Classical Euler Bernoulli Beam Research Articles

The vibration of nanobeam resting on elastic foundation using modified couple stress theory

In this paper, the vibration of nanobeams resting on the Winkler foundation is proposed using the modified couple stress theory. Hamilton’s principle is utilized to construct the governing equations. The size effect of the nanobeam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can capture it because it includes material length scale parameter that a newly developed model has. Once the material length scale parameter is assumed to be zero, the classical Euler-Bernoulli beam theory equation is obtained. Multiple scale method is employed to obtain the result. Simply supported boundary condition is used to study natural frequencies. The influence of material length scale parameter and the Winkler elastic foundation parameter on the fundamental frequencies of the nanobeam is investigated and tabulated. Also, in the present study, Poisson’s ratio is taken as constant. Nanobeam resting on the Winkler foundation which is simply supported is analyzed to illustrate the size effects on the free vibration. Numerical results for the simply supported nanobeam indicate that the first fundamental frequency calculated by the presented model is higher than the classical one. Moreover, it is obtained that the size influence is more substantial for higher vibration modes. The results indicate that the significant importance of the size influences the analysis of nanobeams. The vibration of nanobeam exhibits a hardening spring behavior, and the newly developed models are the beams stiffer than according to the classical beam theory. Modified couple stress theory tends to be more helpful in describing the size-dependent mechanical properties of nanoelectromechanical systems (NEMS).

Edgewise Bending Vibration Analysis of a Rotating Sandwich Beam with Magnetorheological Elastomer Core

The vibration of a rotating sandwich beam with magnetorheological elastomer (MRE) as a core between two elastic layers is theoretically analyzed in this paper. This study is focused on the bending vibration along the edgewise direction of a sandwich beam of rectangular cross-section, which, to the best of our knowledge, has not been addressed yet. The classical Euler–Bernoulli beam theory is used to model the dynamic behavior of the elastic layers. In the modeling, the effect of the MRE layer is considered by incorporating its shear strains and the inertia due to shear deformation and bending motion. The governing equations of motion of the rotating sandwich beam are derived by using the Ritz method and the Lagrange’s equations. The effects of the applied magnetic field, core layer thickness, rotational speed, setting angle and hub radius on the natural frequencies and the corresponding loss factors are investigated parametrically. The results show the significant effect of the magnetic field intensity and the MRE layer thickness on the modal characteristics of the MRE sandwich beam.

A Variation Analysis Method for Linkage Mechanism with Consideration of Joint Clearance and Deformation

As typical transmission mechanisms, linkage mechanisms are widely applied in mechanical engineering field, and accuracy prediction of them has received much more attentions especially from some high-precision application fields. For a certain linkage mechanism system, external forces will also cause linkage deformations and deterministic oriented-drift of joint clearances, which become additional variation sources besides component tolerances. In order to perform a more accurate prediction of linkage mechanism, a variation analysis method is proposed in this study, in which component tolerances, joint clearances and deformations are considered. With an equivalent method, positional tolerance and joint clearances are taken into the present variation analysis model, and serve as circular tolerances. Based on classical Euler-Bernoulli beam theory, linkage deformations are disposed as equivalent deviations and contribute to assembly deviations. A decomposing method of percentage contribution, which contains different variation sources, is presented as well. A case study of four-bar linkage mechanism is illustrated to validate the accuracy of the present method by corresponding FEA simulation and experimental test. Moreover, a case study of a three-loop mechanism is also analyzed for the accuracy and percentage contribution of different variation sources with the present method.

An analytical investigation on large post-buckling behavior of a drilling shaft modeled as a rotating beam with various boundary conditions

Post-buckling behavior of a drilling shaft modeled as a rotating beam subjected to terminal force is investigated in this paper. Three boundary conditions: clamped-clamped, clamped-free, and clamped-hinged are considered. Classical Bernoulli-Euler beam model including shear force is applied to depict the problem. In particular, for clamped-hinged ends boundary condition, the analytical approximate solutions are more difficult to construct, due to asymmetric boundary and configuration. Unlike numerical results in the existing literatures, analytical approximate solutions of the problem above are established via combining the Maclaurin series expansion, orthogonal Chebyshev polynomials, Galerkin and Harmonic balance method in this paper. Compared with numerical solutions obtained by using the shooting method on the exact governing equation, the approximate analytical solutions here show excellent accuracy and rapid convergence. The effect of the system parameters on the dynamic response and stability of system can conveniently be investigated via the present analytical approximate solutions. In conclusion, the analytical approximate solutions presented here are sufficiently precise for a wide range of the maximum angle of the rotating beam.

Flutter of multi-cracked laminated composite beams subjected to a non-conservative compressive load

Free vibration and buckling instability of laminated composite beams and flutter instability of an isotropic beam with different defects have been investigated before, however, still dynamic stability of laminated composite beam with weak section subjected to follower forces has not considered. This paper presents a simple semi-analytical solution to the flutter instability of a laminated composite beam with multi open-edge cracks subjected to a tip follower force. Using classical lamination theory and Euler-Bernoulli beam theory, equation of motion of the beam is derived analytically and characteristic eigenvalue equations of the cracked laminated composite beam are obtained. A Matlab® code is used to solve these equations numerically. After validating the results, a numerical study is performed to investigate the effect of crack depth and stacking sequences on the dynamic stability of the cracked beam.

Geometrically nonlinear beam analysis of composite wind turbine blades based on quadrature element method

A new nonlinear beam theory is proposed for the analysis of composite wind turbine blades. The beam theory is developed by extending classical Euler–Bernoulli beam theory to a generalized Timoshenko beam. Mechanics-based variables are used to describe finite rotation such that the problems of the sequence dependence or spatially discontinuity of rotational variables can be avoided. Furthermore, nonlinear beam theory is implemented using the weak-form quadrature element method. Numerical examples of both non-rotating and rotating beams are given and the comparison with analytical and finite element results shows high computational accuracy and efficiency of the proposed nonlinear quadrature element. A simple parametric study of a virtual 5-MW wind turbine blade shows that bend-twist coupling due to both material anisotropy and geometrical nonlinearity affects the dynamic performance of the blade significantly.

Significance of size dependent and material structure coupling on the characteristics and performance of nanocrystalline micro/nano gyroscopes

Capacitive-based sensing microelectromechanical (MEMS) and nanoelectromechanical (NEMS) gyroscopes have significant advantages over conventional gyroscopes, such as low power consumption, batch fabrication, and possible integration with electronic circuits. However, inadequacies in the modeling of these inertial sensors have presented issues of reliability and functionality of micro-/nano-scale gyroscopes. In this work, a micromechanical model is developed to represent the unique microstructure of nanocrystalline materials and simulate the response of micro-/nano-gyroscope comprising an electrostatically-actuated cantilever beam with a tip mass at the free end. Couple stress and surface elasticity theories are integrated into the classical Euler-Bernoulli beam model in order to derive a size-dependent model. This model is then used to investigate the influence of size-dependent effects on the static pull-in instability, the natural frequencies and the performance output of gyroscopes as the scale decreases from micro-to nano-scale. The simulation results show significant changes in the static pull-in voltage and the natural frequency as the scale of the system is decreased. However, the differential frequency between the two vibration modes of the gyroscope is observed to drastically decrease as the size of the gyroscope is reduced. As such, the frequency-based operation mode may not be an efficient strategy for nano-gyroscopes. The results show that a strong coupling between the surface elasticity and material structure takes place when smaller grain sizes and higher void percentages are considered.

Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams

This paper is pertinent to the analytical solutions for vibration analysis of initially stressed Nonlocal Euler-Bernoulli nano-beams. In order to take into account of small length scale effect, this vibration problem formulation is depending upon both nonlocal Euler-Bernoulli and also Eringen’s nonlocal elasticity theory. The boundary conditions and governing equation are obtained by use of Hamiltonian’s principle. These equations are solved analytically with different initial stresses (both compressive and tensile) and boundary conditions. The effect of small length scale and the initial stress on the fundamental frequency are investigated. The solutions obtained are compared with the ones depending upon both classical Euler-Bernoulli and Timoshenko beam theory to comprehend the responses of nano-beams under the effect of initial stress and small scale in terms of frequencies for both theories. The results supply a better declaration for vibration analysis of nano-beams which are short and stubby with initial stress.

Buckling analysis of bilayer beam-columns with an asymmetric delamination

A general analytical model is developed to evaluate buckling behavior of a bilayer beam-column with a through-the-width asymmetric delamination under simply supported boundary conditions. Both transverse shear deformation and local deformation at delamination tip are taken into account in the model by considering two sublayers in both delaminated and intact regions as two individual shear deformable ones. Interface continuity conditions, which account for effect of interface normal and shear stresses, are also introduced in the intact regions. Furthermore, the present model is capable of dealing with the case of delamination located arbitrary through thickness and along length of beam-columns. In comparisons with the results of classical Euler-Bernoulli beam model, Timoshenko beam model (rigid joint model) and finite element analysis, better agreements are achieved for the present flexible joint model, which captures local deformation at delamination tip due to consideration of interface normal and shear stress components. Parametric studies are conducted to evaluate influences of sublayer thickness ratio, delamination length, delamination location, specimen length, and material parameters on the critical buckling strain with emphasis on transverse shear deformation and local deformation at delamination tip. The developed model can be used with confidence in buckling analysis of layered structures with an arbitrary delamination through the length and thickness.

English

This paper deals with a nonlinear beam model which was published by D.Y.Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.

Comparison of small scale effect theories for buckling analysis of nanobeams

Theories which consider small scale effect have a great importance on analysis in micro and nano scale. In present paper, three kind of nanotubes (Carbon Nanotube (CNT), Boron Nitride Nanotube (BNNT), and Silicon Carbide Nanotube (SiCNT)) are analyzed in case of buckling on two parameters elastic foundation. Three different small scale theories (Nonlocal Elasticity Theory (NET), Surface Elasticity Theory (SET), and Nonlocal Surface Elasticity Theory (NET&SET)) are applied to calculate the buckling loads. Also Classical Euler-Bernoulli Beam Theory (CT) is used to see the effect of small scale effective theories. Comparative results are given for simply supported nanotubes in figures.

Nonlinear analysis of reinforced concrete plane frames exposed to fire using direct stiffness method

This article presents a framework based on the direct stiffness method for nonlinear thermo-mechanical analysis of reinforced concrete plane frames subjected to fire. It accounts for geometric nonlinearity, material nonlinearity, and nonlinear thermal gradients and incorporates two-way coupling between thermal and structural analyses. Force deformation relations are derived from classical Euler–Bernoulli beam theory and are expressed in terms of temperature-dependent stability and bowing functions. This is one of the unique features of proposed framework and allows a coarser spatial discretization to be used as opposed to full finite element–based approaches (such as SAFIR [registered trademark of the software SAFIR developed at the University of Liege]). The cross sections of the structural members are discretized with two-dimensional meshes for thermal analysis while structural analysis utilizes a line element based on direct stiffness method. Equivalent bending and axial rigidities of this line element are computed using several fibers along the length of the member, passing through the nodes of the two-dimensional mesh used for thermal analysis. The total strain at each fiber is decomposed into mechanical, thermal, creep, and transient thermal components. A discrete damage parameter is introduced at fiber level to ensure irreversibility of crushing and cracking in accordance with relevant constitutive laws. Five numerical examples are presented to demonstrate the accuracy and efficacy of the developed framework with respect to theoretical solutions, experimental observations, and some of the existing macro- and micro-finite element–based approaches. It is found that the developed framework can predict the response of reinforced concrete structures very well.

Extended High-Order Theory for Curved Sandwich Panels and Comparison With Elasticity

A new one-dimensional high-order sandwich panel theory for curved panels is presented and compared with the theory of elasticity. The theory accounts for the sandwich core compressibility in the radial direction as well as the core circumferential rigidity. Two distinct core displacement fields are proposed and investigated. One is a logarithmic (it includes terms that are linear, inverse, and logarithmic functions of the radial coordinate). The other is a polynomial (it consists of second and third-order polynomials of the radial coordinate), and it is an extension of the corresponding field for the flat panel. In both formulations, the two thin curved face sheets are assumed to be perfectly bonded to the core and follow the classical Euler–Bernoulli beam assumptions. The relative merits of these two approaches are assessed by comparing the results to an elasticity solution. The case examined is a simply supported curved sandwich panel subjected to a distributed transverse load, for which a closed-form elasticity solution can be formulated. It is shown that the logarithmic formulation is more accurate than the polynomial especially for the stiffer cores and for curved panels of smaller radius.

On the mechanics of functionally graded nanobeams with the account of surface elasticity

In this paper, size-dependent bending, buckling and free vibration responses of functionally graded (FG) nanobeams are investigated using an integrated nonclassical continuum model. The integrated model accounts for the simultaneous effects of the microstructure local rotation and the surface energy in the framework of the nonlocal elasticity. The proposed nonlocal-couple stress-surface elasticity (NLCSSE) model is formulated by incorporating the Eringen nonlocal elasticity theory, modified couple stress theory and surface elasticity theory into the classical Euler–Bernoulli beam model. It is assumed that the material properties of the bulk and surface of the FG nanobeam change continuously through the thickness according to a power law. The size-dependent equations of motion and corresponding boundary conditions are derived utilizing the Hamilton's principle. The proposed model is validated by comparing the obtained results with available benchmark results. Numerical results are presented to reveal the effects of nonlocal parameter, material length scale parameter, surface energy, gradient index, Poisson ratio, thickness and length-to-thickness ratio on the deflection, critical buckling load and natural frequency of a FG simply supported nanobeam.

Efficient Three Dimensional Nonlinear Thermo-Mechanical Analysis of Structures Subjected to Fire

Abstract In this paper, a computationally efficient integrated frame-work is developed for coupled thermo-mechanical analysis of 3D frames. It can account for physical phenomena like large deformations, temperature dependent material degradation, permanent plastic deformations and fire induced spalling which are prevalent at elevated temperatures. The developed frame-work utilizes three way sequential coupling between thermal, mass transport and structural analysis. A two level discretization scheme is incorporated where 1D beam column elements are utilized for structural analysis, the cross-section of these beam-column elements are further discretized into matrix of segments. Aforementioned strategy entails sequential coupling of effects of non-uniform temperature and pore pressure across the cross-section into structural analysis. Subsequently structural analysis is performed with an updated Lagrangean based formulation with force deformation relationships deduced form classical Euler-Bernoulli beam column theory. Critical physical phenomena like cracking, crushing, spalling and transient states of strain in case of concrete and yielding in case of steel are duly accounted. Cross-sectional reduction due to spalling are accounted for by replacing the spalled segments with void segments in the subsequent time steps. Numerical examples of steel and concrete structures subjected to various fire scenarios are presented to demonstrate the accuracy of developed framework. Furthermore, progressive collapse analysis is carried out for concrete and steel 3D subjected to fire.

Effect of temperature-dependent physical properties on nanobeam structures induced by ramp-type heating

The classical Euler-Bernoulli beam theory based upon the nonlocal theory of thermoelasticity is constructed and employed to investigate the vibrating nanobeam due to a ramp-type heating. The thermal conductivity and modulus of elasticity are taken as temperature-dependent functions. The generalized theory of thermoelasticity with Dual-Phase-Lags (DPLs) model is also employed to solve the problem. Analytical and numerical techniques based on Laplace transforms and their inversions are used to calculate the vibrational deflection and temperature of the nanobeam. The effects of the PLs and ramping-time parameters on the lateral vibration, the temperature, the displacement and the flexure moment of the nanobeam are investigated. Some plots have been presented for comparison purposes to estimate the effects of the nonlocal parameter on the field quantities. For the sake of completeness a comparison is also made between the present results and those obtained in the case of temperature-independent thermal and mechanical properties.

A refined model for piezoelectric composite beams

This work presents and compares few simple one-dimensional models for the piezoelectric actuation and detection of beams. The 1D nature, which allows an easy embedding of the model in the classical Euler-Bernoulli beam equations, is obtained by adopting simplifying assumptions along directions of the cross-sectional plane. By changing such assumptions, different models can be built. Their validity is discussed and compared with results of FEM simulations for varying geometries. We show that commonly adopted models fail in a series of practical cases and propose a new model capable of accurately describing wide beams.

Exact stress and deformation analysis in elastoplastic Ramberg–Osgood beam

Elastoplastic deformation and stress distributions in a thin or classical Euler–Bernoulli beam are derived by an exact solution methodology. In the formulations phase, the Hencky deformation theory of plasticity is employed. It is assumed that material of Ramberg–Osgood behavior yields according to von Mises criterion. Using the principals of variational calculus and assuming a tri-axial field of stress, governing equations of elastoplastic beam are extracted. Exact solution of resulting nonlinear integro-differential equations are obtained by using the analytical methods of homotopy and Adomian. In the exemplary study of clamped–clamped beam, the fields of deformation are compared with the results of ABAQUS software. Different graphical representations are provided to show the results of analytical solutions and computer simulations. The level of consistency between the computer simulations and solutions of Adomian and homotopy techniques is assessed. At the end, the capability of classical engineering beam theory for the elastoplastic analysis is assessed.

A discrete Lagrangian based direct approach to macroscopic modelling

A direct discrete Lagrangian based approach, designed at a length scale of interest, to characterize the response of a body is proposed. The main idea is to understand the dynamics of a deformable body via a Lagrangian corresponding to a coupled interaction of rigid particles in the reduced dimension. We argue that the usual practice of describing the laws of a deformable body in the continuum limit is redundant, because for most of the practical problems, analytical solutions are not available. Since continuum limit is not taken, the framework automatically relaxes the requirement of differentiability of field variables. The discrete Lagrangian based approach is illustrated by deriving an equivalent of the Euler–Bernoulli beam model. A few test examples are solved, which demonstrate that the derived non-local model predicts lower deflections in comparison to classical Euler–Bernoulli beam solutions. We have also included crack propagation in thin structures for isotropic and anisotropic cases using the Lagrangian based approach.

Prestressed concrete bridges with corrugated steel webs: Nonlinear analysis and experimental investigation

Concrete bridges with corrugated steel webs and prestressed by both internal and external tendons have emerged as one of the promising bridge forms. In view of the different behaviour of components and the large shear deformation of webs with negligible flexural stiffness, the assumption that plane sections remain plane may no longer be valid, and therefore the classical Euler-Bernoulli and Timoshenko beam models may not be applicable. In the design of this type of bridges, both the ultimate load and ductility should be examined, which requires the estimation of full-range behaviour. An analytical sandwich beam model and its corresponding beam finite element model for geometric and material nonlinear analysis are developed for this type of bridges considering the diaphragm effects. Different rotations are assigned to the flanges and corrugated steel webs to describe the displacements. The model accounts for the interaction between the axial and flexural deformations of the beam, and uses the actual stress-strain curves of materials considering their stress path-dependence. With a nonlinear kinematical theory, complete description of the nonlinear interaction between the external tendons and the beam is obtained. The numerical model proposed is verified by experiments.