Displacement of a composite beam

In summary: So EI = (Eal/Est)*(I/M).If the beam is simply a metal beam glued to an aluminium section, then EI will be the same for both materials. However, if the beam has been transformed (ie: made composite), then EI will be different for the two materials.
  • #1
9988776655
46
0
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.

sumdelta.gif


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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 

Attachments

  • cross section.jpg
    cross section.jpg
    19.2 KB · Views: 486
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 
  • #7
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?
 
  • #8
Both E and I should be the values for the transformed beam.
 

What is displacement of a composite beam?

Displacement of a composite beam refers to the amount of deflection or bending that occurs in a composite beam when a load is applied.

How is displacement calculated in a composite beam?

Displacement in a composite beam can be calculated using various methods such as the moment-area method, the conjugate-beam method, or the virtual work method. These methods involve solving equations and integrating values to determine the displacement at a specific point on the beam.

What factors affect displacement in a composite beam?

The displacement of a composite beam is affected by various factors such as the type and magnitude of the load applied, the material properties of the beam, and the support conditions. The geometry and shape of the beam can also impact displacement.

Why is displacement important in composite beams?

Displacement is an important factor to consider in composite beams as it affects the structural integrity and stability of the beam. Excessive displacement can lead to failure or collapse of the beam, while the desired amount of displacement can provide the necessary flexibility and strength for the beam to support the applied load.

How can displacement be minimized in composite beams?

To minimize displacement in a composite beam, various techniques can be used such as increasing the stiffness of the beam, reducing the load applied, or using materials with higher strength and rigidity. Proper design and analysis techniques can also help to minimize displacement in composite beams.

Similar threads

Replies
9
Views
1K
Replies
15
Views
407
  • Mechanics
Replies
9
Views
1K
  • Classical Physics
Replies
6
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
3
Views
226
  • Materials and Chemical Engineering
Replies
21
Views
1K
  • General Engineering
Replies
8
Views
1K
Replies
33
Views
3K
  • Atomic and Condensed Matter
Replies
0
Views
220
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
Back
Top