<!-- source-page: 401 -->

matrix is proportional to the mass and stiffness matrix. This is known as Rayleigh damping and we have

$$
\boldsymbol {C} = \alpha \boldsymbol {M} + \beta \boldsymbol {K} \tag {10.47}
$$

In the central difference method we can make the approximation that $\beta = 0$ so that

$$
\boldsymbol {C} = \alpha \boldsymbol {M} \tag {10.48}
$$

or

$$
c _ {i i} = a m _ {i i}
$$

where

$$
a = 2 \xi_ {r} \omega_ {r}
$$

in which $\xi_{r}$ and $\omega_{r}$ are the damping factor and circular frequency for the $r^{th}$ mode. This modelling of damping is rather poor since a is fixed for all modes of vibration. Thus if we take r = 1 then the higher modes will be less damped whereas the opposite would be more desirable. This is the price we pay for an otherwise convenient and efficient solution.

# 10.5 Critical time step

In explicit and implicit time integration schemes the selection of an appropriate time step is crucially important. Small time steps are required for accurate and stable solutions whereas for reasons of economy we would prefer large time steps. The analysis of the stability and accuracy characteristics $^{(2)}$ allows us to decide on a suitable time step for the various time stepping schemes. On this basis for the conditionally stable, central difference scheme, the stability considerations are of prime importance and the time step length is limited by the expression

$$
\Delta t \leqslant \frac {2}{\omega_ {\max}} \tag {10.49}
$$

where $\omega_{max}$ is the highest circular frequency of the finite element mesh. This severe time step limit, required for stability, ensures accuracy in practically all modes of vibration. Providing that $\omega_{max}$ represents the maximum nonlinear frequency, (10.49) holds for nonlinear problems. The estimate of the critical time step for conditionally stable schemes apparently necessitates the solution of the eigenvalue problem for the whole system. This is not so. The bound on the highest eigenvalue can be simply obtained by the consideration of an individual element. This is established by an important theorem proposed by Irons $^{(4)}$ which proves that the highest system eigenvalue must always be less than the highest eigenvalue of the individual elements. This allows a very easy estimate of critical time steps (by the above theorem) which will err on the safe side. To avoid the exact evaluation of the highest

<!-- source-page: 402 -->

finite element mesh frequency approximate expressions are usually employed. The most common form for plane strain is

$$
\Delta t \leqslant \mu L \left(\frac {\rho (1 + \nu) (1 - 2 \nu)}{E (1 - \nu)}\right) ^ {1 / 2} \tag {10.50}
$$

where L is the smallest length between any two nodes and $\mu$ is a coefficient dependent on the type of element employed. $^{(5)}$ For problems in which many time steps are used it may be beneficial to calculate the exact highest linear frequency of the finite element mesh prior to the time stepping analysis.

Recall that when an elasto-viscoplastic model is adopted care must be taken not to exceed the critical time step for the Euler scheme in evaluating the viscoplastic strains. (See Section 8.3).

# 10.6 Program DYNPAK

# 10.6.1 Overall structure of DYNPAK

We now present program DYNPAK for the elasto-viscoplastic or geometrically nonlinear, transient dynamic analysis of plane stress, plane strain and axisymmetric problems. The basic structure of the program is shown in Fig. 10.2. Many of the subroutines used in DYNPAK have already been described in earlier chapters.

The algorithm adopted has been presented schematically in Fig. 10.1. The program is written in a dynamically dimensioned form. Efficiency has sometimes been sacrificed for clarity of presentation and the reader may consider ways of making the program more efficient when reviewing this chapter.

Isoparametric 4, 8 and 9-noded quadrilateral elements are included in the program. A special mass lumping procedure $^{(6)}$ has been adopted and separate Gauss–Legendre rules may be adopted in the evaluation of the stiffness and the lumped mass matrices.

Impact and seismic loading can easily be specified. Material nonlinearity is based on elasto-viscoplastic models with Von Mises, Tresca, Mohr-Coulomb or Drucker-Prager yield criteria with isotropic hardening. A total Lagrangian formulation is used to allow for the geometric nonlinear behaviour.

Subroutines GAUSSQ, SFR2 and JACOB2 have already been dealt with and only the remaining routines will be listed and described.

# 10.6.2 Master routine DYNPAK

The master routine organises the calling of the main routines as outlined in Fig. 10.2. In subroutine CONTOL the variables required for dynamic dimensioning are read and a check is made on the maximum available dimensions. Note that the values given in the DIMENSION statement in

<!-- source-page: 403 -->

![](images/page-403_016a82b195acd55b89f16ccfae044bcc87d3595172e60184b80669e135f061e5.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["MASTER DYNPAK"] --> B["CONTOL"]
    A --> C["INPUTD"]
    C --> D["NODXYR"]
    C --> E["GAUSSQ"]
    A --> F["INTIME"]
    A --> G["PREVOS"]
    A --> H["LOADPLf(t)"]
    H --> I["MODPS"]
    H --> J["SFR2"]
    H --> K["JACOB2"]
    A --> L["LUMASS"]
    L --> M["GAUSSQ"]
    L --> N["SFR2"]
    L --> O["JACOB2"]
    A --> P["RESVPL"]
    P --> Q["MODPS"]
    P --> R["SFR2"]
    P --> S["JACOB2"]
    P --> T["JACOBD"]
    A --> U["EXPLIT"]
    U --> V["FUNCTS"]
    U --> W["FUNCTA"]
    U --> X["FUNCTA"]
    A --> Y["RESVPLp(t)"]
    Y --> Z["FWVVP"]
    Y --> AA["YIELDF"]
    A --> AB["OUTDYN"]
    A --> AC["DO LOOP 1, TO NSTEP"]
```
</details>

Fig. 10.2 Flow diagram for program DYNPAK.

<!-- source-page: 404 -->

DYNPAK should agree with the values specified in CONTOL. Subroutines INPUTD, INTIME and PREVOS read the mesh data, the time integration data and data for the previous state of the structure. Subroutines LUMASS and LOADPL generate the lumped mass and applied force vectors respectively. FIXITY deals with fixed boundary nodes. In the time step do loop, EXPLIT performs the direct time integration and RESVPL calculates

$$
\int_ {\Omega} [ \boldsymbol {B} ] _ {n} ^ {T} \boldsymbol {\sigma} _ {n} d \Omega
$$

when an elasto-viscoplastic material model is adopted.

In this version of DYNPAK it should be noted that the maximum dimensions imply that we can solve problems with no more than 50 elements, 200 nodal points, 50 fixed boundary nodes and 600 acceleration ordinates.

Of course, larger problems can be accommodated by increasing the values in CONTOL and also the appropriate dimensions in the DIMENSION statement in the main routine DYNPAK.
<table><tr><td></td><td colspan="7">PROGRAM DYNPAK (INPUT, TAPE5=INPUT, TAPE4, TAPE10, TAPE12, TAPE3, OUTPUT, TAPE6=OUTPUT, TAPE7, TAPE11, TAPE13)</td><td>DYNK</td><td>1</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>2</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>3</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>4</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>5</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>6</td></tr><tr><td>C</td><td colspan="7">DYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAM</td><td>DYNK</td><td>7</td></tr><tr><td>C</td><td colspan="7">DIMENSION ACCEH(600), ACCEV(600), COORD(200,2), DISPL(400), DYNK</td><td>DYNK</td><td>8</td></tr><tr><td>C</td><td colspan="7">FORCE(400), IFPRE(2,200), LNODS(50,9), MATNO(50), DYNK</td><td>DYNK</td><td>9</td></tr><tr><td>C</td><td colspan="7">INTGR(50), NPRQD(10), NGRQS(10), POSGP(4), DYNK</td><td>DYNK</td><td>10</td></tr><tr><td>C</td><td colspan="7">PROPS(10,13), RESID(400), RLOAD(50,18), STRIN(4,450), DYNK</td><td>DYNK</td><td>11</td></tr><tr><td>C</td><td colspan="7">STRSG(4,450), TDISP(400), TEMPE(100), VELOC(400), DYNK</td><td>DYNK</td><td>12</td></tr><tr><td>C</td><td colspan="7">VISTN(4,450), VIVEL(5,450), WEIGP(4), YMASS(400)</td><td>DYNK</td><td>13</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>14</td></tr><tr><td>C</td><td>CALL</td><td>CONTOL</td><td colspan="5">(NDOFN, NELEM, NMATS, NPOIN)</td><td>DYNK</td><td>15</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>16</td></tr><tr><td>C</td><td>CALL</td><td>INPUTD</td><td colspan="5">(COORD, IFPRE, LNODS, MATNO, NCONM, NCRIT, DYNK</td><td>DYNK</td><td>17</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>18</td></tr><tr><td>C</td><td>CALL</td><td>INTIME</td><td colspan="5">(NDIME, NDOFN, NELEM, NGAUM, NGAUS, NLAPS, DYNK</td><td>DYNK</td><td>19</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>20</td></tr><tr><td>C</td><td>CALL</td><td>INTIME</td><td colspan="5">(NMATS, NNODE, NPOIN, NPREV, NSTRE, NTYPE, DYNK</td><td>DYNK</td><td>21</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>22</td></tr><tr><td>C</td><td>CALL</td><td>INTIME</td><td colspan="5">(POSGP, PROPS, WEIGP)</td><td>DYNK</td><td>23</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>24</td></tr><tr><td>C</td><td>CALL</td><td>INTIME</td><td colspan="5">(AALFA, ACCEH, ACCEV, AFACT, AZERO, BEETA, DYNK</td><td>DYNK</td><td>25</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>26</td></tr><tr><td>C</td><td>CALL</td><td>INTIME</td><td colspan="5">(BZERO, DELTA, DTIME, DTEND, GAAMA, IFIXD, DYNK</td><td>DYNK</td><td>27</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>28</td></tr><tr><td>C</td><td>CALL</td><td>PREVOS</td><td colspan="5">IFUNC, INTGR, KSTEP, MITER, NDOFN, NELEM, DYNK</td><td>DYNK</td><td>29</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>30</td></tr><tr><td>C</td><td>CALL</td><td>LOADPL</td><td colspan="5">NGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD, DYNK</td><td>DYNK</td><td>31</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>32</td></tr><tr><td>C</td><td>CALL</td><td>LOADPL</td><td colspan="5">(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNK</td><td>DYNK</td><td>33</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>34</td></tr><tr><td>C</td><td>CALL</td><td>LOADPL</td><td colspan="5">(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNK</td><td>DYNK</td><td>35</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>36</td></tr><tr><td>C</td><td>CALL</td><td>LOADPL</td><td colspan="5">(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNK</td><td>DYNK</td><td>37</td></tr><tr><td>C</td><td colspan="7">DYNK</td><td>DYNK</td><td>38</td></tr><tr><td>C</td><td>CALL</td><td>LOADPL</td><td colspan="5">(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNK</td><td>DYNK</td><td>39</td></tr></table>

<!-- source-page: 405 -->

```txt
C
CALL
FIXITY
(IFPRE, NDOFN, NPOIN, YMASS)
DYNK
40
DYNK
41
DYNK
42
IF(NPREV.NE.O)
.CALL
RESVPL
(COORD, DTIME, LNODS, MATNO, NCRIT, NDIME,
NDOFN, NELEM, NGAUS, NLAPS, NNODE, NMATS,
NPOIN, NSTRE, NTYPE, POSGP, PROPS, RESID,
RLOAD, STRIN, STRSG, TDISP, VISTN, VIVEL,
WEIGP)
DYNK
43
DYNK
44
DYNK
45
DYNK
46
DYNK
47
DYNK
48
DYNK
49
DYNK
50
DYNK
51
CALL
EXPLIT
(ACCEH, ACCEV, AFACT, AZERO, AALFA, BZERO,
DTIME, DTEND, FORCE, IFIXD, IFPRE, IFUNC,
ISTEP, NDOFN, NPOIN, OMEGA, RESID, TDISP,
VELOC, YMASS)
DYNK
52
DYNK
53
DYNK
54
DYNK
55
DYNK
56
CALL
RESVPL
(COORD, DTIME, LNODS, MATNO, NCRIT, NDIME,
NDOFN, NELEM, NGAUS, NLAPS, NNODE, NMATS,
NPOIN, NSTRE, NTYPE, POSGP, PROPS, RESID,
RLOAD, STRIN, STRSG, TDISP, VISTN, VIVEL,
WEIGP)
DYNK
57
DYNK
58
DYNK
59
DYNK
60
DYNK
61
DYNK
62
CALL
OUTDYN
(DISPL, DTIME, ISTEP, NDOFN, NELEM, NGAUS,
NGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD,
NREQS, NTYPE, STRSG, TDISP, VIVEL)
DYNK
63
DYNK
64
DYNK
65
DYNK
66
500 CONTINUE
STOP
END
DYNK
67
DYNK
68
DYNK
69 
```

# 10.6.3 Subroutine BLARGE

This subroutine evaluates the strain-displacement matrix for geometrically nonlinear displacements using the deformation Jacobian matrix $[J_{D}]_{n}$ . Note that for small displacement analysis we pre-set NLAPS = 0.

```txt
SUBROUTINE BLARGE (BMATX, CARTD, DJACM, DLCOD, GPCOD, KGASP, BLAR 1
NLAPS, NNODE, NTYPE, SHAPE) BLAR 2
C******************************************************************************************
BLAR 3
C BLAR 4
C*** LARGE DISPLACEMENT B MATRIX BLAR 5
C BLAR 6
C******************************************************************************************
BLAR 7
DIMENSION BMATX(4,18), CARTD(2,9), DJACM(2,2), DLCOD(2,9), BLAR 8
GPCOD(2,9), SHAPE(9) BLAR 9
NGASH=0 BLAR 10
DO 10 INODE=1, NNODE BLAR 11
MGASH=NGASH+1 BLAR 12
NGASH=MGASH+1 BLAR 13
BMATX(1, MGASH)=CARTD(1, INODE)*DJACM(1,1) BLAR 14
BMATX(1, NGASH)=CARTD(1, INODE)*DJACM(2,1) BLAR 15
BMATX(2, MGASH)=CARTD(2, INODE)*DJACM(1,2) BLAR 16
BMATX(2, NGASH)=CARTD(2, INODE)*DJACM(2,2) BLAR 17
BMATX(3, MGASH)=CARTD(2, INODE)*DJACM(1,1)+CARTD(1, INODE)*DJACM(1,2)BLAR 18
BMATX(3, NGASH)=CARTD(1, INODE)*DJACM(2,2)+CARTD(2, INODE)*DJACM(2,1)BLAR 19
10 CONTINUE BLAR 20
IF(NTYPE.NE.3) RETURN BLAR 21
FMULT=1. BLAR 22
IF(NLAPS.LT.2) GO TO 40 BLAR 23
FMULT=0.0 BLAR 24 
```

<!-- source-page: 406 -->

```asm
DO 20 JNODE=1, NNODE BLAR 25
20 FMULT=FMULT+DLCOD(1, JNODE)*SHAPE(JNODE) BLAR 26
FMULT=FMULT/GPCOD(1, KGASP) BLAR 27
40 NGASH=0 BLAR 28
DO 30 INODE=1, NNODE BLAR 29
MGASH=NGASH+1 BLAR 30
NGASH=MGASH+1 BLAR 31
BMATX(4, MGASH)=SHAPE(INODE)*FMULT/GPCOD(1, KGASP) BLAR 32
30 BMATX(4, NGASH)=0.0 BLAR 33
RETURN BLAR 34
END BLAR 35 
```

BLAR 10-20 Evaluate the complete strain matrix for plane stress/strain problems and the first three rows of the strain matrix for axisymmetric problems.

BLAR 21-33 Evaluate the remainder of the strain matrix for axisymmetric problems, if applicable.

# 10.6.4 Subroutine CONTOL

The purpose of this subroutine is to set the values of variables for the dynamic dimensions which are used elsewhere in the program. If any change in the DIMENSION statement in the master routine is made, then a corresponding change in this subroutine should also be made.

```txt
SUBROUTINE CONTOL (NDOFN, NELEM, NMATS, NPOIN) CONT 1
C*******************************
C
C*** READ CONTROL DATA AND CHECK FOR DIMENSIONS CONT 2
C
C*******************************
C
READ(5,110) NPOIN, NELEM, NDOFN, NMATS CONT 3
IF(NELEM.GT. 50) GO TO 200 CONT 4
IF(NPOIN.GT.200) GO TO 200 CONT 5
IF(NMATS.GT. 10) GO TO 200 CONT 6
GO TO 210 CONT 7
200 WRITE(6,120) CONT 8
STOP CONT 9
120 FORMAT(/'SET DIMENSION EXCEEDED - CONTOL CHECK') CONT 10
110 FORMAT(16I5) CONT 11
210 CONTINUE CONT 12
RETURN CONT 13
END CONT 14
CONT 15
CONT 16
CONT 17
CONT 18 
```

# 10.6.5 Subroutine EXPLIT

This subroutine performs the direct time integration using expressions (10.43) and (10.44) to evaluate the nodal displacements at every time step. Special provisions are made for the first time step.

```txt
SUBROUTINE EXPLIT (ACCEH, ACCEV, AFACT, AZERO, AALFA, BZERO, EXPL 1
. DTIME, DTEND, FORCE, IFIXD, IFPRE, IFUNC, EXPL 2
. ISTEP, NDOFN, NPOIN, OMEGA, RESID, TDISP, EXPL 3
. VELOC, YMASS) EXPL 4
C*************** EXPL 5
C EXPL 6
C *** TIME STEPPING ROUTINE EXPL 7
C EXPL 8
C*************** EXPL 9 
```

<!-- source-page: 407 -->

```csv
DIMENSION YMASS(1), ACCEH(1), TDISP(1), RESID(1), EXPL 10
FORCE(1), ACCEV(1), VELOC(1), IFPRE(2,1)
CFACT=1.0+0.5*AALFA*DTIME
CFACT=1./CFACT
CONS1=2.*CFACT
RCONS=1./CONS1
CONS2=CONS1-1
CONS3=DTIME*DTIME*CFACT
CONS4=-2.0*CONS2*DTIME
IF(ISTEP.GT.1) CONS4=CONS2
NPOSN=0
FACTS=FUNCTS (AZERO, BZERO, DTEND, DTIME, IFUNC, ISTEP, OMEGA)
FACTH=FUNCTA (ACCEH, AFACT, DTEND, DTIME, IFUNC, ISTEP)
FACTV=FUNCTA (ACCEV, AFACT, DTEND, DTIME, IFUNC, ISTEP)
DO 500 IPOIN=1, NPOIN
DO 510 IDOFN=1, NDOFN
FACTT=0.0
IF(IFUNC.NE.0) GO TO 200
IF(IFIXD.EQ.0.AND.IDOFN.EQ.1) FACTT=FACTH
IF(IFIXD.EQ.0.AND.IDOFN.EQ.2) FACTT=FACTV
IF(IFIXD.EQ.1.AND.IDOFN.EQ.2) FACTT=FACTV
IF(IFIXD.EQ.2.AND.IDOFN.EQ.1) FACTT=FACTH
IF(IFPRE(IDOFN, IPOIN).EQ.0) GO TO 200
FACTT=0.0
FACTS=1.0
200 CONTINUE
NPOSN=NPOSN+1
DCURR=TDISP(NPOSN)
RESID(NPOSN)=RESID(NPOSN)-FORCE(NPOSN)*FACTS
TDISP(NPOSN)=-RESID(NPOSN)*CONS3/YMASS(NPOSN)
.-FACTT*CONS3+DCURR*CONS1-VELOC(NPOSN)*CONS4
IF(ISTEP.EQ.1) TDISP(NPOSN)=TDISP(NPOSN)*RCONS
VELOC(NPOSN)=DCURR
510 CONTINUE
500 CONTINUE
RETURN
END
EXPL 10
EXPL 11
EXPL 12
EXPL 13
EXPL 14
EXPL 15
EXPL 16
EXPL 17
EXPL 18
EXPL 19
EXPL 20
EXPL 21
EXPL 22
EXPL 23
EXPL 24
EXPL 25
EXPL 26
EXPL 27
EXPL 28
EXPL 29
EXPL 30
EXPL 31
EXPL 32
EXPL 33
EXPL 34
EXPL 35
EXPL 36
EXPL 37
EXPL 38
EXPL 39
EXPL 40
EXPL 41
EXPL 42
EXPL 43
EXPL 44
EXPL 45
EXPL 46 
```

EXPL 12–19 Evaluate the various time integration constants. After the first time step modify variable CONS4.

EXPL 21 Evaluate the value of the time varying Heaviside or harmonic function for a particular time step.

EXPL 22–23 Evaluate the acceleration ordinates (FACTH for horizontal and FACTV for vertical acceleration respectively) at a particular time step.

EXPL 24–31 The seismic force is only applied for particular degrees of freedom. For IFIXD = 1 only vertical, IFIXD = 2 only horizontal or radial and IFIXD = 0 both components of the acceleration are considered.

EXPL 32-35 Assign appropriate values for restrained boundary nodes.

EXPL 36–40 Evaluate displacements.

EXPL 41 For the first time step modify the displacement.

EXPL 42 Store the current displacements for the next time step.

# 10.6.6 Subroutine FIXITY

This subroutine deals with the restrained degrees of freedom (boundary points). The diagonal mass vector, XMASS, is modified—for restrained

<!-- source-page: 408 -->

degrees of freedom. The component of the XMASS vector is set to a large value such as 1.E30, which artificially makes the displacement zero.

```fortran
SUBROUTINE FIXITY (IFPRE, NDOFN, NPOIN, YMASS) FIXY 1
C********** 1
C 2
C 3
C *** DEALS WITH FIXED BOUNDARY NODES 4
C 5
C********** 6
DIMENSION IFPRE(2,1), YMASS(1) 7
NTOTV=NDOFN*NPOIN 8
IPOSN=0 9
DO 500 IPOIN=1, NPOIN 10
DO 500 IDOFN=1, NDOFN 11
IPOSN=IPOSN+1 12
500 IF(IFPRE(IDOFN, IPOIN).EQ.1) YMASS(IPOSN)=1.E30 13
WRITE(6,900) 14
900 FORMAT(/5X,19HNODAL LUMPED MASSES/) 15
WRITE(6,910) (ITOTV, YMASS(ITOTV), ITOTV=1, NTOTV) 16
910 FORMAT(6(1X, I5, E13.5)) 17
RETURN 18
END 19 
```

# 10.6.7 Subroutine FLOWVP

This routine evaluates the viscoplastic strain rate.

```txt
SUBROUTINE FLOWVP (AVECT, KGAUS, LPROP, NCRIT, NMATS, PROPS, STEFF, VIVEL, YIELD) FLOV 1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DIMENSION AVECT(4), PROPS(NMATS,1), VIVEL(5,1)
IF(STEFF.EQ.0.0) GO TO 90
NSTR1=4
TOLOR=0.01
FDATM=PROPS(LPROP, 6)
HARDS=PROPS(LPROP, 7)
FRICT=PROPS(LPROP, 8)
GAMMA=PROPS(LPROP, 9)
DELTA=PROPS(LPROP,10)
NFLOW=PROPS(LPROP,11)
FRICT=FRICT*0.017453292
IF(NCRIT.EQ.3) FDATM=FDATM*COS(FRICT)
IF(NCRIT.EQ.4) FDATM=6.0*FDATM*COS(FRICT)/
(1.73205080757*(3.0-SIN(FRICT)))
IF(HARDS.GT.0.) FDATM=FDATM+VIVEL(5,KGAUS)*HARDS
IF(FDATM.LT.0.001) FDATM=1.0
FCURR=YIELD-FDATM
FNORM=FCURR/FDATM
IF(FNORM.LT.TOLOR) GO TO 90
IF(NFLOW.EQ.1) GO TO 50
CMULT=GAMMA*(EXP(DELTA*FNORM)-1.0)
GO TO 60
50 CMULT=GAMMA*(FNORM**DELTA)
60 CONTINUE
DO 70 ISTR1=1,NSTR1
70 AVECT(ISTR1)=CMULT*AVECT(ISTR1)
DO 80 ISTR1=1,NSTR1
80 VIVEL(ISTR1,KGAUS)=AVECT(ISTR1)
RETURN
90 DO 100 ISTR1=1,NSTR1
100 VIVEL(ISTR1,KGAUS)=0.
RETURN
FLOV 10 
```

<!-- source-page: 409 -->

# 10.6.8 Function FUNCTA

This function interpolates the accelerogram data for a particular time step. AFACT is the ratio of the accelerogram record time step length to the computational time step length.

```txt
FUNCTION FUNCTA (ACCER, AFACT, DTEND, DTIME, IFUNC, JSTEP) FUNA 1
C*******************************
C
C*** ACCELEROGRAM INTERPOLATION FUNA 2
C
C*******************************
DIMENSION ACCER(1) FUNA 3
IF(IFUNC.NE.0) RETURN FUNA 4
FUNCTA=0.0 FUNA 5
IF(JSTEP.EQ.0.OR.FLOAT(JSTEP)*DTIME.GT.DTEND) RETURN FUNA 6
XGASH=(FLOAT(JSTEP)-1.0)/AFACT+1.0 FUNA 7
MGASH=XGASH FUNA 8
NGASH=MGASH+1 FUNA 9
XGASH=XGASH-FLOAT(MGASH) FUNA 10
FUNCTA=ACCER(MGASH)*(1.0-XGASH)+XGASH*ACCER(NGASH) FUNA 11
RETURN FUNA 12
END FUNA 13
FUNA 14
FUNA 15
FUNA 16
FUNA 17 
```

# 10.6.9 Function FUNCTS

This function sets the value of the time varying function for a particular time step. Heaviside functions $(f(t) = 1.0 \ H(t))$ or harmonic functions, $(f(t) = a - b \sin \omega t)$ can be specified.

```txt
FUNCTION FUNCTS (AZERO,BZERO,DTEND,DTIME,IFUNC,JSTEP,OMEGA) FUNS 1
C************************** FUNS 2
C FUNS 3
C*** HEAVISIDE AND HARMONIC TIME FUNCTION FUNS 4
C FUNS 5
C************************** FUNS 6
IF(IFUNC.EQ.0) RETURN FUNS 7
FUNCTS=0.0 FUNS 8
IF(JSTEP.EQ.0.OR uses FLOAT(JSTEP)*DTIME.GT.DTEND) RETURN FUNS 9
IF(IFUNC.EQ.1) FUNCTS = 1.0 FUNS 10
IF(IFUNC.EQ.2) ARGUM=OMEGA*JSTEP*DTIME FUNS 11
IF(IFUNC.EQ.2) FUNCTS = AZERO + BZERO*SIN(ARGUM) FUNS 12
RETURN FUNS 13
END FUNS 14 
```

# 10.6.10 Subroutine INPUTD

This subroutine reads and writes most of the control parameters, nodal point coordinates, element connectivities, boundary conditions and material properties. It also writes the geometric data onto file 13 for deformation plotting. A similar routine was described in Chapter 6.

```c
SUBROUTINE INPUTD (COORD, IFPRE, LNODS, MATNO, NCONM, NCRIT, NPUT 1
. NDIME, NDOFN, NELEM, NGAUM, NGAUS, NLAPS, NPUT 2
. NMATS, NNODE, NPOIN, NPREV, NSTRE, NTYPE, NPUT 3
. POSGP, PROPS, WEIGP) NPUT 4
C*******************************
C*******************************
C*******************************
C*** DYNPAK INPUT ROUTINE
C
C*******************************
DIMENSION COORD(NPOIN, 1), IFPRE(NDOFN, 1), WEIGP(1), MATNO(1), NPUT 10 
```

<!-- source-page: 410 -->

```csv
READ(5,913) TITLE NPUT 12
913 FORMAT(10A4) NPUT 13
WRITE(6,914) TITLE NPUT 14
914 FORMAT(//,5X,10A4) NPUT 15
C NPUT 16
C*** READ THE FIRST DATA CARD, AND ECHO IT IMMEDIATELY. NPUT 17
C NPUT 18
READ (5,900) NVFIX,NTYPE,NNODE,NPROP,NGAUS,NDIME,NSTRE,NCRIT, NPREV,NCONM,NLAPS,NGAUM,NRADS NPUT 19
WRITE(6,901) NPOIN,NELEM,NVFIX,NTYPE,NNODE,NDOFN,NMATS,NPROP, NGAUS,NDIME,NSTRE,NCRIT,NPREV,NCONM,NLAPS,NGAUM, NRADS NPUT 20
901 FORMAT (/5X,18HCONTROL PARAMETERS/ NPUT 21
/5X,8H NPOIN =,I10,5X,8H NELEM =,I10,5X,8H NVFIX =,I10/ NPUT 22
/5X,8H NTYPE =,I10,5X,8H NNODE =,I10,5X,8H NDOFN =,I10/ NPUT 23
/5X,8H NMATS =,I10,5X,8H NPROP =,I10,5X,8H NGAUS =,I10/ NPUT 24
/5X,8H NDIME =,I10,5X,8H NSTRE =,I10,5X,8H NCRIT =,I10/ NPUT 25
/5X,8H NPREV =,I10,5X,8H NCONM =,I10,5X,8H NLAPS =,I10/ NPUT 26
/5X,8H NGAUM =,I10,5X,8H NRADS =,I10/) NPUT 27
900 FORMAT(16I5) NPUT 28
NPUT 29
NPUT 30
NPUT 31
NPUT 32
NPUT 33
NPUT 34
NPUT 35
NPUT 36
NPUT 37
NPUT 38
NPUT 39
NPUT 40
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NPUT 42
NPUT 43
NPUT 44
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NPUT 680
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NPUT 700
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NPUT 728
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NPUT 737
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NPUT 750
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NPUT 768
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NPUT 777
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NPUT 780
NPUT 781
NPUT 782
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NPUT 786
NPUT 787
NPUT 788
NPUT 789
NPUT 790
NPUT 791
NPUT 792
NPUT 793
NPUT 794
NPUT 795
NPUT 796
NPUT 797
NPUT 798
NPUT 799
NPUT 800
NPUT 801
NPUT 802
NPUT 803
NPUT 804
NPUT 805
NPUT 806
NPUT 807
NPUT 808
NPUT 809
NPUT 810
NPUT 811
NPUT 812
NPUT 813
NPUT 814
NPUT 815
NPUT 816
NPUT 817
NPUT 818
NPUT 819
NPUT 820
NPUT 821
NPUT 822
NPUT 823
NPUT 824
NPUT 825
NPUT 826
NPUT 827
NPUT 828
NPUT 829
NPUT 830
NPUT 831
NPUT 832
NPUT 833
NPUT 834
NPUT 835
NPUT 836
NPUT 837
NPUT 838
NPUT 839
NPUT 840
NPUT 841
NPUT 842
NPUT 843
NPUT 844
NPUT 
```