HW6 - 1 4.18 A beam of circular cross-section radius 3a has...

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1 4.18. A beam of circular cross-section, radius 3 a has two eccentric holes of ra- dius a as shown in Figure P4.18. 2 3 a 2 3 a O 3 a x y a a Figure P4.18 Find the centroid of the section and the second moments of area, I x , I y , I xy about centroidal axes.
2 It follows that A = 9 π a 2 - π a 2 - π a 2 = 7 π a 2 A ¯ x = 9 π a 2 ( 0 ) - π a 2 ( 0 ) - π a 2 parenleftbigg 3 a 2 parenrightbigg = - 3 π a 3 2 A ¯ y = 9 π a 2 ( 0 ) - π a 2 parenleftbigg 3 a 2 parenrightbigg - π a 2 ( 0 ) = - 3 π a 3 2 and hence the centroid has the coordinates ¯ x = ¯ y = - 3 π a 3 2 × 1 7 π a 2 = - 3 a 14 . We can now construct the last two rows of the table, noting for example that ¯ x 3 - ¯ x = 3 a 2 - parenleftbigg - 3 a 14 parenrightbigg = 12 a 7 . The second moments of area are then given by equations (4.36–4.38) as I x = I y = 81 π a 4 4 - π a 4 4 - π a 4 4 + 9 π a 2

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