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13013 m mellogeorgia tech aerospace

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Unformatted text preview: of respecbve internal torques (direcbons of shear stress not typically of interest) Tr 32T ⇢ T⇢ ⌧max = ⌧= = (3- 11) (3- 13) Ip ⇡ d4 Ip Note 2: When determining angle of twist for enbre bar, need to know proper direcbon of twist within each segment. Must respect the sign of the calculated internal torques. TL = 1/30/13 GIp (3- 15) M. Mello/Georgia Tech Aerospace 17 NONUNIFORM TORSION - case I (cont.) STEP 2: Determine the total angle of twist of one end of the bar with respect to the other Internal torque of ith segment (from eqbm.) total angle of twist = n X i=1 i n X Ti L i = Gi ( I p ) i i=1 Shear modulus of the ith segment Length of the ith segment cc Polar moment of Inerba of the ith segment ²་  PROPER SIGNS OF THE INTERNAL TORQUES MUST BE REPSECTED WHEN APPLYING THIS FORMULA 1/30/13 M. Mello/Georgia Tech Aerospace 18 NONUNIFORM TORSION (case II) II. Bar with conbnuously varying cross secbons and constant torque Note 1: When torque is constant, maximum shear stress always occurs at cross secbon with smallest Diameter. 16T ⌧max = (3- 12) ⇡ d3 min DETERMINE ANGLE OF TWIST: TL = (3- 15) AND CAST IT IN ²་  INVOKE GIp DIFFERENTIAL FORM. Constant Torque ZL ZL T dx = d = (3- 21) GIp (x) o 0 1/30/13 Gently tapered prismabc bar under nonuniform torsion Useful as long as polar moment of inerba expression is not too complicated! M. Mello/Georgia Tech Aerospace 19 NONUNIFORM TORSION (case III) III. Bar with conbnuously varying cross secbons and conbnuously varying torque t( x ) ²་  Bar subjected to distributed torque of intensity [t(x)] per along the axis of the per unit distance ZL T ( x) = t(x)dx 0 ²་  Determine shear stress: ⌧ ( x) = ²་  Determine total angle of twist: Z = o L T ( x ) ⇢( x ) Ip ( x) Z d = L 0 T (x)dx GIp (x) t( x ) (3- 22) NOTES/LIMITATIONS: ²་  Analysis in case of nonuniform torsion valid for linear elasbc materials with circular cross secbons, (solid or hollow). ²་  Equabons also only apply to bars with varying cross secbons characterized by gradual diameter changes (taper angle < 10°). ²་  Stress calculabons are valid in regions away form stress concentrabons (such as where diameter abruptly or where concentrated torques are applied) ²་  Stress concentrabons have very ligle effect on twist angle (φ) 2/1/13 M. Mello/Georgia Tech Aerospace 20...
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