What's the easiest way to solve this portal frame with braces?

[FEAnalyst]

TL;DR Summary

How to solve this braced portal frame analytically?

Hi,

I'd like to find the easiest way to solve the following braced portal frame statically (at least for deflection and maybe stress if possible, no need to account for buckling here - reason explained below):

Of course, I could utilize symmetry:

But the problem is still that it's a statically indeterminate structure with a quite high degree of static indeterminacy - I assume no pinned joints.

Now the purpose is just to verify the results of the open-source FEA benchmark (so it's not an actual engineering case or homework). Thus, I will use FEM but I need an analytical solution as well. I was looking for ready-made formulas for such frames in books like Roark's but haven't found anything. Are you aware of such publications ? Those frames seem to be pretty basic after all.

If there are no ready-made formulas, I'm considering using some simplifying assumptions to make it easier. So maybe assuming all pinned joints (I wonder how inaccurate this assumption would be) or even solving the members one by one as individual beams but I'm not sure how to approach that. I'm aware that otherwise likely the only way would be to use the force or displacement method which is not easy and I don't have a civil engineering background.

 

[Lnewqban]

I would treat the braced sections as a non-deformable plane, which only function is transferring forces and moments among the columns and the beam.
Those two planes will suffer some rotation, induced by F and its moments.
Therefore, the ends of the beam and the columns connected to those planes will bend, following the planes rotation, until equilibrium is reached.

This example may help you initiate your hand calculations:
https://mathalino.com/reviewer/strength-materials/problem-707-propped-beam-moment-load

 

[FEAnalyst]

I would treat the braced sections as a non-deformable plane, which only function is transferring forces and moments among the columns and the beam.
Those two planes will suffer some rotation, induced by F and its moments.

Thank you for the reply. Do you mean something like this (assuming that I will only consider half of the frame, also because it's interesting by itself - could represent a workshop crane) and solving only the red and blue beam ?

I'm also thinking about treating just the brace as rigid:

But I'm not sure if it makes sense.

As a result, I will mainly need the deflection of the point loaded with F.

The actual deflection will look like this:

 

[Lnewqban]

Do you mean something like this (assuming that I will only consider half of the frame, also because it's interesting by itself - could represent a workshop crane) and solving only the red and blue beam ?

View attachment 348654

The problem is that the energy of F must deform the whole structure within its geometric limits, which includes making a happy face of the horizontal beam and pushing the top ends of the columns out (harder to do than the represented deformation in your diagram).

I'm also thinking about treating just the brace as rigid:

View attachment 348655
But I'm not sure if it makes sense.

It does, but the read support will have to be freed to also move horizontally some to the left.

As a result, I will mainly need the deflection of the point loaded with F.

The actual deflection will look like this:
View attachment 348657

Symmetry between the two anchored ends of the columns must exist for a centered punctual load.
Please, revisit the approximate deformation shown in the diagram of post #2 above.

 

[hutchphd]

The assumption of a purely vertical load is a very bad idea. The most likely failure mode of this structure is by "racking" where the walls move parallel to each other. This will lead to progressive instability and collapse. Or the racking could be helical. This could be caused by wind or other dynamic sideways loads. If the calculations do not include these failure modes then serious failures impend.
How large/heavy is this structure? Would its collapse threaten life and limb? Enough said.

 

[jrmichler]

If I was masochistic enough to solve this by hand, here's how I would start:
1) Cut the frame in half and use symmetry.
2) Treat the braced portion as a rigid body, and eliminate it by connecting points B and C.
3) The vertical force component at A is equal to the vertical force component at D.
4) There is an unknown horizontal force component at A and an unknown horizontal force component at D.
5) There is an unknown moment at A and an unknown moment at D.
6) Point A has an unknown deflection in the Y direction.
7) Point A has zero deflection in the X direction, and zero angular deflection around the Z axis.
8) Point D has zero deflection in the X and Y directions, and zero angular deflection around the Z axis.
9) Points B and C have equal deflection in the X and Y directions, and equal angular deflection around the Z axis.
10) Solve using simultaneous equations. You will need to calculate the forces and moments at B and C, which will require two more FBD's, one for link AB and one for link CD.
11) Cut at BC, and move link AB to add the braced portion BC. This is simple linear superposition.

I am not masochistic enough to take this problem any further. Although I came close once when analyzing a swimming roll in a calender stack.

 

[FEAnalyst]

@Lnewqban I updated my diagrams to include what you said plus the symmetry condition that I forgot about (although I'm still considering using this for the workshop crane problem where there would be no symmetry). What do you think about those 2 modified approaches? The problem with the updated first approach is that the column is not stiffened by the brace. With the second one - that I'm not sure if I can find deflection formulas for vertical roller support. But it should be essentially half of a double cantilever beam, right? Or maybe I should treat it as half of a simply supported beam...

@hutchphd Thanks but, as I've mentioned in the first post, it's not an actual engineering case. Just a benchmark problem for FEA software so I can use any inputs and it doesn't have to follow the design guidelines.

@jrmichler Thanks, that looks interesting but also a bit complex. I would have to give it a try in practice to follow it to the end.