Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations
Abstract
:1. Introduction
2. Problem Formulation
3. Bending Problem
- (a)
- S–S boundary conditions
- (b)
- C–C boundary conditions
4. Buckling Problem
4.1. Buckling of Perfect Beam
- (a)
- S–S boundary conditions
- (b)
- C–C boundary conditions
4.2. Buckling of Imperfect Beam
- (c)
- S–S boundary conditions
- (d)
- C–C boundary conditions
5. Numerical Analysis
5.1. Results of Nonlinear Bending
5.2. Buckling Analysis
6. Concluding Remarks
- The nonlinear bending deflection due to point load is highly dependent on the application position of the load.
- The hardening structural responses are associated with increasing the elastic foundation constants.
- The buckling strength is improved with an increase in the amplitude of initial imperfection; next, the critical values are continuously decreased with an increase in the initial imperfection amplitude.
- For larger values of amplitude of imperfection, the helicoidal composite beams can enhance the critical buckling loads.
- The layup configurations have a great influence on the nonlinear bending and buckling responses of perfect and imperfect beams.
- The proposed model can be exploited broadly in various engineering applications, such as airplane wings, helicopter blades, wind turbine blades, as well as many others in the aerospace, mechanical, and civil industries.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Designation | Stacking Sequence | Description |
---|---|---|
UD | Unidirectional | |
CP | Cross ply-symmetric | |
QI | Quasi isotropic-symmetric | |
3H | Helicoidal (3°)-symmetric | |
6H | Helicoidal (6°)-symmetric | |
12H | Helicoidal (12°)-symmetric |
Laminate | ||||||
---|---|---|---|---|---|---|
(a) S–S | ||||||
UD | 256.4084 | 861.8419 | 1919.4286 | 321.9250 | 878.2210 | 1926.7082 |
CP | 149.2720 | 501.7343 | 1117.4243 | 187.4135 | 511.2696 | 1121.6623 |
QI | 108.1414 | 363.4857 | 809.5277 | 135.7733 | 370.3937 | 812.5979 |
3H | 232.0188 | 779.8635 | 1736.8525 | 291.3035 | 794.6846 | 1743.4397 |
6H | 188.9145 | 634.9810 | 1414.1813 | 237.1854 | 647.0487 | 1419.5447 |
12H | 137.8104 | 463.2093 | 1031.6245 | 173.0232 | 472.0125 | 1035.5370 |
(b) C–C | ||||||
UD | 883.6214 | 1749.6677 | 3411.9220 | 932.4433 | 1763.0029 | 3424.4339 |
CP | 514.4136 | 1018.5955 | 1986.3019 | 542.8360 | 1026.3588 | 1993.5860 |
QI | 372.6714 | 737.9303 | 1438.9936 | 393.2622 | 743.5545 | 1444.2706 |
3H | 799.5713 | 1583.2392 | 3087.3799 | 843.7493 | 1595.3060 | 3098.7017 |
6H | 651.0276 | 1289.1062 | 2513.8087 | 686.9981 | 1298.9312 | 2523.0271 |
12H | 474.9151 | 940.3841 | 1833.7866 | 501.1551 | 947.5513 | 1840.5113 |
UD | CP | QI | 3H | 6H | 12H | |
---|---|---|---|---|---|---|
358.9829 | 207.2039 | 151.5052 | 316.3949 | 249.3464 | 189.6495 | |
412.8145 | 238.8817 | 174.1865 | 366.2812 | 289.9060 | 219.1237 | |
402.0357 | 237.3041 | 169.3567 | 376.7940 | 312.7668 | 221.6148 | |
260.9025 | 164.7909 | 109.2545 | 291.3420 | 274.9440 | 162.9004 | |
0 | 27.3479 | −1.6466 | 118.7747 | 182.9625 | 48.3833 |
UD | CP | QI | 3H | 6H | 12H | |
---|---|---|---|---|---|---|
1239.6624 | 716.3208 | 523.1438 | 1096.6105 | 869.0241 | 656.4225 | |
1651.2582 | 955.5267 | 696.7461 | 1465.1250 | 1159.6239 | 876.4950 | |
1608.1426 | 949.2164 | 677.4268 | 1507.1759 | 1251.0673 | 886.4591 | |
1043.6099 | 659.1638 | 437.0182 | 1165.3680 | 1099.7761 | 651.6016 | |
0 | 109.3918 | −6.5864 | 475.0987 | 731.8499 | 193.5332 |
(a) A0 = 0 | ||||||
UD | 320.5012 | 364.1790 | 429.6956 | 428.2718 | 471.9496 | 537.4662 |
CP | 186.5846 | 212.0123 | 250.1538 | 249.3249 | 274.7526 | 312.8941 |
QI | 135.1728 | 153.5941 | 181.2261 | 180.6256 | 199.0469 | 226.6788 |
3H | 290.0151 | 329.5383 | 388.8230 | 387.5346 | 427.0577 | 486.3424 |
6H | 236.1363 | 268.3169 | 316.5877 | 315.5387 | 347.7192 | 395.9901 |
12H | 172.2580 | 195.7332 | 230.9461 | 230.1808 | 253.6561 | 288.8689 |
(b) A0 = 4 | ||||||
UD | 107.7706 | 151.4484 | 216.9650 | 215.5412 | 259.2189 | 324.7356 |
CP | 90.0882 | 115.5159 | 153.6574 | 152.8285 | 178.2562 | 216.3977 |
QI | 43.8061 | 62.2274 | 89.8594 | 89.2589 | 107.6802 | 135.3121 |
3H | 216.2941 | 255.8173 | 315.1020 | 313.8136 | 353.3367 | 412.6214 |
6H | 262.3648 | 294.5454 | 342.8162 | 341.7672 | 373.9477 | 422.2186 |
12H | 106.3061 | 129.7814 | 164.9943 | 164.2290 | 187.7042 | 222.9171 |
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Almitani, K.H.; Mohamed, N.; Alazwari, M.A.; Mohamed, S.A.; Eltaher, M.A. Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations. Mathematics 2022, 10, 887. https://doi.org/10.3390/math10060887
Almitani KH, Mohamed N, Alazwari MA, Mohamed SA, Eltaher MA. Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations. Mathematics. 2022; 10(6):887. https://doi.org/10.3390/math10060887
Chicago/Turabian StyleAlmitani, Khalid H., Nazira Mohamed, Mashhour A. Alazwari, Salwa A. Mohamed, and Mohamed A. Eltaher. 2022. "Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations" Mathematics 10, no. 6: 887. https://doi.org/10.3390/math10060887
APA StyleAlmitani, K. H., Mohamed, N., Alazwari, M. A., Mohamed, S. A., & Eltaher, M. A. (2022). Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations. Mathematics, 10(6), 887. https://doi.org/10.3390/math10060887