Calculation of stiffness k 3

[Pretty Girl]

I'm trying to calculate the effective length of the column. The Eurocode 2 mentions the following equation

In that, we have to find out the "k" values so we can feed to the above equation.
The Eurocode 2 mentions this equation to calculate the relative stiffness k.

I noticed the "Background paper to the National Annexes to BS EN 1992-1 PD 6687-1:2010" mentions all the beams connected to the column should be summed up and considered etc.

But I noticed the calculations in design books only consider the beams with lowest moment of inertia. And usually summing up only two weakest beams.

What is the correct way? (picture below)
1. Sum up all the four beams regardless they are different in stiffness?
2. Sum up the two weakest beams? (purple colour beams) (lowest moment of inertia)?
3. Sum up the beams ? (2 connected to the weakest column axis) (still the purple beams in this case)

 

[centondollar]

Is this for a buckling problem? You don't need to calculate stiffness factors - just use FEA to find the buckling load and mode (eigenvalues and eigenmodes).

Equivalent stiffness is usually not straightforward to determine for arbitrary 2D or 3D frames, or for combined structural elements (shear walls, beams, columns, slabs etc) with more or less symmetric layout.

 

[BAretired]

The question is not clear, but I am not familiar with the Eurocode. The z-z axis is not defined, but I assume it is the vertical axis. Nothing in the OP seems to make sense.

 

[Adrdm]

Not sure if the Eurocode has something analogous to the AISC's direct analysis method, for which you don't need to calculate the k values. All members get assigned a k of 1.

Else I would recommend looking into "Structural Stability of Steel: Concepts and applications for structural engineers" by Galambos and Surovek.

 

[Pretty Girl]

@Centondollar
@BAretired
@alejandrodrdm

This is for the effective length calculation of the concrete column. I have edited the post again and included that equation as well.
z-z axis is the short axis of the beam (horizontal axis)

 

[GeorgeTheCivilEngineer]

Surely it's just l_o in the x-x axis, you sum the beam stiffnesses that provide restraint in that direction.

For l_0 in the y-y axis, you sum the other beams. All other things being equal, buckling will occur in the highest l_0.

 

[ThomasH]

I can understand that the question is not clear for those of you who don't have (or know) the Eurocode. In this case I think that if you get the full picture it will be easier.
First, this chapter is called "Simplified criteria for second order effects". So it is approximate methods and not exact.

If I take equation (5.15) and include the preceeding text we get a reference to Figure 5.7 (f). That is a column with rotational springs at both ends. So this is a 2D buckling problem and l[sub]0[/sub] is the buckling length for the specific column in a specified plane.

I have also included a paragraph that comes a bit furter down in the same chapter:

It defines l[sub]0[/sub] in a more general context. And if you rewrite that equation a bit you will hopefully recognize Euler buckling.

 

[Pretty Girl]

Surely it's just l_o in the x-x axis, you sum the beam stiffnesses that provide restraint in that direction.

For l_0 in the y-y axis, you sum the other beams. All other things being equal, buckling will occur in the highest l_0.

Thank you, now it makes sense.

 

[Pretty Girl]

So this is a 2D buckling problem and l0 is the buckling length for the specific column in a specified plane.

Thank you. Now it makes sense.