Bending Analysis of Two Directional Functionally Graded Beams Using A Four-Unknown Shear and Normal Deformation Theory

Year 2018, Volume: 21 Issue: 4, 861 - 874, 01.12.2018

Armağan Karamanlı

https://doi.org/10.2339/politeknik.389616

Cited By: 1

Abstract

The bending behaviour of
two-directional functionally graded beams (FGBs) subjected to various sets of
boundary conditions is investigated by using a shear and normal deformation
theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. A
simply supported conventional FGB problem is studied to validate the developed
code. The comparison studies are performed along with the analytical solutions
and the results from previous studies. The numerical calculations in terms of
maximum dimensionless transverse deflections, dimensionless axial and
transverse shear stresses are performed for various gradation exponents, aspect
ratios (L/h) and sets of boundary conditions. The effects of the gradation
exponents on the accuracy and the robustness of the SSPH method are also
investigated for the two directional functionally graded beams which are having
clamped-free boundary condition.. 

Keywords

Meshless method, Functionally graded beam, SSPH method, Shear and normal deformation theory

References

• [1] Kadoli R., Akhtar K., Ganesan N., “Static analysis of functionally graded beams using higher order shear deformation theory”, Appl Math Model, 32:2509-2525, (2008).
• [2] Li X.F., “A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams”, Journal of Sound and Vibration, 318:1210-1229, (2008).
• [3] Menaa R., Tounsi A., Mouaici F., Mechab I., Zidi M., Bedia E.A.A., “Analytical solutions for static shear correction factor of functionally graded rectangular beams”, Mechanics of Advanced Materials and Structures, 19:641-652, (2012).
• [4] Li S.R., Cao D.F., Wan Z.Q., “Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams”, Appl Math Model, 37:7077-7085, (2013).
• [5] Jing L.L., Ming P.J., Zhang W.P., Fu L.R., Cao Y.P., “Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method”, Compos Struct, 138:192-213, (2016).
• [6] Aydogdu M., Taskin V., “Free vibration analysis of functionally graded beams with simply supported edges”, Materials&Design, 28:1651–1656, (2007).
• [7] Simsek M., “Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories”, Nuc Eng and Des, 240:697–705, (2010).
• [8] Simsek M., “Vibration analysis of a functionally graded beam under a moving mass by using different beam theories”, Compos Struct, 92:904–17, (2010). [9] Pradhan K.K., Chakraverty S., “Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method”, Compos Part B, 51:175–184, (2013).
• [10] Su H., Banerjee J.R., Cheung C.W., “Dynamic stiffness formulation and free vibration analysis of functionally graded beams”, Compos Struct, 106:854–862, (2013).
• [11] Li S.R., Wan Z.G., Zhang J.H., “Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories”, Applied Mathematics and Mechanics, 35:591–606, (2014).
• [12] Aydogdu M., “Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions”, Composite Science and Technology, 67:1096–1104, (2007).
• [13] Akgöz B., Civalek Ö., “Buckling analysis of functionally graded microbeams based on the strain gradient theory”, Acta Mechanica, 224:2185–2201, (2013).
• [14] Aydogdu M., “Semi-inverse method for vibration and buckling of axially functionally graded beams”, Journal of Reinforced Plastics&Composites, 27:683–91, (2008).
• [15] Huang Y., Li X.F., “Buckling analysis of nonuniform and axially graded columns with varying flexural rigidity”, Journal of Engineering Mechanics, 137(1):73–81, (2011).
• [16] Li X.F., Wang B.L., Han J.C., “A higher-order theor y for static and dynamic analyses of functionally graded beams”, Archieve of Applied Mechanics, 80:1197-1212, (2010).
• [17] Vo T.P., Thai H.T., Nguyen T.K., Inam F., Lee J., “Static behaviour of functionally graded sandwich beams using a quasi-3D theory”, Compos Part B, 68:59-74, (2015).
• [18] Filippi M., Carrera E., Zenkour A.M., “Static analyses of FGM beams by various theories and finite elements”, Compos Part B, 72:1-9, (2015).
• [19] Mashat D.S., Carrera E., Zenkour A.M., Khateeb S.A.A., Filippi M., “Free vibration of FGM layered beams by various theories and finite elements”, Compos Part B, 59:269–278, (2014).
• [20] Vo T.P., Thai H.T., Nguyen T.K., Inam F., Lee J., “A quasi-3D theory for vibration and buckling of functionally graded sandwich beams”, Compos Struct, 119:1–12, (2015).
• [21] Mantari J.L., Yarasca J., “A simple and accurate generalized shear deformation theory for beams”, Compos Struct, 134:593–601, (2015).
• [22] Mantari J.L., “A refined theory with stretching effect for the dynamics analysis of advanced composites on elastic foundation”, Mech Mater, 86:31–43, (2015).
• [23] Mantari J.L., “Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells”, Compos Part B, 83:142–152, (2015).
• [24] Nguyen T.K., Vo T.P., Nguyen B.D., Lee J., “An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory”, Compos Struct, 156:238-252, (2016).
• [25] Nemat-Alla M., “Reduction of thermal stresses by developing two-dimensional functionally graded materials”, Int Journal of Solids and Structures, 40:7339–56, (2003).
• [26] Goupee A.J., Vel S.S., “Optimization of natural frequencies of bidirectional functionally graded beams”, Struct Multidisc Optim, 32:473–84, (2006).
• [27] Lü C.F., Chen W.Q., Xu R.Q., Lim C.W., “Semi-analytical elasticity solutions for bidirectional functionally graded beams”, Int Journal of Solids and Structures, 45:258–275, (2008).
• [28] Zhao L., Chen W.Q., Lü C.F., “Symplectic elasticity for two-directional functionally graded materials” Mech Mater, 54:32–42, (2012).
• [29] Simsek M., “Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions”, Compos Struct, 149:304–314, (2016).
• [30] Karamanli A., “Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method”, Compos Struct, 160:653-669, (2017).
• [31] Karamanli A., “Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory”, Compos Struct, 174:70-86, (2017).
• [32] Donning B.M., Liu W.K., “Meshless methods for shear-deformable beams and plates”, Comp Meth in Appl Mech and Eng, 152:47-71, (1998).
• [33] Gu Y.T., Liu G.R., “A local point interpolation method for static and dynamic analysis of thin beams”, Comp Meth in Appl Mech and Eng, 190(42):5515-5528, (2001).
• [34] Ferreira A.J.M., Roque C.M.C., Martins P.A.L.S., “Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates”, Compos Struct, 66:287-293, (2004).
• [35] Ferreira A.J.M., Fasshauer G.E., “Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method”, Comp Meth in Appl Mech and Eng, 196:134-146, (2006).
• [36] Moosavi M.R., Delfanian F., Khelil A., “The orthogonal meshless finite volume method for solving Euler–Bernoulli beam and thin plate problems”, Thin Walled Structures, 49:923-932, (2011).
• [37] Wu C.P., Yang S.W., Wang Y.M., Hu H.T., “A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation”, Mechanics Research Communications, 38:471-476, (2011).
• [38] Roque C.M.C., Figaldo D.S., Ferreira A.J.M., Reddy J.N., “A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method”, Compos Struct, 96:532-537, (2013).
• [39] Qian L.F., Batra R.C., “Design of bidirectional functionally graded plate for optimal natural frequencies”, Journal of Sound and Vibration, 280:415-424, (2005).
• [40] Pilafkan R., Folkow P.D., Darvizeh M., Darvizeh A., “Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method (RPIM)”, European Journal of Mechanics A/Solids, 39:26-34, (2013).
• [41] Yang Y., Kou K.P., Lu V.P., Lam C.C., Zhang C.H., “Free vibration analysis of two-dimensional functionally graded structures by a meshfree boundary–domain integral equation method”, Compos Struct, 110:342-353, (2014).
• [42] Zhang G.M., Batra R.C., “Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems”, Comp Mech, 43:321-340, (2009).
• [43] Batra R.C., Zhang G.M., “SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations”, Comp Mech, 41:527-545, (2008).
• [44] Tsai C.L., Guan Y.L., Batra R.C., Ohanehi D.C., Dillard J.G., Nicoli E., Dillard D.A,. “Comparison of the performance of SSPH and MLS basis functions for two-dimensional linear elastostatics problems including quasistatic crack propagation”, Comp Mech, 51:19-34, (2013).
• [45] Tsai C.L., Guan Y.L., Ohanehi D.C., Dillard J.G., Dillard D.A., Batra R.C., “Analysis of cohesive failure in adhesively bonded joints with the SSPH meshless method”, Int Journal of Adhesion & Adhesives, 51:67-80, (2014).
• [46] Wong S.M., Hon Y.C., Golberg M.A., “Compactly supported radial basis functions for shallow water equations”, Applied Mathematics and Computation, 127:79-101, (2002).