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# RE: question on K factors and P-delta analysis

• To: <seaint(--nospam--at)seaint.org>
• Subject: RE: question on K factors and P-delta analysis
• From: "Eric Ober" <eober(--nospam--at)holbertapple.com>
• Date: Thu, 16 Oct 2003 08:55:20 -0400

```If you check the archives from August 2002 (I believe) you will find a
response to this same question by Geschwinder (thru Charlie Carter)
regarding this issue.  Also, Geschwinder's 2000 T.R. Higgins paper on the
subject explains the issue (available on EPubs on AISC website from
conference proceedings).  I will try to explain my understanding of the
issue and welcome any comments if I am in error.

The gist of the issue is that you must use a K-factor if using the AISC
specifications, regardless of the P-Delta analysis.  The P-Delta analysis
will give the increased moments and axial loads on the rigid frames due to
lateral loads, 'leaning' columns, etc.  However, it does not account for the
impact of the leaning columns on the sway stability of the story.  The AISC
specification is calibrated to rely on use of the K-factor at this time.
for initial out-of-straightness, P-little delta, etc., would be required to
be able to use K=1 (I guess Canadian code provides for this).

Picture a rigid portal frame, equal columns, with equal axial loads, P.
This frame has a sway buckling capacity, say it is 2P, so we have the frame
designed to be exactly critical.  Now add a 'leaning' column, same column
size, attached to the rigid frame by a pinned end beam, with the same
vertical load P applied.  The sway buckling capacity is unchanged, 2P.  To
be able to design the 'leaning' column with K=1, the rigid frame must
provide the lateral restraint at the top of the 'leaning' column.  Sway
instability requires all columns in a story to buckle simultaneously.  This
means that the rigid frame w/ leaning column must resist 3P to be
'sway-stable' but only has capacity for 2P.  If you perform a second order
analysis on both of these frames, the analysis results will be the same as
if you only did a first order analysis (since no deflection).  But the frame
with the leaning column will want to buckle in a sway mode at a lower load,
2P/3 on each column, than the one without the leaning columns.  How do you
account for this without using a modified K-factor (or some equivalent means
- see AISC commentary to LRFD) to reflect the actual buckling capacity of
the rigid frame?

My advice to all, do not use K=1 for rigid frames if using AISC
specification, and don't forget the leaning columns.  Clearly SSRC 3rd
edition was incorrect (else why change it?).

Eric Ober
Holbert Apple Associates

-----Original Message-----
From: Haan, Scott M. [mailto:HaanSM(--nospam--at)ci.anchorage.ak.us]
Sent: Tuesday, October 14, 2003 3:56 PM
To: seaint(--nospam--at)seaint.org
Subject: RE: question on K factors and P-delta analysis

Clifford:

It has been my understanding that the UBC used to specifically say that K
could be taken equal to 1 for seismic resisting moment frames and the
reasoning was that the code had drift limits and required a P-Delta
analysis.  The effective length factor is a way of designing for stability
and if your stability is already accounted for with a P-Delta analysis and
drift limits then K>1 is conservative.

If you are not checking drift and doing a P-Delta analysis then K should be
determined to ensure stability.  If you are designing with drift limits and
a P-delta analysis and want to be conservative you could use a K greater
than 1.

Respectfully,
Scott Haan

-----Original Message-----
From: Clifford Schwinger [mailto:clifford234(--nospam--at)yahoo.com]
Sent: Tuesday, October 07, 2003 5:15 AM
To: seaint(--nospam--at)seaint.org
Subject: question on K factors and P-delta analysis

I've heard people say that when you do a second-order
analysis on a moment frame, you can use a K-factor of
1.0 for the column design.  I'm trying to find an
explanation of this concept, and I am trying to find
out whether the idea of using K=1 when you do a
second-order analysis is correct or not.

I don't understand how doing a second-order analysis
to compute the P-delta moments would allow you to
automatically toss the K-factor out the window (and
use K=1).

As I understand it, the K-factor is a modification you
make to the Euler buckling equation to adjust that
equation to account for column end conditions
different from the ideal pinned-pinned ends assumed
for the "perfect" Euler column.

I have a copy of a book called "Is Your Structure
Suitably Braced".   This publication is a compilation
of papers from a 1993 conference organized by the
SSRC, AISC, AISI and MBMA.  In the back of the book is
a transcript of a panel discussion where someone asked
the following question:

What is your recommendation on the K-factor when a
second-order elastic analysis is performed?  Can K be
taken as 1.0 in all these cases?

Dr. Joseph Yura (one of the panelists) responded as
follows:

"If you are using the AISC Specification, and you are
using a second-order elastic structure analysis, you
must use a K-factor.  The SSRC Third Edition indicated
that you can do a K=1.0.  If you check the Fourth
Edition, you'll see that's been removed and you still
must use a K factor because the checks that we had
done indicated that in high gravity load situations,
that procedure will not cut it."

Surfing the internet for the source of the "you can
use K=1 when you do a second-order analysis" concept,
I found a paper by Dr. Edward Wilson (of CSI?) titled
"Geometric Stiffness and P-Delta Effects".  In that
paper he says:

"...The application of the method of analysis presented
in this chapter should lead to the elimination of the
column effective length (K-) factors, since P-Delta
effects automatically produce the required design
moment amplifications.  Also, the K-factors are
approximate, complicated, and time-consuming to
calculate.  Building codes for concrete and steel now
allow explicit accounting of P-Delta effects as an
alternative to the more involved and approximate
methods of calculating moment magnification factors
for most column designs..."

(Here is the url to the paper:

So now I am really confused.

Section C1.2 in the Third Edition AISC Manual says:

"In structures designed on the basis of elastic
analysis, Mu for beam-columns....shall be determined
from a second order elastic analysis or from the
following approximate second order analysis procedure:
Mu =  B1 x Mnt + B2 x Mlt..."

A similar methodology to the one in the AISC Manual is
specified in ACI 318-02-318 for concrete structures.

The way I see it, engineers have the option of
accounting for P-delta effects by either doing a
second-order analysis or by computing the moment
magnification factors as described in the AISC Manual
or ACI 318, but whichever method is used (for
computing P-delta moments) you still have to use the
appropriate K-factor.

I'm confused by the comment made by Dr. Yura  (see
above) where he said that SSRC Third Edition said to
use K=1 but the SSRC Fourth Edition said to use the
actual K-factor. Was the Third Edition incorrect, or
did something else change when the Fourth Edition was
published.

What is the theory (please explain in simple terms so
that my aging mind can understand) that would allow
you to ever use K=1 for an unbraced moment frame when
you do a second-order analysis?

Here's an example that's reinforcing my "brain-lock"
on this issue:

Let's say you have a single cantilevering "flag-pole"
at the top of the column. You compute the first-order
moment and deflection. You then compute the
second-order moment and deflection, and then design
the cantilevered column using K=2.  Is there any way
you could ever use K=1 on a cantilevered column just
because you did the second-order analysis?

A similar example is where you have a moment frame
with two identical columns with perfectly pinned bases
and connected to an infinitely rigid beam at the top.
Each column has an identical axial load and there is a
single horizontal load applied at the top of one of
the columns acting parallel to the beam span.  This
example is virtually identical to the previous
cantilevered flag-pole example except that the
cantilevering flag-pole columns are upside down. How
could you ever justify using K=1 just because you do a
second-order analysis?

I would be most grateful if someone could explain
where the "you can use K=1 when you do a second-order
analysis" idea came from and whether or not it is a
structural engineering "urban legend".  There must
have been some justification to the K=1 concept if it
was in the SSRC Third Edition.

Thanks.

Cliff Schwinger

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