Deformation Characteristics of Composite Structures

Document Type : Research Paper


1 Department of Systems Engineering, University of Lagos, NIGERIA

2 Bells University of Technology


The composites provide design flexibility because many of them can be moulded into complex shapes. The carbon fibre-reinforced epoxy composites exhibit excellent fatigue tolerance and high specific strength and stiffness which have led to numerous advanced applications ranging from the military and civil aircraft structures to the consumer products. However, the modelling of the beams undergoing the arbitrarily large displacements and rotations, but small strains, is a common problem in the application of these engineering composite systems. This paper presents a nonlinear finite element model which is able to estimate the deformations of the fibre-reinforced epoxy composite beams. The governing equations are based on the Euler-Bernoulli beam theory (EBBT) with a von Kármán type of kinematic nonlinearity. The anisotropic elasticity is employed for the material model of the composite material. Moreover, the characterization of the mechanical properties of the composite material is achieved through a tensile test, while a simple laboratory experiment is used to validate the model. The results reveal that the composite fibre orientation, the type of applied load and boundary condition, affect the deformation characteristics of the composite structures. The nonlinearity is an important factor that should be taken into consideration in the analysis of the fibre-reinforced epoxy composites.


Main Subjects

[1]     Li, Zhi-Min, and Yi-Xi Zhao. "Nonlinear Bending of Shear Deformable Anisotropic Laminated Beams Resting on Two-Parameter Elastic Foundations Based on an Exact Bending Curvature Model." Journal of Engineering Mechanics 141.3, 04014125, 2014.
[2]     Grediac, M., “The use of full-field measurement methods in composite material characterization: interest and limitations”, Composites Part A: applied science and manufacturing, Vol. 5, No. 7, pp.  751-761, 2004.
[3]     Carrera, E., & Giunta, G., Refined beam theories based on a unified formulation. International Journal of Applied Mechanics, Vol. 2, No. 1, pp. 117-143, 2010.
[4]     Carrera, E., Maiarú, M., & Petrolo, M., “Component-wise analysis of laminated anisotropic composites”, International Journal of Solids and Structures, Vol. 49, No. 13, pp. 1839-1851, 2012.
[5]     Carrera, E., Giunta, G., & Petrolo, M., “Carrera Unified Formulation and Refined Beam Theories”, Beam Structures: Classical and Advanced Theories, pp. 45-63, 2011.
[6]     Pagani, A., Petrolo, M., Colonna, G., & Carrera, E., ” Dynamic response of aerospace structures by means of refined beam theories. Aerospace Science and Technology, Vol. 46, pp. 360-373, 2015.
[7]     Filippi, M., Pagani, A., Petrolo, M., Colonna, G., & Carrera, E., “Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials”, Composite Structures, Vol. 132, pp. 1248-1259, 2015.
[8]     Bauchau, O. A., & Hong, C. H., “Nonlinear composite beam theory”, Journal of Applied Mechanics, Vol. 55 No. 1, pp. 156-163, 1988.
[9]     Luo, J. H., Li, L. J., “Theory of elasticity of an anisotropic body for the bending of beams”, Applied Mathematics and Mechanics, Vol. 13, No. 11, pp. 1031-1037, 1992.
[10] Hajianmaleki, M., and Qatu, M. S., “A rigorous beam model for static and vibration analysis of generally laminated composite thick beams and shafts.” Int. J. Veh. Noise Vib., Vol. 8. No. 2, pp. 166–184, 2012.
[11] Hajianmaleki, M., and Qatu, M. S., “Vibrations of straight and curved composite beams: A review.” Compos. Struct., 100 (Jun), pp. 218–232, 2013.
[12] Hodges, D. H., Atilgan, A. R., Cesnik, C. E., & Fulton, M. V., “On a simplified strain energy function for geometrically nonlinear behaviour of anisotropic beams”, Composites Engineering, Vol. 2, No. 5, pp. 513-526, 1992.
[13] Salimi, M., Jamshidian, M., Beheshti, A., & Dolatabadi, A. S., “Bending-Unbending Analysis of Anisotropic Sheet under Plane Strain Condition”, Esteghlal Journal of Engineering, Vol. 26, No. 2, pp. 187-196, 2008.
[14] Vora, M. R., Matlock, H., “A discrete-element analysis for anisotropic skew plates and grids”, Ph.D Thesis, University of Texas at Austin, 1970.
[15] Panak, J. J., Matlock, H., “A Discrete-Element Method of Analysis for Orthogonal Slab and Grid Bridge Floor Systems, No. 56-25 Res Rpt, Center for Highway Research, University of Texas at Austin, 1972.
[16] Noor, A. K., Rarig, P. L., “Three-dimensional solutions of laminated cylinders”, Computer Methods in Applied Mechanics and Engineering, Vol. 3, No. 3, pp. 319-334, 1974.
[17] Malik, M., “Differential quadrature method in computational mechanics: new development and applications”, Ph.D. Dissertation, University of Oklahoma, Oklahoma, 1994.
[18] Malik, M., Bert, C. W., “Differential quadrature analysis of free vibration of symmetric cross-ply laminates with shear deformation and rotatory inertia”, Shock and Vibration, Vol. 2, No. 4, pp.321-338, 1995.
[19] Liew, K. M., Han, J. B., Xiao, Z. M., “Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility”, International Journal of Solids and Structures, Vol. 33, No. 18, pp. 2647-2658, 1996.
[20] Davi, G., “Stress fields in general composite laminates”, AIAA journal, Vol. 34, No. 12, pp. 2604-2608, 1996.
[21] Davı̀, G., Milazzo, A., “Bending stress fields in composite laminate beams by a boundary integral formulation”, Computers & structures, Vol. 71, No. 3, pp. 267-276, 1999.
[22] Milazzo, A., “Interlaminar stresses in laminated composite beam-type structures under shear/bending”, AIAA journal, Vol. 38, No. 4, pp. 687-694, 2000.
[23] Noor, A. K., Burton, W. S., “Stress and free vibration analyses of multi-layered composite plates”, Composite Structures, Vol. 11, No. 3, pp. 183-204, 1989.
[24] Noor, A. K., Peters, J. M., “A posteriori estimates for the shear correction factors in multi-layered composite cylinders”, Journal of engineering mechanics, Vol. 115, No. 6, pp. 1225-1244, 1989.
[25] Noor, A. K., Burton, W. S., Peters, J. M., “Predictor-corrector procedures for stress and free vibration analyses of multilayered composite plates and shells”, Computer Methods in Applied Mechanics and Engineering, Vol. 82, No. 1, pp. 341-363, 1990.
[26] Vel, S. S., Batra, R. C., “Analytical solution for rectangular thick laminated plates subjected to arbitrary boundary conditions”, AIAA journal, Vol. 37, No. 11, pp. 1464-1473. 1999.
[27] Vel, S. S., Batra, R. C., “The generalized plane strain deformations of thick anisotropic composite laminated plates”, International Journal of Solids and Structures, Vol. 37, No. 5, pp. 715-733, 2000.
[28] Machado, S. P., Filipich, C. P.  Rosales, M. B., “Plane Anisotropic Beams with Shear Deformation via a Generalized Solution”, Santa Fe-Paraná, Argentina, October 2002, S. R. Idelsohn, V. E. Sonzogni and A. Cardona (Eds.), Mecánica Computational, Vol. 20, pp. 775-785, 2000.
[29] Machado, S. P., Cortínez, V. H., “Non-linear model for stability of thin-walled composite beams with shear deformation”, Thin-Walled Structures, Vol. 43, No. 10, pp. 1615-1645. 2005.
[30] Filipich, C. P., Rosales, M. B., “Arbitrary precision frequencies of a free rectangular thin plate”, Journal of Sound and Vibration, Vol. 230, No. 3, pp. 521-539, 2000.
[31] Rosales, M. B., Filipich, C. P., “Time integration of non-linear dynamic equations by means of a direct variational method”, Journal of sound and vibration, Vol. 254, No. 4, pp. 763-775, 2002.
[32] Vnučec, Z., “Analysis of the Laminated Composite Plate under Combined Loads”, Analysis, Vol. 2, No. 2, 2005.
[33] Morandini, M., Chierichetti, M., Mantegazza, P., “Characteristic behaviour of prismatic anisotropic beam via generalized eigenvectors”, International Journal of Solids and Structures, Vol. 47, No. 10, pp. 1327-1337, 2010.
[34] Kim, T., Hansen, A. M., Branner, K., “Development of an anisotropic beam finite element for composite wind turbine blades in multibody system”, Renewable Energy, Vol. 59, pp. 172-183, 2013.
[35] Reddy, J. N., & Robbins, D. H., “Theories and computational models for composite laminates”, Applied mechanics reviews, Vol. 47, No. 6, pp. 147-169, 1994.
[36] Varadan, T. K., Bhaskar, K., “Review of different laminate theories for the analysis of composites”, Journal-Aeronautical Society of India, Vol. 49, pp. 202-208, 1997.
[37] Carrera, E., “An assessment of mixed and classical theories for the thermal stress analysis of orthotropic multilayered plates. Journal of Thermal Stresses, Vol. 23, No. 9, pp. 797-831, 2000.
[38] Battini, J. M., Nguyen, Q. H., Hjiaj, M., “Non-linear finite element analysis of composite beams with interlayer slips”, Computers and structures, Vol. 87, No. 13, pp. 904-912, 2009.
[39] Ranzi, G., Dall’Asta, A., Ragni, L., Zona, A., “A geometric nonlinear model for composite beams with partial interaction”, Engineering Structures, Vol. 32, No. 5, pp. 1384-1396, 2010.
[40] Zona, A., Ranzi, G., “Finite element models for nonlinear analysis of steel–concrete composite beams with partial interaction in combined bending and shear”, Finite Elements in Analysis and Design, Vol. 47, No. 2, pp. 98-118, 2011.
[41] Abass, M. K., Elshafei, M. A., “Linear and Nonlinear Finite Element Modelling of Advanced Isotropic and Anisotropic Beams Part I: Euler Bernoulli Theory”, 13th International conference on aerospace sciences & aviation technology, Cairo, Egypt, 2009.
[42] Vanegas, J. D., Patiño, I. D., “Linear and Non-Linear Finite Element Analysis of Shear-Corrected Composites Box Beams”, Latin American Journal of Solids and Structures, Vol. 10, No. 4, pp. 647-673, 2013.
[43] Zhang, Y., and Lin, X., “Nonlinear finite element analyses of steel/FRP-reinforced concrete beams by using a novel composite beam element”, Advances in Structural Engineering, Vol. 16, No. 2, pp. 339-352, 2013.
[44] Rahman, M., Aktaruzzaman, F. N. U., Absar, S., Mitra, A., Hossain, A., “Finite Element Analysis of Polyurethane Based Composite Shafts Under Different Boundary Conditions”, In ASME 2014 International Mechanical Engineering Congress and Exposition (pp. V010T13A014-V010T13A014). American Society of Mechanical Engineers, 2014.
[45] Mahmoud, A. M., “Finite element modeling of steel concrete beam considering double composite action”, Ain Shams Engineering Journal, 2015.
[46] Li, Z. M., Qiao, P., “Buckling and postbuckling behaviour of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression”, Engineering Structures, Vol. 85, pp. 277-292, 2015.
[47] Li, Z. M., Qiao, P.,  Thermal postbuckling analysis of anisotropic laminated beams with different boundary conditions resting on two-parameter elastic foundations”, European Journal of Mechanics-A/Solids, Vol. 54, pp. 30-43, 2015.
[48] Reddy, J. N., “An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics”, OUP Oxford, 2014.
[49] Heinbockel, J. H., Introduction to tensor calculus and continuum mechanics, Vol. 52. 2001.
[50] Gibson, Ronald F., “A simplified analysis of deflections in shear deformable composite sandwich beams”, Journal of sandwich structures and materials, Vol. 13, No. 5, pp. 579-588, 2011.
[51] Bower, A. F., Applied mechanics of solids. CRC press, 2010.
[52] Reddy, J. N., Mechanics of laminated composite plates and shells: theory and analysis. CRC press, 2004.
[53] Jones, R. M., Morgan, H. S., “Analysis of Nonlinear Stress-Strain Behaviour of Fiber-Reinforced Composite Materials”, AIAA Journal, Vol. 15, No. 12, pp. 1669-1676, 1997.
[54] Wang, X., Chung, D. D. L., “Continuous carbon fibre epoxy-matrix composite as a sensor of its own strain”, Smart materials and structures, Vol. 5, No. 6, 796, 1996.
[55] Nye, J. F., “Physical properties of crystals: their representation by tensors and matrices”, Oxford university press, 1985.