On Static Bending, Elastic Buckling and Free Vibration Analysis of Symmetric Functionally Graded Sandwich Beams
Subject Areas : Engineering
1 - Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, India
2 - Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, India
Keywords: FG sandwich beam, Free vibration, Elastic buckling, Hyperbolic shear deformation theory, Static bending,
Abstract :
This article presents Navier type closed-form solutions for static bending, elastic buckling and free vibration analysis of symmetric functionally graded (FG) sandwich beams using a hyperbolic shear deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for a hyperbolic distribution of axial displacement whereas transverse displacement is constant through the thickness i.e effects of thickness stretching are neglected. The present theory gives hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam. The equations of the motion are obtained by using the Hamilton’s principle. Closed-form solutions for static, buckling and vibration analysis of simply supported FG sandwich beams are obtained using Navier’s solution technique. The non-dimensional numerical results are obtained for various power law index and skin-core-skin thickness ratios. The present results are compared with previously published results and found in excellent agreement.
[1] Koizumi M., 1993, The concept of FGM, Ceramic Transaction: Functionally Graded Material 34: 3-10.
[2] Koizumi M., 1997, FGM activities in Japan, Composites Part B 28: 1-4.
[3] 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.
[4] Pompe W., Worch H., Epple M., Friess W., Gelinsky M., Greil P., Hempele U., Scharnweber D., Schulte K., 2003, Functionally graded materials for biomedical applications, Material Science and Engineering: A 362: 40-60.
[5] Schulz U., Peters M., Bach F. W., Tegeder G., 2003, Graded coatings for thermal, wear and corrosion barriers, Material Science and Engineering: A 362: 61-80.
[6] Sankar B. V., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61(5): 689-696.
[7] Zhong Z., Yu T., 2007, Analytical solution of a cantilever functionally graded beam, Composites Science and Technology 67: 481-488.
[8] Daouadji T. H., Henni A. H., Tounsi A., Bedia E. A. A., 2013, Elasticity solution of a cantilever functionally graded beam, Applied Composite Material 20: 1-15.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] Xu Y., Yu T., Zhou D., 2014, Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness, Meccanica 49: 2479-2489.
[14] Bernoulli J., 1694, Curvatura Laminae Elasticae, Acta Eruditorum Lipsiae.
[15] Timoshenko S. P., 1921, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine 41(6): 742-746.
[16] Reddy J. N., 1984, A simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics 51: 745-752.
[17] 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.
[18] Sayyad A. S., Ghugal Y. M., 2017, Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures 171: 486-504.
[19] Sayyad A. S., Ghugal Y. M., 2018, Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures 0(0): 1-20.
[20] Nguyen T. K., Vo T. P., Nguyen B. D., Lee J., 2016, An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory, Composite Structures 156: 238-252.
[21] 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 76: 273-285.
[22] Nguyen T. K., Nguyen B. D., 2015, A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams, Journal of Sandwich Structures and Materials 17: 1-19.
[23] 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(1): 57-66.
[24] Osofero A. I., Vo T. P., Thai H. T., 2014, Bending behaviour of functionally graded sandwich beams using a quasi-3D hyperbolic shear deformation theory, Journal of Engineering Research 19(1): 1-16.
[25] Osofero A. I., Vo T. P., Nguyen T. K., Lee J., 2016, Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi-3D theories, Journal of Sandwich Structures and Materials 18(1): 3-29.
[26] Bennai R., Atmane H. A., Tounsi A., 2015, A new higher-order shear and normal deformation theory for functionally graded sandwich beams, Steel and Composite Structures 19(3): 521-546.
[27] Bouakkaz K., Hadji L., Zouatnia N., Bedia E. A., 2016, An analytical method for free vibration analysis of functionally graded sandwich beams, Wind and Structures 23(1): 59-73.
[28] Giunta G., Crisafulli D., Belouettar S., Carrera E., 2011, Hierarchical theories for the free vibration analysis of functionally graded beams, Composite Structures 94: 68-74.
[29] Giunta G., Crisafulli D., Belouettar S., Carrera E., 2013, A thermomechanical analysis of functionally graded beams via hierarchical modelling, Composite Structures 95: 676-690.
[30] Vo T. P., Thai H. T., Nguyen T. K., Inam F., Lee J., 2015, Static behaviour of functionally graded sandwich beams using a quasi-3D theory, Composites Part B 68: 59-74.
[31] 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.
[32] Vo T. P., Thai H. T., Nguyen T. K., Maheri A., Lee J., 2014, Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory, Engineering Structures 64: 12- 22.
[33] Yarasca J., Mantari J. L., Arciniega R. A., 2016, Hermite–Lagrangian finite element formulation to study functionally graded sandwich beams, Composite Structures 140: 567-581.
[34] Amirani M. C., Khalili S. M. R., Nemati N., 2009, Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Composite Structures 90: 373-379.
[35] Tossapanon P., Wattanasakulpong N., 2016, Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation, Composite Structures 142: 215-225.
[36] Karamanli A., 2017, Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3D shear deformation theory, Composite Structures 174: 70-86.
[37] Mashat D. S., Carrera E., Zenkour A. M., Al Khateeb S. A., Filippi M., 2014, Free vibration of FGM layered beams by various theories and finite elements, Composites Part B 59: 269-278.
[38] Trinh L. C., Vo T. P., Osofero A. I., Lee J., 2016, Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach, Composite Structures 156: 263-275.
[39] Wattanasakulpong N., Prusty B. G., Kelly D. W., Hoffman M., 2012, Free vibration analysis of layered functionally graded beams with experimental validation, Materials and Design 36: 182-190.
[40] Yang Y., Lam C. C., Kou K. P., Iu V. P., 2014, Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method, Composite Structures 117: 32-39.
[41] Sayyad A. S., Ghugal Y. M., 2017, A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates, International Journal of Applied Mechanics 9: 1-36.
[42] Sayyad A. S., Ghugal Y. M., 2018, Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams, Asian Journal of Civil Engineering 19(5): 607-623.
[43] Alipour M. M., Shariyat M., 2013, Analytical zigzag-elasticity transient and forced dynamic stress and displacement response prediction of the annular FGM sandwich plates, Composite Structures 106: 426-445.
[44] Alipour M. M., Shariyat M., 2014, An analytical global–local Taylor transformation-based vibration solution for annular FGM sandwich plates supported by nonuniform elastic foundations, Archives of Civil and Mechanical Engineering 14(1): 6-24.
[45] Alipour M. M., Shariyat M., 2014, Analytical stress analysis of annular FGM sandwich plates with non-uniform shear and normal tractions, employing a zigzag-elasticity plate theory, Aerospace Science and Technology 32(1): 235-259.
[46] Shariyat M., Hosseini S. H., 2015, Accurate eccentric impact analysis of the preloaded SMA composite plates, based on a novel mixed-order hyperbolic global–local theory, Composite Structures 124: 140-151.
[47] Shariyat M., Mozaffari A., Pachenari M. H., 2017, Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory, Applied Mathematical Modelling 43: 421-440.
[48] Soldatos K. P., 1992, A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica 94: 195-200.
[49] Wakashima K., Hirano T., Niino M., 1990, Space applications of advanced structural materials, Proceedings of an International Symposium (ESA SP).