Mechanical & Structural Engineering
Saint-Venant Rectangular Bar Torsion Constant and Peak Shear Stress
Avoid the circular polar-inertia J_p trap on rectangular shafts by computing Saint-Venant torsional stiffness K_tor, twist angle, and midpoint long-side peak shear stress.
Never use circular polar inertia J_p on a flat or rectangular bar in torsion
Circular shafts remain plane under twist, but non-circular rectangular bars (long width a >= short thickness b) warp out of plane — with zero shear stress at the four corners and peak shear stress at the midpoint of the LONG face! Substituting polar inertia J_p = a*b*(a^2 + b^2)/12 into angle-of-twist T*L / (G*J) overstates torsional stiffness by 5x on a 4:1 flat bar.
Test a steel flat bar with width a = 0.08 m (B2, 80 mm), thickness b = 0.02 m (B3, 20 mm), torque T = 320 N*m (B4), shear modulus G = 80 GPa (B5), and length L = 1.0 m (B6). Roark’s peak shear stress formula tau_max = T * (3*a + 1.8*b) / (a^2 * b^2) gives 320 * (0.24 + 0.036) / (0.0064 * 0.0004) = 88.32 / 2.56e-6 Pa = 34.5000 MPa! Saint-Venant torsion constant K_tor = 17.9744 cm^4 vs naive J_p = 90.6667 cm^4 (a 5.04x error!), yielding true twist 1.2751 deg vs naive 0.2528 deg.
| Rectangular Torsion Metric | Saint-Venant / Roark Formula | Sample Output (80 x 20 mm) |
|---|---|---|
| Peak Shear Stress (Long Midpoint) | 10^-6 * T * (3*a + 1.8*b) / (a^2 * b^2) | 34.5000 MPa |
| Torsion Constant K_tor (cm^4) | 10^8 * a * b^3 * (1/3 - 0.21*(b/a)*(1 - b^4/(12*a^4))) | 17.9744 cm^4 |
| Naive Polar Inertia J_p (Bug) | 10^8 * a * b * (a^2 + b^2) / 12 | 90.6667 cm^4 (5.04x too stiff!) |
| True Twist Angle theta (deg) | DEGREES(T * L / ((G_GPa * 10^9) * K_tor_m4)) | 1.2751 deg (vs 0.2528 deg bug) |
Peak Rectangular Wall Shear Stress (MPa) Formula
Automatically orient a_lg = MAX(B2, B3) and b_sh = MIN(B2, B3) inside LET so swapping width and thickness cells never breaks the aspect-ratio assumption a >= b.
=LET(a_lg, MAX(B2, B3), b_sh, MIN(B2, B3), t_nm, B4, IF(OR(b_sh <= 0, t_nm < 0), "Invalid Bar Input", ROUND((10^-6) * t_nm * (3 * a_lg + 1.8 * b_sh) / ((a_lg^2) * (b_sh^2)), 4)))True Twist Angle (Degrees) via Saint-Venant K_tor Formula
Compute K_tor = a * b^3 * (1/3 - 0.21*(b/a)*(1 - b^4/(12*a^4))) in m^4 and convert radians T*L / (G*K_tor) via DEGREES. Test orientation symmetry by swapping B2 = 0.02 and B3 = 0.08: MAX/MIN orientation preserves both 34.5000 MPa and 1.2751 deg.
=LET(a_lg, MAX(B2, B3), b_sh, MIN(B2, B3), t_nm, B4, g_pa, B5 * 10^9, l_m, B6, IF(OR(b_sh <= 0, g_pa <= 0, l_m <= 0), "Invalid Bar Input", LET(k_tor, a_lg * (b_sh^3) * ((1 / 3) - 0.21 * (b_sh / a_lg) * (1 - (b_sh^4) / (12 * (a_lg^4)))), ROUND(DEGREES((t_nm * l_m) / (g_pa * k_tor)), 4))))Try the example
Use the original values or change them to test your understanding.