HOW TO SOLVE THE MOMENT OF INERTIA OF A COMPOSITE BEAM ON PAPER AND MATLAB

Today, I would like to share a simple way to get the area moment of inertia of an I beam. An I beam is shown in the figure above. Let's determine the area moment of inertia of the beam about the axis that passes through its centroid. You can zoom the picture to get a better view

SOLUTION.

 I divided the beam above into three parts M, N, and P.

Please note that:

1 The area of part M is equal to the area of part P.

 2.1ength and breath of part M is W and T respectively

 3. Line X-X is the centroidal axis of the entire beam

. 4. (H-2T) is the length of part N, the central part.

 5. D1 and D2 represent the distance between the centroidal axis of the entire beam and the centroids of part M AND P

From parallel axis theorem,

Í=I+AD^2

Where Í= Area moment of inertia about the centroidal axis of the be

I= Moment of inertia about an axis parallel to the centroidal axis of the beam

A = Area of each beam segment

 D= Distance between the centroid of each segment and the centroidal axis of the entire beam.

The standard formula for calculating The moment of inertia  for a rectangle is given by

1/12*L*B^3……………ii

 L = length of rectangle, B= width of rectangle.   

Therefore, for part M,

Í=     (1/12*T*W^3)* (W*T)* {(H-T)/2} ^2

Since part M and P are equal and same distance from the centroidal axis of the entire beam, it therefore means that Í for part P = Í for part M

For part N, D is = 0 since the centroidal axis of the entire beam lies in this part for part, Therefore

Í= (1/12*T*(H-2T) ^3

To get the area moments of inertia of the entire beam, add all the ‘I’ for the three parts.

That is 

I for part m + I for part p + I for part  n

THE MATLAB FUNCTION FILE BELOW CAN THE PROBLEM

function I =Ibeam(w,h,t)

%this function was written by ngsolve.com to calculate the area moment of inertia of

% inertia of an 'Í'beam 

%input arguments are:

%w: the  respective lengthS of part m and p

%h: the total heightS of the beam

%t: the respective widthS of part m and p

%output argument is:

%I:area moment of inertia

%for the m part of the beam

IPm=1/12.*t.*w.^3;

Am=w.*t;

Dm=(h-t)./2;

Im=IPm+Am*Dm^2;

%where:

%IPm: Moment of inertia about the centroidal axis for part m

%Am: area of part m

%Dm: distance between the centroidal axis of part m and the centroidal

%axis of the entire beam 

%Lets calculate for part p. the terms mean the same as above.

Ipp=1/12.*t.*w.^3;

Ap=w.*t;

Dp=(h-t)./2;

Ip=Ipp+Ap*Dp.^2;

%lets calcute for part o

Io= 1/12.*t*(h-2*t).^2;

I=Im+Ip+Io;

fprintf('An area moment of inertia of %f was gotten',I)