An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Banerjee, J.R.; Ananthapuvirajah, A.

doi:10.1016/j.ijmecsci.2018.10.012

Cited by 39 publications

(38 citation statements)

References 28 publications

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“…As seen in Table 1, the results from the present theory agree well with that provided by Banerjee and Ananthapuvirajah (2019). For this analysis, a reduced model is obtained by taking θ ¼ 0 and making use of only K a for the stiffness matrix and M a and M li for the mass matrix.…”

Section: Validation Studies (Classical Theories)supporting

confidence: 84%

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Mustapha

2020

Journal of Vibration and Control

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A model is proposed for investigating the size-dependent frequency response of arbitrarily oriented microscale frames used in the build-up of lattice structures with micro unit cells. The model employs the Rayleigh–Love, the Timoshenko and the modified couple stress theories to overcome the weaknesses of the conventional theories. Descriptions of the model and finite element implementation are presented. Predictions from the reduced forms of the model agree with published results. The frequency analyses of different microscale frames reveal the influence of material lengthscale, dead weight, lateral inertia and orientation angles. For small aspect ratios, neglecting the lateral inertia effect incurs a substantial error in predicting the frequencies of higher modes, but only marginally affects the lower modes. The resonant frequencies exhibit a sharp drop in the presence of dead weight, but it increases in the presence of material lengthscale.

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Section: Validation Studies (Classical Theories)supporting

confidence: 84%

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Mustapha

2020

Journal of Vibration and Control

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“…In the following, only the equations of motion of the curved beam are presented. Straight Timoshenko beam formulations used here (which will be used later to describe the columns and arch segments) can be found in the literature [10]. The equation of motion of a curved beam considering axial and shear deformation and rotational inertia is written as follows [6]:…”

Section: Model and Formulationsmentioning

confidence: 99%

Free vibration analysis of arch-frames using the dynamic stiffness approach

Bozyiğit¹,

Yeşilce²,

Acikgoz³

2020

Vib. proced.

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The aim of this study is to investigate free vibration characteristics of arch-frames which consist of two columns and an arch. Firstly, an exact formulation of the problem is presented using the Dynamic Stiffness Method (DSM). The end forces and displacements of column elements are obtained analytically using Timoshenko beam theory (TBT). These are then combined with the end forces and displacements of the semi-circular arch, which is modeled with exact curved beam elements that consider axial and shear deformations and rotational inertia. By employing standard assembly and bisection based root finding procedures, exact free vibration analysis of the whole vibrating system is carried out. Then, in an effort to simplify the formulations, an approach based on approximating the arch as assembly of linear straight beam segments is presented. The calculated natural frequencies using DSM for both exact and approximate results are then tabulated for comparison purposes. The mode shapes are also compared. The results show that the proposed model simplification is effective and produces accurate mode frequency and shape estimations.

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“…However, these theories are rarely implemented in the stress analysis of laminated structures as they cannot capture in-plane and transverse stresses and strains of highly heterogeneous laminates [3] with sufficient accuracy. Rather, they are used as a starting point to develop higher-order theories and equivalent single layer zigzag (ZZ) models [4][5][6][7][8][9][10][11][12][13][14][15][16] or in analysing the vibration and buckling behaviours of such structures [17][18][19]. They are also employed to develop couple-stress based models, which can be used to predict the size effect from sub-scale components, i.e.…”

Section: Introductionmentioning

confidence: 99%

A strain-displacement mixed formulation based on the modified couple stress theory for the flexural behaviour of laminated beams

Trinh

Groh

Zucco

2020

Composites Part B: Engineering

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A novel strain-displacement variational formulation for the flexural behaviour of laminated composite beams is presented, which accurately predicts three-dimensional stresses, yet is computationally more efficient than 3D finite element models. A global third-order and layer-wise zigzag profile is assumed for the axial deformation field through the laminate thickness to account for the effect of both stress-channelling and stress localisation. By using this axiomatic kinematic field, a variational formulation is developed based on an equivalent single layer theory such that the number of unknowns is independent of the number of composite layers. The axial and couple stresses are evaluated from the displacement field, while the transverse shear and transverse normal stresses are computed by the interlaminarcontinuous equilibrium conditions within the framework of the modified couple stress theory. Axial and transverse force equilibrium conditions are imposed via two Lagrange multipliers, which correspond to the axial and transverse displacements. Using this mixed variational approach, both displacements and strains are treated as unknown quantities, resulting in more functional freedom to minimise the total strain energy. In this work, two strain-displacementbased models are developed to investigate the effect of couple stress on the flexural behaviour of composite beams. The first model neglects the presence of couple stress, whilst the second includes couple stress with an additional unknown curvature. The differential quadrature method is used to solve the resulting governing and boundary equations for simply-supported and clamped laminated beams. For the simply-supported case, numerical results from this variational formulation for both models agree well with those from a Hellinger-Reissner stressdisplacement mixed model found in the literature and the 3D elasticity solution given by Pagano. For the clamped laminate, the additional curvature associated with the couple stress plays an important role in accurately predicting stresses, which is confirmed by a high-fidelity 3D finite element model.

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An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Cited by 39 publications

References 28 publications

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Free vibration analysis of arch-frames using the dynamic stiffness approach

A strain-displacement mixed formulation based on the modified couple stress theory for the flexural behaviour of laminated beams

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