Displacement of a composite beam

[9988776655]

Lets say I have a beam with a cross section made of half aluminium and half steel glued together. I would like to know how to calculate the deflection of this beam due to bending moment but I am unsure of how to use the virtual work equation to accomplish this.

Should I use a different equation instead? The problem is that EI is constant along the length of the beam but not along the cross section of the beam so I don't know how to integrate. How would I go about integrating?

 

[Studiot]

Is this homework/coursework? I can't see anyone gluing steel and aluminium together for real as they form a corrosion cell.

It is not a virtual work problem.

You should realize that the mechanics of beams works as follows

The applied moment at any section is determined solely by the applied loads and support conditions and does not depend upon the beam properties.

The beam moment of resistance to this applied moment does depend upon the beam properties at any section.

So you need to consider the balance of forces creating this moment of resistance by considering the stress and strain in the steel and aluminium. The usual compatibility condition imposed is that the strains must be equal.

I cannot be more specific since you have not shown how the composite section is built up.

 

[9988776655]

The problem is not not real. I just wanted help with the concept. If the moment is M at the point of interest and the cross section is attached, how do I find the deflection in the downward direction?

I would first transform the aluminium section into steel (see diagram):

a' = a * Eal / Est

moment of inertia, I = (ab^3) / 12 + ab(b/2)^2 + (a'b^3) / 12 + a'b(b/2)^2
simplifying, I = (b^3 / 3) * (a + a')

max stress in steel = Mb/I

max stress in aluminium = (Mb * Eal) / (Est * I)

I am unsure of what to do now. I don't see how hooke's law can help.

 

[Studiot]

You are nearly there.

(I would transform everything to the weaker material, but it doesn't matter the end result will be the same)

The neutral axis of the beam passes through the centroid of the transformed section ie it is not midway up the section.
So you need to calculate the position of the neutral axis.

The elastic curve is then as usual ( the formula or calculation is the same depending upon the supports) taken about the neutral axis, using the moment of inertia of the transfomed section and using the E value for the material transformed to.

 

[9988776655]

Thanks for picking up my mistake in assuming the neutral axis was between the materials.

I think the last step is to use the formula:

strain = stress / Est

Will this give me the displacement? I don't think it will because strain is dimensionless. And this is the strain in the direction of the beam's length. What if i wanted it in the other direction?

 

[Studiot]

There are many methods for obtaining the elastic curve, which is the deviation of the neutral axis from the non bent state, ie the deflection due to bending.

I cannot say which one will suit your problem but you could use the double integration method or simply look up the max deflection from tables.

I'm sorry I didn't initially apprectiate you were after the deflection rather than the stress.

 

[9988776655]

That pretty much brings me back to my original question. Let's say I get an equation using the virtual work theory.

How to I use EI for my beam given that it is a composite beam?

 

[Studiot]

Both E and I should be the values for the transformed beam.