Diagonal Tension Field Webs

Diagonal Tension Field Webs (structures/fem/diagonal-tension-field-webs)

Analyze a plane shear web loaded above its elastic shear-buckling stress with the classical complete-diagonal-tension idealization: the web sheds shear into an inclined tension field and carries the excess load as pure diagonal tension. This leaf computes the tension field ratio, the classical 45 degree tension field angle, the diagonal web tension stress, the flange and end post axial loads pulled in by the field, the rivet shear flows on the flange and end post attachments, and the margin against buckling, in pure Python, stdlib only. It takes the elastic shear-buckling stress tau_cr as an input from the plate-buckling leaf and hands the elastic load distribution below buckling to the torsion-shear-flow leaf; this leaf owns the post-buckled reserve that gates the shear web strength check.

Domain quick reference

• Regime split: below tau_cr the web is elastic and carries the applied shear as shear stress; at tau = tau_cr the web buckles and, above it, the excess (tau - tau_cr) is carried by a diagonal tension field. The tension field ratio k = (tau - tau_cr) / tau is 0 at tau_cr and approaches 1 as the applied shear grows.
• Tension field angle: the classical plane-web value is ALPHA_IDEAL_DEG = 45.0 degrees, from the Kuhn sin(2 alpha) = 1 approximation of the Wagner field orientation; it is constant and continuous across the whole range above tau_cr. A web analyzed with an inclined field takes the angle as an input alpha_deg instead.
• Diagonal web tension stress: sigma_d = (tau - tau_cr) * (cot(alpha) + tan(alpha)) in Pa above tau_cr, zero in the elastic regime. At 45 degrees this equals 2 * (tau - tau_cr), the uniaxial diagonal tension that replaces the excess shear.
• Flange axial load: P_f = (tau - tau_cr) * t * d * cot(alpha) in N, the diagonal-tension component pulled into the flange over the web depth d (d between flanges, t the web thickness, both in m).
• End post axial load: P_e = (tau - tau_cr) * t * d * tan(alpha) in N. At the ideal 45 degree angle the flange and end post loads are equal.
• Rivet shear flows (N/m): flange q = t * (tau_cr + (tau - tau_cr) * tan(alpha)) above buckling, falling back to the elastic q = tau * t below it; end post q = tau * t in both regimes, because the end post carries the full applied shear.
• Margin against buckling: tau_cr / tau, 1.0 at the buckling stress and below 1.0 in the post-buckled field where the web works on its diagonal-tension reserve.
• Units are SI throughout: Pa shear stresses, m dimensions, N loads, N/m shear flow, degrees for angles. The buckling stress tau_cr is always an input; the module hard-codes no material and no allowables.
• FAR-25 frames the airframe shear-web load context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Collect the web state: the applied shear stress tau (Pa), the elastic shear-buckling stress input tau_cr (Pa, from the plate-buckling leaf or the analysis input), the web depth d between flanges (m), the web thickness t (m), and the tension field angle alpha_deg (degrees, default ALPHA_IDEAL_DEG = 45.0).
2. Run the regime check with tension_field_ratio(tau, tau_cr): a ratio of 0.0 means the web is below its buckling stress and has no diagonal-tension reserve to analyze.
3. Take the tension field angle with tension_field_angle(tau, tau_cr) for the classical 45 degree plane-web value, or pass the inclined angle alpha_deg directly into the stress and load functions.
4. Compute the diagonal web tension stress with web_tension_stress(tau, tau_cr, alpha_deg), which returns sigma_d in Pa above buckling and 0.0 below it.
5. Compute the attachment axial loads with flange_axial_load(tau, tau_cr, alpha_deg, depth_m, web_thickness_m) and end_post_load(tau, tau_cr, alpha_deg, depth_m, web_thickness_m), giving P_f and P_e in N.
6. Compute the rivet shear flows with rivet_shear_flow(tau, tau_cr, alpha_deg, web_thickness_m, member) for member "flange" and member "end_post", giving q in N/m for each attachment.
7. Gate the reserve with margin_against_buckling(tau, tau_cr): the web is in the post-buckled field when the margin is below 1.0, and the reserve check compares the diagonal web tension stress and the attachment loads and flows against the allowables.
8. Confirm every result with the deterministic contract test scripts/test_diagonal_tension_field_webs.py (step 8 confirmation).

Worked example

Rectangular shear web 500 mm between end posts x 300 mm depth between flanges x 1.2 mm thick, applied shear tau = 40 MPa and buckling stress input tau_cr = 18 MPa. Running the module functions gives:

• tension_field_ratio(40e6, 18e6) = 0.55.
• tension_field_angle(40e6, 18e6) = 45.0 degrees.
• web_tension_stress(40e6, 18e6, 45.0) = 44.0 MPa (2 * (40 - 18) MPa at 45 degrees).
• flange_axial_load(40e6, 18e6, 45.0, 0.3, 0.0012) = 7920 N; end_post_load(...) = 7920 N.
• rivet_shear_flow(40e6, 18e6, 45.0, 0.0012, "flange") = 48000 N/m (48 N/mm); same for "end_post".
• margin_against_buckling(40e6, 18e6) = 0.45.
• Applied shear force V = tau * t * d = 14400 N; buckling shear force Vcr = tau_cr * t * d = 6480 N; excess shear = 7920 N, which equals the flange axial load at the ideal angle.
• Angle sensitivity on the same web: alpha 38 degrees gives sigma_d = 45.347 MPa, flange load 10.137 kN, end post load 6.188 kN; at alpha 45 degrees the values are 44.0 MPa and 7.92 kN on both attachments.

Verification

• Confirm tension_field_ratio(40e6, 18e6) = 0.55, that the ratio is 0.0 at tau = tau_cr and in the elastic regime, and that it approaches 1 as tau grows.
• Confirm tension_field_angle is 45.0 degrees from just above tau_cr to 1e6 * tau_cr (continuity of the classical angle).
• Confirm web_tension_stress = 44.0 MPa at the worked example, 45.347 MPa at 38 degrees, 0.0 below tau_cr, and that the 30 and 60 degree fields give equal stress (cot + tan symmetry about 45 degrees).
• Confirm flange_axial_load and end_post_load = 7920 N at 45 degrees (the 45 degree symmetry identity), 10137.1 N and 6187.8 N at 38 degrees, 0.0 below tau_cr, and that the flange load is monotone increasing in tau (4320 N at 30 MPa, 15120 N at 60 MPa).
• Confirm both rivet shear flows = 48000 N/m at 45 degrees, that the flange flow below buckling is the elastic tau * t, and that the end post flow is tau * t in both regimes.
• Confirm margin_against_buckling = 0.45 at the worked example, 1.0 at tau = tau_cr and 0.0 at zero applied shear, and that margin * tau reconstructs tau_cr.
• Confirm every negative tau, zero tau_cr, angle outside (0, 90) degrees, non-positive depth or thickness, and any member other than flange or end_post raises ValueError.
• Run the contract test offline: python3 scripts/test_diagonal_tension_field_webs.py (35 tests, deterministic).

Related leaves

• structures/fem/torsion-shear-flow: the elastic Bredt-Batho shear-flow distribution and shear center below buckling; it stops where the web buckles and this leaf takes over above tau_cr.
• structures/fem/plate-buckling: computes the elastic shear-buckling stress tau_cr that this leaf takes as an input; it does not compute the diagonal-tension state above tau_cr.
• structures/fem/buckling-analysis: general Euler column and panel buckling checks of the same structure, adjacent to the post-buckled web reserve computed here.

Pitfalls

• Applying the diagonal-tension reserve below tau_cr: an elastic web below its buckling stress carries the shear as shear stress, and every diagonal-tension quantity here returns 0.0; only the rivet flows keep the elastic tau * t value.
• Reading the flange rivet flow as tau * t above buckling: once the field forms the flange attachment flow is t * (tau_cr + (tau - tau_cr) * tan(alpha)), which equals tau * t only at 45 degrees and exceeds it for alpha above 45 degrees, so the elastic value is unconservative there.
• Swapping the flange and end post roles: the flange picks up the cot(alpha) component of the diagonal tension over the web depth, the end post the tan(alpha) component, and the end post rivets still carry the full applied shear tau * t even when the web is below buckling.
• Quoting sigma_d as a shear stress: the diagonal tension is a uniaxial tensile stress of magnitude 2 * (tau - tau_cr) at 45 degrees, a factor of two above the excess shear that formed it; check it against the tension allowable, not the shear allowable.
• Mixing unit systems: a thickness entered in mm against Pa stresses and m depths silently shifts loads and flows by decades; keep Pa, m, N and N/m together before quoting the reserve.
• Misreading the buckling margin direction: margin_against_buckling below 1.0 does not mean the web fails, it means the web is in the post-buckled diagonal-tension regime; the reserve check then gates the diagonal tension stress and the attachment loads and flows against their allowables.

Behavior contract (gate 3)

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

python3 scripts/test_diagonal_tension_field_webs.py

The test covers the worked 500 x 300 x 1.2 mm web contract (ratio 0.55, sigma_d 44.0 MPa, flange and end post loads 7920 N, rivet flows 48000 N/m, margin 0.45, V 14400 N and Vcr 6480 N with the excess shear reconstructing the flange load), the elastic-regime zeros below tau_cr, the 45 degree symmetry identity for loads and flows, the cot + tan angle symmetry, the 38 degree angle sensitivity anchors (45.347 MPa, 10.137 kN, 6.188 kN), the tension field ratio approach to 1, the margin inverse relation, the flange load monotonicity in tau, the angle continuity across the post-buckling range, run-to-run determinism, and ValueError rejection of every non-physical input in the validation list.

Compliance

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