• Interaction and multiscale mechanics
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• v.2 no.3
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• pp.223-233
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• 2009

This paper presents a new nonlocal stress variational principle approach for the transverse free vibration of an Euler-Bernoulli cantilever nanobeam with an initial axial tension at its free end. The effects of a nanoscale at molecular level unavailable in classical mechanics are investigated and discussed. A sixth-order partial differential governing equation for transverse free vibration is derived via variational principle with nonlocal elastic stress field theory. Analytical solutions for natural frequencies and transverse vibration modes are determined by applying a numerical analysis. Examples conclude that nonlocal stress effect tends to significantly increase stiffness and natural frequencies of a nanobeam. The relationship between natural frequency and nanoscale is also presented and its significance on stiffness enhancement with respect to the classical elasticity theory is discussed in detail. The effect of an initial axial tension, which also tends to enhance the nanobeam stiffness, is also concluded. The model and approach show potential extension to studies in carbon nanotube and the new result is useful for future comparison.

• Coupled systems mechanics
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• v.6 no.3
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• pp.251-272
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• 2017

In this paper, the thermo-mechanical vibration characteristics of functionally graded (FG) nanobeams subjected to three types of thermal loading including uniform, linear and non-linear temperature change are investigated in the framework of third-order shear deformation beam theory which captures both the microstructural and shear deformation effects without the need for any shear correction factors. Material properties of FG nanobeam are assumed to be temperature-dependent and vary gradually along the thickness according to the power-law form. Hence, applying a third-order shear deformation beam theory (TSDBT) with more rigorous kinetics of displacements to anticipate the behaviors of FG nanobeams is more appropriate than using other theories. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. The obtained results are compared with those predicted by the nonlocal Euler-Bernoulli beam theory and nonlocal Timoshenko beam theory and it is revealed that the proposed modeling can accurately predict the vibration responses of FG nanobeams. The obtained results are presented for the thermo-mechanical vibration analysis of the FG nanobeams such as the effects of material graduation, nonlocal parameter, mode number, slenderness ratio and thermal loading in detail. The present study is associated to aerospace, mechanical and nuclear engineering structures which are under thermal loads.

• Steel and Composite Structures
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• v.36 no.3
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• pp.293-305
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• 2020

This article presented a nanoscale modified continuum model to investigate the free vibration of functionally graded (FG) porous nanobeam by using finite element method. The main novelty of this manuscript is presenting effects of four different porosity models on vibration behaviors of nonlocal nanobeam structure including size effect, that not be discussed before The proposed porosity models are, uniform porosity distribution, symmetric with mid-plane, bottom surface distribution and top surface distribution. The nano-scale effect is included in modified model by using the differential nonlocal continuum theory of Eringen that adding the length scale into the constitutive equations as a material parameter constant. The graded material is distributed through the beam thickness by a generalized power law function. The beam is simply supported, and it is assumed to be thin. Therefore, the kinematic assumptions of Euler-Bernoulli beam theory are held. The mathematical model is solved numerically using the finite element method. Results demonstrate effects of porosity type, material gradation, and nanoscale parameters on the free vibration of nanobeam. The proposed model is effective in vibration analysis of NEMS structure manufactured by porous functionally graded materials.

• Geomechanics and Engineering
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• v.23 no.5
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• pp.405-418
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• 2020

Deep excavation may have impact on the adjacent tunnels. It is obvious that the excavation will adversely affect and even damage the existing tunnels if the induced deformation exceeds the capacity of tunnel structures. It hence creates a high necessity to predict tunnel displacement induced by nearby excavation to ensure the safety of tunnel. In this paper, a simplified method to evaluate the heave of the underlying tunnel induced by adjacent excavation is presented and verified by field measurement results. In the proposed model, the tunnel is represented by a series of short beams connected by tensile springs, compressional springs and shear springs, so that the rotational effect and shearing effect of the joints between lining rings can be captured. The proposed method is compared with the previous modelling methods (e.g., Euler-Bernoulli beam, a series of short beams connected only by shear springs) based on a field measured longitudinal deformation of subway tunnels. Results of these case studies show a reasonable agreement between the predictions and observations.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.13 no.7
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• pp.562-569
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• 2003

An iterative modal analysis approach is developed to determine the effect of transverse open cracks on the dynamic behavior of simply supported pipe conveying fluid subject to the moving mass. The equation of motion Is derived by using Lagrange’s equation. The influences of the velocity of moving mass and the velocity of fluid flow and a crack have been studied on the dynamic behavior of a simply supported pipe system by numerical method. The presence of crack results In higher deflections of pipe. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments i.e. the crack is modelled as a rotational spring. Totally. as the velocity of fluid flow and the crack severity are increased, the mid-span deflection of simply supported pipe conveying fluid Is Increased. The time which produce the maximum dynamic deflection of the simply supported pipe Is delayed according to the increment of the crack severity.

• Smart Structures and Systems
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• v.13 no.4
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• pp.659-683
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• 2014

The static analysis of structures with arbitrary cross-section geometry and material lamination via a refined one-dimensional (1D) approach is presented in this paper. Higher-order 1D models with a variable order of expansion for the displacement field are developed on the basis of Carrera Unified Formulation (CUF). Classical Euler-Bernoulli and Timoshenko beam theories are obtained as particular cases of the first-order model. Numerical results of displacement, strain and stress are provided by using the finite element method (FEM) along the longitudinal direction for different configurations in excellent agreement with three-dimensional (3D) finite element solutions. In particular, a layered thin-walled cylinder is considered as first assessment with a laminated conventional cross-section. An atherosclerotic plaque is introduced as a typical structure with arbitrary cross-section geometry and studied for both the homogeneous and nonhomogeneous material cases through the 1D variable kinematic models. The analyses highlight limitations of classical beam theories and the importance of higher-order terms in accurately detecting in-plane cross-section deformation without introducing additional numerical problems. Comparisons with 3D finite element solutions prove that 1D CUF provides remarkable three-dimensional accuracy in the analysis of even short and nonhomogeneous structures with arbitrary geometry through a significant reduction in computational cost.

• Structural Engineering and Mechanics
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• v.30 no.2
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• pp.165-176
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• 2008

In this paper, a new iterative method for solving vehicle-bridge interaction problems is proposed. Iterative methods have advantages over the non-iterative methods in that it is not necessary to update the system matrix for a given wheel location, and the method can be applied for a new type of car or bridge with few or no modifications. In the proposed method, the necessity of system matrices update is eliminated using the equivalent interaction force acting on the bridge, which is obtained iteratively. Ballast stiffness is included in the interaction forces and the geometric compatibility at the contact points are used as convergence criteria. The bridge is considered as an elastic Bernoulli-Euler beam with surface irregularity and ballast stiffness. The moving vehicle is modeled as a multi-axle mass-spring-damper system having many degrees of freedom depending on the number of axles. The pitching effect, which is the interaction effect between the rear and front wheels when a vehicle begins to enter or leave the bridge, is also considered in the formulation including extended ground boundaries having surface irregularity and ballast stiffness. The applicability of the proposed method is illustrated in the numerical studies.

• Smart Structures and Systems
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• v.18 no.5
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• pp.1029-1062
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• 2016

In this research, the nonlinear free vibration analysis of boron-nitride micro ribbon (BNMR) on the Pasternak elastic foundation under electrical, mechanical and thermal loadings using modified strain gradient theory (MSGT) is studied. Employing the von $K{\acute{a}}rm{\acute{a}}n$ nonlinear geometry theory, the nonlinear equations of motion for the graphene micro ribbon (GMR) using Euler-Bernoulli beam model with considering attached mass and size effects based on Hamilton's principle is obtained. These equations are converted into the nonlinear ordinary differential equations by elimination of the time variable using Kantorovich time-averaging method. To determine nonlinear frequency of GMR under various boundary conditions, and considering mass effect, differential quadrature element method (DQEM) is used. Based on modified strain MSGT, the results of the current model are compared with the obtained results by classical and modified couple stress theories (CT and MCST). Furthermore, the effect of various parameters such as material length scale parameter, attached mass, temperature change, piezoelectric coefficient, two parameters of elastic foundations on the natural frequencies of BNMR is investigated. The results show that for all boundary conditions, by increasing the mass intensity in a fixed position, the linear and nonlinear natural frequency of the GMR reduces. In addition, with increasing of material length scale parameter, the frequency ratio decreases. This results can be used to design and control nano/micro devices and nano electronics to avoid resonance phenomenon.

• Structural Engineering and Mechanics
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• v.60 no.5
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• pp.891-902
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• 2016

The beam structure with breathing cracks subjected to harmonic excitations was modeled by FEM based on Euler-Bernoulli theory, and a piecewise dynamical system was deduced. The precise integration method (PIM) was employed to propose an algorithm for analyzing the dynamic responses of the deduced system. This system was first divided into linear sub-systems, between which there are switching points resulted from the breathing cracks. The inhomogeneous terms due to the external excitations were tackled by introducing auxiliary variables to express the harmonic functions, hence the sub-systems are homogeneous. The PIM was then applied to solve the homogeneous sub-systems one by one. During the procedures, a predictor-corrector algorithm was presented to determine the switching points accurately. The presented method can provide solutions with an accuracy to a magnitude of $10^{-12}$ compared with exact solutions obtained by the theories of ordinary differential equations. The PIM results are much more accurate than Newmark ones with the same time step. Moreover, it is found that the PIM can maintain a high level of accuracy even when the time step increases within a relatively wide range.

• Structural Engineering and Mechanics
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• v.64 no.2
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• pp.145-153
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• 2017

In this article, a novel simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded (FG) beams is proposed. The beauty of this theory relies on its 2-unknowns displacement field as the Euler-Bernoulli beam theory, which is even less than the Timoshenko beam theory. A shear correction factor is, therefore, not needed. Equations of motion are obtained via Hamilton's principle. Analytical solutions for the bending and free vibration analysis are given for simply supported beams. Efficacy of the proposed model is shown through illustrative examples for bending and dynamic of FG beams. The numerical results obtained are compared with those of other higher-order shear deformation beam theory results. The results obtained are found to be accurate.