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CONTENTS
Volume 5, Number 1, January 2018
 

Abstract
The present work shows many aspects concerning the use of a numerical wave-based methodology for the computation of the structural response of periodic structures, focusing on cylinders. Taking into account the periodicity of the system, the Bloch-Floquet theorem can be applied leading to an eigenvalue problem, whose solutions are the waves propagation constants and wavemodes of the periodic structure. Two different approaches are presented, instead, for computing the forced response of stiffened structures. The first one, dealing with a Wave Finite Element (WFE) methodology, proved to drastically reduce the problem size in terms of degrees of freedom, with respect to more mature techniques such as the classic FEM. The other approach presented enables the use of the previous technique even when the whole structure can not be considered as periodic. This is the case when two waveguides are connected through one or more joints and/or different waveguides are connected each other. Any approach presented can deal with deterministic excitations and responses in any point. The results show a good agreement with FEM full models. The drastic reduction of DoF (degrees of freedom) is evident, even more when the number of repetitive substructures is high and the substructures itself is modelled in order to get the lowest number of DoF at the boundaries.

Key Words
wave finite element; cylinders dynamics; wave propagation; periodic structures; forced response

Address
Fabrizio Errico, M. Ichchou and O. Bareille: LTDS, Laboratoire de Tribologie et Dynamique des Systems, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, 69134 Écully, France
S. De Rosa and F. Franco: Pasta-Lab, Laboratory for promoting experiences in aeronautical structures and acoustics, Dipartimento
di Ingegneria Industriale Sezione Aerospaziale, Universita\' degli Studi di Napoli \"Federico II\", Via Claudio 21, 80125 Napoli, Italy

Abstract
In this paper, geometric nonlinear bending characteristics of single wall carbon nanotube reinforced composite (SWCNTRC) doubly curved shell panels subjected to uniform transversely loadings are investigated. The nonlinear mathematical model is developed for doubly curved SWCNTRC shell panel on the basis of higher-order shear deformation theory and Green– Lagrange nonlinearity. All nonlinear higher order terms are included in the mathematical model. The effective material properties of SWCNTRC are estimated by using Eshelby-Mori-Tanaka micromechanical approach. The governing equation of the shell panel is obtained using the total potential energy principle and a Newton-Raphson iterative method is employed to compute the nonlinear displacement and stresses. The present results are compared with published literature. The effect of SWCNT volume fraction, width-to-thickness ratio, radius-to-width ratio (R/a), boundary condition, linear and nonlinear deflection, stresses and different types of shell geometry on nonlinear bending response is investigated.

Key Words
SWCNTRC shell panel; micromechanics; nonlinear bending; green-lagrange nonlinearity; HSDT; newton-raphson method

Address
Shivaji G. Chavan and Achchhe Lal: Department of Mechanical Engineering, S.V.N.I.T, Surat, Gujarat 395007, India

Abstract
This work proposes a theoretical and numerical study on the behavior of a tapered shaft rotor made of composite materials by the classical version h and the version p of the finite element method. Hierarchical form functions are used to define the model. The purpose of this paper is to determine the expressions of the kinetic and potential energies of the tree necessary for the results of the equations of motion. A comparison between the version h and the p version of the finite element method of the functions of polynomial and trigonometric hierarchical forms with six degrees of freedom per node, of a composite tapered and cylindrical shaft which rotates at a constant speed about its axis. It is found that when the number of functions of form (the version p) is increased, the solution converges. It is also observed that the conicity of the shaft increases the rigidity with respect to a uniform shaft having the same mechanical properties. The numerical simulation allowed us to determine the natural frequencies and the critical speeds of the composite shaft systems are compared with those available in the literature and the effectiveness of the methods used are discussed.

Key Words
fibre composite multilayer plates; vibration; Structural mechanics; rotordynamics; dynamic analysis

Address
Zahi Rachid and Refassi Kaddour: Laboratory of Mechanics of Structures and Solids LMSS, University of Sidi Bel Abbes, Algeria
Habib Achache: University Dr Yahia Fares Médéa, Algerie

Abstract
A particular type of constant speed helical trajectory, with variable ascension rate, is proposed. Such trajectories are candidates of choice as motion primitives in automatic airplane trajectory planning; they can also be used by airplanes taking off or landing in limited space. The equations of motion for airplanes flying on such trajectories are exactly solvable. Their solution is presented, together with an analysis of the restrictions imposed on the geometrical parameters of the helical paths by the dynamical abilities of an airplane. The physical quantities taken into account are the airplane load factor, its lift coefficient, and the thrust its engines can produce. Formulas are provided for determining all the parameters of trajectories that would be flyable by a particular airplane, the final altitude reached, and the duration of the trajectory. It is shown how to construct speed interval tables, which would appreciably reduce the calculations to be done on board the airplane. Trajectories are characterized by their angle of inclination, their radius, and the rate of change of their inclination. Sample calculations are shown for the Cessna 182, a Silver Fox like unmanned aerial vehicle, and the F-16 Fighting Falcon.

Key Words
airplane helical trajectory; automatic trajectory planning; banked turn; airplane equation of motion; helical arc connection

Address
Gilles Labonte: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada

Abstract
Thermo-mechanical vibration of sandwich beams with a stiff core and face sheets made of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) is investigated within the framework of Timoshenko beam theory. The material properties of FG-CNTRC are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture and are considered to be temperature dependent. The governing equations and boundary conditions are derived by using Hamilton\'s principle and are solved using an efficient semi-analytical technique of the differential transform method (DTM). Comparison between the results of the present work and those available in literature shows the accuracy of this method. A parametric study is conducted to study the effects of carbon nanotube volume fraction, slenderness ratio, core-to-face sheet thickness ratio, and various boundary conditions on free vibration behavior of sandwich beams with FG-CNTRC face sheets. It is explicitly shown that the vibration characteristics of the curved nanosize beams are significantly influenced by the surface density effects.

Key Words
free vibration; sandwich beam; FG-CNTRC; thermal environments

Address
Farzad Ebrahimi and Navid Farazmandnia: Mechanical Engineering Department, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran P.O.B. 16818-34149

Abstract
The composite materials are widely used in aircraft structures. Their relative rigidity/weight gives them an important advantage over the metal structures. The objective of this work is to analyze by the finite element method the mechanical behavior of composite plate type notched with various forms under tensile load. Two basic parameters were taken into consideration. The first, the form of the notch in order to see its effect on the stress and the failure load. The second, we studied the influence of the locale orientation of fiber around the plate\'s notch. These parameters are studied in order to see their effects on the distribution stress and failure load of the plate. The calculation of the failure load is determined numerically with the numerical code ABAQUS using the XFEM (extended Finite Element Modeling) based on the fracture mechanics. The result shows clearly that it is important to optimize the effect of fiber orientation around the notch.

Key Words
CFRP (reinforced carbon fiber polymers); XFEM (extended finite element modeling)

Address
A. Benzaama and T. Tamine: Departement de Génie Maritime, Universite des Sciences et de la Technologie d\' Oran USTOMB, Algeria
M. Mokhtari: Laboratoire de Recherche en Technologie de Fabrication Mecanique, Ecole Nationale Polytechnique,
ENP Oran M.A. Algeria
H. Benzaama: Laboratory of applied Biomechanics and Biomaterials, Ecole Nationale Polytechnique, ENP Oran M.A. Algeria
S. Gouasmi: Laboratoire de Mécanique de Structure et des Solides (LMSS), University of Sidi Bel Abbes, Algeria

Abstract
In this paper, the thermal effect on buckling and free vibration characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form and the material properties are assumed to be temperature-dependent. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on Timoshenko beam theory through Hamilton\'s principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, aspect ratio and mode number on the critical buckling temperature and normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the thermal buckling and vibration behaviour of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Key Words
thermal buckling; Timoshenko beam theory; vibration; functionally graded material; nonlocal elasticity theory

Address
Farzad Ebrahimi: Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin P.O.B. 16818-34149, Iran
Ramin Ebrahimi Fardshad: Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran


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