Seismic Response of Tank-Fluid Systems: State of the Art Review and Dynamic Buckling Analysis of a Steel Tank with the Added Mass Method

PART I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake 
Excitation. 
Keywords: Liquid-storage tanks; dynamic response; earthquake excitation; code provisions; 
buckling of steel shells; elephant’s foot buckling. 
PART II: Numerical Modeling and Dynamic Analyses of a Clamped Steel Tank. 
Keywords: Finite element model, added mass method, time-history analyses, dynamic buckling, 
Budiansky-Roth criterion, fragility curves.
XIII 

 
The earthquakes certainly represent one of the most critical events to the safety of 
industrial plants. In order to estimate the risk associated with the industrial plants, it is of crucial 
importance the knowledge of the vulnerability of each single component of the plant itself. In 
fact the structural collapse of just one of these components can trigger more catastrophic events 
such ase xplosions, fire, dispersion of toxic materials, water shortages, pollution or 
contamination, thus putting in danger the life of people who work in the plant and who live in the 
urban area where the plant is located. A key aspect in the risk analysis of industrial plants is the 
detailed knowledge of each component and sub-system, in terms of design installation and 
operation mode. This step gives a contribution to the ranking of facilities depending on hazard or 
the individuation of critical components that can dramatically increase the seismic risk. To this 
aim it is common to divide the system in a number of sub-systems that have to be analyzed in 
detail up to component level. These are the basic information for the construction of fault-tree 
and sequences of events. A review of the printout of design of the plant and all information 
relative to the boundary conditions is fundamental; at the same time the inspection of the plant is 
recommended to establish the maintenance status of the facilities and upgrade models for 
capacity estimation. 
Industrial facilities show a large number of constructions and structural components. As 
materials are concerned, it is easy to recognize that both reinforced concrete and steel 
constructions are commonly used, even in combination like composite structures. Large 
installations can be characterized by use of pre-stressed members, especially when long spans 
are required. However, it is worth nothing that a large variety of functions have to be 
accomplished by structural components so that the latter can be classified as follows: 
 Building like structures: administration buildings, control buildings, substations, 
warehouses, firehouses, maintenance buildings, and compressor shelters or buildings. 
These are structures having a lateral force resisting system similar to those of building 
systems, such as braced frames, moment resisting frames or shear wall systems. 
 Non-Building like structures: such class of structures covers many industrial 
constructions and self-supporting equipment items found in a typical industrial plant, 
such as tanks, vertical vessels, horizontal vessels and exchangers, stacks and towers.
Foreword 
XIV 
Atmospheric tanks certainly represent the most spread and common component in an 
industrial plant; this is the reason why the present thesis is addressed to the increase of 
knowledge about their methods of analysis and design. Large-capacity ground-supported 
cylindrical tanks are used to store a variety of liquids or liquid-like materials, e.g. water for 
drinking and fire-fighting, petroleum, chemicals, liquefied natural gas and wastes of different 
forms. Satisfactory performance of tanks during strong ground shaking is crucial for modern 
facilities. Tanks that were inadequately designed have suffered extensive damage during past 
earthquakes.
XV 

 
Primary objectives of the thesis are: (a) to provide an overview of the salient aspects of the 
dynamic response of vertical, cylindrical, ground-supported tanks storing a homogeneous liquid; 
(b) to present different methods of analysis and design criteria, especially looking at the different 
codes provisions but also at alternative simplified procedures or innovative approaches given by 
other authors; (c) to set up a finite element model able to correctly represent the seismic behavior 
of the fluid-tank system; (d) investigate the complex phenomenon of buckling by means of 
dynamic numerical simulations and by establishing a buckling criterion; (e) to provide comments 
on the seismic vulnerability of liquid-storage tanks, using the results of dynamic analyses for the 
development of fragility curves. Secondary objectives are: (f) to create a sort of archive, based 
on which a designer can get design information/criteria according to the kind of tank he has to 
deal with; (g) to give a contribution in understanding the efficiency of different ground motion 
intensity measures with respect to the structural response (h) to suggest areas where possible 
future works should be oriented, especially looking at those topics where current design 
guidelines need further development. 
The thesis is organized into two parts: I. Methods of Analysis and Design of Liquid-
Storage Tanks under Earthquake Excitation; II. Numerical Modeling and Dynamic Analyses of a 
Clamped Steel Tank. Part I includes the first three chapters, where all issues relating to methods 
of seismic analysis and design of concrete and steel tanks are discussed in detail. In the second 
Part, including chapters four, five and six, a specific type of tank is chosen and a finite element 
model of it is set up. The results of modal analysis and dynamic simulations using the added 
mass technique for fluid modeling are presented, discussed and compared to those available in 
literature. Furthemore, based on these results, fragility curves are developed. 
The first chapter deals with the theoretical basis of the dynamic behavior of liquid-storage 
tanks. Here, after a brief discussion on the possible failure modes, the governing equation and 
the analytical solution are presented. Then, the most important mechanical models (or analogues) 
used by the various international codes are introduced. The main studies of the most important 
scientist and engineers who investigated the seismic behavior of liquid-storage tanks are listed in 
chronological order. In this way the reader can have an idea of the successive developments on 
these topic and can understand the basis of the current sophisticated models. The modeling
Preface 
XVI 
aspects of the fluid-tank-soil system are covered, but since the subject is very wide and not yet 
fully understood, also suggestions on possible future works are given. 
In the second chapter the main categories of tanks and the most important classification 
criteria are discussed. Here, attention is given to how the various international codes treat the 
different tanks categories. Codes provisions are analyzed and compared in detail. Such 
provisions are mainly related to the analysis and modeling aspects, but also to the design seismic 
forces calculation and verification criteria. The codes reviewed are: ACI 350.3, API 650 and  
Eurocode 8, but also New Zealand guidelines are sometimes mentioned. As the different 
guidelines are illustrated, also a numerical example is worked out through the chapter. The 
chapter ends with an assessment of code guidelines; here all the codes are compared and the 
results of the worked example are discussed. The quality of the different codes provisions is also 
assessed on the basis of numerical results and research works of many different authors. Areas 
where possible developments of the current codes should be oriented are indicated.  
The third chapter is entirely devoted to the buckling phenomenon, which definitely plays a 
fundamental role in steel tanks design due to the small thicknesses used for this class of 
structures. First, an overview on the possible buckling modes with related causes and effects is 
given, especially looking at the correspondence between damages observed in real tanks during 
past earthquakes and analytical studies on simplified models performed by different authors. 
Then, attention is given on how buckling is treated by the various codes and deficiencies on the 
subject are highlighted. Since in the codes there is not a relevant theoretical background, but just 
simple formulas to be applied in order to verify structural elements, a great effort is done in this 
chapter in order to understand what is behind the above mentioned formulas and to relate them to 
the possible buckling modes. The API verification formulas are then applied to a worked 
example. Chapter three also involves very brief notes on possible methods to strengthen tanks 
against “elephant’s foot” buckling. Since these methods make use of innovative techniques that 
are still under verification, this could be an area of interest for future studies. The chapter ends 
with the proposal of a very simple but efficient method to preliminarily design the tank’s 
thickness against elastic-plastic buckling.  
In the fourth chapter the tank model used for numerical computations is presented. All the 
details about geometry, materials, boundary conditions and sources of nonlinearity are 
highlighted. The method used to obtain a good finite element mesh on the tank is discussed. The 
added mass technique used to model the fluid is shown and the added masses are calculated 
explicitly.
Preface 
XVII 
The fifth chapter can be considered as the central chapter of the second part. Here, the type 
of dynamic analysis used to study the buckling phenomenon is explained in detail, from the 
governing equation and the numerical method used to solve it to the criterion chosen to select the 
earthquake accelerograms. The results of the dynamic buckling simulations are presented, 
compared and discussed in details. Prior to the nonlinear dynamic simulations, a modal analysis 
is also performed in this chapter, with the aim of studying the dynamic properties of the model. 
At the end of the chapter, the results obtained from the dynamic analyses of the added mass 
model of the tank are used to develop fragility curves and to understand the efficiency of 
different ground motion intensity measures on the structural response. 
The sixth chapter collects all the conclusions resulting from the analyses carried out in Part 
II. Also suggestions on possible improvements and future developments of this thesis can be 
found in this last chapter.  
Appendix A contains the mathematical formulation of the Bessel’s functions. These 
functions are found in the solution of the fundamental equation that governs the dynamic 
behavior of liquid-storage tanks but since they are not recurrent in typical civil engineering 
problems, it was considered important to give further information about them. Appendix B 
contains the MATLAB code used to calculate the added mass. Appendices C and D contain the 
MATLAB codes used to generate the input files for ABAQUS and run the analyses. Appendix E 
contains the MATLAB codes use to handle and elaborate the output files from ABAQUS. 
As already mentioned, one of the main objectives of the thesis is the comparison between 
the different provisions of the various codes. So, a uniform notation among all the codes has 
been used  in order to facilitate the reader in doing and understanding this comparison. In this 
view a “Notation” Section has been prepared at the end of the thesis. However, the 
corresponding formulas in the codes (in their original notation) are always indicated.  
Finally, it is noted that the subject is very wide and a full coverage of it is not possible due 
to clear reasons of space and time limitation. Therefore, to the opinion of the author, since 
sometimes it is very useful just to know “where to find information” about a specific problem, a 
great importance is given also to the last section “References”.
Part I 
 
Methods of Analysis and Design 
of Liquid-Storage Tanks under 
Earthquake Excitation
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
3 
 
 
1.1 Possible 	failure 	modes	
The complicated deformed configurations of liquid storage tanks and the interaction 
between fluid and structure result in a wide variety of possible failure mechanisms. This section 
discusses the different collapse modes in the light of the performances of existing tanks during 
past earthquakes (Northridge earthquake, [6], and Imperial Valley earthquake, [19]). 
 Shell Buckling Mode. One of the most common forms of damage in steel tanks involves 
outward buckling of the bottom shell courses, a phenomenon known as “elephant’s foot” 
buckling. It usually occurs in tanks with a low height to radius ratio. Initial studies 
claimed that the “elephant’s foot” buckle mechanism results from the combined action of 
vertical compressive stresses exceeding the critical stress and hoop tension close to the 
yield limit. However, Rammerstorfer et al. in [16] attributed the bulge formation to three 
components; the third being the local bending stresses due to the restraints at the tank 
base. Seiler, Wunderlich et al. in [39] and [42] differentiated this phenomenon for slender 
and broad tanks. The “elephant’s foot” buckle often extends around the circumference of 
the tank. Buckling of the lower courses has occasionally resulted in the loss of tank 
contents due to weld or piping fracture and, in some cases, total collapse of the tank. 
Figures 1.1.1 show two examples of “elephant’s foot” buckle for tanks destroyed by an 
earthquake. In Figure 1.1.2 a second kind of buckling is reported: the so called “diamond 
shape” buckling. It is an elastic buckling phenomenon due to the presence of high axial 
compressive stresses. The rocking motion which develops at the base of unanchored 
tanks generates very high compressive axial stresses surrounding the contact zone which 
in turn lead to the “diamond shape” buckle. This kind of damage may also occur well 
above the base of the tank where the hydrodynamic pressure, which leads to an increase 
in the elastic buckling load, is small as compared to its magnitude at the tank base, see 
Figure 1.1.3.
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Damage and collapse of tank roofs. A sloshing motion of the tank contents occurs during 
earthquake motion, as explained in Section 1.2. The actual amplitude of motion at the 
tank circumference has been estimated, on the basis of scratch marks produced by 
floating roofs, to have exceeded several meters in some cases, Hamdan [18]. For full or 
near full tanks, the free sloshing results in an upward pressure distribution on the roof. 
Common design codes do not provide guidance on the seismic design of tank roof 
systems for slosh impact forces. Modern tanks built after 1980 and designed to resist 
“elephant’s foot buckling” or other failure modes may still have inadequate designs for 
roof slosh impact forces. In past earthquakes, damage has frequently occurred to the 
joints between walls and cone roof, with accompanying spillage of tank contents over the 
top of the wall. Extensive buckling of the upper courses of the shell walls has occurred. 
Floating roofs have also sustained extensive damage to support guides from the sloshing 
of contents. Extensive damage to roofs can cause extensive damage to upper course of a 
steel tank. Less common are roof damages due to wind suction, Figure 1.1.5. However, 
roof damage or broken appurtenances, although expensive to repair, usually lead to more 
than a third of total fluid contents loss.  
 Anchorage Failure. Many steel tanks have hold-down bolts, straps or chairs. However, 
these anchors may be insufficient to withstand the total imposed load in large earthquake 
events and still can be damaged. As noted by field inspection, seismic overloads often 
Fig. 1.1.1:  “Elephant’s foot” buckling 
(after [20]-[30]). 
Fig. 1.1.2:  Diamond shape buckling (after [18]). Fig. 1.1.3:  Elastic buckling at the top (after [30]).
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
5 
result in anchor pull-out, stretching or failure. However, failure of an anchor does not 
always lead to loss of tank contents.  
 Tank Support System Failure. Steel and concrete storage tanks supported above grade by 
columns or frames have failed because of the inadequacy of the support system under 
lateral seismic forces, see Figure 1.1.6. Such failure most often leads to complete loss of 
contents. 
 Differential settlements and partial uplifting. The January,1994, Northridge earthquake 
had an immediate impact on the City of Los Angeles Department of Water and Power’s 
(LADWP) water system. Between all types of damages observed, Beverly Glen Tank 
experienced differential settlement varying from 7.5 to 20.3 cm and Coldwater Canyon 
Tank incurred a nearly uniform 10 cm settlement. Moreover, it was estimated that the 
Zelzah Tank uplifted well over 30 cm due to a poor anchorage system. The valve bodies 
on the inlet/outlet lines sheared from vertical displacements caused by tanks settlement or 
uplift.  
 Connecting Pipe Failure. One of the most common cause of loss of tank contents in 
earthquakes has been the fracture of piping at inlet/outlet (I/O) connections to the tank. 
This generally results from large vertical displacements of the tank caused by tank 
buckling, wall uplift or foundation failure. Failure of rigid piping (including cast iron 
valves and fittings, Figure 1.1.4b) that connects adjacent tanks has also been caused by 
relative horizontal displacements of the tanks. Another failure mode has been the 
breaking of pipes that penetrate into the tank from underground due to the relative 
movement of the tank and pipe, Figure 1.1.7. Water leaking from the broken pipe 
connections cause soil erosion and this can undermine the performances of the closest 
tanks. 
        
 
 
 
Fig. 1.1.4:  Examples of (a) flexible piping connection (Dresser couplings) 
and (b) brittle piping connections (cast iron valves and fittings). 
(a) (b)
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
6 
 
 
 
 
 
 
 
 
In Table 1.1.1 from [18], Hamdan F.H. summarizes some of the buckling damage and 
collapse of steel tanks during past earthquakes. It can be seen that unanchored tanks are more 
prone to buckling. Furthermore, there is a very little published data on the seismic-induced 
buckling of concrete tanks. However, observations from available field reports on the 
structural response of tanks during recent earthquake indicate that steel tanks, rather than 
concrete tanks, are more susceptible to damage and eventual collapse. This is one of the 
reason why the present thesis is mainly addressed to the steel tank analysis and design. 
Pre-stressed concrete tanks have become common in the liquid-storage systems over 
the last twenty years. The newer concrete tank walls are reinforced with circumferentially-
wrapped, high-strength pre-stressing steel cables and vertical post-tensioning bars (Figures 
1.1.9). The wall-to-footing connection is a flexible joint utilizing hard rubber bearing pads 
and seismic anchor cables (Section 2.4.3). The system allows limited rotation and movement, 
providing ductility to the joint. The cast-in-place reinforced concrete roof functions as a rigid 
diaphragm. All of the pre-stressed concrete tanks owned by LADWP performed well during 
the Northridge earthquake. Damages were limited to minor spalling of concrete, due to the 
Fig. 1.1.5:  Damage to tank roof caused by 
wind suction (underpressure). 
Fig. 1.1.6:  Elevated tank overturning. 
Fig. 1.1.7:  Water loss due to breaking of tank-
pipe connection. 
Fig. 1.1.8: Tank uplifting.
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
7 
Table 1.1.1: Buckled and collapsed steel
tanks during past earthquakes (after [18]). 
earthquake-induced pounding action, at a roof panel joint (Figure 1.1.10a) and opening of 
narrow gaps between roof panels.  
 
 
 
 
 
       
 
         
 
             
Fig. 1.1.9: Examples of rebar arrangement 
in concrete tanks. 
Fig. 1.1.10: Concrete spalling (a) at the
roof-wall joint (b) at the base.
Fig. 1.1.11: (a) Silo with extensive spalling and 
exposed rebar (b) Close up of exposed rebar. 
(a) (b) 
(a) 
(b)
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
8 
1.2 What’s	behind 	the 	codes 	
1.2.1 System, 	assumptions 	and 	terminology 	
The first system considered is shown in Figure 1.2.1.1. It is a rigid circular cylinder of 
radius R fixed to a rigid base. The tank is filled with a fluid of density ρ to a level H. The fluid is 
assumed to be incompressible and inviscid (or nonviscous). The fluid tank-system is presumed to 
be subjected to a horizontal ground acceleration directed along the x-axis, x null (t). Use is made of a 
cylindrical coordinate system: r, z, ϑ, with origin at the centre of the tank bottom and the z-axis 
vertical.  
 
 
 
 
 
 
 
 
 
 
 
Before investigating the response of flexible tanks, it is desirable to study the 
hydrodynamic forces induced on rigid tanks.  
In all previous studies on this subject it was found convenient to divide these effects into 
two parts: 
 the impulsive effects, which are computed by neglecting the effect of surface waves, i.e., 
by assuming the pressure at the free surface to be zero. The impulsive effects for rigid 
tanks are proportional to the ground acceleration. 
 the convective forces, which are associated with the sloshing of the fluid inside the tank. 
The convective effects depend on the sloshing frequencies of the fluid.  
In order to visualize the problem from a physical point of view, consider first a system for 
which the upper surface of the contained liquid is rigidly capped so that it can’t experience 
vertical motion; in this case, the entire liquid acts in unison with the tank wall as a rigid body. 
Fig. 1.2.1.1:Rigid tank anchored to the foundation.  
ϑ,z,r cylindrical coordinates (after [48]).
Chap. 1: Dynamics of tank-fluid systems 
Part I: Methods of Analysis and Design of Liquid-Storage Tanks under Earthquake Excitation 
9 
For a tank with a free liquid surface, only a portion of the contained liquid in the lower part of 
the tank responds synchronously with the tank wall as if it were rigidly attached to it. The 
remaining part (convective component) experiences a sloshing or rocking motion, which mainly 
depends on the tank dimensions and on the temporal characteristics of the base excitation. The 
convective component of the liquid responds as a continuous system with an infinite number of 
degrees of freedom, each one corresponding to a distinct mode of vibration, as shown in Figure 
1.2.1.2. 
 
 
 
 
 
 
 
 
1.2.2 Governing 	equations 	and 	boundary 	conditions 	
The equation of motion for the fluid, referred to the system shown in Figure 1.2.1.1, is a 
Laplace’s equation, i.e., a second-order partial differential equation belonging to the category of 
elliptic partial differential equations (Appendix A). Written in the cylindrical coordinates r, ϑ,z, it 
takes the form 
null
null
null
nullnull
null
null
1
null
nullnull
nullnull
null
1
null
null
null
null
null
nullnull
null
null
null
null
null
nullnull
null
null0 
(1.2.2.1)
in which ϕ is the velocity potential function. The velocity component of the fluid in the radial, 
tangential and vertical directions are 
null
null
nullnull
nullnull
nullnull
 
(1.2.2.2a)
null
null
nullnull
nullnull
nullnullnull
 
(1.2.2.2b)
null
null
nullnull
nullnull
nullnull
 
(1.2.2.2c)
and the dynamic pressure is related to ϕ by the equation 
Fig. 1.2.1.2:  Radial variation of vertical surface
displacements for first three sloshing modes of vibration in
rigid tanks. ξ=r/R is the dimensionless distance from the
tank vertical axis (after [48]).