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.

FEA Versus IGA in a Two-Node Beam Element Based on Unified and Integrated Method

This paper presents a new concept called Unified and Integrated Method for a shear deformable beam element.In this method,Timoshenko beam theory is unified and integrated in such a way that takes into account the effect of transverse shear and maintains the shear locking free condition at the same time to generate proper behavior in the analysis of thin to thick beams.The unified and integrated method is applied to finite element analysis(FEA)and isogeometric analysis(IGA)on two-node beam element.This method will be used to analyze uniformly loaded beams with various boundary conditions.A shear influence factor of f,which is a function of beam thickness ratio(L/h),is expressed explicitly as control of the transverse shear strain effect.The analysis gives interesting results showing that applying the unified and integrated method in FEA and IGA will yield exact values of DOF’s and displacement function even when using only a single element.Numerical examples demonstrate the validity and efficiency of the unified and integrated methods.

A Comparative Study on the Truck Frame Stiffness with Solid and Beam Element FEA Models

Truck frames should be designed and fabricated with enough rigidity to avoid excessive deflections. Finite element analysis (FEA) plays an important role in all stages of frame designs. While being accurate, 3D solid element FEA models are built upon frame configuration details which are not feasible in the preliminary design stage, partially because of limited available design data of frames and heavy computation costs. This research develops 1D beam element FEA models for simulating frame structures. In this paper, the CAD model of a truck frame is first created. The solid element FEA analysis, which is adopted as the baseline in this study, is subsequently conducted for the stiffness of the frame, Next, beam element FEA analysis is performed for validating the feasibility of the beam element FEA model by comparing the results from the solid and beam element FEA models. It is found that the beam element FEA model can predict the frame stiffness with acceptable accuracy and reduce the computation cost significantly.

A New Isogeometric Finite Element Method for Analyzing Structures

High-performance finite element research has always been a major focus of finite element method studies.This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method.Firstly,the physical field is approximated by uniform B-spline interpolation,while geometry is represented by non-uniform rational B-spline interpolation.By introducing a transformation matrix,elements of types C^(0)and C^(1)are constructed in the isogeometric finite element method.Subsequently,the corresponding calculation formats for one-dimensional bars,beams,and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping.The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis,eliminating the need for mesh generation and maintaining flexibility in element construction.Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method.Finally,the test results of several examples show that:(1)Under the same degree and element node numbers,the constructed elements are almost consistent with the results obtained by traditional finite element method;(2)For bar problems with large local field variations and beam problems with variable cross-sections,high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method;(3)The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom,while as degrees of freedom increase,the computational efficiency between the two is similar.

Divergence angle consideration in energy spread measurement for high-quality relativistic electron beam in laser wakefield acceleration

The thorough exploration of the transverse quality represented by divergence angle has been lacking yet in the energy spread measurement of the relativistic electron beam for laser wakefield acceleration(LWFA). In this work, we fill this gap by numerical simulations based on the experimental data, which indicate that in a C-shape magnet, magnetic field possesses the beam focusing effect, considering that the divergence angle will result in an increase in the full width at half maxima(FWHM) of the electron density distribution in a uniformly isotropic manner, while the length-to-width ratio decreases. This indicates that the energy spread obtained from the electron deflection distance is smaller than the actual value, regardless of the divergence angle. A promising and efficient way to accurately correct the value is presented by considering the divergence angle(for instance, for an electron beam with a length-to-width ratio of 1.12, the energy spread correct from 1.2% to 1.5%), providing a reference for developing the high-quality electron beam source.

Investigation of the Effect of the Force Arm on the Bending Capability of Prestressed Glulam Beam

Prestress enables the Glulam beam could make full use of the compression strength,and then increase the span,but it still could not reduce all drawbacks,such as cross-section weakening and small force arm.To avoid slotting and ensure suitable tension and compression couple,one kind of novel anchor has been proposed,which could meet the bearing capacity requirement.And then the bending test of prestressed Glulam beams with a geometric scale ratio of 1:2 was simulated,to investigate the effect of the force arm on bending capacities,failure modes,and deformation performance.Results show that increasing the force arm could improve the ultimate bending per-formance of the beam significantly,and the anchor arm length has a certain effect on the performance,but it is not obvious.Finally,based on Finite element method analysis,the practice design suggestions have been offered.

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.

3D Finite Element Numerical Simulation of Residual Stresses on Electron Beam Welded BT20 Plates

A three-dimensional finite-element model (FEM) used for calculating electron beam (EB) welding temperature and stresses fields of thin plates of BT20 titanium has been developed in which the nonlinear thermophysical and thermo-mechanical properties of the material has been considered. The welding temperature field, the distributions of residual stresses in as-welded (AW) and electron beam local post-weld heat treatment (EBLPWHT) conditions have been successfully simulated. The results show that: (1) In the weld center, the maximum magnitude of residual tensile stresses of BT20 thin plates of Ti alloy is equal to 60%- 70% of its yield strength σs. (2) The residual tensile stresses in weld center can be even decreased after EBLPWHT and the longitudinal tensile stresses are decreased about 50% compared to joints in AW conditions. (3) The numerical calculating results of residual stresses by using FEM are basically in agreement with the experimental results. Combined with numerical calculating results, the effects of electron beam welding and EBLPWHT on the distribution of welding residual stresses in thin plates of BT20 have been analyzed in detail.

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

Geometric nonlinear dynamic analysis of curved beams using curved beam element

Instead of using the previous straight beam element to approximate the curved beam,in this paper,a curvilinear coordinate is employed to describe the deformations,and a new curved beam element is proposed to model the curved beam.Based on exact nonlinear strain-displacement relation,virtual work principle is used to derive dynamic equations for a rotating curved beam,with the effects of axial extensibility,shear deformation and rotary inertia taken into account.The constant matrices are solved numerically utilizing the Gauss quadrature integration method.Newmark and Newton-Raphson iteration methods are adopted to solve the differential equations of the rigid-flexible coupling system.The present results are compared with those obtained by commercial programs to validate the present finite method.In order to further illustrate the convergence and efficiency characteristics of the present modeling and computation formulation,comparison of the results of the present formulation with those of the ADAMS software are made.Furthermore,the present results obtained from linear formulation are compared with those from nonlinear formulation,and the special dynamic characteristics of the curved beam are concluded by comparison with those of the straight beam.

A fiber-section model based Timoshenko beam element using shear-bending interdependent shape function

A fiber-section model based Timoshenko beam element is proposed in this study that is founded on the nonlinear analysis of frame elements considering axial, flexural, and shear deformations. This model is achieved using a shear-bending interdependent formulation （SBIF）. The shape function of the element is derived from the exact solution of the homogeneous form of the equilibrium equation for the Timoshenko deformation hypothesis.The proposed element is free from shear-locking. The sectional fiber model is constituted with a multi-axial plasticity material model, which is used to simulate the coupled shear-axial nonlinear behavior of each fiber. By imposing deformation compatibility conditions among the fibers, the sectional and elemental resisting forces are calculated. Since the SBIF shape functions are interactive with the shear-corrector factor for different shapes of sections, an iterative procedure is introduced in the nonlinear state determination of the proposed Timoshenko element. In addition, the proposed model tackles the geometric nonlinear problem by adopting a corotational coordinate transformation approach. The derivation procedure of the corotational algorithm of the SBIF Timoshenko element for nonlinear geometrical analysis is presented. Numerical examples confirm that the SBIF Timoshenko element with a fiber-section model has the same accuracy and robustness as the flexibility-based formulation. Finally, the SBIF Timoshenko element is extended and demonstratedin a three-dimensional numerical example.

Discrete Element with Flexible Connector for Dynamic Analysis of 3-D Beam Structures

Combined multi-body dynamics with structural dynamics, a new discrete element with flexible connector, which is applicable for 3-D beam structures, is developed in this paper. Both the generalized elastic coefficient matrix of the flexible connector and the mass matrix of discrete element may be off-diagonal in a general case. The zero-length rigid element is introduced to simulate the node at which multiple elements are jointed together. It may also be effective when the axes of adjacent elements are not in the same line. The examples for eigenvalue calculation show that the model is successful. It can be extended to the geometric nonlinear response analysis.

Simple and universal method in designs of high-efficiency diffractive optical elements for spectrum separation and beam concentration

Diffractive optical elements（DOEs） with spectrum separation and beam concentration（SSBC） functions have important applications in solar cell systems. With the SSBC DOEs, the sunlight radiation is divided into several wave bands so as to be effectively absorbed by photovoltaic materials with different band gaps. A new method is proposed for designing high-efficiency SSBC DOEs, which is physically simple, numerically fast, and universally applicable. The SSBC DOEs are designed by the new design method, and their performances are analyzed by the Fresnel diffraction integral method.The new design method takes two advantages over the previous design method. Firstly, the optical focusing efficiency is heightened by up to 10%. Secondly, focal positions of all the designed wavelengths can be designated arbitrarily and independently. It is believed that the designed SSBC DOEs should have practical applications to solar cell systems.

A new 3-D element formulation on displacement of steel-concrete composite box beam

Slip of a composite box beam may reduce its stiffness, enlarge its deformation and affect its performance. In this work, the governing differential equations and boundary conditions of composite box beams were established. Analytic solutions of combined differential equations were also established. Partial degree of freedom was adopted to establish a new FEA element of three-dimensional beam, taking into account the slip effect. Slip and its first-order derivative were introduced into the nodes of composite box beams as generalized degree of freedom. Stiffness matrix and load array of beam elements were established. A three-dimensional nonlinear calculation program was worked out. The results show that the element is reliable and easy to divide and is suitable for special nonlinear analysis of large-span composite box beams.

Conversion between solid and beam element solutions of finite element method based on meta-modeling theory:development and application to a ramp tunnel structure

In this study, a new method for conversion of solid finite element solution to beam finite element solution is developed based on the meta-modeling theory which constructs a model consistent with continuum mechanics. The proposed method is rigorous and efficient compared to a typical conversion method which merely computes surface integration of solid element nodal stresses to obtain cross-sectional forces. The meta-modeling theory ensures the rigorousness of proposed method by defining a proper distance between beam element and solid element solutions in a function space of continuum mechanics. Results of numerical verification test that is conducted with a simple cantilever beam are used to find the proper distance function for this conversion. Time history analysis of the main tunnel structure of a real ramp tunnel is considered as a numerical example for the proposed conversion method. It is shown that cross-sectional forces are readily computed for solid element solution of the main tunnel structure when it is converted to a beam element solution using the proposed method. Further, envelopes of resultant forces which are of primary importance for the purpose of design, are developed for a given ground motion at the end.

COMPUTATION OF SUPER-CONVERGENT NODAL STRESSES OF TIMOSHENKO BEAM ELEMENTS BY EEP METHOD

The newly proposed element energy projection(EEP) method has been applied to the computation of super_convergent nodal stresses of Timoshenko beam elements.General formulas based on element projection theorem were derived and illustrative numerical examples using two typical elements were given.Both the analysis and examples show that EEP method also works very well for the problems with vector function solutions.The EEP method gives super_convergent nodal stresses,which are well comparable to the nodal displacements in terms of both convergence rate and error magnitude.And in addition,it can overcome the “shear locking” difficulty for stresses even when the displacements are badly affected.This research paves the way for application of the EEP method to general one_dimensional systems of ordinary differential equations.

Finite element analysis of 2D SMA beam bending

A thermomechanical model of a shape memory alloy beam bending under tip force loading is implemented in finite element codes.The constitutive model is a one dimensional model which is based on free energy and motivated by statistical thermodynamics.The particular focus of this paper is on the aspects of finite element modeling and simulation of the inhomogeneous beam bending problem.This paper extends previous work which is based on the small deformation Euler-Bernoulli beam theory and by treating an SMA beam as consisting of multi-layers in a twodimensional model.The flux terms are involved in the heat transfer equation.The simulations can represent both shape memory effect and super-elastic behavior.Different thermal boundary condition effect and load rate effect can also be captured.

A new beam element for analyzing geometrical and physical nonlinearity

Based on Timoshenko＇s beam theory and Vlasov＇s thin-walled member theory, a new model of spatial thin-walled beam element is developed for analyzing geometrical and physical nonlinearity, which incorporates an interior node and independent interpolations of bending angles and warp and takes diversified factors into consideration, such as traverse shear deformation, torsional shear deformation and their coupling, coupling of flexure and torsion, and the second shear stress. The geometrical nonlinear strain is formulated in updated Lagarange （UL） and the corresponding stiffness matrix is derived. The perfectly plastic model is used to account for physical nonlinearity, and the yield rule of von Mises and incremental relationship of Prandtle-Reuss are adopted. Elastoplastic stiffness matrix is obtained by numerical integration based on the finite segment method, and a finite element program is compiled. Numerical examples manifest that the proposed model is accurate and feasible in the analysis of thin-walled structures.

Analytical modeling of sandwich beam for piezoelectric bender elements

Piezoelectric bender elements are widely used as electromechanical sensors and actuators, An analytical sandwich beam model for piezoelectric bender elements was developed based on the first-order shear deformation theory （FSDT）, which assumes a single rotation angle for the whole cross-section and a quadratic distribution function for coupled electric potential in piezoelectric layers, and corrects the effect of transverse shear strain on the electric displacement integration. Free vibration analysis of simplysupported bender elements was carried out and the numerical results showed that, solutions of the present model for various thickness-to-length ratios are compared well with the exact two-dimensional solutions, which presents an efficient and accurate model for analyzing dynamic electromechanical responses of bender elements.

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.