Computing Beam Deflection Using Castiglianos' Theorem

In summary, the conversation is about finding the deflection of a cantilever beam with a length L and a load Q acting downwards. The question specifically asks for the deflection at a point three quarters of the way along the beam. The beam has a constant flexural rigidity EI. The person has tried the problem multiple times and has two different solutions, one being 27QL^3/128EI and the other 11QL^3/384EI, depending on which side of the beam they take their moments from. They are unsure which solution is correct and ask for clarification. The person who responded believes the first solution is correct but asks for the support conditions of the beam. The original person confirms that the beam is
  • #1
Juggler123
83
0
I am trying to find the deflection of the beam as described below,

The beam has an overall length L and a load Q acting downwards at this point, the question is asking for the deflection of the beam at the point three quarters of the way along the beam. The beam has constant flexural rigidity EI.

I have tried this problem a number of times and keep coming up with two different solutions, depending from which side of the beam I take my moments. The solutions I have are;

27QL^3/128EI and 11QL^3/384EI

Could someone please tell me which one is correct? I think it's the first answer but I'm not 100%.
Thanks.
 
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  • #2
It would help to know the support conditions.
 
  • #3
The beam is a cantilever beam so is supported just at one end.
 
  • #4
Well I make your first go to be correct.

Can you show some working?
 

Related to Computing Beam Deflection Using Castiglianos' Theorem

What is Castigliano's Theorem?

Castigliano's Theorem, also known as the Castigliano's Second Theorem, is a method used in structural engineering and mechanics to calculate the deflection of a beam or structure under an applied load. It states that the partial derivative of the strain energy with respect to the forces acting on the structure is equal to the deflection in the direction of the applied load.

How is Castigliano's Theorem used in computing beam deflection?

Castigliano's Theorem is used to calculate the deflection of a beam or structure by taking partial derivatives of the strain energy with respect to the applied loads. This method allows for the calculation of deflection at any point along the beam, as well as the overall deflection of the structure.

What are the assumptions made when using Castigliano's Theorem?

The main assumptions made when using Castigliano's Theorem are that the material is linearly elastic, the structure is statically determinate, and the deflection is small compared to the length of the beam. Additionally, the theorem assumes that the elastic modulus and cross-sectional area of the beam are constant along its length.

What are the limitations of using Castigliano's Theorem?

One of the main limitations of Castigliano's Theorem is that it only applies to linearly elastic materials. It also assumes that the beam is loaded in a single direction and that the deflection is small compared to the length of the beam. Additionally, the theorem does not take into account any effects of shear or axial loads on the beam.

Are there any alternative methods for computing beam deflection?

Yes, there are alternative methods for computing beam deflection such as the method of virtual work, moment-area method, and the conjugate beam method. Each method may have its own set of assumptions and limitations, and the most appropriate method to use will depend on the specific structure and loading conditions.

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