THE UN IVER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 11 DIGITAL COMPUTATION OF TWO-PLY ELASTIC CHARACTERISTICS N. L. Field R., N.- Dodge:~ B.i:,,/er'zog i- ~.; t'~; s -S.: K.., k "'k,' Project Directors: I4 K.",X0G-rk ana R. A. Dodge ORA Project 02957 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1961

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The Tire and Suspension Systems Research Group at The University of Michigan is sponsored by: FIRESTONE TIRE AND RUBBER COMPANY GENERAL TIRE AND RUBBER COMPANY Bo F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UTNITED STATES RUBBER COMPANY iii

TABLE OF CONTENTS Page LIST OF FIGURES vii NOMENCLATURE ix I. FOREWORD 1 II. SUMMARY 3 III, PHYSICAL CONSIDERATIONS 5 IV. EQUATIONS GOVERNING THE ACTION OF A TWO-PLY LAMINATE 7 V. DIGITAL COMPUTER PROGRAM FOR THE SOLUTION OF THE ELASTIC EQUATIONS 13 VI. EXTENSION TO MULTI-PLY LAMINATES 39 VIII. REFERENCES 43 IX. DISTRIBUTION LIST 45

LIST OF FIGURES Figure Page 1. Two plies at right angles. 11 2. Two plies at right angles. 12 3. Flow diagram of computer program. 16 vii

NOMENCLATURE English Letters: aij Elastic constants dependent on both cord angle and elastic properties. E,F,G Elastic constants of a single sheet, Greek Letters: Strain, cx Stress. Subscripts: x,y Co-ordinate directions parallel and perpendicular to the cords. Co-ordinate directions along the bisectors of angles between the cords. Superscripts: *,~** Indicating the actual stress carried by the first and second plies in a two-ply laminate, respectively. Indicating interply stress. +,++ Indicating the sum of externally applied stress and interply stress carried by the first and second plies, respectively, ix

Io FOREWORD The study of the elastic interaction of two similar plies bonded together can be carried on in a very simple way, as indicated in previous reports. When the two plies which are bonded together become dissimilar, as occurs when the cords in one ply are unstressed or in compression, while the cords in the other ply are in a state of tension, then this study becomes considerably more complicated and is generally not possible to do efficiently by hand calculation. As part of the general research effort of this group, it is necessary to determine the important characteristics in the elastic response Of two dissimilar bonded plies, For that reason, it is necessary to consider the solution of the equations governing this structure in some detail, It was not originally intended to present the methods of arriving at conclusions concerning the actions of such dissimilar plies but rather only to present the conclusions themselves, Some interest has been shown in the details of these solutions and for this reason they are presented here, They are not intended at this time to be anything other than a research tool to aid in the study of the load-carrying characteristics of such a structure. 1

II. SUMMARY The elastic action of two dissimilar plies bonded together in a single laminate may be completely described by nine simultaneous linear algebraic equations in nine unknowns. Since solving these equations would not be easy by hand methods, a digital computer program was constructed, permitting us to solve these equations numerically, and inexpensively and quickly. This report presents the details of that program, along with a sample solution illustrating the way the program was actually used,

III. PHYSICAL CONSIDERATIONS The physical effects which take place when two dissimilar plies are bonded together and loaded are discussed in Refo 1. That discussion is also pertinent to this report since, from it, one may see at once that the application of a single load will result in the presence of two interply stresses, as well as general deformation, This means, of course, that normal and shearing effects are completely coupled in such a structure. When the two plies making up a laminate are identical in their properties, shear and normal effects are no longer coupled together along principal, or orthotropic, axes, In these directions, the application of load or stress results in extension without distortion and vice versa. Here, the situation becomes physically quite simple.3

IV. EQUATIONS GOVERNING THE ACTION OF A TWO-PLY LAMINATE The equations governing the action of two bonded dissimilar plies were given as Eqs. (22) of Ref. 2~ These will be repeated here with a slight notation change for completeness. Ply 1 E = [ajj(*Y)]a' + [al2(Y+) ]C+ + [al3(+a ) ]a+ r = [a2l(+cx)]a + [a22(+X) ]4+ + [a23(+o) ]atr = [a31(+oz)Ia + [a32(+o~) ]a + [a33(+oa) ]aCr Ply 2 - = [a'l(-c) ] ]++$-[a2(-Oa) ]a ++[ai3(-c) ]a"++ f, = [a'1(- a) 1re +[a22( -a) gac ++[a23(-X) ]a +j (1) -= [atl(-cO) ]r++ [a 2(-o) ]a +[a33() ]C+ Equations Linking Both Plies 2ao = + cx+ = 2cx = r + + Ci++ -2ag c = + Ad =a+ c **. - In Eqs. (1), the previous notation involving a single and double asterisk has been replaced by a notation using a super-plus and super]-plus-plus, These stresses are taken to mean the sum of the external and interply stress on the first ply, and the difference between the external and interply stress on the _.~, ~ ++ 7 -_. V.++..

second ply, respectively. A conversion equation relating these is given as Eqs. (2). *C~ t + ** t + a' + an = na**- an = an+ (2) rl +O] = - = rl In Eqs. (1), the use of the notation given in Eqs. (2) results in lumping together the interply stress and the stress applied by some external agency. Some care must be taken in interpreting this quantity for various loading cases, For example, when one wishes to apply simultaneously three external stresses qa, aq, and as, the resulting solutions in terms of the a+ and o++ components of each of these stresses do not given any information about the origin of the various interply stresses. To be specific, the number obtained from the solution Eqs. (1) for cr will contain interply stress components arising from the application of both a, and ao The individual contributions of these two will not be separable. Since these equations are linear, superposition always holds, It has been found much more convenient to use these equations by applying one external stress at a time and observing the resulting interply reactions. For example, if one applies only a ca, the resulting solutions or numerical values for Crk and cr++ will represent the q direction normal components of interply stresses generated on each of the two plies due to the application of the -o: alone, Similarly, the numerical values for at and ag will represent the shearing components of interply stress due to the application of the ae alone. 8

If one wishes to determine the total stress state due to the application. of several stresses, effects may be merely added together. The known quantities in Eqs. (1) are usually taken to be a, an and acq along with all the aij and ajj quantities. This means that one must know the elastic characteristics of each of the plies involved in the laminate as well as the stress state applied toithe structure. The resulting unknowns in these nine equations are the three strains e~, Ec, and ~ea, the three stresses on the first ply at, an, and at, and their counterparts on the second ply a++, ++ and a++. If some of these quantities go to zero, a corresponding reduction in the number of equations ensues. The three strains are presumed to be the same since the two plies are bonded tightly together. Using Saint Venant's principle, it is possible to clarify the role of interply stresses in Eqs. (1). First, suppose that certain external stresses a+ and a+ are applied arbitrarily to the edge of a two-ply laminate. A certain portion of this stress ca will then be carried directly by the ply involved while another portion ca will be generated due to an interply stress. For the second ply, the part actually transmitted directly through the ply is given by a* while the contribution from the interply bond is now -a.S This is concluded from Eqs. (2). Similar conclusions could be reached from the same line of reasoning using either the stresses in the j-direction or the shearing stresses. Thus, in the presence of arbitrary edge loads, interply stresses may be generated in the direction in which the loads are applied. Some distance away from the points of application of these loads, the stresses carried by the two plies adjust themselves so that one ply carries a while 9

the other carries T,; so if one postulates that the edge of the structure is loaded with some average stress Ad and solves for the stresses in each of the two plies, the solution is that given by Eqs. (3), Cr= a C = * (3) Similar equations could be written concerning mA and Coe. Thus, some distance away from points of concentrated load application or from free edges, external stresses in the a-direction do not generate interply stresses in that direction, Similarly, external stresses in the 9-direction do not generate interply stresses in that direction and external stresses Gindo not generate shear components of the interply stress o. Thus one may always visualize that interply stresses are generated by external loads in directions different from that of the interply stress. With this concept in mind, one may then use Eqso (1) to study the various elastic characteristics which result from the bonding together of two plies of cord imbedded in rubber. When the elastic characteristics of both plies are the same, and their thicknesses are the same, Eqs. (1) reduce to a set of two equations in two unknowns since the loads and stresses are distributed equally between the two plies. When the materials or thicknesses of the two plies are different, or when the cords of one ply are in a state of tension while the cords of another ply are in a state of compression,, then all nine of Eqso (1) must be considered. Generally, for the application of only one external stress at a0 time, this reduces to seven equations in seven unknowns 10

The computer program developed is applicable only to the proper two-ply laminate. For instance, the two-ply structure shown in Fig. 1 cannot be analyzed by using this program because the cord half angle of Ply 1 is not equal to the negative cord half angle of Ply 2. Ply I Ply 2 2 Ply Structure I i'+! =X111/. Fig. 1. Two plies at right angles. E = 104 X = 104 Ey = 102 = 102 Fxy = 2x104 Fxy = 2x104 Gxy =.5X102 Gxy =.5X102 where the x-direction is parallel to the cords, y-direction perpendicular to cords. However, the two-ply laminate in Fig. 1 can be analyzed if it is noted that the two-ply laminate of Fig. 2 is equivalent to that of Fig. I. 11

Ply I Ply 2 Two Ply Structure -I I / I - - Fig. 2. Two plies at right angles. Ex = 104 E = 102 Ey = 102 E = 104 Fxy = 2x104 Fxy = 2x104 Gxy =.5X102 Gxy = 5x5102 where the x-direction is parallel to the cords, y-direction perpendicular to cords. ~ _ ~~~~~1

V. DIGITAL COMPUTER PROGRAM FOR THE SOLUTION OF THE ELASTIC EQUATIONS The development of the digital computer program for solving Eqs. (1) will become more apparent if the equations are rewritten as shown in Eqs. (la). allag + al2ca] + als3ac + 0 + 0 + ~. + + 0 ai1~+a12a~+a~3a~+ 0 + 0 + 0 0+0 = 0 a2lar + a22a + a230 + 0 + 0 + 0 + 0 - + 0 = o a3la++a32a + a3at + o + o + o + o + O- E = a + ol+ O + alag + a32at+a + a' + ~ + ~ = ~ + + +' ++ t ++ + 0 +0 O + 0 + 0 + O + 0 = 0 O + 0+ 0 + + 0a +aO4 + 0+ O+ O = a 0+ ++ 0 + 0 + + 0 + 5+ + 0 + 0 + ++ + 0+0 0 0 + 0 + + + 0 + 0 + + 0 + 0 + + = c (la).' 15

The results to be obtained from the computer program are defined below. Whenever the denominator of the following ratios approaches zero, the computation recognize this fact and a comment "NO SIGNIFICANCE" is printed. The Moduli XI Extension Modulus -= /ES ETA Extension Modulus -= /Ei XI-ETA Shear Modulus = / XI Extension Cross Modulus = o/E7 ETA Extension Cross Modulus =ao/E XI Deformation Modulus = a/l ETA Deformation Modulus =cT/ XI Shear Cross Modulus =aj/ct ETA Shear Cross Modulus = at I/E (2a) The Stress Ratios For Sigma XI+ For Sigma XI++ Sigma XI = / Sigma XI =CJ l Sigma ETA + ++/ Siga ETA = /l Sigma ETA = +r Sigma XI-ETA = ~/a Sigma XI-ETA = a+/ For Sigma ETA+ For Sigma ETA++ Sigma XI = a/a Sigma XI = a+/C Sigma ETA a= Sigma ETA = a +/a Sigma XI-ETA = /fl Sigma XI-ETA = C+/ 51 For Sigma XI-ETA+ For Sigma XI-ETA++ Sigma XI = Sigma XI Siga XI Sigma ETA = a /l Sigma ETA = Sigma XI-ETA = ~+~/n Sigma XI-ETA ++= /C

The computer program computes the a-coefficients of the Eqs. (la), solves the linear simultaneous equations., and computes the results given by Eqso (2a) and (2b), A flow diagram of the basic parts of the procedure is given in Fig. 3. As input to the program, data must be furnished for the elastic constants, the ply angle, and the stresses which form the right-hand sides of the last three of Eqs. (la), as well as three parameters used in control of the program. The final output of the program is an image of the input data and the results as given by Eqso (2a) and (2b). At the option of the user, it is possible to view a check of the coefficient matrix of Eqs. (la), as well as an accuracy check obtained by back substitution of the solutions into the Eqs. (la) and noting the difference between the computed right-hand sides and those specified. A further option available is to print extra copies of the results. The basic steps of the program are outlined in the flow diagram of Fgo. 30 Data are read into the program in three blocks, io.e. 1o Elastic properties; 2. Ply angle; and 30 Stresses and parameters. The first (MATP) of the three parameters read in the third block controls the option to check the solution of the equations, while the second parameter (COPIES) will be equal to the number of extra copies of the results to be printed. The last parameter TRIG is used at the end of any complete computation to decide upon the next type of computation. Thus it is possible to leave the elastic properties and cord angle unchanged while making a change in the stresses (and control parameters), Alternately, it is possible 15

START Read Elastic Properties NOTE: Whenever data are exhausted, program automatically Read Angle terminates. When MATP is zero, checking of the solution of Eq. ( la) is om mitted. Compute Matrix Coefficients COPIES: the number of extra copies of results to be printed TRIG decides upon the next data-reading sequences upon Read Stresses and the completion of the present pass Parameters MATP, through the program. TRIG and COPIES Solve the Eqs. (la) Compute the Moduli and Stress Ratios Print out the Input Data, Moduli and Stress Ratios NCheck Solution MATP E.O.? of Eqs. (Ia) YES and Print Check Results Is TR IG Equal O, I or2? TRIG= 2 TRIG=I TRIG =O Fig. 3. Flow diagram of computer program. ~16

to leave the elastic constants unchanged and read in a new cord angle; in this case new stress data must be read in. Lastly, the whole sequence may be repeated by reading into the program new elastic constants which must be followed by new angle data which, in turn, must be followed with stress data. A copy of the MAD* program and a typical set of results are included in this report. *MAD (Michigan Algorithm Decoder) is an algebraic statement language designed by members of The University of Michigan Computing Center originally for the IBM 704 computer and now available for IBM 709 and 7090. The main features of MAD are very-high-speed compilation and a very general language~ Programs produced by MAD are not as fast in execution as those produced by some other compilers; however, this disadvantage can be partially overcome by appropriate programming~ 17

S-EPT 1961 090661 733 WEDNESDAY HERZOG CLARK S063F 005 030 $EXECUTE,DUMP,PUNCH LIBRARY SCOMPILE MAD,PUNCH OBJECT,PRINT OBJECT PROP 1 5 57 PM 09/06/61 733 FOR TECHNICAL DISCUSSION OF THIS PROGRAM SEE TECHNICAL REPORT NO. 8 DIGITAL COMPUTATION OF TWO-PLY ELASTIC CHARACTERISTICS JULY 1961 PROGRAM BY B.HERZOG ORIGINAL**** APRIL 1961 REVISED *9** AUGUST 19G1 INFORMATION FOR DATA DECK MAKE UP THE VARIEBLE TRIG DECIDES THE CONSEQUENT DATA READING SEQUENCE NORMAL INITIATION OF THE PROGRAM REQUIRES A COMPLETE SEQUENCE OF DATA AND HENCE NO SELECTION IS REQUIRED AFTER ANY PASS OF THE PROGRAM IT EXPECTS TO READ SOME DATA. IF NO DATA OF ANY KIND IS PRESENT EXECUTION IS AUTOMATICALLY TERMINATED HOWEVER, ANY CONSEQUENT EXECUTION REQUIRES INFORMATION FROM THE PREVIOUS RUN AND THIS INFORMATION IS OBTAINED FORM THE STATEMENT BEGINC2) THEREFORE THERE MUST APPEAR A 0 CZERO), 1, OR 2 FOR TRIG. IF ON THE LAST SWEEP OF THE PROGRAM TRIG WAS 2 THEN ONLY SIGMAS ARE READ TRIG WAS 1 THEN NEW ALPHA AND SIGMAS ARE READ TRIG WAS 0 THEN NEW ELASTIC PROPERTIES, NEW ALPHA AND NEW SIGMAS ARE READ

** IF AT ANY TIME THE C:ROS' ARE NOT PRRA'GED IN ACCORDANC:E ** ** WITH THE EXPECTATIOi!NS THEN THE PROGRCl.AM THROWS ITSELF ** ** OFF THE MACHINE ** EQUIVALENCE (SIGMA SIG *001 DIMENSION R:27), RESULT(9. *002 DIMENtSION MPAT(8'-, DIM), T1? 1 0 T 2(1 -, T31 Cl ), T-4(- 0::i *00.3 -1 c89, D I t RES I D E(8, REULT (-8, TR I G,:'30, MO,-1 -t0 F 0:, KNItiT 0.) I, *003 2NORM 1 ):SS (35),S IG (3:, S I MA (3, I F.'-:-. *1003 VECTOR VALUES 0IM=2 0,10 *004 INTEGER Tf IR I J K,,,MD, KMNT, NORM, SRS, NOD *005 I NTEGER J 1 2,J 3, CF' I ES, TEST, NOR, KKK, KtMt, I.:: F R *00C BOOLEAN MATP *007 READ ELASTIC PROFERTIES FCOR PLY'1) BEE.GIN READ FORMAT FPROP, TEST, E.', EY, F.:Y, XY *008 WHENEVER TEST. NE. E, EXECUTE SYSTEM. *009 FOR PLY2 REiAD FORMPT PROP, TEST, EE:.'P', EYP, FXr'YP, G:F' *010 WHEHEVER TEST. NE. E,EXECUTE SYSTEM. *011 REED RN ANGLE ALPHA IiHICH I THE CHORD HALF RNGLE BEG I N 1 READ FORMAT PROP, TEST, LPHA *.01 2 WHENEVER TEST. NE..;AI,, EXECUTE SY STEM. *013 COMPLFTE THE TRIGONOMETRIC FUNCTIOONS I. lHI CH ARE USED I COMPITING THE COEFFICIENTS OF MAT PLP=RLPHAPFI,180N. *014,-=SIN.:ALP) 01 C=COS. ALP) *01 r S2=S * S *01 7 S3=S2S.0 1: S 4= 2. 2'019 C2=CC * 020 C3= *C2 *021 C4=C2.'C:2 * 022 C2S2=C2.S2 *02.3 CS7=Cw*S3 *024 iS.. = C: * S * 025 C4F 4= 14+S 4 *026 VECTOR VPLUES PI=3.1415927 *027 COMPUTE THE COEFFICIENTS OF SIGNM+ PF'LY(.':l, SEE Ei.UATION. 1 i It SECT!ION 1 V):: OF REPOCiRT E1 =EX *028 E2=EY *029...13 G$ XY *030 F = F "'"Y *0.31 PT(l,l) 1 =F1. E1,E2 MAT=F 1..E 1, E:2 i *032 MAT (2: 2)=F1.'E2, E1 MAT,:1 1 ) =F1. CE2, E1 ) *033 Mi AT,( 3,.3:, =F -. 0) r,11T <22 =F3. (0) *034 T=F12. (0) *035 MPT(:1 2)=T 19

MAT(1 )=T *036 MAT (2, 1)=T MAT(1 0)T *037 K1 =1. /G-2./F *038 K2=2. E 1 -KI *039 K3=K1-2./E2 *040 T=C3S*K2+CS3*K3 *041 MATC1,3)=T MATC2)=T *042 MAT(3, 1)=T MAT(20)=T *043 T=CS3*K2+C3S*K3 *044 MAT(2,3)=T MATC12)=T *045 MATC3,2)-T MATC21)=T *046 COMPUTE THE COEFFICIENTS OF SIGMA++ CPLY(2)), SEE EQUATION ( 1) IN SECTION.IV) OF REPORT E1=EXP *047 E2=EYP *048 G=GXYP *049 F=FXYP *050 MAT (4,4)=F 1..'E 1, E2) MAT (33) =F1. CE 1, E2) *051 MAT C5,5)=F 1. CE2, E 1 MAT(44)=F1. (E2, El) *052 MAT(6,6)=F3. CO) MAT(55)=F3. (0) *053 T=F12. (0) *054 MAT 4,S5)=T MAT34) =T *055 MATC5, 4)=T MAT 43) =T *056 K =1. /G-2./F *057 K2=-2./El +K1 *058 K3=K 1-2./E2 *059 T=C3S*K2-CS3*K3 *060 MATC4,6)=T M AT(35= =T *061 MAT(6,4)=T MAT (53) =T *062 T=CS3*K2-C3S*K3 *063 MATC5, 6)=T MAT(45)=T *064 MAT 6, 5)=T MAT(54)=T *065 READ THE GIVEN SIGMAS,TRIG AND NUMBER OF EXTRA PRINT COPIES SIG(1:),SIG(2> AND SIG(3) ARE THE APPLIED SIGMA EXI, SIGMA ETA AND SIGMA EXI-ETA RESPECTIVELY BEGIN(2) READ FORMRT PROPTEST,SIG(1')...SI3G(),MATPF'TRIGCOPIES *066 WHENEVER TEST. NE. $SEXECUTE SYSTEM. *067 MAT(69)=SIGC1) *068 MAT(79) =S IG(2) *069 MAT C89) =S I G(3) *070 LORD INTO A THROUGH SUBST,FOR I=0;1,I.G.89 *071 SUBST R(I)=MAT(I) *072 SOLVE THE EQUATIONS TRANSFER TO NEXTCSLINEQ. C,9.,1.,O.,T1,T2,T3,T4)) *073 NEXT(1) THROUGH RESFOR I=1,1,I.G.9 *074 20

RES RESULT(I:) =A(1 O*I-1 ) *075 BEGIN CALCULATIONS OF MODULI AND RATI:OS EXECUTE ZERO. (TRIG(1:).. TRIG(27:)) *076 EXECUTE ZERO. CR...R(27)) *077 COMPUTE THE MODULI II WHENEVER. BS.RESULTC7).LE. 1. E-20 *078 TRIG(1 )=10 *079 TRIG(5)=1 0 *080 TRIG(8:)=10 081 OTHERWISE *082 RC1) IS SIGMA EXI/EXI STRAIN R1 )=SIGMA(1 )RESULT(7) *083 R(5) IS SIGMA ETA/EXI STRAIN R C5) =S I GMA (2)'RESULT (7) *084 RC8) IS SIGMA EXI-ETA/EXI STRAIN R (8) =SI GMA (3)/ RESULT C7) *085 END OF CONDITIONAL *086 WHENEVER.ABS.RESULTC8).LE. 1. E-20 *087 TRIG(2)=10 *088 TRIG(4)=10 *089 TRIG(9)=10 *090 OTHERWISE *091 R(2: IS SIGMA ETA/ETA STRAIN R (2)=SIGMAR2) /RESULT(8) *092 R(4) IS SIGMA EXI/ETA STRAIN R(4)=SIGM(1 ) /RESULT(8) *093 R(9) IS SIGMA EXI-ETA/ETA STRAIN R 9)=S I GM(3)/RESULT (8) *094 END OF CONDITIONAL *095 WHENEVER. ABS. RESULT9). LE. 1. E-20 *096 TRIGC3)=10 *097 TRIG ()=1 0 *098 TRIG3C7)=10 *099 OTHERWISE *100 R(.3) IS SIGMA EXI-ETA/EXI-ETA STRAIN R C3) =S I GM (3)/RESULT (9) *101 R(6) IS SIGMA EXI/EXI-ETA STRAIN R (6) =SIGMA(1 ) RESULT 9) *102 R(7) IS SIGMA ETA/EXI-ETA STRAIN R (7)::SIGMA (2)RESULT(9) *103 END OF CONDITIONAL *104 THROUGH INTRIZ,FOR I=:1,1I.G.9 *105 RI=R(I) *106 KR=1 *107 SMALL WHENEVER RI.G.1.E+7.AND.TRIG.E.O *108 RI=RI/10 *109 KR=KR*10 *110 TRANSFER TO SMALL *111 END OF CONDITIONAL *112 IR=RI *1 1 IR(I)=IR*KR *114 INTRIZ CONTINUE *115 COMPUTE STRESS RATIOS K=' * 1 16 THROUGH JOE:FOR J=1llJ.G.6 *117 THROUGH JOE,FOR VALUES OF I=1,2,3 *118 K=K+l *119 21

WHENEVER. BS. SIGMACI).LE. 1. E-20 1 20 TRIGCK)=10 *121 TRANSFER TO JOE *122 END OF CONDITIONAL *123 RCK)=RESULT (J)/SIGMACI) *124 JOE CUNTINUE *125 PRINT RESULTS THROUGH SMELL,FOR ZZ=O1,,ZZ.G.COPIES *126 PRINT FORMAT TITLE *127 PRINT FORMAT TITLEl,EEXP,EYE VPFXFXYPGXGX YP *128 PRINT FORMAT DATA,ALPHRASIGC1),SIGC2),SIGC3.'),TRIGCOPIES *129 PRINT FORMRT TITLE 5 *130 THROUGH MODO,FOR I=1,1,I.G.9 *131 j=10*1 *132 WHENEVER TRIGCI).E.10 *133 K=J+6 * 1 34 J 1=K+1 *135 J2=K+2 *136 J3=K+3 *137 MO (K)=KMNT *138 MOC-(J 1)=KMNTC1) *139 MDCJ 2)=KMNT C2) *140 MDCJ3)=KMNT(3) *141 PRINT FORMAT MDC10*I) *142 MDCK) =NORM *1 43 MDJ 1 )=NORMC1) *144 MDCJ2)=NORMC2) *1 45 MD (J 3) =NORM (3) *146 OTHERWISE *147 PRINT FORMAT MDCJ),IR(I),RCI) *148 MOD END OF CONDIT IONAL *149 PRINT FORMAT TITLEG6 *50 THROUGH STRFOR I10O3, I.G.27 *151 SRS(3)=NODCI) *152 SRS(4)=NODI+ 1) *153 J=I+1 *154 K=I+2 *155 WHENEVER TRIG(I).E.10 *156 SRS(12)=KMN *157 SRS(1 3)KMN(1) *158 SRS(14)=KMN(2) *159 SRS (1 5) =KMN C3) * 160 WHENEVER TRIG(J).E.10 *161 SRS(23)=KMN SRS(24=KMNC 1) *163 SRS 25):KMN<(2) *164 SRS (26) =KMN (3) *165 WHENEVER TRIG K).E.10 *166 SRS 34) =KMN * 167 SRS(35)=KMNC1) *168 SRS C3) KMN 2) *169 SRS (37) =KMN (C3) *170 PRINT FORMAT SRS *171 THROUGH FIX, FOR VALUES OF Z=12'23" 34 *172 SRSCZ)=NOR * 1 73 SRS(Z+ l )-=NOR (1 ) *1 74 SRS (Z+2)=NOR C2) *175 FIX SRS(Z+3)=NOR(C3) *176 TRANSFER TO STR *177 END OF C ONDITIONtL *178 22

PRINT FORMAT SRS,R(K),R(K)t *179 THROUGH FIX(). FOR VALUES OF 2=12,23 *180 SRS (2)=Ni-OR *181 SRS(Z+1)=NORP1) *182 SRS Z+2) =NOR(2 *1 83 FIX(3) SRS (Z +.3) =:NOR (.:..3 *184 TRANSFER TO STR *185 END OF CONDITIONAL *186 WHENEVER TRIaG(K.E.10 *187 SRS(34) K MN *188 SRSf(35)=EMN(1) *189 SRS(36)=KMN(2) *190 SR S (.37) =KMN (: 3::' *1 91 PRINT FORMAT SRSR(J) R R(J) *192 THROUGH FIXC(1)FOR VALUES OF 2=12.34 *193 SRS(Z)=NOR *194 SRS(Z+1)=NOR(1) *195 SRS (+2) =NOR (2) *196 FIX(1 ) SRS (Z+3) =NOR 3: *197 TRANSFER TO STR *198 END OF CONDITIONAL *199 PRINT FORMAT SRS..R(J),R(J::,R (K>,RK) *200 SRS(1 2)=NOR *201 SRS( 13 ) =NR(1 *202 SRS1 4)=NOR(2) *20.3 SRS(1 5)=NOR(3 *204 TRANSFER TO STR *205 OR WHENEVER TRIG(JS.E.10 *206 SRS 23) K MN *207 SRS(24)=KMN:: 1 *208 SRS (25) KMN (2) *209 SRS(26) KMN(3) *21 0 WHENEVER TRIGK).E. 1 *211 SRS (.34)=KMrt 212 SRS(35) =KMN(1 *213 SRS (.36) =KMN'2) *214 SRS(C37) =KMN (3) *21 5 PRINT FORMAT SRS,R(CI)R(I) *216 THROUGH FIX(2),FOR VALUES OF 2=23 34 *217 SRS(Z)=NOR *21 8 SRS Z+1 )=lNOR1: *219 SRS Z+2) = NOR (2) *220 FIX(2) SRS(Z+3)=NORC3) *221 TRANSFER TO STR *222 END OF CONDITIONAL *223 PRINT FORMAT SRS, R(I), R(I:,RCK.K ) R *224 SRSC2.3 =NOR *225 SRS(24)=NOR(1 ) *226 SRS(25) =NOR 2) *227 SRS (26) =NOR (3) *228 TRANSFER TO STR *229 OR WHENEVER TRIG:K).E.10 *230 SRSC34) =KMN 231 SRS(35)=KMNC1) *232 SRS 36) =KMN (2) *233 SRS(37) =KMN(3) *234 PRINT FORMAT SRSR(I),R(I),R(J),R(J) *235 SRS (34)NOR *236 SRS(35)=NOR(1) *237 SRS (.36) =NOR (2) *238

SRSC37)=NORC3:) *239 TRANSFER TO STR *240 END OF CONDITIONAL *241 PRINT FORMAT SRS, RCI),RCI) RCJ)RCJ), RCK),RK) *242 STR CONTINUE *243 IF MATP IS DIFFERENT FROM ZERO THE COEFFICIENT MATRIX ETC. WILL BE PRINTED WHEN EVER MATP *244 COMPUTE RESIDUES TO CHECK SOLUTION THROUGH CHECK,FOR I=:11,I.G.9 *245 T=-MATCI, 10) *246 THROUGH LINE,FOR J=1I1,J.G.9 *247 LINE T=RESULTCJ)*MATCI, J)+ T *248 CHECK RESIDCI):T *249 THE COEFFICIENT MATRIX PRINT OUT PRINT FORMAT TITLE 3 *250 THROUGH DOGFOR I=0,10,I.G.80 *251 DOG PRINT FORMAT CAT MPTCI),... ATCI+5. *252 PRINT FORMAT TITLE 4 *253 THROUGH BIRDFOR I=1 1 I.G.9 *254 BIRD PRINT FORMAT CAT,MPTC1O*I-4.... MPT(1O*I-1),RESULTCI::, RESIDCI:: *255 END OF C ONDITIONAL *256 SMELL CONTINUE *257 TRANSFER TO BEGINTRF.IG) *258 NEXT(2) PRINT FORMAT SING *259 THE SIMOULTANEOUS E.UATIONS APPEAR *.INGULAR* TO SLINEQ. EXECUTE ERROR. *260 INTERNAL FUNCTIONS INTERNAL FUNCTION F.ERA, EB):=C4/EA+S4/E8 *261 1 +C2S2*C.. /G-2. /F) *261 INTERNAL FUNCTION F3. CCOWS)..-=C252*C4. /E 1 +4./E2+8./. F-2. /G) *262 1 +C4S4/G *262 INTERtAL FUNCTION F 12. CUi LLS)=C2S2*1. E 1 +...E2- 1. /G) -C4P S4/F *263 VECTOR VALUE STATEMENTS FOR FORMATS ETC. VECTOR VALUES CAT=-1 HO, 6(E19.8) T. *264 VECTOR VALUES PROP=SC1 F1 9. 5.5 3Ft5. 5. 5 I, I2*$ *265 VECTOR VALUES TITLE=1t 1,S53,1 t 4HTHE INPUT DIAT/1HO, t 18C 1H*:)/. *266 1 *' *266 VECTOR VALUES TITLE1= *267 1 1HO, S1 0, 6HEX =F13.2S S2,4HPSI., S35,HEXP =F13.2, S2,4HPSI., *267 25, *267 31 HOi S 1 0, OHE =F 1.3. 2 S2,4HPSI. S.35, GHE'F =F 1 3.2, 2,4HPSI./ *267 45T, *267 5 1 HO, SO, 6HFXY =F t 3. 2, S2 4HPS I., S35, 6HF','P =F 1. 2, S2,4HPSI..I... *267 C6s *267 7 1 HO S 1 0,HGXY =F13.2,S2,4HPSI.,S 5,6HGXYP =F13.2,S2,4HPSI.. *267 8.$ **267

VECTOR VALUES DATA= *268 lSlHO, S46,7HALPHA =F6.1,S2,7HDEGREES.'HO, S *268 25S10,8HSIGMA1 =F10.2,$, *268 3510, SHSIGMA2 =F1O.2,$ *268 45S10,8HSIGMA3 =FlO.2.'I65, I5//* *268 VECTOR VALUES TITLE3=S1H1,S42,35H THE COEFFICIENT MATRIX AND *269 1 RESULTS *269 21HO,1 8C1H*)///S6,1OHFOR SIGMA+.S47,11HFOR S IGMA++//'*'. *269 VECTOR VALUES TITLE4=1HO//,/S6,11HFOR EPSILON,4 46, *270 112HGIVEN SIGMAS,S 7,13HTHE SOLUTIONS,SG612HTHE RESIDUES//*$ *270 VECTOR VALUES TITLE5=$lH4,S50,18HTHE ELASTIC MODULI *271 1/'1H0,l 1 18C 1*)',,'o*$ *271,VECTOR V'LUES TITLEG=$;1H1,S50O17HTHE STRESS RATIOS *272 1 /1 H,1181 H*)//"*,$ *272 VECTOR VALUES WELL=-1HO,E30.8 E56.8,E2O.8*$ *273 VECTOR VALUES SING=$1H4,15HSINGULAR RETURN*$ *274 VECTOR VALUES SRS= *275 1527HO FOR SIGMA $I *275 25/1H S5,21ClH*),/'/1H,S5 14HSIGMA XI = $, *275 3SF26.4, E60.8 $I *275 45/1H S5,14HSIGMA ETA = S, *275 55F26.4, E60.8 $ *275 6./1H, S5 14HSIGMt XI-ETA = $ *275 7SF26.4, E60.8 ///*$ *275 VECTOR VALUES KMN=-S11,15HNO SIGNIFICANCE $ *276 VECTOR VALUES NOR=SF26.4, E60.8 $ *277 VECTOR VALUES NOD('10)=$ XI+ $ *278 VECTOR VALUES NOD(13)=$ ETA+ s *279 VECTOR VALUES NOD(16)-$ XI-ETA+ s *280 VECTOR VALUES NODC19)=S XI++ $ *281 VECTOR VALUES NOD(22)=S ETA++ s *282 VECTOR VALUES NOD(25)=S XI-ETA++ $ *283 VECTOR VALUES KMNT=SS11,1 5HNO SIGNIFICANCE *$ *284 VECTOR VALUES NORM=-$F20.2,S2,4HPSI.,EG'O.8 *g *285 VECTOR VALUES MDC10)=$1H0,29HXI EXTENSION MODULUS =I20 *286 1, 52,4HPSI.,E60.8 *$ *.286 VECTOR VALUES MDC20)=S1HO229HETA EXTENSION MODULUS =120 *287 1, S2,4HPSI. E60. 8 *S *287 VECTOR VALUES MDC30)=$1HO 29HXI-ETA SHEAR MOFULUS - I20 *288 1, 52,4HPSI, E60.8 *$ *288 VECTOR VALUES MD(40)=$1HO,29HXI EXTENSION CROSS MODULUS =I20 *289 1, S2,4HPSI., E60.8 * *289 VECTOR VALUES MD(50)=t1HO,29HETA EXTENSION CROSS MODULUS =120 *290 1, S2,4HPSI.,EGO.8 *$ *290 VECTOR VALUES MDC6O)=$1HO,29HXI DEFORM1TION MODULLIS =I20 *291 1, S2,4HPSI.,E60.8 *S *291 VECTOR VALUES MD(70)=$1H0,29HETA DEFORMT ION MODULUS =I20 *292 1, S2,4HPSI.,E60. 8 *$ *292 VECTOR VALUES MDC8O=$1HO, 29HXI SHEAR CROSS MODULUS =I20 *293 1, S2,4HPSI.,E60.8 *. *293 VECTOR VALUES MD(90)=$1H OO,29HETA SHEAR CROSS MODULUS =I20 *294 1, S2,4HPSI.,E60.8 *$ *294 VECTOR VALUES M*T=1 0., 10. 10.,0 O. O. -1 10,10. *295 10. 0.,0. 0. 0. -1.*0.01 ~ si-0. 0.0,.r0 010, 10.,., -. *295 2,0.,0., 11., 11.t 11.,,. 11. 1. 11.,0,-t1.,: *295 30.,0. 0.,0.,0., 11. ~ I 1., 11., 0. 0.,-1. 0.,.5,CI. 0..5,0. 0*. 0. ~, 295 40.,0.,12.,.,.5,0.,.,.5,.,00.,0. 12.,.,0..,0.,..5, *295 50., O., O.,12. *295 END OF PROGRAM *296 25

THE IPFLIT DPTP E = 223500.00 PSI. EXE = 12400.00 PSI. El = 970.00 PSI. EYP = 970.00 PSI. F'-, = 44050': 00.00 PSI. F XYP = 1 4000. oo PS I 3 = s:'. co0 PSi. I1 =.308.00 PSI. ALPH - D ~EREES SIGMA1 = 100.00 SIAM-'- U. L! SIGMA3 = 0.00 2 0 THE ELASTIC MN''DULI XI EXTENSION MODULUS- 4 G233 PS I. 0.4623& ilO3 E 04 ETA EXTENSION MODULUS = -0 F SI. -0.00000000oo oo XI-ETP SHEAR MODULUS = -0 PSI. -0. 000000000000 XI EXTENSION CROSS MODULUS = -404 S I 0.240438 E 04 ETA EXTENSION CROSS MOD'L1'0 0 PS I. 0.00000000000 XI DEFORMPTION MODULUS C -:2S5 PSI. -0. 82i56500E 04 ET A DEF 0 R M ATI0 N M 0:'DtlU L- US = -0 PSI. -0. 00000000000 XI SHEAR CROSS MODULUS -'' PSI. 0.000000000000 ETA SHE~P CROSS MODULUS -0 ES I -0.000000000

THE STRESS RATIOS FOR SIGMA XI+ SIGOMA XI I I. U43C 0. 1 0435629E 01 CSIGMA ETA = NO SIGNIFICANCE STIGMP XI-ETA = NO SIONIFICANCE FOR SIGMA ETA+ SIGMA XI - 0.0007 0.691.32327E-03 SIGMA ETA = NO SIGNIFICANCE SI OMP XI-ETA = NO S ION I FICANF FOR SIGMA XI-ETA+ SIGMA XI - 0.520:3 0. 52877135E 00 SIGMP ETA = NO SI3NIFICANC:E STGMA XI-ETA = NO ST3NIFICANCE FOR SIGMP XI++ SIGMA XI - 0. 9564 0.9564370SE 00 SIGMA ETA NO SIGNIFICAN:E SIGMP XI-ETA = NO SIGNTFI CACE FOR SIGMP ETA++ STOMP Xl - -0.0007 -0. 691.32327E-03 SIGMP ETA = NO SIGNIFICANCE SIGMA XI-ETA = NO SIGNIFICANCE FOR SIGMP XI-ETA++ SIGMP XI - -0.5288 -0.52877135E 00 SIGMP ETA = NO SIGNIFICANCE SIGMP XI-ETA = NO SIONIFICPNCE

THE COEFFICIENT MATRIX AND RESULTS FOR SIGMA+ FOR SIGMA++ 0.67479004E-03 -0.41617152E-03 -0.92216979E-03 0.000000000000 0.000000000000 0.000000000000 -0.41617152E-03 0.11880168E-02 0.33234973E-04 0.000000000000 0.000000000000 0.000000000000 -0.92216979E-03 0. 33234973E-04 0.15 5919434E-02 0.000000000000 0.0000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.69177637E-0C- -0.44498913E-03 0.84283503E-03 0.000000000000 0.000000000000 0. 00000000000 -0.4449891 3E-03 0.116691 76E-02 -0. 1 9866139E-04 0.000000000000 0. 000000000000 0.000000000000 0. 84283503E-03 -0.19866139E-04 0.17525107E- 02 0.50000000E 00 0. 000000000000 0.000000000000 0.50000000E 00 0.000000000000 0.000000000000 0.000000000000 0.50000000E 0 0. 000000000000 0.000000000000 0.50000000E 00 0.000000000000 0.000000000000 0.000000000000 0.50000000E 00 0.000000000000 0.000000000000 0.50000000E 00 FOR EPSILON GIVEN SIGMAS THE SOLUTIONS THE RESIDUES -0.09999999E 01 0.000000000000 0.000000000000 0.000000000000 0.10435629E 03 0.13969839E-08 0.000000000000 -0.09999999E 01 0.000000000000 0.000000000000 0.691323528E-01 -0.93132257E-09 0.000000000000 0.000000000000 -0.09999999E 01 0.000000000000 0.52877135E 02 0.000000000000 -0.09999999E 01 0.000000000000 0.000000000000 0.000000000000 0.95643708E 02 -0.46566129E-09 0.000000000000 -0.09999999E 01 0.000000000000 0.000000000000 -0.69132328E-01 -0. 1 3969839E-08 0.000000000000 0.000000000000 -0.09999999E 01 0.000000000000 -0.52877135E 02 0.000000000000 0.000000000000 0.000000000000 0.000000000000 00 0.09999999E 03 0.21 62811 8E-0 1 -0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.00000000000 -0.4159061 6E-01 -0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 -0.12054510E-01 -0. 000000000000

THE INPUT DATA EX = 22-500. 00 PSI-. EXP - = 1 2400.00 PSI. EY 9= 70.00 PSI. EYP =' 70.00 PSI. FXY = 405000.00 PSI. FX.YP = 14000.00 PSI. GXY = 30:. 00 PS I. GXYP = 308.00 PSI. ALPHA = 30. 0 DEGREES SIGMu1 = 0.00 SIGMA2 = 0.00 SIGMA3 = 100.00 1 0 THE ELASTIC MODULI XI EXTENSION MODULUS -0 P I. -0.000000000000 ETA EXTENSION MODULUS - 0 PSI. 0. 000000000000 X I-ETA SHEAR MODULUS = 4743 PSI. 0.47436296E 04 XI EXTENSION CROSS MODULUS - O PSI. 0.000000000000 ETA EXTENSION CROSS MODULUS = - PSI,. -0.000000000000 XI DEFORMATION MODULUS =0 PSI. 0.000000000000 ETA DEFORMATION MODULUS = 0 PSI. 0.000000000000 XI SHEAR CROSS MODULUS = -E295 PSI. -0.82956494E 04 ETA SHEAR CROSS MODULUS = 2.555.3 PSI. 0.255538.50E 05

THE STRESS RT IOS FOR SIGMA XI+ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 1.6659 0.16658786E 01 FOR SIGMA ETA+ * *** * **** ** ***** SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 0.5862 0.58615185E 00 FOR SIGMA XI-ETA+ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICRNCE SIGMA XI-ETA - 1.0852 0.10851838E 01 0 FOR SIGMA XI++ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = -1.6659 -0.16658786E 01 FOR SIGMA ETA++ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = -0.3862 -0.58615185E 00 FOR SIGMA XI-ETA++ SIGMA XI = NO SIGNIFICANCE SIGMA ETA NO SIGNIFICANCE SIGMA XI-ETA = 0.9148 0.91481622E 00

THE IUPUT DPTA EX = 223500.00 PSI. EXF = 12400.00 PSI. EY = 970.00 PSI. EYP = 970.00 PSI. FXY = 405000.00 PSI. FXYF = 14000.00 PSI. GXY 309.00 PSI. GX''Y P = 309.00 PSI. ALPH 45.0 DEGREES SIGM1 = 0.00 SIGMA2 = 0.00 SIGMA3 = 100.00 0 0 THE ELASTIC MODULI XI EXTENSION MODULUS - -0 PSI. -0.000000000000 ETR EXTENSION MODULUS - -0 PSI. -0.000000000000 XI-ETA SHEAR MODULUS 6585 PSI. 0.65858535E 04 XI EXTENSION CROSS MODULUS -0 PSI. -0. 000000000000 ETA EXTENSION CROSS MODULUS = -0 PSI. -0. 000000000000 XI DEFORMATION MODULUS 0 PSI. 0. 000000000000 ETA DEFORMATION MODULULS - 0 PSI. 0.000000000 XI SHEAR CROSS MODULUS -1512G PSI. -0.15126802E 05 ETA SHEAR CROSS MODULUS - -15126 PSI. -0.15126801E 05

THE STRESS RATIOS FOR SIGMA XI+ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 0.9936 O.99358482E 00 FOR SIGMA ETP+ SIGMA XI NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 0.9936 0.99358482E 00 FOR SIGMA XI-ETA+ SIGMA XI = NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 1.1263 O.11262149E 01 R) FOR SIGMA XI++ SIGMA* XI NOSIGNIFICANCE SIGMA XIE NO SIGNIFICANCE SIGMR ETR = NO SIGNIFICeHNCE SIGMAI XI-ETA = -0.9936 -0.99358482E 00 FOR SIGMA ETA++ SIGMA XI = NO SIGNIFICANCE SIGMR ETA = NO SIGNIFICANCE SIGMA XI-ETAI = -0.99 36 -0.99358482E 00 FOR SIGMA XI-ETA++ SIGMA XI NO SIGNIFICANCE SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = 0.8737 O.87372506E 00

THE COEFFICIENT MATRIX AND RESULTS FOR-SIGMAr+ FOR SIGMA++ 0.10693042E-02 -0. 55407233E-03 -0.51322675E-03 0.000000000000 0.000000000000 0.000000000000 -0.55407233E-03 0.10693042E-02 -0.513222675E-03 0.000000000000 0.000000000000 0.000000000000 -0.51322675E-03 -0.51322675E-03 0.10403403E-02 0.000000000000 0.000000000000 0.000000000000 0. 000000000000 0.000000000000 0. 000000000000 0.10538672E-02 -0.56950932E-03 0.47514132E-03 0.000000000000 0.000000000000 0.000000000000 -0. 5950932E-0:3 0.10538672E-02 0.475141332E-03 0. 000000000000 0.000000000000 0. 000000000000 0.475141 32E-03 0.47514132E-03 0. 12544300E-02 0.50000000E 00 0.000000000000 0.000000000000 0. 500OOOO0E 00 0.000000000000 0.00000000000 0.000000000000 0.500OO0OE 00 0.000000000000 0.000000000000 0.50000000E 00 0.000000000001 0.000000000000 0.000000000000 0.50000000E 00 0.000000000000 0.000000000000 0.50OOOOOOE 00 FOR EPSILON GIVEN SIGMAS THE SOLUTIONS THE RESIDUES -0.09999999E 01 0. 00000000000 0.000000000000 0.000000000000 0.99358482E 02 -0.17462298E-0? 0.000000000000 -0.09999999E 01 0.000000000000 0.000000000000 0.99358482E 02 -0.13969839E-0S 0.000000000000 0.000000000000 -0.09999999E 01 0.000000000000 0.11262749E 03 -0.93132257E-09 -0. 09999999E 01 0. 000000000000 0.000000000000 0.000000000000 -0.99358482E 02 0.29103830E-08 0.000000000000 -0.0'999999E 01 0.000000000000 0.000000000000 -0.99358482E 02 0.32596290E-08 0.000000000000 0.000000000000 -0.09999999E 01 0.000000000000 0.87372506E 02 -0.93132257E-09 0.000000000000 0.000000000000 0. 000000000000 0. 000000000000 -0. 66107824E-02 0. 000000000000 0.000000000000 0.000000000000 0.000000000000 0.000000000000 -0.66107828E-02 0.00000000000 0.000000000000 0. 000000000000 0. 000000000000 0.09939999E 03 0. 151 84060E- - 1 0.000000000000

THE INPUT DATA EX = 200000.00 PSI. EXP = 200000.00 PSI. EY 1125.00 PSI. EYP = 1125.00 PSI. FXY = 421000.00 PSI. FXYP = 421000.00 PSI. GXY = 775.00 PSI. GXYP = 775.00 PSI.. ALPHA.= 30.0 DEGREES SIGMAI = 100.00 SIGMi2 = 0.00 SIGMA3 = 0.00 2 0 THE ELASTIC MODULI XI EXTENSION MODULUS = 12489 PSI. 0.12489467E 05 ETA EXTENSION MODULUS - -0 PSI. -0. 0000000000 XI-ETA SHEAR MODULUS - -0 PSI. -0.0000000000 XI EXTENSION CROSS MODULUS = -4638 PSI. -0.46384025E 04 ETA EXTENSIONCROSS MODULUS 0 PSI. 0.000000000000 XI DEFORMATION MODULUS = -69905066 PSI. -0.71582787E 11 ETA DEFORMATION MODULUS = -0 PSI. -0.000000000000 XI SHEAR CROSS MODULUS -0 PSI. 0.000000000000 ETA SHEAR CROSS MODULUS - -O PSI. -0.000000000000 -

THE STRESS RATIOS ********** *** ********************************************************************************************************** FOR SIGMA XI+ ***** ****** ********* SIGMA XI = 1.0000 0.09999999E 01 SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = NO SIGNIFICANCE FOR SIGMA ETA+ SIGMA XI = O.0000 0.1 1920928E-08 SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA =NO SIGNIFICANCE FOR SIGMA XI-ETA+ *********+*********** SIGMA XI = 0.4692 0.46915075E 00 SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = NO SIGNIFICANCE FOR SIGMA XI++ SIGMA XI = 1.0000 0.99999999E 00 SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = NO SIGNIFICANCE FOR SIGMA ETA++ SIGMA XI = -0.0000 -0.11920928E-08 SIGMA ETP = NO SIGNIFIICANCE SIGMA XI-ETA = NO SIGNIFICANCE FOR SIGMA XI-ETA++ ******************** SIGMA XI = -0.4692 -0.46915075E 00 SIGMA ETA = NO SIGNIFICANCE SIGMA XI-ETA = NO SIGNIFICANCE

THE COEFFICIENT MATRIX AND RESULTS FOR SIGMR+ FOR SIGMA++ CO. 2994':'41 27 qE-.03 -0. 7581 5873E-04 -0. 4675.:,3696E-0.3 OO. CIOOOio00 0O. 000000000000 0. 000000000000 -0.75815873E-04 0.74135721 E- 03 0.2':' 7.._321EC-O.3 0. O. 000C00OOOOi O 0.000000000000 0.000000000000 -0. 4675369E- 3 -0. 2 -7. -.3 3 2 1 E- - O.!'. -! 9-. ID 9 c C, I-! O O i Cli 11;O 0. 1000000000000Ii ii 0. 000000000000. i 00000 00000 0. i000i]0000i:000-.i O. 0 3ii!ii-OOC00 -Ciitl- 294127'E - 0. -0. 751 587_3E- 04 0. 46753696E-03 0. Oo0Ooooo0000 0. 00000000 i. 00 i f -: 8 55'- 73E - 04 O. 741 35721 E- 03 0. 2979 3 32 1 E- 03 0. 00000 t'00000 00. i 0 l 00 0 I.1!' 1-i E E l 0 l. 4 _:' 7 5 3 600 9 E - 0.-"3 i. 29793.321 E-03 0.996560 1 9E-03 0.5000000 E0 Ci 0.0 CI 0000 0000CC 0 0 000000 i 0' 0 Ci O U OC 0.. L5Cf ]00E L.00 0Ci. CCi00000000000 i 0.0000 00 00 I. 00OOC ooI000ii' - O'i'' s 0:. 50 COOOi i OE00 I. Li i'-ioriCt O. 3 I Oi i LiOiO OL!'.OIO 0. 500001i000E 00 0. 000000000000 o. i00001:C10 Cli00 0.- i i f!Tl OCI, 0' i 0 i i. 5 C_:! I:! iI I0 iE!0 E S0:i0 0 L C, I O. - C i]OOi.iI ii,OIO ti.50 OO00OE 00 FOR EPSILON ITEN SIGMlAS T.HE SOLUTIONS THE RESIiUES -0.09999999E 01 0.O0iD000O00O 0. 000000000000.OOOOC0000000 0. 09999999E 03 -0.000000000000 0. O0000000 -C.09999999E 01 Ci 1. 000i00C0i0 LlOOO000Ci00000 0. 11 920929E-0- O. 2328.30R64E-09 O. 000000000000 0. I':O00-.:9't.'.0 - T.'.'C9 9E 0 1 0.i 0O00o000 i. 4.91 5075E C02 0. 4656.1 29E-09 -0. 09999999E 01 0. _00000000000 0. CIOi 1Oi00I00.IUOO i. O i, Oii!0000 00. 0.999'99999E 02 -. 698491 93E-09 0. 000000000000 -0. 099399999'e'- 001 O.00000000000 O. CC0000i000000 -. 11 920 92t9E-06 0. 2328.3064E-09 0.000000000000 0. Oi0 0 i00Oi,1 00 f -"0 -0. 09-,0 i1 i i.0'i 00000001. iF- -0.4. ii 415075E 2 0. 13969839 E- Cf 0.000000000000.0'00000 0.0000C00 C 000 COOOO0C Ci]O 0. 0'::9 99:-90 03 0.80067466E-202 -i 00~00000000diOOt0 01. 000000000000 0. 0 00000000000 Ci. CO 0CCICI0, IO ~ C'.'000f0000000 - -1. 21 559 1 46E-O 1 -0. 0000000000 0. 000000000000 0. 0000OO 1-iO iO I. COCiO-OOOi]_-000 f't -'10l. O OOOOOOI-IOf ] -0. _1. 3:6'1 839E-08 -c. OOOi0OOOC.OO0

6 SEPT 1961 090661 733 WEDNESDAY 5 57 PM 6 01 PM HERZOG CLARK S063F ELAPSED TIMES IN MINUTES S 17. TQTAL.7 EXECUTE 2.8 COMPILE ALL USERS HIOT~'. THE 704 WILL BE REPLACED BY A 709 COMPUTER AS OF SEPT. 18. SEE NOTICE ON THE BULLETIN BOARD.

VI. EXTENSION TO MULTI-PLY LAMINATES The equations writen here, and the resulting computer program, may be used for a study of laminates: involving even numbers of plies such as four, six, or eight, provided that each alternate pair takes on the same distribution of elastic properties as the first. This would quite generally be the case in a structure where the application of torsion or twist caused-the cords in one direction to go into one stress state while those in the other direction went into a stress state of opposite sign. The reason that this same set of equations may be used is simply that here each pair of plies acts as a unit in exactly the same way as the first pair of plies. Thus, the solution for each pair is exactly the same as the solution for the first pair. When a number of plies are bonded together and each of the plies has a different set of elastic characteristics, it is necessary to write equations similar to Eqs. (1) but now embracing the entire structure. For example, in a four-ply structure made up so that each ply is different from the others, either: in angle or in elastic characteristics, three equations would have to be written for each of the four plies, similar to the first three of Eqs. (1). In addition, a set of three equations would be necessary linking together the over-all average stresses and the stresses carried by each separate ply such as the last three of Eqs. (1). One would then be faced with a set of fifteen equations in the fifteen unknowns, such as given by Eqs6 (4). 39

Ply 1 E = [al1(aOl) ]a + [a12(o1) ]carll-. + [al3(Oej) ]arll c~ = [a21(cl1) ]a1 + [a22(cl) ]az1 + [a23(c~el) ]aqi e1] ~ = [a31(cl) ]aj1 + [a32(Col) ]a1 + [a33(cl) ]Ca0r Ply 2 ~ = [all(a2C) ]c6r2 + [a12(0a2) ]OC2 + [a13(Ca2) ]cxra2 ~ = [a21(ca2) ]C2 + [a22(a2) ]%cx2 + [a23(ca2) ]Cx.2 ~n = [a31(a2) ]af2 + [a32(ca2) ]] 2 + [a33(a2) ]af2 Ply 3 ~ = [a11(aO3) ]Cry3 + [al2(a3) ]a3 + [al3(o3) ]CaT3 = [a2l(a3) ]Co 3 + [a22(a3) ]a 3 + [a23(a3) ]ICn3 Etn = [a31(a3) ]a3 + [a32(aC3) ]a3 + [a33(a3) ]aqv Ply 4 E: f [ajll(c4) ]c4 + [a12(04) ]C4 + [a(3(a4) ]+a 14 En = [a2l(c4) ]a4 + [a22(o4) ]an4 + [a23(,c4) ]a4 ETn = [a31(a4) ]4 + [a32(oC4) ]Cq4 + [a33(Co4) ]an4 q(hz + h2 + h3 + h4) = h1, + h2a~2 + h3as3 + h4U54 ar(hz + h2 + h3 + h4) = hlal + h2CY=2 + h3sas + h+4T4 acx(hj + h2 + h3 + h4) = hlarll + h2a c2 + h3c3 + h4a4 In Eqs. (4), it has been assumed. that each ply lies at some arbitrary angle given by the symbol a with a subscript used to denote the particular ply in question, Similarly, it is most convenient to denote the stresses in the various plies by means of a subscript showing direction, such as ~ and r, 40

followed. by a number indicating the number of the ply involved. These fifteen equations in fifteen unknowns could be solved in just exactly the same manner used. here to solve nine equations in nine unknowns. This could. undoubtedly be done most conveniently by means of some standard program for the solution of a large number of simultaneous algebraic equations on a digital computer. In any event, the equations are determinate, could be solved, and are linear. Thus, answers from them could be obtained for quite general four-ply structure, and for a structure made up of any number of plies such as six, eight, or any larger number. In dealing with laminates involving odd numbers of plies, exactly the same line of reasoning may be used as just given for even numbers of plies. Here, however, it will be necessary simply to write the appropriate equations similar to the first three of Eqs. (1) for each of the plies involved in the structure. Finally, it will be necessary to construct a set of equations similar to the last three of Eqs. (1) involving a relation between the average stress on the structure and the stresses in the individual plies. Within these rules, a set of equations can now be constructed which will allow determination of the interply stresses as well as the loads carried by each of the individual plies in the structure, From the nature of these equations, it may be seen that the number of them necessary to describe a general, anisotropic, structure having n plies is 3(l+n). Since the number of equations only increases linearly with the number of plies, it should be relatively easy to study multi-ply structures

involving combinations of wire and textile materials in a very complete fashion. In addition, the effect of using varying cord angles in multi-ply laminates can be easily seen, 42

VIII REFERENCES 10 S. K. Clark, Interply Stresses and Load Distribution in Cord-Rubber Laminates, The Univ. of Michigan, Office of Research Administration, Technical Report 02957-8-T, Ann Arbor, Michigan. 2. So K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, The Univ. of Michigan, Office of Research Administration, Technical Report 02957-3-T, Ann Arbor, Michigan. 43

i

IX. DISTRIBUTION LIST Names No. of Copies The General Tire and Rubber Company Akron, Ohio 6 The Firestone Tire and Rubber Company Akron, Ohio 6 B. F. Goodrich Tire Company Akron, Ohio 6 Goodyear Tire and Rubber Company Akron, Ohio 6 United States Rubber Company Detroit, Michigan 6 S. So Attwood 1 R. A. Dodge 1 The University of Michigan ORA File 1 Se K. Clark 1 Project File 10

UNIVERSITY OF MICHIGAN 3 9015 02827 6056