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# Re: P-Delta

• To: seaint(--nospam--at)seaint.org
• Subject: Re: P-Delta
• From: Charley Hamilton <chamilto(--nospam--at)uci.edu>
• Date: Tue, 06 Apr 2004 09:27:26 -0700

```When looking at individual frames in the small model, with only
```
```the wind loads applied in one direction (also the innate mass load, which
would be the "P" of P-delta), the horizontal reactions at the base are
greater than the applied story forces for each direction.  I modeled one of
the end frames using Visual Analysis 4.0 and did not get the same results as
I did with RAM despite using the P-delta analysis.  The reactions added up
```
to the applied loads in VA.
```
As they should.

I contacted RAM to see if I was interpreting
```
```their program incorrectly and the representative stated that since I used
the P-delta calculation, the building is no longer in static equilibrium and
will have base horizontal reactions greater than the applied lateral loads.
```
```
The only way that the building can be out of static equilibrium
is if it is accelerating (i.e. collapsing).  Consider two
scenarios (pardon the ASCII art):

| P                    | P                   | P                    | P
H   v                      v                 H   v                      v
---> ________________________                ---> ________________________
|                      |              <--D-->|                      |
|                      |                     |                      |
|                      |                    |                      |
|                      |                   |                      |
|                      |                  |                      |
|                      |                 |                      |
|                      |                |                      |
|                      |               |                      |
|                      |              |                      |
^                      ^              ^                      ^
Ax  --->                Bx --->       Ax  --->                Bx --->
^                      ^              ^                      ^
| Ay                   |  By          | Ay                   |  By

In each case, the equations of equilibrium are:

Sum(Fx) = Ax + Bx +  H = 0
Sum(Fy) = Ay + By - 2P = 0
Sum(M)  = 0

only the third equation varies for the wo different cases.

for the "underformed" case, we take:

Sum(M @ A) = -H*h - P*L + By*L = 0

for the "P-D" case, we take:

Sum(M @ A) = -H*h - P*D - P*(D+L) + By*L = 0
^^^^^     ^^^

Note the underlined portions.  Equilibrium is *still* satisfied,
but the actual values of Ax, Ay, Bx, By may change.  The physics
is the same whether you substitute a leaning column or a wet
noodle.  Equilibrium is still satisfied, as I believe Allen pointed
out in his email.  If you get -100k to stabilize the leaning column,
```
then you had better get +100k at the bottom of the leaner to satisfy equilibrium. Therefore, your "extra" base shear is -100+100=0.
```Each *column* may see more (absolute value) base shear than you expect,
but the total base shear still has to add to the total lateral load.

If the members can't support the demands imposed on them, then
you may see a lack of equilibrium as collapse ensues.
Is it possible that the RAM rep was trying to tell you that
situations where the equilibrium equations are not satisfied
indicates instability in the structure?  I would agree with
that assertion, altough the description of the response from
RAM was somewhat less than an ideal explanation of that
concept if that was the rep's intent.

If what Jason describes is correct --- that the program is only
reporting base shear for the LFRS columns --- then it's no wonder
you were confused.  I would be confused as well.  It sounds like
the numbers are the ill-begotten child of trying to satisfy the
code requirements that the LFRS *only* carries lateral load while
still trying to satisfy physics, which doesn't care what the code
says.  If this interpretation is correct, this is not an ill
reflection on RAM.  They're trying to satisfy both masters.  Of
course, it'd be nice if there was some sort of comment (e.g. in
the manuals) on this matter.  And also if the fellow on the phone
could explain this better.

Best of luck.

Charley

--
Charles Hamilton, PhD EIT               Faculty Fellow
Department of Civil and                 Phone: 949.824.3752
Environmental Engineering           FAX:   949.824.2117
University of California, Irvine        Email: chamilto(--nospam--at)uci.edu

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