Tools › Structures › Beams › Section Properties › Example
Timoshenko and Goodier: torsion constants of the square, circular and tubular sections
| Tool | Section Properties |
| Source | Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 10, torsion of rectangular bars (table of the factors k for a/b = 1) and of circular shafts. |
| Level | Undergraduate |
Problem
Find the Saint-Venant torsion constant J of a 2 × 2 in square bar, of a solid circular bar of 1 in radius and of a tube with outer radius 2 in and inner radius 1.5 in, and locate the shear centre of the square.
Inputs to Enter
- Square: polygon (0, 0), (2, 0), (2, 2), (0, 2) in; grid 0.25 in
- Circle r = 1 in and tube r_o = 2 in, r_i = 1.5 in drawn as 180-gon / 120-gon templates
Answers
- Square: J = 0.1406 a⁴ = 2.2496 in⁴ (Timoshenko and Goodier k = 0.1406; tool within 1 %); shear centre at the centre (1, 1) in
- Circle: J = π r⁴ / 2 = 1.5708 in⁴ (within 1 %); warping constant C_w = 0
- Tube: J = π (r_o⁴ − r_i⁴) / 2 = 17.18 in⁴ (within 2 %); C_w = 0, shear centre at the centre
- Thin I-section b = 10, t_f = 0.4, h = 20, t_w = 0.4 in: C_w = t_f b³ h_0² / 24 = 6,403 in⁶ with h_0 = 19.6 in (thin-walled formula; FEM warping solution within 3 %)
Notes
Library checks (test_section_warping.php and the torsion reference shapes of the changelog: square 0.1406, circle π/2, tube 1.4726 per unit) rather than a dimensioned textbook problem; the tool's default section is its own. J from the stress-function grid converges from below and J_ω from the warping function from above.