Moments of area first created 04/11/05 - last modified xx/xx/xx Page Author: Ty Harness
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Figure 1 - Rectangular Cross Section with y measured from the XX axis

The first moment of area is better known as the centroid, centre of gravity or the neutral axis. It's the point where you could consider a body to be balanced. At school, did you ever try to balance your exercise book on the point of a pair of compasses. It's not always obvious where the centroid lies in complex shapes and quite often lies outside the actual body, take for example angle iron or channel section.

Mathematically the first moment of area is the area multiplied by distance from your chosen point or axis. So for a rectangular area measuring distance parallel from the XX axis then the equation is derived as follows.
$ A = b*d$

$delta A = b*dy$

$A*Y_c = int_0^d y*b*dy $

$A*Y_c = [(b*y^2)/2 ]_0^d = ((b*d^2)/2) - ((b*0^2)/2) $

$b*d*Y_c = (b*d^2)/2 = b*d*d/2 $

$Y_c = d/2 $

There's no surprise the centroid lies in the centre of a rectangular area and if you repeat the process measuring distance parallel from the YY axis you'll get:

$X_c = b/2$


The second moment of area is equal to the area multiplied by distance squared from your chosen axis or point. So for a rectangular area measuring distance from the XX axis then the equation is derived as follows.

$delta A = b*dy$

$delta I_(x x) = y^2* delta A= y^2*b*dy$

We then allow dy to become an infinitesimal small strip of area and use and summate all those strips from y = 0 to y = d.

$I_(x x) = int_0^d y^2*b*dy $

$I_(x x) = [(b*y^3)/3 ]_0^d = ((b*d^3)/3) - ((b*0^3)/3) $

$I_(x x) = (b*d^3)/3 $

and again if repeat about YY axis then

$I_(y y) = (d*b^3)/3 $
I mentioned you could find the second moment of area about any axis so let's repeat it about the centroid or neutral axis. $I_(x x) = int_(-d/2)^(+d/2) y^2*b*dy $

$I_(x x) = [(b*y^3)/3 ]_(-d/2)^(+d/2) =2* ((b*(d/2)^3)/3) =(2* b*d^3)/24 $

$I_(Y_c Y_c) = (b*d^3)/12 $

repeating in the other direction

$I_(X_c X_c) = (d*b^3)/12 $

The second moment of area is Area multiplied by distance squared which is also better known as the parallel axis theorem. In the above example INA is either IYcYc or IXcXc and h can respectively be y or x.

$I = I_(N A) + A*h^2$

Reworking the above from the XX axis then

$I_(X X) = (b*d^3)/12 + b*d*(d/2)^2$

which is of course is again

$I_(X X) = (b*d^3)/3 $

so the INA will always be the minimum value.




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