Torsion Shear Flow

Torsion Shear Flow (structures/fem/torsion-shear-flow)

Compute the torsional response of shafts, fuselage and wing-box sections: polar second moment for solid and tube sections, the Saint-Venant torsion constant for open thin sections, the Bredt-Batho closed-section shear flow q = T / (2 A_m), the closed-section twist rate, the shear stress and torsional stress margin, and the multi-cell shear-flow distribution for a two-cell section under an applied torque. This leaf implements the standard torsion methodology in pure Python, stdlib only. It pairs with the bar-truss and frame solution leaves for overall stiffness checks and with the buckling leaves for the load-carry limit of the same sections.

Domain quick reference

• Closed single cell (Bredt-Batho): A_m is the area enclosed by the mid-line of the section (rectangular box: (w - t) * (h - t), or the documented input A_m directly). Shear flow q = T / (2 * A_m) in N/m.
• Closed twist rate: d(theta)/dx = q / (2 * A_m * G) times the closed integral of ds/t, evaluated as T / (4 * A_m**2 * G) * sum(side_length_i / thickness_i) in rad/m.
• Closed shear stress: tau = q / t_min in Pa, with the margin tau_allow / tau - 1.
• Open thin sections (Saint-Venant): one thin rectangle of width b and thickness t has J = b * t3 / 3; a built-up open section sums b_i * t_i3 / 3 over independent rectangles (conservative open-section model, junction stiffening ignored). Twist rate = T / (G * J); surface shear stress tau = T * t_max / J uses the FULL thickness of the thickest element.
• Two-cell closed section: equal twist-rate compatibility (q1*(S1 + S12) - q2S12) / A1 == (q2(S2 + S12) - q1S12) / A2 plus torque balance T = 2A1q1 + 2A2*q2, solved by Cramer's rule, where S_i is the outer wall length/thickness integral of cell i and S12 that of the shared wall.
• Shafts: solid circular shaft J = pir**4/2; tube J = pi(ro4 - ri4)/2. Shear stress at the surface tau = T*r/J.
• Units are SI throughout: N m torque, m dimensions, Pa modulus, rad/m twist. The shear modulus G and the allowable shear stress are always inputs; the module hard-codes no material.
• FAR-25 and CS-25 frame the airframe torque-load context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Classify the section: solid or tube shaft, open thin built-up section, or closed single-cell or two-cell section. Each class uses a different J or shear-flow model; do not mix them.
2. Solid or tube shaft: polar_j_solid(radius) or polar_j_tube(radius_outer, radius_inner) for J, then tau = T*r/J by hand and torsion_margin(tau, tau_allow).
3. Open thin section: saint_venant_j_rectangle(width, thickness) per element, or saint_venant_j_open(elements) over the list of (width, thickness) pairs of the built-up section.
4. Closed single cell: fix the mid-line enclosed area A_m, then bredt_shear_flow(T, A_m) for q and closed_twist_rate(T, G, side_lengths, thicknesses, A_m) for the twist rate (A_m may be omitted for a rectangular outline listed as (a, b, a, b)).
5. Closed-section stress and margin: tau = q / t_min, then torsion_margin(tau, tau_allow) to gate the section against the allowable shear stress.
6. Two-cell section: multi_cell_shear_flow(T, [A1, A2], [S1, S2], S12, G) returns the dict {q1, q2, twist_rate} with q in N/m and the twist rate in rad/m.
7. Confirm the deterministic checks with the contract test scripts/test_torsion_shear_flow.py.

Worked example

Rectangular box 0.5 m x 0.3 m (mid-line enclosed area A_m = 0.5 * 0.3 = 0.15 m2), uniform wall 0.002 m, G = 27e9 Pa, T = 100 kN m.

• Shear flow: bredt_shear_flow(1e5, 0.15) = 333333 N/m, within 1% of the prep bound 333333 N/m.
• Twist rate: closed_twist_rate(1e5, 27e9, [0.5, 0.3, 0.5, 0.3], [0.002]*4) = 0.03292 rad/m: sum s/t = 1.6 / 0.002 = 800, and 1e5 / (4 * 0.0225 * 27e9) * 800 = 0.03292.
• Shear stress: tau = q / t_min = 333333 / 0.002 = 1.667e8 Pa; margin at tau_allow 2.0e8: torsion_margin(1.667e8, 2.0e8) = 0.2.
• Solid shaft r = 0.05 m: J = 9.817e-6 m4. Tube ro = 0.05, ri = 0.04: J = 5.796e-6 m4. Open rectangle 0.1 m x 0.003 m: J = 9.0e-10 m4.
• Two-cell box (total width 0.6 m, height 0.3 m, divider 0.2 m from the left, uniform wall 0.002 m, G = 27e9 Pa, T = 50 kN m): A1 = 0.06 m2, A2 = 0.12 m2, S1 = 350, S2 = 550, S12 = 150; multi_cell_shear_flow(5e4, [0.06, 0.12], [350, 550], 150, 27e9) = q1 = 126263 N/m, q2 = 145202 N/m, twist rate 0.012763 rad/m; torque balance 2A1q1 + 2A2q2 reconstructs 50000 N m and the two cell twist computations agree to better than 1e-6.

Verification

• Confirm polar_j_solid(0.05) = 9.817e-6 m4 and polar_j_tube(0.05, 0.04) = 5.796e-6 m4; a tube with ri = 0 degenerates to the solid shaft value.
• Confirm saint_venant_j_rectangle(0.1, 0.003) = 9.0e-10 m4 and that a built-up open section equals the per-rectangle sum.
• Confirm bredt_shear_flow(1e5, 0.15) = 333333 N/m and closed_twist_rate at the worked box = 0.03292 rad/m, each within 1% of the prep bound.
• Confirm torsion_margin(1.667e8, 2.0e8) = 0.2, that the margin is 0 at the allowable and negative above it, and that it decreases as the running torque grows.
• Confirm multi_cell_shear_flow reconstructs the applied torque from q1 and q2, gives cell twist computations that agree within 1e-6, and degenerates to the single-cell q and twist for symmetric cells.
• Confirm every non-positive torque, modulus, dimension, area, wall integral and allowable stress, every empty or mismatched list, and every reversed tube radius raises ValueError.
• Run the contract test offline: python3 scripts/test_torsion_shear_flow.py (33 tests, deterministic).

Related leaves

• structures/fem/truss-analysis: bar truss solution methods for the axial load paths of the same airframe; it does not model torsion of closed sections.
• structures/fem/beam-frame-analysis: beam and frame bending response under applied loads, the companion stiffness check to torsion.
• vehicle-design/structures-integration/wing-box-sizing: sizes the wing-box skin and web from the lift-induced load path of the box, adjacent to but distinct from the torque shear-flow distribution here.
• structures/fem/plate-buckling: panel stability of the same walls under combined load, the post-check that can cap the allowable shear stress used in the margin.
• structures/fem/cylindrical-shell-buckling: stability of fuselage barrel sections that also carry torque shear flow.

Pitfalls

• Using the polar J of the solid shaft for an open thin section: J = pir**4/2 belongs to axisymmetric shaft torsion; an open channel or slit tube must use the Saint-Venant bt**3/3 sum, which gives a far lower torsional stiffness for the same material volume.
• Building A_m from the gross outer rectangle: the Bredt area is the mid-line enclosed area (0.15 m2 for the 0.5 x 0.3 box in the example, not 0.5*0.3 inflated by wall thickness twice over), so gross dimensions understate q and the twist rate.
• Treating an open section as a closed loop: a slit tube has no Bredt-Batho circulation, so q = T / (2 A_m) does not apply; its twist comes from the open-section J and is much larger than the closed envelope of the same shape.
• Double-counting the shared wall in the two-cell model: S12 enters each cell circulation with the opposite sign as (q1 - q2) flow, so it cancels when q1 = q2 and must never be added to both S1 and S2 as a positive path.
• Reading the margin sign backwards: margin = tau_allow / tau - 1 is positive below the allowable, zero at it, and negative past it; a negative margin at the worked torque means the wall or the allowable must change before the section passes.
• Mixing unit systems: kN m torque against Pa modulus and m dimensions silently shifts q and twist by decades; keep N m, m and Pa together before quoting a margin.

Behavior contract (gate 3)

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

python3 scripts/test_torsion_shear_flow.py

The test covers the worked box contract (q = 333333 N/m and twist 0.03292 rad/m within 1% of the prep bounds, tau = 1.667e8 Pa, margin 0.2), the polar J truth table with the ri = 0 degeneracy, the Saint-Venant J of a rectangle and of a built-up open channel, the uniform-wall Bredt identity for the closed twist rate, the torque doubling and area scaling laws, the two-cell solve (q1 = 126263 N/m, q2 = 145202 N/m, twist 0.012763 rad/m, torque reconstruction to 50000 N m, cell twist compatibility within 1e-6, symmetric-cell degeneracy to the single-cell result), dict key exactness, run-to-run determinism and ValueError rejection of every non-physical input listed in the validation list.

Compliance

• FAR-25 and CS-25 are referenced, not reproduced: standards-map.yaml marks them gated: false and reference-only: true; only the summary paraphrase above is used, never standard text.
• compliance: STANDARDS-REF, gated: false.