Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation
Abstract
:1. Introduction
2. Statement of the Problem
3. Determination of Natural Frequencies and Eigenmodes
3.1. Analytic Solution
3.2. Formulation of Finite Element Method
4. Results and Discussion
4.1. Graphical and Tabular Representations
4.2. Validation of the Results
5. Conclusions
- The eigenvalues obtained in terms of the slenderness ratio for the RB depend on the geometry, unlike the EBB where eigenvalues do not depend on the slenderness ratio.
- The behavior of a beam under different conditions, such as the presence of rotatory inertia or placement on an elastic foundation, impacts its natural frequencies.
- For smaller stiffness parameters, the beam undergoes rigid body modes without significant internal deformation.
- The inclusion of rotational inertia had a minimal effect on the fundamental mode frequency, but it had a significant impact on the higher frequency modes.
- Placing the Winkler elastic foundation under the beam caused an increase in stiffness, leading to higher frequencies as the elastic foundation stiffness increased.
- A detailed tabular and graphical analysis proved that the vibration frequencies and mode shapes are more affected by the linear spring stiffness compared to rotary spring stiffness.
- Unlike independent beams, beams on an elastic foundation require higher frequencies to vibrate. Thus, by controlling the elastic foundation parameter, one can adjust the vibrating frequency to minimize collateral damage to the vibrating structure.
- While comparing results with the existing ones in the literature, it has been observed that the finite element scheme provided the best accuracy for obtaining the mode shapes of the beam structure.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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BC | (Clamped-Clamped Beam) | |||
RB-AM | 250.696553 | 670.332080 | 1257.418531 | 1957.632175 |
RB-FEM | 250.508697 | 668.569075 | 1250.974018 | 1952.105425 |
PE | 0.07 | 0.2 | 0.5 | 0.2 |
EBB-FEM | 253.570332 | 698.649266 | 1368.685838 | 2260.407107 |
BC | ||||
RB-AM | 244.547215 | 627.425812 | 1108.8354005 | 11,607.889441 |
RB-FEM | 244.381088 | 626.097545 | 1105.006215 | 1600.50013598 |
PE | 0.06 | 0.2 | 0.3 | 0.4 |
EBB-FEM | 247.081229 | 648.3165172 | 1170.690514 | 1732.887219 |
BC | ||||
RB-AM | 0.939639 | 3.597750 | 242.763465 | 637.058042 |
RB-FEM | 0.939654 | 3.595103 | 242.07727 | 633.471148 |
PE | 0.01 | 0.07 | 0.2 | 0.5 |
EBB-FEM | 0.9401511 | 3.63815 | 253.7942605 | 699.31880034 |
BC | ||||
RB-AM | 0 | 0.359903 | 242.612375 | 636.908474 |
RB-FEM | 0.094067 | 0.359129 | 241.926479 | 633.3220740 |
PE | 0 | 0.2 | 0.2 | 0.5 |
EBB-FEM | 0.09731147 | 0.364811 | 253.638632 | 699.161921 |
BC | (Free-Free Beam) | |||
RB-AM | 0 | 0 | 242.6108484 | 636.906962 |
RB-FEM | 0 | 0 | 241.924691 | 633.315889 |
PE | 0 | 0 | 0.2 | 0.5 |
EBB-FEM | 0 | 0 | 253.637057 | 699.1906165 |
K | |||||
---|---|---|---|---|---|
RB-AM | 244.635620 | 627.458744 | 1108.853233 | 1607.90124959 | |
RB-FEM | 244.469433 | 626.1304062 | 1105.023973 | 1600.51188801 | |
PE | 0.06 | 0.2 | 0.3 | 0.4 | |
EBB-FEM | 247.1705 | 648.350565 | 1170.709371 | 1732.899957 | |
245.429833 | 627.755057 | 1109.013621 | 1608.0075189 | ||
245.263102 | 626.426070 | 1105.1837765 | 1600.617654 | ||
0.06 | 0.2 | 0.3 | 0.4 | ||
247.973024 | 648.656915 | 1170.879057 | 1733.01459964 | ||
254.75624506 | 631.803299 | 1111.738550003 | 1610.2197737 | ||
253.0629107 | 629.375057 | 1106.7805309 | 1601.674927 | ||
0.6 | 0.3 | 0.4 | 0.5 | ||
255.859337 | 651.712493 | 1172.574581 | 1734.160598 | ||
321.011724 | 659.535432 | 1226.516780 | 1619.654366 | ||
320.793113 | 658.136708 | 1222.621996 | 1612.209306 | ||
0.06 | 0.2 | 0.3 | 0.4 | ||
324.34210031 | 681.515201 | 1189.396883 | 1745.579210 | ||
696.079274 | 890.563168 | 1264.837768 | 1711.141649 | ||
701.049992 | 896.218158 | 1270.112079 | 1713.973989 | ||
0.7 | 0.6 | 0.4 | 0.1 | ||
708.897576 | 928.332346 | 1346.106578 | 1855.905373 |
BC | ||||
RB-AM | 247.876023 | 663.070974 | 1244.440889 | 1948.406590 |
RB-FEM | 247.692291 | 661.344524 | 1238.119361 | 1933.149164 |
PE | 0.07 | 0.2 | 0.5 | 0.7 |
EBB-FEM | 250.685988 | 690.7803537 | 1353.424206 | 1933.149164 |
BC | ||||
RB-AM | 118.095539 | 439.042478 | 939.30890 | 1578.319082 |
RB-FEM | 118.024116 | 438.040036 | 934.620135 | 1566.555535 |
PE | 0.06 | 0.2 | 0.4 | 0.7 |
EBB-FEM | 119.183343 | 454.989152 | 1014.407906 | 1796.176890 |
BC | ||||
RB-AM | 110.868207 | 431.774070 | 931.988756 | 1571.619022 |
RB-FEM | 110.801167 | 430.788243 | 927.610823 | 1559.904574 |
PE | 0.06 | 0.2 | 0.4 | 0.7 |
EBB-FEM | 111.889149 | 447.456074 | 1006.476980 | 1788.564665 |
BC | (Supported-Supported beam) | |||
RB-AM | 110.860491 | 431.766562 | 931.981563 | 1571.612216 |
RB-FEM | 110.793383 | 430.779004 | 927.586263 | 1559.814971 |
[31] | 110.867126 | 431.85454 | 932.340065 | —– |
PE | 0.06 | 0.2 | 0.3 | 0.7 |
EBB-FEM | 111.881330 | 447.448294 | 1006.469162 | 1788.556629 |
[31] | 111.888296 | 447.553217 | 1006.994779 | —– |
K | |||||
---|---|---|---|---|---|
RB-AM | 111.073523 | 431.8216718 | 932.009064 | 1571.629866 | |
RB-FEM | 110.996365 | 430.835737 | 927.631036 | 1559.915337 | |
PE | 0.06 | 0.2 | 0.4 | 0.7 | |
EBB-FEM | 112.086260 | 447.505404 | 1006.498912 | 1788.577007 | |
112.562480 | 432.249847 | 932.191820 | 1571.727457 | ||
112.737932 | 431.262934 | 927.812933 | 1560.012200 | ||
0.1 | 0.2 | 0.2 | 0.7 | ||
113.844930 | 447.794913 | 1006.696279 | 1778.688080 | ||
128.943498 | 436.508501 | 934.017415 | 1572.703052 | ||
128.865529 | 435.512864 | 929.629948 | 1560.980497 | ||
0.06 | 0.2 | 0.4 | 0.7 | ||
130.120866 | 452.362476 | 1008.667823 | 1789.798432 | ||
235.877972 | 477.008461 | 952.080831 | 1582.425719 | ||
235.735339 | 475.919341 | 947.608469 | 1570.630622 | ||
0.06 | 0.2 | 0.4 | 0.7 | ||
238.735339 | 494.333492 | 1028.175362 | 1800.864299 | ||
667.650974 | 772.987265 | 1116.758071 | 1676.553113 | ||
667.247203 | 771.222161 | 1111.511651 | 1664.055472 | ||
0.06 | 0.2 | 0.4 | 0.7 | ||
673.799540 | 801.064028 | 1206.019299 | 1907.996384 |
BC | ||||
RB-AM | 0.939662 | 110.809968 | 431.603421 | 931.7944980 |
RB-FEM | 0.939663 | 110.742948 | 430.617197 | 927.411308 |
PE | 0.001 | 0.6 | 0.2 | 0.4 |
EBB-FEM | 0.939852 | 111.8306027 | 447.292920 | 100.640580 |
BC | ||||
RB-AM | 0.297149 | 110.802923 | 431.601704 | 931.794248 |
RB-FEM | 0.297230 | 110.735907 | 430.615483 | 927.410579 |
PE | 0.02 | 0.06 | 0.2 | 0.4 |
EBB-FEM | 0.297394 | 111.823493 | 447.2291140 | 100.640501 |
BC | ||||
RB-AM | 0.0939667 | 110.802218 | 431.601533 | 931.794167 |
RB-FEM | 0.093957 | 110.735202 | 430.615312 | 927.410506 |
PE | 0.01 | 0.06 | 0.2 | 0.4 |
EBB-FEM | 0.098001 | 111.822788 | 447.290962 | 100.640493 |
BC | ||||
RB-AM | 0.0297149 | 110.802148 | 431.601515 | 931.794167 |
RB-FEM | 0.029710 | 110.735133 | 430.615295 | 927.410498 |
PE | 0.01 | 0.06 | 0.2 | 0.4 |
EBB-FEM | 0.033645 | 111.822711 | 447.290944 | 100.640493 |
BC | ||||
RB-AM | 0 | 110.802140 | 431.601513 | 931.794166 |
RB-FEM | 0.001565 | 110.735124 | 430.615293 | 927.410498 |
PE | 0 | 0.06 | 0.2 | 0.4 |
EBB-FEM | 0.033058 | 111.8227109 | 447.290943 | 100.640493 |
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Kanwal, G.; Nawaz, R.; Ahmed, N. Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation. Buildings 2023, 13, 1457. https://doi.org/10.3390/buildings13061457
Kanwal G, Nawaz R, Ahmed N. Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation. Buildings. 2023; 13(6):1457. https://doi.org/10.3390/buildings13061457
Chicago/Turabian StyleKanwal, Gulnaz, Rab Nawaz, and Naveed Ahmed. 2023. "Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation" Buildings 13, no. 6: 1457. https://doi.org/10.3390/buildings13061457