The centroid locaon is quite obvious centroid

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Unformatted text preview: 2 Q x = y 1 A1 + y 2 A2 = M. Mello/Georgia Tech Aerospace t2 ) t2 ) 20 Some final notes on Centroids •  When a composite area divided into only 2 parts, the centroid lies along line joining centroids C1 and C2. •  Methodology for calcula\ng centroid can be applied to parts with “holes” •  Handle the absence of an area by subtrac\on •  Or treat the blue regions above as posi\ve areas and the gaps as nega\ve areas 2/22/13 M. Mello/Georgia Tech Aerospace 21 PROBLEM 4.3- 11 “DISTRIBUTED LIFT LOAD ACTING ON AIRPLANE WING” FREE- BODY DIAGRAMS/STATIC CALCULATIONS: 1600N/m M c V DETERMINE: (A)  SHEAR FORCE AT INBOARD END OF WING (B)  BENDING MOMENT AT INBOARD END OF WING SOLUTION: X 2.6m 2.6m 1m Fv = 0 V + 0.5(1600 V= 900N/m 900)(2.6) + (900)(5.2) + 0.5(900)(1) = 0 6.04kN (minus sign means guessed wrong sign for shear force) X Mc = 0 M + 0.5(700)(2.6)[2.6/3] + (900)(5.2)[5.2/2] + 0.5(900)(1)[5.2 + 1/3] = 0 M = 1.54 ⇥ 104 N m 2/22/13 M. Mello/Georgia Tech Aerospace 22 Homework problem 2/22/13 M. Mello/Georgia Tech Aerospace 23 Moment- Curvature Rela\onship Beam cross sec\on •  x- z plane coincident with neutral surface •  Beam axis running out of the page •  Normal stress distribu\on •  Beam axis running along x ²་  Balance of moments within the beam demands that the moment resultant induced by σx must equal the bending moment M. ²་  Assuming a posi\ve bending moment, we expect a compressive stress for y > 0 along the beam. ²་  The element of force σxdA ac\ng on the element dA is in the nega\ve direc\on when σx < 0. ²་  Hence, dM = 2/22/13 x ydA M. Mello/Georgia Tech Aerospace 24 Moment- Curvature Rela\onship (cont.) •  Integra\ng over all elements over the en\re cross sec\on: M = Z x ydA A Z •  Subs\tu\ng for stress x = E y then leads to: M = E y 2 dA A M = EI zz (5- 10) Z E y 2 dA •  Recall, = 1/⇢ and so equivalently we may write: M = ⇢A What is this? Z •  MOMENT OF INERTIA: Izz = y 2 dA (5- 11) (unique property of the cross sec\onal area) A •  Izz = moment of iner\a of the cross sec\onal area with respect to the z axis (neutral axis) •  We can carry this quan\ty as the variable I in the moment- curvatu...
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