Moving Loads on an Elastic Half-Plane With Hysteretic Damping

2001 ◽  
Vol 68 (6) ◽  
pp. 915-922 ◽  
Author(s):  
A. Verruijt ◽  
C. Cornejo Co´rdova
Keyword(s):  
Wave Velocity ◽  
Half Plane ◽  
High Speed ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
Moving Loads ◽  
Hysteretic Damping ◽  
Loading And Unloading ◽  
Dynamic Amplification

A closed-form solution is presented for the problem of a moving point load on an elastic half-plane with hysteretic damping. The problem has been studied in order to investigate the dynamic amplification of stresses and displacements if the velocity of the load approaches the Rayleigh wave velocity or the shear wave velocity in the elastic medium. This is relevant for the construction of high speed railway lines on relatively soft soils. Hysteretic damping is introduced as pseudo-viscous damping, assuming that the damping in a full cycle of loading and unloading is independent of the frequency.

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

2020 ◽  
Vol 223 (2) ◽  
pp. 1446-1459
Author(s):  
Xi Feng ◽  
Haiming Zhang
Keyword(s):  
Point Source ◽  
Closed Form ◽  
High Speed ◽  
Moving Load ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
High Speed Rail ◽  
Lamb’S Problem ◽  
Homogeneous Half Space

SUMMARY In 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.

Diffraction by a half-plane perpendicular to the distinguished axis of a general gyrotropic medium

1977 ◽  
Vol 55 (4) ◽  
pp. 305-324 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Diffraction Problem ◽  
Plane Waves ◽  
Closed Form Solution ◽  
Basic Feature ◽  
Form Solution ◽  
Electromagnetic Plane Wave ◽  
Edge Diffraction ◽  
Special Cases

An exact, closed-form solution is found for the following half-plane diffraction problem: (I) The medium surrounding the half-plane is both electrically and magnetically gyrotropic. (II) The scattering half-plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The incident electromagnetic plane wave propagates in a direction normal to the edge of the half-plane.The formulation of the problem leads to a coupled pair of Wiener–Hopf equations. These had previously been thought insoluble by quadratures, but yield to a newly discovered technique : the Wiener–Hopf–Hilbert method. A basic feature of the problem is its two-mode character i.e. plane waves of both modes are necessary for the spectral representation of the solution. The coupling of these modes is purely due to edge diffraction, there being no reflection coupling. The solution obtained is simple in that the Fourier transforms of the field components are just algebraic functions. Properties of the solution are investigated in some special cases.

Scattering by hard and soft parallel half-planes

1981 ◽  
Vol 59 (12) ◽  
pp. 1879-1885 ◽  
Author(s):  
R. A. Hurd ◽  
E. Lüneburg
Keyword(s):  
Plane Wave ◽  
Closed Form ◽  
Half Plane ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Normal Derivative ◽  
Form Solution ◽  
The Other ◽  
Total Field

We consider the diffraction of a scalar plane wave by two parallel half-planes. On one half-plane the total field vanishes whilst on the other its normal derivative is zero. This is a new canonical diffraction problem and we give an exact closed-form solution to it. The problem has applications to the design of acoustic barriers.

Determination of site period for NZS1170.5:2004

2014 ◽  
Vol 47 (1) ◽  
pp. 28-40 ◽  
Author(s):  
Tam Larkin ◽  
Chris Van Houtte
Keyword(s):  
Shear Wave ◽  
Wave Velocity ◽  
Shear Wave Velocity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Site Classification ◽  
Lumped Mass ◽  
Mass System ◽  
Site Classes ◽  
Site Period

The fundamental site period, T, is a key parameter for site classification in NZS 1170.5:2004. Many sites in New Zealand will fall into site classes C and D, where the boundary between the site classes is T = 0.6 seconds. NZS 1170.5 offers several methods of determining site classification. The intent of this paper is to expand on NZS 1170.5 and guide practising engineers towards more accurate and efficient methods for determining site period. We review methods to calculate the shear-wave velocity, then give specific examples for calculating the site period for five types of soil profile (uniform layer, shear-wave velocity increasing as a power of depth, shear modulus increasing linearly with depth, two-layer profile and three-layer profile). We find that NZS 1170.5 clause 3.1.3.7 for calculating site period at layered sites is unconservative and inconsistent with two other well-accepted methods for calculating site period. We consider the most accurate and efficient method of calculating site period for layered sites is to represent the profile as a lumped mass system, then calculate the fundamental frequency from the eigenvalues of the system. The successive application of the two-layer closed form solution is also considered an acceptable method.

Generalized and Simplified Linear Closed-Form Solution of an Active Tensegrity Unit in the Form of Octahedral Cell

2014 ◽  
Vol 969 ◽  
pp. 192-198
Author(s):  
Stanislav Kmeť ◽  
Peter Platko
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
Cable System ◽  
Form Analysis ◽  
Tensile Forces

Results of the generalized and simplified linear closed form solution of an active or adaptive tensegrity unit, as well as its numerical analysis using finite element method are presented in the paper. The shape of the unit is an octahedral cell with a square base and it is formed by thirteen members (four bottom and four top cables, four edge struts and one central strut). The central strut is designed as an actuator that allows for an adjustment of the shape of the unit which leads to changes of tensile forces in the cables. Due to the diagonal symmetry of the 3D tensegrity unit the closed-form analysis is based on the 2D solution of the equivalent planar biconvex cable system with one central strut under a vertical point load.

The Sliding With Coulomb Friction of a Rigid Indentor Over a Power-Law Inhomogeneous Linearly Viscoelastic Half-Plane

1984 ◽  
Vol 51 (2) ◽  
pp. 289-293 ◽  
Author(s):  
J. R. Walton
Keyword(s):  
Shear Modulus ◽  
Half Plane ◽  
Power Law ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Plane Boundary ◽  
Physical Parameters ◽  
Title Problem ◽  
Rigid Indentor

In a previous paper, the title problem was solved for a homogeneous power-law linearly viscoelastic half-plane. Such material has a constant Poisson’s ratio and a shear modulus with a power-law dependence on time. In this paper, the shear modulus is assumed also to have a power-law dependence on depth from the half-plane boundary. As in the earlier paper, only a quasi-static analysis is presented, that is, the enertial terms in the equations of motion are not retained and the indentor is assumed to slide with constant speed. The resulting boundary value problem is reduced to a generalized Abel integral equation. A simple closed-form solution is obtained from which all relevant physical parameters are easily computed.

Moving Load on a Fluid-Solid Interface: Supersonic Regime

1973 ◽  
Vol 40 (1) ◽  
pp. 137-142 ◽  
Author(s):  
T. C. Kennedy ◽  
G. Herrmann
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Moving Load ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
Steady State Response ◽  
Entire Space ◽  
State Response ◽  
Supersonic Regime

The steady-state response of a semi-infinite solid with an overlying semi-infinite fluid subjected at the plane interface to a moving point load is determined for supersonic load velocities. The exact, closed-form solution valid for the entire space is presented. Some numerical results for the displacements at the interface are calculated and compared to the results obtained when no fluid is present.

Nonhertzian Contact: Geometric and Computational Issues

2005 ◽  
Author(s):  
Leon M. Keer
Keyword(s):  
High Speed ◽  
Surface Characterization ◽  
Closed Form Solution ◽  
Form Solution ◽  
Computer Hardware ◽  
Future Research ◽  
Inelastic Behavior ◽  
Materials Development ◽  
Technological Advances ◽  
Computational Speed

This talk will focus on some of the consequences that arise due to improvements that were made in such areas as surface characterization, materials development, power requirements and other technological advances, but primarily advances in computer hardware and software. These have led analysis from the arena of a closed-form solution of a single smooth contact (Hertz theory) to the semi-analytical analysis of rough contact, in which there may occur a large number of contacts due to the presence of many asperities. This paper will limit itself to the following aspects involved with contact that is not Hertzian: geometrical issues, friction, and finally, purely numerical issues that may relate to inelastic behavior. To illustrate how the development of high-speed computers enabled the solution of relatively complex problems, two specific examples are given. The first is the case of rough contact, which is solved by a combination of fast Fourier transform and some computer-enhanced methods. The second example is indentation of an inelastic body and calculation of residual stresses. Future research in contact mechanics will involve calculations at even smaller length scales that depend upon the ever-increasing computational speed and development of sensors to investigate materials at these scales.

Resonance and Cancellation in Torsional Vibration of Monosymmetric I-Sections Under Moving Loads

2018 ◽  
Vol 18 (09) ◽  
pp. 1850111 ◽  
Author(s):  
Y. B. Yang ◽  
Mei Li ◽  
Bin Zhang ◽  
Yuntian Wu ◽  
Judy P. Yang
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Moving Loads ◽  
Coupled Vibration ◽  
Wheel Load ◽  
Severe Condition ◽  
Resonance Response ◽  
Variation Of Constants ◽  
Eccentric Wheel ◽  
Sine Functions

This paper is concerned with the lateral and torsional coupled vibration of monosymmetric I-beams under moving loads. To this end, a train is modeled as two subsystems of eccentric wheel loads of constant intervals to account for the front and rear wheels. By assuming the lateral and torsional displacements to be restrained at the two ends of the beam, both the lateral and torsional displacements are approximated by a series of sine functions. The method of variation of constants is adopted to derive the closed-form solution. For the most severe condition when the last wheel load is acting on the beam, both the conditions of resonance and cancellation are identified. Once the condition of cancellation is enforced, the resonance response can always be suppressed, which represents the optimal design for the beam. Since the condition for suppressing the torsional resonance is exactly the same as that for the vertical resonance, this offers a great advantage in the design of monosymmetric I-beams, as no distinction needs to be made between the suppression of vertical or torsional resonance.

Closed-Form Solution of the Frictional Sliding Contact Problem for an Orthotropic Elastic Half-Plane Indented by a Wedge-Shaped Punch

2014 ◽  
Vol 618 ◽  
pp. 203-225 ◽  
Author(s):  
Aysegul Kucuksucu ◽  
Mehmet A. Guler ◽  
Ahmet Avci
Keyword(s):  
Contact Problem ◽  
Half Plane ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Contact Problems ◽  
Form Solution ◽  
Orthotropic Materials ◽  
Principal Axes ◽  
The Coefficient Of Friction ◽  
Processing Techniques

In this paper, the frictional contact problem of a homogeneous orthotropic material in contact with a wedge-shaped punch is considered. Materials can behave anisotropically depending on the nature of the processing techniques; hence it is necessary to develop an efficient method to solve the contact problems for orthotropic materials. The aim of this work is to develop a solution method for the contact mechanics problems arising from a rigid wedge-shaped punch sliding over a homogeneous orthotropic half-plane. In the formulation of the plane contact problem, it is assumed that the principal axes of orthotropy are parallel and perpendicular to the contact. Four independent engineering constants , , , are replaced by a stiffness parameter, , a stiffness ratio, a shear parameter, , and an effective Poisson’s ratio, . The corresponding mixed boundary problem is reduced to a singular integral equation using Fourier transform and solved analytically. In the parametric analysis, the effects of the material orthotropy parameters and the coefficient of friction on the contact stress distributions are investigated.

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