Nonlinear Plane Frame Analysis 2

[Charlie Delta Whiskey]

I am attempting to program a second order (geometrically nonlinear) plane frame analysis based on the Direct Analysis Method in AISC (reduced stiffness, notional loads, etc.) to use for designing Moment Frames. I already programmed a first order plane frame analysis program in C++ that works perfectly. My question is when doing a second order analysis, how should I apply the loads. I could apply the loads all at once and do Newton-Raphson iterations until I converge on a solution for the displacements, OR I could apply the gravity loads all at once and the lateral loads incrementally like a pushover analysis does, OR I could apply all of the loads incrementally. Computational time is only a little bit of an issue because the moment frames we work with are pretty small BUT I do run 14 or so load combinations and I check several different designs (meaning different W-Shapes). That means if I am checking 20 designs that would be 20 x 14 = 280 analyses and that usually takes my program about 25 seconds or so. Any input would be great because I really am just sort of flying by the seat of my pants on this project and I couldn't find any information that I thought was relevant to my specific project.

 

[JoshPlumSE]

I believe applying the lateral load incrementally will probably be the most efficient way to do it.

But, keep in mind that you'll want to include some convergence controls like:
Definition of how quickly the load increment increases.
Definition of how close you have to be to define convergence / divergence.

 

[Charlie Delta Whiskey]

When you say "definition of how quickly the load increment increases" do you mean something like how many increments you will divide the load into?

 

[JoshPlumSE]

Yes, that's part of it. But, there's a different concept as well.

If you're apply a 10 kip lateral load. Instead of applying it in 10 increments of 1 kip each, the programs I've seen do something more like applying a 0.5 kips at first, then an addition 1 kip, then 2 more kips.... et cetera. But, then the user can control the rate of change of this load increment. You can flatten that out so it's more uniform, or you can make it increase more quickly.

 

[Whoopdedo_LSU]

I use the DAM method in most cases when it comes to lateral loads.
Run the load cases with both vertical and lateral loads. I don't worry about 2nd order unless serviceability is an issue.
If you are trying to meet a sway requirement, don't worry about the reduced stiffness.

 

[klaus.]

well keep in mind
second order theory is kind of 'linear' theory
to do a nonlinear calculation is more than second order

 

[IDS]

Klaus - why do you say that?

Geometric non-linearity isn't everything, but it isn't linear.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[klaus.]

IDS..
I say that because it is true
Second order Theory is geometrically linear theory because the deformation term is only linear in the equations
But no need to discuss this here also my University time is a long time ago

 

[IDS]

Second order Theory is geometrically linear theory because the deformation term is only linear in the equations

But it isn't linear. Deflections are highly non-linear as you approach the buckling load.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[klaus.]

that's correct...thats why you cannot use second order Theory for buckling problems
you need nonlinear approach....

Second order theory ( P Delta ) is actually a method for doing 'hand' calculation
Using a computer and program we can use nonlinear theory (third order Theory)

 

[dik]

Doug:

As buckling approaches, the stiffness ==> 0. That's why it doesn't work so well.

Dik

 

[Charlie Delta Whiskey]

I am basing my program after Dr. Richard Balling's Book, "Computer Structural Analysis", which I used when I took his class at BYU. I believe it is quite similar to the method laid out in Matrix Structural Analysis by Mcguire, Gallagher, and Ziemian from Bucknell. I add a nonlinear part to the stiffness components which depend on the current displacements and add those to the components of the stiffness matrix which are linear (with respect to the loads). So I am able to capture buckling (if the stiffness matrix is singular) and therefore am doing a nonlinear analysis. Forgive me if I equated second order analysis and nonlinear analysis in my original post.

 

[JoshPlumSE]

The terms 2nd order and non-linear and such can be defined differently by different people.

In this case, we're talking about a geometrically non-linear analysis. By that I mean we are taking into account the deformation of the structure in the analysis and we are NOT assuming that the initial stiffness matrix of the structure remains valid through the entire loading.

A true/ rigorous geometrically non-linear analysis would be valid for a wide range of structures with varying degrees of slenderness. I believe that's mostly what Charlie Delta Whiskey was referring to. I could be wrong though. That usually requires multiple iterations of both loading and stiffness.

Now, many structural programs will save some computational effort by going with a dumbed down way of accounting for geometric non-linearity. This would be the P-Delta analysis that we engineers usually refer to. And, as Klaus has said, a P-Delta analysis is really a way of turning this deformation into a linear effect. This can usually be done with great accuracy for moment frame and braced frame structures. But, it is not as effect for tall structures like cell towers or such.

In RISA, for example, the stiffness matrix remains based on the original undeformed structure. But, the program adds secondary shears to the structure to account for the deformation. So, you do get a non-linear effect, but it's still based on a linear stiffness matrix. I usually refer to this as the "Secondary Shears" method of accounting for P-Delta analysis.

SAP2000, for another example, will usually use a single step modification to their stiffness matrix. This is valid when vertical and lateral loading are essentially uncoupled. Since you end up using a single adjustment to the stiffness matrix, this can also be referred to as a way of linear-izing the geometric non-linearity.

So, if the dumbed down P-Delta analysis gives you good results, when would you want to go with a rigorous / true geometrically non-linear analysis? I tend to look at this as small vs large deflection theory. In small deflection theory the assumption is that the initial deflection is small relative to the dimensions of the structure. P-Delta analysis is used when small deflection is essentially valid, but when you want a slightly greater accuracy.... A rigorous 2nd order analysis would be used when you expect large deflections.

 

[Charlie Delta Whiskey]

An update on the original post, I tried going the easy route and applying all the loads at once and it doesn't converge with 100 or 1000 iterations (unless I make the lateral loads small, then it will converge in 2-4 iterations). I will try applying the lateral loads incrementally and let you engineers know how it goes.

 

[dik]

Home computers have come a long way... my first rigid frame program, I wrote, ran in 64K with disc swapping... current i7 processors have lots of 'snort' to easily handle the work described... two reasons I can see for not implementing this is the cost to recode and also the effects on existing code... way back, I did some simple modifications to some source in Fortan... and 'broke' the existing code. I've also encountered some code that I've wondered why I did it that way... only to find out after modifying it to something better... it didn't work.

Dik

 

[Charlie Delta Whiskey]

Yeah I am amazed at how fast I can run 1000's of simulations (given they are usually just 1 bay, 1 story frames). 14 different load combinations + 7 or so to cover lateral loads coming in from either direction, dozens of different W-Shape combinations, hopefully no more than 5 or so Newton-Raphson iterations, and now maybe 10 or so load-steps, all in the time it takes me to go grab some water from the fridge.

 

[IDS]

As buckling approaches, the stiffness ==> 0. That's why it doesn't work so well.

That's why what doesn't work so well?

I don't particularly like the term "second order effects", but I take it to mean all effects related to changes in the geometry of the structure under load, not a particular way of calculating them.

I pretty much agree with all of what Josh said, except I wouldn't call calculating geometric non-linearity by adjusting the stiffness matrix a dumbed down approach. It provides an excellent and quick estimate of geometric effects for constant axial load, linear material properties, and small deflections. Obviously if the deflections are sufficiently large, or properties are non-linear, or if the deflections affect the axial load significantly, then some iteration is required.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JoshPlumSE]

I pretty much agree with all of what Josh said, except I wouldn't call calculating geometric non-linearity by adjusting the stiffness matrix a dumbed down approach.

I can agree with that.... the procedure isn't really "dumb". Just simplified and less complex. Though I do think it is important to distinguish between these valid P-Delta procedures and a "rigorous" consideration of geometric non-linearity. Mostly because I have seen lots of attempts to use these simplified procedures in cases where they have questionable accuracy.

 

[IDS]

Josh - agreed.

I think we are off topic anyway, since the question relates to an analysis which does not converge if all the loads are applied in the first iteration.

The general principle that should be applied in any non-linear, non-elastic analysis is that the loads should be applied in the same sequence as they will be applied to the actual structure, so in this case apply the dead loads then increment transverse loads.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

I recently discovered this text book available for free download:

Matrix Structural Analysis, 2nd Edition, by McGuire, Gallagher, and Ziemian.

https://digitalcommons.bucknell.edu/books/7/

I thought it covered non-linear geometric effects well. It also has links to an open-source frame analysis package (MASTAN2).



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/