16.20 HANDOUT #3 Fall, 2002 Review of “Basic” Torsion Theory SOLID CROSS-SECTIONS (St. Venant Theory) φ = 0 on contour (free boundary) Τ = − 2 ∫ ∫ φ dxdy ∇2φ = 2Gk ∂φ = σ yz ∂x ∂φ = − σ xz ∂y • Rigid rotation of cross-section • Free to warp • σxz, σyz only nonzero stresses dα Τ = dz GJ GJ = torsional rigidity J = torsion constant Stress resultant = τ = σ 2zx + σ 2zy OPEN, THIN-WALLED SECTIONS (Membrane Analogy) Same governing equation and B.C. for torsion and pressurized membrane ∇ 2w = − pi N w = 0 on contour Analogy: Membrane Torsion w → φ pi → -k N → 1 2G → ∂φ = σ zy ∂x → ∂φ = − σ zx ∂y ∂w ∂x ∂w ∂y Volume = ∫∫ wdxdy − → Τ 2 Apply to a narrow rectangular cross-section bh 3 J = 3 (local axes) apply to: CLOSED, THICK-WALLED SECTIONS φ = C1 on one boundary φ = C2 on one boundary τ res ≈ 2T x = σ yz J ( σ xz = 0) ∫ τds = 2AGk on any closed boundary THIN-WALLED CLOSED SECTIONS τ res constant through thickness ∫ τds = 2GKA “shear flow”: q = τ t A = enclosed area Bredt’s formula τ resultant = T 2 At 4 A2 J = ds ∫ t Note: Free-to-warp assumptions violated near end constraints for all torsion problems.

16.20 Handout #3: Review of Basic Torsion Theory

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