Replacing column by equivalent force and couple

[wee VooDoo]

The Problem
I've been given the beam shown below to carry out the Hardy Cross method (Moment Distribution) too. I have no problems with the method but I'm struggling with part b, which states "Replace column by an equivalent force and couple at point C"
http://img696.imageshack.us/img696/1705/problemat.jpg

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Excuse the crude photoshopping! The column is 1unit in length and infinitely stiff.

Part a of the question was work out stiffness and distribution factors which were easy.
But part b states "Replace column by an equivalent force and couple at point C" and I can't do the rest of the question without this. I've tried and failed many times.

I've tried replacing with what I think is right, and replacing by guessing but I don't think I understand what to do. I tried inserting a column straight underneath with the same length and the 10kN load acting to the left. But that would double the moment around C so I'm stumped.

Any help pointing me in the right direction is very appreciated. Feel stupid that I can do the hard bits but can't figure out the couple! lol

 

[SteamKing]

What the problem is asking you to do is replace the horizontal force at D with an equivalent force and moment at C such that equilibrium is maintained. In other words, the force at C will have the same direction and magnitude as the force at D. The moment acting at C due to the force at D should be easy to determine (not guess).

 

[wee VooDoo]

Thanks SteamKing. I know it sounded stupid but I just could not figure it out for the life of me. Too much studying frying my head! Thanks again!

 

[pongo38]

another to look at it is: Add a 10 kN to the right at C, and another 10 kN force at C to the left. In other words nothing has changed. Now you can regard the rightwards force at D and the leftwards force at C as a moment (at C if you like, it doesn't matter where). The remainder is the rightwards force at C.

 

[DeeNos]

Hello,

Thank you for sharing your problem with me. From what I understand, you are having difficulty replacing the column in the beam with an equivalent force and couple at point C. This can be a challenging task, but I will try my best to guide you in the right direction.

Firstly, let's define what an equivalent force and couple mean. An equivalent force is a single force that has the same effect as a system of forces acting on a body. Similarly, an equivalent couple is a single couple that has the same effect as a system of couples acting on a body.

In this case, we want to replace the column at point C with an equivalent force and couple. To do this, we need to consider the forces and moments acting on the column.

At point C, there is a 10kN load acting downwards and a moment of 25kNm acting clockwise. To replace this with an equivalent force and couple, we need to find a force that has the same effect as the 10kN load and a couple that has the same effect as the 25kNm moment.

To find the equivalent force, we can use the principle of moments. We know that the moment of the 10kN load about point C is 10kN x 1m = 10kNm. Therefore, the equivalent force can be found by dividing this moment by the length of the column, which is 1m. This gives us an equivalent force of 10kN acting downwards at point C.

To find the equivalent couple, we can use the principle of moments again. We know that the moment of the 25kNm moment about point C is 25kNm. Therefore, the equivalent couple can be found by taking this moment and dividing it by the distance from point C to the end of the column, which is 0.5m. This gives us an equivalent couple of 50kNm acting clockwise at point C.

So, to summarize, the equivalent force and couple at point C are 10kN downwards and 50kNm clockwise, respectively. You can now use these equivalent forces and moments in your calculations for the Hardy Cross method.

I hope this helps you understand how to replace a column with an equivalent force and couple. If you have any further questions, please don't hesitate to ask.

Best of luck with your calculations!

Sincerely,