73 
 
REFERENCES 
 
[1]  C. M. Wang, J. N. Reddy and K. H. Lee, “Shear deformable 
beam       and plates -Relationship with classical solutions” Oxford: 
Elsevier Science Ltd, 2000. 
[2]  S. Timoshenko, “On the correction for shear of the 
differential equation for transverse vibrations of prismatic bars,” 
Philosoph. Mag, vol. 6, p. 742–746, 1921.  
[3]  M. Levinson, “A new rectangular beam theory,” J. Sound 
Vibration, vol. 74, no. 1, p. 81–87, 1981.  
[4]  W. B. Bickford, “ A consistent higher order beam theory,” 
Development of Theoretical and Applied Mechanics, SECTAM, vol. 
11, pp. 137-150, 1982.  
[5]  J. N. Reddy, “A simple higher-order theory for laminated 
composite plates,” J. Appl. Mech, vol. 51, no. 4, p. 745–752, 1984.  
[6]  P. R. Heyliger and J. N. Reddy, “A higher order beam finite 
element for bending and vibration problems,” J. Sound Vibration, 
vol. 126, no. 2, p. 309–326, 1988.  
[7]  M. Touratier, “An efficient standard plate theory,” Int. J. Eng. 
Sci., vol. 29, no. 8, p. 901–916, 1991.  
[8]  V. Z. Vlasov and U. N. Leont’ev, Beams, Plates and Shells on 
Elastic foundation, Jerusalem: (Translated from Russian) Israel 
Program for Scientific Translation Ltd, 1996.  
[9]  M. Stein, “Vibration of beams and plate strips with three-
dimensional flexibility,” Transaction ASME Journal of Applied 
Mechanics, vol. 56, no. 1, p. 228–231, 1989.  
74 
 
[10]  Y. M. Ghugal and R. P. Shimpi, “ A review of refined shear 
deformation theories for isotropic and anisotropic laminated beams,” 
Journal of Reinforced Plastics and Composites, vol. 21, p. 775–813, 
2002.  
[11]  Y. M. Ghugal and R. Sharma, “A hyperbolic shear deformation 
theory for flexure and free vibration of thick beams,” Int. J. Comput. 
Methods, vol. 6, no. 4, p. 585–604, 2009.  
[12]  Eurocode 2: Design of concrete structures Part 1-1: General 
rules and rules for buildings NEN-EN-1992-1-1. 
[13]  ACI 318-08 “Building Code Requirements for Structural 
Concrete and Commentary”. 
[14]  R. Patel, S. K. Dubey and K. K. Pathak, “Effect of depth span 
ratio on the behaviour of the beams,” Int J Adv Struct Eng, vol. 56, no. 
6, 2014.  
[15]  J. Bernoulli, “Problema novum ad cujus solutionem 
Mathematici invitantur,” (A new problem to whose solution 
mathematicians are invited.) Acta Eruditorum , 1696. 
 
[16]  A. Hrennikoff, “Solution of Problems in Elasticity by the Frame 
Work Method,” Journal of Applied Mechanics, vol. 8, no. 4, pp. 169-175, 
1941.  
[17]  D. McHenry, “A Lattice Analogy for the Solution of Plane Stress 
Problems,” Journal of Institution of Civil Engineers, vol. 21, pp. 59-82, 
1943.  
[18]  R. Courant, “Variational Methods for the Solution of Problems of 
Equilibrium and Vibrations,” Bulletin of the American Mathematical 
Society, vol. 49, pp. 1-23, 1943.  
75 
 
[19]  S. Levy, “Computation of Influence Coefficients for Aircraft 
Structures with Discontinuities and Sweepback,” Journal of 
Aeronautical Sciences, vol. 14, no. 10, pp. 547-560, 1947.  
[20]  S. Levy, “Structural Analysis and Influence Coefficients for Delta 
Wings,” Journal of Aeronautical Sciences, vol. 20, no. 7, pp. 449-454, 
1953.  
[21]  J. H. Argyris and S. Kelsey, Energy Theorems and Structural 
Analysis, London: Butterworths,, 1960.  
[22]  M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, 
“Stiffness and Deflection Analysis of Complex Structures,” Journal of 
Aeronautical Sciences, vol. 23, no. 9, p. 805–824, 1956.  
[23]  R. W. Clough, “The Finite Element Method in Plane Stress 
Analysis,” in Proceedings,American Society of Civil Engineers, 2nd 
Conference on Electronic Computation, Pittsburgh, 1960.  
[24]  R. J. Melosh, “A Stiffness Matrix for the Analysis of Thin Plates 
in Bending,” Journal of the Aerospace Sciences, vol. 28, no. 1, pp. 34-
42, 1961.  
[25]  P. E. Grafton and D. R. Strome, “Analysis of Axisymmetric 
Shells by the Direct Stiffness Method,” Journal of the American Institute 
of Aeronautics and Astronautics, vol. 1, no. 10, p. 2342–2347, 1963.  
[26]  H. C. Martin, “Plane Elasticity Problems and the Direct Stiffness 
Method,” The Trend in Engineering, vol. 13, pp. 5-19, 1961.  
[27]  R. H. P. J. Gallagher and P. P. Bijlaard, “Stress Analysis of 
Heated Complex,” Journal of the American Rocket Society, vol. 32, pp. 
700-707, 1962.  
  
76 
 
 
 
[28]  R. J. Melosh, “Structural Analysis of Solids,” Journal of the 
Structural Division, Proceedings of the American Society of Civil 
Engineers, pp. 205-223, 1963.  
[29]  J. H. Argyris, “Recent Advances in Matrix Methods of Structural 
Analysis,” Progress in Aeronautical Science, vol. 4, 1964.  
[30]  R. W. Clough and Y. Rashid, “Finite Element Analysis of 
Axisymmetric Solids,” Journal of the Engineering Mechanics Division, 
Proceedings of the American Society of Civil Engineers, vol. 91, p. 71–
85, 1965.  
[31]  E. L. Wilson, “Structural Analysis of Axisymmetric Solids,” 
Journal of the American Institute of Aeronautics and Astronautics, vol. 
3, no. 12, p. 2269–2274, 1965.  
[32]  J. S. Archer, “Consistent Matrix Formulations for Structural 
Analysis Using FiniteElement Techniques,” Journal of the American 
Institute of Aeronautics and Astronautics, vol. 3, no. 10, p. 1910–1918, 
1965.  
[33]  B. A. Szabo and G. C. Lee, “Derivation of Stiffness Matrices for 
Problems in Plane Elasticity by Galerkin’s Method,” International 
Journal of Numerical Methods in Engineering, vol. 1, p. 301–310, 1969.  
[34]  
 
O. C. Zienkiewicz and C. J. Parekh, “Transient Field Problems: 
Two-Dimensional and Three-Dimensional Analysis by Isoparametric 
Finite Elements,” International Journal of Numerical Methods in 
Engineering, vol. 2, no. 1, p. 61–71, 1970. 
[35]  G. R. Cowper, “The shear coefficient in Timoshenko’s beam 
theory,” ASME J. appl. Mech., vol. 33, pp. 335-340 , 1966.  
77 
 
 
 
[36]  G. R. Cowper, “On the accuracy of Timoshenko’s beam theory,” 
ASCE Journal of Engineering Mechanics Division, vol. 94, no. 6, 1968.  
[37]  H. Darijani and H. Mohammadabadi, “A new deformation beam 
theory for static and dynamic analysis of microbeams,” International 
journal of mechanical sciences, vol. 89, pp. 31-39, 2014.  
[38]  R. J.N., An Introduction to the Finite Element Method, New 
York: McGraw-Hill, 1993.  
[39]  R. J.N, “On locking-free shear deformable beam finite elements,” 
Comput. Methods Appl. Mech. Engrg., vol. 149, pp. 113-132, 1997.  
[40]  R. B. McCalley, “Rotatory Inertia Correction for Mass Matrices,” 
General Electric Knolls Atomic Power Laboratory, Schenectady, NY, 
1963. 
[41]  T. J. Hughes, R. Taylor and W. Kanoknukulchoii, “A simple and 
efficient plate element for bending,” Int. J. Numer. Meth. Engng, vol. 11, 
pp. 1529-1543, 1977.  
[42]  T. Kant and A. Gupta, “A finite element model for a higher order 
shear deformable beam theory,” Journal of Sound and Vibration, vol. 
125, no. 2, pp. 193-202, 1988.  
[43]  J. Petrolito, “Stiffness analysis of beams using higher-order 
theory,” Coput.Structures, vol. 55, no. 1, p. 33–39, 1995.  
[44]  M. Eisenberger, “An exact high order beam element,” 
Coput.Structures, vol. 81, p. 147–152, 2003.  
[45]  Y. Liu, “A beam element using Touratier theory,” 
Int.J.Comput.Methods Eeng. Sci.Mech, vol. 11, p. 142–145, 2010.  
78 
 
 
[46]  
 
K. P. Soldatos, “A transverse shear deformation theory for 
homogeneous monoclinic plates,” Acta Mech, vol. 94, no. 3, p. 195–220, 
1992.  
[47]  J. N. Reddy,  “Unified finite elements based on the classical and 
shear deformation theories of beams and axisymmetric circular plates,” 
Commun. Numer. Methods Engrg., 1997.  
[48]  J. N. Reddy,  “Exact relationships between the bending solutions 
of the Euler-Bernoulli beam theory and shear deformable beam theories,” 
Int. J. Solids Struct., 1997.  
[49]  L. G. R. and Y. T. Gu, An Inroduction to MeshFree Methods and 
Their Progamming, Dordrecht: Springer, 2005.  
[50]  P. Seshu, Textbook of Finite Element Analysis, New Delhi: PHI 
Learning, 2012.  
[51]  S. S. Bhavikatti, Finite Element Analysis, New Delhi: New Age 
International, 2005.  
[52]  O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method-
Volume 1: The Basis, Oxford: Butterworth-Heinemann, 2000.  
[53]  S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 
Singapore: McGraw-Hill, 1970.  
[54]  K. Amar, Introduction to Finite Element Analysis Using 
MATLAB and ABAQUS, Boca Raton: CRC Press, 2013.  
[55]  M. Mohammad, M. Z. Bin Jumaat, M. Chemrouk, A. Ghasemi, 
S. J. S. Hakim, and R. Najmeh, “An experimental investigation of the 
stress-strain distribution in high strength concrete deep beams,” Procedia 
Eng., vol. 14, pp. 2141–2150, 2011.