Contents [show]
The problem:
Determine the second moments of area, Ixx and Iyy, for the sections indicated in Figures 2a to 2e.
Tip: It is given that the total height and the area of the 457 x 152 x 52 UB section are: h = 449.8mm and A = 66.6cm2, respectively. Furthermore, the second moments of area are Ixx = 21400cm4 and Iyy = 645cm4.
The solution:
a) i) y1 and y2
y1 = 360 + 7.5 – 273.75 = 93.75mm
y2 = 273.75 – 180 = 93.75mm
ii) Ixx
Ixx = (240 x 153)/12 + 240 x 15 x 93.752 + (10 x 3603)/12 + 360 x 10 x 93.752 = 102.23 x 106 mm4
iii) Iyy
Iyy = (15 x 2403)/12 + (360 x 103)/12 = 17.31 x 106 mm4
b) i) y1, y2 and y3
y1 = 15 + 400 + 7.5 – 234.66 = 187.84mm
y2 = 234.66 – (15+200) = 19.66mm
y3 = 234.66 – 7.5 = 227.16mm
ii) Ixx
Ixx = (240 x 153)/12 + 240 x 15 x 187.842 + (8 x 4003)/12 + 400 x 8 x 19.662 + (180 x 153)/12 + 180 x 15 x 227.162 = 310.37 x 106 mm4
iii) Iyy
Iyy = (15 x 2403)/12 + (400 x 83)/12 + (15 x 1803)/12 = 24.59 x 106 mm4
c) i) y1, y2, y3 and y4
y1 = 15 + 300 + 12 + 5 - 216.63 = 112.37mm
y2 = 12 + 300 + 6 - 216.63 = 101.37mm
y3 = 216.63 – (12+150) = 54.63mm
y4 = 216.63 – 6 = 210.63mm
ii) Ixx
Ixx = (350 x 103)/12 + 350 x 10 x 112.372 + (200 x 123)/12 + 200 x 12 x 101.372 + (8 x 3003)/12 + 8 x 300 x 54.632 + (200 x 123)/12 + 200 x 12 x 210.632= 200.58 x 106 mm4
iii) Iyy
Iyy = (10 x 3503)/12 + 2 x (12 x 2003)/12 + (300 x 83)/12 = 51.74 x 106 mm4
d) i) x1, x2, y1, y2 and y3
x1 = 110 – 7.5 = 102.50mm
x2 = 110 – 7.5 = 102.50mm
y1 = 449.8 + 4 -303.51 = 150.29mm
y2 = 449.8 - 41 -303.51 = 105.29mm
y3 = 303.51 – (449.8/2) = 78.61mm
ii) Ixx
Ixx = (220 x 83)/12 + 220 x 8 x 150.292 + 2 x (15 x 823)/12 + 15 x 82 x 105.292 + (21400 x104) + 6660 x 78.612= 323.57 x 106 mm4
iii) Iyy
Iyy = (8 x 2203)/12 + 2 x [(82 x 153)/12 + (82 x 15 x 102.52)] + 645 x 104 = 39.44 x 106 mm4
e) i) x1, x2, y1, y2 and y3
x1 = 1200/2 + 5 = 605mm
x2 = 1200/2 + 5 = 605mm
y1 = 12 + 500 + 10 – 330.56 = 191.44mm
y2 = 330.56 – (12 +250) = 68.56mm
y3 = 330.56 – 6 = 324.56mm
ii) Ixx
Ixx = (1420 x203)/12 + 1420 x 20 x 191.442 + 2 x (10 x 5003)/12 + 10 x 500 x 68.562 + (1220 x123)/12 + 1220 x 12 x 324.562= 2839.47 x 106 mm4
iii) Iyy
Iyy = (20 x 14203)/12 + 2 x [(500 x 103)/12 + (10 x 500 x 6052)] + (12 x 12203)/12 = 10248.30 x 106 mm4
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