Islamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330A Problem of Axisymmetric Vibration of Nonlocal Microstretch Thermoelastic Circular Plate with Thermomechanical Sources11366421310.22034/jsm.2019.664213ENRKumarDepartment of Mathematices, Kurukshetra University, Kurukshetra, Haryana, IndiaRRaniDepartment of Mathematices, Choudhary Devilal University, Sirsa, Haryana, IndiaAMiglaniDepartment of Mathematices, Choudhary Devilal University, Sirsa, Haryana, IndiaJournal Article20180901In the present manuscript, we investigated a two dimensional axisymmetric problem of nonlocal microstretch thermoelastic circular plate subjected to thermomechanical sources. An eigenvalue approach is proposed to analyze the problem. Laplace and Hankel transforms are used to obtain the transformed solutions for the displacements, microrotation, microstretch, temperature distribution and stresses. The results are obtained in the physical domain by applying the numerical inversion technique of transforms. The results of the physical quantities have been obtained numerically and illustrated graphically. The results show the effect of nonlocal in the cases of Lord Shulman (LS), Green Lindsay (GL) and coupled thermoelasticity (CT) on all the physical quantities. http://jsm.iau-arak.ac.ir/article_664213_77b256d640d75f4c41ee056225009cad.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Size-Dependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load142566421210.22034/jsm.2019.664212ENMShahrokhiFaculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, IranEJomehzadehFaculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, IranMRezaeizadehFaculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, IranJournal Article20180902In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular micro-plates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates. http://jsm.iau-arak.ac.ir/article_664212_755559e9217f228cc090f1d527bf5d27.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Bending Behavior of Sandwich Plates with Aggregated CNT-Reinforced Face Sheets263866421410.22034/jsm.2019.664214ENMMirzaalianDepartment of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, IranFAghadavoudiDepartment of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, IranRMoradi-DastjerdiYoung Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, IranJournal Article20180904The main aim of this paper is to investigate bending behavior in sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets with considering the effects of carbon nanotube (CNT) aggregation. The sandwich plates are assumed resting on Winkler-Pasternak elastic foundation and a mesh-free method based on first order shear deformation theory (FSDT) is developed to analyze the deflection of sandwich plates. In the face sheets, volume fraction of CNTs and their clusters are considered to be changed along the thickness. To estimate the material properties of the nanocomposite, Eshelby-Mori-Tanaka approach is applied. In the mesh-free analysis, moving least squares (MLS) shape functions are employed to approximate the displacement field and transformation method is used for imposition of essential boundary conditions. The effects of CNT volume fraction, distribution and degree of aggregation, and also boundary conditions and geometric dimensions are investigated on the bending behavior of the sandwich plates. It is observed that in the same value of cluster volume, FG distribution of clusters leads to less deflection in these structures. http://jsm.iau-arak.ac.ir/article_664214_ffde374ff33e440f13237b4aee54e77d.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Size-Dependent Forced Vibration Analysis of Three Nonlocal Strain Gradient Beam Models with Surface Effects Subjected to Moving Harmonic Loads395966421510.22034/jsm.2019.664215ENKRajabiDepartment of Mechanical Engineering, College of Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran0000-0003-0785-3471ShHosseini HashemiSchool of Mechanical Engineering , Iran University of Science and Technology, Tehran, IranA.RNezamabadiDepartment of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, IranJournal Article20180906The forced vibration behaviors are examined for nonlocal strain gradient nanobeams with surface effects subjected to a moving harmonic load travelling with a constant velocity in terms of three beam models namely, the Euler-Bernoulli, Timoshenko and modified Timoshenko beam models. The modification for nonlocal strain gradient Timoshenko nanobeams is exerted to the constitutive equations by exclusion of the nonlocality in the shear constitutive relation. Some analytical closed-form solutions for three nonlocal strain gradient beam models with simply supported boundary conditions are derived by using the Galerkin discretization method in conjunction with the Laplace transform method. The effects of the three beam models, the nonlocal and material length scale parameters, the velocity and excitation frequency of the moving harmonic load on the dynamic behaviors of nanobeams are discussed in some detail. Specifically, the critical velocities are examined in some detail. Numerical results have shown that the aforementioned parameters are very important factors for determining the dynamic behavior of the nanobeams accurately.http://jsm.iau-arak.ac.ir/article_664215_f406244df25e8c9f901e372f0ccd934a.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Numerical Analysis of Composite Beams under Impact by a Rigid Particle607766421710.22034/jsm.2019.664217ENNAkbariDepartment of Aerospace Engineering, Shahid Sattari Aeronautical University of Science Technology, Tehran, IranBChabsangDepartmen of Mechanical Engineering, Amirkabir University of Technology, Tehran, IranJournal Article20180910Analysis of a laminated composite beam under impact by a rigid particle is investigated. The importance of this project is to simulate the impact of objects on small scale aerial structures. The stresses are considered uni- axial bending with no torsion loading. The first order shear deformation theory is used to simulate the beam. After obtaining kinematic and potential energy for a laminated composite beam, the motion equations, boundary conditions and initial conditions are obtained by using Hamilton’s principle. The deformation of beam is considered large so these equations are nonlinear. Then by using the numerical methods such as generalize differential quadrature (GDQ) and Newmark methods, the equations will be converted in to a set of nonlinear algebraic equations. These nonlinear equations are solved by numerical methods such as Newton- Raphson. By solving the equations, the displacement of beam and rotation of cross section in terms of time for different number of points of beam for variety of orientation angle of layers are obtained. Then the displacements of impacted point of beam, stresses and contact forces in different times for variety of orientation of layers for different situations of impact are compared.http://jsm.iau-arak.ac.ir/article_664217_048f8f8fd934ca28ce393e232caca260.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Smart Flat Membrane Sheet Vibration-Based Energy Harvesters789066421910.22034/jsm.2019.664219ENYShahbaziArchitecture and Urbanism Department, Tabriz Islamic Art University, Tabriz, Iran0000-0001-9454-1729Journal Article20180920The dynamic responses of membrane are completely dependent on Pre-tensioned forces which are applied over a boundary of arbitrary curvilinear shape. In most practical cases, the dynamic responses of membrane structures are undesirable. Whilst they can be designed as vibration-based energy harvesters. In this paper a smart flat membrane sheet (SFMS) model for vibration-based energy harvester is proposed. The SFMS is made of an orthotropic polyvinylidene fluoride (PVDF) flat layer that has piezoelectricity effect. For this aim, polarization vector of PVDF layer is considered parallel to the applied electric field intensity vector. Electrodynamics governing equations of transverse motion of SFMS including active and modified pre-tensioned force are exploited. Transverse displacement component is expanded by the separable form corresponding to the axial and transverse and the linear ODE of motion based on generalized shape coefficients is obtained using Galerkin method. Finally, the explicit relation between forced vibration of SFMS and current and voltage harvesting are obtained. Numerical energy harvesting analyses were developed for an orthotropic rectangle SFMS and the voltage as function of the time is calculated based on different resistances. Parametric simulation shows a 1 m length and 0.5 width SFMS has ability to produce a peak to peak voltage about of 30 <em>mV</em>.http://jsm.iau-arak.ac.ir/article_664219_e2dc2db5c94bbb75247e6130e475b41f.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Fracture Parameters for Cracked Cylincal Shells9110466422110.22034/jsm.2019.664221ENMKadriLaboratoire de Mécanique Appliquée, Université des Sciences et de la Technologie d’Oran , AlgeriaASahliLaboratoire de Recherche des Technologies Industrielles, Université Ibn Khaldoun de Tiaret, AlgeriaSSahliUniversité d'Oran 2 Mohamed Ben Ahmed, AlgeriaJournal Article20180927In this paper, 2D boundary element stress analysis is carried out to obtain the <em>T</em>-stress for multiple internal edge cracks in thick-walled cylinders for a wide range of cylinder radius ratios and relative crack depth. The <em>T</em>-stress, together with the stress intensity factor <em>K</em>, provides amore reliable two-parameter prediction of fracture in linear elastic fracture mechanics. <em>T</em>-stress weight functions are then derived from the <em>T</em>-stress solutions for two reference load conditions corresponding to the cases when the cracked cylinder is subject to a uniform and to a linear applied stress variation on the crack faces. The derived weight functions are then verified for several non-linear load conditions. Using the BEM results as reference <em>T</em>-stress solutions; the <em>T</em>-stress weight functions for thick-walled cylinder have also been derived. Excellent agreements between the BEM results and weight function predictions are obtained. The weight functions derived are suitable for obtaining <em>T</em>-stress solutions for the corresponding cracked thick-walled cylinder under any complex stress fields. Results of the study show that the two dimensional BEM analysis, together with weight function method, can be used to provide a quick and accurate estimate of <em>T</em>-stress for 2-D crack problems.http://jsm.iau-arak.ac.ir/article_664221_46fbd1e945c36c4866982234067fab64.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Experimental and Numerical Free Vibration Analysis of Hybrid Stiffened Fiber Metal Laminated Circular Cylindrical Shell10511966422310.22034/jsm.2019.664223ENANazariDepartment of Aerospace Engineering, Aerospace Research Institute, Tehran, IranKMalekzadehMalekAshtar University, Tehran, IranA.ANaderiFaculty of Mechanical Engineering, Emam Ali University, Tehran, IranJournal Article20181101The modal testing has proven to be an effective and non-destructive test method for estimation of the dynamic stiffness and damping constant. The aim of the present paper is to investigate the modal response of stiffened Fiber Metal Laminated (FML) circular cylindrical shells using experimental and numerical techniques. For this purpose, three types of FML-stiffened shells are fabricated by a specially-designed method and the burning examination is used to determine the mechanical properties of them. Then, modal tests are conducted to investigate the vibration and damping characteristics of the FML-stiffened shells. A 3D finite element model is built using ABAQUS software to predict the modal characteristics of the FML-stiffened circular cylindrical shells with free-free ends. Finally, the achievements from the numerical and experimental analyses are compared with each other and good agreement has been obtained. Modal analyses of the FML-stiffened circular cylindrical shells are investigated for the first time in this paper. Thus, the results obtained from this study are novel and can be used as a benchmark for further studies. http://jsm.iau-arak.ac.ir/article_664223_4d2474cd47cc4ae85b951c2da6a95a29.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Effect of Thermal Environment on Vibration Analysis of Partially Cracked Thin Isotropic Plate Submerged in Fluid12014366422410.22034/jsm.2019.664224ENShashankSoniNational Institute of Technology, Raipur, Chhattisgarh 492010, IndiaN.KJainNational Institute of Technology, Raipur, Chhattisgarh 492010, IndiaP.VJoshiIndian Institute of Information Technology, Nagpur, Maharashtra, 440006, IndiaJournal Article20181103Based on a non classical plate theory, an analytical model is proposed for the first time to analyze free vibration problem of partially cracked thin isotropic submerged plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory and the modified couple stress theory. The crack terms are formulated using simplified line spring model whereas the effect of thermal environment is introduced using thermal moments and in-plane forces. The influence of fluidic medium is incorporated in governing equation in form fluids forces associated with inertial effects of its surrounding fluids. Applying the Galerkin’s method, the derived governing equation of motion is reformulated into well known Duffing equation. The governing equation for cracked isotropic plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack, temperature rise and internal material length scale parameter. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The variation in natural frequency of the cracked plate with uniform rise in temperature is studied considering various parameters such as crack length, fluid level and internal material length scale parameter. Furthermore the variation of the natural frequency with plate thickness is also established.http://jsm.iau-arak.ac.ir/article_664224_ae26c477e230049487273d0578d3f782.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Free Vibration Analysis of Functionally Graded Piezoelectric Material Beam by a Modified Mesh Free Method14415466422510.22034/jsm.2019.664225ENMForoutanDepartment of Mechanical Engineering, Razi University, Kermanshah, IranShSharafiDepartment of Mechanical Engineering, Razi University, Kermanshah, IranSMohammadiDepartment of Mechanical Engineering, Razi University, Kermanshah, IranJournal Article20181109A mesh-free method based on moving least squares approximation (MLS) and weak form of governing equations including two dimensional equations of motion and Maxwell’s equation is used to analyze the free vibration of functionally graded piezoelectric material (FGPM) beams. Material properties in beam are determined using a power law distribution. Essential boundary conditions are imposed by the transformation method. The mesh-free method is verified by comparison with a finite element method (FEM) which performed for FGPM beams. Comparisons showed that this model has a good accuracy. After validation of the presented model, a parametric study was carried out to investigate the effect of mechanical and electrical boundary conditions, slenderness ratio and distribution of constituent materials on natural frequencies of FGPM beams. It is concluded that slenderness ratio has more significant effect on lower frequencies. On the other hand higher frequencies are affected by the volume fraction power index much more than lower frequencies.http://jsm.iau-arak.ac.ir/article_664225_f7ff55f5c6c68e516a641ac40d5b8ff5.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Effect of Micropolarity on the Propagation of Shear Waves in a Piezoelectric Layered Structure15516566422610.22034/jsm.2019.664226ENRKumarDepartment of Mathematics, Kurukshetra University, Kurukshetra 136119, IndiaKSinghDepartment of Mathematics, Lovely Professional University, Phagwara(Research Scholar Punjab Technical University, Jalandhar), IndiaD.SPathaniaDepartment of Mathematics, Guru Nanak Dev Engineering College, Ludhiana, IndiaJournal Article20181110This paper studies the propagation of shear waves in a composite structure consisting of a piezoelectric layer perfectly bonded over a micropolar elastic half space. The general dispersion equations for the existence of shear waves are obtained analytically in the closed form. Some particular cases have been discussed and in one special case the relation obtained is in agreement with existing results of the classical –Love wave equation. The micropolar and piezoelectric effects on the phase velocity are obtained for electrically open and mechanically free structure. To illustrate the utility of the problem numerical computations are carried out by considering PZT-4 as a piezoelectric and aluminium epoxy as micropolar elastic material. It is observed that the micropolarity present in the half space influence the phase velocity significantly in a particular region. The micropolar effects on the phase velocity in the piezoelectric coupled structure can be used to design high performance acoustic wave devices.http://jsm.iau-arak.ac.ir/article_664226_28a0be85fcc2c62de7ff4ce5041339f9.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330On Static Bending, Elastic Buckling and Free Vibration Analysis of Symmetric Functionally Graded Sandwich Beams16618066422710.22034/jsm.2019.664227ENA.SSayyadDepartment of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, IndiaP.VAvhadDepartment of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, IndiaJournal Article20181201This article presents Navier type closed-form solutions for static bending, elastic buckling and free vibration analysis of symmetric functionally graded (FG) sandwich beams using a hyperbolic shear deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for a hyperbolic distribution of axial displacement whereas transverse displacement is constant through the thickness i.e effects of thickness stretching are neglected. The present theory gives hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam. The equations of the motion are obtained by using the Hamilton’s principle. Closed-form solutions for static, buckling and vibration analysis of simply supported FG sandwich beams are obtained using Navier’s solution technique. The non-dimensional numerical results are obtained for various power law index and skin-core-skin thickness ratios. The present results are compared with previously published results and found in excellent agreement.http://jsm.iau-arak.ac.ir/article_664227_008bb7e3e1a8208dd0730276194d4920.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Study of the Effect of an Open Transverse Crack on the Vibratory Behavior of Rotors Using the h-p Version of the Finite Element Method18120066422810.22034/jsm.2019.664228ENFAhmedIS2M Laboratory, Faculty of Technology, University of Tlemcen, AlgeriaHAbdelhamidIS2M Laboratory, Faculty of Technology, University of Tlemcen, AlgeriaBBrahimIS2M Laboratory, Faculty of Technology, University of Tlemcen, AlgeriaSAhmedIS2M Laboratory, Faculty of Technology, University of Tlemcen, AlgeriaJournal Article20181205In this paper, we use the hybrid h-p version of the finite element method to study the effect of an open transverse crack on the vibratory behavior of rotors, the one-dimensional finite element Euler-Bernoulli beam is used for modeling the rotor, the shape functions used are the Hermite cubic functions coupled to the special Legendre polynomials of Rodrigues. The global matrices of the equation of motion of the cracked rotor are derived by the application of the Lagrange equation taking into account the local variation in the shaft’s stiffness due to the presence of the crack, and the stiffness of the cracked element of the shaft are determined using the time-varying stiffness method. Numerical results generated by a program developed in MATLAB show the rapidity of the convergence of the h-p version of FEM compared to the classical version, after the validation of our results with theoretical and experimental results and other obtained with the simulator ANSYS Workbench, a parametric study was provided to show the influence of the depth and position of the crack on the vibratory behavior of a symmetrical and asymmetrical rotor.http://jsm.iau-arak.ac.ir/article_664228_6a267353940b8cec5e04fcfe0b4b68bd.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Influence of the Imperfect Interface on Love-Type Mechanical Wave in a FGPM Layer20120966422910.22034/jsm.2019.664229ENSChaudharyMadanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, IndiaASinghalMadanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, India---
Indian Institute of Technology (ISM) Dhanbad, Jharkhand, IndiaS.ASahuIndian Institute of Technology (ISM) Dhanbad, Jharkhand, IndiaJournal Article20181210In this study, we consider the propagation of the Love-type wave in piezoelectric gradient covering layer on an elastic half-space having an imperfect interface between them. Dispersion relation has been obtained in the form of determinant for both electrically open and short cases. The effects of different material gradient coefficients of functionally graded piezoelectric material (FGPM) and imperfect boundary on the phase velocity of Love-type waves are discussed. Also, the influence of mechanically and electrically imperfect interface on the surface wave phase velocity is obtained and shown graphically. The dispersion curves are plotted and the effects of material properties of both FGPM and orthotropic material are studied. Moreover, dispersion relation of the considered microstructure depends substantially on the material gradient coefficients and width of the guiding plate. Numerical results are highlighted graphically and are validated with existing literature. The present study is the prior attempt to show the interfacial imperfection influence with the considered structure on wave phase velocity. The outcomes are widely applicable and useful for the development and characterization of Love-type mechanical waves in FGPM-layered media, SAW devices and other piezoelectric devices.http://jsm.iau-arak.ac.ir/article_664229_3388e38bf7dda6770544755eb14f871e.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Influence of Addendum Modification Factor on Root Stresses in Normal Contact Ratio Asymmetric Spur Gears21022166423010.22034/jsm.2019.664230ENR PrabhuSekarMechanical Engineering Department, Motilal Nehru National Institute of Technology, Allahabad, India0000-0003-4506-9987RRavivarmanResearch Scholar Department of Mechanical Engineering, Pondicherry Engineering college, Pondicherry, IndiaJournal Article20190102Tooth root crack is considered as one of the crucial causes of failure in the gearing system and it occurs at the tooth root due to an excessive bending stress developed in the root region. The modern power transmission gear drives demand high bending load capacity, increased contact load capacity, low weight, reduced noise and longer life. These subsequent conditions are satisfied by the aid of precisely designed asymmetric tooth profile which turns out to be a suitable alternate for symmetric spur gears in applications like aerospace, automotive, gear pump and wind turbine industries. In all step up and step down gear drives (gear ratio > 1), the pinion (smaller in size) is treated as a vulnerable one than gear (larger in size) which is primarily due to the development of maximum root stress in the pinion tooth. This paper presents an idea to improve the bending load capacity of asymmetric spur gear drive system by achieving the same stresses between the asymmetric pinion and gear fillet regions which can be accomplished by providing an appropriate addendum modification. For this modified addendum the pinion and gear teeth proportion equations have been derived. In addition, the addendum modification factors required for a balanced maximum fillet stress condition has been determined through FEM for different parameters like drive side pressure angle, number of teeth and gear ratio. The bending load capacity of the simulated addendum modified asymmetric spur gear drives were observed to be prevalent (very nearly 7%) to that of uncorrected asymmetric gear drives.http://jsm.iau-arak.ac.ir/article_664230_a6f4516631c85986a43f55c971fff78f.pdfIslamic Azad University Arak BranchJournal of Solid Mechanics2008-350511120190330Nonlinear Modeling of Bolted Lap Jointed Structure with Large Amplitude Vibration of Timoshenko Beams22223566423110.22034/jsm.2019.664231ENMJamal-OmidiDepartment of Aerospace Engineering, Space Research Institute, Malek Ashtar University of Technology, Tehran, IranFAdelDepartment of Aerospace Engineering, Space Research Institute, Malek Ashtar University of Technology, Tehran, IranJournal Article20190110This paper aims at investigating the nonlinear behavior of a system which is consisting of two free-free beams which are connected by a nonlinear joint. The nonlinear system is modelled as an in-extensional beam with Timoshenko beam theory. In addition, large amplitude vibration assumption is taken into account in order to obtain exact results. The nonlinear assumption in the system necessities existence of the curvature-related and inertia-related nonlinearities. The nonlinear partial differential equations of motion for the longitudinal, transverse, and rotation are derived using the Hamilton’s principle. A set of coupled nonlinear ordinary differential equations are further obtained with the aid of Galerkin method. The frequency-response curves are presented in the section of numerical results to demonstrate the effect of the different dimensionless parameters. It is shown that the nonlinear bolted-lap joint structure exhibits a hardening-type behavior. Furthermore, it is found that by adding a nonlinear spring the system exhibits a stronger hardening-type behavior. In addition, it is found that the system shows nonlinear behavior even in the absence of the nonlinear spring due to the nonlocal nonlinearity assumption. Moreover, it is shown that considering different engineering beam theories lead to different results and it is found that the Euler-Bernoulli beam theory over-predict the resonance frequency of the structure by ignoring rotary inertia and shear deformation.http://jsm.iau-arak.ac.ir/article_664231_66dcb4ba96892cb177c7bc9fda142bd5.pdf