• Structural Engineering and Mechanics
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• v.59 no.3
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• pp.503-526
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• 2016

In-plane and out-of-plane free vibration analysis of Timoshenko curved beams is addressed based on the isogeometric method, and an effective scheme to avoid numerical locking in both of the two patterns is proposed in this paper. The isogeometric computational model takes into account the effects of shear deformation, rotary inertia and axis extensibility of curved beams, and is applicable for uniform circular beams, and more complicated variable curvature and cross-section beams as illustrated by numerical examples. Meanwhile, it is shown that, the $C^{p-1}$-continuous NURBS elements remarkably have higher accuracy than the finite elements with the same number of degrees of freedom. Nevertheless, for in-plane or out-of-plane vibration analysis of Timoshenko curved beams, the NURBS-based isogeometric method also exhibits locking effect to some extent. To eliminate numerical locking, the selective reduced one-point integration and $\bar{B}$ projection element based on stiffness ratio is devised to achieve locking free analysis for in-plane and out-of-plane models, respectively. The suggested integral schemes for moderately slender models obtain accurate results in both dominated and non-dominated regions of locking effect. Moreover, this strategy is effective for beam structures with different slenderness. Finally, the influence factors of structural parameters of curved beams on their natural frequency are scrutinized.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2007.04a
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• pp.105-110
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• 2007

The differential equations governing free, in-plane vibrations of circular curved beams with elastic springs at beth ends, including the effects of axial deformation, rotatory inertia and shear defamation. are solved numerically using the corresponding boundary conditions. The lowest three natural frequencies are calculated over a wide range of non-dimensional system parameters, the radial, tangential and rotational spring parameters, the subtended angle, the slenderness ratio and the shear parameter.

• Structural Engineering and Mechanics
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• v.84 no.1
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• pp.77-83
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• 2022

In the last ten years, many researchers have studied the vibrations of carbon nanotubes using different beam theories. The nano- and micro-scale systems have wavy shape and there is a demand for a powerful tool to mathematically model waviness of those systems. In accordance with the above mentioned lack for the modeling of the waviness of the curved tiny structure, a novel approach is employed by implementing the Timoshenko-beam model. Owing to the small size of the micro beam, these structures are very appropriate for designing small instruments. The vibrations of double walled carbon nanotubes (DWCNTs) are developed using the Timoshenko-beam model in conjunction with the wave propagation approach under support conditions to calculate the fundamental frequencies of DWCNTs. The frequency influence is observed with different parameters. Vibrations of the double walled carbon nanotubes are investigated in order to find their vibrational modes with frequencies. The aspect ratios and half axial wave mode with small length are investigated. It is calculated that these frequencies and ratios are dependent upon the length scale and aspect ratio.

• Steel and Composite Structures
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• v.25 no.2
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• pp.141-155
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• 2017

This paper presents an investigation into the magneto-thermo-mechanical vibration and damping of a viscoelastic functionally graded-carbon nanotubes (FG-CNTs)-reinforced curved microbeam based on Timoshenko beam and strain gradient theories. The structure is surrounded by a viscoelastic medium which is simulated with spring, damper and shear elements. The effective temperature-dependent material properties of the CNTs-reinforced composite beam are obtained using the extended rule of mixture. The structure is assumed to be subjected to a longitudinal magnetic field. The governing equations of motion are derived using Hamilton's principle and solved by employing differential quadrature method (DQM). The effect of various parameter like volume percent and distribution type of CNTs, temperature change, magnetic field, boundary conditions, material length scale parameter, central angle, viscoelastic medium and structural damping on the vibration and damping behaviors of the nanocomposite curved microbeam is examined. The results show that with increasing volume percent of CNTs and considering magnetic field, material length scale parameter and viscoelastic medium, the frequency of the system increases and critically damped situation occurs at higher values of damper constant. In addition, the structure with FGX distribution type of CNTs has the highest stiffness. It is also observed that increasing temperature, structural damping and central angle of curved microbeam decreases the frequency of the system.

• Advances in nano research
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• v.7 no.4
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• pp.249-263
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• 2019

In the current paper, an exact solution method is carried out for analyzing the thermo-mechanical vibration of curved FG nano-beams subjected to uniform thermal environmental conditions, by considering porosity distribution via nonlocal strain gradient beam theory for the first time. Nonlocal strain gradient elasticity theory is adopted to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field is considered. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Material properties of curved porous FG nanobeam are assumed to be temperature-dependent and are supposed to vary through the thickness direction of beam which modeled via modified power-law rule. Since variation of pores along the thickness direction influences the mechanical and physical properties, porosity play a key role in the mechanical response of curved FG nano-structures. The governing equations and related boundary condition of curved porous FG nanobeam under temperature field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved nanobeam supposed to thermal loading. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, porosity volume fractions, thermal effect, gradient index, opening angle and aspect ratio on the natural frequency of curved FG porous nanobeam are successfully discussed. It is concluded that these parameters play key roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

• Smart Structures and Systems
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• v.20 no.3
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• pp.351-368
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• 2017

To peruse the free vibration of curved functionally graded piezoelectric (FGP) nanosize beam in thermal environment, nonlocal elasticity theory is applied for modeling the nano scale effect. The governing equations are obtained via the energy method. Analytically Navier solution is employed to solve the governing equations for simply supported boundary conditions. Solving these equations enables us to estimate the natural frequency for curved FGP nanobeam under the effect of a uniform temperature change and external electric voltage. The results determined are verified by comparing the results by available ones in literature. The effects of various parameters such as nonlocality, uniform temperature changes, external electric voltage, gradient index, opening angle and aspect ratio of curved FGP nanobeam on the natural frequency are successfully discussed. The results revealed that the natural frequency of curved FGP nanobeam is significantly influenced by these effects.

• Steel and Composite Structures
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• v.36 no.2
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• pp.179-186
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• 2020

In this paper, the forced resonance vibration of porous functionally graded (FG) curved nanobeam is examined. In order to capture the hardening and softening mechanisms of nanostructure, the nonlocal strain gradient theory is employed to build the size-dependent model. Using the Timoshenko beam theory together with the Hamilton principle, the equations of motion for the curved nanobeam are derived. Then, Navier series are used in order to obtain the dynamical deflections of the porous FG curved nanobeam with simply-supported ends. It is found that the resonance position of the nanobeam is very sensitive to the nonlocal and strain gradient parameters, material variation, porosity coefficient, as well as geometrical conditions. The results indicate that the resonance position is postponed by increasing the strain gradient parameter, while the nonlocal parameter has the opposite effect on the results. Furthermore, increasing the opening angle or length-to-thickness ratio will result in resonance position moves to lower-load frequency.

• Smart Structures and Systems
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• v.20 no.6
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• pp.709-728
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• 2017

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

• Steel and Composite Structures
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• v.46 no.6
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• pp.759-772
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• 2023

This manuscript presents a comprehensive mathematical model to investigate buckling stability and postbuckling response of bio-inspired composite beams with helicoidal orientations. The higher order shear deformation theory as well as the Timoshenko beam theories are exploited to include the shear influence. The equilibrium nonlinear integro-differential equations of helicoidal composite beams are derived in detail using the energy conservation principle. Differential integral quadrature method (DIQM) is employed to discretize the nonlinear system of differential equations and solve them via the Newton iterative method then obtain the response of helicoidal composite beam. Numerical calculations are carried out to check the validity of the present solution methodology and to quantify the effects of helicoidal rotation angle, elastic foundation constants, beam theories, geometric and material properties on buckling, postbuckling of bio-inspired helicoidal composite beams. The developed model can be employed in design and analysis of curved helicoidal composite beam used in aerospace and naval structures.

• Advances in nano research
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• v.13 no.6
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• pp.599-617
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• 2022

In the present research, for the first time, the vibrational as well as buckling characteristics of a three-layered curved nanobeam including a core made of functionally graded (FG) material and two layers of smart material-piezo-magneto-electric-resting on a Winkler Pasternak elastic foundation are examined. The displacement field for the nanobeam is chosen via Timoshenko beam theory. Also, the size dependency is taken into account by using nonlocal strain gradient theory, aka NSGT. Then, by employing Hamilton's principle, energy procedure, the governing equations together with the boundary conditions are achieved. The solution procedure is a numerical solution called generalized differential quadrature method, or GDQM. The accuracy and reliability of the formulation alongside solution method is examined by using other published articles. Lastly, the parameter which can alter and affect the buckling or vocational behavior of the curved nanobeam is investigated in details.