Stability of columns with original defects under periodic transient loadings

To study the influence of original defects on the dynamic stability of the columns under periodic transient loadings,the approximate solution method and the Fourier method of the stable periodic solution are adopted while considering the influence of original defects on columns.The dynamic stability of the columns under periodic transient loadings is analyzed theoretically.Through the study of different deflections,the dynamic instability of the columns is obtained by Maple software.The results of theoretical analysis show that the larger the original defects,the greater the unstable area,the stable solution amplitude of columns and the risk of instability caused by parametric resonance will be.The damping of columns is a vital factor in reducing dynamic instability at the same original defects.On the basis of the Mathieu-Hill equation,the relationship between the original defects and deflection is deduced,and the dynamic instability region of the columns under different original defects is obtained.Therefore,reducing the original defects of columns can further enhance the dynamic stability of the compressed columns in practical engineering.

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.

Analytical solution of ground-borne vibration due to a spatially periodic harmonic moving load in a tunnel embedded in layered soil

In this study,we propose a novel coupled periodic tunnel–soil analytical model for predicting ground-borne vibrations caused by vibration sources in tunnels.The problem of a multilayered soil overlying a semi-infinite half-space was solved using the transfer matrix method.To account for the interactions between the soil layer and tunnel structure,the transformation characteristics between cylindrical waves and plane waves were considered and used to convert the corresponding wave potentials into forms in terms of the Cartesian or cylindrical coordinate system.The induced ground-borne vibration was obtained analytically by applying a spatially periodic harmonic moving load to the tunnel invert.The accuracy and efficiency of the proposed model were verified by comparing the results under a moving constant and harmonic load with those from previous studies.Subsequently,the response characteristics under a spatially periodic harmonic moving load were identified,and the effects of a wide range of factors on the responses were systematically investigated.The numerical results showed that moving and Doppler effects can be caused by a spatially periodic harmonic moving load.The critical frequency and frequency bandwidth of the response are affected by the load type,frequency,velocity,and wavenumber in one periodicity length.Increasing the tunnel depth is an efficient way to reduce ground-borne vibrations.The effect of vibration amplification on the free surface should be considered to avoid excessive vibration levels that disturb residents.

Mass transport in a thin layer of power-law fluid in an Eulerian coordinate system

The mass transport velocity in a thin layer of muddy fluid is studied theoretically. The mud motion is driven by a periodic pressure load on the free surface, and the mud is described by a power-law model. Based on the key assumptions of the shallowness and the small deformation, a perturbation analysis is conducted up to the second order to find the mean Eulerian velocity in an Eulerian coordinate system. The numerical iteration method is adopted to solve these non-linear equations of the leading order. From the numerical results, both the first-order flow fields and the second-order mass transport velocities are examined. The verifications are made by comparing the numerical results with experimental results in the literature, and a good agreement is confirmed.