Stability of elevated liquid-filled conical tanks under seismic loading,Part I—theory

Conical steel shells are widely used as water containments for elevated tanks. However, the current codes for design of water structures do not specify any procedure for handling the seismic design of such structures. In this paper, a numerical model is developed for studying the stability of liquid-filled conical tanks subjected to seismic loading. The model involves a previously formulated consistent shell element with geometric and material non-linearities included. A boundary element formulation is derived to obtain the hydrodynamic pressure resulting from both the horizontal and the vertical components of seismic motion acting on a conical tank which is prevented from rocking. The boundary element formulation leads to a fluid added-mass matrix which is incorporated with the shell element formulation to perform non-linear dynamic stability analysis of such tanks subjected to both horizontal and vertical components of ground motion. Although, the formulation was developed for conical vessels, it is general and can be easily modified to study the stability of any liquid-filled shell of revolution subjected to seismic loading. The accuracy of fluid added-mass formulation was verified by performing the free vibration analysis of liquid-filled cylindrical tanks and comparing the results to those available in the literature. © 1997 John Wiley & Sons, Ltd.     

Dielectrophoretic,thermal instability in a spherical shell of fluid

Abstract

This paper is concerned with the dielectrophoretic instability of a spherical shell of fluid. A dielectric fluid, contained in a spherical shell, with rigid boundaries is subjected to a simultaneous radial temperature gradient and radial a.c. electric field. Through the dependence of the dielectric constant on temperature, the fluid experiences a body force somewhat analogous to that of gravity acting on a fluid with density variations. Linear perturbation theory and the assumption of exchange of stabilities lead to an eighth order differential equation in radial dependence of the perturbation temperature. The solution to this equation, satisfying appropriate boundary conditions, yields a critical value of the electrical Rayleigh number and corresponding critical wave number at which convective motion begins. The dependence of each critical number is presented as a function of the gap size and temperature gradient. In the limit of zero shell thickness both the critical Rayleigh number and critical wave number agree with results for the case in the infinite plane problem.     

2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves

The problem of harmonic wave diffraction by tunnels in an infinite poroelastic saturated soil obeying Biot's theory is studied numerically under conditions of plane strain and the effect of poroelasticity on the response is assessed through some parametric studies. The method is based on the theory of Mei and Foda, which considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer at the free boundaries. Both problems are solved numerically by the boundary element method in the frequency domain. Results dealing with the response of a circular tunnel with and without an elastic concrete liner in an infinite poroelastic medium to incident harmonic P and SV plane waves are provided and compared against analytical ones as well as to those corresponding to linear elastic soil behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

Scattering of elastic wave by a cylindrical shell deeply embeded in saturated soils

The compression of soil grain and pore fluid as well as viscid coupling of pore fluid and soil skeleton is considered, the scattering problem of incident plane P1 wave （fast compressional wave） by an infinite cylindrical shell deeply embedded in isotropic saturated soils is studied by adopting the amended Biot model, amplitude equations about potential functions of scattering and refracting fields are obtained, and the effect of dimensionless frequencies and shell thickness on the back-scattering spectra and dynamic stress concentration factors of two types of cylindrical shells with high and low rigidity are numerically computed and analyzed.     

Diversification of skeletal microstructures of organisms through the interval from the latest Precambrian to the Early Cambrian

From the latest Precambrian to the Early Cambrian was the most important phase throughout the evolutionary history of life on the earth. The diversification of latest Precambrian multicellular organisms was obvious and some of them began to form primary s…     

饱和弹性土体中圆柱壳体对弹性波的散射

考虑土颗粒、孔隙流体的压缩性及孔隙流体与土骨架之间的粘性耦合作用,采用修正的Biot模型,研究了入射平面P1波（快压缩波）在饱和土中不透水的圆柱壳体上的散射问题, 得到了散射和折射波场势函数的幅值方程. 通过数值计算, 分析了无量纲频率及厚度对高、低硬度两类圆柱壳体的背向散射谱和动应力集中因子的影响.      

Dynamic plate-soil interaction - finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil

An integral method to calculate the solution of a homogeneous or layered soil due to a harmonic point load is described. An infinite plate at the surface of the soil can be introduced in this integration in wavenumber domain, too. Finite structures on the soil are calculated by a combined finite element and boundary element method, which makes use of the point load solution of the soil. The compliance functions for a vertical point load and some vibration modes are calculated for realistic parameters of the plate and the soil and for a wide range of frequencies. The influence of the stiffness of the soil and the foundation is investigated, showing that the soil mainly affects the low-frequent response whereas the structural properties are more important at higher frequencies. A rigid approximation of flexible plates is only found at low frequencies, if the elastic length is used as the radius of a rigid disk. At higher frequencies, a characteristic behaviour of the flexible plate of approximately is observed, what is in clear contrast to the compliance of rigid foundations. A plate on a visco-elastic support (Winkler soil) shows similar displacements as a plate on a homogeneous half-space, but the maximal stresses between the plate and the soil are considerably smaller which is found to be more realistic for a plate on a layered soil. For practical applications, a normalized diagram and some explicit formulas of the exact and the approximate solutions of an infinite plate on a homogeneous half-space are given, which is a useful model to represent the soil-structure interaction of flexible foundations.     

Two-dimensional dynamic analysis of concrete gravity and embankment dams including hydrodynamic effects

An analysis procedure in the frequency domain is developed for determining the earthquake response of two-dimensional concrete gravity and embankment dams including hydrodynamic effects; responses of the elastic dams and compressible water are assumed linear. The dam and fluid domain are treated as substructures and modelled with finite elements. The only geometric restriction is that an infinite fluid domain must maintain a constant depth beyond some point in the upstream direction. For such an infinite uniform region, a finite element discretization over the depth is combined with a continuum representation in the upstream direction. The fluid domain model approximately accounts for interaction between the fluid and underlying foundation medium through a damping boundary condition applied along the reservoir bottom, while the dam foundation is assumed rigid. Several examples are presented to demonstrate the accuracy of the fluid domain model and to illustrate dam responses obtained from the analysis procedure.     

无限元法在深覆盖层土石坝动力分析中的应用

以复杂地质条件的西藏高原地区旁多水电站实际工程为背景,探讨了有限元法与无限元法在深覆盖层悬挂式防渗墙土石坝结构非线性动力分析中的差异,分析和比较了无限元法和有限元法在西藏高原旁多水电站土石坝结构三维模型地震动响应的计算结果,验证了引入无限元法模拟西藏高原地区地基中复杂地质条件下的无限域或半无限域问题的可靠性和精确性。结果表明:在西藏高原地区复杂地质条件下,无限元法相比有限元法能量弥散现象较为明显。通过分析得到了西藏高原具有深覆盖层坝体结构地震动响应的规律性分析结果,为无限元法在西藏高原地区此类工程中的应用提供参考。     

Time-domain boundary element method in viscoelasticity with application to a spherical cavity

The weighted-residual technique, the indirect boundary element method, the truncated indirect boundary element method and the direct boundary element method can be used to analyse nonlinear soil-structure interaction in the time domain. They are illustrated and compared by using the one-dimensional dynamic problem of the spherical cavity in an infinite space. For realistic time steps, all formulations lead to accurate results, but the weighted-residual technique and the truncated indirect boundary element method are much more efficient than the direct boundary element method in the time domain. Hysteretic damping leads to noncausal behaviour, which can, however, be neglected from a practical point of view.     

Dynamic buckling and seismic fragility of anchored steel tanks by the added mass method

Buckling plays a fundamental role in the design of steel tanks because of the small thicknesses of the walls of this class of structures. The first part of the paper presents a review of this phenomenon for liquid‐containing circular cylindrical steel tanks that are fully anchored at the base, considering the different buckling modes and especially the secondary buckling occurring in the top part of the tank. A case study based on a cylindrical tank is then introduced in order to investigate various aspects of dynamic buckling. The finite element model of the case study tank is set‐up using the added mass method for fluid modelling. The influence of pre‐stress states caused by hydrostatic pressure and self‐weight on the natural periods of the structure is first studied and it is found that this influence is very small as far as the global behaviour of the tanks is considered, while it is important for local, shell‐type, vibration modes. In the following, the efficiency and sufficiency of different ground motion intensity measures is analysed by means of cloud analysis with a set of 40 recorded accelerograms. In particular, the peak ground displacement has been found being the most efficient and sufficient intensity measure so far as the maximum relative displacement of the tank walls is concerned. Finally, incremental nonlinear time‐history analyses are performed considering the case study structure under recorded earthquake ground motions in order to identify the critical buckling loads and to derive fragility curves for the buckling limit state. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing density-dependent groundwater models: two-dimensional steady state unstable convection in infinite,finite and inclined porous layers

This study proposes the use of several problems of unstable steady state convection with variable fluid density in a porous layer of infinite horizontal extent as two-dimensional (2-D) test cases for density-dependent groundwater flow and solute transport simulators. Unlike existing density-dependent model benchmarks, these problems have well-defined stability criteria that are determined analytically. These analytical stability indicators can be compared with numerical model results to test the ability of a code to accurately simulate buoyancy driven flow and diffusion. The basic analytical solution is for a horizontally infinite fluid-filled porous layer in which fluid density decreases with depth. The proposed test problems include unstable convection in an infinite horizontal box, in a finite horizontal box, and in an infinite inclined box. A dimensionless Rayleigh number incorporating properties of the fluid and the porous media determines the stability of the layer in each case. Testing the ability of numerical codes to match both the critical Rayleigh number at which convection occurs and the wavelength of convection cells is an addition to the benchmark problems currently in use. The proposed test problems are modelled in 2-D using the SUTRA [SUTRA––A model for saturated–unsaturated variable-density ground-water flow with solute or energy transport. US Geological Survey Water-Resources Investigations Report, 02-4231, 2002. 250 p] density-dependent groundwater flow and solute transport code. For the case of an infinite horizontal box, SUTRA results show a distinct change from stable to unstable behaviour around the theoretical critical Rayleigh number of 4π² and the simulated wavelength of unstable convection agrees with that predicted by the analytical solution. The effects of finite layer aspect ratio and inclination on stability indicators are also tested and numerical results are in excellent agreement with theoretical stability criteria and with numerical results previously reported in traditional fluid mechanics literature.     

The dynamics of offshore gravity platforms: Some insights afforded by a two degree of freedom model

This paper presents some findings of an investigation into the dynamic behaviour of offshore gravity platforms excited by waves. Results obtained from a finite element model are presented in the form of dynamic magnification factor curves for two structures typical of current concrete platform designs. In order to facilitate interpretation of these results a simple two degree of freedom model is developed and the equivalent results presented. These are found to be very similar to those obtained from the more detailed idealization, and it is therefore concluded that the principal mechanisms involved in the dynamic behaviour of offshore gravity structures are adequately represented in the simple model. Several useful insights into this behaviour are then achieved by closer examination of the two degree of freedom configuration.     

The effects of reservoir geometry on the seismic response of gravity dams

Conventional seismic analysis of gravity dams assumes that the behaviour of the dam–water–soil system can be represented using a 2‐D model since dam vertical contraction joints between blocks allow them to vibrate independently from each other. The 2‐D model assumes the reservoir to be infinite and of constant width, which is not true for certain types of reservoirs. In this paper, a boundary element method (BEM) model in the frequency domain is used to investigate the influence of the reservoir geometry on the hydrodynamic dam response. Important conceptual conclusions about the dam–reservoir system behaviour are obtained using this model. The results show that the reservoir shape influences the seismic response of the dam, making it necessary to account for 3‐D effects in order to obtain accurate results. In particular, the 3‐D pressure and displacement responses can be substantially larger than those computed with the 2‐D model. Copyright © 2007 John Wiley & Sons, Ltd.     

The patterns of high-degree thermal free convection and its features in a spherical shell

Introduction Mantle convection is thought to be the most effective way of heat transportation in the earth and the source of driving lithospheric plates (Elasser, 1971). The velocity field of plate motion can be split into poloidal and toroidal parts, which are corresponding to the vertical (i.e. radial) and horizontal motions, respectively, in global model. The toroidal component is manifested in the existences of transform faults and the poloidal one in the presences of convergence and diver…     

An asymptotic model for resistive instability in the Earth's outer core

Abstract

Numerical work indicates that resistive instability may be the dominant mode of instability in the Earth's outer core for realistic core parameter regimes. In this paper, we assume that the Elsasser number is large in order to obtain an asymptotic analysis of resistive instability in an electrically conducting fluid confined to a rotating cylindrical shell of infinite extent in the axial direction. The dimensionless equations of motion are linearized about an ambient magnetic field which is purely azimuthal and depends only on the cylindrical radial variable. Applying the theory of ordinary differential equations with a large parameter, we obtain an asymptotic approximation to the solution. Relatively simple analytic expressions for the complex frequencies are obtained by applying the boundary conditions for insulating boundaries at the cylindrical sidewalls and then assuming that the ambient magnetic field vanishes at one or both of those sidewalls. The results appear to be consistent with previous numerical work.     

节点刚度对单层柱面网壳地震响应及内力的影响分析

论文首先采用壳单元Shell163建立了单层柱面网壳模型,同时利用壳单元模拟了焊接空心球节点,并通过计算验证了模拟方法的合理性。然后,通过调节空心球节点的厚度,考察了节点刚度变化对单层柱面网壳地震响应的影响,分析结果表明,在强震作用下节点刚度变化对网壳节点响应与杆件内力的影响均较大,建议在进行此类结构的动力分析时考虑节点刚度的影响。     

Electrical conduction and formation factor in isotropic porous media

A macroscopic form of Ohm's law is obtained for isotropic porous media saturated with an electrically conductive fluid by using volumetric averaging concepts. Closure of the macroscopic charge transport equation is aided by approximative modelling of the average geometric structures of three different types of isotropic porous media, namely foamlike materials, granular media and crossflow over prismatic bundles. Modelling of the microscopic charge transport necessitated the introduction of a representative interstitial flux of charge carriers and required quantification of the geometric tortuosity applicable to transport phenomena in general. Deterministic expressions for the formation factor are obtained and compare favourably with experimental results.     

Coupling method of finite and infinite elements for strip foundation wave problems

Starting from two-dimensional wave equations and making use of Galerkin weighted residual approximations, discretized formulations for wave problems of a visco-elastic foundation have been derived. With considerations of geometrical and mechanical characteristics of a semi-infinite domain, a frequency-dependent compatible infinite element is also presented. Finally, by coupling the infinite elements with ordinary finite elements the system is used for simulation of propagating waves in a semi-infinite foundation. This model is not only suitable for simulations of complicated variations of geometrical conditions, but also for describing the unbounded behaviour of arbitrary multiple layers. Examples given indicate the model has excellent computational accuracy and feasibility for analysing the effects on foundation response due to the existence of faults or any other soft layers.     

Infinite elements for elastodynamics

An axisymmetric infinite element and a three-dimensional infinite element are developed to solve three-dimensional elastic wave propagation problems in unbounded media. The elements are capable of transmitting Rayleigh, shear and compressional waves in the frequency domain. A scheme to integrate numerically the characteristic matrices of the elements is formulated based upon Gauss—Laguerre quadrature. Finally, the axisymmetric infinite element is used to find the compliance functions of a rigid circular plate subjected to harmonic loading on a semi-infinite medium. By using infinite elements, the size of the near field may be kept small. Consequently, the system is characterized by relatively few degrees of freedom, thus providing the analyst with an inexpensive solution.