Beam Finite Element Model Research Articles

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.

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.

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.

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.

Stress-strain state of steel corrugated I-beams prestressed by flange drawing

The material distribution in structures is an urgent scientific and technical problem. Corrugated and prestressed steel structures are efficient and reliable.Purpose: The aim of this paper is to study the operation of steel I-beam with a thin corrugated wall, prestressed by flange drawing.Methodology/approach: Finite element modeling of beams with corrugated and flat walls (models with flat wall, transversely corrugated wall, downward corrugation at an angle of inclination 20 degrees, ascending corrugation at an angle of inclination 20 degrees). The proposed method is used for prestressing the lower beam flange using negative temperature effects. Numerical experiments in LIRA-SAPR software allow to obtain the stress distribution isofields and deformation patterns. A comparative analysis of the stress-strain state.Research findings: The normal stress distribution over flanges of corrugated beams with and without prestress under the external load has low difference. Prestress produces significant deflections of structures and the greatest deflection of the beam with ascending corrugations.

Digital Twin Implementation for Three-Dimensional Rotordynamic Response via Physics-Informed LSTM Neural Networks

Abstract The rotating assemblies of critical machinery are complex dynamical systems and rotordynamic model response prediction inaccuracy risks machinery failure leading to high production losses. Jeffcott, Euler beam, and high-fidelity 3D solid finite element models are frequently utilized for rotordynamic analyses. Even though the 3D rotor has the higher accuracy, beam models are still widely used in industrial applications. To improve prediction accuracy of the lower-fidelity Jeffcott and beam models, a rotordynamics physics-informed neural network (R-PINN) is proposed. This models physics-informed long short-term memory (LSTM) neural networks that utilize partial or limited measured data, by incorporating physical laws. This approach enables the creation of a Digital Twin, which can produce additional data and help remove noise and outliers. In the current study, two R-PINNs are introduced to validate the superior capability of the model for both low- and high-fidelity physics. Random noise of 10% is introduced into the measured data produced by the Digital Twin to replicate real-world noisy measurements. The result shows that both low- and high-fidelity physics R-PINNs can achieve high accuracy even with high noise data, thereby increasing the robustness of the model. The results clearly demonstrate the ability of the proposed R-PINN algorithm to enhance an Euler beam model's predicted response to the level of accuracy of a 3D solid element model's predicted response, the latter acting as a surrogate for test measurements in an actual application.

Accurate free and forced vibration behavior prediction of functionally graded sandwich beams with variable cross-section: A finite element assessment

The current investigation proposes an enhanced finite element beam model to assess the dynamic behavior of functionally graded sandwich beams with variable cross-sections. By integrating the shear effect, this beam model integrates both C0 and C1 continuities to generate elementary matrices using an efficient three-unknown shear deformation beam theory. The model focuses on advanced sandwich beams with functionally graded material face-sheets and metallic cores, exploring various geometric arrangements. The originality of this study addresses variable cross-sections in sandwich beams, evaluating the influence of material composition and geometric configurations on free and forced vibration responses, including different boundary conditions. The methodology uses the eigenvalue method for examining free vibration and applies Newmark’s algorithm for solving forced vibration problems. The current model expands the scope of analysis compared to previous models that primarily focus on uniform cross-sections and conventional material distributions. The developed model incorporates non-uniform geometrics, material gradient parameters, and advanced finite element analysis to enhance precision and efficacy through detailed parametric investigations and comparisons with existing literature. This contribution significantly improves the mechanical modeling of composite sandwich structures, particularly those incorporating functionally graded materials and complex geometrical aspects.

A Nonlinear Beam Finite Element with Bending–Torsion Coupling Formulation for Dynamic Analysis with Geometric Nonlinearities

Vibration analysis of wing-box structures is a crucial aspect of the aeronautic design to avoid aeroelastic effects during normal flight operations. The deformation of a wing structure can induce nonlinear couplings, causing a different dynamic behavior from the linear counterpart, and nonlinear effects should be considered for more realistic simulations. Moreover, composite materials and aeroelastic tailoring require new simulation tools to include bending–torsion coupling effects. In this research, a beam finite element with bending–torsion coupling formulation is used to investigate the effects of the deflection of beam structures with different aspect ratios. The nonlinear effects are included in the finite element formulation. The geometrical effect is considered, applying a deformation dependent transformation matrix. Stiffness effects are introduced in the stiffness matrix with Hamilton’s Principle and a perturbation approach. The results obtained with the beam finite element model are compared with numerical and experimental evidence.

Stiffness-driven design and optimization of a 3D-printed composite prosthetic foot: A beam finite Element-Based framework

Composite foot prostheses are traditionally produced via lamination, a process that grants high structural efficiency. However, it is an expensive and time-consuming process. Production rate and customizability are thus limited. Additive manufacturing of composites can be a potential solution to these limitations. This work presents a tool to design and optimize Continuous Fiber-Reinforced Additively Manufactured (CFRAM) prosthetic feet using beam Finite Element (FE) modeling. This optimization tool was developed for weight minimization and obtaining a CFRAM prosthesis design matching up to three static stiffness parameters. The design variables were defined through parametrizing the geometry of the prosthesis designed and using the composite structure parameters. Thanks to the versatility of the tool, solutions to multiple optimization and design cases were used to assess different design concepts, such as the shape of the prosthesis (C-shape or J-shape). Also, the tool successfully duplicated the stiffness characteristics of an assumed laminated prosthesis. Finally, the sources of inaccuracy associated with the beam FE modeling approach were identified through a comparison with plane stress FE analysis.

Nonlinear Modeling and Vibration Response Analysis of a Dual-Rotor System with an Inter-Shaft Graphite Seal

The inter-shaft graphite seal is a common structure in the dual-rotor system, and may generate nonlinear lateral excitation when there is large relative angular motion at the seal position. This paper represents the first attempt to study the nonlinear behaviors of a dual-rotor system with inter-shaft graphite seal excitations. First, an excitation model of the inter-shaft graphite seal is proposed by the analytical method. This model is then introduced into the beam finite element model of a dual rotor system, and the dynamic equations of the whole system are finally obtained. The vibration responses under the effect of the inter-shaft seal are clarified by analyzing 3D waterfall plots, rotor orbits, and time and frequency domain waveforms. The results show that the inter-shaft seal can lead to vibration coupling between HP and LP rotors. The axial spring stiffness and contact end face friction coefficient of the graphite seal have a significant effect on rotor vibration. When those two parameters are small, only coupling vibration phenomena can be observed, i.e., the rotation frequency of one rotor can be observed in another rotor vibration. With the increase in axial spring stiffness or contact end face friction coefficient, multiple modes of LP and HP rotors are excited, which dominates the vibration behaviors of the dual rotor system.

Advanced Fiber Beam Finite Element Model for Neural Network Training in Vibration-Based Bridge Monitoring

Recent advancements in civil infrastructure monitoring have witnessed the increasingly high-performance sensor technologies and data-driven algorithms, opening up new possibilities for assessing structural conditions. In recent years, there has been a growing interest in leveraging the potential of Artificial Intelligence for civil infrastructure monitoring. One promising approach is the use of computational models to train and test data-driven algorithms aiming to tackle damage detection problems. To enhance the effectiveness of such procedures based on simulated data, this study proposes a high-performance beam finite element model for training a neural network model able to predict the dynamic response of the structure and for generating various damage scenarios. Compared to 2D and 3D finite element models, the advanced fiber beam model offers superior computational efficiency while accurately capturing the nonlinear behavior of structural elements. Specifically, a force-based beam finite element based on a damage-plasticity model is implemented to describe damage and degradation of materials in reinforced concrete girders. Through the simulation of the dynamic structural response under withe noise excitation, a neural network model representing the structure in the undamaged conditions is obtained. The prediction error of such network model is investigated as a suitable measure for the definition of a damage indicator able to detect the presence of damage (concrete cracks and reinforcement yielding). The integration of an advanced fiber beam model, accurate constitutive law and neural network models shows promising potential in the monitoring of existing bridges.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

The coupled band gap of the double beam with attached periodic spring-mass structure

This study presents a novel investigation into the phenomenon of coupled band gaps within double beam structures and achieves the construction of beam configurations featuring inherent coupled band gaps. Initially, the identification of coupled band gaps in double beam structures is discussed, and it is engineered within the low-frequency range by adjusting the thickness ratio of the two beams. Subsequently, double beam models are constructed using various methods. A comparative analysis between the Spectral Element Method and the Plane Wave Expand method validates the existence of coupled band gaps. Furthermore, through systematic exploration using the Plane Wave Expand method, we investigate the effect of different parameters on coupled band gaps. The attenuation properties within the identified coupled band gap are experimentally validated using double beam Finite Element models and a dedicated experiment for forced vibration testing. Consequently, controlled variations in thickness ratio confirm well-defined coupled band gaps exist in conventionally thick beam structures. Within this specific frequency range (254–279 Hz) for the double beam with thicknesses 0.01 m and 0.002 m, respectively, average attenuation rates of −3.79 dB and −11.16 dB are recorded correspondingly. It significantly contributes to applying theories on coupled band gaps to large-scale architectural frameworks.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

Refined beam finite element model for thin-walled multi-cell box girders considering distortion and secondary distortional moment deformation effect

In this paper, a novel one-dimensional refined beam finite element model (FEM) accounting for non-uniform distortional warping and secondary distortional moment deformation effect (SDMDE) is formulated for the distortional analysis of thin-walled multi-cell box girders with cantilevered flanges. The extra distortional angle is introduced to incorporate SDMDE, and the validity of which is proved through rigorous theoretical derivation. The in-plane and out-plane deformations at any point around the edge of transverse section built upon the distortional center are unified in the same form and described in detail. The governing differential equations in terms of two generalized displacements, distortional angle χ and primary distortional angle curvature Θ, are derived based on the principle of minimum potential energy (MPE) and solved exactly so as to provide non-polynomial shape functions for the establishment of element stiffness matrix and nodal load vector. A series of multi-cell rectangular hollow section (RHS) girders are analyzed and particular extensions are given to single-cell and multi-cell bridge girders with cantilevered flanges. Comparisons to the results generated from corresponding ones provided by pioneering work or shell FEM analyses not only validate the correctness and reliability of the present beam element formulations but also highlight the significance of the inclusion of SDMDE when the cantilevered flanges are arranged.

Simplified nonlinear analysis of doubly corrugated cold‐formed steel arches

AbstractDoubly corrugated steel arches are made by pressing transverse corrugations on trapezoidal cold‐formed steel sheets. Several studies show that these transverse corrugations may have a significant detrimental effect on the stiffness and load‐bearing capacity of the original trapezoidal profile. This paper presents an experimental campaign aimed at investigating the effects of transverse corrugations on the behaviour of arches. The campaign includes reduced tests on flat sheets and sheets with a single corrugation, as well as full tests on arches with spans ranging from 10 to 20 m. Subsequently, a simplified design procedure based on nonlinear 2D beam finite element models is proposed. The models account for the transverse corrugation effects by using intermediate rotational springs and equivalent beam section areas, which are calibrated from the results of the reduced experiments. A brief sensitivity study of the finite element results with respect to different modelling factors is conducted, and the different effects of the corrugation assessed. Finally, the simplified finite element models are validated by comparing the experimental and numerical force‐displacement curves, which show reasonably good agreement. However, further research is needed to refine and extend the proposed procedure.

Centrifugal and gyroscopic effects on dynamic response of rotating cantilever beams under step loading

In this paper, a simplex finite element model for beams (FEMB) with a complete formulation is presented to study the dynamics of rotating cantilever beams subjected to distributed external loads. In our approach, the finite element method (FEM) and Timoshenko's beam theory is used. We will, particularly, examine the centrifugal and gyroscopic effects on the linear vibration of a rotating cantilever beam in stationary regime. In this model, extension, bending, and torsion degrees of freedom (DOF) are combined using a parameterization of the 3D motion of the beam by Euler's three-angle sequence. That allows identifying all the terms of the gyroscopic coupling in a more compact equations system. To ensure convergence, a sufficient number of finite elements are required in this model. The considered beam in the numerical simulation is pre-twisted and linearly tapered with a rectangular section. Results show that the dynamic centrifugal effect decreases the natural frequencies of the beam. Furthermore, the gyroscopic coupling induces rapid extension and torsion beatings when the beam undergoes a step bending load. Without damping, these beatings can persist and cause material fatigue.

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.

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.