Static deflection of fully coupled composite Timoshenko beams: an exact analytical solution. (English) Zbl 1475.74076

Summary: The purpose of this paper is to present the exact analytical solution for the static deflection analysis of fully coupled composite Timoshenko beams. The system of governing equations and the boundary conditions are derived from variational principles. Using the method of direct integration, the exact analytical solution of the static deflection of a Timoshenko beam is obtained by solving this system of differential equations in terms of transverse displacements and cross-sectional rotations. Static deflection analyses of Timoshenko beams, subject to various boundary conditions and uniformly distributed and tip loads, are performed and the results are compared to those obtained from classical Euler-Bernoulli theory by using different values of length-to-thickness ratio. In addition, it is shown that for the case of a cantilevered composite beam subject to tip loads, the proposed exact analytical solution is equivalent to the exact solution from the intrinsic formulation. The Chebyshev collocation method is also employed to validate the obtained exact analytical solution. In the proposed formulation the stiffness properties of the composite beam are expressed by engineering constants, therefore is not limited by the cross-sectional shape of the beam, type of material and thus can be utilised for engineering applications and design purposes. The exact analytical solution can also be used as a benchmark for validating results obtained from various numerical methods.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74E30 Composite and mixture properties
74G05 Explicit solutions of equilibrium problems in solid mechanics
Full Text: DOI


[1] Adamek, V.; Valeš, F., Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load, Eur. J. Mech. A Solids, 49, 373-381 (2015) · Zbl 1406.74364
[2] Antes, H., Fundamental solution and integral equations for Timoshenko beams, Comput. Struct., 81, 6, 383-396 (2003)
[3] Berdichevskii, V., On the energy of an elastic rod, J. Appl. Math. Mech., 45, 4, 518-529 (1981) · Zbl 0519.73043
[4] Bernstein, D. S., Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas-Revised and Expanded Edition (2018), Princeton University Press
[5] Blasques, J., Optimal Design of Laminated Composite Beams (2011), DCAMM Special Repport (S134)
[6] Cesnik, C. E.S.; Hodges, D. H., VABS: a new concept for composite rotor blade cross-sectional modeling, J. Am. Helicopter Soc., 42, 1, 27-38 (1997)
[7] Doeva, O., Masjedi, P., Weaver, P.M., 2020. Exact solution for the deflection of composite beams under non-uniformly distributed loads. In: AIAA Scitech 2020 Forum. p. 0245.
[8] Ghugal, Y.; Shimpi, R., A review of refined shear deformation theories for isotropic and anisotropic laminated beams, J. Reinf. Plast. Compos., 20, 3, 255-272 (2001)
[9] Giavotto, V.; Borri, M.; Mantegazza, P.; Ghiringhelli, G.; Carmaschi, V.; Maffioli, G. C.; Mussi, F., Anisotropic beam theory and applications, Comput. Struct., 16, 1-4, 403-413 (1983) · Zbl 0499.73066
[10] Gordaninejad, F.; Bert, C. W., A new theory for bending of thick sandwich beams, Int. J. Mech. Sci., 31, 11-12, 925-934 (1989)
[11] Hill, G. F.J.; Weaver, P. M., Analysis of anisotropic prismatic sections, Aeronaut. J., 108, 1082, 197-205 (2004)
[12] Hodges, D. H., Nonlinear Composite Beam Theory (2006), American Institute of Aeronautics and Astronautics
[13] Hodges, D.; Yu, W., A rigorous, engineer-friendly approach for modelling realistic, composite rotor blades, Wind Energy Int. J. Prog. Appl. Wind Power Convers. Technol., 10, 2, 179-193 (2007)
[14] Hutchinson, J. R., On Timoshenko beams of rectangular cross-section, J. Appl. Mech., 71, 3, 359-367 (2004) · Zbl 1111.74448
[15] Jelenić, G.; Papa, E., Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order, Arch. Appl. Mech., 81, 2, 171-183 (2011) · Zbl 1271.74262
[16] Khdeir, A.; Reddy, J., An exact solution for the bending of thin and thick cross-ply laminated beams, Compos. Struct., 37, 2, 195-203 (1997)
[17] Khdeir, A.; Reddy, J., Jordan canonical form solution for thermally induced deformations of cross-ply laminated composite beams, J. Therm. Stresses, 22, 3, 331-346 (1999)
[18] Kim, J. S.; Cho, M.; Smith, E. C., An asymptotic analysis of composite beams with kinematically corrected end effects, Int. J. Solids Struct., 45, 7-8, 1954-1977 (2008) · Zbl 1152.74027
[19] Lee, S. Y.; Kuo, Y., Static analysis of nonuniform Timoshenko beams, Comput. Struct., 46, 5, 813-820 (1993) · Zbl 0773.73048
[20] Li, X.-F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams, J. Sound Vib., 318, 4-5, 1210-1229 (2008)
[21] Masjedi, P. K.; Maheri, A., Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation, Eur. J. Mech. A Solids, 66, 329-340 (2017) · Zbl 1406.74304
[22] Masjedi, P. K.; Maheri, A.; Weaver, P. M., Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation, Appl. Math. Model., 76, 938-957 (2019)
[23] Masjedi, P. K.; Ovesy, H., Chebyshev collocation method for static intrinsic equations of geometrically exact beams, Int. J. Solids Struct., 54, 183-191 (2015)
[24] Masjedi, P. K.; Ovesy, H. R., Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations, Acta Mech., 226, 6, 1689-1706 (2015) · Zbl 1325.74086
[25] Masjedi, P. K.; Weaver, P. M., Analytical solution for the fully coupled static response of variable stiffness composite beams, Appl. Math. Model., 81, 16-36 (2020)
[26] Mason, J. C.; Handscomb, D. C., Chebyshev Polynomials (2002), Chapman and Hall/CRC · Zbl 1015.33001
[27] Mechab, I.; Tounsi, A.; Benatta, M.; Adda bedia, E., Deformation of short composite beam using refined theories, J. Math. Anal. Appl., 346, 2, 468-479 (2008) · Zbl 1141.74031
[28] Morandini, M.; Chierichetti, M.; Mantegazza, P., Characteristic behavior of prismatic anisotropic beam via generalized eigenvectors, Int. J. Solids Struct., 47, 10, 1327-1337 (2010) · Zbl 1193.74079
[29] Nguyen, N.-D.; Nguyen, T.-K.; Vo, T. P.; Thai, H.-T., Ritz-based analytical solutions for bending, buckling and vibration behavior of laminated composite beams, Int. J. Struct. Stab. Dyn., 18, 11, Article 1850130 pp. (2018)
[30] Pai, P. F., High-fidelity sectional analysis of warping functions, stiffness values and wave properties of beams, Eng. Struct., 67, 77-95 (2014)
[31] Pei, Y.; Geng, P.; Li, L., A modified uncoupled lower-order theory for FG beams, Arch. Appl. Mech., 89, 4, 755-768 (2019)
[32] Phuong, N. T.B.; Tu, T. M.; Phuong, H. T.; Van Long, N., Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position, J. Sci. Technol. Civil Eng. NUCE, 13, 1, 33-45 (2019)
[33] Pydah, A.; Batra, R., Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams, Compos. Struct., 172, 45-60 (2017)
[34] Romano, F., Deflections of Timoshenko beam with varying cross-section, Int. J. Mech. Sci., 38, 8-9, 1017-1035 (1996) · Zbl 0869.73035
[35] Sayyad, A. S.; Ghugal, Y. M., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Compos. Struct., 171, 486-504 (2017)
[36] Sidhardh, S.; Ray, M., Exact solution for size-dependent elastic response in laminated beams considering generalized first strain gradient elasticity, Compos. Struct., 204, 31-42 (2018)
[37] Timoshenko, S. P., LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Lond. Edinburgh Dublin Phil. Mag. J. Sci., 41, 245, 744-746 (1921)
[38] Timoshenko, S. P., X. On the transverse vibrations of bars of uniform cross-section, Lond. Edinburgh Dublin Phil. Mag. J. Sci., 43, 253, 125-131 (1922)
[39] Timoshenko, S. P.; Gere, J. M., Theory of Elastic Stability (2009), Courier Corporation
[40] Viet, N.; Zaki, W.; Umer, R., Bending models for superelastic shape memory alloy laminated composite cantilever beams with elastic core layer, Composites B, 147, 86-103 (2018)
[41] Viet, N. V.; Zaki, W.; Umer, R., Bending theory for laminated composite cantilever beams with multiple embedded shape memory alloy layers, J. Intell. Mater. Syst. Struct., 30, 10, 1549-1568 (2019)
[42] Wang, C. M., Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions, J. Eng. Mech., 121, 6, 763-765 (1995)
[43] Wang, C.; Lam, K.; He, X., Exact solutions for Timoshenko beams on elastic foundations using Green’s functions, Mech. Struct. & Mach., 26, 1, 101-113 (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.