Stochastic Modeling of Geometric Imperfectionsin Aboveground Storage Tanks for Probabilistic

Buckling Capacity EstimationSabarethinam Kameshwar, S.M.ASCE1; and Jamie E. Padgett, A.M.ASCE2

Abstract: This study develops a methodology to model random geometric imperfections in welded cylindrical aboveground storage tanks(ASTs). Precise modeling of the imperfections in ASTs is important for quantitative assessment of buckling strength of ASTs under variousexternal loads, such as wind and storm surge, because buckling phenomenon is highly sensitive to stochastic imperfections. Even thoughbuckling of ASTs due to extreme winds during hurricanes has been reported as a common failure mode, existing literature lacks models forrandom geometric imperfections to facilitate probabilistic buckling capacity estimation of ASTs under such external loadings. Current studieson buckling behavior of ASTs use deterministic imperfection models, neglecting the inherent uncertainties present in the imperfections andthe response of ASTs. Usually, studies use eigen or buckling mode shapes to depict imperfections which are most detrimental to the bucklingstrength of ASTs; furthermore, the amplitude of these imperfections is also random but is often deterministically chosen as a multiple of theshell thickness. Such imperfection modeling methods may lead to conservative buckling load estimates and are unable to facilitate prob-abilistic capacity estimation, which is important for reliability assessment or future calibration of existing design guidelines per the loadresistance factor design (LRFD) philosophy. Therefore, this study develops a stochastic model for global imperfections in welded cylindricalASTs using a two-dimensional Fourier series expression. The coefficients of the Fourier series are modeled as random variables, which arebased on imperfection data measured from similar structures in Germany and Australia. The proposed imperfection modeling scheme isapplied to different sample tanks to assess the influence of imperfections on the wind buckling capacity. The results show presenceof significant uncertainty in buckling capacity. Furthermore, useful insights are gained on the effects of imperfection magnitude on themean and variance of the capacity. In the future, observations from this study can be used to probabilistically determine the knockdownfactors for buckling design of ASTs or to support regional risk assessment of existing portfolios of ASTs. DOI: 10.1061/AJRUA6.0000846.© 2015 American Society of Civil Engineers.

Introduction

Aboveground storage tanks (ASTs) are often used to store crudeoil, petrochemical products, and other hazardous material in indus-trial facilities. Safety of these structures during extreme events suchas hurricanes should be ensured to protect the surrounding environ-ment from the catastrophic consequences of a spill. However, sev-eral tank failures have been observed during past events such asHurricane Katrina; flotation of tanks and shell buckling due to ex-cess wind pressure have been observed to be the most commonmodes of tank failure during hurricanes (Godoy 2007). Bucklingof the shell affects postevent tank functionality, for example, byhindering floating roof operations. Furthermore, in extreme cases,buckling can lead to rupture of the tank shell and consequent loss ofcontents causing environmental pollution. Because buckling of thinshell cylindrical structures is a serious problem, several studieshave addressed the buckling load capacity estimation of these

structures under axial and circumferential loads (Batterman 1965;Forrestal and Herrmann 1965; Karman and Tsien 2003). However,these studies were performed on smaller shells used for aerospaceapplications. Recent studies have investigated buckling capacity oflarge-scale tanks, typical of energy production or industrial facili-ties, using finite-element analysis. For example, Flores and Godoy(1998) perform bifurcation-buckling analyses (eigenvalue bucklinganalyses) to estimate the buckling strength of cylindrical shellstructures subjected to hurricane winds. Chen and Rotter (2012)use linear-bifurcation analysis and nonlinear-geometric analysisto study buckling of tanks and silos under wind pressure. All ofthese studies, on small-scale and large-scale cylindrical shells, ac-knowledge the sensitivity of buckling loads to the imperfections inthe cylindrical shells. Moreover, experimental tests have shown thatthe realistic buckling strength differs from these theoretical or ana-lytical buckling strength estimates because of the imperfect shapeof the tank (Teng and Rotter 2006).

For small shells, studies have used existing databases of exper-imentally measured imperfections, such as the one by Arboczand Abramovich (1979), to develop imperfection models and as-sess probabilistic buckling capacity of small shells under axialloads (Schenk and Schuëller 2003). In this study by Schenkand Schuëller, the imperfections are modeled as a nonhomogene-ous Gaussian random field by decomposing the covariance matrixof the imperfections. Other covariance decomposition methods(Davis 1987) and spectral methods (Shinozuka and Deodatis1996) can also be used to model imperfections. However, suchmethods require a large databank of imperfections to developa generalized imperfection model which may be applied to

1Graduate Research Assistant, Dept. of Civil and Environmental Engi-neering, Rice Univ., 6100 Main St., MS-318, Houston, TX 77005. E-mail:sk56@rice.edu

2Associate Professor, Dept. of Civil and Environmental Engineering,Rice Univ., 6100 Main St., MS-318, Houston, TX 77005 (correspondingauthor). E-mail: jamie.padgett@rice.edu

Note. This manuscript was submitted on February 3, 2015; approved onAugust 7, 2015; published online on September 16, 2015. Discussion per-iod open until February 16, 2016; separate discussions must be submittedfor individual papers. This paper is part of the ASCE-ASME Journal ofRisk and Uncertainty in Engineering Systems, Part A: Civil Engineer-ing, © ASCE, C4015005(9)/$25.00.

© ASCE C4015005-1 ASCE-ASME J. Risk Uncertainty Eng. Syst., Part A: Civ. Eng.

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cylindrical shells of different dimensions, but currently, such anextensive database is not available for aboveground storage tanks.Therefore, for large-scale tanks and silos, such generalized imper-fection models are not available in the literature. Thus, existingstudies deterministically model the imperfections in tank shellsto evaluate the buckling capacities of tanks and silos. Particularly,global imperfection in the shells of large-scale tanks and siloshave been modeled using eigenmode shapes and buckling modeshapes of these structures. Portela and Godoy (2005a, b) modeledwind pressure around the circumference of a case-study tankbased on experimental data and estimated the buckling load capac-ity of the tank with the first few buckling mode shapes as imper-fections using geometric nonlinear analyses. Similarly, Chen andRotter (2012) also used eigenmode shapes and buckling modeshape based imperfections to assess wind buckling resistanceof tanks with different aspect ratios. Beyond the shape of imper-fections, the magnitude of imperfection is also uncertain; there-fore, in the studies mentioned previously (Portela and Godoy2005a, b), the first few buckling modes were scaled to differentlevels, and sensitivity to magnitude of imperfection was studied.However, in reality, the shape and magnitude of imperfections areuncertain and may not be similar to the buckling or eigenmodeshapes of the tank. Furthermore, imperfections modeled via theeigenmode and buckling mode shapes were found to be the mostdetrimental to the buckling strength of ASTs in comparison toother forms of imperfections such as ovalization of the shelland combination of shell ovalization with local imperfection inthe shape of a rectangle (Greiner and Derler 1995). Hence, forreliability assessment of ASTs or to evaluate uncertainty in thebuckling capacity of ASTs, the adoption of imperfectionsperfectly correlated to the buckling mode shape, as done in thepast, is undesirable. Recent studies have experimentally measuredthe imperfections in tanks and silos (Hornung and Saal 2002;Teng et al. 2005). Availability of imperfection data offers alterna-tives to using eigenmode shapes for imperfection modeling inbuckling analyses, such as the use of Fourier representation ofthe imperfections, spectral simulations, covariance decomposition,and other stochastic simulation methods. However, application ofthese methods to derive a generalized imperfection model requiresa large database of imperfections from tanks, which is not avail-able yet.

To address the lack of probabilistic imperfection models whichmay support reliability assessment of ASTs or help assess the un-certainty in buckling capacity for various loads, this study proposesa probabilistic global imperfection modeling scheme. For this pur-pose, global imperfection data measured from tanks (Hornung andSaal 2002) and silos (Teng et al. 2005) are gathered. First, Fourieranalysis of the imperfection data is performed to uncover thepatterns in imperfections which are used to derive a generalizedprocedure to generate random instances of global imperfectionsfor ASTs. The proposed imperfection modeling scheme is de-scribed in the following sections along with selection of imperfec-tion magnitude and details of a normalization procedure whichexpands the scope of applicability of the proposed imperfectionmodeling scheme to tanks of different dimensions. Furthermore,an application of the proposed imperfection modeling scheme ispresented in this paper where few representative tanks from theHouston Ship Channel region are modeled, and samples of tankimperfections are generated to facilitate Monte Carlo simulationof shell buckling attributable to wind pressure. The results ofthe Monte Carlo simulations are used to estimate the mean andstandard deviation of the buckling strength of the ASTs, and theobserved trends are discussed in this paper.

Characterization of Imperfections

A central objective of this study is to develop a generalized globalimperfection model for ASTs. For this purpose, all existing pub-lished global imperfection data from tanks and silos are analyzedto detect any common features among the imperfections in differenttanks and silos. The common features are used to develop an im-perfection modeling scheme which also incorporates randomnessto the imperfections. To this end, imperfections measured fromtanks in Germany (Hornung 2000) are analyzed, and in additionto observations gathered from the analysis of tanks in Germany,this study also incorporates the observations from analysis of im-perfections on silos in Australia (Teng et al. 2005) to propose animperfection modeling scheme. Hornung and Saal (2002) inspectedfour oil storage tanks in Germany with unstiffened shell made ofunalloyed or low alloy steel for local and global imperfections. Thefour tanks had different dimensions; the heights of the first twotanks are 13.29 m, 17.07 m, respectively, whereas the third andfourth tanks are 10.00 m high, and their radii are 5.00, 35.00,5.75, and 7.00 m, respectively. However, complete imperfectionmeasurements were only obtained for Tanks 3 and 4 by Hornung(2000); therefore, imperfections measured from these two tanks areanalyzed in this study. For each of these two tanks, the imperfec-tions were measured at 640 locations using tachymetry, with a pre-cision of 2 mm. These imperfection measurements were adjustedfor eccentricity of the tanks; for details, refer to Hornung andSaal (2002).

One-dimensional and two-dimensional Fourier analysis are per-formed on imperfection data to detect any systemic patterns in theimperfections. First, one-dimensional Fourier analysis of imperfec-tions measured around the circumference of the tank is performed.For this purpose, a full Fourier representation shown in the follow-ing equation is used

IðθÞ ¼XNn¼0

ðan cos nθþ bn sin nθÞ ð1Þ

where an and bn are Fourier coefficients, θ is the angle aroundthe circumference of the tank (0 to 2π), N is the number of Fouriercoefficients, and IðθÞ is the value of imperfection at angle θ.Figs. 1(a and b) show the magnitude of the Fourier coefficientsof the imperfections measured around the circumference of thetanks for the nth harmonic, An ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2n þ b2n

p, for Tanks 3 and 4,

respectively. In Fig. 1(a), it can be observed that for lower harmon-ics, the magnitude of the coefficients is high. Furthermore, nearn ¼ 6 and n ¼ 12, the coefficients have peaks; i.e., in the neigh-borhood of n ¼ 6 and n ¼ 12, the magnitude of the coefficientsdecreases with increase or decrease in the value of n. Similarly,in Fig. 1(b), peaks can be clearly identified around n ¼ 7 andn ¼ 14. In both Figs. 1(a and b), peaks corresponding to higherharmonics have lower value. The harmonics around which peaksare observed correspond to the number of panels in the circumfer-ential direction or a multiple of the number of panels. Tank 3 has sixpanels, whereas Tank 4 has seven panels; therefore, peaks areobserved near harmonics that are multiples of six and seven,respectively. Figs. 2(a and b) show the magnitude of Fourier coef-ficients of the full Fourier series representation, shown in Eq. (1), ofimperfections measured along the meridians of Tanks 3 and 4. Un-like imperfections measured around the circumference, the Fouriercoefficients for imperfections measured along the meridians havevery small peaks near harmonic n ¼ 6, which corresponds to thenumber of panels along the height of the tank.

One-dimensional Fourier analysis provides several valuable ob-servations on the correlation of imperfections with the number of

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panels in the tank. However, it ignores the correlation between theimperfections in the circumferential and meridional directions.Therefore, a two-dimensional (2D) Fourier analysis was also per-formed on the imperfections using the following half cosine waveexpression:

Iðθ; zÞ ¼XMm¼0

XNn¼0

cos

�mπzH

�ðCmn cos nθþDmn sin nθÞ ð2Þ

In the above expression, H is the height of the tank, Cmnand Dmn are the Fourier coefficients, and M and N are the numberof harmonics in the meridional and circumferential direction,respectively. The amplitude of the Fourier coefficients, Amn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C2mn þD2

mn

p, for imperfections in Tank 3 is shown in Fig. 3.

Similar to one-dimensional Fourier analysis, lower harmonicsin both directions, circumferential and meridonial, have largemagnitudes. On close inspection, it can be observed that the co-efficient Amn assumes a large value where either m or n corre-sponds to a multiple of the number of panels in the meridionalor circumferential direction. Even though one-dimensional Fourieranalysis of imperfections along the meridonal direction does notshow a prominent trend, as it was observed for imperfections in

circumferential, the 2D Fourier analysis shows a trend relatingamplitude of coefficients in both of the directions. Similar trendsare also observed from two-dimensional Fourier analysis of imper-fections on Tank 4.

Teng et al. (2005) analyzed imperfections on three silos inAustralia. The observations in this research on the imperfectionsobtained from tanks in Germany are similar to the observationsby Teng et al. on silos in Australia—namely, the peaks in the am-plitude of one-dimensional Fourier coefficients of circumferentialimperfections are observed near harmonics that are multiples ofnumber of panels along the circumference; a less prominent trendis observed in one-dimensional Fourier coefficients of imperfec-tions in the meridonal direction, and a higher magnitude oftwo-dimensional Fourier coefficients is observed for harmonicsthat correspond to the number of panels in either direction. Thesimilarities in the imperfections of the two shell structures situatedin two different continents suggest that the imperfections arisebecause of manufacturing process of these shell structures. Thisconclusion is consistent with Teng and Rotter (2006) where theysuggest that specific imperfection patterns may arise in shellstructures because of the manufacturing process. Based on the ob-servations of one-dimensional and two-dimensional Fourier series

Fig. 1. Fourier coefficient amplitude in circumferential direction

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analysis on imperfections in large-scale shell structures, fromthis study and from Teng et al. (2005), the following section willpropose a new probabilistic imperfection modeling scheme forlarge-scale shell structures.

Proposed Imperfection Modeling Scheme

The observations from Fourier analysis of imperfections in tankand silo shells are used in this section to develop a probabilisticimperfection modeling scheme. The proposed imperfection mod-eling scheme uses a two-dimensional Fourier series representation,which was also used by Teng et al. (2005) to represent imperfec-tions on a few silos in Australia. However, this study suggests themagnitude of the Fourier coefficients based on the observations onFourier coefficients from the previous section such that the Fourierseries representation can be generalized for ASTs of different di-mensions. The first observation is that the coefficients correspond-ing to lower harmonics have high values in circumferential andmeridonal directions, but the magnitude of the coefficients de-creases for larger harmonics; to represent this observation, a func-tion that decays the magnitude of the coefficients for largerharmonics is required. Furthermore, an oscillatory pattern is ob-served in the magnitude coefficients with higher values near har-monics corresponding to the number of panels in either direction.This observation can be represented by a periodic function with aperiod corresponding to the number of panels. To represent thesepatterns probabilistically, Eq. (3) is proposed to evaluate the mag-nitude of Fourier coefficients; however, it is acknowledged that

Fig. 2. Fourier coefficient amplitude in meridonial direction

Fig. 3. Amplitude of Fourier coefficients (2D) of Tank 3

© ASCE C4015005-4 ASCE-ASME J. Risk Uncertainty Eng. Syst., Part A: Civ. Eng.

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other functional forms could also be explored to model the pat-terns observed in the magnitude of the Fourier coefficients

Amn ¼ jNð0,1Þje−mα−nβ�c1þ

���� cos�mπpm

�������

c2þ���� cos

�nπpc

������

ð3Þ

In the above equation, Nð0,1Þ is a normally distributed randomvariable with zero mean and unit standard deviation; pm and pc arethe number of panels in meridonal and circumferential direction; c1and c2 are non-negative constants; α and β are decay coefficientsin the meridonal and circumferential direction, respectively. InEq. (3), the cosine terms are the periodic functions that accountfor the oscillatory pattern observed in the Fourier coefficients;the constants c1 and c2 are added to the cosine terms to preventthe coefficients from assuming null values. The exponential termswith negative coefficients decrease the magnitude of the coeffi-cients for larger harmonics; i.e., the exponential terms serve asthe decay function. The decay coefficients, α and β, assume smallpositive values to facilitate the decay of the coefficients for largerharmonics; the magnitude of the decay coefficient determines therate of decrease of the coefficients. The exponential functions andthe cosines together form an envelope function, which emulates thepatterns observed from the Fourier coefficients of the imperfec-tions. The standard normal variable adds randomness to the coef-ficients; it is acknowledged that other distributions may also beused to add uncertainty in the magnitude of coefficients. The co-efficients from Eq. (3) are used in the following two-dimensionalFourier series

Fðθ; zÞ ¼XMm¼0

XNn¼0

Amn cosðmθþ ϕÞ cos�nπzH

�ð4Þ

where Fðθ; zÞ is the unscaled imperfection at angle θ and height z;ϕ is the phase angle, which is assumed to be a uniformly distributedrandom variable varying between 0.7–2.4 radians based on the ob-servations from imperfection on tanks and silos.

Eq. (4) qualitatively represents the imperfections that can be ob-served in tanks and silos and offers consistency with the systematictrends observed from Fourier analysis of tank imperfection dataacross different tank dimensions and locations. However, the mag-nitude of the imperfections is still unknown; therefore, Eq. (4) mustbe scaled appropriately to obtain the imperfections. To scale theimperfections for tanks of all dimensions, this study normalizesthe imperfection magnitude by the square root of the surface areaof the cylindrical shell (

ffiffiffiffiffiffiffiffiffiffiffiπDH

pwhere D is the diameter of the

tank). The magnitude of normalized imperfection of the tanks de-scribed in the previous section was also analyzed and observed tolie within 5.0 × 10−4 and 3.0 × 10−3. Since the magnitude of nor-malized imperfection is obtained from a few imperfection measure-ments, it is assumed to be a uniformly distributed variable withinthe previously mentioned range. Therefore, the scaled imperfectioncan be obtained as

Iðθ; zÞ ¼ Rsign

ffiffiffiffiffiffiffiffiffiffiffiπDH

pUFðθ; zÞ=maxjFðθ; zÞj ð5Þ

In the above equation, U is a uniformly distributed random var-iable, between 5.0 × 10−4 and 3.0 × 10−3, representing the normal-ized imperfection magnitude, and Rsign is a random variable whichtakes values –1 and 1 with equal probability. The variable Rsign ran-domly changes the direction of the imperfections from outward toinward. The imperfection model proposed in Eq. (5) is based on thetrends observed in the imperfections on two ASTs in Germany andthree silos in Australia. Even though the number of observationsused to derive the model is limited, the observations are consistentacross different continents instilling confidence in the model whichemulates these consistent trends. In addition to providing a gener-alization of the patterns observed in the imperfections, the proposedmodel also incorporates uncertainty in the imperfection model. Fur-thermore, the proposed model can be easily updated as more im-perfection measurements become available. In the next section, togain more confidence on the proposed imperfection modelingscheme, the effects of the proposed imperfection modeling methodon the buckling capacity of a tank are compared against the effectsof imperfections modeled via spectral simulations. Furthermore,the imperfection model presented in this section is used to assessthe uncertainties in the buckling strength of tanks subjected to windpressure.

Effect of Imperfections

The quantitative effects of imperfections on the buckling capacityof ASTs are unknown; however, qualitatively, past studies suggest adecrease in the buckling capacity of ASTs. To quantify the effectsof imperfections on the buckling performance of ASTs, this studyevaluates the mean and standard deviation of the buckling capacityof empty ASTs and compares it with conventionally used bucklingcapacity estimates. For this purpose, four tanks are selected basedon an inventory analysis of ASTs in the Houston Ship Channel,named A, B, C, and D. The four tanks are representative of therange of tank dimensions and aspect ratios observed in the ShipChannel region. The four tanks selected for analysis haverafter-supported conical roofs and their properties such as theirdimensions, number of shell courses (pm), number of panels incircumferential direction (pc), the thickness of each shell course,and inclination of the roof cone with respect to the base of the tankare provided in Table 1. These tanks are modeled in LS-DYNA(Hallquist 2007) using shell and beam elements; four-node shellelement are used for the cylindrical portion of the tank shell,three-node triangular elements are used for the conical roof, andbeam elements are used to model the rafters supporting the roof.The arrangement of rafters in the roof is based on Portela andGodoy (2005a), and the roof is assumed to have a shell thicknessof 7.9 mm. To verify the modeling procedure adopted in this study,one of the tanks studied by Virella et al. (2003) was modeled, andthe natural periods of the tank were compared with the values re-ported by Virella et al. (2003); the eigenvalues of the tank evaluated

Table 1. Dimensions of Selected Tanks

Tank H (m) D (m)Roof cone inclination

(degrees) pm pc Shell thickness (mm)

Tank A 13.11 30.48 10.50 5 6 12.7; 9.5; 7.9; 7.9; 7.9Tank B 9.00 6.00 10.00 5 4 Constant 7.0 mm thicknessTank C 16.80 13.50 10.00 7 5 9.24; 7.84; 7.0; 7.0; 7.0; 7.0; 7.0Tank D 12.60 42.00 10.00 7 9 21.0; 18.0; 16.1; 16.1; 11.9; 11.2; 11.2

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in this study were within 5% of the eigenvalues reported by Virellaet al. (2003).

Furthermore, to gain more confidence in the proposed imperfec-tion modeling procedure, the wind buckling capacity of Tank 3from Hornung and Saal (2002) is evaluated using the proposed im-perfection model and compared with the buckling capacity of thetank with imperfections modeled using a two-dimensional spectralsimulation approach (Shinozuka and Deodatis 1996). In this re-search, the spectral simulation method uses the power spectral den-sity directly obtained from imperfections in Tank 3. Therefore, theimperfections generated using spectral simulation will closely em-ulate the characteristics of imperfections on Tank 3. Conversely, theproposed imperfection modeling scheme, which can be applied totanks of different dimensions, is generalized based on the system-atic observation gathered from imperfection samples. Hence, thiscomparison may highlight the effectiveness of the proposed imper-fection modeling scheme to represent the effects of imperfectionson the tank’s buckling capacity. For the selected tank, with a diam-eter of 11.5 m and a height of 10.0 m, the imperfections have beenmeasured and reported in Hornung (2000); further details on thetank are available in Hornung and Saal (2002). To implementthe spectral simulation approach, the experimentally measured im-perfections obtained from the tank are smoothed, and the powerspectral density of the imperfections is obtained which is usedto generate 500 sample imperfections; for details, refer to Shino-zuka and Deodatis (1996). Spectral simulation could only be imple-mented because of the availability of imperfection measurementsfrom the tank of interest, which are required to inform the powerspectral density function. However, in the absence of imperfectionmeasurements from tanks, the spectral simulation will be difficultto apply. Instances of 500 imperfections are also generated usingthe method proposed in this study, using Eqs. (3)–(5), with c1 ¼c2 ¼ 1.0 and α ¼ β ¼ 1=6. The wind buckling capacity of the tankis evaluated using instances of imperfections generated using thetwo methods under a generalized wind pressure distribution de-scribed in Eurocode EN 1993-4-1 (CEN 2007), which is basedon wind tunnel tests of tanks of different aspect ratios. The windpressure distribution per EN 1993-4-1 is

PðθÞ ¼ λð−0.54þ 0.16ðD=LÞþ f0.28þ 0.04ðD=LÞgcosθþf1.04− 0.20ðD=LÞg cos2θþf0.36− 0.05ðD=LÞg cos3θ− f0.14− 0.05ðD=LÞgcos4θÞ ð6Þ

In the above equation, λ is reference pressure which can be ad-justed to change the magnitude of wind pressure, and L is the totalheight of the tank, including the height of the roof. Using the in-stances of imperfections generated via spectral simulations, themean and the coefficient of variation (COV) of the wind-bucklingcapacity are obtained as 9.84 kPa and 11.84%, respectively. Themean and the COVof the buckling wind pressure of the tank withimperfections obtained from the proposed imperfection model are9.94 kPa and 27.39%, respectively. The mean wind bucklingcapacities obtained with the two different imperfection modelingschemes are very close to each other. However, the two COVsare quite different; this is in part because of the large range ofnormalized imperfection adopted in the proposed scheme—U inEq. (3) and partly because of the fact that the spectral simulationis based only on one instance of imperfection, which leads to a verynarrow range of imperfection magnitude. However, as the range ofU in Eq. (3) was restricted to the range of normalized imperfectionobserved solely in Tank 3, 1.4 × 10−3 to 1.9 × 10−3, the COV de-creased to 16.00% and the mean buckling pressure slightly de-creased to 9.21 kPa. Therefore, the results from this comparison

suggest that the two imperfection modeling methods yield rela-tively consistent buckling capacity estimates for a given tank.The comparison lends confidence to the proposed model becausesimilar results are obtained for a single tank using the proposedmethod and using the spectral simulation method described inShinozuka and Deodatis (1996). However, this paper goes a stepbeyond methods in the existing literature by proposing a potentialform of imperfection model that reflects the overall systematictrends observed from past measurements but is generalized andcan be applied to ASTs of different dimensions. Therefore, themodel proposed in this study is used for probabilistic bucklingstrength analysis of tanks.

Using the proposed imperfection model, the uncertainty inthe wind buckling capacity of the four tanks described in Table 1is evaluated. The first tank, i.e., Tank A, has also been studied byPortela and Godoy (2005a) wherein they assess the pressure dis-tribution around the tank and estimate its buckling strength usinglinear eigenvalue analysis and nonlinear geometric analysis. Thewind pressure distribution obtained by Portela and Godoy is alsoused in this study to assess the buckling strength of the tank and isgiven by the following equation:

PðθÞ ¼ λð−0.2055þ 0.2943 cos θþ 0.4897 cos 2θ

þ 0.2624 cos 3θ − 0.0353 cos 4θ − 0.0092 cos 5θ

þ 0.0778 cos 6θþ 0.0263 cos 7θÞ ð7Þ

With the above wind pressure, this study first assesses the per-formance of Tank A using the first eigenmode shape as the imper-fection with different amplitudes—multiples of minimum shellthickness (T). Since large imperfection magnitudes were observedin the tanks analyzed in the previous sections, this study considersvery large magnitudes of imperfections, up to 6.5T, which is incontrast to existing studies such as Portela and Godoy (2005a),which only consider magnitudes only up to 2.0T. Figs. 4 and 5show the load displacement curve for Tank A with differentamplitudes of eigenmode shape imperfections for positive imper-fections and negative imperfections, respectively, obtained usingthe arc-length solution scheme in LS-DYNA after subjecting thetank to gravity loads. Positive imperfection implies that the location

Fig. 4. Load displacement curve for eigenmode shape imperfection(positive)

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with maximum imperfection amplitude is protruding radiallyoutside the tank, i.e., away from the center of the tank. However,positive imperfection does not imply that imperfections at all thelocations on the tank shell are protruding radially outside; at otherlocations, imperfections can be protruding radially outside or radi-ally inside towards the center of the tank, forming a depression onthe surface of the shell. Similarly, tanks with negative imperfectionshave a depression, pointing radially inward towards the center ofthe tank, at the location with maximum imperfection amplitude. Asshown from Figs. 4 and 5, a clear buckling point is identifiable onlyfor an imperfection magnitude of 0.5T; the buckling pressure istabulated in Table 2.

Results from Figs. 4 and 5 show that the use of eigenmodeshapes as imperfections does not yield a clear buckling load forlarge magnitudes of imperfections. However, such large magni-tudes are observed in ASTs, and buckling load estimates of ASTswith such magnitudes of imperfections are required. Therefore, theeffect of different magnitudes of imperfections on the bucklingstrength is explored using the imperfection modeling procedure de-scribed in Eqs. (3)–(5); furthermore, effect of variation in imper-fection modeling parameters is also studied. Table 2 providesthe results of buckling capacity estimation as the imperfection mag-nitude and imperfection modeling parameters are varied; for eachcombination of parameters, 500 simulations are performed, whichleads to convergence in mean and COV within �2.5%, and thebuckling analysis is performed using nonlinear geometric analysis.

At low magnitude of imperfection such as at 0.5T, variation of im-perfection modeling parameters does not cause significant variationin the mean buckling load and COV. The use of eigenmode shapeimperfections with 0.5T amplitude yields buckling loads of1.28 kPa and 1.81 kPa for outward and inward directions, respec-tively, which are highly conservative in comparison to the bucklingloads obtained using the proposed scheme, given in Table 2. As themagnitude of imperfections is increased to 2.0T, no significantchange is observed in the mean and COVof the buckling strength.However, as the magnitude of imperfection is increased further to3.5T, the mean buckling load increases for all combinations of im-perfection modeling parameters, whereas the COVs do not changesignificantly. At higher imperfection magnitudes of 5.0T and6.5T, the mean buckling strength increases for all parametercombinations, whereas a slight increase in COV is observed forsome parameter combinations. Also, as the imperfection modelingparameters are varied, only slight variations are observed in themean capacity for all the magnitudes of imperfection; for each im-perfection amplitude, the mean capacity estimates obtained usingdifferent combinations of parameters lie within 10% of each other.However, the modeling parameters seem to have a larger effect onthe COV, especially at higher imperfection magnitudes. Resultsfrom Table 2 show that parameters c1 and c2 do not have a signifi-cant effect on the mean and COV at all imperfection magnitudes.However, the decay coefficients, α and β, have a slight impact onthe COV. At higher imperfection magnitudes, 5.0T and 6.5T,smaller decay coefficients lead to a reduced COV, whereas largervalues lead to a large COV. This observation may be attributed tothe characteristics of the imperfection; imperfection samples gen-erated using low decay coefficients have larger contributions fromhigher harmonics, whereas instances generated using higher decaycoefficients have more prominent out-of-roundness imperfectionswhich emanate from lower harmonics. However, based on the ob-servations from imperfections on tanks, this study suggests valuesof the decay coefficients close to the reciprocal of the number ofpanel in each direction. Future work may further look into the ef-fects of modeling parameters on the mean and variance of bucklingstrength. Overall, observations from Table 2 may suggest that alarger imperfection may improve the buckling capacity of the tank,but the increase in mean capacity may be accompanied by anincrease in the coefficient of variation, which may decrease theoverall reliability of the tank under wind loads. Therefore, it is im-portant to estimate the uncertainty in the buckling capacity in ad-dition to evaluating the mean buckling capacity.

The analysis presented previously provides several insightsinto the effects of imperfection magnitude and modeling parame-ters on the buckling capacity of a tank. Probabilistic bucklingcapacity estimates can be used for reliability analysis or for prob-abilistic calibration of design codes. Therefore, uncertainty in thebuckling load capacity of tanks representative of the Houston Ship

Fig. 5. Load displacement curve for eigenmode shape imperfection(negative)

Table 2. Variation of Buckling Capacity due to Change in Imperfection Modeling Parameters

Imperfectionmagnitude

Buckling pressure (kPa)

Test 1 c1 ¼ c2 ¼ 1.0,α ¼ β ¼ 1=10

Test 2 c1 ¼ c2 ¼ 0.5,α ¼ β ¼ 1=5

Test 3 c1 ¼ c2 ¼ 0.5,α¼β¼ 1=20

Test 4 c1 ¼ c2 ¼ 2.0,α ¼ β ¼ 1=5

Test 5 c1 ¼ c2 ¼ 2.0,α ¼ β ¼ 1=20

Eigenmodeshape

(positive)

Eigenmodeshape

(negative)

0.5T 3.28 (10.23%) 3.27 (10.25%) 3.28 (10.37%) 3.30 (10.83%) 3.26 (9.58%) 1.28 1.812.0T 3.24 (9.04%) 3.21 (11.60%) 3.29 (9.98%) 3.22 (11.98%) 3.30 (9.74%) — —3.5T 3.66 (9.39%) 3.55 (11.54%) 3.77 (7.95%) 3.58 (11.77%) 3.83 (8.62%) — —5.0T 4.33 (11.44%) 4.26 (16.45%) 4.50 (10.38%) 4.23 (13.72%) 4.52 (10.26%) — —6.5T 5.09 (12.39%) 5.02 (16.68%) 5.29 (10.67%) 5.05 (16.35%) 5.38 (10.46%) — —

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Channel’s tanks, described in Table 1, is quantified in this researchusing Monte Carlo simulations. Mean and COV of the bucklingcapacity of the four ASTs are assessed with high confidence;i.e., convergence in mean and COV values is observed within�2.5%. For this purpose, over 500 instances of imperfectionsare generated for each tank using Eqs. (3)–(5) with parameters pro-vided in Table 3. The exact wind pressure around Tank A, given inEq. (7), is used; however, since the exact wind pressure distributionis unknown for Tanks B, C, and D, the generalized pressure dis-tribution described in Eq. (6) is used. Using the nonlinear geometricanalysis and the wind pressure distributions in Eqs. (6) and (7),buckling loads are evaluated for the four tanks for all instancesof imperfections. Table 4 provides the probabilistic estimates ofbuckling pressure and compares the probabilistic estimates withthe buckling pressure obtained from the eigenvalue buckling analy-sis. The comparison clearly shows that only for Tank B the estimateof buckling pressure obtained from the eigenvalue buckling analy-sis is comparable to the mean obtained from Monte Carlo simula-tions; for all other tanks, the eigenvalue buckling loads are highlyconservative. Moreover, for all four tanks, significant uncertainty isobserved in the buckling capacity, ranging from 14 to 37%, whichcannot be estimated via the eigenvalue buckling analysis. Apartfrom showing the conservative bias in existing buckling load analy-sis methods, these results highlight the significant amount of un-certainty in the buckling capacity of ASTs. Quantification ofthese uncertainties is necessary for reliability analysis and forcalibration of existing design codes per the load resistance factordesign (LRFD) philosophy. Because variation is observed in COVattributable to differences in tanks sizes, further research is neededto efficiently estimate the mean and variance of buckling capacityfor tanks of different dimensions. The uncertainty analysis pre-sented herein can be used for risk assessment for a portfolio oftanks, such as the one in the Houston Ship Channel. Furthermore,the uncertainty estimates can be used to probabilistically assess thedesign knockdown factor, which reduces the buckling capacity ofshells to account for uncertainty in buckling capacity attributable toimperfections.

Conclusions

An imperfection modeling scheme is proposed in this studybased on imperfections observed in actual full-size tanks and silos.Fourier analysis of imperfections on the two tanks from Germany

and the results of imperfection analysis on silos in Australia showseveral interesting features; it is observed that the amplitude ofthe Fourier coefficients is related to the number of panels alongthe circumference and axis of the tank. Because the observationson imperfections are consistent across continents, this study con-cludes that the imperfections observed in the tanks may be attrib-uted to the manufacturing process. A generalized imperfectionmodel, using a two-dimensional Fourier series, is proposed inwhich the magnitude of Fourier coefficients is selected to reflectthe systematic observations obtained from the analysis of imperfec-tions. Furthermore, randomness is included in the coefficients ofthe Fourier series which introduces randomness in the resultingimperfections. The effects of imperfections generated usingthe proposed imperfection modeling scheme are comparedwith the effects of imperfections generated via spectral simulations.The comparison shows that the two imperfection modelingschemes have similar effects on buckling capacity, although theproposed modeling scheme affords flexibility in a generalizedapplication to a broad range of ASTs where empirical data onimperfections from the field may be limited to otherwise informstochastic models.

Furthermore, realizations of imperfections generated using theproposed scheme are used to determine the effect of imperfectionson the wind buckling capacity of ASTs representative of tanks inthe Houston Ship channel. As the parameters of the imperfectionmodel and the imperfection magnitude are varied, several interest-ing trends are observed. As the imperfection magnitude increases,buckling is not observed if the imperfections are modeled usingeigenmode shapes, as typically performed. However, using the pro-posed imperfection modeling scheme, it is observed that for smallimperfection magnitudes, an increase in amplitude of imperfectionsleads to a small change in the mean buckling capacity, whereas forlarger amplitude of imperfections, a further increase in the magni-tude actually increases the mean buckling capacity. However, insome cases, the increase in mean buckling capacity is accompaniedby increase in the variance of the buckling capacity, which mayultimately decrease the reliability of the tank. In addition to study-ing the effects of imperfection magnitude, this study also comparedthe buckling strength using eigenmode shapes as imperfectionsand the proposed imperfection modeling scheme which showsthat using eigenmode shapes as imperfections leads to a veryconservative buckling load estimate. Additionally, the bucklingloads were also obtained using linear eigenvalue buckling analysisand compared with the probabilistic capacity estimates. The eigen-value buckling analysis provides a highly conservative estimate ofbuckling loads for all the tanks. Moreover, such an analysis doesnot provide any insight into the uncertainty associated with thebuckling capacity which is significant for all the tanks as shownby the results of Monte Carlo simulations using the proposedimperfection model.

This study has proposed a probabilistic imperfection modelbased on real imperfection data from tanks and silos, which is com-pletely lacking in the literature. Furthermore, the probabilisticnature of the proposed imperfection model helped uncover the levelof uncertainty in buckling strength of ASTs subjected to windloads, which is an essential input for reliability estimation of tanksand future calibration of design codes to a prospective LRFD phi-losophy applied to tanks. Uncertainty estimates can also be used toprobabilistically estimate the knockdown factors used for designingtank shells. Beyond these opportunities, future work shall also fo-cus on efficient evaluation of uncertainty in buckling capacity of aportfolio of tanks to support regional risk assessment under hurri-cane wind and surge loads.

Table 3. Parameters for Imperfection Simulation

Tank M N α β c1 c2

Tank A 40 80 1=5 1=5 1.0 1.0Tank B 40 80 1=5 1=4 1.0 1.0Tank C 40 80 1=7 1=5 1.0 1.0Tank D 40 80 1=7 1=9 1.0 1.0

Table 4. Buckling Pressure for Tanks

TankEigenvalue buckling

pressure (kPa)

Buckling pressure (kPa)

Mean COV (%)

Tank A 2.52 5.78 36.29Tank B 28.95 37.34 13.85Tank C 4.71 8.09 27.48Tank D 4.94 8.88 30.43

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Acknowledgments

The authors would like to acknowledge the support for this researchby the Houston Endowment and Shell Center for Sustainability.Any opinions, findings, and conclusions or recommendationsexpressed in this paper are those of the authors and do not neces-sarily reflect the views of the funding agencies. The authors wouldalso like to acknowledge computational facilities provided by DataAnalysis and Visualization Cyberinfrastructure [National ScienceFoundation (NSF) grant OCI-0959097].

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