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This paper analyzes the nonlinear dynamic response of truncated conical shells reinforced with carbon nanotubes with functional graded ceramic-metal matrix subjected to harmonic excitation. Carbon nanotubes are distributed with three different patterns along the length and thickness of the conical shell. The matrix material of the shell is considered to be a combination of metal and ceramic, whose properties change as a power function along the thickness of the shell. In order to analyze the dynamic of this system, firstly, the nonlinear dynamic equations of the conical shell are derived based on the first order shear deformation theory and von Karman's strain-displacement relations. Then, with the help of Galerkin discretization method, partial differential equations of the system are converted into time-dependent ordinary differential equations. Adams-Bashforth numerical method is used to solve the system of nonlinear differential equations. Finally, a parametric study is presented to investigate the effects of some parameters of the system, such as the power index, volume fraction and distribution pattern of carbon nanotubes, the geometric characteristics of the shell, and amplitude of the excitation force on the nonlinear dynamic response of the conical shell. In order to validate, the results of this article are compared and presented with the results of previous valid references.

This Paper deals with the large amplitude frequency behavior of porous bi-directional functionally graded beams subjected to various boundary conditions which are simply supported, clamped-simply supported, clamped-clamped, and clamped-free utilizing Reddy third-order shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam change according to power and exponential law in both directions. The equations of motion and associated boundary conditions are derived by means of Hamilton’s principle. A generalized differential quadrature method in conjunction with a direct numerical iteration method is selected to solve the system of equations. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Timoshenko beam theory. The results of extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratio, porosity coefficient, Tapered ratio, shear and elastic foundation parameters, and boundary conditions on the Large amplitude vibration frequencies of the bi-directional functionally graded beams. The results reveal that non-linear frequencies increase with the rise of elastic foundation and tapered coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non-dimensional frequencies. The results of this study, while carefully examining the frequencies of variable cross-sectional functionally graded beams, are effective in the optimal design of bi-directional beams and are very effective in predicting and detecting failure modes of these beams.

In this paper, the free vibrations of functionally graded porous beams with simple boundary conditions on an elastic foundation in a thermal environment were studied using the theory of third-order shear deformation. The properties of the material are temperature dependent and continuously change in the direction of the thickness of the beam and according to the power law distribution of the volume fraction of the material constituents. The uniform porosity distribution at the cross section is examined. Hamilton's principle was used to obtain the governing equations of motion. In order to discretize these equations, the generalized differential quadrature method has been used. Here, the effect of various parameters such as heat field type, temperature difference, power law index, porosity volume fraction, slenderness ratio and elastic foundation parameters on the natural frequencies of a functionally graded porous beam was studied for simple boundary conditions. The results, in addition to showing these effects on the thermomechanical behavior of the beam, also confirm the accuracy of the numerical method used.

Bolt connections often loosen under environmental loading conditions and system vibrations, which can lead to disaster risks during its operation. In this study, the behavior of nonlinear vibrations of an aluminum single-lap joint has been studied analytically and experimentally. Accordingly, considering the effects of nonlinear behavior at the bole joint, a nonlinear two-degree of freedom model for this type of connection is proposed. Then, in order to determine the unknown parameters of the proposed model, the vibrational and dynamic properties of this structure have been estimated using experimental modal analysis and model updating method. Finally, the effect of amplitude of the excitation force and the preload force of the bolts on the dynamic behavior of these systems has been studied analytically. Examination of amplitude-frequency curves shows that reducing the preload force of the bolts reduces the natural frequency and also distorts the amplitude-frequency curve to the left side, which indicates the softening nonlinear behavior of the system with decreasing applied bolt preload force. In addition, the comparison of the theoretical and experimental natural frequencies shows that the proposed model predicts the vibrational characteristics of these systems with a good accuracy and using the proposed model can study the dynamic behavior of these systems for different parameters.

In this paper, Frequency analysis of cracked porous functionally graded beams which are resting on elastic foundation based on reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with model of exponential law. Governing equations obtained with the aid of third order shear deformation theory and by considering elastic foundation effects within hamilton’s principle. Due to complicating closed form solution of equations, differential equations solved utilizing Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results, comparisons are made with solutions which are available for other papers. This study shows that the difference between the results of this paper and the results of others is negligible. Finally the effects of geometrical parameters, exponential law indexes, crack location and depth, elastic foundation and porosity on natural frequencies of functionally graded beams are studied. The results of this paper and effects of these parameters can be used in the optimum design of functional graded beams and crack prediction, detection and monitoring techniques.

In this paper, frequency analysis of Bi-directional Porous functionally graded beams with variable cross section which are resting on elastic foundation based on Reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with two models of exponential and volume fraction power law. Governing equation which is obtained with the aid of third order shear deformation theory and by considering elastic foundation effect in conjunction with Hamilton’s principle. Due to intrinsic closed form solution of equations, differential equations solved with using Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results comparisons are made with solutions which are available for other papers. This study reveals that the difference between the results of this paper and the results of others is negligible. Eventually the effects of geometrical parameters, power and exponential law indexes and elastic foundation coefficients on natural frequencies of Bi-directional FGM beams is studied. The results reveal that non dimensional frequencies increase with the rise of elastic foundation coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non dimensional frequencies.The results of this study can be used in optimal design, vibration control and detection of failure structure of functional graded structures

In the standard finite element method, the edges of the adjacent elements are aligned to each other, and the corner of an element does not located on the edges of another one. If this constraint is violated it is said that the mesh is non-conforming and the use of such meshes in the finite element method requires specific techniques. In the present paper, a new method is suggested for the use of non-conforming meshes which are generated using quadtree algorithm. Quadtree is a data structure that has a very fast recursive algorithm and can be used to divide a two-dimensional domain into sub-regions or elements. The elements produced by this method are non-conforming and are not usable in the standard finite element method. In general, six types of elements with different arrangement of nodes may be appeared in a quadtree mesh and the conventional methods which are using the polynomials cannot be used to derive the shape functions. In the present paper, a new idea is proposed to obtain the shape functions of such elements. In this method, a boundary value problem is defined and its solution is used as the shape functions of the non-conforming elements. To evaluate the applicability and accuracy of the proposed method, two numerical examples are solved and the results are presented. The results show that the proposed method can be used to effectively apply the non-conforming meshes in the finite element method.

Flow structure at water intake is very complicated and three-dimensional. Due to this the application of one and two-dimensional equations can not sufficiently explain the flow structure at water intake. In this study, the simulation of flow structure at a 45 degree water intake was investigated numerically and experimentally. Here, the characteristic of separation zone and velocity distribution at 45 degree water intake were investigated using Standard k-ε and RNG k-ε models. It was found that the Standard k-ε model enables to estimate the flow and separation zone at water intake. Also, the location of separate zone at the inlet of water intake was studied using experimental and numerical approach. It was found that the separation zone at 45 degree water intake occurs at the downstream of the inlet.