9.356339321107032e-13 <= 1e-10`. The largest observed free force-residual component is
`5.9604644775390625e-7 N`, and the largest free moment-residual component is
`2.0861625671386719e-7 N*m`. Verdict: **pass**.

### 3. displacement direction and rotation sign

For free-end local `Pz=-1.0e6 N`, the documented Euler-Bernoulli solution is

`w(L)=Pz*L^3/(3*E*Iy)`,  `theta_y(L)=-Pz*L^2/(2*E*Iy)`.

The expected values are `-0.0190476266666697 m` and `+0.00285714400000046 rad`.
HDF5 gives tip `UZ=-0.019047626666677083 m` and `URY=+0.0028571440000013902 rad`, with relative
errors `3.87e-13` and `3.27e-13`, below the analytical `1e-9` criterion. All non-root `UZ` values
are negative and monotonically increase in magnitude toward the loaded tip; all non-root `URY`
values are positive, as required by `theta_y=-dw/dx`. Verdict: **pass**.

### 4. expected zero and uncoupled symmetry

The fixed-root displacement row is exactly zero in all six components. Across all nodes,
`UX`, `UY`, `URX`, and `URZ` are exactly zero. The only nonzero kinematic components are `UZ` and
`URY`, and the only physical constrained reactions are `RF3` and `RM2`. In the element recovery,
`epsilon0`, `kappa_x`, and `kappa_z`, and the corresponding `N`, `T`, and `Mz`, are exactly zero.
This is the documented uncoupled local-`z` bending symmetry, with no axial, torsional, or
cross-plane leakage. Verdict: **pass**.

### 5. element force, adjacent endpoints, and boundary balance

The HDF5 `end_force_local` rows are outward endpoint actions in
`[FX,FY,FZ,MX,MY,MZ]`; `section_resultant` rows are positive-local-`x` section cuts in
`[N,T,My,Mz]`. With no distributed load,

`F_X=n*N`, `M_X=n*T`, `M_Y=n*My`, `M_Z=n*Mz`,
`F_Y=-n*dMz/dx`, and `F_Z=n*dMy/dx`, where `n=-1` at `xi=-1` and `n=+1` at `xi=+1`.

Observed section `My` is positive and decreases linearly from
`10000000.000005048 N*m` at the root to `2.4286118949223834e-7 N*m` at the free end. The maximum
positive-face `My` mismatch between adjacent unloaded endpoints is
`2.73110345005989e-7 N*m`, normalized to `2.73110345005989e-14` against the model moment scale.
The comparison ledger independently records `endpoint_consistency_passed=true` without averaging.

Adjacent outward actions cancel. The maximum interior `FZ_right+FZ_left` magnitude is
`5.364418029785156e-7 N` (`5.364418029785156e-13` normalized); the maximum interior
`MY_right+MY_left` magnitude is `2.682209014892578e-7 N*m`
(`2.682209014892578e-14` normalized). Both satisfy the documented end-sign/residual criteria.

At the root, the first element action is exactly the constrained reaction evidence:
`FZ=+1000000.0000008196 N`, `MY=-10000000.000005051 N*m`. At the free boundary, the last element
has `FZ=-999999.9999998808 N` and `MY=-5.9604644775390625e-8 N*m`, balancing the applied end force
and the zero applied end moment to normalized residual scale. Verdict: **pass**.

### 6. local/global mapping and section-force signs

Every stored local-axis matrix is the identity. Therefore the global `-Z` load is local
`Pz=-1.0e6 N`, `UZ=w<0`, `URY=theta_y>0`, and the positive-face section resultant is
`My=-Pz*(L-x)>0`. The observed outward signs are `FZ>0, MY<0` at left endpoints and
`FZ<0, MY>0` at right endpoints, except for the physically zero free-end moment residue. These
values satisfy the documented `theta_y=-w'`, `My=-E*Iy*w''`, outward-normal, and
positive-face-section-cut conventions. Verdict: **pass**.

### 7. stress location, unit, and sign sanity

The stress contract is

`S11(xi,y,z)=E*(epsilon0 + z*kappa_y - y*kappa_z)`,

where `x1=y`, `x2=z`, the coordinate system is beam local, the unit is `force/length^2`, and the
location is a section point at each of two Gauss points. The reference input has no section
points, so HDF5 correctly contains 20 ordered `fesa-default` centroid rows
`(x1,x2)=(0,0)`, one at each Gauss point of ten elements. This model has pure bending with
`epsilon0=0`; consequently all 20 observed centroid `S11` values are exactly `0 Pa`.

Nonzero location/sign evidence comes only from the approved analytical/unit portfolio, not from
an Abaqus stress comparison. `EulerBeam3D.RecoversSectionPointAndDefaultCentroidS11` passed with
`epsilon0=0.01`, `kappa_y=0.02 1/m`, `kappa_z=-0.03 1/m`, `E=2.1e11 Pa`: the formula gives
`1.575e9 Pa` at `(y,z)=(0.25,-0.5)`, `8.4e8 Pa` at `(-0.4,0.3)`, and `2.1e9 Pa` at the default
centroid. The test enforces the formula at both Gauss points with normalized `1e-12` evidence.
Abaqus beam stress comparison remains explicitly N/A. Verdict: **pass**.

### 8. nonfinite, rigid-body, abnormal-magnitude, and energy symptoms

All mandatory numeric HDF5 rows inspected here are finite; `comparison.json` also records no
nonfinite row among the 176 compared rows and the 20 stress rows are finite. The fixed root is
exactly zero, factorization/solution completed, the normalized free residual is `9.36e-13`, and
the displacement field is smooth, so there is no rigid-body-mode symptom.

For this one-load linear case, the recoverable strain energy is
`U=0.5*F^T*d=9523.81333333854 N*m`, which is finite and positive. The ratios
`|UZ_tip|/L=0.00190476266666771` and `|URY_tip|=0.00285714400000139 rad` agree with the analytical
solution and show no abnormal magnitude relative to the documented small-displacement/rotation
model. The targeted rank/energy test also passed the six-rigid-mode, rank-six, and positive
deformation-energy checks. Verdict: **pass**.

### 9. model coverage

The approved B33 bundle is one identity-axis local-`z` bending cantilever. It directly covers the
end-to-end parser/solver/HDF5 path, global equilibrium, reaction sign, `UZ/URY`, `My/FZ`, endpoint
continuity, and the centroid stress fallback. It does not by itself cover axial, torsion, local-`y`
bending, rotated space, nonzero fiber stress, prescribed displacement, or the formulation-only
line-load kernel. The targeted analytical/unit portfolio supplies the documented complementary
coverage:

| coverage | targeted passing evidence | criterion |
| --- | --- | --- |
| axial, torsion, both bending planes | `EulerBeam3D.AnalyticalAxialTorsionAndTwoPlaneBendingRecover` and `ResultRecovery.MatchesAxialTorsionAndTwoPlaneEndSigns` | analytical relative `1e-9`; signed recovery contract |
| rotated local/global mapping | `EulerBeam3D.RotatedTransformPreservesWorkAndEnergy` | transform/work/energy normalized `1e-12` |
| constant local line-load kernel | `EulerBeam3D.ConstantLineLoadMatchesAllSignedComponents` | all 12 signed components normalized `1e-12`; `*DLOAD` remains outside CLI scope |
| rigid modes, rank, and energy | `EulerBeam3D.HasSixRigidModesRankSixAndPositiveDeformationEnergy` | rigid residual `1e-10`, rank six, positive deformation energy |
| prescribed displacement and residual | `ResultRecovery.ComputesResidualReactionForNonzeroPrescription`, `ResultRecovery.EnforcesNormalizedFreeResidual` | partition/reaction and normalized residual `1e-10` |
| result identity and continuity | `ResultRecovery.KeepsEndActionSectionAndGaussResultsDistinct`, `ResultRecovery.RequiresInteriorEndpointConsistencyWithoutAveraging` | distinct locations and no-average consistency |
| S11 location/sign/default | `EulerBeam3D.RecoversSectionPointAndDefaultCentroidS11`, `ResultRecovery.OrdersStressPointsAndDefaultCentroid` | formula/schema normalized `1e-12` |

The exact acceptance command passed all 17 selected tests. The single reference model plus this
analytical portfolio covers every documented physical expectation without attributing unsupported
coverage to the legacy CSV bundle. Verdict: **pass**.

## Failure Classification

- classification: `N/A`
- primary_failure: `N/A`
- evidence: all documented physics checks passed; no equilibrium, reaction, displacement,
  symmetry, element-force, stress-location, rigid-body, nonfinite, coverage, contract, or
  environment failure was found
- correction_handoff: `N/A`

## Evaluation Verdict

- verdict: `pass-for-release-agent`
- reason: the exact reference gate and artifact identity are valid; force and origin-moment
  equilibrium, constrained reaction consistency, true free residual, deformation signs, expected
  zeros, element force balance, local/global and section-force signs, S11 schema/analytical sanity,
  finite/energy/mode checks, and complementary model coverage all satisfy their documented criteria
- release_approval: `not granted by this report`

## Handoff Recommendation

| target_agent | reason | required_input |
| --- | --- | --- |
| Release Agent | All documented physical checks passed. | This report, the Step 26 reference-verification report, exact build-local HDF5/comparison identities, targeted CTest evidence, and the limitations below. |

## No-Change Assertion

- source_files_modified: `false`
- test_files_modified: `false`
- cmake_files_modified: `false`
- requirements_modified: `false`
- formulations_modified: `false`
- numerical_review_modified: `false`
- io_contract_modified: `false`
- reference_model_contract_modified: `false`
- reference_verification_report_modified: `false`
- reference_artifacts_modified: `false`
- tolerance_policies_modified: `false`
- Abaqus_or_other_reference_solver_executed: `false`
- owned_report_created: `true`
- phase_index_step27_modified: `true`
- notes: HDF5, comparison JSON, and legacy reference artifacts were inspected read-only; the only
  generated files were the ignored build-local evidence regenerated by the exact approved CTest.

## Open Issues

- Non-blocking coverage limitation: the approved Abaqus bundle is one identity-axis local-`z`
  bending cantilever. Axial, torsion, local-`y`, rotated, prescribed-displacement, line-load, and
  nonzero stress checks rely on the approved analytical/unit portfolio; no broader Abaqus reference
  coverage is claimed.
- Non-blocking stress limitation: the B33 bundle has no section points, so its physical `S11`
  evidence is the correct zero centroid result. Nonzero fiber location/sign evidence is analytical;
  Abaqus beam stress comparison remains N/A.
- Non-blocking output limitation: HDF5 has no strain-energy dataset by contract. The positive energy
  value in this report is calculated from `0.5*F^T*d` and is supported by the rank/energy unit test.
- Known formulation limitations remain: Euler-Bernoulli deep-beam applicability, transverse and
  torsional shear stress, warping, `I12!=0`, B31/Timoshenko behavior, and CLI `*DLOAD` are outside V0.
- No open issue blocks Release Agent review.
