How Do You Calculate the Second Moment of Area for Complex Shapes?

[Raita]

Hi All,

I am having a doubt regarding second Moment of Area. I know how to determine the basic formula like the example below:

But, when it become more complicated like picture below:

I don't know how to determine the I formula.

The Question is how to find the second moment of area formula for the picture above which is use to find the deflection of the beam?

Thanks

 

[Mapes]

Hi Raita, welcome to PF. I can't see a way to solve this besides splitting the beam into three segments lengthwise and solving for the deflection of each individual segment.

 

[Raita]

Hi Raita, welcome to PF. I can't see a way to solve this besides splitting the beam into three segments lengthwise and solving for the deflection of each individual segment.

Hi Mapes,

Thanks for the replied and is good to be here in PF. Let's say when a force and reaction force is acting on the beam structure like below:

If we need to find the maximum deflection occurs in beam structure which is in the middle of the beam. According what you said, it need to be split into three segments like below:

If i spit it out, is that able to find the maximum deflection occurs in the center of the beam? And how about the forces? Once it spitted, is that the forces will be separate like the picture above? Thanks

 

[Mapes]

Right, and now you would write the equilibrium equations for each segment to get the forces and moments at each connection point. Then you could either link the known beam deflection equations or solve the beam equation $EI(d^4w/dx^4)=0$ while applying all the boundary conditions.

 

[Raita]

Right, and now you would write the equilibrium equations for each segment to get the forces and moments at each connection point. Then you could either link the known beam deflection equations or solve the beam equation $EI(d^4w/dx^4)=0$ while applying all the boundary conditions.

Do you have any examples to show me? i not really understand about the equilibrium equations for each segment and the link between the deflection equations.

 

[Mapes]

Let's take the first segment on the left. The external load is R_a, an upward force. For the segment to be in static equilibrium, the sum of the forces in the x and y directions must each be zero, and the moments around any point must be zero: $\Sigma F_x=0$, $\Sigma F_y=0$, $\Sigma M_O=0$. Now consider the possible internal forces that must exist at the connection with segment two: a possible vertical force, a possible horizontal force, and a possible moment. Can you find the magnitudes of these forces and moment by applying the static equilibrium equations? If so, you can calculate the deflection at the connection by using the beam equation $EI(d^4w/dx^4)=0$, along with the boundary conditions (discussed http://en.wikipedia.org/wiki/Euler–Bernoulli_beam_equation#Boundary_considerations", for example). Does this help? This technique may also be discussed in mechanics of materials texts (e.g., Johnston and Beer), but I don't have it on hand.

 

[Raita]

Thanks mapes, now i get it what you mean. Give me sometime to digest and i will try to solve the problems.