<!-- source-page: 371 -->

```csv
DO 30 ILAYR=1,NLAYR
KGAUS=KGAUS+1
JLAYR=ILAYR+1
C
C*** EVALUATE Z-COORDINATES FOR CURRENT LAYER
C
DEPT1=DEPTH(ILAYR)
DEPT2=DEPTH(JLAYR)
CONS3=(DEPT2+DEPT1)*(DEPT2**2-DEPT1**2)/4.0
C
C*** EVALUATE ELASTO-PLASTIC D MATRIX FOR CURRENT LAYER
C
CALL MDMPA(DPLAN,DSHER,LPROP,MMATS,PROPS,1,0)
IF(IINCS.EQ.1)GO TO 40
IF(EPSTN(KGAUS).EQ.0.0)GO TO 40
DO 50 ISTRE=1,5
50 SGTOT(ISTRE)=STRSG(ISTRE,KGAUS)
CALL INVMP(DEVIA,NCrit,SINT3,STEFF,SGTOT,THETA,VARJ2,YIELD)
CALL FLOWMP(ABETA,AVECT,DEVIA,DPLAN,DVECT,HARDS,NCrit,SINT3,
STEFF,THETA,VARJ2)
DO 60 ISTRE=1,3
DO 60 JSTRE=1,3
60 DPLAN(ISTRE,JSTRE)=DPLAN(ISTRE,JSTRE)-ABETA*DVECT(ISTRE)*
.DVECT(JSTRE)
40 CONTINUE
C
C*** SUM D MATRIX OVER ELEMENT DEPTH
C
DO 70 ISTRE=1,3
DO 70 JSTRE=1,3
70 DFLEF(ISTRE,JSTRE)=DFLEF(ISTRE,JSTRE)+CONS3*DPLAN(ISTRE,JSTRE)
30 CONTINUE
GO TO 200
C
C*** ZERO D MATRIX FOR SHEAR
C
100 DO 80 ISTRE=1,2
DO 80 JSTRE=1,2
80 DSHES(ISTRE,JSTRE)=0.0
C
C*** EVALUATE ELASTIC D MATRIX
C
CALL MDMPA(DPLAN,DSHER,LPROP,MMATS,PROPS,0,1)
C
C*** LOOP AROUND LAYERS
C
DO 90 ILAYR=1,NLAYR
JLAYR=ILAYR+1
C
C*** EVALUATE Z-COORDINATES FOR CURRENT LAYER
C
DEPT1=DEPTH(ILAYR)
DEPT2=DEPTH(JLAYR)
CONS4=DEPT2-DEPT1
C
C*** SUM D MATRIX OVER ELEMENT DEPTH
C
DO 110 ISTRE=1,2
DO 110 JSTRE=1,2
110 DSHES(ISTRE,JSTRE)=DSHES(ISTRE,JSTRE)+CONS4*Dsher(ISTRE,JSTRE)
90 CONTINUE
200 CONTINUE
RETURN
END
LAYR 27
LAYR 28
LAYR 29
LAYR 30
LAYR 31
LAYR 32
LAYR 33
LAYR 34
LAYR 35
LAYR 36
LAYR 37
LAYR 38
LAYR 39
LAYR 40
LAYR 41
LAYR 42
LAYR 43
LAYR 44
LAYR 45
LAYR 46
LAYR 47
LAYR 48
LAYR 49
LAYR 50
LAYR 51
LAYR 52
LAYR 53
LAYR 54
LAYR 55
LAYR 56
LAYR 57
LAYR 58
LAYR 59
LAYR 60
LAYR 61
LAYR 62
LAYR 63
LAYR 64
LAYR 65
LAYR 66
LAYR 67
LAYR 68
LAYR 69
LAYR 70
LAYR 71
LAYR 72
LAYR 73
LAYR 74
LAYR 75
LAYR 76
LAYR 77
LAYR 78
LAYR 79
LAYR 80
LAYR 81
LAYR 82
LAYR 83
LAYR 84
LAYR 85
LAYR 86
LAYR 87
LAYR 88
LAYR 89
LAYR 90 
```

<!-- source-page: 372 -->

LAYR 10 If JFFLE is zero $D_f'$ is not evaluated. If it is one $D_s'$ is not evaluated.

LAYR 15-17 Initializes $D_{f}'$ .

LAYR 21 Starts the summation loop to form DFLEF, i.e.

$$
\hat {D} _ {f} = \sum_ {i = 1} ^ {n} \frac {1}{4} (z _ {i + 1} + z _ {i}) (z _ {i + 1} ^ {2} - z _ {i} ^ {2}) D _ {f} ^ {\prime}.
$$

LAYR 22 Increases the counter for Gauss points in each layer by 1. It is needed to use the effective plastic strain (EPSTN) stresses (STRSG) calculated in RESMPA.

LAYR 27-29 Forms $\frac{1}{4} (z_{i + 1} + z_i)(z_{i + 1}^2 -z_i^2)$ .

LAYR 33-45 Calls MDMPA to get DPLAN and $D_{ep'}$ is formed using INVMP and FLOWMP.

LAYR 49–51 DFLEF is formed.

LAYR 57-59 DSHES is initialised.

LAYR 63 Calls MDMPA to form DSHER.

LAYR 67–74 Starts the summation loop and the integrating constant for DSHES is evaluated, i.e.

$$
\hat {\boldsymbol {D}} _ {s} = \sum_ {i = 1} ^ {n} (z _ {i + 1} - z _ {i}) \boldsymbol {D} _ {s}.
$$

LAYR 78-81 DSHES is formed.

# 9.6.6 Subroutine MDMPA

This subroutine evaluates $D_{f}'$ and $D_{s}'$ .
```txt
SUBROUTINE MDMPA (DPLAN,DSHER,LPROP,MMATS,PROPS, MODL 1
IFPLA,IFSHE) MODL 2
C**************************MODL 3
C
C*** CALCULATES MATRIX OF ELASTIC RIGIDITIES FOR EACH LAYER
C*** OF MINDLIN PLATE
C
C**************************MODL 8
DIMENSION DPLAN(3,3),DSHER(2,2), MODL 9
PROPS(MMATS,8) MODL 10
YOUNG=PROPS(LPROP,1) MODL 11
POISS=PROPS(LPROP,2) MODL 12
THICK=PROPS(LPROP,3) MODL 13
C*** FORM DPLAN
IF(IFPLA.EQ.0) GO TO 10
DO 1 IROWS=1,3
DO 1 JCOLS=1,3
1 DPLAN(IROWS,JCOLS)=0.0
CONST=YOUNG/(1.0-POISS*POISS) MODL 18
DPLAN(1,1)=CONST
DPLAN(2,2)=CONST
DPLAN(1,2)=CONST*POISS
MODL 20
MODL 21
MODL .22 
```

<!-- source-page: 373 -->

```csv
DPLAN(2,1)=CONST*POISS
DPLAN(3,3)=CONST*(1.0-POISS)/2.0
C*** FORM DSHER
10 IF(IFSHE.EQ.0) RETURN
DO 3 IROWS=1,2
DO 3 JCOLS=1,2
3 DSHER(IROWS,JCOLS)=0.0
DSHER(1,1)=YOUNG/(2.4+2.4*POISS)
DSHER(2,2)=YOUNG/(2.4+2.4*POISS)
RETURN
END
MODL 23
MODL 24
MODL 25
MODL 26
MODL 27
MODL 28
MODL 29
MODL 30
MODL 31
MODL 32
MODL 33 
```

# 9.6.7 Subroutine OUTMPA

This subroutine outputs nodal displacements and reactions and also the Gauss point stress resultants and the stresses within each layer. It is very similar to subroutine OUTMP which was described in Section 9.5.7. Statements OUTP 1-3 are replaced by OUTL 1-3 and statements OUTP 56-66 are replaced by statements OUTL 56-67.

```fortran
SUBROUTINE OUTMPA (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, OUTL 1
. NGAUS,NLAPS,NOFIX,NUOTP,NPOIN,NVFIX, OUTL 2
. STRSG,TDISP,TREAC) OUTL 3
C******************************************************************************************OUTL 4
C OUTL 5
C*** OUTPUT DISPLACEMENTS,REACTIONS AND GAUSS POINT STRESSES OUTL 6
C*** IN EACH LAYER FOR EP MINDLIN PLATE ANALYSIS OUTL 7
C OUTL 8
C******************************************************************************************OUTL 9
DIMENSION EPSTN(MTOTG),GPCOD(2,9),NOFIX(MVFIX),NUOTP(2), OUTL 10
. STRSG(5,MTOTG),TDISP(MTOTV),TREAC(MVFIX,3) OUTL 11
KOUTP=NOUTP(1) OUTL 12
IF(IITER.GT.1) KOUTP=NOUTP(2) OUTL 13
C OUTL 14
C*** OUTPUT DISPLACEMENTS OUTL 15
C OUTL 16
IF(KOUTP.LT.1) GO TO 10 OUTL 17
WRITE(6,900) OUTL 18
900 FORMAT(1H0,5X,13HDISPLACEMENTS) OUTL 19
WRITE(6,950) OUTL 20
950 FORMAT(1H0,6X,4HNODE,6X,5HDISP.,8X,7HXZ-ROT.,7X,7HYZ-ROT.) OUTL 21
DO 20 IPOIN=1,NPOIN OUTL 22
NGASH=IPOIN*3 OUTL 23
NGISH=NGASH-3+1 OUTL 24
20 WRITE(6,910) IPOIN,(TDISP(IGASH),IGASH=NGISH,NGASH) OUTL 25
910 FORMAT(I10,3E14.6) OUTL 26
10 CONTINUE OUTL 27
C OUTL 28
C*** OUTPUT REACTIONS OUTL 29
C OUTL 30
IF(KOUTP.LT.2) GO TO 30 OUTL 31
WRITE(6,920) OUTL 32
920 FORMAT(1H0,5X,9HREACTIONS) OUTL 33
WRITE(6,960) OUTL 34
960 FORMAT(1H0,6X,4HNODE,6X,5HFORCE,3X,9HXZ-MOMENT,5X,9HYZ-MOMENT) OUTL 35
DO 40 IVFIX=1,NVFIX OUTL 36
40 WRITE(6,910) NOFIX(IVFIX),(TREAC(IVFIX,IDOFN),IDOFN=1,3) OUTL 37
30 CONTINUE OUTL 38
C OUTL 39
C*** OUTPUT STRESSES OUTL 40 
```

<!-- source-page: 374 -->

```csv
C
IF(KOUTP.LT.3) GO TO 50
REWIND 3
WRITE(6,970)
970 FORMAT(1H0,5X,8HSTRESSES)
WRITE(6,980)
980 FORMAT(1H0,4HG.P.,2X,8HX-COORD.,2X,8HY-COORD.,3X,8HX-MOMENT,4X,
.8HY-MOMENT,3X,9HXY-MOMENT,3X,
.13HEFF.PL.STRAIN)
KGAUS=0
DO 60 IELEM=1,NELEM
READ(3)GPCOD
KELGS=0
WRITE(6,930)IELEM
930 FORMAT(1H0,5X,13HELEMENT NO. =,I5)
DO 60 IGAUS=1,NGAUS
DO 60 JGAUS=1,NGAUS
KELGS=KELGS+1
DO 60 ILAYR=1,NLAPS
KGAUS=KGAUS+1
WRITE(6,940)KELGS,(GPCOD(IDIME,KELGS),IDIME=1,2),
.(STRSG(ISTRE,KGAUS),ISTRE=1,3),EPSTN(KGAUS)
940 FORMAT(I5,2F10.4,6E12.5)
60 CONTINUE
50 CONTINUE
RETURN
END
OUTL 41
OUTL 42
OUTL 43
OUTL 44
OUTL 45
OUTL 46
OUTL 47
OUTL 48
OUTL 49
OUTL 50
OUTL 51
OUTL 52
OUTL 53
OUTL 54
OUTL 55
OUTL 56
OUTL 57
OUTL 58
OUTL 59
OUTL 60
OUTL 61
OUTL 62
OUTL 63
OUTL 64
OUTL 65
OUTL 66
OUTL 67 
```

# 9.6.8 Subroutine RESMPA

This routine evaluates the residual forces for the layered Mindlin plate. It is very similar to RESMP described in Section 9.5.10.

```csv
SUBROUTINE RESMPA (ASDIS,COORD,EFFST,ELOAD,EPSTN,LNODS, RESL 1
. MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV, RESL 2
. NCRIT,NELEM,NEVAB,NGAUS,NNODE,NLAPS, RESL 3
. PROPS,STRSG) RESL 4
C************************** RESL 5
C RESL 6
C*** EVALUATES EQUIVALENT NODAL FORCES FOR THE STRESSES RESL 7
C*** IN LAYERED MINDLIN PLATES DURING EP ANALYSIS RESL 8
C RESL 9
C************************** RESL 10
DIMENSION ASDIS(MTOTV),AVECT(5),CARTD(2,9), RESL 11
. COORD(MPOIN,2),DERIV(2,9),DESIG(5),DEVIA(4), RESL 12
. DEPTH(26),DVECT(5), RESL 13
. EFFST(MTOTG),ELCOD(2,9), RESL 14
. ELDIS(3,9),ELOAD(MELEM,27),EPSTN(MTOTG),GPCOD(2,9), RESL 15
. LNODS(MELEM,9),MATNO(MELEM),POSGP(4), RESL 16
. PROPS(MMATS,8),SGTOT(5),SHAPE(9),SIEMA(5), RESL 17
. STRES(5),STRSG(5,MTOTG),TOSPB(5),WEIGP(4), RESL 18
. DPLAN(3,3),DSHER(2,2),BFLEI(3,3),BSHEI(2,3), RESL 19
. DUMMY(3,3),FORCE(3),DGRAD(6) RESL 20
NTIME=1 RESL 21
DO 10 IELEM=1,NELEM RESL 22
DO 10 IEVAB=1,NEVAB RESL 23
10 ELOAD(IELEM,IEVAB)=0.0 RESL 24
KGAUS=0 RESL 25
LGAUS=0 RESL 26
DO 20 IELEM=1,NELEM RESL 27
LPROP=MATNO(IELEM) RESL 28 
```

<!-- source-page: 375 -->

```txt
C
C*** COMPUTE COORDINATE AND INCREMENTAL DISPLACEMENTS OF THE
C ELEMENT NODAL POINTS
C
DO 190 INODE =1, NNODE
LNODE=IABS(LNODS(IELEM, INODE))
NPOSN=(LNODE-1)*3
DO 30 IDOFN=1,3
NPOSN=NPOSN+1
30 ELDIS(IDOFN, INODE)=ASDIS(NPOSN)
DO 180 IDIME=1,2
180 ELCOD(IDIME, INODE)=COORD(LNODE, IDIME)
190 CONTINUE
KGASP=0
CALL DEMPA(DEPTH, LPROP, MMATS, NLAPS, PROPS)
CALL MDMPA (DPLAN, DSHER, LPROP, MMATS, PROPS, 1, 1)
CALL GAUSSQ (NGAUS, POSGP, WEIGP)
DO 40 IGAUS=1, NGAUS
DO 40 JGAUS=1, NGAUS
EXISP=POSGP(IGAUS)
ETASP=POSGP(JGAUS)
CALL SFR2 (DERIV, ETASP, EXISP, NNODE, SHAPE)
KGASP=KGASP+1
CALL JACOB2 (CARTD, DERIV, DJACB, ELCOD, GPCOD, IELEM, KGASP, NNODE, SHAPE)
DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS)
DO 400 ISTRE=1,3
400 TOSPB(ISTRE)=0.0
DO 410 ILAYR=1, NLAPS
BRING=1.0
KGAUS=KGAUS+1
JLAYR=ILAYR+1
DEPT1=DEPTH(ILAYR)
DEPT2=DEPTH(JLAYR)
CONST=0.5*(DEPT2+DEPT1)
CALL GRADMP (CARTD, DGRAD, ELDIS, 3, NNODE)
CALL STRMPA (CARTD, CONST, DPLAN, DGRAD, DSHER, ELDIS, NNODE, SHAPE, STRES, 1, 0)
PREYS=PROPS(LPROP, 6)+EPSTN(KGAUS)*PROPS(LPROP, 7)
DO 150 ISTRE=1,3
DESIG(ISTRE)=STRES(ISTRE)
150 SIGMA(ISTRE)=STRSG(ISTRE, KGAUS)+STRES(ISTRE)
CALL INVMP (DEVIA, NCRIT, SINT3, STEFF, SIGMA, THETA, VARJ2, YIELD)
ESPRE=EFFST(KGAUS)-PREYS
IF(ESPRE.GE.0.0) GO TO 50
ESCUR=YIELD-PREYS
IF(ESCUR.LE.0.0) GO TO 60
RFACT=ESCUR/(YIELD-EFFST(KGAUS))
GO TO 70
50 ESCUR=YIELD-EFFST(KGAUS)
IF(ESCUR.LE.0.0) GO TO 60
RFACT=1.0
70 MSTEP=ESCUR*8.0/PROPS(LPROP, 6)+1.0
ASTEP=MSTEP
REDUC=1.0-RFACT
DO 80 ISTRE=1,3
SGTOT(ISTRE)=STRSG(ISTRE, KGAUS)+REDUC*STRES(ISTRE)
80 STRES(ISTRE)=RFACT*STRES(ISTRE)/ASTEP
DO 90 ISTEP=1, MSTEP
CALL INVMP (DEVIA, NCRIT, SINT3, STEFF, SGTOT, THETA, VARJ2, YIELD)
HARDS=PROPS(LPROP, 7)
CALL FLOWMP (ABETA, AVECT, DEVIA, DPLAN, DVECT, HARDS, 
```

<!-- source-page: 376 -->

```csv
NCRIT,SINT3,STEFF,THETA,VARJ2) RESL 94
AGASH=0.0 RESL 95
DO 100 ISTRE=1,3 RESL 96
100 AGASH=AGASH+AVECT(ISTRE)*STRES(ISTRE) RESL 97
DLAMD=AGASH*ABETA RESL 98
IF(DLAMD.LT.0.0) DLAMD=0.0 RESL 99
BGASH=0.0 RESL 100
DO 110 ISTRE=1,3 RESL 101
BGASH=BGASH+AVECT(ISTRE)*SGTOT(ISTRE) RESL 102
110 SGTOT(ISTRE)=SGTOT(ISTRE)+STRES(ISTRE)-DLAMD*DVECT(ISTRE) RESL 103
90 EPSTN(KGAUS)=EPSTN(KGAUS)+DLAMD*BGASH/YIELD RESL 104
DO 120 ISTRE=1,3 RESL 105
120 DESIG(ISTRE)=SGTOT(ISTRE)-STRSG(ISTRE,KGAUS) RESL 106
CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,SGTOT,THETA,VARJ2,YIELD) RESL 107
CURYS=PROPS(LPROP,6)+EPSTN(KGAUS)*PROPS(LPROP,7) RESL 108
IF(YIELD.GT.CURYS) BRING=CURYS/YIELD RESL 109
60 DO 130 ISTRE=1,3 RESL 110
SGTOT(ISTRE)=BRING*(STRSG(ISTRE,KGAUS)+DESIG(ISTRE)) RESL 111
130 STRSG(ISTRE,KGAUS)=SGTOT(ISTRE) RESL 112
EFFST(KGAUS)=BRING*YIELD RESL 113
CONSA=(DEPT2**2-DEPT1**2)/2.0 RESL 114
DO 440 ISTRE=1,3 RESL 115
440 TOSPB(ISTRE)=TOSPB(ISTRE)+SGTOT(ISTRE)*CONSA RESL 116
410 CONTINUE RESL 117
DO 430 ISTRE=1,3 RESL 118
430 SGTOT(ISTRE)=TOSPB(ISTRE) RESL 119
C RESL 120
C*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THE RESL 121
C ELEMENT NODES RESL 122
DO 140 INODE=1,NNODE RESL 123
C*** ZERO FORCE VECTOR RESL 124
CALL VZERO (3,FORCE) RESL 125
CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, RESL 126
0, 1, 0) RESL 127
FORCE(2)=(BFLEI(1,2)*SGTOT(1)+BFLEI(3,2)*SGTOT(3))*DAREA RESL 128
+FORCE(2) RESL 129
FORCE(3)=(BFLEI(2,3)*SGTOT(2)+BFLEI(3,3)*SGTOT(3))*DAREA RESL 130
+FORCE(3) RESL 131
IPOSN=(INODE-1)*3+1 RESL 132
DO 135 IDOFN=2,3 RESL 133
IPOSN=IPOSN+1 RESL 134
135 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN) RESL 135
140 CONTINUE RESL 136
40 CONTINUE RESL 137
C RESL 138
C*** CALCULATE FORCES ASSOCIATED WITH SHEAR DEFORMATION RESL 139
C RESL 140
NGAUM=NGAUS-1 RESL 141
CALL GAUSSQ (NGAUM,POSGP,WEIGP) RESL 142
C RESL 143
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION RESL 144
C RESL 145
KGASP=0 RESL 146
DO 300 IGAUS=1,NGAUM RESL 147
DO 300 JGAUS=1,NGAUM RESL 148
EXISP=POSGP(IGAUS) RESL 149
ETASP=POSGP(JGAUS) RESL 150
CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) RESL 151
KGASP=KGASP+1 RESL 152
CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, RESL 153
KGASP,NNODE,SHAPE) RESL 154
DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) RESL 155
DO 610 ISTRE=4,5 RESL 156
610 TOSPB(ISTRE)=0.0 RESL 157
RESL 158 
```

<!-- source-page: 377 -->

```csv
C
C*** LOOP AROUND LAYRS
C
DO 600 ILAYR=1,NLAPS
LGAUS=LGAUS+1
JLAYR=ILAYR+1
DEPT1=DEPTH(ILAYR)
DEPT2=DEPTH(JLAYR)
CONST=1.0
CALL GRADMP (CARTD,DGRAD,ELDIS, 3,NNODE)
CALL STRMPA (CARTD,CONST,DPLAN,DGRAD,DSHER,ELDIS, NNODE,SHAPE,STRES, 0, 1)
DO 310 ISTRE=4,5
SGTOT(ISTRE)=STRSG(ISTRE,LGAUS)+STRES(ISTRE)
310 STRSG(ISTRE,LGAUS)=SGTOT(ISTRE)
CONSB=DEPT2-DEPT1
DO 620 ISTRE=4,5
620 TOSPB(ISTRE)=TOSPB(ISTRE)+SGTOT(ISTRE)*CONSB
600 CONTINUE
DO 605 ISTRE=4,5
605 SGTOT(ISTRE)=TOSPB(ISTRE)
C
C*** CALCULATE THE EQUIVALENT NODAL FORCES
C
DO 320 INODE=1,NNODE
C*** ZERO FORCE VECTOR
CALL VZERO(3,FORCE)
CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, 0, 0, 1)
FORCE(1)=(BSHEI(1,1)*SGTOT(4)+BSHEI(2,1)*SGTOT(5))*DAREA
+FORCE(1)
FORCE(2)=(BSHEI(1,2)*SGTOT(4))*DAREA+FORCE(2)
FORCE(3)=(BSHEI(2,3)*SGTOT(5))*DAREA+FORCE(3)
IPOSN=(INODE-1)*3
DO 315 IDOFN=1,3
IPOSN=IPOSN+1
315 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN)
320 CONTINUE
300 CONTINUE
20 CONTINUE
RETURN
END
RESL 159
RESL 160
RESL 161
RESL 162
RESL 163
RESL 164
RESL 165
RESL 166
RESL 167
RESL 168
RESL 169
RESL 170
RESL 171
RESL 172
RESL 173
RESL 174
RESL 175
RESL 176
RESL 177
RESL 178
RESL 179
RESL 180
RESL 181
RESL 182
RESL 183
RESL 184
RESL 185
RESL 186
RESL 187
RESL 188
RESL 189
RESL 190
RESL 191
RESL 192
RESL 193
RESL 194
RESL 195
RESL 196
RESL 197
RESL 198
RESL 199
RESL 200 
```

# 9.6.9 Subroutine STIFMPA

This routine evaluates the stiffness matrices for layered elasto-plastic Mindlin plate elements.

```csv
SUBROUTINE STIMPA (COORD, EPSTN, IINCS, LNODS, MATNO, MELEM, STFL 1
. MEVAB, MMATS, MPOIN, MTOTG, NCRIT, NELEM, STFL 2
. NEVAB, NGAUS, NNODE, NLAPS, PROPS, STRSG) STFL 3
C**************************STFL 4
C STFL 5
C*** EVALUATE STIFFNESS MATRICES FOR LAYREED ELASTO-PLASTIC STFL 6
C*** MINDLIN PLATE ELEMENTS STFL 7
C STFL 8
C**************************STFL 9
DIMENSION CARTD(2,9), COORD(MPOIN,2), STFL 10
. DERIV(2,9), DEPTH(26), ELCOD(2,9), STFL 11
. EPSTN(MTOTG), ESTIF(27,27), GPCOD(2,9), LNODS(MELEM,9), STFL 12
. MATNO(MELEM), POSGP(4), PROPS(MMATS,8), SHAPE(9), STFL 13
. STRSG(5, MTOTG), WEIGP(4), STFL 14
. DFLEX(3,3), DSHER(2,2), BFLEI(3,3), BFLEJ(3,3), STFL 15 
```

<!-- source-page: 378 -->

```asm
BSHEI(2,3),BSHEJ(2,3),DUMMY(3,3)
STFL 16
REWIND 1
STFL 17
REWIND 3
STFL 18
KGAUS=0
STFL 19
C
C*** LOOP OVER EACH ELEMENT
STFL 20
C
DO 70 IELEM=1,NELEM
STFL 21
LPROP=MATNO(IELEM)
STFL 22
C
C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS
STFL 23
C
DO 10 INODE=1,NNODE
STFL 24
LNODE=LNODS(IELEM,INODE)
STFL 25
LNODE=IABS(LNODE)
STFL 26
DO 10 IDIME=1,2
STFL 27
10 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME)
STFL 28
C
C*** INITIALIZE THE ELEMENT STIFFNESS MATRIX
STFL 29
C
DO 20 IEVAB=1,NEVAB
STFL 30
DO 20 JEVAB=1,NEVAB
STFL 31
20 ESTIF(IEVAB,JEVAB)=0.0
STFL 32
CALL DEMPA(DEPTH,LPROP,MMATS,NLAPS,PROPS)
STFL 33
C
C*** EVALUATE PART OF STIFFNESS MATRIX
STFL 34
C
ASSOCIATED WITH BENDING DEFORMATION
STFL 35
C
KGASP=0
STFL 36
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 37
C
C
C*** SET UP GAUSSIAN INTEGRATION CONSTANTS
STFL 38
C
STFL 39
C
STFL 40
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 41
C
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 42
C
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 43
C
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 44
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 45
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 46
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 47
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 48
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 49
C
C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION
STFL 50
C
CALL GAUSSQ (NGAUS,POSGP,WEIGP)
STFL 51
C
DO 50 IGAUS=1,NGAUS
STFL 52
DO 50 JGAUS=1,NGAUS
STFL 53
KGASP=KGASP+1
STFL 54
EXISP=POSGP(IGAUS)
STFL 55
ETASP=POSGP(JGAUS)
STFL 56
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 57
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 58
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 59
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 60
C
CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE)
STFL 61
CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,
KGASP,NNODE,SHAPE)
STFL 62
DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS)
STFL 63
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 64
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 65
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 66
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 67
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 68
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 69
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 70
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 71
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 72
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 73
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 74
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 75
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 76
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 77
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 78
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 79
C
C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC
STFL 80 
```

<!-- source-page: 379 -->

```txt
50 CONTINUE
C
C*** EVALUATE PART OF STIFFNESS MATRIX
C ASSOCIATED WITH SHEAR DEFORMATION
C
KGASP=0
NGAUM=NGAUS-1
C
C*** ENTER LOOPS FOR AREA INTEGRATION
C
C
C*** SET UP GAUSSIAN INTEGRATION CONSTANTS
C
CALL GAUSSQ (NGAUM,POSGP,WEIGP)
DO 51 IGAUS=1,NGAUM
DO 51 JGAUS=1,NGAUM
KGASP=KGASP+1
EXISP=POSGP(IGAUS)
ETASP=POSGP(JGAUS)
C
C*** EVALUATE THE SHAPE FUNCTIONS, ELEMENTAL AREA,ETC
C
CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE)
CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,
KGASP,NNODE,SHAPE)
DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS)
C
C*** EVALUATE THE B AND DB MATRICES
C
CALL LAYMPA (DEPTH,DFLEX,DSHER,EPSTN,IINCS,KGAUS,LPROP,
MMATS,MTOTG,NCRIT,NLAPS,PROPS,STRSG,0)
C
C*** EVALUATE ELEMENT STIFFNESSES
C
DO 31 INODE=1,NNODE
CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE,
0, 0, 1)
DO 31 JNODE=INODE,NNODE
CALL BMATPB (BFLEJ,DUMMY,BSHEJ,CARTD,JNODE,SHAPE,
0, 0, 1)
31 CALL SUBMP (BSHEI,BSHEJ,DAREA,DSHER,ESTIF,INODE,
JNODE, 3, 2, 3)
51 CONTINUE
C
C*** CONSTRUCT THE LOWER TRIANGLE OF THE STIFFNESS MATRIX
C
DO 60 IEVAB=1,NEVAB
DO 60 JEVAB=IEVAB,NEVAB
60 ESTIF(JEVAB,IEVAB)=ESTIF(IEVAB,JEVAB)
C
C*** STORE THE STIFFNESS MATRIX,STRESS MATRIX AND SAMPLING POINT
C COORDINATES FOR EACH ELEMENT ON DISC FILE
C
WRITE(1) ESTIF
WRITE(3) GPCOD
70 CONTINUE
RETURN
END
STFL 81
STFL 82
STFL 83
STFL 84
STFL 85
STFL 86
STFL 87
STFL 88
STFL 89
STFL 90
STFL 91
STFL 92
STFL 93
STFL 94
STFL 95
STFL 96
STFL 97
STFL 98
STFL 99
STFL 100
STFL 101
STFL 102
STFL 103
STFL 104
STFL 105
STFL 106
STFL 107
STFL 108
STFL 109
STFL 110
STFL 111
STFL 112
STFL 113
STFL 114
STFL 115
STFL 116
STFL 117
STFL 118
STFL 119
STFL 120
STFL 121
STFL 122
STFL 123
STFL 124
STFL 125
STFL 126
STFL 127
STFL 128
STFL 129
STFL 130
STFL 131
STFL 132
STFL 133
STFL 134
STFL 135
STFL 136
STFL 137
STFL 138
STFL 139 
```

# 9.6.10 Subroutine STRMPA

This subroutine evaluates the stresses within each layer.

<!-- source-page: 380 -->

```csv
SUBROUTINE STRMPA (CARTD,CONST,DFLEX,DGRAD,DSHER,ELDIS,NNODE, STRL 1
SHAPE,STRES,IFFLE,IFSHE) STRL 2
C**********STRRL 3
C STRL 4
C*** EVALUATES STRESSES FOR MINDLIN PLATE STRL 5
C STRL 6
C**********STRRL 7
DIMENSION CARTD(2,9),DFLEX(3,3),DGRAD(6),DSHER(2,2), STRL 8
ELDIS(3,9),SHAPE(9),STRES(5) STRL 9
C*** ZERO STRESS VECTOR STRL 10
CALL VZERO (5,STRES) STRL 11
C*** EVALUATE ROTATIONS AT GAUSS POINT, IF NEEDED STRL 12
IF(IFSHE.EQ.0) GOTO 50 STRL 13
XZROT=0.0 STRL 14
YZROT=0.0 STRL 15
DO 30 INODE=1,NNODE STRL 16
XZROT=XZROT+SHAPE(INODE)*ELDIS(2,INODE) STRL 17
30 YZROT=YZROT+SHAPE(INODE)*ELDIS(3,INODE) STRL 18
C*** EVALUATE BENDING STRESS RESULTANTS STRL 19
50 IF(IFFLE.EQ.0) GOTO 60 STRL 20
EFLXX=-DGRAD(2)*CONST STRL 21
EFLYY=-DGRAD(6)*CONST STRL 22
EFLXY=-(DGRAD(3)+DGRAD(5))*CONST STRL 23
STRES(1)=DFLEX(1,1)*EFLXX+DFLEX(1,2)*EFLYY STRL 24
STRES(2)=DFLEX(2,1)*EFLXX+DFLEX(2,2)*EFLYY STRL 25
STRES(3)=DFLEX(3,3)*EFLXY STRL 26
C*** EVALUATE SHEAR STRESS RESULTANTS STRL 27
60 IF(IFSHE.EQ.0) RETURN STRL 28
ESHXX=DGRAD(1)-XZROT STRL 29
ESHYY=DGRAD(4)-YZROT STRL 30
STRES(4)=DSHER(1,1)*ESHXX STRL 31
STRES(5)=DSHER(2,2)*ESHYY STRL 32
RETURN STRL 33
END STRL 34 
```

# 9.7 Examples

To test the program, the elasto-plastic analysis of a simply supported plate is performed and 9 noded and Heterosis elements are used. The geometry, material properties of the plate are shown in Fig. 9.6.

![](images/page-380_949931b2d11daf9bfc7dc0bcf8a47e8fa08c56f5adfa3422e63f364437764ab0.jpg)

<details>
<summary>text_image</summary>

y
L
x
L
= 1.0, E = 10.92, ν = 0.3, t = 0.01, q = 1.0, σ₀ = 1600.0
</details>

Fig. 9.6 Geometry and material properties of simply supported square plate.