• Structural Engineering and Mechanics
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• 제45권4호
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• pp.569-594
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• 2013

The free vibration of axially functionally graded tapered arches including shear deformation and rotatory inertia are studied through solving the governing differential equation of motion. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal arches with hinged-hinged, hinged-clamped and clamped-clamped end restraints. In this study Differential Quadrature element of lowest order (DQEL) or Lagrangian Interpolation technique is applied to solve the problems. Three general taper types for rectangular section are considered. The lowest four natural frequencies are calculated and compared with the published results.

• Advances in nano research
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• 제6권3호
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• pp.257-278
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• 2018

Vibration of axially functionally graded nano-rods and beams is investigated. It is assumed that the material properties change along the rod and beam length. The Ritz method with algebraic polynomials is used in the formulation of the problems. Stress gradient elasticity theory is utilized in order to include the nonlocal effects. Frequencies are obtained for different boundary conditions, geometrical and material properties. Nonlocal parameter is assumed as changing linearly or quadratically along the length of the nanostructure. Frequencies are compared to constant nonlocal parameter cases and considerable differences are observed between constant and variable nonlocal parameter cases. Mode shapes in various cases are depicted in order to explain the effects of axial grading.

• Advances in nano research
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• 제13권6호
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• pp.545-559
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• 2022

This paper studies the buckling response of nonuniform functionally graded micro-sized tubes according to the high-order tube theory (HOTT) and classical beam theory (CBT) in addition to nonlocal strain gradient theory. The microtube is made of axially functionally graded material (AFGM). Both inner and outer tube radiuses are changed along the tube length; the microtube is the truncated conical type of tube. The nonlinear partial differential (PD) the formulations are obtained on the basis of the energy conservation method. Then, the linear and nonlinear results are computed via a powerful numerical approach. Finally, the impact of various parameters on the stability of axially functionally graded (AFG) microtube regarding the buckling analysis is discussed.

• Structural Engineering and Mechanics
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• 제40권4호
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• pp.583-594
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• 2011

In the present study, free vibration of an axially functionally graded (AFG) pile embedded in Winkler-Pasternak elastic foundation is analyzed within the framework of the Euler-Bernoulli beam theory. The material properties of the pile vary continuously in the axial direction according to the power-law form. The frequency equation is obtained by using Lagrange's equations. The unknown functions denoting the transverse deflections of the AFG pile is expressed in modal form. In this study, the effects of material variations, the parameters of the elastic foundation on the fundamental frequencies are examined. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

• Steel and Composite Structures
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• 제33권1호
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• pp.133-142
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• 2019

In the present study, microstructure-dependent static stability analysis of inhomogeneous tapered micro-columns is performed. It is considered that the micro column is made of functionally graded materials and has a variable cross-section. The material and geometrical properties of micro column vary continuously throughout the axial direction. Euler-Bernoulli beam and modified couple stress theories are used to model the nonhomogeneous micro column with variable cross section. Rayleigh-Ritz solution method is implemented to obtain the critical buckling loads for various parameters. A detailed parametric study is performed to examine the influences of taper ratio, material gradation, length scale parameter, and boundary conditions. The validity of the present results is demonstrated by comparing them with some related results available in the literature. It can be emphasized that the size-dependency on the critical buckling loads is more prominent for bigger length scale parameter-to-thickness ratio and changes in the material gradation and taper ratio affect significantly the values of critical buckling loads.

• Smart Structures and Systems
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• 제25권6호
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• pp.669-678
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• 2020

This paper focuses on the free vibration analysis of axially functionally graded (FG) Euler-Bernoulli beams. The material properties of the beams are assumed to obey the linear law distribution. The complexities in solving differential equation of transverse vibration of composite beams which limit the analytical solution to some special cases are overcome using the Differential Transformation Method (DTM). Natural frequencies and corresponding normalized mode shapes are calculated. Validation targets are experimental data or finite element results. Different parameters such as reinforcement distribution, ratio of the reinforcement Young's modulus to the matrix Young's modulus and ratio of the reinforcement density to the matrix density are taken into investigation. The delivered results prove the capability and the robustness of the applied method. The studied parameters are demonstrated to be very crucial for the normalized natural frequencies and mode shapes.

• Structural Engineering and Mechanics
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• 제85권1호
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• pp.135-145
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• 2023

This study presents the free vibrational responses of bi-directional axially graded cylindrical shell panels using 3D graded finite element approximation under a temperature field. The cylindrical shell panel is graded in two directions and made of metal-ceramic materials. To extract material properties, the Voigt model is combined with a Power-law material distribution. Convergence and validation studies are performed on the developed computational model to ensure its accuracy and effectiveness. Furthermore, a parametric study is performed to evaluate the developed model, which demonstrates that geometrical parameters, imperfect materials (porosity), support conditions, and surface temperature all have a significant impact on the free vibration responses of a bi-directional axially graded cylindrical shell panel in a thermal environment.

• Steel and Composite Structures
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• 제28권6호
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• pp.749-758
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• 2018

Free axial vibration of axially functionally graded (AFG) nanorods is studied by focusing on the inertia of lateral motions and shear stiffness effects. To this end, Bishop's theory considering the inertia of the lateral motions and shear stiffness effects and the nonlocal theory considering the small scale effect are used. The material properties are assumed to change continuously through the length of the AFG nanorod according to a power-law distribution. Then, nonlocal governing equation of motion and boundary conditions are derived by implementing the Hamilton's principle. The governing equation is solved using the harmonic differential quadrature method (HDQM), After that, the first five axial natural frequencies of the AFG nanorod with clamped-clamped end condition are obtained. In the next step, effects of various parameters like the length of the AFG nanorod, the diameter of the AFG nanorod, material properties, and the nonlocal parameter value on natural frequencies are investigated. Results of the present study can be useful in more accurate design of nano-electro-mechanical systems in which nanotubes are used.

• Steel and Composite Structures
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• 제42권5호
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• pp.649-655
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• 2022

In presented paper, moving load problem of simply supported axially functionally graded (AFG) beam is investigated under temperature rising based on the first shear beam theory. The material properties of beam vary along the axial direction. Material properties of the beam are considered as temperature-dependent. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the moving load problem, the Newmark average acceleration method is used in the time history. In the numerical examples, the effects of material graduation, temperature rising and velocity of moving load on the dynamic responses ofAFG beam are presented and discussed.

• Geomechanics and Engineering
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• 제36권2호
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• pp.99-109
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• 2024

Studying the dynamic behavior of axially moving cylindrical shells in hygro-thermal environments has important theoretical and engineering value for aircraft design. Therefore, in this paper, considering hygro-thermal effect, the nonlinear forced vibration of an axially moving cylindrical shell made of functionally graded materials (FGM) is studied. It is assumed that the material properties vary continuously along the thickness and contain pores. The Donnell thin shell theory is used to derive the motion equations of FGM cylindrical shells with hygro-thermal loads. Under the four sides clamped (CCCC) boundary conditions, the Gallekin method and multi-scale method are used for nonlinear analysis. The effects of power law index, porosity coefficient, temperature rise, moisture concentration, axial velocity, prestress, damping and external excitation amplitude on nonlinear forced vibration are explored through parametric research. It can be found that, the changes in temperature and humidity have a significant effect. Increasing in temperature and humidity will cause the resonance position to shift to the left and increase the resonance amplitude.