Beam Structures: Classical and Advanced Theories

Beam Structures: Classical and Advanced Theories

Volumn , Issue , 2011, Pages

  Carrera, Erasmo ^a   Giunta, Gaetano ^b   Petrolo, Marco ^a  

^a POLITECNICO DI TORINO   (Italy)

^b LUXEMBOURG INSTITUTE OF SCIENCE AND TECHNOLOGY   (Luxembourg)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84891579567     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9781119978565     Document Type: Book

References (193)

1. Berdichevsky VL, Armanios E, and Badir A 1992 Theory of anisotropic thin-walled closedcross-section beams. Composites Engineering, 2(5-7), 411-432.
2. Bernoulli D 1751 De vibrationibus et sono laminarum elasticarum. In: Commentarii Academiae Scientiarum Imperialis Petropolitanae, Petropoli.
3. Carrera E 1995 A class of two dimensional theories for multilayered plates analysis. Atti Accademia delle Scienze di Torino, Memorie Scienze Fisiche, 19-20, 49-87.
4. Carrera E 2002 Theories and finite elements for multilayered, anisotropic, composite plates and shells. Archives of Computational Methods in Engineering, 9(2), 87-140.
5. Carrera E 2003 Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering, 10(3), 216-296.
6. Carrera E, Brischetto S, and Robaldo A 2008 Variable kinematic model for the analysis of functionally graded material plates. AIAA Journal, 46, 194-203.
7. Carrera E and Giunta G 2010 Refined beam theories based on a unified formulation. International Journal of Applied Mechanics, 2(1), 117-143.
8. Carrera E, Giunta G, Nali P, and Petrolo M 2010a Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
9. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502 DOI: 10.3233/SAV-2010-0528.
10. Carrera E, Petrolo M, and Varello A 2011 Advanced beam formulations for free vibration analysis of conventional and joined wings. Journal of Aerospace Engineering. In press. DOI: 10.1061/(ASCE)AS.1943-5525.0000130.
11. Carrera E and Petrolo M 2010 Refined beam elements with only displacement variables and plate/shell capabilities. Submitted.
12. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78. DOI: 10.1115/1.4002207.
13. Cowper GR 1966 The shear coefficient in Timoshenko's beam theory. Journal of Applied Mechanics, 33(2), 335-340.
14. de Saint-Venant A 1856a Mémoire sur la flexion des prismes. Journal de Mathématiques pures et appliqués, 1, 89-189.
15. de Saint-Venant A 1856b Mémoire sur la torsion des prismes. Académie des Sciences de l'Institut Impérial de Frances, 14, 233-560.
16. Dinis P, Camotim D, and Silvestre N 2006 GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily branched open cross-sections. Thin-Walled Structures, 44(1), 20-38.
17. Dong SB, Kosmatka JB, and Lin HC 2001 On Saint-Venant's problem for an inhomogeneous, anisotropic cylinder-Part I: Methodology for Saint-Venant solutions. Journal of Applied Mechanics, 68(3), 376-381.
18. El Fatmi R 2002 On the structural behavior and the Saint Venant solution in the exact beam theory: application to laminated composite beams. Computers & Structures, 80(16-17), 1441-1456.
19. El Fatmi R 2007a Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory. International Journal of Solids and Structures, 44(18-19), 5912-5929.
20. El Fatmi R 2007b Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications. International Journal of Solids and Structures, 44(18-19), 5930-5952.
21. El Fatmi R 2007c A non-uniform warping theory for beams. Comptes Rendus Mecanique, 335, 476-474.
22. El Fatmi R and Zenzri H 2004 A numerical method for the exact elastic beam theory: applications to homogeneous and composite beams. International Journal of Solids and Structures, 41, 2521-2537.
23. Euler L 1744 De curvis elasticis. In: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti. Bousquet.
24. Gruttmann F and Wagner W 2001 Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections. Computational Mechanics, 27, 199-207.
25. Gruttmann F, Sauer R, and Wagner W 1999 Shear stresses in prismatic beams with arbitrary cross-sections. International Journal for Numerical Methods in Engineering, 45, 865-889.
26. Ieşan 1986 On Saint-Venant's problem. Archive for Rational Mechanics and Analysis, 91, 363-373.
27. Kanok-Nukulchai W and Shik Shin Y 1984 Versatile and improved higher-order beam elements. Journal of Structural Engineering, 110, 2234-2249.
28. Kapania K and Raciti S 1989a Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling. AIAA Journal, 27(7), 923-935.
29. Kapania K and Raciti S 1989b Recent advances in analysis of laminated beams and plates, part II: Vibrations and wave propagation. AIAA Journal, 27(7), 935-946.
30. Kosmatka JB, Lin HC, and Dong SB 2001 On Saint-Venant's problem for an inhomogeneous, anisotropic cylinder-Part II: Cross-sectional properties. Journal of Applied Mechanics, 68(3), 382-391.
31. Krayterman BL and Krayterman, AB 1987 Generalized nonuniform torsion of beams and frames. Journal of Structural Engineering, 113, 1772-1787.
32. Krishna Murty AV 1985 On the shear deformation theory for dynamic analysis of beams. Journal of Sound and Vibration, 101(1), 1-12.
33. Ladéveze P and Simmonds J 1996 De nouveaux concepts en théorie des poutres pour des charges et des géométries quelconques. Comptes Rendus de l'Academie des Sciences. Paris, 332, 445-462.
34. Ladéveze P and Simmonds J 1998 New concepts for linear beam theory with arbitrary geometry and loading. European Journal of Mechanics-A/Solids, 17(3), 377-402.
35. Ladéveze P, Sanchez P, and Simmonds J 2004 Beamlike (Saint-Venant) solutions for fully anisotropic elastic tubes of arbitrary closed cross section. International Journal of Solids and Structures, 41(7), 1925-1944.
36. Librescu L and Song O 1992 On the static aeroelastic tailoring of composite aircraft swept wings modelled as thin-walled beam structures. Composites Engineering, 2, 497-512.
37. Lin HC and Dong SB 2006 On the Almansi-Michell problems for an inhomogeneous, anisotropic cylinder. Journal of Mechanics, 22(1), 51-57.
38. Lin HC, Dong SB, and Kosmatka JB 2001 On Saint-Venant's problem for an inhomogeneous, anisotropic cylinder-Part III: End effects. Journal of Applied Mechanics, 68(3), 392-398.
39. Mechab I, Tounsi A, Benatta MA, and Bedia EA 2008 Deformation of short composite beam using refined theories. Journal of Mathematical Analysis and Applications, 346, 468-479.
40. Mucichescu DT 1984 Bounds for stiffness of prismatic beams. Journal of Structural Engineering, 110, 1410-1414.
41. VV Novozhilov 1961 Theory of elasticity Pergamon
42. Pai PF and Schulz MJ 1999 Shear correction factors and an energy consistent beam theory. International Journal of Solids and Structures, 36, 1523-1540.
43. Popescu B and Hodges DH 2000 On asymptotically correct Timoshenko-like anisotropic beam theory. International Journal of Solids and Structures, 37, 535-558.
44. Qin Z and Librescu L 2002 On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations. Composite Structures, 56, 345-358.
45. Schardt R 1966 Eine Erweiterung der technischen Biegetheorie zur berechnung prismatischer Faltwerke. Der Stahlbau, 35, 161-171.
46. Schardt R 1989 Verallgemeinerte technische Biegetheorie. Springer.
47. Schardt R 1994 Generalized beam theory-an adequate method for coupled stability problems. Thin-Walled Structures, 19, 161-180.
48. Silvestre N 2002 Second-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Structures, 40(9), 791-820.
49. Silvestre N 2003 GBT buckling analysis of pultruded FRP lipped channel members. Computers & Structures, 81(18-19), 1889-1904.
50. Silvestre N 2007 Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes. Thin-Walled Structures, 45(2), 185-198.
51. Silvestre N and Camotim D 2002 First-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Structures, 40(9), 791-820.
52. Sokolnikoff IS 1956 Mathematical Theory of Elasticity. McGraw-Hill.
53. Timoshenko SP 1921 On the corrections for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41, 744-746.
54. Timoshenko SP 1922 On the transverse vibrations of bars of uniform cross section. Philosophical Magazine, 43, 125-131.
55. Timoshenko SP and Goodier JN 1970 Theory of elasticity. McGraw-Hill.
56. Volovoi VV and Hodges DH 2000 Theory of anisotropic thin-walled beams. Journal of Applied Mechanics, 67, 453-459.
57. Volovoi VV, Hodges DH, Berdichevsky VL, and Sutyrin VG 1999 Asymptotic theory for static behavior of elastic anisotropic I-beams. International Journal of Solid Structures, 36, 1017-1043.
58. Wagner W and Gruttmann F 2002 A displacement method for the analysis of flexural shear stresses in thin-walled isotropic composite beams. Computers & Structures, 80, 1843-1851.
59. Washizu, K 1968 Variational methods in elasticity and plasticity. Pergamon.
60. Yu W and Hodges DH 2004 Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. Journal of Applied Mechanics, 71, 15-23.
61. Yu W and Hodges DH 2005 Generalized Timoshenko theory of the variational asymptotic beam sectional analysis. Journal of the American Helicopter Society, 50(1), 46-55.
62. Yu W, Hodges DH, Volovoi VV, and Cesnik CES 2002a On Timoshenko-like modeling f initially curved and twisted composite beams. International Journal of Solids and Structures, 39, 5101-5121.
63. Yu W, Volovoi VV, Hodges DH, and Hong X 2002b Validation of the variational asymptotic beam sectional analysis (VABS). AIAA Journal, 40, 2105-2113.
64. Washizu, K 1968 Variational methods in elasticity and plasticity Pergamon.
65. Ballarini R 2003 The Da Vinci-Euler-Bernoulli beam theory? Mechanical Engineering Magazine. Available Online, http://www.memagazine.org/contents/current/webonly/webex418.html.
66. Carrera E and Petrolo M 2010 Guidelines and recommendations to construct theories for metallic and composite plates. AIAA Journal, 48(12), 2852-2866.
67. Cowper GR 1966 The shear coefficient in Timoshenko's beam theory. Journal of Applied Mechanics, 33(2), 335-340.
68. Euler L 1744 De curvis elasticis. In: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti. Bousquet.
69. Gruttmann F, Sauer R, and Wagner W 1999 Shear stresses in prismatic beams with arbitrary cross-sections. International Journal for Numerical Methods in Engineering, 45, 865-889.
70. Gruttmann F and Wagner W 2001 Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections. Computational Mechanics, 27, 199-207.
71. Kapania K and Raciti S 1989 Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling. AIAA Journal, 27(7), 923-935.
72. Pai PF and Schulz MJ 1999 Shear correction factors and an energy consistent beam theory. International Journal of Solids and Structures, 36, 1523-1540.
73. Reti, L 1974 The unknown Leonardo. McGraw-Hill.
74. Timoshenko SP 1921 On the corrections for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41, 744-746.
75. Timoshenko SP 1922 On the transverse vibrations of bars of uniform cross section. Philosophical Magazine, 43, 125-131.
76. Carrera E and Giunta G 2010 Refined beam theories based on a unified formulation. International Journal of Applied Mechanics, 2(1), 117-143.
77. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2), DOI: 10.1115/1.4002207.
78. Carrera E, Giunta G, Nali P, and Petrolo M 2010 Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
79. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502. DOI: 10.3233/SAV-2010-0528.
80. Kapania K and Raciti S 1989 Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling. AIAA Journal, 27(7), 923-935.
81. Novozhilov VV 1961 Theory of elasticity. Pergamon.
82. Schardt R 1989 Verallgemeinerte technische biegetheorie. Springer.
83. Schardt R 1994 Generalized beam theory-an adequate method for coupled stability problems. Thin-Walled Structures, 19, 161-180.
84. Silvestre N 2007 Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes. Thin-Walled Structures, 45(2), 185-198.
85. Washizu K 1968 Variational methods in elasticity and plasticity. Pergamon.
86. Carrera E and Brischetto S 2008a Analysis of thickness locking in classical, refined and mixed multilayered plate theories. Composite Structures, 82(4), 549-562.
87. Carrera E and Brischetto S 2008b Analysis of thickness locking in classical, refined and mixed theories for layered shells. Composite Structures, 85(1), 83-90.
88. Carrera E and Giunta G 2010 Refined beam theories based on a unified formulation. International Journal of Applied Mechanics, 2(1), 117-143.
89. Carrera E and Petrolo M 2010 Refined beam elements with only displacement variables and plate/shell capabilities. Submitted.
90. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2). DOI: 10.1115/1.4002207.
91. Carrera E, Giunta G, Nali P, and Petrolo M 2010a Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
92. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502. DOI: 10.3233/SAV-2010-0528.
93. Carrera E, Petrolo M, and Varello A 2011. Advanced beam formulations for free vibration analysis of conventional and joined wings. Journal of Aerospace Engineering, In Press. DOI: 10.1061/(ASCE)AS.1943-5525.0000130.
94. Carrera E and Giunta G. 2007 Hierarchical closed form solutions for plates bent by localized transverse loadings. Journal of Zhejiang University SCIENCE A, 8(7), 1026-1037.
95. Carrera E and Giunta G. 2008 Hierarchical models for failure analysis of plates bent by distributed and localized transverse loadings. Journal of Zhejiang University SCIENCE A, 9(5), 600-613.
96. Carrera E and Nali P 2009 Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement. International Journal for Numerical Methods in Engineering, 80(4), 403-424.
97. Carrera E, Giunta G, Nali P, and Petrolo M 2010a Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
98. Carrera E, Giunta G, and Petrolo M 2010 A modern and compact way to formulate classical and advanced beam theories. In: Developments in Computational Structures Technology, Ch. 4. DOI: 10.4203/csets.25.4.
99. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502. DOI: 10.3233/SAV-2010-0528.
100. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2). DOI: 10.1115/1.4002207.
101. Giunta G, Belouettar S, and Carrera E 2010 Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures, 17(8), 622-635.
102. Giunta G, Biscani F, Carrera E, and Belouettar S 2011 Analysis of thin-walled beams via a one-dimensional unified formulation. International Journal of Applied Mechanics. In Press.
103. Carrera E and Nali P 2009 Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement. International Joournal for Numerical Methods in Engineering, 80(4), 403-424.
104. Carrera E, Giunta G, Nali P, and Petrolo M 2010 Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
105. Carrera E, Giunta G, and Petrolo M 2010 A modern and compact way to formulate classical and advanced beam theories. In: Developments in Computational Structures Technology, Ch. 4. DOI: 10.4203/csets.25.4.
106. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502. DOI: 10.3233/SAV-2010-0528.
107. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2), DOI: 10.1115/1.4002207.
108. Giunta G, Belouettar S, and Carrera E 2010 Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Material Science and Engineering, 10(1).
109. Giunta G, Belouettar S, and Carrera E 2010 Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures, 17(8), 622-635.
110. Giunta G, Biscani F, Carrera E, and Belouettar S 2011 Analysis of thin-walled beams via a one-dimensional unified formulation. International Journal of Applied Mechanics. In Press.
111. Bathe K 1996 Finite element procedure. Prentice Hall.
112. Carrera E and Nali P 2009 Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement. International Journal for Numerical Methods in Engineering, 80(4), 403-424.
113. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2), DOI: 10.1115/1.4002207.
114. Carrera E, Giunta G, Nali P, and Petrolo M 2010a Refined beam elements with arbitrary cross-section geometries. Computers & Structures, 88(5-6), 283-293. DOI: 10.1016/j.compstruc.2009.11.002.
115. Carrera E, Petrolo M, and Nali P 2011 Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock and Vibrations, 18(3), 485-502. DOI: 10.3233/SAV-2010-0528.
116. Carrera E, Giunta G, and Petrolo M 2010c A modern and compact way to formulate classical and advanced beam theories. In: Developments in Computational Structures Technology, Ch. 4. DOI: 10.4203/csets.25.4.
117. Cinefra M, Carrera E, and Nali P 2010 MITC technique extended to variable kinematic multilayered plate elements. Composite Structures, 92, 1888-1895.
118. Oñate E 2009 Structural Analysis with the Finite Element Method: Linear Statics, Volume 1. Springer.
119. Reddy, JN 1997 On locking-free shear deformable beam finite elements. Computer Methods in Applied Mechanics and Engineering, 149, 113-132.
120. Carrera E, Giunta G, and Petrolo M 2010a A modern and compact way to formulate classical and advanced beam theories. In: Developments in Computational Structures Technology, Ch. 4. DOI: 10.4203/csets.25.4.
121. Carrera E, Giorcelli E, Mattiazzo G, and Petrolo M 2010b Refined beam models for static and dynamic analysis of wings and rotor blades. In: Proceedings of CST2010, The Tenth International Conference on Computational Structures Technology, Valencia, Spain, 14-17 September.
122. Carrera E, Petrolo M, and Varello A 2011 Advanced beam formulations for free vibration analysis of conventional and joined wings. Journal of Aerospace Engineering. In press. DOI: 10.1061/(ASCE)AS.1943-5525.0000130.
123. Giunta G, Biscani F, Carrera E, and Belouettar S 2011 Analysis of thin-walled beams via a one-dimensional unified formulation. International Journal of Applied Mechanics. In Press.
124. Gruttmann F and Wagner W 2001 Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections. Computational Mechanics, 27, 199-207.
125. Gruttmann F, Sauer R, and Wagner W 1999 Shear stresses in prismatic beams with arbitrary cross-sections. International Journal for Numerical Methods in Engineering 45, 865-889.
126. Petrolo M, Zappino E, and Carrera E 2011 Refined free vibration analysis of onedimensional structures with compact and bridge-like cross-sections. Submitted.
127. Bleich F 1952 Buckling Strength of Metal Structures. McGraw-Hill.
128. Getting Started with MSC. Nastran User's Guide. MSC. Software Corporation, Santa Ana, CA.
129. Goodier JN 1942 Flexural-torsional buckling of bars of open section under bending, eccentric thrust or torsional loads. Cornell University Engineering Experimental Station Bulletin, 28.
130. Simitses GJ 1986 An Introduction to the Elastic Stability of Structures. Prentice Hall.
131. Timoshenko SP 1945 Theory of bending, torsion and buckling of thin-walled members of open cross section. Journal of the Franklin Institute, 239(3), 201-219.
132. Timoshenko SP and Gere JM 1961 Theory of Elastic Stability. Dover.
133. Timoshenko SP and Goodier JN 1970 Theory of elasticity. McGraw-Hill.
134. Vlasov VZ 1959 Thin-walled elastic beams. Fizmatgiz.
135. Wagner H 1929 Verdrehung und Knickung von offenen Profilen. The 25th Anniversary of the Technische Hochschule, Danzig, 1929.
136. Wagner H 1936 Torsion and buckling of open sections. Technical Memorandum No. 807, National Advisory Committee for Aeronautics, USA, 1936.
137. Aboudi J 1991 Mechanics of composite materials: a unified micromechanical approach. Elsevier.
138. Aboudi J, Pindera MJ, and Arnold SM 1994 Elastic response of metal matrix composites with tailored microstructures to thermal gradients. International Journal of Solids and Structures, 31(10), 1393-1428.
139. Aboudi J, Pindera MJ, and Arnold SM 1996 Thermoelastic theory for the response of materials functionally graded in two directions. International Journal of Solids and Structures, 33(7), 931-966.
140. Aboudi J, Pindera MJ, and Arnold SM 1999 Higher-order theory for functionally graded materials. Composites Part B: Engineering, 30(8), 777-832.
141. Bhattacharyya M, Kapuria S, and Kumar AN 2007 On the stress to strain transfer ratio and elastic deflection behavior for Al/SiC functionally graded material. Mechanics of Advanced Materials and Structures, 14(4), 295-302.
142. Birman V and Byrd LW 2007 Modelling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, 60(5), 7.
143. Chakraborty A, Gopalakrishnan S, and Reddy JN 2003 A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45(3), 519-539.
144. Ding JH, Huang DJ, and Chen WQ 2007 Elasticity solutions for plane anisotropic functionally graded beams. International Journal of Solids and Structures, 44(1), 176-196.
145. Finot M, Suresh S, Bull C, and Sampath S 1996 Curvature changes during thermal cycling of a compositionally graded Ni-Al2O3 multi-layered material. Materials Science and Engineering A, 205(1-2), 59-71.
146. Getting Started with MSC. Nastran. User's Guide. MSC .Software Corporation, Santa Ana, CA.
147. Giunta G, Belouettar S, and Carrera E 2010 Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures, 17(8), 622-635.
148. Hill R 1965 A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13(4), 213-222.
149. Kadoli R, Akhtara K, and Ganesanb N 2008 Static analysis of functionally graded beams using higher order shear deformation theory. Applied Mathematical Modelling, 32(12), 2509-2525.
150. Kapuria S, Bhattacharyya M, and Kumar AN 2008 Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation. Composite Structures, 82(3), 390-402.
151. Koizumi M 1997 FGM activities in Japan. Composites Part B: Engineering, 28(1-2), 1-4.
152. Li XF 2008 A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. Journal of Sound and Vibration, 318(4-5), 1210-1229.
153. Lü CF, Chen WQ, Xu RQ, and Lim CW 2008 Semi-analytical elasticity solutions for bidirectional functionally graded beams. International Journal of Solids and Structures, 45(1), 258-275.
154. Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, and Ford RG 1999 Functionally Graded Materials: Design, Processing and Applications. Kluwer Academic.
155. Mori T and Tanaka K 1973 Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21(5), 571-574.
156. Nemat-Nasser S and Hori M 1993 Micromechanics: overall properties of heterogeneous materials. North-Holland.
157. Sankar BV 2001 An elasticity solution for functionally graded beams . Composites Science and Technology, 61(5), 689-696.
158. Silverman K 1964 Orthotropic beams under polynomial loads. In: Proceedings of the ASCE, Journal of the Engineering Mechanics Division, 90, 293-319.
159. Suresh S and Mortensen A 1998 Fundamentals of Functional Graded Materials. IOM Communications Limited, London.
160. Tomota Y, Kuroki K, Mori T, and Tamura I 1976 Tensile deformation of two-ductile-phase alloys: flow curves of α-γ Fe-Cr-Ni alloys. Materials Science and Engineering, 24(1), 85-94.
161. Wakashima HT and Tsukamoto H 1990 Micromechanical approach to the thermomechanics of ceramic-metal gradient materials. In: 1st Symposium on Functionally Gradient Materials, The Functionally Gradient Materials Forum, Sendai, Japan, 19-26.
162. Watanabe R, Nishida T, and Hirai T 2003 Present status of research on design and processing of functionally graded materials. Metals and Materials International, 9(6), 513-519.
163. Zhu H and Sankar BV 2004 A combined Fourier Series-Galerkin method for the analysis of functionally graded beams. Journal of Applied Mechanics, 71(3) 421-424.
164. Zuiker JR and Dvorak GJ 1994 The effective properties of functionally graded composites - I. Extension of the Mori-Tanaka method to linearly varying fields. Composites Engineering, 4(1) 19-35.
165. Abaqus Theory Manual, version 6.5. Hibbit and Karlson and Sorensen Inc, Pawtucket, RI.
166. Bank RE 1983 The efficient implementation of local mesh refinement algorithms. In: Adaptive Computational Methods for Partial Differential Equations, 74-81. SIAM.
167. Bathe K 1996 Finite element procedure. Prentice Hall.
168. Ben Dhia H 1998 Multiscale mechanical problems: the Arlequin method. Comptes Rendus de l'Academie des Sciences Series IIB Mechanics Physics Astronomy, 326(12), 899-904.
169. Ben Dhia H 1999 Numerical modelling of multiscale problems: the Arlequin method. CD Proceedings of ECCM'99, Munich.
170. Ben Dhia H and Rateau G 2005 The Arlequin method as a flexible engineering tool. International Journal for Numerical Methods in Engineering, 62(11), 1442-1462.
171. Biscani F, Giunta G, Belouettar S, and Carrera E 2011 Variable kinematic beam elements coupled via Arlequin method. Composite Structures, 93(2), 697-708.
172. Blanco PJ, Feijoo RA, and Urquiza SA 2008 A variational approach for coupling kinematically incompatible structural models. Computer Methods in Applied Mechanics and Engineering, 197, 1577-1620.
173. Brezzi F and Marini LD 2005 The three-field formulation for elasticity problems. GAMM Mitteilungen, 28(1), 124-153.
174. Carrera E and Giunta G 2010 Refined beam theories based on a unified formulation. International Journal of Applied Mechanics, 2(1), 117-143.
175. Fish J 1992 The s-version of the finite element method. Computers & Structures, 43(3), 539-547.
176. Fish J and Markolefas S 1993 Adaptive s-method for linear elastostatics. Computational Methods in Applied Mechanics and Engineering, 103, 363-396.
177. Fish J, Pan L, Belsky V, and Gomaa S 1996 Unstructured multigrid method for shells. International Journal for Numerical Methods in Engineering, 39(7), 1181-1197.
178. Hu H, Belouettar S, Potier-Ferry M, and Daya EM 2008 Multi-scale modelling of sandwich structures using the Arlequin method Part I: Linear modelling. Finite Elements in Analysis and Design, 45(1), 37-51.
179. Hu H, Belouettar S, Potier-Ferry M, Daya EM, and Makradi A 2010 Multi-scale nonlinear modelling of sandwich structures using the Arlequin method. Composite Structures, 92(2), 515-522.
180. Möes N, Dolbow J, and Belytschko T 1999 A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1), 131-150.
181. Park KC and Felippa CA 2000 A variational principle for the formulation of partitioned structural systems. International Journal for Numerical Methods in Engineering, 47(1-3), 395-418.
182. Reddy JN 2004 Mechanics of Laminated Composite Plates and Shells. CRC Press.
183. Szabó BA and Babuška I 1992 Finite element analysis. John Wiley & Sons, Inc.
184. Carrera E and Petrolo M 2010 Guidelines and recommendations to construct theories for metallic and composite plates. AIAA Journal, 48(12), 2852-2866.
185. Carrera E and Petrolo M 2011 On the effectiveness of higher-order terms in refined beam theories. Journal of Applied Mechanics, 78(2), DOI: 10.1115/1.4002207.
186. Cicala P 1965 Systematic approximation approach to linear shell theory, Levrotto e Bella.
187. Euler L 1744 De curvis elasticis. In: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti, Bousquet.
188. Gol'denweizer AL 1962 Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity. Prikladnaya Matematika i Mekhanika 26, 1000-1025.
189. Kapania K and Raciti S 1989 Recent advances in analysis of laminated beams and plates, part I: Shear effects and buckling. AIAA Journal, 27(7), 923-935.
190. Timoshenko SP 1921 On the corrections for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41, 744-746.
191. Timoshenko SP 1922 On the transverse vibrations of bars of uniform cross section. Philosophical Magazine, 43, 125-131.
192. Washizu, W 1968 Variational methods in elasticity and plasticity. Pergamon.
193. Yu Wand Hodges DH 2005. Generalized Timoshenko theory of the variational asymptotic beam sectional analysis. Journal of the American Helicopter Society, 50(1), 46-55.