Displacement of the node, superposition of forces

[margareta]

Homework Statement

Hi,
I was wondering if anybody has an idea how to divide the forces on this structure to get the dispacement of the node 4.

2. Formulas
I would like to solve this task by superpositioning forces working on different elements of the construction. (note different EI)
using basic values for basic beams:

The Attempt at a Solution

? should it be devided like this?
but then, how would i consider displacement at the node 2?

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[Nidum]

Proceed in six stages :

(1) Treat the vertical beam initially as just a lever which is applying a bending moment to the horizontal beam at the point where the two join .

(2) Work out the slope of the horizontal beam at the point where the two join .

(3) From that slope work out the displacement of the bottom of the vertical beam using geometry .

(4) Now treat the vertical beam as a cantilever fixed at the top to some fictional rigid structure .

(5) Work out the deflection caused by the load applied at the bottom end .

(6) Add the deflection from (3) to the deflection from (5) to get the total deflection .

 

[margareta]

ok. I understand, however I have a question how to get the slope of the horizontal beam. Can I divide this beam into 4 elements like this:

 

[Nidum]

You can use superposition for this simple problem .

Work out the slope due to the distributed load and to the lever moment separately and add them together . Pay attention to signs when doing this .

 

[margareta]

I guess I made some wrong superposition assumptions considering case 3,4. Could u please tell me, what kind of assumptions should i do to get the final displacement in this middle node. This task has to be solved using superposition of forces

 

[Nidum]

Your M as shown in top diagram is turning the wrong way .

You need to work out the slope at the joining point not the vertical deflection .

 

[margareta]

Yes. I agree. On the top diagram it is just the exaple of how forces work in case of my beam. In case 2 i take the moment with correct sign.
i will try to find the formulas for calculating the change of the slope depending on the position of x. cause i need it for my case 3, where i want to calculate the angle in the middle of the beam.