Y0 x0 x 0 2x 0 y 0 2 similarly y y0 2 results

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Unformatted text preview: = R dA ①  COMPUTE (A) i.e., the area under the curve ◆ Zb Zb ✓ x2 h1 A= f (x)dx = dx b2 0 0 A= 2bh 3 ②  COMPUTE 1st moments Qx, Qy in order to determine : x= 2/22/13 Qy A M. Mello/Georgia Tech Aerospace y= Qx A 9 Qy x= A Z Example 12- 1 (cont.) b Qy = xdA Z0 b ✓ Qy = xh 1 ◆ x2 b2 h dx = b2 4 0 b2 h 4 2bh 3 x= x= Qy A 3b 8 y •  Now how to determine ? •  One way is to take an area element dA as a horizontal strip r y of height dy and and width x = b 1 Zb h y •  But there is an easier way… Q x = dA (con\nue to integrate over x) 2 0 (SEE SLIDE 5) 2/22/13 M. Mello/Georgia Tech Aerospace 10 Qx = y= y= y= Z b 0 ✓ y h1 2 ◆ x2 4bh2 dx = b2 15 Example 12- 1 (finish) Qx A 4bh2 15 2bh 3 2h 5 2/22/13 Qx A ✓ ◆ 3b 2h , CENTROID COORDINATES: 85 (in agreement with appendix D case 17) M. Mello/Georgia Tech Aerospace 11 2/22/13 M. Mello/Georgia Tech Aerospace 12 2/22/13 M. Mello/Georgia Tech Aerospace 13 2/22/13 M. Mello/Georgia Tech Aerospace 14 2/22/13 M. Mello/Georgia Tech Aerospace 15 2/22/13 M. Mello/Georgia Tech Aerospace 16 2/22/13 M. Mello/Georgia Tech Aerospace 17 CENTROIDS OF COMPOSITE AREAS In some cases the centroid loca\on is quite obvious: Centroid at center of symmetry centroid Area with one axis of symmetry Area with two axes of symmetry (centroid At the intersec\on) Area symmetric about a point •  AREAS AND FIRST MOMENTS OF COMPOSITE AREAS CALCULATED BY SUMMING THE CORRESPONDING PROPERTIES OF THE COMPONENT PARTS. Pn Pn y Ai x i Ai i i P=1 y = P=1 i x= n n Ai i=1 Ai i=1 •  COMPOSITE AREA DIVIDED INTO A TOTAL OF n PARTS •  Ai IS THE AREA OF THE ith PART 2/22/13 M. Mello/Georgia Tech Aerospace 18 EXAMPLE (sec\on 12.3) A1 A2 Coordinate system imposed L- BRACKET STEP 1: Calculate area and centroid loca\ons of each rectangular sec\on th Area of i sec\on A1 = bt A2 = ( c 2/22/13 t) t x- coord. of centroid t x1 = 2 c+t x2 = 2 M. Mello/Georgia Tech Aerospace y- coord. of centroid b y1 = 2 t y2 = 2 19 EXAMPLE (sec\on 12.3) (cont.) STEP 2: Calculate total area A = A1 + A2 A = bt + (c t)t A = t(b + c t) Step 3: Compute 1st moments of the composite area (numerators) and then calculate x , y Qy Qx P2 P2 x i Ai y Ai i i x = P=1 y = P=1 i n n Ai i=1 i=1 Ai Qy bt + c2 x= = A 2(b + c Qx bt + ct y= = A 2(b + c 2/22/13 t2 t) t2 t) t2 (b + ct 2 t Qy = x1 A1 + x2 A2 = (bt + c2...
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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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