Examples of calculating moment of inertia of compound shape. Ixx, Iyy, Ixy


Composite Section

Solution:  

RECTANGLE

Tan(θ)=6/8=0.75

θ=-arctan(0.75)= -37° (Clockwise rotation is generally assumed negative)

I ̅xx=  (Iaa+Ibb)/2+((Ibb-Iaa)/2)×cos⁡2θ-Iabxsin2θ = 384.8

I ̅yy=(Iaa+Ibb)/2-(Ibb-Iaa)/2×cos⁡2θ+Iab sin⁡(2θ)=295.2

I ̅xy=((Ibb-Iaa)/2)*sin⁡2θ + Iab*cos⁡2θ=-153.6

Ixx=I ̅xx+d^2×A=384.8+(5.8)^2×60=2403.2
Iyy=I ̅yy+d^2×A=295.2+(5.25)^2×60=1948.95
Ixy=I ̅xy+X ̅Y ̅×A=-153.6+5.8×5.25×60=1673.4

TRIANGLE

Ixx=(b×h^3)/12=(6×8^3)/12=256
Iyy=(8×6^3)/12=144
Ixy=(b^2×h^2)/24=96

RECTANGLE+TRIANGLE

Ixx=2659.2
Iyy= 2092.95
Ixy=1769.4

 

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