ho3 - ∂ w y → Volume wdxdy = ∫∫ − Τ 2 Apply to a...

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MIT - 16.20 Fall, 2002 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 2 φ = 2Gk ∂φ = σ ∂φ = − σ xz x yz y • Rigid rotation of cross-section • Free to warp σ xz , σ yz only nonzero stresses d α Τ = dz GJ GJ = torsional rigidity J = torsion constant Stress resultant = τ = σ zx zy 2 σ + 2 OPEN, THIN-WALLED SECTIONS ( Membrane Analogy ) Same governing equation and B.C. for torsion and pressurized membrane 2 w = p i N w = 0 on contour Paul A. Lagace © 2002 Handout 3-1
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MIT - 16.20 Fall, 2002 Analogy: Torsion Membrane φ w - k p i N 1 2G w x = φ σ x zy = φ σ y zx
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Unformatted text preview: ∂ w y → Volume wdxdy = ∫∫ − Τ 2 Apply to a narrow rectangular cross-section bh 3 J = 3 2T (local axes ) τ res ≈ J x = σ yz ( σ xz = 0 ) apply to: CLOSED, THICK-WALLED SECTIONS φ = C 1 on one boundary φ = C 2 on one boundary Paul A. Lagace © 2002 Handout 3-2 MIT - 16.20 Fall, 2002 ∫ τ ds = 2AGk on any closed boundary THIN-WALLED CLOSED SECTIONS τ res constant through thickness τ ds = 2GKA ∫ “shear flow”: q = τ t A = enclosed area Bredt’s formula T τ resultant = 2 At 4 A 2 J = t ds ∫ Note : Free-to-warp assumptions violated near end constraints for all torsion problems. Paul A. Lagace © 2002 Handout 3-3...
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This note was uploaded on 02/03/2012 for the course AERO 16.20 taught by Professor Paullagace during the Fall '02 term at MIT.

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ho3 - ∂ w y → Volume wdxdy = ∫∫ − Τ 2 Apply to a...

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