• Steel and Composite Structures
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• v.51 no.1
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• pp.73-91
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• 2024

This paper has focused on presenting vibration analysis of trapezoidal sandwich plates with a damaged core and FG wavy CNT-reinforced face sheets. A damage model is introduced to provide an analytical description of an irreversible rheological process that causes the decay of the mechanical properties, in terms of engineering constants. An isotropic damage is considered for the core of the sandwich structure. The classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes' effective aspect ratio as the filler content increases. The First-order shear deformation theory of plate is utilized to establish governing partial differential equations and boundary conditions for the trapezoidal plate. The governing equations together with related boundary conditions are discretized using a mapping-generalized differential quadrature (GDQ) method in spatial domain. Then natural frequencies of the trapezoidal sandwich plates are obtained using GDQ method. Validity of the current study is evaluated by comparing its numerical results with those available in the literature. After demonstrating the convergence and accuracy of the method, different parametric studies for laminated trapezoidal structure including carbon nanotubes waviness (0≤w≤1), CNT aspect ratio (0≤AR≤4000), face sheet to core thickness ratio (0.1 ≤ ${\frac{h_f}{h_c}}$ ≤ 0.5), trapezoidal side angles (30° ≤ α, β ≤ 90°) and damaged parameter (0 ≤ D < 1) are carried out. It is explicated that the damaged core and weight fraction, carbon nanotubes (CNTs) waviness and CNT aspect ratio can significantly affect the vibrational behavior of the sandwich structure. Results show that by increasing the values of waviness index (w), normalized natural frequency of the structure decreases, and the straight CNT (w=0) gives the highest frequency. For an overall comprehension on vibration of laminated trapezoidal plates, some selected vibration mode shapes were graphically represented in this study.

• Structural Engineering and Mechanics
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• v.63 no.1
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• pp.89-102
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• 2017

In this work, we present a theoretical framework to study the thermovisco-elastic responses of homogeneous, isotropic and perfectly conducting medium subjected to inclined load. Based on recently developed generalized thermoelasticity theory with fractional order strain, the two-dimensional governing equations are obtained in the context of generalized magnetothermo-viscoelasticity theory without energy dissipation. The Kelvin-Voigt model of linear viscoelasticity is employed to describe the viscoelastic nature of the material. The resulting formulation of the field equations is solved analytically in the Laplace and Fourier transform domain. On the application of inclined load at the surface of half-space, the analytical expressions for the normal displacement, strain, temperature, normal stress and tangential stress are derived in the joint-transformed domain. To restore the fields in physical domain, an appropriate numerical algorithm is used for the inversion of the Laplace and Fourier transforms. Finally, we have demonstrated the effect of magnetic field, viscosity, mechanical relaxation time, fractional order parameter and time on the physical fields in graphical form for copper material. Some special cases have also been deduced from the present investigation.

• Advances in nano research
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• v.8 no.2
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• pp.115-134
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• 2020

In this paper, the free vibrations analysis of the nanoplates made of three-directional functionally graded material (TDFGM) with small scale effects is presented. To study the small-scale effects on natural frequency, modified strain gradient theory (MSGT) has been used. Material properties of the nanoplate follow an arbitrary function that changes in three directions along the length, width and thickness of the plate. The equilibrium equations and boundary conditions of nanoplate are obtained using the Hamilton's principle. The generalized differential quadrature method (GDQM) is used to solve the governing equations and different boundary conditions for obtaining the natural frequency of nanoplate made of three-directional functionally graded material. The present model can be transformed into a couple stress plate model or a classic plate model if two or all parameters of the length scales set to zero. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants and the aspect ratio with different boundary conditions on the free vibrations of nanoplates. To the best of the researchers' knowledge, in the literature, there is no study carried out into MSGT for free vibration analysis of FGM nanoplate with arbitrary functions.

• Journal of Korean Association for Spatial Structures
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• v.4 no.1 s.11
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• pp.39-48
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• 2004

Generally, a structural system with large inextensional deformations, or in other words, non-strained deformation is called as 'Unstable Structure', Truss-linked structures, cable structures, membrane structures and movable structures as foldable space structures etc, are included in this category. In this paper, a dynamic analysis method for unstable structural systems is presented. Governing equations for dynamic analysis of unstable truss structures with inextensional displacements are derived. Because of singularity of inverse matrixin in practical analysis of unstable structure, the generalized inverse matrix is Introduced to resolve the singular problem. Also, the RREF technique is used to get the inextensional displacement mode. Two unstable truss structures are analyzed by using presented method. Damping is not considered. From the given results, it is known that proposed method is useful to figure out the dynamic behavior of unstable truss structures.

• Structural Engineering and Mechanics
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• v.51 no.6
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• pp.939-953
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• 2014

Using the infinitesimal theory of elasticity and analytical formulation based on the first-order shear deformation theory (FSDT) is presented for axisymmetric thick-walled cylinders made of functionally graded materials under internal and/or external uniform pressure. The material is assumed to be isotropic heterogeneous with constant Poisson's ratio and radially exponentially varying elastic modulus. At first, general governing equations of the FGM thick cylinders are derived by assumptions of the FSDT. Then the obtained equations are solved under the generalized clamped-clamped conditions. The results are compared with the findings of both FSDT and finite element method (FEM).

• Structural Engineering and Mechanics
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• v.64 no.6
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• pp.683-693
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• 2017

According to a generalized nonlocal strain gradient theory (NSGT), dynamic modeling and free vibrational analysis of nanoporous inhomogeneous nanoplates is presented. The present model incorporates two scale coefficients to examine vibration behavior of nanoplates much accurately. Porosity-dependent material properties of the nanoplate are defined via a modified power-law function. The nanoplate is resting on a viscoelastic substrate and is subjected to hygro-thermal environment and in-plane linearly varying mechanical loads. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton's principle. These equations are solved for hinged nanoplates via Galerkin's method. Obtained results show the importance of hygro-thermal loading, viscoelastic medium, in-plane bending load, gradient index, nonlocal parameter, strain gradient parameter and porosities on vibrational characteristics of size-dependent FG nanoplates.

• Proceedings of the KSR Conference
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• 2005.05a
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• pp.536-543
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• 2005

Derivation procedures of exact dynamic stiffness matrices of thin-walled curved beams subjected to axial forces are rigorously presented for the spatial free vibration analysis. An exact dynamic stiffness matrix is established from governing equations for a uniform curved beam element with nonsymmetric thin-walled cross section. Firstly this numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. Thus, displacement functions of dispalcement parameters are exactly derived and finally exact stiffness matrices are determined using element force-displacement relationships. The natural frequencies of the nonsymmetric thin-walled curved beam are evaluated and compared with analytical solutions or results by ABAQUS's shell elements in order to demonstrate the validity of this study.

• Proceedings of the KSR Conference
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• 2000.11a
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• pp.358-365
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• 2000

For the general case of loading conditions and boundary conditions, it is very difficult to obtain closed form solutions for buckling loads and natural frequencies of thin-walled structures because its behaviour is very complex due to the coupling effect of bending and torsional behaviour. In consequence, most of previous finite element formulations are introduce approximate displacement fields to use shape functions as Hermitian polynomials, and so on. The Purpose of this study is to presents a consistent derivation of exact dynamic stiffness matrices of thin-walled straight beams, to be used ill tile free vibration analysis, in which almost types of boundary conditions are exist An exact dynamic element stiffness matrix is established from governing equations for a uniform beam element of nonsymmetric thin-walled cross section. This numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. The natural frequency is evaluated for the thin-walled straight beam structure, and the results are compared with analytic solutions in order to verify the accuracy of this study.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2000.06a
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• pp.731-736
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• 2000

The vibration of an axially moving string is studied when the string has geometric non-linearity and translating acceleration. Based upon the von karman strain theory, The equation for the longitudinal vibration is linear and uncoupled, while the equation for the transverse vibration is non-linear and coupled between the longitudinal and transverse deflections. The governing equations are discretized by using the Galerkin approximation. With the discretized nonlinear equations, the time responses are investigated by using the generalized-${\alpha}$ method.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2001.10a
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• pp.345-352
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• 2001

Derivation procedures of exact static element stiffness matrix of shear deformable thin-walled straight beams are rigorously presented for the spatial buckling analysis. An exact static element stiffness matrix is established from governing equations for a uniform beam element with nonsymmetric thin-walled cross section. First this numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. Thus, the displacement functions of dispalcement parameters are exactly derived and finally exact stiffness matrices are determined using member force-displacement relationships. The buckling loads are evaluated and compared with analytic solutions or results of the analysis using ABAQUS' shell elements for the thin-walled straight beam structure in order to demonstrate the validity of this study.