Rounding Issues

[DaveVikingPE]

Topic to all:

In my recent work, I've had a need to compute light gauge steel weak axis section modulii.

Since I'm cheap, I'm looking at the 1962 AISI light gauge steel design manual for guidance. They have some truly elegant examples solving for XX and YY section properties of light gauge channels and studs (channels with stiffener lips).

HOWEVER, boy do they round their numbers a lot. Boy, I tried to repeat their examples in a spreadsheet (after working through by hand - with a calculator) and I can't repeat their numbers unless I round, in my opinion, arbitrarily and too much.

I almost felt like getting out a slide rule (I have one, mostly as a knick-knack)...

I was able to repeat, to my satisfaction, the various examples only through "hit-r-miss" using the TRUNC() and CEILING() functions.

Thoughts?

 

[IRstuff]

1962, they would have been using slide rules, no?

so they would have been limited to 2-3 significant figures..

TTFN

 

[DaveVikingPE]

Definitely. ...and the SR-71 was designed and built by slide rule users, amazing as it sounds. Just imagine inverting a matrix back then.

So, since we have these fancy computers these days we don't have to "limit" ourselves to "conservative" precision. ...or do we?

The TRUNC() function (Excel) used in conjunction with the CEILING() function seems to round and cut-off numbers to my satisfaction- mimicing the example in the 1962 manual.

A non-life-threatening issue of this is that the section properties given in SSMA tables (Steel Stud Manufactures Association) issued circa 2001-2002 differ little (added gauge thicknesses, radically changed shape designations - yes, but moment of inertia, no) from the values given in the 1962 tables. Hence, the standard shapes/section properties/similar are based on "antiquated" (but absolutely sound) computatations.

The crux of the issue is that a spreadsheet-based tool to compute section properties results in values that differ by as much as 3-5%. Is that a problem? Maybe. Is it worth dwelling on? I think so (which is why we have these forums) because a) such curiosities should be discussed among engineers who are expected to use computers, b) the engineering judgement behind rounding decisions is not documented in the example's narrative - is it rounding or is it a numerical error? If this is the *standard* then rounding should be thoroughly explained. And finally, c) confidence in both the computer and the standard is built by testing well-known examples.

 

[IRstuff]

Seems like that's more of a question of how much design margin to put it.

Certainly no one of sound mind would have design margin limited to the round-off error. Most building codes are based on some sort of worst-case design, in which case, a 3-5% tolerance is buried in the design margin and in the usage of reasonable performance envelopes. Is building lumber ultimate strength repeatable to 3-5%?


You may wish to continue this thread in either the Civil or Mechanical engineering forums, btw.

 

[DaveVikingPE]

Actually, I want this thread to continue in the spreadsheets forum since it is relevant to spreadsheets and all engineering, scientific, economic, etc. disciplines. I don't consider this a design issue per se because I know the limits of design, etc. If thrown out to the engineering forums at-large, this issue would be rightly treated as trivial ("a 2x4 is really a 1.5x3.5, but only nominally so...&quot. Here, where we discuss spreadsheets, it's not trivial. Point: Excel will round automatically to whatever its default precision may be; it'll round to however the user chooses to format the numbers - not taking into account significant figures. It takes unusual effort, i.e., using CEILING() and TRUNC(), to repeat calculations in spreadsheet form resulting in industry-wide standards. I don't mind the unusual effort - it's the unusualness that irks me. Also, I'm not going to blindly accept numbers given to me if I cannot independently arrive at them using the same methods the standards folks used.

A 3-5% error - if it is an error - is still the wrong answer (this doesn't apply to strength values, especially in timber since those are determined by lab tests and subsequent statistical analyses, etc. - I'm concerned with properties that are arrived at solely by arithmetic, based on a set dimension: if the standard light-gauge channel has a thickness of .0451 inches, then by gosh my spreadsheet-computed value for a section's area had better reflect a thickness of .0451 inches! - NOTE: I recognize that there is variance in individual pieces, I am not concerned with physical specimens, only with the "ideal" standardized section upon which the computed section properties are based). Hence, if the industry standard - recognized by those at the top of the field - is filled with unrepeatable numbers, what confidence do I have in the standard? OR - what confidence do I have in the spreadsheet? Should I set a ceiling on computations and then truncate the numbers in every calculation or what?

Again, I'm limiting this issue to the spreadsheets forum ONLY. Another point: simply formatting numbers in Excel to, say, four decimal places, is not the same as truncating the numbers to four or less decimal places.

 

[reichertc]

In general, I carry the extra significant digits all the way through the calculation, and then round off my final number so that it looks pretty and is useful and measurable.

If I had even tried this in one of my labs in university, I would still be repeating courses trying to graduate.

The technically correct way to do it is to round off at each step, factoring in error. If I multiply 5 times 5 do I really get 25? No, I get 5+/-0.4999... times 5+/-0.4999... equals 25+/-???. Carry your error through all your calculations, and round off as required.

I promptly discarded this technique as soon as I graduated. It hurt my brain to think about it, and I never seemed to be able to do it correctly. At least that's what the lab assistants said.

If I ever need to do something that critical, I'll dig out my old notes...

 

[IRstuff]

My personal opinion is that rounding at each step is a school exercise(torture). It seems to be a guaranteed method for slowing increasing the error in the final calculation at each intermediate step.

There was a good example in one the threads somewhere in this site talking about rounding after a unit conversion, such as converting from 2.54 cm to 1.00 in. From the signficant figure argument, this is the correct procedure, but from a relative error perspective, you've doubled the apparent probable error. If you do this sort of thing at every step, you'll eventually wind up with a result that significantly more error than what it should have been had you carried the extra digits.

TTFN

 

[austim]

IRstuff,

In 1962 we were using slide rules, yes. But not only slide rules.

In 1957 our bridge design office had a number of hand operated mechanical calculators capable of addition, subtraction, multiplication and division to about 10 figure accuracy. There was a cunning way in which we could extract square roots with them, but all trig functions had to be read from tables - my copy of Chambers six figure mathematical tables is still in my bookshelf; higher accuracy was available. We also had an electro/mechanical calculator which clattered away and disturbed every one in the office.

By 1962 an electronic calculator (very expensive) was introduced to the office, and matrix arithmetic was also appearing in some of the brighter engineers' calculations (up to 6*6 max).

We did not invert matrices with them, since matrices were not generally used in structural engineering at that time. Hence the popularity and regular usage of computational methods such as the Hardy-Cross method of Moment Distribution, which permitted analysis of multistorey multibay space frames by successive approximation.

I would be quite astonished if the SR-71 designers did not have more advanced calculators than we did.

 

[IRstuff]

Austim,

No doubt that there were and always are people with better calculators, etc. But certainly, your average person, yours truly included, would have had only access to a really slick (?!) slide rule and a trig table, but even when I was using slide rules in school, you didn't have enough precision from the basic calculation on the slide rule to get beyond 3 to 4 digits on the slide and for interpolation on trig tables.

But, unless you were one of the lucky ones to use a serious calculator or computer, every time you had to go to the slide rule, you took a hit from rounding.

The point is that this is no longer necessary nor is it necessary to cripple one's calculations by rounding at each step. Every time you round, you lose some accuracy, so why take the hit?

Tools such as Excel and Mathcad, even my calculator, etc., carry 14 or 15 digits of precision throughout the calculation. There is no loss here, only the ability to maintain as high an accuracy as possible through the calculation process.

TTFN