Equivalent Beam Model Research Articles

Dinamička analiza posmičnih stijena s čeličnom ispunom

In this paper, the Timoshenko beam model (continuous system model) is originally adapted for dynamic analysis of steel plate shear wall (SPSW) systems. Dynamic characteristics for the first three modes are found by solving differential equation of the equivalent Timoshenko beam model using the differential transformation method. Dynamic characteristics are tabulated for quick and practical calculation. With the help of the dynamic characteristics, the response spectrum analysis of such buildings is performed. The continuous system calculation model has been used in the literature for free vibration analysis of such systems. Using the approach developed in this study, it is possible to calculate not only natural periods, but also the base shear force, maximum storey displacement, and maximum storey drift ratio. The differential transformation method is used in this study for solving the differential equation written according to the continuous system calculation model. To investigate the suitability of the method presented in the study, an example taken from the literature is solved and the results are evaluated. The results show that the method presented can be used in the preliminary design stage.

Study on tool deflection compensation method based on cutting force observer for orbital drilling of CFRP/Ti stacks

Orbital drilling is an alternative hole-making machining process that combines drilling and milling, usually performed with specific devices and industrial robots to make holes in aircraft composite structures. In order to eliminate the adverse effects on the diameter precision of the tool deflection in the orbital drilling process of CFRP/Ti stacks, a novel method that fuses cutting force model prediction and feed-axis disturbance observation is proposed to estimate the radial cutting force. The tool deflection caused by the radial cutting force is calculated based on the equivalent diameter cantilever beam model. Then the tool path is corrected to compensate for the tool deflection and prevent borehole diameter deviation by adjusting the tool eccentricity in real-time. The orbital drilling experiments demonstrate that the geometrical error of the hole was reduced by 50%, and the manufacturing precision requirements of IT9 level can be satisfied after one single shot even the cutting force was increased to twice the initial value caused by tool wear, the proposed method effectively improved the hole-making precision of CFRP/Ti stacks.

Evaluating elongated grains with diffuse ultrasonic double scattering and rectangular transducers.

Diffuse scattering of ultrasound by the microstructure of polycrystal specimens can be used to evaluate grain size and grain elongation. The existing diffuse scattering models mostly dealt with circular transducers whose symmetrical sound field is insensitive to the asymmetric elongated grain. The sound field of a rectangular transducer provides a new perspective for acquiring additional information. First, the existing single scattering response (SSR) and double scattering response (DSR) models are modified for a rectangular transducer, where the sound field of a rectangular transducer is equivalent to that of an elliptical transducer in the far-field. Therefore, an equivalent single Gaussian beam model is derived using amplitude-equivalent and beamwidth-equivalent coefficients. Then, the spatial correlation function of elongated grains is transformed into the wavenumber domain, giving rise to the SSR and DSR of a rectangular transducer that reveals the interaction effect of an asymmetric sound field and elongated grains on ultrasonic backscattering. The experimental results show that the sizes of elongated grains in a cold-rolled aluminum are evaluated as 1086 ± 8, 90 ± 4, and 10 ± 1 μm in the x, y, and z directions, where the exact values are 1184.2 ± 11.9, 80.7 ± 5.2, and 8.3 ± 0.5 μm according to metallographic measurements.

A Novel Approach for Design and Simulation of Equivalent Continuum Model of Lattice Structures Used in Tunnels and Caverns

Developing a continuum model of a lattice structure means expressing a complicated structure into a simpler one, and this is done by determining equivalent stiffness of the original structure, which can further be used to transform into desired simpler geometry for multiple benefits. Objective of this paper is to find a reliable method for determination of equivalent stiffness of lattice girder, comparing all the techniques suggested by various researchers. In present work the actual lattice girder structures and their equivalents were analysed using Finite Element Method tool Strand7. The primary focus is on lattice girder, which is a primary rock support system employed in tunnels and caverns. The equivalent stiffness so obtained is used as input parameter in numerical analysis tool STRAND7 for analysis of equivalent lattice beam. The maximum deflection and bending stress at centre of equivalent lattice beam were calculated and compared. In an attempt to develop a more accurate equivalent beam model a novel approach which is modification of the Method-B has been developed by authors. This Noval approach is the most convenient approach to develop an equivalent section with an extremely uncompromised representation of stress development and deformation characteristic. In this approach, the focus was made on the physical resemblance of a proposed equivalent beam with the actual lattice girder, leading us to adopt a beam of I-cross-section. This approach yielded similar stress and deflections in the equivalent beam as that of the actual lattice girder.

An equivalent mechanical model investigating endplates deflection for PEM fuel cell stack

The aim of this study is to obtain the deflection curve equations of endplates with one to five clamping belts which allows investigating endplates deflection for uniform contact pressure distribution. Based on an equivalent mechanical model for a large fuel cell stack, the effects of the thicknesses of endplates, numbers, and positions of clamping belts are discussed, and the optimal thickness of endplate with different clamping belts is obtained, and moreover the optimal position of intermediate and outer clamping belts on the endplates. Finally, a three-dimensional finite element analysis (FEA) of a fuel cell stack clamping with steel belts and nonlinear contact elements is compared to what the equivalent mechanical beam model predicts. The result of this study shows that the equivalent mechanical model gives good prediction accuracy for the deflection behavior of endplates and the clamping force of the fuel cell stack, which is effective and helpful for the design of a large fuel cell stack assembly.

Modeling beam-like planar structures by a one-dimensional continuum: an analytical-numerical method

In this paper, beam-like structures, macroscopically behaving as planar Timoshenko beams, are considered. Planar frames, made by periodic assemblies of micro-beams and columns, are taken as examples of these structures and the effectiveness of the equivalent beam model in describing their mechanical behavior, is investigated. The Timoshenko beam (coarse model) is formulated via the direct one-dimensional approach, by considering rigid cross-sections and flexible axis-line, while its constitutive laws is determined through a homogenization procedure. An identification algorithm for evaluation of the constitutive constants is illustrated, based on Finite Element analyses of the cell of the periodic system. The inertial properties of the equivalent model are instead analytically identified under the hypothesis the masses are lumped at the joints. The advantages in using the equivalent model are discussed with reference to the linear static and dynamic responses of some planar frames, taken as case-studies, for which both analytical and numerical tools are used. Numerical results, obtained by the equivalent model, are compared with Finite Element analyses on planar frames (fine models), considering both symmetric and not-symmetric layouts, in order to show to effectiveness of the proposed algorithm. A comparison with analytical results is carried out to validate the limits of applicability of the method.

Estimation of the Mechanical Parameters for a Reduced Coupled Flexural–Torsional Beam Model of a Tall Building by a Sub-Structure Approach

The use of equivalent beam models to estimate the dynamical characteristics of complex tall buildings has been investigated by several authors. The main reason is the structural response estimation to stochastic loads, such as wind and earthquake, using a reduced number of degrees of freedom, which reduces the computational costs and therefore gives the designer an effective tool to explore a number of possible structural solutions. In this paper, a novel approach to calibrate the mechanical and dynamical features of a complete 3D Timoshenko beam, i.e., describing bending, shear and torsional behavior, is proposed. This approach is based on explicitly considering the sub-structures of the tall building. In particular, the frames, shear walls and lattice sub-systems are modeled as equivalent beams, constrained by means of rigid diaphragms at different floors. The overall dynamic features of the tall building are obtained by equating the deformation energy of an equivalent sandwich beam with that of the selected sub-structures. Finally, the 3D Timoshenko equivalent beam parameters are calibrated by minimizing a suitable function of modal natural frequencies and static displacements. The closed form modal solution of the equivalent beam model is used to obtain the response to stochastic loads.

Design and optimization of repeatable locking/unlocking device for magnetically suspended control moment gyro

Due to the air gap between stator and rotor, magnetically suspended control moment gyro (MSCMG) should be equipped with locking/unlocking device (LUD) to sustain launch vibration and impact. Current LUDs are not repeatable, reliable and compact enough for gyro applications. In this paper, a repeatable locking/unlocking device (RLUD) with several elastic brackets as clamping/releasing mechanism is proposed. Based on the equivalent beam model, static and dynamic analyses of the brackets are carried out, and the maximum stress, the locking/unlocking force and the resonance frequency are obtained. Subsequently, the optimization design using non-linear programming by quadratic Lagrangian (NLPQL) algorithm is implemented, and structural parameters of the optimized brackets are calculated. The maximum stress and first resonance frequency are simulated through finite element method (FEM), and the results are in good agreement with those of the elastic beam analyses. Finally, a MSCMG rotor system with the designed RLUD is fabricated. The environmental mechanics tests including sine sweep and random vibration are performed, which verifies the locking protection effect for MSCMG rotor.

Equivalent Beam Model for Spatial Repetitive Lattice Structures with Hysteretic Nonlinear Joints

This study presented an equivalent modeling method for analysis of the fundamental harmonic response of the spatial repetitive lattice structures with hysteretic nonlinear joints. Firstly, the joint was modelled as a spatial nonlinear spring-damper system with bilinear hysteresis, the equivalent linear stiffness and viscous damping coefficients of the hysteretic joint were derived using the describing function method. Subsequently, high-precision displacement and rotation fields that consider the warping and distortion of the cross section of the lattice structure were presented for the repeating element and the equivalent continuum modeling method based on the principle of energy equivalence was adopted to obtain an 8-DOFs spatial beam element for the repeating element of the lattice structure. Afterwards, the equations of motion of the equivalent beam model for the lattice structure were established in the frequency domain and solved by using the Newton-Raphson method. In the numerical studies, the joints with axial nonlinearity and rotational nonlinearity were considered, and the influences of joint parameters and excitation amplitude on the dynamic response of the lattice structure were investigated. The correctness of the presented modeling method was verified by comparison with the nonlinear finite element model of the original structure.

Design finite element method based models of floating dry docks strength

The features of global strength modelling of floating dry docks using finite element method are considered. Comparative analysis of two- and three-dimensional models was performed considering the interaction of the floating dry dock and the ship. To solve the problem of reducing the complexity of creating and the size of the finite element model, it is proposed to use the elements of a volumetric orthotropic body to model the main transverse beams of the pontoon. Hydrostatic elastic base of floating dry dock is represented as spring elements. The model of the dock support device includes spring and gap elements. The vessel is considered in the equivalent beam model. Results were obtained on such effects as redistribution of buoyant force due to deformation of the dock, incomplete inclusion of the towers in the general longitudinal bending of the dock, the effect of ship stiffness not only on the longitudinal, but also on the transverse bending of the dock.

Combined backbone application on numerical simulations and a model experiment of a 20,000 TEU container ship

For the large-scale development and large opening characteristics of container ships, the characteristics of springing in waves has become more complex. It is thus urgent to find reasonable equivalent principles to solve the dilemma of simultaneously simulating vertical, horizontal and torsional vibrations in model tests. Meanwhile, to avoid the complexity and time-consuming nature of full three-dimensional modelling, a large-scale combined beam model is adopted to simulate three-dimensional coupled vibrations and the positions of the neutral axis and torsion centre. Combined with the finite element method, a 3D equivalent modelling method of the combined model is analysed in detail, and the stiffness influence of transverse bulkheads on the coupled vibrations mode is revealed. Furthermore, the elastic modal responses, vibration principles and waves load in the fluid-structure interaction vibrations are assessed and investigated by the three-dimensional linear hydroelastic mechanics method in the frequency- and time-domains. Then, the scheme and load measurement method in the model test in the wave basin are put forward, and the experimental data and numerical calculation results are compared and found to be consistent. The proposed the equivalent combined beam model is of great significance for model tests and numerical analyses of very large container ships.

Vibration performance assessment of a long-span steel footbridge

To meet the basic requirements of long spans, steel footbridges are generally lightweight structures with low stiffness. Moreover, current trends in innovative structural design have led to more vibration problems related to resonance or quasi-resonance of footbridge structures at typical pedestrian walking frequencies. The present authors apply a feasible methodology to assess the vibration performance of footbridges based on an equivalent beam model developed from a detailed finite-element model and experimental measurements. Free- and induced-vibration tests are performed on a long-span inverted-queen-post-truss steel footbridge located in Rio de Janeiro, which exhibits vibration problems. Pedestrian walking simulations are performed with the simplified footbridge model using a typical dynamic load given from a Fourier series, as well as with a biodynamical formulation that considers human–structure interactions. The equivalent beam model provides a practical means of investigating corrective intervention strategies for the problem of excessive footbridge vibration using tuned mass dampers, and allows the in-service footbridge performance to be assessed based on current standards and design guidelines.

Aeroelastic vibration analysis of wind turbine blade with Gurney flap

In this paper, based on the parametric design of 3 D blades including typical cross-section and natural frequency calculation of the equivalent model, an integral method for aerodynamic performance and aeroelastic vibration analysis of blade with Gurney flap is proposed. Parametric design and unstructured grid are used to pre-process the blades with/without the Gurney flap. The discrete aerodynamic performance parameters distribution including the lift, drag, and torsion coefficients calculated by Fluent is fitted by the nonlinear least square method based on the trust-region algorithm, and the natural frequencies of the rotating blade are accurately solved by the equivalent thin-walled beam model and Green’s functions. Based on the aerodynamic performance coefficients and natural frequencies obtained by the accurate calculation above, the aeroelastic response equation of typical cross-section considering local aerodynamic damping matrix is established, and the vibration response of blade in flap and torsion direction is further described. From the analysis results, it can be seen that the Gurney flap structure can not only bring higher lift performance to the blade, but also can reduce the amplitude and vibration range of aeroelastic vibration, improve the aeroelastic stability of the blade, and prove the effectiveness of Gurney flap.

Cracked Equivalent Beam Models for Assessing Tunneling-Induced Damage in Masonry Buildings

This paper proposes simple numerical models that simulate the influence of tunneling-induced ground movements on masonry buildings founded on shallow strip footings. The focus is on developing an improved understanding of the impact of structural discontinuities due to pre-existing cracks and joints between adjacent buildings. The soil–structure system is idealized with equivalent Timoshenko beams connected to a homogenous elastic half-space with rigid links. The flexibility of the equivalent beam at the location of the crack or building interface is determined using linear-elastic fracture mechanics. The proposed models are first applied to a generic numerical example. Then, they are used to interpret the detailed displacement and strain measurements gathered from a masonry church during nearby tunneling. The results show that pre-existing cracks can have variable influence on the soil–structure interaction depending on their extent and position relative to the building and settlement trough. For the investigated case study, the proposed models illustrate how pre-existing large cracks in the hogging zone localized deformations and how the interaction between adjacent buildings drastically altered response mechanisms.

Shear and flexural factors for static analysis of homogenized beam models of planar frames

One-dimensional equivalent beam models, embedded in a two-dimensional space, are developed in this work for static analysis of beam-like structures, such as planar frames (fine models). Extensible shear and Timoshenko beam models (coarse models) are formulated in the framework of the direct one-dimensional approach, under the assumption of rigid cross-section, while their constitutive laws are determined through a homogenization procedure. A linear constitutive equation is obtained, also taking into account for the presence of bracing elements in the frame, with axial and shear forces coupled with bending. The warping of the cross-section, caused both by shear and bending, is taken into account, in an approximate way, by resorting to the concept of shear and flexural factors. These factors are introduced to correct the constitutive coefficients of the equivalent beam. They are determined by equating the elastic potential energy of the fine model and that of the coarse models, once the same macroscopic displacement is given. The limits of applicability of the corrected homogenized beam models are discussed with reference to the linear static response of some planar frames, taken as case-studies. Numerical results obtained by the homogenized models are compared with finite element analyses on planar frames, in order to show the effectiveness of the procedure.

Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model

The main goal of this paper is to develop an equivalent nonlinear beam model (ENBM) for forced vibration analysis of the nonlinear beamlike truss (NBT) and its numerical implementation for describing the nonlinear behaviors of the periodic NBT. For the equivalent dynamic modeling method based on the energy equivalence principle of the truss structures, most of the existing approaches focus on studying the linear characteristic which cannot satisfy the nonlinear dynamic analytical requirement. Here, we extend the equivalent dynamic modeling to nonlinear forced vibration responses of the large NBT civil engineering structure with two pinned ends. We introduce the geometric nonlinearity into the equivalent dynamic model using the von Karman nonlinear strain-displacement relationship to imitate the nonlinear behaviors of the beamlike truss. On the basis of the Hamilton principle, the fourth-order governing partial differential equations of motion of the ENBM are obtained and solved utilizing the Galerkin method. It is shown that the ENBM accurately captures the nonlinear dynamic response of the beamlike truss subjected to external load by using the first four order modes. In order to showcase the efficiency and accuracy of the ENBM, time histories, phase maps and fast Fourier transform frequency spectra are investigated for the NBT and the ENBM under different external excitation magnitudes and frequencies. Comparisons between results of the ENBM and those obtained from the finite element simulations of the NBT show good agreements and identify the periodic motions of them. Furthermore, the effect of the external loading and damping parameters on the frequency-response and force-response curves is discussed. In addition, computing time of the proposed ENBM are compared with that of the full-scale finite element model in ANSYS so as to highlight the significant less computational cost of the equivalent nonlinear dynamic modeling. Particularly, the proposed ENBM can achieve the nonlinear mechanism of the NBT in analytical method and design the low-order control law conveniently.

Equivalent Dynamic Model for Triangular Prism Mast with the Tape-Spring Hinges

The development of space mission requires the lightweight, large deploying-to-folding ratios and high-precision deployable structures. The tape-spring hinge can achieve a deflection of 180 deg within a 5% elastic deformation range. For one triangular prism mast (TPM) with the tape-spring hinges, the equivalent beam model for the TPM is derived using an energy equivalence method. The node deflections’, geometrical parameters’, and material properties’ relationship between the TPM and equivalent model are built based on the structural features. The strain and kinematic energy equations of the key points are derived by the equivalent model. The stiffness and mass matrices of the equivalent model are obtained. In addition, the modal of the TPM in free vibration modes is analyzed based on the Timoshenko continuum beam theory, and natural frequencies are solved. Two TPM units are developed to validate the precision of the continuum beam equivalent model. A model test is conducted at the cantilever position, and the finite element (FE) model of the two TPM units is modified. Subsequently, the modeling approach of the two TPM units is applied to establish the FE model of the 10 TPM units. Moreover, the accuracy of the continuum beam equivalent theoretical model is verified.

Experimental and analytical investigation on the in-plane mechanical property of discretely connected precast RC floor diaphragm

Discretely connected precast reinforced concrete floor diaphragm (DCPCD) is a new-type untopped precast concrete floor system. Using slab joint connectors, instead of cast-in-situ concrete overlap layers on precast slab, to transmit slab joints inner forces caused by vertical and horizontal loads eliminates the need for wet works in construction site, thus leading to economical designs and high efficiency in installation. This paper presents the development and experimental evaluation of DCPCD subjected to in-plane low-cyclic reversed loading. Performance of the tested specimen was evaluated in terms of failure mode, in-plane deformation pattern, hysteretic curves, stiffness, ductility and energy dissipation capacity. The results show that DCPCD had high in-plane stiffness and good integrity, and the beam-to-slab connections and slab joints perform well during the test process. The ductility of the specimen is acceptable, while the energy dissipation is not that satisfactory, confirming that the elastic design method of precast RC diaphragms is rational. The shear and axial force distribution regularity of the tested specimen was analyzed, and the results indicate that internal force distribution is similar to that of the simply supported beam. The equivalent beam model theory was modified based on the measured tensile (compressive) stiffness of slab joint connectors and the actual load effect in the floor et al. It is demonstrated that the in-plane deformations determined by the modified calculation method agree well with the test results. Hence, the proposed in-plane stiffness analysis method is reliable and can be used for designing buildings adopting DCPCD.

Buckling of Planar Micro-Structured Beams

In this paper, a Timoshenko beam model is formulated for buckling analysis of periodic micro-structured beams, uniformly compressed. These are planar grid beams, whose micro-structure consists of a square lattice of equal fibers, modeled as Timoshenko micro-beams. The equivalent beam model is derived in the framework of a direct one-dimensional approach and its constitutive law, including the effect of prestress of the longitudinal fibers, is deduced through a homogenization approach. Accordingly, micro–macro constitutive relations are obtained through an energy equivalence between a cell of the periodic model and a segment of the equivalent beam. The model also accounts for warping of the micro-structure, via the introduction of elastic and geometric corrective factors of the constitutive coefficients. A survey of the buckling behavior of sample grid beams is presented to validate the effectiveness and limits of the equivalent model. To this purpose, results supplied by the exact analyses of the equivalent beam are compared with those given by finite element models of bi-dimensional frames.

Characterization of an Equivalent Coupled Flexural-Torsional Beam Model for the Analysis of Tall Buildings under Stochastic Actions

AbstractAn equivalent beam model to estimate the three-dimensional response of tall buildings to random loads is proposed in this paper. The model fully describes coupling between bending and torsi...