In this contribution, results of elastostatic analysis of spatial composite beam structures are presented using our new beam finite element of double symmetric cross-section made of a Functionally Graded Material (FGM). Material properties of the real beams vary continuously in the longitudinal direction while variation with respect to the transversal and lateral directions is assumed to be symmetric in a continuous or discontinuous manner. Continuously longitudinal varying spatial Winkler elastic foundations (except the torsional foundation) and the effect of axial and shear forces are considered as well. Homogenization of spatially varying material properties to effective quantities with a longitudinal variation is done by the multilayer method (MLM). For the homogenized beam finite element the local stiffness matrix is established by means of the transfer matrix method. By the conventional finite element procedure, the global element stiffness matrix and the global system of equation for the beam structure are established for calculation of the global displacement vector. The secondary variables (internal forces and moments) are then calculated by means of the transfer relations on the real beams. Further, the mechanical stress in the real beams are calculated. Finally, the numerical experiments are carried out concerning the elastic-static analysis of the single FGM beams and beam structures in order to show the possibilities of our approach.

[1] Birman, V., Byrd, L.W. Modeling and analysis of functionally graded materials and structures. Applied mechanics reviews (2007) 60: 195-216.

[2] Ying, J., Lu, C.F., Chen, W.Q. Two -dimensional elasticity solutions for functionally graded beams resting on elastic foundation. Composite Structures (2008) 84: 209-219.

[3] Benatta, M.A, Mechab, I., Tounsi, A., Adda Bedia, E.A. Static analysis of functionally graded short beams including warping and shear deformation effects. Computational Materials Science (2008)44: 765-773.

[4] Kadoli, R., Akhtar, K., Ganesan, N. Static analysis of functionally graded beams using high order shear deformation theory. Applied Mathematical Modelling (2008) 32: 2509-2525

[5] Giunta, G., Belouettar, S., Carrera, E. Analysis of FGM Beams by Means of Classical and Advanced Theories. Mechanics of Advanced Materials and Structures (2010) 17: 622-635.

[6] Kang. Y.A., Li, X.F. Large Deflections of a Non-linear Cantilever Functionally Graded Beam. Journal of Reinforced Plastics and Composites (2010) 29: 1761-1774.

[7] Huang, Y., Li, X.F. Buckling Analysis of Nonuniform and Axially Graded Columns with Varying Flexural Rigidity. Journal of Engineering Mechanics - ASCE (2011) 137: 73-81.

[8] Asghari, M., Rahaeifard, M., Kahrobaiyan M.H., Ahmadian, M.T. The modified couple stress functionally graded Timoshenko beam formulation. Material and Design (2011) 32: 1435-1443

[9] Kocaturk, T., Simsek, M., Akbas, S.D. Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material. Science and Engineering of Composite Materials (2011) 18: 21-34.

[10] Mohanty, C.S., Dash, R.R., Rout, T. Parametric instability of a functionally graded Timoshenko beam on Winkler’s elastic foundation. Nuclear Engineering and Design (2011) 241: 2698-2715.

[11] Ma. L.S., Lee, D.W. Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading. European Journal of Mechanics A/Solids (2012) 31: 13-20.

[12] 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 (2012) 19: 641-652.

[13] Zhou, Li, S.R., Wan, Z.Q., Zhang, P. Relationship between Bending Solutions of FGM Timoshenko Beams and Those of Homogenous Euler-Bernoulli Beams. Applied Mechanics and Materials: Progress in Structures, PTS 1-4 (2012) 166-169: 2831-2836.

[14] Soleimani, A., Saadatfar, M. Numerical Study of Large Deflection of Functionally Graded Beam with geometry Nonlinearity. Advanced Material Research: MEMS, Nano and Smart Systems, PTS 1-6 (2012) 403-408: 4226-4230

[15] Birsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D. deformation analysis of functionally graded beams by the direct approach. Composites: Part B (2012) 43: 1315-1328

[16] Mohanty, S.C., Dash, R.R., Rout, T. Static and Dynamic Stability of Functionally Graded Timoshenko Beam. International Journal of Structural Stability and Dynamics (2012) 12.

[17] Zhao, L., Chen, W.Q., Lu, C.F. Symplectic elasticity for bi-directional functionally graded materials. Mechanics of Materials (2012) 54: 32-42.

[18] Esfahani, S.E., Kiani, Y., Eslami, M.R. Non-linear stability thermal analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations. International Journal of Mechanical Science (2013) 69: 10-20

[19] Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V., Sahmani, S. Size-dependent bending, buckling and free vibration of functionally graded Timoshenko nicrobeams based on the most general strain gradient theory. Composite Structure (2013) 100: 385-397.

[20] Zhang, Da-Guang. Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Composite Structures (2013) 100: 121-126.

[21] Li, S.R., Cao, D.F, Wan, Z.Q. Bending solutions of FGM Timoshenko beams from those of the homogenous Euler-Bernoulli beams. Applied Mathematical Modelling (2013) 37: 7077-7085.

[22] Zamanzadeh M., Rezazadeh G., Jafarsadeghi-poornaki, I., Shabani, R. Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes. Applied Mathematical Modelling (2013) 37: 6964-6978.

[23] Mao, Y.Q., Ai, S.G., Fang, D.N., Fu, Y.M, Chen, C.P. Elasto-plastic analysis of micro FGM beam basing on mechanism-based strain gradient plasticity theory. Composite Structures (2013) 101: 168-179.

[24] Abbasnejad, B., Rezazadeh, G., Shabani, R. Stability Analysis of a Capacitive FGM Micro-Beam using Modified Couple Stress Theory. ACTA Mechanica Solida Sinica (2013) 26: 427-440.

[25] Akgoz, B., Civalek, O. Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mechanica (2013) 224: 2185-2201.

[26] Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L. A novel size-dependent functionally graded curved microbeam model based on the strain gradient elasticity theory. Composite Structures (2013) 106: 374-392.

[27] Zhang. D.G. Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Meccanica (2014) 49: 283-293.

[28] Li, Y.L., Meguid, S.A., Fu, Y.M., Xu, D.L. Nonlinear analysis of thermally and electrically actuated functionally graded material microbeam. Proceedings of the Royal Society a Mathematical Physical and Engineering sciences (2014) 470.

[29] Shen H-S., Wang, Z-X. Nonlinear analysis of shear deformable FGM beams resting on elastic foundation in thermal environments. International Journal of Mechanical Sciences (2014) 81: 195-206.

[30] Hadji, L., Daouadji, T.H., Tounsi, A., Bedia, E.A. A higher order shear deformation theory for static and free vibration of FGM beam. Steel and Composite Structures (2014) 16: 507-519.

[31] Nguyen, D.K., Gan, B.S., Trinh, T.H. Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material. Structural Engineering and Mechanics (2014) 49: 727-743.

[32] Zhang. D.G., Zhou, H.M. Nonlinear Bending and Thermal Post-Buckling Analysis of FGM Beams Resting on Nonlinear Elastic Foundation. CMES - Computer Modelling in Engineering & Science (2014) 100: 201-222.

[33] Sitar. M., Kosel, F., Brojan, M. Large deflections of nonlinearly elastic functionally graded composite beams. Archives of Civil and Mechanical Engineering (2014) 14: 700-709.

[34] Cai, K., Gao, D.Y., Qin, Q.H. Postbuckling analysis of a nonlinear beam with axial functionally graded material. Journal of Engineering Mathematics (2014) 88: 121-136.

[35] Chu, P., Li, X.-F., Wang, Z.-G., Lee, K.Y. Double cantilever beam model for functionally graded materials based on two-dimensional theory of elasticity. Engineering Fracture Mechanics (2015) 135: 232-244.

[36] Filippi, M., Carrera, E., Zenkour, A.M. Static analyses of FGM beams by various theories and finite elements. Composites: Part B (2015) 72: 1-9.

[37] Chakraborty, A., Gopalakrishnan, S., Reddy, J.N. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences (2003) 45: 519-539.

[38] Alshorbagy A.E., Eltaher, M.A., Mahmoud F.F. Free vibration characteristics of a functionally graded beam by finite element. Applied Mathematical Modelling (2011) 35: 412-425.

[39] Murin, J., Aminbaghai M., Kutis, V. Exact solution of the bending vibration problem of the FGM beam with variation of material properties. Engineering Structures (2010) 32: 1631-1640.

[40] Aminbaghai, M., Murin, J., Kutis V. Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force. Engineering Structures (2012)34: 314-329.

[41] Murin, J., Aminbaghai, M., Kutis, V., Hrabovsky, J. Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic Winkler foundation. Engineering Structures (2013) 49: 234-247.

[42] Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V., Kugler St. Modal analysis of the FGM beams with effect of the shear correction function. Composites: Part B (2013) 45:1575-1582.

[43] Kutis, V., Murin, J., Belak, R., Paulech, J. Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials. Computers and Structures (2011)89: 1192 - 1205.

[44] Rubin, H. Analytische Berechnung von Stäben und Stabwerken mit stetiger Veränderlichkeit von Querschnitt, elastischer Bettung und Massenbelegung nach Theorie erster und zweiter Ordnung, Baustatik - Baupraxis 7. Berichte der 7. Fachtagung "Baustatik - Baupraxis" Aachen/Deutschland 18.-19. März 1999. Balkema 1999, Abb., Tab.S.135-145.

[45] Rubin, H. Solution of differential equations of arbitrary order with polynomial coefficients and application to a statics problem ZAMM (1996)76: 105-117.

[46] S. Wolfram MATHEMATICA 5, Wolfram research, Inc., 2003.

[47] Altenbach, H., Altenbach, J., Kissing, W. Mechanics of composite structural elements. Springer Verlag, (2003).

[48] Halpin, J.C., Kardos, J.L. The Halpin-Tsai equations. A review, Polymer Engineering and Science. (1976) 16: 344-352.

[49] Reuter, T., Dvorak, G.J. Micromechanical models for graded composite materials: Ii.Thermomechanical loading. J. of the Mechanics and Physics of Solids (1998) 46:1655-1673.

[50] Murin, J., Kutis, V. Improved mixture rules for composite (FGMs) sandwich beam finite element.In Computational Plasticity IX. Fundamentals and Applications. Barcelona, Spain, (2007): 647-650.

[51] Alshorbagy, A.E., Eltaher, M.A., Mahmoud F.F. Free vibration of a functionally graded beam by finite element method. Applied Mathematical Modelling (2010) 35: 412 - 425.

[52] Simsek, M. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures (2010) 92: 904-917.

[53] Rout, T. On the dynamic stability of functionally graded material under parametric excitation.PhD thesis. National Institute of Technology Rourkela, India. (2012).

[54] Kutis, V., Murin, J., Belak, R., Paulech, J. Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials. Computers and Structures (2010) 89: 1192-1205.

[55] Murin, J., Kugler, S., Aminbaghai, M., Hrabovsky, J., Kutis, V., Paulech, J. Homogenization of material properties of the FGM beam and shells finite elements. In.: 11th World Congress on Computational mechanics (WCCM XI), Barcelona, 2014.

[56] Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V., Paulech, J., Kugler, S. A new FGM beam finite element for modal analysis. In.: 11th World Congress on Computational mechanics (WCCM XI), Barcelona, 2014.

[57] Murin, J., Kugler, S., Aminbaghai, M., Hrabovsky, J., Kutis, V., Paulech, J. Homogenization of material properties of the FGM beam and shell finite elements. In.: 11th World Congress on Computational mechanics (WCCM XI), Barcelona, 2014.

[58] Murín, J., Hrabovský, J., Kutiš, V., Paulech, J.Shear correction function derivation for the FGM beams. In: 2nd International Conference on Multi-scale Computational Methods for solid and Fluids. 10. 6- 12. 6.2015, Sarajevo, Bosnia and Hercegovina, (2015).

[59] Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V. Kugler, S. Effect of the Shear Correction Function in the FGM Beams Modal Analysis. In Proceedings of the 15th European Conference on Composite Materials. 24-28 June 2012, Venice, Italy, (2012) ISBN 978-88-88785-33-2.

[60] ANSYS Swanson Analysis System, Inc., 201 Johnson Road, Houston, PA 15342/1300, USA.