Structural Safety 70 (2018) 48–58

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Structural Safety

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Storm surge fragility assessment of above ground storage tanks

https://doi.org/10.1016/j.strusafe.2017.10.0020167-4730/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: sabarethinam.kameshwar@gmail.com (S. Kameshwar).

Sabarethinam Kameshwar ⇑, Jamie E. PadgettDepartment of Civil and Environmental Engineering, Rice University, 6100 Main St. MS-318, Houston, TX 77005, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 July 2016Received in revised form 24 April 2017Accepted 7 October 2017Available online 19 October 2017

Keywords:Above ground storage tanksSurgeBucklingFlotationFragilityMetamodel

This study proposes a dual layer metamodel based approach to develop parameterized fragility functionsfor above ground storage tanks (ASTs) subjected to hurricane induced storm surge. ASTs are extensivelyused in petrochemical facilities for storing large volumes of hazardous substances. Failure of ASTs due tostorm surge may lead to spills that can cause severe environmental damage and considerable economicloss. A significant number of ASTs are located in coastal areas which are susceptible to hurricanes, such asin the Houston Ship Channel, Texas. However, tank design guidelines are deficient in addressing preven-tion of surge related failures. Although their vulnerability has been exposed during past hurricanes, theliterature lacks studies on performance analysis of ASTs during storm surge events. In this light, a methodis presented to derive fragility functions for the most important failure modes of ASTs – flotation andbuckling – in addition to the system fragility, considered as series system of failure modes. For this pur-pose, a novel dual layer metamodel based approach is proposed where a limited number of simulationsare used to train the first binary classifier, which predicts failure of tanks, upon which the second meta-model is trained; the final metamodel is used to derive parameterized fragility functions. This approachsignificantly reduces the number of limit state evaluations, which may require costly finite element sim-ulations, and enables accurate fragility assessment to capture the nonlinear behavior of tanks under surgeloading, while also considering the correlation between failure modes during system fragility modeling.Results indicated that the fragility estimates of a typical tank obtained with the dual layer metamodelcompare well with those derived by high fidelity methods such as Monte Carlo Simulations. In orderto demonstrate the application of the parameterized fragility functions to study the effect of variationin design and construction parameters, fragilities of four case study tanks are evaluated. The results high-light the effect of parameter variation on the fragilities and offer insights into the influence of alternativedesign impacts on tank vulnerability. For example, anchoring tanks significantly reduces the probabilityof flotation; however, anchoring leads to buckling dominated failures.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Aboveground storage tanks (ASTs) are often used for bulk stor-age of hazardous materials, including a variety of fuels and chem-icals at industrial sites such as petrochemical and oil & gasfacilities. ASTs are typically thin walled steel shells constructedas vertical cylindrical structures. This shape allows ASTs to sustainlarge internal liquid pressure using a relatively thin shell. Conse-quently, ASTs are very light in weight and susceptible to flotationduring storm surge induced by hurricane events. Furthermore,the thin walls make ASTs susceptible to buckling failures due toexternal forces such as wind and storm surge. The vulnerabilityof ASTs has been observed during past hurricanes such as Katrinaand Rita where several tanks failed due to hurricane wind and

surge, spilling over 26500 m3 of petroleum products into the envi-ronment [1]. In one of the cases, flotation failure of just one tanklead to a 4000 m3 spill leaving over 1700 homes un-inhabitableat a cost over $330 million in clean up and litigation [2]. The vul-nerability of tanks to storm surge was exposed again during subse-quent hurricane events such as Ike and Gustav [3,4]. Tank failurescan lead to catastrophic oil spills which not only adversely affectthe surrounding environment [5,6] but also impact the physicaland mental wellbeing of surrounding communities [7].

In order to prevent tank failures during extreme hazards, suchas earthquakes, design codes like API 620 [8] and API 650 [9] pro-vide standards for designing ASTs. API 650 provides extensiveguidelines to prevent failures due to earthquakes and hurricanewinds. For example, anchorage is prescribed to prevent uplift dur-ing earthquakes and the use of stiffening rings and thicker wallsare recommended to prevent shell buckling due to strong winds.However, API 650 and other similar design codes provide no

S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58 49

mandatory guidelines to prevent failures due to storm surge andfloods. Any measures to prevent surge and flood related failuresare left at the discretion of the purchaser. Furthermore, most stateregulations fail to impose any additional requirements for suchprevention, with few exceptions such as the state of Colorado[10] which requires a method to be declared for flotation preven-tion during flood conditions in order to obtain a permit.

Literature also has largely focused on seismic performanceassessment of ATSs with limited investigation of ASTs’ perfor-mance during hurricane winds and tsunamis. A large number ofexperimental and analytical studies have been conducted tounderstand the seismic performance of ASTs [11–14]. Furthermore,studies have used data on past performance of AST under earth-quakes to derive empirical fragility curves using probit analysisand convolved the fragilities with seismic hazard curves to esti-mate risk [15,16]. For hurricanes, studies have primarily focusedon deterministic wind buckling capacity estimation of ASTs [17–20], while some studies propose new designs for AST componentssuch wind girders [21]. Similarly, few deterministic studies existon behavior of ASTs during tsunamis where forces acting on ASTsare evaluated and deterministic performance analyses are con-ducted [22,23].

While risk assessment methods exist for ASTs subjected toearthquakes and several deterministic performance assessmentstudies are present for ASTs under hurricane wind and tsunamis,there is a large gap in the literature in understanding and modelingthe performance of ASTs subjected to hurricane induced stormsurge. The literature lacks studies that adequately address thesafety of ASTs subjected to storm surge. Godoy [1] reports the per-formance of ASTs during hurricanes Katrina and Rita and attributestank failure to flotation and buckling. A recent study by Kamesh-war and Padgett [24] performed fragility assessment of a casestudy tank under storm surge considering flotation and bucklingfailure. However, the methodology used by Kameshwar and Pad-gett [24] was not tailored to support fragility assessment of a regio-nal portfolio of tanks or to study effect of variation in designparameters on fragility; furthermore, a system fragility formula-tion considering multiple failure modes is also lacking. In view ofthe compelling evidence of AST vulnerability to storm surge, con-sequences of tank failure, and lack of research on storm surge per-formance assessment of ASTs, this study develops a method toderive fragility functions for ASTs subjected to storm surge. Herein,fragility functions are developed for flotation, buckling, and systemfailure of ASTs. The fragility functions express the probability offailure given the geometry and design parameters of ASTs, in addi-tion to storm surge inundation. These parameterized fragility func-tions will be helpful in understanding the failure mechanisms asthe geometry and the tanks’ features are varied. Additionally, theresulting probabilistic models for AST performance can be usedin the future to assess the benefits of large-scale regional coastalprotection systems.

The next sections of the paper present the steps involved indeveloping the parameterized fragility functions. Using reconnais-sance reports, section two identifies the potential failure modes ofASTs under storm surge to be studied. In section three, the resis-tance and demands for flotation failure, and demands for bucklingfailure are evaluated using closed form equations while finite ele-ment simulations are used to determine buckling resistance. Usingthe capacities and demands, classifiers are trained to identify tankfailures, which are used further to derive the fragility functionsdescribed in sections four and five. In order to demonstrate theapplication of the fragility functions, the performance of four casestudy tanks is evaluated and the insights obtained from the resultsare discussed in section six. Finally, section seven summarizes theconclusions and the main contributions.

2. Failure modes

A reconnaissance report on performance of ASTs in the states ofTexas and Louisiana, USA, during hurricanes Katrina and Rita byGodoy [1] attributes tank failures to tank dislocation due to flota-tion, global shell buckling caused by strong winds, and debrisimpact. Cozzani et al. [25] have analyzed over 272 flood relatedaccidents in industries and identified ASTs as one of the most vul-nerable components. They identify floatation, debris impact, col-lapse of tank due to water pressure, i.e. buckling, and collapse offloating roof as possible causes of tank failure. Based on the perfor-mance of ASTs during Katrina, Rita, and the 2012 floods in Coloradothe Regional Response Team 6 fact sheet #103 [26] (RecommendedBest Practices for Flood Preparedness) also identifies flotation,buckling due to floods and storm surge, and debris impact aspotential failure modes. Furthermore, analysis of a case study tanksubjected to storm surge by Kameshwar and Padgett [24] alsofound flotation and surge buckling to be important failure modes.In addition to flotation, sliding and wave impact may also damagetanks during a hurricane. However, as a first step towards under-standing the storm surge performance of ASTs, this study will pri-marily focus on fragility assessment for the two most importantfailure modes (based on prevalence from past reconnaissance):flotation and storm surge buckling.

Flotation failures are caused when the uplift created by thesurge, due to buoyancy forces, is greater than the self-weight ofthe tank. Some tanks may be fitted with anchors to prevent flota-tion; however, such tanks may also float if the uplift forces over-come the combined resistance due to the anchors and the tank’sself-weight. A buoyant tank may float away from its position andspill its contents as it settles at a different place or hits other tanksnearby. Additionally, spills may be caused by ruptured pipelines, adirect consequence of AST flotation. Spills, due to flotation failureof tanks, may adversely affect the environment and accrue costsdue to clean up and litigation. Considering the consequences of aspill, initiation of flotation is considered a failure in this study, eventhough not every buoyant AST may result in a spill. Shell bucklingof tanks, in extreme cases, may also lead to rupture of the tankshell resulting in a spill. Therefore, initiation of buckling is alsoconsidered as a failure. Shell buckling due to storm surge occurswhen the hydrostatic water pressure exceeds the load resistingcapacity of the tank. Usually, ASTs are primarily designed to with-stand large internal pressures and external wind pressure; how-ever, they have limited capacity against storm surge. Overall,failure of ASTs due to either of the failure modes will affect thepost-hurricane functionality of tanks leading to economic losscaused by delay in restarting operations, and in severe cases bothfailure modes can lead to spills. Therefore, the system failure ofASTs is modeled with a series system assumption; i.e. ASTs areassumed to fail if they float and/or buckle.

3. Load and resistance models

3.1. Flotation failure

The buoyancy forces exerted on a tank due to storm surge areevaluated using the Archimedes principle; i.e., the buoyancy forceequals the weight of the water displaced. Therefore, the buoyancyforce on a tank prior to flotation is evaluated as:

Ff ¼ 1000qwgpD2S

4; S < H ð1aÞ

Ff ¼ 1000qwgpD2H

4; S > H ð1bÞ

Table 1Distributions for variables.

Parameter Distribution Reference

D (m) Deterministic –H (m) Deterministic –ts, tb, tr (m) Deterministic based on H, D and ql –qw Uniform (1.020, 1.029) Assumedqs Uniform (7.750, 8.050) Assumedql Uniform* AssumedL (m) Uniform (0.0, 0.9H) Assumedfs (MPa) Lognormal* [28]fc (MPa) Lognormal* [29]c1, hef, d (mm) Deterministic –e1 Normal (0.99, 0.18) [27]e2 Normal (1.04, 0.26) [27]e3 Normal (0.96, 0.19) [27]

* Tank specific parameters should be used.

50 S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58

where qw is the relative density of sea water (i.e., the ratio of thedensity of sea water to the density of water – 1000 kg/m3), D is tankdiameter, S is the inundation level, g is acceleration due to gravity,and H is the height of the tank. Typically, tank height, (H), is greaterthan the surge height; therefore, Eq. (1a) is used most of the times.The above equations are applicable before the tank floats; once thetank is afloat, the buoyancy force will be equal to the self-weight ofthe tank.

Anchors, if any, and the self-weight of the tank provide resis-tance against the buoyancy forces. Therefore, net resistance againstflotation (Rf) is evaluated as the sum of self-weight and anchorresistance:

Rf ¼ p DHts þ D2tb4

þ D2tr4

!1000qsg þ 1000qlpD

2Lg4

þ Ra ð2Þ

In the above equation, ts refers to the thickness of the tank’sshell, tb is shell thickness of the tank’s base, tr is the thickness ofroof shell, qs is the relative density of steel, L is the level of the liq-uid stored in the tank, the relative density of the stored liquid is ql,and Ra is the resistance provided by anchors. Strength of cast inplace anchors is governed by several failure modes – yielding ofthe anchor, concrete cone failure, and side face blow out. The fail-ure mode with least strength dictates the strength of an anchor.Overall resistance provided by the anchors and the capacity asso-ciated with each of the anchor failure modes can be evaluatedusing the following equation [27]:

Ra ¼ na �minfRs;Rcc1;Rcc2;Rcbg ð3aÞ

RS ¼ ASf ut ð3bÞ

Rcc1 ¼ e116:8h1:5ef f

0:5c ð3cÞ

Rcc2 ¼ e2AC;N

A0C;N

WS;NRcc1 ð3dÞ

Rcb ¼ e320:9c0:751 A0:5h f 0:75c ð3eÞ

na is the number of anchors; Rs, Rcc1, Rcc2, and Rcb are anchor yieldstrength, concrete cone failure strength when sufficient edge dis-tance is available, concrete cone failure strength in absence of suf-ficient edge distance, and side face blowout strength respectively.As is anchor bolt area which is calculated using the diameter ofthe anchor bolt – d (in mm) as pd2/4; fut refers to ultimate steelstrength which is calculated as 1.9 times steel strength (fs in N/mm2); hef (mm) represents embedment depth of anchor and fc (in

N/mm2) is concrete strength; AC;N=A0C;N is the ratio of the projected

areas of the concrete cone for anchors with limited and large edgedistance respectively, wS,N � 1 is a modification factor; c1 (in mm)represents edge distance, and Ah (in mm2) is the area of the anchorhead. In Eqs. (3c), (3d), and (3e), e1, e2 and e3 are bias removal factorsmodeled as normal random variables; distribution parameters aredescribed in Table 1. Some parameters in Eq. (1) through 5 are takenas deterministic variables such as D, H, As, hef, c1, and na; while sev-eral variables may be considered as random variables in the analy-sis, such as qw, qs, ql, L, fs, fc. Tank diameter, height, and shellthicknesses are considered deterministic since these parametersare assumed to be known for each tank. For random variables,Table 1 shows the distribution considered for the parameters; lowerand upper bounds for uniform random variables and mean andstandard deviation for normal random variables are shown inparenthesis. Herein, relative density of sea water and steel are con-sidered as uniformly distributed variables in order to account forthe small natural variations in these values. These random variablesare assigned uniform distribution due to lack of data on their actual

probability density function (PDF). Based on the contents of a tank,the relative density of the stored liquid may be known apriori andthe variation in its density due to factors like temperature can betaken into account, in absence of such data the relative density ofthe stored liquid is assumed to be uniformly distributed. Liquidlevel (L) is also considered as a random variable since ascertainingthe level of contents stored in each tank at the time of a hurricanemay not be possible. Therefore, the liquid level in a tank is assumedto be a uniformly distributed variable ranging between zero and90% of tank height (H). For several variables, uniform distributionis chosen since it is unbiased – gives equal weight to each valuein the range; with more information, the distribution type may beupdated. Steel and concrete strength can be assumed to follow alognormal distribution for a given mean and standard deviationvalue.

3.2. Buckling failure

Excessive water pressure created by storm surge may bucklethe tank shell. A triangular hydrostatic pressure distribution isadopted herein to estimate the pressure demand on the tank shell.For a given depth of inundation (S), the maximum pressure isexerted at the base of the tank equal to 1000qwgS, and no pressureis exerted on the portions of the shell above the water surface.Since acceleration due to gravity (g) is constant and qw is a randomvariable, surge height, i.e. the depth of inundation, serves as anintensity measure for surge buckling demand on ASTs. It isacknowledged that additional water pressure may also arise dueto the flow velocity of the incoming surge. However, ADCIRC sim-ulations of Hurricane Ike and several synthetic storms for the TexasGulf Coast region show that over the Houston Ship Channel area,which has over 4000 ASTs, the velocity of oncoming surge overland is very low [30]. Hence, the above emphasis on hydrostaticloading is reasonable.

Resistance against buckling is provided by structural action ofthe tank shell (the thicker the tank’s shell the grater the resistance)and structural members such as stiffening rings. The liquid storedin the tank creates a static pressure opposing the storm surge’shydrostatic pressure, thereby providing additional resistanceagainst buckling. This study performs finite element simulationsto obtain the uncertain buckling response of tanks; therefore, thisstudy does not adopt any existing stochastic model for the buck-ling response but instead simulates and derives the behavior. Inorder to evaluate the buckling capacity of ASTs, they are modeledin a finite element software, LS-DYNA [31]. In order to model thetanks, the thickness of shell courses of ATSs and section propertiesof the stiffening rings, if required, are decided based on API 650 [9].

S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58 51

These design details – thickness of shells and stiffening ring prop-erties – also depend on the height, diameter, relative density of liq-uid stored, and the material used for construction of the tank. Thetank and roof shells are modeled with four node and three nodeshell elements respectively and the stiffening rings are modeledas beam elements. Hydrostatic water pressure created by stormsurge is directly applied on the outer surface of tanks’ walls andsimilarly, the pressure created by the stored liquid is also directlyapplied to the inside of the tanks’ wall. Since the buckling demandis expressed in terms of inundation depth at each tank, the capacityof ASTs is also expressed in terms of surge height as the maximumsurge height or inundation depth that the tank can resist beforebuckling (Scr). In order to evaluate the buckling capacity of ASTs,the surge height is increased at each time step and an equilibriumanalysis is performed. Buckling is identified when the eigen valuesof the system become negative and the corresponding surge heightrepresents the inundation level beyond which the tank will buckle;i.e. surge height capacity of the tank (Scr).

In the above procedure, using efficient sampling strategies,several parameters are varied such as liquid level in the tank, rel-ative density of stored liquid, strength of material used for tankshell, tank height, and tank diameter. Several variables are con-sidered as deterministic parameters such as the tank heightand diameter, while uncertainty may be propagated throughothers. Furthermore, the uncertainty in buckling capacity of ASTsdue to the presence of geometric imperfections on the shells ofASTs is also considered since geometric imperfections may signif-icantly affect the buckling capacity of tanks [32]. For this pur-pose, imperfections are modeled using a two-dimensionalFourier series expression where the coefficients are treated asrandom variables, which are modeled based on imperfectionsmeasured on tanks and silos in Germany and Australia [33].Using the geometric imperfections modeling scheme outlinedby Kameshwar and Padgett [33] and the above mentioned finiteelement analysis procedure, the critical surge height causingbuckling (Scr) is evaluated.

4. Fragility analysis

In order to facilitate fragility assessment of a portfolio of ASTs ina region, this study aims to develop parameterized fragility func-tions that can estimate failure probability for any tank with a givenset of parameters such as tank height, diameter, and relative den-sity of contents. Conventional fragility assessment methods such asfirst order reliability methods can provide great insight into theimportance of variables but the conventional methods cannot beused for efficiently developing parameterized fragility functions.Furthermore, gradient based methods may not be best suited forproblems where the limit state equation is noisy and evaluationof the limit state equation involves finite element simulations;such as in the case of buckling analysis where stochastic imperfec-tions on the tank shells introduce uncertainty in buckling capacityestimate, even if all variables are fixed. Monte Carlo simulations(MCS) can handle such limit states but MCS is computationallyexpensive, especially when evaluation of the limit state requiresexpensive finite element analysis. The computational cost of apply-ing MCS for fragility assessment of a portfolio of ASTs will be pro-hibitively expensive. Therefore, in order to efficiently address thesechallenges, this study uses a metamodel-based procedure to deriveparameterized fragility functions for a portfolio of ASTs. Ametamodel-based approach requires a large number of limit stateevaluations upfront; however, once the metamodels are trained,fragilities for thousands of ASTs can be evaluated without anyadditional limit state evaluations, which may need finite elementsimulations.

4.1. Fragility assessment methodology

The first step in the metamodel-based fragility assessment, likeconventional methods, is formulation of the limit state equation.Next, key deterministic and stochastic variables, on which the fra-gility functions need to be parameterized, are identified, such astank dimensions and relative density of contents. Using the antic-ipated range of the selected variables, Latin Hypercube Sampling(LHS) [34] is performed to span the space of these parameters uni-formly and efficiently. For each parameter combination generatedby LHS, the state of the system is determined, i.e. safe or failed, byevaluating the limit state function using closed form equations forflotation analysis and using finite element simulations for bucklingfailure. In the next step, metamodels are trained to identify fail-ures, given the set of input parameters. This study considers threewidely used classifiers: support vector machines (SVM), randomforest (RF), and logistic regression (LR); however, others may alsobe used. Support vector machines project the space of input vari-ables into a higher dimensional space, using kernels, where classi-fication is easier [35]. Random forest is an ensemble of logicdecision trees, each trained on a subset of the parameters space.For classification of test data, the classification decision of each treein the ensemble is collectively used to determine the majority out-come for the test data [36]. Logistic regression is a discriminativemodel and it also provides the probability of belonging to one ofthe two classes, i.e. failed or safe in this study [37]. All of the threemodels are trained and their predictive accuracy is evaluated on atest data set. Typically, the size of the test data set is about 10–20%of the training data set and the test data set is not used for trainingthe metamodels. Based on performance, i.e. their predictive accu-racy, the best metamodel is chosen.

Even if a metamodel has good predictive accuracy, its qualitymust be further assessed to avoid over-fitted models and ascertainthe level of uncertainty around the predictions of the metamodel;therefore, the quality of the chosen metamodel is assessed further.For example, in the case of SVM, the number of support vectors is animportant criterion that reveals the quality of the model. In addi-tion to accuracy, the confidence bounds of logistic regression pre-dictions provide additional information on the goodness of fit ofthe trained model and the uncertainty around the predictions ofthe model. Over-fitted models can be improved by retraining themetamodel with a different set of training parameters. Models withlarge uncertainty around the predictions may be improved with alarger training data set since a larger training data set would reducethe uncertainty around the parameter estimates of the metamodel.For example, in a logistic regression model, the standard errors,which represent the uncertainty in the parameter estimates,decreasewith increase in the number of training data points. There-fore, a larger training data set may be used when limit state evalu-ations are computationally inexpensive as in the case of flotationanalysis. In the case of buckling failure, the size of the training dataset may be limited by the computational cost of the finite elementsimulations required to evaluate the limit state. In such cases, thisstudy proposes a dual layer metamodel strategy where the firstmetamodel, which must have good accuracy, may be used as aproxy of the limit state. Thus, a larger set of training data can beobtained, without significant computational cost, upon which thesecond metamodel can be trained. This approach leads to modelswith less uncertaintywithout loss of accuracy, compared to the firstmetamodel; the approach is demonstrated herein for AST bucklingfragility assessment. The dual layer approach is based on the ideathat binary limit state system output, i.e. fail or safe, is enough fortraining a binary classifier. Therefore, a metamodel with high clas-sification accuracy, whichmay have considerable uncertainty in theprediction, may be used to train a second metamodel. Since a largetraining set can be obtained using the first metamodel, the second

52 S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58

metamodel has significantly less uncertainty around the predictionand it retains the accuracy of the first metamodel. The final meta-models may be used to propagate uncertainties in variables andestimate the fragility, using MCS or by numerical integration.Fig. 1 summarizes the above mentioned procedure in a flowchart.The flowchart also suggests possible actions when metamodelsoverfit or have low accuracy. In the following sub-sections, thismethodology is used to evaluate the fragility of ASTs for the twoindividual failure modes and the tank system.

4.2. Flotation fragility: un-anchored tanks

The limit state equation for un-anchored tanks can be obtainedas the difference of Eqs. (2) and (1) and substituting Ra = 0, sinceun-anchored ASTs are considered. The limit state equation can bewritten as:

gfunac ¼1000 p DHts þD2tb4

þD2tr4

!qsgþ

pD2Lqlg4

�minqwgpD

2S4

;qwgpD

2H4

( )" #

ð4Þ

In the above equation, tank diameter, tank height, relative den-sity of stored liquid, and surge level are chosen to parameterizethe fragility function. Additionally, product design stress (Sd), whichis themaximumallowable stress on the shell coursesdue to the stor-age of products in the tanks, is also used to parameterize the fragilityfunction. The product design stress value determines the materialused for the shell courses; a large value of Sd decreases the shellthickness (ts) which would reduce the self-weight of the tank.Ranges of all the variables used to parameterize the fragility func-tions are shown in Table 2. Tank height is usually correlated to thediameter of the tank; therefore, based on the analysis of portfolio

Fig. 1. Dual layer fragility as

of tanks in the Houston Ship Channel, which has over 4000 ASTs,lower and upper bounds of aspect ratios (ratio of height to diameter)areobtainedas expðð1� 2 lnDÞ=4Þandexpð3:07� 0:95 lnDÞ respec-tively. For a given diameter, the aspect ratio is randomly chosenwithin the corresponding range of aspect ratio. With the range ofparameters shown in Table 2, Latin Hypercube Sampling is per-formed to generate 1.0 � 104 samples. For each combination ofparameters, tanks are designed as per API 650 and the thickness ofthe base plate and the shell courses (tr, tb, ts) are obtained and stiff-ening rings are designed, if required. Other variables such as qw

and qs that are not used to parameterize the fragility functions arerandomly varied as per distributions shown in Table 1. For eachparameter combination created by LHS, the limit state is evaluatedand the tanks are classified as either failed or safe. Following thesame procedure, a test set containing 1.0 � 103 samples is gener-ated. The metamodels are trained using the set of 1.0 � 104 param-eter combinations of D, H, L, ql, and Sd. The accuracy of the threemetamodels, SVM, RF, and LR, for the test set is 98.9%, 98.3% and99.9% respectively. The relatively simple limit state equation in Eq.(4) is easily emulated by all of themetamodels; therefore, high accu-racies are observed. Basedon the accuracy, logistic regression is cho-sen to evaluate the probability of flotation, which describes theprobability of anevent (i.e. flotation) for a given set of parameters as:

PðFlotation D;H; L; S;qlj Þ ¼ 11þ expð�lðD;H; L; S;ql; SdÞÞ

ð5Þ

In the above equation, lð�Þ is the logit function, which may be apolynomial function. In this study, step-wise logistic regression isperformed in order to include the most influential parameter com-binations in the logit function. For flotation of un-anchored tanks,the logit function of the trained LR model is:

sessment methodology.

Table 2Range of variables used to parameterize fragility functions.

Parameter Lower bound Upper bound

D (m) 5.0 70.0ql 0.5 1.0L (m) 0.0 0.9HS (m) 0.0 10.0Sd (MPa) 137.0 196.0d (m) 6.35 � 10�3 6.35 � 10�2

c1 (m) 0.05 0.6hef (m) 0.15 1.0fc (MPa) 20.0 40.0fy (MPa) 250.0 550.0sp (m) 0.25 4.0

S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58 53

lunancðD;H; L; S;qlÞ ¼ �8:67þ 0:43D� 0:64H � 0:10L� 3:14ql

þ 39:53S� 38:47Lql � 4:47� 10�3D2 ð6ÞSd is not included in the logit function shown above. Sd only

influences the thickness of the tank shell, which influences theself-weight of the tank. In comparison to the buoyancy forcesand the weight of the liquid stored in the tank, the self-weight ofthe ASTs is negligible; therefore, its effect is not significant and isnot included in the logit function. Even though, the logistic regres-sion model predicts with high accuracy, uncertainty in the failureprobability estimates of the LR model is further assessed byinspecting the confidence bounds of the fragility predictions for atypical tank and comparing the fragility of the tank with MCS.For this purpose, the fragility a typical tank with 10.0 m height,15.0 m diameter containing a liquid with relative density of 0.75is evaluated. The liquid level is varied as a uniformly distributedrandom variable and the failure probability is calculated at differ-ent surge heights. Uncertainty in liquid height is propagatedthrough Eq. (5) by numerically integrating the fragility functionwith the probability density function (PDF) of liquid height ðf LÞ:

PðFlotation D;H; S;qlj Þ ¼Z

PðFloatation D;H; l; S;qlj Þf LðlÞdl ð7Þ

Fig. 2 compares the failure probabilities obtained using MCSand logistic regression; furthermore, the lower and upper boundsof the 95% confidence interval for the predictions obtained usinglogistic regression are also shown. The confidence interval bounddirectly shows the range within which the actual failure probabil-

Fig. 2. Validation of the logistic regression based un-anchored fragility function forrepresentative tank.

ity may lie; indirectly, it shows the magnitude of standard errorsassociated with the coefficients. Small standard errors are desir-able since it ensures that uncertainties contributed by the logisticregression model are low and uncertainty added due to metamod-eling is minimal. The close agreement between the MCS performedwith 1.0 � 105 simulations and LR results as well as the extremelynarrow confidence intervals observed in Fig. 2 instill confidence inthe chosen model; therefore, the logistic regression model withlogit function shown in Eq. (6) is chosen to model the flotation fra-gility of un-anchored tanks. Additionally, the prediction errors inthe final logistic regression model, i.e. the rate of false positivesand false negatives, are included as model uncertainty for fragilityassessment. For this purpose, the false positive and negative ratesof all the metamodels, for different damage modes such as flotationand buckling, are obtained at various surge levels. Next, MonteCarlo simulations are performed with the logistic regression modelwith 10.0 � 103 simulations to obtain the failure probability foreach storm surge level. Within each simulation, the positiveresponse (i.e. failure) is changed to a negative response if a uni-formly distributed random number is less than the false positiverate. Similarly, a negative response is changed to a positiveresponse if the uniformly distributed random number is less thanthe false negative rate. This procedure allows the propagation ofuncertainties due to metamodel error in the fragility estimates.Using this approach, the flotation fragility curve, for the un-anchored case study tank, is obtained and is shown in Fig. 2 asLR – MCS. Since the logistic regression (LR) models have high accu-racy and low bias, the LR – MCS curve and the LR – Point estimatesare very close to each other.

4.3. Flotation fragility: Anchored tanks

The parameterized fragility function for flotation failure ofanchored tanks is derived following a similar procedure. The limitstate equation for anchored tanks is:

gfac ¼ 1000 p DHts þ D2tb4

þ D2tr4

!gqs þ

pD2Lqlg4

"

þRa �minqwgpD

2S4

;qwgpD

2H4

( )#ð8Þ

The equation shown above is similar to Eq. (4) with an addi-tional term for the resistance provided by anchors, Ra, which canbe evaluated using Eq. (3a). For anchored tanks, the fragility func-tions are parametrized on d, c1, hef, fc, fy, sp, and na in addition to S,D, H, L, ql, and Sd. Using the procedure described in the sectionabove, the three metamodels are trained using a data set contain-ing 1.0 � 104 parameter combinations. The accuracy of the threemetamodels, SVM, RF, and LR, on a test data set containing 1.0 �103 parameter combinations is: 94.4%, 93.4%, and 97.3% respec-tively. Since LR has the best accuracy, it is chosen for further good-ness of fit assessments and eventual fragility modeling. The qualityof the logistic regression model is further assessed by inspectingthe 95% confidence bounds of the predictions for a case study tankand the predictions are also compared against MCS. Fig. 3 showsthe confidence bounds and the comparison of the fragility curvewith MCS. Close agreement between the fragility curve obtainedfrom MCS and LR highlights the accuracy of the logistic regressionmodels; however, the LR model has very wide 95% confidenceinterval. The wide confidence interval is indicative of large uncer-tainty associated with the coefficients for the terms in the logitfunction. This uncertainty can be primarily attributed to theuncertainty in the strength of the anchors, contributed by the biasremoval factors e1, e2, and e3. The standard error of the coefficientscan be considered as standard deviation estimates; if large

Fig. 3. Assessment of the logistic regression model for representative anchoredtank flotation fragility.

54 S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58

uncertainties are present, a larger number of simulations will berequired to estimate the variance accurately. Therefore, to reducethe width of the confidence bounds for the logistic regressionmodel a larger training data set containing 1.0 � 105 is samplesused. Since the limit state function in Eq. (8) can be evaluatedeasily in closed form, training the logistic regression model with1.0 � 105 data points is feasible. The accuracy of the LR modeltrained with 1.0 � 105 data points is 97.3%. For the new LR model,Fig. 4 shows the confidence bounds and the comparison with MCS.The decrease in the width of the confidence interval can be clearlyobserved in Fig. 4 and the accuracy of the models is also evidentfrom the close agreement between MCS and LR fragility curves.Furthermore, the fragility curve obtained by propagating theprediction errors in the logistic regression model (LR – MCS inFig. 4) also closely matches the fragility estimates from MCS andthe logistic regression estimate, which does not include theprediction errors. This comparison further instills confidence inthe fragility estimates obtained from the logistic regressionmodel. The logit function obtained to predict the flotation ofanchored tanks contains 62 terms, for brevity the logit functionis not shown here.

Fig. 4. Validation of representative anchored tank flotation fragility model.

4.4. Buckling fragility

The limit state equation for buckling failure can be written asthe difference between the critical inundation causing buckling,i.e. capacity, and the depth of inundation from the base of the tank.Therefore, the limit state equation can be expressed as:

gbuck ¼ Scr � S ð9ÞThe above equation is an implicit limit state equation that is

influenced by parameters such as tank height, diameter, liquidheight, relative density of stored liquid, and design stress of thematerial used to construct the tank; additionally, the geometricimperfections also add uncertainty to Scr. Therefore, S, D, H, L, ql,and Sd are used to parameterize the fragility function. Since eachlimit state evaluation requires non-linear finite element bucklinganalysis a small training set of 1800 parameter combinations isgenerated via LHS. For each parameter combination, a tank is fullydesigned and modeled in LS-DYNA and the limit state is evaluated.An additional set of parameter combinations containing 200 sam-ples is generated using LHS for testing the metamodels, i.e. to eval-uate the accuracy of the trained metamodels. Accuracy of SVM, RFand LR are: 94.9%, 93.9% and 95.4% respectively. Following the pro-cedure used for flotation fragility, the most accurate model, i.e. LRfor buckling, is chosen for further assessment. In Fig. 5, the fragilitycurve of the representative tank is plotted along with the 95% con-fidence bounds and it is compared with a tank specific fragilityestimate (TS – point estimate in Fig. 5). The procedure outlinedby Kameshwar and Padgett [24] is adopted to produce the tankspecific estimates. The 95% confidence interval for the chosen LRmodel, shown in Fig. 5, can be observed to be very wide; in orderto improve the confidence interval, a larger set of training data isrequired.

The computational cost of evaluating the buckling limit stateprohibits obtaining a sufficiently large training data set. Since thetrained LR model has good accuracy, it is used as a proxy for thelimit state, thereby avoiding expensive finite element simulations.Using this approach, a large training set containing 1.0 � 104

parameter combinations is generated and a second logistic regres-sion model is trained on the output of the first LR model. The pre-dictions of the resulting LR model for a case study tank arecompared with the tank specific fragility estimate and the confi-dence intervals are also shown in Fig. 6. The second LR modelhas the same accuracy of 95.4% and narrower confidence boundshighlighting the reduced uncertainty in the coefficient estimates.

Fig. 5. Assessment of the logistic regression model for buckling fragility.

Fig. 6. Validation of buckling fragility model.

S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58 55

The fragility curve obtained by propagating the prediction errors inthe second logistic regression model also closely matches the fragi-lity curves obtained from the logistic regression model withoutpropagating uncertainty in predictions. This observation can beattributed to the relatively high accuracy and the unbiased predic-tions of the final logistic regression model. Furthermore, overall, agood agreement is observed between the generalized fragilitymodel and the tank specific fragility estimate.

As per Kameshwar and Padgett [24] tank surge buckling specificfragility estimates are obtained by performing 200 finite elementbuckling simulations for a specific tank considering random geo-metric imperfections with varying levels of liquid level in the tank,while all other parameters are kept constant. The variation inresulting critical surge buckling heights (Scr) can be attributed tothe liquid level and random geometric imperfections. Therefore,a regression model is developed for the specific tank; the tankspecific regression model predicts Scr as a function of liquid level.For the specific tank considered in Fig. 6, the tank specific regres-sion model which predicts Scr has R2 = 0.99, mean average percent-age error of 3%, and root mean square error (RMSE) of 0.18 m,which is very low in comparison to Scr values. The performancemetrics show that the tank specific regression model is accurate.Therefore, considering the RMSE of the regression model as asource of uncertainty, the 95% confidence bounds of the tank speci-fic fragility estimates are added to Fig. 6. The point estimatesobtained from the parameterized logistic regression fragility modelcan be observed to lie within the 95% confidence bounds of thetank specific fragility estimate. Furthermore, the confidence inter-vals of the parameterized logistic regression model shown in Fig. 6reflect the uncertainty in the coefficients of the terms in the logitfunction for the buckling fragility model so the confidence boundsonly represent the uncertainties emanating due to model fitting.Therefore, the observation that the tank specific fragility estimateslie outside the parameterized LR confidence interval does not nec-essarily indicate that the considered AST is an extreme outlier orthat the LR model is overly inaccurate. The differences in fragilityestimates from the LR model and the tank specific estimatebetween 3.5 m to 4.5 m surge arise due to the slight differencesin the fragility estimates from the logistic regression model andthe tank specific estimate. These differences are further com-pounded by the uncertainties in the buckling response emanatingfrom the geometric imperfections and the level of liquid stored inthe tank, which affect both of the fragility estimates. Moreover, itmust be noted that the generalized fragility function is applicable

for a wide range of tank dimensions and parameters while the tankspecific estimates are only applicable for only one tank. This dis-cussion also highlights the need to identify the inundation levelswhere the fragility estimates must be predicted most accurately.These ranges of inundation levels may be informed by convolutionof stochastic models of hurricane storm surge with the fragility ofthe above ground storage tanks (ASTs). However, at present, suchanalysis is not feasible because stochastic models of storm surgefor the Houston Ship Channel region, which is the case study regionfor this study, are not available. However, in the future as stochas-tic models for hurricane storm surge become available, regions ofthe fragility curve that need to be modeled most accurately maybe identified. The logit function obtained for buckling failure is

lbuckðD;H; L; S;ql; SdÞ ¼ �46:72þ 4:55Dþ 0:87H � 3:10S

� 16:76ql þ 1:81Lþ 6:50� 10�2Sd

� 2:08� 10�2DH þ 1:50SH � 0:40HL

� 0:18D2 � 0:30H2 � 4:71� 10�2H2S

þ 1:5� 10�2H2Lþ 3:06� 10�3D3

þ 9:8� 10�3H3 � 1:85� 10�5D4 ð10ÞThe above equation includes design stress (Sd) in the logit func-

tion, which was not included for flotation fragility, since it controlsthe thickness of the tank shell, which contributes significantly tothe buckling resistance of ASTs. This observation shows the abilityof the selected approach to include the most influential parametersin the fragility function. While this methodology for evaluatingparameterized fragility functions provides one key contributionof the paper, its application to understand the influence of designparameter variation also offers insight for practical purposes aslater discussed in the paper.

5. System fragility

This study considers the two failure modes in series since tankfailure in either mode may influence the immediate functionalityof the tanks and, in the worst case, a spill may result. System fra-gility functions are developed separately for anchored and un-anchored tanks. System fragility assessment would entail evalua-tion of both flotation and buckling limit state functions. However,buckling limit state evaluation requires computationally expensivefinite element simulations. Therefore, the final metamodels usedfor fragility assessment of individual modes are used as proxy fortheir respective limit states. For assessment of buckling failures,the depth of inundation with respect to the base of tank is consid-ered as the intensity measure for fragility assessment. For tanksthat do not become buoyant, the depth of inundation for bucklingis same as the depth of water at the tank’s location. However, whenthe tank becomes buoyant, for buckling assessment, the waterheight above the base of the tank is used. Metamodels are trainedon a data set containing 1.0 � 104 sample generated by LHS toidentify cases, which have failed in either of the modes or bothof the modes. For un-anchored and anchored tanks, logistic regres-sion is found to be the most accurate model with 99.8% and 95.9%accuracy respectively with narrow confidence intervals. Sincelogistic regression models are selected for fragility assessment,the failure probability is directly predicted using the logit function.In the above mentioned procedure, correlation between failuremodes is inherently considered in the system failure outcome.The correlation between different failure modes arises becausethe values of the common input parameters for Monte Carlo simu-lations and logistic regression models; for both of the failuremodes inputs are identical, i.e. they are fully correlated. Addition-ally, correlations in the failure modes arise due to increasing like-

56 S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58

lihood of failure for both, flotation and buckling, failure modeswith increasing surge height. The logistic regression model forthe system fragility is trained on the correlated response of LRmodels for flotation and buckling. Therefore, the LR model for sys-tem fragility, which is trained on the correlated responses, alsolearns the correlations in the failure modes along with the systembehavior. Thus, the approach proposed herein automatically con-siders correlations between failure modes while the predicting sys-tem response and fragility. Application and accuracy of the systemfragility functions are demonstrated on four case study tanks in thenext section.

6. Results and discussion

Four ASTs are selected to demonstrate the applicability of thefragility functions to a study the effect of variation in parametersand possible application to a regional portfolio of ASTs. The tanks’dimensions cover a wide range of tank dimensions observed intank farms and industrial facilities. Table 3 shows the dimensionsof the tanks and the assumed relative density of the contentsstored in each of the tanks. Fragility curves for flotation, buckling,and tank system are plotted for all four tanks for un-anchored andanchored cases. For anchored tanks, 25.4 mm diameter anchorbolts with 30 cm embedment and 10 cm edge distance are pro-vided at 0.6 m spacing. Anchor spacing is kept constant for allthe tanks to enable consistent comparison of the flotation fragilitycurves. Fig. 7a to d show the fragility curves for tanks 1 through 4respectively, un-anchored and anchored. The four figures clearlyshow that for un-anchored tanks, system failure is dominated byflotation failure. Due to the low self-weight of ASTs, flotation fail-ure commences at a very low inundation depth; on the other hand,

Table 3Case study tank dimensions and content relative densities.

Tank # D (m) H (m) ql

1 30.48 13.11 0.62 6.00 9.00 0.83 13.50 16.80 0.74 42.00 12.60 0.9

Fig. 7. Fragility estimates for case study tanks: a

due to the inherent buckling resistance of ASTs buckling does notcommence until around 3.0 m of inundation. Due to this disparityin ASTs’ inherent resistance for different failure modes, flotationfailure completely dominates the system behavior of un-anchored tanks.

Comparison of flotation failure probability among un-anchoredand anchored tanks shows that anchoring the tanks drasticallydecreases the probability of flotation. For the given arrangementof anchors, anchoring is most effective for small diameter tanks –2 and 3. Since the anchors are spaced at an equal distance for allfour tanks, the total resistance provided by the anchors increaseslinearly with an increase in diameter while the uplift forces see aquadratic increase. Therefore, for large diameter tanks, the anchorsmust be closely spaced for a drastic decrease in flotation probabil-ity. Nevertheless, anchoring the tanks leads to a decrease in flota-tion failure probability. Due to the decrease in flotation probabilityof anchored tanks, the buckling failure probability exceeds theflotation probability at large surge heights, as in the case of tank4. However, failure of small diameter anchored tanks, such as tanks2 and 3, is dominated by buckling failure at all surge heights.Therefore, the system fragility is primarily dominated by bucklingfailure, which can be clearly seen from the comparison of bucklingand system fragility curves in Fig. 7b and c. For large diameteranchored tanks, both of the failure modes influence the system fra-gility. Observations form Fig. 7a and d show that for large diameteranchored ASTs, flotation dominates at low surge heights and athigh surge heights buckling is dominant. This trends in can beattributed to the anchor spacing and the inherent buckling resis-tance in ASTs. At smaller anchor spacing, the system failure fortanks 1 and 4 would be dominated completely by buckling failureas in the case of tanks 2 and 3.

In order to further study the effect of the anchor spacing crite-rion on the system fragility, all of the four tanks are installed withanchors to prevent flotation up to 6.0 m of surge with a fill levelequal to 25% of the tank’s height. Additionally, the maximumanchor spacing is limited to 1.5 m to obtain a practical numberof anchors for all tanks (a limit that is reached for tank 2). This cri-terion results in anchor spacing of 0.47 m, 1.5 m, 1.46 m, and 0.45m for tanks one to four, respectively. The system fragility resultsfor these anchor spacing values are shown in Fig. 8. For large tanks,tank 1 and 4, the new anchor spacing selection criterion leads to

) tank 1, b) tank 2, c) tank 3 and d) tank 4.

Fig. 8. Fragility estimates for case study tanks: a) tank 1, b) tank 2, c) tank 3 and d) tank 4 for different anchor spacing values.

S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58 57

smaller anchor spacing values; therefore, a small decrease in fragi-lity can be observed in Fig. 8a and d, with respect to Fig. 7a and d.Since the new anchor spacing criterion only leads to a small changein anchor spacing, the same trends in the system fragility, i.e. dom-inance of buckling fragility, are observed in Fig. 7a and d andFig. 8a and d. For tanks 2 and 3, the smaller tanks, the new anchorselection criterion provides larger anchor spacing that leads to anincrease in the flotation fragility in Fig. 8b and c, with respect tofragility curves shown in Fig. 7b and c. Even though, flotation fra-gility is observed to increase for these two small tanks, the systemfragility is still dominated by buckling failure for tank 2. However,for tank 3, both buckling and flotation contribute to system failure,which is in contrast with the trends observed in Fig. 7c where 0.6m anchor spacing is provided.

These observations suggest that for anchored tanks, dependingon the anchor spacing, flotation may dominate at low surge heightsand buckling may dominate the system behavior at large heights,above 4.0 m. Such high surges can be expected in regions housingASTs such as the Houston Ship Channel [38], which has over 4000tanks. Moreover, climate change adds uncertainty to the frequencyand magnitude of future storm events. Some studies predict anincrease in the intensity and frequency of storms [39] while otherstudies predict a decrease in the frequency of tropical storms alongwith an increase in the intensity of hurricanes and storms [40]. Inorder to prepare for such uncertain circumstances, the fragilityfunctions developed in this study would help understand the fail-ure modes and facilitate risk assessment for the regional portfolioof tanks.

7. Conclusions

A fragility assessment methodology has been proposed for hur-ricane induced storm surge fragility assessment of above groundstorage tanks, which is lacking in the literature. Herein, flotationand shell buckling are identified as the most common failuremodes of ASTs during storm surge. For each failure mode, proba-bilistic load and resistance models are developed and the limitstate function is evaluated using closed form equations and finiteelement models. Fragility functions are developed using a noveltwo layer metamodel based approach. For this purpose, differentmetamodels – support vector machine, random forest, and logisticregression – are first trained. Next, the best metamodel, based on

prediction accuracy, is further assessed for its goodness. In thisstudy, for all failure modes, logistic regression models were foundto be most accurate; therefore, the logistic regression models werefurther evaluated based on their 95% confidence interval. In thecase of un-anchored tanks, the first logistic regression model isfound to be sufficient; while for anchored tanks, a larger trainingdata set containing 1.0 � 105 samples was required to obtain a sat-isfactory model. However, in order to obtain an satisfactory logisticregression model for buckling failure with narrow confidenceintervals a second metamodel is trained on the first metamodel,which has good accuracy. System fragility functions for un-anchored and anchored tanks are also evaluated considering thecorrelation between the failure modes. This approach required1.0 � 104, 1.0 � 105, and 2.0 � 103 limit state evaluations for train-ing the metamodels for un-anchored flotation, anchored flotationand buckling failure respectively; however, once the metamodelsare trained, uncertainty propagation and fragility assessment canbe performed without additional simulations. The results obtainedfrom the metamodels developed in this study are verified for a casestudy tank against high confidence estimates such as Monte Carlosimulations. Such metamodels are very useful when the limit statefunction evaluation is expensive as in the case of buckling wherefinite element analysis is required for each limit state evaluation.Moreover, the fragility functions developed are parameterized onAST parameters such as tank dimensions and contents of the tanks.Therefore, the fragility functions developed herein can facilitateefficient fragility assessment of a regional portfolio of ASTs.

In order to demonstrate the applicability of the fragility func-tions developed in this paper, and gain insight on the influenceof design details on tank failure modes and vulnerability, thefragilities of four tanks are evaluated. The four tanks have differentdimensions and are assumed to store liquids with relative densityvarying from 0.5 to 1.0. For each tank, un-anchored and anchoredconditions are considered and system fragility curves are evaluatedalong with fragility curves for individual failure modes. The resultsprovide several insights regarding the response of ASTs understorm surge. In un-anchored condition, all tanks have higher prob-ability of flotation than buckling; consequently, the system fragi-lity is completely dominated by flotation failure. However, whenthe tanks are anchored the probability of flotation reduces signifi-cantly. Since the same anchor spacing was used for all of the tanks,anchors were most effective for small diameter tanks. This resultindicates that for large diameter tanks the anchor spacing should

58 S. Kameshwar, J.E. Padgett / Structural Safety 70 (2018) 48–58

be kept very small in order to prevent flotation. Depending on theeffectiveness of the anchors, the system response is influenced byeither or both of the failure modes. Generally, flotation dominatesthe response at low surge heights and buckling dominates the fail-ure at higher surge heights. This general trend is also observedwhen tanks are anchored to prevent flotation at 6.0 m inundationwith a 25% fill level. Although, this study has only assessed the per-formance of a few tanks, the derived fragility functions can be usedfor risk assessment of an entire portfolio of tanks. Future work willfocus on risk assessment of ASTs in the Houston Ship Channelregion. Transition of failure modes for anchored tanks from flota-tion to buckling failure suggests that anchoring alone may not besufficient to prevent failure of ASTs; therefore, future work shouldalso focus on alternative methods to reduce the system fragility oftanks in surge prone regions. Additionally, in the future, as stochas-tic models for hurricane storm surge in the HSC become available,regions of the fragility curve that need to be modeled most accu-rately may be identified by convolving the fragility functionsdeveloped herein with the stochastic storm surge model.

Acknowledgement

The authors would like to acknowledge the support for thisresearch by the Houston Endowment and Shell Center for Sustain-ability. Any opinions, findings, and conclusions or recommenda-tions expressed herein are those of the authors and do notnecessarily reflect the views of the funding agencies. The authorswould also like to acknowledge computational facilities providedby Data Analysis and Visualization Cyberinfrastructure (NSF GrantOCI-0959097).

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