Out of Plane Bending - Frame Structure 1

[RaptorEIT]

All,

I have a conceptual question regarding calculating the support reactions for a multi-bay frame supporting an out-of-plane load. Please see the attached sketch. All the literature I could find deals with loads in the same plane as the frame.

Without modeling, I am looking for a way to calculate the support reactions for a plane frame subjected to an out-of-plane load. Looking at the sketch, the best approach I can come up with in solving for the support reactions is to assume 1) the deflection of member AB = deflection of member BC, using the stiffness of each element I can solve for the reaction at support A and leftover load at joint C. Then I can repeat the process by assuming 2) the deflection of member DC = deflection of member CE. The assumption though is that Joint C cannot transversely deflect. Since Joint C is not transversely braced however, I know that the calculation is not entirely correct but approximate.

My question: How would you approach this problem? Does anyone know of the "correct" way to solve this type of loading problem. If the load was in the same plane as the frame I would use slope-deflection method and solve for reactions, but since it is not in the same plane, I am not entirely sure on the approach.

Thank you

 

[Celt83]

moment distribution

For the moment distribution piece:
[ul]
[li]solve it once with a horizontal restraint - and get the Rx reaction at the restraint[/li]
[li]keeping the restraint impose a unit deflection in the direction of your load, get the fixed end moments, and perform another moment distribution - get restraint reaction, call it Rx'[/li]
[li]for the real condition of no restraint -> Rx - alpha*Rx' = 0, solve this for alpha[/li]
[li]final joint moments are the first pass moments, M, plus alpha*the second step moments - M = M + alpha*M'[/li]
[li]frame rigid deflection is unit displacement*alpha[/li]
[/ul]

if you allow the top beams to deform axially you need to do a final pass with slope-deflection to get the individual joint deflections

Edit:
well I can't read. my process above is for in plane loading.

Look at the top beam line as a continuous beam supported by springs at each column location with a spring stiffness equivalent to fixed-free cantilever deflection for a unit end deflection assuming fixed bases. Once you have the reactions for the springs look at each column individually.

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

 

[MIStructE_IRE]

If this were real..I’d portalise or brace in the plane of the applied load.

If I couldn’t do that for whatever reason, I’d probably just design the first column as a pure cantilever. The contribution from the remaining frame would be minimal anyway.

If this were not real, and a purely theoretical exam question, I’d use Celt’s edited method above.

 

[Celt83]

moment distribution can be used for the continuous beam on spring support approach also.
General process attached:link
the L in the k formula should be L^3

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

 

[JStephen]

One approximation: Take one load case with a load at each end of that top beam in the same direction, find approximate solution to that. Second load case, do the same with the loads in opposite direction, so net torsion from the top. Then combine the two to get your load case. The "approximate solution" might be assuming top beam is rigid in bending but has zero torsional resistance, which would just give you four cantilevers with force proportional to distance.

I've seen reverences to a design guide for handrail that is a similar situation.

 

[IDS]

My question: How would you approach this problem?

I would model it in a 3D frame analysis program.

But if you must have a more simplified approach you could model the top member (B-G) as a continuous beam with spring supports at the column locations:
- Calculate translational stiffness from the cantilever deflection equation applied to the vertical members.
- Calculate rotational stiffness from the torsional stiffness of the vertical members.

That ignores the effect of torsion in the top member, but that would be conservative, and the effect would be small anyway.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[RaptorEIT]

@Celt83: Thank you for the link, I will look into it.

@MIStructE_IRE: I agree. In the real world, for the sake of time and simplicity, I would conservatively assume the first column as a cantilever taking the entire load. Just wanted to educate myself on the theoretical approach to this type of load scenario.

@IDS: I analyzed the model in STAAD to verify the reactions. The model showed that the reaction at support A was around 19.8 kips out of the total 20 kip load. Seems it isn't worth the time to calculate the exact results for the out-of-plane loading.