Theories and Analysis of Functionally Graded Beams
Abstract
:1. Introduction
1.1. Preliminary Comments
1.2. Functionally Graded Structures
1.2.1. Background
1.2.2. FGM Material Models
1.3. Modified Couple Stress Effects
1.3.1. Background
1.3.2. The Strain Energy Functional
2. Classical Theory of Beams (CBT)
2.1. Kinematics
2.2. Equations of Equilibrium
2.3. Governing Equations in Terms of Displacements
2.4. Exact Solutions
2.4.1. General Solution
2.4.2. Pinned-Hinged Beams
2.4.3. Pinned-Pinned Beams
2.4.4. Numerical Results
2.4.5. Clamped Beams
3. First-Order Theory of Beams (TBT)
3.1. Preliminary Comments
3.2. Displacements and Strains
3.3. Equations of Equilibrium
3.4. Governing Equations in Terms of Displacements
Beam Constitutive Equations
3.5. Exact Solutions
3.5.1. General Solution
3.5.2. Pinned-Pinned Beams
4. The Third-Order Beam Theory
4.1. Preliminary Comments
4.2. Kinematics
4.3. Equations of Equilibrium
4.4. Beam Constitutive Relations
4.5. Beam Stiffness Coefficients for FGM Beams
4.6. Equilibrium Equations in Terms of the Displacements
4.7. Exact Solutions for Bending
5. Summary
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
References
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n | -CBT | -TBT | w-CBT | w-TBT | ||
---|---|---|---|---|---|---|
0.0 | 0.00000 | 0.00000 | 0.5208 | 0.5210 | 0.01657 | 0.01657 |
1.0 | 0.09973 | 0.09973 | 1.0014 | 1.0016 | 0.03062 | 0.03062 |
2.0 | 0.20118 | 0.20118 | 1.2635 | 1.2638 | 0.03722 | 0.03722 |
3.0 | 0.26301 | 0.26301 | 1.4261 | 1.4265 | 0.04148 | 0.04148 |
4.0 | 0.29297 | 0.29297 | 1.5440 | 1.5445 | 0.04495 | 0.04495 |
5.0 | 0.30523 | 0.30523 | 1.6415 | 1.6420 | 0.04806 | 0.04806 |
6.0 | 0.30850 | 0.30850 | 1.7286 | 1.7292 | 0.05097 | 0.05097 |
10.0 | 0.29367 | 0.29367 | 2.0305 | 2.0312 | 0.06131 | 0.06131 |
20.0 | 0.24302 | 0.24302 | 2.6047 | 2.6056 | 0.08080 | 0.08080 |
n | w-CBT | w-TBT | -CBT | -TBT | -CBT | -TBT |
0.0 | 0.06510 | 0.06517 | 0.04167 | 0.04193 | 0.05208 | 0.05338 |
1.0 | 0.12517 | 0.12529 | 0.08011 | 0.08058 | 0.10014 | 0.10250 |
2.0 | 0.15793 | 0.15809 | 0.10108 | 0.10173 | 0.12635 | 0.12960 |
3.0 | 0.17827 | 0.17847 | 0.11409 | 0.11409 | 0.14261 | 0.14661 |
4.0 | 0.19300 | 0.19324 | 0.12352 | 0.12445 | 0.15440 | 0.15905 |
5.0 | 0.20518 | 0.20544 | 0.13132 | 0.13236 | 0.16415 | 0.16935 |
6.0 | 0.21608 | 0.21636 | 0.13829 | 0.13943 | 0.17286 | 0.17855 |
10.0 | 0.25382 | 0.25417 | 0.16244 | 0.16387 | 0.20305 | 0.21020 |
20.0 | 0.32559 | 0.32605 | 0.20838 | 0.21020 | 0.26047 | 0.26957 |
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Reddy, J.N.; Ruocco, E.; Loya, J.A.; Neves, A.M.A. Theories and Analysis of Functionally Graded Beams. Appl. Sci. 2021, 11, 7159. https://doi.org/10.3390/app11157159
Reddy JN, Ruocco E, Loya JA, Neves AMA. Theories and Analysis of Functionally Graded Beams. Applied Sciences. 2021; 11(15):7159. https://doi.org/10.3390/app11157159
Chicago/Turabian StyleReddy, J. N., Eugenio Ruocco, Jose A. Loya, and Ana M. A. Neves. 2021. "Theories and Analysis of Functionally Graded Beams" Applied Sciences 11, no. 15: 7159. https://doi.org/10.3390/app11157159