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Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams

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Abstract

A simple modified exponential shear deformation theory (ESDT) is developed and applied for the bending, buckling, and free vibration analyses of functionally graded beams with different boundary conditions. The properties of functionally graded material are assumed to vary through the thickness direction according to power law (P-FGM) and exponential law (E-FGM). The present theory is different from existing theories, because in the present theory, the transverse displacement consists of bending and shear components to understand the contribution of transverse displacement due to bending and due to shear separately. The developed theory accounts for higher order variation of transverse shear stress through the thickness of the beam, and satisfies the traction-free conditions on the top and bottom surfaces of the beam. The theory appropriately represents the strain energy of shear deformation without using shear coefficient. Equations of motion and associated boundary conditions are derived from Hamilton’s principle. Closed-form solutions for various boundary conditions are obtained, and the numerical results are compared with those available in the literature. The present study contributes some new results on the P-FGM and E-FGM beams for the reference of future research in this area.

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References

  • Alshorbagy, A. E., Eltaher, M. A., & Mahmoud, F. F. (2011). Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35, 412–425.

    Article  MathSciNet  MATH  Google Scholar 

  • Atmane, H. A., Tounsi, A., Meftah, S. A., & Belhadj, H. A. (2010). Free vibration behaviour of exponential functionally graded beams with varying cross-section. Journal of Vibration and Control, 17, 311–318.

    Article  MATH  Google Scholar 

  • Aydogdu, M., & Taskin, V. (2007). Free vibration analysis of functionally graded beams with simply supported edges. Materials and Design, 28, 1651–1656.

    Article  Google Scholar 

  • Bernoulli, J. (1694). Curvatura laminae elasticae. Acta Eruditorum Lipsiae, 3(6), 262–276.

    Google Scholar 

  • Bourada, M., Kaci, A., Houari, M. S. A., & Tounsi, A. (2015). A new simple shear and normal deformations theory for functionally graded beams. Steel and Composite Structures, 18(2), 409–423.

    Article  Google Scholar 

  • Chu, P., Li, X. F., Wu, J. X., & Lee, K. Y. (2015). Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending. Acta Mechanica, 226, 2235–2253.

    Article  MathSciNet  MATH  Google Scholar 

  • Daouadji, T. H., Henni, A. H., Tounsi, A., & Bedia, E. A. A. (2013). Elasticity solution of a cantilever functionally graded beam. Applied Composite Materials, 20, 1–15.

    Article  Google Scholar 

  • Ding, J. H., Huang, D. J., & Chen, W. Q. (2007). Elasticity solutions for plane anisotropic functionally graded beams. International Journal of Solids and Structures, 44(1), 176–196.

    Article  MathSciNet  MATH  Google Scholar 

  • Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (pp. 1–322). Lausanne and Geneva: Apud Marcum-Michaelem Bousquet & Socio.

    MATH  Google Scholar 

  • Filippi, M., Carrera, E., & Zenkour, A. M. (2015). Static analyses of FGM beams by various theories and finite elements. Composites Part B Engineering, 72, 1–9.

    Article  Google Scholar 

  • Frikha, A., Hajlaoui, A., Wali, M., & Dammak, F. (2016). A new higher order C 0 mixed beam element for FGM beams analysis. Composites Part B Engineering, 106, 181–189.

    Article  Google Scholar 

  • Ghumare, S. M., & Sayyad, A. S. (2017). A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams. Latin American Journal of Solids and Structures, 14, 1–19.

    Article  Google Scholar 

  • Giunta, G., Belouettar, S., & Carrera, E. (2010a). Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures, 17(8), 622–635.

    Article  Google Scholar 

  • Giunta, G., Belouettar, S., & Carrera, E. (2010b). Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Materials Science and Engineering, 10(2010), 1–10.

    Google Scholar 

  • Giunta, G., Crisafulli, D., Belouettar, S., & Carrera, E. (2011). Hierarchical theories for the free vibration analysis of functionally graded beams. Composite Structures, 94(1), 68–74.

    Article  Google Scholar 

  • Hadji, L., Daouadji, T. H., Meziane, M. A. A., Tlidji, Y., & Bedia, E. A. A. (2016a). Analysis of functionally graded beam using a new first-order shear deformation theory. Structural Engineering and Mechanics, 57, 315–325.

    Article  Google Scholar 

  • Hadji, L., Khelifa, Z., & Bedia, E. A. A. (2016b). A new higher order shear deformation model for functionally graded beams. KSCE Journal of Civil Engineering, 20(5), 1835–1841.

    Article  Google Scholar 

  • Huang, D. J., Ding, J. H., & Chen, W. Q. (2009). Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading. Science in China Series G, 52(8), 1244–1256.

    Article  Google Scholar 

  • Kahya, V., & Turan, M. (2017). Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Composites Part B Engineering, 109, 108–115.

    Article  Google Scholar 

  • Koizumi, M. (1993). The concept of FGM. Ceramic Transaction Functionally Graded Materials, 34, 3–10.

    Google Scholar 

  • Koizumi, M. (1997). FGM activities in Japan. Composites Part B Engineering, 28, 1–4.

    Article  Google Scholar 

  • Koochaki, G. R. (2011). Free vibration analysis of functionally graded beams. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 74, 514–517.

    Google Scholar 

  • Li, S. R., & Batra, R. C. (2013). Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Composite Structures, 95, 5–9.

    Article  Google Scholar 

  • Li, X. F., Wang, B. L., & Han, J. C. (2010). A higher-order theory for static and dynamic analyses of functionally graded beams. Archives of Applied Mechanics, 80, 1197–1212.

    Article  MATH  Google Scholar 

  • Muller, E., Drasar, C., Schilz, J., & Kaysser, W. A. (2003). Functionally graded materials for sensor and energy applications. Materials Science and Engineering A, 362, 17–39.

    Article  Google Scholar 

  • Nguyen, T. K., Nguyen, T. P., Vo, T. P., & Thai, H. T. (2015). Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Composites Part B Engineering, 76, 273–285.

    Article  Google Scholar 

  • Nguyen, T. K., Vo, T. P., & Thai, H. T. (2013). Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites Part B Engineering, 55, 147–157.

    Article  Google Scholar 

  • Pendhari, S. S., Kant, T., Desai, Y. M., & Subbaiah, C. V. (2010). On deformation of functionally graded narrow beams under transverse loads. International Journal of Mechanics and Materials in Design, 6, 269–282.

    Article  Google Scholar 

  • Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., et al. (2003). Functionally graded materials for biomedical applications. Materials Science and Engineering A, 362, 40–60.

    Article  Google Scholar 

  • Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASME Journal of Applied Mechanics, 51, 745–752.

    Article  MATH  Google Scholar 

  • Sankar, B. V. (2001). An elasticity solution for functionally graded beams. Composites Science and Technology, 61(5), 689–696.

    Article  Google Scholar 

  • Sayyad, A. S., & Ghugal, Y. M. (2015). On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Composite Structures, 129, 177–201.

    Article  Google Scholar 

  • Sayyad, A. S., & Ghugal, Y. M. (2017a). Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Composite Structures, 171, 486–504.

    Article  Google Scholar 

  • Sayyad, A. S., & Ghugal, Y. M. (2017b). A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. International Journal of Applied Mechanics, 9(1), 1–36.

    Article  Google Scholar 

  • Schulz, U., Peters, M., Bach, F. W., & Tegeder, G. (2003). Graded coatings for thermal, wear and corrosion barriers. Materials Science and Engineering A, 362, 61–80.

    Article  Google Scholar 

  • Simsek, M. (2010). Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design, 240, 697–705.

    Article  Google Scholar 

  • Sina, S. A., Navazi, H. M., & Haddadpour, H. (2009). An analytical method for free vibration analysis of functionally graded beams. Materials and Design, 30, 741–747.

    Article  Google Scholar 

  • Thai, H. T., & Vo, T. P. (2012). Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. International Journal of Mechanical Sciences, 62, 57–66.

    Article  Google Scholar 

  • Timoshenko, S. P. (1921). On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41, 742–746.

    Google Scholar 

  • Vo, T. P., Thai, H. T., Nguyen, T. K., & Inam, F. (2014a). Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica, 49, 155–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Vo, T. P., Thai, H. T., Nguyen, T. K., Inam, F., & Lee, J. (2015). A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Composite Structures, 119, 1–12.

    Article  Google Scholar 

  • Vo, T. P., Thai, H. T., Nguyen, T. K., Maheri, A., & Lee, J. (2014b). Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Engineering Structures, 64, 12–22.

    Article  Google Scholar 

  • Xu, Y., Yu, T., & Zhou, D. (2014). Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness. Meccanica, 49, 2479–2489.

    Article  MathSciNet  MATH  Google Scholar 

  • Ying, J., Lu, C. F., & Chen, W. Q. (2008). Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 84, 209–219.

    Article  Google Scholar 

  • Zhong, Z., & Yu, T. (2007). Analytical solution of a cantilever functionally graded beam. Composites Science and Technology, 67(3–4), 481–488.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon, Maharashtra, 423601, India

    Atteshamuddin S. Sayyad

  2. Department of Applied Mechanics, Government College of Engineering, Karad, Maharashtra, 415124, India

    Yuwaraj M. Ghugal

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  1. Atteshamuddin S. Sayyad
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  2. Yuwaraj M. Ghugal
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Correspondence to Atteshamuddin S. Sayyad.

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Sayyad, A.S., Ghugal, Y.M. Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams. Asian J Civ Eng 19, 607–623 (2018). https://doi.org/10.1007/s42107-018-0046-z

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  • DOI: https://doi.org/10.1007/s42107-018-0046-z

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