Seismic pressures on rigid cantilever walls retaining elastic continuously non-homogeneous soil: An exact solution

The dynamic response of an elastic continuously nonhomogeneous soil layer over bedrock retained by a pair of rigid cantilever walls to a horizontal seismic motion and the associated seismic pressure acting on these walls are determined analytically–numerically. The soil non-homogeneity is described by a shear modulus increasing nonlinearly with depth. The problem is solved in the frequency domain under conditions of plane strain and its exact solution is obtained analytically. This is accomplished with the aid of Fourier series along the horizontal direction and solution of the resulting system of two ordinary differential equations with variable coefficients by the method of Frobenius in power series. Due to the complexity of the various analytical expressions, the final results are determined numerically. These results include seismic pressures, resultant horizontal forces and bending moments acting on the walls. The solution of the problem involving a single retaining wall can be obtained as a special case by assuming the distance between the two walls to be very large. Results are presented in terms of numerical values and graphs using suitable dimensionless quantities. The effect of soil non-homogeneity on the system response is assessed through comparisons for typical sets of the parameters involved.     

Twisting of a non-homogeneous sphere

Summary Effect of non-homogeneity of shear modulus on the stresses in a twisted sphere by surface couple has been exhibited in this paper.     

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract

Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 10⁵, Prandtl number P = 1, and Rayleigh number R between 1 |MX 10⁴ and 4 |MX 10⁴ (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane.

The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 10⁴, which straddles the critical wave number m = 14 predicted by linear theory. These are modes which peak near the equator. The spectrum shows a second strong peak at m = 0, which represents the differential rotation driven by the peak convective modes. As R is increased, the amplitude of low wave numbers increases relative to high wave numbers as convection fills in in high and middle latitudes, and as the longitudinal scale of equatorial convection grows. By R = 3 |MX 10⁴, m = 8 is the peak convective mode. There is a clear minimum in the total kinetic energy at middle latitudes relative to low and high, well into the nonlinear regime, representing the continued dominance of equatorial and polar modes found in the linear case. The kinetic energy spectrum for m > 0 is maintained primarily by buoyancy work in each mode, but with substantial nonlinear transfer of kinetic energy from the peak modes to both lower and higher wave numbers.

For R = 1 to 2 |MX 10⁴, the differential rotation takes the form of an equatorial acceleration, with angular velocity generally decreasing with latitude away from the equator (as on the sun) and decreasing inwards. By R = 4 |MX 10⁴, this equatorial profile has completely reversed, with angular velocity increasing with depth and latitude. Also, a polar vortex which has positive rotation relative to the reference frame (no evidence of which has been seen on the sun) builds up as soon as polar modes become important. Meridional circulation is quite weak relative to differential rotation at R = 10⁴, but grows relative to it as R is increased. This circulation takes the farm of a single cell of large latitudinal extent in equatorial regions, with upward flow near the equator, together with a series of narrower cells in high latitudes. It is maintained primarily by axisymmetric buoyancy forces. The differential rotation is maintained at all R primarily by Reynolds stresses, rather than meridional circulation. Angular momentum transport toward the equator for R = 1–2 |MX 10⁴ maintains the equatorial acceleration while radially inward transport maintains the opposite profile at R = 4 |MX 10⁴.

The total heat flux out the top of the convective shell always shows two peaks for the range of R studied, one at the equator and the other near the poles (no significant variation with latitude is seen on the sun), while heat flux in at the bottom shows only a polar peak at large R. The meridional circulation and convective cells transport heat toward the equator to maintain this difference.

The helicity of the convection plus the differential rotation produced by it suggest the system may be capable of driving a field reversing dynamo, but the toroidal field may migrate with lime in each cycle toward the poles and equator, rather than just toward the equator as apparently occurs on the sun.

We finally outline additions to the physics of the model to make it more realistic for solar application.     

Calculation of free field response spectrum of a non-homogeneous soil deposit from bed rock response spectrum

In this paper, a method to determine free field response spectrum of a soil deposit from a specified bed rock response spectrum is presented. This method treats the earthquake motion as if it was a stationary Gaussian random process but to account for the non-stationary character, an approximate method is used. The soil deposit is assumed to have mechanical properties (strength, shear modulus, etc.) increasing with some power exponent of depth. This layer overlies either a compliant elastic half space or a layer with shear modulus increasing linearly with depth.To demonstrate the validity and usefulness of this approach, two examples are presented. The first one consists in the transfer, from bed rock to free surface of a given soil profile, of the response spectrum derived from an accelerogram used to generate the Eurocode 8 response spectrum. The obtained free field response spectrum is compared to the one obtained by using a wave propagation program. The second example consists in the validation of this method with experimental records.     

不均匀场地土液化引起的地下管道上浮反应研究

本文采用非线性增量有限元迭代法,对不均匀场地土液化引起的地下管道的上浮反应进行了研究,考虑了液化土3种不均匀的情况,给出了一些计算结果。     

Time-harmonic response of transversely isotropic multilayered half-space in a cylindrical coordinate system

With the aid of the analytical layer-element method, a comprehensive analytical derivation of the response of transversely isotropic multilayered half-space subjected to time-harmonic excitations is presented in a cylindrical coordinate system. Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the Fourier expansion, Hankel and Laplace integral transform, analytical layer-elements for a finite layer and a half-space are derived. Considering the continuity conditions on adjacent layers׳ interfaces and the boundary conditions, the global stiffness matrix equations for multilayered half-space are assembled and solved. Finally, some numerical examples are given to make a comparison with the existing solution and to demonstrate the influence of parameters on the dynamic response of the medium.     

Liquefaction-induced deformation of earthen embankments on non-homogeneous soil deposits under sequential ground motions

Damage of embankments during earthquakes is widely attributed to the liquefaction of foundation soil. Previous studies have investigated the dynamic response of embankments by mainly considering uniform sand foundation and a single earthquake event. However, the foundation of an embankment consists of many sublayers of soil from liquefiable sand to relatively impermeable layer, and during earthquakes a mainshock may trigger numerous aftershocks within a short time which may have the potential to cause additional damage to soil structures. Accordingly, the investigation of liquefaction-induced deformation of earthen embankments on various liquefiable foundation conditions under mainshock–aftershock sequential ground motions is carried out by a series of dynamic centrifuge tests in this study. The liquefiable foundation includes uniform sand profile, continuous layered soil profile, and non-homogeneous soil profiles. Effects of various foundation conditions on embankment deformations are compared and analyzed. From the test results, it is found that the embankment resting on non-homogeneous soil deposits suffer more damage compared to the uniform sand foundation of same relative density. The test results also suggest that the sequential ground motions have a significant effect on the accumulated deformation of embankment.     

Vibration isolation by trenches in continuously nonhomogeneous soil by the BEM

Vibration isolation of structures from ground-transmitted waves by open trenches in isotropic, linearly elastic or viscoelastic soil with a shear modulus varying continuously with depth is numerically studied. Both an exponential and a linear shear modulus variation with depth are used in this work. Waves produced by the harmonic motion of a rigid surface machine foundation are considered. The problem is solved by the frequency domain boundary element method employing the Green's function of Kausel-Peek-Hull for a thin layered half-space. Thus only the trench perimeter and the soil-foundation interface need essentially to be discretized. The proposed methodology is first tested for accuracy by solving two Rayleigh wave propagation problems in nonhomogeneous soil with known analytical solutions and/or for which experimental results are available. Then the method is applied to vibration isolation problems and the effect of the inhomogeneity on the wave screening effectiveness of trenches is studied.     

Torsion of a non-homogeneous solid of revolution

Summary Solution has been obtained in the case of torsion of a solid of revolution of non-homogeneous material. The problem of a twisted sphere has been deduced from the general solution. The effect of non-homogeneity has been illustrated graphically.     

Centrifuge model tests on liquefaction-induced settlement and pore water migration in non-homogeneous soil deposits

This paper presents the results of dynamic centrifuge model tests conducted to investigate the liquefaction mechanism in non-homogeneous soil deposits. Four types of model tests were conducted: one model test involved a uniform soil deposit; one involved continuous layered soil deposit; and two involved discontinuous layered soil deposits. Non-homogeneity in the tests was incorporated by including periodically distributed discontinuous silty sand patches. It was found that more excess pore water pressure (EPWP) remains for a longer period of time in the discontinuous region in non-homogeneous soil deposits compared with the continuous layered and uniform soil deposits. The generation of pore water pressure ceases the supply of a new mass of water after seismic excitation; therefore the dissipation of EPWP becomes the dominant factor for settlement after seismic excitation. The rapid dissipation of EPWP through the discontinuous part in the non-homogeneous soil deposits manifests as a larger settlement in the discontinuous part, causing non-uniform settlements.     

Intensity of waves in a randomly non-homogeneous layered medium

The intensity of shear waves in the model of a multilayered stratum, the material properties of which are random functions, is considered. The solutions for displacements and stresses are obtained for one layer and then the formulation is extended to a multilayered stratum through transfer matrices. The solution for a random medium has been compared with the solution for the homogeneous medium. The analysis indicates that the stochastic inhomogeneities are likely to increase the damping in a significant way.     

Deformation and stresses in a non-homogeneous earth-model with a rigid core

Summary Small strain theory has been applied to find out the deformation and stresses in the interior of the earth considered as a self-gravitating non-homogeneous isotropic sphere of heterogeneous density distribution. Stresses have been compared at different layers due to different desity distributions.     

土-结构相互作用问题中的放大效应研究

在土-结构动力相互作用问题的研究中,当土层与结构的自振频率相接近时,可能会出现类共振,对此进行了土-框架结构相互作用的振动台试验,将试验结果与刚性地基情况进行对比,选取的指标包括加速度、层间位移及应变峰值。结果表明:考虑土-结构相互作用的影响后,加速度、层间位移和框架应变均较刚性地基时有大幅的增加,应变峰值比最大可达到7倍,加速度峰值比最大可达5倍多,而位移峰值比最大约3倍左右。     

One dimensional elastic wave propagation in a non-homogeneous conical rod

Summary A one dimensional problem concerning the wave propagation in a non-homogeneous conical rod having its cross-section directly proportional to the square of the distance from a chosen point is considered in this paper. One end of the rod is subjected to a pressure step while the other end is stress free and the pulse shape is examined when the propagation speed of the wave has a particular continuous distribution over the entire rod.     

Some problems of plane strain in a non-homogeneous isotropic cylinder

Summary The expressions for stresses and extension have been obtained in an isotropic circular cylinder in a state of plane strain, the elastic constants and the density of the material varying exponentially and the results have been compared with those in the homogeneous case.     

ASH-source in an elastic half-space with a non-homogeneous surface layer

Summary The response of an elastic half-space with a non-homogeneous surface layer due to aSH-source operating in the non-homogeneous layer of an elastic half-space have been studied. The elastic and physical constants vary exponentially in the surface layer. The surface displacement consists of several integrals. On approximate evaluation of these integrals, they are identified as direct or reflected pulses.     

Forced vibration of a thin non-homogeneous circular plate having a central hole

Summary This paper deals with the forced vibration produced in a thin non-homogeneous circular plate having a central hole when a periodic force is acting on the internal boundary.     

Ray tracing for continuously rotated local coordinates belonging to a specified anisotropy

Conventional ray tracing for arbitrarily anisotropic and heterogeneous media is expressed in terms of 21 elastic moduli belonging to a fixed, global, Cartesian coordinate system. Our principle objective is to obtain a new ray-tracing formulation, which takes advantage of the fact that the number of independent elastic moduli is often less than 21, and that the anisotropy thus has a simpler nature locally, as is the case for transversely isotropic and orthorhombic media. We have expressed material properties and ray-tracing quantities (e.g., ray-velocity and slowness vectors) in a local anisotropy coordinate system with axes changing directions continuously within the model. In this manner, ray tracing is formulated in terms of the minimum number of required elastic parameters, e.g., four and nine parameters for P-wave propagation in transversely isotropic and orthorhombic media, plus a number of parameters specifying the rotation matrix connecting local and global coordinates. In particular, we parameterize this rotation matrix by one, two, or three Euler angles. In the ray-tracing equations, the slowness vector differentiated with respect to traveltime is related explicitly to the corresponding differentiated slowness vector for non-varying rotation and the cross product of the ray-velocity and slowness vectors. Our formulation is advantageous with respect to user-friendliness, efficiency, and memory usage. Another important aspect is that the anisotropic symmetry properties are conserved when material properties are determined in arbitrary points by linear interpolation, spline function evaluation, or by other means.