Curved Beam Analysis

Curved Beam Analysis (structures/fem/curved-beam-analysis)

Use when the task is the stress check of a curved member: a frame segment, torque link arc or clevis arc whose section bends about the center of curvature. Where the straight Euler-Bernoulli beam law puts the extreme fiber stress at M c / I with a linear gradient, a curved member shifts its neutral axis inward of the centroid and stresses its fibers hyperbolically, so the inner fiber runs hotter than any straight-beam reading of the same section. This leaf applies the Winkler curved-beam correction: it computes the neutral-axis radius of the radial section from the exact closed form A / integral(dA / rho), the neutral-axis-eccentricity e = r_c - r_n, the inner and outer fiber stresses sigma = M (r_n - r_fiber) / (A e r_fiber), the combined axial-plus-bending stresses, and the curved-beam amplification over the straight-beam value, then rates the gating fiber stress against the allowable. It pairs with the straight-member FEM leaves, and it owns the frame-corner, torque-link and clevis-arc stress check that no straight-beam or lug leaf covers.

Domain quick reference

• Neutral-axis radius of a curved bar with a rectangular radial section spanning inner radius r_i to outer radius r_o: r_n = (r_o - r_i) / ln(r_o / r_i), the exact closed form of A / integral(dA / rho); the width out of plane cancels. The neutral axis always sits strictly between r_i and the centroidal radius r_c = (r_i + r_o) / 2.
• Neutral-axis radius of a circular solid or tube section whose material annulus (inner radius a_i, outer radius a_o) is centered at distance r_c from the center of curvature: r_n = (sqrt(r_c^2 - a_i^2) + sqrt(r_c^2 - a_o^2)) / 2, from the disk integrals integral(dA / rho) = 2 pi (r_c - sqrt(r_c^2 - a^2)); a_i = 0.0 reduces the tube to the solid round section, r_n = (r_c + sqrt(r_c^2 - a_o^2)) / 2. Geometry requires a_o < r_c.
• Neutral-axis eccentricity: e = r_c - r_n, the inward shift of the neutral axis toward the center of curvature, positive for every physical curved beam.
• Winkler curved-beam bending stress at fiber radius r_fiber: sigma = M (r_n - r_fiber) / (A e r_fiber), hyperbolic across the depth. Sign convention: a positive moment opens the arc, tending to straighten the member, so the inner fiber (r_fiber < r_n) is in tension, sigma > 0, and the outer fiber is in compression, sigma < 0; a negative moment reverses both signs.
• Straight Euler-Bernoulli baseline of the same rectangular section: sigma_straight = 6 M / (A h), from M c / I with c = h / 2 and I = A h^2 / 12. The curved-beam amplification is the ratio of the Winkler inner-fiber stress to this value and grows with the depth-to-radius ratio h / r_c.
• Combined fiber stress with axial load: sigma + P / A, tension positive.
• Stress verdict: ratio = |sigma| / allowable, pass when ratio <= 1.0, margin = allowable - |sigma|.
• FAR-25 (25.301-25.307) sets the certification context for structure loads and strength; the allowable used here comes from the design allowables of the material, not from reproduced standard text.

Workflow

1. Gather the curved-member data at the checked section: the inner and outer radii of the rectangular radial section (r_i, r_o), or the section center radius r_c with the material annulus radii (a_i, a_o) for a solid-round or circular-tube section, the section depth h and area A, the centroidal radius r_c = (r_i + r_o) / 2, the applied bending moment M (positive opens the arc), the axial load P when present, and the material allowable.
2. Compute the neutral-axis radius of the section: call neutral_axis_radius_rect(r_i, r_o) for the rectangular radial section, or neutral_axis_radius_circular_tube(r_c, a_i, a_o) for a circular section (pass a_i = 0.0 for the solid round). The closed forms are A / integral(dA / rho); the result is independent of the out-of-plane width of a rectangle and lies strictly between r_i and r_c.
3. Take the neutral-axis eccentricity with eccentricity(r_centroid, r_n): e = r_c - r_n, the inward shift of the neutral axis toward the center of curvature, positive for every physical curved member.
4. Resolve the Winkler fiber stresses with curved_beam_stress(moment, area, e, r_n, r_fiber) at the inner fiber r_i and the outer fiber r_o of the section, and form the straight Euler-Bernoulli baseline of the same section with straight_beam_stress_rect(moment, area, depth) to expose the curved-beam amplification, the inner-fiber stress divided by the straight reading.
5. Add the axial contribution when the member carries axial load with combined_axial_stress(bending_stress, axial_force, area), tension positive, at each fiber.
6. Rate the gating inner-fiber combined stress against the allowable with stress_verdict(sigma, allowable) and read the ratio, verdict and margin from the returned dict.

Worked example

Torque-link / frame-corner style curved member, 7075-T6-like aluminum alloy with the design allowable tension set at 500 MPa for the check. Rectangular radial section: r_i = 60 mm, r_o = 100 mm, depth h = 40 mm, width b = 40 mm, area A = 1600 mm^2, centroidal radius r_c = 80 mm, axial tension P = 25 kN. Real module outputs:

• neutral_axis_radius_rect(60.0, 100.0) = 78.30461 mm: the neutral axis sits inward of the centroid by eccentricity(80.0, 78.30461) = 1.69539 mm, about 4.2 percent of the depth toward the center of curvature. The closed form agrees with a 400000-bin quadrature of A / integral(dA / rho) to relative difference 1.45e-13.
• Circular sections: neutral_axis_radius_circular_tube(80.0, 10.0, 20.0) = 78.41610 mm (e = 1.58390 mm), and the solid round neutral_axis_radius_circular_tube(80.0, 0.0, 20.0) = 78.72983 mm (e = 1.27017 mm).
• Load case 1, M = 1.2e6 N mm (moderate in-service moment): inner fiber curved_beam_stress = 134.95848 MPa tension, outer fiber = -95.97509 MPa compression. The straight Euler-Bernoulli baseline of the same section is straight_beam_stress_rect = 112.5 MPa: the curved-beam correction amplifies the inner fiber by 1.19963 and runs the outer fiber at 0.85311 of the straight reading, the classic Winkler signature. Adding the axial tension P / A = 15.625 MPa: combined inner = 150.58348 MPa, combined outer = -80.35009 MPa. Against the 500 MPa allowable: ratio 0.30117, verdict pass, margin 349.41652 MPa.
• Load case 2, limit moment M = 4.0e6 N mm with the same axial load: inner fiber = 449.86161 MPa, combined inner = 465.48661 MPa, ratio 0.93097, verdict pass, margin 34.51339 MPa.
• Load case 3, ultimate moment M = 4.5e6 N mm: inner fiber = 506.09432 MPa, combined inner = 521.71932 MPa, ratio 1.04344, verdict fail, margin -21.71932 MPa. Read as straight (6 M / (A h) = 421.875 MPa combined) the same member would still pass, which is the point of the correction: the curved-beam inner-fiber stress is the gating number at a frame corner or torque link arc.

Pitfalls

• Reading a curved member with the straight law: the straight Euler-Bernoulli baseline of the worked example passes load case 3 while the Winkler inner-fiber stress fails it; always rate the curved inner fiber, never 6 M / (A h), for a frame segment or arc.
• Signing the moment backwards: a positive moment opens the arc, so the inner fiber goes into tension and the outer fiber into compression; a negative moment reverses both. Flipping the sign convention swaps which fiber the verdict gates on.
• Taking the neutral axis at the centroid: e is the whole point of the correction; setting r_n = r_c would divide by zero in the Winkler stress and erase the hyperbolic gradient.
• Using the tube closed form on an invalid section: the annulus outer radius must stay below r_c (the section must clear the center of curvature) and a_i must be 0 or positive, or the function raises ValueError; feeding a_o >= r_c silently corrupts the square root instead if you bypass the guard.
• Trusting a raw 2-D grid quadrature of A / integral(dA / rho) for a circular section: an independent cell-centered grid over the section annulus reproduces the circular closed forms only to about one percent (grid and measure sensitive; the prep quadrature note of 1e-6 class was not independently reproducible in Euclidean section measure). Use the closed forms, which are the engineering standard for circular-section curved beams and match the spec anchors to 1e-5. The rectangular closed form, by contrast, is exact to the 1e-13 class against quadrature.
• Mixing units: radii, area and moment must share one consistent system. In the worked example mm and MPa are used with moment in N mm and area in mm^2, so N mm / mm^3 = N / mm^2 = MPa; mixing N m with mm^2 sections silently scales every stress by 1000.

Verification

• The contract test asserts the real module outputs above inside the spec tolerances: geometry anchors to 1e-5, fiber stresses to 1e-4, ratios and margins to 1e-3, and the straight baseline 112.5 MPa exactly.
• Identities verified: the rectangular closed form equals a 400000-bin quadrature of A / integral(dA / rho) to 1.45e-13 (better than the 1e-9 class), the neutral-axis radius is independent of the out-of-plane width, r_n lies strictly between r_i and r_c, the eccentricity is positive for physical sections, the inner-fiber magnitude exceeds the outer-fiber magnitude for the same moment, the curved-beam amplification grows with the depth-to-radius ratio (1.04333, 1.09046, 1.14220, 1.19963 for h = 10, 20, 30, 40 mm at r_c = 80 mm), a_i = 0.0 in the tube function reproduces the solid-section closed form, a negative moment flips both fiber signs exactly, zero moment gives zero stress, and the verdict flips from pass to fail exactly at ratio 1.0.
• ValueError rejection: non-physical inputs raise ValueError in every function (non-positive radii, r_o <= r_i, negative or equal annulus radii, a_o >= r_c, non-positive area, eccentricity, neutral-axis radius, fiber radius, depth or allowable).
• Determinism: no random numbers anywhere; identical floats run to run. Run the contract test offline: python3 scripts/test_curved_beam_analysis.py (35 tests, deterministic, well under one second).

Related leaves

• skills/structures/fem/beam-frame-analysis: the straight-member FEM sibling; its Euler-Bernoulli element and the cantilever closed forms carry no curvature radius, no neutral-axis shift and no fiber-stress law, so a curved frame corner belongs here.
• skills/structures/fem/beam-vibration: straight continuous-member natural frequencies only, no curved stress state.
• skills/structures/fem/lug-joint-analysis: pin and lug bearing and tearout checks of the clevis fitting, not the curved-member bending stress of the arc that feeds it.
• skills/structures/composites/peel-stress-bonded-joints: an adhesive Winkler-foundation beam model of a flat single-lap bondline; it takes no curvature radius and no fiber stress, so it is a different Winkler model, not a competitor for curved members.

Contract test

Run from the repository root:

python3 skills/structures/fem/curved-beam-analysis/scripts/test_curved_beam_analysis.py

The stdlib unittest contract test runs fully offline and covers the worked-example anchors of all three load cases (steps 1 to 6 of the workflow), the circular tube and solid-round closed forms, the rectangular quadrature identity to the 1e-9 class, the honest cell-centered grid quadrature agreement of the circular closed form to about one percent, the width-independence and neutral-axis-position identities, the amplification ladder, the fiber stress sign and zero behaviors, the axial contribution identities, the exact verdict boundary at ratio 1.0 with the fixed pass and fail strings, and the ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: FAR-25 is public-domain US government work (17 U.S.C. 105) but standards-map.yaml marks it gated: false and reference-only: true, so only the summary paraphrase above is used, never standard text. The design allowable of the worked example is a paraphrased 7075-T6-like value, not a quoted table.
• The Winkler curved-beam relations and the closed forms are standard engineering methodology, summary-only.
• compliance: STANDARDS-REF, gated: false.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_curved_beam_analysis.py

The test covers the worked-example geometry and all three load case stresses inside the spec bounds, the circular tube and solid-round neutral-axis closed forms, the rectangular quadrature identity, the honest tube-grid quadrature agreement, the width independence and neutral-axis-between-r_i-and-r_c identities, the amplification growth with h / r_c, the inner-fiber-over-outer-fiber magnitude law, exact sign flip for a negative moment, zero stress at zero moment, the axial-tension and P / A identities, the verdict flip exactly at ratio 1.0 with the fixed pass and fail strings and the exact dict keys, and ValueError rejection of every non-physical input, all deterministic and offline in under a second.