Seismic Response of Pipeline Systems in a Soil Liquefaction Envir

Old Dominion University ODU Digital Commons Civil & Environmental Engineering Theses & Dissertations Civil & Environmental Engineering Winter 1992 Seismic Response of Pipeline Systems in a Soil Liquefaction Environment Hongzhi Zhang Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/cee_etds Part of the Civil Engineering Commons Recommended Citation Zhang, Hongzhi "Seismic Response of Pipeline Systems in a Soil Liquefaction Environment" (1992) Doctor of Philosophy (PhD), Dissertation, Civil & Environmental Engineering, Old Dominion University, DOI: 10.25777/168t-tt44 https://digitalcommons.odu.edu/cee_etds/93 This Dissertation is brought to you for free and open access by the Civil & Environmental Engineering at ODU Digital Commons It has been accepted for inclusion in Civil & Environmental Engineering Theses & Dissertations by an authorized administrator of ODU Digital Commons For more information, please contact digitalcommons@odu.edu SEISMIC RESPONSE OF PIPELINE SYSTEMS IN A SOIL LIQUEFACTION ENVIRONMENT by HONGZHI ZHANG December 1992 Department of Civil Engineering Old Dominion University Norfolk, Virginia 23529 Reproduced with permission of the copyright owner Further reproduction prohibited without permission Seismic Response of Buried Pipeline System in a Soil Liquefaction Environment by Hongzhi Zhang B.E., July 1969, Beijing Building Material Institute, Bering, China M.E., July 1984, Beijing Municipal Engineering Research Institute, Beijing, China A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY CIVIL ENGINEERING OLD DOMINION UNIVERSITY December 1992 Approved by QDr J Mark D o rrep aa Dorrepaal Dr Due T Nguyen Dr Zia Razzaq Reproduced with permission of the copyright owner Further reproduction prohibited without permission This research is dedicated to my mother GUEI-RONG XIAO iii Reproduced with permission of the copyright owner Further reproduction prohibited without permission Acknowledgement I wish to express my highest appreciation to my advisor, Dr Leon R.L Wang, for his technical advice and moral support throughout this research The appreciation is also extended to Dr J Mark Dorrepaal, Dr Due T Nguyen, Dr Isao Ishibashi and Dr Zia Razzaq for their suggestions and comments on my dissertation Special thanks are extended to my wife, Ping Sun, whose love, encouragement and support are invaluable The support from the faculty, Earthquake Engineering Research Group and my colleague graduate students of the Civil Engineering Department is also acknowledged iv Reproduced with permission of the copyright owner Further reproduction prohibited without permission Seismic Response of Buried Pipeline System in a Soil Liquefaction Environment Hongzhi Zhang Old Dominion University Advisor: Dr Leon R.L Wang Abstract This research is a study of the general seismic response behavior of buried pipeline systems during a soil liquefaction process To aid the design of buried pipelines in a soil liquefaction environment, the purpose of this research is to provide the basic dynamic seismic response of different pipeline systems Several important parameters such as pipe diameter, buried depth, additional mass and the size of the liquefiable soil zone have been introduced The pipeline systems under study are cross-types, T-types and straight pipelines, with or without a manhole, buried in a soil liquefiable zone Time-varying soil spring constants are used for the analysis of the soil liquefaction process The equation of motion includes nonlinear geometric and material damping terms The pipe body is assumed to be elastic A computer program based on the finite element method has been developed The mode superposition method is used to solve the equation of motion of the pipeline The required eigenvalues and eigenvectors are calculated by subspace iterations A few uncoupled modal equations of motion are solved by a step-by-step numerical integration method This dissertation presents the background, formulation, v Reproduced with permission of the copyright owner Further reproduction prohibited without permission verification of the developed program, numerical results, conclusions of seismic response of buried pipeline systems under a soil liquefaction environment and suggestions of future research to aid seismic design of pipeline systems vi Reproduced with permission of the copyright owner Further reproduction prohibited without permission TABLE OF CONTENTS PAGE LIST OF TA B L E S .x LIST OF F IG U R E S xii Chapter I Introduction .1 1.1 Background 1.2 Brief Review of Studies on Seismic Response of Pipelines .3 1.3 Assumptions and Limitations .6 1.4 Objectives and Scope Chapter II Finite Element Formulation 11 2.1 Equations of Motion in Matrix Form 11 2.2 Mass Matrix 2.3 Damping Matrix 13 12 2.3.1 Geometric Damping 13 2.3.2 Material D am ping 15 2.4 Stiffness M a trix 15 2.5 Earthquake Input 18 Chapter III Method of Solution 29 3.1 General Remarks 29 3.2 Modal Superposition M ethod 30 vii Reproduced with permission of the copyright owner Further reproduction prohibited without permission 3.3 Subspace Iteration Method 31 3.4 Step-by-Step Numerical Integration M ethod 33 3.5 Computer Program Developed for this Dissertation Research 34 3.6 Verification of the Developed Computer Program 36 3.6.1 Verification of eigenvalues and eigenvectors of a straight pipeline with different boundary conditions 36 3.6.2 Minimum number of modes 37 3.6.3 Check of structural symmetry and the effect of axial loads 38 3.6.4 Comparison to a buried pipeline experiment 39 Chapter IV Parametric Study On Straight Pipelines 62 4.1 General Remarks .62 4.2 Parametric Study 63 4.2.1 The relationship of maximum response topipe diameters 64 4.2.2 The size of the soil liquefiable z o n e 65 4.2.3 The relationship between maximum response via depth of pipelines 66 4.2.4 The effect of the soil spring value in the non-liquefiable zone 67 4.2.5 The influence to the maximum response of a straight pipeline with different amount of additional soilmass 68 4.3 Discussions 68 Chapter V ‘T ”-type and Cross-type Pipeline Systems 79 viii Reproduced with permission of the copyright owner Further reproduction prohibited without permission i “T ”.type of Pipeline System 79 5.1.1 General Remarks 79 5.1.2 Verification of the Computer Program for “T ”-type pipelines 80 5.1.3 Parametric study of “T ’-type pipeline 82 5.1.3.1 Effects of the diameter ratios between the mains and the branches 82 5.1.3.2 Effects of the size of the soil liquefiable zone 84 5.1.3.3 The effects on the seismic response of different soil spring values in the non-liquefiablezo n e 5.1.3.4 85 Effects of the direction of the input earthquake waves 86 5.2 Cross-type of Pipeline Systems 87 5.2.1 General Remarks 87 5.2.2 Verification of the Computer Program for Cross-type of Pipeline 87 5.2.3 Parametric study of Cross-type pipeline 89 5.2.3.1 Effect on the response analysis from the main and branch diameter ratio 89 5.2.3.2 Effect of the size of the soil liquefiable zone 91 5.2.3.3 Effect of the soil spring value in the non-liquefiable zone 91 5.3 Discussion 92 ix Reproduced with permission of the copyright owner Further reproduction prohibited without permission FILE: STRA FORTRAN A1 OLD DOMINION UNIVERSITY 700 1002 1005 1006 1007 1008 1010 1020 1030 1035 101*0 1050 1060 1070 RETURN FORMAT (10F 0) FORMAT (2X, 10E10.3) FORMAT ( X E ) FORMAT( / / / 2X.'STOP,NC LARGER THAN # OF MASS DOF') F O R M A T (///.2 X ,'D O F EXCITED BY UNIT STARTING ITERATION VECTORS') FORMAT (1H 1,2X , I T E R A T I O N NUMBER', I A) FORMAT(2X,'PROJECTION OF A (MATRIX A R )') FORMAT (2X.'PROJECTION OF B (MATR IX BR)') FORMAT(2X,' EIGENVALUES OF AR-LAMDA*BR') FORMAT (2X, ' AR AND BR AFTER JACOBI 01 AGONAL IZATIO N') FORMAT (2X ,'RELATIVE TOLERANCE REACHED ON EIGENVALUES') FORMAT (2X, 'CONVERGENCE REACHED FOR RTOL ‘ , E10.it) FORMAT (2X 'N O CONVERGENCE AFTER MAX # OF ITERATIONS ' , / / , X , 'ACCEPT CURRENT ITERATION V A L U E S X , ' NO STURM SEQ CHECK') 1100 FORMAT (2X ,'T HE CALC EIGENVALUES ARE') 1115 FORMAT(2X,'PRINT ERROR NORMS ON THE EIGENVALUES') 1110 FORMAT (2X ,'T HE CALC EIGENVECTORS ARE’ ) 1120 FORMAT (2X.'CHECK APPLIED AT SHI F T ' , E 2 14) 1130 FORMAT(2X,'THERE ARE' , 14 , ' EIGENVALUES MISSING') 1140 FORMAT (2X,'WE FOUND THE LOWEST', 14 , ' E1GENVALUES') END S U B R O U T IN E MULTSP ( T T B R R , M A X A , N N N W M ) C * TO E VA LU ATE PRODUCT OF B T I M E S RR AND STORED IM P L IC IT R E A L *8 (A -H ,0 -Z ) D I M E N S I O N T T ( ) , B ( ) , R R ( ) ,M A X A ( ) 10 20 40 I F (NWM G T NN) DO 10 = , NN T T ( I ) =B ( I ) * R R ( I ) R ETURN GO T O DO 40 1== , NN TT ( I ) = DO 100 = , NN KL=MAXA ( I ) KU=MAXA (1 + 1) -1 I = 1+ 100 21 220 200 CC=RR ( I ) DO 100 KK=KL,KU 11=11-1 TT ( I I ) =TT ( I I ) +B (KK) *CC IF (NN EQ 1) RETURN DO 200 I = ,NN KL=MAXA ( I ) +1 KU=MAXA (1 + 1) -1 IF (KU-KL) 0 , , I 1=1 AA=0 DO 220 KK=KL, KU 11=11-1 AA=AA+B (KK) *RR ( I I) TT ( I ) = T T ( I ) + AA CONTINUE IN T T STR17610 STR17620 S T R I 763O STR17640 S T R 17650 STR17660 STR17670 STR17680 STR17690 STR17700 STR17710 STR17720 STR17730 STR17740 STR17750 STR17760 STR17770 STR17780 STR17790 S TR 17800 STR17810 STR17820 STR 17830 STR17840 S TR 17850 STR17860 STR17870 STR17880 STR STR17900 STR17910 STR17920 STR17930 STR17940 STR17950 STR17960 STR17970 STR17980 STR17990 STR18000 STR18010 STR18020 STR18030 STR18040 STR18050 STR18060 STR 18070 STR18080 STR18090 STR18100 S T R 10 STR18120 STR18130 STR STR18150 Reproduced with permission of the copyright owner Further reproduction prohibited without permission FILE: STRA FORTRAN A1 OLD DOMINION UNIVERSITY C STR18160 STR18170 STR18180 STR 18 190 STR18200 STR18210 STR 18220 STR18230 STR182L0 STR18250 STRI8260 STR 18270 STR18280 STR18290 STR18300 STRI IO STR18320 STR18330 STR 1831*0 STR18350 STRI O STR18370 STRI 8 O STR18390 STR 181(00 STR 181*10 STR181*20 STR181*30 STR1 M C THIS IS THE PROGRAMFOR SOLVING THE EIGENVALUES AND EIGENVECTORS STRl81*50 C OF A PIPELINE SYSTEM WITH A MANHOLE (COORD I NATE SYSTEM) STRl81*60 STR1 Bk'JO C STR 181*80 IM PLICIT REAL* ( A - H , -Z ) STRI L90 DIMENSION XN (101) YN (101) , PL (100) ,TD (100) ,AE(100) , DA (100) ,TH(100) STR18500 STRI 8510 Zl (100) , OK (100) , AK (100) ,A M (1 0 ,6 ,6 ) , B M (1 0 , ,6 ) , T ( 0 , , ) , s t r 18520 TTR ( 0 , , ) ,GM(300) ,GK (300,300) , P K ( 0 , ) , S K ( 0 , , ) , s t r 18530 AM M{100,6,6) ,AMT ( 0 , , ) ,TK ( 0 , , ) ,TKT ( 0 , , ) , STR 1851*0 T K K ( 0 , , ) ,PM(100) , 1* A (10000) , B (1000) , MAXA (301) ,R (300,1*) , STR18550 I ROWL (300) , E (100) , OB ( 0 ) , AC (1 0 ) , ADDM (100) ,Z0 (100) , PWM (100) , STR18560 STR18570 GC (300) , RT ( ,300) , DM ( 0 ) , DMD ( , ) OK (5 0 ) ,DKD ( ) STR18580 DC ( , 30 ) , DCD ( , ) PN ( 0 ) , DPN ( ) STR18590 DOC=60.0 STR18600 c dcc=doc/6 o o STR18610 DCC=1.0 STR18620 OKO=927.0127l* STR18630 RAT=OS/OKO STR 1861*0 C RAT=1.0 STR18650 XN (1) = STR18660 YN (1) = STR18670 DO 111 1=2,101 STR18680 XN ( I ) =XN( I - ) + 0 STR18690 YN ( I ) =YN ( - ) STR18700 111 CONTINUE RETURN END C* * * * * * * * ! ' ! * * * * * * * * * # ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE A8 CC(A,B,MAXA,N,NTERMS,NP1, IROWL) IMPLICIT RE A L*8(A -H ,0 -Z ) ' DIMENSION A (NTERMS) , MAXA(NP1 ) , B (N) , I ROWL (N) MAXA (1) = J DO 10 = ,NP1 MAXA ( I ) =MAXA (I - ) + 10 CONTINUE C L=MAXA (NP1) DO 11 = ,N B (I) =2.0 11 CONTINUE DO 1= , NTERMS A ( I)= 0 CONTINUE A (I) =1.0 DO 1= , NTERMS A ( I ) =A (I - 1) + CONTINUE DO 1= , N IROWL ( I ) =1 CONTINUE RETURN END SUBROUTINE ABCD (OS,GK.GM.GC,A,B,MAXA,NN.NTERMS,NP1, IROWL.NF5) permission of the copyright owner Further reproduction prohibited without permission FILE: STRA FORTRAN Al OLD DOMINION UNIVERSITY DO 1)2 1=1,1 0 DA ( I ) =000 TH (l)=DCC ZO(0 = ' E (l)= 2100000.0 112 CONTINUE AMAX=0.4 D = 0 D I =120.0 H = 0 20 =2 1T = 15*0 ERFA=0.7 ZO3 = ETA=370.0 GAMA=0.944 Z0**= F E = T P = C CCRX=0.12297E07 C CCRY=0.53419E06 C DRATI0 = C ******** HM=Z02* ( D * * - D I * * ) *H*ATAN (1 0) / 0 / C DO 333 1= 1.1 0 C A IT = *IT C333 K ( I ) = K ( I ) * ( - A I T ) C * * * TO CALCULATE THE LENGTH AND THE ANGLE OF EACH ELEMENT DO 1=1,100 P L ( I ) = S Q R T ( ( X N ( I + ) - X N ( | ) ) * * + (YN (l + l) -YN ( I ) ) * * ) CONTINUE DO 1= 1,1 0 T D (I)= A S IN ((Y N (I+ )-Y N (|))/P L (1 )) CONTINUE DO 107 = 1.1 0 C W R I T E ( ,* ) I ,PL ( I ) ,TD ( I ) 107 CONTINUE C * * * TO TO CALCULATE THE MASS PER UNIT LENGTH (KG/CM) I F ( (O S /9 23 *0 - L E (1 / 9 - ) ) GO TO 853 A D D =10.*0S /927.0 3^ *1 0.0 GO TO 85^ 853 ADD=10 851* DO 1= ,3 AE ( I ) = * A T A N ( 0) *DA ( I ) *TH ( I ) AC (1) =ATAN (1 0) * A ( I ) * * Zl ( l ) = l » * A T A N ( ) * D A ( l ) * * * T H ( l ) / C MASS IS KG/CM**3/CM PWM ( I ) = (ZO ( I ) *AE ( l)+ Z *A C ( ) ) / 10 0 C ADDM ( I ) = (E T A *6 * Z * 453) / ( * * ) *AC ( I ) PM ( I ) = (1 O+ADD) *PWM ( I ) / C PM ( I ) = (PWM ( I ) +ADDM ( I ) ) CONTINUE DO 88 1=31 ,7 AE ( I ) =4.0*ATAN ( ) * D A ( I ) * T H ( I ) AC ( I ) =ATAN (1 ) *DA ( I ) * * STR18710 STR18720 STR18730 STR18740 STR18750 STR 187 60 STR18770 STR18780 STR18790 STR18800 STR18810 STR18820 STR18830 STR18840 STR18850 STR18860 STR18870 STR18880 STR18890 STR18900 STR18910 STR18920 STR18930 STR18940 STR18950 STR18960 STR18970 ST RI O STR18990 STR19000 STR19010 STR19020 STRI9030 STR19040 STR19050 STR19060 STR19070 STR19080 STR19090 STR19100 STR19110 STR19120 STR19130 STR19140 STR19150 STR19160 STR19170 STR19180 STR19190 STR19200 STR19210 STR19220 STR19230 STR19240 STR19250 Reproduced with permission of the copyright owner Further reproduction prohibited without permission F IL E : STRA FORTRAN A1 OLD DOMINION UNIVERSITY Zl ( l ) = * A T A N ( ) * D A ( l ) * * * T H ( l ) / STR19260 STR19270 MASS IS KG/CM**3/CM STR19280 PWM ( I ) = (ZO ( I ) *AE (I )+Z *A C ( ) ) / 10 0 STR19290 C ADDM ( I ) = (ET A*62 * Z * 3) / ( * * ) *AC ( I ) STR19300 PM( I ) = ( 0+ADD*RAT)*PWM( I ) / STR19310 C P M (l) = (PWM(l)+ADDMO)) STR19320 88 CONTINUE STR19330 DO 89 1=71,100 STR19340 AE ( I ) =4.0*ATAN ( ) *0 A ( I ) *TH ( I ) STR19350 AC ( I ) =ATAN ( ) * D A ( I ) * * STR19360 ZI ( I ) =1*.0*ATAN (1 ) *DA ( I ) ** * T H ( I ) / C MASS IS KG/CM**3/CM STR19370 STR19380 PWM ( I ) = (ZO ( I ) *AE ( I ) +Z03*AC ( I ) ) / 10 0 STRI 9390 C ADDM( I ) = (ET A*62 * Z * 453) / (3 * * ) *AC ( I ) STR19400 PM( I ) = ( O+ADD)*PWM ( I ) / STR19410 C PM( I ) = (PWM( I ) +ADDM( I ) ) STR19420 89 CONTINUE C * * * TO ESTABLISH THE ELEMENT MASS MATRIX FOR STIFFNESS MATRIX OF SOIL STR19430 STR19440 DO = 1,1 0 STR19450 DO J = ,6 STR19460 DO K =1,6 STR19470 AM (I , J ,K ) = STR19480 CONTINUE STR19490 DO 10 1=1,100 STR19500 A M (I,1 ,1 )= 0 STR19510 A M ( I , , ) =AM ( , , ) STR19520 AM ( , >U) = STR19530 AM ( , , ) = * STR19540 AM ( 1, , ) =AM(1, , ) STR 19550 AM (I , , ) = 2 * P L ( I ) STR19560 Art( 1, ) = —AM( 1, , ) STR 19570 A M O , , ) =5**.0 STR19580 AM(1, , ) = - - * P L ( I ) STR19590 AM ( , , ) “ “ AM(1, , ) STR 19600 AM ( , ) = * P L ( I ) * * STRI IO AM ( , , ) =AM(1, , ) STR19620 AM ( , ) = - - * P L ( I ) * * STR19630 10 CONTINUE STR19640 DO 110 1=1,100 STR 19650 DO 110 J = , STR19660 DO 110 K=1, J - l STR19670 AM(1 , J ,K )= A M (! ,K , J) STR19680 110 CONTINUE STR19690 DO 11 1=1,100 STR19700 DO 11 J = , STR19710 DO 11 K=1,6 STR19720 BM (I , J , K) =AM (I ,J , K) C WRITE ( , * ) 'BMC , I , J , K , ' ) = ' ,BM(I ,J ,K ) STR19730 STR19740 11 CONTINUE DO 16 1=1,100 STR19750 STR19760 C * * * TO ESTABLISH THE TRANSFORMATION MATRIX FOR EACH ELEMENT STR19770 DO 16 J = , • STR19780 00 16 K=1 ,6 STR19790 T ( l , J ,K ) = STR19800 16 CONTINUE C Reproduced with permission of the copyright owner Further reproduction prohibited without permission FIL E: STRA FORTRAN A1 OLD DOMINION UNIVERSITY DO 17 = 1,100 T ( 1, , ) =COS (TD ( I ) ) T ( I ) = - S I N ( T D ( l ) ) T ( 1,2', 1) = - T ( , , ) T ( I , , ) =T ( 1, , ) T ( l , ) = 1.0 T ( 1, , ) =T ( 1, , ) T ( I , , ) =T ( 1, , ) T ( 1, , ) =T ( 1, , ) T ( 1, ) =T ( , ) T ( 1, , ) = 17 CONTINUE C*ft* TO GET THE TRANSPOSE MATRIX OF THE TRANSFORMATION MATRIX DO 18 = 1,100 DO 18 J = ,6 DO 18 K=1,6 TTR (I , J , K) =T ( I , K, J) 18 CONTINUE ************** C*ft* TO ESTABLISH THE STIFFNESS MATRIX OF PIPE ELEMENT DO 40 1=1,100 DO 40 J = ,6 DO 40 K= 1, P K ( l.J K ) =0.0 40 CONTINUE DO 41 1=1,100 PK { 1, , ) =AE ( I ) *PL ( I ) * * / Z I ( I ) 41 CONTINUE DO 45 1=1,100 PK ( , , ) = - P K ( , , ) PK ( I , , ) = 2.0 PK ( I , , ) = * P L ( I ) PK ( I , ) =-PK (I , , ) PK ( I , ) = P K ( I , , ) PK ( I , , ) = * P L ( I ) * * PK ( , , ) =~PK (1 , , ) PK ( 1, ) = * P K (I , , ) PK ( ) =PK ( , ) PK ( , , ) = P K ( , , ) PK ( 1, , ) — P K ( , , ) PK ( , , ) = P K ( ) 45 CONTINUE DO 46 1=1,100 DO 46 J = , DO 46 K=1, J—1 PK (I , J ,K) =PK (I , K , J) 46 CONTINUE DO 47 1=1,100 DO 47 J = ,6 DO 47 K=1, PK ( I , J , K ) = E ( I ) * Z I ( l ) / ( P L ( l ) * * ) * P K ( l J , K ) C WRITE ( , * ) , P K ( ‘ , I , J , K , ‘ ) = I , P K ( I , J , K ) 47 CONTINUE Cftft* TO USE THE LUMPED-MASS MATRIX FOR EACH ELEMENT DO 12 1=1,100 STR19810 STR19820 STR19830 STR1?840 STR19850 STR19860 STR19870 STR19880 STR19890 STR19900 STR19910 STR19920 STR19930 STR19940 STR19950 STR19960 STR19970 STR19980 STR19990 STR20000 STR20010 STR20020 STR20O30 STR20040 STR20050 STR20060 STR20070 STR20080 STR20090 STR20100 STR20110 STR20120 STR20130 STR20140 STR20150 STR20160 STR20170 STR20180 STR20190 STR20200 STR20210 STR20220 STR20230 STR20240 STR20250 STR20260 STR20270 STR20280 STR20290 STR20300 STR20310 STR20320 STR20330 STR20340 STR20350 Reproduced with permission of the copyright owner Further reproduction prohibited without permission FILE: STRA FORTRAN A1 OLD DOMINION UNIVERSITY DO 12 AMM (1 , J , J) =PM (1) *PL (1) 12 CONTINUE C* * * TO ASSEMBLE THE GLOBAL MASS MATRIX WITH LUMPEDMASS METHOD DO 21 1=1.300 GM (1) = 21 CONTINUE C DO 22 1=1,2 C G M (l,l)= A M M (1 ,l,l)/2 C 22 CONTINUE DO 23 K=1,9 L=K -1 DO 23 1=1,2 GM (1 +3*L ) = (AMM (K 1+ , 1+2) +AMM (K+l , , 1)) / 23 CONTINUE DO 25 =298,299 C WITHOUT " / " MEANS FIXED END G M(I)=AMM(100, -2 1-297) 25 CONTINUE C THIS THE MASS OF MANHOLE C GM(151/=GH(15D+HM C GM (152) =GM (152) +HM DO 525 1= 1,3 0 C WRITE ( , * ) ,G M (I) 525 CONTINUE DO 26 1=1,300 GC (1) = 26 CONTINUE C##### DENSITY*V-SHEAR*i».0*B*L/NQ OF ELEMENTS C0 N=0 0 * 13508 *1» * * 0 / 0 / 5 Ct t m t t FIRST IS A (BOT) +A (TOP) ; 2ND IS TWO SIDE AREA CX=CON* ( * + ) *DCC Z t tm ittt IN ( - ) : IS THE POISSON RATIO OF SATURATED SAND CY=CON* (2 * + l* / • 1^ 159265/ (1 • - ) * ) *DCC CU M M CALCULATE THE GEOMETRIC DAMPING C DAMPING IS PROPORTIONAL TO SHEAR VELOCITY & V(S) =SQRT(G/ZO) 1F (RAT.LE.1 / 9 ) GO TO 579 FCX=CX*SQRT (RAT) FCY=CY*SQRT (RAT) GO TO 975 579 FCX=0.0 FCY=0.0 975 DKX=0.101771E06*DCC DKY=0.1L627E06*DCC C STIFFNESS IS PROPORTIONAL TO SHEAR MODULUS OF SOIL FKX=DKX*RAT FKY=DKY*RAT 00 27 K=1,3 L=K-1 GC(3*L+1)=CX GC (3*L+2) =CY 27 CONTINUE DO 127 K=31,7 L=K-1 GC(3*L+1)=FCX Reproduced with permission of the copyright owner Further reproduction prohibited without permission STR203&0 STR20370 STR20380 STR20390 STR2OA00 STR20A10 STR20L20 STR20l