Restrained Warping

Restrained Warping (structures/fem/restrained-warping)

Use when the load case is restrained (non-uniform) torsion of a thin-walled open doubly symmetric I-section: the built-in or fork-held ends block the free warping of the flanges, so the torque is carried by differential flange bending (a bimoment) as well as by Saint-Venant shear, and the twist follows the fourth-order equation ECwtheta'''' - GJtheta'' = 0 rather than the free uniform twist rate. This leaf implements the classical warping-torsion model (Vlasov sectorial theory for an I-section with its shear center at the centroid and the web on the sectorial zero line) in pure Python, stdlib only. It pairs with structures/fem/torsion-shear-flow, the free-warping complement that reports the uniform Saint-Venant twist rate and the closed-section shear flow, and with structures/fem/shear-center-analysis, the transverse-shear complement that walks a contour under a transverse force. Deterministic, offline, SI units (metres, newtons, pascals).

Domain quick reference

• Open-section Saint-Venant constant: J = (2bt_f3 + h_w*t_w3)/3, the sum b_it_i**3/3 over the two flanges and the web, with h_w = d - 2t_f the web depth.
• Warping constant: h = d - t_f (flange mid-line spacing), I_y = 2b**3t_f/12 (flange pair about the web plane), Cw = I_yh**2/4. The direct sectorial integral t_fh2*b3/24 reproduces Cw to roundoff (the web sits on the sectorial zero line, so its own minor term contributes nothing).
• Sectorial coordinate: omega(s) = integral of rho ds from the web mid-line with rho = h/2 the flange lever arm; omega is linear along each flange, zero on the web, and omega_tip = h*b/4 at the flange tip. Sectorial modulus W_w = Cw/omega_tip.
• Non-uniform torsion ODE: ECwtheta'''' - GJtheta'' = 0 on the torque-free span, with decay parameter k = sqrt(GJ/(ECw)) (1/m), the characteristic 1/k over which the restraint decays, and dimensionless u = kL (cantilever) and v = kL/2 (fork half-beam).
• Bimoment: B(z) = -ECwtheta''(z) in N m^2, the flange counter-bending couple B = M_fh. Warping normal stress at the flange tip: sigma_w = Bomega_tip/Cw.
• Torque partition: T_sv(z) = GJtheta'(z), T_w(z) = -ECwtheta'''(z) (the z-derivative of the bimoment), with T_sv + T_w equal to the torque carried at every station.
• Fixed-end cantilever (root z = 0 built in, point torque T at the free tip): theta(z) = (T/(GJk))(kz - sinh(kz) + tanh(u)(cosh(kz) - 1)), identically satisfying theta(0) = 0, theta'(0) = 0, theta''(L) = 0 and GJtheta'(L) - ECwtheta'''(L) = T. Closed-form consequences: theta(L) = (TL/(GJ))(1 - tanh(u)/u), |B(0)| = Ttanh(u)/k, T_sv(L) = T(1 - 1/cosh(u)), T_w(L) = T/cosh(u), T_sv(0) = 0 and T_w(0) = T.
• Fork-supported beam (fork ends hold theta = theta'' = 0, torque T at midspan through a rigid collar): half-beam x in [0, L/2], theta(x) = (T/(2GJ))(x - sinh(kx)/(kcosh(v))), mirrored across the symmetry plane theta'(L/2) = 0. Closed-form consequences: theta(L/2) = (TL/(4GJ))(1 - tanh(v)/v), B(L/2) = (T/(2k))tanh(v), B(0) = 0, and per half at the fork face T_sv = (T/2)(1 - 1/cosh(v)) with T_w = (T/2)/cosh(v), while at the collar T_sv = 0 and T_w = T/2.
• Free Saint-Venant baselines appear only inside the twist ratios (never as a deliverable): cantilever free twist TL/(GJ) and fork free midspan twist TL/(4G*J); the ratios 1 - tanh(u)/u and 1 - tanh(v)/v quantify the stiffening of the restraint.

Workflow

1. Fix the section and the material: the I-section dims (d, b, t_f, t_w) in that order and the isotropic moduli E, G, with the thin-walled open geometry d > 2*t_f. Evaluate the open-section Saint-Venant constant J with i_section_saint_venant_j.
2. Compute the warping section constants: i_section_warping_constant for Cw = I_yh**2/4, i_section_sectorial_max for the sectorial coordinate omega_tip = hb/4 at the flange tip, and the sectorial modulus Cw/omega_tip. Verify the identity Cw = t_fh**2b**3/24.
3. Get the decay parameter of the non-uniform torsion equation: torsion_decay_parameter for k = sqrt(GJ/(ECw)), the decay length 1/k and the dimensionless products u = kL and v = kL/2.
4. Solve the built-in cantilever response: fixed_end_response for the restrained tip twist, tip twist rate, root bimoment and tip torque split, then fixed_end_twist, fixed_end_twist_rate and fixed_end_bimoment for the span profile, with torque_components showing the Saint-Venant share climbing from 0% at the built-in root to T*(1 - 1/cosh(u)) at the free tip.
5. Solve the fork-supported beam response: fork_response for the restrained midspan twist, the midspan bimoment peak and the per-half torque components at the fork face and the collar, then fork_twist and fork_bimoment for the symmetric span profile.
6. Evaluate the warping normal stress at the flange tip: warping_stress on the peak bimoment (the root for the cantilever, the midspan collar for the fork beam), sigma_w = B*omega_tip/Cw.
7. Quantify the restraint and confirm determinism: compare the restrained twist against the free Saint-Venant baseline through the twist_ratio keys of the response dicts, then confirm the deterministic checks with the contract test scripts/test_restrained_warping.py.

Worked example

I-beam d = 0.3 m, b = 0.15 m, t_f = 0.01 m, t_w = 0.006 m (a spar-like open section), aluminium E = 70 GPa, G = 27 GPa, span L = 3 m. All values below are real outputs of scripts/restrained_warping_logic.py (stdlib math, exit 0).

• Section constants: h = 0.29 m, h_w = 0.28 m; J = 1.2016e-07 m^4 with GJ = 3244.32 N m^2; I_y = 5.625e-06 m^4, Cw = 1.18265625e-07 m^6 with ECw = 8278.59375 N m^4 (the sectorial integral t_fh**2b*3/24 matches to a 1.32348898008e-23 m^6 residual); omega_tip = 0.010875 m^2 with sectorial modulus W_w = 1.0875e-05 m^4; k = 0.626013294436 1/m, decay length 1/k = 1.59741016507 m (more than half the span, so u = kL = 1.87803988331 is a strongly restrained regime, not a local end effect).
• Fixed-end cantilever, torque T = 500 N m at the free tip: restrained tip twist theta(L) = 0.227407224589 rad against the free baseline TL/(GJ) = 0.462346500962 rad, twist ratio 1 - tanh(u)/u = 0.49185453792: the built-in root cuts the twist to 49.2% of the free value. Twist rate peaks at the tip at 0.108066620404 rad/m (zero at the root, where warping is blocked); the Saint-Venant surface shear there is tau = Gttheta' = 29177987.509 Pa on the flange (t = t_f) and 17506792.5054 Pa on the web (t = t_w).
• Bimoment at the root B(0) = -762.21819312 N m^2 (magnitude Ttanh(u)/k) and the tip bimoment B(L) = 3.54696309006e-13 N m^2, zero to roundoff: the tip warps freely. Warping normal stress at the flange tip at the root sigma_w = B(0)omega_tip/Cw = -70089029.2524 Pa (-70.09 MPa, compression at the positive-omega tip; the two tips of each flange pair carry equal and opposite stress), which equals Eomega_tip|theta''(0)|, the flange-bending picture of the bimoment.
• Torque split (T = 500 N m carried everywhere): at the root T_sv = 0 and T_w = 500 N m (all warping torque: the root blocks warping, so the torque arrives as differential flange bending); at the tip T_sv(L) = T*(1 - 1/cosh(u)) = 350.602697909 N m (70.1%) and T_w(L) = T/cosh(u) = 149.397302091 N m (29.9%). Span table (real module rows; theta rad, theta' rad/m, B N m^2, sigma_w Pa, T_sv N m, T_w N m): z = 0.000: theta = 0.000000e+00, rate = 0.000000e+00, B = -7.622182e+02, sigma_w = -7.008903e+07, T_sv = 0.000000e+00, T_w = 5.000000e+02 z = 0.750: theta = 2.208046e-02, rate = 5.431743e-02, B = -4.588543e+02, sigma_w = -4.219350e+07, T_sv = 1.762231e+02, T_w = 3.237769e+02 z = 1.500: theta = 7.591534e-02, rate = 8.622829e-02, B = -2.585118e+02, sigma_w = -2.377120e+07, T_sv = 2.797522e+02, T_w = 2.202478e+02 z = 2.250: theta = 1.476402e-01, rate = 1.028972e-01, B = -1.162102e+02, sigma_w = -1.068600e+07, T_sv = 3.338314e+02, T_w = 1.661686e+02 z = 3.000: theta = 2.274072e-01, rate = 1.080666e-01, B = 3.546963e-13, sigma_w = 3.261575e-08, T_sv = 3.506027e+02, T_w = 1.493973e+02 The Saint-Venant share climbs monotonically from 0% at the root to 70.1% at the tip while the bimoment magnitude decays hyperbolically from the root peak.
• Fork-supported beam, torque T = 1000 N m at midspan through a rigid collar (both forks hold theta = 0 and let the section warp): restrained midspan twist theta(L/2) = 0.0502830037738 rad against the free baseline TL/(4G*J) = 0.231173250481 rad, twist ratio 1 - tanh(v)/v = 0.217512206405: the collar restraint cuts the midspan twist to 21.8% of the free value, far stronger than the cantilever case because the collar blocks warping at the very section of maximum twist.
• Bimoment: it vanishes at the forks (B(0) = 0 N m^2, warping free) and peaks at the collar at B(L/2) = (T/(2*k))*tanh(v) = 586.865845196 N m^2, with sigma_w = 53964675.4204 Pa (53.96 MPa) at the flange tip there.
• Torque split (each half carries T/2 = 500 N m): at the fork face T_sv = (T/2)*(1 - 1/cosh(v)) = 160.842723176 N m (32.2%) and T_w = (T/2)/cosh(v) = 339.157276824 N m (67.8%); at the collar T_sv = 0 and T_w = 500 N m per half: the applied torque enters the beam entirely as warping torque at the midspan collar. Span table (real module rows): z = 0.000: theta = 0.000000e+00, B = 0.000000e+00, sigma_w = 0.000000e+00, T_sv = 1.608427e+02, T_w = 3.391573e+02 z = 0.750: theta = 3.427006e-02, B = 2.638170e+02, sigma_w = 2.425903e+07, T_sv = 1.227691e+02, T_w = 3.772309e+02 z = 1.500: theta = 5.028300e-02, B = 5.868658e+02, sigma_w = 5.396468e+07, T_sv = 0.000000e+00, T_w = 5.000000e+02 z = 2.250: theta = 3.427006e-02, B = 2.638170e+02, sigma_w = 2.425903e+07, T_sv = 1.227691e+02, T_w = 3.772309e+02 z = 3.000: theta = 0.000000e+00, B = 0.000000e+00, sigma_w = 0.000000e+00, T_sv = 1.608427e+02, T_w = 3.391573e+02 The profile is symmetric about midspan: theta and the bimoment mirror, and each half twists toward its own fork.
• Residual and identity checks (real module runs): the cantilever ODE residual max |ECwtheta'''' - GJtheta''| over 501 stations is zero to machine roundoff (spec anchor 1.137e-13, bound 1e-9), as is the torque partition residual max |T_sv + T_w - T|; the cantilever boundary residuals are |theta(0)| = 0.0, |theta'(0)| = 0.0, |theta''(L)| = 4.284e-17 1/m^2 and |GJtheta'(L) - ECwtheta'''(L)
  • T| = 0.0; the fork torque at the fork section and at the midspan face of the half-beam both equal 5.000000e+02 N m against T/2 = 500; the mirror remainder |theta(L/2) - theta(L/2 + 0.006)| = 1.274e-06 rad equals theta''(L/2)*dz**2/2, the second-order remainder of the symmetry.

Verification

• Confirm the section constants of the anchor I-beam: J = 1.2016e-07 m^4, Cw = 1.18265625e-07 m^6, omega_tip = 0.010875 m^2, W_w = 1.0875e-05 m^4 and k = 0.626013294436 1/m, with the identity |Cw - t_fh**2b**3/24| below 1e-20 m^6.
• Confirm the restraint twist ratios: fixed_end_response twist_ratio = 0.49185453792 equals 1 - tanh(u)/u, fork_response twist_ratio = 0.217512206405 equals 1 - tanh(v)/v, each below the free baseline.
• Confirm the torque partition: torque_components sums T_sv + T_w to the carried torque at every sampled station, T_sv equals GJtheta'(z) from the twist-rate profile, the root split is 0/500 N m and the tip split is 350.602697909/149.397302091 N m for the cantilever.
• Confirm the boundary conditions: |theta(0)| and |theta'(0)| below 1e-9 for the cantilever root and |B(L)| below 1e-9 N m^2 at its free tip; |fork_twist(0)|, |fork_bimoment(0)| and the fork midspan twist rate below 1e-9, with fork_twist(z) equal to fork_twist(L - z).
• Confirm the flange-bending identity of the warping stress: |sigma_w| = Bomega_tip/Cw equals Eomega_tip*|theta''| at the root and midspan peaks (anchor -70089029.2524 Pa and 53964675.4204 Pa).
• Confirm every non-physical input raises ValueError: non-positive section dims, d <= 2*t_f, non-positive E, G, Cw, J, zero span, zero torque in the response functions, and non-positive omega_tip or Cw in warping_stress; wrong function arity raises TypeError.
• Run the contract test offline: python3 scripts/test_restrained_warping.py (35 tests, deterministic, no RNG, math only).

Related leaves

• structures/fem/torsion-shear-flow: the free-warping complement of this pack. It returns the uniform Saint-Venant twist rate T/(G*J), the closed-section shear flow of the Bredt-Batho circulation and the torsional stress margin; this leaf returns the restrained response that torsion-shear-flow never touches (no warping constant, no bimoment, no theta'' warping-stress term in its claim).
• structures/fem/shear-center-analysis: the transverse-shear complement that walks the contour under a transverse force V and builds the V*Q/I shear-flow distribution to locate the shear-center offset; it applies no torque, while this leaf applies torque with the shear center at the centroid.
• structures/fem/beam-column-analysis: elastic compression and bending of columns, the axial companion; torsional-flexure instability under axial load is outside both leaves.
• structures/fem/buckling-analysis: elastic buckling eigenvalues, a stability check, not a torque response.
• structures/fem/beam-frame-analysis: member-end bending and frame response under applied loads, adjacent stiffness checks in the same airframe.

Pitfalls

• Reading the free twist rate as the restrained response: the free Saint-Venant baselines TL/(GJ) and TL/(4G*J) exist in this leaf only as denominators of the twist ratios; the deliverable is the restrained twist, 49.2% of free for the cantilever and 21.8% for the fork beam in the worked example.
• Flipping the root torque split: at the built-in root the full torque arrives as warping torque (T_sv = 0, T_w = T) because the root blocks warping and the torque becomes differential flange bending; quoting the Saint-Venant share there as anything but zero misstates the load path. At the free tip the shares are T*(1 - 1/cosh(u)) and T/cosh(u).
• Using the full-section minor inertia for Cw: the thin-walled sectorial model keeps only the flange pair in I_y = 2b**3t_f/12 because the web lies on the sectorial zero line; the AISC-style addition of the web's own t_w*3h_w/12 term stays below 0.1% here and belongs to a different model.
• Applying the doubly symmetric closed form to a channel, Z, angle or hat section: their shear center sits off the web line and the sectorial coordinates need the shear-center-offset integration, a stated extension outside this contract.
• Forgetting the mirror in the fork case: the half-beam closed form only covers x in [0, L/2]; the full span is theta(min(z, L - z)), which is why the fork table rows repeat symmetrically about midspan.
• Sign of the warping stress: sigma_w = B*omega_tip/Cw is signed, and the two flange tips of each flange pair carry equal and opposite stress; quote the sign relative to the positive-omega tip.
• Mixing unit systems: kN m torque against Pa moduli and metre dimensions silently shifts k, the twist and the bimoment by decades; keep N m, m and Pa together before quoting a result.

Behavior contract (gate 3)

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

python3 scripts/test_restrained_warping.py

The test covers the worked-example contract (section constants within 1e-9 relative, k = 0.626013294436, restrained tip twist 0.227407224589 rad with twist ratio 0.49185453792 for the cantilever and 0.0502830037738 rad with ratio 0.217512206405 for the fork beam), the open-section rectangle-sum identity for J and the sectorial integral identity for Cw, the closed-form bimoment and torque-split expressions at the root, tip, fork face and collar, the five anchor span rows of both cases, the torque partition and ODE residual over 501 stations, the boundary conditions, the flange-bending stress identity, fork symmetry and the second-order mirror remainder, run-to-run determinism and ValueError rejection of every non-physical input in the validation list. All numeric asserts are order-safe tolerances, so the file passes under both /usr/bin/python3 and the pre-push hook interpreter.

Compliance

• FAR-25 (25.301 applied loads, 25.303 1.5 ultimate factor) and CS-25 (25.301/25.303) set the load context of the torque cases, summarized only; standards-map.yaml marks both reference-only and gated: false. No standard text is reproduced.
• compliance: STANDARDS-REF, gated: false.