Expert: Arshad Khan - 3/7/2007

Question
I have an assignment that it is "three-hinged beam" with a point load (say 50 kips right at the hinge point). This is basically, simple supported beam (pinned both ends) and has a hinge in the middle. Spans are 20 feet equal both sides of the hinge. My problem was to develop a moment diagram of this beam. From this moment diagram, I was supposed to find out tension force right at the joint location. In the real life, this is a two W-beam (steel beam) of 20 feet long of each, and has and end plate welded to their ends (at mid-span of 40 feet thus becomes the hinge), then they are bolted thru together.

   When you apply the 50 kip load, the beam will sag, thus creating tension force in the beam as well as in the bolts. You may wonder how on the earth I ended up with this system. In the big picture, this is a part of a steel framing system that girders above ground floor are running above the column. Due to an explosion (!!!) on the ground floor, you may loose a column. Therefore, I came up with an idea that at the top of the column, girders are connected with thru bolts via end plates. Once you loose a column on the ground floor, the column above will turn into transfer column, and girders will sag coming into tension action. How to calculate the moment (of course there should not be moment since it is hinge) and then find out other forces (i.e. tension in both beams, tension in the bolts etc.) is the real question for me. It would be truly appreciated, if you can help me on this issue. I would like to thank you in advance, as always you have been very helpful.  

Answer
Hi Adam,

First you have to ask what the boundary conditions are- i.e. teh columns at either end. Are they fully rigid, in that they cannot bend/move horizontally inwards? If so, the only way to work this out easily is to assume that the middle will sag say 100mm, then by simple resolution of forces you can work out the tension in the beams. Then you will need to check how much elongation (strain) you get from this tension and see whether the geometry still works, as the strain will cause a further sag- however, the more the sag, the less will be the axial tension in the beams, so you will raech a point of equilibrium- that will be your solution!

However, if the columns/support positions can shift inwards, then you need to take that into consideration as well. A frame analysis package should be able to work this out quite accurately.

Hope that helps!

Kind regards,

Arshad.