A weak form quadrature element method for nonlinear free vibrations of Timoshenko beams

2016 ◽  
Vol 33 (1) ◽  
pp. 274-287 ◽  
Author(s):  
Minmao Liao ◽  
Hongzhi Zhong
Keyword(s):  
Slenderness Ratio ◽  
Weak Form ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Radius Of Gyration ◽  
Element Formulation ◽  
Timoshenko Beams ◽  
Content Type ◽  
Quadrature Element Method ◽  
Element Method

Purpose – The purpose of this paper is to highlight the implementation of a recently developed weak form quadrature element method for nonlinear free vibrations of Timoshenko beams subjected to three different boundary conditions. Design/methodology/approach – The design of the paper is based on considering the geometrically nonlinear effects of axial strain, bending curvature, and shear strain. Then the quadrature element formulation of the beam is introduced. Findings – The efficiency of the method is demonstrated by a convergence study. Ratios of the nonlinear fundamental frequencies to the corresponding linear frequencies are extracted. Their variations with the ratio of amplitude to radius of gyration and the slenderness ratio are examined. The effects of the nonlinearity on higher order frequencies and mode shapes are also investigated. Originality/value – The computed results show fast convergence and compare well with available literature results.

Application of a Weak Form Quadrature Element Method to Nonlinear Free Vibrations of Thin Rectangular Plates

2016 ◽  
Vol 16 (01) ◽  
pp. 1640001 ◽  
Author(s):  
Minmao Liao ◽  
Hongzhi Zhong
Keyword(s):  
Plate Theory ◽  
Weak Form ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Integration Scheme ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Quadrature Element Method ◽  
Numerical Integration Scheme ◽  
Element Method

Geometrically nonlinear free vibrations of thin rectangular plates are studied using the recently developed weak form quadrature element method (QEM). The nonlinear von Karman plate theory is employed to express the strain-displacement relations. The weak form description of the plate is formulated on the basis of the variational principle. The integrals involved in the variational description are evaluated by an efficient numerical integration scheme, and the partial derivatives at the integration points are approximated by differential quadrature analogs. A system of algebraic equations is eventually derived, and the nonlinear frequencies and mode shapes are extracted from solving the equations. The efficiency of the method is demonstrated by a convergence study. The accuracy of the method is illustrated by comparing the computed nonlinear to linear frequency ratios with those available in the literature. The influences of the nonlinearity on higher order frequencies and mode shapes are exhibited as well.

scholarly journals Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Zhiqiang Shen ◽  
Hongzhi Zhong
Keyword(s):  
Degrees Of Freedom ◽  
Vibrational Analysis ◽  
Weak Form ◽  
Composite Beams ◽  
Free Vibrations ◽  
Computational Accuracy ◽  
Direct Evaluation ◽  
Quadrature Element Method ◽  
Partially Composite ◽  
Element Method

Deformation of partially composite beams under distributed loading and free vibrations of partially composite beams under various boundary conditions are examined in this paper. The weak-form quadrature element method, which is characterized by direct evaluation of the integrals involved in the variational description of a problem, is used. One quadrature element is normally sufficient for a partially composite beam regardless of the magnitude of the shear connection stiffness. The number of integration points in a quadrature element is adjustable in accordance with convergence requirement. Results are compared with those of various finite element formulations. It is shown that the weak form quadrature element solution for partially composite beams is free of slip locking, and high computational accuracy is achieved with smaller number of degrees of freedom. Besides, it is found that longitudinal inertia of motion cannot be simply neglected in assessment of dynamic behavior of partially composite beams.

Earthquake response of linear-elastic arch-frames using exact curved beam formulations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Baran Bozyigit
Keyword(s):  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Opening Angle ◽  
Earthquake Response ◽  
Response Analysis ◽  
Curved Beam ◽  
Curved Beams ◽  
Base Shear ◽  
Content Type

PurposeThis study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.Design/methodology/approachThe dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.FindingsThe displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.Originality/valueThe earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

Stability Analysis of Cylindrical Shells Resting on Winkler Elastic Foundation Using Semi-Analytical Quadrature Element Method

2019 ◽  
Vol 821 ◽  
pp. 459-464
Author(s):  
Qi Gao Hu ◽  
Xu Dong Hu ◽  
Zhi Qiang Shen ◽  
Liang Yun Tao ◽  
Ze Tan
Keyword(s):  
Elastic Foundation ◽  
Cylindrical Shells ◽  
Longitudinal Direction ◽  
External Pressure ◽  
Element Formulation ◽  
Quadrature Element Method ◽  
Advanced Composite Materials ◽  
Winkler Elastic Foundation ◽  
Parametric Studies ◽  
Element Method

The buried pipelines or vessels and other similar structures made of homogeneous or advanced composite materials are commonly used in civil engineering and biotechnology. The radial stability problem of these structures was widely studied using the cylindrical shell model over the past years. In this paper, the linear stability of cylindrical shells resting on Winkler elastic foundation under uniformly distributed external pressure was analyzed with semi-analytical quadrature element method (QEM). As for the longitudinal direction, the radial deflection of shell was approximated by the quadrature element formulation. While the analytic trigonometric function was adopted for description of radial deflection in circumferential direction. The Numerical results of critical buckling load were compared with the semi-analytical FEM. It is found that the semi-analytical QEM possesses higher computational efficiency and applicability than semi-analytical FEM. Then, the effects of the shell length, radius, and thickness on the critical buckling pressures are systematically investigated through the parametric studies.

scholarly journals Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983445
Author(s):  
Ma’en S Sari ◽  
Wael G Al-Kouz ◽  
Rafat Al-Waked
Keyword(s):  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Engineering Structures ◽  
Axial Forces ◽  
Double Beam ◽  
Tensile Forces ◽  
Torsional Coupling ◽  
The Stability

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

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