Second draft: August 30, 2004

Here I update my original analysis for my upgraded software. Updates are in green and preceded by exclamation points (!). I've made a few minor updates to these August 30 notes.

Stu,

Here are my results for the analysis of your structure with the weights factored in. I had to make a few approximations, mostly to deal with software problems I ran into.

The approximations:

- A strut's tendons now all meet at one point in the center of the strut
instead of being offset a few inches from the center of the strut.
A problem with my software forced me to do this. I'd never tried
doing the external load computations with offsets factored in and
it turned out it couldn't handle them.
! No longer necessary to do this.

- I fixed the three base points (A), (C) and (D). The iterations I
use to compute external loads will not converge otherwise.
I've run into this problem with other people's models before.
When I tested the software originally I worked with domes where
the base points were fixed. In my previous calculation for this
model, I just fixed six of the coordinate values to make the
model determinate in space, but for some reason I haven't figured
out, more needs to be done if the external load computations
are to converge. Fixing three points means that now
nine coordinate values must be fixed and that the forces for
tendons (A)-(C), (C)-(D) and (A)-(D) can't be computed.
! No longer necessary to do this, and actually, if I'd been clever, ! I could have avoided fixing the three points with the old software.

- I gave the three base points common Z values. They were already
pretty close. This allowed me to have simple force-of-gravity
vectors going in the -Z direction. I made sure tendon lengths
didn't change except the base tendons must have changed length
a very slight amount.
! I redid this so that the three ground contact points have a ! common Z value of zero. This required adding three BasicVecs, ! and defining three new points using them along with required ! constraints and rewriting tcalc so any Control could go in the ! objective function.

- Another approximation comes with the way the software is designed. As
far as external load analysis is concerned, the software is made for
*a priori* design rather than the *ex post* analysis we're trying to do
here. It assumes the weight of the members distort the structure
slightly. When I do an ex post analysis, I use your final
configuration as the zero-gravity configuration in which the effect of
member weights has not been figured in. When the effect of
member weights is figured in, the model is distored away from the
zero-gravity configuration which means it is distorted away from
your final configuration. The inelasticity of the tendons (1%
elongation at 10000 pounds is what I read off your table) makes
this a reasonable approximation.
! This still stands. Displacements due to the applied force of gravity ! are now stated below. The postulated average prestress ! force of 1371.36 pounds means an extension of (1371.36/10000)% ! or 0.137136%.

All told, I think these are all reasonable approximations and, as far as member force computations are concerned, didn't distort the model within the tolerances we're operating at.

Finally, I must say that I think in my previous calculations I mis-stated the strut member forces due to a software problem. I used a convex mapping so the ends of the struts extended past where the tendons connected. When I got rid of this extension (which should be purely cosmetic as far these calculations are concerned), the strut member forces changed. So, I don't think the force calculations are handled correctly when the extensions are present. The tendon member forces weren't affected.

! This has been fixed. When initial forces are calculated within ! the exogenous load analyzer for complex hubs they are no longer ! affected by the length of strut extensions. The updated values ! are stated below. However I was unable to duplicate the ! original stage2.forces values for tendons. The discrepancy was ! only at the third decimal place and so wouldn't affect Stu. ! stage2.forces has been updated and the original values ! (which I can't duplicate) are in stage2_old.forces.

So I had a little fudge factor that I could play with to get my computations in line with your results. The fudge factor is the scale factor I use to convert from model units to pounds. I used that so that my result for the (A)-(B) force got fairly close to your value of 2450 pounds. For an ex post analysis, there was really no other way to proceed since somehow one must decide how much is due to the zero-gravity pre-stressing and how much is due to the member weights. For your structure, the effect due to the pre-stressing predominates, so factoring in the weights doesn't make a whole lot of difference.

So my scale factor is .087 model units per pound, so you would divide all the values below by .087 to get the corresponding pound value.

! Revised text: ! ! The next step is to choose the level of zero-gravity prestress. ! The program provides relative values roughly (sometimes exactly) ! scaled to be comparable to the member length values. The analysis ! of exogenous forces requires these to be rescaled to yield the ! actual prestress in pounds. Your numbers for member forces represent ! final values with prestress and the effect of the weight of the ! members factored in. ! So my procedure is to choose a level of ! prestress such that when the weight of the members is factored ! in I get as close to your final numbers as possible. As ! far as testing my software is concerned, it would be better if ! we could take measurements on your structure in a zero-gravity ! environment to get prestress values and then difference these ! with the measurements taken on Earth to see the difference gravity ! makes. These differences could then be compared with the ! software's predictions and we might learn something. As it is, ! since I think the prestress forces are very ! large in relation to gravitiational effects, I think the effects ! of gravity are going to be miniscule in comparison with ! measurement uncertainty so we're not going to learn a whole ! lot with this exercise except that we will get reaction values for the ! three footings, so you could measure those to see how well the ! software predicts how the member weights will be distributed ! over the three footings. Another approach would be to hang a large ! weight on the structure, measure the effects and compare these ! measurements with my software's predictions. ! So I chose a scale factor of 0.10 model units per pound, which ! means I'm guessing the average tendon prestress is ! 137.136/00.10 or 1371.36 pounds.

The chart below summarizes the zero-gravity member forces with the new approximations. These are comparable to the figures I gave you last time and haven't changed significantly except for the strut values due to the problem I mentioned above. Except for the strut figures and scale changes, you can view any discrepancies between these figures and the ones I sent you before as an indicator of the effect of the various approximations I introduced above. Also, forces for the base tendons are missing due to another problem I mentioned above:

(A)-A: -400.254 (B)-B: -285.459 (C)-C: -105.037 (D)-D: -399.626 (A)-(B): 204.161 (A)-D: 185.618 A-(B): 330.975 A-B: 68.8091 A-(C): 66.7029 A-C: 59.1437 (B)-D: 118.398 B-C: 88.0917 B-D: 148.015 C-(D): 229.898 C-D: 167.092

Average tendon force (in model units): 151.537

! The chart below summarizes the zero-gravity member forces ! (taken from stage2b.forces). Since the updated software lists ! all values strut by strut, each tendon force now appears twice ! in the list since each tendon is connected to two struts. ! The two listings are differentiated by the suffixes [0] and [1]. ! The values for each of the two listings is the same though. ! The values for the strut prestresses no longer are erroneously ! affected by extension lengths: ! ! (A)+-A+: -413.453 (A)-(B)[0]: 231.253 (A)-(D)[0]: 147.734 ! (A)-D[0]: 165.109 (A)-(C)[0]: 38.3543 A-(B)[0]: 353.323 ! A-B[0]: 77.8802 A-(C)[0]: 62.4145 A-C[0]: 58.8924 ! (B)+-B+: -285.96 (A)-(B)[1]: 231.253 A-(B)[1]: 353.323 ! A-B[1]: 77.8802 (B)-D[0]: 122.038 B-C[0]: 78.7765 ! B-D[0]: 145.14 (C)+-C+: -99.6572 (A)-(C)[1]: 38.3543 ! A-(C)[1]: 62.4145 A-C[1]: 58.8924 B-C[1]: 78.7765 ! (C)-(D)[0]: 66.5478 C-(D)[0]: 215.605 C-D[0]: 156.831 ! (D)+-D+: -369.331 (A)-(D)[1]: 147.734 (A)-D[1]: 165.109 ! (B)-D[1]: 122.038 B-D[1]: 145.14 (C)-(D)[1]: 66.5478 ! C-(D)[1]: 215.605 C-D[1]: 156.831 ! ! Average tendon force magnitude: 137.136 ! Average strut force magnitude: -292.1

With the member weights factored in, the values are:

(A)-A: -422.853 (B)-B: -294.331 (C)-C: -125.965 (D)-D: -409.799 (A)-(B): 215.071 (A)-D: 182.758 A-(B): 349.648 A-B: 63.5981 A-(C): 49.4357 A-C: 74.1759 (B)-D: 127.819 B-C: 77.4189 B-D: 160.099 C-(D): 224.071 C-D: 165.25

Average tendon force (in model units): 153.577

So the average tendon force increases by (153.577 - 151.537)/.087 = 23.4483 pounds which seems reasonable but is fairly insignificant in comparison with the zero-gravity forces present. Although on average there is an increase, for several tendons the member force decreases when the member weights are factored in, C-D for example.

! With the weight of the members factored in, the values are: ! (A)+-A+: -445.024 (A)-(B)[0]: 248.305 (A)-(D)[0]: 153.461 ! (A)-D[0]: 160.345 (A)-(C)[0]: 42.7853 A-(B)[0]: 382.314 ! A-B[0]: 75.9373 A-(C)[0]: 43.331 A-C[0]: 73.8147 ! (B)+-B+: -302.339 (A)-(B)[1]: 248.305 A-(B)[1]: 382.314 ! A-B[1]: 75.9373 (B)-D[0]: 134.353 B-C[0]: 66.8532 ! B-D[0]: 161.767 (C)+-C+: -120.323 (A)-(C)[1]: 42.7853 ! A-(C)[1]: 43.331 A-C[1]: 73.8147 B-C[1]: 66.8532 !(C)-(D)[0]: 79.0208 C-(D)[0]: 203.545 C-D[0]: 152.242 ! (D)+-D+: -379.115 (A)-(D)[1]: 153.461 (A)-D[1]: 160.345 ! (B)-D[1]: 134.353 B-D[1]: 161.767 (C)-(D)[1]: 79.0208 ! C-(D)[1]: 203.545 C-D[1]: 152.242 ! Average tendon force magnitude: 141.291 ! Average strut force magnitude: -311.7 ! So the average tendon force increases by ! (141.291 - 137.136)/0.10 = 41.55 pounds which seems reasonable ! but is fairly insignificant in comparison with the prestress ! forces present (average of 1371.36 pounds for the tendons). ! Although on average there is an increase, for ! several tendons the member force decreases when the member weights ! are factored in, C-D for example. ! The displacements (in inches) of the strut end points due to the ! gravitational force is: ! (A): 0.00250559 A: 0.0293124 ! (B): 0.7838 B: 0.19432 ! (C): 0.0460378 C: 0.319292 ! (D): 0.0598514 D: 0.705225 ! The reaction forces at the three support points. The calculations ! are done assuming the strut is supported on the rim of the cylinder. ! (A): 29.1391/0.10 = 291.391 pounds ! (C): 42.0937/0.10 = 420.937 pounds ! (D): 21.4773/0.10 = 214.773 pounds ! Total = 927.101 pounds

Overall, I don't think I've told you much more. The effect due to the member weights is so insignificant for this structure, I don't think there's much in your measurements that could tell me whether I'm right or wrong on this score. I would call the last set of values my best guess though. It almost seems that the noise introduced by the approximations outweighs any changes due to factoring in the weights. The best scenario would be if your measurements were accurate enough that it could be shown that factoring in the member weights seemed to close any discrepancy observed with the first set of calculations I gave you.

Perhaps I could do better if I used the previous calculations as my base somehow and used these calculations to indicate the changes introduced by factoring in the member weights. I would calculate my deltas in pounds and choose a scale factor for the previous set of results so that the (A)-(B) value comes out right when the deltas are added in. Hopefully the other members would come closer to their actuals as well.

So I hope I haven't worn out your eyes. I've certainly done a job on mine. I'd be glad to give any clarifications for the above if you need them or address any reservations you might have.

! The new scale factor of 0.10 instead of 0.087 is a lot easier to ! work with in doing comparisons and looking through everything, ! the numbers seem very close. It sure took me awhile to get this right.

Bob