The Rayleigh–Ritz method is a classical method that has been widely used to investigate dynamic, static and buckling behavior, i.e., the natural frequencies, mode shapes, moments, stresses, critical buckling loads of vibrating structures and to solve boundary value problems. Different developments in the method have taken place from time to time. This paper presents a comprehensive literature review on the application of the Rayleigh–Ritz method to analyze vibration, static and buckling characteristics of beams, shells and plates using different theories.
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2.
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3.
AliRAtwalSJ (1980) Predictions of natural frequencies of vibration of rectangular plates with rectangular cut-outs. Computers and Structures12: 819–823.
4.
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5.
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7.
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8.
AnjomshoaA (2013) Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory. Meccanica48: 1337–1353.
9.
ArshadSHNaeemMNSultanaN (2007) Frequency analysis of functionally graded material cylindrical shells with various volume fraction laws. Journal of Mechanical Engineering Science221 Part C: 1483–1495.
10.
AvalosDLarrondoHLauraPAA (1989) Free and forced vibrations of a simply supported a rectangular plate partially embedded in an elastic foundation. Applied Acoustics28: 295–300.
11.
AvalosDRLarrondoHALauraPAA (1999) Analysis of vibrating rectangular anisotropic plate with free edge holes. Journal of Sound and Vibration222: 691–695.
12.
BambillDVLauraPAABergmannAet al. (1991) Fundamental frequency of transverse vibration of symmetrically stepped simply supported rectangular plates. Journal of Sound and Vibration150(1): 167–169.
13.
BeheraLChakravertyS (2014a) Effect of scaling effect parameters on the vibration characteristics of nanoplates. Journal of Vibration and Control. Epub ahead of print 3 September 2014. DOI: 10.1177/1077546314547376.
14.
BeheraLChakravertyS (2014b) Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials. Applied Nanoscience4: 347–358.
15.
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16.
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17.
BhardwajNGuptaAPChoongKK (2006) Vibration of rectangular orthotropic quarter elliptic plates with simply-supported curved boundary and other complicated effects. Tamkang Journal of Science and Engineering9(1): 1–18.
18.
BhardwajNGuptaAPChoongKK (2007) Effect of elastic foundation on the vibration of rectangular orthotropic elliptic plates with varying thickness. Meccanica42: 341–358.
19.
BhardwajNGuptaAPChoongKK (2008) Asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness with same boundary conditions. Shock and Vibration15: 599–617.
20.
BhardwajNGuptaAPChoongKKet al. (2012) Transverse vibrations of clamped and simply-supported circular plates with two dimensional thickness variation. Shock and Vibration19: 273–285.
21.
BhatRB (1985a) Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh–Ritz method. Journal of Sound and Vibration102: 493–499.
22.
BhatRB (1985b) Plate deflections using orthogonal polynomials. Journal of Engineering Mechanics111: 1301–1309.
23.
BhatRB (1986) Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh–Ritz method. Journal of Sound and Vibration105(2): 199–210.
24.
BhatRB (1987) Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables. Journal of Sound and Vibration114: 65–71.
25.
BhatRB (1996) Effect of normal mode contents in assumed deflection shapes in Rayleigh–Ritz method. Journal of Sound and Vibration189(3): 407–419.
26.
BhatRB (2015) Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions. Advances in Mechanical Engineering. Epub ahead of print 17 April 2015. DOI: 10.1177/1687814015578355.
27.
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28.
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29.
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30.
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31.
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32.
ChakravertySBeheraL (2014) Free vibration of rectangular nanoplates using Rayleigh–Ritz method. Physica E: Low-dimensional Systems and Nanostructures56: 357–363.
33.
ChakravertySPetytM (1997) Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials. Applied Mathematical Modelling21: 399–417.
34.
ChakravertySPetytM (1999) Free vibration analysis of elliptic and circular plates having rectangular orthotropy. Structural Engineering Mechanics7: 53–67.
35.
ChakravertySPradhanKK (2014) Free vibration of functionally graded thin rectangular plates resting on Winkler elastic foundation with general boundary conditions using Rayleigh–Ritz method. International Journal of Applied Mechanics. Epub ahead of print 24 June 2014. DOI: 10.1142/S1758825114500434.
36.
ChakravertySBhatRBStiharuI (2001) Free vibration of annular elliptic plates using boundary characteristic orthogonal polynomials as shape functions in the Rayleigh–Ritz method. Journal of Sound and vibration241(3): 524–539.
37.
ChakravertySJindalRAgarwalVK (2007a) Effect of non-homogeneity on natural frequencies of vibration of elliptic plates. Meccanica42(6): 585–599.
38.
ChakravertySJindalRAgarwalVK (2007b) Vibration of nonhomogeneous orthotropic elliptic and circular plates with variable thickness. Journal of Vibration and Acoustics129: 256–259.
39.
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40.
ChenHSChenYYangJ (2003) Bending and vibration characteristics of a strengthened plate under various boundary conditions. Engineering Structures25: 1157–1168.
41.
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42.
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43.
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44.
CupicalP (1997) Calculation of the natural frequencies of composite plates by the Rayleigh–Ritz method with orthogonal polynomials. Journal of Sound and Vibration201(3): 385–387.
45.
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46.
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47.
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48.
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49.
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50.
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51.
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52.
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53.
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54.
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55.
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56.
FasanaAMearchesielloS (2001) Rayleigh–Ritz analysis of sandwich beams. Journal of Sound and Vibration241(1): 643–652.
57.
GeannakakesGN (1995) natural frequencies of arbitrarily shaped plates using the Rayleigh–Ritz method together with natural co-ordinate regions and normalized characteristic orthogonal polynomials. Journal of Sound and Vibration182(3): 441–478.
58.
GrossiROBhatRB (1995) Natural frequencies of edge restrained tapered rectangular plates. Journal of Sound and Vibration185(2): 335–343.
59.
GuptaAKKaurH (2008) Study of the effect of thermal gradient on free vibration of clamped visco-elastic rectangular plate with linearly thickness variation in both directions. Meccanica43(4): 449–458.
60.
GuptaAKKhannaA (2007) Vibration of visco-elastic rectangular plate with linearly varying thickness in both directions. Journal of Sound and Vibration301: 450–457.
61.
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62.
GuptaAKKumarL (2009) Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic of variable thickness. Meccanica44: 507–518.
63.
GuptaAPBhardwajN (2004) Vibration of rectangular orthotropic elliptic plates of quadratically varying thickness resting on elastic foundation. Journal of Vibration and Acoustics126: 132–140.
64.
GuptaAPBhardwajN (2005) Free vibration of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundation. Applied Mathematical Modelling29: 137–157.
65.
Hashemi ShHTaherHRDOmediM (2008) 3D free vibration analysis of annular plates on Pasternak elastic foundation via p-Ritz method. Journal of Sound and Vibration311: 114–1140.
66.
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67.
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68.
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69.
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70.
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75.
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78.
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81.
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83.
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91.
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92.
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93.
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95.
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97.
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98.
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100.
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104.
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105.
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106.
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107.
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108.
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109.
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110.
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111.
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112.
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113.
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114.
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115.
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116.
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117.
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118.
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119.
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120.
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121.
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122.
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123.
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124.
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125.
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126.
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127.
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128.
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129.
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130.
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131.
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132.
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135.
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136.
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137.
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138.
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139.
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141.
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142.
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169.
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170.
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171.
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