Computing Beam Deflection Using Castiglianos' Theorem

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The discussion focuses on calculating the deflection of a cantilever beam with a downward load Q, specifically at a point three-quarters along its length L. Two potential solutions have been proposed: 27QL^3/128EI and 11QL^3/384EI. The correct answer is confirmed to be the first option, 27QL^3/128EI, based on the beam's support conditions. The user is encouraged to provide their calculations for further verification. Understanding the application of Castigliano's theorem is essential for accurate deflection analysis.
Juggler123
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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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It would help to know the support conditions.
 
The beam is a cantilever beam so is supported just at one end.
 
Well I make your first go to be correct.

Can you show some working?
 

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