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And z axes are termed principal centroidal

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Unformatted text preview: troid. (when the beam material is linear elas\c (obeys Hooke’s law) and when no axial forces act on the cross sec\on. ) M. Mello/Georgia Tech Aerospace 4 net Fy q (y ) dy dx ²་  We typically consider some load distribu\on q(x) along a beam… ་  and rarely (if ever) consider problems with two orthogonal load distribu\ons like this ²་  nonetheless, the no\on of two load distribu\ons serves to establish the centroid coordinates (x, y ) q ( x) AREA = 2/22/13 R q (x)dx AREA = R dFx = q (y )dy q (y )dy dFy = q (x)dx y REVIEW OF CENTROIDS x M. Mello/Georgia Tech Aerospace 5 REVIEW OF CENTROIDS dFy = q (x)dx •  Ver\cal load on differen\al element: dFy = q (x)dx net Fy •  where q(x) is a load intensity in say, (N/m) •  Consider moment ac\ng on differen\al element (with respect to origin): dM = xq (x)dx Z Z •  Integrate for total moment: dM = xq (x)dx R net then, Fy · x = xq (x)dx R net but, Fy = q (x)dx dy dx R R xq (x)dx q (x)dx dA and so, x = Area •  Centroid (abscissa) defini\on recovered: x = q ( x) dA and so, •  Likewise can show that, y = x AREA = 2/22/13 R q (x)dx dx R R yq (y )dx q (y )dy R R xdA dA •  And so centroid (ordinate) defini\on recovered: R ydA y = R dA M. Mello/Georgia Tech Aerospace 6 CENTROID FOR AREAS BOUNDED BY INTEGRABLE FUNCTIONS x= Qy A = y= Qy A = R R xdA dA R R ydA dA Centroids equal to “first moment” divided by the Area (mathema\cal defini\on) R xdA R x= dA R x0 R xf (x)dx xdA 0 R x= = R x0 dA f (x)dx 0 Consider the simplest case of a rectangular profile: q ( x) f ( x ) = y0 y0 (x, y ) dx 2/22/13 x0 x R x0 xy0 dx 0 R x0 x= y0 dx 0 x 2 y0 x0 x= 0 = 2x 0 y 0 2 Similarly, y= y0 2 (results as expected) M. Mello/Georgia Tech Aerospace 7 CENTROID FOR AREAS WITH IRREGULAR BOUNDARIES x= LEVERAGE THE DISCRETIZED L.H.S VERSION OF THESE EQNS. FOR Qx, Qy Pn x i Ai i x = P=1 n Ai i=1 Pn y Ai i y = P=1 i n Ai i=1 2/22/13 x y •  •  •  •  •  Z Z Qy A dA = dA = Z Z xdA = Qy ydA = Qx Divide geometric figure into discrete elements (rectangles) Area of ith element = ΔAi n = total number of discre\zed elements x is the the x coordinate of the centroid of the ith element i y i is the the y coordinate of the centroid of the ith element M. Mello/Georgia Tech Aerospace 8 Example 12- 1 Given: Parabolic semisegment: ✓ ◆ 2 y = f (x) = h 1 x2 b Determine: centroid loca\on SOLUTION: R Qy R xdA x = A = dA R ydA Qy y = A...
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