Two-node Beam Element Research Articles

Homogenized Stiffness Matrix of Two-Node Elements through Experimental Flexibility tests

Homogenization is a numerical technique to obtain the equivalent mediated response of a complex material. However, there are some scenarios in which the complexity of the physical phenomena present in the real structure makes it difficult to generate a faithful Representative Volume Element (RVE). In these cases, it may turn easier and more reliable to perform a homogenization directly by experimental test results. For this reason, we present a procedure to build-up the whole stiffness matrix starting from experiments. This approach is here discussed to get a homogenised two-node beam element. The procedure requires some flexibility measurements, thus allowing fewer measurements if compared to stiffness-based approach. The method is verified with some experiments carried on a frame structure. The comparisons with a Finite Element model build up with stiffness matrix assembly demonstrates the validity and robustness of the proposed procedure.

A Quasi-3D theory for bending, vibration and buckling analysis of FG-CNTRC and GPLRC curved beams

This paper proposes a finite element model based on Quasi-3D theory, which includes both normal and shear effects, to study the bending, vibration, and buckling responses of FG-CNTRC and GPLRC curved beams. A two-node beam element satisfying C1 continuity requirement is utilized to compute displacements, critical buckling loads, and natural frequencies for beams with various boundary conditions. To evaluate the precision of the suggested model, various numerical examples are conducted. Next, a comprehensive parameter study is carried out to study the effects of distribution patterns and fractional volume of reinforced materials, open angles, aspect ratios, and boundary conditions. Numerous numerical results can serve as benchmarks for subsequent investigations.

Static and buckling behaviors analysis of FG beams using a three unknowns finite element based on enhanced Timoshenko theory

A two-node beam element based on enhanced Timoshenko beam theory with three unknowns is developed for static and buckling behaviors of functionally graded beams. The model considers a quadratic variation of shear strain through the depth, and satisfies zero-shear strain at the top and bottom surfaces of the beam. The concept of a physical neutral axis is introduced to avoid stretching-bending phenomenon. The elementary matrices are derived utilizing the minimum potential energy. The accuracy and performance of the present model are verified through a series of examples. Furthermore, the effect of different parameters on behaviors of beams is discussed.

Dynamic finite element modeling of axially functionally graded Timoshenko microbeams under a moving mass

This paper presents a finite element model for dynamic analysis of axial functionally graded (AFG) microbeams subjected to amovingmass. Thematerial properties of themicrobeams are considered to vary in the axial direction by a power-law function, and they are evaluated byMori-Tanaka scheme. The first-order shear deformation theory is employed in conjunction with the modified couple stress theory to establish the differential equations of motion for the AFG microbeams. A two-node beam element with six degrees of freedom was derived and used in combination with Newmark method to solve the equations of motion and to compute the dynamic response of the microbeams. Numerical investigations are carried out on AFG microbeam with simply supported ends. The proposed method is validated by comparing with results in previous work. The effects of dimension-less material length scale parameters, the material distribution and the moving mass parameters on the dynamic characteristics of the AFG microbeams are studied and discussed in detail.

Analysis of isotropic stiffened plate structure

The analysis of stiffened plate has been carried out using finite element method. The study is divided into static and dynamic analyses. Primarily, free vibration frequencies of the stiffened plate have been determined followed by determination of deflection and von Mises stress for the stiffened plate subjected to the unit uniformly distributed load (1 kN/m2). Further, deflection and von Mises stress are determined, when the plate structure is subjected to hydrostatic loading. The analyses, static and dynamic, have been done using ANSYS Workbench 15.0. A single flat stiffener is used to stiffen the plate structure and the results are evaluated by varying the stiffener geometry, keeping the volume of material almost same as the unstiffened plate. Stiffened plate structure is very popular in the structural engineering domain and has a wide range of engineering applications from ship to aerospace structures. The material used to model stiffener is extracted from the primary plate for keeping the same volume of material. The plate structure is stiffened to reduce the out-of-plane bending even if the same (approximately) amount of material is utilised. Eight-noded plate bending and two-noded beam elements are used to model the plate and stiffener, respectively. The obtained results are compared with the published results and the effect of varying stiffener geometry is further examined. It may be concluded that the response of the stiffened plates are better compared to unstiffened plate even if the same or even less quantity of material is utilised in stiffening. The present findings are useful for the designers when the plate structure is influenced by the surrounding fluid.

Dynamics of functionally graded beams carrying a moving load with influence of porosities and partial foundation support

The novelty of the present work is to study the simultaneous influence of porosities and partial Pasternak foundation support on dynamics of functionally graded (FG) beams carrying a moving load. The beams are made from an open-cell steel foam with symmetric and asymmetric porosity distributions in the thickness direction. Based on a refined third-order shear deformation theory, a two-node beam element with ten degrees of freedom is derived and employed to construct the discretized equation of motion for the beams. Dynamic characteristics, including the time histories for mid-span deflection, dynamic magnification factor (DMF) and the stress distribution, are computed with the aid of the Newmark method. The numerical result reveals that the foundation supporting length has an important role on the dynamics of the beams, and the dependence of the DMF upon the porosity coefficient is governed by the foundation supporting length. It is also found that the asymmetric porosity distribution has more impact on the dynamic response of the beams than the symmetric one does, and the difference between the DMFs obtained from the two porosity distributions is more significant for the beam with a higher porosity coefficient. The effects of the porosities, the foundation support and the moving load velocity on the dynamic behavior of the beams are examined in detail and highlighted

Bending, buckling and free vibration behaviours of 2D functionally graded curved beams

The flexural, free vibration and buckling responses of 2D-FG curved beams with various shear deformation theories are presented by using finite element model. Their material properties via both directions (length and thickness) are varied with power-law distribution. A two-node beam element satisfying C1 continuity requirement is employed to solve the problems. Various problems including isotropic, 1D- and 2D-FG curved beams are analysed and the obtained results are verified with those available in the literature. Comprehensive parameter examinations are carried out to depict the effects of gradation indexes in both directions, open angles, end conditions and aspect ratios on structural behaviours of 2D-FG curved beams.

Static and Dynamic Stability Analyses of Functionally Graded Beam with Inclined Cracks

The focus of this paper is to examine the static and dynamic instabilities of functionally graded beam that contains multiple inclined cracks under the influence of an axial force comprising both static and time-varying harmonic components. The elasticity modulus and mass density of the functionally graded beam are assumed to vary exponentially along its thickness direction. Local stiffness matrix model-based finite element analysis (FEA) is conducted to determine the bending stiffness and tensile stiffness of the section with a crack, and the coupled effect of tensile and bending loadings. Two-node beam elements with three degrees-of-freedom per node are utilized. By combining the Euler–Bernoulli beam theory with Lagrange method, we derive the governing equations that describe the static and dynamic instabilities of a functionally graded beam with multiple inclined cracks. These equations can be solved as eigenvalue problems to obtain the natural frequency and static critical buckling load of the beam. Furthermore, to investigate the dynamic instability of the system, we use the Bolotin method to determine the boundary between the regions of instability and stability based on the same governing equations. By adopting this approach, the study comprehensively investigates the impacts of crack position, inclination angle, and length, as well as elasticity modulus ratio, static and dynamic load factors on both static and dynamic stabilities of a cracked functionally graded beam to gain valuable insights into the stability and performance of cracked functionally graded structures.

Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams

ABSTRACT This paper presents the hygro-thermal vibration analysis of bidirectional functionally graded (BDFG) porous nanobeams resting on elastic foundation based on a nonlocal strain gradient theory and the finite element method. The effects of temperature and moisture on structures are assumed to cause tension load in the plane and do not change the material's mechanical properties. Nanobeams are made from functionally graded (FG) materials which the mechanical properties change along with the thickness (z-axis) and length (x-axis) of nanobeams according to the power-law model. The porosity distribution is considered to be even and uneven. For the first time, a two-node beam element with nine degrees of freedom at each node is developed to analyze the hygro-thermal vibration behavior in the nanoscale. The elastic foundation is used in this work to be the Winkler-Pasternak type. Applying Hamilton's principle based on the refined higher-order shear deformation beam theory (RBT) and the~nonlocal strain gradient model, governing equations of nanobeams are derived. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. The nonlocal coefficient makes the nanobeam softer, while the strain gradient coefficient makes it stiffer. In addition, the effects of the nonlocal coefficient, porosity coefficient, foundation stiffness, boundary conditions on the natural frequency of nanobeams are investigated in detail.

Vibration and wave propagation in functionally graded beams with inclined cracks

This paper proposes an FEA method to study the vibration and wave propagation in functionally graded material (FGM) beams with multiple inclined cracks by introducing the local flexibility matrix caused by the cracks. The bending and tensile stiffnesses of the inclined cracks and their interaction are taken into consideration. Two-node (three degrees of freedom per node) beam element is used. The inclined cracks are equivalent to transversal cracks to calculate the additional flexibility matrix of the inclined cracked FGM beam element. The material properties of the FGM beam are supposed to satisfy the exponential law along its thickness. The governing equations of motion for the FGM beam with multiple inclined cracks are deduced in the frame of Euler-Bernoulli theory and Lagrange function, and numerically solved by the Newmark average acceleration method. Both theoretical and numerical comparisons are presented to validate the present method for the inclined cracked FGM beam with varying boundary conditions, arbitrary inclined crack angles as well as varying inclined crack lengths. The effects of the location, length, inclined angle and number of the inclined cracks as well as material property ratio on the fundamental frequencies and wave propagation characteristics of the inclined cracked cantilever FGM beams are comprehensively investigated.

Dynamics of hierarchical beam lattice structures by an exact reduced-order dynamic-stiffness model

An exact reduced-order dynamic-stiffness model is proposed to study the dynamics of two-dimensional hierarchical beam lattices. Upon modelling every member of the lattice by a single exact two-node beam element, the model retains a few primary nodes of the unit cell and involves an exact dynamic condensation of the nodal degrees of freedom of the lattice structure internal to the unit cell and connected to the primary nodes. It is shown that the modes of the reduced-order model exhibit two orthogonality conditions, with dynamic-stiffness terms pertinent to the condensed lattice structures. The orthogonality conditions are the basis to decouple the motion equations and derive the exact modal response to external loading, in both time and frequency domains, under the assumption of proportional damping. The modal response is obtained in simple and elegant analytical form for the whole lattice, including the condensed lattice structures internal to the unit cells. Moreover, it can be calculated very expeditiously thanks to an exact representation of the eigenfunctions based on frequency dependent shape functions, which applies regardless of geometry and complexity of the hierarchical lattice structure. Numerical results confirm the exactness of the proposed reduced-order dynamic-stiffness model by comparison with standard finite-element solutions in ABAQUS.

Influence of temperature and porosities on free vibration of a three-phase bidirectional functionally graded sandwich beam

Understanding the influence of practical factors on natural frequencies of beams play an important role in design of this structure. This article explores the influence of temperature and porosities on vibration characteristics of a sandwich beam made from a three-phase bidirectional functionally graded material (BFGSW beam) for the first time. The sandwich beam composed from a homogeneous core and two face layers made of a three-phase composite material. The material properties are graded in both the beam axis and thickness by power-law functions, and they are evaluated by using the Voigt model. Heat loading with uniform temperature rise and even distribution of porosities are also considered. The expressions of the elastic strain energy and the kinetic energy for the beam as well as the strain energy due to temperature are obtained in the framework of the hyperbolic shear deformation theory. A two-node beam element with eight degrees of freedom has been proposed and used to establish the discretized equation of motion for the beam. The proposed method is validated by comparing the fundamental frequency parameters with two previous works. The beam with simply supported ends are used in numerical studies. The convergence of the derived beam element is represented by evaluating the fundamental frequency parameters. The effects of the temperature rise, the porosity volume fraction, the material indexes, the length to height ratio as well as the beam layer thickness ratio on the vibration characteristics are investigated and discussed in detail. It is concluded that the BFGSW beam parameters, the temperature rise and porous parameter play an important role in the natural frequency, which is helpful for the design of beam – like structures with the desired natural frequency.

Dynamic response of FG-CNTRC beams subjected to a moving mass

This article presents the forced vibration of composite beams reinforced by single-walled carbon nanotubes (SWCNTs) and subjected to a moving mass. Considering the distribution of carbon nanotubes such as uniform (UD-CNT), functionally graded Λ (FGΛ-CNT) and X (FGX-CNT), three different beams are studied. Based on a third-order shear deformation theory (TSDT), the motion equations of the beams are derived using Hamilton's principle. Including mass interaction forces, the motion equations are transformed into a finite element equation in which a two-node beam element with eight degrees of freedom is utilized. To improve the efficiency of the beam element, the transverse shear rotation is employed as an independent variable in the derivation of the beam element. The vibration characteristics, including the dynamic magnification factors and the time histories for mid-span deflections are computed by using the Newmark method. Numerical result reveal that the vibration of the beams is clearly influenced of the CNT reinforcement, and the dynamic magnification is significantly decreased by increasing the CNT volume fraction. It is also shown that the FGX-CNT beam is the best in dynamic resistance in terms of the lowest dynamic deflection and dynamic magnification factors. The effects of the total volume fraction and the moving load velocity on the dynamic behaviour of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) beams are examined in detail and highlighted.

Efficient beam element model for analysis of composite beam with partial shear connectivity

A two-node beam element formulation with eight degrees of freedom was developed and implemented into MATLAB to simulate the nonlinear behavior of the composite. The beam element includes the reinforced concrete slab, the concrete reinforcement, the steel beam, and the shear connectors. One dimension-softening model was used to simulate the concrete, the bilinear isotropic plastic model was used to simulate the steel, and a multi-linear spring was used to simulate the partial shear connectivity. Experimental results were used to validate the numerical model, which was able to closely predict the experimental response by a 2% difference. The model was able to simulate several material nonlinearities such as the plasticity of the steel beam and cracking and crushing of concrete slab. A parametric study was performed to investigate the effects of strength of shear connector, concrete slab thickness, steel beam depth, and the concrete compressive strength on the overall response of steel–concrete composite beams. It was found that the variation in shear connector strength significantly affects the response of the composite beams. In addition, the composite beam stiffness and strength increased with the increase of the steel beam depth, significantly.

Dynamic modeling of a fish tail actuated by IPMC actuator based on the absolute nodal coordinate formulation

Ionic polymer metal composite (IPMC) is a smart material with low driving voltage, high energy conversion, light weight and simple structure. It is suitable for driving the tail fin of small bionic fish. Actuators made of this material are quite flexible, and they are laborious to accurately describe the complex deformation of structures. Absolute nodal coordinate formulation (ANCF) has great merits in describing large deformation and large displacement problems. In this paper, the feasibility of ANCF in dynamic modeling of a fish tail actuated by IPMC actuator will be explored for the first time. The tail fin is simplified into two forms: a pure flexible IPMC beam and a hybrid beam that consists of an IPMC beam and a fixed rigid beam. The ANCF one-dimensional two-node beam element is adopted. Considering the influence of deformation curvature, the exact expression of generalized elastic force is obtained. The proposed dynamic model can describe the large deformation problem of the flexible beam. The dynamic responses of the pure IPMC beam tail fin and the hybrid tail fin driven by square wave voltage are calculated. The displacement changes of end nodes are compared, and the swing law under different driving voltage amplitude, frequency, and fluid resistance is analyzed. It is found that the driving efficiency of IPMC can be improved by increasing the driving voltage amplitude, reducing the voltage frequency, and reducing the fluid resistance in a certain range. This research would be beneficial to the study of the large swing motion mechanism of bionic robotic fish.

Free and Forced Vibration Analyses of Functionally Graded Graphene-Nanoplatelet-Reinforced Beams Based on the Finite Element Method.

The finite element method (FEM) is used to investigate the free and forced vibration characteristics of functionally graded graphene-nanoplatelet-reinforced composite (FG-GPLRC) beams. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary continuously along the beam thickness according to a linear, parabolic, or uniform pattern. For the FG-GPLRC beam, the modified Halpin–Tsai micromechanics model is used to calculate the effective Young’s modulus, and the rule of mixture is used to determine the effective Poisson’s ratio and mass density. Based on the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory, finite element formulations are derived to analyze the free and forced vibration characteristics of FG-GPLRC beams. A two-node beam element with six degrees of freedom is adopted to discretize the beam, and the corresponding stiffness matrix and mass matrix containing information on the variation of material properties can be derived. On this basis, the natural frequencies and the response amplitudes under external forces are calculated by the FEM. The performance of the proposed FEM is assessed, with some numerical results obtained by layering method and available in published literature. The comparison results show that the proposed FEM is capable of analyzing an FG-GPLRC beam. A detailed parametric investigation is carried out to study the effects of GPL weight fraction, distribution pattern, and dimensions on the free and forced vibration responses of the beam. Numerical results show that the above-mentioned effects play an important role with respect to the vibration behaviors of the beam.

Size-dependent behavior of a MEMS microbeam under electrostatic actuation

The size-dependent behavior of a silicon microbeam with an axial force in MEMS is studied using a nonlinear finite element procedure. Based on a refined third-order shear deformation theory and the modified couple stress theory (MCST), nonlinear differential equations of motion for the beam are derived from Hamilton’s principle, and they are transferred to a discretized form using a two-node beam element. Newton-Raphson based iterative procedure is used in conjunction with Newmark method to obtain the pull-in voltages and deflections of a clamped-clamped microbeam under electrostatic actuation. The influence of the axial force, applied voltage and material length scale parameter on the behavior of the beam is studied in detail and highlighted.

Free vibration of axially loaded zigzag and armchair nanobeams using doublet mechanics

Based on doublet mechanics theory and various beam theories, this paper presents free vibration of axially loaded zigzag and armchair nanobeams. A two-noded higher order beam element is used to solve the problems of nanobeams with various boundary conditions. The verification studies in terms of fundamental frequencies and buckling loads are performed by comparing the results by the present model with Molecular Dynamic Simulations, Doublet Mechanics, and Eringen’s nonlocal theory. The effects of material length scale parameter, slenderness ratio, nanotube model and boundary conditions on the fundamental frequencies of axially loaded nanobeams are investigated in details. Softening material behavior is detected for zigzag and armchair nanobeams. Having stiffer boundary conditions lead to the effect of material length scale parameter to be observed more prominently.

Wave propagation in stress-driven nonlocal Rayleigh beam lattices

This paper focuses on small-size planar beam lattices, where size effects are modelled by the stress-driven nonlocal elasticity theory in conjunction with the Rayleigh beam theory. The purpose is to propose two novel computational approaches for elastic wave propagation analysis. In a first dynamic-stiffness approach, every lattice member is modelled by a unique two-node beam element, the exact dynamic-stiffness matrix of which is built solving, in concise analytical form, the stress-driven differential equations of motion. In a second finite-element approach, every lattice member is discretized by an increasingly refined mesh of two-node beam elements; in this case, the stiffness and mass matrices of the lattice member are obtained from shape functions built based on the exact solutions of the stress-driven differential equations for static equilibrium. Advantages of the two approaches are compared and discussed. Dispersion curves are calculated for a typical planar lattice, highlighting the role of nonlocality.

On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory

In the present paper, a novel refined shear deformation beam theory is proposed and applied, for the first time, to investigate the bending behavior of functionally graded (FG) sandwich curved beam. The present theory is exploited to satisfy parabolic variation of shear stress distribution along the thickness direction thereby obviating the use of any shear correction factors. The material properties of FG sandwich beam change continuously from one surface to another according to a power-law function. Three common configurations of FG beams are considered for the study, namely: (a) single layer FG beam; (b) sandwich beam with FG face sheets and homogeneous core and (c) sandwich beams with homogeneous face sheets and FG core. The governing equations derived herein are solved by employing the finite element method using a two-noded beam element, developed for this purpose. A wide range of comparative studies is carried-out to establish the robustness and the performance of the present finite element model. A comparison study shows that the proposed model is: (a) accurate and comparable with the literature; b) of fast rate of convergence to the reference solution; c) excellent in terms of numerical stability and d) valid for both thin and thick FG sandwich curved beams. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, material distributions, length-to-thickness ratio, face-to-core- thickness ratio, loadings and boundary conditions on the static response of FG curved sandwich beam. New referential results for thin and thick FG sandwich curved beam are offered, in which can be used to establish benchmarks for future research.