Hand calculations for a slender beam-column structure?

[godisadj]

Hi there,

I am currently carrying out a buckling finite element analysis of a simple, slender beam-column structure. I am after the force-deflection curve for the structure when it is loaded at each one of 3 different locations (A, B and C).

The structure is pin jointed at the supports and a quick sketch of the structure is provided below:

http://i165.photobucket.com/albums/u57/godisadj_2007/frame.gif

I'm wondering if anyone can suggest some typical hand calculations or theory which would help verify my FE results. The results are partially verified by a (very) simple implementation of the Euler buckling equation for the column member when the structure is loaded at point B, but obviously this does not take into consideration the effect of the conjoined beam, or indeed the load eccentricities when loaded at points A or C.

Any help or insight would be much appreciated.

Thanks.

 

[desertfox]

HI godisadj

Just wondering if you could analyse your beam /column
using Castigliano's Theorem.

regards

desertfox

 

[Boiler8019]

For a hand calc, use the moment distribution method to give you your end moments. Then use the conjugate beam method to determine your slope & deflection.

 

[csd72]

1. your effective column length should be in the realm of 0.85L. Conservatively take 1.0 (wont be any more than this because it is a braced frame)

2.Find out your end moment by moment distribution.

3.Find out deflections of beam as pin supported. Reduce this by the restraint moment obtained in 2. Use design charts in steel designers manual or code.

Why us FEM for something so simple?

 

[godisadj]

Thanks for your responses.

I am using FEM as I am not familiar with the above methods.

Please forgive my ignorance, but as I understand, since the structure is statically indeterminate, i.e the degree of indeterminancy is 1 (4 unknown reaction forces minus 3 equations of equilibrium), in order to formulate the primary system for use with the moment distribution method I must release one reaction force, and replace it with a redundant force. This leads me to believe I must replace either the upper or lower pin support with a roller support, but wouldnt replacing either one result in a structure that is unstable?

csd72, you mention that the frame is "braced". Does this mean that my frame is braced against sway and I do not need to take joint translation into account? But surely the joint at B will indeed translate when a force is applied to either A, B, or C.

Im new to this method, so thanks for staying with me this far!

Cheers.

 

[Boiler8019]

The methods I mentioned are analysis procedures for indeterminite structures.

 

[swearingen]

I'm sorry to be blunt - but if you can't do that problem by hand, you really shouldn't be using an FEA program yet.

If you "heard" it on the internet, it's guilty until proven innocent.

 

[csd72]

godisadj,

My reference to braced frame is a steel code reference referring to the sway behaviour of the frame.

The frame will have minimal translation of joint B. Not zero as there will be axial force in the horizontal member that will change its length, but this is usually ignored.

This is a very fundamental problem, and you really should get a structures text to understand the theory (there may be one in your local public library).

 

[rb1957]

godisadj,

a singly redundant structure is easily solved by deflection methods; i prefer "unit load" method myself.

the vertical member would be a beam column if you're applying a moment at the intersection of the two beams ... the horizontal member is the stiffer loadpath for horizontal loads (stiffer than bending of the vertical member), and the vertical member would similarly react vertical loads ... this also indicates the end reactions of the beams (the horizontal member reacts horizontal loads, so its vertical reaction at the supported end should be very small (unless you're applying moments).

I'd use Bruhn "analysis of Flight Vehicle Structures" for both these methods.

 

[UcfSE]

You can use a mechanics of materials approach: Euler buckling of columns with linearly elastic end constraints. This basically models a column similar to the pinned-pinned Euler column but with elastic translational and rotational supports instead of rigid supports. Boresi and Schmidt have a simple chapter on stability of columns that outlines this procedure. Depending on just how simple you want to be, there probably is no simple hand calculation that will capture everything you want.

 

[rb1957]

If the two supports are pinned, and if the members are beams, and if the beams are "welded" together at the load point, then there is a single redundancy. The primary reaction is obviously a vertical reaction at the lower end of the vertical (loaded) member. There is a secondary reaction (both vertical and lateral) that will act along the line connecting the two supports acting at both supports (as a load/reaction pair); this can be shown with a force vector diagram (for a three force body). In the "real" world I think this is almost certainly negligible.