Simplified analysis of unanchored tanks

Local uplift of the tank wall is perhaps the most important characteristic of the seismic behaviour of unanchored liquid storage tanks made out of steel. Such uplift is necessary, because unanchored tanks rely primarily on the weight of the liquid resting on an uplifted portion of the base plate to balance the overturning moments that occur during strong shaking. A simplified method of analysis for static lateral loads is developed, based on the assumption that the restraining action of the base plate can be modelled with equivalent, non-linear springs. This assumption, together with a Fourier decomposition of the displacements, simplifies the problem to the extent that it can be solved on a personal computer. The solutions are compared with those from experiments and those from current U.S. design analysis methods.     

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.     

Dynamic response of rigid tanks with inhomogeneous liquids

A study of the response to horizontal ground shaking of a rigid cylindrical tank containing an inviscid liquid with a continuous vertical variation in density is presented. In addition to the free vibrational sloshing characteristics of the liquid, the responses examined include the vertical displacements at the free surface, and the impulsive and convective components of the hydrodynamic wall pressures and associated tank forces. The equations of motion for the system are formulated for an arbitrary variation in liquid density but the solutions presented are for a density that increases exponentially from top to bottom. Comprehensive numerical data are included which elucidate the underlying response mechanisms and the effects and relative importance of the various parameters involved. The solution for the continuous density variation considered herein is also compared with a previously reported solution in which the liquid was modelled as a multi-layered, discrete system.     

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.     

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.     

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。     

Finite element analysis of the seismic response of anchored and unanchored liquid storage tanks

The seismic response of liquid-filled cylindrical storage tanks has been investigated using finite element techniques implemented in the general purpose structural analysis computer code ANSYS. Both added mass concepts and displacement-based fluid finite elements were employed to allow for the effects of the liquid. Simplified response spectrum modal analyses of a tank making use of the axisymmetric harmonic displacement patterns of the principal modes of deformation were found to give accurate predictions of the tank behaviour with a rigidly anchored base. Time history analyses of three-dimensional finite element models of unanchored and flexibly anchored tanks, with gap conditions between the tank base and the supporting floor to allow lift-off of the base, indicated that stresses in the tank and resultant loads on the floor can be much greater than for a rigidly restrained tank. These results demonstrate the importance of carefully considering the restraint conditions when performing seismic design calculations on storage tanks.     

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.     

核电结构PCS水箱液动压力分析等效模型

中国核电厂抗震设计规范推荐采用的Housner模型不适用于复杂形状核电储液结构的流固耦合分析。对于AP1000和CAP1400核电站屏蔽厂房顶部非能动安全壳冷却系统重力水箱(简称PCS水箱),基于圆柱形水箱的Housner等效质量-弹簧模型,通过引入水箱体积修正参数,提出PCS水箱的三维等效质量-弹簧模型。采用有限元软件ADINA建立水箱结构流固耦合整体有限元模型以进行模态分析,计算分析PCS水箱和对应环形水箱在不同尺寸和液体深度条件下的液体晃动自振特性。对比整体有限元模型与三维等效质量-弹簧模型计算结果发现,提出的PCS水箱三维等效质量-弹簧模型能给出其内液体晃动各阶振型的液动压力合理估计值,适用于具有复杂形状的PCS水箱液动压力分析。本文的等效模型方法可推广应用于其他复杂形状水箱的液动压力分析。     

On the seismic behavior of cylindrical base-isolated liquid storage tanks excited by long-period ground motions

A common effective method to reduce the seismic response of liquid storage tanks is to isolate them at base using base-isolation systems. It has been observed that in many earthquakes, the foregoing systems significantly affect on the whole system response reduction. However, in exceptional cases of excitation by long-period shaking, the base-isolation systems could have adverse effects. Such earthquakes could cause tank damage due to excessive liquid sloshing. Therefore, the numerical seismic response of liquid storage tanks isolated by bilinear hysteretic bearing elements is investigated under long-period ground motions in this research. For this purpose, finite shell elements for the tank structure and boundary elements for the liquid region are employed. Subsequently, fluid–structure equations of motion are coupled with governing equation of base-isolation system, to represent the whole system behavior. The governing equations of motion of the whole system are solved by an iterative and step-by-step algorithm to evaluate the response of the whole system to the horizontal component of three ground motions. The variations of seismic shear forces, liquid sloshing heights, and tank wall radial displacements are plotted under various system parameters such as the tank geometry aspect ratio (height to radius), and the flexibility of the isolation system, to critically examine the effects of various system parameters on the effectiveness of the base-isolation systems against long-period ground motions. From these analyses, it may be concluded that with the installation of this type of base-isolation system in liquid tanks, the dynamic response of tanks during seismic ground motions can be considerably reduced. Moreover, in the special case of long-period ground motions, the seismic response of base-isolated tanks may be controlled by the isolation system only at particular conditions of slender and broad tanks. For the case of medium tanks, remarkable attentions would be required to be devoted to the design of base-isolation systems expected to experience long-period ground motions.     

Sloshing behaviours of rectangular and cylindrical liquid tanks subjected to harmonic and seismic excitations

A numerical and experimental study on the sloshing behaviours of cylindrical and rectangular liquid tanks is addressed. A three‐dimensional boundary element method for space with the second‐order Taylor series expansion in time is established to simulate the sloshing phenomenon and its related physical quantities inside a liquid tank subjected to horizontal harmonic oscillations or recorded earthquake excitations. The small‐scale model experiments are carried out to verify some results of numerical methods in this study. The comparisons between numerical and experimental results show that the numerical method is reliable for both kinds of ground excitations. Finally, the water wave and the base shear force of a rectangular tank due to harmonic excitation are also presented at different frequencies. A huge cylindrical water tank subjected to a recorded earthquake excitation is used for application and discussion. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic analysis of column-supported hyperboloidal shells

The dynamic response of hyperboloidal shells on discrete column supports is studied using a curved rotational shell finite element. In this finite element formulation, the displacement field over each element domain is approximated by polynomial functions in which the coefficients of the linear terms correspond to the nodal values of the displacements and the higher order terms vanish at the nodal circles. The stiffness and mass matrices associated with the equations of motion are derived from Hamilton's variational principle and include the effects of transverse shearing deformation and rotatory inertia. Since the formulation, as such, involves a great many degrees of freedom because of the use of higher order displacement functions, the kinematic condensation technique is employed to reduce the order of the dynamic problem The dynamic analysis indicates the importance of realistically modelling the base region of the shell. Studies on a prototype tower indicates that the base flexibility reduces the natural frequencies of the shell and increases the displacements near the base. The magnitude of this reduction, which could be significant, depends primarily on the tangential stiffnesses of the supporting columns and is hardly affected by the thickened ring beam at the base.     

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.     

跨越断层地下管道地震反应研究

本文采用圆柱壳单元模拟地下管道,用有限元模拟手段建立了管道-土相互作用分析模型,采用非线性接触问题分析方法分析了管道因断层运动而产生的反应,通过大量数值计算,对影响管道地震反应的各种因素进行了研究,如断层位错量、管道跨越角、断层运动方式、管道埋设深度、管道初始轴向力、断层裂缝宽度、管道径厚比等,得到了一些规律性的结论。     

钢结构建筑钢板外部震动下承载力有限元分析

传统钢结构建筑钢板外部震动下承载力分析方法,是基于总体钢板结构以及受力特征,获取载荷同振动频率间的关系,实现承载力的分析,并未对钢板同混凝土间的应力-应变关系进行分析,导致分析结果存在较高的偏差。提出新的钢结构建筑钢板外部震动下承载力有限元分析方法,将钢结构建筑钢板横截面简化成混凝土的矩形截面和波纹钢板的工字形梁截面,采用ANSYS有限元软件中的CONTACI2接触单元仿真分析钢板同混凝土两者的影响,分析混凝土及钢板的应力-应变关系、界面模块的应力-滑移关系,采用力平衡迭代法获取外部震动下钢板荷载增量;在上述基础工作上采用有限元软件,对外部震动下钢结构建筑钢板实施有限元接触单元建模及承载力分析。实验结果表明,该方法能够实现钢结构建筑钢板外部震动下承载力有限元分析,并且分析结果具有准确性高和效率高的优点。     

The effect of seismic uplift on the shell stresses of liquid‐storage tanks

Previous theoretical studies have shown that tank uplift, that is, separation of the tank base from the foundation, generally reduces the base shear and the base moment. However, there is a paucity of experimental investigations concerning the effect of uplift on the tank wall stresses, which is the principal parameter that controls the seismic design of liquid‐storage tanks. This paper reports a series of shake table experiments on a polyvinyl chloride model tank containing water. A comparison of the seismic behaviour of the tank with and without anchorage is described. Stochastically generated ground motions, based on a Japanese design spectrum, and three tank aspect ratios (liquid‐height/radius) are considered. Measurements were made of the stresses at the outer shell of the tank, the tank wall acceleration and the horizontal displacement at the top of the tank. While the top displacement and the tank shell acceleration increased when uplift was allowed, axial compressive stresses decreased by between 35% and 64% with tank uplift. The effect of uplift on the hoop stresses was variable depending on the aspect ratio. A comparison of experimental values with a numerical model is provided. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Stresses in column-supported hyperboloidal shells subject to seismic loading

The computation of stresses within a finite element displacement method analysis of a shell of revolution is considered. The common procedure of applying the kinematic and constitutive laws to the displacement functions is examined and justified for models where the displacements are represented by high-order polynomial expansions. Also, two alternative computational formats within this technique are explored. The influence of the column-supported base condition on a hyperboloidal shell of revolution is studied with respect to the stresses calculated from a response spectrum analysis. These studies emphasize the importance of accurately modelling the base region of a column-supported shell such as a hyperbolic cooling tower.     

单层柱面网壳弹塑性地震反应特征

本文对网壳结构的弹塑性抗震性能进行了探讨。根据结构弹性有限元理论，结合网壳结构受力特点，推导出网壳结构单元的弹塑性矩阵表达式。然后对单层柱面网壳结构的弹塑性地震反应进行了具体分析，给出了节点位移和杆件应力变化规律，并讨论了矢跨比变化对网壳弹塑性性能的影响，从而揭示出这类网壳的弹塑性地震反应特点。     

Radial vibration of a cylindrical shell having transverse isotropy

Stresses and displacements in a transversely isotropic cylindrical shell when it vibrates radially under prescribed boundary loadings have been obtained with the help of finite Hankel transforms [1].