Chapter 7b, Torsion. τ = 0. τ T. T τ D'' A'' C'' B'' 180 -rotation around axis C'' B'' D'' A'' A'' D'' 180 -rotation upside-down C'' B''

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1 Chpter 7b, orsion τ τ τ ' D' B' C' '' B'' B'' D'' C'' 18 -rottion round xis C'' B'' '' D'' C'' '' 18 -rottion upside-down D'' stright lines in the cross section (cross sectionl projection) remin stright the whole cross section (cross sectionl projection) rottes b the sme mount θ ( x,, ) θ ( x) β x

2 Wrping Function Sint-Vennt theor,w,v x,u τ x d τ x θ β dθ dx,w '(+v,+w) θ O -v w (,),v O + r, r w rθ, v r rθ u βψ (, ), v θ β x, w θ β x Ψ Ψ (, ) is the so-clled wrping function u ε x x v w w v, ε, ε, γ + v u γ x + β( ) x w u γ x + β ( + ) x τ x Gβ( ), τ x Gβ ( + )

3 comptibilit reltionship utomtic (strted from continuous, differentible displcements) equilibrium equtions σ τ x x τ + + x x τx σ τ + + x τ τ x σ + + x τ x τ + + x τ x τx G + β ( ) + ( + ) Ψ Ψ + Ψ boundr conditions O,w d τ x -d ds s τ x n d,v d d boundr norml unit vector n n j + n j ds ds d d boundr tngent unit vector s s j + s j + ds ds

4 sher trction d d j + d d( τ x j + τx ) d d d d dn, i.e., ( τ x j + τx ) ( j ) τx τ x ds ds ds ds d d d d d s, i.e., ( τ x j + τ x ) ( j + ) i( τx τ x ) ds ds ds ds τ d d x x ds τ ds τ x Gβ( ) τ x Gβ ( + ) d d d d + ds ds ds ds d d 1 d ( + ) ds ds ds torque ( τ τ ) d x x G ( β + + ) d G ( β ) d + G β J J ( + ) d I + I

5 Summr Sint-Vennt wrping function, Ψ sher stress τ x Gβ( ), τ x Gβ ( + ) comptibilit reltionship utomtic equilibrium equtions Ψ boundr conditions τ d d x x ds τ ds d d 1 d ( + ) ds ds ds torque ( τ τ ) d x x G ( β ) d + G β J Prndtl stress function, φ sher stress comptibilit reltionship equilibrium equtions φ τ x, τ x φ Gβ utomtic φ boundr conditions dφ torque φd

6 Sint-Vennt Wrping Function for n Ellipticl Cross Section,w -τ x O τ x b,v equilibrium eqution wrping function Ψ Ψ sher stresses τ x Gβ( ) Gβ( 1) τ x Gβ ( + ) Gβ ( + 1) boundr condition τx τx d d + f(, ) 1 b τx τ + x 1 1 f f df d + d d + d b d d b

7 b b + b b Ψ + b sher stress components Gβ τ x Gβ( ) + b Gβb τ x Gβ ( + ) + b sher stress Gβ Gβb Gβ x x ( ) ( b ) τ τ + τ b + b + b τ τ ( + ) ' b' τ / / Gβ b + b, ', b b' b mximum sher stress ' b' b τ mx{ τ } mx x Gβ b + b

8 torque ( τx τx ) d Gβb Gβ Gβb Gβ ( + ) d I + I + b + b + b + b I πb, b I π b π b Gβ GβJ + b t J π b t + b Gβ + b π b τ x, π b τ x, π / / b τ, nd τ mx π b πb circulr cross section ( b) Jt π J + ττ + ' b' π r τ, where J r +

9 Prndtl Stress Function φ φ (, ) sher stress equilibrium equtions φ τ x, τ x φ σ τ x x τ + + x x τx σ τ + + x τ τ x σ + + x τ x τ + + x τ x + τ x φ φ utomtic! comptibilit reltionship three displcements! indirect solution using the Sint-Vennt wrping function τ x Gβ( ) nd τ x Gβ ( + ) τ x τx Ψ Ψ Gβ( 1 1) Gβ φ τ x, τ x φ φ Gβ

10 boundr conditions τ d d x x ds τ ds φ φ d + d dφ φ constnt for solid cross section with closed loop boundr line one cn choose φ torque ( x x ) d φ d φ τ τ d D d O d τ x τ x B d C For solid cross section with closed loop boundr line nd φ on the boundr: φ d D B φ d d D B D B B [ ] d φ φ d φ d d φ d C C C φ d B D φ d d B D B D D [ ] C d φ φ d φ d d φ d C C C φd

11 If φ φ on s : ( φ φ ) d When the cross section is bordered b multiple discontinuous boundr lines, φd cnnot be used directl. s 1 s ssume tht the he ctul cross section 1 is divided into solid cross sections 1 nd bordered b s 1 nd s, respectivel, nd both s 1 nd s re constnt contour lines of the sme stress function φ: s 1 s 1 s 1 s φ on s nd φφ on 1 s φd, 1 1 ( φ φ ) d φd ( φ φ ) d 1 1 φ d + φ

12 Prndtl Stress Function for n Ellipticl Cross Section boundr eqution boundr condition + b 1 φ φ C (1 ) b comptibilit reltionship 1 1 φ C( + ) Gβ b C Gβ b + b Gβ b φ (1 ) + b b φ Gβ τ x + b nd φ Gβb τ x + b torque Gβ b φ d (1 ) d b b + Gβ b 1 1 Gβ b 1 π b 1 πb ( I ) ( ) I πb + b b + b b b πgβ b + b Gβ + b π b τ x nd π b τ x π b

13 Prndtl Stress Function for n Equilterl ringle Cross Section boundr equtions boundr condition φ φ ( + )( + )( + ), where is n unnown constnt to be determined from φ ( + )[9 ( ) ] φ ( ) φ G β. φ ( ) φ 6 G β Gβ 18

14 torque / / / φd φdd / Gβ 8 Gβ φ ( + )[9 ( ) ] 5 9 stress φ [9 ( ) ] ( )( ) 5 τ x τ x 9 ( ) ( + )( ) ( ) 5 τ x + φ φ τ x ( + ) [9 ( ) ] τ x ( + ) 5

15 orsionl Stiffness β GJ t prismtic br of ellipticl cross section: πg b + b Sint-Vennt's pproximte formul for prismtic brs of rbitrr cross section SV G π J prismtic br of ellipticl cross section: π b 1 J I + I π b( + b ) SV Gπ b πg b 1 π π b( + b ) + b prismtic br of equilterl tringle cross section: J 5 19 SV G 9 G π J 8π.197G

16 G.17G β 8 prismtic br of squre cross section: J I + I 6 SV G π.15g.11g thin prismtic br of rectngulr cross section ( >> b): b J I + I b 1 SV Gb π.gb.gb

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