LECTURE+28+COE-3001-A

Min principal stresses are known s 2

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Unformatted text preview: as either σ1 or σ2 (by comparing to values obtained using (6- 17)) 3/25/13 M. Mello/Georgia Tech Aerospace 6 Resolving the double angle ambiguity: method 2 y Method 2: •  The only angle that sa]sfies BOT(6- 13a) AND (6- 13b) is Θp1 (maximum principal stress direc]on) •  Sign combinaJon of Cosine and Sine uniquely determine the quadrant cos 2✓p1 = sin 2✓p1 x y (6- 13a) 2R ⌧xy = R 1 cos ✓ 0 0 cos ✓ 1 0 sin ✓ 1 (6- 13b) 1 cos ✓ 0 1 sin ✓ 0 0 sin ✓ 1 x 0 cos ✓ 1 1 sin ✓ 0 ²་  The 3rd principal stress: •  Finally, note that there is a 3rd principal stress which we tend to overlook when focusing on 2D plane stress. •  We should also strictly include the normal stress ac]ng on the z- face of the element (for plane stress σ3=0) •  However in some instances we my have σ3=k (constant). •  This is then classified as generalized plane stress •  END RECAP OF LECTURE 27 3/25/13 M. Mello/Georgia Tech Aerospace 7 BEGIN LECTURE 28 x1 = x y1 = x + 2 + 2 ⌧x 1 y 1 = y + y y x y 2 (6- 5a) cos 2✓ (6- 5b) 2 2 x cos 2✓ + ⌧xy sin 2✓ y x ⌧xy sin 2✓ sin 2✓ + ⌧xy cos 2✓ Resta]ng the transforma]on equa]ons here for the sake if convenience and easy recall (6- 6) ²་  Determine planes of maximum shear stress: Differen]ate (6- 6) and set equal to zero! ²་  maximum shear stress (with respect to Θ) and how to determine the direc]on of maximum shear stress: x y tan 2✓s = (6- 20) 2⌧xy ²་  DEFINES ORIENTATION OF PLANES OF MAX, POS. AND NEG. SHEAR STRESS ²་  (6- 20) yields one angle between 0° and 90° and a second angle between 90° and 180° 3/25/13 M. Mello/Georgia Tech Aerospace 8 DETERMINING PLANES OF MAXIMUM SHEAR STRESS ²་  maximum shear stress and how to determine the direc]on of maximum shear stress: x y tan 2✓s = 2⌧xy •  DEFINES ORIENTATION OF PLANES OF MAX, POS. AND NEG. SHEAR STRESS •  (6- 20) yields one angle between 0° and 90° and a second angle between 90° and 180° x y 2⌧xy tan 2✓s = (6- 11) ⌧ WITH tan 2✓p = ²་  COMPARE: (6- 20) 2 xy x y 1 tan ✓s = (6- 21) tan ✓p ✓s = ✓p ± 45 (6- 22) •  Planes of maximum shear stress occur at 45° to the principal planes defined by Θp1, Θp2 3/25/13 M. Mello/Georgia Tech Aerospace 9 PLANES OF MAXIMUM...
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