Frame Deflection: Solve for B Deflection + Reactions @ A & D in 3-4 Lines

[zzinfinity]

I had this as an exam question in my first Structural Analysis course, and couldn't figure it out. The frame is supported by 2 cantilevers at points A and D. The top member is perfectly rigid, that is EI=infinity. E is constant. Solve for the deflection at point B and the reactions at A and D.

Could someone give me an idea of how you could approach this? My Professor said it could be solved in 3 or 4 lines but I sure don't see how. Thanks

 

[PhanthomJay]

The horizontal load distributes to each column in accordance with the relative stiffnesses of each (the column with the higher EI hogs a greater share of the load since deflections at B and C must be equal). Also, because of the infinite beam stiffness, there can be no rotation of joints A and C, so each column is fixed at the base and "guided" at the top. I assume that you can then use beam tables for reactions and deflections?
Note I am assuming rigid connections between beam and column, and not pin connections. This is not that fully clear from the sketch.

 

[zzinfinity]

Thanks for your response! However I made a mistake. The frame is supported by 2 pin supports, not cantilevers. Yes the connections between the beam and columns are rigid so the angle between the beam and columns is 90 degrees. The only table provided have the fixed-end moment equations, but I thought the FEM in each member would be zero since there is no load applied any where along the members.

The way I tried to approach this problem was the slope-deflection method but didn't now how to deal with the infinite EI.When you say more of the load goes to the column with the higher stiffness, is it a direct proportion? Is the load on the right column exactly 3 times the load on the left column?

Thanks!

 

[Sadeq]

yes it is direct proportion,because you need to know the lateral forces always distribute according to stiffness.

 

[PhanthomJay]

Thanks for your response! However I made a mistake. The frame is supported by 2 pin supports, not cantilevers.

Oh OK!

Yes the connections between the beam and columns are rigid so the angle between the beam and columns is 90 degrees. The only table provided have the fixed-end moment equations, but I thought the FEM in each member would be zero since there is no load applied any where along the members.

There are loads applied at the ends, right? They produce moments along the members.

The way I tried to approach this problem was the slope-deflection method but didn't now how to deal with the infinite EI.

Draw a sketch of the pinned-pinned frame with the rigid support at B and C. Superimpose the deflected shape of the frame onto that. Note that for member AB, there is deflection but no rotation at B; and there is rotation but no deflection at A. Looks like a simple cantilever as if B was fixed and A was free, yes? Use your beam table for deflections of a simple cantilever with a point load applied at its end.

When you say more of the load goes to the column with the higher stiffness, is it a direct proportion? Is the load on the right column exactly 3 times the load on the left column?

Thanks!

As Sadeq has already noted, yes (1/4 of the lateral load to one column and 3/4 to the other).