Simple Two Variable Refined Theory for Shear Deformable Isotropic Rectangular Beams

Document Type: Research Paper


Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India


In this paper, a displacement-based, variationally consistent, two variable refined theory for shear deformable beams is presented. The beam is assumed to be of linearly elastic, homogeneous, isotropic material and has a uniform rectangular cross-section. In this theory, the beam axial displacement and beam transverse displacement consist of bending components and shearing components. The assumed displacement field of this theory is such that, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. This theory utilizes linear strain-displacement relations. The displacement functions give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. Hence the shear correction factor is not required. Hamilton’s principle is utilized to derive governing differential equations and variationally consistent boundary conditions. This theory involves only two governing differential equations of fourth-order. These governing equations are only inertially coupled for the case of dynamics and are decoupled for the case of statics. This theory is simple and has a strong resemblance with the Bernoulli-Euler beam theory. To demonstrate the efficacy of the present theory, illustrative examples pertain to the static bending and free vibrations of shear deformable isotropic rectangular beams are presented.


Main Subjects

[1] Carrera, E., Giunta, G., Petrolo, M., Beam Structures: Classical and Advanced Theories, A John Wiley & Sons Ltd Publication, New Delhi, 2011, 1-42.

[2] Timoshenko, S.P., Gere, J.M., Mechanics of Materials, Van Nostrand Reinhold Company, New York, 1973, 190-288.

[3] Shames, I.H., Dym, C.L., Energy and Finite Element Methods in Structural Mechanics, Hemisphere Publishing Corporation, Washington, 1985, 185-204.

[4] Shimpi, R.P., Shetty, R.A., Guha, A., A Simple Single Variable Shear Deformation Theory for a Rectangular Beam, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231(24), 2017, 4576-4591.

[5] Timoshenko, S.P., On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars, Philosophical Magazine, 41, 1921, 744-746.

[6] Timoshenko, S.P., On the Transverse Vibrations of Bars of Uniform Cross-section, Philosophical Magazine, 43(253), 1922, 125-131.

[7] Allen, H.G., Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, 1969, 1-147.

[8] Simão, P.D., Influence of Shear Deformations on the Buckling of Columns using the Generalized Beam Theory and Energy Principles, European Journal of Mechanics - A/Solids, 61, 2017, 216-234.

[9] Kiendl, J., Auricchio, F., Hughes, T.J.R., Reali, A., Single-variable Formulations and Isogeometric Discretizations for Shear Deformable Beams, Computer Methods in Applied Mechanics and Engineering, 284, 2015, 988-1004.

[10] Saoud, K.S., Le Grognec, P., An Enriched 1D Finite Element for the Buckling Analysis of Sandwich Beam-columns, Computational Mechanics, 57(6), 2016, 887-900.

[11] Ghugal, Y.M., Shimpi, R.P., A Review of Refined Shear Deformation Theories for Isotropic and Anisotropic Laminated Beams, Journal of Reinforced Plastics and Composites, 20(3), 2001, 255-272.

[12] Gjelsvik, A., Stability of Built-up Columns, ASCE Journal of Engineering Mechanics, 117(6), 1991, 1331-1345.

[13] Kalochairetis, K.E., Gantes, C.J., Elastic Buckling Load of Multi-story Frames Consisting of Timoshenko Members, Journal of Constructional Steel Research, 71, 2012, 231-244.

[14] Gantes, C.J., Kalochairetis, K.E., Axially and Transversely Loaded Timoshenko and Laced Built-up Columns with Arbitrary Supports, Journal of Constructional Steel Research, 77, 2012, 95-106.

[15] Chan, K.T., Lai, K.F., Stephen, N.G., Young, K., A New Method to Determine the Shear Coefficient of Timoshenko Beam Theory, Journal of Sound and Vibration, 330(14), 2011, 3488-3497.

[16] Cowper, G.R., The Shear Coefficient in Timoshenko’s Beam Theory, ASME Journal of Applied Mechanics, 33(2), 1966, 335-340.

[17] Jensen, J.J., On the Shear Coefficient in Timoshenko's Beam Theory, Journal of Sound and Vibration, 87(4), 1983, 621-635.

[18] Hutchinson, J.R., Shear Coefficients for Timoshenko Beam Theory, ASME Journal of Applied Mechanics, 68(1), 2001, 87-92.

[19] Stephen, N.G., Levinson, M., A Second Order Beam Theory, Journal of Sound and Vibration, 67(3), 1979, 293-305.

[20] Levinson, M., A New Rectangular Beam Theory, Journal of Sound and Vibration, 74(1), 1981, 81-87.

[21] Levinson, M., Further Results of a New Beam Theory, Journal of Sound and Vibration, 77, 1981, 440-444.

[22] Rehfield, L.W., Murthy, P.L.N., Toward a New Engineering Theory of Bending-Fundamentals, AIAA Journal, 20(5), 1982, 693-699.

[23] Levinson, M., On Bickford's consistent higher order beam theory, Mechanics Research Communications, 12(1), 1985, 01-09.

[24] Heyliger, P.R., Reddy, J.N., A Higher Order Beam Finite Element for Bending and Vibration Problems, Journal of Sound and Vibration, 126(2), 1988, 309-326.

[25] Kant, T., Gupta, A., A Finite Element Model for a Higher-order Shear-deformable Beam Theory, Journal of Sound and Vibration, 125(2), 1988, 193-202.

[26] Kant, T., Manjunath, B.S., Refined Theories for Composite and Sandwich Beams with C0 Finite Elements, Computers & Structures, 33(3), 1989, 755-764.

[27] Soldatos, K.P., Elishakoff, I., A Transverse Shear and Normal Deformable Orthotropic Beam Theory, Journal of Sound and Vibration, 155, 1992, 528-533.

[28] Karama, M., Afaq, K.S., Mistou, S., Mechanical Behaviour of Laminated Composite Beam by the New Multi-layered Laminated Composite Structures Model with Transverse Shear Stress Continuity, International Journal of Solids and Structures, 40(6), 2003, 1525-1546.

[29] 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, 44(2), 2008, 765-773.

[30] Benatta, M.A., Tounsi, A., Mechab, I., Bachir Bouiadjra, M., Mathematical Solution for Bending of Short Hybrid Composite Beams with Variable Fibers Spacing, Applied Mathematics and Computation, 212(2), 2009, 337-348.

[31] Mahi, A., Adda Bedia,, E.A., Tounsi, A., Mechab, I., An Analytical Method for Temperature-dependent Free Vibration Analysis of Functionally Graded Beams with General Boundary Conditions, Composite Structures, 92(8), 2010, 1877-1887.

[32] Shi, G., Voyiadjis, G.Z., A Sixth-order Theory of Shear Deformable Beams with Variational Consistent Boundary Conditions, ASME Journal of Applied Mechanics, 78(2), 2011, 021019.

[33] Karttunen, A.T., von Hertzen, R., Variational Formulation of the Static Levinson Beam Theory, Mechanics Research Communications, 66, 2015, 15-19.

[34] Mantari, J.L., Canales, F.G., A Unified Quasi-3D HSDT for the Bending Analysis of Laminated Beams, Aerospace Science and Technology, 54, 2016, 267-275.

[35] Canales, F.G., Mantari, J.L., Buckling and Free Vibration of Laminated Beams with Arbitrary Boundary Conditions using a Refined HSDT, Composites Part B: Engineering, 100, 2016, 136-145.

[36] Wang, C.M., Reddy, J.N., Lee, K.H., Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier Science Ltd, New York, 2000, 1-7.

[37] Shimpi, R.P., Refined Plate Theory and Its Variants, AIAA Journal, 40(1), 2002, 137-146.

[38] Shimpi, R.P., Patel, H.G., Arya, H., New First-order Shear Deformation Plate Theories, ASME Journal of Applied Mechanics, 74(3), 2007, 523-533.

[39] Murty, A.V.K., Vibrations of Short Beams, AIAA Journal, 8(1), 1970, 34-38.

[40] Murty, A.V.K., Analysis of Short Beams, AIAA Journal, 8, 1970, 2098-2100.

[41] Murty, A.V.K., Toward a Consistent Beam Theory, AIAA Journal, 22(6), 1984, 811-816.

[42] Murty, A.V.K., On the Shear Deformation Theory for Dynamic Analysis of Beams, Journal of Sound and Vibration, 101(1), 1985, 1-12.

[43] Shimpi, R.P., Patel, H.G., Free Vibrations of Plate using Two Variable Refined Plate Theory, Journal of Sound and Vibration, 296(4-5), 2006, 979-999.

[44] Shimpi, R.P., Patel, H.G., A Two Variable Refined Plate Theory for Orthotropic Plate Analysis, International Journal of Solids and Structures, 43(22-23), 2006, 6783-6799.

[45] El Meiche, N., Tounsi, A., Ziane, N., Mechab, I., Adda Bedia, E.A., A New Hyperbolic Shear Deformation Theory for Buckling and Vibration of Functionally Graded Sandwich Plate, International Journal of Mechanical Sciences, 53(4), 2011, 237-247.

[46] Daouadji, T.H., Henni, A.H., Tounsi, A., Adda Bedia, E.A., A New Hyperbolic Shear Deformation Theory for Bending Analysis of Functionally Graded Plates, Modelling and Simulation in Engineering, 2012, 2012, 29.

[47] Daouadji, T.H., Tounsi, A., Adda Bedia, E.A., A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates, Advances in Applied Mathematics and Mechanics, 5(3), 2013, 351-364.

[48] Sayyad, A.S., Ghugal, Y.M., Naik, N.S., Bending Analysis of Laminated Composite and Sandwich Beams According to Refined Trigonometric Beam Theory, Curved and Layered Structures, 2(1), 2015, 279-289.

[49] Sayyad, A.S., Ghugal, Y.M., Shinde, P.N., Stress Analysis of Laminated Composite and Soft Core Sandwich Beams using a Simple Higher Order Shear Deformation Theory, Journal of Serbian Society for Computational Mechanics, 9(1), 2015, 15-35.

[50] Shimpi, R.P., Guruprasad, P.J., Pakhare, K.S., Single Variable New First-order Shear Deformation Theory for Isotropic Plates, Latin American Journal of Solids and Structures, 15(10), 2018, 1-25.

[51] Bathe, K.J., Finite Element Procedures, Prentice Hall, New Jersey, 1996, 116-120, 338-484.

[52] Kabir, H.R.H., A Shear-locking Free Robust Isoparametric Three-node Triangular Finite Element for Moderately-thick and Thin Arbitrarily Laminated Plates, Computers & Structures, 57(4), 1995, 589-597.

[53] Reddy, J.N., On Locking-free Shear Deformable Beam Finite Elements, Computer Methods in Applied Mechanics and Engineering, 149(1-4), 1997, 113-132.

[54] Ainsworth, M., Pinchedez, K., The hp-MITC Finite Element Method for the Reissner–Mindlin Plate Problem, Journal of Computational and Applied Mathematics, 148(2), 2002, 429-462.

[55] Reddy, J.N., Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons Inc, New York, 2002, 112-114, 177-203.

[56] Hao-jiang, D., De-jin, H., Hui-ming, W., Analytical Solution for Fixed-end Beam Subjected to Uniform Load, Journal of Zhejiang University-SCIENCE, 6(8), 2005, 779-783.

[57] Venkatraman, B., Patel, S.A., Structural Mechanics with Introductions to Elasticity and Plasticity, McGraw-Hill Book Company, New York, 1970, 158-165.

[58] Srinivas, S., Rao, A.K., Rao, C.J., Flexure of Simply Supported Thick Homogeneous and Laminated Rectangular Plates, Zeitschrift für Angewandte Mathematik und Mechanik, 49(8), 1969, 449-458.

[59] Pagano, N.J., Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composite Materials, 4(1), 1970, 20-34.

[60] Touratier, M., An Efficient Standard Plate Theory, International Journal of Engineering Science, 29(8), 1991, 901-916.

[61] Sayyad, A.S., Comparison of Various Refined Beam Theories for the Bending and Free Vibration Analysis of Thick Beams, Applied and Computational Mechanics, 5, 2011, 217-230.