Euler-Bernoulli Beam Element Research Articles

Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.

A comparative study of geometrical curvature expressions for the large displacement analysis of spatial absolute nodal coordinate formulation Euler–Bernoulli beam motion

While the geometrical curvature expressions have become widely used in the beams based on absolute nodal coordinate formulation, they have not yet been compared and sufficiently verified, particularly for the spatial beam motion. In this paper, two geometrical curvature expressions for the large displacement analysis of spatial beam motion are compared and their fundamental differences are highlighted: the exact curvature model and general curvature model. In order to examine numerically the ability of two curvature expressions in the large displacement analysis, a three-dimensional absolute nodal coordinate formulation Euler–Bernoulli beam element is selected as the candidate. Several static and dynamical benchmarks are performed to assess their accuracy and differences. It is shown that the exact curvature model can be effectively and efficiently used in static applications, particularly in the highly nonlinear static analysis. In dynamics problems, the accuracy of two curvature models is equivalent, while the exact curvature model is more time-consuming. These conclusions can be a recommendation for the choice of geometrical curvature expressions in some certain practical examples.

Modeling planar pantographic sheets using a nonlinear Euler–Bernoulli beam element based on B‐spline functions

AbstractAn undeformed pantographic sheet consists of two orthogonal arrays of straight fibers interconnected by internal pins. In this paper, we model the fibers of this lattice‐like sheet as nonlinear Euler–Bernoulli beams and use B‐spline functions for their finite element discretization. Using the concept of one‐dimensional generalized force laws, we show how different models for the pins can be introduced in the model. Finally, the simulation of a tensile test is presented.

Multiobjective optimization of rotor-bearing systems with an investigation of goal programming approach

With ever-increasing demand for higher flexibility and higher speeds of operation of high-speed rotating machinery, study of their unbalance responses and stability are of paramount importance. For easier maintenance and transport, weight of such systems is also a critical criterion. In the present study, optimum design parameters of a rotor-bearing system are found by employing multiple multiobjective optimization schemes for simultaneously minimizing unbalance response, maximizing stability limit speed, and minimizing system weight. This is an improvement over previous systems where simple equal weighting of multiple objective functions is carried out to reduce computational challenges. The shaft is assumed to be hollow and modeled as an assembly of 10 two-noded Euler–Bernoulli beam elements with an isotropic disc mounted on it. Genetic algorithm and its multiobjective variant (NSGA-II) are used to find the global optima. The technique of goal programming is identified as a more practical and less computationally challenging alternative to Pareto-based optimization, while yielding an approximate optimum. Complexities arising due to gyroscopic effects and dynamic nature of anisotropic fluid film bearings are also considered, along with practical upper and lower bounds of design variables and maximum von Mises stress developed in the system as constraints.

Isogeometric Analysis of Euler-Bernoulli Beam Element

Abstract: The Isogeometric analysis is a computational geometry based on a series of polynomial functions (Non-Uniform Rational B-Spline, NURBS) which are assembled to represent the exact geometry. In the Isogeometric analysis, the curvature geometry of the beam element can be represented exactly. The conventional beam element can be developed by using the Isogeometric approach which is based on Euler-Bernoulli principle which is under the assumption that the dimension of the beam cross section is small compared with the length of the beam. The geometric shape of the beam and the shape functions formulation of the element can be formulated by using the Isogeometric approach. This paper highlights the application of the NURBS for the Euler-Bernoulli beam element in the context of finite element analysis. Examples are given to verify the effectiveness of the Isogeometric approach in static and free vibration problems.

On the molecular mechanics of single layer graphene sheets

The molecular structural mechanics (MSM) method is developed by applying beam elements to model bonded interactions between carbon atoms in the atomic lattices of single-layer graphene sheets (SLGSs). The novelty of the approach developed in this paper lies in the accurate adjustment of the geometric and material parameters of Bernoulli–Euler beam elements to simulate reference mechanical moduli (2D Young’s modulus, Poisson’s ratio, and bending rigidity modulus) of graphene. The MSM method with an advanced geometric and material parameter set of Bernoulli–Euler beam elements is implemented by means of the commercial MSC.Marc finite element (FE) code. We also employ the standard molecular mechanics (MM) method using the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909). We implemented this force field in the homemade PIONER FE code using a modified parameter set which reproduces the mechanical moduli of graphene reasonably well. Computer simulations show that the free vibration frequencies and modes of SLGSs obtained using the standard MM and MSM methods converge. However, the buckling forces of compressed SLGSs obtained by the two methods provide acceptable convergence only for the lowest values of the critical forces.

Fracture toughness of hierarchical self-similar honeycombs

The influence of self-similar hierarchy on the brittle fracture behavior of a two-dimensional honeycomb is investigated. The honeycomb walls are modeled as Bernoulli-Euler beam elements rigidly connected at the nodal points, and zero, first and second order hierarchies are addressed. The stress state in the vicinity of a semi-infinite crack is determined, and the fracture toughness is calculated in terms of tensile strength of parent material. The growth of hierarchical order leads to increase in the number of nodal degrees of freedom to be taken into account, and the computational cost of the calculations is reduced by employing a novel analysis method based on the discrete Fourier transform.An increase in hierarchical order enhances the fracture toughness after optimization of geometrical parameters at fixed relative density. This effect becomes more pronounced for honeycombs of lower relative density.

Modeling and analysis of sliding joints with clearances in flexible multibody systems

The modeling of the sliding joint with clearance between a flexible beam and a rigid hole is investigated in this paper. The flexible beam is discretized using the three-dimensional curved Euler–Bernoulli beam element of the Absolute Nodal Coordinate Formulation, while the motion of the rigid hole is described by the Cartesian coordinates. Moreover, the cross sections of both the flexible beam and the rigid hole are assumed to be circular. The existing joints with clearances are mainly rigid joints with small clearances, and the contact detection algorithm adopted can solve only one pair of potential contact points within one section. In order to model the contact problem in the sliding joint with clearance, a new contact detection method based on the intersection of the rigid hole’s cross section and the flexible beam is proposed, which yields a two-dimensional contact detection problem. Based on the common-normal concept, the ellipse–circle contact detection problem within the hole’s cross section can be solved. The potential contact point on the hole’s cross section will be determined, and the closest point projection on the beam’s neutral axis can be defined further. The proposed contact detection method can deal with the sliding joint with large clearance and the multiple-point contact problem within one section. In addition, the penalty method is adopted to model the frictionless contact between the flexible beam and the rigid hole. Finally, two numerical examples about sliding joints with clearances, one with an initially curved beam under gravity and the other with a straight beam under zero gravity, are presented to demonstrate the influence of the clearance of sliding joint on the dynamic performance of flexible multibody systems.

Performance evaluation of a reticulated shell with atmospheric corrosion damage

The study on damage evaluation induced by atmospheric corrosion of engineering structures has attracted more and more international concern over the past three decades. However, the effects of atmospheric corrosion on reticulated shells have not been systematically investigated. In this regard, the performance assessment of a reticulated shell subjected to atmospheric corrosion damage is actively conducted in this study. The atmospheric corrosion model of shell elements is first presented, and a refined exponential model for estimating the corrosion depth of steel materials is developed by using the pattern recognition technique. The sensitivity of stiffness matrix to element thickness is established by using Euler–Bernoulli beam element. The sensitivity of mass matrix to element thickness is developed based on lumped mass assumption. Then, the expression of natural frequency sensitivity to element thickness and mass is derived by considering the section loss induced by both the inner and outer surface corrosion. In addition, the explicit expression of frequency sensitivity to mass of spherical joints is also established in detail. The nonlinear static structural analysis is conducted to evaluate effects of atmospheric corrosion on the stress of structural elements. A real reticulated shell constructed in northern China is taken as the example structure to examine the feasibility of the proposed approach and to assess the potential damage caused by atmospheric corrosion to the structure.

An extension of a discrete-module-beam-bending-based hydroelasticity method for a flexible structure with complex geometric features

The discrete-module-beam-bending based hydroelasticity method proposed by Lu et al. (2016) deals with the hydroelastic response of a flexible structure by discretising it into several rigid submodules which are connected by Euler-Bernoulli beam elements. A stiffness matrix is determined in terms of the geometrical and physical properties of the flexible structure for each beam element and it has an analytical form for simple geometric features such as a rectangular cross section unchanged along the longitudinal direction. However, the analytical stiffness matrix may not exist for a flexible structure with complex geometric features. In the present study, with the aid of finite element method, the discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features.

Study on Vibration Characteristics of Fan Shaft of Geared Turbofan Engine With Sudden Imbalance Caused by Blade Off

Based on the requirements of the dynamic design of geared turbofan (GTF) engines, the vibration characteristic of the fan shaft is investigated. The effect of sudden imbalance caused by blade off, the time-varying meshing stiffness, and meshing errors on the vibration characteristics is fully considered in the dynamic model. The improved Euler–Bernoulli beam element considering the effects of shear deformation is employed and a coupled relationship between the gear–shaft–bearing–casing is established. Under windmilling condition with a single blade completely lost, the vibration characteristics of the fan shaft of the turbofan engine with and without a gearbox system are compared. The effect of the gear system on the vibration of the fan shaft under different rotating speeds is examined. The results show that the orbit of the fan shaft center in the turbofan engine with the gearbox system exhibits a multifrequency whirling motion and has a stable limit cycle. Under windmilling condition, the meshing frequency and the modal frequency have multiple intersection points. The critical speed is dense and the peak value of the transient vibration of the GTF engine gearbox shows a wave rise with an increase in speed. The results of this study could provide a reference for the parameter design and optimization of GTF engines.

The multiple slope discontinuity beam element for nonlinear analysis of RC framed structures

The seismic nonlinear response of reinforced concrete structures permits to identify critical zones of an existing structure and to better plan its rehabilitation process. It is obtained by performing finite element analysis using numerical models classifiable into two categories: lumped plasticity models and distributed plasticity models. The present work is devoted to the implementation, in a finite element environment, of an elastoplastic Euler–Bernoulli beam element showing possible slope discontinuities at any position along the beam span, in the framework of a modified lumped plasticity. The differential equation of an Euler–Bernoulli beam element under static loads in presence of multiple discontinuities in the slope function was already solved by Biondi and Caddemi (Int J Solids Struct 42(9):3027–3044, 2005, Eur J Mech A Solids 26(5):789–809, 2007), who also found solutions in closed form. These solutions are now implemented in the new beam element respecting a thermodynamical approach, from which the state equations and flow rules are derived. State equations and flow rules are rewritten in a discrete manner to match up with the Newton–Raphson iterative solutions of the discretized loading process. A classic elastic predictor phase is followed by a plastic corrector phase in the case of activation of the inelastic phenomenon. The corrector phase is based on the evaluation of return bending moments by employing the closest point projection method under the hypothesis of associated plasticity in the bending moment planes of a Bresler’s type activation domain. Shape functions and stiffness matrix for the new element are derived. Numerical examples are furnished to validate the proposed beam element.

A dynamic finite element method for the estimation of cable tension

Cable supported structures have been widely used in civil engineering. Cable tension estimation has great importance in cable supported structures\' analysis, ranging from design to construction and from inspection to maintenance. Even though the Bernoulli-Euler beam element is commonly used in the traditional finite element method for calculation of frequency and cable tension estimation, many elements must be meshed to achieve accurate results, leading to expensive computation. To improve the accuracy and efficiency, a dynamic finite element method for estimation of cable tension is proposed. In this method, following the dynamic stiffness matrix method, frequency-dependent shape functions are adopted to derive the stiffness and mass matrices of an exact beam element that can be used for natural frequency calculation and cable tension estimation. An iterative algorithm is used for the exact beam element to determine both the exact natural frequencies and the cable tension. Illustrative examples show that, compared with the cable tension estimation method using the conventional beam element, the proposed method has a distinct advantage regarding the accuracy and the computational time.

Development of a Unified Rock Bolt Model in Discontinuous Deformation Analysis

In this paper, a unified rock bolt model is proposed and incorporated into the two-dimensional discontinuous deformation analysis. In the model, the bolt shank is discretized into a finite number of (modified) Euler–Bernoulli beam elements with the degrees of freedom represented at the end nodes, while the face plate is treated as solid blocks. The rock mass and the bolt shank deform independently, but interact with each other through a few anchored points. The interactions between the rock mass and the face plate are handled via general contact algorithm. Different types of rock bolts (e.g., Expansion Shell, fully grouted rebar, Split Set, cone bolt, Roofex, Garford and D-bolt) can be realized by specifying the corresponding constitutive model for the tangential behavior of the anchored points. Four failure modes, namely tensile failure and shear failure of the bolt shank, debonding along the bolt/rock interface and loss of the face plate, are available in the analysis procedure. The performance of a typical conventional rock bolt (fully grouted rebar) and a typical energy-absorbing rock bolt (D-bolt) under the scenarios of suspending loosened blocks and rock dilation is investigated using the proposed model. The reliability of the proposed model is verified by comparing the simulation results with theoretical predictions and experimental observations. The proposed model could be used to reveal the mechanism of each type of rock bolt in realistic scenarios and to provide a numerical way for presenting the detailed profile about the behavior of bolts, in particular at intermediate loading stages.

Data-driven reduced-order model of microtubule mechanics.

A beam element is constructed for microtubules based upon data reduction of the results from atomistic simulation of the carbon backbone chain of αβ-tubulin dimers. The database of mechanical responses to various types of loads from atomistic simulation is reduced to dominant modes. The dominant modes are subsequently used to construct the stiffness matrix of a beam element that captures the anisotropic behavior and deformation mode coupling that arises from a microtubule's spiral structure. In contrast to standard Euler-Bernoulli or Timoshenko beam elements, the link between forces and node displacements results not from hypothesized deformation behavior, but directly from the data obtained by molecular scale simulation. Differences between the resulting microtubule data-driven beam model (MTDDBM) and standard beam elements are presented, with a focus on coupling of bending, stretch, shear deformations. The MTDDBM is just as economical to use as a standard beam element, and allows accurate reconstruction of the mechanical behavior of structures within a cell as exemplified in a simple model of a component element of the mitotic spindle.

Meso-scale modeling of heat transport in a heterogeneous cemented geomaterial by lattice element method

The simulation of heat transport in a heterogeneous cemented geomaterial using lattice element method is the focus of this paper. The proposed method represents a heterogeneous cemented medium with the inter-connected Euler–Bernoulli beam elements for transmitting heat and mechanical loads. The mechanical equilibrium is assessed with minimizing the potential energy and in a meanwhile the conducted heat between solids is calculated based on modified thermal discrete element method. A validation study for heat transfer is carried out with the existing finite element method. In order to generate the heterogeneity, the random distribution or image processing techniques are implemented and subsequently the effective thermal conductivity (ETC) is determined. The effect of controlling parameters, such as mesh size, randomness factor, voids, heterogeneity and applied external mechanical loads, on calculated ETC is studied. Finally, with application of the proposed coupled thermo-mechanical lattice element, the ETC of three rock samples is determined and compared to the experimental data. The proposed method is able to model the heat transport in a heterogeneous cemented geomaterial and predict the ETC, which matches the experimental results.

Multi-objective optimization of laminated composite beam structures using NSGA-II algorithm

The paper deals with the multi-objective optimization problems of laminated composite beam structures. The objective function is to minimize the weight of the whole laminated composite beam and maximize the natural frequency. In particular, the simultaneous use of all the design variables such as fiber volume fractions, thickness and fiber orientation angles of layers is conducted, in which the fiber volume fractions are taken as continuous design variables with the constraint on manufacturing process while the thickness and fiber orientation angles are considered as discrete variables. The beam structure is subjected to the constraint in the natural frequency which must be greater than or equal to a predetermined frequency. For free vibration analysis of the structure, the finite element method is used with the two-node Bernoulli-Euler beam element. For solving the multi-objective optimization problem, the nondominated sorting genetic algorithm II (NSGA-II) is employed. The reliability and effectiveness of the proposed approach are demonstrated through three numerical examples by comparing the current results with those of previous studies in the literature.

NONLINEAR DYNAMIC DAMAGE EVOLUTION OF A HIGHWAY BRIDGE DUE TO ITS DYNAMIC INTERACTION WITH RANDOM FORMS OF IRREGULARITIES AND MOVING VEHICLES

The effects of vehicle-structure interaction have a significant importance on the dynamic responses of both systems with several applications within the highways field. If the vehicle-irregularities-bridge dynamic interaction is capable to produce accumulated strains that cause damage to the structure, the dynamic responses are affected. By crossing a highway bridge with any speed, the vehicle is subjected to the highway irregularities. The movement of a vehicle on a bridge is already a dynamic action on the structure. However, the highway irregularities tend to excite the vehicle dynamically which in turns triggers additional vibrations in the highway bridge structure apart from those produced by their own movement, increasing the bridge’s damage evolution. This modifies the dynamic responses of the structure, increasing the magnitude and the oscillations particularly at critical speeds of the vehicle, capable to provoke some resonance. Apart from changing the displacements, velocities and accelerations responses, the damage alters the structural natural frequencies of vibration. Such effects are not possible to be analysed with linear dynamic models. The nonlinearities occur by the fact that the forces no longer linearly depend from the displacements when damage occurs. This work aims to evaluate the nonlinear dynamic damage evolution of a reinforced concrete highway bridge through the Finite Element Method, on which the degree of damage is altered over time by the dynamic interaction with random irregularities and moving vehicles due to the stiffness loss of the structure by Damage Mechanics. The highway irregularities are represented by random functions. Euler-Bernoulli beam elements with Hermite cubic interpolation functions are used for the bridge model. The Mazars Damage Constitutive Model is implemented with the condition of stress inversion due to vibration. The continuum damage mechanics is considered dynamically. Therefore, the damage is evaluated in each layer of the structure cross section for each iteration within each time step. The structural damping is defined by the Rayleigh method with updated coefficients due to damage. The equations of motion are obtained by nonlinear dynamic equilibrium and numerically integrated in time using the Newmark Method together with the Newton-Raphson iterative Method. This proposal seeks to contribute to the study of the health monitoring and the structural integrity of damaged highway bridges structures. Keywords: Damage Mechanics, Dynamic Interaction, Finite Element Method, Nonlinear Dynamics, Computational Modeling

Vibrations of Flexible Strip on Viscoelastic Halfspace

The dynamic response of rigid and flexible foundations on the soil has been subject of extensive study in the past decades. A hybrid method using a combined finite element method (FEM) and boundary element method (BEM) is the most common method used for solving this problem. The objective of this paper is to present an effective frequency domain method to obtain the dynamic response of a flexible strip foundation resting on a viscoelastic halfspace. The foundation is treated with the spectral element method (SEM), while the soil is modelled using the integral transform method (ITM). Both SEM and ITM are based on the analytical solution of the Lame-equations in the frequency domain and therefore are suitable for combining. The solution is obtained in the transformed space-frequency or wavenumber-frequency domain using the Fourier transformation. The study is performed as a 2D plane-strain analysis, assuming that the foundation cross-section behaves as an Euler-Bernoulli beam and that there is no sliding between the foundation and the soil, nor discontinuities in terms of the displacement field. The vertical displacements field of the foundation is described by a set of modal functions corresponding to free vibration mode shapes of a SEM Euler-Bernoulli beam element. The coupling between the foundation and the soil is achieved using the modal soil impedance functions, which are determined by using the ITM. The displacements of the coupled foundation-soil system are solved by the modal superposition method. The accuracy of the proposed method is assessed by comparing the obtained results with the results obtained by a commercial software package SASSI2000. The comparison shows that the presented method is accurate and less costly in terms of computational effort, especially in the high frequency range. The presented method can be easily extended to provide the solution of the response of a flexible strip on a layered halfspace due to a horizontal and vertical excitation.