the imperfections often follow some sort of sinusoidal curve with a max, amplitude of the order L/1000, where L is the length of the brace. However, if you are doing this for design purposes, we generally neglect the buckling of the braces, i.e. they resist zero compression. As for transferring of forces, i thought you applied and analysed using transient displacement-time-history loading, not force. If the acceleration doesn't convert to a force, then there is obviously a problem in the software or the input procedure.

It is a pity I cannot help. It is an on going research topic for me now though I am not devoting enough time to it.

Dwaikat has mentioned one of the factors to consider. The other factor is to consider that the axial load has a small eccentricity e that is generally less than h/20 where his the member size in the possible plane.

Dear Mahmud

Thanks for the reply. I am doing analysis and I have to consider imperfection. I know how to apply imperfection to a single column but my problem is I don't know how to apply it to braces in a two story frame. The other problem that I have is that previously I used Statik Riks method in a push over analysis for the same frame and My problem was that instead of first floor buckling, I have buckling in second floor of my frame. Now that I am using Dynamic implicit, I have the same problem and I dont really know how to fix this.

Dear Anaechi

You mean that instead of applying imperfection, I use an axial eccentricity of h/20? How can it be done when I have a seismic displacement control amplitude applied to a frame?

The dynamic stiffness approach to vibration of a frame provides a solution irrespective of the type of control applied (Force, displacement etc). Could you refer to my 2002 conference paper on my Researchgate profile that is available for download. In that article I derived the dynamic stiffness matrix of a beam-column element in local axes with the x-axis corresponding to the beam axis. What is additionally needed is the dynamic stiffness matrix of a beam-column frame element.

Your problem can thereafter be solved if you can derive the dynamic stiffness matrix of a beam-column with initial imperfections consisting of an initial curvature plus a small eccentricity of the axial load. Both types of imperfections are needed to obtain a reasonable result. I hope I have clarified myself!

There are two methods of handling the dynamic analysis of a frame with a consideration of axial force: The lumped mass approach and the distributed mass approach. The former is more straight forward and involves the use of stability stiffness matrix (functions) for columns and braces and ordinary stiffness matrix for beams. Of course it is only accurate if the vibration frequency is less than the fundamental frequency.

Let me now show how displacement control can be applied.

Consider the dynamic equilibrium equation for a frame with 5 degrees of freedom , 2 supports (corresponding to d4 and d5) and one prescribed displacement d2, and 2 unknown displacements d1, d3.

Using the Lump mass approach

[ K - M*w^2]d = P

k11 k12 k13 k14 k15 ] [ d1 = p1 (1)

k21 k22 k23 k24 k25 ] [ d2 = p2 (2)

k31 k32 k33 k34 k35 ] [ d3 = p3 (3)

k41 k42 k43 k44 k45 ] [ d4 = p4 (4)

k51 k52 k53 k54 k55 ] [ d5 = p5 (5)

where kii = kii - M_1 *w^2 if di is a translation otherwise kii = kii

Now p1 = p2 = p3 = 0 (No dynamic forces) and that creates an apparent problem. How do we obtain d1, d3 given that d2 is known?

d4 = d = 0 They are supports.

Ignoring Eq. 2 ( because d2 is known) Eqs. (1) and (3) can then be written as

k11 k13 ] [ d1] = p1 - k12*d2 (4)

k31 k33 ] [ d3] = p3 - k32*d2 (5)

Eqs. (4) and (5) can then be solved for d1 and d3 and thereafter the support reactions P4 and p5 can be computed from Eqs. (4) and (5).

Of course actual number of dofs are far more than 5. but the same principle applies

You need to consider the imperfections in all possible "worse" scenarios, including out-of-plane buckling, so apply imperfections out-of plane too, and you can make two cases, case 1-when imperfections alternate (thus creating double curvature along the diagonal) and case 2- when they dont alternate. I would guess maybe to check upon the rigidity of the joints between the two floors, if the columns connect at a hinge (in case you tried to simulate a sort of truss-action for the CBF) then maybe it affect your results compared to having a continuous rigid connection through the columns.

Columns, struts, braces are usually square except where geometrical conditions dictate otherwise. There are 3 types of buckling: (1)In plane (2) out of plane and then (3) lateral buckling. Lateral buckling occurs in narrow relatively deep beams, so it is ruled out for a bracing element in a frame. The bracing element does not carry much distributed load and hence buckling will occur in the plane in which it has the smaller x-sectional dimension. Bearing in mind my previous explanations the secret to the analysis is to develop a stability stiffness matrix for a bar with imperfections. I have already mentioned the two common imperfections. Initial curvature was used by Perry-Robertson to develop a more realistic buckling load of columns. BS 8110 1997 specifies that a column should be designed with a minimum axial load eccentricity of h/20 where h is x-sectional dimension of column.

Amaechi Anyaegbunam , why can we rule out lateral buckling in braces? Depending on the shape of the cross section (might be non-symmetric like an L), one might have lateral, or torsional, or even lateral-torsional buckling of these braces. Also, what if the braces are X (or cross), you might end up with sort of internal point load in the bracing, depending on how it is configured.

lateral and torsional buckling are quite complex phenomena. We should proceed from the simple to the complex. If every problem is to be solved exactly, then Euler-Bernoulli beam theory should have been discarded a long time ago in favor of the more complex and better Timoshenko beam.

Dear Aria

Thank you very much for the helpful websites you provided here.

Maybe this paper can help you (there tcl draft files for the modelling):

Modeling of the seismic response of concentrically braced steel frames using the OpenSees analysis environment

• Advanced Steel Construction 2(3):242
• Antonio Agüero
• Carmen Izvernari
• Robert Tremblay