6.2.6 Dome Step 6: Make adjustments to fix problems
The same clearance goals that are used for the
4ν t-octahedron spherical truss in
Section 8.2.3
seem appropriate for this structure.
With these thresholds, eight member pairs are singled out as having
poor clearances. The poor clearances are mostly between pairs of struts.
Table 6.25
enumerates the member pairs involved and the corresponding clearances.
In addition, the solution exhibits a substantial range in
member forces in the tendons, from a minimum of 0.7076 (#143) to
a maximum of 5.5859 (#99).
Member Pair
Preliminary Clearance
Final Clearance
7-20
0.1373
0.1994
13-17
0.1596
0.1855
13-24
0.1587
0.1811
17-24
0.1591
0.1894
18-20
0.1403
0.1927
21-23
0.1741
0.1901
18-44
0.1101
0.1519
20-31
0.1106
0.1616
Table 6.25: 6ν T-Octahedron Dome:
Preliminary and Final Values for Problem Clearances
The interference problem is the most fundamental one. A range
of tendon forces can be dealt with at construction time
by using different materials depending on the relative force
for the tendons, though in some situations it
might be worthwhile to see what can be done to moderate the
range of forces at design time. The forces are greatest in
the convergence and interlayer tendons and smallest in the
binding tendons and guys.
The interference problem can mostly be attributed to the
low frequency of the model. At lower frequencies, the
inward-pointing tripods whose peaks are the
inner convergence tendon triangles tend to be shallow.
This means the non-adjacent component members approach each other too
closely in the vicinity of the convergence triangle.
The interference problem can be fixed by decreasing the lengths of the outer
binding tendons which constrain the extent of the base of the tripod.
Since the binding tendons are all weighted members of the objective
function in this model, this means increasing the weight corresponding
to the outer binding tendon in question. The outer binding
tendon to select is the one which most parallels the strut with
the clearance problem. Increasing the weight on this tendon
gives the strut a steeper trajectory
on its path from the outer to the inner layer and thus keeps it from
approaching nearby tendons and struts at the convergence too closely.
Table 6.26 lists the outer-binding tendon
corresponding to each strut with an interference problem and the
new value selected for the tendon's weight.
Strut #
Outer Binding Tendon #
Revised Weight
18
138
0.4038
20
140
0.4597
13
133
0.4038
24
144
0.3371
21
141
0.3678
17
137
0.3371
Table 6.26: 6ν T-Octahedron Dome:
Member Weight Adjustments
The revised model is brought to convergence using three iterations
with the exact method in conjunction with Fletcher-Reeves.
The derivatives of
the objective function with respect to the independent
coordinate values are all less than
and all clearances are above their respective thresholds.
Tables 6.27 to
6.31 show the values for the
final lengths and relative forces.
As before, excluded members are marked with ‡.
Tables 6.32 and
6.33
show the final values for the coordinates of the basic points.
Figures 6.11 and
6.12 show how the
final structure appears as viewed from the side and base of
the structure respectively. For clarity, interlayer tendons have been
excluded8
and members in the background have been eliminated by
truncation. For reference, selected points are labeled.
Member #
Relative Force
Member #
Relative Force
1
-10.0652
13
-10.1760
2
-10.0700
14
-9.9202
3
-9.8692
15
-10.1476
4
-11.3735
16
-6.9104
5
-9.7762
17
-9.8347
6
-9.9708
18
-10.6229
7
-10.4024
19
-6.8948
8
-10.0965
20
-9.9960
9
-9.8578
21
-10.0265
10
-10.0713
22
-6.9341
11
-9.9785
23
-9.5009
12
-11.2464
24
-10.1033
Table 6.27: 6ν T-Octahedron Dome:
Final Strut Forces
Member #
Length
Relative Force
Member #
Length
Relative Force
25
2.21528
4.43056
49
2.03547
2.84966
26
2.21563
4.43127
50
2.04556
2.86379
27
2.23160
4.46319
51
2.05261
2.87366
28
2.34655
4.69309
52
2.11527
2.96138
29
2.38618
4.77235
53
2.08314
2.91640
30
2.49013
4.98025
54
2.14563
3.00388
31
2.27745
4.55489
55
2.04900
2.86860
32
2.29331
4.58661
56
2.03644
2.85101
33
2.32354
4.64708
57
1.66652
2.33313
34
2.38195
4.76390
58
2.06601
2.89242
35
2.49956
4.99913
59
2.15096
3.01134
36
2.37208
4.74416
60
2.11199
2.95678
37
2.26419
4.52838
61
2.03412
2.84777
38
2.29568
4.59136
62
1.64781
2.30693
39
2.26971
4.53941
63
2.03600
2.85040
40
2.25878
4.51755
64‡
N/A
N/A
41
2.24921
4.49841
65
2.03254
2.84556
42
2.23545
4.47090
66
2.04716
2.86602
43
2.25887
4.51773
67‡
N/A
N/A
44
2.24099
4.48198
68
2.04286
2.86000
45
2.25145
4.50290
69
2.03439
2.84814
46
2.22034
4.44068
70‡
N/A
N/A
47
2.23455
4.46910
71
2.04209
2.85893
48
2.21944
4.43888
72
2.03221
2.84510
Table 6.28: 6ν T-Octahedron Dome:
Final Primary and Secondary Interlayer Tendon Lengths and Forces
Member #
Relative Force
Member #
Relative Force
73
4.94358
97
4.80745
74
5.26772
98
5.13742
75
5.16773
99
5.60404
76
4.99886
100
4.16775
77
4.65185
101
4.90834
78
3.74185
102
4.01986
79
5.18694
103
5.00085
80
5.03322
104
4.84606
81
4.93701
105
5.26904
82
5.14960
106
4.86884
83
3.71641
107
3.88380
84
4.87812
108
4.23798
85
5.03940
109
5.26776
86
5.07807
110
5.32991
87
5.16635
111
4.92827
88‡
N/A
112
4.59887
89
5.01478
113
5.07911
90
5.46741
114
5.78452
91‡
N/A
115
5.09214
92
4.28841
116
4.80643
93
4.59298
117
5.27596
94‡
N/A
118
4.72593
95
4.69291
119
5.04795
96
4.92845
120
5.43235
Table 6.29: 6ν T-Octahedron Dome:
Final Inner and Outer Convergence Tendon Forces
Member #
Length
Relative Force
Member #
Length
Relative Force
121
2.71468
0.83201
145
1.89573
1.39443
122
2.63323
0.80705
146
1.83588
1.35041
123
2.66060
0.71632
147
2.01047
1.29907
124
1.83931
0.91966
148
1.21969
1.46362
125
2.46934
0.75682
149
1.28634
1.29525
126
2.30250
0.96601
150
1.28603
1.29493
127
2.54956
0.78140
151
1.28778
0.94725
128
2.61512
0.70407
152
1.65590
1.06997
129
2.59438
1.08847
153
1.28393
0.94442
130
2.56952
0.78752
154
1.25425
1.26293
131
2.18508
0.91675
155
1.27167
1.28048
132
1.85150
0.92575
156
1.22455
1.46946
133
2.63419
1.06381
157
1.75026
1.13094
134
2.60938
1.09477
158
1.32695
0.97606
135
2.73434
0.83804
159
1.39698
1.02757
136‡
N/A
N/A
160
2.90432
1.09132
137
2.69610
0.90895
161
2.54133
0.95493
138
2.72853
1.10191
162
1.84911
1.19481
139‡
N/A
N/A
163
2.94704
0.79343
140
2.54147
1.16838
164
2.52654
0.77435
141
2.69328
0.99054
165
1.79527
1.32054
142‡
N/A
N/A
166
2.87212
0.99419
143
2.66242
0.71680
167
2.67465
0.81974
144
2.68231
0.90430
168
1.89872
1.39663
Table 6.30: 6ν T-Octahedron Dome:
Final Outer and Inner Binding Tendon Lengths and Forces
Member #
Length
Relative Force
175
2.13957
0.85583
176
2.03774
0.81510
177
2.20311
0.88124
178
2.06426
0.82570
179
2.23546
0.89419
180
2.00217
0.80087
Table 6.31: 6ν T-Octahedron Dome:
Final Guy Lengths and Forces
Point
Coordinates
1.80813
0.75948
3.17349
1.37576
1.63043
2.94000
-0.38534
2.33141
2.97035
0.93791
1.01456
3.59498
1.08789
-0.26100
3.66069
0.00032
0.20887
3.95059
-1.37989
2.26180
2.89276
-1.03831
1.21623
3.55502
-0.98267
0.24115
3.76980
0.13756
-0.21195
3.96804
1.10801
-1.24300
3.47290
2.34501
-0.88144
2.93375
-2.16855
0.81829
3.08325
-0.80566
-1.01805
3.78455
1.36294
-2.31666
2.73597
-2.65858
1.30563
2.36051
-2.80252
0.32432
2.48821
-1.69815
-0.70421
3.46056
-1.33769
-1.58608
3.15663
0.38282
-2.35663
2.93031
0.77488
-2.93589
2.21564
-3.73094
-0.08603
0.04784
-3.10232
-1.76219
1.09539
-1.60996
-3.20168
1.04252
Table 6.32: 6ν T-Octahedron Dome:
Final Inner Coordinate Values
Point
Coordinates
3.14866
0.86815
3.60831
1.46480
2.92782
3.57404
0.49843
2.97345
3.82708
1.12403
1.16198
5.26616
1.52593
0.25847
5.41497
0.54407
0.39653
5.54496
-2.11381
2.32487
4.18070
-1.49949
1.78849
4.75941
-1.19400
-0.20205
5.60735
-0.20625
-0.35308
5.64651
1.60916
-1.87445
4.55170
2.49145
-1.52781
4.23324
-2.36532
1.39524
4.45003
-0.82018
-1.11302
5.43305
2.16560
-2.42225
3.92697
-3.97029
1.88345
2.29814
-3.70848
-0.43912
3.29623
-3.04238
-0.81407
3.94099
-1.41738
-2.90865
3.69479
-0.43534
-3.06546
3.79964
1.37815
-4.22827
2.17849
-4.47657
1.26415
1.69803
-3.50421
-1.41803
3.29141
-0.95692
-3.59216
3.12842
-4.79594
-0.71196
1.73878
-3.05833
-3.26445
2.55366
-0.47757
-4.87734
1.58579
Table 6.33: 6ν T-Octahedron Dome:
Final Outer Coordinate Values
Figure 6.11: 6ν T-Octahedron Dome: Side View
Figure 6.12: 6ν T-Octahedron Dome: Base View
8
In Figure 6.11 the interlayer tendons
at the base are included.
In Figure 6.12 guys are also excluded.