Mass Timber & Structural Wood Design, NDS Hankinson Grain-Angle Bearing Formula
Hankinson Wood Grain-Angle Bearing Strength Formula
Calculate NDS Hankinson Inclined-to-Grain Bearing Strength F_theta (MPa), Retained Parallel Strength (%), Allowable Joint Bearing Load (kN), and Naive Linear Error (MPa).
Never linearly interpolate timber strength between 0° and 90°—wood crushes sideways like bundled drinking straws at 35°!
Structural Douglas-Fir, Southern Pine, and Glulam timber is strongly **orthotropic**: cellulose tracheid fibers carry `3.5x–5x` higher compression **Parallel to Grain `F_c`** (MPa, `0°` end-grain column bearing) than **Perpendicular to Grain `F_{c,perp}`** (MPa, `90°` sill-plate side crushing)! When a heavy-timber roof truss rafter meets a tie-beam birdsmouth notch or split-ring bolt at **Load-to-Grain Angle `theta`** (`0° <= theta <= 90°`), naive linear interpolation (`F_c - (F_c - F_{c,perp}) * (theta / 90)`) dangerously **overpredicts strength by `+33%` to `+52%`**! Under the **AWC National Design Specification (NDS) / Eurocode 5 Hankinson Equation**, inclined bearing strength follows the harmonic trig surface **`F_theta = (F_c * F_{c,perp}) / (F_c * SIN(theta)^2 + F_{c,perp} * COS(theta)^2)`**, multiplying Contact Notch Area `A_b` (mm²) for **Allowable Joint Bearing Load `(F_theta * A_b) / 1,000`** (`kN`)!
Verify on a Douglas-Fir Glulam Truss Heel Notch—Parallel-to-Grain_Strength `F_c` = 13.50 MPa (B2), Perpendicular-to-Grain_Strength `F_{c,perp}` = 3.80 MPa (B3), Rafter_Chord_Angle `theta` = 35.0° (B4), and Birdsmouth_Bearing_Area `A_b` = 9,600 mm² (`120 x 80` mm, B5): Hankinson Inclined Strength drops to `7.34` MPa (E2 = `54.35%` of `F_c` in E3), Allowable Joint Load equals `70.44` kN (E4), whereas Naive Linear Interpolation falsely predicts `9.73` MPa (E5 = `+32.6%` unsafe overprediction)!
| AWC NDS Hankinson Timber KPI (Fc=13.5 MPa, Fc_perp=3.8 MPa, theta=35°, Ab=9,600 mm²) | Formula | Verified Output |
|---|---|---|
| Hankinson Strength F_theta (MPa) | (Fc * Fcp) / (Fc*SIN(rad)^2 + Fcp*COS(rad)^2) | 7.34 MPa (4.63 MPa @ steep 60° pitch) |
| Retained Parallel Strength (%) | (Hankinson_F_theta / Parallel_Fc) * 100 | 54.35% (34.31% @ steep 60° pitch) |
| Allowable Notch Bearing Load (kN) | (Hankinson_F_theta_MPa * Area_mm2) / 1000 | 70.44 kN (44.47 kN @ steep 60° pitch) |
| Naive Linear Interpolation (MPa) | Fc - (Fc - Fcp) * (Angle_deg / 90) (UNSAFE!) | 9.73 MPa (+32.6% unsafe overprediction!) |
AWC NDS Hankinson Orthotropic Grain-Angle Bearing Formula
Validate `Perp_Strength F_{c,perp} > 0.5` MPa, `Parallel F_c > F_{c,perp}`, `Grain_Angle theta` in `[0°, 90°]`, and `Bearing_Area A_b > 100` mm²! In a shared structural-calculation workbook, a timber detailer can `@mention` the engineer of record in a whole-cell comment on `E2` to verify whether wet-service factor `C_M` has already been applied to `B2:B3`.
=LET(fc_p, B2, fp_p, B3, ag_d, B4, ab_m, B5, IF(OR(fp_p <= 0.5, fc_p <= fp_p, ag_d < 0, ag_d > 90, ab_m <= 100), "Invalid Timber Input", ROUND(fc_p - (fc_p - fp_p) * (ag_d / 90), 2)))Steep 60° A-Frame Strut Side-Grain Crushing Check (Set Grain Angle B4 = 60°)
Steepen the timber strut angle to `theta` B4 = 60°: Hankinson Strength E2 plunges to `4.63` MPa (`34.31%` retained in E3, Joint Capacity = `44.47` kN in E4)—nearly `2.40` MPa below Naive Linear `7.03` MPa (E5), requiring a steel shoe washer or widened bearing seat!
=IF(ISTEXT(E5), E5, LET(rd_a, B4 * (PI() / 180), ROUND((B2 * B3) / (B2 * (SIN(rd_a) ^ 2) + B3 * (COS(rd_a) ^ 2)), 2)))Try the example
Use the original values or change them to test your understanding.