Egieerig, 013, 5, 543-550 doi:10.436/eg.013.56065 Published Olie Jue 013 (http://www.scirp.org/joural/eg) Fiite Elemet Modelig of Seismic Respose of Field Fabricated Liquefied Natural Gas (LNG) Spherical Storage Vessels Oludele Adeyefa, Oluleke Oluwole Departmet of Mechaical Egieerig, Uiversity of Ibada, Ibada, Nigeria Email: deleadeyefa@yahoo.com Received April 5, 013; revised May 6, 013; accepted May 16, 013 Copyright 013 Oludele Adeyefa, Oluleke Oluwole. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. ABSTRACT All real physical structures behave dyamically whe subjected to loads or displacemets. This research paper, therefore, presets seismic respose of field fabricated liquefied atural gas spherical storage vessels usig fiite elemet aalysis. The seismic aalysis procedure used represets a practical approach i quatifyig the respose of spherical storage vessel with its cotet whe it is subjected to seismic loadig. I the fiite elemet method approach, six degrees of freedom per ode is used for legs/colum of the spherical storage taks. Lumped mass procedure is employed to determie system mass matrix of the structure. Computer programme code is developed for the resultig matrix equatio form fiite elemet aalysis of the structure usig FORTRAN 90 programmig laguage. The modelig of the seismic load utilies the groud acceleratio curve of a site. From the results of the modal aalysis, the system is ucoupled thereby gives way to the applicatio of Newmark s method. Newmark s method as oe of the widely used time-step approach for the seismic respose is applied. The developed programme codig is validated with aalytical results (P > 0.5). It shows that the approach i this research work ca be successfully used i determie the stability of large spherical storage vessels agaist seismic loadigs whe base acceleratio spectral of the site are kow. This approach gives better results tha the static-force approach which gives coservative results. While the approach used i this research treats seismic loads as time evet, static-force approach assumed that the full groud force due to seismic motio is applied istataeously. Keywords: FEM; LNG; Seismic; FORTRAN; Newmark; UBC; ASME 1. Itroductio I more ad more egieerig situatios today, it is ecessary to obtai approximate umerical solutios to problems rather tha exact closed-form solutios. It is discovered that may of the egieerig problems faced by egieers i today s world preset o simple aalytical solutios. Though, desig egieer may have bee able to write dow the goverig equatios ad boudary coditios for the problem, yet he sees immediately that o simple aalytical solutio ca be foud. The difficulty i these types of egieerig problems lies i the fact that either the geometry or some other feature of the problem is irregular or arbitrary. Aalytical solutios to problems of this type seldom exist; yet these are the kids of problems that desig egieers are called upo to solve. The resourcefuless of the aalyst usually comes to the rescue ad provides several alteratives to overcome this dilemma. Oe possibility is to simplify the difficulties ad reduce the problem to oe that ca be hadled. Sometimes this procedure works; but, more ofte tha ot, it leads to serious iaccuracies or wrog aswers. Now that high-speed computers are widely available, a more viable alterative is to retai the complexities of the problem ad fid a approximate umerical solutio. Fiite elemet method is oe the widely used approximate umerical method ad its model of a problem gives a piecewise approximatio to the goverig equatios. The basic premise of the fiite elemet method is that a solutio regio ca be aalytically modeled or approximated by replacig it with a assemblage of discrete elemets. Sice these elemets ca be put together i a variety of ways, they ca be used to represet ex- Copyright 013 SciRes.
544 O. ADEYEFA, O. OLUWOLE ceedigly complex shapes. Though, the fiite elemet method ca be used to solve a very large umber of complex problems, there are still some practical egieerig problems that are difficult to address because we lack a adequate theory of failure, or because we lack appropriate material data. This is ot a fiite elemet problem, but is of serious cocer to ay egieer who wats to supplat testig with aalysis. The use of aalysis usually permits faster desig turaroud, the exploratio of widely varyig eviromets, ad the use of optimiatio tools. The seismic aalysis procedures outlie i Uiversal Buildig Code (UBC) is static-force procedures, which assume that the etire seismic force due to groud motio is applied istataeously. This assumptio is coservative but greatly simplifies the calculatio procedures. I reality earth quakes are time-depedet evets ad the full force is ot realied istataeously. The UBC allows, ad i some cases requires, that a dyamic aalysis be performed i lieu of the static force method. Although much more sophisticated, ofte the seismic loadigs are reduced sigificatly.. Fiite Elemet Methodology The most commo ad effective approach for seismic aalysis of liear structural systems is the mode superpositio method. After a set of orthogoal vectors have bee evaluated, this method reduces the large set of global equilibrium equatios to a relatively small umber of ucoupled secod order differetial equatios. The umerical solutio of those equatios ivolves greatly reduced computatioal time..1. Model Assumptios Below are the assumptios made i the model i this research work: 1) The storage tak is full of water, hydrotest, for the worst case sceario. This gives maximum compressive load/force per leg support. ) I determiig the system mass matrix, lumped sum mass matrix is employed. 3) The oly exterally applied load/force is the weight of the cotet (tak full of water) ad shell. Exterally applied force o each of the leg is the sum of the weight of cotet (tak full of water) ad shell divided by the umber of legs. While there is base acceleratio applied to the support due to seismic effect Figure 1. 4) Seismic effect o oe of the storage tak support legs is represetative of the effect of seismic load o the etire storage tak structure. If oe of the legs fails due to seismic load, the etire structure fails Figure. Figure 1. Typical vibratio model with compressive force ad earth/support motio. Figure. Showig sectio of spherical tak with leg support... Dyamic Equilibrium Takig ito accout the equivalet static force, the geeral equatios of equilibrium for a structure is as follows Ku F F 0 (1) where {F} is the force vector represetig the exteral applied forces as a fuctio of time. {F} dy, is the assembled force vector for the complete structure due to iertia ad dampig forces. [K] is the system static stiffess matrix ad {u} is odal displacemet. For may structural systems, the approximatio of liear structural behaviour is made to covert the physiccal equilibrium statemet, Equatio (1), to the followig set of secod-order, liear, differetial equatios, geeral equatio of motio M u C u K u F () dy where M is the mass matrix (lumped or cosistet), C is a Copyright 013 SciRes.
O. ADEYEFA, O. OLUWOLE 545 viscous dampig matrix (which is ormally selected to approximate eergy dissipatio i the real structure) ad K is the static stiffess matrix for the system of structural elemets. The time-depedet vectors, u, u ad u are ode acceleratios, velocities ad displacemets respectively..3. System Stiffess Matrix To determie the stiffess matrix for the beam-colum support of spherical storage vessels. It is assumed that a typical member leg ca be thought of a 3-D space frame havig four actios: two bedig actios, a twistig actio, ad a axial actio. Thus, the displacemet of each ode is described by three traslatioal ad three rotatioal compoets of displacemet, givig six degrees of freedom at each urestraied ode. Correspodig to these degrees of freedom are six odal loads. The otatios use for the displacemet ad force vectors at each ode are, respectively, u u v w (3a) T x y f f f f m m m T (3b) e x y x y O the local level, for each member, the forces are related to the displacemets by the partitioed matrices Fe ku (4) k a 0 0 0 0 0 a 0 0 0 0 0 b1 0 0 0 b 0 b1 0 0 0 b c1 0 c 0 0 0 c1 0 c 0 d1 0 0 0 0 0 d1 0 0 c3 0 0 0 c 0 c4 0 b3 0 b 0 0 0 b4 a1 0 0 0 0 0 b1 0 0 0 b c1 0 c 0 d1 0 0 c3 0 b 3 1 1 (5) where EA 1EI 1EI a1 b1 c 3 1 3 1 6EI 4EI 6EI y b 3 c c 3 b c EI y 4 b4, EI L e GI x d L e 4EI y E is the elastic modulus, G is the modulus of rigidity, L e is the legth of beam-colum elemet, I x, I y, ad I are the secod momet of iertia about x, y, ad respectively..4. System Mass Matrix It is useful to realie that usig lumped mass method to geerate the system mass matrix, the methodology does ot ivolve derivatives of the shape fuctios. We ca therefore be more lax about the choice of shape fuctio for the determiatio of system mass matrix tha for the stiffess matrix. I may applicatios, it is preferable to L e use a lumped mass (ad dampig) approximatio where the oly oero terms are o the diagoal. The simplest mass model is to cosider oly the ger traslatioal iertias, which are obtaied simply by dividig the total mass by the umber of odes ad placig this value of mass at the ode. Thus, the diagoal terms for the 3-D frame ca be give as 1 m AL e 1 1 1 0 0 0 1 1 1 0 0 0 (6) where ρ is material desity of beam elemet, A is the beam-colum cross sectioal area ad L e is the legth of beam-colum ad [M] is the mass matrix. These eglect the rotatioal iertias of the flexural actios. Geerally, may researchers i the field of fiite elemet methods had proved that the cotributios of the rotatioal iertia of the flexural actios to the system mass matrix are egligible ad matrix Equatio (6) above is quite accurate especially whe the elemets are small. There is, however, a very importat circumstace whe a ero diagoal mass is uacceptable ad reasoable o- Copyright 013 SciRes.
546 O. ADEYEFA, O. OLUWOLE ero values are eeded. I the preset research, a ero diagoal mass is uacceptable because this leads to computer programme error whe applyig Newmark s method for the seismic respose aalysis. Therefore, Equatio (6) is the modified to give equatio below with o-ero diagoal. 1 m AL e (7) 1 1 1 1 1 1 L where e. 40 where is typically take as uity. This scheme has the merit of correctly givig the traslatioal iertias. Therefore, estimate the effective legth L A/π as basically the radius of a disk of the same area as the triagle. Agai, α is typically take as uity..5. Free Udamped Vibratio Mode The structure is ot excited exterally i free vibratio mode i.e. o force or support motio acts o it. So, uder the coditio of free motio, dyamic aalysis ca be carried out ad the importat properties like atural frequecies ad mode shapes correspodig to the atural frequecy ca be obtaied. Sice free vibratio mode is cosidered, the structure is ot uder the ifluece of ay exteral force. Hece, the force ad dampig force vectors i stiffess equatios are take as ero. By takig the above coditio ito cosideratio, the stiffess equatio ca be represeted as, M u K u (8) 0 Thus u u (9) substitutig Equatio (9) ito Equatio (8) gives K M u 0 (10) This is liear eigevalue problem whose solutios, whe arraged i order of magitude of the eigevalues, are approximatio to the correspodig atural frequecies ad ormal modes of the vibratio of the system. To covert the geeralied eigevalue problem to the stadard form, the followig steps eed to be take: The geeral equatio is writte as or or Rearragig Equatio (11) gives K M u 0 (11) K u Mu 1 M K u u (11a) (11b) K M u u 1 1 Equatio (11c) may be rewritte as K M I u 1 1 (11c) 0 (1) where [I] is a idetity matrix The oly problem with Equatio (1) is that although [K] ad [M] are symmetric, the product [K] 1 [M] is geerally ot symmetric. To preserve symmetry, the Cholesky s method may be used [1]. The Cholesky method decomposes a square, symmetric matrix to the product of a upper triagular matrix ad the traspose of the upper triagular matrix. Applyig the Cholesky decompositio to the matrix [K] gives K CC T (13) Substitutig Equatio (13) ito (1) ad covertig it ito the stadard form gives (Griffiths ad Smith, 1991) 1 C M C I u 1 T 1 0 (14) Equatio (14) is i the form of a stadard eigevalue problem. It ca be expressed more closely to Equatio (11) if it is rewritte as where S Iu 0 (15) 1 1 T S C M C ad 1 Sice the matrices i eigevalue problem ofte become very large, the matrices may be coverted to a simpler form usig Householder s method before solvig for the eigevalues [1]. Householder s method coverts a symmetric matrix ito a tridiagoal matrix. A tridiagoal matrix has o-ero elemets oly o the diagoal plus or mius oe colum []. Solutios are obtaied usig FORTRAN 90 programmig laguage. The output of the program cosists of the atural frequecies of the system with the correspodig mode shapes. Sice the matrices i eigevalue problem ofte become very large, the matrices may be coverted to a simpler form usig Householder s method before solvig for the eigevalues [1]. Householder s method coverts a symmetric matrix ito a tridiagoal matrix. A tridiagoal matrix has o-ero elemets oly o the diagoal plus or mius oe colum []. Solutios are obtaied usig FORTRAN 90 programmig laguage. The output of the program cosists of the atural frequecies of the system with the correspodig mode shapes..6. Trasformatio of Modal Equatio The dyamic force equilibrium Equatio () ca be re- Copyright 013 SciRes.
O. ADEYEFA, O. OLUWOLE 547 writte i the followig form as a set of N secod order differetial equatios: M u C u K u F f g t j (16) J j1 All possible types of time-depedet loadig, icludig wid, wave ad seismic, ca be represeted by a sum of J space vectors f j, which are ot a fuctio of time, ad J time fuctios g(t) j. The umber of dyamic degrees of freedom is equal to the umber of lumped masses i the system. The fudametal mathematical method that is used to solve Equatio (16) is the separatio of variables. This approach assumes the solutio ca be expressed i the followig form: u t Y t (17a) where is a N by N matrix cotaiig N spatial vectors that are ot a fuctio of time, ad Y(t) is vector cotaiig N fuctios of time. From Equatio (17a), it follows that: u t Y t (17b) u t Y t (17c) Before solutio, it is require that the space fuctios satisfy the followig mass ad stiffess orthogoality coditios: T T M I ad K (18) where I is a diagoal uit matrix ad Ω is a diagoal matrix i which the diagoal terms are. The term has the uits of radia per secod. After substitutio of Equatios (17) ito Equatio (16) T ad the pre-multiplicatio by, the followig matrix of N equatios is produced: J j I Y d Y Y F P g t (19) j1 For three-dimesioal seismic motio, this equatio ca be writte as: T where Pj f j ad are defied as the modal participatio factors for load fuctio j. The term P j is associated with the th mode. There is oe set of N modal participatio factors for each spatial load coditio f j. For all real structures, the N by N matrix is ot diagoal; however, to ucouple the modal equatios, it is ecessary to assume classical dampig where there is o couplig betwee modes. Therefore, the diagoal terms of the modal dampig are defied by: d (0) where is defied as the ratio of the dampig i mode to the critical dampig of the mode [3]. A typical j j ucoupled modal equatio for liear structural systems is of the followig form: y t y t y t F P g t (1) j j j1 For three-dimesioal seismic motio, this equatio ca be writte as: yt y t yt () Put Put Put x gx y gy g where the three-directioal modal participatio factors, or i this case seismic excitatio factors, are defied by T Pj M j i which j is equal to x, y, or ad is equal to the mode umber. By examiig Equatio () above, the dampig term costat ca be writte as c (3) It has bee show that dampig ratio of a structure is proportioal to the structural stiffess. Therefore, this is the adopted i the preset work ad Equatio (3) the becomes where c is the mode dampig ratio. J (4).7. Step by Step Numerical Itegratio Procedure The step by step aalysis is based o the powerful iterative techique developed by [4] kow as Newmark s parameter method. The acceleratio at the ed of a time step is estimated, ad the velocity ad displacemet are the calculated by Pa (5) u, u, ad u represet the displacemet, velocity, ad acceleratio of ay poit. The suffixes i ad i + 1 represetig the eds of the time itervals i ad i + 1. The above expressios are substituted i the geeral equatios of motio ad a ew estimate for the acceleratio at the ed of the time step i + 1 is determied. The process is repeated util successive values of the acceleratio agree withi a specified tolerace. The parameter β i the Newmark s equatio govers the ifluece of the acceleratio at the ed of the time iterval i + 1 o i + 1 the displacemet at that istat. Furthermore, the value selected for β determies the variatio of the acceleratio durig the iterval Δt from i to i + 1, the method becomes a liear acceleratio assumptio [4]. A β value of 1/4 represets a costat acceleratio throughout the iterval ad β = 1/8 may be iterpreted as a step fuctio havig acceleratio u i over the first half of the time iterval ad u through the last Copyright 013 SciRes.
548 O. ADEYEFA, O. OLUWOLE half [4]..8. Participatig Mass Ratio This requiremet is based o a uit base acceleratio i a particular directio ad calculatig the base shear due to that load. The steady state solutio for this case ivolves o dampig or elastic forces; therefore, the modal respose equatios for a uit base acceleratio i the x-directio ca be writte as: y P (6) The ode poit iertia forces i the x-directio for that mode are by defiitio: f Mu t M y P M (7) x x x The resistig base shear i the x-directio for mode is the sum of all ode poit x forces. Or: V P I M P (8) T x x x x The total base shear i the x-directio, icludig N modes, will be: V x N Px 1 (9) For a uit base acceleratio i ay directio, the exact base shear must be equal to the sum of all mass compoets i that directio. Therefore, the participatig mass ratio is defied as the participatig mass divided by the total mass i that directio. Or: Z X Y mass mass mass N Px 1 M x N Py 1 M y N P 1 M (30a) (30b) (30c) A examiatio of values givig by the factors i Equatios (30) gives desig egieer a idicatio of the value of the base shear associated with each mode. The agle with respect to the x-axis of the base shear associated with the first mode is give by: P 1 1x 1 ta P 1y 3. Seismic Respose Aalysis (31) Case 1: To validate the computer program developed, modal aalysis is performed o a -degree of freedom structural system subjected to free udamped vibratio with the followig characteristics ad the results are compared with the aalytical method. M 0 0 1, K= 6 1 Case : Spherical pressure vessel leg support with the characteristics below is subjected to the seismic loadig. The leg colum is carbo steel, cold hollow circular pipe of outside diameter 33.9mm. The seismic loadig i this model is base acceleratio of Table 1. Compressive vertical Load/force per leg (N) = 3465.0; Momet of iertia about x-axis, I x (m 4 ) =8600 10 8 ; Momet of iertia about y-axis, I y (m 4 ) =14300 10 8 ; Momet of iertia about -axis, I (m 4 ) =14300 10 8 ; gth of the support (m) =.0; Modulus of Elasticity, E (N/m ) =06.8 10 9 ; Poisso Ratio, v = 0.3; Modulus of Rigidity, G (N/m ) = E/((1 + v)); Material Desity (Kg/m 3 ) = 7850; Structural Dampig proportioal costat = 0.5. 4. Results ad Discussios The results of the Case 1 i Table show that the FOR- TRAN codig developed i this research give better results of the atural frequecy of free udamped vibratio cosidered. I case two cosidered, Figure 3 gives the variatio of atural frequecies agaist for differet modes. Mode 7 gives highest atural frequecy of value of over 4500 H while the least of all is mode 1 which gives atural frequecy with value of ero H. Figures 3-6 give relative respose of the secod mode of vibratio for acceleratio, velocity ad displacemet. I the computer simulatio for the Case, the same value of dampig ratio is assumed for all modes. The structural system uder cosideratio has the same value for the dampig costat i all modes. Also, the results show above for the case two simulatio cosidered seismic effect i oe directio aloe the vertical directio. I reality, the effects of base acceleratio i differet directios at the site o the structural system have to be cosidered ad superimposed. By doig this, the desig Figure 3. Showig frequecy (H) agaist mode. Copyright 013 SciRes.
O. ADEYEFA, O. OLUWOLE 549 Figure 4. Showig relative acceleratio respose of secod mode. Figure 6. Showig relative displacemet respose of secod mode. egieer will be able to have better uderstatig of the behavioral or respose patter of the structure to the seismic loadig. Figure 5. Showig relative velocity respose of secod mode. Table 1. Showig typical earthquake groud acceleratio with time. Time (Sec) Mode Acceleratio Spectrum (g) Time (Sec) Acceleratio Spectrum (g) 0 0.068 0.14 0.0611 0.01 0.05914 0.15 0.06088 0.0 0.00503 0.16 0.060709 0.03 0.075961 0.17 0.06653 0.04 0.067458 0.18 0.060541 0.05 0.065777 0.19 0.060319 0.06 0.063504 0.0 0.060005 0.07 0.061549 0.1 0.059668 0.08 0.060357 0. 0.05944 0.09 0.060173 0.3 0.05983 0.1 0.06095 0.4 0.059559 0.11 0.061601 0.5 0.05983 0.1 0.061857 0.6 0.060157 0.13 0.061563 Table. Comparig FEA ad aalytical solutios. Natural Frequecy (H) (FEA) Natural Frequecy (H) (Aalytical) Percetage Error (%) 1 0.5 0.5 0.0 0.356 0.356 0.0 5. Coclusios ad Recommedatios The use of fiite elemet method i the seismic respose aalysis of field fabricated spherical pressure vessels caot be over emphasied. This is also i lie with the recommedatio from UBC which allows desig egieer to use other method such as fiite elemet method i carryig out the seismic respose of a structure. This is because; it is believed that static-force used by UBC code is coservative, ad to get accurate respose of ay structure subjectig to seismic load the use of fiite elemet aalysis is desirable. While it is oted that i ASME code sectio VIII Div 1, recommedatio is made to cosider other loads apart from iteral ad exteral pressures. Such other loads are wid, local ad seismic yet it silet o procedures for determiig the effects of such loads o pressure vessels. The decisio, therefore, lies i the hads of desig egieer to use ay suitable method available at his disposal i aalyig effects of such loads o pressure vessels. Oe of such methods available to desig egieers is fiite elemet method. If this method is properly applied, it will give soud ad safe desig. This research work has shed light o the methodology of applyig fiite elemet method to seismic respose aalysis of field fabricated spherical pressure vessels. The seismic aalysis procedure used represets a practical approach i quatifyig the respose of spherical storage vessel with its cotet whe it is subjected to seismic loadig. The result shows that the approach i this research work ca be successfully used i determie the stability of large spherical storage vessels agaist seismic loadigs whe base acceleratio spectral of the site is kow. This approach gives better results tha the static-force approach which gives coservative results. Copyright 013 SciRes.
550 O. ADEYEFA, O. OLUWOLE REFERENCES [1] D. V. Griffiths ad I. M. Smith, Numerical Methods for Egieers, Blackwell Scietific Publicatios, Lodo, 1991. [] W. H. Press, Numerical Recipes i C: The Art of Scietific Computig, Cambridge Uiversity Press, Cambridge, 199. [3] L. W. Edward, Three-Dimesioal Static ad Dyamic Aalysis of Structures, A Physical Approach with Emphasis o Earthquake Egieerig, Computers ad Structures Ic., Berkeley, 00. [4] N. M. Newmark, A Method of Computatio for Structural Dyamics, ASCE Joural of the Egieerig Mechaics Divisio, Vol. 85, 1959, pp. 67-94. Copyright 013 SciRes.