A review of displacement and stress based refined theories for isotropic and anisotropic laminated beams is presented. Various equivalent single layer and layerwise theories for laminated beams are discussed together with their merits and demerits. Exact elasticity solutions for the beam problems are cited, wherever available. Various critical issues, related with beam theories, based on the literature reviewed are presented.
[Journal of Applied Mathematics and Mechanics, Vol. 32, No. 4, pp. 704-718.]
2.
2. Kil’chevskiy, N. A., 1965, “Fundamentals of the Analytical Mechanics of Shells,”NASA TT F-292, Washington, D.C., pp. 80-172.
3.
3. Donnell, L. H., 1976. Beams, Plates and Shells, McGraw-Hill Book Company, New York, 453 p.
4.
4. Vlasov, V. Z. and Leont’ev, U. N., 1960, Beams, Plates and Shells on Elastic Foundations, Translated from Russian by Barouch, A., and edited by Pelz, T.; Israel Program for Scientific Translations Ltd., Jerusalem, Chapter 1, pp. 1-8.
5.
5. Sayir, M. and Mitropoulos, C., 1980, “On Elementary Theories of Linear Elastic Beams, Plates and Shells,”Zeitschrift für Angewandte Mathematik und Physik, Vol. 31, No. 1, pp. 1-55.
6.
6. Rankine, W. J. M., 1858, A Manual of Applied Mechanics, R. Griffin and Company Ltd., London, U.K., pp. 342-344.
7.
7. Bresse, J. A. C., 1859, Cours de Mecanique Applique, Paris: Mallet-bacheleier, (1866 2nd ed.), Gauthier-Villars, Paris.
8.
8. Rebello, C. A., Bert, C. W. and Gordaninejad, F., 1983, “Vibration of Bimodular Sandwich Beams with Thick Facings: A New Theory and Experimental Results,”Journal of Sound and Vibration,Vol. 90, No. 3, pp. 381-397.
9.
9. Lord Rayleigh, 1880, Theory of Sound, Macmillan Publishers, London, U.K.
10.
10. Timoshenko, S. P., 1921, “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars,”Philosophical Magazine, Series 6, Vol. 41, pp. 742-746.
11.
11. Kruszewski, E. T., 1949, “Effect of Transverse Shear and Rotatory Inertia on the Natural Frequency of a Uniform Beam,”NACA TN 1909.
12.
12. Dengler, M. A. and Goland, M., 1951, “Transverse Impact of Long Beams Including Inertia and Shear Effects,”ASME Proceedings of 1st U.S. National Congress of Applied Mechanics, New York, pp. 179-186.
13.
13. Dym, C. L. and Shames, I. H., 1973, Solid Mechanics: A Variational Approach, McGraw-Hill Book Co., New York, 556 p.
14.
14. Mindlin, R. D. and Deresiewicz, H., 1954, “Timoshenko’s Shear Coefficient for Flexural Vibrations of Beams,”Proceedings of 2nd U.S. National Congress of Applied Mechanics, ASME Publication, New York, pp. 175-178.
15.
15. Cowper, G. R., 1966, “The Shear Coefficients in Timoshenko Beam Theory,”ASME Journal of Applied Mechanincs, Vol. 33, pp. 335-340.
16.
16. Cowper, G. R., 1968, “On the Accuracy of Timoshenko’s Beam Theory,”ASCE Journal of the Engineering Mechanics Division, Vol. 94, No. EM 6, pp. 1447-1453.
17.
17. Murty, A. V. K., 1970, “Vibration of Short Beams,”AIAA Journal, Vol. 8, pp. 34-38.
18.
18. Murty, A. V. K., 1970, “Analysis of Short Beams,”AIAA Journal, Vol. 8, pp. 2098-2100.
19.
19. Kaneko, T., 1975, “On Timoshenko’s Correction for Shear in Vibrating Beams,”Journal of Physics D: Applied Physics, Vol. 8, No. 16, pp. 1927-1936.
20.
20. Hutchinson, J. R., 1981, “Transverse Vibrations of Beams, Exact Versus Approximate Solutions,”Trans. ASME, Journal of Applied Mechanics, Vol. 48, No. 4, pp. 923-928.
21.
21. Hutchinson, J. R. and Zillmer, S. D., 1986. “On the Transverse Vibration of Beams of Rectangular Cross-section,”Trans. ASME, Journal of Applied Mechanics, Vol. 53, No. 1, pp. 39-44.
22.
22. Rychter, Z., 1987, “On the Shear Coefficient in Beam Bending,”Mechanics Research Communications, Vol. 14, No. 5/6, pp. 379-385.
23.
23. Stephen, N. G. and Levinson, M., 1979, “A Second Order Beam Theory,”Journal of Sound and Vibration, Vol. 67, pp. 293-305.
24.
24. Rychter, Z., 1988, “An Engineering Theory for Beam Bending,”Ingenieur-Archiv, Vol. 58, No. 1, pp. 25-34.
25.
25. Renton, J. D.1991, “Generalised Beam Theory Applied to Shear Stiffness,”International Journal of Solids and Structures, Vol. 27, No. 15, pp. 1955-1967.
26.
26. Fan, H. and Widera, G. E. O., 1991, “Refined Engineering Beam Theory Based on the Asymptotic Expansion Approach,”AIAA Journal, Vol. 29, No. 3, pp. 444-449.
27.
27. Kathnelson, A. N., 1996, “Improved Engineering Theory for Uniform Beams,”Acta Mechanica, Note, Vol. 114, No. 1-4, pp. 225-229.
28.
28. Zaslavsky, A., 1980, “On the Limitations of the Shearing Stress Formula,”The International Journal of Mechanical Engineering Education, Vol. 8, pp. 13-19.
29.
29. Lo, K. H., Christensen, R. M. and Wu, E. M., 1977, “A Higher Order Theory for Plate Deformations, Part 1: Homogeneous Plates,”ASME Journal of Applied Mechanics, Vol. 44, pp. 663-668.
30.
30. Lo, K. H., Christensen, R. M. and Wu, E. M., 1977, “A Higher Order Theory for Plate Deformations, Part 2: Laminated Plates,”ASME Journal of Applied Mechanics, Vol. 44, pp. 669-676.
31.
31. Soler, A. I., 1968, “Higher Order Effects in Thick Rectangular Elastic Beams,”International Journal of Solids and Structures, Vol. 4, No. 7, pp. 723-739.
32.
32. Tsai, H. and Soler, A. I., 1970, “Approximate Theory for Locally Loaded Plane Orthotropic Beams,”International Journal of Solids and Structures, Vol. 6, No. 8, pp. 1055-1068.
33.
33. Essenburg, F., 1975, “On the Significance of the Inclusion of the Effect of Transverse Normal Strain in Problems Involving Beams with Surface Constraint,”ASME Journal of Applied Mechanics, Vol. 97, pp. 127-132.
34.
34. Leech, C. M., 1977, “Beam Theories: A Variational Approach,”The International Journal of Mechanical Engineering Education, Vol. 5, No. 1, pp. 81-87.
35.
35. Levinson, M., 1981, “A New Rectangular Beam Theory,”Journal of Sound and Vibration, Vol. 74, pp. 81-87.
36.
36. Levinson, M., 1981, “Further Results of a New Beam Theory,”Journal of Sound and Vibration,Vol. 77, pp. 440-444.
37.
37. Levinson, M., 1985, “On Bickford’s Consistent Higher Order Beam Theory,”Mechanics Research Communications, Vol. 12, pp. 1-9.
38.
38. Levinson, M., 1986, “Consistent and Inconsistent Higher Order Beam and Plate Theories: Some Surprising Comparisons,”Proceedings of Euromech Colloquium 219 on Refined Dynamical Theories of Beams, Plates and Shells and Their Applications, Kassel, F.R. Germany, pp. 122-130.
39.
39. Bickford, W. B., 1982, “A Consistent Higher Order Beam Theory,”Developments in Theoretical and Applied Mechanics, SECTAM, 11, pp. 137-150.
40.
40. Rychter, Z., 1987, “On the Accuracy of a Beam Theory,”Mechanics of Research Communications,Vol. 14, No. 2, pp. 99-105.
41.
41. Rychter, Z., 1988, “A Simple and Accurate Beam Theory,”Acta Mechanica, Vol. 75, pp. 57-62.
42.
42. Petrolito, J., 1995, “Stiffness Analysis of Beams Using A Higher-Order Theory,”Computers and Structures, Vol. 55, No. 1, pp. 33-39.
43.
43. Rehfield, L. W. and Murthy, P. L. N., 1982, “Toward a New Engineering Theory of Bending: Fundamentals,”AIAA Journal, Vol. 20, pp. 693-699.
44.
44. Rychter, Z., 1987, “An Error Estimate for Solutions in Beam Theory,”ZAMM. Zeitschrift für Angewandte Mathematik und Mechanic, Vol. 67, No. 3, pp. 205-207.
45.
45. Baluch, M. H., Azad, A. K. and Khidir, M. A., 1984, “Technical Theory of Beam with Normal Strain,”Journal of the Engineering Mechanics, Proceedings of ASCE, Vol. 110, pp. 1233-1237.
46.
46. Valisetty, R. R., 1990, “Refined Bending Theory for Beams of Circular Cross-section,”Proceedings of ASCE, Journal of Engineering Mechanics, Technical Note, Vol. 116, No. 9. pp. 2072-2079.
47.
47. Krishna Murty, A. V., 1984, “Toward a Consistent Beam Theory,”AIAA Journal, Vol. 22, pp. 811-816.
48.
48. Bhimaraddi, A. and Chandrashekhara, K., 1993, “Observations on Higher-Order Beam Theory,”Journal of Aerospace Engineering, Proceedings of ASCE, Technical Note, Vol. 6, pp. 408-413.
49.
49. Irretier, H., 1986, “Refined Effects in Beam Theories and their Influence on the Natural Frequencies of Beams,”Proc. Euromech Colloquium 219, on Refined Dynamical Theories of Beams, Plates, and Shells and their Applications. Edited by Elishakoff, I. and Irretier, H., Springer-Verlag, Berlin, pp. 163-179.
50.
50. Heyliger, P. R. and Reddy, J. N., 1988, “A Higher Order Beam Finite Element for Bending and Vibration Problems,”Journal of Sound and Vibration, Vol. 126, No. 2, pp. 309-326.
51.
51. Kant, T. and Gupta, A., 1988, “A Finite Element Model for a Higher Order Shear Deformable Beam Theory,”Journal of Sound and Vibration, Vol. 125, No. 2, pp. 193-202.
52.
52. Irschik, H., 1991, “Analogy Between Refined Beam Theories and the Bernoulli-Euler Theory,”International Journal of Solids and Structures, Vol. 28, No. 9, pp. 1105-1112.
53.
53. Donnell, L. H., 1952, “Bending of Rectangular Beams,”ASME Journal of Applied Mechanics, Vol. 74, p. 123.
54.
54. Boley, B. A. and Tolins, I. S., 1956, “On the Stresses and Deflections of Rectangular Beams,”ASME Journal of Applied Mechanics, Vol. 23, pp. 339-342.
55.
55. Levy, M., 1877, “Memoire sur la Theorie des Plaques Elastiques Planes,”Journal des Mathmatiques Pures et Appliquees, Vol. 30, pp. 219-306.
56.
56. Stein, M., 1989, “Vibration of Beams and Plate Strips with Three-Dimensional Flexibility,”Trans. ASME, Journal of Applied Mechanics, Vol. 56, No. 1, pp. 228-231.
57.
57. Love, A. E. H., 1944, The Mathematical Theory of Elasticity, 4th ed., Dover Publications, New York, p. 462.
58.
58. Rao, B. S., 1989, A Refined Beam Theory, M.Tech. Thesis, Dept. Aero. Engg., IIT. Powai, Mumbai, 29 p.
59.
59. Ghugal, Y. M. and Shimpi, R. P., 2000, “A Trigonometric Shear Deformation Theory for Flexure and Free Vibration of Isotropic Thick Beams, Structural Enigineering Convention, SEC-2000, IIT Bombay, India.
60.
60. Lekhnitskii, S. G., 1957, Anisotropic Plates, 2nd ed., Moscow, translated by Tsai, S. W. and Cheron, T., Gordon and Breach Science Publishers, New York.
61.
61. Timoshenko, S. P. and Goodier, J. N., 1970, Theory of Elasticity, McGraw-Hill, 3rd Int. Ed., Singapore.
62.
62. Iyengar, K. T. S. R. and Prabhakara, M. K., 1968, “Analysis of Continuous Beams: A Three-Dimensional Elasticity Solution,”International Journal of Engineering Science, Vol. 6, pp. 193-208.
63.
63. Neou, Y. C., 1957, “A Direct Method for Determining Airy Polynomial Stress Functions,”Trans. ASME, Journal of Applied Mechanics, Vol. 24, pp. 387-390.
64.
64. Niedenfuhr, F. W., 1957, “On Choosing Stress Functions in Rectangular Coordinates,”Journal of Aeronautical Sciences, Vol. 24, pp. 460-461.
65.
65. Hashin, Z., 1967, “Plane Anisotropic Beams,”ASME Journal of Applied Mechanics, Vol. 34, pp. 257-262.
66.
66. Whitney, J. M., 1987, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing. Co., Inc., Lancaster, PA.
67.
67. Tauchert, T. R., 1975, “On the Validity of Elementary Bending Theory for Anisotropic Elastic Slabs,”Journal of Composite Materials, Vol. 9, pp. 207-214.
68.
68. Krajcinovic, D., 1972, “Sandwich Beam Analysis,”Trans. ASME, Journal of Applied Mechanics,Vol. 39, No. 3, pp. 773-778.
69.
69. Ojalvo, I. U., 1977, “Departures from Classical Beam Theory in Laminated, Sandwich, and Short Beams,”AIAA J., Vol. 15, No. 10, pp. 1518-1521.
70.
70. Swift, G. W. and Heller, R. A., 1974, “Layered Beam Analysis,”Journal of the Engineering Mechanics Division, Proceedings of ASCE, Vol. 100, pp. 267-282.
71.
71. Bert, C. W., 1973, “Simplified Analysis of Static Shear Factor for Beams of Nonhomogeneous Cross Section,”Journal of Composite Materials, Vol. 7, pp. 525-529.
72.
72. Dharmarajan, S. and McCutchen, H. Jr., 1973, “Shear Coefficient for Orthotropic Beams,”Journal of Composite Materials, Vol. 7, pp. 530-535.
73.
73. Gordaninejad, F. and Bert, C. W., 1989, “A New Theory for Bending of Thick Sandwich Beams,”International Journal of Mechanical Sciences, Vol. 31, No. 11/12, pp. 925-934.
74.
74. Ambartsumyan, S. A., 1970, Theory of Anisotropic Plates, J. E. Ashton, ed., Technomic Publishing Co., Inc., Lancaster, PA.
75.
75. Krishna Murty, A. V. and Shimpi, R. P., 1974, “Vibration of Laminated Beams,”Journal of Sound and Vibration, Vol. 36, pp. 273-284.
76.
76. Krishna Murty, A. V., 1985, “On the Shear Deformation Theory for Dynamic Analysis of Beams,”Journal of Sound and Vibration, Vol. 101, pp. 1-12.
77.
77. Silverman, I. K., 1980, “Flexure of Laminated Beams,”Journal of the Structural Division, Proceedings of ASCE, Vol. 106, pp. 711-725.
78.
78. Hu, M. Z., Kolsky, H. and Pipkin, A. C., 1985, “Bending Theory for Fiber Reinforced Beams,”Journal of Composite Materials, Vol. 19, pp. 235-249.
79.
79. Khdeir, A. A. and Reddy, J. N., 1997, “An Exact Solution for the Bending of Thin and Thick Cross-ply Laminated Beams,”Composite Structures, Vol. 37, No. 2, pp. 195-203.
80.
80. Soldatos, K. P. and Watson, P., 1997, “A General Theory for the Accurate Stress Analysis of Homogeneous and Laminated Composite Beams,”International Journal of Solids and Structures, Vol. 34, No. 22, pp. 2857-2885.
81.
81. Kant, T. and Manjunatha, B. S., 1989, “Refined Theories for Composite and Sandwich Beams with C0 Finite Elements,”Computers and Structures, Vol. 33, pp. 755-764.
82.
82. Manjunatha, B. S. and Kant, T., 1993a, “New Theories for Symmetric/Unsymmetric Composite and Sandwich Beams with C0 Finite Elements,”Composite Structures, Vol. 23, pp. 61-73.
83.
83. Manjunatha, B. S. and Kant, T., 1993b, “Different Numerical Techniques for the Estimation of Multiaxial Stresses in Symmetric/Unsymmetric Composite and Sandwich Beams with Refined Theories,”Journal of Reinforced Plastics and Composites, Vol. 12, pp. 2-27.
84.
84. Maiti, D. K. and Sinha, P. K., 1994, “Bending and Free Vibration Analysis of Shear Deformable Laminated Composite Beams by Finite Element Method,”Composite Structures, Vol. 29, pp. 421-431.
85.
85. Vinayak, R. U., Prathap, G. and Naganarayana, B. P., 1996, “Beam Elements Based on a Higher Order Theory—I: Formulation and Analysis of Performance,”Computers and Structures, Vol. 58, pp. 775-789.
86.
86. Soldatos, K. P. and Elishakoff, I., 1992, “A Transverse Shear and Normal Deformable Orthotropic Beam Theory,”Journal of Sound and Vibration, Vol. 154, No. 3, pp. 528-533.
87.
87. Murakami, H., Reissner, E. and Yamakawa, J., 1996, “Anisotropic Beam Theories with Shear Deformation,”Trans. ASME, Journal of Applied Mechanics, Vol. 63, pp. 660-668.
88.
88. Reddy, J. N., 1987, “A Generalization of Two Dimensional Theories of Laminated Composite Plates,”Communications in Applied Numerical Methods, Vol. 3, pp. 173-180.
89.
89. Lu, X. and Liu, D., 1992, “An Interlaminar Shear Stress Continuity Theory for both Thin and Thick Composite Laminates,”ASME Journal Applied Mechanics, Vol. 59, pp. 502-509.
90.
90. Davalos, J. F., Kim, Y. and Barbero, E. J., 1994, “Analysis of Laminated Beams with a Layerwise Constant Shear Theory,”Composite Structures, Vol. 28, pp. 241-253.
91.
91. Muskhelishvili, N. I., 1963, Some Basic Problems of the Mathematical Theory of Elasticity, English Translation of 4th Ed., Noordhoff, Groningen, The Netherlands, Chap. 25.
92.
92. Silverman, I. K., 1964, “Orthotropic Beams under Polynomial Loads,”Journal of the Engineering Mechanics Division, Proceedings of ASCE, Vol. 90, pp. 293-319.
93.
93. Gerstner, R. W., 1968, “Stresses in a Composite Cantilever,”Journal of Composite Materials, Vol. 2, pp. 498-501.
94.
94. Pagano, N. J., 1969, “Exact Solution for Composite Laminates in Cylindrical Bending,”Journal of Composite Materials, Vol. 3, pp. 398-411.
95.
95. Pagano, N. J., 1970, “Influence of Shear Coupling in Cylindrical Bending of Anisotropic Laminates,”Journal of Composite Materials, Vol. 4, pp. 330-343.
96.
96. Rao, K. M. and Ghosh, B. G., 1979, “Exact Analysis of Unsymmetric Laminated Beam,”Journal of the Structural Division, ASCE, Vol. 105, pp. 2313-2325.
97.
97. Cheng, S., Wei, X. and Jiang, T., 1989, “Stress Distribution and Deformation of Adhesive-bonded Laminated Composite Beams,”Journal of Engineering Mechanics, ASCE, Vol. 115, pp. 1150-1162.
98.
98. Holt, P. J. and Webber, J. B. H., 1982, “Exact Solutions to Some Honeycomb Sandwich Beam, Plate and Shell Problems,”The Journal of Strain Analysis for Engineering Design, Vol. 17, No. 1, pp. 1-8.
