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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (4) : 863-889     https://doi.org/10.1007/s11709-019-0523-9
RESEARCH ARTICLE
Optimal design of steel skeletal structures using the enhanced genetic algorithm methodology
Tugrul TALASLIOGLU()
Department of Civil Engineering, Osmaniye Korkut Ata University, Osmaniye 80000, Turkey
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Abstract

This study concerns with the design optimization of steel skeletal structures thereby utilizing both a real-life specification provisions and ready steel profiles named hot-rolled I sections. For this purpose, the enhanced genetic algorithm methodology named EGAwMP is utilized as an optimization tool. The evolutionary search mechanism of EGAwMP is constituted on the basis of generational genetic algorithm (GGA). The exploration capacity of EGAwMP is improved in a way of dividing an entire population into sub-populations and using of a radial basis neural network for dynamically adjustment of EGAwMP’s genetic operator parameters. In order to improve the exploitation capability of EGAwMP, the proposed neural network implementation is also utilized for prediction of more accurate design variables associating with a new design strategy, design codes of which are based on the provisions of LRFD_AISC V3 specification. EGAwMP is applied to determine the real-life ready steel profiles for the optimal design of skeletal structures with 105, 200, 444, and 942 members. EGAwMP accomplishes to increase the quality degrees of optimum designations Furthermore, the importance of using the real-life steel profiles and design codes is also demonstrated. Consequently, EGAwMP is suggested as a design optimization tool for the real-life steel skeletal structures.

Keywords design optimization      genetic algorithm      multiple populations      neural network     
Corresponding Authors: Tugrul TALASLIOGLU   
Just Accepted Date: 07 March 2019   Online First Date: 23 April 2019    Issue Date: 10 July 2019
 Cite this article:   
Tugrul TALASLIOGLU. Optimal design of steel skeletal structures using the enhanced genetic algorithm methodology[J]. Front. Struct. Civ. Eng., 2019, 13(4): 863-889.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0523-9
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I4/863
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Fig.1  A pseudo code for EGAwMP[29]
operator name parameter name and its abbreviation method static
parameter value
dynamic
parameter value
- minimum of feasible solutions obtained previously MinFeasPrev - 1E9
- application of design strategy Apply_Des_Str 1(yes) or 0(No)
- number of outer generation Pa rOGN - 10 -
- number of inner generation Pa rIGN - 50100 -
- number of sub-population Pa rSPN - 10 -
- sub-population size Pa rSPS - 10 1 10*
- number of design variables, their upper and lower bounds ParND, ParUDV, ParLDV depends on design problem - -
selection
(stochastic universal sampling)
insertion rate Pa rSelIR - - (0<<1)*
insertion method fitness based selection - -
pressure ParSelP - - (1<<2)*
ranking method non-linear ranking - -
generation gap Pa rSelGG - - (1<<2)*
crossover
(single point crossover)
crossover rate ParCrosCR - - (0<<1)*
mutation
(single point mutation)
mutation rate ParMutMR - - (0<<1)*
migration migration interval Pa rMigMI - 1 -
migration rate Pa rMigMR - - (0<<1)*
migration topology Pa rMigMT neighborhood (1) and ring(2) - 1 or 2
migration selection best individual - -
competition competition interval Pa rCompCI - 1 -
competition rate Pa rCompCR - - (0<<1)*
number of sub-population Pa rCompNSC
for competition
- - (1 10)*
Tab.1  Genetic operators and their related parameters for binary-coded design variables (see Fig. 1 for ParAll)
Fig.2  (a) Front view and (b)schematic view of w-shaped profiles corresponding to optimum designation of planar frame with 105-bar
Fig.3  Trend steps (a) for migration (b)for competition and (c) trend line for feasible designs stored in best result (planar frame with 105-bar)
Fig.4  Fig. 4 The convergence history for average of best result and 10 runs (planar frame with 105-bar)
Fig.5  (a)Maximum displacements and (b)unities values corresponding to optimum design (a-b) (planar frame with 105-bar)
design variable number the worst unfeasible solution optimum designation CBO
Kaveh
WOA
Kaveh
EWOA
Kaveh
EGAwMP ignoring NN
1 W150X131
(W6X8.5)2
W360X162
(W14X109)
W610X155
(W24X104)
W360X134
(W14X90)
W360X147
(W14X99)
W690X140
(W27X94)
2 W150X13
(W6X8.5)
W690X265
(W27X178)
W1000X249
(W40X167)
W760X257
(W30X173)
W690X240
(W27X161)
W1000X584
(W40X392)
3 W150X13
(W6X8.5)
W690X125
(W27X84)
W690X125
(W27X84)
W310X117
(W12X79)
W690X125
(W27X84)
W760X284
(W30X191)
4 W150X13
(W6X8.5)
W610X140
(W24X94)
W690X170
(W27X114)
W690X170
(W27X114)
W610X155
(W24X104)
W760X173
(W30X116)
5 W150X13
(W6X8.5)
W530X85
(W21X57)
W530X101
(W21X68)
W360X101
(W14X68)
W530X101
(W21X68)
W610X415
(W24X279)
6 W150X13
(W6X8.5)
W460X128
(W18X86)
W760X134
(W30X90)
W760X134
(W30X90)
W460X128
(W18X86)
W690X350
(W27X235)
7 W150X13
(W6X8.5)
W530X66
(W21X44)
W200X71
(W8X48)
W530X72
(W21X48)
W530X72
(W21X48)
W250X131
(W10X88)
8 W150X13
(W6X8.5)
W360X110
(W14X74)
W530X101
(W21X68)
W360X101
(W14X68)
W360X101
(W14X68)
W530X272
(W21X182)
9 W150X13
(W6X8.5)
W200X31.3
(W8X21)
W360X509
(W14X34)
W200X35.9
(W8X24)
W200X46.1
(W8X31)
W360X91
(W14X61)
10 W150X13
(W6X8.5)
W250X58
(W10X39)
W200X52
(W8X35)
W360X72
(W14X48)
W250X67
(W10X45)
W250X80
(W10X54)
11 W150X13
(W6X8.5)
W530X66
(W21X44)
W530X74
(W21X50)
W530X66
(W21X44)
W530X66
(W21X44)
W530X72
(W21X48)
NPJ 61 0 0 0 0 0
NPM 104 0 0 0 0 0
Vol. in3
(mm3)
81072.13647
(1328534288.9506)
311428.9544
(5103406208.7852)
331146.8073
(5426523924.6811)
313325.6633
(5134487697.795)
311766.9822
(5108945490. 6464)
543141.5701
(8900495670.2892)
Density (lb/in3) - - - - - -
Weight kg
(lb)
22950.3576 (10410.1071) 39745.7186
(87624.3105)
42544.7870
(93795.1999)
40211.6893
(88651.5999)
39956.9518
(88089.9999)
69729.4518 (153727.1269)
Av. W.,kg - 46578.9213 N/A N/A N/A 90546.0256
Std. Dv.,kg. - 60456.2589 N/A N/A N/A 120589.1568
NA ParOGN*Pa rIGN*Pa rSPS*ParSPN3
- 21180 9520 19060 19940 50000
Tab.2  Optimum design comparison for planar frame with 105-bar
Fig.6  (a) Front view and (b) schematic view of w-shaped profiles corresponding to optimum designation of planar structure with 200-bar (Design case I) and (c) (Design case II)
Fig.7  Fig. 7 (a) Trend steps for migration; (b) for competition; (c) trend line for feasible designs stored in best result (planar structure with 200-bar and Design case I)
Fig.8  The convergence history for average of best result and 10 runs (planar structure with 200-bar and Design case I)
Fig.9  Maximum displacements and unity values corresponding to feasible design with higher quality obtained when ParOGN=7 (a-b) and optimum design (c-d) (planar structure with 200-bar and Design case I)
design variable number the worst unfeasible solution feasible solution with higher quality obtained when ParOGN=9 optimum designation design variables of continuous type obtained in literature EGAwMP ignoring NN
Lee and Geem [2004] (mm2) Farshi and Aliniazizi [2010] (mm2) Lamberti [2008] (mm2)
1 W150X13
(1630 mm2)
W200X26.6 W200X26.6 80.8385 94.8385 94.6449 W460X113
2 W150X13 W150X13.5 W150X13.5 655.2890 609.6762 606.4504 W310X28.3
3 W150X13 W150X13.5 W150X13.5 68.9676 64.5160 64.5160 W460X113
4 W150X13 W310X28.3 W310X28.3 68.9676 64.5160 64.5160 W460X113
5 W150X13 W200X22.5 W200X22.5 1249.6104 1254.9007 1251.6104 W130X28.1
6 W150X13 W150X29.8 W150X29.8 173.2899 191.5480 191.0963 W460X113
7 W150X13 W150X22.5 W150X22.5 67.2256 64.5160 64.5160 W150X22.5
8 W150X13 W150X13.5 W100X19.3 1918.1251 2003.9959 2002.5766 W410X38.8
9 W150X13 W150X13.5 W150X13.5 84.4514 64.5160 64.5160 W200X19.3
10 W150X13 W310X23.8 W130X23.8 2698.7687 2648.5108 2647.7366 W200X26.6
11 W150X13 W410X46.1 W100X19.3 255.9349 260.5801 260.2575 W460X113
12 W150X13 W200X15 W200X15 284.9026 124.7739 123.9997 W250X49.1
13 W150X13 W250X32.7 W150X29.8 3346.6384 3502.5091 3502.0575 W250X32.7
14 W150X13 W150X29.8 W150X29.8 123.3545 64.5160 64.5160 W460X158
15 W150X13 W250X38.5 W250X38.5 4026.443 4147.6691 4147.2175 W150X22.5
16 W150X13 W150X22.5 W150X22.5 451.2249 370.6444 370.1928 W460X113
17 W150X13 W100X19.3 W100X19.3 74.7095 86.3869 85.4837 W150X13.5
18 W150X13 W200X52 W460X52 5009.2157 5144.3122 5143.6026 W460X113
19 W150X13 W150X29.8 W150X29.8 64.5160 64.5160 64.5160 W460X113
20 W150X13 W360X57.8 W360X57.8 5695.4079 5789.4722 5788.7626 W460X113
21 W150X13 W150X29.8 W100X19.3 450.7087 455.0313 454.7087 W460X113
22 W150X13 W200X19.3 W200X19.3 1004.0625 271.9349 271.0962 W200X41.7
23 W150X13 W360X44 W360X44 7084.2438 559.6763 7010.6956 W460X113
24 W150X13 W200X26.6 W200X26.6 84.9675 64.5160 64.5160 W150X22.5
25 W150X13 W250X49.1 W250X58 7838.1778 7656.3717 7655.8556 W460X113
26 W150X13 W460X52 W310X44.5 1056.3204 667.6760 667.3535 W460X113
27 W150X13 W200X86 W360X79 3227.8645 4312.8300 4312.1204 W460X113
28 W150X13 W250X101 W250X101 6035.1492 6974.2441 6973.0828 W460X113
29 W150X13 W460X113 W460X113 9736.1095 8927.6595 8925.0789 W460X113
30 - - W250X49.1 - - - -
NPJ 54 0 0 01 01 01 0
NPM 141 0 0 01 01 01 0
Vol. in3
(mm3)
88719.986
(1453860088)
266213.729
(4362461414)
235856.362
(3864993299)
89919.082
(1473509751)
89952.544
(1474058095)
89915.194
(1473446038)
626226.270
(1.0262E+10)
Density (lb/in3) - - - 0.283 0.283 0.283
Weight kg
(lb)
11625.134 (25629.033) 34342.2943
(75711.798)
30506.4606
(67255.233)
11542.6101
(25447.1 )
11546.9051
(25456.57)
11542.4561
(25446.76)
80588.6341
(177667.525)
Av. W.,kg - 64489.324 N/A N/A N/A 338978.9251
Std. Dv.,kg. - 25258.125 N/A N/A N/A 109456.0256
NA Pa rOGN*Pa rIGN*Pa rSPS*Pa rSPN 2
- 412702 N/A N/A N/A 100000
Tab.3  Optimum design comparison for planar structure with 200-bar (Design case I)
Fig.10  Trend Steps for Migration (a), for Competition (b) and Trend Line for Feasible Designs Stored in Best Result (c) (Planar Structure with 200-bar and Design Case II)
Fig.11  The Convergence History for Average of Best Result and 10 Runs (200-bar Planar Structure, Design Case II)
Fig.12  Maximum displacements and unity values corresponding to optimum design (a-b) (planar structure with 200-bar and Design case II)
design variable number the worst unfeasible solution optimum designation continuous values of design variables obtained by
Lamberti [2008] (mm2)
EGAwMP
ignoring NN
1 W150X13 (1630 mm2) W150X13.5 94.6449 W410X67
2 W150X13 W250X17.9 606.4504 W690X323
3 W150X13 W150X18 64.5160 W760X257
4 W150X13 W150X13 64.5160 W360X39
5 W150X13 W150X22.5 1251.6104 W920X420
6 W150X13 W360X44 191.0963 W530X74
7 W150X13 W250X28.4 64.5160 W200X59
8 W150X13 W200X46.1 2002.5766 W530X101
9 W150X13 W310X44.5 64.5160 W360X162
10 W150X13 W410X60 2647.7366 W310X74
11 W150X13 W130X23.8 260.2575 W760X220
12 W150X13 W150X13.5 123.9997 W610X217
13 W150X13 W250X80 3502.0575 W360X262
14 W150X13 W130X23.8 64.5160 W920X238
15 W150X13 W360X72 4147.2175 W610X455
16 W150X13 W310X44.5 370.1928 W200X52
17 W150X13 W150X18 85.4837 W150X22.5
18 W150X13 W310X86 5143.6026 W460X144
19 W150X13 W410X38.8 64.5160 W530X196
20 W150X13 W250X101 5788.7626 W840X392
21 W150X13 W150X29.8 454.7087 W310X143
22 W150X13 W200X26.6 271.0962 W460X421
23 W150X13 W760X147 7010.6956 W250X73
24 W150X13 W150X18 64.5160 W310X86
25 W150X13 W840X210 7655.8556 W410X114
26 W150X13 W200X26.6 667.3535 W760X173
27 W150X13 W530X101 4312.1204 W760X389
28 W150X13 W360X147 6973.0828 W690X289
29 W150X13 W360X196 8925.0789 W460X286
NPJ 62 0 01 0
NPM 166 0 01 0
Vol. in3
(mm3)
88719.986
(1453860088)
360166.209
(5902066717)
40786.067
(668363890)
1234119.301
(2.022359E+10)
Density
(lb/in3)
- - 0.283 -
Weight kg
(lb)
11625.134 (25629.033) 45825.4961
(101027.925)
11542.4571
(25446.7631)
166450.1700
(366959.810)
Av. W., kg - 87124.364 N/A 298765.984
Std. Dv., kg. - 26897.147 N/A 117863.478
NA ParOGN*Pa rIGN*Pa rSPS*ParSPN2
- 386002 N/A 100000
Tab.4  Optimum design comparison for planar structure with 200-bar (Design case II)
Fig.13  (a) Plan and (b)perspective view along with (c) schematic view of w-shaped profiles corresponding to optimum designation of spatial structure with 444-bar
Fig.14  Trend steps for migration, (b)for competition (c) trend line for feasible designs stored in best result (spatial structure with 444-bar)
Fig.15  The convergence history for average of best result and 10 runs (spatial structure with 444-bar)
Fig.16  Maximum displacements and unity values corresponding to feasible design with higher quality obtained when ParOGN=5 (a-b) and optimum design (c-d) (spatial structure with 444-bar)
design variable number the worst unfeasible solution feasible solution with higher quality obtained when ParOGN=5 optimum designation EGAwMP ignoring NN Lamberti and Pappalettere [2004]
1 W150X13
(1630 mm2)
W200X46.1 W200X46.1 W610X140 N/A
2 W150X13 W310X79 W310X79 W760X147 N/A
3 W150X13 W250X17.9 W250X17.9 W530X248 N/A
4 W150X13 W310X38.7 W310X38.7 W200X59 N/A
5 W150X13 W310X60 W310X60 W460X193 N/A
6 W150X13 W150X13.5 W150X18 W310X97 N/A
7 W150X13 W100X19.3 W100X19.3 W410X38.8 N/A
8 W150X13 W310X86 W310X86 W1000X321 N/A
9 W150X13 W200X26.6 W200X26.6 W310X143 N/A
10 W150X13 W150X18 W150X18 W410X85 N/A
11 W150X13 W200X71 W200X71 W920X390 N/A
12 W150X13 W200X19.3 W200X19.3 W690X265 N/A
13 W150X13 W760X173 W200X22.5 W200X22.5 N/A
14 W150X13 W760X161 W310X60 W250X73 N/A
15 W150X13 W760X147 W150X13.5 W130X23.8 N/A
16 W150X13 W760X173 W150X18 W310X32.7 N/A
17 W150X13 W760X185 W310X86 W200X35.9 N/A
18 W150X13 W920X289 W150X13 W920X313 N/A
19 W150X13 W920X345 W150X13 W1000X415 N/A
20 W150X13 W920X381 W310X79 W610X82 N/A
21 W150X13 W920X345 W150X18 W410X53 N/A
22 W150X13 W920X787 W200X19.3 W250X49.1 N/A
23 W150X13 W920X787 W310X60 W250X149 N/A
24 W150X13 W920X970 W310X52 W410X53 N/A
25 W150X13 W920X970 W150X13.5 W760X147 N/A
26 W150X13 W920X1191 W150X13.5 W250X131 N/A
27 W150X13 W920X1191 W150X13 W920X253 N/A
28 W150X13 W920X1191 W250X17.9 W530X196 N/A
29 - - W310X67 - -
NPJ 102 0 0 01
NPM 44 0 0 01
Vol. in3
mm3
157770.717
(2585398850.269)
3399400.012
(5.570618557E+10)
363402.273
(5955096314.484)
1610614.822
(2.639324817E+10)
-
Density
(lb/in3)
- - - - N/A
Weight kg
(lb)
20672.972
(45576.101)
436162.167
(961572.980)
46779.091
(103130.242)
211765.818
(47606.849)
9202.080
(20287.125)
Av. W., kg - 208115.3811 387689.4789 N/A
Std. Dv., kg. - 631561.6557 173478.0123 N/A
NA ParOGN*Pa rIGN*Pa rSPS*ParSPN2
- 442002 100000 N/A
Tab.5  Optimum design comparison for spatial truss structure with 444-bar
Fig.17  (a) Plan and (b)perspective view along with schematic view of w-shaped profiles (c) corresponding to optimum designation of spatial structure with 942-bar
section no. X-direction Y-direction Z-direction
6672.232 N
(1500 lbf) (1.5 kipf)
4448.221 N
(1000 lbf) (1.0 kipf)
4448.221 N
(1000 lbf)
(1.0 kipf)
-13344.665 N (-3000 lbf)
(-3.0 kips)
-26689.329 N (-6000 lbf)
(-6.0 kips)
-40033.995 N (-9000 lbf)
(-9.0 kips)
section 1
with 5 levels
1,4
5,8
.
.
21,24
2,3
6,7
.
.
22,23
all nodes
(1,2,….,232)
all nodes
(1,2,….,232)
section 2
with 7 levels
25,26,27,31,32
33,34,35,39,40
.
.
81,82,83,87,88
28,29,30
36,37,38
.
.
84,85,86
all nodes
(1,2,….,232)
all nodes
(1,2,….,232)
section 3
with 11 levels
89,90,91,98,99,100
101,102,103,110,111,112
.
.
221,222,223,230,231,232
92,93,94,95,96,97
104,105,106,107,108,109
.
.
224,225,226,227,228,229
all nodes
(1,2,….,232)
all nodes
(1,2,….,232)
Tab.6  Joint load values and their distribution in x, y and z directions according to node numbers
Fig.18  Trend steps for migration (a), for competition (b) and trend line for feasible designs stored in best result (c) (spatial structure with 942-bar)
Fig.19  The convergence history for average of best result and 10 runs (spatial structure with 942-bar)
Fig.20  Maximum displacements and unity values corresponding to feasible design with higher quality obtained when ParOGN=7 (a-b) and optimum design (c-d) (spatial structure with 942-bar)
design variable number the worst unfeasible solution feasible solution with higher quality obtained when ParOGN=7 optimum designation Erbatur and et al. (2000) Hasancebi and Erbatur (2002b) Hasancebi (2008) EGAwMP
ignoring NN
1 W150X13 W200X52 W150X13.5 W250X32.7 W150X13.5 W150X13.5 W530X196
2 W150X13 W150X13.5 W150X13.5 W150X13.5 W150X13.5 W200X15 W360X314
3 W150X13 W150X13.5 W150X13.5 W150X13.5 W150X13.5 W150X13.5 W460X286
4 W150X13 W150X29.8 W150X29.8 W150X22.5 W150X22.5 W150X22.5 W250X44.8
5 W150X13 W150X13 W150X13 W150X13.5 W150X13.5 W150X13.5 W310X67
6 W150X13 W150X29.8 W150X29.8 W130X28.1 W150X22.5 W150X22.5 W610X140
7 W150X13 W150X29.8 W150X29.8 W130X23.8 W150X22.5 W150X22.5 W760X147
8 W150X13 W150X13 W150X13 W360X32.9 W150X13.5 W150X13.5 W360X134
9 W150X13 W200X35.9 W200X35.9 W460X74 W150X29.8 W150X29.8 W460X68
10 W150X13 W200X35.9 W200X35.9 W200X35.9 W200X35.9 W150X37.1 W360X72
11 W150X13 W150X29.8 W150X29.8 W150X22.5 W150X22.5 W150X22.5 W310X179
12 W150X13 W150X13 W150X13 W150X13.5 W150X13.5 W150X13.5 W460X106
13 W150X13 W310X38.7 W310X38.7 W250X32.7 W150X29.8 W150X29.8 W840X193
14 W150X13 W150X29.8 W150X29.8 W150X22.5 W150X22.5 W150X22.5 W310X44.5
15 W150X13 W100X19.3 W100X19.3 W130X23.8 W100X19.3 W100X19.3 W460X213
16 W150X13 W150X13.5 W150X13.5 W150X13.5 W150X22.5 W150X13.5 W460X235
17 W150X13 W200X35.9 W200X35.9 W200X41.7 W200X41.7 W200X41.7 W310X158
18 W150X13 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W760X147
19 W150X13 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W410X53
20 W150X13 W150X13 W150X13 W150X13.5 W150X13.5 W150X13.5 W360X51
21 W150X13 W410X100 W310X67 W200X52 W200X52 W200X52 W760X134
22 W150X13 W150X22.5 W150X22.5 W150X29.8 W150X29.8 W150X29.8 W610X125
23 W150X13 W250X67 W250X67 W200X46.1 W150X37.1 W200X35.9 W610X174
24 W150X13 W310X52 W310X52 W310X60 W200X52 W250X67 W840X176
25 W150X13 W360X91 W360X91 W200X86 W250X73 W200X86 W360X57.8
26 W150X13 W310X52 W310X52 W250X49.1 W200X46.1 W200X46.1 W760X314
27 W150X13 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W840X299
28 W150X13 W200X35.9 W200X35.9 W310X38.7 W200X35.9 W200X35.9 W410X75
29 W150X13 W150X29.8 W150X29.8 W200X35.9 W360X39 W150X37.1 W310X44.5
30 W150X13 W100X19.3 W100X19.3 W360X32.9 W200X31.3 W250X32.7 W920X201
31 W150X13 W250X149 W310X129 W250X101 W310X129 W360X134 W1000X554
32 W150X13 W150X29.8 W150X29.8 W200X35.9 W150X29.8 W150X29.8 W530X92
33 W150X13 W150X29.8 W150X29.8 W150X22.5 W150X29.8 W150X22.5 W920X253
34 W150X13 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W310X79
35 W150X13 W150X13 W150X13 W150X13.5 W150X13.5 W150X13.5 W460X384
36 W150X13 W200X52 W150X13 W150X13.5 W150X13.5 W150X13.5 W1000X222
37 W150X13 W840X176 W360X162 W610X155 W360X147 W360X147 W840X299
38 W150X13 W200X35.9 W200X35.9 W200X35.9 W200X35.9 W200X35.9 W200X71
39 W150X13 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W150X22.5 W360X162
40 W150X13 W150X29.8 W150X29.8 W150X29.8 W150X29.8 W150X29.8 W360X72
41 W150X13 W150X13 W150X13 W150X13.5 W150X13.5 W150X29.8 W360X314
42 W150X13 W150X13 W150X13 W100X19.3 W150X13.5 W200X15 W250X67
43 W150X13 W610X155 W610X155 W310X202 W610X195 W610X195 W360X216
44 W150X13 W200X35.9 W200X35.9 W200X46.1 W200X46.1 W200X46.1 W250X44.8
45 W150X13 W150X22.5 W200X35.9 W150X22.5 W150X22.5 W150X22.5 W760X185
46 W150X13 W200X35.9 W200X35.9 W200X35.9 W200X35.9 W200X35.9 W460X193
47 W150X13 W100X19.3 W100X19.3 W200X26.6 W100X19.3 W100X19.3 W460X384
48 W150X13 W150X13.5 W150X13.5 W150X29.8 W150X13.5 W150X13.5 W360X72
49 W150X13 W360X216 W360X216 W360X216 W360X216 W360X216 W250X149
50 W150X13 W200X46.1 W200X46.1 W200X46.1 W200X46.1 W200X46.1 W920X313
51 W150X13 W200X35.9 W200X35.9 W150X29.8 W200X41.7 W310X44.5 W410X85
52 W150X13 W200X35.9 W200X35.9 W200X46.1 W200X35.9 W200X35.9 W460X97
53 W150X13 W250X89 W250X89 W360X91 W250X89 W310X97 W310X342
54 W150X13 W410X114 W310X86 W200X71 W610X101 W530X109 W610X498
55 W150X13 W360X196 W360X196 W360X179 W360X196 W360X196 W250X67
56 W150X13 W200X52 W200X52 W200X46.1 W200X52 W200X46.1 W460X315
57 W150X13 W310X97 W310X97 W250X149 W310X117 W310X107 W840X527
58 W150X13 W200X52 W200X52 W250X49.1 W200X35.9 W200X41.7 W530X300
59 W150X13 W200X52 W200X52 W250X49.1 W200X52 W200X46.1 W310X97
60 - - W200X46.1 - - - -
NPJ 169 0 0 0 0 0 0
NPM 721 0 0 2 4 3 0
Vol. in3
mm3
460235.486
(7.541E09)
1348647.146
(2.210E+10)
1299029.831
(2.128E+10)
1395862.387
(2.287E10)
1359183.630
(2.227E10)
1337873.767
(2.192E10)
5885911.467
(9.645E+10)
Density
(lb/in3)
- - - - - - -
Weight kg
(lb)
60305.459
(132950.779)
173045.1131
(381499.170)
167573.0363
(369435.306)
178864.876
(394329.552)
172241.498
(379727.503)
171437.346
(377954.651)
756852.954
(1668575.145)
Av. W., kg 382025.024 N/A N/A 1758653.874
Std. Dv., kg. 97561.025 N/A N/A 547962.150
NA ParOGN*Pa rIGN*Pa rSPS*ParSPN1
399001 100000
Tab.7  Optimum design comparison for spatial structure with 942-bar
Ag: : gross cross sectional area λ : slenderness parameter
Fy : yield stress S : elastic section modules
Fcr: : critical stress Mn : nominal flexural strength
K: : effective length factor Mr : limiting buckling moment
L : un-braced member length Mp : plastic bending moment
Q : reduction factor Pn : nominal axial strength
h : clear distance Cb : bending coefficient
Aw : area of web Cw : warping coefficient
t : plate thickness Vn : nominal shear strength
b : plate width
  
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