Hello, everyone,
we are going to solve our first example, example # 1.
We have x and y axes.
But we have a rectangular section.
With width=0.40m.
And the height is 0.60 m.
This rectangle.
is
Apart from the y-axis by distance of 0.20 m.
It is required to estimate the following for the given rectangle.
IX.
Kx , radius of gyration at x.
axis, and b.
We want to estimte the moment of inertia at y axis ...
and the radius of gyration for the y axis.
We have our X and Y.
External axes.
As we know that I x = the Ixg
+ the area *(y^2).
We have, for the moment of inertia
about the cg, for a rectangle section, we have b*
* h^3/12.
Then we have this
x axis , we have to add the product of area multiplied by y^2
first our Ix g=b*h^3/12= (0.40*
0.60^3/12, which will give us 0.0072
m4
Our area is (0.40*0.60), which is 0.24 m2
For the item A* y^2
y̅ ^2
we have.
A=b*h, and y̅ =h/2
All raised to the power of 2.
So we have.
b*h, which is (0.40*0.60)*(h/2)^2
h/2=0.30
0.30^2
divided by 4, will give us 0.0216 m4,
so our required moment of inertia about x axis,
will be the summation of these two items,
first item is Ixg=
=0.0072+
A*y̅^2, which is =0.0216
adding both
will give us 0.0288
m4
While for the expression for the radius of gyration for x, we have K^2 x=
Ix/area, our Ix already obtained =0.0288
divided by the area
so K^2x=3/25
we can get the kx, by taking the sqrt of 3
/25
which is =0.3464 m
For the moment of inertia is m^4 and the area is m^2, we take
If we divide by each other, we have m2,
we take the sqrt of m^2, will give us
We proceed for the moment of inertia for the y,the ...
moment of inertia for the y,
=Iycg for y
+ Area*x̅  ^2
The moment of inertia at g=h*b^3/12
our h=0.6
our b=0.4 raised to 3
all divided by 12
will give us 0.0032 m^4
Our area still the same =0.24m^2
For A*x̅ ^2
(0.24)
by
our x̅
=0.2+0.50*(0.40), which will
give as a total  of 0.40m
We raise it to the power of 2.
will be obtain 0.0384 m^4
So, our Iy is the summation of ...
these two terms, the first term
0.0032m^4
+ our A*x̅ ^2=0.0384m^4, it will give us 0.0416 m^4.
So our Ky is sqrt of Iy/area.
our Iy=(0.0416/0.24)
All under the square root, will give
0.416 m, thanks a lot and see you in the next
Example.
