stress analyses in single-deck and double-deck oating...

Scientia Iranica B (2017) 24(2), 727{739

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Research Note

Stress analyses in single-deck and double-deck oatingroofs subjected to earthquake ground motions

F.G. Golzara, R. Shabania;� and S. Tariverdilob

a. Department of Mechanical Engineering, Faculty of Engineering, Urmia University, Urmia, Iran.b. Department of Civil Engineering, Faculty of Engineering, Urmia University, Urmia, Iran.

Received 29 June 2014; received in revised form 25 November 2015; accepted 18 July 2016

KEYWORDSCylindrical tank;Sloshing response;Earthquake;Floating roof.

Abstract. This paper investigates the stress patterns in circular single- and double-deck oating roofs excited by earthquake ground motions. Variational formulation is employedto derive the governing equations. Response of the oating roofs to di�erent classes ofground motions including near-�eld and far-�eld records is evaluated. Results indicatethat far-�eld earthquakes produce larger wave elevation but smaller bending stresses in the oating roof. Moreover, the bending stresses in double-deck roofs emerge in a smooth andapproximately symmetric pattern with a peak around the mid-span radius while the peakvalues of bending stress in single-deck oating roofs occur at the vicinity of deck center andat the deck-pontoon interface. The results can have practical implications in the designprocess of oating-roof cylindrical tanks.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Floating roofed storage tanks are employed to storedi�erent liquids in petroleum and petrochemical in-dustries. Floating roofs eliminate evaporation bycovering the exposed liquid surface and thus reducethe material loss and air pollution. In addition, theyreduce the possibility of �re hazard in highly ammablehydrocarbon vapors due to sources like earthquakes,static electricity, cigarette smoking, etc. [1]. Takinginto account the past major earthquakes, it has beenfound that the liquid sloshing could destroy the oatingroofs [2]. For instance, during the 2003 Tokachi-Oki earthquake, seven oil storage tanks of oatingroof type located at Tomakomi, Hokkaido, Japan wereseriously damaged. Damages included the sinking ofthe oating roofs, which led one of the seven tanksto a whole surface �re. Therefore, the inclusion

*. Corresponding author.E-mail address: [email protected] (R. Shabani)

of uid-structure coupling in design and analysis of oating roofs subjected to ground motions is impor-tant.

Employing variational principle, Sakai et al. [3]calculated the natural frequencies of oating roofs.Isshiki and Nagata [4] derived the governing equationsof a oating plate by assuming Hamilton's principlefor plate and Kelvin's principle for water. Amabili andKwak [5] utilized Hankel transformation to analyze thevibrational properties of circular plates with di�erentforms of uniform boundary conditions. Investigationswere continued by Matsui, who employed Fourier-Bessel series to derive the governing equations ofmotion for a base excited storage tank with double-deck [6] and single-deck [7] oating roofs. Matsui [6,7]investigated the impact of di�erent exural modes onthe response metrics including roof de ection, liquidpressure imposed on the tank wall, and bending stresseswithin the roof. In a shaking table test, Nagaya etal. [8] investigated the vibrational characteristics ofroofs and showed the strong dependency of damping

728 F.G. Golzar et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 727{739

ratio on roof type. Epstein [9] utilized shooting methodand showed the e�ect of structural parameters on thestresses in single-deck oating roofs. Added massfactors have been used to explore the e�ect of uidon natural frequencies of axisymmetric circular plateswith di�erent boundary conditions [10]. Sun et al. [11]carried out a �nite element method in a single-deck oating roof subjected to water loading. Shabani etal. [12] used a variational approach to evaluate theequilibrium conditions between the elastic and buoyantforces and depicted the contribution of membraneand exural constituents to the overall roof sti�ness.Using extended Hamilton's principle, Golzar et al. [13]introduced a set of results predicting behavior of tankswith double-deck roofs when excited by di�erent typesof earthquakes. Shabani and Golzar [14] extractedthe governing nonlinear equation of a oating roof ofa seismically excited cylindrical tank. They showedthe suppressing e�ect of nonlinearities on the sloshingheights for a wide range of tank dimensions. In anexperimental study, De Angelis et al. [15] studiedthe response of base isolated cylindrical tanks withand without roofs to earthquake excitations. Theyshowed that the base isolators and oating roofs couldreduce the wall pressure and oscillation amplitudessigni�cantly. Later, Shabani [16] investigated theinduced stresses in single-deck oating roofs on seismicstorage tanks. He depicted two critical locations in thedeck roofs: one near the center of the roof and the otheralong the deck-pontoon interface. He concluded thatthe near-source ground motions produce larger stressesat the inner critical radius of the deck, but far-�eldground motions lead to larger stresses in deck-pontooninterface.

In some references [6,13], the double-deck roofshave been investigated and in other reports [7,16], thestress patterns in single-deck roofs are reported. How-ever, the present paper compares the stress pattern ofthe single- and double-deck oating roofs and discussesthe changes in this pattern due to change in the decktype. Applying the Hamilton's variational principle tothe oating roof and contained liquid, simultaneously,the responses of roofed storage tanks to di�erent earth-quakes, including near-source and far-�eld records,are investigated. Direct numerical integration methodis employed to calculate the de ection of roof andcorresponding stresses. Stress patterns produced indouble-deck roofs are found and compared with thesingle-deck roof results qualitatively and quantitatively.Moreover, the e�ect of earthquake type is studied fora practical range of tank dimensions. The results mayprove useful in design considerations.

2. Mathematical model

Figure 1 illustrates a cylindrical tank of radius, R,

Figure 1. Cylindrical liquid storage tank with (a)single-deck and (b) double-deck oating roofs.

�lled to a height, H, with an incompressible liquidof density, �, and equipped with an elastic oatingroof. The conventional oating roofs can be categorizedinto two types: single-deck and double-deck. Single-deck roofs (Figure 1(a)) are composed of a circum-ferential buoyant ring (pontoon) with relatively largesti�ness surrounding an inner circular plate (deck)with small thickness and sti�ness. Double-deck roofs(Figure 1(b)), on the other hand, consist of upper andlower decks connected by structural rims. For changing uid level in the tank, the oating roof should stayin contact with contained uid. Therefore, there isa relatively large gap between the side wall and the oating roof edge. On the other hand, to protect vaporescape through oating roof perimeter, special sealsare employed, where single- and double-wiper seals andliquid-mounted foam seals are the most common types.These seals could also operate as damping element toprevent the oating roofs from side wall impacts. Inthis study, e�ect of the lateral side wall forces on the oating roofs is ignored as this study only considersvertical vibration of the oating deck due to uidsloshing. In the mathematical modeling, the followingassumptions have been made:

1. The tank has a rigid at bottom and a rigidcylindrical wall;

2. The tank is excited by the unidirectional groundexcitation, �xg;

F.G. Golzar et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 727{739 729

3. The contained liquid is an inviscid and incompress-ible uid in an irrotational motion;

4. The oating roofs deform in bending only;

5. The oating roofs are always in contact with thecontained liquid;

6. E�ect of the lateral side wall forces on the oatingroofs is ignored;

7. The proposed model does not account for verticalcomponent of the seismic excitation.

Laplacian equation, together with the impermeabilityand kinematic boundary conditions, may be used tofully de�ne the movement of the liquid inside theenclosed tank volume:

r2� = 0; 0�r�R; 0��<2�; 0�z�H; (1)

@�@z

= 0; z = 0; (2)

@�@r

= _xg cos � r = R: (3)

Imposing these conditions, the velocity potential isobtained in the following form:

�(r; z; �) = r _xg cos � + �0;

�0 =1Xk=1

AkJ1� "krR

�J1("k)

cosh� "zR

�cosh

� "kHR

� cos �;

k = 1; 2; 3; ::: (4)

where Ak is the kth unknown time-dependent shapefactors and "k is the kth eigenvalue of the velocitypotential obtained from the characteristic equationJ 01("k) = 0. Lateral vibration of the oating roof can be

considered as a weighted summation of free vibrationmode shapes of a free-edge circular plate:

�(r; �; t) =1Xi=1

Bi(t)Xi(r) cos �; (5)

where Bi denotes the generalized coordinates and Xidenotes the radial mode shapes consisting of rigid andelastic modes [17]:

X1(r) =rR;

Xi(r) =h�iJ1

��irR

�+ �iI1

��irR

�i;

i = 2; 3; � � � (6)

�i represents the radial eigenvalues of the free-edgecircular plate. J1 denotes the Bessel function of the�rst kind of order one and I1 denotes the modi�edBessel function of the �rst kind of order one. Bearingin mind that the roof and liquid stay in complete con-tact, the assembled Lagrangian of the coupled system,accounting for the kinetic and potential energies, iswritten in form of the sum of the roof LagrangianLR and liquid Lagrangian Ll [18] as shown in Box I.Making use of the Hamiltonian principle, the equationof motion of roof is obtained in the following matrixform (see Appendix A):

MR[P ]n

�Bo

+ (DR[Q] + �Lg[U ])fBg

+ �L[T ]n

_Ao

+ �L�xgfFg = 0; (8)�T t� n _B

o� [S]fAg = 0: (9)

Elements of matrices P , Q, U , F , S and T are speci�edas follows:

L =LR + Ll =RZ

r=0

2�Z�=0

8>>>>>>>>>><>>>>>>>>>>:12MR _�2 � 1

2DR

266666666664

�@2�@r2

�2

+�

1r@�@r

+1r2@2�@�2

�2

+2��@2�@r2

��1r@�@r

+1r2@2�@�2

�+2(� � 1)

�1r@2�@r@�

� 1r2@�@�

�

377777777775

9>>>>>>>>>>=>>>>>>>>>>;rdrd�

+RZ

r=0

2�Z�=0

�L��1

2

�@�0

@z

��0 +

@�@t

�� 12g�2�rdrd�; (7)

where MR and DR denote the mass and rigidity of the roof.Box I


Table 1. Seismic data used in this study [13].

Input ground motion Kobe(1995)

Imperial valley(1940)

Tokachi-oki(2003)

Tohoku(2011)

Duration (s) 40 40 290 300PGA (g) 0.4862 0.3130 0.0800 0.1397Epicentral distance (km) 8.7 13 227 364

Pij =RZ

0

XiXjrdr;

Qij =RZ

0

X 00i X 00j rdr

+RZ

0

�1rX 0i� 1

r2Xi

��1rX 0j� 1

r2Xj

�(3�2�)rdr

+RZ

0

"X 00i

�1rX 0j � 1

r2Xj

�+X 00j

�1rX 0i � 1

r2Xi

�#�rdr;

Uij =RZ

0

XiXjrdr;

Tij =RZ

r=0

XiJ1("jr=R)J1("j)

rdr;

Fi =RZ

r=0

rXirdr;

Sjj =R2"j tanh

�"jHR

� 1� 1

"2j

!: (10)

Combining Eqs. (8) and (9) yields the following equa-tion [13]:�

MR[P ] + �L[T ]�S�1� [T t]

�n �Bo

+ (DR[Q]+�Lg[U ]) fBg+�L�xgfFg=0: (11)

This equation suggests that both dynamic and staticcouplings exist between the roof and oscillating uid,resulting in a coupled set of equations. The massmatrix in Eq. (11) is composed of two parts: MR[P ]accounting for the presence of oating plate and

�L[t][S�1][T t] representing the added inertial e�ectof the liquid. Likewise, sti�ness matrix is composedof DR[Q] accounting for the plate sti�ness and�Lg[U ] representing the added sti�ness due to uidco-vibration.

By truncating Eqs. (4) and (5) to limited numbersn and m, respectively, the obtained set of equationsmay be solved to yield the generalized coordinates and,consequently, plate de ection and liquid velocity po-tential. Moreover, radial and circumferential bendingmoments resulting in the bending stresses in roof cross-section are calculated by the following formulae [13]:

Mr = �DR

�@2�@r2 + �

�1r@�@r

+1r2@2�@�2

��; (12)

M� = �DR

��1r@�@r

+1r2@2�@�2

�+ �

@2�@r2

�: (13)

3. Excitation characteristics

To study the dynamic response of a seismically excitedroofed tank, the following earthquakes data were usedas base excitations [13]: 090 components of 1995Kobe earthquake at Japan recorded at Nishi-Akashistation (KOBE/NIS000), 180 components of 1940 Im-perial Valley earthquake recorded at El Centro sta-tion (IMPVALL/I-ELC180), EW component of 2003Tokachi-oki earthquake recorded at Tomakomai sta-tion (HKD1290309260450EW), and EW component of2011 Tohoku earthquake (Eastern Japan earthquake)recorded at Chiba station (CHB0091103111446). Ta-ble 1 shows the duration, peak ground acceleration,and epicentral distance of earthquake records. Timehistories and Fourier decomposition of the records areshown in Figures 2 and 3, respectively.

Kobe and Imperial Valley earthquakes have sim-ilar frequency contents with spectral peaks at lowperiods and are classi�ed as near-source earthquakes.However, Tokachi-oki and Tohoku with broad bandfrequency content are classi�ed as long-period, far-�eldrecords. Near-source earthquakes act in a shorter timeinterval, but impose larger accelerations to the struc-tures compared to long-period, far-�eld earthquakes.

4. Simulation results

In this section, recordings of the aforementioned earth-quake ground motions are used as base excitation input


Figure 2. Time histories of ground motions [13]: (a) Kobe, (b) Imperial Valley, (c) Tokachi-oki, and (d) Tohoku.

Figure 3. Fast Fourier Transform (FFT) of ground motions [13]: (a) Kobe, (b) Imperial Valley, (c) Tokachi-oki, and (d)Tohoku.

for a wide range of tank dimensions. It should bementioned that in actual structures, some form ofdamping will emerge, owing to sources such as viscousliquid-wall interaction or/and friction between roof andwall. To account for the energy dissipation in thesystem, damping matrix C is added to Eq. (11):�

MR[P ] + �L[T ]�S�1� [T t]

�n �Bo

+ [C]n

_Bo

+ (DR[Q]+�Lg[U ]) fBg+�LxgfFg=0: (14)

The sti�ness proportional damping is utilized with themodal damping ratio depicted in Table 2. The selected

value (0.005) is in accordance with the experimental�ndings of Nagaya et al. [8], where it was also statedthat the sti�ness proportional damping was appropri-ate for double-deck roofs. In numerical simulations,Newmark's method is employed to solve the governingequation [13].

4.1. Numerical results for double-deck oatingroof

In this section, recordings of the aforementionedearthquakes are used as base excitation for a wide rangeof tank dimensions with the geometrical and materialproperties shown in Table 2.


Table 2. Geometrical and material properties of the tankwith a double-deck oating roof.

Contained liquid height, H 15 mRoof areal mass density, MR 120 kg/m2

Roof bending rigidity, DR 1:612�105 kN.mRoof thickness, hR 800 mmLiquid density, �L 800 kg/m3

Damping ratio, double-deck, � 0.01

E�ective cross sectionalcoe�cient, Ae�

1050 mm2

Poisson's ratio, � 0.3

Number of modes adopted forliquid sloshing, m

33

Number of modes adopted forroof vibration, n

5

Response of double-deck oating roof to the near-�eld excitation of Kobe is shown in Figure 4. Similarbehavior is witnessed when the Imperial Valley excita-tion is used (it is not shown here). Maximum radialstress in the roof occurs during the major earthquakeperiod (Figure 4(b)) while the roof oscillation continuestill the end of the excitation (Figure 4(a)).

Time histories of roof de ection and radial stressin a tank subjected to far-�eld excitations of Tokachi-oki and Tohoku earthquakes are shown in Figure 5(a),(b), (c), and (d), respectively. Response of the tanksubjected to Tokachi-oki shows that both metrics reachtheir peak values simultaneously with the input excita-tion and gradually fade, oscillating in the fundamentalmode of the coupled system (Figure 5(a) and (b)). Itis shown that the roof de ection is larger; nevertheless,smaller stresses are witnessed in the roof cross section.Figure 5(c) and (d) show the response of a tanksubjected to Tohoku earthquake. Due to the record'srich frequency content in low and high periods, oatingroof behavior in this case shows similarities with bothnear-source records of Kobe and Imperial Valley andfar-�eld, long period record of Tokachi-oki. It isobserved that the radial stress decreases swiftly afterthe major earthquake duration has passed while roofmaintains its high-amplitude oscillation till the end ofthe record, like the phenomenon seen in near-sourceexcitations.

Peak values of stresses are slightly higher thanthose of the Tokachi-oki excitation, which are at-tributed to the greater share of the second mode inthe oscillation. In conclusion, the vibration of roof-edge is dominated by the fundamental period of thesystem [13] while the oscillation of bending stresses is

Figure 4. Time history of double-deck roof due tonear-�eld Kobe excitation: (a) Roof edge de ection, and(b) induced radial stress.

well a�ected by the frequency content of the excitationrecord. The richer the content of the excitation at lowerperiods, the higher the contribution of higher modes.

Quantitative di�erence between the results ofnear-source and far-�eld excitations is highlighted inFigure 6, where the peak value diagrams in the previous�gures are shown altogether. It is observed thatvertical roof de ection is far larger in the case of far-�eld excitations as a direct result of rich frequencycontent around fundamental period (Figure 3). Onthe other hand, near-source earthquakes impose largerbending stresses as the participation of second naturalmode in the stresses is much greater due to their richcontent at lower periods (Figure 3).

Figure 7 compares the peak values of de ectionand stresses for tanks with di�erent radii, which couldbe considered as response spectra of these parameters.Note that in evaluating these results, all the propertiesare kept intact except for the radius of the tankand roof. Stronger response spectrum of far-�eldearthquakes renders the produced roof de ection largerand more radius-dependent than the case of near-�eldearthquakes, where the absolute value of de ection isapproximately invariant for all tank radii. On thecontrary, bending stresses caused by near-source exci-tations are considerably larger in the majority of radiusspan as a consequence of stronger frequency contentat lower periods, which gives rise to the second-mode


Figure 5. Time history of roof edge de ection and radial stress due to far-�eld excitations: (a) and (b) Tokachi-oki, (c)and (d) Tohoku.

Figure 6. Comparison of e�ects of earthquakes on aspeci�c tank (H = 15, R = 25): (a) Roof de ection and(b) radial bending stress.

contribution in the stresses. As it was observed, thesecond mode of oscillation had a crucial contributionto the bending stresses. This makes the absolutevalue of stress highly radius-dependent since any small

Figure 7. E�ect of earthquakes on tanks with di�erentdimensions: (a) Roof de ection and (b) radial bendingstress.

change in the radius may result in the overlapping ofperiods with frequency peaks of excitation inputs, thusboosting the bending stress. It is concluded that in


designing the oating roofed cylindrical tanks, far-�eldearthquakes should be considered for determining thefree-board while the bending strength of the roof shouldbe set great enough to endure near-source earthquakes.

4.2. Numerical results for single-deck oatingroof

This section is devoted to the numerical analysis ofsingle-deck oating roofs subjected to the 4 seismicrecords described in the previous sections. Table 3shows the numerical data used in the calculations.

4.2.1. Note on radial mode shapesRadial mode shapes of the single-deck roof di�er fromthose of the double-deck due to its radius-dependentbending characteristics. The mode shapes of thedouble-deck are those of a circular plate with uniformgeometrical and material properties while in the single-deck roof, deck plate and pontoon possess di�erentcharacteristics rendering their exural behavior con-siderably di�erent. Deck plate behaves as an elasticplate, but the relatively large bending sti�ness of thepontoon prevents it from undergoing radius-dependentelastic deformation. Thus, the behavior of deck in theradial direction, relative to the pontoon, is analogous tothe behavior of a cantilever to its �xed end. This makesthe interface radius susceptible to high bending stressesand, potentially, a failure point. To extract the modeshapes, Eq. (6) has to be modi�ed to account for thezero curvature of the pontoon, and zero relative de ec-tion and slope in the deck-pontoon interface. Second-and third-mode shapes are illustrated in Figure 8 forthe sake of comparison.

Response of the single-deck storage tank to near-source excitation of Kobe is depicted in Figure 9.The edge-de ection of the oating roof is similar tothat of the double-deck type in terms of amplitude(Figure 9(a)).

Dynamic behavior of single-deck roof subjected to

Table 3. Geometrical and material properties of the tankwith a single-deck oating roof.

Tank height, H 15 mTank radius, R 25 mPontoon width, WP 3 mDeck areal mass density, Md 58 kg/m2

Pontoon areal mass density, MP 173 kg/m2

Pontoon bending rigidity, DP 2:7�105 kN.mDeck plate thickness, hD 4.5 mmLiquid density, � 800 kg/m3

Damping ratio, � 0.005Poisson's ratio, � 0.3Adopted modes for roof vibration, n 29Adopted modes for liquid sloshing, m 33

Figure 8. Comparison of radial mode shapes forsingle-deck and double-deck roofs: (a) 2nd mode and (b)3rd mode.

Figure 9. Time history of single-deck roof due tonear-�eld Kobe excitation: (a) Roof edge de ection and(b) radial stress.


Figure 10. Time history of single-deck roof due tofar-�eld excitation of Tohoku: (a) Roof edge de ection,and (b) radial stress.

Far-Field excitation of Tohoku is shown in Figure 10.Similar behavior is witnessed when the Tokachi-okiexcitation is used (it is not shown here). Despitethe near-source excitations, here, the amplitude ofbending stress shows a low rate of dissipation (Fig-ure 10(b)). Roof shows high-amplitude oscillatorymotion till the end of the earthquake (Figure 10(a)),producing considerable bending stresses in the roofcross section. Stresses induced within the roof sharecharacteristics of both near-source excitations of Kobeand Imperial Valley and far-�eld Tokachi-oki as theTohoku earthquake possesses rich frequency content atboth high and low periods.

For the sake of comparison, vertical de ectionand radial stresses in the deck plate, caused by thementioned 4 ground motions, are plotted altogetherin Figure 11. Similar to the tanks with double-deckroof, far-�eld long-period earthquakes produce largerroof de ections compared to near-source earthquakes.Stresses in the pontoon were assumed zero and, thus,are not plotted here. Again, it is seen that near-source ground motions of Kobe and Imperial Valleyproduce large stresses in the central critical radiuswhile far-�eld long-period earthquakes of Tokahci-okiand Tohoku result in large stresses along the deck-pontoon interface. Moreover, Tohoku earthquake im-

Figure 11. Comparison of e�ects of earthquakes on theresponses of single-deck oating roof: (a) Roof de ection,and (b) raidal bending stress.

poses considerably large stresses at the central criticalradius as its frequency content is rich at low periods aswell as high periods, activating higher modes of verticalde ection of the deck plate.

Figure 12 compares the peak values of stresseswithin the single-deck oating roof with di�erent radii,which could be considered as response spectra ofstresses. The radius of the roof and tank is swept from20 to 40 meters to include the typical range of tanksused in the petroleum industries. Radial stresses atthe two most vulnerable radii are plotted with respectto the radius of the tank. Strong content of thenear-source earthquakes of Kobe and Imperial Valleyresults in the higher modes of the coupled system beingactivated and, thus, the stresses at the central criticalradius, which are signi�cantly in uenced by the highermodes and take larger values compared to the caseof far-�eld excitations. The contrary is valid for thestresses occurring along the deck-pontoon interface; theoscillation of these stresses is majorly controlled by thefundamental period of the system, i.e., far-�eld earth-quakes, which possess strong content at high periodsand render greater values at this radius. The stressesalong the deck-pontoon interface show an overall de-


Figure 12. E�ect of earthquakes on tanks with di�erentdimensions: (a) Radial stress at central critical radius and(b) radial stress at deck-pontoon interface.

scending trend with respect to radius. Consequently,for su�ciently wide tanks, deck-pontoon interface is nolonger a high-risk critical point compared to the centralcritical radius. In conclusion, the design criteria forthe strength of the deck plate may be divided into twoparts; strength at the central radii, which should bedetermined by the characteristics of near-source groundmotions; and bending strength along its connectionwith annular pontoon, which should be predicted bythe probable hazards of far-�eld quakes.

4.3. Comparison of stress patterns insingle-deck and double-deck roofs

Dissimilarities in stress patterns occurring in single-deck and double-deck oating roofs are presented inFigure 13. In case of near-source ground motions,double-deck roofs experience more destructive stressescompared to single-deck roofs and the most vulnerablepoints are located at r � 0:35R. On the other hand,when the tank is subjected to far-�eld long-periodground motions, the values of bending stress in double-deck roofs are of the same scale as those of single-deckroofs. It could be stated that design criteria for thedouble-deck roofs are controlled by the requirements ofnear-source quakes; but, deck strength in single-deckroofs has to be determined by accounting for both near-source and far-�eld seismic hazards.

5. Conclusion

This paper studied the stress analyses in oating

Figure 13. Stress patterns in single-deck and double-deck roofs: (a) Kobe, (b) Imperial Valley, (c) Tokachi-oki, and (d)Tohoku.


roofs subjected to near-source and long-period far-�eld ground motion records. Two types of oatingroofs, single-deck and double-deck, were utilized in thesimulations. It was revealed that higher modes havesigni�cant impact on the stresses, especially at centralparts of single-deck roof. Near-source excitationsproved more destructive in case of double-deck roofs,while single-deck roofs were prone to failure whenexcited by far-�eld ground motions as well as near-source ones. Results suggest that near-source groundmotions produce larger stresses in the case of double-deck roofs and also at the inner critical radius of single-deck roofs, but far-�eld ground motions lead to largerstresses in deck-pontoon interface. It is concluded thatfree-board design should be carried out taking intoaccount long-period earthquakes while bending stressevaluation requires consideration of both near-sourceand far-�eld hazards.

References

1. Changa, J.I. and Linb, C-C. \A study of storage tankaccidents", Journal of Loss Protection in the ProcessIndustries, 19(1), pp. 51-59 (2006).

2. Hatayama, K., Zama, S., Nishi, H., Yamada, M.,Hirokawa, M. and Inoue, R. \The damages of oilstorage tanks during the 2003 Tokachi-oki earthquakeand the long period ground motions", Proceedings ofthe JSCE-AIJ Joint Symposium on Huge SubductionEarthquakes-Wide Area Strong Ground Motion Predic-tion, pp. 7-18 (2005).

3. Sakai, F., Nishimura, M. and Ogawa, H. \Sloshingbehavior of oating-roof oil storage tanks", Comput.Struct., 19(1-2), pp. 183-192 (1984).

4. Isshiki, H. and Nagata, S. \Variational principlesrelated to motions of an elastic plate oating on a watersurface", Proceeding of the 11th International O�shoreand Polar Engineering Conference, Stavanger, Norway,pp. 190-197 (2001).

5. Amabili, M. and Kwak, M.K. \Free vibration ofcircular plates coupled with liquids: revising the lambproblem", J. Fluids Struct., 10(7), pp. 743-761 (1996).

6. Matsui, T. \Sloshing in a cylindrical liquid storagetank with a oating roof under seismic excitation",ASME J. Pressure Vessel Technol., 129(4), pp. 557-566 (2007).

7. Matsui, T. \Sloshing in a cylindrical liquid storagetank with a single deck type oating roof under seis-mic excitation", ASME J. Pressure Vessel Technol.,131(2), pp. 557-566 (2009).

8. Nagaya, T., Matsui, T. and Wakasa, T. \Model testson sloshing of a oating roof in a cylindrical liquidstorage tank under seismic excitation", Proceedings of

the ASME Pressure Vessels and Piping Division Con-ference, San Antonio, Texas, Illinois, USA, PVP2008-61675 (2008).

9. Epstein, H.I. \Stress and displacements for oat-ing pan roofs", Comput. Struct., 15(4), pp. 433-438(1982).

10. Kwak, M.K. and Kim, K.C. \Axisymmetric vibrationof circular plates in contact with uid", J. Sound Vib.,146(3), pp. 381-389 (1991).

11. Sun, X., Liu, Y., Wang, J., Cen, Z. \Stress andde ection analyses of oating roofs based on a load-modifying method", Int. J. Press. Vessels Pip., 85,pp. 728-738 (2008).

12. Shabani, R., Tariverdilo, S., Salarie, H. and Reza-zadeh, G. \Importance of exural and membranesti�nesses in large de ection analysis of oating roofs",Appl. Math. Modell., 34, pp. 2426-2436 (2010).

13. Golzar, F.G., Shabani, R., Tariverdilo, H. and Reza-zadeh, G. \Sloshing response of oating roofed liquidstorage tanks subjected to earthquakes of di�erenttypes", ASME J. Pressure Vessel Technol., 134(5),051801-13 (2012).

14. Shabani, R. and Golzar, F.G. \Large de ection anal-ysis of oating roofs subjected to earthquake groundmotions", Nonlinear Anal. Real World Appl., 13, pp.2034-2048 (2012).

15. De Angelis, M., Giannini, R. and Paolacci, F. \Ex-perimental investigation on the seismic response of asteel liquid storage tank equipped with oating roofby shaking table tests", Earthquake Eng. Struct. Dyn.,39, pp. 377-396 (2009).

16. Shabani, R., \Stress patterns in single deck oat-ing roofs subjected to ground motion accelerations",International Journal of Engineering-Transactions C:Aspects, 26(12), pp. 1495-1504 (2013).

17. Thomson, W.T. and Dahleh, M.D., Theory of Vibra-tion with Applications, Prentice Hall, Chap. 11 (1998).

18. Meirovitch, L., Principles and Techniques of Vibra-tions, Prentice Hall, Chap. 7 (1997).

Appendix A

Equations of motionSubstituting Eqs. (4) and (5) into Eq. (7), the La-grangian is calculated by Eq. (A.1) as shown inBox A.1. Referring to the principle of variation,time integral of variation of Lagrangian vanishes in anydesired time interval:

t2Zt1

�Ldt = 0; (A.2)

which requires the multiplicands of both unknowns �A

Box A.1


Tabriz, Iran and his MS degree in the �eld of AppliedMechanics from Urmia University, Iran, in 2012. Heis now a PhD Candidate in the �eld of StructuralMechanics in University of Canterbury, New Zealand.His research interests include liquid-coupled vibration,stability in mems structures, and earthquake inducedstructural vibrations.

Rasoul Shabani received his BS degree in MechanicalEngineering from KNT University of Technology in1993. He received his MS degree in 1996 and his PhDin 2006 from Sharif University of Technology, Iran.

He is now a Professor in the Mechanical EngineeringDepartment at Urmia University. His research interestincludes uid structure interaction, chaotic vibrationsand nonlinear dynamics, and control of MEMS.

Saeid Tariverdilo received his BS degree in CivilEngineering in 1994 and his MS and PhD degrees in1997 and 2001 in Structural Engineering from SharifUniversity of Technology, Iran. He is now a Professor inthe Civil engineering Department at Urmia University.His research interest includes uid structure interac-tion.

stress analyses in single-deck and double-deck oating...

Documents