Final Project

Problem description

Make User Manual for the 1D shallow truss problem with given Fortran program

- Theoretical background

- Structure of program

- Explanation for individual subroutine

- Applications (or numerical examples)

- Discussion (or analysis)

Flow charts

1. Euler method (incremental method)

2. subroutine for iteration in N-R method

3. Combined incremental/N-R method

Appendix: Fortran program

From Non-linear FE analysis of solids and structures by M.A. Crisfield

PROGRAM NONLTCCC FROM 2.5.1C PERFORMS NON-LIN. INCREMENTAL/ITERATIVE SOLN. FOR TRUSSC NV = NO. OF VARIABLES (4 OR 5)C QFI = FIXED LOAD VECTORC IBC = B. COND. COUNTER (0=FREE, 1=FIXED)C Z = Z CO-ORDS OF NODESC QINC = INC. LOAD VECTORC PT = TOTAL DISP. VECTORC QEX = TOTAL (EXTERNAL) LOAD VECTORC AKTS = STR. TAN. STIFF. MATRIXC AKTE = ELE. TAN. STIFF. MATRIXC FI = INTERNAL FORCESC D = DIAGONAL PIVOTS FROM LDL(TRAN) FACTORISATIONC ID14S = VAR. NOS. (1-4) AT WHICH LINEAR EARTHED SPRINGSC AK14S = EQUIV. LINEAR SPRING STIFFNESSC AK15 = LIN SPRING STIFF. BETWEEN VARBS. 1 AND 5 (IF NV=5)C GM = OUT-OF-BALANCE FORCESC REAC = REACTIONSC X = X CO-ORDSC ARGUMENTS IN COMMON/DAT2/ AND ARRAY X NOT USED FOR SHALLOW TRUSSC IMPLICIT DOUBLE PRECISION (A-H,O-Z)c COMMON /DAT/ X(2),Z(2),E,ARA,AL,ID14S(4),AK14S(4),NDSP,ANIT,AK15c COMMON /DAT2/ ARN,POISS,ALN,ITYEL

COMMON /DAT/ X(2),Z(2),E,ARA,AL,ID14S(4),AK14S(4) COMMON /DAT2/ ARN,POISS,ALN,ANIT,AK15

common/DAT21/ITYEL,NDSP DIMENSION QFI(5),IBC(5),QINC(5),PT(5),AKTE(4,4) DIMENSION FI(5),D(5),QEX(5),GM(5),AKTS(25),REAC(5)C IRE=5 IWR=6

OPEN (UNIT=5,FILE='D:\work\FEM-code\inpt13.dat') OPEN (UNIT=6,FILE='D:\work\FEM-code\out13.dat')C CALL INPUT(E,ARA,AL,QFI,X,Z,ANIT,IBC,IRE,IWR,AK14S,ID14S,NDSP, 1 NV,AK15, 2 POISS,ITYEL)C ARGUMENTS IN LINE ABOVE NOT USED FOR SHALLOW TRUSSC BELOW RELEVANT TO DEEP TRUSS BUT LEAVE FOR SHALLOW TRUSS ALN=AL ARN=ARAC READ (IRE,*) FACI,NINC,IWRIT WRITE (IWR,1000) FACI,NINC,IWRIT

1000 FORMAT(/,1X,'INCREMENTAL LOAD FACTOR = ',G13.5,/,1X, 1 'NO. OF INCS. (NINC) = ',I5,/,1X, 2 'WRITE CONTROL (IWRIT) = ',I5,/,3X, 3 '0 = LIMITED ; 1 = FULL') READ (IRE,*) BETOK,ITERTY WRITE (IWR,1003) BETOK,ITERTY 1003 FORMAT(/,1X,'CONV. TOL FACTOR, BETOK = ',G13.5,/, 1 1X,'ITERATIVE SOLN. TYPE, ITERTY = ',I5,/, 2 5X,'= 1, FULL N-R; = 2, MOD. N-R')C AN=ANIT FACT=0.D0 DO 5 I=1,NV GM(I)=0.D0 5 PT(I)=0.D0CC DO 100 INC=1,NINC FACT=FACT+FACIC FACI IS INC LOAD FACTOR, FACT IS TOTAL WRITE (IWR,1001) INC,FACT 1001 FORMAT(//,1X,'INC = ',I5,' LD. FACTOR = ',G12.5) DO 10 I=1,NV QEX(I)=FACT*QFI(I) QINC(I)=FACI*QFI(I)C USE BELOW COMMENT LINE INSTEAD OF ABOVE TO INCLUDEC ALLOWANCE FOR PREVIOUS O.B. FORCESC QINC(I)=FACI*QFI(I)+GM(I) 10 CONTINUECC BELOW FORMS EL. TAN. STIFF MATRIX AKTE CALL ELEMENT(FI,AKTE,AN,X,Z,PT,E,ARA,AL,IWRIT,IWR,2, 1 ITYEL,ALN,ARN)C ARGUMENTS IN LINE ABOVE NOT USED FOR SHALLOW TRUSSC BELOW PUTS EL. STIFF. AKTE IN STRUCT. STIFF. AKTSC AND ADDS EFFECT OF VARIOUS LINEAR SPRINGS CALL ELSTRUC(AKTE,AKTS,NV,AK15,ID14S,AK14S,NDSP,FI,PT, 1 2,IWRIT,IWR)CC CALL BCON(AKTS,IBC,NV,QINC,IWRIT,IWR)C ABOVE APPLIES B. CONDITIONS CALL CROUT(AKTS,D,NV,IWRIT,IWR)C ABOVE FORMS LDL(TRAN) FACTORISATION INTO AKTS AND DC BELOW CHECKS FOR NEGATIVE PIVOTS NEG=0 DO 30 I=1,NV IF (D(I).LT.0.D0) NEG=NEG+1 30 CONTINUE IF (NEG.GT.0) WRITE (IWR,1007) NEG 1007 FORMAT(/,1X,'*** WARNING, NO. OF NEG. PIVOTS = ',I5)C CALL SOLVCR(AKTS,D,QINC,NV,IWRIT,IWR)C ABOVE SOLVES EQNS. AND GETS INC. DISPS IN QINC DO 20 I=1,NV IF (IBC(I).EQ.0) THEN PT(I)=PT(I)+QINC(I) ELSE

PT(I)=QEX(I) ENDIF 20 CONTINUEC ABOVE UP-DATES TOTAL DISPS.C WRITE (IWR,1002) (PT(I),I=1,NV) 1002 FORMAT(/,1X,'TOTAL DISPS. AFTER TAN. SOLN ARE',/,1X,7G13.5)CC BELOW ITERATES TO EQUILIBRIUM CALL ITER(PT,AN,BETOK,QEX,IBC,IWRIT,IWR,AKTS,D,ITERTY,NV, 1 GM,FI,REAC)C WRITE (IWR,1004) (PT(I),I=1,NV) 1004 FORMAT(/,1X,'FINAL TOTAL DISPLACEMENTS ARE',/,1X,6G12.5) WRITE (IWR,1006) (REAC(I),I=1,NV) 1006 FORMAT(/,1X,'FINAL REACTIONS ARE',/,1X,5G12.5) WRITE (IWR,1005) AN 1005 FORMAT(/,1X,'AXIAL FORCE IN BAR IS ',G12.5)C 100 CONTINUEC STOP 'NONLTC' ENDccc SUBROUTINE ELEMENT(FI,AKT,AN,X,Z,P,E,ARA,AL,IWRIT,IWR,IMOD, 1 IDUM,ADUM1,ADUM2)CC FOR GENERAL TRUSS ELEMENT (2.2.1)C IMOD=1 COMPUTES INT. LD. VECT. FIC IMOD=2 COMPUTES TAN. STIFF. AKTC IMOD=3 COMPUTES BOTHCC AN=INPUT TOTAL FORCE IN BARC Z=INPUT=Z CO-ORD VECTOR; X=INPUT=X CO-ORDSC P=INPUT=TOTAL DISP. VECTORC AL=INPUT=LENGTH OF ELEMENTC E=INPUT=YOUNG'S MOD: ARN=INPUT=CURRENT AREACC IF IWRIT.NE.0 WRITES OUT FI AND/OR AKT ON CHANNEL IWRC IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION AKT(4,4),FI(4),Z(2),P(4),X(2)C

EA=E*ARAEAL=EA/ALZ21=Z(2)-Z(1)W21=P(4)-P(3)BET=(Z21+W21)/AL

C IF (IMOD.NE.2) THENC COMPUTES INT. FORCE VECTOR (2.17) FI(1)=-1.D0 FI(2)=-1.D0 FI(3)=-BET FI(4)=BET

DO 1 I=1,4

1 FI(I)=AN*FI(I)C IF (IWRIT.NE.0) THEN WRITE (IWR,1000) (FI(I),I=1,4) 1000 FORMAT(/,1X,'INT. FORCE VECT. FOR TRUSS EL IS ',/,1X,4G13.5) ENDIFC ENDIFC IF (IMOD.NE.1) THENC COMPUTES TAN STIFF. MATRIX (UPPER TRIANGLE)C SEE (2.23) AKT(1,1)=1.D0 AKT(1,2)=-1.D0

AKT(1,3)=BETAKT(1,4)=-BET

AKT(2,2)=1.D0AKT(2,3)=-BETAKT(2,4)=BET

AKT(3,3)=BET*BET AKT(3,4)=-AKT(3,3) AKT(4,4)=BET*BET

DO 12 I=1,4DO 12 J=1,4

12 AKT(I,J)=EAL*AKT(I,J)

CANL=AN/ALAKT(3,3)=AKT(3,3)+ANLAKT(3,4)=AKT(3,4)-ANLAKT(4,4)=AKT(4,4)+ANL

C IF (IWRIT.NE.0) THEN WRITE (IWR,1001) 1001 FORMAT(/,1X,'TAN. STIFF. MATRIX FOR TRUSS EL. IS',/) DO 14 I=1,4 14 WRITE (IWR,67) (AKT(I,J),J=1,4) 67 FORMAT(1X,7G13.5) ENDIFC ENDIF

RETURN END

Ccc SUBROUTINE INPUT(E,ARA,AL,QFI,X,Z,ANIT,IBC,IRE,IWR,AK14S,ID14S, 1 NDSP,NV,AK15, 2 ADUM1,IDUM)CC READS INPUT FOR DEEP TRUSS ELEMENT (3.9.2)C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION X(2),Z(2),QFI(5),IBC(5),AK14S(4),ID14S(4)c DIMENSION X(2),Z(2),QFI(NV),IBC(NV),AK14S(4),ID14S(4)C

READ (IRE,*) NV,EA,AL,ANITE=EAARA=1.D0

WRITE (IWR,1000) NV,EA,AL,ANIT 1000 FORMAT(/,1X,'NV = NO. OF VARBLS. = ',I5,/,1X, 1 'EA = ',G13.5,/,1X, 2 'AL = EL. LENGTH = ',G13.5,/,1X, 3 'ANIT = INIT. FORCE = ',G13.5) IF (NV.NE.4.AND.NV.NE.5) STOP 'INPUT 1000' READ (IRE,*) Z(1),Z(2) WRITE (IWR,1001) Z(1),Z(2) 1001 FORMAT(/,1X,'Z CO-ORD OF NODE 1 = ',G13.5,1X, 1 'Z CO-ORD OF NODE 2 = ',G13.5) READ (IRE,*) (QFI(I),I=1,NV) WRITE (IWR,1002) (QFI(I),I=1,NV) 1002 FORMAT(/,1X,'FIXED LOAD OR DISP. VECTOR,QFI = ',/,1X,5G13.5) WRITE (IWR,1008) 1008 FORMAT(/,1X,'IF IBC(I) - SEE BELOW - = 0, VARIABLE = A LOAD',/,1X, 2 'IF IBC(I) - SEE BELOW - =-1, VARIABLE = A DISP.') READ (IRE,*) (IBC(I),I=1,NV) WRITE (IWR,1003) (IBC(I),I=1,NV) 1003 FORMAT(/,1X,'BOUND. COND. COUNTER, IBC',/,1X, 1 '= 0, FREE: = 1, REST. TO ZERO: =-1 REST. TO NON-ZERO',/,1X, 2 5I5) READ (IRE,*) NDSP IF (NDSP.NE.0) THEN READ (IRE,*) (ID14S(I),I=1,NDSP) READ (IRE,*) (AK14S(I),I=1,NDSP) DO 40 I=1,NDSP WRITE (IWR,1004) AK14S(I),ID14S(I) 1004 FORMAT(/,1X,'LINEAR SPRING OF STIFFNESS ',G13.5,/,1X, 1 'ADDED AT VAR. NO. ',I5) 40 CONTINUE ENDIFC IF (NV.EQ.5) THEN READ (IRE,*) AK15 WRITE (IWR,1005) AK15 1005 FORMAT(/,1X,'LINEAR SPRING BETWEEN VARBLS. 1 AND 5 OF STIFF ', 1 G13.5) ENDIFC RETURN ENDccc SUBROUTINE FORCE(AN,ANIT,E,ARA,AL,X,Z,P,IWRIT,IWR, 1 IDUM,ADUM1,ADUM2,ADUM3)CC FROM 2.2.3

IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION Z(2),P(4),X(2)

EA=E*ARAEAL=EA/ALU21=P(2)-P(1)W21=P(4)-P(3)

Z21=Z(2)-Z(1)AN=U21+(Z21*W21/AL)+0.5D0*(W21*W21/AL)

AN=EAL*AN+ANIT

IF(IWRIT.NE.0) WRITE(IWR,1000) ANC 1000 FORMAT(/,1X,'AXIAL FORCE AN = ',G13.5) RETURN ENDcc============================================c common subroutinesc SUBROUTINE ELSTRUC(AKTE,AKTS,NV,AK15,ID14S,AK14S,NDSP,FI,PT, 1 IMOD,IWRIT,IWR)CC FROM 2.2.4C FOR IMOD=2 OR 3C PUTS EL-STIFF MATRIX AKTE(4,4) INTO STRUCT. STIFF AKTS(NV,NV)C IF NV=5, ALSO ADDS IN LINEAR SPRING AK15 BETWEEN VARBLS. 1&5C ALSO ADDS IN NDSP EARTHED LINEAR SPRINGS FOR VARBLS. 1-4C USING PROPERTIES IN AK14S(4) AND DEGS. OF F. IN IDSPS(4)C THROUGHOUT ONLY WORKS WITH UPPER TRIANGLEC FOR IMOD=1 OR 3C MODIFIES INTERNAL FORCE VECT., FI, TO INCLUDE EFFECTS FROMC VARIOUS LINEAR SPRINGS USING TOTAL DISPS., PT.C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION AKTE(4,4),AKTS(NV,NV),ID14S(4),AK14S(4) DIMENSION FI(NV),PT(NV)C IF (IMOD.NE.2) THENC MODIFY FORCES IF (NDSP.NE.0) THENC FOR EARTHED SPRINGS DO 40 I=1,NDSP IDS=ID14S(I) FI(IDS)=FI(IDS)+AK14S(I)*PT(IDS) 40 CONTINUE ENDIFC IF (NV.EQ.5) THENC MODIFY FOR SPRING BETWEEN VARBLS. 1 AND 5 FI(1)=FI(1)+AK15*(PT(1)-PT(5)) FI(5)=AK15*(-PT(1)+PT(5)) ENDIFC IF (IWRIT.NE.0) WRITE (IWR,1002) FI 1002 FORMAT(/,1X,'STR. INT. FORCE VECT IS'/,1X,5G13.4)C ENDIFC IF (IMOD.NE.1) THENC WORK ON STIFFNESS MATRIX DO 10 I=1,NV DO 11 J=1,NV 11 AKTS(I,J)=0.D0 10 CONTINUE

C DO 20 I=1,4 DO 21 J=I,4 21 AKTS(I,J)=AKTE(I,J) 20 CONTINUECC SPRING BETWEEN VARBLS. 1&5 IF (NV.EQ.5) THEN AKTS(1,1)=AKTS(1,1)+AK15 AKTS(1,5)=AKTS(1,5)-AK15 AKTS(5,5)=AKTS(5,5)+AK15 ENDIFCC EARTHED SPRINGS FOR VARBLS. 1-4. IF (NDSP.NE.0) THEN DO 30 I=1,NDSP IDS=ID14S(I) AKTS(IDS,IDS)=AKTS(IDS,IDS)+AK14S(I) 30 CONTINUE ENDIFC IF (IWRIT.NE.0) THEN WRITE (IWR,1001) 1001 FORMAT(/,1X,'FULL STRUCT. TAN. STIFF. IS ',/) DO 50 I=1,NV 50 WRITE (IWR,67) (AKTS(I,J),J=1,NV) 67 FORMAT(1X,7G13.5) ENDIFC ENDIFC RETURN ENDccc SUBROUTINE BCON(AK,IBC,N,F,IWRIT,IWR)CC FROM 2.2.5C APPLIES B. CONDS. TO MATRIX AK AS WELL ASC ALTERING 'LOAD VECTOR', F FOR PRESC. DISPS.C BY SETTING DIAG=1. AND ROW AND COL TO ZERO IF REST.C USES COUNTER IBC WHICH IS O IF FREE, 1 IF REST. TO ZERO,C -1 IF REST. TO NON-ZERO VALUE.C ON ENTRY F HAS LOADS FOR FREE VARIABLES AND DISPS. FORC REST. (POSSIBLY ZERO) VARIABLES.C ON EXIT THE LATTER ARE UNCHANGED BUT LOADS ARE ALTEREDC IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION IBC(N) DIMENSION AK(N,N),F(N)C IPRS=0 DO 10 I=1,N II=IBC(I) IF (II.LT.0) IPRS=1 IF (II.NE.0) AK(I,I)=1.D0 IF (I.EQ.N) GO TO 10 DO 20 J=I+1,N

JJ=IBC(J) IF (II.EQ.0.AND.JJ.EQ.0) GO TO 20C ABOVE BOTH FREE, BELOW BOTH REST. IF (II.NE.0.AND.JJ.NE.0) GO TO 25C BELOW I REST OR PRESC. IF (II.NE.0) THEN F(J)=F(J)-AK(I,J)*F(I)C BELOW J REST OR PRESC ELSE F(I)=F(I)-AK(I,J)*F(J) ENDIF 25 AK(I,J)=0.D0 20 CONTINUE 10 CONTINUEC IF (IWRIT.NE.0) THEN WRITE (IWR,1000) 1000 FORMAT(/,1X,'STIFF. MAT. AFTER B. CONDS. IS',/) DO 30 I=1,N 30 WRITE (IWR,67) (AK(I,J),J=1,N) IF (IPRS.EQ.1) WRITE (IWR,1001) F 1001 FORMAT(/,1X,'MODIFIED LOAD VECTOR AFTER B CONDS. IS ',/,1X,5G13.4) 67 FORMAT(1X,7G12.5) ENDIFC RETURN ENDccc SUBROUTINE CROUT(AK,D,N,IWRIT,IWR)CC FROM 2.2.6C INPUTS AK(N,N); OUTPUTS UPPER TRIANGLE IN AK AND DIAGC PIVOTS IN D(N)C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION AK(N,N),D(N)C D(1)=AK(1,1) DO 1 J=2,N DO 2 I=1,J-1 A=AK(I,J) IF (I.EQ.1) GO TO 2 DO 3 L=1,I-1 3 A=A-AK(L,J)*AK(L,I) 2 AK(I,J)=A DO 4 I=1,J-1 4 AK(I,J)=AK(I,J)/AK(I,I) DO 5 L=1,J-1 5 AK(J,J)=AK(J,J)-AK(L,J)*AK(L,J)*AK(L,L) 1 D(J)=AK(J,J)C IF (IWRIT.NE.0) THEN WRITE (IWR,1000) 1000 FORMAT(/,1X,'FACTORISED MATRIX IS',/) DO 10 I=1,N 10 WRITE (IWR,67) (AK(I,J),J=1,N) 67 FORMAT(1X,7G12.5)

WRITE (IWR,1001) 1001 FORMAT(/,1X,'DIAG. PIVOTS ARE',/) WRITE (IWR,67) (D(I),I=1,N) ENDIFC RETURN ENDccc

SUBROUTINE SOLVCR(AK,D,Q,N,IWRIT,IWR)CC APPLIES FORWARD AND BACK CROUT SUBS ON Q (2.2.7)C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION AK(N,N),D(N),Q(N)CC FORWARD SUBS. DO 1 J=2,N DO 2 L=1,J-1 2 Q(J)=Q(J)-AK(L,J)*Q(L) 1 CONTINUE IF (IWRIT.NE.0) THEN WRITE (IWR,1000) (Q(I),I=1,N) 1000 FORMAT(/,1X,'DISP. INCS AFTER FORWARD SUBS. ARE',/, 1 1X,7G12.5) ENDIFCC BACK SUBS. DO 3 I=1,N 3 Q(I)=Q(I)/D(I)C DO 4 JJ=2,N J=N+2-JJ DO 5 L=1,J-1 5 Q(L)=Q(L)-AK(L,J)*Q(J) 4 CONTINUEC IF (IWRIT.NE.0) THEN WRITE (IWR,1001) (Q(I),I=1,N) 1001 FORMAT(/,1X,'DISP INCS. AFTER BACKWARD SUBS. ARE',/, 1 1X,7G12.5) ENDIFC RETURN ENDccc SUBROUTINE ITER(PT,AN,BETOK,QEX,IBC,IWRIT,IWR,AKTS,D,ITERTY,NV, 1 GM,FI,REAC)CC FROM 2.4.2C INPUTS PREDICTOR DISPS. PT(NV) AND EXT. FORCE VECTOR QEX(NV)C ALSO BETOK = CONV. TOL, IBC = B.CON COUNTERC ITERATES TO EQUILIBRIUM: OUTPUTS NEW PT AND FORCE IN BAR,ANC IF ITERTY (INPUT) = 1 USES FULL N-R, = 2 USES MOD N-RC IN LATTER CASE, AKTS AND D INPUT AS CROUT FACTORS (D=PIVOTS)

C LOCAL ARRAY IS AKTE = EL. STIFF. MATRIXC ARGUMENTS IN COMMON/DAT2/ AND ARRAY X NOT USED FOR SHALLOW TRUSSC IMPLICIT DOUBLE PRECISION (A-H,O-Z)c COMMON /DAT/ X(2),Z(2),E,ARA,AL,ID14S(4),AK14S(4),NDSP,ANIT,AK15c COMMON /DAT2/ ARN,POISS,ALN,ITYEL

COMMON /DAT/ X(2),Z(2),E,ARA,AL,ID14S(4),AK14S(4) COMMON /DAT2/ ARN,POISS,ALN,ANIT,AK15

common/DAT21/ITYEL,NDSP DIMENSION PT(NV),QEX(NV),IBC(NV),REAC(NV) DIMENSION FI(NV),GM(NV),AKTS(NV,NV),D(NV),AKTE(4,4)C SMALL=0.1D-2 NITMAX=16 IMOD=1 IF (ITERTY.EQ.1) IMOD=3C DO 100 ITE=1,NITMAXC IF (IWRIT.EQ.1) WRITE (IWR,1005) ITE 1005 FORMAT(/,1X,'ITERATIVE LOOP WITH ITE = ',I5)C BELOW CALCS FORCE IN BAR (AN) CALL FORCE(AN,ANIT,E,ARA,AL,X,Z,PT,IWRIT,IWR, 1 ITYEL,ARN,ALN,POISS)C ABOVE ARGUMENTS NOT USED FOR SHALLOW-TRUSSCC ABOVE CALCS FORCE IN BAR, AN: BELOW TAN STIFF AKTC (IF NR) AND INT. FORCE VECT. FI CALL ELEMENT(FI,AKTE,AN,X,Z,PT,E,ARA,AL,IWRIT,IWR,IMOD, 1 ITYEL,ALN,ARN)C ABOVE ARGUMENTS NOT USED FOR SHALLOW TRUSSCC BELOW PUTS EL. STIFF. MAT., AKTE, IN STR. STIFF., AKTS ANDC ADDS IN EFFECTS OF VARIOUS LINEAR SPRINGS (IF NR)C ALSO MODIFIES INT. FORCE VECT. FI FOR SPRING EFFECTS CALL ELSTRUC(AKTE,AKTS,NV,AK15,ID14S,AK14S,NDSP,FI,PT, 1 IMOD,IWRIT,IWR)CC BELOW FORMS GM=OUT-OF-BALANCE FORCE VECTORC AND REACTION VECTOR DO 10 I=1,NV GM(I)=0.D0 REAC(I)=FI(I) IF (IBC(I).EQ.0) THEN GM(I)=QEX(I)-FI(I) ENDIF 10 CONTINUE 67 FORMAT(6G13.5) 47 FORMAT(5I5)CC OVERWRITE SPRING REACTION TERMS IF (NDSP.NE.0) THEN DO 50 I=1,NDSP 50 REAC(ID14S(I))=-AK14S(I)*PT(ID14S(I)) ENDIFCC BELOW CHECKS CONVERGENCE FNORM=0.D0

GNORM=0.D0 RNORM=0.D0 IDSP=0 DO 20 I=1,NV IF (IBC(I).EQ.0) FNORM=FNORM+QEX(I)*QEX(I) IF (IBC(I).EQ.-1) IDSP=1 RNORM=RNORM+REAC(I)*REAC(I) 20 GNORM=GNORM+GM(I)*GM(I) FNORM=DSQRT(FNORM) GNORM=DSQRT(GNORM) RNORM=DSQRT(RNORM) BAS=MAX(FNORM,SMALL)C BELOW FOR DISP. CONTROL IF (IDSP.EQ.1) BAS=MAX(RNORM,SMALL) BET=GNORM/BAS ITEM=ITE-1 WRITE (IWR,1001) ITEM,BET 1001 FORMAT(/,1X,'ITERN. NO. = ',I5,' CONV. FAC. = ',G13.5) IF (IWRIT.EQ.1) WRITE (IWR,1003) (GM(I),I=1,NV) 1003 FORMAT(/,1X,'OUT-OF-BAL. FORCE VECTOR = ',/,1X,4G13.5) IF (BET.LE.BETOK) GO TO 200C IF (ITERTY.EQ.1) THEN CALL BCON(AKTS,IBC,NV,GM,IWRIT,IWR)C ABOVE APPLIES B. CONDITIONS CALL CROUT(AKTS,D,NV,IWRIT,IWR)C ABOVE FORMS LDL(TRAN) FACTORISATION INTO AKTS AND D ENDIFC CALL SOLVCR(AKTS,D,GM,NV,IWRIT,IWR)C ABOVE GETS ITER. DISP. CHANGE IN GMC DO 30 I=1,NV IF (IBC(I).EQ.0) THEN PT(I)=PT(I)+GM(I) ELSE PT(I)=QEX(I) ENDIF 30 CONTINUEC ABOVE UP-DATES DISPS. IF (IWRIT.EQ.1) WRITE (IWR,1004) (PT(I),I=1,NV) 1004 FORMAT(/,1X,'TOTAL DISPS ARE',/,1X,6G13.5)C 100 CONTINUEC WRITE (IWR,1002) 1002 FORMAT(/,1X,'FAILED TO CONVERGE ****') STOP 'ITER 100'C 200 CONTINUE RETURN ENDccc

////Main subroutine for Incremental method

PROGRAM NONLTACC PERFORMS NON-LIN. INCREMENTAL SOLN. FOR TRUSS (2.3.1)C NV = NO. OF VARIABLES (4 OR 5)C QFI = FIXED LOAD VECTORC IBC = B. COND. COUNTER (0=FREE, 1=FIXED)C Z = Z CO-ORDS OF NODESC QINC = INC. LOAD VECTORC PT = TOTAL DISP. VECTORC AKTS = STRUCT. TAN. STIFF. MATRIXC AKTE = ELEMENT TAN. STIFF. MATRIXC FI (NOT USED HERE) = INTERNAL FORCESC D = DIAGONAL PIVOTS FROM LDL(TRAN) FACTORISATIONC ID14S = VAR. NOS.(1-4) AT WHICH LIN EARTHED SPRINGSC AK14S = EQUIV. LINEAR SPRING STIFFNESSESC AK15 = LINEAR SPRING STIFF BETWEEN VARBLS. 1 AND 5 (IF NV=5)C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION QFI(5),IBC(5),Z(2),QINC(5),PT(5),AKTE(4,4) DIMENSION FI(5),D(5),AK14S(4),ID14S(4),AKTS(25),X(2)C ARRAY X ABOVE NOT USED FOR SHALLOW TRUSSC IRE=5 IWR=6 OPEN (UNIT=5,FILE=' ') OPEN (UNIT=6,FILE=' ')C CALL INPUT(E,ARA,AL,QFI,X,Z,ANIT,IBC,IRE,IWR,AK14S,ID14S,NDSP, 1 NV,AK15, 2 POISS,ITYEL)C ARGUMENTS IN LINE ABOVE NOT USED FOR SHALLOW TRUSSC BELOW RELEVANT TO DEEP TRUSS BUT LEAVE FOR SHALLOW TRUSS ALN=AL ARN=ARAC READ (IRE,*) FACI,NINC,IWRIT WRITE (IWR,1000) FACI,NINC,IWRIT 1000 FORMAT(/,1X,'INCREMENTAL LOAD FACTOR = ',G13.5,/,1X, 1 'NO. OF INCS. (NINC) = ',I5,/,1X, 2 'WRITE CONTROL (IWRIT) = ',I5,/,3X, 3 '0 = LIMITED ; 1 = FULL')C AN=ANIT FACT=0.D0 DO 5 I=1,NV 5 PT(I)=0.D0C

C DO 100 INC=1,NINC FACT=FACT+FACI WRITE (IWR,1001) INC,FACT 1001 FORMAT(//,1X,'INC = ',I5,' LD. FACTOR = ',G13.5) DO 10 I=1,NV 10 QINC(I)=FACI*QFI(I)CC BELOW FORMS EL. TAN. STIFF MATRIX AKT CALL ELEMENT(FI,AKTE,AN,X,Z,PT,E,ARA,AL,IWRIT,IWR,2, 1 ITYEL,ALN,ARN)C ARGUMENTS IN LINE ABOVE NOT USED FOR SHALLOW TRUSSC CALL ELSTRUC(AKTE,AKTS,NV,AK15,ID14S,AK14S,NDSP,FI,PT, 1 2,IWRIT,IWR)C ABOVE PUTS EL.STIFF AKTE IN STRUC STIFF AKTS ANDC ADDS EFFECTS OF VARIOUS LINEAR SPRINGSC CALL BCON(AKTS,IBC,NV,QINC,IWRIT,IWR)C ABOVE APPLIES B. CONDITIONS CALL CROUT(AKTS,D,NV,IWRIT,IWR)C ABOVE FORMS LDL(TRAN) FACTORISATION INTO AKT AND D CALL SOLVCR(AKTS,D,QINC,NV,IWRIT,IWR)C ABOVE SOLVES EQNS. AND GETS INC. DISPS IN QINC DO 20 I=1,NV 20 PT(I)=PT(I)+QINC(I)C ABOVE UP-DATES TOTAL DISPS.C WRITE (6,1002) (PT(I),I=1,NV) 1002 FORMAT(/,1X,'TOTAL DISPS. ARE',/,1X,5G13.5)CC BELOW FORMS TOTAL FORCE IN BAR CALL FORCE(AN,ANIT,E,ARA,AL,X,Z,PT,IWRIT,IWR, 1 ITYEL,ARN,ALN,POISS)C ABOVE ARGUMENTS NOT USED FOR SHALLOW TRUSSC 100 CONTINUEC STOP 'NONLTA' END

///Main subroutine for N-R method

PROGRAM NONLTBCC FROM 2.4.1C PERFORMS NEWTON-RAPHSON ITERATION FROM STARTING PREDICTOR, PTC NV = NO. OF VARIABLES (4 OR 5)C QFI = FIXED LOAD VECTORC IBC = B. COND. COUNTER (0=FREE, 1=FIXED)C Z = Z CO-ORDS OF NODESC PT = TOTAL DISP. VECTORC ID14S = VAR. NOS.(1-4) AT WHICH LINEAR EARTHED SPRINGSC AK14S = EQUIV. LINEAR SPRING STIFFNESSESC QFI = TOTAL LOAD VECTORC AKTS = STRUCT. STIFF. MATRIXC AK15 = LIN SPRING STIFF. BETWEE VARBS. 1 AND 5 (IF NV=5)C FI = INTERNAL FORCE VECTORC GM = OUT-OF-BALANCE FORCE VECTORC REAC = REACTIONSC X = X CO-ORDSC ARGUMENTS IN COMMON/DAT2/ AND ARRAY X NOT USED FOR SHALLOW TRUSSC IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /DAT/ X(2),Z(2),E,ARA,AL,ID14S(4),AK14S(4),NDSP,ANIT,AK15 COMMON /DAT2/ ARN,POISS,ALN,ITYEL DIMENSION QFI(5),IBC(5),PT(5),AKTS(25),D(5),GM(5),FI(5) DIMENSION REAC(5)C IRE=5 IWR=6 OPEN (UNIT=5,FILE=' ') OPEN (UNIT=6,FILE=' ')C CALL INPUT(E,ARA,AL,QFI,X,Z,ANIT,IBC,IRE,IWR,AK14S,ID14S,NDSP, 1 NV,AK15, 2 POISS,ITYEL)C ARGUMENTS IN LINE ABOVE NOT USED FOR SHALLOW TRUSSC BELOW RELEVANT TO DEEP TRUSS BUT LEAVE FOR SHALLOW TRUSS ALN=AL ARN=ARAC READ (IRE,*) (PT(I),I=1,NV) WRITE (IWR,2000) (PT(I),I=1,NV) 2000 FORMAT(/,1X,'STARTING PREDICTOR DISPS ARE',/,1X,6G12.5) READ (IRE,*) BETOK,IWRIT WRITE (IWR,2001) BETOK,IWRIT 2001 FORMAT(/,1X,'CONV. TOL FACTOR, BETOK = ',G12.5,/,1X, 2 'DIAGNOSTIC WRITE CONTROL (IWRIT) = ',I5,/,3X, 3 '0 = NO ; 1 = YES')C SET TO NEWTON-RAPHSON ITERATIONS

ITERTY=1C CALL ITER(PT,AN,BETOK,QFI,IBC,IWRIT,IWR,AKTS,D,ITERTY,NV, 1 GM,FI,REAC)C WRITE (IWR,1004) (PT(I),I=1,NV) 1004 FORMAT(/,1X,'FINAL TOTAL DISPLACEMENTS ARE',/,1X,5G12.5) WRITE (IWR,1006) (REAC(I),I=1,NV) 1006 FORMAT(/,1X,'FINAL REACTIONS ARE',/,1X,5G12.5) WRITE (IWR,1005) AN 1005 FORMAT(/,1X,'AXIAL FORCE IN BAR IS ',G12.5) STOP 'NONLTB' END