******* Pipes-Pagano Approximation Gr/Ep ******* ******************************************************************************** ENGINEERING ELASTIC CONSTANTS ******************************************************************************** E1= 0.2100E+07 E2= 0.2100E+07 E3= 0.2000E+08 G12= 0.8500E+06 G13= 0.8500E+06 G23= 0.8500E+06 NU12= 0.2100E+00 NU13= 0.2205E-01 NU23= 0.2205E-01 NU21= 0.2100E+00 NU31= 0.2100E+00 NU32= 0.2100E+00 ******************************************************************************** STIFFNESSES CALCULATED FROM ENGINERRING CONSTANTS ******************************************************************************** C11= 0.2213E+07 C22= 0.2213E+07 C33= 0.2024E+08 C12= 0.4771E+06 C13= 0.5648E+06 C23= 0.5648E+06 C44= 0.8500E+06 C55= 0.8500E+06 C66= 0.8500E+06 ******************************************************************************** STIFFNESSES CALCULATED FROM COMPLIANCES WHICH WERE CALC. FROM ENGR. CONSTANTS ******************************************************************************** C11= 0.2213E+07 C22= 0.2213E+07 C33= 0.2024E+08 C12= 0.4771E+06 C13= 0.5648E+06 C23= 0.5648E+06 C44= 0.8500E+06 C55= 0.8500E+06 C66= 0.8500E+06 ******************************************************************************** LAYER WITH CRACK, ANGLE (DEG.)= 0.300E+02 LAYER WITHOUT CRACK, ANGLE (DEG.)= 0.600E+02 BELOW ARE STIFFNESSES FOR CRACKED/UNCRACKED LAYERS ******************************************************************************** ******************************************************************************** 0.3359E+07 0.4990E+06 0.3925E+07 0.0000E+00 0.1963E+07 0.0000E+00 0.4990E+06 0.2213E+07 0.5429E+06 0.0000E+00 0.3799E+05 0.0000E+00 0.3925E+07 0.5429E+06 0.1237E+08 0.0000E+00 0.5842E+07 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.8500E+06 0.0000E+00 0.0000E+00 0.1963E+07 0.3799E+05 0.5842E+07 0.0000E+00 0.4210E+07 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.8500E+06 ******************************************************************************** 0.1237E+08 0.5429E+06 0.3925E+07 0.0000E+00 0.5842E+07 0.0000E+00 0.5429E+06 0.2213E+07 0.4990E+06 0.0000E+00 0.3799E+05 0.0000E+00 0.3925E+07 0.4990E+06 0.3359E+07 0.0000E+00 0.1963E+07 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.8500E+06 0.0000E+00 0.0000E+00 0.5842E+07 0.3799E+05 0.1963E+07 0.0000E+00 0.4210E+07 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.8500E+06 ******************************************************************************** ******************************************************************************** MATRICES FOR EIGEN-VALUE PROBLEM FOR CRACKED/UNCRACKED LAYERS ARE WRITTEN BELOW [A]/[AP] ******************************************************************************** 0.0000E+00 0.4272E+00 0.0000E+00 -0.5858E+00 0.0000E+00 0.2767E+00 -0.4470E-01 0.0000E+00 -0.1639E+01 0.0000E+00 -0.2603E+01 0.0000E+00 0.0000E+00 -0.3143E+00 0.0000E+00 0.2767E+00 0.0000E+00 -0.1994E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 ******************************************************************************** 0.0000E+00 0.2449E+00 0.0000E+00 -0.2775E+00 0.0000E+00 0.1621E+00 -0.4470E-01 0.0000E+00 -0.1587E+01 0.0000E+00 -0.2603E+01 0.0000E+00 0.0000E+00 -0.5448E+00 0.0000E+00 0.1621E+00 0.0000E+00 -0.3478E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00 0.0000E+00 ******************************************************************************** W(1) 0.119E-06 0.136E+01 Z(1)-0.365E-07-0.362E+00 Z(2) 0.681E+00 0.000E+00 Z(3)-0.169E-07 0.236E+00 Z(4)-0.266E+00 0.336E-07 Z(5)-0.423E-07-0.499E+00 Z(6) 0.173E+00 0.478E-07 W(2) 0.119E-06-0.136E+01 Z(1)-0.365E-07 0.362E+00 Z(2) 0.681E+00 0.000E+00 Z(3)-0.169E-07-0.236E+00 Z(4)-0.266E+00-0.336E-07 Z(5)-0.423E-07 0.499E+00 Z(6) 0.173E+00-0.478E-07 W(3) 0.000E+00 0.987E+00 Z(1)-0.108E-07 0.551E+00 Z(2)-0.299E+00-0.181E-06 Z(3) 0.763E-07-0.317E+00 Z(4) 0.558E+00 0.000E+00 Z(5)-0.119E-06 0.303E+00 Z(6)-0.321E+00-0.836E-07 W(4) 0.000E+00-0.987E+00 Z(1)-0.108E-07-0.551E+00 Z(2)-0.299E+00 0.181E-06 Z(3) 0.763E-07 0.317E+00 Z(4) 0.558E+00 0.000E+00 Z(5)-0.119E-06-0.303E+00 Z(6)-0.321E+00 0.836E-07 W(5) 0.298E-07 0.240E+00 Z(1) 0.184E-07 0.112E+00 Z(2) 0.318E-01 0.548E-09 Z(3) 0.267E-07 0.203E+00 Z(4) 0.467E+00 0.161E-08 Z(5)-0.269E-07-0.132E+00 Z(6) 0.842E+00 0.000E+00 W(6) 0.298E-07-0.240E+00 Z(1) 0.184E-07-0.112E+00 Z(2) 0.318E-01-0.548E-09 Z(3) 0.267E-07-0.203E+00 Z(4) 0.467E+00-0.161E-08 Z(5)-0.269E-07 0.132E+00 Z(6) 0.842E+00 0.000E+00 ******************************************************************************** WP(1)-0.298E-07 0.116E+01 ZP(1) 0.512E-09-0.221E+00 ZP(2) 0.595E+00 0.000E+00 ZP(3) 0.984E-07 0.413E+00 ZP(4)-0.191E+00-0.407E-07 ZP(5)-0.444E-07-0.513E+00 ZP(6) 0.356E+00 0.213E-08 WP(2)-0.298E-07-0.116E+01 ZP(1) 0.512E-09 0.221E+00 ZP(2) 0.595E+00 0.000E+00 ZP(3) 0.984E-07-0.413E+00 ZP(4)-0.191E+00 0.407E-07 ZP(5)-0.444E-07 0.513E+00 ZP(6) 0.356E+00-0.213E-08 WP(3)-0.119E-06 0.935E+00 ZP(1) 0.641E-07-0.283E+00 ZP(2) 0.401E+00 0.111E-06 ZP(3) 0.133E-07 0.475E+00 ZP(4)-0.303E+00-0.112E-06 ZP(5) 0.599E-07-0.429E+00 ZP(6) 0.508E+00 0.000E+00 WP(4)-0.119E-06-0.935E+00 ZP(1) 0.641E-07 0.283E+00 ZP(2) 0.401E+00-0.111E-06 ZP(3) 0.133E-07-0.475E+00 ZP(4)-0.303E+00 0.112E-06 ZP(5) 0.599E-07 0.429E+00 ZP(6) 0.508E+00 0.000E+00 WP(5) 0.000E+00 0.394E+00 ZP(1) 0.485E-07 0.304E+00 ZP(2) 0.527E-01 0.513E-08 ZP(3) 0.143E-07 0.198E+00 ZP(4) 0.772E+00 0.000E+00 ZP(5)-0.897E-07-0.134E+00 ZP(6) 0.501E+00 0.254E-07 WP(6) 0.000E+00-0.394E+00 ZP(1) 0.485E-07-0.304E+00 ZP(2) 0.527E-01-0.513E-08 ZP(3) 0.143E-07-0.198E+00 ZP(4) 0.772E+00 0.000E+00 ZP(5)-0.897E-07 0.134E+00 ZP(6) 0.501E+00-0.254E-07 ******************************************************************************** SUM FOR EQUATION NO.1 =-0.1342E+01-0.2250E+01 CRACKED LAYER SUM FOR EQUATION NO.1 = 0.3000E+00 0.0000E+00 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.2 =-0.1312E+01-0.7822E-01 CRACKED LAYER SUM FOR EQUATION NO.2 =-0.6250E-01 0.1429E+00 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.3 =-0.2361E+01-0.6500E+01 CRACKED LAYER SUM FOR EQUATION NO.3 =-0.9479E-02-0.2500E+00 UNCRACKED LAYER ******************************************************************************** EIGEN-VALUE P(1)= 0.1192E-06 0.1364E+01 CRACKED LAYER NORMALIZED EIGEN-VECTOR V(1,1)= 0.5685E-07 0.4494E+00 V(2,1)=-0.8440E+00 0.7155E-07 V(3,1)= 0.8081E-07-0.2927E+00 ******************************************************************************** EIGEN-VALUE PP(1)=-0.2980E-07 0.1160E+01 LAYER NO CRACK NORMALIZED EIGEN-VECTOR VP(1,1)=-0.6228E-07 0.2919E+00 VP(2,1)=-0.7858E+00 0.6806E-07 VP(3,1)= 0.3265E-08-0.5453E+00 ******************************************************************************** ******************************************************************************** SUM FOR EQUATION NO.1 = 0.6621E+00-0.5000E+00 CRACKED LAYER SUM FOR EQUATION NO.1 = 0.5244E+00 0.2500E+00 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.2 = 0.3750E+00 0.7017E-01 CRACKED LAYER SUM FOR EQUATION NO.2 = 0.2500E+00 0.9524E-01 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.3 = 0.1086E+01-0.1000E+01 CRACKED LAYER SUM FOR EQUATION NO.3 = 0.1215E+00 0.2500E+00 UNCRACKED LAYER ******************************************************************************** EIGEN-VALUE P(2)= 0.0000E+00 0.9869E+00 CRACKED LAYER NORMALIZED EIGEN-VECTOR V(1,2)= 0.0000E+00-0.7842E+00 V(2,2)= 0.4263E+00 0.1673E-06 V(3,2)=-0.1175E-06 0.4508E+00 ******************************************************************************** EIGEN-VALUE PP(2)=-0.1192E-06 0.9350E+00 LAYER NO CRACK NORMALIZED EIGEN-VECTOR VP(1,2)=-0.1535E-06 0.4143E+00 VP(2,2)=-0.5870E+00-0.8200E-07 VP(3,2)= 0.0000E+00-0.6956E+00 ******************************************************************************** ******************************************************************************** SUM FOR EQUATION NO.1 = 0.1021E+00 0.2500E+00 CRACKED LAYER SUM FOR EQUATION NO.1 =-0.9761E-02-0.2500E+00 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.2 =-0.3125E-01 0.2162E-01 CRACKED LAYER SUM FOR EQUATION NO.2 =-0.1875E+00 0.2150E+00 UNCRACKED LAYER ******************************************************************************** SUM FOR EQUATION NO.3 = 0.1784E+00 0.5234E+00 CRACKED LAYER SUM FOR EQUATION NO.3 =-0.4233E-01-0.2969E+00 UNCRACKED LAYER ******************************************************************************** EIGEN-VALUE P(3)= 0.2980E-07 0.2405E+00 CRACKED LAYER NORMALIZED EIGEN-VECTOR V(1,3)= 0.1652E-08-0.4805E+00 V(2,3)=-0.1362E+00 0.2763E-07 V(3,3)= 0.0000E+00-0.8664E+00 ******************************************************************************** EIGEN-VALUE PP(3)= 0.0000E+00 0.3943E+00 LAYER NO CRACK NORMALIZED EIGEN-VECTOR VP(1,3)= 0.0000E+00-0.8299E+00 VP(2,3)=-0.1438E+00 0.9639E-07 VP(3,3)= 0.2726E-07-0.5390E+00 ******************************************************************************** T111-0.571E+05 0.589E+00 T112-0.145E+05-0.371E+00 T113 0.976E+06-0.115E+00 T121 0.483E-01 0.382E+06 T122 0.000E+00-0.667E+06 T123 0.140E-02-0.408E+06 T131-0.280E+06 0.995E+00 T132 0.675E+06-0.671E+00 T133 0.170E+07-0.208E+00 T211 0.483E-01 0.382E+06 T212 0.000E+00-0.667E+06 T213 0.140E-02-0.408E+06 T221-0.167E+07 0.204E+00 T222 0.731E+06 0.307E+00 T223-0.184E+06 0.466E-01 T231-0.998E-01-0.123E+07 T232-0.240E+00 0.741E+06 T233-0.910E-02-0.764E+06 T311-0.280E+06 0.995E+00 T312 0.675E+06-0.671E+00 T313 0.170E+07-0.208E+00 T321-0.998E-01-0.123E+07 T322-0.240E+00 0.741E+06 T323-0.910E-02-0.764E+06 T331 0.900E+06 0.174E+01 T332-0.751E+06-0.134E+01 T333 0.318E+07-0.386E+00 ******************************************************************************** TP111 0.777E+05-0.357E+00 TP112-0.290E+05-0.846E+00 TP113 0.267E+07 0.945E-01 TP121-0.529E-01 0.248E+06 TP122-0.130E+00 0.352E+06 TP123 0.000E+00-0.705E+06 TP131-0.214E+06-0.299E+00 TP132-0.377E+06-0.652E+00 TP133 0.179E+07 0.248E-01 TP211-0.529E-01 0.248E+06 TP212-0.130E+00 0.352E+06 TP213 0.000E+00-0.705E+06 TP221-0.144E+07 0.158E+00 TP222-0.989E+06-0.147E+00 TP223-0.200E+06 0.219E+00 TP231-0.444E-01-0.124E+07 TP232 0.125E+00-0.106E+07 TP233-0.913E-02-0.506E+06 TP311-0.214E+06-0.299E+00 TP312-0.377E+06-0.652E+00 TP313 0.179E+07 0.248E-01 TP321-0.444E-01-0.124E+07 TP322 0.125E+00-0.106E+07 TP323-0.913E-02-0.506E+06 TP331 0.107E+07-0.576E-01 TP332 0.113E+07-0.141E+00 TP333 0.128E+07 0.842E-01 ******************************************************************************** L=1 T1 =-0.134E+01-0.244E+01 T2 =-0.138E+01-0.782E-01 T3 =-0.950E+01 0.314E+01 L=1 T1P= 0.300E+00-0.156E-01 T2P=-0.250E+00 0.143E+00 T3P=-0.500E+00 0.154E+00 ******************************************************************************** L=2 T1 = 0.662E+00-0.625E+00 T2 = 0.312E+00 0.702E-01 T3 =-0.188E+00-0.100E+01 L=2 T1P= 0.524E+00 0.344E+00 T2P= 0.000E+00 0.952E-01 T3P= 0.250E+00-0.184E-01 ******************************************************************************** L=3 T1 = 0.102E+00 0.250E+00 T2 = 0.000E+00 0.216E-01 T3 = 0.125E+00-0.213E-01 L=3 T1P=-0.976E-02-0.250E+00 T2P=-0.172E+00 0.215E+00 T3P=-0.297E+00 0.232E+00 ******************************************************************************** ******************************************************************************** IK= 1 K= 0.000000100 DET= 0.136527944E+01 IK= 2 K= 0.010000099 DET= 0.137244642E+01 IK= 3 K= 0.020000100 DET= 0.113367558E+01 IK= 4 K= 0.030000100 DET= 0.394028568E+01 IK= 5 K= 0.040000100 DET= 0.959274197E+01 IK= 6 K= 0.050000101 DET= 0.191899776E+01 IK= 7 K= 0.060000099 DET= 0.338678026E+01 IK= 8 K= 0.070000097 DET= 0.547674465E+01 IK= 9 K= 0.080000095 DET= 0.829996300E+01 IK= 10 K= 0.090000093 DET= 0.119604135E+01 IK= 11 K= 0.100000091 DET= 0.165506470E+01 IK= 12 K= 0.110000089 DET= 0.221471810E+01 IK= 13 K= 0.120000087 DET= 0.288059092E+01 IK= 14 K= 0.130000085 DET= 0.365576696E+01 IK= 15 K= 0.140000090 DET= 0.454039669E+01 IK= 16 K= 0.150000095 DET= 0.553135157E+01 IK= 17 K= 0.160000101 DET= 0.662194633E+01 IK= 18 K= 0.170000106 DET= 0.780176258E+01 IK= 19 K= 0.180000111 DET= 0.905663681E+01 IK= 20 K= 0.190000117 DET= 0.103687227E+01 IK= 21 K= 0.200000122 DET= 0.117168188E+01 IK= 22 K= 0.210000128 DET= 0.130767834E+01 IK= 23 K= 0.220000133 DET= 0.144222093E+01 IK= 24 K= 0.230000138 DET= 0.157252097E+01 IK= 25 K= 0.240000144 DET= 0.169574451E+01 IK= 26 K= 0.250000149 DET= 0.180912900E+01 IK= 27 K= 0.260000139 DET= 0.191010261E+01 IK= 28 K= 0.270000130 DET= 0.199643397E+01 IK= 29 K= 0.280000120 DET= 0.206636286E+01 IK= 30 K= 0.290000111 DET= 0.211874866E+01 IK= 31 K= 0.300000101 DET= 0.215320516E+01 IK= 32 K= 0.310000092 DET= 0.217022133E+01 IK= 33 K= 0.320000082 DET= 0.217127228E+01 IK= 34 K= 0.330000073 DET= 0.215889716E+01 IK= 35 K= 0.340000063 DET= 0.213676071E+01 IK= 36 K= 0.350000054 DET= 0.210965490E+01 IK= 37 K= 0.360000044 DET= 0.208349180E+01 IK= 38 K= 0.370000035 DET= 0.206524515E+01 IK= 39 K= 0.380000025 DET= 0.206282020E+01 IK= 40 K= 0.390000015 DET= 0.208492565E+01 IK= 41 K= 0.400000006 DET= 0.214086103E+01 IK= 42 K= 0.409999996 DET= 0.224030995E+01 IK= 43 K= 0.419999987 DET= 0.239303732E+01 IK= 44 K= 0.429999977 DET= 0.260861182E+01 IK= 45 K= 0.439999968 DET= 0.289608192E+01 IK= 46 K= 0.449999958 DET= 0.326366305E+01 IK= 47 K= 0.459999949 DET= 0.371836567E+01 IK= 48 K= 0.469999939 DET= 0.426572895E+01 IK= 49 K= 0.479999930 DET= 0.490946150E+01 IK= 50 K= 0.489999920 DET= 0.565121841E+01 IK= 51 K= 0.499999911 DET= 0.649036121E+01 IK= 52 K= 0.509999931 DET= 0.742378712E+01 IK= 53 K= 0.519999921 DET= 0.844581223E+01 IK= 54 K= 0.529999912 DET= 0.954814243E+01 IK= 55 K= 0.539999902 DET= 0.107199025E+01 IK= 56 K= 0.549999893 DET= 0.119477403E+01 IK= 57 K= 0.559999883 DET= 0.132160306E+01 IK= 58 K= 0.569999874 DET= 0.145071149E+01 IK= 59 K= 0.579999864 DET= 0.158016348E+01 IK= 60 K= 0.589999855 DET= 0.170789099E+01 IK= 61 K= 0.599999845 DET= 0.183174741E+01 IK= 62 K= 0.609999835 DET= 0.194954336E+01 IK= 63 K= 0.619999826 DET= 0.205910301E+01 IK= 64 K= 0.629999816 DET= 0.215832114E+01 IK= 65 K= 0.639999807 DET= 0.224520898E+01 IK= 66 K= 0.649999797 DET= 0.231795001E+01 IK= 67 K= 0.659999788 DET= 0.237494254E+01 IK= 68 K= 0.669999778 DET= 0.241485023E+01 IK= 69 K= 0.679999769 DET= 0.243662715E+01 IK= 70 K= 0.689999759 DET= 0.243955779E+01 IK= 71 K= 0.699999750 DET= 0.242327523E+01 IK= 72 K= 0.709999740 DET= 0.238777184E+01 IK= 73 K= 0.719999731 DET= 0.233340836E+01 IK= 74 K= 0.729999721 DET= 0.226089811E+01 IK= 75 K= 0.739999712 DET= 0.217131400E+01 IK= 76 K= 0.749999702 DET= 0.206603956E+01 IK= 77 K= 0.759999692 DET= 0.194676089E+01 IK= 78 K= 0.769999683 DET= 0.181541419E+01 IK= 79 K= 0.779999673 DET= 0.167415714E+01 IK= 80 K= 0.789999664 DET= 0.152530503E+01 IK= 81 K= 0.799999654 DET= 0.137129712E+01 IK= 82 K= 0.809999645 DET= 0.121463013E+01 IK= 83 K= 0.819999635 DET= 0.105781484E+01 IK= 84 K= 0.829999626 DET= 0.903318214E+01 IK= 85 K= 0.839999616 DET= 0.753514481E+01 IK= 86 K= 0.849999607 DET= 0.610637903E+01 IK= 87 K= 0.859999597 DET= 0.476736498E+01 IK= 88 K= 0.869999588 DET= 0.353634810E+01 IK= 89 K= 0.879999578 DET= 0.242895865E+01 IK= 90 K= 0.889999568 DET= 0.145793414E+01 IK= 91 K= 0.899999559 DET= 0.632856750E+01 IK= 92 K= 0.909999549 DET=-0.400438738E+01 IK= 93 K= 0.919999540 DET=-0.558041573E+01 IK= 94 K= 0.929999530 DET=-0.922015953E+01 IK= 95 K= 0.939999521 DET=-0.113652253E+01 IK= 96 K= 0.949999511 DET=-0.120979357E+01 IK= 97 K= 0.959999502 DET=-0.115375137E+01 IK= 98 K= 0.969999492 DET=-0.983940029E+01 IK= 99 K= 0.979999483 DET=-0.719462776E+01 IK= 100 K= 0.989999473 DET=-0.382868004E+01 BETWEEN 0.0 AND 0.1 ONLY 2 ROOT/S WERE FOUND APPROXIMATION FOR THREE ROOTS FOR K BETWEEN 0.0 AND 1.0 KR(1)= 0.899999559 KR(2)= 0.000000000 KR(3)= 0.000000000