Order Shear Deformation Beam Theory Research Articles

Investigation on internal resonance of fluid conveying pipes with initial geometric imperfection

This paper shows numerical research on the 1:2 internal resonance and chaotic bifurcation phenomena of the graphene platelets reinforced metal foams (GPLRMF) pipes, with emphasis on the effect of initial geometric imperfection. To this end, combining the high order shear deformation beam theory and Kelvin-Voigt viscoelasticity, a mathematical model that considers geometric imperfection is established. The fluid in pipeline is supposed to be incompressible and inviscid. Subsequently, the Galerkin discretization and multiscale method are utilized for solving the nonlinear governing equations in the case of 1:2 internal resonance. And the nonlinear dynamic behavior of the fluid conveying pipes can be analyzed through the frequency amplitude diagrams, bifurcation curves, time course curves, phase trajectories, and Poincaré maps. The double jump phenomena are excited to demonstrate the dynamic characteristics of the pipes with 1:2 internal resonance. In the case of small disturbance coefficients, various internal resonance and chaos phenomena can be captured in the pipeline system. Research shows that the coupled effects between initial geometric imperfection and nonlinear geometric relationships is non-monotonic. That is to say, for bimodal internal resonance, when the initial geometric imperfection increase, the unstable region will be compressed, and the response amplitude will increase.

The theoretical and experimental buckling analysis of a nanocomposite beams reinforced by nanorods made of recycled materials

AbstractIn this article, the numerical and experimental results for buckling analysis of a nanocomposite beams reinforced by nanorods are studied and compared. The Chebyshev–Ritz method is used to calculate the buckling numerically and compare it with the buckling test. Epoxy resin, glass's fiber, and nanorods are used to make composites. These nanorods are made using recycled materials. Also, bending and tensile tests are used to obtain the properties of these composites. In addition, the effect of various parameters such as Young's modulus, temperature change, Poisson's ratio, and boundary conditions, the strain gradient parameter, and different shape function is investigated. The addition of nanorods increases the strength and stiffness of composites and increases the critical buckling load. Actually, the made nanorods are used as a reinforcement in matrix, thus in the made nanocomposites, fracture and buckling tests are done and the results indicate that the inclusion of nanorods (0.32%) increases the critical buckling load by 16.4%. A facile hydrothermal method to synthesis the nano rodes has a less cost with the other methods, thus by adding a small percentage of it to the composite, the stiffness of the structure increases. According to the results and the significant increase in Young's modulus and critical buckling load and the cost‐effectiveness of carbon nanorods (CNRs) made from recycled materials, these environmentally friendly reinforcements can be used in various industries such as machine body construction, automobile manufacturing, shipbuilding and construction of type's turbines.Highlights Using potato peels and hydrothermal methods, CNRs produced. These CNRs are useful, and production is cheap and no harm environmental. Buckling sample include of glass fiber, without and with CNRs and epoxy resin. Third order shear deformation beam theory and Chebyshev–Ritz are used Theoretical and experimental buckling critical load are compared.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

Various homogenisation schemes for vibration characteristics of axially FG core multilayered microbeams with metal foam face layers based on third order shear deformation theory

This study assesses the significance of homogenisation models in theoretically analysing the vibrations of three-layered microbeams with a core composed of axially functionally graded (AFG) material and with upper and lower face layers composed of metal foam material, especially when the third order shear deformation beam theory and the modified couple stress theory are used. The material of the core is assumed to vary following a power gradation profile. The range of homogenisation schemes, including the Voigt model, Reuss model, Voigt-Reuss-Hill model, Tamura model, Mori-Tanaka model, lower and upper bounds of Hashin-Shtrikman, and the local representative volume element (LRVE) model, are utilised to approximate the material properties of the AFG core. The upper and lower layers of metal foams are modelled using the closed-cell and open-cell solid porosity models. The coupled motion equations are solved for the natural frequencies of the system for the transverse, rotational and axial directions. A software-based finite element analysis (FEA) study of two simplified macrobeam systems was undertaken and the agreement between the FEA and numerical results, using the presented theoretical method, is found to be excellent. It has been confirmed that the homogenisation schemes have a notable effect on the mechanical properties of the AFG core, and hence on the coupled translational-rotational motion's natural frequencies. The largest difference in natural frequencies between different homogenisation schemes is found with a power gradation constant of approximately 0.53. A larger disparity of natural frequencies between different homogenisation methods was seen with an increase in the thickness of the AFG core and the microbeam's length-scale parameter of multilayered porous microbeam. The results of this study highlight the significance of homogenisation scheme employed when investigating the vibrations of a multilayered microbeam with an AFG core and metal foam porous face layers.

Porosities-dependent wave propagation in bi-directional functionally graded cantilever beam with higher-order shear model

The contribution provided in this paper is to investigate the dynamic response of bidirectional graded material beams with porosities. Higher order shear deformation beam theory for wave propagation in porous functionally graded cantilevered beam are developed, taking into account the bidirectional distribution, which is mainly represented in the density and the Young’s modulus. The material properties of the porous FG beam are graded through the thickness and width using a power law distribution. The governing equations of the wave propagation in the porous FG beam are derived by employing the Hamilton’s principle. The analytic dispersion relation of the bidirectional porous FG beam is obtained by solving an eigenvalue problem. Two porosities models approximating the even and uneven distributions of porosity are considered. The results obtained for bidirectional FG beam were compared with those reported in the literature, and the results were very agreement. The effects of the volume fraction distributions, the number of wave propagation, various parameters and the porosity type on dynamic of functionally graded beam are discussed in details. The numerical results show that the power law index, number of wave, wave velocity, and porosity distribution model affect the dynamic behavior of the beam significantly. The results show that the uneven porosity predicts higher natural frequencies than the even porosity.

Re-examination of nonlinear vibration, nonlinear bending and thermal postbuckling of porous sandwich beams reinforced by graphene platelets

This paper re-examines the nonlinear free vibration, nonlinear bending and thermal postbuckling behaviors of porous sandwich beams with graphene platelets (GPL) reinforcements rested on elastic foundations. We remove the equivalent isotropic model (EIM) and introduce an inhomogeneous model. The shear modulus together with the Young’s moduli for porous GPLRC core is determined by a generic Halpin–Tsai model with porosity. Mechanical properties of both porous GPLRC core and metal face sheets are temperature dependent. Based on the framework of higher order shear deformation beam theory, the motion equations of porous sandwich beams with GPL reinforcements are established. In the modeling, the von Kármán kinematic nonlinearity, beam-foundation interaction and thermal effect are considered. By employing the two-step perturbation method, the analytical solutions are obtained for the nonlinear vibration and nonlinear bending as well as thermal postbuckling problems. Numerical investigations are performed for comparing the results obtained from the EIM and the present model. The outcomes reveal that the EIM is invalid for linear free vibration, thermal buckling and postbuckling analyses of porous sandwich beams with GPL reinforcements, but the EIM is valid in some cases for nonlinear bending and nonlinear vibration analyses of the same beam.

Nitsche-based isogeometric approach for free vibration analysis of laminated plate with multiple stiffeners and cutouts

On the basis of the first order shear deformation and Timoshenko beam theories, a general Nitsche-based isogeometric analysis (IGA) is developed for studying the free vibration behaviors of composite plates with/without stiffeners and cutouts. Firstly, the laminated plate with/without cutout is described by several non-overlapped NURBS patches. The stiffeners are placed on the common edges of adjacent patches and share the same control points with these sub-patch edges. Then, different patches are stuck together by the means of Nitsche method. Eventually, the proposed method is implemented for analyzing the free vibration of composite plates with/without stiffener and cutout. First of all, it avoids the discretization of the stiffener and the identification of plate elements where stiffener nodes are located at, that are required in finite element method and IGA. In addition, it can handle with laminated plate with multiple stiffeners and cutouts conveniently. At first, four meshing schemes with different element sizes are implemented to validate the good convergence and accuracy of the present method in case of laminated plate with stiffener. Then, rectangular, skew and elliptical plates without stiffener and cutout are considered and the obtained natural frequencies and mode shapes agree well with those reported in literature. Furthermore, results from composite plates with/without stiffener and cutout under different boundary conditions are analyzed by considering different lay-up numbers and orientations of fiber, shapes and numbers of cutout, orientation angles and numbers of stiffener.

Vibration analysis of laminated composite higher order beams under varying axial loads

The article develops a computational mathematical model to predict the dynamic response of composite laminated beam under variable axial load using a third higher order shear deformation beam theory, for the first time. The geometrical kinematic relations of displacements are portrayed with higher parabolic shear deformation beam theory. Constitutive equations of composite beam are used on the basis of plane stress problem. The variable axial load is distributed along the axial direction by: constant, linear, and parabolic functions. The equations of motion and associated boundary conditions are derived using Hamilton's principle. Then, the variable coefficients-differential equations of motion are discretized in spatial direction by using numerical differential quadrature method (DQM). The proposed model is verified with previous works available in literature. Parametric analyses are developed to present the influence of axial load type, orthotropy ratio, slenderness ratio, fiber orientations, and boundary conditions on the natural frequencies of composite beams. The present enhanced model can be used specially in designing of spacecrafts, naval, automotive, helicopter, the wind turbine, musical instruments, and civil structures subjected to the variable axial loads.

Thermal postbuckling analysis of sandwich beams with functionally graded auxetic GRMMC core on elastic foundations

This article deals with thermal postbuckling of sandwich beams caused by uniform temperature rise and resting on elastic foundations. The sandwich beam is composed of auxetic graphene-reinforced metal matrix composite (GRMMC) core with negative Poisson's ratio (NPR) and two metal face sheets. The volume fraction of graphene through the thickness domain of the GRMMC core is arranged as a functionally graded (FG) pattern. The thermo-mechanical properties of both metal face sheets and the GRMMC core are assumed to be temperature dependent. The governing equations of the auxetic sandwich beam are formulated based on the higher order shear deformation beam theory coupled with von Kármán-type kinematic nonlinearity. The thermal effects and the beam-foundation interaction are also taken into account in the modeling. Thermal postbuckling solutions are obtained by employing a two-step perturbation approach. The effects of the FG pattern, the face sheet-to-core-to-face sheet thickness ratio, and foundation stiffness on thermal postbuckling behavior of sandwich beams with FG auxetic GRMMC core are studied in detail.

Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory

The goal of this manuscript is to develop a nonclassical size dependent model to study and analyze the dynamic behaviour of the perforated Reddy nanobeam under moving load including the length scale and microstructure effects, that not considered before. The kinematic assumption of the third order shear deformation beam theory in conjunction with modified continuum constitutive equation of nonlocal strain gradient (NLSG) elasticity are proposed to derive the equation of motion of nanobeam included size scale (nonlocal) and microstructure (strain gradient) effects. Mathematical expressions for the equivalent geometrical parameters due to the perforation process of regular squared pattern are developed. Based on the virtual work principle, the governing equations of motion of perforated Reddy nanobeams are derived. Based on Navier's approach, an analytical solution procedure is developed to obtain free and forced vibration response under moving load. The developed methodology is verified and checked with previous works. Impacts of perforation, moving load velocity, microstructure parameter and nonlocal size scale effects on the dynamic and vibration responses of perforated Reddy nanobeam structures have been investigated in a wide context. The obtained results are supportive for the design of MEMS/NEMS structures such as frequency filters, resonators, relay switches, accelerometers, and mass flow sensors, with perforation.

Experimental, analytical and numerical investigation of pultruded GFRP composite beams infilled with hybrid FRP reinforced concrete

This paper presents the experimental, analytical and numerical analyses on the behavior of the pultruded GFRP composite beams infilled with hybrid fiber reinforced concrete under four-point loading. The examined experimental variables included: (1) effect of pultruded GFRP box profile, (2) effect of conventional steel bars, GFRP bars and hybrid bars, (3) effect of externally GFRP wrapping. Pursuant to this goal, a total of 9 medium-sized beam specimens were constructed. The experimental tests were simulated using finite element models with ABAQUS program. A novel analytical method for analyzing the flexure behavior of beams has been presented. Applying the first order shear deformation beam theory and introducing auxiliary functions, the equation of motion is derived using the Hamiltonian approach. Initial stiffness, ductility, energy dissipation capacity were compared based on the load–displacement relationship. The pultruded profile significantly enhanced the behavior of traditional reinforced concrete beam and GFRP composite wrapping also considerably improved the behavior of the pultruded profile infilled with reinforced concrete beam. More importantly is that hybrid reinforcements are recommended for all cases. Moreover, detailed damage analyses are provided.

Examination of thermal postbuckling of temperature dependent FG-GRMMC laminated beams with negative Poisson’s ratio on elastic foundations

This paper examines the in-plane negative Poisson’s ratio (NPR) effect on the thermal postbuckling of graphene-reinforced metal matrix composite (GRMMC) laminated beams subjected to a uniform temperature rise and supported by an elastic foundation. GRMMC layers having different graphene volume fractions can facilitate the formation of functionally graded (FG) patterns of the beams. The temperature-dependent material properties of GRMMC layers can be estimated by a nonlinear function of temperature from the results of the molecular dynamics simulations. The governing equations for the beam thermal postbuckling problem can be derived based on a higher order shear deformation beam theory and solved by applying the two-step perturbation approach. The von Kármán geometric nonlinearity, the elastic foundation support and the thermal effect are considered in the modeling. Numerical investigations are carried out for (10/-10/10/-10/10)S and (10/-10/10)S GRMMC beams with in-plane NPR. The results show that the buckling temperature of (10/-10/10)S UD beam is slightly higher than that of (10/-10/10/-10/10)S UD beam. The thermal postbuckling capacity for the (10/-10/10)S FG-X beam becomes much higher than that of the (10/-10/10/-10/10)S FG-X beam, when the beam deflection is sufficiently large.

Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics

This paper presents a comprehensive study on bending, vibration and buckling behaviours of the zigzag and armchair nanobeams. Based on a third order shear deformation beam theory and the doublet mechanics, a finite element model is proposed and employed to solve the problems of zigzag and armchair nanobeams with various boundary conditions for the first time. The verification is performed by comparing the numerical results with those from the previous studies with respect to various boundary conditions. A number of numerical examples on zigzag and armchair nanobeams with four boundary conditions have been carried out. The effects of material length scale parameters, aspect ratio, nanobeam model and boundary conditions on the displacements, natural frequencies and buckling loads of nanobeams are investigated in details. Some new results, which are not available in open literature, are provided as references for the future studies.

Snap-through analysis of thermally postbuckled graphene platelet reinforced composite beam

Present investigation deals with the snap-through phenomenon in a thermally pre-buckled or post-buckled reinforced composites beam. Composite laminated beam is reinforced with graphene platelets (GPLs) where the weight fraction of GPLs may vary in each layer. The governing equations of the beam are obtained by means of the first order shear deformation beam theory and the von-Kármán type of kinematic assumptions. The conventional Ritz method with Chebyshev polynomials is used to discretize the governing equations and establish a set of non-linear equations dealing with the response of a beam under the action of thermal and mechanical loads. Furthermore, the case of uniform temperature elevation in the beam and uniform lateral pressure is considered. At first, the response of the beam under uniform temperature elevation is obtained. Depending on the magnitude of temperature rise, beam may remain flat or undergo thermal post-buckling. Using the displacement control strategy, the non-linear eigenvalue problem is solved. After that the response of the pre-buckled or post-buckled beam subjected to uniform lateral pressure in large deflection is evaluated. Due to the possibility of snap-through phenomenon, the cylindrical arch-length technique is employed. At first, two comparison studies are provided to assure the validity and efficiency of the developed formulation. After that, novel numerical results are given to discuss the effects of temperature change, weight fraction of GPLs, distribution pattern of GPLs and boundary conditions. It is shown that the limit loads of the snapping phenomenon and the intensity of the snap-through phenomenon are highly dependent to the mentioned factors.

Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam

This work focused on the novel numerical tool for the bending responses of carbon nanotube reinforced composites (CNTRC) beams. The higher order shear deformation beam theory (HSDT) is used to determine strain-displacement relationships. A new exponential function was introduced into the carbon nanotube (CNT) volume fraction equation to show the effect of the CNT distribution on the CNTRC beams through displacements and stresses. To determine the mechanical properties of CNTRCs, the rule of the mixture was employed by assuming that the single-walled carbon nanotubes (SWCNTs)are aligned and distributed in the matrix. The governing equations were derived by Hamilton's principle, and the mathematical models presented in this work are numerically provided to verify the accuracy of the present theory. The effects of aspect ratio (l/d), CNT volume fraction (Vcnt), and the order of exponent (n) on the displacement and stresses are presented and discussed in detail. Based on the analytical results. It turns out that the increase of the exponent degree (n) makes the X-beam stiffer and the exponential CNTs distribution plays an indispensable role to improve the mechanical properties of the CNTRC beams.

Optimal weight for buckling of FG beam under variable axial load using Pareto optimality

In real situation, magnitude, distribution and/or location of axial loads causing a buckling are not known precisely, due to a lack of loading under different operational conditions. Thus, functionally graded structures have to be designed optimally in order to realize their potential. For this reason, this article investigates buckling of functionally graded (FG) beam under variable axial load and optimizes its wight to get the maximum buckling load. The gradation of beam through the thickness direction is described by different function distributions (i.e.; symmetric power and sigmoidal functions). Equations of equilibrium are developed on the basis of higher order shear deformation beam theory by using Hamilton’s principles. Load distribution through the axial direction is portrayed by step, linear and parabolical continuous functions. Numerical differential quadrature method (DQM) is used to solve the equilibrium equations and get buckling loads. Multiobjective optimization procedure is performed to design FG beams under varying axial load for maximum buckling load and minimum weight objectives. Pareto front representation of the best light weight design of a beam under variable axial load that maximizes the critical buckling load parameter is predicted through optimization process. The proposed model is efficient in analysis, design and optimizing an isotropic and FG beam under variable in-plane load.

Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects

A novel approach for the vibration and buckling analysis of nanobeams with surface stress effects is presented in this paper. The material model is derived in which the bulk core of a beam is assumed to be a single crystalline material and have certain crystallographic directions. The beams are assumed to be covered by atomic layers with the same crystallographic directions on the top and bottom surfaces. A generic third order shear deformable beam theory is employed to derive the total energy functional of the beams. Minimizing the total potential energy functional by applying the p-Ritz method, the eigenvalue equations for the buckling and vibration of nanobeams with surface stress effects are obtained. The buckling and vibration behaviours of the beams considering the surface stress effects are discussed in detail and are compared with the results obtained from the first order beam theory and the Reddy's third order beam theory. It is found that in certain cases, both the first order and the Reddy's third order beam theories may not be able to capture the surface effect accurately on the vibration and buckling behaviours of the nanobeams. The effects of surface stresses on the buckling and vibration behaviours for nanobeams made of Ni single crystals obtained by the generic third order beam theory showed opposite trends to the ones obtained by the first order or the Reddy's third beam theory.

Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory

In this work, thermally induced dynamic behaviors of functionally graded flexoelectric nanobeams (FGFNs) are analyzed theoretically while considering the neutral surface concept and the von Karman nonlinearity induced by thermal environment. The temperature field is assumed to vary only in the thickness direction by solving a simple steady state heat transfer equation and to be constant in the plane of the beam. The temperature-dependent material properties of FGFNs are assumed to vary continuously throughout the thickness according to a power-law form. For such FGFNs, the nonlocal simplified strain gradient elasticity theory to capture the effect of size-dependent and, the higher order shear deformation beam theory to account for rotary inertia and transverse shear strains are adopted to formulate the governing equations and associated boundary conditions, which are solved by using a two-step perturbation method. In the numerical part, comparison study is also performed to verify the present theoretical model and the parametric analysis is systematically studied. It is also found that the thermally induced bending amplitude, nonlinear frequency and frequency ratio depend enormously on the material distribution profile, the flexoelectricity, the size-dependent effect and the imposed temperature field.

Nonlinear Vibration of Temperature-Dependent FG-CNTRC Laminated Beams with Negative Poisson’s Ratio

Laminated beams made of nanocomposite materials have been used in many industrial sectors. This paper reports a study on the vibration behavior of laminated beams when experiencing the large amplitude vibration. The beams are made of perfectly bonded carbon nanotube-reinforced composite (CNTRC) layers. The novel constructions of CNTRC laminated beams with out-of-plane maximum negative Poisson’s ratio (NPR) are proposed. The volume fraction of CNT may change across the beam thickness which results in a piece-wise pattern. The material properties of the CNTRC layers are temperature-dependent and can be estimated by the extended rule of mixture model. The beams are considered to rest on a two-parameter elastic foundation and under differential thermal environmental conditions. The higher order shear deformation beam theory is applied to derive the motion equations of the nonlinear vibration of FG-CNTRC laminated beams. These equations include the influencing factors such as the geometrical nonlinearity in the von Kármán sense, the thermal effects and the beam–foundation interaction. The nonlinear vibration solutions can be obtained by employing a two-step perturbation approach. The nonlinear vibration characteristics of FG-CNTRC laminated beams under different sets of loading conditions and thermal environmental conditions are discussed in detail through a series of parametric studies. Numerical results show that the NPR has a significant effect on the large amplitude vibration characteristics of CNTRC laminated beams.

Experimental and theoretical investigation on flexure performance of pultruded GFRP composite beams with damage analyses

Fiber reinforced polymer (FRP) composite materials produced by pultrusion technique attract the attention of researchers due to their superior properties. Limited studies are available in the literature where experimental and theoretical investigations on the FRP composites are carried out together. In this study, the flexure performances of pultruded glass FRP (P-GFRP) composite beams were investigated experimentally and theoretically. Theoretical approaches for flexural analysis of P-GFRP composite beams were developed with the help of variational methods. Kinematic relations of composite beams were defined based on high order shear deformation beam theory. Effective material properties of composite beams were obtained by using mixture rule model. The differential field equations were transformed to functional with Gâteaux differential method. The element matrix with a total of 10 degrees of freedom was obtained by using the mixed finite element method (MFEM). Moreover, classical finite element modeling (FEM) was performed with the help of ABAQUS program. Various beam specimens with different fiber orientations were extracted from the P-GFRP box profile. Tensile and burnout tests were performed to determine the mechanical properties of the obtained P-GFRP composite beam specimens. Three-point bending test was utilized to investigate flexure performance. Kinetic, macro and micro mechanical damage analyzes were performed to better understand the behavior of the P-GFRP composite beams. Experimental results, theoretical solution results and FEM simulation results were compared and numerical values ​​were found to be very close to each other.