Beam Finite Element Research Articles

Free vibration analysis of thin to thick straight or curved beams by a solid-3D beam finite element method

A solid-3D beam finite element method is presented for the vibration analysis of thin to thick beam-like structures. This method is efficient for a spatial beam and considers all mechanical effects: membrane, bending and torsion. The structure is discretized with solid elements and contains several elements throughout the cross-section. Displacement fields of first-order or higher-order beam theories are met by directly applying kinematic relations on the solid finite element model, throughout the cross-section. This leads to a set of linear equations applied at slave nodes. Only the master nodes are kept in the final model, resulting in a model size reduction. The effectiveness of this method is demonstrated through two numerical examples, including a straight and a curved beam. The modal shapes and natural frequencies obtained by the proposed method are compared with the complete solid model and classical beam model. The size of these solid-beam models is slightly larger than that of an approach with classical beam element.

Vibration analysis of functionally graded microbeams with a moving mass based on Timoshenko beam theory

Vibration analysis of functionally graded (FG) microbeams carrying a moving mass is carried out in the framework of Timoshenko beam theory. The beam material properties are considered to be graded in the thickness by a power-law function, and they are estimated by Mori-Tanaka scheme. The influence of the microsize effect is captured with the aid of the modified couple stress theory (MCST). A finite beam element is formulated and used to establish the discretized equation of motion for the beams. Vibration characteristics, including the natural frequencies, the time histories for mid-span deflection, the dynamic magnification factor and the stress distribution, are computed for a simply supported beam. The obtained result reveals that both the dynamic deflection and dynamic magnification factor are overestimated by ignoring the microsize effect. The effects of the material distribution, the moving mass velocity and the length scale parameter are studied in detail and highlighted.

A new locking-free finite element for N-layer composite beams with interlayer slips and finger joints

A new locking-free finite element for N-layer composite beams with interlayer slips and finger joints

DAMAGE DETECTION IN A STEEL BEAM USING VIBRATION RESPONSE

Changes in vibration signals, natural frequency and modal parameters were used for damage detection in a steel beam. Several tests were performed on simply supported steel beams having open transverse cracks with varying depths and locations. Frequencies for the first three vibration modes were generated from the analytical, numerical, and experimental models. Analytical frequencies were calculated assuming Euler-Bernoulli beam and solving partial differential equations. Vibration signals were collected using a portable digital vibrometer and analyzed through Fast Fourier Transform. The finite element beam models were calibrated using experimental results. The graphical plots of the normalized frequency from the experimental study and FE models showed similar pattern for the first mode. The nodal points were identified as the locations where the damaged frequency equals the frequency of an intact beam. The existence of damage in the beam specimen was confirmed by comparing the natural frequency of a beam at its intact and damaged states. Equations relating normalized frequency with varying crack depth and location were proposed.

Modelling and Analysis of Expansion Joints' Effect on Dynamic Performance of Railway Rigid Overhead System.

This study focuses on developing a comprehensive model of a rigid overhead system, which includes essential components such as the suspension structure, positioning clamp, and expansion joint. The modelling approach utilizes finite element theory and beam elements to accurately represent the displacement, stiffness, and mass characteristics of the system. The models also incorporate the suspension structure and positioning line clamp, which play crucial roles in suspending and positioning the busbar. Various suspension structures and positioning line clamps are evaluated based on their dynamic characteristics. The expansion joint, responsible for connecting different anchor sections of the rigid overhead system, undergoes a detailed analysis. Different assembly scenarios, including ideal and deflected assembly conditions, are considered. To simulate the dynamic behaviour of the expansion joint, additional beams are introduced into the system model. The primary finding of the analysis is that the maximum stresses observed in the constructed expansion joint model, under different temperature conditions and normal/deflected assembly conditions, remain within the permissible stress limits of the material. This indicates a high level of safety. However, certain areas exhibit stress concentration, particularly at the sliding block B and sliding rod A positions. This stress concentration is primarily attributed to the unique assembly form of the expansion joint. To improve stress distribution and enhance service reliability, the analysis suggests optimizing the installation deflection angle and geometric design of the expansion joint. Furthermore, the concentrated mass at the expansion joint significantly impacts the current collection quality of the pantograph-overhead system. Mitigating this negative impact can be achieved by reducing the mass of the expansion joint.

A novel finite element for nonlinear static and dynamic analyses of helical cables

Cables and wire ropes are structural components designed for high axial strength and relatively low torsion and flexure stiffness. Given the complex geometry of helical cables, analytical models for predicting their mechanical behavior include many assumptions and limitations. Three-dimensional finite elements can be used, but its practical use in Engineering applications with hundreds or thousands of meters long is hampered. On the other hand, beam finite elements do not consider the complexity of the cable construction, leading to wrong results, and therefore they are not an option neither, in general. To circumvent this problem, a new finite element is designed and implemented, aiming to incorporate the behavior of a 3D cable modeled using solid elements into a 1D beam element. The designed element is able to combine the accuracy of solid models with the practical implementation of line elements in nonlinear regime. In comparison with the cable modelled with solid elements, the proposed equivalent element yielded deviations of about 4% in the strain energy, when submitted to a variety of load conditions. High accuracy was also obtained in comparing axial, torsional, and bending stiffness of the equivalent element as function of the cable helix angle. Stiffness coupling terms, however, showed lower accuracy. Regarding the dynamic behavior, modal analysis yielded errors below 10% when comparing natural frequencies. Cable’s response under harmonic loading was also accurately reproduced by the introduced element.

A locking free virtual element formulation for Timoshenko beams

The virtual element method (VEM) provides new ways of deriving discretization for problems in structural and solid mechanics, starting with the contribution by Beirão da Veiga et al. (2013) for elastic solids. Interestingly, the virtual element method allows also to revisit the construction of different elements which have the same shape as finite elements. This is even true for one-dimensional structures like trusses and beams. Here we study a virtual element development of the Timoshenko beam which surprisingly leads to a straight forward formulation of locking free and in the linear range even exact Timoshenko beam elements. These elements can be easily incorporated into classical finite element codes since they have the same number of unknowns as finite elements for beams. The formulation allows to compute nonlinear structural problems undergoing large deflections and rotations using the formulation provided in Reissner (1972).

DISCRETE TIMOSHENKO BEAM MODEL FOR MODELING BEAMS WITH NON-UNIFORM CROSS-SECTIONAL AREA, CURVATURE, AND LARGE DEFLECTION

The Timoshenko beam theory is widely used in various fields due to its ability to accurately model the behavior of beams with specific characteristics. However, when it comes to analyzing large deflection beams, the Timoshenko beam theory becomes less precise. This paper introduces a proposed discrete Timoshenko beam model to predict the mechanical behavior of slender beams subjected to large deflections under various loadings. A mathematical analysis was developed and validated using finite element analysis in conjunction with a compliant mechanism. The modeling results demonstrate that the force-displacement behavior of a bistable mechanism can be accurately captured by the discrete Timoshenko beam model. In particular, when comparing the discrete Timoshenko beam model and finite element analysis, the maximum and minimum forces exhibited a strong agreement, with a percentage difference of less than 3% and 5%, respectively. In terms of the total elastic strain energy and maximum principal stress, the deviations between the discrete Timoshenko beam model and finite element analysis were approximately 2% and 7%, respectively. The proposed model can be integrated into an optimization algorithm for designing compliant mechanisms.

Modified beam modeling of powder bed fusion manufactured lattice structures

In this work a modified beam approach to model the compressive behavior of additively manufactured BCC and reinforced BCC lattice structures is proposed. Three-dimensional beam models are of interest because of their reduced computational time, with respect to numerical models that involves solid mesh elements, but they are not able to capture the behavior of lattice structures in proximity of the unit cell nodes, that presents higher stiffness due to the struts intersection and consequent material overlapping. In order to overcome this drawback when using beam finite elements, a modified beam approach is used in this work considering each lattice cell strut subdivided in a central flexible portion and two rigid extremities. A numerical procedure is implemented to tune the rigid extremities length of the strut by means of numerical simulations that aims to reproduce accurately the compressive behavior of the lattice cell solid model. The experimental campaign conducted on different powder bed fusion manufactured lattice structures is presented and experimental results are provided along with a numerical-experimental correlation that proves the accuracy of the method.

Robotic check of a subassembly, and its simulation

This paper discusses a quality inspection process of a subassembly of a battery cover, which is performed with an industrial robot equipped with an intelligent end-effector. The subassembly has a plastic pin and a torsion spring part. During intended use of the unit, the necessary force to actuate the pin is determined with measurements and simulations. In the measurement a self-devised intelligent end-effector is equipped with a microcontroller and a beam type load cell in order to measure the spring force. The main aim is to make an automatic decision without operator intervention at the quality check process, which is related to the quality of the subassembly. In the simulation deformation of the spring part is modeled with geometrically nonlinear 2D beam finite elements in the course of assembly and intended use. Co-rotational approach is used to consider the large displacements and rotations with small strains. Penalty method is applied to treat the contact between the spring and pin parts.

A fiber beam element based on plastic and damage models for prestressed concrete structures

Modeling of prestressed concrete beams and their degrading processes are relevant issues in the assessment of existing structures and infrastructures, particularly for bridges. This work focuses on the formulation and validation of a fiber beam finite element for prestressed concrete structural elements that accounts for nonlinear material behavior and time-dependent phenomena. The element is a 3D Timoshenko beam that follows a force-based approach. A fiber discretization of the cross-section is used to model concrete, steel reinforcements and prestressing cables based on damage and plasticity for concrete, and plasticity for steel through three-dimensional constitutive laws. The steel fibers are fully bonded. Strain-softening in the concrete requires an appropriate regularization procedure. The tendons are treated as additional fibers with prestressing applied as initial strain of the relevant fibers. No additional discretization is required except for the description of the tendons’ profile. Time-dependent phenomena such as creep, shrinkage and cable relaxation are taken into account. The proposed model is implemented in the OpenSees computational framework to carry out specific validation tests and demonstrate the element potential.

Nonlinear analysis on the impact wear characteristics of casing based on drillstring dynamics

Casing integrity is significant for subsequent drilling and production and prediction of casing wear accurately helps to reduce casing damage. Thus, this study aims to represent the casing impact wear characteristics more intuitively and accurately during drilling. The vibrating drillstring when drilling will collide with casing frequently and increase the casing wear. The drillstring nonlinear dynamic model is established using beam finite element method in this work, and the drillstring dynamic characteristics are obtained using generalized-α method. Subsequently, the friction coefficient and wear factor are obtained through the casing wear test. On this basis, the casing impact wear characteristic combined with the casing wear efficiency model is described, and the residual strength of casing is analyzed. In addition, the effects of drilling depth, weight on bit (WOB), rotation speed and the eccentric distance of the drillstring on the casing wear are discussed. We hope that our research will aid in designing drilling parameters and reducing casing damage.

Data-driven modelling of the multiaxial yield behaviour of nanoporous metals

Nanoporous metals, built out of complex ligament networks, can be produced with an additional level of hierarchy. The resulting complexity of the structure makes modelling of the mechanical behaviour computationally expensive and time consuming. In addition, multiaxial stresses occur in the higher hierarchy ligaments. Therefore, knowledge of the multiaxial material behaviour, including the 6D yield surface, is required. Surrogate models, predicting the mechanical behaviour of the lower level of hierarchy, represented by finite element beam models, are a promising approach to overcome such challenges, when existing analytical models are not able to describe the material behaviour. Therefore, as a first step, we studied the elastic behaviour and the yield surfaces of representative volume elements with idealised diamond and Kelvin structure in finite element simulations. The yield surfaces showed pronounced anisotropy and could not be described by the Deshpande-Fleck model for isotropic solid foams. Instead, we used data-driven and hybrid artificial neural networks, as well as data-driven support vector machines and compared them regarding their potential for the prediction of yield surfaces. All considered methods were well suited and resulted in relative errors <4.5%. Support vector machines showed the best generalisation and accuracy in 6D stress space and are suitable for extrapolation outside the range of training data.

Evaluation of the effects of the seismic wave incidence angle on soil-structure interactions via advanced finite beam elements

This work investigates the interactions between the soil and a superstructure subjected to various seismic waves using variable-kinematics finite beam elements. Artificial boundary conditions based on springs and dashpots with appropriate elastic and viscous coefficients were adopted to ensure the absorption of the incident waves and simulate the infinite domain condition. The one-dimensional finite element formulation is obtained with a unified formalism that enables arbitrary structural models to be adopted. Lagrange-type expansions have been used in this work to approximate the beam kinematics and facilitate the application of artificial boundary conditions. Both pressure and shear waves with different incidence angles have been considered, and the dynamic responses calculated with the current approach have been compared with analytical solutions when available. The results have demonstrated the accuracy and potentialities of the methodology and provided reasonable confidence for future applications.

Stability analysis of stiffness reduction models on rotary- and non-rotary straightened W-shapes

This paper presents a parametric study of stiffness reduction models on rotary- and non-rotary straightened hot-rolled steel W-shapes to examine if the models for non-rotary straightened sections are appropriate for stability analysis of rotary-straightened members. Four stiffness reduction models were investigated, which include (1) a conventional model adopted by AISC 360, which is known as the CRC model, (2) a model fitting the column strength curves provided in Eurocode 3, and models fitting cross-section distributed plasticity curves incorporating (3) the ECCS residual stress pattern and (4) a residual stress pattern of rotary-straightened sections. Beam finite element models were created and second-order inelastic analyses were conducted. A range of different cross-section geometries typically used for beam–columns was investigated, which includes various aspect ratios and flange-to-web area ratios. Columns and beam–columns were investigated with uniaxial bending about the major and minor axes with several axial utilization ratios. The effects of the stiffness reduction models on the stability limit state are discussed. The results of the parametric study are presented along with comparisons of the stiffness reduction values for hot-rolled steel members, which provide further understanding of the stiffness reduction model’s influence on the stability analysis of rotary straightened hot-rolled steel members.

Prediction and control of fracture paths in disordered architected materials using graph neural networks

Architected materials typically rely on regular periodic patterns to achieve improved mechanical properties such as stiffness or fracture toughness. Here we introduce a class of irregular cellular materials with engineered topological and geometrical disorder, which represents a shift from conventional designs. We first develop a graph learning model for predicting the fracture path in these architected materials. The model employs a graph convolution for spatial message passing and a gated recurrent unit architecture for temporal dependence. Once trained on data gleaned from experimentally validated elastoplastic beam finite element analyses, the learned model produces accurate predictions overcoming the need for expensive finite element calculations. We finally leverage the trained model in combination with a downstream optimization scheme to generate optimal architectures that maximize the crack path length and, hence, the associated fracture energy.

Free form shape optimization of three dimensional beams using cross section analysis

We introduce an efficient method for shape optimization of three dimensional beams that is based on free-form design of the beam’s cross section. The cross section boundary of each beam segment is parameterized by a closed, unclamped B-spline curve, whose control points’ coordinates serve as design variables. The response of the beam is evaluated globally as a one-dimensional entity using 3D beam finite elements, while each cross section is modeled locally using cross section analysis following anisotropic beam theory. A fully consistent analytical sensitivity analysis is derived to account for the relation between the control points and the beam cross-sectional properties and design updates are performed using gradient-based optimization. The proposed method is applied to several test cases, aiming at minimization of compliance or maximization of the fundamental frequency. The numerical results and a comparison to other methods for beam optimization show that the method can generate highly competitive designs. A consistent quantitative comparison to high resolution continuum topology optimization shows that a performance which is only slightly inferior can be achieved, for only a small fraction of the computational effort.

Mechanism of interconnected synchronized switch damping for vibration control of blades

The Synchronized Switch Damping (SSD) is regarded as a promising alternative to mitigate the vibration of thin-walled structures in aero-engines, especially for blades or bladed disks. The common manner is to shunt the switch circuit independently to a single piezoelectric structure. This paper is aimed at exploring a novel way of using the SSD, i.e., the SSD is interconnected between two piezoelectric structures or substructures. The damping mechanism, performance, and effective range of the interconnected SSD are studied numerically and experimentally. First, based on a dual cantilever beam finite element model, the time domain and frequency domain modeling and solving methods of the interconnected SSD are deduced and validated. Then, the influence of the amplitude and phase relationship on the damping effect of the interconnected SSD is numerically studied and compared with the shunted SSD. A self-sensing SSD control board is developed, and experimental studies are carried out. The results show that the interconnected SSD establishes an additional energy channel between the corresponding piezoelectric structures. When the amplitudes of the two cantilever beams are different, the interconnected SSD balances the vibration level of each beam. When the amplitudes of the two cantilever beams are the same, if the appropriate interconnection manner is selected according to the phase, the resonance peak can be reduced by more than 30%. When the vibration is in-phase/out-of-phase, the damping generated by the interconnected SSD in a cross/parallel manner is even more significant than the shunted SSD. Furthermore, this novel connection scheme reduces the number of SSD circuits in half. Finally, for engineering applications, we implement the proposed damping technology to the finite element model of a typical dummy bladed disk. A piezoelectric damping ratio of 13.7% is achieved when the amount of piezo material is only 10% of blade mass. Compared with traditional friction dampers, the major advancements of the interconnected SSD are: (A) it can reduce the vibration level of blades without friction interface; (B) the space constraint is overcome, i.e., the vibration energy is not necessarily dissipated independently in one sector or through physically adjacent blades, and instead, the dissipation and transfer of vibrational energy can be realized between any blade pair. If a specific gating circuit is adopted to adjust the interconnection manner of the SSD, vibration mitigation under variable working conditions with different engine orders will be expected; (C) designers do not need to worry about the annoying nonlinearities related to working conditions anymore.

Impact of the rotor blades elasticity on the loads and wake of the large IEA 15-MW wind turbine

This work presents an investigation of the aeroelastic effects occurring on the IEA 15-MW reference wind turbine in a turbulent atmospheric boundary layer. Large Eddy simulation is carried out using an actuator line method coupled to the finite element beam solver BeamDyn. The results show that the blades experience significant displacements, leading to an important variation of the loads along the blade span. The turbulence also interacts with the blades natural frequencies, resulting in a variation of the spectra of the various loads. The impact of the flexibility is also evaluated on the wake by comparing the undeformed configuration with a statically and dynamically deformed rotor. It appears that the mean deformation plays an important role in the wake development due to changes in the load distribution, but that the dynamic effects have minimal impact on the wake behavior.

Optimization of Curvilinear Stiffener Beam Structures Simulated by Beam Finite Elements with Coupled Bending-Torsion Formulation.

This research presents the application of a beam finite element, specifically derived for simulating bending-torsion coupling in equivalent box-beam structures with curvilinear stiffeners. The stiffener path was simulated and optimized to obtain an expected coupling effect with respect to four typical static load cases, including geometric constraints related to the additive manufacturing production method. The selected load condition was applied to the centroid of the beam section, and the structure performance was consequently determined. A variation in load position up to one-fourth of the beam width was considered for investigating the stiffener path variation corresponding to a minimum bending-torsion coupling effect. The results demonstrated the capability of such a beam finite element to correctly represent the static behavior of beam structures with curvilinear stiffeners and show the possibility to uncouple its bending-torsion behavior using a specific stiffener orientation. The simulation of a laser powder bed fusion process showed new opportunities for the application of this technology to stiffened panel manufacturing.