OOPS, sorry. It was late and I was lazy leading me to
jump to a wrong conclusion. The original example (see below)
was for SOLID cylinders, NOT thin walled cylinders. For a
better understanding of the mechanics involved, refer to Chapter 3
of Physical Design of Yagi's by David Leeson, W6QHS / W6NL
or any good mechanical engineering text.
The SECTION MODULUS for a hollow cylindrical element is
S = Pi X ( D^4 - d^4) / 32 D where D is o.d. and d is i.d.
Bending Stress (lb/in^2) = M / S where M is the Bending Moment (in-lb)
The Maximum Allowable Bending Stress is equal to the Yield Strength (Y)
of the material (6061-T6 is 35,000 psi, 6063-T5 is 15,000 psi). For
these materials, there is a 2.3 : 1 advantage to the stronger material.
The Maximum Allowable Bending MOMENT, M = Y times S
To compare M for different element sizes, you can compute the
section modulus for each size and wall thickness. This is best
done in a chart or spread sheet, a good "exercise for the student".
For D = 2 in. and d = 1.88 in. (0.60 in. wall thickness),
D^4 - d^4 = 16 - 12.5 = 3.5
For D = 1 in and d = 0.88 in.
D^4 - d^4 = 1 - 0.6 = 0.4
The ratio of S(2) to S(1) = 8.75 which means that a 2 inch tube
is 8.75 times as strong (in bending moment) as a 1 inch tube
assuming 0.058 wall thickness (rounded to 0.06).
A 2 inch tube is 2.4 times as strong (in bending moment)
as a 1.5 inch tube and 4.3 times as strong (in bending moment)
as a 1.25 inch tube assuming 0.058 wall thickness (rounded to 0.06).
Outside Diameter plays a role in computing the actual force that
a tube can withstand, reducing this effect by one level. Note -
Mechanical Engineers - please feel free to step in and clarify this
point!
Bottom Line: Bending Strength increases non-linearly as a
complicated exponential function of diameter versus the linear
relationship to yield strength. In other words, a small increase
in diameter yields a much greater increase in strength than a
similar percentage increase in yield strength of the material.
de Tom N4KG (an Electrical Engineering playing with Mechanics)
>>>>>>>>>>>>>>>>>>
On Thu, 30 Aug 2001 n4kg@juno.com writes:
>
> Yep, NN4T had problems with 2 inch o.d. tapered to 1-1/4 and
> he didn't even have to consider ICE. His 2 inch o.d. 42 ft
> verticals with couplers seem quite rigid with only one set of guys.
>
> Remember, maximum bending moment of a thin cylinder
> goes up as the 4th power of the diameter so a 2 inch diameter
> section will be 16* times stronger than a 1 inch diameter section.
> A double walled section will be less than twice as strong as
> a single walled section of equal wall thickness.
>
*(actually 8.75 as shown above)
> Tom N4KG
>
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