POLITECNICO DI TORINO

Dipartimento di Ingegneria dell'Ambiente, del Territorio e delle Infrastrutture

Corso di Laurea Magistrale in Ingegneria Del Petrolio

Sessione di Laurea novembre 2024

Tesi di Laurea Magistrale

Advantages and disadvantages of the simplified models to evaluate the seismic
performance of steel storage tanks

Relatore: Condidato:
Prof. SURACE CECILIA BOUAKLI JUGURTA

Relatore esterno:
Dott: VASQUEZ MUNOZ LUZ ELIZABETH

ACADEMIC YEAR 2023-2024

Turin, 12/11/2024
Acknowledgements

At the end of this work, I would like to thank all those who participated in its realization.

First, I would like to warmly thank my supervisor Mrs. SURACE CECILIA for proposing this

thesis to me, for taking the time and effort to help me and her advice during all the stages of

this work. I would like to extend my sincere appreciation to my external supervisor VASQUEZ

MUNOZ LUZ ELIZABETH for her valuable suggestions, her mentorship has added depth to

my understanding of the subject matter, contributing significantly to the overall refinement of

this thesis.

I am grateful to my family for their constant encouragement, understanding and support

during the hard time of this academic endeavor.

iii
Abstract

Aboveground vertical cylindrical steel storage tanks are critical structures used for storing

various types of liquids, some of which may be toxic or flammable. These tanks are

susceptible to a range of natural events, including wind and earthquakes. The thesis provides a

comprehensive understanding of performance and seismic response of aboveground steel

storage tanks with uniformly supported flat bottoms. It emphasizes the crucial need for special

attention to the tank design and resilience to ensure safety, especially in the face of potentially

catastrophic events such as earthquakes. The research delves into the different failure modes

experienced by these tanks during ground shaking, including elephant's foot buckling,

diamond-shaped wall buckling, base plate failure, anchor bolts failure, roof damage, and

piping connection failure.

The seismic behavior of storage tanks is complex, involving hydrodynamic pressure from the

oscillating liquid. This pressure creates an overturning moment at the tank bottom, leading to

potential buckling. Sloshing of the liquid near the top of the tank can cause severe damage,

particularly in roofed tanks with insufficient freeboard or floating roofs. Unanchored tanks are

vulnerable to base uplift, resulting in substantial plastic deformation and potential fracture or

fatigue failure.

Various simplified models, including the Housner model, Veletsos model, Yang model,

Wozniak and Mitchell model, Haroun and Housner model, and Malhotra model, have been

developed to analyze the seismic response of steel storage tanks. These models aim to capture

the complex interactions between the liquid, tank walls, and foundation, providing insights

into the tank's behavior during earthquakes.

iv
It is of paramount importance in engineering to understand the dynamic responses of these

structures under a strong seismic excitation to mitigate the risk of catastrophic collapse and

potential spillage. Such incidents could lead to significant socio-economic and environmental

consequences. Storage tanks behavior during ground acceleration is of great interest because

of the complexity of its response.

Consequently, this topic has attracted the attention of numerous researchers over the past

decades. The objective of this thesis is to provide a concise overview of some of the most

utilized simplified models, along with their pros and cons, for assessing the seismic behavior

of steel storage tanks under seismic excitation.

v
Table of Contents

Acknowledgements .............................................................................................................. iii

Abstract ................................................................................................................................ iv

Table of Contents ................................................................................................................. vi

List of figures ..................................................................................................................... viii

Preface ...................................................................................................................................1

CHAPTER 1 ..........................................................................................................................3

1.1 Introduction ..................................................................................................................4

1.2 Examples of past disastrous earthquakes .......................................................................6

1.3 Assessment of damage to steel storage tanks due to earthquakes ...................................9

1.3.1 Buckling at the Bottom of Tank Wall .....................................................................9

1.3.2 Buckling at the top of tank and roof damage......................................................... 11

1.3.3 Base plate uplifting .............................................................................................. 13

1.3.4 Tank sliding and piping system failure ................................................................. 14

1.3.5 Anchorage Failure ................................................................................................ 15

1.3.6 Foundation Settlement.......................................................................................... 16

1.4 Damage States for Liquid Storage Tanks: ...................................................................16

1.5 Conclusion .................................................................................................................. 17

CHAPTER 2 ........................................................................................................................ 19

2.1 Introduction ................................................................................................................ 20

2.2 Existing Models for Evaluating Seismic Performance ................................................. 21

2.2.1 Housner Model (Mass-Spring Model) ..................................................................21

2.2.2 Veletsos Model for Flexible Tanks ....................................................................... 25

2.2.3 Yang Model for Flexible Tanks ............................................................................ 26

2.2.4 Wozniak and Mitchell 1978 (unanchored tanks) ................................................... 26

2.2.5 Haroun and Housner ............................................................................................ 27

vi
 2.2.6 Velestos Model for Flexible Tanks ....................................................................... 28

2.2.7 Malhotra model for Base Uplift ............................................................................ 29

2.2.8 Model of Base Isolated Tank ................................................................................ 30

2.2.9 Joystick Model .....................................................................................................32

2.2.10 Vathi Model for Base Plate Uplift ......................................................................33

2.2.11 3D simplified model ........................................................................................... 34

2.3 Conclusion .................................................................................................................. 35

CHAPTER 3 ........................................................................................................................ 37

3.1 Introduction ................................................................................................................ 38

3.2 Comparative Analysis of Seismic Analysis Models ..................................................... 38

3.3 Comclusion................................................................................................................. 43

CONCLUSIONS .................................................................................................................. 44

References ............................................................................................................................ 46

vii
List of Figures

FIGURE 1-1: EXAMPLE OF ABOVE-GROUND STORAGE TANK (COURTESY OF SIS GMBH).............5

FIGURE 1-2: EXAMPLE OF BELOWGROUND STORAGE TANK (COURTESY OF D&H GROUP ...........5

FIGURE 1-3: EXAMPLE OF ANCHOR (COURTESY VATHI & KARAMANOS) ...................................5

FIGURE 1-4: EXAMPLE OF UNANCHORED STORAGE TANK (COURTESY OF D&H GROUP).............5

FIGURE 1-5: TANK FARM WITH SOME TANKS DESTROYED BY FIRE IN TÜPRAS REFINERY [6]........7

FIGURE 1-6: THERMAL BUCKLING DUE TO HEAT RADIATION [6] ................................................7

FIGURE 1-7: DIAMOND-SHAPED BUCKLING (COURTESY OF N R I OF FIRE AND DISASTER,

JAPAN) ............................................................................................................................8

FIGURE 1-8: OIL SPILLAGE TO THE PORT [7]..............................................................................8

FIGURE 1-9: OIL SPILLAGE DUE TO BOTTOM PLATE RUPTURE [4] ...............................................9

FIGURE 1-10: DAMAGE TO PIPING CONNECTIONS [4] .................................................................9

FIGURE 1-11: ELEPHANT’S FOOT BUCKLING OF THE LOWER OF THE TANK [11] ......................... 11

FIGURE 1-12: DIAMOND-SHAPED BUCKLING [15].................................................................... 11

FIGURE 1-13: DIFFERENT TYPES OF SLOSHING BEHAVIOR OF THE FREE-LIQUID SURFACE OF A

TANK EXCITED BY HORIZONTAL ACCELERATION.[16] ...................................................... 11

FIGURE 1-14: FLOATING ROOF WITH SEALING SYSTEM.[19]..................................................... 13

FIGURE 1-15: (RIGHT) BASE PLATE UPLIFT PHENOMENA IN UNANCHORED TANKS, (LEFT) THE

LOCATIONS WHERE PLASTIC HINGES DEVELOP [21] ......................................................... 14

FIGURE 1-16: ANCHORAGE FAILURE OF A TANK DURING HANSHIN-AWAJI EARTHQUAKE [26] ..15

FIGURE 1-17: ANCHOR ROD FAILURE DURING PERU EARTHQUAKE 1995 [27] ........................... 15

FIGURE 2-1: HOUSNER MODEL (HOUSNER 1957).................................................................... 22

FIGURE 2-2: SIMPLIFIED HOUSNER MODEL (HOUSNER 1963) .................................................. 23

viii
FIGURE 2-3: DIFFERENT HYDRODYNAMIC PRESSURES INDUCED BY A GROUND EXCITATION ON

TANK [35]...................................................................................................................... 24

FIGURE 2-4: MECHANICAL MODEL FOR FLEXIBLE TANK [41 – 42]............................................ 28

FIGURE 2-5: UPLIFTING PLATE (A) AXISYMMETRIC, (B) ASYMMETRIC [44] .............................. 30

FIGURE 2-6: LIQUID STORAGE TANK ISOLATED BY PROPOSED METHOD .................................... 31

FIGURE 2-7: MODEL OF BASE ISOLATED TANK ........................................................................ 31

FIGURE 2-8: JOYSTICK MODEL (LEFT), AND ITS DEFLECTED SHAPE (RIGHT) .............................. 33

FIGURE 2-9: SIMPLIFIED MODEL OF ANCHORED TANK (LEFT), AND UNANCHORED TANK (RIGHT)

.....................................................................................................................................34

ix
Preface

Steel storage tanks have been an integral part of various industries, including oil and gas,

petrochemicals, and energy production. These tanks play a crucial role in handling a wide

variety of liquids in large quantities, such as refined petroleum products, water, and other

liquid substances. However, these ground-supported steel storage tanks are vulnerable to

various natural phenomena like wind, snow, and notably earthquakes. The forces exerted by

these events present significant challenges to the tanks’ structural integrity, potentially leading

to structural failure and disruption in industrial processes. Among these, seismic loads pose

the most substantial threat to the structural integrity and safety of storage tanks.

Tank failure can lead to catastrophic consequences, including uncontrollable fires if the water

supply is cut off, explosion risks, and pollution due to the spillage of toxic chemicals. Such

incidents could result in the disruption of the supply of essential products and energy

resources which may lead to substantial socioeconomic consequences.

Therefore, addressing the seismic vulnerability of storage tanks is imperative to mitigate these

risks and protect human life, property, and the environment.

Implementing robust seismic design standards and integrating advanced structural engineering

solutions are vital for enhancing the resilience of these tanks against seismic events. This

necessitates careful consideration of design factors and adherence to stringent safety

protocols.

The aim of this thesis, titled “Advantages and Disadvantages of the Simplified Models to

Evaluate the Seismic Performance of Steel Storage Tanks,” is to conduct a thorough analysis

of the most commonly used simplified models for evaluating the seismic performance of steel

storage tanks under seismic loads.

1
The thesis is structured into six chapters. Chapter 1 lays the foundation with a general

introduction to the behavior of these tanks during past earthquakes. Chapter 2 explores the

core topic, discussing the most relevant existing simplified models used for evaluating such

behavior. Chapter 3 weighs the pros and cons of these chosen models, providing a balanced

view of their advantages and disadvantages. Finally, the conclusions bring the thesis to a close

by presenting the results obtained from the study and drawing conclusions based on these

findings.

By addressing these objectives, this thesis aims to contribute valuable insights into the seismic

vulnerability and structural integrity of above-ground cylindrical steel storage tanks, thereby

informing strategies for enhancing their resilience and mitigating potential risks.

2
 CHAPTER 1
INTRODUCTION

3
1.1 Introduction

Steel storage tanks are structures used to store large volumes of liquids substances in many

process industries, such as oil refineries and petrochemical plants, particularly those of

vertical cylindrical configuration. These are commonly used due to their ease of fabrication,

erection, and maintenance. These tanks often contain toxic, flammable, and explosive

substances or fuel that are crucial for recovery after a catastrophic event. Damage to these

facilities poses significant risks, extending beyond material loss to the potential loss of human

life and long-lasting environmental impact. Understanding how these tanks interact with their

foundation and containments during earthquakes is a complex analytical task. [1]

The specifications outlined by the facility owner can impact the selection of a liquid storage

tank and the nature of the stored substance often leads to choosing a suitable tank type. We

can classify storage tanks in numerous ways. A common classification is based on their

position concerning the ground level: they can be either above or below ground (Figures 1-1

and 1-2), which may also differ based on whether the tank's base plate rests on the ground or a

supporting structure and whether it is anchored or unanchored (Figures 1-3 and 1-4).

This thesis is primarily concerned with aboveground tanks that have uniformly supported flat

bottoms. Most tanks possess a vertical cylindrical body, the cylinder is constructed from

curved plates that are welded together. The walls can either be the same thickness all the way

up or have different thicknesses at different heights. These structures offer several benefits:

they are less complex to construct, can accommodate larger volumes, and are more

economical. Aboveground storage tanks are often preferred because of their ease of

inspection, maintenance, and cleaning access. Another approach to categorizing storage tanks

is by the operating pressure. Tanks that operate at a pressure slightly above atmospheric

pressure are known as atmospheric tanks. On the other hand, tanks designed to contain gases

4
or liquids at a significantly different from the ambient pressure are termed pressure vessels or

high-pressure tanks. [2]

Figure 1-1: Example of above-ground storage Figure 1-2: Example of belowground storage
tank (Courtesy of SIS GmbH) tank (Courtesy of D&H Group

Figure 1-3: Example of anchor (Courtesy Figure 1-4: Example of unanchored storage
Vathi & Karamanos) tank (Courtesy of D&H Group)

In some cases, steel storage tanks, especially those of a vertical cylindrical configuration, may

require a top closure and can be differentiated in numerous ways, with one significant method

being based on their roof designs. Fixed-roof tanks are distinguished by a shallow cone roof

deck that mimics a flat surface, typically constructed from steel plates, offering a robust and

durable cover for the tank.

5
Conical roof tanks, the most used for storing large volumes of fluid, are designed with roof

rafters and support columns for additional stability. However, these features may not be

present in tanks with minimal diameters.

Umbrella-roof tanks bear a striking resemblance to cone-roof tanks. However, with a critical

difference - the roof is shaped like an umbrella, eliminating the need for support columns

extending to the bottom of the tank and offering a clear internal space.

Dome-roof tanks feature a roof that mirrors a spherical surface, providing a distinct aesthetic

and functional advantage with enhanced strength and resistance to external pressures.

Floating-roof tanks are equipped with a cover that floats on the surface of the liquid stored

within, minimizing evaporation losses, and making an excellent choice for storing volatile

liquids. The floating roof adjusts its height with the liquid level, thereby reducing vapor space

above the liquid level and decreasing evaporation. [3]

Given the crucial role of liquid storage tanks in various industries, it is imperative to

understand their vulnerability to natural loads such as wind and earthquakes. This thesis

underscores the need for special attention to the design and performance of these tanks under

to ensure their resilience and safety. With the potentially catastrophic consequences of tank

failure, particularly during earthquakes, and the complex behavior of these structures during

earthquakes, researchers have significantly increased interest in investigating the seismic

response of ground-supported steel storage tanks.

1.2 Examples of past disastrous earthquakes

During a ground shaking, the tanks can experience different types of failure modes, each

contributing to the overall complexity of their behavior. Specifically, the traditional elephant's

6
foot buckling is the most often seen collapse mechanism in aboveground steel storage tanks,

the tank wall's diamond-shaped buckling close to the base, failure of base plate (uplifting),

anchor bolts failure, roof damage and piping connection failure are other well-known tank

failure scenarios such as failures of base anchoring in the case of anchored tank. [4]

The 1999 Kocaeli Earthquake in Turkey, with a magnitude of 7.6, had a significant impact on

industrial facilities, particularly the Tüpras refinery. The seismic event led to substantial

damage, with approximately 20 tanks in the refinery farm being damaged or destroyed by fire.

A major fire was ignited in the tanks that contained naphtha, the sloshing motion of the

containment generated by the earthquake caused the floating roof to rub against the walls and

created sparks, instantly igniting the liquid. The fire then spread to the crude oil tanks,

damaging 30 of the 45 tanks. Furthermore, the intense heat from the burning tanks caused

thermal buckling of a fixed roof tank (Figures 1-5 and 1-6). In addition to economic losses,

large quantities of toxic materials were released into the environment. [5]

Figure 1-5: Tank farm with some tanks Figure 1-6: Thermal buckling due to heat
destroyed by fire in Tüpras refinery [6] radiation [6]

Three notable earthquakes in Japan offer important new information for the investigation of

behavior of storage tanks during seismic events. The Showa Oil refinery suffered significant

damage because of the 1964 Niigata Earthquake. In particular, the combustible vapors ignited

in 12 tanks as a result of mechanical shoe seals colliding with the tank wall.
7
The 1978 Miyagi Earthquake demonstrated another form of damage. The seismic activity

caused an uplift of the bottom plate of the tanks. The uplift caused a significant plastic strain

in the shell-bottom joint weld, which resulted in a catastrophic failure. The tank contents

subsequently spilled into the port (Figure 1-8). However, unlike the Niigata incident, no fire

ensued.

The 1995 Kobe Earthquake presented yet another pattern of damage. Tanks exhibited

diamond-shaped and “elephant foot” buckling of the tank shell (Figure 1-7). Additionally,

some tanks were inclined due to soil liquefaction. Despite these deformations, no leaks or

fires were reported. [7]

Figure 1-7: Diamond-Shaped Buckling Figure 1-8: Oil spillage to the port [7]
(Courtesy of N R I of Fire and Disaster,
Japan)

In the USA, California, a state that is known for its high seismic activity, steel storage tanks

experienced significant damage during the May 1983 Coalinga Earthquake. Typical damage

observed across various tank sites include Elephant’s Foot Buckling at the base of the tank.

Riveted joint tanks suffered severe damage with buckled top courses and ripped joints,

leading to extensive oil spills. Tanks with floating roofs showed damage to roof pontoons.

Broad tanks experienced a rupture in the bottom plate due to the uplift of the base plate. Other

sites reported damage to piping connections and punctures due to internal frame impact.

8
These damages, while less catastrophic than complete structural failure, can still lead to

significant operational disruptions, financial losses, and environmental impact. [4]

Figure 1-9: Oil spillage due to bottom plate Figure 1-10: Damage to piping connections
rupture [4] [4]

1.3 Assessment of damage to steel storage tanks due to

earthquakes

Aboveground steel storage tanks can suffer minor to moderate damage in certain

circumstances depending on the intensity of the earthquake and the tanks configuration. In the

case of unanchored broad tanks, the overturning moment can cause a partial uplifting of the

base plate from the foundation, and the consequences can lead to damage to any connected

pipping. One of the most common forms of damage is the buckling of the tank wall,

which can take several forms. Furthermore, accessories surrounding the tank, such as the fire-

fighting system, inlet/outlet piping and maintenance stairs, are also vulnerable to damage. [8]

1.3.1 Buckling at the Bottom of Tank Wall

9
The behavior of a storage tank is relatively straightforward under static condition. However,

its dynamic response during seismic loading is quite complex. When subjected to seismic

excitation, the liquid inside the tank begins to oscillate, creating hydrodynamic pressure. This

pressure generates an overturning moment at the tank bottom, which can lead to the formation

of the Elephant Foot Buckling due to the increase on the axial stress in the lower course of the

tank. [9]

This Elephant Foot Buckling (EFB) is an elastic-plastic buckling, characterized by an outward

bulge near the base of the tank (Figure 1-11). EFB is a critical failure mode in storage tank,

even though the tank is not completely collapsed, typically formed due to the uplift of the

base plate. This phenomenon results from the combined effects of tensile hoop stress and

compressive meridional stress. When these stresses exceed the critical threshold, EFB occurs.

[10] However, the formation of the EFB is highly sensitive to initial shell imperfections, but

this sensitivity decreases with internal pressure. At low levels of internal pressure, the shell

fails through elastic buckling, but with a notable increase in strength. As internal pressure

increases, the strength gains are smaller, and the buckling becomes more axisymmetric and

less affected by initial imperfections. [11] However, if an EFB forms around a nozzle or

manhole, it is more likely to cause a leak. [12]

Another observed type of buckling is the Diamond-Shaped Buckling (DSB), which is less

common than EFB. DSB is an elastic deformation characterized by a diamond-like pattern on

the bottom course of the tank, occurs relatively at low hoop stress levels and is highly

sensitive to internal pressure and initial imperfection in the tank shell. [13] DSB is typically

observed in slender stainless tanks, the ratio of the radius to shell thickness is low, most

slender tall tanks are anchored to the foundation. [14] Figure 1-12 shows such a buckling.

10
 Figure 1-11: Elephant’s foot buckling of the Figure 1-12: Diamond-shaped buckling [15]
lower of the tank [11]

1.3.2 Buckling at the top of tank and roof damage

Sloshing refers to the movement of the liquid near the top of the tank due to the excitation of

the convective mass, which is characterized by a long period (6 to 10 seconds). In roofed

tanks with insufficient freeboard or tanks with floating roofs, this sloshing can cause severe

damage due to the interaction between the upper part of the tank and the sloshing liquid.

This behavior can be categorized into three types based on the intensity of the oscillations and

the shape of the liquid’s surface.

(a) (b) (c)

Figure 1-13: Different types of sloshing behavior of the free-liquid surface of a tank excited
by horizontal acceleration.[16]

11
Linear Sloshing (Case a): In this scenario, the liquid experiences minor oscillations, and its

surface remains flat. This is a perfectly linear

Weakly Nonlinear Sloshing (Case b): Here, the liquid undergoes oscillations of varying

magnitude, and its surface is no longer flat.

Nonlinear Sloshing (Case c): In this case, the liquid exhibits a strongly nonlinear motion at

the surface, primarily due to rapid velocity changes associated with hydrodynamic pressure

impacts near the free liquid surface. This highly nonlinear fluid behavior necessitates the use

of sophisticated computational methods.

In the case of nonlinear waves, deriving the sloshing phenomenon using an analytical method

is challenging. Therefore, numerical simulation becomes essential for investigating

parameters such as sloshing, maximum height, and periods of resonance. [17]

In the case of aboveground steel storage tanks equipped with a floating roof, the floating roof

is designed to rise and fall with the liquid level, minimizing vapor space and reducing fire

risk. However, sloshing can cause damage to the floating roof, lead to wear and tear on the

roof's seals and joints, compromising its ability to effectively contain the tank's contents. In

more severe cases, the roof can sink into the tank, due to mechanical failure. This sinking can

render the floating roof inoperable, leading to increased vapor emissions and a higher risk of

fire and explosion. Roof damage can also result in spilling of the tank's contents, posing

environmental and safety hazards. [18]

12
 Figure 1-14: Floating roof with sealing system.[19]

1.3.3 Base plate uplifting

The seismic response of aboveground vertical cylindrical steel storage tanks is a complex

phenomenon. These tanks, often constructed without anchoring, are in simple contact with the

foundation. When subjected to strong seismic loading, they may experience a base uplifting

mechanism, which is a highly nonlinear phenomenon where a portion of the base plate is

uplifted and separated from the support foundation. [20]

The base uplift can lead to substantial plastic deformation in the vicinity of the welded

connection between the tank shell and the bottom plate, posing a risk of fracture or fatigue

failure of the welded joint under repeated cycles of the uplift. Since the base plate is held

down by the hydrostatic pressure of the tank contents, the base weld is subject to high stresses

and fracture may result. Additionally, the tank's uplift can cause an increase in compressive

meridional stress and hoop stress in the tank walls, which may result in elephant foot buckling

damage, tearing, and failure of pipe connections. [21]

13
 However, tanks resting on a flexible soil foundation do not experience a significant increase

in axial compressive stress in the tank wall, instead, lard foundation penetration may occur.

Consequently, tanks that are resting on flexible foundation are less susceptible to EFB but

more prone to uneven foundation settlement. [22]

Overall, the seismic performance of unanchored liquid storage tanks is dominated by the

uplift phenomenon, this response underscores the need for careful consideration in the design

and assessment of such tanks to mitigate potential structural failures during seismic events.

Negative pressure Positive pressure

Plastic hinge 1
Weld
Plastic hinge 2

Base plate
Location 1 Location 2

Figure 1-15: (right) base plate uplift phenomena in unanchored tanks, (left) the locations
where plastic hinges develop [21]

1.3.4 Tank sliding and piping system failure

When unanchored or not adequately anchored and subjected to strong ground motion, the tank

is susceptible to horizontal and vertical movements; these movements can lead to several

types of damage. A tank's horizontal (sliding) movement can lead to breaks in inlet and outlet

piping connections that are not designed to accommodate such motions. This inability to

absorb the Tank's movements due to insufficient flexibility is a common cause of product loss

from storage tanks during earthquakes. This type of damage can be effectively mitigated by

14
implementing flexible connections, a solution that ensures the piping systems remain intact

and functional, even during the Tank's movement. [23]

1.3.5 Anchorage Failure

Anchorage failure in steel storage tanks during earthquakes is very important because it can

cause serious damage to the structure. The rocking motion induced by the overturning

moment during an earthquake generates tensile forces in these anchor bolts. This motion can

lead to significant stress, potentially causing pulling forces that can rupture the tank wall.

Additionally, some tanks experience weld ruptures at the joint between the bottom course and

the annular plate, which is integral to the anchorage system. [24]

The figures below highlights anchorage failure in storage tanks subjected to strong seismic

excitation. Frequently, excessive inelastic strain demands were imposed on the anchor bolts,

resulting in their fracture or pull-out from the concrete pads. [25]

Figure 1-16: anchorage failure of a tank Figure 1-17: Anchor rod failure during Peru
during Hanshin-Awaji earthquake [26] earthquake 1995 [27]

15
1.3.6 Foundation Settlement

The performance of cylindrical storage tanks also depends on the quality and stability of their

foundation. The uneven settlement of a liquid storage tank during an earthquake can result

from several factors. One primary cause is the heterogeneous nature of the foundation of the

soil beneath the tank. Variations in soil composition and density can lead to differential

settlement, where some foundation areas settle more than others.

Uneven settlement can lead to a tilt or differential settlement of the base, which imparts

additional stress to the tank wall, which can cause the tank shell to buckle. [2]

However, the foundation flexibility can reduce the compressive stress developed at the tank's

base. [4]

1.4 Damage States for Liquid Storage Tanks

In assessing the seismic damage of liquid storage tanks, it is essential to define various

damage states based on the severity of the damage. According to studies [28], four distinct

levels of damage can be identified: Level 0, indicating no damage; Level 1, representing

minor (non-severe) damage; Level 2, indicating major damage without loss of containment;

and Level 3, signifying major damage with loss of containment.

A risk study on Natech accidents, which are industrial accidents triggered by natural events

such as earthquakes, was conducted by Krausmann et al. (2011) [29]. This study focuses on

the ranking of seismic damage commonly observed in reservoirs based on severity. The

seismic damage is categorized into two groups: whether there was a liquid spill or not. Such

16
classifications and studies are vital for improving safety measures and mitigating the risks

associated with seismic events affecting liquid storage tanks.

Tank failure without any leakage

Severity Failure mode
EFB
Minor Anchors elongation (tensile)
Sloshing damage
Anchor failure
Well connection failure
Moderate
Roof damage due to sloshing
Failure of supporting columns

Table: Different failure modes without contents spill [29]

Tank failure with leakage

Severity
Minor Failure of inlet/outlet piping connections
Moderate Spill from tank top
Catastrophic Tank collapse and tilting

Table: Different failure modes with contents spill [29]

1.5 Conclusion

In conclusion, this chapter has examined the critical aspects of the seismic response of steel

storage tanks, emphasizing the various failure mechanisms and damage states that these tanks

may experience during earthquakes. Steel storage tanks, particularly those with vertical

cylindrical configurations, are crucial for storing large volumes of liquids in industries such as

17
petrochemical and oil refining, where their structural integrity is vital to both operational

safety and environmental protection.

The review of past earthquake case studies has shown the devastating consequences of tank

failure. The observed damage states, such as elephant foot buckling, diamond-shaped

buckling, roof damage, piping connection failure, and base plate uplifting, highlight the

critical need for cautious design and assessment to ensure the tanks' resilience and safety.

Consequently, this detailed analysis underscores the necessity for implementing robust safety

measures and innovative design approaches to mitigate the destructive effects of earthquakes

on above-ground steel storage tanks.

18
 CHAPTER 2

DESCRIPTION OF THE SIMPLIFIED MODELS

19
2.1 Introduction

Early research on the response behavior of steel storage tanks subjected to horizontal ground

motion, predominantly focused on anchored tank with rigid walls, and resting on rigid support

foundation. Extensive investigations have been conducted on evaluating the seismic response

of aboveground steel storage tanks. Analyzing the seismic behavior of these structures is

complex due to different interactions that take place between the liquid, the tank walls and the

foundation. Horizontal ground acceleration generates both impulsive and convective

hydrodynamic pressures. Similarly, vertical ground acceleration also induces both impulsive

and convective hydrodynamic pressures. The determination of the hydrodynamic forces is a

crucial step in the seismic design of structures such as storage tanks.

During seismic events, the liquid within a storage tank plays an essential role, especially since

these tanks can be full or empty at the moment of the seismic event. The ground accelerations

induce inertial forces on the liquid which interacts with the tank walls, leading to

overpressures or hydrodynamic depressions along the walls and at its base.

Research on the behavior of liquid storage tanks and their interactions have led to the

development of simplified models. These models are designed to simulate the dynamics of

fluid movement within the tanks and the fluid-structure-foundation interactions. One of the

first to provide a solution to this problem was Jacobsen (1949) [30], who determined the

hydrodynamic pressures on a rigid cylindrical tank, anchored to the rigid foundation and

subjected to horizontal acceleration using a simplified analytical method.

In the late 1950s and early 1960s, Housner published two works, Housner (1957) [31] and

Housner (1963) [32], in which he formulated the simplified analytical method, which is still

employed today by practical engineers and current tank seismic design code for estimating the

response of a cylindrical liquid storage tank under seismic excitation. He analytically solved
20
Laplace’s equation for the fundamental mode of rectangular and cylindrical rigid tanks resting

on rigid foundations and subjected to horizontal excitation. His study concluded that the

hydrodynamic pressure of the liquid in a rigid tank can be divided into two components: an

impulsive component and a convective component.

Following the development of Housner’s model, a significant volume of research was

conducted, leading to the creation of various simplified models. These studies were carried

out by many researchers, including Westergrade, Haroun, Velestos, and Malhotra, among

others. In this chapter, we will present the developed simplified models for the seismic

analysis of liquid storage tanks.

2.2 Existing Models for Evaluating Seismic Performance

2.2.1 Housner Model (Mass-Spring Model)

Developed by George Housner, it is a simple and widely adopted mechanical model for

analyzing the seismic response of liquid storage tanks under a ground excitation. The toral

response of the liquid withing a storage tank can be modeled as a mechanical mass-spring

system. This model uses the decomposition of hydrodynamic pressure into impulsive and

convective pressure. The upper portion of the liquid has a sloshing motion has a long

vibration period of 6 to 10 seconds, while the rest of the fluid that moves together with the

tank, (i.e. impulsive motion) has a shorter period of 0.1 to 0.2 seconds. [33]. The primary

objective of this model is to individually calculate the seismic responses of the SDGF

systems. Once these responses are determined, they are combined to obtain the overall tank

base shear and overturning moment.

21
Impulsive pressures, which are directly associated with the inertial forces generated by the

impulsive movements of the tank walls, are directly proportional to the acceleration of these

walls. [31] The proportion of the liquid mass participating in the convective motion is

contingent upon the ratio of the free surface height to the tank diameter of the tank. [16]

Impulsive pressure is represented by a lumped mass that is rigidly attached to the tank walls

through a rigid link, its mass 𝒎𝒊 and height 𝒉𝒊 are defined accordingly to simulate the same

lateral forces and overturning moment as the impulsive liquid pressure, this impulsive mass

moves in unison with the tank wall, contributing primarily to the hydrodynamic pressure on

the tank wall. In contrast, convective pressure is illustrated by a series of masses connected to

the walls by springs elastically attached to the tank wall. These masses decrease in size to

represent different fundamental sloshing modes. The masses are fixed at levels above the base

plate, corresponding to the height of their respective centers of pressure. The components

were subsequently modeled as an equivalent single degree of freedom (SDOF) oscillators.

Figure 2-1 shows the mechanical model of Housner.

𝑚5 ℎ5

𝑚3 ℎ3

𝑚11
H ℎ1
R
𝑚0
ℎ0

Figure 2-1: Housner Model (Housner 1957)

As illustrated in the figure, only the odd-numbered convective masses are present, which

correspond to the antisymmetric convective modes. The even-numbered symmetrical

22
convective modes are absent due to the rigidity of the tank wall; their participation factor is

zero.

Moreover, the mass attributed to the first convective mode significantly outweighs the other

modes (𝒎𝟑 and 𝒎𝟓 ), thereby making it the predominant mode in the convective response

[34]. This dominance of the first convective mode is also emphasized in Housner’s 1963

paper. Housner justified the retention of only the first convective mode in his model by

highlighting its significant role in response to seismic excitation.

d
𝑚𝑐1
ℎ𝑐
H
𝑚𝑖1

ℎ𝑖

Figure 2-2: Simplified Housner Model (Housner 1963)

Figure above on left shows the free surface of the tank deformation of the first convective

mode. On the right it shows the simplified model using mass-spring for impulsive and

convective modes. The convective mode characterized by a convective mass denoted as 𝒎𝒄

and an associated stiffness 𝒌𝒄 , the spring stiffness 𝒌𝒄 is calculated so that the frequency of the

mass-spring system matches the fundamental vibration frequency f the convective response.

The convective mass 𝒎𝒄 and its position 𝒉𝒄 relative to the tank base are determined to ensure

that the lateral force and overturning moment exerted by the mechanical oscillator at the tank

base match those of the oscillating convective mode. The aim of the model is to calculate the

seismic responses of the Single Degree of Freedom (SDOF) systems independently. The

23
maximum lateral base shear and overturning moment at the tank base are derived by summing

the contributions from both the impulsive and convective response modes.

Figure 2-3: Different hydrodynamic pressures induced by a ground excitation on tank [35]
.

24
2.2.2 Veletsos Model for Flexible Tanks

During strong earthquake events, such as those in Alaska, liquid storage tanks often exhibited

poor performance, prompting researchers to develop new procedures for seismic analysis that

account for the flexibility of the tank wall. These procedures are inherently more complex

than those for rigid tanks due to the simultaneous dynamic response of the liquid and the tank

wall's motion.

Previous analytical studies primarily considered tanks as rigid, focusing on the dynamic

behavior of the contained liquid. However, significant post-earthquake damage revealed that

this rigid assumption could lead to an underestimation of the seismic response. This

highlighted the importance of considering the flexibility of the storage tank and its interaction

with the liquid. [36]

When accounting for the flexibility of the tank wall, the impulsive portion can experience

accelerations significantly higher than the peak ground acceleration (PGA). Consequently, the

calculated base shear and overturning moment assuming a rigid tank may yield non-

conservative results. In contrast, the convective portion, or sloshing response, remains

unaffected by the wall's flexibility. [37]

Veletsos proposed one of the first analytical methods that included tank wall flexibility in

1974. [38] Veletsos introduced a straightforward method for analyzing and evaluating the

dynamic response of storage tanks subjected to horizontal ground motion. This method does

not consider convective forces, which must be determined separately using Housner's

procedure for rigid tanks. The tank system is treated as a single degree of freedom. Several

assumptions underpin this approach: the tank's cylindrical cross-section remains circular

throughout the analysis, the tank deflects in a predetermined configuration at any given time,

and the ratio of the height of the liquid to the tank's radius is less than 1.2
25
2.2.3 Yang Model for Flexible Tanks

Yang in 1976 [39], made significant contributions to the understanding of fluid-structure

interactions in storage tanks by incorporating tank wall flexibility into his studies. He

modeled the tank as a beam, which undergoes various shape modes under dynamic loading.

Yang's approach included several key assumptions: the tank's cross-section remained circular,

the deflection exhibited a specific height-wise distribution, and only the impulsive mode was

considered, as the convective mode was unaffected by tank wall flexibility. This model,

represented as a single degree of freedom system where the cross-section retains its shape,

allows for a focused analysis of the impulsive pressure response. The mathematical

formulations can be found in Yang's original document, which provides analytical functions

for critical parameters such as maximum hydrodynamic pressure on the tank wall, overturning

moments, and maximum base shear resulting from this hydrodynamic pressure. The results

reveal notable differences when compared to scenarios involving a rigid tank wall,

highlighting the importance of accounting for wall flexibility in seismic design

considerations.

2.2.4 Wozniak and Mitchell 1978 (unanchored tanks)

Unanchored tanks, characterized by their bottom plates not being fixed to the foundation,

present unique challenges in design and behavior, particularly under dynamic loading

conditions. Unlike anchored tanks, which benefit from the stability provided by fixed

foundations, unanchored tanks experience significant overturning moments due to

hydrodynamic pressures induced by the earthquake, leading to a partial base uplift. This uplift

26
results in nonlinear fluid-shell-soil interactions and an increase in the maximum axial

compression forces within the tank wall. Both dynamic tests and static tilt tests have been

conducted to understand these dynamics, revealing that the uplift mechanism largely governs

the response of these structures.

Wozniak and Mitchell (1978) outlined fundamental principles for designing unanchored

tanks. Their uplift model attributes resistance to overturning moments to a fraction of the

fluid's weight, using a small deflection theory to calculate the critical width at which the

bottom plate loses contact with the ground. This calculation is based on the assumption of two

plastic hinges forming at the plate-shell junction and along the uplifted section.

Additionally, Wozniak and Mitchell (1978) developed a quasi-static beam model for the

bottom plate of tanks, which considers the bending of the bottom plate while neglecting its

membrane action. This model assumes small uplift displacement, minimal uplift length

compared to the tank's radius and models the uplifted bottom plate as a series of beams with

unit width and constant length along the circumference. The beam rests on a solid foundation

and is subjected to uniform loading. At the ultimate state, two plastic hinges form: one at the

junction of the tank shell and bottom plate, and another some distance inward from the shell.

The model assumes the bottom plate membrane force to be zero and neglects shear force. [40]

2.2.5 Haroun and Housner

Recent advancements in mechanical models for analyzing the response of the tank-fluid

system under horizontal excitation have addressed the impact of wall flexibility on the seismic

response of storage tanks. Researchers Haroun and Housner developed a model treating the

system as a single degree of freedom, dividing the impulsive mass into two components: a

27
rigid impulsive mass, rigidly attached to the tank wall, and a flexible impulsive mass,

connected via a spring with an elastic constant k, This approach allows for a precise

evaluation of the tank's behavior, providing insights into the dynamic interactions between the

fluid and the structure. The tank wall is represented using finite element modeling, while the

liquid inside is analyzed as a continuous medium through boundary solution methods. The

model accounts for significant coupling of shell and liquid motions primarily in the impulsive

response mode. The dynamic response of the tank-liquid system is represented by distinct

masses (mc, mf, mr), and their corresponding heights with natural frequencies and damping

ratios. These parameters enable the model to accurately replicate the actual behavior of the

tank-liquid system [41 – 42]. This model, depicted in Figures 2-4

Figure 2-4: Mechanical model for flexible tank [41 – 42].

2.2.6 Velestos Model for Flexible Tanks

In 1984, Velestos developed a procedure to determine the hydrodynamic forces in a tank

subjected to acceleration. The tank is anchored to a rigid foundation, and the flexibility of the

wall is considered. In this framework, the tank wall is analyzed as a uniform cantilever beam.

Assumptions include that the liquid is incompressible, inviscid, and irrotational, the tank is

28
anchored to the foundation, and both the liquid and tank motions are within the linearly elastic

range. The tank wall flexibility influences only the impulsive response.

Given the boundary conditions require the tank bottom and liquid velocities to be equal, with

the vertical component velocity being zero. The radial velocities of the liquid must match the

tank wall, and for flexible tanks, the coupling of the deformable wall and liquid motions is

considered. At the liquid's free surface, the impulsive pressure assumes zero hydrodynamic

pressure. This analysis is more accurate than that of Housner. [43]

2.2.7 Malhotra model for Base Uplift

Unanchored tanks are commonly used in the field, primarily for economic reasons, yet their

behavior presents unique challenges, particularly due to the nonlinear effects associated with

bottom plate uplift mechanisms. This complexity has pushed interest among researchers to

develop realistic models that can effectively simulate the uplift behavior of these tanks.

Malhotra 1995 [44] presented a method for analyzing the uplift behavior of the base plate of

cylindrical tanks under seismic loading addressing both axisymmetric and asymmetric

conditions. The partial uplift of the base plate is a highly nonlinear phenomenon driven by

several factors, the continuously changing contact area between the plate and foundation, the

plastic yielding of the plate material and the membrane action associated with significant

deflection of the plate. The problem considered is a circular plate with radius R and uniform

thickness simulating the base of an uplifting cylindrical tank resting on a rigid foundation.

The plate is subjected to a uniform lateral pressure P due to the hydrostatic pressure of the

tank’s contents and a uniform lateral line load W representing the dead weight of the tank

wall.

29
The plate is constrained against radial displacement and rotation by elastic constraints for the

axisymmetric case, the relationship between the total upward force and the plates

displacement is derived using large deflection plate theory with the corresponding equations

presented in Malhotra’s paper. In the asymmetric case the overturning moment and rigid-body

rotation of the plate boundary are analyzed using a beam model, which represents the uplifted

portion of the plate as a series of independent beams.

A comparison of the beam model with the actual model reveals that for moderate uplift the

width of the uplifted region predicted by the beam model agrees well with the actual model.

However, at very large uplift values discrepancies arise primarily due to differences in the

radial membrane stresses computed by the two models.

Figure 2-5: uplifting plate (a) Axisymmetric,

(b) Asymmetric [44]

2.2.8 Model of Base Isolated Tank

Seismic isolation of liquid storage tanks has been relatively underexplored, with only a

limited number of studies addressing this approach. The primary objective of implementing

an isolation system is to enhance the tanks' ability to dissipate seismic energy. By isolating the

tank from the foundation, the system helps reduce the forces transmitted during an

earthquake, thereby minimizing the risk of damage. It has been shown that isolation can

significantly reduce hydrodynamic base shears, overturning moments, and axial compressive
30
stress in the tank wall without notably increasing the vertical displacements of the liquid

surface due to sloshing. [45] See figure 2-6

Figure 2-6: Liquid storage tank isolated by proposed method

Malhotra proposed an isolation system for storage tanks, where the tank’s base plate rests

directly on the soil, and soft rubber bearings are used below the tank wall. This model,

depicted in Figure 21, is similar to the fixed base system but includes isolation bearings

beneath the tank to achieve a more realistic representation. The analysis parameters for this

system can be derived from results published by Haroun and Housner [46].

Figure 2-7: Model of base isolated tank

31
2.2.9 Joystick Model

Bakalis et a1 [47] developed a 3D surrogate model for liquid storage tanks. This model

considers all translational components of the ground motion, ensuring its applicability to both

anchored and unanchored tanks through static or dynamic analysis methods. Following the

approach of Velestos and Tang (1990), the model decomposes the hydrodynamic pressure

acting on the tank into impulsive and convective components, though only the impulsive

component is considered in the analysis since the convective component does not

significantly affect the overall response of the tank.

The proposed "joystick model" is based on a beam-column element that represents the

impulsive mass of the tank. The tank itself is supported by a rigid beam-spoke system resting

on point or edge springs. To validate the uplift mechanism, a detailed finite element (FE)

model using ABAQUS is developed. Results from the FE model and the joystick model,

specifically regarding uplift and separation length, show good agreement for a certain aspect

ratio of the tanks considered in the analysis. However, the joystick model slightly

underestimates the separation length for tanks with low aspect ratio tanks, though the error

remains acceptable given the model's simplicity.

In terms of plastic rotation, the joystick model exhibits strong agreement with the EC8

guidelines, with only a 15% difference between the two curves.

32
 Figure 2-8: Joystick model (left), and its deflected shape (right)

2.2.10 Vathi Model for Base Plate Uplift

In seismic events, one of the primary failure modes observed in steel storage tanks is partial

base uplift, which can cause significant structural issues. When the base of the tank

experiences an uplift, it can lead to plastic deformation at the connection between the tank

shell and the bottom plate, potentially resulting in fractures and the loss of stored liquid.

To address this issue, Maria Vathi et al. [48] developed a simplified model for dynamic

analysis of storage tanks, particularly focusing on the effects of uplift.

For anchored tanks, the model considers the tank as a spring-mass system, incorporating the

hydrodynamic response of the tank-liquid system, including both convective and impulsive

motions of the stored liquid. In the case of unanchored tanks, the same configuration is

applied, but with the addition of tank rotation (rocking) due to uplift. This is modeled by

adding a rotational spring at the base of the tank to account for the rotational motion induced

by uplift forces.

33
 Figure 2-9: simplified model of anchored tank (left), and unanchored tank (right)

The model includes two degrees of freedom: horizontal motion and rotational motion, while

the convective motion is considered negligible in influencing the overturning moment.

A nonlinear static analysis is used to calculate key parameters, such as the maximum local

strain at the welded connection, with the results supported by a finite element model (FEM)

for determining two critical uplifting parameters: the uplifting length and the uplifting size.

2.2.11 3D simplified model

Colombo et al. [49] developed a simplified 3D model for the seismic analysis of unanchored

tanks. This model provides accurate estimations of both the rocking resistance of the base

plate and the stress distribution on the tank wall. Notably, the simplified nonlinear model

proposed in this study can account for the nonlinear moment-rotation relationship and the

flexibility of the foundation, making it suitable for the 3D dynamic analysis of storage tanks,

allowing for a more accurate representation of the tank’s behavior during an earthquake.

The model builds upon a previous model by Malhotra, a key feature of this model is its ability

to consider the two horizontal components of ground acceleration. The model's advantages
34
include its reliable estimation of compressive axial stress on the tank wall and its ease

implementation in dynamic analysis with earthquake time-histories, making it a practical tool

for engineers.

A recent study [50], focused on the damage caused to the shell-base connection due to

significant plastic rotation of the base plate. According to current design guidelines, such as

EC8 and NZSEE, this rotation is required to be limited to 0.2 radians.

The research highlights the differences in fatigue life capacity between stainless steel and

carbon steel tanks. Previous studies predominantly addressed carbon steel tanks. Carbon steel

connections withstand a higher number of cycles before crack initiation under small strain

amplitudes. Conversely, under large strain amplitudes, stainless steel connections exhibit

greater resistance to crack initiation.

Additionally, the study finds that connections with thicker base plates tend to fail after fewer

cycles compared to those with thinner base plates. However, this issue can be mitigated by

using materials with higher ductility. Enhancing the ductility of the base plates significantly

improves the fatigue life of the connections.

2.3 Conclusion

In conclusion, this chapter has provided a comprehensive overview of previous research on

the seismic response of steel storage tanks subjected to horizontal ground motion. It discussed

various simplified models that are commonly used to simulate the dynamic behavior of

aboveground steel tanks during seismic events, emphasizing the complex interactions between

the liquid, tank walls, and foundation. Notably, the chapter highlighted widely recognized

models, such as the Housner model, which plays a key role in seismic design provisions for

35
storage tanks. Each model considers different aspects of the seismic response. Furthermore,

the models examine potential failure modes, including the risk of partial base uplift and early

failure of the foundation (EFB) during seismic excitation. Overall, this chapter underscores

the importance of understanding these interactions and failure mechanisms to improve the

seismic resilience of steel storage tanks.

36
 CHAPTER 3

STRENGTHS AND LIMITATIONS OF CHOSEN

MODELS

37
3.1 Introduction

Steel storage tanks are particularly vulnerable to seismic events, necessitating improvements

in safety measures to mitigate the risk of potential damage. Many theoretical and

experimental studies have been done recently to address this issue. Building on the analytical

findings of researchers such as Veletsos, Haroun, Malhotra, and many others, various

simplified models were developed to analyze the responses of both anchored and unanchored

tanks. These models have strengths and limitations, which will be discussed in this chapter.

While they offer simplified approaches that make it easier to analyze the dynamic responses

of storage tanks, certain assumptions and approximations within these models may impact

their accuracy.

3.2 Comparative Analysis of Seismic Analysis Models

Housner made several assumptions for this model, only one horizontal acceleration is

considered, the tank wall is considered rigid, the liquid within the tank is assumed

maintaining contact with the tank’s shell, and the base is attached to a rigid foundation, which

prevents the base plate from uplifting. However, Experimental studies conducted by Clough

et al. (1979) [51], Clough and Niwa (1979) [52], Shih (1981) [53], and Manos and Clough

(1982) [54] demonstrated that the axial compressive stress at the shell bottom of unanchored

tanks exceeds that of anchored tanks under similar loading conditions. Additionally, Natsiavas

and Babcock (1988) [55] highlighted significant differences in hydrodynamic loading

between anchored and unanchored tanks, so this may lead to non-conservative analysis of

anchored tanks. As the primary concern in the seismic design of liquid storage tanks is the

prevention of tank wall buckling. So, to address this, it is essential to monitor the compressive

and hoop stresses in the tank wall, particularly at the bottom course, as these stresses are

38
critical to avoiding buckling. Additionally, the determination of liquid sloshing height are

vital for establishing the minimum freeboard required between the filling liquid and the roof

of the tank.

Housner's method, using a response spectrum, is widely used to determine the seismic

response of a storage tank. Until the 1960s, tank wall flexibility was neglected in seismic

response analysis, which focused solely on fluid dynamic behavior. It is important to note that

this model currently considers only one horizontal direction of base excitation, neglecting the

effects of both horizontal and vertical components of ground acceleration.

Various standards and codes for the construction and seismic analysis of cylindrical steel

storage tanks, including API 650 [56] and Eurocode 8 [57]. API 650 uses a mechanical model

first developed by Housner in 1963 [58], with modifications by Wozniak and Mitchell in

1978. Studies show little difference in the parameters of these models for rigid and flexible

tank walls, particularly for impulsive and convective modes.

Eurocode 8, part 4 which covers seismic design and analysis for liquid storage tanks, accepts

the mechanical model by Velestos and Yang (1977) [59]

In the context of unanchored tanks, various analytical models have been developed to analyze

the nonlinear uplift mechanisms of the base plate. These models have been incorporated into

numerous standards and design provisions. Notably, Wozniak and Mitchell (1978) and

Clough (1977) introduced quasi-static models that simplify the description of uplift

mechanisms, making them suitable for practical design applications

The proposed models by Malhotra offer several advantages, notably its ability to account for

the nonlinear effects of both membrane action and material yielding, making it suitable for

analyzing the dynamic response of unanchored tanks. Additionally, Malhotra's method

considers the effects of load reversals, and the energy dissipation associated with yielding,

39
providing an accurate and efficient approach for analyzing asymmetrically uplifted plates.

However, the method has some limitations, primarily its complexity and time-consuming

nature, especially when used for dynamic response analyses, where solutions require

numerous integration steps. Despite these challenges, the method remains valuable for

detailed and accurate analysis of uplift phenomena in cylindrical tanks under seismic loading.

The beam model significantly reduces computational effort, requiring at least an order of

magnitude less than the plate model. It offers a more cost-effective and realistic solution by

incorporating both material and geometric nonlinearities in the analysis, representing an

improvement over the previously proposed method. A major limitation of the beam model is

its inability to fully capture the circumferential membrane stresses, which are critical for

accurately modeling the behavior of the plate, especially under large uplift. The beam model

is most effective when analyzing uplift values that are no greater than 1% of the plate radius.

Sliding Isolation Model is effective in minimizing base shear and the displacement of the

tanks. Moreover, these systems can potentially lower costs in areas such as the foundation,

anchorage, and materials for the tank, possibly offsetting or even surpassing the additional

expenses for isolation bearings, flexible membranes, and base stiffeners.

However, there are some disadvantages. Implementing a flexible membrane between the tank

wall and the base plate to prevent spills is complex. Additionally, out-of-round deformation of

the tank wall near the base necessitates the use of a stiffener ring. Furthermore, this model

only considers the responses of convective and impulsive modes under horizontal

accelerations, which may not fully capture all dynamic behaviors during seismic events.

The Joystick Model provides several advantages, including the ability to incorporate

anchorage effects, deliver reasonable accuracy with good computational efficiency, and

perform 3D analysis of liquid storage tanks subjected to multiple components of ground

40
motion. Additionally, it can be easily implemented using general-purpose structural analysis

software like ABAQUS [60]

However, a notable limitation of the joystick model is its inability to address complex

hydrodynamic effects and fluid–structure interactions. This omission may lead to minor errors

in the calculation of certain parameters, particularly where these interactions are significant.

Model of Vethi, while this simplified model offers several advantages, such as its

applicability to nonlinear dynamic analysis and its ability to compute local strain, it also has

notable disadvantages. The model’s reliance on numerical simulations and finite element

models means it requires considerable computational resources and specialized expertise.

Moreover, the simplifications made in the model, such as neglecting the flexibility of the tank

wall, may not fully capture the intricate behavior of tanks during seismic events, potentially

leading to inaccuracies in certain scenarios where more complex interactions occur.

The 3D Model, model also addresses the interaction between the tank and its foundation,

considering both the rocking resistance of the liquid-loaded base plate and the hysteretic

damping effect. These effects are modeled using elastic nonlinear springs and equivalent

rotational linear viscous dampers, respectively.

Furthermore, the model can estimate the contact length between the tank wall perimeter and

the foundation during partial uplift, providing crucial insights into the maximum compressive

axial stress on the tank wall. This is achieved by estimating the angle of the arc in contact

between the tank wall and the foundation during partial base uplift. a key factor in

understanding the tank’s response to seismic loads.

In conclusion, the proposed model offers reliable estimations of both the rocking resistance of

the base plate and the stress distribution on the tank wall, considering the soil-foundation-

structure interaction. It also shows good agreement with the results obtained from the finite

41
element model and Malhotra's model. Overall, the simplified nonlinear elastic model offers a

reliable and efficient approach for the dynamic analysis of unanchored tanks, particularly in

estimating the stress distribution on the tank wall and the interaction with the foundation

during seismic events.

However, with advancements in computational capabilities and the increased availability of

commercial finite element analysis (FEA) software, numerous researchers have shown a

growing interest in this subject. Simplified methodology for risk-targeted seismic

performance assessment for the evaluation of unanchored liquid storage tanks, focusing on the

elephant foot buckling (EFB) failure mode. Their approach employs a pushover-based

analysis to couple the seismic demand, derived from spectral acceleration at the impulsive

period, with the limit-state capacity of the tank wall, defined by stress criteria. The analysis is

based on a refined 3D finite element (FE) model of the tank, developed using Abaqus

Software, which accounts for nonlinearities such as base uplifting and sliding, as well as both

horizontal and vertical components of ground motion and the resulting hydrodynamic

pressures.

One of the key advantages of this methodology is its computational efficiency, especially

compared to dynamic analyses of similarly refined 3D nonlinear tank models. However, the

proposed approach has several limitations. Notably, it focuses exclusively on verifying the

EFB failure mode, neglects soil-structure interaction effects and assumes a rigid foundation.

Additionally, the pushover analysis cannot account for fatigue from cyclic loading, which

may affect tank performance under repeated seismic events.

In conclusion, while the methodology offers a promising approach for assessing the seismic

performance of tanks, further research is needed to determine its applicability for evaluating

other failure modes, such as base uplifting, sliding, and top wall buckling, as well as to

incorporate soil-structure interaction and fatigue effects. [61]

42
3.3 Comclusion

The interaction between the tank and its foundation plays a critical role in determining its

dynamic response. The flexibility of both the tank wall and the supporting foundation can

significantly influence the overall system response, primarily due to the dominant impulsive

response. Understanding these interactions is essential for accurately modeling the dynamic

behavior of storage tanks, thereby improving design practices and enhancing structural

resilience in seismic events.

43
CONCLUSIONS

The thesis provides a comprehensive understanding of the performance, and seismic response

of aboveground steel storage tanks under seismic excitation. The research delves into the

different failure modes experienced by these tanks during past earthquakes, including

elephant's foot buckling, diamond-shaped wall buckling, base plate uplifting, anchor bolts

failure, roof damage, and piping connection failure.

Dynamic studies on the seismic response of unanchored tanks have highlighted the

complexity of their behavior under earthquake loading, because of the complicated

interactions between the fluid, the tank shell, and the foundation. The current models

available are not fully satisfactory, these models may effectively capture many aspects of tank

behavior; however, there is a need for future research to develop more sophisticated finite

element models. Such models should incorporate material and geometric nonlinearities to

provide a deeper understanding of the complex interactions involved.

Looking towards for future research on the seismic performance of steel storage tanks,

potential work could focus on developing advanced models in order to:

1. Investigate the effects of incorporating both horizontal and vertical components of

ground motion in the seismic analysis of storage tanks

2. Develop advanced analytical and numerical models that can accurately capture the

complex fluid-structure-soil interactions, including the effects of base uplift and

sliding, to provide more comprehensive and reliable analysis.

3. Explore the influence of soil-structure interaction on the seismic response of storage

tanks, particularly for tanks resting on flexible soil foundations

4. Examine the long-term fatigue behavior of tank-foundation connections under

repeated seismic loading cycles, as this can lead to damage and failures
44
 5. Investigate the effectiveness of seismic isolation systems for storage tanks, including

the use of base isolation, and expand the understanding of their performance under

various seismic loading conditions

6. Conduct further experimental and numerical studies to validate and refine the existing

simplified models, ensuring that they can accurately capture the complex nonlinear

behavior of storage tanks subjected to seismic excitations.

Another avenue for future investigation could involve the development of simplified

methodologies for risk-targeted seismic performance assessment, particularly in evaluating

unanchored liquid storage tanks with a focus on possible failure modes This could include

exploring innovative analysis techniques, such as pushover-based approaches, to couple

seismic demand with the limit-state capacity of tank walls to define stress criteria and enhance

the overall understanding of tank behavior during seismic events.

With the advancement in computational capabilities and the availability of commercial finite

element analysis software, there is a growing opportunity for further research in this domain.

45
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