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# R: P-Delta

• To: "INTERNET:seaint(--nospam--at)seaint.org" <seaint(--nospam--at)seaint.org>
• Subject: R: P-Delta
• From: Mark Gilligan <MarkKGilligan(--nospam--at)compuserve.com>
• Date: Wed, 7 Apr 2004 03:17:17 -0400

```We know that static or dynamic equilibrium is always satisfied.

I believe that part of the confusion has to do with how you define the
system.  While you may get some additional shear in those members that are
part of what we think of as being the lateral system, there is no extra
shear in the total structure.  When considering p-delta effects the gravity
columns have a horizontal as well as a vertical component.  It just so
happens that this horizontal component does not result in any bending shear
in the gravity members because of the slope of these gravity columns.  The
trick is to look at the whole structure and look at the reactions. (This
assumes static equilibrium.  Dynamic equilibrium  requires consideration of
inertial and damping

It is possible to compute an additional force to be applied to the"lateral
resisting members"  to approximate the p-delta effect.  In this case the
balancing base shear exists in the phantom gravity columns that were not
included in the model.  Here again the secret is to look at the total
system.

Mark Gilligan

__________________________________
To expand on what Charlie Carter said, let me try to illustrate the
situation with a simplified example. Imagine a one-bay frame consisting of
a
gravity column (pinned at both ends), a frame column (flag pole, pinned at
the top and fixed at the bottom) and a beam spanning between them. There is
a vertical load, P, applied to the top of the gravity column. Now imagine
that the top of the frame column is displaced some distance. The beam pulls
the top of the gravity column a distance Delta to match the frame column's
displacement. Assuming that the frame column is strong enough not to
buckle,
this system will be in static equilibrium with a deflection at the tops of
the columns equal to some value, Delta. The gravity column is now "leaning"
on the frame column; the gravity column is pushing on the frame column, and
the frame column is pushing back on the gvavity column, keeping the gravity
column from just tipping over. The leaning gravity column is inducing a
lateral force onto the frame column equal to P times Delta divided by the
column height. This can be determined by taking a free-body diagram of the
gravity column and taking moments about the top joint; the vertical
reaction
at the base is equal to the applied load P, which creates a couple equal to
P x Delta. In order to be in equilibrium there must be a force couple
acting
in the other direction; this is produced by a horizontal reaction at the
base, times the story height. Now look at the entire system. In order to
remain in equilibrium, since there is a horizontal force at the base of the
gravity column there must be an equal and opposite reaction at the base of
the frame column. So even though there is no externally applied horizontal
force, there is a horizontal shear induced into the frame system due to the
P-delta effects. The "extra" base shear is the shear created by the P-delta
effects.

This is an oversimplification of what RAM Frame is doing, but hopefully it
illustrates the concept.
RAM International

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