Moving Point Load Research Articles

Analyzing dynamic response of nonlocal strain gradient porous beams under moving load and thermal environment

This research presents dynamic response analysis of a porous functionally graded (FG) nanobeam under a moving point load. The nanobeam formulation has been established with the use of a higher-order refined beam model and nonlocal strain gradient theory (NSGT) including two scale factors named nonlocal and strain gradient factors. The porous FG material has been modeled via modified power-law functions which contain porosity volume according to even or uneven porosity dispersions. Moreover, graded nonlocality has been considered in order to provide a better modeling of size effects for FG nano-size structures. The governing equations of the nanobeam have been discretized with the use of differential quadrature method (DQM) and inverse Laplace transform approach has been utilized to calculate the dynamic deflections. The main findings of the present research indicate the influences of nonlocal strain gradient factors, moving load speed, pore amount, porosity distribution and elastic medium on dynamic deflection of FG nanobeams.

Nonlinear vibrations of a thin isotropic circular plate subjected to moving point loads

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff’s theory and then incorporate von Kármán nonlinearity and employ Hamilton’s principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies.

An anisotropic layered poroelastic half-space subjected to a moving point load

The dynamic responses of an anisotropic layered poroelastic half-space subjected to a moving point load are studied based on the Stroh formalism and Fourier transform. The material in each layer of the half-space is anisotropic and poroelastic one. Utilizing the boundary conditions at the half-space surface (permeable or impermeable) and the continuity conditions between each layer, the three-dimensional (3D) solutions for the displacements, the pore pressure, the stresses and the fluid fluxes are obtained. Numerical examples show the validity and elegance of the present solution. For some special cases, our solutions agree with the known results. For the anisotropic case, the 3D dynamic Green's functions for the multi-layered poroelastic half-space show clearly the effect of material properties on the displacement and stress distributions.

Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

Here, we formulate and study nonlinear vibrations of a thin, polar-orthotropic circular plate subjected to a circularly moving point load at a fixed radius and constant angular velocity. The governing equations and boundary conditions are obtained following Kirchhoff’s plate theory, incorporating von Kármán nonlinearity, and employing extended Hamilton’s principle. The external damping is introduced via Rayleigh dissipation function. The governing equations are solved using mode summation procedure. The mode shapes and the natural frequencies of the polar-orthotropic circular plate are found using Frobenius series method. Mode summation procedure results in coupled nonlinear ordinary differential equations. These equations are solved using the Runge–Kutta method for time response and method of harmonic balance with the arc continuation method for frequency response. The spectrum of the undamped linear vibration response of isotropic and polar-orthotropic plates exhibits natural frequencies of plates and angular velocity of the rotating load. The damped response contains the frequency of the angular velocity of the rotating load only. The nonlinear transverse vibrations of the undamped and damped plate due to rotating point load reveal frequency rich spectrums. The frequency response function shows strong modal interactions.

Exact closed-form solutions for Lamb’s problem—II: a moving point load

SUMMARYIn this paper, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb’s problem. Starting with the integral solutions of Bakker et al., we followed the method developed by Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker et al., which confirms the correctness of our formulae. The solution obtained in this paper may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.

Dynamic Behavior of a Bidirectional Functionally Graded Sandwich Beam under Nonuniform Motion of a Moving Load

A bidirectional functionally graded Sandwich (BFGSW) beam model made from three distinct materials is proposed and its dynamic behavior due to nonuniform motion of a moving point load is investigated for the first time. The beam consists of three layers, a homogeneous core, and two functionally graded face sheets with material properties varying in both the thickness and longitudinal directions by power gradation laws. Based on the first-order shear deformation beam theory, a finite beam element is derived and employed in computing dynamic response of the beam. The element which used the shear correction factor is simple with the stiffness and mass matrices evaluated analytically. The numerical result reveals that the material distribution plays an important role in the dynamic response of the beam, and the beam can be designed to meet the desired dynamic magnification factor by appropriately choosing the material grading indexes. A parametric study is carried out to highlight the effects of the material distribution, the beam layer thickness and aspect ratios, and the moving load speed on the dynamic characteristics. The influence of acceleration and deceleration of the moving load on the dynamic behavior of the beam is also examined and highlighted.

Dynamic response of plates resting on a fractional viscoelastic foundation and subjected to a moving load

The dynamic behavior of the thin plates resting on a fractionally damped viscoelastic foundation subjected to a moving point load is investigated. This work explores the application of fractional calculus in the modeling of a viscoelastic foundation. A parametric study is executed to determine the impacts of the fractional-order derivative, foundation parameters, velocity and acceleration of the moving load on the dynamic behavior of the plate. The results show that the damping of the foundation system increases with increasing the order of the fractional derivative, which leads to a decrease in the dynamic response.

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

The dynamic behaviour of an Euler–Bernoulli beam resting on the fractionally damped viscoelastic foundation subjected to a moving point load is investigated. The fractional-order derivative-based Kelvin–Voigt model describes the rheological properties of the viscoelastic foundation. The Riemann–Liouville fractional derivative model is applied for a fractional derivative order. The modal superposition method and Triangular strip matrix approach are applied to solve the fractional differential equation of motion. The dependence of the modal convergence on the system parameters is studied. The influences of (a) the fractional order of derivative, (b) the speed of the moving point load and (c) the foundation parameters on the dynamic response of the system are studied and conclusions are drawn. The damping of the beam-foundation system increases with increasing the order of derivative, leading to a decrease in the dynamic amplification factor. The results are compared with those using the classical integer-order derivative-based foundation model. The classical foundation model over-predicts the damping and under-predicts the dynamic deflections and stresses. The results of the classical (integer-order) foundation model are verified with literature.

An accurate differential quadrature procedure for the numerical solution of the moving load problem

In moving load-type problems, the moving point load is modeled mathematically by a time-dependent Dirac-delta function. A key and difficult step in solving this type of problems using a point discrete method such as the differential quadrature method is the discretization of the Dirac-delta function in a simple and accurate manner. This paper is conducted to facilitate this step and to present a new way to do this task. In this way, the Dirac-delta function is approximated by orthogonal polynomials such as the Legendre and Chebyshev polynomials. Unlike the original Dirac-delta function, which is a generalized singularity function, the resulting approximation function is a non-singular function which can be discretized simply and efficiently. The proposed procedure is applied herein to solve the moving load problem in beams and rectangular plates. Comparisons with available analytical and numerical solutions prove that the proposed approach is highly accurate and efficient.

Dynamic Response of Fractionally Damped Viscoelastic Plates Subjected to a Moving Point Load

AbstractThe dynamic response of fractionally damped viscoelastic plates subjected to a moving point load is investigated. In order to capture the viscoelastic dynamic behavior more accurately, the material is modeled using the fractionally damped Kelvin–Voigt model (rather than the integer-type viscoelastic model). The Riemann–Liouville fractional derivative of order 0 < α ≤ 1 is applied. Galerkin's method and Newton–Raphson technique are used to evaluate the natural frequencies and corresponding damping coefficients. The structure is subject to a moving point load, traveling at different speeds. The modal summation technique is applied to generate the dynamic response of the plate. The influence of the order of the fractional derivative on the free and transient vibrations is studied for different velocities of the moving load. The results are compared with those using the classical integer-type Kelvin–Voigt viscoelastic model. The results show that an increase in the order of the fractional derivative causes a significant decrease in the maximum dynamic amplification factor, especially in the “dynamic zone” of the normalized sweep time. The dynamic behavior of the plate is verified with ansys.

A second-order asymptotic model for Rayleigh waves on a linearly elastic half plane

Abstract We derive a second-order correction to an existing leading-order model for surface waves in linear elasticity. The same hyperbolic–elliptic equation form is obtained with a correction term added to the surface boundary condition. The validity of the correction term is shown by re-examining problems which the leading-order model has been applied to previously, namely a harmonic forcing, a moving point load and a periodic array of compressional resonators.

Vibration of layered saturated ground with a tunnel subjected to an underground moving load

The transmission of vibrations within multi-layered saturated ground subjected to an underground moving load in a tunnel is investigated theoretically. A two-dimensional model of a layered saturated half-space with an embedded beam is established first. The beam simulating the tunnel is located between two horizontal soil layers with a moving point load acting upon it. The half-space soil is assumed as Biot’s poroelastic medium and the transmission and reflection matrices (TRM) method is applied to consider the wave propagation in the layered soil. Combined with the surface boundary conditions and the continuity conditions between the soil layers and the beam, the governing equations of the ground-beam coupled system are solved by the Fourier transform. Finally, the dynamic response of the layered saturated ground in the time-space domain is obtained by using the fast Fourier transform (FFT) method. Three different soil cases are selected to study the influences of the soft/hard interlayer on dynamic response. The displacement as well as the velocity and acceleration of the surface vibration are analyzed. The displacement amplitude spectra at different depths are presented to find the attenuation law of the ground vibration along depth. The numerical results show that both the stiffness and the embedded depth of the interlayer have impacts on the critical velocity of the coupled system and the surface vibration. When the vibration propagates from the tunnel to ground surface, the high-frequency components decrease and the low-frequency components increase.

Identifying damage in a bridge by analysing rotation response to a moving load

This article proposes a bridge damage detection method using direct rotation measurements. Initially, numerical analyses are carried out on a one-dimensional (1D) simply supported beam model loaded with a single moving point load to investigate the sensitivity of rotation as a main parameter for damage identification. As a result of this study, the difference in rotation measurements due to a single moving point load obtained for healthy and damaged states is proposed as a damage indicator. A relatively simple laboratory experiment is conducted on a 3-m long simply supported beam structure to validate the results obtained from the numerical analysis. The case of multi-axle vehicles is investigated through numerical analyses of a 1D bridge model and a theoretical basis for damage detection is presented. Finally, a sophisticated 3D dynamic finite element model of a 20-m long simply supported bridge structure is developed by an independent team of researchers and used to test the robustness of the proposed damage detection methodology in a series of blind tests. Rotations from an extensive range of damage scenarios were provided to the main team who applied their methods without prior knowledge of the extent or location of the damage. Results from the blind test simulations demonstrate that the proposed methodology provides a reasonable indication of the bridge condition for all test scenarios.

An analytical solution to investigate the dynamic impact of a moving surface load on a shallowly-buried tunnel

In order to investigate the dynamic impact of a moving surface load on a shallow-buried tunnel, an analytical model of a tunnel embedded in an elastic half-space was proposed. The half-space and the tunnel structure were modeled as visco-elastic media and the moving surface load was simplified as a moving point load on the ground surface. Based on the fundamental solution for the isotropic elastic half-space system in Cartesian and cylindrical coordinates, the dynamic response of a shallowly-buried tunnel in a half-space generated by a moving surface load was obtained. The transformations between plane wave and cylindrical wave functions were used to facilitate the application of boundary conditions at the ground surface and the tunnel interface. It was found that the vibration of the shallowly-buried tunnel increases significantly as the load moving speed increases, and reaches a maximum value at a critical load velocity. The tunnel vibration can be greatly reduced as the buried depth increases, and can satisfy the requirement of vibration specification (ISO 04866-2010) after it exceeds the critical depth. The critical depth increases exponentially with the increase of the moving speed of the surface load.

An exact solution to the problems of flexo-poro-elastic structures rested on elastic beds acted upon by moving loads

This paper aims to examine flexural vibrations of fully saturated poroelastic structureson an elastic bed subjected to moving point loads via an analytical solution. Using aflexural beam model in conjunction with the Biot’s poro-elasticity theory, the equationsof motion of the porous structure are derived. Using assumed mode method and Laplacetransform, the explicit expressions of displacement and pore pressure are obtained carefully.For a particular case, the predicted results are also compared with those of another workand a reasonably good agreement is achieved. The influences of the moving load velocity,permeability ratio, transverse stiffness of the foundation, viscosity of the pore fluid, andporosity on the maximum elasto-dynamic fields and pore pressure are conclusively discussed.The velocity pertinent to the maximum possible dynamic response is graphically determinedand the roles of influential parameters on this crucial factor are displayed. The present modelcould be easily extended to multi-layered poroelastic structures under moving loads.

Strain gradient based dynamic response analysis of heterogeneous cylindrical microshells with porosities under a moving load

Forced vibration of a porous functionally graded (FG) cylindrical microshell due to a moving point load with constant velocity is studied for the first time. Through the thickness of microshell, there are even-type or uneven-type porosities. Therefore, material properties of the microshell become porosity-dependent and are described via modified power-law function. For micro-scale shells, small size effects due to non-uniform strain field can be considered via strain gradient theory (SGT). At first, the governing equations of the microshell are converted to new equations in Laplace domain. Then, time response of the microshell will be obtained implementing inverse Laplace transform technique. It will be demonstrated that forced vibration characteristics of a FG microshell rely on the velocity of moving load, strain gradients, porosity percentage, material/porosity distribution and its geometry.

Indian railway track analysis for displacement and vibration pattern estimation

This paper presents the dynamic response of the Indian Railway track. Two track models are considered for the dynamic response in terms of vertical displacement and acceleration at different wheel speeds, keeping the moving point load at constant magnitude. The rail is treated as a beam either on viscoelastic foundation or on the discrete elastic support system. The governing equation is implemented in finite element analysis using ANSYS 14.0. For the validation of result from system equation are compared with those available in published literature and the maximum deviation for displacement at the midpoint of rail is found to be within 5 %. Different wheel speed generates variation in displacement and acceleration of the rail track. The study can be viewed as the foundation for the comparison of FEA based simulation of rail track to specify its dynamic response useful to provide better safety and comfort to commuters.

The Effects of Moving Load on the Hydroelastic Response of a Very Large Floating Structure

The hydroelastic response of a very large floating structure in regular waves suffering an external moving point load is considered. The linearized velocity potential theory is adopted to describe the fluid flow. To take into account the coupled effects of the structure deformation and fluid motion, the structure is divided into multiple segments and connected by an elastic beam. Then through adding a stiffness matrix arising from the elastic beam into the multiple bodies coupled motion equations, the hydroelastic response is considered. By applying the Fourier transform to the obtained frequency domain coefficients, the motion equation is transformed into the time domain and the external point load is further considered. The accuracy and effectiveness of the proposed method are verified through the comparison with experimental results. Finally, extensive results are provided, and the effects of the moving point load on the hydroelastic response of the very large floating structure are investigated in detail.

A time domain discrete-module-beam-bending-based hydroelasticity method for the transient response of very large floating structures under unsteady external loads

A time domain method is developed for estimating the transient response of very large floating structures under unsteady external loads. The hybrid frequency-time domain approach based on Cummins equations is adopted. First, the discrete-module-beam-bending-based hydroelasticity method in which a flexible structure is discretised into several rigid submodules connected by beam elements is used to establish the equations of motion for a flexible structure in frequency domain. The equations of motion in frequency domain are transformed into time domain following the idea of Cummins equations. The unsteady external loads at any point of the flexible structure are transferred to the centre of gravity of each submodule using the hydrostatic analysis of a beam in calm water. Good agreements are obtained between the numerical and experimental data for a weight drop test and a moving point load test of a continuous flexible structure in calm water, which validates the present time domain method. Finally, the time domain method is used for simulating the transient response of an interconnected flexible structure under the combination of wave and unsteady external loads, which highlights the significant effects of moving point loads on both the vertical displacement and bending moment of the structure.

Vibration suspension of Euler-Bernoulli-von Kármán beam subjected to oppositely moving loads by optimizing the placements of visco-elastic dampers

AbstractPossibilities of control and suspension of bending vibrations of a geometrically nonlinear Euler‐Bernoulli beam subjected to two oppositely moving point loads in given finite time are considered. It is assumed that the beam undergoes large deformations and the nonlinear von Kármán strains are considered. The suspension is carried out by means of optimizing the placements of visco‐elastic dampers under the beam. By applying the modified Bubnov‐Galerkin procedure, it becomes possible to avoid linearization of the state equation. The validation of the theory is carried out on the example of a finite simply supported beam. It is observed that by optimizing the dampers placements, both the maximal absolute value of the beam transverse displacements and the vibration reduction time can be reduced.