I am going to try to do this and keep the math simple.
Maximum loads are different in a 3 guy and a 4 guy system.
In a 4 guy system the worst case is when the wind is straight down the guy
wire.
Then one guy wire takes the full load of the wind on the tower. In this
case the
guy tension is proportional to the wind load,and the Horizontal Component
of the
Guy Tension is
Horizontal Component of the Guy Tension = 1 * Wind Load
If the wind is coming exactly between two guy wires in a 4 guy system, each
guy wire
must support 1/2 the load, but the guy wire is not pulling inline with the
load, it is 45
degrees off, so the guy tension is proportional to 1/2 the wind load
divided by the
cosine of the angle between the wind load vector and the guy tension
vector: The
horizontal component of the guy tension is:
Horizontal Component of the Guy Tension = Wind load / (2 * cos (45 degrees))
which is
Horizontal Component of the Guy Tension = 0.707 * Wind Load
For a 4 guy system, the worst loading on the guys is when the wind is
exactly inline
with one up wind guy anchor.
In a three guy system, the worst case for the guy is not when the wind is
in line
with the guy. The worst case is when the wind is exactly 30 degrees off one
side
of the guy like this:
Top View of Tower
\
\____
/
/
^
Wind blowing to the top of the page from the bottom.
The guy at the top of the sketch isn't doing anything. The guy going to the
right is
not pulling into the wind, it is only balancing the offset tension from the
guy running
down and to the left. It is the only one doing anything to resist the wind
force, but it
is pulling off at a 30 degree angle. But there is only one, so the
horizontal component
of the guy tension is:
Horizontal Component of the Guy Tension = Wind Load /(1* Cos (30 degrees))
which is
Horizontal Component of the Guy Tension = 1.154 * Wind Load
Now if we look at a wind coming from the left in the little sketch, we see
that
we have two guys working for us, but they are pulling 60 degrees off. Using
the
equation above we get:
Horizontal Component of the Guy Tension = Wind Load /(2* Cos (60 degrees))
or
Horizontal Component of the Guy Tension = 1 * Wind Load
Thus a 4 guy system places a smaller worst case load on the guy wires and guy
anchors than does a 3 guy system.
Now to go a little further into Tower Engineering 101:
There are two basic considerations in a guyed tower: The guy wires have to
resist
the wind, and the tower has to resist the guy wire loads so it doesn't
buckle. The
tower buckling is caused by the fact the guy wires run down hill so any
pull on them
pushes down on the tower.
So what is the down load on the tower in different wind conditions for 3
guy system
and a 4 guy system?
Glad you asked - I'll try to explain. ;-)
The simplest case to understand is the wind coming right inline with the
guy anchor.
The wind is trying to push the tower over and (if we have a pined tower
base) the only
thing holding it is the pull on the guy wire. If the anchor is at 80% of
the tower height,
the guy is running up from the anchor (assuming no sag) at an angle defined
as:
Horizontal Guy Angle = arc Cotan (.80) = 90 degrees - arctan (.80) = 51.3
degrees.
if you go out to 100% then
Horizontal Guy Angle = 90 degrees - arctan (1) = 45 degrees.
Now the Horizontal Component of the Guy Tension required to resist the wind
load is
related to the Guy Tension and the Horizontal Guy Angle in the following way:
Horizontal Component of the Guy Tension = Guy Tension * cosine ( Horizontal
Guy Angle )
or another way to say it is:
Horizontal Component of the Guy Tension
Guy Tension = ----------------------------------------------------------------
cosine ( Horizontal Guy Angle )
And the Guy Tension is what pulls down on the tower. The Down Force on the
Tower is:
Down Force on the Tower = Guy Tension * sine ( Horizontal Guy Angle )
Bear with me we are going to do a little high school algebra and combine
the equations.
( And the equations are going to get very wide so watch out for word
wrapping!!!)
Horizontal Component of the Guy
Tension
Down Force on the Tower =
---------------------------------------------------------------- * sine (
Horizontal Guy Angle )
cosine ( Horizontal Guy
Angle )
or
sine ( Horizontal Guy Angle )
Down Force on the Tower = Horizontal Component of the Guy Tension *
------------------------------------------------
cosine ( Horizontal Guy Angle )
Now a Sine / Cosine is the Tangent function so
Down Force on the Tower = Horizontal Component of the Guy Tension *
tangent ( Horizontal Guy Angle )
But Horizontal Component of the Guy Tension is dependant on the wind force
and
the angle between the wind and the guy wire from above.
In the 4 guy system case the wind is straight down the guy line. So we only
have
one guy "working" for us and the Down Force on the Tower is
Down Force on the Tower = (1 * Wind Load) * tangent ( Horizontal Guy Angle )
In the 4 guy system if we have the wind coming between the guys then we have
two guys working for us, as above, and each one contributes to the
download on
the tower:
Down Force on the Tower = 2 * ( 0.707 * Wind Load) * tangent ( Horizontal
Guy Angle )
or
Down Force on the Tower = 1.414 * Wind Load * tangent ( Horizontal Guy
Angle )
So the worst case tower compressive load in 4 guy system is when the wind
is right
between two guy anchors.
In a 3 guy system we have to look at the problem for winds in 30 degree
increments.
Obviously the inline case is the same as the 4 guy system.
The 30 degree off case has two guys that have tension because of the wind
load.
The guy running down and to the left in the sketch above has:
Horizontal Component of the Guy Tension = 1.154 * Wind Load
so for this one guy:
Down Force on the Tower = 1.154 * Wind Load * tangent ( Horizontal Guy
Angle )
Now the Horizontal Component of the Guy Tension in the guy running off to
the right
is related to the tension on the guy taking the wind load as
Horizontal Component of the Second Guy Tension =
Horizontal Component of the First Guy Tension * Sin ( 30 degrees ) =
0.5 * Horizontal Component of the First Guy Tension
>From this we know that the second guy adds an additional load to the tower of:
Down Force on the Tower = 0.5 * 1.154 * Wind Load * tangent ( Horizontal
Guy Angle )
So adding the two together we get
Total Down Force on the Tower = ( 1+ 0.5 ) * 1.154 * Wind Load * tangent (
Horizontal Guy Angle )
or
Total Down Force on the Tower = 1.731 * Wind Load * tangent ( Horizontal
Guy Angle )
Now if we swing the wind around so it comes from the left, then we have two
guys
"working" for us, but they are pulling at a 60 degree offset.
>From our work above we know that EACH guy has a Horizontal Component of the
Guy Tension of:
Horizontal Component of the Guy Tension = Wind Load /(2* Cos (60 degrees))
or
Horizontal Component of the Guy Tension = 1 * Wind Load
Again we know that the Down Force on the Tower for each guy is:
Down Force on the Tower = Horizontal Component of the Guy Tension *
tangent ( Horizontal Guy Angle )
or
Down Force on the Tower = 1 * Wind Load * tangent ( Horizontal Guy Angle )
So for our two guys, the Total Down Force on the Tower is
Total Down Force on the Tower = 2 * Wind Load * tangent ( Horizontal Guy
Angle )
So a 4 guy system will place a lower maximum load on the guy wires and a
lower maximum download load on the tower than a 3 guy system.
Now I know the sharper of you are thinking that I am ignoring the weight of
the
guy wires the wind load on the guy wires.
You are correct. However, I have calculated in these factors many times and
I find
that they amount to less than 3% of the total loads in a typical 100'
amateur installation.
For shorter installations they make even less difference.
I hope this had been somewhat informative and gives a little insight into
the technical
issues of tower design.
You can have a lot of fun when you start to take into consideration guy
wire sag, guy
wire wind load, lateral deflection of the tower, and ice loading. It gets
real complicated, real
fast. Then for the tower over 500' you get to look at dynamic response and
oscillations
of the whole tower system in varying wind profiles. For that you need a
computer and
specialized software. Those I leave to the real professionals.
Steven H. Sawyers PE
ARRL Volunteer Consulting Engineer
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