FINITE DIFFERENCE APPROXIMATION TO WAVE EQUATION TO SIMULATE SLOSHING OF GROUND SUPPORTED TANKS FOR EARTHQUAKE LOADINGS

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1 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 FINIE DIFFEENCE APPOXIAION O WAVE EQUAION O SIUAE SOSHING OF GOUND SUPPOED ANKS FO EAHQUAKE OADINGS Z.Z.A. aeed *,. askaramaaraa and K.K. Wiesundara Department of Civil Engineering, Faculty of Engineering, University of Peradeniya, Peradeniya, 4, Sri anka. * amzzano@gmail.com, P: Astract: Slosing is a liquid viration pysical penomenon wic causes wen liquid storage tank is suected to external loading. aor effects due to slosing are iger impact pressure on tank walls and over spillage of liquid. erefore, tis study was aimed to investigate te slosing pressure of ground supported rigid cylindrical tanks under eartquake loading. In tis study, wave equation was used to convert te pysical penomenon to a matematical model and nonlinear terms were approximated. Finite difference metod was used to solve te matematical model for te simulation of slosing in frequency domain for D analysis. Input motions of eartquake loading were otained from te average Fourier spectrum of seven eartquake records. Here, te liquid was assumed to e inviscid, incompressile and irrotational. ased on te results otained using te generated finite difference code, te aspect ratio of te tank and frequency of ground motion affects te slosing pressure. Keywords: eartquake; finite difference; frequency domain; ground supported tanks; slosing; 1. Introduction Slosing is a pysical penomenon wic is caused wen liquid tanks are suected to movement or viration y external loading. Higer impact pressure on tank walls and spill over of liquid are maor effects of slosing. Higer impact pressure will cause instaility to te structure wic leads to structural failures wereas tanks wit acids, fuel will cause severe environmental prolems due to over spilling. ese effects sould e analysed to avoid damages. Analytical, numerical and experimental tecniques were used in previous studies. ass-spring model (alotra, P., 1997), potential flow teory (Zang, H. and Sun,., 14), aplace and ernoulli equations (uiz,.o. et al, 15) are some of te models used to study slosing in past researces. e present work focuses on developing a general finite difference code for te simulation of slosing pressure of ground supported tanks for eartquake loadings. In tis study, wave equation in frequency domain was used to develop a matematical model and finite difference metod was applied to solve te matematical model.. Formulation of atematical odel for Slosing A scematic diagram of a D cylindrical tank and te coordinate system as sown in Figure 1 was considered trougout te study. e tank was assumed to e rigid, fixed at te ottom and top as free surface. e liquid was assumed as inviscid, incompressile and irrotational. y Fig 1: Scematic diagram of D cylindrical tank x

2 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 Wave equation for pressure was considered, P P 1 P x y c t (1) Here, t, P, c are time, pressure and P-wave velocity in te water respectively. ut, pressure can e defined as a simple armonic motion for slosing. P () i t P e From equation (1) and (), wave equation in frequency domain can e derived as in equation (3) P P P x y c (3) code was written using atla ased on te following procedure. D rectilinear mes wit te size of nx ny and aving constant grid spacing of and along te x and y directions accordingly was considered. e terms were grouped y grid location as sown in Figure to solve te equation (6) implicitly. Were te suscripts i and were used as local references to discrete locations on te grid. P P P i1, i 1, P c Pi, 1 Pi, Pi, 1 (6) Since slosing occurs due to external loading, a source term was introduced to equation (3) wic implies equation (4). Here is external loading as pressure function. P P P x y c (4) Equation (4) was converted as finite difference equation y applying central finite difference metod as follows: Fig : Grid locations P P P P P P x x, y x, y x x, y x, y y x, y x, y y ( x) ( y) P c x, y x, y Equation (5) represents te matematical model. Here, frequency of external loading. 3. Solution for te atematical odel o solve accurately te and (5) slosing as ω is te finite difference equation easily, a finite difference e equation (6) was solved for te coefficient set as sown elow wic operates on P. e coefficient sets represents te amount of contriution to pressure coming from te adacent nodes to te considered point as in Figure 3. C, C, C, C,,,,, represents te contriutions from top left corner, top rigt corner, ottom left corner, ottom rigt corner, top, ottom, rigt, left and middle sides respectively.

3 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 C C () Coefficient set for te ottom oundary, c C Fig 3: Symolic notations for te different cardinal directions. Equation (6) was rearranged as follows and coefficient sets for te internal nodes were otained. P P c ( ) i 1, Pi 1, Pi, 1 Pi, 1 i, C Coefficient set for te rigt oundary, c (7) It can e noted tat tere is no contriution coming from adacent corner nodes to te considered node. Coefficient sets for te top, ottom, rigt side and left side oundaries and for te corners also were derived as follows. Coefficient set for te top oundary, c g Coefficient set for te left oundary, c

4 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 Coefficient set oundary, for te top rigt corner Coefficient set oundary, for te ottom left corner 1 c 4 g Coefficient set for te ottom rigt corner oundary, 1 c 4 Coefficient oundary, set for te top left 1 c 4 g corner 1 c 4 Equation (7) can e furter modified into matrix form as follows: KP 1 P K (8) [K] can e otained from te aforementioned coefficient sets. [] can e found y sustituting external loading as pressure. In tis study, te ottom of te tank was considered as fixed and te top surface was free. us, external loading was assumed to e transferred to te tank from te two rigid side oundaries. Using te equation (8) slosing pressure at eac node can e found. A atla code was generated to do te simplifications quickly and easily for te aforementioned simulation procedure. e input parameters w (frequency) and a (amplitude) can e otained from Fourier spectrum of eartquake loading. 4. Analysis for te Possile Eartquake oading of Sri anka Analysis was performed for te selected case studies in ale 1 y applying possile level of eartquake loading in Sri anka.

5 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE Case Studies In order to find out te effect of aspect ratio in slosing pressure four case studies were considered. ale 1: Case studies Aspect atio (D/H) Fig 4: Details of tank Diameter (m) Heigt (m) 4. Selection of Eartquake oading D According to Euro code 8, minimum of seven eartquake records are needed to find te average spectrum for analysis. Eartquake records were otained from PEEC (Pacific Eartquake Engineering esearc Centre) network as accelerograms and response spectra of tose records were otained. Seven accelerograms were selected in suc a way tat te average response spectrum of te selected eartquake records matced wit te availale design response spectrum of Sri anka (Uduweriya. et al, 13) as sown in Figure 5. H Spectral acceleration (g) Design response spectrum ean Period (s) Fig 5: Average response spectrum 4.3 Applying Eartquake oadings to te Code Collected eartquake data were accelerograms. It is required to find out te frequency and amplitude of tose data since tose are important input parameters in te generated finite difference code for te simulation. Since te eartquake data contain discrete values, to find te frequency and amplitude, Fast Fourier transform algoritm was used. atla code was generated and used for tis computation and tose values were verified from te results otained using te software SeismoSignal. Figure 6 sows te average Fourier spectrum wic was used for te analysis. Using te amplitude values of eac frequency from average Fourier spectrum, eartquake loading was calculated as pressure value. ese pressure values were sustituted to te external load matrix () in te code. Fourier Amplitude/ (m/s²) Frequency/ (Hz) Fig 6: Average Fourier spectrum

6 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 Since, te ase was assumed to e fixed and top surface to e free, external load was considered as transferring from te two rigid side walls. Since te walls were treated as rigid te same amount of external loadings applicale for tose oundaries. Input values for frequency and amplitude for matla code were otained from Figure esults and Discussion e generated code will give te slosing at eac node of mesed tank as output. is code is needed to e run required numer of times y canging te size of mes in order to get accurate values. equired mes size to get converged accurate results was varied wit te size of tanks. Figure 7 sows te variation of slosing pressure wit mes size for case study (D=6m and H=3m) esults otained for te case study at te peak point of average Fourier spectrum (1.196 Hz and.894 m/s) is sown in Figure 8. Size of te mes to get te converged answer for te case study 1 was.5m and for case studies,3 and 4 was.5m. Slosing Pressure Heigt (m) (kpa) Heigt (m) (a) Diameter (m) (kpa) (kpa) aximum Slosing Pressure (kpa) es Size (m) Fig 7: Variation of slosing pressure wit mes size Diameter (m) () Fig 8: Variation of slosing pressure across te tank (H=3, D=6m) From Figure 8, iger effect due to slosing pressure occurs at te side walls of te tank wereas at te middle slosing pressure is zero. Figure 9 sows te variation of normalized maximum slosing pressure (normalized wit respect to static pressure of eac tank) wit te frequency of input motion from te aove analysis.

7 aximum Slosing Pressure/ Static Pressure (y) aximum Slosing Pressure/Static Pressure e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE Case Study 1 Case Study.8.6 Case Study 3 Case Study Frequency (Hz) Fig 9: Variation of normalized slosing pressure wit frequency According to Figure 9, slosing pressure effect is iger for te frequency range of.1hz 3Hz. Wen frequency increases, waves interfere destructively. us eyond 3Hz slosing effect is inappreciale y =.389x -.51x +.45 Fig 1: Variation of normalized slosing pressure wit aspect ratio Figure 1 sows te variation of normalized slosing pressure wit aspect ratio for te possile eartquake saking of Sri anka. e variation sows a second order polynomial trend. 5. Conclusions and ecommendations Actual Plot rend ine Aspect atio (x) Generated finite difference code can e used to simulate slosing pressure of ground supported rigid cylindrical tanks wit fixed ottom and free top surface. From te results otained, Slosing pressure is dominant for te frequency range of.1 Hz 3 Hz irrespective of aspect ratio. eyond tis limit te effect of slosing is insignificant. Slosing pressure increases wit aspect ratio. ut for te aspect ratios less tan 1 slosing effect is inappreciale. erefore, slosing effect for te tanks wit aspect ratio more tan 1 sould e necessarily cecked for te eartquake loading witin te frequency range of.1 Hz 3 Hz to avoid te ground supported water tank failures due to slosing in Sri anka. Acknowledgement e autors would like to tank te Department of Civil Engineering, University of Peradeniya for giving an opportunity to carry out tis study. eferences [1]. alotra, P. (1997). New metod for seismic isolation of liquid-storage tanks. Eartquake engineering and structural dynamics, 6,

8 e 7 t International Conference on Sustainale uilt Environment, Earl s egency Hotel, Kandy, Sri anka from 16 t to 18 t Decemer 16 ICSE16-81 []. Zang, H, & Sun,. (14). Numerical simulation of slosing in D rectangular tanks ased on te prediction of free surface. (H. Kar, Ed.) atematical prolems in engineering, 14, pp. 1-1 [3]. uiz,.o, opez-garcia, D, & aflanidis, A.A. (15). An efficient computational procedure for te dynamic analysis of liquid storage tanks. Engineering structures, 85, 6-1 [4]. Uduweriya, S., Wiesundara, K.K and Dissanayake, P.. (13) 'Seismic risk in Colomo - Proailistic approac', SAI esearc Symposium on Engineering Advancements 13,