Equivalent Beam Element Research Articles

Nonlinear wire rope isolator with magnetic negative stiffness

The high-static-low-dynamic-stiffness (HSLDS) isolators have been widely employed to achieve low-frequency vibration isolation. However, traditional HSLDS isolators are prone to nonlinear jump phenomena due to the introduction of nonlinear stiffness characteristics, which will cause the performance deterioration under high excitation levels. To overcome this issue, inspired by the dry friction characteristic of wire ropes, the wire ropes are employed to design HSLDS isolators to suppress nonlinear jump phenomenon by introducing hysteretic damping in this paper. Therefore, a nonlinear wire rope isolator with magnetic negative stiffness is proposed to achieve HSLDS with hysteretic damping. The elastic force of wire rope and the magnetic force are modeled to obtain nonlinear restoring force, and then the dynamic model of the proposed isolator is established to investigate isolation performance. The equivalent beam element method is proposed to model wire ropes for obtaining dynamic characteristics by considering tensile and bending stiffnesses. Then, numerical simulation is performed to investigate the isolation performance of the wire rope isolators with different magnetic negative stiffnesses, and demonstrates that the proposed isolator can work in a higher excitation level than the isolator without dry friction of wire ropes. Moreover, experiments are carried out to show that the onset isolation frequency of the proposed isolator can be reduced by 66.7% compared with the wire rope isolator without magnetic negative stiffness. Therefore, it is demonstrated that the proposed wire rope isolator with magnetic negative stiffness has the preferable performance in low frequency isolation, and can improve the adaptability under a higher excitation level.

Heterogeneous Beam Element for Multiscale Modeling of Non-prismatic Composite Beam-like Structures

Anisotropy, heterogeneity, and geometry make full system-level three-dimensional (3D) finite element analysis (FEA) simulation of real composite beam-like structures, such as wind blades, impossible due to limitations on computing resources. A popular approach is to cut the structure into multiple cross-sections, from which the beam properties based on the Euler-Bernoulli or Timoshenko beam models are extracted using two-dimensional (2D) cross-sectional analyses. These properties are then utilized in beam elements for system-level analyses. However, the mechanics behind 2D cross-sectional analyses are based on prismatic beams, limiting the accuracy of this approach on non-prismatic structures, especially in predicting stresses. In this work, we present a new multiscale method based on Mechanics of Structure Genome to analyze beam-like non-prismatic composite structures. A beam-like structure is homogenized into a series of 3-node Heterogeneous Beam Elements (HBE) with effective beam element stiffness matrices, which are then used as input for one-dimensional (1D) beam analysis using Abaqus User Element subroutine (UEL). Using the macroscopic beam analysis results as input, we can also perform dehomogenization to predict the stresses and strains in the original structure. We use three examples, a prismatic composite beam, an isotropic homogeneous tapered beam, and a composite tapered beam, to demonstrate the capability of HBE and show its advantages over the traditional cross-sectional analysis approach. HBE can capture macroscopic behavior and detailed stresses due to non-prismatic geometry. HBE provides a new concept to model non-prismatic composite beams by modeling heterogeneous beam-like structures as equivalent beam elements rather than beam properties of material points of the reference line.

Beam based rotordynamics modelling for preloaded Hirth, Curvic and butt couplings

Gas turbine and other machinery rotating assemblies are frequently manufactured as multiple components for cost and component alignment reasons. Butt joint, Hirth, and Curvic coupling are widely used for this purpose. Localized joint flexibility in these preloaded couplings introduce non-beamlike behavior which affects rotordynamic critical speeds, imbalance response and stability, rendering a conventional beam model inadequate. 3D solid finite element models of the couplings with Greenwood Williamson (GW) asperity interface features provide accurate representations of the couplings, however computational costs are impractical for use in an industrial design setting, which are limited to beam element models. A novel modeling approach for the coupling is developed that derives equivalent beam element Young's modulus and shear form factor properties, that replicate the bending behavior of the high fidelity 3D solid models, including GW based interface asperity stiffness. The equivalent beam models for butt, Hirth and Curvic couplings are validated using measured natural frequencies as a benchmark, for a range of through-bolt preloads. The equivalent beam model and the 3D solid model in this correlation incorporate GW contact models derived with experimentally measured surface roughness parameters. An Ecoupling sensitivity study for GW surface roughness parameters was conducted and showed a significant level of sensitivity. The effect of the coupling on an industrial class rotor's critical speed is included to illustrate usage of the approach.

End diaphragm and interior cross frame modeling in steel tub girder superstructures

Composite steel tub girders are proven to be an economical choice for long span bridges. However, detailed analysis of such superstructures requires extensive Finite Element (FE) modeling that includes many internal and external diaphragms. This paper is an attempt to study diaphragms and interior cross frames in tub girders and present simplified FE modeling details for the tub girder superstructures under gravity and seismic loadings incorporating accurate end diaphragm and interior cross frame stiffness equations. The simplified model is applicable to design of new tub girder bridges as well as load rating, health monitoring and retrofitting of existing bridges which require more sophisticated models. To this end, first a parametric study using 18 different FE models was carried out to better understand the behavior of tub girder superstructure and its components such as end diaphragms, the composite concrete deck and interior cross frames. Second, the stiffness equations for the end diaphragms and interior cross frames were derived analytically. These were associated with equivalent beam elements, which were used in combination with girder/deck elements to arrive at the simplified model. The resultant FE modeling was validated against detailed 3D FE models under gravity and seismic loads as well as the fundamental modes of vibration. Results indicated that the modeling approach is able to efficiently simulate different component behaviors and estimate the static and dynamic demands on the girders with remarkable accuracy.

Influence on dynamic behaviour of single layer graphene by Stone wales and pinhole defects

ABSTRACT This paper presents a computational procedure for the determination of the dynamic behaviour of graphene with different types of defects. The lattice of graphene is modelled using the molecular structural mechanics (MSM) approach, where the C–C covalent bonds are replaced by equivalent beam elements. Cantilever and bridged boundary conditions are applied for the analysis. Four types of Stone–Wales (SW) defects such as S-W (555-8), S-W (555-888-3), S-W (555-77-8) and S-W (888-3), and two types of pinhole defects with 6 and 24 elements eliminated are examined on armchair, zigzag and chiral type of graphene sheet. The effect of the structural length of the sheet, chirality and defect type on the vibrational properties of graphene sheets is investigated. The computed results reveal that SW defects produce a high frequency to that of pristine graphene, whereas the effect of pinhole defects is significant as compared to SW defects. The computed results will be useful in nano-resonator-based sensor applications.

Mechanical properties of graphene nanoplatelets containing random structural defects

Graphene nanoplatelets (GNPs) consist of a small number of graphene sheets connected through van der Waals forces and, like graphene, offer exceptional mechanical, thermal and electric properties. In this work, GNPs are considered as potential reinforcements in composites and their equivalent mechanical properties are computed. Similarly to graphene, the mechanical behavior of GNPs is greatly affected by the presence of structural defects and therefore, a parametric investigation is conducted herein. Three types of planar defects are considered: Stone–Wales, single vacancy and double vacancy defects. The examined parameters include the defect type, density, distribution as well as the number of graphene layers. The Molecular Structural Mechanics approach is employed to simulate the lattice of each graphene sheet, where the C–C covalent bonds are modeled by energetically equivalent beam elements. The van der Waals interactions between two carbon atoms belonging to different graphene sheets are modeled using truss elements with mechanical properties based on Lennard-Jones potential. Equivalent properties are computed by employing a homogenization-like procedure and random fields of the stiffness tensor are obtained through the moving window technique. A severe reduction of the stiffness of GNPs is observed for vacancy defects, which also lead to considerable variability. Inplane behavior, which differs greatly from out of plane behavior, is shown to be much more susceptible to the effect of defects.

A Substructure Condensed Approach for Kinetostatic Modeling of Compliant Mechanisms with Complex Topology.

Compliant mechanisms with complex topology have previously been employed in various precision devices due to the superiorities of high precision and compact size. In this paper, a substructure condensed approach for kinetostatic analysis of complex compliant mechanisms is proposed to provide concise solutions. In detail, the explicit relationships between the theoretical stiffness matrix, element stiffness matrix, and element transfer matrix for the common flexible beam element are first derived based on the energy conservation law. The transfer matrices for three types of serial–parallel substructures are then developed by combining the equilibrium equations of nodal forces with the transfer matrix approach, so that each branch chain can be condensed into an equivalent beam element. Based on the derived three types of transfer matrices, a kinetostatic model describing only the force-displacement relationship of the input/output nodes is established. Finally, two typical precision positioning platforms with complex topology are employed to demonstrate the conciseness and efficiency of this modeling approach. The superiority of this modeling approach is that the input/output stiffness, coupling stiffness, and input/output displacement relations of compliant mechanisms with multiple actuation forces and complex substructures can be simultaneously obtained in concise and explicit matrix forms, which is distinct from the traditional compliance matrix approach.

Research on single-layer grid structures considering joint connection failure based on MDEM

The traditional numerical simulation method ignores the volume and stiffness of the joints in single-layer grid structures. Such a simplification is not conducive to observing the failure of joint connections. This paper proposes a novel type of numerical simulation method—the multi-scale discrete element method (MDEM) to simulate the joint connection failure of single-layer grid structures. The joint and the part of the rod that is prone to plastic failure are divided by the discrete solid element method (DSEM) as a microscopic model. The other parts of the rods are simulated by the equivalent beam element method (EBEM) as a macrocosmic model, and all elements are connected with parallel bond contact. Moreover, to ensure the cooperative work of different scale models, the spring stiffness at the interface between DSEM and EBEM is derived based on the energy equivalence principle, and the effectiveness of MDEM is verified by the impact test of single-layer grid structures. Finally, the joint connection failure of single-layer grid structures is effectively simulated by MDEM in the PFC3D software, and the error between MDEM and the finite element method (FEM) is less than 4.07%.

A two-step method for delamination detection in composite laminates using experience-based learning algorithm

Delamination in composite laminates reduces the structural stiffness and thus causes changes in the vibration responses of the laminates. Therefore, it is feasible to employ dynamic characteristics (such as natural frequencies and mode shapes) for delamination detection by using an optimization method. In the present study, a two-step method is proposed for the delamination detection in composite laminates using an experience-based learning algorithm. In the first step, one-dimensional equivalent through-thickness beam elements are employed to model the composite laminated beam and potential delamination locations are identified. In the second step, a typical three-dimensional finite mesh is utilized for the beam’s modeling and the detailed delamination information (including the delamination location, size, and interface layer) is detected. This two-step method combines the advantages of the two different modeling techniques and is able to significantly reduce the computational cost without reducing detection accuracy. The proposed method is applied for an eight-layer quasi-isotropic symmetric (0/-45/45/90)s composited beam with different delamination situations to verify its effectiveness and robustness. The performance of the two-step method is demonstrated by comparing with the one-step method and other three state-of-the-art algorithms (CMFOA, PSO, and SSA). Moreover, the influence of artificial noise on the accuracy of the detection performance is also investigated. Both numerical and experimental results confirm the superiority of the proposed method for delamination detection in composite laminates especially for the prediction of delamination interface.

Multiscale analysis and mechanical characterization of open-cell foams by simplified FE modeling

Foam materials are experiencing a great diffusion in the industrial field because of their numerous properties and adaptability to different purposes. At the same time, research efforts are addressed to define new fields of application and methodologies of foam simulation, capable of combining their behavior at macroscale to phenomena at microscale, i.e. ligaments level. In this paper, a FE beam modeling strategy for open-cell foams acting on different scales of analysis is described. At first, on the microscale, the definition of the fundamental Kelvin unit cell is outlined with a particular characterization of the intersection zone between ligaments, that is a critical issue to obtain an accurate simulation instrument. The repetition of the unit cell within the material domain allows to obtain the overall foam material and operate at the macroscale. The elastic properties at macroscale are derived by means of FE analysis and compared with both solid model and experimental results. The proposed simplified FE modeling, using equivalent beam elements having suitable mechanical characteristics, demonstrates high accuracy levels and substantially relieve the computational efforts required by the commonly employed solid models.

The influence of the frequency content of ground motion on the nonlinear dynamic response and seismic vulnerability of historical masonry towers

Observations made on the response of historical masonry towers during past earthquakes indicate that in addition to the intensity of the ground shaking, the frequency content of shaking also affects the seismic performance of these monuments. To evaluate this phenomenon, the influence of the mean period of the ground motion (Tm), as a frequency content indicator, on the seismic behavior of the towers is assessed. To this end, first, the vulnerability of four towers with different aspect ratios and vibration periods (Ts) are evaluated by means of the Incremental Dynamic Analysis (IDA). For this purpose, 37 ground motion records, corresponding to the stations located in sites with different types of soil, are utilized. The nonlinear time history analyzes of the towers are carried out using the OPENSEES software by means of an Equivalent Beam Element with fiber sections. In order to investigate the effects of the frequency content of the ground motion on the seismic response of the towers, for every tower, the variation of the PGA of the ground motion and the induced internal force in the tower at the point of failure are plotted against the period ratio (Tm/Ts). According to the analysis results, it is found that the failure PGA increases as the period ratio becomes smaller. It is also noted that the induced shear in the tower exhibits a similar trend.

Atomistic finite element modeling of superlubricity in long double walled carbon nanotubes

The question of how much the nanotubes are really superlubricious when they are at macro lengths, is still intriguing. In this research, multiscale modeling of wall separation for various double walled carbon nanotube pairs has been done. VDW forces are modeled by a 3D truss element based on Lennard-Jones potentials, while C–C bonds are modeled by a deformable and equivalent beam elements. In the steady pullout process, we divided DWCNT into 3 main effective segments and characterized their effects separately; (a) free and pulled out length, (b) end atom VDWs, and (c) overlapping region. The effects of each aforementioned segments are separated by doing a proper finite element coding. For the zigzag/zigzag pairs, the end atoms are responsible for the average load while the overlapping region answers the load fluctuation during pullout. Otherwise, for the incommensurate case of zigzag/armchair pair, both end atoms and overlapping loads are constant. Finally for the armchair/armchair DWCNT, the end atom and the overlapping loads both are fluctuating, however at different phases that makes its overall pullout force constant. The methodology and coding introduced in this numerical effort makes it possible to precisely determine the average pullout load for long nanotubes with lengths in the order of hundreds of nanometers in short computational time.

The Concept modeling method: An approach to optimize the structural dynamics of a vehicle body

Although the concept modeling method has already been proposed in the literature, there is still very limited knowledge about the validation and the application of this method for vehicle body design. This paper substantially increases this limited knowledge by developing a concept model for predicting and optimizing the structural dynamics of a vehicle body-in-white and validating this concept model against a detailed finite element model. The geometry and parameters of the concept model are extracted from its detailed finite element model. The major members and panels of the detailed finite element model are replaced by their equivalent beam and shell elements models. The joints of the concept model are represented by stiffness and mass matrices extracted from the detailed finite element model using the Guyan Reduction Method. The developed concept model is validated by comparing its structural dynamics, including the resonant frequencies and the vibration mode shapes, with the original detailed finite element model and the experimental results. The simplicity and small size of the concept model enable it to easily enhance the structural dynamics of the body-in-white by optimizing the cross-sections of the load-carrying members of the structure. The optimization in this case increased the resonant frequencies of the body-in-white while reducing the total mass by about 6 kg. The results prove that the concept modeling method can significantly enhance the body-in-white structural dynamics by reducing the complexity of the model and allowing the focus for the optimization to be on the main members of the structure at the development stage when the final design parameters are not well known and have not been fixed.

Modeling of anisotropic elastic properties of multi-walled zigzag carbon nanotubes

Non-linear three dimensional finite element model of double wall carbon nanotube has been developed to evaluate its anisotropic elastic properties. Carbon nanotube was modeled as a frame-like structure and its elastic properties have been considered in the sense of the average over volume (effective) response on the axial and radial loadings. The principal bonds between two adjacent atoms were simulated using equivalent pseudo-rectangular beam elements which exhibit different flexural rigidity along the two principal centroid axes of the cross-section. The simulation of the interlayer van der Waals force with intrinsic nonlinearity and complicated applying region has been performed. Resulting Young's moduli of double walled carbon nanotube of diameter of 0.7–2 nm was found to vary from 0.2 to 0.5 TPa in axial direction and from 1.4 to 43.3 GPa in radial direction. The radial Young's modulus exponentially decays with the tube diameter.

Effects of equivalent beam element on the in-plane shear performance of 3D stochastic fibrous networks

Abstract The stochastic fibrous network structure in papers, nonwovens, and fibrous tiles have been used and studied widely. The connections in stochastic fibrous networks not only transmit loads between fibers but are also crucial to the mechanical performance of the networks. In this study, a finite element model for three-dimensional (3D) stochastic fibrous networks is built and the connections are treated as equivalent beam elements. Subsequently, the in-plane shear performance of 3D stochastic fibrous networks is investigated. The stress–strain curve and failure analysis obtained from the finite element model agree with the experimental results, thus validating the finite element model. Our simulation suggests that the connections between fibers are crucial on the macromechanical performance of the networks, especially when the damage accumulation is dominated by connections. Flexible connections increase the energy absorption capacity of the material significantly. The diameter of the connecting beam not only affects the strength and modulus of the network, but also changes the elasto-plastic behavior of the network.

Numerical model for predicting the structural response of composite UHPC–concrete members considering the bond strength at the interface

In this study, an improved finite element (FE) model was developed for the prediction of the structural behaviour of reinforced concrete members strengthened with ultrahigh-performance concrete (UHPC). A concrete damage model and an implicit solver in LS-DYNA were adopted in the numerical simulation. The model was calibrated and validated using experimental data. Accurately representing the interfacial bond characteristics of composite UHPC–concrete members was the primary challenge in developing the modelling technique. A novel technique using equivalent beam elements at the interface between UHPC and normal strength concrete (NSC) substrate was proposed for this purpose. The material properties of the equivalent beam elements were defined to represent the equivalent bond characteristics of NSC. The developed FE model was found to be able to effectively and efficiently predict the structural response of composite UHPC–concrete members with good accuracy.

Effect of temperature on vibrations and buckling behavior of carbon nanotube-based mass sensors using a new temperature-dependent structural model

In last decade, structural mechanics (SM) approach has been one of the most known and effective tools for the determination of the mechanical properties of carbon nanotubes (CNTs) such as elastic modulus, shear modulus, natural frequency, and critical axial buckling strain. Considering increasing applications of CNTs as resonators at low and high temperatures, a new temperature-dependent structural mechanics model is presented here. The proposed model can be utilized to predict the effect of temperature on the mechanical response of CNTs as functional resonators. Mechanical and geometrical properties of equivalent beam element are correlated to the CNT's coefficient of thermal expansion (CTE). Owing to elaborate series of finite element simulations, the frequency shift of a CNTs-based mass sensor with different attached mass and aspect ratio in temperature range of 0–1600 K is calculated. Afterwards, the critical buckling temperature of CNTs-based mass sensor due to increasing the axial force by changing ambient temperature is obtained. Results show that decreasing the first natural frequency is clearly appeared by attaching a mass more than about 1 zg on the CNTs structure. In other words, CNTs can be used to detect particles with mass above 1 zg. Moreover, it is shown that the variation of CTE versus temperature is a critical parameter that influences the vibration and buckling behavior of CNTs-based mass sensors. Increasing temperature leads to decreasing natural frequency due to rising axial forces along the CNTs axis. As a result, the natural frequency approaches to zero when the temperature rises up to the critical buckling temperature. In addition, increase in temperature over the critical buckling temperature changes the mode shape of CNT vibration to the next mode shape.

Experimental and numerical investigation of the possibilities for the structural health monitoring of railway axles based on acceleration measurements

In this article, numerical and experimental investigations are carried out to assess the possible use of vibration measurements to identify the presence of a fatigue crack in railway axles, detecting components of axle vibration occurring at frequencies that are integer multiples of the axle’s frequency of revolution (N×Rev components). A model of a cracked axle is defined using the Timoshenko beam finite elements incorporating an equivalent beam element having cross-sectional area and moments which are periodically changing with the angular position of the axle turn to reflect the crack breathing mechanics. Two levels of validation are considered for this model: first, verification with numerical results from a solid model of railway cracked axle (solid model was built before using the solid finite element model) and, second, validation against experimental results obtained from full-scale tests performed on cracked axles using a rotating bending fatigue test rig available at the labs of Politecnico di Milano. To consider the effect of disturbances arising from train–track interaction, track irregularity and wheel out-of-roundness, a multi-body model of a complete railway vehicle is defined, incorporating one axle modelled as a flexible cracked axle. This model is used to evaluate the N×Rev components of axle vibration occurring in the presence of axle cracks having different positions along the axle and different depths, in combination with different sources of disturbance. The results show that the 2×Rev and 3×Rev components of the horizontal axle-box acceleration are well correlated to the size of the crack and are almost insensitive to the effect of all disturbances considered in this study. Hence, they can be used for the continuous monitoring of axel integrity. Finally, a criterion for crack detection is defined and applied to the experimental and numerical results, showing the possibility of detecting cracks with sizes in the order of 8% of the total section area.

Electromechanical properties of Boron Nitride Nanotube: Atomistic bond potential and equivalent mechanical energy approach

We present a molecular mechanics-based approach to model the isolated zigzag BNNTs bonds by an (energy-)equivalent piezoelectric beam element. The harmonic DREIDING force field is employed for bonded and nonbonded interatomic interactions. To investigate the effectiveness of the proposed method we predict the elastic modulus and piezoelectric coefficients of BNNTs and demonstrate good accuracy compared to quantum mechanics predictions. Subsequently, we study the influence of some input parameters such as the tube diameter, aspect ratio and chirality. Finally, our approach is validated by comparison to data from the literature.

Determination of random material properties of graphene sheets with different types of defects

Abstract This paper presents a computational procedure for the determination of the stochastic material properties of graphene with different types and density of defects. The lattice of graphene is modeled using the molecular structural mechanics (MSM) approach, which is a continuum based nanoscale modeling technique, where the C-C covalent bonds are replaced by energetically equivalent beam elements. Random fields describing the spatial variation of the anisotropic elasticity tensor of defective graphene sheets are determined using the moving window method and Monte Carlo (MC) simulation. Three types of randomly dispersed defects are examined, namely Stone-Wales (SW), single vacancy (SV) and double vacancy (DV). The effect of window size, defect type and density on the random elastic properties of graphene sheets of area 100 × 100 n m 2 is investigated. The computed results reveal that vacancy defects can reduce the axial stiffness of graphene by approximately 60% with respect to that of pristine graphene, whereas the effect of SW defects is less significant. The computed random elasticity tensors can be assigned to equivalent continuum stochastic finite elements used in the framework of continuum modeling of graphene structures.