On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams
Abstract
:1. Introduction
2. Functionally Graded Beams
3. Development of the Modified Timoshenko Beam Element
3.1. Displacement Field and Strain Energy
3.2. A New Modified Timoshenko Beam Element
4. Results and Discussion
4.1. Validation Study
4.2. Numerical Results
4.2.1. The Effects of the Length-to-Thickness Ratio on the Bending of FG Beams
4.2.2. The Effects of the Power-Law Index on the Bending of FG Beams
5. Conclusions
- -
- When the length-to-thickness ratio increases, the deflections in the FG beams increase rapidly.
- -
- When the power-law index increases, the deflections in the FG beams increase.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Element Type | Number of Elements | Li et al. [14] | Thai et al. [17] | ||||||
---|---|---|---|---|---|---|---|---|---|
2 | 12 | 18 | 24 | 32 | |||||
5 | 0 | Proposed | 3.1657 | 3.1657 | 3.1657 | 3.1657 | 3.1657 | 3.1657 | 3.1654 |
Timoshenko | 0.8624 | 2.9836 | 3.0823 | 3.1183 | 3.1388 | - | - | ||
1 | Proposed | 6.2599 | 6.2599 | 6.2599 | 6.2599 | 6.2599 | 6.2599 | 6.2594 | |
Timoshenko | 1.5105 | 5.8417 | 6.0675 | 6.1503 | 6.1979 | - | - | ||
5 | Proposed | 9.6483 | 9.6483 | 9.6483 | 9.6483 | 9.6483 | 9.7802 | 9.8281 | |
Timoshenko | 3.0362 | 9.3314 | 9.5914 | 9.6854 | 9.7391 | - | - | ||
10 | Proposed | 10.7195 | 10.7195 | 10.7195 | 10.7195 | 10.7195 | 10.8979 | 10.9381 | |
Timoshenko | 3.6940 | 10.4678 | 10.7257 | 10.8187 | 10.8716 | - | - | ||
20 | 0 | Proposed | 2.8963 | 2.8963 | 2.8963 | 2.8963 | 2.8963 | 2.8962 | 2.8962 |
Timoshenko | 0.0702 | 1.5237 | 2.0700 | 2.3657 | 2.5720 | - | - | ||
1 | Proposed | 5.8049 | 5.8049 | 5.8049 | 5.8049 | 5.8049 | 5.8049 | 5.8049 | |
Timoshenko | 0.1190 | 2.8056 | 3.9391 | 4.5847 | 5.0495 | - | - | ||
5 | Proposed | 8.8069 | 8.8069 | 8.8069 | 8.8069 | 8.8069 | 8.8151 | 8.8182 | |
Timoshenko | 0.2573 | 5.0532 | 6.6292 | 7.4377 | 7.9846 | - | - | ||
10 | Proposed | 9.6767 | 9.6767 | 9.6767 | 9.6767 | 9.6767 | 9.6879 | 9.6905 | |
Timoshenko | 0.3229 | 5.8740 | 7.5244 | 8.3413 | 8.8829 |
4 | 0.0730 | 0.0899 | 0.1057 | 0.1327 | 0.1678 | 0.2110 | 0.2429 |
5 | 0.1200 | 0.1488 | 0.1759 | 0.2218 | 0.2799 | 0.3456 | 0.3933 |
8 | 0.3919 | 0.4908 | 0.5847 | 0.7428 | 0.9341 | 1.1199 | 1.2511 |
10 | 0.7204 | 0.9052 | 1.0809 | 1.3762 | 1.7288 | 2.0540 | 2.2813 |
15 | 2.2815 | 2.8769 | 3.4441 | 4.3960 | 5.5159 | 6.4881 | 7.1588 |
20 | 5.2837 | 6.6714 | 7.9947 | 10.2137 | 12.8103 | 15.0106 | 16.5204 |
4 | 0.2369 | 0.2973 | 0.3546 | 0.4510 | 0.5668 | 0.6762 | 0.7531 |
5 | 0.4403 | 0.5539 | 0.6620 | 0.8435 | 1.0593 | 1.2541 | 1.3898 |
8 | 1.7036 | 2.1501 | 2.5757 | 3.2896 | 4.1265 | 4.8413 | 5.3327 |
10 | 3.2823 | 4.1459 | 4.9695 | 6.3504 | 7.9639 | 9.3224 | 10.2532 |
15 | 10.9280 | 13.8141 | 16.5681 | 21.1837 | 26.5592 | 31.0191 | 34.0639 |
20 | 25.7791 | 32.5966 | 39.1035 | 50.0069 | 62.6906 | 73.1583 | 80.2955 |
4 | 2.0954 | 2.6409 | 3.1604 | 4.0324 | 5.0606 | 5.9612 | 6.5837 |
5 | 4.0027 | 5.0511 | 6.0504 | 7.7266 | 9.6927 | 11.3764 | 12.5345 |
8 | 15.9962 | 20.2147 | 24.2395 | 30.9858 | 38.8524 | 45.4154 | 49.9019 |
10 | 31.0628 | 39.2680 | 47.0982 | 60.2206 | 75.5009 | 88.1690 | 96.8159 |
15 | 104.2373 | 131.8167 | 158.1407 | 202.2495 | 253.5398 | 295.7938 | 324.5911 |
20 | 246.5837 | 311.8631 | 374.1753 | 478.5807 | 599.9251 | 699.6670 | 767.6072 |
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Ghazwani, M.H.; Vinh, P.V.V. On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics 2025, 13, 73. https://doi.org/10.3390/math13010073
Ghazwani MH, Vinh PVV. On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics. 2025; 13(1):73. https://doi.org/10.3390/math13010073
Chicago/Turabian StyleGhazwani, Mofareh Hassan, and Pham V. V. Vinh. 2025. "On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams" Mathematics 13, no. 1: 73. https://doi.org/10.3390/math13010073
APA StyleGhazwani, M. H., & Vinh, P. V. V. (2025). On the Development of a Modified Timoshenko Beam Element for the Bending Analysis of Functionally Graded Beams. Mathematics, 13(1), 73. https://doi.org/10.3390/math13010073