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A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free Vibration of Isotropic and Anisotropic Beams and Plates

A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free... The first goal of this work is to present a literature review regarding the use of several sets of admissible functions in the Ritz method. The papers reviewed deal mainly with the analysis of buckling and free vibration of isotropic and anisotropic beams and plates. Theoretically, in order to obtain a correct solution, the set of admissible functions must not violate the essential or geometric boundary conditions and should also be linearly independent and complete. However, in practice, some of the sets of functions proposed in the literature present a bad numerical behavior, namely in terms of convergence, computational time and stability. Thus, a second goal of the present work is to compare the performance of several sets of functions in terms of these three features. To achieve this objective, the free vibration analysis of a fully clamped rectangular plate is carried out using six different sets of functions, along with the study of the convergence of natural frequencies and mode shapes, the computational time and the numerical stability. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Computational Methods in Engineering Springer Journals

A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free Vibration of Isotropic and Anisotropic Beams and Plates

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Publisher
Springer Journals
Copyright
Copyright © 2017 by CIMNE, Barcelona, Spain
Subject
Engineering; Mathematical and Computational Engineering
ISSN
1134-3060
eISSN
1886-1784
DOI
10.1007/s11831-017-9214-7
Publisher site
See Article on Publisher Site

Abstract

The first goal of this work is to present a literature review regarding the use of several sets of admissible functions in the Ritz method. The papers reviewed deal mainly with the analysis of buckling and free vibration of isotropic and anisotropic beams and plates. Theoretically, in order to obtain a correct solution, the set of admissible functions must not violate the essential or geometric boundary conditions and should also be linearly independent and complete. However, in practice, some of the sets of functions proposed in the literature present a bad numerical behavior, namely in terms of convergence, computational time and stability. Thus, a second goal of the present work is to compare the performance of several sets of functions in terms of these three features. To achieve this objective, the free vibration analysis of a fully clamped rectangular plate is carried out using six different sets of functions, along with the study of the convergence of natural frequencies and mode shapes, the computational time and the numerical stability.

Journal

Archives of Computational Methods in Engineering – Springer Journals

Published: Mar 24, 2017

References

  • Continuous crack modeling in piezoelectrically driven vibrations of an Euler-Bernoulli beam
    Afshari, M; Inman, DJ
  • Analytical treatment of the free vibration of single-walled carbon nanotubes based on the nonlocal Flugge shell theory
    Ansari, R; Rouhi, H
  • Nonlocal Flugge shell model for thermal buckling of multi-walled carbon nanotubes with layerwise boundary conditions
    Ansari, R; Rouhi, H
  • Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh-Ritz technique
    Ansari, R; Sahmani, S; Rouhi, H
  • Rayleigh-Ritz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions
    Ansari, R; Sahmani, S; Rouhi, H
  • Ritz-based static analysis method for fiber reinforced plastic rib core skew bridge superstructure
    Aref, AJ; Alampalli, S; He, Y
  • Effects of exponential volume fraction law on the natural frequencies of FGM cylindrical shells under various boundary conditions
    Arshad, SH; Naeem, MN; Sultana, N; Iqbal, Z; Shah, AG
  • Analysis of anisotropic plates II
    Ashton, JE
  • Natural modes of free-free anisotropic plates
    Ashton, JE
  • Anisotropic plate analysis-boundary conditions
    Ashton, JE
  • The natural modes of vibration of boron-epoxy plates
    Ashton, JE; Anderson, JD
  • Analysis of anisotropic plates
    Ashton, JE; Waddoups, ME
  • Coupled vibrations of a partially fluid-filled cylindrical container with an internal body including the effect of free surface waves
    Askari, E; Daneshmand, F; Amabili, M
  • Free vibration analysis of a hanged clamped-free cylindrical shell partially submerged in fluid: the effect of external wall, internal shaft, and flat bottom
    Bae, CH; Kwak, MK; Koo, JR
  • Natural frequencies of thin, rectangular plates with holes or orthotropic “patches” carrying an elastically mounted mass
    Bambill, DV; Felix, DH; Rossit, CA
  • Buckling and lateral vibration of rectangular plates subject to inplane loads-A Ritz approach
    Bassily, SF; Dickinson, SM
  • On the use of beam functions for problems of plates involving free edges
    Bassily, SF; Dickinson, SM
  • Free vibrations of unsymmetrically laminated anisotropic plates with clamped edges
    Bert, CW; Mayberry, BL
  • Damping analysis of laminated beams and plates using the Ritz method
    Berthelot, JM
  • Damping analysis of unidirectional glass and Kevlar fibre composites
    Berthelot, JM; Sefrani, Y
  • Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method
    Bhat, RB
  • Plate deflections using orthogonal polynomials
    Bhat, RB
  • Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method
    Bhat, RB
  • Beam characteristic orthogonal polynomials with fractional power increments
    Bhat, RB
  • Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables
    Bhat, RB
  • Rayleigh-Ritz method with separate deflection expressions for structural segments
    Bhat, RB
  • Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness
    Bhat, RB; Laura, PAA; Gutierrez, RG; Cortinez, VH; Sanzi, HC
  • Formulas for natural frequency and mode shape
    Blevins, RD
  • Detection and monitoring of side crack in a rectangular plate using mobility
    Bose, T; Mohanty, AR
  • Free vibration of rectangular isotropic plates with and without a concentrated mass
    Chai, GB
  • Buckling of generally laminated composite plates with various edge support conditions
    Chai, GB
  • Frequency analysis of a S-C-S-C plate carrying a concentrated mass
    Chai, GB
  • Frequency analysis of rectangular isotropic plates carrying a concentrated mass
    Chai, GB
  • The effect of varying the support conditions on the buckling of laminated composite plates
    Chai, GB; Khong, PW
  • Tension effects on the natural frequencies of centre-loaded clamped beams
    Chai, GB; Low, KH; Lim, TM
  • Recent research on vibration of structures using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method
    Chakraverty, S; Bhat, RB; Stiharu, I
  • The free vibrations of tapered rectangular plates using a new set of beam functions with the Rayleigh-Ritz method
    Cheung, KY; Zhou, D
  • Eigenfrequencies of tapered rectangular plates with intermediate line supports
    Cheung, YK; Zhou, D
  • The free vibrations of rectangular composite plates with point-supports using static beam functions
    Cheung, YK; Zhou, D
  • Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions
    Cheung, YK; Zhou, D
  • Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints
    Cheung, YK; Zhou, D
  • Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports
    Cheung, YK; Zhou, D
  • Vibration analysis of symmetrically laminated rectangular plates with intermediate line supports
    Cheung, YK; Zhou, D
  • Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions
    Cheung, YK; Zhou, D
  • Vibration characteristics of a cantilever plate with attached spring-mass system
    Chiba, M; Sugimoto, T
  • Approximate study of the free vibrations of a cantilever anisotropic plate carrying a concentrated mass
    Ciancio, PM; Rossit, CA; Laura, PAA
  • Variational methods for the solution of problems of equilibrium and vibrations
    Courant, R
  • Calculation of the natural frequencies of composite plates by the Rayleigh-Ritz method with orthogonal polynomials
    Cupial, P
  • A layer-wise analysis for free vibrations of thick composite spherical panels
    Dasgupta, A; Huang, KH
  • Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh-Ritz technique
    Deobald, LR; Gibson, RF
  • The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh’s method
    Dickinson, SM
  • On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates
    Dickinson, SM; Blasio, A
  • On the use of simply supported plate functions in the Rayleigh-Ritz method applied to the flexural vibration of rectangular plates
    Dickinson, SM; Li, EKH
  • On asymptotic approximations to beam model shapes
    Dowell, EH
  • On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves
    Eringen, AC
  • On nonlocal elasticity
    Eringen, AC; Edelen, DGB
  • Rayleigh-Ritz analysis of sandwich beams
    Fasana, A; Marchesiello, S
  • Single-layer plate theories applied to the flexural vibration of completely free thick laminates
    Frederiksen, PS
  • Modal analysis of delaminated composite plates using the finite element method and damage detection via combined Ritz/2D-wavelet analysis
    Gallego, A; Moreno-García, P; Casanova, CF
  • Chladni figures and the Tacoma bridge: motivating PDE eigenvalue problems via vibrating plates
    Gander, M; Kwok, F
  • From Euler, Ritz, and Galerkin to modern computing
    Gander, M; Wanner, G
  • Hierarchical Bernoulli-Euler beam finite elements
    Ganesan, N; Engels, RC
  • Natural frequencies and biplanar response of generalized rotating spindle systems
    Gartner, JR; Cobb, EC
  • Improved numerical computation of uniform beam characteristic values and characteristic functions
    Gartner, JR; Olgac, N
  • Natural frequencies of edge restrained tapered rectangular plates
    Grossi, RO; Bhat, RB
  • The frequency of flexural vibration of rectangular orthotropic plates with clamped or supported edges
    Hearmon, RFS
  • Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads
    Hosseini-Hashemi, S; Karimi, M; Rokni, DH
  • Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method
    Hosseini-Hashemi, S; Karimi, M; Taher, HRD
  • Vibration of angle-ply laminated plates with twist by Rayleigh-Ritz procedure
    Hu, XX; Sakiyama, T; Lim, CW; Xiong, Y; Matsuda, H; Morita, C
  • Vibration analysis of rectangular plates with side cracks via the Ritz method
    Huang, CS; Leissa, AW
  • Vibrations of rectangular plates with internal cracks or slits
    Huang, CS; Leissa, AW; Chan, CW
  • Accurate vibration analysis of thick, cracked rectangular plates
    Huang, CS; Leissa, AW; Li, RS
  • Vibrations of cracked rectangular FGM thick plates
    Huang, CS; McGee, OG; Chang, MJ
  • Three-dimensional vibration analyses of functionally graded material rectangular plates with through internal cracks
    Huang, CS; Yang, PJ; Chang, MJ
  • Three-dimensional vibrations of cracked rectangular parallelepipeds of functionally graded material
    Huang, CS; McGee, OG; Wang, KP
  • Dynamics, load balancing, and modal control of piezoelectric tube actuators
    Hulzen, JR; Schitter, G; Hof, PMJV; Eijk, J
  • Comments on “The historical bases of the Rayleigh and Ritz methods”
    Ilanko, S
  • Material characterization for orthotropic shells using modal analysis and Rayleigh-Ritz models
    Ip, KH; Tse, PC; Lai, TC
  • Frequency analysis of multiple layered cylindrical shells under lateral pressure with asymmetric boundary conditions
    Isvandzibaei, MR; Jamaluddin, H; Raja Hamzah, R
  • Free vibration of multiple rectangular plates coupled with a liquid
    Jeong, KH; Kang, HS
  • The Rayleigh-Ritz solution to estimate vibration characteristics of building floors
    Kato, Y; Honma, T
  • On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method
    Kim, CS; Young, PG; Dickinson, SM
  • Temperature dependent vibration analysis of functionally graded rectangular plates
    Kim, YW
  • Buckling of rectangular orthotropic plates subjected to biaxial normal forces
    Kollar, LP; Veres, IA
  • Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory
    Kong, S
  • Free vibration analysis of a hung clamped-free cylindrical shell partially submerged in fluid
    Kwak, MK; Koo, JR; Bae, CH
  • Parameter estimation of orthotropic plates by bayesian sensitivity analysis
    Lai, TC; Ip, KH
  • Natural frequencies of rectangular stepped plates using polynomial beam functions with subsectioning
    Lam, KY; Amrutharaj, G
  • Analysis of clamped laminated plates subjected to conventional blast
    Lam, KY; Chun, L
  • Vibration study on plates with stiffened openings using orthogonal polynomials and partitioning method
    Lam, KY; Hung, KC
  • Effects of boundary conditions on frequencies of a multi-layered cylindrical shell
    Lam, KY; Loy, CT
  • Vibration analysis of plates with cutouts by the modified Rayleigh-Ritz method
    Lam, KY; Hung, KC; Chow, ST
  • The natural frequency of elastic plates with void by Ritz-method
    Lee, D-C; Wang, C-S; Lee, L-T
  • Free vibration of rectangular plates on elastic point supports with the application of a new type of admissible function
    Lee, LT; Lee, DC
  • The free vibration of rectangular plates
    Leissa, AW
  • The historical bases of the Rayleigh and Ritz methods
    Leissa, AW
  • Convergence of the Ritz method
    Leissa, AW; Shihada, SM
  • Integration of beam functions
    Leung, AYT
  • Recurrent integration of beam functions
    Leung, AYT
  • Method of domain decomposition in vibrations of mixed edge anisotropic plates
    Liew, KM; Hung, KC; Lim, MK
  • Roles of domain decomposition method in plate vibrations: Treatment of mixed discontinuous periphery boundaries
    Liew, KM; Hung, KC; Lim, MK
  • On the use of the domain decomposition method for vibration of symmetric laminates having discontinuities at the same edge
    Liew, KM; Hung, KC; Lim, MK
  • Vibration of Mindlin plates using boundary characteristic orthogonal polynomials
    Liew, KM; Hung, KC; Lim, MK
  • Rayleigh-Ritz vibration analysis of thick plates by a simple higher order theory
    Lim, SP; Senthilnathan, NR; Lee, KH
  • XXIV. On the calculation of Chladni’s figures for a square plate
    Rayleigh, L
  • Experimental and analytical study of the frequencies of an S-C-S-C plate carrying a concentrated mass
    Low, KH; Chai, GB; Ng, CK
  • Experimental and analytical investigations of vibration frequencies for centre-loaded beams
    Low, KH; Lim, TM; Chai, GB
  • A comparative study of vibrating loaded plates between the Rayleigh-Ritz and experimental methods
    Low, KH; Chai, GB; Tan, GS
  • Comparisons of experimental and theoretical frequencies for rectangular plates with various boundary conditions and added masses
    Low, KH; Chai, GB; Lim, TM; Sue, SC
  • Vibration of cylindrical shells with ring support
    Loy, CT; Lam, KY
  • Modal vibration damping of anisotropic FRP laminates using the Rayleigh-Ritz energy minimization scheme
    Maheri, MR; Adams, RD
  • Bending of an elastic rectangular clamped plate: exact versus ‘engineering’ solutions
    Meleshko, VV
  • Ritz-type dynamic analysis of cross-ply laminated circular cylinders subjected to different boundary conditions
    Messina, A; Soldatos, KP
  • Further simplest-expression integrals involving beam eigenfunctions and derivatives
    Morales, CA; Ramírez, JF
  • Vibration tests in CFRP plates for damage detection via non-parametric signal analysis
    Moreno-García, P; Castro, E; Romo-Melo, L; Gallego, A; Roldán, A
  • A new technique to optimize the use of mode shape derivatives to localize damage in laminated composite plates
    Moreno-García, P; Araújo dos Santos, JV; Lopes, H
  • Boundary conditioning technique for structural tuning
    Muthukumaran, P; Bhat, R; Stiharu, I
  • Boundary conditioning for structural tuning using fuzzy logic approach
    Muthukumaran, P; Demirli, K; Stiharu, I; Bhat, R
  • On the use of orthogonal polynomials in the study of anisotropic plates
    Nallim, LG; Grossi, RO
  • An analytical-numerical approach to simulate the dynamic behaviour of arbitrarily laminated composite plates
    Nallim, LG; Oller, S
  • Statical and dynamical behaviour of thin fibre reinforced composite laminates with different shapes
    Nallim, LG; Martinez, SO; Grossi, RO
  • Accurate calculation methods for natural frequencies of plates with special attention to the higher modes
    Oosterhout, G; Hoogt, P; Spiering, R
  • Buckling of anisotropic composite plates under stress gradient
    Pandey, MD; Sherbourne, AN
  • Stability analysis of inhomogeneous, fibrous composite plates
    Pandey, MD; Sherbourne, AN
  • Contour-plot simulation of vibrational and buckling mode shapes of composite plates
    Pao, YC; Peterson, KA
  • Three-dimensional analytical modeling of vibration behavior of piezoceramic cylindrical shells
    Parashar, SK; Kumar, A
  • Structural-acoustic model of a rectangular plate-cavity system with an attached distributed mass and internal sound source: theory and experiment
    Pirnat, M; Cepon, G; Boltezar, M
  • Natural frequencies of uniform and non-uniform rectangular cantilever plates
    Plunkett, R
  • A variational Ritz formulation for vibration analysis of thick quadrilateral laminated plates
    Rango, RF; Bellomo, FJ; Nallim, LG
  • A simple higher-order theory for laminated composite plates
    Reddy, JN
  • Boundary characterization of microstructures through thermo-mechanical testing
    Rinaldi, G; Packirisamy, M; Stiharu, I
  • Quantitative boundary support characterization for cantilever MEMS
    Rinaldi, G; Packirisamy, M; Stiharu, I
  • Boundary characterization of MEMS structures through electro-mechanical testing
    Rinaldi, G; Packirisamy, M; Stiharu, I
  • Theorie der transversalschwingungen einer quadratische platte mit freien rändern
    Ritz, W
  • Über eine neue methode zur lösung gewisser variationsprobleme der mathematischen physik
    Ritz, W
  • Rayleigh-Ritz vibrational analysis of multiwalled carbon nanotubes based on the nonlocal Flugge shell theory
    Rouhi, H; Bazdid-Vahdati, M; Ansari, R
  • Free vibration and buckling analysis of beams with a modified couple-stress theory
    Santos, JVA; Reddy, JN
  • On the benefits of exposing mathematics majors to the Rayleigh-Ritz procedure
    Savoye, P
  • Calculation of integrals involving characteristic beam functions
    Sharma, CB
  • Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh-Ritz method
    Smith, ST; Bradford, MA; Oehlers, DJ
  • Vibration studies of cross-ply laminated shear deformable circular cylinders on the basis of orthogonal polynomials
    Soldatos, KP; Messina, A
  • The influence of boundary conditions and transverse shear on the vibration of angle-ply laminated plates, circular cylinders and cylindrical panels
    Soldatos, KP; Messina, A
  • Vibration analyses of symmetrically laminated composite cylindrical shells with arbitrary boundaries conditions via rayleigh-ritz method
    Song, X; Han, Q; Zhai, J
  • Traveling wave analysis of rotating cross-ply laminated cylindrical shells with arbitrary boundaries conditions via rayleigh-ritz method
    Song, X; Zhai, J; Chen, Y; Han, Q
  • Vibration studies of rotating cylindrical shells with arbitrary edges using characteristic orthogonal polynomials in the rayleigh-ritz method
    Sun, S; Cao, D; Han, Q
  • Vibration problems in engineering
    Timoshenko, S
  • Theory of plates and shells
    Timoshenko, S; Woinowsky-Krieger, S
  • The vibration of rectangular plates
    Warburton, GB
  • Vibrations of rectangular plates with elastically restrained edges
    Warburton, GB; Edney, SL
  • Richard Courant and the finite element method: a further look
    Williamson, F
  • Couple stress based strain gradient theory for elasticity
    Yang, F; Chong, ACM; Lam, DCC; Tong, P
  • Kirchhoff plate theory-based electromechanically-coupled analytical model considering inertia and stiffness effects of a surface-bonded piezoelectric patch
    Yoon, H; Youn, BD; Kim, HS
  • Vibration of rectangular plates by the Ritz method
    Young, D
  • Application of a three-dimensional shell theory to the free vibration of shells arbitrarily deep in one direction
    Young, PG
  • On the free flexural vibration of rectangular plates with straight or curved internal line supports
    Young, PG; Dickinson, SM
  • The application of a type of new admissible function to the vibration of rectangular plates
    Zhou, D
  • Natural frequencies of elastically restrained rectangular plates using a set of static beam functions in the Rayleigh-Ritz method
    Zhou, D
  • Natural frequencies of rectangular plates using a set of static beam functions in Rayleigh-Ritz method
    Zhou, D
  • Vibrations of Mindlin rectangular plates with elastically restrained edges using static Timoshenko beam functions with the Rayleigh-Ritz method
    Zhou, D
  • Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions
    Zhou, D
  • The free vibration of a type of tapered beams
    Zhou, D; Cheung, YK
  • Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions
    Zhou, D; Cheung, YK
  • Vibration analysis of rectangular Mindlin plates with internal line supports using static Timoshenko beam functions
    Zhou, D; Lo, SH; Au, FTK; Cheung, YK