The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses

2021 ◽  
pp. 27-34
Author(s):  
Ahmed Babahammou ◽  
Rhali Benamar
Keyword(s):  
Ritz Method ◽  
Supporting Point ◽  
Elastically Restrained

SHEAR BUCKLING OF CORRUGATED PLATES WITH EDGES ELASTICALLY RESTRAINED AGAINST ROTATION

2004 ◽  
Vol 04 (01) ◽  
pp. 89-104 ◽  
Author(s):  
CHAWALIT MACHIMDAMRONG ◽  
EIICHI WATANABE ◽  
TOMOAKI UTSUNOMIYA
Keyword(s):  
Ritz Method ◽  
Transition Curve ◽  
Mindlin Plate ◽  
Box Girders ◽  
Plate Buckling ◽  
Rotational Restraint ◽  
Simple Support ◽  
Shear Buckling ◽  
Corrugated Plates ◽  
Elastically Restrained

This paper presents an estimation of the elastic shear buckling capacity of corrugated plates with edges elastically restrained against rotation. The corrugated plate possesses higher shear buckling capacity compared to an unfolded flat plate. It has been used to replace the concrete web in PC box girders in recent bridge constructions in Japan. In this study, the corrugated plate is modelled as an orthotropic Mindlin plate. Elastically rotational restraint on boundary edges is taken into account in the form of rotational springs in the analysis. The prediction of buckling capacities of corrugated plates is carried out by using the Rayleigh–Ritz method, which was proved to be consistent with those as predicted by existing formulas for the limiting cases of simply-supported and clamped edges. The present study covers the more general case of elastically rotational restraint on the boundary edges showing transition curve of plate buckling capacities from the case of simple support to the case of clamped support.

Power Transmission of Two Coupled Rectangular Plates with Elastically Restrained Coupling Edge

2010 ◽  
Vol 450 ◽  
pp. 71-74 ◽  
Author(s):  
Yue Hua Chen ◽  
Guo Yong Jin ◽  
Jing Tao Du ◽  
Zhi Gang Liu
Keyword(s):  
Fourier Series ◽  
Power Transmission ◽  
Ritz Method ◽  
Internal Force ◽  
Rectangular Plates ◽  
Bending Moments ◽  
Transverse Shearing ◽  
Force Components ◽  
Elastically Restrained ◽  
Plane Shearing

A model of two coupled rectangular plates with elastically restrained coupling edge by using the Fourier series and Rayleigh-Ritz method is employed to analyze the power transmission and dynamic response, in which the flexural and in-plane vibrations are considered simultaneously. The contributions and effects of vibrational and internal force components of both flexural and in-plane vibration to the power transmission are investigated numerically. It is shown that the transverse shearing forces, twist moments and in-plane shearing forces have little influence on the power transmission, while the bending moments and the in-plane longitudinal forces play an important role in power transmission.

scholarly journals Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
María Virginia Quintana ◽  
Ricardo Oscar Grossi
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
Free Vibrations ◽  
Rectangular Domain ◽  
Internal Line ◽  
Thin Plates ◽  
Mode Shapes ◽  
Approximate Analytical Solutions ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

scholarly journals Buckling of elastically restrained nonlocal carbon nanotubes under concentrated and uniformly distributed axial loads

2019 ◽  
Vol 10 (1) ◽  
pp. 145-152
Author(s):  
Mouafo Teifouet Armand Robinson ◽  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Numerical Solutions ◽  
Ritz Method ◽  
Small Scale ◽  
Axial Loads ◽  
Weak Formulation ◽  
Compressive Loads ◽  
Method Of Solution ◽  
Elastically Restrained ◽  
Elastic Restraints

Abstract. Buckling of elastically restrained carbon nanotubes is studied subject to a combination of uniformly distributed and concentrated compressive loads. Governing equations are based on the nonlocal model of carbon nanotubes. Weak formulation of the problem is formulated and the Rayleigh quotients are obtained for distributed and concentrated axial loads. Numerical solutions are obtained by Rayleigh–Ritz method using orthogonal Chebyshev polynomials. The method of solution is verified by checking against results available in the literature. The effect of the elastic restraints on the buckling load is studied by counter plots in term of small-scale parameter and the spring constants.

Dynamic Analysis of Two Parallel Rectangular Plates Coupled with Mechanical Links

2021 ◽  
pp. 2150157
Author(s):  
Yufei Zhang ◽  
Jingtao Du
Keyword(s):  
Boundary Conditions ◽  
Dynamic Performance ◽  
Ritz Method ◽  
Forced Response ◽  
Rectangular Plates ◽  
Engineering Applications ◽  
Energy Principle ◽  
Structural Dynamic ◽  
Elastically Restrained ◽  
Plate System

The plate-type structures are classical configurations in many engineering applications. A comprehensive understanding of the structural dynamic mechanisms is of great significance. An analytical modeling approach is established and applied to investigate the dynamic characteristics of a plate system. The model encompasses two parallel elastically restrained rectangular plates coupled through mechanical links. The linear stiffness parameters are used to simulate various structural boundary conditions and mechanical links. The wave propagation of the plate structure is considered based on the improved Fourier series method. And the theoretical formulations for the dynamic performance of the plate system are obtained by employing the energy principle and Rayleigh–Ritz method. The stability and efficiency of the proposed model are firstly validated for the plate system with classical boundary conditions by comparing the results obtained from FEM software. Subsequently, the boundary restraining parameters are analyzed to figure out their effects on the modal characteristics of the plate system. In addition, the influence of mechanical link distributions on the forced response properties of the plate system is presented and discussed. Numerical results show that the importance of both the boundary conditions and the mechanical link distributions on the dynamic behavior of the plate system. The obtained results of the dynamic investigation and parametric analysis of the plate system can be useful for the further work of vibration and noise control technology of engineering applications.

EXPLICIT LOCAL BUCKLING ANALYSIS OF ROTATIONALLY RESTRAINED COMPOSITE PLATES UNDER BIAXIAL LOADING

2007 ◽  
Vol 07 (03) ◽  
pp. 487-517 ◽  
Author(s):  
PIZHONG QIAO ◽  
LUYANG SHAN
Keyword(s):  
Composite Structures ◽  
Biaxial Loading ◽  
Ritz Method ◽  
Local Buckling ◽  
Composite Plates ◽  
Biaxial Compression ◽  
Thin Walled Structures ◽  
Special Cases ◽  
Rotational Restraint ◽  
Elastically Restrained

A variational formulation of the Ritz method is used to establish an eigenvalue problem for the local buckling behavior of composite plates elastically restrained (R) along their four edges (the RRRR plates) and subjected to biaxial compression, and the explicit solution in terms of the rotational restraint stiffness (k) is presented. Based on the different boundary and loading conditions, the explicit local buckling solution for the rotationally restrained plates is simplified to several special cases (e.g. the SSSS, SSCC, CCSS, CCCC, SSRR, RRSS, CCRR, and RRCC plates) under biaxial compression (and further reduced to uniaxial compression) with a combination of simply-supported (S), clamped (C), and/or restrained (R) edge conditions. The deformation shape function is presented by using the unique harmonic functions in both the axes to account for the effect of elastic rotational restraint stiffness (k) along the four edges of the orthotropic plate. A parametric study is conducted to evaluate the influences of the loading ratio (α), the rotational restraint stiffness (k), the aspect ratio (γ), and the flexural-orthotropy parameters (α OR and β OR ) on the local buckling stress resultants of various rotationally restrained plates, and design plots with respect to these parameters are provided. The present explicit local buckling solution of the elastically restrained composite plates and the associated design plots can be employed to facilitate design analysis of composite structures (e.g. stiffened panels, thin-walled structures, and honeycomb cores).

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