center of rigidity calcution

[grandstructures]

Does anyone have a good example for calculating the center of rigidity in a braced frame building with rigid diaphragm?
thanks

 

[rscassar]

The 'Seismic Design Handbook' edited by Farzad Naeim, Chapter 8 'Seismic Design of Floor Diaphragms' co-authored by Farzad Naeim and R. Rao Boppana has an apt design example.

What I like to do is use a frame analysis package to calculate the distribution of force to each element by modelling the diaphragm as an infinitely stiff beam which is supported by elastic springs (each elastic springs relates to lateral force resisting element in N/mm or kips/in).

By knowing the reaction distribution, you can calculate where the centre of rigidity is.

I have attached the important calculations from the Naeim paper.

 

[WillisV]

Center of rigidity calc can be performed as kikflip's attachment shows - those calcs can be seen in many places -one particularly good source is the Masonry Designer's Guide:

http://www.cement.org/bookstore/profile.asp?itemid=lT305

If you are specifically asking about the braced frame portion, you just have to determine the relative stiffnesses of the frames. You can determine the stiffness of a frame in a software analysis package (of course if you were using one of those you can find the center of rigidity without doing hand calcs as well..). If you want to determine relative stiffnesses of braced frames by hand you can do so with virtual work.

Using virtual work, ignoring the elastic shortening of the beam and columns and only looking at the braces, expressions for stiffness are as follows:

For a single diagonal:

k=cos^2(theta)*(AE/L)

For a chevron configuration:

k = 2*cos^2(theta)*(AE/L)

where theta is angle from horizontal to the brace, and AE/L are the brace properties.

Of course for C.O.R. calcs only the relative (not actual) stiffnesses matter so you can multiply the above by 1000 to make the numbers easier to work with if you wish.

 

[Lion06]

It gets a little more complicated when you have orthogonal frames that share a column.

For the astronomical number of load combinations with all of the torsional load cases just do a couple hand checks of some very basic cases to get comfortable with the software and then let it do the lateral load distribution.