3D Euler-Bernoulli Beam Kernel Research Brief

Metadata

What theory is sufficient for a straight, prismatic, two-node 3D Euler-Bernoulli beam kernel?
Which terms must appear in the local stiffness matrix for axial, torsion, and two uncoupled bending planes?
Which implementation checks can verify the kernel without running Abaqus or creating reference CSV artifacts?
Which input cases must be rejected before matrix construction?

Source Inventory

source_type	title	author_or_org	URL_or_identifier	access_date	reliability_tier	notes
textbook	Concepts and Applications of Finite Element Analysis	Cook, Malkus, Plesha, Witt	ISBN 978-0-471-35605-9	2026-06-12	Tier 2	Beam element stiffness and coordinate transformation background.
textbook	A First Course in the Finite Element Method	Daryl L. Logan	ISBN 978-1-305-63511-1	2026-06-12	Tier 2	Direct-stiffness beam and frame examples.
textbook	Matrix Structural Analysis	William McGuire, Richard H. Gallagher, Ronald D. Ziemian	ISBN 978-0-471-12379-7	2026-06-12	Tier 2	Space-frame element local stiffness and transformation conventions.
project requirement	3D Euler-Bernoulli Beam Kernel Requirements	FESA	docs/requirements/euler-beam-3d.md	2026-06-12	Tier 1 project contract	Defines approved kernel scope and exclusions.

Verified fact: a 3D straight prismatic Euler-Bernoulli beam frame element can be assembled from one axial response, one torsional response, and two uncoupled cubic-Hermite bending responses in the element local frame.
Verified fact: for a local element axis x, bending displacement in local y couples with rotation about local z and uses EIz; bending displacement in local z couples with rotation about local y and uses EIy.
Verified fact: a two-node space-frame element with six DOFs per node has a 12-entry vector. The FESA requirement fixes per-node order as ux, uy, uz, rx, ry, rz.
Verified fact: the unconstrained element local stiffness is symmetric positive semidefinite and has six rigid body modes before constraints are applied.
Project contract: no shear deformation, warping, end releases, offsets, mass, geometric stiffness, nonlinear kinematics, parser integration, HDF5 output, or reference artifact generation is allowed in this kernel increment.

benchmark_id	source	benchmark_type	physics	target_quantities	artifact_needs	applicability
EB3D-BENCH-001	project/formulation	code verification	local matrix algebra	symmetry of all `K(i,j)`	C++ unit test only	Does not verify solver assembly.
EB3D-BENCH-002	direct stiffness theory	analytical	axial extension	`EA/L` terms and equal/opposite axial end forces	C++ unit test only	Axis-aligned local response.
EB3D-BENCH-003	direct stiffness theory	analytical	torsion	`GJ/L` terms and equal/opposite torsional end moments	C++ unit test only	Saint-Venant torsion constant assumed supplied.
EB3D-BENCH-004	beam theory	analytical	bending in local `x-y`	`12EIz/L^3`, `6EIz/L^2`, `4EIz/L`, `2EIz/L` terms	C++ unit test only	Euler-Bernoulli small-displacement bending.
EB3D-BENCH-005	beam theory	analytical	bending in local `x-z`	`12EIy/L^3`, `6EIy/L^2`, `4EIy/L`, `2EIy/L` terms	C++ unit test only	Euler-Bernoulli small-displacement bending.
EB3D-BENCH-006	matrix structural analysis	invariant	rigid body motion	local end forces are near zero for rigid translations/rotations	C++ unit test only	Numerical tolerance needed.
EB3D-BENCH-007	matrix structural analysis	invariant	coordinate transform	axis-aligned global stiffness equals local stiffness when basis is identity	C++ unit test only	Does not validate parser orientation input.

code_verification: local stiffness entries, symmetry, end-force recovery as K*u, invalid input handling, and transformation invariants can be checked by deterministic C++ unit tests.
solution_verification: future cantilever axial, torsion, and bending reference models can compare displacements, reactions, and internal forces after parser/assembly/HDF5 integration exists.
validation: no physical experiment validation is in scope for this kernel increment.
reference_comparison: future reference comparison requires Abaqus-generated artifacts under reference/<model-id>/; this phase only defines the contract and must not create those files.

linear_or_nonlinear: linear only.
deformation: small displacement and small rotation.
element_type: straight two-node prismatic Euler-Bernoulli beam.
material_model: constant linear elastic E and G.
geometry: nonzero length, no offsets, no curved beams.
boundary_conditions: not applied inside the kernel.
loads: no distributed or equivalent nodal load vector in this increment.
coordinate_system: right-handed local Cartesian basis and global Cartesian basis.
units: user-consistent; no conversion.

Orientation vector parallel or nearly parallel to the beam axis must be rejected because the local y direction is undefined.
Zero or near-zero length must be rejected before stiffness coefficient computation.
Nonpositive or nonfinite E, G, A, J, Iy, or Iz must be rejected.
Very slender beams can produce ill-conditioned local stiffness matrices; kernel tests should avoid relying on a condition-number estimate as a pass/fail criterion.
Exact parser keyword mapping for future beam input remains open and belongs to the I/O contract, not the kernel implementation.

Define the local stiffness matrix with named axial, torsion, EIz, and EIy coefficients.
Define the local-to-global transform convention so global stiffness and force recovery are unambiguous.

Review symmetry, rigid body modes, sign convention, coefficient dimensions, and invalid geometry handling.

Future reference models should include axial cantilever, torsion cantilever, two bending cantilevers, and one skew transform case.

Unit tests should cover representative local entries, symmetry, K*u force recovery, invalid constants, axis-aligned transform identity, rotated symmetry, rigid translation, and axial end forces.