Good morning. Andy here from Back of the Envelope -- the place to be for all your p-delta soul-crushing info.

Just kidding. No but seriously. Over the weekend, as I was working on a RISA Floor/3d model for a looming deadline, I repeatedly ran into the dreaded “P-delta diverging” error. It drove me crazy.

So today, I am going to cover:

• What is P-delta

• Why it is important

• What is “P-delta divergence”

• Why it happens and how to resolve it

(Estimated reading time = 4 minutes and 30 seconds)

What is P-delta

Quick refresher.

Say you have a column.

“P” is the axial load applied to the column.

“Delta” is the lateral displacement on one end of the column (caused by whatever reason. E.g., earthquake load or miscellaneous small displacement that just happens in a complex 3d model).

P-delta effect is essentially the extra (aka secondary) moment and shear due to this displacement.

Here is a nice picture from RISA explaining it:

(Source: RISA Help File)

Why is P-delta important?

Because if we don’t account for that secondary moment in the design, we could be under-designing the column.

For example, if your DC ratio for axial load is already high, a little extra moment can put your column into the overstress territory, which is obviously not good.

What is P-delta divergence

As you can see from the picture above, the secondary shear will cause more lateral displacement, which means… more p-delta effect.

In other words:

Lateral displacement -> secondary shear -> more lateral displacement -> and so on

In practical terms, if we were to do this by hand (don’t):

1/ Apply the load and apply a displacement

2/ Determine the secondary shear caused by p-delta

3/ Apply that shear and determine the new displacement and new secondary shear

4/ Rinse and repeat until the new displacement is so small that it won’t make any more difference -- aka the solution converges.

RISA does all this for you in the backend.

But it gets tricky (and soul-crushing) when the solution cannot converge. In other words, the “new displacement” in step 3 is not getting any smaller on each iteration – hence the “P-delta divergence” error.

Why does it happen

Because you screwed up, and your model is broken.

Jk, but actually, yes the model is broken-ish. But how?

If you think about it: What could cause the new displacement to get larger at each iteration?

One reason could be that the column is not restrained by other members. So one end of the column is either moving or rotating out whack -- aka “instability.”

Another reason could be because the applied load is causing the member to buckle (remember Euler’s formula from school?). So a solution literally cannot be derived because the column is unstable.

How to resolve it

“How do I get rid of this error asap so I can move on with the rest of my life!” I asked myself.

RISA has a few helpful info to help you debug this (link here and here and here).

They created all of these because, apparently, it is a very common issue.(“I know I’m not the only one,” sings Sam Smith when he found out that other people were also having p-delta divergence issues).

Long story short, there are 3 steps:

1/ Run the load combination without p-delta and see what happens to the deflected shape (difficulty = easy-ish)

This helped me find my first issue.

The top of my columns did not have rotation stiffness since it’s pinned all around.

The result = whacky deflected shape.

(See the long lines at the bottom? Those are some beams deflecting miles into the center of earth.)

Easy fix -- go back to RISA Floor and change the top of columns to "fixed". This will hopefully resolve the entire issue.

If not, move on to step 2.

2/ Turn on p-delta, but run load combo with only a fraction of the load (difficulty = medium)

In other words, to debug: Create a self-weight dead-load-only combo. Change the load factor to something small like 0.1 to see if the divergence error goes away.

If it does, then it means there is a buckling issue somewhere.

You can potentially find this by looking at the deflected shape in the plan view.

For example, here is a top view of a truss buckling out-of-plane:

(Source: RISA.com)

3/ Look at the coordinate of the error and hypothesize (difficulty = sucks)

Sometimes the lateral displacement is so tiny that the deflected shape from steps 1 or 2 tells you literally nothing.

And because you know nothing at this point, Jon Snow, the only thing you could really do is locate the coordinate (RISA tells you where it is) and just “look around it” to see if you can figure out what’s wrong.

Luckily for me, after fidgeting for hours with this, I was able to see that one of my columns in the area was not restrained in one of the directions.

Adding a beam where my arrow is showing resolved the issue. And the soul has been uncrushed, for now.

And that’s all – thanks for reading!

Hope this is helpful (at least maybe someday).

If p-delta divergence ever occurs to you… good luck, but never give up, never surrender. You got this.

Andy
Back of the Envelope/Structural Engineer HQ