LECTURE+11+COE-3001-A

Zl zl t dx d

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Unformatted text preview: 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 elasVc materials with circular cross secVons, (solid or hollow). ²་  EquaVons also only apply to bars with varying cross secVons characterized by gradual diameter changes (taper angle < 10°). ²་  Stress calculaVons are valid in regions away form stress concentraVons (such as where diameter abruptly changes or where concentrated torques are applied) ²་  Stress concentraVons have very likle eﬀect on twist angle (φ) 2/1/13 M. Mello/Georgia Tech Aerospace 5 STRESSES AND STRAINS IN PURE SHEAR STRESSES ACTING ON A STRESS ELEMENT CUT FROM A BAR IN TORSION ELEMENT IN STATE OF PURE SHEAR ²་  only stresses are shear stress τ on the four side faces ²་  lengths of the sides of the element do not change NOTE/RECALL ORIGINAL SIGN CONVENTON: ²་  ⌧ > 0 shear force acts in posiVve direcVon on a posiVve face (a posiVve face has unit normal directed along posiVve direcVon of a coordinate axis) ²་  ⌧ < 0 shear force acts in negaVve direcVon on a posiVve face ²་  In the case above, all shear stresses acVng on the stress element are posiVve ²་  DirecVon of torques chosen so that resultant shear stresses are posiVve. ²་  THIS IS ALL ABOUT TO CHANGE 2/1/13 M. Mello/Georgia Tech Aerospace 6 STRESSES ACTING ON INCLINED PLANES X Y ⌧✓ A0 sec ✓ Free body diagram ELEMENT IN PURE SHEAR STRESSES ON TRIANGULAR STRESS ELEMENT FORCES ON TRIANGULAR STRESS ELEMENT ²་  REMEMBER THAT THE ELEMENTS ARE REALLY 3D A0 / cos ✓ = A0 sec ✓ NO STRESS ON FRONT A0 AND REAR (Z) FACES ²་  NOW RESET SIGN CONVENTON FOR NORMAL AND SHEAR STRESSES ON INCLINED SURFACES •  Diﬀerent than sign convenVon for ordinary shear stresses (τ) that act on sides of rectangular elements oriented to a set of xy axes. •  NORMAL STRESSES (σΘ > 0) IN TENSION •  SHEAR STRESSES (τΘ >0) IF TENDENCY IS TO ROTATE ELEMENT CCW 2/1/13 M. Mello/Georgia Tech Aerospace 7 STRESSES ON INCLINED SURFACES FOR AN ELEMENT IN PURE SHEAR (cont.) X Y X 0...
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