At 09:16 AM 4/11/2005, Steve Maki wrote:
>Jim Jarvis wrote:
>
> > THAT SAID, a few short comments:
>
>
>Since I was the one who led it astray (don't they all go astray?) - my
>point is that unless I'm mistaken, adding a sensible guy system to a
>small self support tower will generally increase load capability by a
>large amount - keeping in mind long boom twisting issues of course.
>
>So the oft repeated flat out statements that self supporters are
>compromised by guys should not be left unchallenged because it's
>a wrong headed thing to say.
>
>You may argue on the insurance and legal issues all you want,
>I'm only talking about the strength of the tower.
>
>Steve K8LX
OK... some real numbers.. off the Rohn BX data sheets which I found on the
web. Whether or not this is a tower anyone would actually use, or
contemplate guying is sort of immaterial. It's representative...
First off.. the limiting load on the actual lattice work sections is
determined by.... drum roll.... buckling failure of the vertical members
within the lattice. These are about 12.5" long, and have a radius of
gyration of 0.505 inches (for the bottom section), for a slenderness ratio
of 24.8.
The allowable leg load for this tower section (BX-8) is 20,250 pounds. To
compare to straight compression failure, the cross sectional area is .6043
square inches, the material is spec'd as 45,000 psi yield cold rolled steel
for a compressive failure limit of 27,194 pounds. That's almost 50% more
than the failure load due to buckling.
So, raw compression strength of the steel is NOT the issue here.
Now... let's look at some guying strategies...
Free standing, with the nominal 6 square ft antenna and 20psf wind on a BX-64..
The total load in the leg is 19,260 pounds (fairly close to the max
allowable, eh?), of which 19,100 pounds is the wind load and 160 pounds is
the weight of the structure.
(37,770 ft pound moment, 1.978 ft moment arm at base).
Now, let's add three guys...
Let's say we use 3/16" EHS as the guy material, and put the guy anchors 60%
the tower height out. (so the angle between guy and tower is 30
degrees...). The guys will be about 74 ft long (64/cos(30))
The breaking strength is about 4000 pounds, it weighs 73 pounds/1000 ft.
Each guy will weigh about 5-6 pounds, so we can probably neglect the guy
weight. (it's small compared to 19,000 pounds)
Now, it's been oft asserted that the guys should be tensioned to 10% of
breaking strength. Sure, in this case, where you are using the guys as
"safety cables" you might tension them less, but let's start there... The
tension is, therefore, 400 pounds, per cable.
The additional downforce is now 400*3*.866 pounds or about 1040 pounds. In
the no wind situation, that's no big deal. To the tower, it's just like it
weighs 1500 pounds instead of 476, and divided over the three legs,
compared to the leg max allowable of 20,250, it's pretty small.
Now, let's look at putting some wind load on the system. It's a tapered
tower, and the calculations would be somewhat involved (because it's really
an elastic system, both the tower and the guys would stretch, etc.)
However, we'll approximate, using Rohn's moment calculation of 37,770 ft
pounds. At the top of the 64 ft tower, this is about 590 pounds. So,
assuming the wind comes from the direction of the guy, the guy is going to
have to resist the 590 pounds.. assuming no deflection (so the angles
remain the same) requiring a tension of 590/sin(30) = 1180 pounds. That's
an increased downforce of about 1400 pounds, for a total of 2440
pounds. Yes, it won't be quite that much, because the downwind guys will
relax a bit, etc.
-------------------
Some load will still be transmitted as a bending moment to the base,
loading the downwind leg. Whipping out the handbook (always dangerous, but
it will give us a rough idea...)
For a uniformly loaded canteliever beam, the maximum moment is at the base,
and is 1/2 w*l^2 (w is the load per unit length)... conveniently, Rohn has
calculated this for us, and it's the 37,770 ft lbs.
Now lets assume that it's uniformly loaded, and the base is fixed, but the
(top free end) is supported by a reaction force (from the guys). The peak
moment is still at the base, and is 1/8 w*l^2, or one quarter of the
unsupported case. That's about 9440 ft lbs. Using the same 1.987 ft
baseline, that's a vertical load (on one leg) at the bottom of the section
of 4752 lbs. Plus 158 pounds for the weight of the tower, plus about 800
pounds for the resultant of the guys. We're up to around 5700 lbs.
One would need to go through this for all the sections (because the max
allowable for the sections varies from 5300 at the top to 20,250 at the
bottom).
So, if it were a uniform load (which it's not) and if the tower were of
uniform stiffness (which it is not), it looks superficially safe. At least
at the rated capacity...
Of course, if you're just going to load it with the rated capacity, why
bother putting guys on it. That tower is only rated with 6 square feet at
20 psf (70 mi/hr). If you think that by putting guys on it, you can safely
handle 10 square feet at 90 mi/hr, you're dreaming... That's 2.5 times the
wind load at the top... what was before a moment from the antenna of 8000
ft pounds is now 20,000 ftlb. And, your guys are going to be exerting a
downforce of about 4500 pounds.
All bets are off for a casual analysis.
-----
All this foregoing analysis makes the fatal assumption that the structure
is a rigid body (which is what all those guy and tower compression
calculations assume), but it is NOT. It will flex.. Different parts are
going to bend differently, and the stresses will divide in a way not
intended by the mfr. If you look at the chart of expected loads vs the
strength, there's not a huge margin anywhere in the system (it's a well
balanced design from that standpoint.) The allowable loads at the splices
neatly match the allowable loads on the vertical members at that
joint. The load on the leg is typically about 90-95% of the allowable
load on the leg.
This isn't some sort of design where you're working at 20-30% of the
ultimate capacity. Even with a simple rigid body analysis, you're pretty
close to the edge. I haven't even looked at the loads on the diagonal
braces (those look like they are loaded about about 30% of allowable), or
the effect of torsion (clearly this is important, given the big warning
about avoiding antennas with large twisting moments at the top of the sheet).
Torsion raises the specter of eccentric loads on the vertical
legs. Eccentric loads on columns dramatically reduce the maximum allowable
load. The existing design assumes that the legs are loaded symmetrically
and axially... Let this be a warning to those who would hang something off
a leg (or bend the rolled profile or drill holes )that violates that
assumption... there's not a lot of design margin.
I suspect that if you start digging into the analysis, and actually looking
at the distribution of loads as it flexes and is restrained by the guys,
you'll find a serious column buckling problem in the middle.
----
This analysis is truly a back of the envelope calculation... anyone is
welcome to refine it, correct it, or provide counter examples...
Jim, W6RMK
_______________________________________________
See: http://www.mscomputer.com for "Self Supporting Towers", "Wireless Weather
Stations", and lot's more. Call Toll Free, 1-800-333-9041 with any questions
and ask for Sherman, W2FLA.
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