Dynamic Magnification Factor Research Articles

Dynamic behaviour of an inclined FG-CNTRC sandwich beam under a moving mass

This paper studies dynamic behavior of an inclined sandwich beam with a homogeneous core and functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets under a moving mass. The effective properties of the face sheets are estimated by the extended rule of mixture. Three types of carbon nanotube distribution, namely uniform distribution (UD), functionally graded ( ) and V (FG-V) distributions, are considered. Based on the first-order shear deformation theory, a finite element formulation is formulated by using hierarchical functions to interpolate the displacements and rotation. Using the derived formulation, dynamic response the sandwich beam is computed with the aid of the Newmark method. The obtained result reveals that the inclined angle has a significant influence on the response of the beam, and the dynamic magnification factor decreases for the beam associated with a larger inclined angle. The effects of various parameter, including the nanotube volume fraction, the type of carbon nanotube distribution, the layer thickness ratio and the moving mass velocity on dynamic behavior of the sandwich beam are examined and highlighted.

Wind Load Model and Response Spectra of Overbridges Subjected to Train-Induced Wind Loads

The vibration of pedestrian overbridges due to train-induced wind loads (TIWLs) during the passage of a high-speed train threatens the overbridge safety and causes pedestrian discomfort. To incorporate the structural dynamic magnification effect on overbridges under TIWLs in the design process, a framework to calculate the structural dynamic response, equivalent static TIWLs and response spectra of overbridges is established. A new simplified TIWL model with a “three-segment” wind pressure time history described by harmonic functions is proposed. This model facilitates the acquisition of analytical solutions for the structural dynamic response, streamlining the analysis process. The dynamic response and equivalent static TIWLs based on dynamic magnification factors (DMFs) of overbridges are obtained from the analytical method. The DMFs are primarily affected by the structural natural frequency and the characteristic frequency of TIWLs. Furthermore, based on the analytical solution of the overbridge dynamic response, parametric analysis is performed, and the response spectra representing the dynamic displacement and acceleration of overbridges are developed. The response spectra can be used to quickly calculate the displacement, internal force, and acceleration of the overbridges during engineering applications, which is more convenient but slightly less accurate than the analytical method.

Vibration response of viscoelastic nanobeams including cutouts under moving load

This manuscript presents a mathematical model and numerical solution to predict vibration responses of nonlocal strain gradient (NLS) perforated viscoelastic nanobeam under dynamic moving loads. The microstructure and size length scales are considered in the frame of NLS theory. The viscoelastic damping of the structure is considered by linear Kelvin Voigt viscoelastic model. Timoshenko and Euler Bernoulli beam theories are developed to consider both thin and thick structure response. The moving load is portrayed by point load and harmonic type. The dynamic equations of motion incorporating viscoelasticity and size dependency effects is derived by using Hamilton principle. A differential quadrature method (DQM) with Chebyshev–Gauss–Lobatto formula is used to discretize the space domain and solve the model numerically. The obtained results are validated and compared with available literature. Numerical studies are conducted to show the influence of the material viscosity parameter, perforation parameters, shear deformation effect, nonlocal strain gradient, moving load velocity, and beams slenderness ratio on the dynamic behavior of viscoelastic perforated nanobeams under moving load. The material viscosity parameter can effectively control the magnitude and stability of the magnification factor profiles. Higher filling ratios result in larger values of the dynamic magnification factor, whereas larger numbers of hole rows lead to smaller values. The proposed procedure is supportive in the analysis and design of perforated viscoelastic NEMS structures under moving load.

Vibration analysis of functionally graded microbeams with a moving mass based on Timoshenko beam theory

Vibration analysis of functionally graded (FG) microbeams carrying a moving mass is carried out in the framework of Timoshenko beam theory. The beam material properties are considered to be graded in the thickness by a power-law function, and they are estimated by Mori-Tanaka scheme. The influence of the microsize effect is captured with the aid of the modified couple stress theory (MCST). A finite beam element is formulated and used to establish the discretized equation of motion for the beams. Vibration characteristics, including the natural frequencies, the time histories for mid-span deflection, the dynamic magnification factor and the stress distribution, are computed for a simply supported beam. The obtained result reveals that both the dynamic deflection and dynamic magnification factor are overestimated by ignoring the microsize effect. The effects of the material distribution, the moving mass velocity and the length scale parameter are studied in detail and highlighted.

Dynamics of functionally graded beams carrying a moving load with influence of porosities and partial foundation support

The novelty of the present work is to study the simultaneous influence of porosities and partial Pasternak foundation support on dynamics of functionally graded (FG) beams carrying a moving load. The beams are made from an open-cell steel foam with symmetric and asymmetric porosity distributions in the thickness direction. Based on a refined third-order shear deformation theory, a two-node beam element with ten degrees of freedom is derived and employed to construct the discretized equation of motion for the beams. Dynamic characteristics, including the time histories for mid-span deflection, dynamic magnification factor (DMF) and the stress distribution, are computed with the aid of the Newmark method. The numerical result reveals that the foundation supporting length has an important role on the dynamics of the beams, and the dependence of the DMF upon the porosity coefficient is governed by the foundation supporting length. It is also found that the asymmetric porosity distribution has more impact on the dynamic response of the beams than the symmetric one does, and the difference between the DMFs obtained from the two porosity distributions is more significant for the beam with a higher porosity coefficient. The effects of the porosities, the foundation support and the moving load velocity on the dynamic behavior of the beams are examined in detail and highlighted

Multi-objective optimal design of double tuned mass dampers for structural vibration control

A double tuned mass damper (DTMD) for suppressing oscillations of civil structures is proposed in this study. DTMD is a combination of an undamped TMD and a smaller TMD. The impact of parameters on the essential characteristics, as well as the vibration absorption capacity of DTMD, is investigated. Using genetic algorithms (GA), the optimum parameters of DTMD are determined by minimizing the peak dynamic magnification factor of structural responses for a wide range of excitation frequencies. The effectiveness and robustness of DTMD are also compared with those of the optimized TMD having a similar weight as the DTMD. Furthermore, multi-objective optimization designs of DTMD (for both two-objective and three-objective) are also developed here. This study indicates that the DTMD is more effective than a single TMD. If keeping a similar efficiency to that of an optimized TMD, the optimum DTMD has a broader domain for choosing the frequency and damping ratio. In this sense, a DTMD is much more robust than a single TMD.

Size-dependent vibrations of bi-directional functionally graded porous beams under moving loads incorporating thickness effect

This article analyzes the forced and free vibrations of bi-directional functionally graded (BDFG) porous nanobeams under moving loads by considering the thickness effect based on the nonlocal strain gradient theory (NSGT). It is assumed that the material properties of the system are graded across the transverse and axial directions according to power and exponential laws, respectively. Various porosity models, including even, uneven, and asymmetric porosity distribution patterns, are considered to model the porosity effects. The dynamic equation of the system is derived by implementing the physical neutral surface concept. Analytical and numerical approaches are adopted to acquire the vibration response of the system. Mathematical closed-form expressions are provided for various dynamic phenomena, such as two types of cancelation, two types of resonance, maximum resonance, cancelation disappearance, and resonance disappearance. In addition, for the first time in this article, it is demonstrated that BDFG porous nanobeams under moving loads are prone to experience the double cancelation phenomenon. Several comparison studies with published data are conducted to validate the results. Also, comprehensive parametric studies are carried out to elucidate the impacts of system parameters such as thickness power index, axial gradient parameter, scale parameter ratio, axial force, porosity coefficient, and porosity distribution pattern on the dynamic magnification factor, critical load speed, and maximum free-response. The outcomes established that by considering the thickness effect, the stiffening behavior of the system is amplified depending on the slenderness ratio. Also, it is concluded that, among the considered porous models, the critical load speed for the system with an even porosity distribution is more dependent on the thickness power index.

Dynamic finite element analysis of microbeams under a moving mass based on refined third-order shear deformation theory

The dynamic behavior of microbeams with rectangular cross section under action of a moving mass is studied in the present paper in the framework of a refined higher-order shear deformation beam theory. The modified couple stress theory (MCST) with only one additional scale parameter is adopted to describe the influence of the microsize effect on the dynamic response of the microbeams. A finite element formulation with ten degrees of freedom is formulated and used to establish the discretized equation of motion for the microbeams. The formulation, taking into account the influence of the inertial effect, the Coriolis and centrifugal forces resulted from the mass moving, is derived from the expressions of the elastic and kinetic energies of the microbeams. The accuracy and efficiency of the derived formulation are confirmed by comparing the result obtained in the present work with the published data. The dynamic response of the microbeams with simply supported ends, such as the curves for dimensionless mid-span deflection-moving time relationship, the dynamic magnification factors (DMFs) and the thickness distribution of stresses are assessed by using an implicit Newmark method. The obtained numerical results reveal that the material length scale parameter which is introduced in the MCST has an important role on the dynamic behavior of the microbeams, and the DMF obtained from the theory incoporating the MCST is considerably lower than that using the conventional beam theory. It is also shown that the amplitude of both the axial stress and shear stress is considerably decreased by the increase of the material length scale parameter. A numerical study is carried out to highlight the influence of various parameters such as the moving mass velocity and the the ratio of the total beam length to its height on the dynamic response of the microbeams.

A study of impact induced stress due low velocity impact on laminated composite skewed hypar shell roof

The doubly ruled hypar shell, being one of the most feasible shell roof configurations, enjoys industrial preference for covering large column free areas. This class of shells is unique as the only curvature here is the cross curvature and these do not admit easy closed form solution particularly when the boundary conditions are complicated. Laminated composite, an innate choice to different industrial sectors for its huge specific strength and stiffness, good weathering resistance, is now being extensively employed in civil engineering. However the low transverse shear strength of the composite shell impelled the researchers to study the response of the same under the action of impact loads. In the present study, a finite element code is functioned to investigate the impact induced stress history generated in simply supported laminated composite hypar shell for different impact velocities of a spherical solid striker. Contact behavior is described by modified Hertzian contact law where as time dependent equations are solved using Newmark’s time integration algorithm in present analysis. A practicing engineer often has to select a particular shell option consisting stacking sequence among a number of possibilities. The present paper discusses the behaviour of composite hypars under low velocity impact and different stresses induced due to the same from engineering standpoint to propose a selection guide line among the available options and to proposes values of practical parameters like the equivalent static loads and dynamic magnification factors for designing such shells through static simplification of the problems.

Dynamic response of FG-CNTRC beams subjected to a moving mass

This article presents the forced vibration of composite beams reinforced by single-walled carbon nanotubes (SWCNTs) and subjected to a moving mass. Considering the distribution of carbon nanotubes such as uniform (UD-CNT), functionally graded Λ (FGΛ-CNT) and X (FGX-CNT), three different beams are studied. Based on a third-order shear deformation theory (TSDT), the motion equations of the beams are derived using Hamilton's principle. Including mass interaction forces, the motion equations are transformed into a finite element equation in which a two-node beam element with eight degrees of freedom is utilized. To improve the efficiency of the beam element, the transverse shear rotation is employed as an independent variable in the derivation of the beam element. The vibration characteristics, including the dynamic magnification factors and the time histories for mid-span deflections are computed by using the Newmark method. Numerical result reveal that the vibration of the beams is clearly influenced of the CNT reinforcement, and the dynamic magnification is significantly decreased by increasing the CNT volume fraction. It is also shown that the FGX-CNT beam is the best in dynamic resistance in terms of the lowest dynamic deflection and dynamic magnification factors. The effects of the total volume fraction and the moving load velocity on the dynamic behaviour of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) beams are examined in detail and highlighted.

Dynamic Response of Slope Inertia-Based Timoshenko Beam under a Moving Load

In this paper, the dynamic response of a simply supported beam subjected to a moving load is reinvestigated. Based on a new beam theory, slope inertia-based Timoshenko (SIBT), the governing equations of motion of the beam are derived. An analytical solution is presented by using a coupled Fourier and Laplace–Carson integral transformation method. The finite element solution is also developed and compared with the analytical solution. Then, a comparative study of three beam models based on the SIBT, Euler–Bernoulli and Timoshenko, subjected to a moving load, is presented. The results show that for slender beams, the dynamic responses calculated by the three theories have marginal differences. However, as the ratio of the cross-sectional size to beam length increases, the dynamic magnification factors for the mid-span displacement obtained by the SIBT and Timoshenko beams become larger than those obtained by the Euler–Bernoulli beams. Furthermore, until the ratio is greater than 1/3, the difference between the calculated results of the SIBT and Timoshenko beams becomes apparent.

Numerical Investigation of the Dynamic Responses of Fibre-Reinforced Polymer Composite Bridge Beam Subjected to Moving Vehicle.

In modern industry, heavy traditional materials are being substituted with light and strong fibre-reinforced polymer composite materials. Bridges and railroads made of composite laminates are considerably affected by traffic loads. Therefore, it is very important to analyse this effect which would find practical applications in engineering designs. This paper explains the theoretical formulation that governs the dynamic response of a composite beam subjected to a moving load. The governing equations for the dynamic effect on the laminated composite bridge beam are explained here. The main theories in the micro–macro modelling of composite laminates are also described in the paper. Within the macro modelling, the Classical Laminate and Shear Deformation Laminate Theory of beams are presented. The symmetric cross-ply laminated bridge, made of boron/epoxy is under consideration. The computational two-dimensional model of the vehicle is adopted. The governing equations for the dynamic effect on the laminated composite bridge beam are explained. The calculation of the time response of the bridge for the characteristic speeds of the vehicle is performed in the environment of the MATLAB software. The maximum dynamic magnification factor for the dynamic analysis of a composite beam is found.

Experimental study of wave impact on a vertical wall with overhanging horizontal cantilever slab and structural response analysis

Two-dimensional model test of a vertical wall with overhanging horizontal cantilever slab has been carried out with the scale factor of 1:30. Different influences of wave condition and structural geometry on wave load are investigated. Wave pressure of cantilever slab highly depends on structural clearance and reaches the peak when clearance close to 0.2 times wave height. Larger wave steepness is responsible for larger wave pressure for vertical wall. With the increasing length of slab, wave forces are reduced on slab while enlarged on vertical wall. Besides, a negative correlation between rise frequency and rise time is observed and corresponding empirical formula has been proposed. Through finite element analysis, the structural response of a simple plate under wave load has been studied. In general, the cut-off frequency has a dramatic effect on the dynamic magnification factor, where the actual dynamic response keeps stable when cut-off frequency larger than 4 times structural natural frequency. Finally, it is adoptable to determine the adequacy of the structure under wave load by considering the quasi-static wave load and corresponding impact factor from 2.0 to 6.0.

Experimental investigation of geogrid-reinforced sand cushions for rock sheds against rockfall impact

Sand cushions for rock sheds have been used to protect roads and railways from rockfall hazard. However, thick sand cushions are often needed, which lead to high construction cost. This paper investigates the use of geogrid-reinforced sand cushions to improve impact resistance as well as reduce required thickness of the sand cushion. Impact tests were conducted in the laboratory to investigate the influence of position and layer number of geogrids on the impact responses of the sand cushions under different impact energy levels. The experimental results indicate that the geogrid placed at an appropriate position in the sand cushion could allow the penetration of an impact object, distribute the impact force, reduce the distributed vertical stress and the vibration of the slab, and increase the impact duration. The test results in this study show that placing the geogrid at one-third height of the sand cushion resulted in the optimum impact resistance. In addition, the upper sand layer within the geogrid-reinforced sand cushion should have a sufficient thickness to provide good energy dissipation ability. Furthermore, the sand cushion reinforced with multiple geogrid layers should be used carefully especially when a thin cushion is used to resist high impact energy rockfall. Moreover, dynamic magnification factors are proposed to correct the vertical stresses calculated by static analysis.

Seismic response analysis of loess site under far-field bedrock ground motion of the Wenchuan earthquake.

In this paper, considering the far-field seismic input, an accelerogram recorded in the bedrock at Wuquan Mountain in Lanzhou city during the 2008 Wenchuan Ms8.0 earthquake was selected, and numerical dynamic analyses were conducted. The one-dimensional equivalent linear method was implemented to estimate the ground motion effects in the loess regions. Thereafter, slope topographic effects on ground motion were studied by applying the dynamic finite-element method. The results revealed the relationship between the PGA amplification coefficients and the soil layer thickness, which confirmed that the dynamic response of the sites had obvious nonlinear characteristics. The results also showed that there was an obvious difference in the dynamic magnification factor between the short-period and long-period structures. Moreover, it was found that the amplification coefficient of the observation point at the free surface was greater than the point inside the soil at the same depth, which mainly occurred in the upper slope. Through this study, the quantitative assessment of ground motion effects in loess regions can be approximately estimated, and the amplification mechanism of the far-field ground motion mechanism can be further explained. In addition to the refraction and reflection theory of seismic waves, the resonance phenomenon may help explain the slope topographic effect through spectrum analysis.

Dynamic behaviour of bidirectional functionally graded sandwich beams under a moving mass with partial foundation supporting effect

Dynamic analysis of bidirectional functionally graded sandwich beams under a moving mass with the effect of partial support by a Pasternak foundation is presented on the basis of a quasi-3D theory. The face layers of the sandwich beams are made of bidirectional functionally graded material (FGM), while the core is axially FGM. The material properties of the skin layers are varied smoothly in both the axial and transverse directions by power gradation laws, and they are evaluated by both Voigt and Mori–Tanaka micromechanical models. A finite element formulation is derived and employed to construct the equation of motion of the beams. Dynamic characteristics, including the dynamic deflections, dynamic magnification factors and stress distribution, are computed with the aid of the Newmark method. The numerical results reveal that the ratio of the foundation supporting part to the total beam length plays an important role in the dynamic response of the beams. The influence of the micromechanical model on the dynamic response of the beams is found to be dependent on the foundation stiffness and the power-law indexes. The effects of the material gradation, the foundation and loading parameters on the dynamic behaviour of the beams are examined in detail and highlighted.

Numerical Investigation on the Transverse Vibration of Prestressed Large-Span Beams with Unbonded Internal Straight Tendon.

This paper deals with the effect of the prestress load on the free and forced dynamic behavior and vertical vibration of the prestressed beams. The analysis applies both the analytical frequency equation and the finite element method (FEM) using ABAQUS software to predict the fundamental natural frequency (FNF) of the simply supported unbonded prestressed beams. The energy method has been employed to derive the effective prestressing load to determine the eccentricity effect. In regard to the forced response of the prestressed beam, a moving point load with a constant value and various velocities and excitation frequencies is applied. Extensive parametric studies are carried out taking into account different factors including prestress load, eccentricity, concrete ratio, span-to-depth ratio, velocity, and frequency of the moving load. The comparison of the FNFs obtained by the formula with those obtained from FEM models indicates that the results are in a good agreement. This convergence demonstrates that the proposed formulation can predict the FNF of the eccentrically prestressed beams with high reliability. The time-histories curves for midspan displacement of the unbonded prestressed beams and the dynamic magnification factors are also evaluated. The results illustrate that the aforementioned factors have an indispensable contribution to the beam dynamic behavior.

Experimental study on an overhead crane passing a rail track joint

Purpose. To confirm the theoretical conclusions that determining the load from the wheel impact with the joint of rails in the metal structures of overhead cranes used in the technological processes at ore mining and processing mills and integrated works and moving along the rail tracks requires that the dynamic magnification factor should take into account the trolley position. Methodology. To assess the load to which the crane metal structures are exposed the electric strain measurement method using direct bridge circuit was chosen. To calibrate the strain measurement system, the direct method was used, under which calibration is performed directly on the structure on which experimental studies will be carried out in the future. Findings. It was established that the dynamic magnification factor in the metal structure of the overhead crane had the following values when passing the last rail joints: 1.54 (with the trolley in the middle of the bridge), 2.46 (with the trolley at 0.25 of the crane span from the end beam), 3.33 (with the trolley in the extreme position). So, with the trolley being at a distance of 0.25 of the crane span from the end beam, the dynamic magnification factor is 74%, and with the trolley in the middle of the bridge, it is 46% of the dynamic magnification factor if the trolley is in its extreme position. Originality. The scientific novelty consists in the first experimental confirmation of the results obtained in the theoretical studies on the overhead crane passing the joints of the rail track with regard to the crane trolley position, which leads to changes in the stiffness of the main beam in the interval between the trolley and the final beam. Practical value. The results obtained enable calculations on the crane bridge metal structure during the design and repair of the main and end beams taking into account the value of the dynamic magnification factor, which allows increasing the reliability and durability of the crane metal structure as a whole.

Numerical simulation of transient processes in propeller shaft straining under ice loads

Object and purpose of research. The object of the research is the shaftline of an icebreaker and ice-going vessels, the purpose is to develop an algorithm for determining the loads for calculating the strength of the shafts of icebreakers and icegoing vessels under ice loads in a nonlinear dynamic setting and determining the dynamic magnification factor.
 Materials and methods. The study is based on finite element method (FEM).
 Main results. As a result of numerical analysis, the magnification factors of ice loads acting on the propeller shaft when vessel moves in an ice field up to 4 m thick are investigated.
 Conclusion. The research results can be used to calculate the fatigue strength of the icebreakers and ice-going vessels shaftlines.

Design and performance evaluation of inerter-based tuned mass dampers for a ground acceleration excited structure

This paper investigates the design and performance of three configurations of inerter–based tuned mass dampers (TMDs) for the control of undamped single-degree-of-freedom structures subjected to ground acceleration. Inerter-based TMDs utilize the mass effects provided by an inerter in combination with spring and damping elements in otherwise conventional TMDs. The three configurations of inerter-based TMDs investigated have been developed by replacing the dashpot in traditional TMDs with different configurations connecting together a spring, a damper and an inerter. H2 and H-infinity optimum design values are presented through an exact optimization and a numerical optimization, respectively. Each device is also optimally designed considering 44 individual earthquake records. Utilizing individual, average, H2, and H-infinity designs, the performance evaluation of the devices is presented. Similar performance for the three inerter-based TMDs is observed. Compared to the TMD, the inerter-based TMDs can provide a 7% reduction in RMS response to a white noise ground motion, a 25% reduction in the maximum dynamic magnification factor, and an average 7% reduction in the RMS of the primary structure's displacement subjected to earthquake records.