verifying the stiffness of a pipe, al vs. steel

huangyhg

verifying the stiffness of a pipe, al vs. steel
first off, i'm not a structural engineer, but i'm an electrical engineer and we install equipment on poles. we recently changed our pole design from steel to aluminum and found a drastic difference in stiffness. after researching the numbers, i believe i've found the info i'm looking for. if i can get verification that i did this correctly, that'd be a huge help.
original pipe: 12' long, sch 40 3" nominal
youngs modulus of elasticity (e):
- 29.5 million psi for steel
- 10 million psi for aluminum
this immediately tells me the aluminum pole of the same dimension is going to be 3 times less stiff. (right?)
i found the moment of inertia (i) values for various size pipe. by calculating i*e, i found that i'd need a 4" nominal pipe at sch 80 to get the same i*e as a 3" sch 40 steel pipe. 4" sch 40 would be a lot better than the 3", but still short of the stiffness of the steel.
i also found a deflection formula on wikipedia for cantilever deflection, which i think is applicable.
cantilever deflection = (force * length^3) / (3 * e * i)
this seemed to be helpful for comparing the different results.
attached is a screen cap of a table i created in excel. if i'm on the right track, that would be a huge help. basically, we need to verify what we need to do in aluminum to match the performance of the steel.

here's the table as a jpg.. .might be easier to view.  
you were going great untill you mentioned "cantilever" ... is this a pole (vertical) or a cantilever (horizontal) ?
a pole's critical load is governed by ei, see euler column, so making the two columns (steel and al) have the same ei is one thing to check.  ei also governs displacement, so that's matched too.
i would also check the stress in the pole, al has a lower allowable than steel (at least in my business, but you may have lower strength steel (annealled ?).  look up your allowable stresses ... a column stress is going to be governed by area (is weight a factor ??), bending stress (lateral load on your pole) is governed by r/i ...
good luck
i have taken a look at your spread sheet, you are on the right track, but here are a few comments about your calculations.
nominal diameter, correct.
schedule, correct.
o.d. & i.d. & wall thickness - correct.
weight... not correct for steel, and not too important.
inside area... doesn't matter structurally.
moment of inertia, very important, and wrong.
the values you produced match the answer from using correct formula, but they differ from values found in the steel construction manual.  use these values for i instead.
3 sch.40 = 2.85
3 sch.80 = 3.70
3.5 sch.40 = 4.52
3.5 sch.80 = 5.94
4 sch. 40 = 6.82
4 sch. 80 = 9.12
at this time i do not know the reason for the difference between the values.  use the same values for both steel and aluminum pipe unless you hear differently from someone more familiar with aluminum, but it seems that they have exactly the same dimensions.
it is correct to assume a cantilever deflection equation if the base is fixed (no rotation or translation in any direction) and the loads considered are lateral (such as wind loads or seismic).  with a pole design, the gravity loads are relatively light and the controlling factor is likely the lateral loads.  so your deflection equation is correct, but you need to modify the i values which will increase your total deflection at the free end.  
question:  will you need to cut any holes in the side wall for any reason, or is it simply welded to a base?  
the next thing to look at will be this connection at the base.  is everything strong enough here to handle the forces?  base plate?  weld?  etc.  
i am unaware of how you connect your pole to a foundation, if you use a base plate and embedded anchor bolts, please be aware that there are potential galvanic reactions when you have bare steel and aluminum in contact with one another.  the resulting corrosion will compromise the connection and could fail given time and conditions.  so, you may need to use aluminum anchor bolts as well, but someone with more experience in this area may need to offer their advice.  however, you may not connect to your foundation in that fashion and this may be an unnecessary rabbit trail.

thanks. that's helpful.
i thought the cantilever equation would be useful, as to illustrate the stiffness in a practical example. i'm not sure, though, that the equation i found takes into account the weight of the cantilever itself, but i figured for a vertical pole, it may not matter (except for the momentum the pole has when swaying).
our poles are simple: metal welded base plate, 12' main pole with 2' on top for an antenna. the pole has a box with a battery mounted about 3' up, a solar panel about 10' up and an antenna up top.
i didn't go to the trouble to calculate torsional or wind loads, since i knew that the steel pole met our needs. so, the exercise is to try to identify an equivalent ei in aluminum as to steel.
i wasn't able to find a text that instructed me to compare ei, i just somewhat figured it. so, thanks much for helping me to verify that!!!  
danronb,
the weights in the table were from an al pipe chart. that was just copy and pasting a chart, so since i didn't need it for the ie calculation, i didn't bother adding the steel values.
i'm going to try to attach an image of the newly constructed pole to this post. yes, pole is welded to a plate and has some ports welded into it. the fabricator did an excellent job on the welds. but, yes, we do have the aluminum plate attached to the steel j-bolts in the foundation.
the ei will give you a good stiffness comparison, but realize that your loads will likely be higher than 25 lbs. for high wind events (50 year wind) with the solar panel on the pole.  
wind speeds and forces depend on your location, terrain, elevation, etc. ... but you could simply use 100 lbs. as a starting estimate. (may be higher or lower)
but, deflection in a pole like this is the least of your concerns (although interesting), you need to know if it will fail at the base or not.
you will need to consider both the pole base, weld connection, and the base plate designs.
intuition says, no problem, but the math is fun to do anyway.  
moment at the base due to 100 lbs. at 12' off the ground is 1200 lb-ft.  convert to lb-inches and get 14,400 lb-in.  convert to kip-in and get 14.4 k-in.  1 kip = 1,000 lbs.
the yield stress of schedule 40 steel pipe is 35 kips per square inch (35 ksi).  to give ourselves a safety factor for allowable bending stress we divide this value by 1.67 (this is the allowable stress design approach, asd).
so our original bending capacity was 21 ksi.
the 3" sch. 40 pipe has a section modulus of 1.63 (again, steel table values differ from equations.)
section modulus x bending capacity = allowable moment.
1.63 in^3 x 21 ksi = 34.16 k-in.
34.1 k-in > 14.4 k-in... pipe base is ok.
it looks as though the original pipe could have easily handled a force of 100 lbs and gives us a level of comfort that the original design was sufficient.
the bending stresses for aluminum will be different and the accepted safety factor could be different as well, not familiar with this material.
aluminum alloy 6061-t6 has a yield strength of 40 ksi... greater than our 35 ksi on the steel.  that's a good start, so even though it is not as stiff, it is still a strong and light material.
i would say, based on a quick inspection, that the aluminum pole of the same size (3" sch. 40) may be strong enough to survive the wind loads... but the deflections will be greater than steel and that may not be acceptable for your equipment.
so, that's interesting. though the 6061 would be less rigid, it'd have a higher yield strength.
you're assumption about our poles is correct, the pole seems plenty strong enough, but the swaying isn't great for the long term on the equipment. though the equipment can handle it just fine, i fear that prolonged movement will effect long term robustness and tightness of bolted connections and wiring connections inside the box.
we have some plans to brace the installed poles, and will hopefully use these calculations to determine a course of action for our next set of poles. if we want to keep using al, looks like we're going to have to go to at least sch 80 4", or a tapered pole.
the other option is to just go back to steel, but the al poles are pretty darn cool (and light!!!).

tapered poles would be expensive, unless they're readily available.
poles waving in the breeze like corn stalks tend to make the passer-bys worried !
al corrodes more than steel and weldability sucks (tho' sucks less for 6061).
also i thought your i calc was right, = pi/64*(d^4-d^4), don't know why damronb derated them by about 5% ?
a simple bracing technique for this pole would be three guy lines in tension.  it would be less material than compression knee braces.  but, in the end, it would likely cost less to simply go with a stiffer pole to reduce the labor and material of installing additional foundations and cables for the tension system.  
another stiffening option, but not as practical to employ, you can fill the pole cavity with concrete slurry.  it would increase both the i and e values, but not easy to accomplish if your pole arrives in the field pre-assembled with the cap i saw in the photo.  
just brain storming with you.