sap3pr: a fortran program for calculating equivalent nodal

JL& O R N L / T M - 6 Q 9 1

SAP3PR: A Fortran Program for Calculating Equivalent Nodal Loads

Resulting from Pressure on the Faces of 8- to 20-Node Isoparametric

Elements

D. N. Fanning

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OAK RIDGE NATIONAL LABORATORY * OPERATED BY U N I O N C A R B I D E C O R P O R A T I O N • FOR THE DEPARTMENT OF ENERGY

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ORNL/TM-6091 Dist. Category UC-77

Contract No. W-7405-eng-26

Engineering Technology Division

HTGR BASE TECHNOLOGY PROGRAM Prestressed Concrete Nuclear Pressure

Vessel Development (189a 01331)

SAP3PR: A FORTRAN PROGRAM FOR CALCULATING EQUIVALENT NODAL LOADS RESULTING FROM PRESSURE ON THE FACES

OF 8- TO 20-N0DE ISOPARAMETRIC ELEMENTS

Date P u b l i s h e d - A p r i l , 1978

Prepared by the OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830

operated by UNION CARBIDE CORPORATION

for the DEPARTMENT OF ENERGY

D. N. Fanning

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iii

CONTENTS

Page

ABSTRACT 1 INTRODUCTION 1

PROGRAM INPUT 1

PROGRAM OPERATION 7

PROGRAM LISTING 1° CONCLUSION J-6

REFERENCES I6

APPENDIX

SAP3PR: A FORTRAN PROGRAM FOR CALCULATING EQUIVALENT NODAL LOADS RESULTING FROM PRESSURE ON THE FACES

OF 8- TO 2G-NODE ISOPARAMETRIC ELEMENTS

D. N. Fanning

ABSTRACT

A computer program is described for calculating the equivalent nodal loads resulting from distributed loads on a three-dimensional finite-element model. Included is a listing of the computer program, a description of the input data, and an example of the output.

INTRODUCTION

Pressure loads or distributed loads comprise at least part of the load on most structures analyzed using three-dimensional finite-element computer programs. Unfortunately, not all finite-element codes allow the user to specify pressure loads directly (e.g., NONSAP and ADINA). For codes such as these, only loads acting at the nodes are acceptable as input. Therefore, before the effects of a pressure load can be analyzed, it is necessary to convert the distributed loads into equiva-lent nodal loads. The computer program SAP3PR was written expressly for this purpose. SAP3PR was written for the Digital Equipment Corporation PDP-10 using the Fortran language; with minor modifications tb? program can be executed on the IBM 360 or 370 computer. A listing is presented in the next section, and examples of the input to and output from the program are given in the appendix.

PROGRAM INPUT f

The input cards for the program are of two types: model definition cards and load case definition cards. The model definition cards define the geometry of the finite-element analytical model, while the load case cards describe the pressure loadings. The format (in parentheses), variable names, and variable definitions are given for each card.

2

1. Model Definition Cards

Model Control Card (4I5.4X.L1)

Columns

1-5 6 - 1 0

11-15 16-20

25

Variable

NUMNP NUMEL NLOAD NUMLTM

NONSAP

Node Cards (35X.3F10.0)

Columns Variable

36-45 4&-55 56-65

XY(1,I) XY(2,I) XY(3,I)

Definition

Number of nodes Number of elements Number of load ctses Maximum number of pressure distributions fo.r any load case Flag indicating type of format = T for NONSAP1 or ADINA2 format = F for STATIC-SAP format3

Definition

X-coordinate of the Ith node Y-coordinate of the Ith node Z-coordinate of the Ith node

Node cards must be input consecutively and in ascending order.

Element Cards (/8I5/:.2i5) or (/20I4)

Columns Variable Definition

LN0DE(1,I) Node 1 of element I LN0DE(2,I) Node 2 of element I LN0DE(3,I) Node 3 of element I LN0DE(4,1) Node 4 of element I LN0DE(5,1) Node 5 of element I LN0DE(6,I) Node 6 of element I LN0DE(7,1) Node 7 of element I LN0DE(8,I) Node 8 of element I LN0DE(9,I) Node 9 of element I LNODE(10,1) Node 10 of element I

3

Element Cards (/8I5/12I5) or (/20I4) (continued)


LNODE(ll.I) Node 11 of element I LN0DE(12,I) Node 12 of element I LN0DE(13,I) Node 13 of element I LN0DE(14,I) Node 14 of element I LN0DE(15,I) Node 15 of element I LN0DZ(16,I) Node 16 of element I LN0DE(17,I) Node 17 of element I LN0DE(18,I) Node 18 of element I LN0DE(19,I) Node 19 of element I LN0DE(20,I) Node 20 of element I

The first format listed for ths element cards is the format used by NONSAP and ADINA (NONSAP = T on the model control card); the second format is for STATIC-SAP (NONSAP = F on the model control card). The element cards must be input in ascending order with no elements omitted. The sequence of nodes defining the variable node isoparametric element is shown in Fig. 1; the finite-element codes NONSAP, ADINA, and STATIC-SAP (a modified version) use this .umbering scheme for their three-dimensional isoparametric element. If a nodal sequence different from Fig. 1 is needed (such as for the 20-node element of the unmodified ver-sion of STATIC-SAP), appropriate changes could easily be made in SAP3PR.

2. Load Case Definition Cards

Input NLOAD groups of load case definition cards.

Load Control Card (415)


1-5 NFACE Number of element faces which are pressurized Order of numerical integration to be used (default = 3)

6-10 INT

4

i ih\; [ j ' . , / "i-i-K,

Fig. 1. Three-dimensional finite-element node numbering convention.

Load Control Card (415) (continued)


1 1 - 1 5

16-20

NCUR

NUMLT

Nr.iuber of the load curve to which this load case corresponds

Number of pressure distributions

In general, the pressure loads should be calculated using the same order of integration as will be used in the finite-element analysis. If

5

the pressurized element faces are flat and rectangular and the pressure is constant or varies linearly, INT can be reduced to 2 with no loss of accuracy. NCUR is the number of the load curve in the finite-element analysis to which this load ca*e corresponds; NCUR is not used in SAP3PR but is placed on the nodal load cards output by SAP3PR.

Pressure Cards (8F10.0)


1-10 PRESR(1,I) Magnitude of the 1th pressure distribution at face node 1

11-20 PRESR(2,1) Magnitude of the Ith pressure distribution at face node 2

21-30 PRESR(3,I) Magnitude of the Ith pressure distribution at face node 3

31—AO FRESR(4,I) Magnitude of the Ith pressure distribution at face node 4





There should be NUMLF pressure cards, one for each pressure distri-bution. To properly specify surface pressures, it is necessary to se-quence the six faces and the eight nodal points (N0F1, N0F2 N0F8) on each face. The six faces of the element are defined according to Fig. 1 as follows: face 1 (R = 1), face 2 (S = 1), face 3 (T = 1), face 4 (R = —1), face 5 (S = —1), face 6 (T = —1). Each face contains either NI or N7, which will be used as the first of the eight face nodes. Figure 2 shows the arrangement' of the face nodes. Using Fig. 2 in conjunction

6

N0F2

N 0 F 6

N0F3

O R N L DWG 77 19448

NOF1

NOF7

NOF8

N O F 1

Fig. 2. Node numbering convention of a three-dimensional element face.

i.th Fig. 1 results in the following nodal sequence for each face:

FACE N0F1 N0F2 N0F3 N0F4 N0F5 N0F6 N0F7 N0F8

1 NI N4 N8 N5 N12 N20 N16 N17 2 NI N5 N6 N2 N17 N13 N18 N9 3 NI N2 N3 N4 N9 N10 Nil N12 4 N7 N3 N2 N6 N19 N10 N18 N14 5 N7 N8 N4 N3 N15 N20 Nil N19 6 N7 N6 N5 N8 N14 N13 N16 N15

Face Card (315)

Columns

1-5 6-10

11-15

Variable

LM NUMF

LODTYP

Definition

Number of the pressurized element Number of the face which is pressurized

Number of the pressure distribu-tion which acts on this face (default = 1)

There should be NFACE face cards.

7

PROGRAM OPERATION

Equivalent nodal forces due to distributed loads are given by the following equation (Ref. 4):

{F} = - J[N] T {p} dV ,

with the integration being over the volume of the loaded element. The matrix [N] is composed of the shape functions as follows:

nl,3i-2 = hi ni,3i-l = °

nl,3i = ° n2,3i-2 = ° n_ _. , = h. 2,3i-l x

n_ = 0 2 , 3 i

n3,3i-2 = ° n3,3i-l = °

(i = 1, 2, , m ) ,

n_ ,. = n. 3,3i i

where h. is the shape function for node i, n. , is the term in the jth i J row and kth column of [N], and m is thp. number of nodes the element has. The terms of the vector {p} are the components of the distributed load in the global coordinate directions. Since {p} is zero except on the face which is pressurized, it is only necessary to integrate over the surface of the pressurized face, not the volume of the element. Numeri-cal integration is used after the face has been mapped into the local two-dimensional u, v-space as shown in Fig. 3. Assuming that one of the global coordinates (x, y, or z) is a function of the other two, the

h fu , v)

v

4 ORNL DAG 77 19444

+ 1

+ 1

oo

Fig. 3. Mapping of a pressurized face into a local two-dimensional space.

9

following expression can be obtained:

{F} =f l fl [N]T {P} du dv ,

where

r ^

3u 3v 3u 3v

i—> i 3x 3z 3z 3x L p p = I h l ~3v — "3u 3v y n

3y 3x _ _3x ^ 3u 9v 3u 3v v. J

and p^ is the magnitude of the pressure, a function of u and v

r i

pn - E Pi hi ( u» v) r

I The matrix [N] is composed of the shape functions for the nodes on the pressuvized face.

The program expects the input data to be in a disk file named F0R2l.DAT. After reading the model control card, the program sets up the pointers necessary for the vector storage and checks to see if suf-ficient space is available in blank common. The output, which is com-patible with ADINA, NONSAP, or STATIC-SAP, depending upon the value of the variable NONSAP on card 1, is written into a disk file named F0R22.DAT.

10

PROGRAH LISTING

C * M M M * * * M M M O * * M * * < * U * M M t * M « * » * * * * « * * M * » * * * * * * * M » » * » * t 10 c 20 C DRIVER FOR PROGRAM WHICH CALCULATES NODAL LOADS EQUIVALENT 30 C TO PRESSURE LOADS ON e-20 NODE 3-D ISOPARAMETRIC ELEMENTS «0 C SO C * * * * * * * * * * * * * * * * * * * * * * * * * * • » * * * * » » • • • * » • • * * « * * • • * » « * • * » * » * • • » • * • » • 6 0

LOGICAL NONSAP 70 CO (IRON A (40000) AO BTCT = <40000 90

C 100 C INPUT 110 C NOMNE - NO. OF NOCES 120 C NOBEL - NO. OF ELEMENTS 130 C NLOAD - NO. OF LCAC CASES 1U0 C N0HL1M - BAXIMUM NC. OF PRESSORE DISTRIBUTIONS 150 C FOR ANY ICAD CASE 160 C NONSAP - FLAG INDICATING FORMAT 170 C = 1 FOR •NCNSAP' OR •ADINA* FORMAT 100 C = F FOR 'STATIC-SAP' FORMAT 190 C 200

READ(21,5000)NCHNP,NO BEL,NLOAD,NUMLTN,NONSAP 210 5000 FORMAT (<415,4X, 11) 220

N1 = 1 230 N2 = NI • 2*NLCAD*3*NUHNP 2«0 N3 = N2 + 2*NLOAD*3 250 NU = N3 • 2*3*NCMNP 260 N5 = NU • 2*8 + BUMLTM 270 N6 = N5 + 21+NCBEL 290 • NLAST = N6 + NLOAD - 1 290 IF (NLAS1 .LE. HTOT)GC TO 10 300 TYPE 50 10,NLAS1 310

5010 FOBHAT(IH ,'MTCT MOST EE INCREASED TO',16) 320 STOP 330

10 TYPE 5020,NLAS1 3U0 5020 FOBHAT (1HO,' MICT MUSI BE AT LEAST',16) 350

CALL SAP3PB(A(N1),A(N2),A(N3),A(NU),A(N5),A(N6), 360 1 NOMNP,NOEEL,NLOAD,NU MLTM,NONSAP) 361 STCP 370 END 380

SUBROUTINE SAP3ER (F,FSOfl,XY,PRESRr LNODE,NC, SAP3P 10 1 NOMKE.NUMEL,NLOAD,NUMLTM,NONSAP) SAP3P 11

C 4 * * * * * * * ^ * + * * * * * * ^ * * * * . * * * * * * * * * * * * * * * * * * * » * * S A p 3 p 2 0 C SAP3P 30 C PROGRAM TO CALCULATE NODAL LOADS EQUIVALENT TO PRESSURE LOADS SAP3P UO C ON 8-20 NODE 3-C ISOPARAMETRIC ELEMENTS SAP3P 50 C SAP3P 60 C + 7 0 C SAE3P 80 C CEFINITION OF VARIAELES SAP3P 90 C SAP3 100 C F(I,J,K) - RESULTING LOAD IN 'J'TH DIRECTION ON NODE »K« FOR SAP3 110 C LCAD CASE 'I' SAP3 120 C IDGEN - FLAG FOR DEGENERATE EDGES OF AN ELEMENT SAP3 130 C LH - ELEMENT SHICH IS LOADED SAP3 140 C LNCDE(I,J) - •I'TH NODE OF THE 'J'TH ELEMENT SAP3 150 C LODTIP - NUMBER CF PRESSORE DISTRIBUTION ON THE FACE SAP3 160 C NLOAD - NUMBER CF LOAC CASES SAP3 170 C NODE(I) - •I'TH NODE CF LOADED FACE SAP3 180 C NONSAP - FLAG INDICATING FORHAT SAP3 190 C = 1 FOR 'NONSAP* OR * ADINA' FORMAT SAP3 200 C = F FOR 'STATIC-SAP' FORMAT SAP3 210

11

f* w NU BEL - NUMBER CF ELEHENTS SAP3 220 c NUHF - NUMBER CF FACE WHICH IS LOADED SAP3 230 c NUHLTH - HAXIHUM SUHBER OP PRESSURE DISTRIBUTIONS SAP3 240 c FOR AN? LCAD CASE SAP3 250 c NUNNP - NUMBER CF NODES SAP3 260 c XY (I,J) - * I'TH CCCBDINATE OF THE 'J'TH NODE SAP3 270 c SAP3 280 c 290

c c c

1SFLICIT REAl»e (A-H.C-Z) SAPS LOGICAL NONSAP,OUTPUT SAP3 DIME NSICN FP (3(8) , NODE (8) , IDGENI(U), SAP3 1 I06EN2 (4) , NZEBO(3), IH(I(4), NOF(8,6) SAP3 DIMENSION F (NIOAD, 3,NUHNP) , 5AP3 1 FSUM (3.NLOAD), XY (3,NUMNP) , PBESR (8, NUBLTM) , LNODE (21, NUMEL) , SAP3 2 NC(NLCAD) SAPS DATA IDGEN1/1, 2 ,3,4/ ,IDGEN2/2,3, 4,1/,NZERO/5,6,7.8/,IHH/7,8,5,6/ DATA NOI/1,4,8,5,12,20,16,17,

1 1,5,6,2,17,13,18,9, 2 1,2,3,4,9, 10,11,12, 3 7,3,2,6,19,10,18,11, 4 7,8,4,3,15,20,11,19, 5 7,6,5,8,14,12,16,15/

INEUT NCDAL COORDINATES

BEAD (21,5000) ( (XY (J,I) , J=1,3) ,I = 1,N0HNP) 5000 FOFMAT (35X,3F10.0»

INPUT ELEMENTS

IF (.NOT. NONSAP)READ (21,5010) ((LNODE (J,I) ,J=1,20) ,1=1,NOBEL) 5010 FOBHAT(/20I4)

IF (NCNSAP)READ (21,5020) ( (LNODE(J,I) ,J=1,20),1=1,NUHEL) 5020 FORMAT (/8I5/12I5)

C c c

SAP3 SAPS SAP3 SAP3 SAPS SAPS SAPS SAPS SAP3 SAPS SAPS SAPS SAPS SAP3 SAP3 S A 53 SAP3 SAP3 SAP3

C * * * » > . * • * * » » » * » » » * * » » » » » » * » » » » » » * « * » * * » « » » » » * » » » » » » » • » » » * » • » » » » » » # » 5 j p 3 C LOOP OVER ALL 1CAD CASES SAF3

10

20 C c c c c c c

DO 110 LOAD = I.NLOAT DO 10 I = 1 <NUMNP DO 10 J = 1,3 F (LOAD,J,I) = O.ODO DO 2 0 J = 1,3 FSUfl (J , LOAD) = O.ODO

INPUT HFACE - NC. OP EI.EHENT FACES UNDER PRESSURE INT - ORDER OF INTEGRATION (DEFAULT = 3) NCOS - NC. CF LOAE CURVE (USED AS INPUT TO NONSAP) HUMLI - NC. OF PRESSURE DISTRIBUTIONS

READ (21,5030)NFACE,INT,NCUfl,NUHLT 5030 FOBHAT (415)

IF (NUML1 -LE. NUHLTH)GO TO 30 TYPE 5010,LOAD

5010 FOBHAT (1H0,"»*» ERROR »»« TOO MANY PRESSURE', 1 • DISTRIBUTIONS SPECIPIED FOR LOAD CASE*,It) NC (LOAD) = NCUE IF(INT .E0- 0)INT = 3

INPUT NODAL VALUES OF PRESSURE DISTRIBUTIONS

BEAD(21,5050) ((PRESR (I,J) ,1=1,8) ,J=1,NUHLT) 5050 FORHAT(8F10.0)

DO 90 I = 1,NPACE

INPOT LM - NUflEER OF ELEMENT HHICH XS LOADED NUMF - NUMEER OF FACE HHICH IS LOADED LCDTYP - NUREER OF PRESSURE DISTRIBUTION (DEFAULT = 1)

30

C c c

c c c c c

SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 S i \P3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAPS SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAP3 SAPS SAP3 SAP3 SAP3 SAP3 SAP3

300 310 320 321 330 331 332 340 350 351 352 353 3 5 4 355 360 370 380 390 <100 m o 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 681 690 700 710 720 730 740 750 760 770 780 790 800 810

12

READ (21,5060) 1t, NtJHF ,LODTYP 5060 FORMAT (315)

IF (LODTYF .EC- 0)LODIYP = 1 IF (IB.LH.NOBEL .AND. SUHF.LE.6 . AHD.

1 LODTYP» LE.NCflLT)GC 10 40 TYPE 5070,1.(1, NCEF,LOCTYP

5070 FORMAT 1 1 H 0 E B B O F ***' ,315) STCP DO 50 J = 1,8 40

50 C C c

NODE (J) = LNCDE(KOF(J,HUHF),LH)

CHECK FCR DEGENERATE EDGES

-OR. NODE (IDGEN2 (11) ) ) ) GO TO 60

12 = 0 IDGEN = 0 DO 60 II = 1,1 IF ( (NODE (IHH (11) ) .EC. 0)

1 (NODE(IDGEN1(II)) .NE. NODE (NZERO(11)) = 0 1 2 = 1 2 + 1 IDGEN = II

60 CONTINUE IF (12 .II. 2)GC TO 7C TYPE 5080,NUKF.IH,NODE

50a0 FORMAT (1H0,**** EHROB *»» FACE',I2,» OF ELEMENT*,14, 1 • HAS TOO fl ANY DEGENERATE N0DES*/8I5) STOP

C C C

CAICOLATE NODAI LOADS CN THIS FACE

70 CAIL FOBCE(FP,JY,NODE,INT,ERESR(1,LCDTYP) ,IDGEN)

ADD CONTRIBUTION OE TEIS FACE C c c

DO 80 J = 1,8 NODEI = NODE(J) IF(SCDEI -HQ. 0)GO TO 80 F(LOAD,1,NODEI) = F(LCAD,1,NODEI) • FP(1,J) F(LOAD,2,NODEI) = FfICAD,2,NODEI) • FP(2,J) F(LCAD,3,NODEI) = F(LCAD,3,NODEI) • FP(3,J)

80 CONTINCE 90 CONTINUE

IF (.NOT. NONSAf)GO TC 110 C **»»»**#**»******»*****»«*****»*.»*«<.»*** *»»»*»*»*•*»*»**** C PUNCH CARDS FOR USE AS INPUT TC •NCNSAP* OR •ADINA' C •»•***»*»*••****•***•***»******•***«*•*•••••***•*•«•*

12 = 0 DO 100 II = 1,NUHNP DO 100 J = 1,3 FSUH (J,LOAD) = FSUH (J,LOAD) «• F(LOAD,J,I1) IF (DABS (F(LOAD,J,I1) ) .LE. 5.0D-2JGOTO 100 1 2 = 1 2 • 1 HRITE(22,5090)I1,J,NC (LOAD),F(LOAD,J,11)

5090 FORMAT(315,F10.1) 100 CONTINUE

TYPE 5100,12,NC (LOAD),(F50M (J, LOAD) ,J=1,3) 5100 FORMAT(1HO,15,* LOAD CARDS FOR LOAD CUBV^'.IS/

1 1H ,«SUMJt =',F14.1,» SUHY =",F14.1,* SUBZ =• 110 CONTINUE

IF (NCNSAP) RETURN Q a * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * • « * « * * * * * * • • * * * * « « * • * * *

C PUNCH CARDS FOB USE AS INPUT TO 'STATIC-SAP*

12 = 0 DO 130 I = 1,NQHNP DO 130 LOAD = 1 ,NL0AC DO 120 J = 1,3

SAP3 820 SAP3 830 SAP3 840 SAP3 850 SAP3 860 SAP3 861 SAP3 870 SAP3 880 SAP3 890 SAP3 900 SAP3 910 SAP3 920 SAP3 930 SAP3 9U0 SAP3 950 SAP3 960 SAP3 970 SAP3 980 SAP3 S81 SAP3 990 SAP 1000 SAE 1010 SAP 1020 SAP 1030 SAP 1040 SAP 1050 SAP 1051 SAP 1060 SAP 1070 SAP 1080 SAP 1090 SAP 1100 SAP 1110 SAP 1120 SAP 1130 SAP 1140 SAP 1150 SAP 1160 SAP 1170 SAP 1180 SAP 1190 SAP 1200 SAP 1210 SAP 1220

*»*****SAP 1230 SAP 1240

•»»»»**SAP 1250

SAP 1260 SAP 1270 SAP 1280 SAP 1290 SAP 1300 SAP 1310 SAP 1320 SAP 1330 SAP 1340 SAP 1350 SAE 1360

,F14.1)SAE 1361 SAP 1370 SAP 1380

• • • • » * * S & p 1 3 9 0 SAP 1400

***»***SAP 1410 SAP 1420 SAP 1430 SAP 1440 SAP 1450

13

120 FSOH(J,lCAD) = FSCH (J,LOAD) + F{LOAD,J,I) SAP 1060 OOTPOT=.FALSE. SAP 1470 IP (DABS (F(LOAD,1,1)) . GT.5.0D-2 .OB. SAP 10R0

1 DABS ( F ( L O A D , 2 , I ) ) . G T . 5 . 0 D - 2 . O R . SAP 1 0 8 1 2 DABS (F(LOAD,3,I)).GT.5.0D-2)0UTP0T=.TR0E. SAP HI82 IF (OOTPOT) -5AP 1090 1 HBITE (22,5110) I,NC (ICAD),F(LOAD,1,1),F(LOAD,2,I),F(LOAD,3,I) SAP 1091

5110 FOPHAT (2l5,3r1C.1) SAF 1500 IF(OOTFOT)12 = 12 • 1 SAP 1510

130 CONTINOE SAP 1520 TYPE 5120,12, (HC<J), (FSOM(I,J) ,1 = 1,3),J=1,NL0AD) SAE 1530

5120 FORHAT (1H0,'TOTAL NOBIEB 0? LOADED NODES =',I5// SAP 1500 1 1HO,'LO»E CASE SUMX SOHY S0HZ<//SSP 1501 2 < 1H ,19,3F10.1)) SAP 1502 EETOBN SAP 1550 END SAP 1560

SOBROOTINE FORCE (F,XY,BODE,INTrPRES,IDGEN1 FORCE 10 c •**••*»•»••*•*••*•**•••**•*•••*•***••a*********************»*»****FORCE 20 C FCBCE 30 C S0EBC0T1NE TO CALCOLATE NODAL LOADS EOE TO PRESSORE ON THE FOBCE 00 C SOBFACE OF A 3-D ISOEARABETRIC ELEHENT OP THE 'SERENDIPITY* FOBCE 50 C FARILY FOBCE 60 C FOPCE 70 C A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i » i * f o r c e no

IN ELICIT BEAL»8 (A-H,C-Z) FCBCE 90 BEAL*8 J FOBC 100 DIHFNSICN XY (3,1) ,F (3,8) , H (9) , P(8, 2) . FORC 110 1 J(3,2),PL(3),PRES(8),NODE(8) FOBC 111 DO 10 11 = 1,8 FORC 120 DO 10 12 = 1,3 FOBC 130

10 F(1 2 , 11) = O.OrO FCBC 100 DO 30 I NT 1 = 1.INT FCBC 150 CALL GAOSS (B,W1 ,INT,INT1) FOBC 160 DO 30 INT2 = 1,INT FOBC 170 CALL GACSS (S,H,INT,INT2) FOBC 180

C FOBC 190 C CALCULATE JACOEIAN * J * UNO SHAFE FUNCTIONS 'H' FOBC 200 C FOBC 210

CALL JACCB(R,S,H,P,J,XY,NOCE,IDGEN) FOBC 220 FLCAD = O.ODO FCBC 2 30 CO 2C I = 1,8 FOBC 200 IF (NODE (I) .NE. 0) FLCAE = FLOAD • H(I)*PRES(I) FCEC 250

20 CONTINOH FOBC 260 *F = W*«1*PLCAC FOfiC 270 FL (1) = (J{3,1)*J(2,2)-J(2,1)*J(3,2)) F0BC2fl0 PL (2) = (J(1,1)»J(3,2)-J(3,1)»J(1,2)) FOBC 290 PL (3) = (J(2,1)»J<1,2)-J(1,1) *J(2,2)) FCBC 300 DO 30 11 = 1,8 FOEC 310 IF (NODE (11) -EC- 0)GO TO 30 FOFC 320 DO 30 12 = 1,3 FOBC 330 F ( 12,11) = H (I1)*PL(I2)*VF • F(I2,I1) ?OiiC 300

30 CONTINOE -OBC 350 BETDBN FCBC 3*0 ENC FORC 370

SOBBOOTINE JACCE (B,S,H,P,J,XY,NODE,IDGEN) JACOB 10 c ************************* ********************************** »**»***JAC0B 20 C JACOB 30 C SUBROUTINE TO CALCULATE SHAPE FUNCTIONS, PABTIAL DERIVATIVES JACOB 00 C OF SHAPE FDNCTICNS, AND THE JACOBIAN AT THE POINT (R,S) JACOB SO C JACOB 60 C * * * * * * * * * * * * * * » » * * * * • » « * * • * * * * * * * * * * * * * * * * * * * * • * * * * * » * * * * * * * « * » » » » J A C O B 70

14

c c c

10

20

30

C C C

c c c

1

UO

50

60

70 eo

IMPLICIT REAL*8 (A-H,C-Z) JACOB 80 BEAL*8 J JACOB 90 DIMENSION H<8),P (8, 2) , J (3, 2),XY(3,1),NODE (8), JACO 100

IHH(4),HHOD (4),IHOD(4) ,IEFECT (2,«) JACO 101 EQUIVALENCE (HHOD (1),SH),(HHOD(2),RF),(HHOD(3) ,SP),(HnOD (4 ),BH) JACO 110 DATA IHH/7,8,5,6/,IHCE/2,1,2,1/,IEFECT/1,2,2,3,3,4,4,1/ JACO 120 RP s 1.0D0 • R JACO 130 BM = 1.0D0 - R JACO 140 B2 = 1.0D0 - R*R JACO 150 SP = 1.0D0 • S JACO 160 sn = 1.0D0 - S JACO 17C S2 = 1.0D0 - S«S JACO 180

FUNCTIONS 1 - <1

R(1) = 0.25DC*5F*SP H(2) = 0.25D0*RH*SP H (3) = 0.25D0*BH*-SH H(4) = 0.25D0»BF»SH P(l,1) = 0.25DCSP P(2,1) = -P(1,1) P(3,1) = -0.25t0*SH P(4,1) = -P(3,1) P(1,2) = 0.25DO*RP P (2,2) - 0.25D0«RH P(3,2) = -P(2,2) F(4,2) = -P (1 ,2) IF(NODE (5).EQ.0) GO IC 10 H(5) = 0.5DO+R2»SP P (5, 1) = -R*SP P(5,2) = 0.5C0*B2 IF (NODE (6).EC.0) GO TC 20 H(6) = C.5D0*RH*S2 P(6,1) = -0.5DOS2 F (6,2) = -S»BM IF (NODE (7).EQ.0) GO TC 30 H(7) = C.5D0*R2*SM P(7,1) = -R*SN P(7,2) = -0-5CC*82 IF(NODE(8).EC.0) GO IC 00 H(8) = 0.5D0+RP+S2 P (8,1) = 0.5t0»S2 P (8,2) = -S*RP

HOCIFT FUNCTIONS 5 - 8 IF ELEMENT IS DEGENERATE

IF (IDGEH .EQ. C)GO TC 60 IH = IHH(IDGEN) NMCD = IMOD (ICSEN) TEMP = HHOD(I3GEN) *0.EDO H (IH) = TEHP*B(IH) DO 50 12 = 1,2 P (IH,I2) = TEME*P(IH,I2) P (IH,NHOD) = 2.0D0*P (1H,NH0D)

MODIFY FUNCTIONS 1 - 4 IF ANT OF NODES 5 - 8 ARE PRESENT

DO 80 II = 5,8 IF (NODE (II) .EC. 0)GC TO 80 DO 70 12 = 1,2 IH = IEFECT(12,11-4) H(IH) = H(IK) - 0.5D0*F(II) DO 70 13 = 1,2 P(IH,I3) = P(IH,I2» - 0.5DO*P(11,13) CONTINUE

C C C

EVALUATE JACCBIAN AT (R,S)

JACO 190 JACO 200 JACO 210 JACO 220 JACO 230 JACO 240 JACO 250 JA 0 260 JACO 270 JACO 280 JACO 290 JACO 300 JACO 310 JACO 320 JACO 330 JACO 340 JACO 350 JACO 360 JACO 370 JACO 380 JACO 390 JACO 400 JACO 410 JACO 420 JACO 430 JACO 440 JACO 450 JACO 460 JACO 470 JACO 480 JACO 490 JACO 500 JACO 510 JACO 520 JACO 530 JACO 540 JACO 550 JACO 560 JftCO 570 JACO 580 JACO 590 JACO 600 JACO 610 JACO 620 JACO 630 JACO 640 JACO 650 JACO 660 JACO 670 JACO 680 JACO 690 JACO 700 JACO 710 JACO 720 JACO 730 JACO 740

15

DO 1 1 0 1 1 = 1 , 3 JACO 7 5 0 DO 110 12 = 1 , 2 JACO 7 6 0 TE HE = O.ODO JACO 7 7 0 DO 9 0 1 3 = 1 , 4 JACO 7 8 0

9 0 TEHP = TEKP • X 1 ( I 1 , N C D E ( I 3 ) ) * P ( I 3 , I 2 ) JACO 7 9 0 DO 1 0 0 1 3 = 5 , 8 JACO 8 0 0 I F ( N O D I ( I 3 ) . N E . O ) T E H E = TEHP • X Y ( 1 1 , N O D E ( 1 3 ) ) * P ( 1 3 , 1 2 ) JACO 8 1 0

1 0 0 CONTINUE JACO 8 2 0 1 1 0 J ( 1 1 , 1 2 ) = TEHE JACO 8 3 0

BETUBN JACO 8 4 0 END JACO 8 5 0

C C C C C C

SUBROUTINE GAUSS ( C , W , I N T , I E ) GAUSS GAUSS GAUSS GAUSS GAUSS GAUSS

GAUSS POINT

PCINT ANE WEIGHTING NUHEER ' I E *

VALUE FOR INTEGRATION ORDER ' I N T * AT

I H F L I C I T BEAL*8 DIMENSION DATA WT/

( A - H . C - Z ) H T ( S , 9 ) ,

2 . 0 D 0 , 2 * 1 . 0 D 0 ,

. 5 5 5 5 5 5 5 5 5 5 5 5 5 S 6 D O , 6*C .ODO,

. 3 4 7 8 5 4 8 4 5 1 3 7 4 5 4 D 0 , . 3 4 7 8 5 4 6 4 5 1 3 7 4 5 4 D 0 , . 2 3 6 9 2 6 E 8 5 0 5 6 1 8 9 D 0 , . ' 1 7 8 6 2 8 6 7 0 4 9 9 3 6 6 D 0 , . 1 7 1 3 2 4 4 9 2 3 7 9 1 7 0 D 0 , . 4 6 7 9 1 3 9 3 4 5 7 2 6 5 1 D O ,

3 * 0 . 0 D 0 , . 1 2 9 4 8 4 9 6 6 1 6 8 8 7 0 D 0 , • 4 1 7 9 5 9 1 8 3 6 7 3 4 6 9 D 0 , - 1 2 9 4 8 4 9 6 6 1 6 8 6 7 O D O , . 1 0 1 2 2 8 5 3 6 2 9 0 3 7 6 D 0 , - 3 6 2 6 6 3 "783378 36 2D0 , . 2 2 2 3 8 1 C 3 4 4 5 3 3 7 4 D 0 , . 0 8 1 2 7 4 3 8 8 3 6 1 5 7 4 D O , . 3 1 2 3 4 7 C 7 7 C 4 0 0 C 3 D O ,

AB ( 9 , 9 ) 8 * 0 . O D O , 7 * 0 . 0 D 0 ,

. 8 3 8 8 8 8 8 8 8 8 8 8 8 8 9 D O ,

. 6 5 2 1 4 5 1 5 4 8 6 2 5 4 6 D 0 , 5 * 0 . O D O ,

. 4 7 8 6 2 8 6 7 0 4 9 9 3 6 6 D 0 , . 2 3 6 9 2 6 8 8 5 0 5 6 1 8 9 D 0 , . 3 6 0 7 6 1 5 7 3 0 4 8 1 3 9 0 0 , . 3 6 0 7 6 1 5 7 3 0 4 8 1 3 9 D 0 ,

. 2 7 9 7 0 5 3 9 1 4 8 9 2 7 7 D O , . 3 8 1 8 3 O O 5 0 5 C 5 1 1 9 D O ,

2 * 0 . O D O , . 2 2 2 3 8 1 0 3 4 4 5 3 3 7 4 D O , . 3 6 2 6 8 3 7 S 3 3 7 8 3 6 2 D 0 , . 1 0 1 2 2 8 5 3 6 2 9 0 3 7 6 D O , . 1 8 0 6 4 8 1 6 0 6 9 4 8 5 7 D O , . 3 3 0 2 3 9 3 5 5 0 0 1 2 6 0 D 0 , . 1 8 0 6 4 8 1 6 0 6 9 4 8 5 7 D O ,

5 7 7 3 5 0 2 6 9 1 8 9 6 2 6 D 0 , O.ODO,

2 6 0 6 1 0 6 9 6 4 0 2 9 3 5 D 0 , DATA HE/ 9*C .ODO,

- u 5 7 7 3 5 0 2 6 9 1 8 9 6 2 6 D 0 , - . 7 7 4 5 9 6 6 6 9 2 4 1 4 e 3 D O ,

6 * 0 . O D O , - . 8 6 1 1 3 6 3 1 1 5 9 4 C 5 3 D O , - . 3 3 9 9 8 1 0 4 3 5 8 4 8 5 6 0 0 ,

. 8 6 1 1 3 6 3 1 1 S 9 4 C 5 3 D 0 , 5 * 0 . O D O , - . 9 0 6 1 7 9 C U 5 9 3 8 6 6 4 D 0 , - . 5 3 8 4 6 9 3 1 0 1 0 5 6 8 3 D 0 ,

. . 5 3 8 4 6 9 3 1 0 1 0 5 6 e 3 D 0 , . 9 0 6 1 7 9 8 4 5 9 3 8 6 6 4 D 0 , 9 3 2 4 6 9 5 1 4 2 0 3 1 E 2 D 0 , - . 6 6 1 2 0 9 3 8 6 4 6 6 2 6 5 D 0 , •

. . 2 3 8 6 1 9 1 8 6 0 8 3 1 S 7 D 0 , . 6 6 1 2 0 9 3 8 6 4 6 6 2 6 5 D 0 , 3 * 0 . O D O ,

GAUSS GAUSS GAUS GAUS

. 5 5 5 5 5 5 S 5 5 5 5 5 5 5 6 0 0 , G A U S GAUS

. 6 5 2 1 4 5 1 5 4 8 6 2 5 4 6C0,GADS GAUS

. 5 6 8 8 8 8 8 8 8 8 8 8 8 8 9 D 0 , G A U S 4*0 .OBO,GAUS

. 4 6 7 9 1 3 9 3 4 5 7 2 6 9 1 D 0 , GAUS , 17 13 24 4 S2379170DO,GAUS

GAUS . 3 8 1 8 3 0 0 5 0 5 0 5 1 1 9 E 0 , G A U S . 2 7 9 7 0 5 3 9 1 4 8 9 2 7 7 D 0 , G A U S

GAUS . 3 1 3 7 0 6 6 4 5 8 7 7 8 8 7 D 0 , G A U S . 3 1 3 7 0 6 6 4 5 8 7 7 8 8 7 D 0 , G A O S

0.OCO,GAUS . 2 6 0 6 1 0 6 ^ 6 4 0 2 9 3 500 ,GAUS . 3 1 2 3 4 7 0 7 7 0 4 0 0 0 3 E 0 , G A U S . O 8 1 2 7 ' » 3 8 8 3 6 1 5 7 « E 0 / G A 0 S

GAUS 7*0 .ODO,GAOS

. 7 7 4 5 9 6 6 6 9 2 4 1 4 8 3 D 0 , G A U S GAUS

• 3 3 9 9 8 1 0 4 3 5 8 4 8 5 6 D C , G A U S GAUS

0 . 0 D 0 , G A U S 4*0 .ODO,GAUS

• • 2 3 8 6 1 9 1 8 6 0 8 3 1 9 7 D 0 , G A U S . 9 3 7 . 4 6 9 5 1 4 2 0 3 1 5 2 B 0 , G A U S

GAUS 9 4 9 1 0 7 9 1 2 3 4 2 7 E 9 D 0 , - . 7 4 1 5 3 1 1 6 5 5 9 9 3 9 4 D 0 , - . 40 58 4 5 1 5 1 3 7 7 3 9 7 0 0 , GADS

O.ODO, . 4 0 5 8 4 5 1 5 1 3 7 7 3 9 7 D 0 , . 7 4 1 5 3 1 1 8 5 5 9 9 3 9 4 C 0 , G A U S . 9 4 9 1 0 7 9 1 2 3 4 2 7 5 9 D 0 , 2 * 0 . O D O , GAUS

9 6 02 8 9 8 56 4 9 7 5 3 6D 0 , - . 7 9 66 66 4 7 7 4 1 3 6 2 7 U 0 , - . 5 2 55 3 2 4 0 9 9 1 6 32 9 DO,GAUS - . 1 8 3 4 3 4 6 4 2 4 9 5 6 5 0 D 0 , . 1 8 3 4 3 4 6 4 2 4 9 5 6 5 0 D 0 , . 5 2 5 5 3 2 4 0 9 9 1 6 3 2 9 E 0 , G A U S

. 7 9 6 6 6 6 4 7 7 4 1 3 C 2 7 D 0 , . 9 6 0 2 8 9 8 5 6 4 9 7 5 3 6 D 0 , O.ODO,GAUS 9 6 8 1 6 O 2 3 9 5 0 7 6 2 6 D 0 , - . 8 3 6 0 31 1 O 7 3 2 6 6 3 6 D O , - . 6 1 3 3 7 1 4 3 2 7 O 0 5 9 0 C 0 , G A U S

- . 3 2 4 2 5 3 I I 2 3 4 C 3 8 C 9 D 0 , O.ODO, . 3 2 4 2 5 3 4 2 3 4 0 3 8 0 9 D C , G A U S . 6 1 3 3 7 1 4 3 2 7 0 0 5 S O D O , - 8 3 6 0 3 1 1 0 7 3 2 6 6 3 6 D 0 , „ 9 6 8 1 r t 0 2 3 9 5 0 7 6 2 6 C O / G A U S

GAUS GAUS GAUS GAUS GAUS

C = A B ( I P , I N T ) W = NT ( I P , I N T ) RETURN END

10 20 3 0 40 5 0 60 7 0 80 90

100 101 102 1 0 3 104 1 0 5 1 0 6 107 1 0 8 109 1 0 9 1 0 9 1 0 9 109 109 1 0 9 109 1 0 9 109 109 110 111 112 1 1 3 114 1 1 5 116 117 118 119 119 119 119 119 1 1 9 119 119 119 1 1 9 119 120 130 140 150 160

16

CONCLUSION

SAP3PR provides an easy way of calculating equivalent nodal loads resulting from distributed pressure loadings. The code can be easily modified to calculate nodal loads for elements with a nodal ordering which differs from that used by ADINA, NONSAP, or STATIC-SAP.

REFERENCES

1. Klaus-Jurgen Bathe, E. L. Wilson, and R. H. Iding, NONSAP: A Struc-tural Analysis Program for Static and Dynamic Response of Nonlinear Systems, Report UC SESM 74-3, University of California, Berkeley (February 1974).

2. Klaus-Jiirgen Bathe, ADINA: A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis, Report No. 82448-1, Massachu-setts Institute of Technology (1975).

3. Finite Element Analysis of Mine Structures, Final Report to Denver Mining Research Center, U.S. Department of the Interior, Bureau of Mines, Contract H0110231 (September 1972).

4. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science, p. 22, McGraw-Hill, London, 1971.

17

APPENDIX

This appendix contains an example of the input necessary to run SAP3PR and the resulting output. The finite-element model used for this example consists of four 20-node elements as shown "n Fig. A.l. Equiva- FA.l lent nodal loads are calculated for two load cases - e for a uniform pressure of 1000 in the negative x-direction and tb-. other for a pressure in the negative y-direction which varies linearly with respect to the z-direction (Fig. A.2). FA.2

The input is shown first followed by the output to the teletype and disk file FOR22.DAT. [Note that the NONSAP = T format option is being used.] The input for load case 1 requires only one pressure card since the pressure on each element face is the same constant 1000. Load case 2 needs four pressure cards because the pressure distribution on all four elements differ. The summation of forces given by the teletype output is useful as a check on the input data. The load cards are out-put on unit 22.

18

ORNL IJWCi / / 1444/

Fig. A.l. Finite-element model of example structure.

i >1 : \ I l nV . < I Vi/l/l

1000

I O A n CASt 1 I O A O t :A5f ?

4000

Fig. A.2. Pressure loadings on example structure.

19

t X A H P L E O P I S P O T

56 4 2 1 1 0 2 1 1 3 0 1 4 0 0 c 1 0 6 0 1 7 0 0 8 0 0 9 1 0

10 1 1 11 0 1 12 0 0 13 1 0 14 1 1 15 0 1 16 0 0 17 1 0 18 0 1 19 0 0 20 0 0 21 1 0 22 1 1 23 0 1 24 0 0 25 1 0 26 1 1 27 0 1 28 0 0 29 1 0 30 0 1 31 0 0 32 0 0 33 1 0 34 1 1 35 0 1 36 0 0 37 1 0 38 1 1 39 0 1 40 0 0 41 1 0 42 0 1 43 0 0 44 0 0 45 1 0 46 1 1 47 0 1 48 0 0 49 1 0 50 1 1 51 0 1 52 0 0 53 1 0 54 0 1 55 0 0 56 0 0 1 1 2 3 5 6 7 2

4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 . 0 0 . 0 1.0 1.0 0 . 0 0 . 5 1.0 0 . 5 0 . 0 0 . 0 1 - 0 1.0 0 . 0 0 . 0 1.0 1.0 0 . 0 0 . 5 1.0 0 . 5 0 . 0 0 . 0 1.0 1 . 0 0 . 0 0 . 0 1.0 1.0 0 . 0 0 . 5 1.0 0 . 5 0 . 0 0 . 0 1 . 0 1 . 0 0 . 0 0 . 0 1 - 0 1.0 0 . 0 0 . 5 1 . 0 0 . . 5 0 . 0 0 . 0 1..0 1.0 0 . 0 0 . 0 1.0 1 . 0 0 . 0 0 . 5 1.0 0 . 5

1.0 4.0 0.0 4.0 0.0 4.0 1.0 4.0 0.5 4.0 0.0 4.0 0.5 4.0 1.0 4.0 1.0 3.5 0.0 3.5 0.0 3.5 1.0 3.5 1.0 3.0 0.0 3.0 0.0 3.0 1.0 3.0 0.5 3.0 0.0 3.0 0.5 3.0 1.0 3.0 1.0 2.5 0.0 2.5 0.0 2.5 1.0 2.5 1.0 2-0 0.0 2.0 0.0 2.0 1.0 2.0 0.5 2.0 0 . 0 2.0 0.5 2.0 1.0 2.0 1.0 1.5 0 . 0 1.5 0 . 0 1.5 1.0 1.5 1.0 1.0 0 . 0 1.0 0 . 0 1.0 1.0 1.0 0.5 1.0 0 . 0 1.0 0.5 1.0 1.0 1.0 1-0 0-5 0 . 0 0.5 0 . 0 0.5 1.0 0.5 1.0 0 . 0 0 . 0 0.0 0 . 0 0 . 0 1.0 0 . 0 0.5 0 . 0 0 . 0 0 . 0 0.5 0 . 0 1.0 0.0

1 13 14 17 18

1

15 19

16 20 10 11 12

20

13 11 15 16 25 26 27 28 17 18 19 50 29 30 31 32 21 22 23 24 3 1 1

24

25 26 27 28 37 38 39 40 29 30 31 32 41 42 43 44 33 34 35 36

4 1 1 37 38 39 40 49 50 51 52 Ul 42 43 44 53 54 55 56 45 46 47 48

4 2 1 1 1000 .0 1000.0 1000.0 1000 . 0 1000.0 1000 . 0 1 5 1 2 5 1 3 5 1 a 5 1 <4 2 5 4

0 . 0 0.0 1000.0 1000 . 0 0.0 500 . 0 1000 .0 1000.0 200C.0 2000 . 0 1000.0 1500 . 0 2000 . 0 2000.0 3000.0 3000 . 0 2000.0 2500 . 0 3000 .0 3000.0 4000.0 4000 . 0 3000.0 3500 . 0 1 1 1 2 1 2 3 1 3 4 1 4

1000.0

1000.0 2000.0 3000.0 4000.0

1000.0

500.0 1500.0 2500.0 3500.0

21

2XAHPLE OF TELETYPE OUTPDT

HTOT »0ST EE AT LEAST 1170

23 LOAD CAFDS FCR LOAD COBVE 1 SOI!X = -UCOO.O SOB? = - 0 . 0 SOMZ

23 LOAD CASTS FCR LCAC COBVE S SOHX = - 0 . 0 SOHY = - 8 0 0 0 - 0 SDH Z

22

EXAMPLE OP UNIT 22 OUTPUT

3 1 1 83.3 1 1 €3.3

7 1 1 -333.3 11 1 1 -333.3 12 1 1 -333-3 15 1 1 166.7 16 1 1 166.7 19 1 1 -666.7 23 1 1 -233.3 24 1 1 -333.3 27 1 1 166.7 2 e 1 166.7 31 1 1 -€66.7 35 1 1 -333.3 36 1 1 -233-3 3? 1 1 166.7 40 1 1 166..7 13 1 1 -666.7 47 1 1 -333.. 3 «8 1 1 -333.3 5", 1 1 83.3 52 1 1 83.3 55 1 1 -333.. 3 1 2 5 55» 6 4 2 5 55.. 6 8 2 5 -111. i '•) 2 5 -166.7 12 2 5 -166.7 13 2 5 166.7 16 2 5 166.7 20 2 5 -666.7 21 2 5 -500.0 24 2 5 -500.0 25 2 5 333. 3 28 2 c 233.3 32 2 5 -1233.. 3 33 2 5 -833.3 36 2 5 -633.3 37 2 5 500. C 40 2 5 500.0 44 2 5 -2COO.O 45 2 5 -1166.7 48 2 5 -1166.7 49 2 5 277.8 52 2 5 277.. B 56 2 5 -1222.2

sap3pr: a fortran program for calculating equivalent nodal

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