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LECTURE+10+COE-3001-A

# Is therefore maximum at the outer surface

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Unformatted text preview: TENSILE FAILURE OF A CIRCULAR CYLINDER LOADED IN PURE TORSION TENSILE AND COMPRESSIVE STRESSES ACTING ON A STRESS ELEMENT ORIENTED AT 45° TO THE LONGITUDINAL AXIS 1/30/13 ²་  MATERIAL WEAKER IN TENSION THAN IN SHEAR ²་  FAILURE OCCURS IN TENSION ALONG A HELIX INCLINED AT 45° TO THE LONGITUDINAL AXIS THIS IS ALSO EXACTLY HOW CHALK FAILS! (WHEN YOU TWIST IT) hgp://www.youtube.com/watch?v=wJk9- Gi5DEw&playnext=1&list=PL267E59746C5E3C45&feature=results_main (pure Torsion Test fast forward to 1:40 ) M. Mello/Georgia Tech Aerospace 11 RELATION BETWEEN APPLIED TORQUE AND RESULTING SHEAR STRESS r Applied torque resulbng shear stress shear force acbng on diﬀerenbal element of area A ⌧ ⇢dA = dM moment of the diﬀerenbal shear force ✓ ◆ ⌧max 2 ⇢ (subsbtuted for the shear stress) dM = ⇢ dA ⌧max ⇢dA = dM r r Z Z ⌧max 2 (3- 8) resultant moment (equal to the torque T) T= dM = ⇢ dA r A A what is this interesbng integral?? ⌧ dA = dV 1/30/13 M. Mello/Georgia Tech Aerospace 12 RELATION BETWEEN APPLIED TORQUE AND RESULTING SHEAR STRESS (cont.) I. DEFINE POLAR MOMENT OF INERTIA: dA = (⇢d )d⇢ Z ⇢2 dA (3- 9) Ip = A ⇢d d⇢ Z 2⇡ Z r2 Ip = ⇢2 ⇢d⇢d ⇢ r1 0 ⇡4 ⇡ d4 (3- 10) Polar moment of inerba of a circular cross secbon Ip = 2 r = 32 II. THE POLAR MOMENT OF INERTIA PROVIDES A CRITICAL LINK BETWEEN THE APPLIED TORQUE AND THE SHEAR STRESS WITHIN THE CIRCULAR BAR ⇢ Z Z III. Finally, since ⌧ = ⌧max ⌧max r ✓ ◆ T= dM = ⇢2 dA (so we know what ⇢ 16T r this is now) A A it follows that ⌧ = r ⇡ d3 ⌧max ⇢ 16T T= Ip or, ⌧ = r d/2 ⇡ d3 Tr 32T ⇢ T⇢ 16T ⌧max = Hence, ⌧ = = ⌧max = (3- 11) (3- 12) (3- 13) Ip ⇡ d4 Ip ⇡ d3 1/30/13 M. Mello/Georgia Tech Aerospace 13 RELATION BETWEEN APPLIED TORQUE AND THE ANGLE OF TWIST d =⇢ ²་  RECALL THE FUNDAMENTAL REALTIONSHIP: dx d ²་  This naturally reminds us that , ⌧ = G = G⇢ dx T⇢ ²་  Subsbtute into ⌧ = (3- 13) (from previous page) Ip d T ²་  To obtain: = (3- 14) (rate of change of twist angle (φ) with respect to axial dx GIp distance length (x)) •  Proporbonal to applied torque (T) •  Inversely proporbonal to (GIp) = [torsional rigidity] d = constant = ²་  Recall for a bar in pure torsion:...
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