Beam Finite Element Research Articles

Beam finite element models with node dependent kinematics and arbitrary displacement fields over the cross-section

The present paper proposes new, improved one-dimensional finite elements that use the node-dependent kinematics method, i.e. each node may adopt a different expansion over the cross-section without using a special coupling method. Furthermore, the presented method is able to choose independent models for each of the three displacement variables. In particular, Taylor-based and Lagrange-based functions have been chosen as expansions for the cross-sections. These two outcomes are possible through the use of the Carrera Unified Formulation, which subdivides the three-dimensional displacement field into a cross-section domain and an axis domain. Starting from the principle of virtual displacements, the governing equations and finite element matrices are obtained. In numerical results, compact and thin-walled beams have been studied and subjected to various loading conditions, including bending and torsion-bending. The selected cases are compared with existing literature or with high-refined CUF-based solutions. The accuracy of the models presented is assessed for both displacements and stress components. The results show that the choice of the ‘best’ computational models in terms of accuracy vs degree of freedom is problem-dependent. Finally, the Node-dependent kinematics method permits the selection of the most suitable expansions for each cross-section.

Geometrically nonlinear analysis of composite beams based on global–local superposition

A composite beam finite element is designed to capture through-thickness effects, specifically normal stress and strain and transverse shear, in the context of geometrically nonlinear analyses. The starting point for the formulation is a similar element already proposed for linear analyzes based on a global–local superposition approach, where local functions are defined in each layer of the laminate, and global functions are defined along the thickness. The consistency of the kinematic hypotheses is guaranteed by imposing the continuity equations of displacements through the thickness, the force balance equations along the thickness, directly or indirectly, by imposing the continuity of transverse stresses, and by applying the boundary conditions on the lower and upper surfaces of the elements. In the context of nonlinear analyzes, the imposition of continuity of displacements is straightforward. However, the continuity of the transverse stresses needs to be carefully imposed, as the relevant stresses are the second order Piola-Kirchhoff stresses and the strains are the Green-Lagrange strains, consistent with the total Lagrangian approach used. The constitutive equations are written in incremental form and a detailed analysis is conducted to ensure that the stresses and strains involved are physically consistent across the different reference frames employed. In order to assess the accuracy of the numerical model implemented, a unique semi-analytical technique is developed to obtain the response of asymmetrical laminated beams under compression.

On buckling of compact I-section beams subjected to biaxial bending

This paper aims at providing a new insight into the lateral-torsional buckling behavior of compact I-section steel beams subjected to biaxial bending, without axial force, a subject seldom treated in the literature. This investigation is based on both analytical and numerical results, the latter obtained using a two-node geometrically exact beam finite element developed by one of the authors (Gonçalves, 2019), which has all the required features. Several geometrically non-linear analysis (GNA) types are used to characterize the buckling behavior, namely with geometric imperfections (GNIA), with material non-linearity (GMNA) and combining material non-linearity and imperfections (GMNIA). Different boundary conditions along the two principal planes are allowed and particular attention is paid to cases which involve the same first-order bending moment diagrams but different support schemes. As a consequence, it is shown that the buckling behavior depends not only on the slenderness and the first-order moment diagrams, but also on the supports. On the basis of results obtained concerning the application of the Eurocode 3 provisions for different support conditions, it is shown that unsafe results can be obtained but may be resolved using the recommendations presented in the paper.

Rapid dynamic load estimation procedure for lifting propellers in forward flight

AbstractLifting propellers operate at oblique inflow and thus encounter severe dynamic loads during forward flight, impacting structural integrity, fatigue, and vibration. Numerical optimization approaches consider aerodynamic, structural mechanical, and aeroacoustic aspects within preliminary design. To also account for dynamic loads during forward flight, a novel procedure allows their rapid estimation. Based on steady-state simulations combining strip theory and finite beam elements, aerodynamic excitation, damping, and stiffness are defined in the frequency domain. Loads are derived through a linear inflow model and quasi-steady aerodynamics. Damping and stiffness loads are linearized and transferred into matrix form to calculate the frequency response. The computationally expensive need for simulations in the time domain is thus avoided. Applicability extends to both fixed and variable pitch lifting propellers utilized in large multicopters for cargo or passenger transportation. Comparisons to time-marching simulations show good agreement with deviations of approximately 10 %. The analytical derivation yields physical insights to understand and reduce dynamic loads and their magnification due to resonance.

Lateral Response Analysis of a Large‐Diameter Pile Under Combined Horizontal Dynamic and Axial Static Loads in Nonhomogeneous Soil

ABSTRACTThe large diameter piles are widely used in structures such as offshore wind turbines due to their superior lateral load‐bearing capacity. To explore the lateral response of a large‐diameter pile under combined horizontal dynamic and axial static loads in nonhomogeneous soil, a simplified analytical model of the pile–soil interaction is developed. This model represents the pile as a Timoshenko beam resting on the Pasternak foundation, incorporating the double‐shear effect by considering both pile and soil shear. The governing matrix equations for the pile elements are derived from the principle of virtual work. Further, the pile's lateral deformations and internal forces are obtained using the modified finite beam element method (FBEM) and then validated through existing analytical solutions. Finally, the contribution of various properties of pile, soil, and applied load to the pile's lateral vibration response are performed. It is found that both pile and soil shear effects significantly impact the lateral dynamic response of a large‐diameter pile. Additionally, in nonhomogeneous soil, decreasing surface soil strength and dimensionless frequency lead to increased lateral displacements and bending moments of the pile, which are significantly affected by the P‐Δ effect under increasing axial load.

First-order GBT for tapered regular convex polygonal tubes

This paper presents an accurate and efficient first-order Generalized Beam Theory (GBT) for linearly tapered regular convex polygonal tubes, such as those widely employed in the construction industry. Even though tapered members require a significantly involved formulation, it is shown that it is possible to enforce the standard GBT assumptions exactly, without additional simplifications, a key aspect that (i) is essential for the accuracy and computational performance of the formulation and (ii) allows identifying the deformed configurations pertaining to inextensible deformation. Consequently, very accurate solutions are achieved even for complex cases, such as tubes with a high taper angle and undergoing localized deformation. The GBT deformation modes for the prismatic case are directly used, meaning that the proposed approach for tapered tubes does not require a specific GBT cross-section analysis procedure. All expressions are presented in a straightforward vector-matrix format, for implementation purposes. The excellent performance of the resulting displacement-based beam finite element and the advantages of the GBT modal decomposition features are highlighted through several numerical examples, where results obtained with refined shell finite element models are used for comparison purposes.

Finite Element Beam Propagation Method Based on Coordinate Transformation for Optimal Design of Optical Waveguide Devices

Finite Element Beam Propagation Method Based on Coordinate Transformation for Optimal Design of Optical Waveguide Devices

Global Sensitivity Analysis and Surrogate Models for Evaluation of Limit States in Steel Truss Structures

This article presents the global sensitivity analysis of the serviceability limit state of a steel truss using Monte Carlo simulations. The focus is on the probabilistic assessment of deflection, with failure probability defined as the likelihood of exceeding the deflection limit. Deflection is computed using the beam finite element method. A surrogate model is introduced to reduce computational costs. By integrating the surrogate and original models, significant CPU cost reductions are achieved. Furthermore, classical Sobol sensitivity analysis is used to examine the model outputs and analyze the significance of member loading and stiffness on the deflection. This study advances the use of surrogate models in global sensitivity analysis, enhancing computational efficiency and the understanding of interactions between input variables in the reliability assessment of steel truss structures.

A generalized Timoshenko beam with embedded rotation discontinuity coupled with a 3D macroelement to assess the vulnerability of reinforced concrete frame structures

A generalized finite element beam with an embedded rotation discontinuity coupled with a 3D macroelement is proposed to assess, till complete failure (no stress transfer), the vulnerability of symmetrically reinforced concrete frame structures subjected to static (monotonic, cyclic) or dynamic loading. The beam follows the Timoshenko beam theory and its sectional behavior is described in terms of generalized forces and generalized strains. The beam response up to the peak is described by a macroelement, based on plasticity theory, that adopts a 3D failure criterion expressed in terms of axial force, shear force and bending moment. The Embedded Finite Element Method is then adopted to reproduce bending dominated failure, with a global cohesive model that links the cohesive moment to a rotational jump. The formulation allows for remedy of localization phenomena and significant reduction of the necessary computational time. The performance of the proposed simplified strategy is illustrated by comparison with experimental results.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Few-mode optical fiber-based flexible pressure feedback tentacle doped with fluorescence temperature pointer

Abstract In this paper, a flexible fiber pressure feedback whisker is proposed, which consists of a water droplet shaped polydimethylsiloxane (PDMS) elastomer with an embedded balloon shaped few mode fiber. The mechanical sensing performance of the device was analyzed and optimized using a combination of finite element method and beam propagation method (BPM). The built-in cladding corroded few-mode fiber increases pressure sensitivity by more than four times. The collection efficiency of fluorescence signal is improved by cladding corrosion. The PDMS elastomer was doped with upconversion nanoparticles NaYF4:Yb, Er@NaYF4 in order to achieve temperature measurement by fluorescence intensity ratio technology. The combination of fluorescence signal and interference spectrum can not only achieve real-time and accurate pressure detection at different temperatures, but also incorporate fluorescent materials into flexible bionic skin for temperature self-compensation, which has potential application value for the development of bionic fiber micro-nano sensing and control devices.

Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams

The study focuses on the free vibration analysis of beams made of axially functionally graded materials (AFGM) with curvilinear variable cross-sections along their length. The beams encompass various shapes, including concave and convex conic sections, with axial material properties varying according to polynomial and exponential laws. The equations of motion are derived using Hamilton’s principle within the framework of Timoshenko beam theory. These governing equations, subjected to various boundary conditions, are solved using the differential transform method (DTM). The proposed solution technique is validated by comparing computed natural frequencies with the existing literature and results obtained using three-dimensional finite element analysis in ABAQUS. The incorporation of material gradients into the beam finite element models was achieved using the user-defined material subroutine (UMAT). Additionally, a comprehensive study is conducted to examine the influence of various factors on the natural frequencies of functionally graded beams. These factors include parameters of material laws, types of variable beam shapes, slenderness ratio, and specific boundary conditions. This study provides a thorough understanding of the modal dynamics of the considered beams, offering valuable insights into the behavior of FGM structures.

Mesh objective characteristic element length for higher-order finite beam elements

The use of fracture energy regularization techniques can effectively mitigate the mesh dependency of numerical solutions caused by the strain softening behavior of quasi-brittle materials. However, the successful regularization depends on the correct estimation of the crack bandwidth in Finite Element solutions. This paper aims to present an enhanced crack band formulation to overcome the strain localization instability especially for the higher-order elements developed in the framework of Carrera Unified Formulation (CUF). Besides, a modified Mazars damage method incorporating fracture energy regularization is employed to describe the nonlinear damage behavior of the concrete. To evaluate the efficiency of the proposed crack band formulation, three experimental concrete benchmarks are selected for the numerical damage analysis. By comparing numerical and experimental results, the proposed method can guarantee mesh objectivity despite varying finite element numbers and orders, indicating perseved fracture energy consumption within proposed higher-order beam models.

Active vibration control of a large space telescope based on H∞ control with unilateral and constrained input

The development of the telescope with large aperture is driven by the increasing demand for higher imaging resolution and imaging area. The large membrane diffraction space telescope is particularly attractive due to their lightweight nature, ease of folding and unfolding, and high precision imaging capabilities. However, the flexible and large structure of these telescope makes them prone to long-time, low-frequency vibrations in space. These vibrations can degrade the imaging quality and potentially cause damage to the instrument in the space telescope. Therefore, it is crucial to develop effective active vibration control methods to suppress these structural vibrations. This paper proposes a control method that utilizes cables as actuators. Since cables can only withstand tension and have limited output tension, this constraint is taken into account during the control design process. Firstly, a dynamics model of the membrane diffraction space telescope with hinge flexibility is developed by the beam theory and finite element method. Next, an active controller is designed by combining H∞ control theory and Linear Matrix Inequalities (LMI). Additionally, the optimal positions of cable actuators are determined using the Gram controllability criterion and a particle swarm algorithm. Finally, the effectiveness of the control method is verified through numerical simulations. The simulation results demonstrate that the hinge flexibility significantly influences the structural vibration behavior of the telescope. Furthermore, the proposed control method demonstrates the ability to effectively suppress the vibrations of the telescope while also exhibiting robustness to changes in natural frequencies of the structure.

Optimizing Coupled Fluid-Structure Simulations for Nuclear-Relevant Geometries

Abstract The numerical simulation of fluid-structure interactions (FSI) has gained interest to study flow-induced vibrations. Nevertheless, the high computational resources required by such simulations can represent a significant limitation for their application to industrial configurations. Therefore, simplified modeling approaches, when physically applicable, can represent an interesting compromise. This can be the case of slender structures (tubes, rods) often encountered in nuclear power plants. In this paper, an Euler–Bernoulli beam finite element model is implemented inside the computational fluid dynamics (CFD) code code_Saturne. With the goal of finding CFD methods less expensive than large eddy simulations (LES), unsteady Reynolds Navier–Stokes (URANS) and hybrid URANS/LES approaches are considered. The resulting fluid-structure model is able to calculate the vibration response of cantilever beams under a fluid flow, avoiding the necessity of CFD-finite element method (FEM) code coupling. The first part of the paper describes the model and its implementation: it allows to perform 2-way explicit fluid-structure coupling, using the Arbitrary Lagrangian-Eulerian approach to account for the structure deformations. Validation test cases are presented in the second part: first, the model is validated in terms of frequency, added mass, and damping for a cylinder vibrating in static air and water; then, the model is validated toward the vortex-induced resonance and lock-in mechanisms for a cylinder subjected to water cross-flow. The model is then applied to a real experimental configuration of two in-line cylinders in water cross-flow: the calculated vibrations are found to be in good agreement with the experimental measurements.

On the Shear Force Redistribution in Composite Steel and Concrete Beams with Web Openings

As far as research into the field of composite steel and concrete beams with web openings has shown, the initiation of a complex stress state due to the introduction of web openings is, unfortunately, inevitable. As a logical consequence, estimation of the resistance becomes complex as well. Aiming to provide a more convenient approach, especially for evaluation of the bending resistance to the Vierendeel effect with the presence of high shear forces, a marked simplification with respect to quantification of the shear force redistribution between T-sections is presented. This novel approach stands on strong fundamentals of experimental evidence and parametric finite element (FE) investigation. Concretely, within the experimental part of this investigation, the load response of four composite beam samples was recorded and profoundly analyzed. Subsequently, a detailed FE analysis in the software ANSYS 2024 R1 (ANSYS Inc., Canonsburg, PA, USA), exploiting its parametric design language (APDL), was conducted. Finally, concise data compilation and their comprehensible interpretation were enabled by employing the software MATLAB 2024a (The MathWorks, Inc., Natick, MA, USA). In view of this treatment, a cornerstone for successive assessment of the obtained data was laid. Attempting to identify the key factors influencing the shear force proportion between the tees, integration of the shear stresses along the vertical sections with regular spacing along the FE beam models was conducted. By virtue of this concept, not only the magnitudes of the shear forces for the tees were obtained but also the leading importance of the opening depth was demonstrated. Furthermore, a complete picture of the shear force behavior along the composite perforated beams was derived. On this basis, a novel method representing a simpler and more accurate form of estimation of the shear force proportion for T-sections was proposed.

Reduced-order modelling of floating offshore wind turbine: Aero-hydro-elastic stability analysis

A reduced-order aero-hydro-elastic analysis tool for performing the stability analysis on floating offshore wind turbines is presented. In this work, a high-order non-linear aero-elastic model initially developed for onshore wind turbines is extended by utilizing a super-element body, which models the floating platform, the mooring system and the hydrodynamic contributions. The turbine structure is modelled by a Finite beam Element Method, and the aerodynamic loads are modelled by the Blade Element Momentum method coupled with a Beddoes-Leishman type dynamic stall model in a state-space formulation. The linearization is performed around steady state equilibrium at any given mean wind speeds, rotor speeds and collective blade pitch angles utilizing Coleman transformation to remove the periodic terms. The order reduction is based on two projections to reduce the number of structural states and aerodynamic states separately using the structural modal project matrix and the aerodynamic shape functions. Concurrently, we investigate the effect of unsteady aerodynamic states on the stability analysis of a floating wind turbine. The results show the low-order aero-hydro-elastic model expands the scope of numerical tools available for conducting structural modal analysis and aero-hydro-elastic stability analysis on floating wind turbines.

High-performance beam finite element for predictive response in monitoring existing bridges

High-performance beam finite element for predictive response in monitoring existing bridges

Contact formulations for analysis of micropolar media with finite continuum beam elements

Microscale beam-like structures are standard components of micromechanical systems in many devices. However, the small dimensionality affects their deformation characteristics and leads to misinterpretations of the results. Presenting such size dependency is possible with advanced continuum descriptions via introducing additional variables compared to the classical one. Hence, the problems where contact occurs require the readjustment of boundary conditions considering these extra parameters. That burdens the already challenging task of contact problem calculations and restricts most demonstration examples to two-dimensional problems and geometrical linearity. The resolution of the imposed restrictions within finite element modeling further emphasizes the usage of advanced media in design and facilitates its application to various problems, which is the aim of this paper. The delivery of that task is as follows. Firstly, we present the kinematics and material descriptions for the micropolar media. The authors propose to use a newly developed continuum-based micropolar beam formulation to avoid an overwhelming computational burden and, at the same time, deal with nonlinear stress–strain relations. Secondly, the work develops a contact approach within the micropolar theory from two-dimensional to three-dimensional elasticity, although the contact is considered frictionless. Finally, it compares two existing contact formulations, including the developed one, for the contact beam problems, using the examples of two collinear sliding beams’ bending.

Large displacement model for the nonlinear behaviour of the face plate component

This paper deals with the face plate component in steel joints and addresses the full characterization of the nonlinear behaviour of the face plate component. An equivalent beam strip model is proposed that tackles the connection of a beam to the column web of open I-sections in a minor axis joint or the face of tubular columns, covering endplate or fin plate joint typologies. Closed-form analytical solutions are obtained for the elastic large displacement and the elastic-plastic large displacement behaviour of the equivalent beam strip. Criteria for the establishment of the design resistance using the continuous strength method are also proposed. It was concluded that the model is easy to apply, was validated against a large parametric study using finite element beam models, leads to accurate solutions and demonstrates the need to consider membrane effects in design.