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.     

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.     

钢管混凝土拱桥弹性动力稳定性能研究

基于运动稳定性理论,以1座实际的钢管混凝土肋拱桥为例,利用改进的时间冻结法(动态特征值法)求得结构在地震波作用下的动态稳定系数时间历程,研究了不同输入方向和阻尼比对动力稳定临界系数的影响,并探讨了地震波作用下拱桥的稳定安全系数,对钢管混凝土拱桥的弹性动力稳定性能作出了评估。     

Tank damage during the may 1983 coalinga earthquake

The magnitude M = 6-5 Coalinga earthquake of 2 May 1983 caused intense ground shaking throughout the epicentral region. Unanchored cylindrical ground supported tanks located at six sites within this oil producing area were damaged; damages included elephant's foot buckling at the base of three moderate sized tanks, joint rupture and top shell buckling in one large old rivetted tank, bottom plate rupture of a relatively new welded tank and damage to the floating roofs of 11 tanks. Also oil spilled over the top of many tanks and secondary damages occurred in pipe connections, ladders, etc. In this paper an estimate is made of the intensity of ground motion at each of the tank sites, based on strong motion records made during the main shock and the strongest aftershock. Then response parameters specified by current codes are correlated with the damages observed at each tank site. Based on this comparison, it is concluded that current U.S. practice under-estimates the sloshing response of tanks with floating roofs and does not adequately address the uplifting mechanism of unanchored ground supported tanks.     

地震作用下立式储液罐罐壁"象足"变形仿真分析

基于ANSYS软件建立了考虑液体晃动和罐底提离立式储液罐有限元模型,分别进行了水平地震和竖向地震作用下罐壁"象足"变形分析.分析表明:立式储液罐罐壁"象足"变形主要是由罐壁纵向压应力超过临界应力而产生的局部屈曲破坏,并非强度破坏.因罐底提离导致的罐底与基础反复撞击加大了作用在罐壁上的应力,使罐壁底部"象足"变形不断发展,最终导致罐壁撕裂.在完全相同地震加速度作用下,水平地震作用比竖向地震作用罐壁更早更容易进入屈曲状态,产生"象足"变形.     

地震作用下立式储液罐罐壁“象足”变形仿真分析

基于ANSYS软件建立了考虑液体晃动和罐底提离立式储液罐有限元模型,分别进行了水平地震和竖向地震作用下罐壁“象足”变形分析。分析表明:立式储液罐罐壁“象足”变形主要是由罐壁纵向压应力超过临界应力而产生的局部屈曲破坏,并非强度破坏。因罐底提离导致的罐底与基础反复撞击加大了作用在罐壁上的应力,使罐壁底部“象足”变形不断发展,最终导致罐壁撕裂。在完全相同地震加速度作用下,水平地震作用比竖向地震作用罐壁更早更容易进入屈曲状态,产生“象足”变形。     

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.     

Buckling of cylindrical liquid-storage tanks under earthquake loading

The earthquake response behaviour of a cylindrical wine storage tank similar to many that were damaged in Livermore, California during the January 1980 earthquake was studied on the University of California shaking table. Tests of the 9.5 ft diameter by 20 ft high tank, with simulated earthquake accelerations up to 0.95 g, induced buckling patterns similar to those observed after the actual earthquake. Observed peak axial compression stresses in the test tank wall were substantially higher than those assumed in typical design standards, demonstrating the need for further study of the buckling problem in tanks free to uplift during earthquake excitation.     

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.     

Fluid–structure interaction analysis of 3-D rectangular tanks by a variationally coupled BEM–FEM and comparison with test results

A variationally coupled BEM–FEM is developed which can be used to analyse dynamic response, including free-surface sloshing motion, of 3-D rectangular liquid storage tanks subjected to horizontal ground excitation. The tank structure is modelled by the finite element method and the fluid region by the indirect boundary element method. By minimizing a single Lagrange function defined for the entire system, the governing equation with symmetric coefficient matrices is obtained. To verify the newly developed method, the analysis results are compared with the shaking-table test data of a 3-D rectangular tank model and with the solutions by the direct BEM–FEM. Analytical studies are conducted on the dynamic behaviour of 3-D rectangular tanks using the method developed. In particular, the characteristics of the sloshing response, the effect of the rigidity of adjacent walls on the dynamic response of the tanks and the orthogonal effects are investigated. © 1998 John Wiley & Sons, Ltd.     

腹板开槽冷弯薄壁卷边槽钢抗火性能有限元分析模型

位于冷弯薄壁型钢构件腹板的槽孔,可延长构件两边的热量传递路线,减弱冷弯薄壁钢构件作为墙板龙骨引起的冷桥效应。常温下冷弯薄壁构件的屈曲破坏模态已非常复杂,包括整体屈曲、局部屈曲和畸变屈曲等,且不同的屈曲破坏模态可能会相互影响。火灾下冷弯薄壁构件的破坏机理将更为复杂,根据常规的稳定理论难以求解其火灾下的临界荷载。本文利用有限元方法对火灾下腹板开槽冷弯薄壁卷边槽钢的受力性能进行了模拟,研究了单元类型、材料模型、初始几何缺陷以及求解方法等因素对分析结果的影响,为同类构件抗火性能的有限元模拟分析提供了建模依据。     

考虑桩土作用的储罐模拟地震振动台试验

为研究考虑桩土相互作用的储液罐的动力响应及长周期地震波对储液晃动、储罐提离的影响,根据量纲分析法设计了桩-土-储罐模型进行了振动台试验。试验中采用4条基岩波、4条地表波进行振动台试验。试验显示基岩波与地表波输入时,体系变化规律基本一致,其结果表明:土体地表加速度被放大,且输入加速度峰值增加,地表加速度放大倍数减小;一般地震波时,随着输入加速度峰值的增加,储液晃动波高大致呈线性增加。长周期地震波下则为非线性增加,且晃动波较大。此外,液体产生的晃动波高与储罐类型相关。细高型储罐产生的波高稍大;储罐提离高度随着输入加速度峰值的增加呈非线性增长。长周期地震波激励下,储罐提离高度小于一般地震波时的提离高度。细高型储罐在长短周期地震波激励下,提离高度较为接近,而一般储罐在两种地震波激励下,提离高度相差较大。细高型储罐提离高度大于一般储罐的提离高度。建议在储罐设计时考虑长周期地震波的影响。     

强震下四边钢柱支承单层柱面网壳弹塑性地震响应研究

采用集中塑性铰理论和SAP2000结构分析软件,对某体育练习馆（钢柱周边支承单层柱面网壳）整体结构进行了强震下弹塑性地震响应分析。分析中考虑了几何和材料双重非线性影响,获得了节点位移响应、杆件塑性铰的分布特征和结构的整体变形与失效形态,并评定了整体结构在强震下的极限承载力与失效类型。结果表明：该结构在强震下的失效界限地震加速度峰值为1260gal,最大竖向变形为短向跨度的1/163,满足＂避难与救灾建筑结构＂的抗震性能设防要求;结构的失效类型为动力失稳破坏,临界失效时出现塑性铰的杆件较少,结构塑性发展程度不充分;由整体稳定控制的单层柱面网壳在满足稳定承载力的要求下具有较大的抗震潜能。     

Sloshing analysis of rectangular tanks with a submerged structure by using small-amplitude water wave theory

A new sloshing analysis method for rectangular tank systems with a submerged structure are proposed by using the velocity potential and the linear water wave theory. The velocity potential functions are obtained by decomposing the surface wave into a wall-induced wave, reflected and transmitted waves, and a scattered wave. A simplified method using a response spectrum for zero damping is also proposed. The results of the simplified method are in good agreement with those of the analytical method. The sloshing response of the fluid-structure system is found to be very sensitive to the characteristics of the ground motion and the configuration of the system. Under typical earthquakes, the submerged structure shows a tendency to decrease sloshing amplitude, hydrodynamic pressure, and base shear, while it shows a tendency to increase the overturning moment. For the ground excitation dominated by low-frequency contents, the sloshing response increases significantly and the contribution of the higher sloshing modes increases. Copyright © 1999 John Wiley & Sons, Ltd.     

Method for solving dynamic equilibrium equations based on displacement excitation

Mode superposition is a widely used method for solving the dynamic equilibrium equation in structural dynamic analysis. However, the accuracy of this method may be reduced when the dynamic equilibrium equations are set up using displacement excitation. A new method for developing solutions for dynamic equilibrium equations based on displacement excitation is introduced. The dynamic equilibrium equation is decomposed into two parts, namely displacement excitation and velocity excitation, and precise integration and mode superposition methods are combined to solve the equation. Ritz vectors are then used to calculate the static response of the truncated modes of the structure, and a method for determining the number of participating modes is obtained. Using multi-degree-of-freedom systems as two computational examples, the differences in the structural responses obtained from the displacement excitation and acceleration excitation are compared and analyzed. It is shown that the new solution method generates consistent accuracy between the displacement excitation and acceleration excitation.     

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.     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

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.     

Stability of columns subjected to earthquake support motion

The problem of stability of an initially straight, simple column which is subjected to dynamic overload due to the vertical component of ground acceleration during an earthquake is considered. The time history of this component of earthquake, its maximum absolute amplitude and a Liapunov's function are used to define a sufficient condition under which the column will always be stable. It is shown for the case of a simple column that this condition implies that the column will be stable as long as the maximum load which the column is subjected to is less than the Euler buckling load during an earthquake.     

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.