Higher-order Beam Model Research Articles

Modeling and Analysis of Multilayered Elastic Beam Using Spectral Finite Element Method

Multilayered elastic structures are widely used in engineering applications. In this paper, a spectral finite element model (SFEM) is developed to predict the dynamic behavior of a multilayered beam structure. First, a higher-order multilayered beam model is derived. Each layer is modeled as a Timoshenko beam, in which both shear deformation and rotational inertia are considered. By allowing different rotation in each layer, the overall sectional warping effect is included as well. A set of fully coupled governing equations presented in a compact form and associated boundary conditions are obtained by the application of Hamilton's principle. Second, a semi-analytical solution of these equations is determined and used in formulating the SFEM. The SFEM predictions are validated against the nastran results and other results in literature. Compared to the conventional FEM (CFEM), a very small number of elements are required in the SFEM for comparable accuracy, which substantially reduce the computing time, especially for simulations of high-frequency wave propagations. Finally, the SFEM is used to predict the lamb wave responses in multilayered beams. Wave propagation characteristics in both undamaged and damaged cases are well captured. In summary, the SFEM can accurately and efficiently predict the behavior of multilayered beams and serve as a framework to conduct wave propagation prediction and damage diagnostic analysis in structural health monitoring (SHM) applications.

Surface and shear energy effects on vibrations of magnetically affected beam-like nanostructures carrying direct currents

Free vibration of current-carrying nano-scaled beams immersed in a magnetic field is of huge interest. To bridge this scientific gap, Rayleigh, Timoshenko, and higher-order beam models accounting for the surface energy are employed and their equations of motion are established appropriately. For spatial discretization of the deformations fields, a meshless approach is exploited. The effects of surface and shear deformation, electric current, magnetic field strength, and geometric parameters of the nanobeam on the first ten natural frequencies are examined and discussed. The critical values of slenderness ratio, nanobeam's diameter, electric current, and magnetic field strength, which are corresponding to the dynamically unstable nanobeams, are graphically identified. The obtained results indicate that the discrepancies between the frequencies by considering the surface effect and those evaluated without consideration of the surface energy would increase notably at the above-mentioned critical values.

A Thermal Stress Analysis of Three-Dimensional Beams by Refined One-Dimensional Models and Strong Form Solutions

This paper investigates the mechanical behaviour of three-dimensional beams subjected to thermal stresses.The temperature field is obtained by exactly solving Fourier's heat conduction equation and, as classically done by a staggered solution approach, it is considered as an external load within the mechanical analysis.Several higher-order beam models are derived thanks to a compact notation for the a-priori approximation of the displacement field upon the cross-section.The governing differential equations and boundary conditions are obtained in a compact nuclearform using the Principle of Virtual Displacement.The final form does not depend upon the order of approximation of the displacement fieldover the cross-section (this latter being a free parameter of the proposed modelling approach).The obtained problem is solved by means of two strong form solutions: an analytical Navier-type solution andpoint collocation (using Wendland's radial basis functions).Isotropic, functionally graded and laminated beams are considered.Results are validated towards three-dimensional FEM solution obtained by ANSYS.The proposed models yield accurate results characterised by smooth stresses thanks to the used solution methods.Furthermore, computational costs are very attractive when compared to the reference three-dimensional solutions.

Hierarchical Beam Finite Elements Based Upon a Variables Separation Method

A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

Buckling of thin-walled structures through a higher order beam model

A higher order beam model for the buckling analysis of thin-walled structures is presented in this paper.The model considers an enrichment of the displacement field so as to accurately represent the three-dimensional behaviour of thin-walled structures. The definition of an uncoupled set of deformation modes allows a meaningful definition of hierarchical higher order solutions, which are useful for the linear buckling analysis of thin-walled structures. A criterion for the definition of local and global buckling modes, as well as possible interaction between modes is put forward. A comparison between the results obtained with the higher order beam model and results obtained from a shell finite element model implemented in Abaqus allows to conclude not only the efficiency of the beam model but also its simplicity of use.

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials

This paper presents a new class of refined beam theories for static and dynamic analysis of composite structures. These beam models are obtained by implementing higher-order expansions of Chebyshev polynomials for the three components of the displacement field over the beam cross-section. The Carrera Unified Formulation (CUF) is adopted to obtain higher-order beam models. The governing equations are written in terms of fundamental nuclei, which are independent of the choice of the expansion order and the interpolating polynomials. Static and free vibration analysis of laminated beams and thin-walled boxes has been carried out. Results obtained with the novel Chebyshev Expansion (CE) model have been compared with those available in the literature. For comparison, Taylor-like Expansion (TE) and Lagrange Expansion (LE) CUF models, commercial codes, analytical and experimental data are exploited. The performances of refined beam models in terms of computational cost and accuracy in comparison to the reference solutions have been assessed. The analysis performed has pointed out the high level of accuracy reached by the refined beam models with lower computational costs than 2D and 3D Finite Elements.

Column buckling of magnetically affected stocky nanowires carrying electric current

Axial load-bearing capacity of current carrying nanowires (CCNWs) acted upon by a longitudinal magnetic field is of high interest. By adopting Gurtin–Murdoch surface elasticity theory, the governing equations of the nanostructure are constructed based on the Timoshenko and higher-order beam models. To solve these equations for critical compressive load, a meshfree approach is exploited and the weak formulations for the proposed models are obtained. The predicted buckling loads are compared with those of assume mode method and a remarkable confirmation is reported. The role of influential factors on buckling load of the nanostructure is carefully addressed and discussed. The obtained results reveal that the surface energy effect becomes important in buckling behavior of slender CCNWs, particularly for high electric currents and magnetic field strengths. For higher electric currents, relative discrepancies between the results of Timoshenko and higher-order beam models increase with a higher rate as the slenderness ratio magnifies.

Free vibration analysis of civil engineering structures by component-wise models

Abstract Higher-order beam models are used in this paper to carry out free vibration analysis of civil engineering structures. Refined kinematic fields are developed using the Carrera Unified Formulation (CUF), which allows for the implementation of any-order theory without the need for ad hoc formulations. The principle of virtual displacements in conjunction with the finite element method (FEM) is used to formulate stiffness and mass matrices in terms of fundamental nuclei. The nuclei depend neither on the adopted class of beam theory nor on the FEM approximation along the beam axis. This paper focuses on a particular class of CUF models that makes use of Lagrange polynomials to discretize cross-sectional displacement variables. This class of models are referred to as component-wise (CW) in recent works. According to the CW approach, each structural component (e.g. columns, walls, frame members, and floors) can be modeled by means of the same 1D formulation. A number of typical civil engineering structures (e.g. simple beams, arches, truss structures, and complete industrial and civil buildings) are analyzed and CW results are compared to classical beam theories (Euler–Bernoulli and Timoshenko), refined beam models based on Taylor-like expansions of the displacements on the cross-section, and classical solid/shell FEM solutions from the commercial code MSC Nastran. The results highlight the enhanced capabilities of the proposed formulation. It is in fact demonstrated that CW models are able to replicate 3D solid results with very low computational efforts.

A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation

A new class of beam finite elements is proposed in a three-dimensional fully parameterized absolute nodal coordinate formulation, in which the distortion of the beam cross section can be characterized. The linear, second-order, third-order, and fourth-order models of beam cross section are proposed based on the Pascal triangle polynomials. It is shown that Poisson locking can be eliminated with the proposed higher-order beam models, and the warping displacement of a square beam is well described in the fourth-order beam model. The accuracy of the proposed beam elements and the influence of cross-section distortion on structure deformation and dynamics are examined through several numerical examples. We find that the proposed higher-order models can capture more accurately the structure deformation such as cross-section distortion including warping, compared to the existing beam models in the absolute nodal coordinate formulation.

Influence of Non-Structural Localized Inertia on Free Vibration Response of Thin-Walled Structures by Variable Kinematic Beam Formulations

Variable kinematic beam theories are used in this paper to carry out vibration analysis of isotropic thin-walled structures subjected to non-structural localized inertia. Arbitrarily enriched displacement fields for beams are hierarchically obtained by using the Carrera Unified Formulation (CUF). According to CUF, kinematic fields can be formulated either as truncated Taylor-like expansion series of the generalized unknowns or by using only pure translational variables by locally discretizing the beam cross-section through Lagrange polynomials. The resulting theories were, respectively, referred to as TE (Taylor Expansion) and LE (Lagrange Expansion) in recent works. If the finite element method is used, as in the case of the present work, stiffness and mass elemental matrices for both TE and LE beam models can be written in terms of the same fundamental nuclei. The fundamental nucleus of the mass matrix is opportunely modified in this paper in order to account for non-structural localized masses. Several beams are analysed and the results are compared to those from classical beam theories, 2D plate/shell, and 3D solid models from a commercial FEM code. The analyses demonstrate the ineffectiveness of classical theories in dealing with torsional, coupling, and local effects that may occur when localized inertia is considered. Thus the adoption of higher-order beam models is mandatory. The results highlight the efficiency of the proposed models and, in particular, the enhanced capabilities of LE modelling approach, which is able to reproduce solid-like analysis with very low computational costs.

Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory

In the present investigation, a numerical analysis is conducted to predict size-dependent nonlinear free vibration characteristics of third-order shear deformable microbeams made of functionally graded materials (FGMs). For this purpose, the modified strain gradient elasticity theory and von Karman geometric nonlinearity are implemented into the classical third-order shear deformation beam theory to develop a nonclassical higher-order beam model including three additional length scale parameters to capture size effect efficiently. It is assumed that the material properties of the FGM microbeams are evaluated by the Mori–Tanaka homogenization technique. On the basis of the Hamilton’s principle, the size-dependent nonlinear governing differential equations of motion and associated boundary conditions are derived and then discretized along various end supports by employing generalized differential quadrature (GDQ) method. A direct iterative process corresponding to both positive and negative deflection cycles is adopted. Secondly, a parametric study is performed to demonstrate the influences of the values of dimensionless length scale parameter, material property gradient index and length to thickness aspect ratio on the linear and nonlinear natural frequencies of FGM microbeams.

Effect of higher-order deformation gradients on buckling of single-walled carbon nanotubes

A single-walled carbon nanotube (SWCNT) is treated as a tubular Bernoulli–Euler beam and the higher-order deformation gradients are involved to establish a higher-order multiscale beam model. The higher-order Cauchy–Born rule is used to calculate the deformation of bond vectors in the representative cell and the strain energy density is equivalent to the energy per unit surface area, calculated from the Brenner potential. On the basis of the classical Bernoulli–Euler beam theory, the second-order deformation gradients with respect to the axial direction are also considered. The total strain energy is expressed as an integral equation, in which all parameters are obtained by calculating the constitutive response around the circumference. The physical meaning of these parameters is discussed in detail. The global buckling of SWCNTs is studied, and the critical buckling force is obtained as the analytical formula for different boundary conditions. The critical buckling force is plotted against the tube chirality, tube radius and tube length, and it is discovered that the contribution of the higher-order terms rapidly becomes large when the tube radius or length is small enough.

A Thermo-Mechanical Analysis of Isotropic and Composite Beams via Collocation with Radial Basis Functions

In this article, the mechanical behavior of beams subjected to thermal loads is investigated. The temperature field is obtained by exactly solving Fourier's heat conduction equation and it is considered as an external load within the mechanical analysis. Several higher-order beam models as well as Timoshenko's classical theory are derived thanks to a compact notation for the a priori approximation of the displacement field upon the cross-section. The governing differential equations and boundary conditions are obtained in a compact nucleal form that does not depend upon the displacements’ expansion order. The latter can be regarded as a free parameter of the formulation. A meshless strong-form solution based upon collocation with Wendland's radial basis functions is adopted. Isotropic and laminated orthotropic beams are investigated. Results are validated toward an analytical Navier-type solution and three-dimensional FEM results. It is shown that good accuracy can be obtained.

Comparison of various shear deformation theories for free vibration of laminated composite beams with general lay-ups

This study deals with the numerical evaluations of various higher-order shear deformation beam theories. Free vibration analysis of generally laminated composite beams with arbitrary boundary conditions is performed by using the spectral finite element method. A unified type of assumed displacement field is considered and the influences of shear deformation, rotary inertia and Poisson effect are included in the formulation. The spectral element matrix is derived by rigorous use of the analytical solutions of the governing differential equations of the laminated beam in free vibration. The application of the spectral element matrix to study the free vibration characteristics of the laminated beams is demonstrated by adopting the Wittrick–Williams algorithm. Numerical results for the natural frequencies of the illustrative examples, which are found by various higher-order shear deformation beam models, are compared with each other and with the available solutions in the literature.

Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams

This paper covers a large variety of theoretical generic beam models including some small length scale terms. Strain gradient elasticity and Eringen’s nonlocal elasticity models are applied to beam mechanics including Euler–Bernoulli, Timoshenko and higher-order shear beam models. The buckling and vibration behaviour of these generalized shear beam models is investigated for pinned–pinned boundary conditions. The variational formulation of these enriched beam models is given leading to consistent variationally-based boundary conditions. The paper first starts with the axial behaviour of gradient or nonlocal elasticity bars. The beam behaviour is then analyzed using a unified framework, where the kinematics classification is presented from a generalized gradient constitutive law. It is shown that higher-order shear beam models can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. We show the kinematics equivalence between Bickford–Reddy higher-order shear beam model and Shi–Voyiadjis higher-order shear beam model, even if both models are statically not equivalent (from the stress calculation). This equivalence is highlighted on buckling and vibrations results. The model valid for macrostructures is generalized for micro or nanostructures using some nonlocal and gradient theories to account for small scale effects, in the axial and in the bending directions. We both use the Eringen’s based integral theory and the gradient theory to derive the buckling and vibration differential equations. These two theories can be connected using a generalized hybrid nonlocal law. Eringen’s model is compared to a stress gradient model, whereas the gradient elasticity theory is typically a strain gradient theory. The nonlocal framework is also developed in a variational consistent framework, for bending, vibrations and buckling configurations. The nonlocality is shown to be equivalent to higher-order inertia modelling for the dynamics analysis. Buckling and vibrations solutions are presented for the nonlocal higher-order beam/column models with pinned–pinned boundary conditions. We finally analyse the main characteristics of both nonlocal and gradient theories to capture the small scale effects for micro and nanostructures. Stiffening or softening effect of gradient or nonlocal elasticity models are discussed for the buckling and the vibrations analyses.

PLATE BUCKLING ANALYSIS USING A GENERAL HIGHER-ORDER SHEAR DEFORMATION THEORY

The buckling of higher-order shear plates is studied in this paper with a unified formalism. It is shown that usual higher-order shear plate models can be classified as gradient elasticity Mindlin plate models, by augmenting the constitutive law with the shear strain gradient. These equivalences are useful for a hierarchical classification of usual plate theories comprising Kirchhoff plate theory, Mindlin plate theory and third-order shear plate theories. The same conclusions were derived by Challamel [Mech. Res. Commun.38 (2011) 388] for higher-order shear beam models. A consistent variational presentation is derived for all generic plate theories, leading to meaningful buckling solutions. In particular, the variationally-based boundary conditions are obtained for general loading configurations. The buckling of the isotropic or orthotropic composite plates is then investigated analytically for simply supported plates under uniaxial or hydrostatic in-plane loading. An analytical buckling formula is derived that is common to all higher-order shear plate models. It is shown that cubic-based interpolation models for the displacement field are kinematically equivalent, and lead to the same buckling load results. This conclusion concerns for instance the plate models of Reddy [J. Appl. Mech.51 (1984) 745] or the one of Shi [Int. J. Solids Struct.44 (2007) 4299] even though these models are statically distinct (leading to different stress calculations along the cross-section). Finally, a numerical sensitivity study is made.

A size-dependent shear deformation beam model based on the strain gradient elasticity theory

A new size-dependent higher-order shear deformation beam model is developed based on modified strain gradient theory. The model captures both the microstructural and shear deformation effects without the need for any shear correction factors. The governing equations and boundary conditions are derived by using Hamilton’s principle. The static bending and free vibration behavior of simply supported microbeams are investigated. Analytical solutions including Poisson effect for deflections under point and uniform loads and for first three natural frequencies are obtained by Navier solution. The results are compared with other beam theories and other classical and non-classical models. A detailed parametric study is carried out to show the influences of thickness-to-material length scale parameter ratio, slenderness ratio and shear deformation on deflections and natural frequencies of microbeams. It is observed that effect of shear deformation becomes more significant for both smaller slenderness ratios and higher modes.

A higher-order beam model for tubes

Abstract A refined beam model is proposed for tubes to improve the prediction of the transverse shear deformation and study its influence on the mechanical response of tubes. The current beam model adopts a Laurent series expansion form for the displacement field, instead of the conventional Taylor series expansion form used by most higher-order beam models. This new form can capture the geometric features of tubes very well and, more importantly, shear stress can be vanished readily on the inner and outer surfaces. Both a generalized theory and a simplified third-order model are presented. It is found that the transverse shear stress distribution of the third-order model is very close to that predicted by the Saint-Venant solution. Moreover, the static bending, wave propagation, and free vibration problems are studied, whose results are also compared with those obtained from other beam models and three-dimensional elasticity solutions to show the advantage of the current model.

Buckling of Generic Higher-Order Shear Beam/Columns with Elastic Connections: Local and Nonlocal Formulation

AbstractIn this paper, the buckling behavior of generic higher-order shear beam models is investigated in a unified framework. This paper shows that most higher-order shear beam models developed in the literature (polynomial, sinusoidal, exponential shear strain distribution assumptions over the cross section) can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. The governing equations of the bending/buckling problem are obtained from a variational approach, leading to a generic sixth-order differential equation. Buckling solutions are presented for usual archetypal boundary conditions such as pinned-pinned, clamped-free, clamped-hinge, and clamped-clamped boundary conditions. The results are then extended to general boundary conditions based on generalized linear elastic connection law including vertical and rotational stiffness boundary conditions. Engineering analytical solutions are derived in a dimensionless fo...