Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey

Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature

Reduced order modeling for nonlinear structural analysis using Gaussian process regression

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Karman assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

A mixed ritz-dq method for forced vibration of functionally graded beams carrying moving loads

A weak form quadrature element method for plane elasticity problems

A weak-form quadrature element method is presented to study the flexural vibrations of an eccentric annular Mindlin plate. Typical combinations of boundary conditions are considered and the natural frequencies are obtained for both thin and moderately thick plates. All results are verified using the commercial computer code ANSYS. Excellent agreement is reached in all cases. Comparison of the present predictions with other available results for thin plates is also made.

Flexural vibration analysis of an eccentric annular Mindlin plate

Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load

The recently proposed weak form quadrature element method (QEM) is applied to flexural and vibrational analysis of thin plates. The integrals involved in the variational description of a thin plate are evaluated by an efficient numerical scheme and the partial derivatives at the integration sampling points are then approximated using differential quadrature analogs. Neither the grid pattern nor the number of nodes is fixed, being adjustable according to convergence need. The C1 continuity conditions characterizing the thin plate theory are discussed and the robustness of the weak form quadrature element for thin plates against shape distortion is examined. Examples are presented and comparisons with analytical solutions and the results of the finite element method are made to demonstrate the convergence and computational efficiency of the weak form quadrature element method. It is shown that the present formulation is applicable to thin plates with varying thickness as well as uniform plates.

Analysis of thin plates by the weak form quadrature element method

This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.

Hongzhi Zhong

Papers

A weak form quadrature element method for plane elasticity problems

Flexural vibration analysis of an eccentric annular Mindlin plate

Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load

Analysis of thin plates by the weak form quadrature element method

Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method