Hydrodynamic effects in rigid tanks containing layered liquids

As a supplement to a recently reported study, the hydrodynamic wall pressures and the associated tank forces induced by horizontal ground shaking in a rigid, vertical, circular cylindrical tank containing liquid layers of different thicknesses and mass densities are examined, and comprehensive numerical solutions are presented for two-layered and some three-layered systems which elucidate the underlying response mechanisms and the effects of the various parameters involved. Both the impulsive and convective actions are studied. Additionally, solutions are presented for multi-layered systems approximating a liquid with an exponential, continuous variation in density, and the interrelationship of the solutions for the continuous system and its discretized, layered approximation is discussed.     

Response of tanks to vertical seismic excitations

A method for analyzing the earthquake response of elastic, cylindrical liquid storage tanks under vertical excitations is presented. The method is based on superposition of the free axisymmetrical vibrational modes obtained numerically by the finite element method. The validity of these modes has been checked analytically and the formulation of the load vector has been confirmed by a static analysis. Two forms of ground excitations have been used: step functions and recorded seismic components. The radial and axial displacements are computed and the corresponding stresses are presented. Both fixed and partly fixed tanks are considered to evaluate the effect of base fixation on tank behaviour. Finally, tank response under the simultaneous action of both vertical and lateral excitations is calculated to evaluate the relative importance of the vertical component of ground acceleration on the overall seismic behaviour of liquid storage tanks.     

Seismic analysis of base-isolated liquid storage tanks using the BE–FE–BE coupling technique

A three-dimensional soil–structure–liquid interaction problem is numerically simulated in order to analyze the dynamic behavior of a base-isolated liquid storage tank subjected to seismic ground motion. A dynamic analysis of a liquid storage tank is carried out using a hybrid formulation, which combines the finite shell elements for structures and the boundary elements for liquid and soil. The system is composed of three parts: the liquid–structure interaction part, the soil–foundation interaction part, and the base-isolation part. In the liquid–structure interaction part, the tank structure is modeled using the finite elements and the liquid is modeled using the internal boundary elements, which satisfy the free surface boundary condition. In the soil–foundation interaction part, the foundation is modeled using the finite elements and the half-space soil media are modeled using the external boundary elements, which satisfy the radiation condition in the infinite domain. Finally, above two parts are connected with the base-isolation system to solve the system's behavior. Numerical examples are presented to demonstrate the accuracy of the developed method, and an earthquake response analysis is carried out to demonstrate the applicability of the developed technique. The properties of a real LNG tank located in the west coast of Korea are used. The effects of the ground and the base-isolation system on the behavior of the tank are analyzed.     

Seismic design of flexible cylindrical liquid storage tanks

Seismic response of cylindrical storage tanks anchored to rigid base slabs is considered. Finite elements are used for the liquid and tank wall, idealized as a thin shell. For steel tanks of practical dimensions, design charts are presented for natural frequencies, maximum shear and overturning moment on the foundation, and maximum stress resultants in the tank wall. Furthermore, an analytical expression for the superelevation of the free surface is presented.     

大型储液罐摩擦摆基底隔震控制分析

针对弹性钢制圆柱储液罐,基于Haroun-Housner模型,将连续流体质量等效为3种集中质量,分别为:对流质量、脉冲质量和刚性质量,与这些集中质量连接的相应刚度取值依赖于储罐壁和流体质量.在水平地震激励下,在储罐底部加摩擦单摆支座,给出了简化的液体 - 储罐-隔震支座的力学分析模型,建立了摩擦摆支座基底隔震体系的振动控制方程,并利用Newmark逐步积分法对控制方程进行了数值求解,研究了摩擦摆支座基底隔震的储液罐地震反应,验证了FPB隔震的有效性.     

竖向地震作用对储液罐地震响应的影响分析

地震作用下大型储液罐的安全问题日益引起重视。基于ANSYS软件建立储罐液体耦合有限元模型,考虑罐底非线性接触效应,以El-Centro南北向和竖直向记录地震波为输入,研究水平激励以及水平和竖向同时激励两种工况下储罐的动力响应。研究结果表明,两种工况下靠近罐底1.2m处均发生了"象足"变形,竖向激励下水平相对位移增加了14%。竖向激励使得罐壁环向应力和轴向压应力均有不同程度的增加。竖向地震激励对液面的竖向晃动影响较小。储液罐底板在地震作用下发生了竖向提离和永久滑移,竖向激励时增长幅度均在10%左右。同时罐体基底剪力在竖向地震作用下也有所增大。储罐抗震设计时应考虑竖向地震分量的影响,研究结论可为立式储罐的抗震设计提供一定的参考和依据。     

液压阻尼系统控制浮放立式储罐提离失稳

立式圆柱储罐在油田、石油化工等企业用途广泛，多用作存贮易燃、易爆介质，一旦遭遇震害，后果将十分严重。本文针对浮放立式储罐地震响应的提离失稳问题，采用液压阻尼系统（HDS）用以控制储罐的提离反应，建立了安装HDS储罐的力学分析模型并进行了数值计算，对安装HDS后储罐的提离反应进行了分析。仿真计算结果表明：HDS可以有效地减小提离应力，从而地震烈度可以降低1度进行储罐的提离设计。     

Soil-structure interaction effects for laterally excited liquid storage tanks

A comprehensive study is made of the effects of soil-structure interaction on the response of liquid containing, upright, circular cylindrical tanks subjected to a horizontal component of ground shaking. A simple, physically motivated method of analysis is employed which elucidates the effects and relative importance of the principal actions involved. Both the impulsive and convective actions of the liquid are examined. The interrelationship of the tank responses to horizontal and rocking actions of the foundation is established, and the well known mechanical model for laterally excited, rigid tanks supported on a non-deformable medium is generalized to permit consideration of the effects of tank and ground flexibilities and base rocking. Critical responses are evaluated for harmonic and seismic excitations over wide ranges of tank proportions and soil stiffnesses, and the results are presented in a form convenient for use in practical applications. In addition to a precise method of analysis, an approximate, hand-computation method is presented with which the effects of the primary parameters may be evaluated readily. The soil-structure interaction effects in the latter approach are provided for by modifying the natural frequency and damping of the tank-liquid system and evaluating its response to the prescribed free-field ground motion considering the tank to be rigidly supported at the base. The requisite modifications may be determined from information presented herein. It is shown that soil-structure interaction may reduce significantly the impulsive components of response but that it has a negligible effect on the convective components.     

The effect of seasonally frozen soil on stochastic response of elevated water tank under random excitation

Significant seismic events have occurred around the world during winter months in regions where cold temperatures cause ground freezing. Current seismic design practice does not address the effects of cold temperatures in the seasonally frozen areas. Since many elevated water tank structures in cold regions are located in seismic active zones, determining the effect of seasonally frozen soil on the stochastic response of elevated water tank structures subjected to random seismic excitation is an important structural consideration. A three dimensional finite element model, which considers viscous boundaries, was built up to obtain the stochastic seismic behavior of an elevated water tank–fluid–soil interaction system for frozen soil condition. For this model, the power spectral density function represents random ground motion applied to each support point of the three dimensional finite element model of the elevated water tank–fluid–soil interaction system. Numerical results show that the soil temperature affects the seismic response of the elevated water tank; whereas the variation in the thickness of the frozen soil causes insignificant changes on the response. In addition, the effect of the variation in water tank’s fullness on the stochastic response of the coupled system is investigated in the study. As a result, the seasonal frost changes the foundation soil stiffness and may impact seismic behavior of the water tank.     

Investigation of sloshing damping in baffled rectangular tanks subjected to the dynamic excitation

In this research, an analytical model is developed to estimate the hydrodynamic damping ratio of liquid sloshing for wall bounded baffles using the velocity potential formulation and linear wave theory. Here, an analytical solution approach and experimental investigations are conducted for describing the hydrodynamic damping which is provided by vertical and horizontal baffles in partially filled rectangular liquid tanks. In order to evaluate the accuracy of the analytical solution which is developed in present work, a series of experiments are carried out with a rectangular liquid tank excited by harmonic oscillation. The parametric study is conducted on the damping efficiencies of both vertical and horizontal baffles with various dimensions and locations. According to the results of the present investigations, the hydrodynamic damping is significantly affected by the size and location of baffles. Furthermore, the validity of the developed analytical approach as well as the effectiveness of various baffle configurations are discussed. Finally, a simple approach is proposed for estimating the damping ratios of the baffles during earthquake motions.     

Investigation of nonlinear sloshing effects in seismically excited tanks

Nonlinear behavior of liquid sloshing inside a partially filled rectangular tank is investigated. The nonlinearity in the numerical modeling of the liquid sloshing originates from the nonlinear terms of the governing equations of the fluid flow and the liquid free surface motion as a not known boundary condition. The numerical simulations are performed for both linear and nonlinear conditions. The computed results using linear conditions are compared with readily available exact solution. In order to verify the results of the nonlinear numerical solution, a series of the shaking table tests on rectangular tank were conducted. Having verified linear and nonlinear numerical models, they are used for computation of near wall sloshing height at a series of real scale tanks (with various dimensions) under the both harmonic and earthquake base excitation. Finally, the nonlinear effects on liquid sloshing modeling are discussed and the practical limitations of the linear solution in evaluating the response of seismically excited liquids are also addressed.     

Seismic response of base-isolated liquid storage tanks considering fluid–structure–soil interaction in time domain

The seismic response analysis of a base-isolated liquid storage tank on a half-space was examined using a coupling method that combines the finite elements and boundary elements. The coupled dynamic system that considers the base isolation system and soil–structure interaction effect is formulated in time domain to evaluate accurately the seismic response of a liquid storage tank. Finite elements for a structure and boundary elements for liquid are coupled using equilibrium and compatibility conditions. The base isolation system is modeled using the biaxial hysteretic element. The homogeneous half-space is idealized using the simple spring-dashpot model with frequency-independent coefficients. Some numerical examples are presented to demonstrate accuracy and applicability of the developed method.Consequently, a general numerical algorithm that can analyze the dynamic response of base-isolated liquid storage tanks on homogeneous half-space is developed in three-dimensional coordinates and dynamic response analysis is performed in time domain.     

地震作用下储罐与管道连接波纹管的动力响应

针对储罐与管道连接这个抗震薄弱环节进行研究，考虑了储罐与地基、管道与地基的相互弹性作用及流固耦合作用，使计算模型比较符合工程实际情况。将储罐罐壁看作为刚体，将波纹管部分和管道部分分别用旋转锥壳单元和空间梁单元离散化，通过分析得到波纹管与储罐连接接合面、波纹管与管道连接接合面不同单元之间的位移协调约束方程，并用罚函数法进行处理。根据流体力学速度势理论和有限元法的基本理论，利用哈密尔顿变分原理推导出储罐与管道连接波纹管系统动力分析方程，编制了系统动力分析有限元程序，计算了垂直地震激励不同场地土地基条件下储罐与管道连接波纹管位移响应。     

A simplified seismic analysis of rigid base liquid storage tanks under vertical exicitation with soil-structure interaction

A study is carried out to evaluate dynamic response of an elastic circular cylindrical tank having a rigid base under a vertical excitation taking into consideration the interaction with the foundation soil. At first, the soil is represented by frequency-independent parameters. Two coupled differential equations, governing the motion of the shell and the base, are solved using a step by step integration technique. The hydrodynamic pressures, acting on the shell and on the base, are derived from a velocity potential function which satisfies the Laplace equation and the appropriate boundary conditions. The response of the simplified model of a tank having a rigid base on a stiff foundation soil is compated to that obtained elsewhere to check the accuracy of the present model. Reasonable agreement is found between the maximum wall displacement and the associated stresses with those found by a more elaborate model. The interaction of the tank and the soil reduces the response than that calculated under the assumption of a rigid foundation soil. A parametric study to examine the effects of the height-to-radius ratio of the tank, and the effects of the shear wave velocity of the soil on the response is conducted. Varieties of foundation models are used to assess the sensitivity of the response to the variation in the soil parameters. Finally, a more representative solution for the problem in the frequency domain is obtained where the soil is appropriately modelled by frequency-dependent parameters. The transfer functions of the response of the tank wall and of the relative base motion are evaluated, and a comparison between the frequency-dependent and the frequency-independent solutions is made.     

Temporal observations of a seasonal snowpack using upward‐looking GPR

An increase of the spatial and temporal resolution of snowpack measurements in Alpine or Arctic regions will improve the predictability of flood and avalanche hazards and increase the spatial validity of snowpack simulation models. In the winter season 2009, we installed a ground‐penetrating radar (GPR) system beneath the snowpack to measure snowpack conditions above the antennas. In comparison with modulated frequency systems, GPR systems consist of a much simpler technology, are commercially available and therefore are cheaper. The radar observed the temporal alternation of the snow height over more than 2·5 months. The presented data showed that with moved antennas, it is possible to record the snow height with an uncertainty of less than 8% in comparison with the probed snow depth. Three persistent melt crusts, which formed at the snow surface and were buried by further new snow events, were used as reflecting tracers to follow the snow cover evolution and to determine the strain rates of underlaying layers between adjacent measurements. The height in two‐way travel time of each layer changed over time, which is a cumulative effect of settlement and variation of wave speed in response to densification and liquid water content. The infiltration of liquid water with depth during melt processes was clearly observed during one event. All recorded reflections appeared in concordance with the physical principles (e.g. in phase structure), and one can assume that distinct density steps above a certain threshold result in reflections in the radargram. The accuracy of the used impulse radar system in determining the snow water equivalent is in good agreement with previous studies, which used continuous wave radar systems. The results of this pilot study encourage further investigations with radar measurements using the described test arrangement on a daily basis for continuous destruction‐free monitoring of the snow cover. Copyright © 2010 John Wiley & Sons, Ltd.     

Wavelet‐based non‐stationary seismic rocking response of flexibly supported tanks

The non‐stationary rocking response of liquid storage tanks under seismic base excitations including soil interaction has been developed based on the wavelet domain random vibration theory. The ground motion has been characterized through statistical functionals of wavelet coefficients of the ground acceleration history. The tank–liquid–foundation system is modelled as a multi‐degree‐of‐freedom (MDOF) system with both lateral and rocking motions of vibration of the foundation. The impulsive and convective modes of vibration of the liquid in the tank have been considered. The wavelet domain coupled dynamic equations are formulated and then solved to get the expressions of instantaneous power spectral density function (PSDF) in terms of functionals of input wavelet coefficients. The moments of the instantaneous PSDF are used to obtain the stochastic responses of the tank in the form of coefficients of hydrodynamic pressure, base shear and overturning base moment for the largest expected peak responses. Parametric variations are carried out to study the effects of various governing parameters like height of liquid in the tank, height–radius ratio of the tank, ratio of total liquid mass to mass of foundation, and shear wave velocity in the soil medium, on the responses of the tank. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic analysis of flexible rectangular fluid containers subjected to horizontal ground motion

In this paper, an analytical method is proposed to determine the dynamic response of 3‐D rectangular liquid storage tanks with four flexible walls, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The velocity potential in which boundary conditions are satisfied is solved by the method of separation of variables using the principle of superposition. The impulsive pressure distribution is then computed. Solutions based on 3‐D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that evaluates convective pressure and surface displacements in a similar rigid tank. The variation of dynamic response characteristics with respect to different tank parameters is investigated. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts, and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a simplified but sufficiently accurate design procedure is developed to improve code formulas for the seismic design of liquid storage tanks. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐stationary seismic response of tanks with soil interaction by wavelets

A wavelet‐based random vibration theory has been developed for the non‐stationary seismic response of liquid storage tanks including soil interaction. The ground motion process has been characterized via estimates of statistical functionals of wavelet coefficients obtained from a single time history of ground accelerations. The tank–liquid–soil system has been modelled as a two‐degree‐of‐freedom (2‐DOF) system. The wavelet domain equations have been formulated and the wavelet coefficients of the required response state are obtained by solving two linear simultaneous algebraic equations. The explicit expression for the instantaneous power spectral density function (PSDF) in terms of the functionals of the input wavelet coefficients has been obtained. The moments of this PSDF are used to estimate the expected pseudo‐spectral acceleration (PSA) response of the tank. Parametric variations are carried out to study the effects of tank height, foundation natural frequency, shear wave velocity of soil and ratio of the mass of tank (including liquid) to the mass of foundation on the PSA responses of tanks. Copyright © 2001 John Wiley & Sons, Ltd.     

Water pressures on rigid gravity dams with finite reservoir during earthquakes

An approximate analytical solution is presented for earthquake-induced hydrodynamic pressures on rigid gravity dams with a finite reservoir and incompressible fluid. Using the Trefftz-Mikhlin method, the solution is constructed with function expansions of solutions of the problem's governing equation which satisfy boundary conditions at the bottom and free surface. Unknown coefficients of the linear combinations are obtained from a continuous least-squares treatment of the remaining boundary conditions at the upstream dam face and reservoir wall. Numerical results are presented for different geometries of the dam-water and wall-water interfaces. Out-of-phase motion at the end of the reservoir is considered. When the upstream dam face and reservoir wall are vertical, the known solution for in-phase dam and wall movement is recovered.     

Spatially coupled vibrations of a suspension bridge under random highway traffic

The problem of spatial dynamic response of a suspension bridge to the passage of trains of concentrated forces with random values is considered. The arrival of forces at the bridge is assumed to constitute a Poisson process of events. Such an excitation process is an appropriate model of vehicular traffic loads acting on the bridge. The bridge is idealized by a single-span thin-walled beam underslung to two cables. The response of the bridge in the space-time domain is described by a coupled system of non-linear, integro-differential equations. The dynamic influence functions of vertical and horizontal deflections at each cross-section point are obtained for the linear case. Cumulants and probability density functions of response are determined. Numerical methods have been used to develop a computer-oriented algorithm aimed at the numerical solution of the problem. As examples, numerical results for a particular bridge with some practical load cases are presented and illustrated by graphs.