Equivalent Beam Model Research Articles

Equivalent modeling of LSTS and frequency calculation based on dynamic stiffness method

Large space truss structures (LSTS) are quickly becoming a vital type of space structure due to the fast expansion of the outer space field. The equivalent modeling approach has recently attracted the attention of researchers and engineers due to the finite element method’s limitations. In order to solve the equivalent beam model (EBM) in terms of frequency, the Wittrick-Williams algorithm based on dynamic stiffness matrix and the equivalent modeling procedure of large space truss structure are introduced in this study. The frequencies obtained by the equivalent model are good agreement with that obtaied by the finite element model, demonstrating the efficiency of the frequency solution method and equivalent modeling.

Experimental Research on the Floating Amount of Shield Tunnel Based on the Innovative Cumulative Floating Amount Calculation Method

The study of shield tunnel segment flotation is crucial for controlling the precision of underground excavation projects. Based on Winkler’s beam foundation theory, the load structure method, and the equivalent continuous beam model, and by considering the mechanical and spatial conditions that cause segment flotation, a novel theoretical calculation method for cumulative flotation is proposed using a simplified equivalent stiffness model of the tunnel. Additionally, a new concept of “equivalent flotation force” is introduced. The rationality and applicability of this theoretical calculation method are verified by comparing it with on-site construction data from the Yuanjiang River Crossing Tunnel Project in Changde, Hunan Province, China. The experimental results demonstrate that the theoretical calculation closely approximates the surface deformation monitoring data of the tunnel alignment in the eastern section of the project, and their deformation patterns are similar. Near the starting shaft, there is significant settlement influenced by stratum loss due to smaller tunnel flotation, with greater settlement occurring in the upper part. However, at approximately 45 m into both sections, they enter a deformation stability zone showing significant correlation in longitudinal deformation. Through comparison and verification of on-site experiments and theoretical model analysis, we preliminarily elucidate the feasibility of this innovative cumulative flotation theoretical calculation method which provides an important theoretical basis for assessing segment flotation issues in subsequent tunnel shield construction evaluations.

Equivalent Continuum Modeling for Flexible Slender Quadrilateral Truss Structure

A space in-orbit service simulation experiment platform is a type of equipment platform that allows spacecraft such as satellites and deep-space explorers to be adequately ground tested before launch. The function of the crane system is to drive the target spacecraft to perform a large-scale movement. This study focuses on the dynamics of a space in-orbit service simulation experiment platform with suspension rope and column quadrilateral truss structure as connecting devices. A space in-orbit service simulation experiment platform with a column quadrilateral truss structure as a connecting device is studied, modeled as a crane system–column quadrilateral truss structure–target spacecraft system. For the column quadrilateral truss structure, the equivalent beam model is used to make it equivalent based on the Timoshenko beam theory. The required equivalent stiffness parameters are determined and adjusted. The relative error between the finite element model and the corrected equivalent beam model of the column quadrilateral truss structure is no more than 4.7%. The results indicate that the accuracy of the modified equivalent beam model is sufficient. The improved equivalent beam model has excellent precision according to numerical calculations, and the derived equivalent stiffness parameters may be employed directly in dynamic modeling.

Analytical method for distortional buckling load of cold-formed steel members with perforated web under bending moment

In order to reduce weight or meet specific functional requirements, cold-formed steel (CFS) members with perforated webs are widely used in practical engineering. However, these web holes, especially the un-stiffened ones, can result in stress concentrations and section deficiencies, leading to a reduction of the distortional buckling load. Existing specifications, such as the American Iron and Steel Institute (AISI 2016) and Australia New Zealand Standards (AS/NZS 4600:2018), do not fully consider this impact. By simplifying the perforated web into an equivalent beam model and modifying the calculation method of the web rotational stiffness, this paper proposes an analytical method for the distortional buckling load of perforated CFS members with arbitrary hole parameters, including the hole spacing, location, and geometric shape. The proposed method is verified by finite element analysis (FEA) and experiment results in the literature. The FEA result indicates that holes near the compressed flange or with smaller spacing can lead to more reduction in the distortional buckling load. Compared to existing methods, the calculation accuracy of the proposed method has been improved by 2% to 4%.

An improved equivalent beam model of large periodic beam-like space truss structures

Space truss structures are essential components for space-based remote sensing loads with high spatial and temporal resolutions. To achieve high-precision vibration control, an accurate and efficient dynamics model is essential. In addition to the current equivalent beam model (EBM) based on the classical continuum theory, an improved equivalent beam model (IEBM) is proposed that considers the impact of the distinction between trusses and beams on torsional and shear deformations, as well as the impact of shear deformation on flexural rigidity. According to the displacement expressions of spatial beams, torsional, shear, and bending correction coefficients are introduced to derive expressions of strain energy and kinetic energy. The energy equivalence principle is then utilized to calculate the elasticity and inertia matrices, and dynamics equations are established using the finite element method. Subsequently, an IEBM is constructed by employing the particle swarm optimization approach to determine the correction coefficients with the truss natural frequency as the optimization target. The natural vibration characteristics of the structure are estimated for various material properties. Compared with the full-scale finite element model, the EBM reaches a maximum error of 80% for a low modulus of elasticity, while the maximum error of the IEBM is less than 2% for any given parameters, indicating its superior accuracy to the EBM.

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.

Accurate Estimation of Inter-Story Drift Ratio in Multistory Framed Buildings Using a Novel Continuous Beam Model

This study presents a novel method for accurately predicting the dynamic behavior of multistory frame buildings under earthquake ground motion. The proposed method allows approximately estimating the inter-story drift ratio, a crucial parameter strongly associated with building damage, its distribution along the building height, and its maximum value location. An equivalent continuous beam model with a rotation at the base, consisting of a combination of a shear beam and a flexural beam, is proposed to achieve this. This model derives closed-form solutions for the building’s dynamic characteristics. The lateral deformations along the height of frame buildings subjected to a given earthquake load, particularly the inter-story drift ratio profiles, and the maximum inter-story drift ratio parameter, are investigated. The proposed continuous model requires two dimensionless parameters: the lateral stiffness ratio (α) and the rotation at the base (θ), representing the drift ratio of the first story. For the expression of the lateral stiffness ratio (α) coefficient, a simple equation is also proposed using the beam-to-column stiffness ratio (ρ, or Blume coefficient) associated with the framed (discrete) system. Various building models are employed to validate the proposed method, demonstrating its applicability to both high-rise and low-rise building configurations. With the results obtained, it is shown that the proposed continuous model can be used not only for high-rise or multistory building models but also for low-rise building models.

Internal Force Calculation Method for a Complex Curved Beam Under a Vertical Load

In design, curved beams are popular owing to their aesthetic appearance. They are found to be in a state of coupled bending, shear and torsion when subjected to vertical loads. This complex behavior makes it difficult to perform force analysis in structural design. In this article, a set of equations are derived to calculate the internal forces at any position of a curved beam that consists of two orthogonal straight segments and one circular arc axis segment under a vertical load based on the force method. To simplify the calculation in practice, a simplified internal force analysis method is further developed by transforming the arc section into an equivalent straight beam model. The simplified formula is derived by fitting the internal force of 729 different curved beams, and it can be used quickly and accurately for hand calculation of internal force in a curved beam. Finally, the effectiveness of the proposed methods is verified by three engineering examples. The results show that the proposed force method can reasonably estimate internal forces at different locations of a complex curved beam, and the proposed simplified method can provide a convenient and practical approach to calculating the internal force distribution of curved beams.

Active control of large space antenna truss structures using the equivalent beam model

Active control of large space antenna truss structures using the equivalent beam model

Equivalent plate dynamic modeling of space periodic truss structures

Antenna structures used in spaceflight always contain nonlinear joints distributed along their cross section; hence, an equivalent beam model (EBM) that uses a point to describe the cross section is not ideally suited to depicting connection nonlinearity in the structure. To address this difficulty, a novel orthotropic equivalent plate model (EPM) of a space periodic truss structure is proposed. Firstly, utilizing the Mindlin plate hypothesis, strain in the periodic truss element is expanded into a binary high-order Taylor series, and corrections are introduced to normal strain. Secondly, employing the principle of energy equivalence, the equivalent elastic and inertial matrices of the EPM are calculated for in-plane and out-of-plane vibration. Finally, the static condensation method is utilized to reduce the dimensions of the equivalent model to establish an orthotropic equivalent Mindlin plate dynamic model, and its inherent characteristics are analyzed. The simulations show that when in-plane shear strain is expanded to the third order, the EPM can more accurately represent torsional vibration of the periodic truss structure. By introducing normal strain correction coefficients in the longitudinal direction, lateral bending vibrations of the periodic truss structure can be simulated more precisely. Compared with the EBM, the EPM can more accurately depict mechanical characteristics along the cross section. The EPM broadens the scope of continuum equivalent modeling of periodic trusses, and is more useful in engineering practice. In addition, the frequencies and mode shapes of the analytical EPM are consistent with a model using the finite element method, which supports the accuracy and effectiveness of the EPM. The analytical EPM provides a convenient way to study the nonlinear connections and vibration control of antenna structures with multiple joints along their cross section.

An effective equivalent stiffness model combined with equivalent beam model to predict the contact pressure distribution for a large PEM fuel cell stack

The contact pressure distribution plays a key role to the proton exchange membrane (PEM) fuel cell performance. The purpose of this study is to propose an equivalent stiffness model combined with an equivalent beam model to predict the contact pressure distribution for a large fuel cell stack. With the established equivalent stiffness model, the equivalent values of contact pressure is revealed. Moreover, these equivalent values are extended by an iterative calculation for contact pressure distribution with an established equivalent beam model. Finally, the presented model is compared to a three-dimensional FEA model. This equivalent stiffness model has high accuracy only with a 4.1% tolerance and the correlation coefficient of FEA and equivalent beam model for the contact pressure distribution at transverse and longitudinal direction is 0.976 and 0.998. This presented equivalent model is effective and can provide a direction for future work on high-performance fuel cell stack design and optimization.

Equivalent Beam Model and Improved Structure Design of Large Space Antenna Truss With Geometric Nonlinearity

AbstractThis paper proposes a extend linear equivalent method that can extend the linear equivalent micropolar beam model to the nonlinear equivalent micropolar beam model to analyze nonlinear vibration for large space truss structure, and also proposes a lattice enhancement method to improve the cantilever truss for the buckling problem that exists in space cantilever truss structures. The nonlinear equivalent model is obtained by introducing a corotating coordinate system into the linear equivalent beam model. Since the instability of the fixed root end of the cantilever truss is the main reason for buckling, the method of strengthening the longeron of the truss lattice by lattice is proposed. The accuracy of the equivalent geometric nonlinear model and the effectiveness of the improved cantilever truss structure are verified by four numerical simulation examples. The methods proposed in this paper provide some reference for studying the dynamics analysis of large space trusses and the design of structures.

<b>Seismic Design Method for Shield Tunnels in Ground Conditions Subject to Change in the Longitudinal Direction</b>

The authors of this paper investigated the effect of seismic motion on shield tunnels that are located in areas with varying ground conditions. This research revealed that the larger the inclination angle of a boundary between a basic layer and a soft layer becomes, the larger the sectional forces, such as axial force, bending moment and shear force become. In addition, the sectional force obtained by an equivalent stiffness beam model was found to be larger than that obtained by a beam-spring model. This research also confirmed that results using a beam-spring model when the inclination angle is 5.9 degrees satisfy verification requirements.

An Intuitive Derivation of Beam Models of Arbitrary Order

This article presents a new beam model that employs a recursive derivation procedure that enables the user to set the order of the governing differential equations as an input parameter, without the need for ad hoc assumptions or methodologies. This article employs a novel system of kinematic variables, section constants, and section functions that facilitate the development of higher-order beam models that retain a clear philosophical link to classical beam models such as Euler–Bernoulli beam theory and Timoshenko beam theory. The present beam model is a type of equivalent single layer beam model, wherein section constants are used to model the global stiffness characteristics of the beam, and section functions are used to recover sectional fields of displacements, strains, and stresses. The present beam model is solved for several example beams, and the results are compared to the results of finite element analyses. It is shown that the present beam model can accurately predict the deformed shapes and stress fields of each of the example beams. This article also reveals an interesting peculiarity of the elastic potential energy that pertains to any unidimensional beam model that is governed by differential equations that are of finite order.

Applications of VAM-based equivalent beam model on effective performance of composite sandwich beam with M-shaped folded core

The issue of facesheet-core debonding is expected to be addressed by increasing the bonding platform of the M-shaped folded core. To investigate its effective performance, the three-dimensional traditional equivalent beam (3D-TEB) model and the one-dimensional equivalent beam model (1D-EBM) of the M-shaped folded sandwich beam were established based on the variational asymptotic method. The expression of the original 3D elastic problem was given in variational form. By minimizing the energy functional, the asymptotic solutions of the warping functions and the Euler beam model (zeroth-order approximation) were obtained. Through the static and dynamic analysis of a three-dimensional folded sandwich beam and two equivalent beam models, the effectiveness of the 1D-EBM instead of the original 3D-TEB for static and dynamic analysis was verified. The influences of the location and frequency of the excitation on the dynamic performance of the CSB-MFC were investigated. Compared with 3D-FEM and 3D-TEM, the novelties of cell tailorability and local field recovery within the unit cell are very important features, which significantly facilitates the parametrical analysis of the CSB-MFC. The results showed that the change in the top width of the bonding platform had a particularly significant impact on the torsional stiffness. The wall thickness mainly affected the torsional performance, and the increase in the core height primarily improved the torsional and bending stiffness, resulting in the gradual increase in the fundamental frequency. • The VAM-based 1D-EBM is established to analyze the global response. • The accuracy of the model is verified by static and dynamic examples. • The influence of cell geometric parameters is systematically analyzed. • The dynamic performance of CSB-MFC is investigated based on the equivalent model. • The reduced DOFs of 1D-EBM result in high calculation efficiency.

Modeling and analysis of a metal rubber vibration isolation system considering the nonlinear stiffness characteristics.

In this study, the complete design process of a metal rubber isolator using a numerical method applied in automobile underwater gliders (AUGs) is researched. A ring-like metal rubber isolator that has the potential to reduce the vibration of the AUGs is proposed. In the numerical design method, the equivalent cantilever beam model is used to identify the mechanical properties of the metal rubber isolator, whose accuracy is verified by experiment. The static stiffness and dynamic stiffness are gained through a fitting method, considering different mass loadings. Different material parameters of a metal rubber isolator, such as relative densities, wire diameters, wire spiral pitch diameters, and mass loading from the power system, greatly influence the mechanical properties and stiffness characteristic. The vibration isolation performance of a metal rubber isolator applied in the AUG is evaluated by a nonlinear single degree of freedom dynamic model, which is solved by the harmonic balance method. The linear stiffness component of the metal rubber isolator is the main influence factor for the vibration isolation performance. The nonlinear stiffness component could shift the isolation frequency.

Equivalent Nonlinear Beam Model for Static and Free Vibration Analysis of the Beamlike Truss

Equivalent Nonlinear Beam Model for Static and Free Vibration Analysis of the Beamlike Truss

Arrangement of Belleville Springs on Endplates Combined with Optimal Cross-Sectional Shape in PEMFC Stack Using Equivalent Beam Modeling and FEA

A set of Belleville springs integrated into an endplate plays a key role in a proton exchange membrane fuel cell (PEMFC) stack, which makes the applied assembly force smoother, resulting from the absorbed vibration and thermal expansion. The appropriate arrangement of Belleville springs is important in PEMFC stack design. The aim of this study is to establish an equivalent beam model to optimize the numbers and positions of Belleville springs to minimize endplate deformation. Based on this, a finite element analysis (FEA) model of the PEMFC stack is proposed to further optimize the cross-sectional shape of the endplate. For the endplate with two, three and four groups of Belleville springs, its optimal positions correspond to 0.17lin, 0.27lin and 0.5lin (lin is the equal distance between steel belts). In addition, the low thickness should be 2/3 of the high thickness of the curved endplate for a uniform contact pressure distribution as well as the high-volume-specific power. However, the curvature radius of the endplate arc is negative to the uniformity of the contact pressure distribution, and particularly the internal cells of the PEMFC stack. This study provides a design direction for endplates combined with Belleville springs in large fuel cell stacks clamped with steel belts.

Calculation of Building Settlement Induced by Tunneling Based on an Equivalent Beam Model

Calculation of Building Settlement Induced by Tunneling Based on an Equivalent Beam Model

Fragility analysis of floor micro vibrations induced by internal vehicles in high technology factories

This study explores the micro vibrations in long span floors induced by internal moving vehicles or automated guided vehicles (AGVs) in high technology fabrication factories (or fabs) by using a bivariate fragility assessment model that considers both the vehicle weight and moving speed. Dynamic time history analyses of an equivalent sub-structural multiple span continuous beam model with moving forces generated by a modified Kanai-Tajimi power spectral density function are performed. The maximum root-mean-square (RMS) floor velocity response of each pair of AGV weight and speed are obtained and in turn used to determine their best fitting median plane and standard deviation for constructing the graphical fragility surface. Probabilistic assessments of floor micro vibrations that exceed a specific vibration level for various vehicle weights and speeds can be directly determined from the fragility surfaces. Instead of using a deterministic approach, probability-based surfaces are a promising alternative for assessing floor micro vibrations that exceed a specific vibration level for various random AGV movements. Moreover, the fragility surfaces can be used to determine the appropriate weight and speed of the AGV or assist in the conceptual design of vibration-sensitive production floor systems based on a given exceedance probability of the desired vibration level.