Beam Finite Element Research Articles

Piezoelectric Beam Finite Element Model and Its Reduction and Control

Abstract The paper deals with the development of the finite element method (FEM) model of piezoelectric beam elements, where the piezoelectric layers are located on the outer surfaces of the beam core, which is made of functionally graded material. The created FEM model of piezoelectric beam structure is reduced using the modal truncation method, which is one of model order reduction (MOR) method. The results obtain from reduced state-space model are compared with results obtain from finite element model. MOR state-space model is also used in the design of the linear quadratic regulator (LQR). Created reduced state-space model with feedback with the LQR controller is analysed and compared with the results from FEM model.

Distortional buckling of compressed cold‐formed lipped C channels via buckling mode shapes

AbstractDistortional buckling is a characteristic failure mode of compressed C channels with an intermediate unbraced length in the range of 4 to 7 times the web depth. The usual codified design method to calculate the buckling resistance by Eurocode EN 1993‐1‐3 proceeds through the determination of an effective cross‐section with reduced thicknesses of the stiffener zones working against distortional buckling. To calculate the slenderness of the compressed stiffener zones Eurocode gives simple formulas, applicable separately for end stiffeners and intermediate stiffeners. The identification of the part of the cross‐section to be used in these formulas can be doubtful. It can be demonstrated that these formulas will predict unreliable critical stresses being often on the unsafe side. EN 1993‐1‐3 allows to use more reliable values based on FEA or other approaches without giving a detailed, applicable design methodology.This paper presents a method which is able to assign cross‐section buckling modes calculated with the help of finite strip analysis to compressed stiffener zones of a compressed general C section. Assignment is based on the determination of deformation energy balance. The proposed approach finds the zones of the section which tend to buckle during distortional buckling and the zones which are responsible to restrain this buckling. Additionally, it gives guidance on how to determine the exact area where the calculated thickness reduction shall be applied around the stiffener. For the thickness reduction the buckling curve given in the actual EN 1993‐1‐3 will be used.The proposed method is part of a larger research project which aims to provide a complete Eurocode compatible design methodology considering local, distortional and global buckling of general shaped cold‐formed members subject to 3D loading and support condition, to be implemented in computer software programs working with 7DOF beam finite elements..

A superconvergent elastically coupled double beam element for analysis of adhesively bonded lap joints

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan (“Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element,” Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers.

Computational analysis of reinforced concrete structures subjected to fire using a multilayered finite element formulation

Fire is a critical risk in reinforced concrete (RC) structures and appropriate structural resistance against it has to be ensured. In this contribution, an approach using corotational layered beam finite elements is employed in which the cross-section temperature is derived from a low-cost closed form model, as opposed to the more commonly used fully computational thermal analysis. The effect of geometrical and material nonlinearities (constitutive behavior fitted to experimental data for concrete and steel), material degradation as a function of temperature rise, and the contributions of thermal, transient, and creep strains are incorporated in the structural analysis. The computational results are favorably compared to experimental data from the literature for an RC beam and for a larger RC frame. Taking benefit of the layered beam formulation offering local insight into the cross-sectional and material behavior, the relationship between the structural degradation and data extracted from the cross-sectional behavior is successfully established. Noteworthy originalities of the contribution are the use of ultimate strain and its evolution as a function of temperature for both materials and the explanation of the observed structural response in fire conditions from cross-sectional data.

Nonlinear random seismic response analysis of the double-trough aqueduct based on fiber beam element model

As an important lifeline project, large reinforced concrete aqueduct structures sometimes pass through areas with high seismic intensity. Therefore, the aqueduct structure is susceptible to earthquake damage or destruction during the water conveyance process, resulting in interruption of water conveyance and major safety accidents. In fact, because concrete is a multiphase composite material, it has obvious nonlinear and random characteristics. At the same time, seismic excitation is also naturally random due to differences in frequency spectrum and duration. Based on this, this paper uses the concrete elastoplastic random damage constitutive relationship and the random seismic excitation model to carry out the random nonlinear seismic response analysis of the double-trough aqueduct structure. The thin-walled characteristics of the girder section of the aqueduct structure is significantly different from that of the general bridge structure, and there is a certain error in the simulation of the seismic response of the aqueduct with traditional beam elements. Therefore, this paper combines the refined fiber beam element model, compiles the TCL language program suitable for the OpenSEES open analysis platform, and establishes the refined fiber beam finite element model of the aqueduct structure based on the random damage constitutive relationship of the concrete. The study of the response law and failure mechanism of aqueduct structures considering the coupling of concrete nonlinearity and seismic excitation randomness provides an important theoretical basis for the seismic optimization design of large-scale aqueduct structures.

Geometrically exact beam element with predefined stress resultant fields for nonlinear analysis of FG curved beams with discontinuous stiffness

Based on the geometrically exact beam theory, a novel force-based beam finite element formulation is proposed in this paper for geometrically nonlinear analysis of functional graded (FG) curved beams with discontinuous stiffness. In this formulation, the distributions of stress resultants including axial force, shear force and bending moment along the beam’s axis are treated as the unknown fields of the beam, and the specific forms of these unknown fields are predefined according to the equilibrium differential equations of geometrically exact beam. The element equation system and its linearization are detailed for implementation of incremental/iterative solution scheme. Numerical examples are presented to validate the formulation proposed in this study. As revealed by the numerical results, the proposed element has excellent performance and can achieve high solution accuracy in geometrically nonlinear analysis of curved beams, especially for those with discontinuous stiffness. Furthermore, using the proposed element, an investigation is conducted into the nonlinear stability of the FG circular arch, and the impact of FG material parameters on the level of nonlinear stability is discussed.

A multi-director continuum beam finite element for efficient analysis of multi-layer strand cables

This study presents a multi-director continuum beam finite element developed for efficient analysis of multi-layer strand cables. The beam element is derived from the continuum mechanics based beam formulation incorporating multi-directors and cross-sectional elements to describe the helical geometries of wires that constitute the strand cables. The main advantage of the present continuum beam formulation is that complex geometries of strand cable and inter-wire motions can be addressed in the framework of the beam kinematics regardless of the numbers of layers and subordinate helical wires. This approach enables simple modeling together with efficient analysis, and allows both geometric and material nonlinearities to be considered. In addition to its excellent accuracy, the proposed beam element model requires a significantly reduced number of degrees of freedom compared to conventional three-dimensional solid finite element models for cable analysis. Through several numerical examples, the predictive ability of the proposed beam model is demonstrated.

Experimental study of lag-twist coupling concept for rotor blade application

A novel passive twist morphing concept is examined for helicopter blades. The concept is demonstrated using a thin-walled rectangular composite beam created with symmetric layup to obtain bend-twist property. The twist of a rotor blade is proposed to be actuated though a movable mass at the blade tip which is able to provide a range of lagwise bending moment during rotation as a result of the centrifugal force. First a set of static bending test is performed which provides detailed characterisation of the deformation and strain distribution of the composite beam subjected to a number of bending loads. The results of the experiment fully verify numerical predictions including finite element approach (FE) and beam cross sectional analysis. A series of simulations are then conducted using the verified numerical model to demonstrate how the desired twist can be effectively achieved by manipulating the size and location of the mass.

Combining shell and GBT-based finite elements: Vibration and dynamic analysis

This paper extends the approach previously proposed by the authors Manta et al. (2020, 2021) to the dynamic case, more precisely to the calculation of natural frequencies (including pre-loaded members) and the linear time-history response of thin-walled members and frames with complex geometry, undergoing global–local–distortional deformation. This approach relies on a combination of standard shell and GBT-based (beam) finite elements, where the prismatic beam parts are modelled with standard GBT-based elements with a minimum number of deformation modes (hence a minimum number of DOFs), while the geometrically complex zones (perforations, tapering, joints) are handled using shell elements. Three numerical examples are presented to demonstrate the capabilities and potential of the proposed approach. These examples concern (i) a lipped channel cantilever with two long holes, (ii) a lipped channel cantilever with a tapered segment, subjected to pre-loading, and (iii) an L-shaped frame with I-section members and a tapered joint. For comparison and validation purposes, full shell finite element model solutions are provided. In all examples, it is concluded that the proposed approach leads to very accurate results and a significant DOF economy with respect to full shell finite element models.

Rapid Prototyping of Inertial MEMS Devices through Structural Optimization.

In this paper, we propose a novel design and optimization environment for inertial MEMS devices based on a computationally efficient schematization of the structure at the a device level. This allows us to obtain a flexible and efficient design optimization tool, particularly useful for rapid device prototyping. The presented design environment—feMEMSlite—handles the parametric generation of the structure geometry, the simulation of its dynamic behavior, and a gradient-based layout optimization. The methodology addresses the design of general inertial MEMS devices employing suspended proof masses, in which the focus is typically on the dynamics associated with the first vibration modes. In particular, the proposed design tool is tested on a triaxial beating-heart MEMS gyroscope, an industrially relevant and adequately complex example. The sensor layout is schematized by treating the proof masses as rigid bodies, discretizing flexural springs by Timoshenko beam finite elements, and accounting for electrostatic softening effects by additional negative spring constants. The MEMS device is then optimized according to two possible formulations of the optimization problem, including typical design requirements from the MEMS industry, with particular focus on the tuning of the structural eigenfrequencies and on the maximization of the response to external angular rates. The validity of the proposed approach is then assessed through a comparison with full FEM schematizations: rapidly prototyped layouts at the device level show a good performance when simulated with more complex models and therefore require only minor adjustments to accomplish the subsequent physical-level design.

Static analysis of wind turbine blade using 1D FE model

The study presents a unique technique to determine the static response of wind turbine (WT) blade. A 1D Finite element (FE) beam model of WT blade is developed using thin-walled beam theory coupled with PreComp tool used to compute crosssectional stiffness with elastic coupling effects. A realistic 9.2 m long, WT blade is developed using different aerofoils with fourth order polynomial variation for twisting angle of blade span. Three different aerfoil sections NREL S818, NREL S825, and NACA 2412 are employed in the current study. For validation, the results of 1D model developed using MATLAB are compared with that of 3D WT blade model which is analyzed in ANSYS using NuMAD..

A finite element approach for the static and vibration analyses of functionally graded material viscoelastic sandwich beams with nonlinear material behavior

A beam finite element model is proposed for the static and free vibration analyses of FGM sandwich beams with viscoelastic nonlinear material behavior. In the analysis, zigzag theory is adopted for the displacement fields. The damping results from the shear properties of the viscoelastic layer and its low stiffness. Timoshenko 1st order and Reddy’s higher order shear models are implemented for static and vibration behaviors. Various viscoelastic frequency-dependent laws are considered. The resulting stiffness matrix is nonlinear and is frequency dependent. Solutions are possible according to a powerful asymptotic method combined with recent method for the power series terms. The efficiency of the present model is proven through simulations using 3D volume elements of Abaqus code, where new module for viscoelastic FGM materials is now available. The effects of power law index and boundary conditions on static, vibration and the damping properties of the viscoelastic sandwich FGM beam are investigated successfully. It is shown that the beam behavior is very sensitive on the loss factor. In vibration, the damping properties are nonlinearly power law index dependent. The boundary conditions have an incidence on the vibration modes. The cantilever case is particular and interesting that needs an optimization process.

FEM computational models of shear walls with openings

This study is devoted to the comparison of shear wall computational models and the juxtaposition of displacement and stress fields. Two types of numerical models are compared. The first type concerns discretization of the geometry by using the beam finite elements model, and the second type-with plane finite elements. The displacements and the internal forces are used as the key indicators for the comparison of the mechanical behaviour. Some conclusions have been drawn, as well as recommendations have been made on the application of the juxtaposed type computational models and their practical application in real life.

Hyperbolic‐Like Structure with Negative Poisson's Ratio: Deformation Mechanism and Structural Design

Mechanical metamaterials with negative Poisson's ratio (NPR) effect possess important practical applications in aerospace, nanotechnology, and other fields. The microstructure and its deformation mode directly determine the NPR ability and mechanical performance of auxetic metamaterials. Herein, inspired by the double‐negative‐index ceramic aerogels, a novel NPR structural model is proposed in which symmetric reentrant oblique ligaments and short transversal ligaments constitute the main framework inside a square, which is different from the traditional reentrant structure due to the hyperbolic‐like skeleton and deformation modes. The influence of structural parameters on the NPR effect is investigated using the Timoshenko beam model and finite element simulations. The structural model is then extended to design multiple reentrant structures and assembled structures, respectively. The multiple reentrant structures include multiple pairs of symmetrically hyperbolic‐like skeletons, and the distribution of horizontal ligaments can be regulated to achieve a better NPR effect. For assembled structures under compression, the local rotational deformation and buckling result in structural instability, demonstrated by inflection points in the NPR curves. Moreover, one special assembled structure is found for which there is a linear relationship between the compressive strain and NPR value. The insights should be significant for designing mechanical metamaterials with the NPR effect.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

Efficient strong Unified Formulation for stress analysis of non-prismatic beam structures

The Unified Formulation (UF) has gained attention as a powerful tool for efficient design of structural components. Due to the inherent flexibility of its kinematics representation, arbitrary shape functions can be selected in different dimensions to achieve a high-fidelity characterisation of structural response under load. Despite this merit, the classical isoparametric description of UF limits the application to prismatic structures. The weak-form anisoparametric approach adopted to overcome this limitation in a recent work by Patni et al. proves to be versatile yet computationally challenging owing to the expensive computation of its UF stiffness matrix by means of full volume integrals. We propose a strong-form anisoparametric UF (SUF) based on the Serendipity Lagrange Expansion (SLE) cross-sectional finite element and differential quadrature beam element. The main objective of the SUF is to achieve an efficient computation of the UF stiffness matrix by restricting Gauss operations to the variable cross-sections of non-prismatic structures in a discrete sense, thus eliminating the need for full volume integrals. When assessed against weak-form based UF, ABAQUS FE and analytical solutions, the static analysis of non-prismatic beam-like structures under different loads by the SUF is shown to be accurate, numerically stable, and computationally more efficient than state-of-the-art methods.

Identifying the transient milling force coefficient of a slender end-milling cutter with vibrations

Milling is a complex process during the coupling effect for multiple physical fields, especially for slender end-milling cutters. Because of the weak rigidity of its process system, it is easily affected by the physical behavior such as the milling force. In addition, it produces large vibrations and deformations, which affects the machining quality. To accurately predict the milling force of a slender end-milling cutter, a method to identify the milling force coefficients while considering vibrations is proposed. When considering the small diameter of the milling cutter edge, it is difficult to install sensors to measure the vibration. By combining the measured frequency response function of the cutter handle end and the cutter’s Timoshenko beam finite element model, the interface dynamics parameters between the cutter and the cutter handle are identified. This is based on the receptance coupling substructure analysis, and the cutter edge’s frequency response function is established. When considering the measured milling force data and the state equation of the cutter edge vibration, the real-time vibration of the cutter edge during the milling process is predicted. In addition, the undeformed cutting thickness during the superposition of the cutter edge vibration is obtained. The milling force coefficients are identified based on the undeformed cutting thickness of the coupled vibration, and the influence of the process parameters on the variation of the milling force coefficient is examined. The prediction accuracy and the average milling force coefficient without the vibrations are compared to verify the accuracy.

Transport of heat and material properties in continuous casting of steel

Liquid steel solidifies within a containment of a continuous casting machine and moves in axial direction as the solidified shell is withdrawn towards the end of the machine. A mechanical model for computing the deformation of the solidifying steel in the roll containment is developed based on the beam theory and on a ferro-static pressure load. The material properties of the steel at high temperatures result from high temperature constitutive equations including creep for the solidified shell between the rolls. The temperature distribution within the solidified shell is computed based on the thermal boundary conditions as in the secondary cooling zone water is spread onto the surface of the steel shell and heat is removed. The solidified steel shell thickness increases along the machine containment up to the point of total solidification. A suitable discretization of the region by the finite beam elements is used to get a suitable mechanical model. The ferro-static pressure acts inside of the solidified shell and causes some bulging of the shell between the roll containment. As there is a slow axial motion of the solidifying shell the numerical modeling of the transport e.g. of heat and of the creep strain, which is computed in each time step, has to be as accurate as possible as a lot of time steps are necessary. Different numerical strategies have been analysed with respect to the accuracy of the computation results.

Design and Research on Modification Method of Finite Element Dynamic Model of Concrete Beam Based on Convolutional Neural Network

Because of the uncertainty of the measurement results, the dynamic inverse problem equation is called the stochastic model correction equation. In order to make this equation or method can be used smoothly in actual engineering, it is aimed at low modal orders and small samples in actual engineering. Condition, this paper proposes a concrete beam finite element dynamic model correction method based on convolutional neural network technology. This method trains the convolutional neural network algorithm by expanding the data set, so that the accuracy of the finite element dynamic model correction method is obtained. Improve, and the method is feasible in actual engineering.

A geometrically exact beam finite element for curved thin-walled bars with deformable cross-section

This paper presents a geometrically exact beam finite element for curved and twisted thin-walled bars undergoing large displacements and finite rotations, combined with cross-section in-plane and out-of-plane (warping) deformation. The underlying formulation is based on a previous work by the authors Gonçalves et al. (2010), which is now improved and extended for the pre-curved/twisted case. Cross-section deformation is allowed for by adding kinematic DOFs, the so-called “cross-section deformation modes”, which are obtained using Generalised Beam Theory. All expressions required to implement the proposed finite element are derived. For validation purposes, several numerical examples are presented and discussed. In each case, the results obtained with the proposed element and refined shell finite element models are compared, and an excellent agreement is found. In particular, it is demonstrated that the proposed element predicts very accurately the spatial behaviour of curved beams undergoing complex coupling between large displacements, finite rotations and cross-section deformation.