Question

For the composite area below, it is known that Ix = 1.26 x 106 in4, 1y = 6.55 x 105 in, and Ixy = -1.02 x 10% in4. a) Compute the moments of inertia Ixı, Iyı, and product of inertia Izry, if 0 = 30°. Compute the principal angle 0, and the corresponding principal moments of inertia Ixp and lyp:

Answer 1

Answer Given that and 1.26x 106 in 6 Ir Ty = 615 5x105 int Try = - 1.02 X 105 in 4 0 300 the moment of Iz! Ixt 14 da - Ty cos20 - Tuy Sin 20 2 1:26X1D6+655X10% -2681066.55X/DOCOS 2130) - (+1,028105 ) x Bosin2 (30) 2 1915.000 + 605000 cos 60 + 102.000 sin bo - 957500 + 15)250 +8833453 T = 1197084.59 in Tyl: Txt Jy - Ty-Jy cos 20+ I ry Si no 21.26x10676155X105 1:26x10-6155X/0Scos 60+ -1.02810) Sin 60 1915000 - 605000 COS60 - 102000 sin bo Iy' => 717915.409 ine

Idyl - In-ly since + Ixy Coszo 2 1.26 X 106-6.55x106 sind +/- 1.02 X 105) cos 2(30) = 605000 sin 60 - 1.02 x 105 cos 66 Try = 210972.658 inh (6) The principle angle tanzop - - 2 Inuy tan20p tan 2 Ep - (- 1.02 %105) 1.268186_6155x105 102.000 605000 0.16 859514 tan! (0.16 859514) tanzop 20p 2 Op 9.5698 Op 4.5678 [Op = 417849°

Irup Intly In ly cos20p - Iry Sin 20p 126x10676155X10S 120*106_6155%80%COS/4,5898) + 102 x 10° - -+ Sin 1965688) - 3p = 95000 + 60 5000 cos(9.6898)+ 102000 Sin 19.5688) 1, 272.747.75 in 4 ] - Ixt ly Jy ly cosobe + Isey Sin20p - 2 1:26X16+655X105 1.26X16+6155x16 1:26X106 655X10 cos 19.5698)+Hio2x10 2 Sin (9.5698) 1915000 605.000 COS[3.5698) + (-10 2000 Sin 1915698) 642, 252.2503 in 4