Applications of Energy-Based Evaluation Method of Element Importance to Grid Structures
Abstract
:1. Introduction
2. Evaluation Method of Element Importance
2.1. Hypothesis
- Only linear elasticity and geometric nonlinearity are introduced in the analysis, while material damage and structural stability are not taken into account.
- The topological relationship will be modified by the conceptual removal method in the numerical simulation for the initial damage. The additional load and dynamic effects induced by falling and explosion are not included.
- In the event of an accident, the structure is rarely damaged by earthquake and wind loads simultaneously. As a result, only the vertical nodal load is taken into the evaluation.
- The external loads at the nodes are not affected by the failure of elements in the grid structures.
2.2. Element Importance Coefficient
2.3. Flow Chart of the Method
3. Element Importance of Square Pyramid Space Grid Structures
4. Influence of Load Pattern and Geometry
4.1. Effects of Load Patterns
4.2. Influence of the Support Conditions
4.3. Influence of the Span
4.4. Influence of Sizes of the Basic Pyramid Grid
5. Effects of the Spatial Grid Topology
5.1. Diagonal Square Pyramid Grids
5.2. Bi-Orthogonal Lattice Grids
5.3. Bi-Orthogonal-Diagonal Lattice Grids
6. Conclusions
- (1)
- For square pyramid grid structures, the results show that the lower and top chords are more important than the diagonal web members. The importance coefficients of the elements are different under different load patterns, and the order of importance of the elements is also changed when subjected to horizontal loads.
- (2)
- The change of the support condition reduces the importance of the upper and lower chords in the center area for square pyramid grid structures. The effects of the structural span and the size of the basic pyramid grid can be ignored for the location of the most important members.
- (3)
- There are different distribution characteristics for the element importance coefficients for grid structures with different topologies. The location of important elements for the upper chords, the lower chords and the web members are dependent on the relationship between the arrangement of these members and support sides.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Element Number | αi | Element Number | αi | Element Number | αi |
---|---|---|---|---|---|
27 | 0.00437 | 228 | 0.00649 | 407 | 0.00113 |
29 | 0.00779 | 230 | 0.00766 | 409 | 0.00104 |
31 | 0.01024 | 245 | 0.00905 | 410 | 0.00106 |
33 | 0.01112 | 246 | 0.00615 | 411 | 0.00160 |
47 | 0.00720 | 247 | 0.01391 | 413 | 0.00145 |
49 | 0.00929 | 248 | 0.00647 | 414 | 0.00180 |
50 | 0.01339 | 249 | 0.01672 | 415 | 0.00204 |
51 | 0.01036 | 263 | 0.01308 | 416 | 0.00126 |
52 | 0.01802 | 265 | 0.01569 | 417 | 0.00162 |
54 | 0.01972 | 266 | 0.02062 | 418 | 0.00223 |
70 | 0.01762 | 267 | 0.01658 | 419 | 0.00229 |
72 | 0.01982 | 268 | 0.02510 | 420 | 0.00155 |
73 | 0.02406 | 284 | 0.02501 | 449 | 0.00106 |
75 | 0.02645 | 286 | 0.02652 | 453 | 0.00120 |
93 | 0.02722 | 287 | 0.03066 | 455 | 0.00101 |
96 | 0.02999 | 305 | 0.03256 | 457 | 0.00117 |
Element Number | αi | Element Number | αi | Element Number | αi |
---|---|---|---|---|---|
47 | 0.0275 | 261 | 0.0380 | 483 | 0.0075 |
66 | 0.0352 | 263 | 0.0390 | 522 | 0.0069 |
68 | 0.0465 | 278 | 0.0307 | 523 | 0.0093 |
70 | 0.0377 | 280 | 0.0532 | 524 | 0.0089 |
87 | 0.0439 | 282 | 0.0553 | 561 | 0.0072 |
89 | 0.0587 | 284 | 0.0338 | 562 | 0.0075 |
91 | 0.0481 | 297 | 0.0347 | 563 | 0.0103 |
108 | 0.0468 | 299 | 0.0608 | 564 | 0.0103 |
110 | 0.0628 | 301 | 0.0638 | 584 | 0.0066 |
112 | 0.0516 | 303 | 0.0394 | 544 | 0.0066 |
Element Number | αi | Element Number | αi | Element Number | αi |
---|---|---|---|---|---|
70 | 0.0267 | 116 | 0.0243 | 306 | 0.0314 |
72 | 0.0240 | 282 | 0.0329 | 308 | 0.0286 |
73 | 0.0247 | 284 | 0.0337 | 418 | 0.0030 |
75 | 0.0249 | 285 | 0.0263 | 422 | 0.0032 |
89 | 0.0307 | 286 | 0.0268 | 483 | 0.0031 |
91 | 0.0358 | 287 | 0.0297 | 522 | 0.0026 |
93 | 0.0325 | 289 | 0.0268 | 523 | 0.0039 |
94 | 0.0275 | 299 | 0.0292 | 524 | 0.0034 |
96 | 0.0281 | 301 | 0.0393 | 561 | 0.0030 |
110 | 0.0333 | 303 | 0.0406 | 562 | 0.0030 |
112 | 0.0391 | 304 | 0.0276 | 563 | 0.0043 |
114 | 0.0356 | 305 | 0.0327 | 564 | 0.0042 |
Element Number | αi | Element Number | αi | Element Number | αi |
---|---|---|---|---|---|
54 | 0.0010 | 404 | 0.0194 | 416 | 0.0164 |
73 | 0.0012 | 405 | 0.0215 | 417 | 0.0219 |
75 | 0.0014 | 406 | 0.0205 | 418 | 0.0214 |
93 | 0.0014 | 407 | 0.0169 | 420 | 0.0161 |
96 | 0.0016 | 408 | 0.0182 | 445 | 0.0221 |
268 | 0.0012 | 409 | 0.0217 | 448 | 0.0183 |
284 | 0.0012 | 410 | 0.0208 | 449 | 0.0210 |
286 | 0.0013 | 411 | 0.0159 | 452 | 0.0168 |
287 | 0.0015 | 412 | 0.0170 | 453 | 0.0208 |
305 | 0.0016 | 413 | 0.0218 | 456 | 0.0160 |
401 | 0.0181 | 414 | 0.0212 | 457 | 0.0208 |
403 | 0.0189 | 415 | 0.0155 | 460 | 0.0156 |
Element Number | Two Edges | Four Edges | Element Number | Two Edges | Four Edges | Element Number | Two Edges | Four Edges |
---|---|---|---|---|---|---|---|---|
2 | 0.0053 | 0 | 96 | 0.0029 | 0.0300 | 298 | 0.0913 | 0.0004 |
4 | 0.0087 | 0.0004 | 106 | 0.0068 | 0.0031 | 300 | 0.0548 | 0.0065 |
6 | 0.0093 | 0.0013 | 241 | 0.0475 | 0.0002 | 302 | 0.0519 | 0.0166 |
8 | 0.0093 | 0.0022 | 260 | 0.0697 | 0.0003 | 304 | 0.0509 | 0.0265 |
10 | 0.0093 | 0.0029 | 262 | 0.0402 | 0.0052 | 306 | 0.0506 | 0.0326 |
12 | 0.0093 | 0.0031 | 279 | 0.0856 | 0.0004 | 402 | 0.0064 | 0 |
64 | 0.0033 | 0.0022 | 281 | 0.0509 | 0.0061 | 406 | 0.0031 | 0.0003 |
85 | 0.0058 | 0.0029 | 283 | 0.0481 | 0.0157 | 407 | 0.0030 | 0.0011 |
92 | 0.0029 | 0.0198 | 285 | 0.0473 | 0.0250 | 441 | 0.0041 | 0.0005 |
94 | 0.0029 | 0.0272 | 287 | 0.0470 | 0.0307 | 442 | 0.0048 | 0.0004 |
Number of Basic Pyramid Grids | 8 × 8 | 8 × 10 | 8 × 12 | 10 × 10 | 10 × 12 |
---|---|---|---|---|---|
Upper chord | 0.0446 | 0.0520 | 0.0511 | 0.0300 | 0.0346 |
Lower chord | 0.0472 | 0.0554 | 0.0546 | 0.0326 | 0.0377 |
Web member | 0.0052 | 0.0050 | 0.0046 | 0.0023 | 0.0023 |
Number of Basic Pyramid Grids | 10 × 16 | 10 × 12 | 10 × 10 | 10 × 8 |
---|---|---|---|---|
Upper chord | 0.0228 | 0.0270 | 0.0300 | 0.0361 |
Lower chord | 0.0234 | 0.0286 | 0.0326 | 0.0394 |
Web member | 0.0014 | 0.0018 | 0.0023 | 0.0035 |
Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
Cross-section | Φ60 × 4 | Φ76 × 4 | Φ89 × 4 | Φ114 × 4 | Φ140 × 4 | Φ159 × 6 | Φ159 × 8 |
Element Number | Element Number | Element Number | |||
---|---|---|---|---|---|
9 | 0.0106 | 145 | 0.0119 | 502 | 0.0559 |
12 | 0.0145 | 165 | 0.0105 | 503 | 0.0673 |
37 | 0.0135 | 442 | 0.0180 | 521 | 0.0686 |
55 | 0.0126 | 444 | 0.0407 | 627 | 0.0184 |
57 | 0.0257 | 446 | 0.0414 | 629 | 0.0228 |
59 | 0.0242 | 461 | 0.0241 | 631 | 0.0363 |
77 | 0.0244 | 463 | 0.0489 | 633 | 0.0226 |
79 | 0.0261 | 465 | 0.0527 | 635 | 0.0385 |
99 | 0.0347 | 479 | 0.0280 | 661 | 0.0010 |
101 | 0.0192 | 481 | 0.0296 | 669 | 0.0250 |
103 | 0.0128 | 482 | 0.0542 | 671 | 0.0129 |
121 | 0.0148 | 483 | 0.0304 | 673 | 0.0275 |
123 | 0.0102 | 484 | 0.0621 | 675 | 0.0174 |
143 | 0.0142 | 500 | 0.0558 | 713 | 0.0101 |
Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
Cross-section | Φ60 × 4 | Φ76 × 4 | Φ89 × 4 | Φ114 × 4 | Φ140 × 4 | Φ159 × 6 | Φ159 × 8 | Φ180 × 8 |
Upper Chord Number | Lower Chord Number | Web Member Number | |||
---|---|---|---|---|---|
8 | 0.0146 | 73 | 0.0693 | 295 | 0.0667 |
10 | 0.0259 | 75 | 0.0855 | 313 | 0.0610 |
12 | 0.0313 | 93 | 0.0703 | 316 | 0.0754 |
29 | 0.0276 | 96 | 0.0863 | 458 | 0.0257 |
31 | 0.0464 | 251 | 0.0247 | 460 | 0.0311 |
33 | 0.0552 | 253 | 0.0298 | 481 | 0.0183 |
47 | 0.0108 | 270 | 0.0244 | 675 | 0.0174 |
50 | 0.0347 | 272 | 0.0412 | 676 | 0.0293 |
52 | 0.0616 | 274 | 0.0493 | 677 | 0.0327 |
54 | 0.0774 | 290 | 0.0294 | 687 | 0.0199 |
70 | 0.0379 | 292 | 0.0331 | 688 | 0.0203 |
72 | 0.0360 | 293 | 0.0528 | 721 | 0.2497 |
Number | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
Cross-section | Φ60 × 4 | Φ76 × 4 | Φ89 × 4 | Φ114 × 4 | Φ140 × 4 | Φ159 × 6 |
Upper Chord Number | Lower Chord Number | Web Member Number | |||
---|---|---|---|---|---|
1 | 0.0109 | 314 | 0.0169 | 685 | 0.0130 |
201 | 0.0175 | 315 | 0.0103 | 686 | 0.0118 |
202 | 0.0204 | 324 | 0.0143 | 695 | 0.0165 |
203 | 0.0152 | 463 | 0.0123 | 696 | 0.0145 |
212 | 0.0253 | 472 | 0.0165 | 697 | 0.0114 |
213 | 0.0205 | 481 | 0.0148 | 705 | 0.0176 |
214 | 0.0134 | 543 | 0.0127 | 706 | 0.0162 |
223 | 0.0187 | 552 | 0.0131 | 715 | 0.0190 |
224 | 0.0129 | 613 | 0.0155 | 767 | 0.0149 |
234 | 0.0104 | 614 | 0.0136 | 768 | 0.0187 |
302 | 0.0237 | 615 | 0.0105 | 769 | 0.0202 |
303 | 0.0223 | 623 | 0.0170 | 778 | 0.0101 |
304 | 0.0155 | 624 | 0.0153 | 779 | 0.0104 |
313 | 0.0229 | 633 | 0.0186 | 855 | 0.0345 |
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Wang, L.; Feng, J.; Xu, Y.; Zhuang, L.; Zhang, Q.; Cai, J. Applications of Energy-Based Evaluation Method of Element Importance to Grid Structures. Symmetry 2022, 14, 370. https://doi.org/10.3390/sym14020370
Wang L, Feng J, Xu Y, Zhuang L, Zhang Q, Cai J. Applications of Energy-Based Evaluation Method of Element Importance to Grid Structures. Symmetry. 2022; 14(2):370. https://doi.org/10.3390/sym14020370
Chicago/Turabian StyleWang, Liwu, Jian Feng, Yixiang Xu, Liping Zhuang, Qian Zhang, and Jianguo Cai. 2022. "Applications of Energy-Based Evaluation Method of Element Importance to Grid Structures" Symmetry 14, no. 2: 370. https://doi.org/10.3390/sym14020370
APA StyleWang, L., Feng, J., Xu, Y., Zhuang, L., Zhang, Q., & Cai, J. (2022). Applications of Energy-Based Evaluation Method of Element Importance to Grid Structures. Symmetry, 14(2), 370. https://doi.org/10.3390/sym14020370