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I have to solve this question, and I've tried multiple things with the angle and displacement but I just cannot figure it out.

I started with a M/EI diagram and i went from there.

Any tips on how to tackle these problems?

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I think the aim of this duet of exercises is to make you understand that you can break down problems and superimpose solutions.

infinite stiffness

In the first case what you do is you have a simply supported beam with moment in one support the beam BC.

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$\theta_1$ is the angle at the support. (note that Moment is negative and is equal to 40*1.5)

Since you know the angle and AB does not bend then you only need to calculate the $\theta_1 \cdot 1.5[m]$.

You should find that $\delta\approx 1.6875 mm$

Finite stiffness

In the finite stiffness case you will have the same deflection due to beam BC, and you need to superimpose the deflection of the beam $AB$.

AB is a cantilever beam with an end load.

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you'd need to add the deflection $\delta_{max} \approx 1.125mm$ to the deflection due to beam BC(=1.6875 mm).

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