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Unformatted text preview: POSITIVE SHEAR STRESS x tan 2✓s = y (6- 20) 2⌧xy cos(2✓s1 ) = ⌧xy R= 2 s ✓s ✓ sin(2✓s1 = ( x 2 y ◆2 3/25/13 2 (6- 23a) x y 2R (6- 23b) Equa]ons define planes of max. pos. shear stress +⌧2 cos 2✓p1 = sin 2✓p1 y) x ⌧xy R xy x 2R ⌧xy = R y (6- 13a) ✓ s1 = ✓ p 1 45 (6- 24) (6- 13b) M. Mello/Georgia Tech Aerospace 10 MAXIMUM SHEAR STRESS (MAGNITUDE) cos(2✓s1 ) = •  SUBSTITUTE EXPRESSIONS: sin(2✓s1 = ⌧xy R (6- 23a) x y (6- 23b) 2R •  INTO SHEAR STRESS TRANSFORMATION EQUATION: ⌧x1 1 = y s✓ •  YIELDS: ⌧max = x y 2 ◆2 x 2 y sin 2✓ + ⌧xy cos 2 ✓ (6- 6) + ⌧xy (6- 25) (note this = R on previous page) 2 ⌧ •  ANOTER USEFUL REALTION FOR max : •  RECALL: 1 2 3/25/13 = = x x + 2 + 2 y y + s✓ s✓ x y 2 x y 2 ◆2 ◆2 2 +⌧xy 2 +⌧xy (6- 14) subtract (6- 16) M. Mello/Georgia Tech Aerospace ⌧max = 1 2 2 (6- 26) 11 And finally, normal stresses on planes of maximum posiJve shear cos(2✓s1 ) = •  SUBSTITUTE EXPRESSIONS: sin(2✓s1 = •  INTO : x1 = •  TO OBTAIN : x + 2 aver y = + x x y 2 + 2 y ⌧xy R (6- 23a) x y 2R cos 2✓ + ⌧xy sin 2✓ (6- 23b) (6- 5a) (6- 27) •  Stress denoted average only because it takes the average of σx and σy and not because it represents some sort of average approxima]on •  IN PARTICULAR CASES OF UNIAXIAL AND BIAXIAL STRESS, THE PLANES OF MAX. SHEAR STRESS OCCUR AT 45° TO THE X AND Y AXES. •  IN THE CASE OF PURE SHEAR (TORSION) THE MAXIMUM SHEAR STRESSES OCCUR ON THE x and y PLANES 3/25/13 M. Mello/Georgia Tech Aerospace 12 IN- PLANE SHEAR STRESSES •  SO FAR WE HAVE ONLY FOCUSED ON IN- PLANE SHEAR STRESSES •  i.e., shear stress in the xy plane and max shear stress within planes rotated about z axis (figure b) •  As was just noted on the previous page, in the case of uniaxial and biaxial stress states the max. in- plane shear stress is obtained by rota]ng the xyz coordinate system by 45° about the z axis Plane of maximum shear (τmax)z1 along plane rotated by 45° about the z axis z = z1 Principal stresses within plane of rota]on about z axis: 1 6= 0, aver = 3/25/13 x + 2 y 2 6= 0 Normal stress ac]ng on planes of max. pos. shear stress M. Mello/Georgia Tech Aerospace 45° (⌧max )z1 = ± 1 2 2 where 1 , 2 are principal stres...
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This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Institute of Technology.

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