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RE: APPENDIX D - EQUATION (D-5)

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Christopher,

You are correct that this effect can happen with or without the edge
effects for small embedment depths.  But, keep in mind that the An/An0
ratio CAN become less than one when edge effects come into play.

Let's look at two examples.  Both will deal with a one anchor bolt
anchorage and a four bolt anchorage with a square pattern in a 12 inch
spacing.

For example #1 one, consider a big expansive concrete footing such that
edge distances will not become a factor.  In both the single anchor case
and the four anchor case (we will assume that all four anchors are engaged
in tension), An0 will be the same as a function of the embedment depth
(hef).  An0 per secion D.5.2.1 is equal to 9*hef^2.  This is regardless of
the number of anchors.  For the case of a single anchor, then An is always
going to be equal to An0 (or 9*hef^2).  Thus, An/An0 will always be 1 for
a single anchor where edge distances don't effect things.  For the 4
anchor group, An will be larger than An0.  And in this case will be equal
to (2*1.5*hef + 12 in)^2 or 9*hef^2 + 72*hef + 144.  Thus, for various
embedment depth, An, An0, and the ratio will be as follows:

hef	An		An0		An/An0
3"	441 sq in	81 sq in	5.444
4"	576		144		4
5"	729		225		3.24
10"	1764		900		1.96
20"	5184		36000		1.44
etc

It should be apparent, that the An/An0 ratio will asymptotically approach
1 (although only really at embedment depths that start to approach
unrealistic lengths).  So far for both the single anchor and 4 anchor
group, this is inline with your comments.  In the case of the single
anchor, the overall strength of the anchor in pullout will increase since
An/An0 stays equal to one the whole time and the Nb terms increases as a
function of hef^1.5 (assuming that you choose not to use the alternative
equation for Nb...D-8).  In the case of the four anchor group, the overall
strength of the anchorage group will increase as the An/An0 ratio is
decreasing as a funciton of 1/hef (in truth, hef/hef^2 since the
hef^2/hef^2 terms cancel each other and the constant/hef^2 term will
slowly have little effect) but the Nb term is increasing as a function of
hef^1.5.  Thus, for the group of bolts the overal strength in effect will
increase as a function of hef^.5.

Thus, you are correct in that when edge distances DON'T effect things, the
overall strength will increase.

Now, consider the case where edge distances will effect things.  Let's
take as our example a pedestal of 24" by 24" (per the question that
started this thread).

First, look at the case of a single anchor.  This will start of with the
An/An0 ratio being 1 just like before.  But, when the embedment depth
reaches 8", then An will remain at a constant value of 24"*24" or 576 sq
in.  While this occurs, An0 will continue increase as a function of hef^2.
Thus, the An/An0 ratio at an embedment depth of 8" will begin to decrease
from the value of 1 at a rate of 1/hef^2.  Thus, the overall anchorage
strength will DECREASE in since the Nb term only increases as a function
of hef^1.5, which will get over-powered by the decrease in the An/An0
term.

In the case of the 4 anchor group, the edge distance will be 6 inches on
each side (keeping our 12 in spacing).  Again, to begin with, the An/An0
ratio will behave just like if edge distances did not effect things
(because they don't effect things until a "critical" embedment depth is
reached).  Once the embedment depth reaches 4", An will again become a
constant value of 24in*24in or 576 sq in.  Again, An0 will continue to
increase as a function of hef^2.  And just like the single anchor case,
the An/An0 ratio will decrease as a function hef^2 after the embedment
depth has reached 4".  And again, the overall strength will decrease as
the Nb term only increases as a function of hef^1.5, which means the
overall pullout strength of the anchor group will decrease as a function
of 1/hef^.5.

And, as I mentioned in my last post, this all makes sense to me.  The
overall pullout strength is in essence a function of how much concrete can
be "engaged" by the concrete.  When edge distances don't effect things,
then gradually overall pullout strength will increase as the embedment
depth increases since more concrete will be "engaged".  When edge
distances DO effect things, the overall pullout strength will decrease
since in effect by the empirical equation the amount of "engaged" concrete
will decrease.  Now, personally, I would probably argue that maybe the
overall pullout strength should stay constant once the since in reality,
the amount of concrete (An) that is engaged will remain constant at some
point.  But, then I have not read enough on the theory or testing to know
if my belief is anywhere near reality.

HTH,

Scott
Ypsilanti, MI


On Tue, 15 Jul 2003, Chris Banbury wrote:

> As best I can tell this effect occurs with or without the edge effect as John has noted in his initial post.  The An/Ano ratio decreases, approaching unity (1) for large embedment depths.  At large embedment depths the projected failure surface for a single bolt is nearly identical to the projected failure surface of a closely spaced bolt group.  At shallow embedments the the failure surface of a 4-bolt pattern is almost 4 times the failure surface of a single bolt.  Hence the unexpected effect.  Equation (D-5) should still behave properly however since Nb increases so dramatically with embedment depth and overides the reduction in the An/Ano ratio.  The capacity of the bolt group increases with embedment depth when the edge distances are held constant.
>
>
> Christopher A. Banbury, PE
> Vice President
> Nicholson Engineering Associates, Inc.
> PO Box 12230, Brooksville, FL 34603
> 7468 Horse Lake RD, Brooksville, FL 34601
> (352) 799-0170 (o)         (352) 754-9167 (f)
> www.nicholson-engineering.com
>



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