Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

Isogeometric Analysis of Longitudinal Displacement of a Simplified Tunnel Model Based on Elastic Foundation Beam

Serious uneven settlement of the tunnel may directly cause safety problems.At this stage,the deformation of the tunnel is predicted and analyzed mainly by numerical simulation,while the commonly used finite element method(FEM)uses low-order continuous elements.Therefore,the accuracy of tunnel settlement prediction is not enough.In this paper,a method is proposed to study the vertical deformation of the tunnel by using the combination of isogeometric analysis(IGA)and Bézier extraction operator.Compared with the traditional IGA method,this method can be easily integrated into the existing FEM framework,and ensure the same accuracy.A numerical example of an elastic foundation beam subjected to uniformly distributed load and an engineering example of an equivalent elastic foundation beamof the tunnel are given.The results show that the solution of the IGA method is closer to the theoretical solution of the initial-parameter method than the FEM,and the accuracy and reliability of the proposedmodel are verified.Moreover,it not only provides some theoretical support for the longitudinal design of the tunnel,but also provides a new way for the application and popularization of IGA in tunnel engineering.

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded(FG)beams supported by a viscoelastic Winkler-Pasternak foundation is presented.The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation,which were not considered in most literature on this subject,and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven(ε-D)and stress-driven(σ-D)two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered,which can address both the stiffness softening and toughing effects due to scale reduction.The generalized differential quadrature method(GDQM)is used to solve the complex eigenvalue problem.After verifying the solution procedure,a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained.Subsequently,the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the eff ects of the Winkler coeffi cient, Pasternak coeffi cient and damping coeffi cient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary diff erential equations with variable coeffi cients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verifi ed through two numerical examples. The infl uences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with diff erent taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

Dynamic response of a beam on a Pasternak foundation and under a moving load

The dynamic response of an infinite beam placed on a Pasternak foundation when the system was subjected to a moving load was investigated.We used the double Fourier transform and its inversion to solve the formulations of the problem.A closed form analytic solution of the beam was obtained by the theorem of residues.We selected a numerical example to illustrate the dynamic response of the beam on Pasternak and Winkler foundations,respectively.We discuss the effect of the moving load velocity on the dynamic displacement response of the beam.The maximum deflection of the beam increases slightly with increased load velocity but increases significantly with reduced shear modulus of subgrade at a given velocity.The maximum deflection of a beam resting on a Pasternak foundation is much smaller than that of a beam on a Winkler foundation.

Design parameter optimization of beam foundation on soft soil layer with nonlinear finite element

Finite element method was performed to investigate the influences of beam stiffness, foundation width and cushion thickness on the beating capacity of beam foundation on underlying weak laminated clay. The comparison between numerical results and results from field test including plate-bearing test and foundation settlement observation shows reasonable agreement. According to the numerical results, the beam width, length, cross section and cushion thickness were optimized. The results show that the stresses in subgrade soil decrease greatly with increasing the cushion thickness and width of foundation. However, the foundation settlement and influencing depth of displacement also increase correspondingly under conditions of relatively thinner cushion thickness. For the foundations on underlying weak layer, increasing foundation width merely might be inadequate for improving the bearing capacity, and the appropriate width and cushion thickness depend on the response of subgrade. A comparison between rigid and flexible beams was also discussed. The influence of a flexible beam foundation on subgrade is relatively smaller under the same loading conditions, and the flexible beam foundation appears more adaptable to various subgrades. The proposed flexible beam foundation was adopted in engineering. According to the calculation results, beam width of 2.4 m and cushion thickness of 0.8 m are proposed, and a flexible beam foundation is applied in the optimized design, which is confirmed reasonable by the actual engineering.

Vibration of axially moving beam supported by viscoelastic foundation

In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton＇s principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.

Vibration suppression of magnetostrictive laminated beams resting on viscoelastic foundation

The vibration suppression analysis of a simply-supported laminated composite beam with magnetostrictive layers resting on visco-Pasternak’s foundation is presented.The constant gain distributed controller of the velocity feedback is utilized for the purpose of vibration damping.The formulation of displacement field is proposed according to Euler-Bernoulli’s classical beam theory(ECBT),Timoshenko’s first-order beam theory(TFBT),Reddy’s third-order shear deformation beam theory,and the simple sinusoidal shear deformation beam theory.Hamilton’s principle is utilized to give the equations of motion and then to describe the vibration of the current beam.Based on Navier’s approach,the solution of the dynamic system is obtained.The effects of the material properties,the modes,the thickness ratios,the lamination schemes,the magnitudes of the feedback coefficient,the position of magnetostrictive layers at the structure,and the foundation modules are extensively studied and discussed.

Solving Nonlinear Differential Equation Governing on the Rigid Beams on Viscoelastic Foundation by AGM

In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by a simple and innovative approach, which has been called Akbari-Ganji＇s method （AGM）. AGM is a very suitable computational process and is usable for solving various nonlinear differential equations. Moreover, using AGM which solving a set of algebraic equations, complicated nonlinear equations can easily be solved without any mathematical operations. Also, the damping ratio and energy lost per cycle for three cycles have been investigated. Furthermore, comparisons have been made between the obtained results by numerical method （Runk45） and AGM. Results showed the high accuracy of AGM. The results also showed that by increasing the amount of initial amplitude of vibration （A）, the value of damping ratio will be increased, and the energy lost per cycle decreases by increasing the number of cycle. It is concluded that AGM is a reliable and precise approach for solving differential equations. On the other hand, it is better to say that AGM is able to solve linear and nonlinear differential equations directly in most of the situations. This means that the final solution can be obtained without any dimensionless procedure Therefore, AGM can be considered as a significant progress in nonlinear sciences.

A semi-infinite beam theoretical model on predicting rock slope subsidence induced by underground mining

When the mining goaf is close to the cliff,rock slope subsidence induced by underground mining is significantly affected by its boundary conditions.In this study,an analytical method is proposed by considering the key strata as a semi-infinite Euler-Bernoulli beam rested on a Winkler foundation with a local subsidence area.The analytical solutions of deflection are derived by analyzing the boundary and continuity conditions of the cliff.Then,the analytical solutions are verified by the results from experimental tests,FEM and InSAR,respectively.After that,the influence of changing parameters on deflections is studied with sensitivity analysis.The results show that the distance between goaf and cliff significantly affects the deflection of semi-infinite beam.The response of semi-infinite beam is obviously determined by the length of goaf and the bending stiffness of beam.The comparisons between semi-infinite beam and infinite beam illustrate the ascendancy of the improved model in such problems.

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

Free vibration and buckling analysis of polymeric composite beams reinforced by functionally graded bamboo fbers

Natural fibers have been extensively researched as reinforcement materials in polymers on account of their environmental and economic advantages in comparison with synthetic fibers in the recent years.Bamboo fibers are renowned for their good mechanical properties,abundance,and short cycle growth.As beams are one of the fundamental structural components and are susceptible to mechanical loads in engineering applications,this paper performs a study on the free vibration and buckling responses of bamboo fiber reinforced composite(BFRC)beams on the elastic foundation.Three different functionally graded(FG)layouts and a uniform one are the considered distributions for unidirectional long bamboo fibers across the thickness.The elastic properties of the composite are determined with the law of mixture.Employing Hamilton’s principle,the governing equations of motion are obtained.The generalized differential quadrature method(GDQM)is then applied to the equations to obtain the results.The achieved outcomes exhibit that the natural frequency and buckling load values vary as the fiber volume fractions and distributions,elastic foundation stiffness values,and boundary conditions(BCs)and slenderness ratio of the beam change.Furthermore,a comparative study is conducted between the derived analysis outcomes for BFRC and homogenous polymer beams to examine the effectiveness of bamboo fibers as reinforcement materials,demonstrating the significant enhancements in both vibration and buckling responses,with the exception of natural frequencies for cantilever beams on the Pasternak foundation with the FG-◇fiber distribution.Eventually,the obtained analysis results of BFRC beams are also compared with those for carbon nanotube reinforced composite(CNTRC)beams found in the literature,indicating that the buckling loads and natural frequencies of BFRC beams are lower than those of CNTRC beams.

Temperature stress analysis for bi-modulus beam placed on Winkler foundation

The materials with different moduli in tension and compression are called bi-modulus materials. Graphene is such a kind of materials with the highest strength and the thinnest thickness. In this paper, the mechanical response of the bi-modulus beam subjected to the temperature effect and placed on the Winkler foundation is studied. The differential equations about the neutral axis position and undetermined parameters of the normal strain of the bi-modulus foundation beam are established. Then, the analytical expressions of the normal stress, bending moment, and displacement of the foundation beam are derived. Simultaneously, a calculation procedure based on the finite element method （FEM） is developed to obtain the temperature stress of the bi-modulus struc- tures. It is shown that the obtained bi-modulus solutions can recover the classical modulus solution, and the results obtained by the analytical expressions, the present FEM proce- dure, and the traditional FEM software are consistent, which verifies the accuracy and reliability of the present analytical model and procedure. Finally, the difference between the bi-modulus results and the classical same modulus results is discussed, and several reasonable suggestions for calculating and optimizing the certain bi-modulus member in practical engineering are presented.

Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.

GENERAL SOLUTION FOR DYNAMICAL PROBLEM OF INFINITE BEAM ON AN ELASTIC FOUNDATION

Based on the theory of Eider-Bernoulli beam and Winkler assumption for elastic foundation, a mathematical model is presented. By using Fourier transformation for space variable, Laplace transformation for time variable and convolution theorem for their inverse transformations, a general solution for dynamical problem of infinite beam on an elastic foundation is obtained. Finally, the cases of free vibration,impulsive response and moving load are also discussed.

Analysis of wave propagation in a functionally graded nanobeam resting on visco-Pasternak＇s foundation

Wave propagation analysis for a functionally graded nanobeam with rectangular cross-section resting on visco-Pasternak＇s foundation is studied in this paper. Timoshenko＇s beam model and nonlocal elasticity theory are employed for formulation of the problem. The equations of motion are derived using Hamilton＇s principals by calculating kinetic energy, strain energy and work due to viscoelastic foundation. The effects of various parameters such as wavenumber, non-homogeneous index, nonlocal parameter and three parameters of foundation are performed on the phase velocity of the nanobeam. The obtained results indicate that some parameters such as non-homogeneous index, nonlocal parameter and wavenumber have significant effect on the response of the system.

DYNAMIC ANALYSIS OF ARREST OF BUCKLE PROPAGATION ON A BEAM ON A NONLINEAR ELASTIC FOUNDATION BY FEM

Based on the dynamic governing equation of propagating buckle on a beam on a nonlinear elastic foundation, this paper deals with an important problem of buckle arrest by combining the FEM with a time integration technique. A new conclusion completely different from that by the quasi-static analysis about the buckle arrestor design is drawn. This shows that the inertia of the beam cannot be ignored in the analysis under consideration, especially when the buckle propagation is suddenly stopped by the arrestors.

A Unified Dimensionless Parameter for Finite Element Mesh for Beams Resting on Elastic Foundation

Discretising a structure into elements is a key step in finite element(FE)analysis.The discretised geometry used to formulate an FE model can greatly affect accuracy and validity.This paper presents a unified dimensionless parameter to generate a mesh of cubic FEs for the analysis of very long beams resting on an elastic foundation.A uniform beam resting on elastic foundation with various values of flexural stiffness and elastic supporting coefficients subject to static load and moving load is used to illustrate the application of the proposed parameter.The numerical results show that(a)Even if the values of the flexural stiffness of the beam and elastic supporting coefficient of the elastic foundation are different,the same proposed parameter“s”can ensure the same accuracy of the FE solution,but the accuracy may differ for use of the same element length;(b)The proposed dimensionless parameter“s”can indeed be used as a unified index to generate the mesh for a beam resting on elastic foundation,whereas the use of the same element length as a criterion may be misleading;(c)The errors between the FE and analytical solutions for the maximum vertical displacement,shear force and bending moment of the beam increase with the dimensionless parameter“s”;and(d)For the given allowable errors for the vertical displacement,shear force and bending moment of the beam under static load and moving load,the corresponding values of the proposed parameter are provided to guide the mesh generation.