THE UNIVERSITY OF MI CHI GAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 15 TABLES OF ELASTIC CONSTANTS OF ORTHOTROPIC LAMINATES S. K. lCark N. L. Field -. Project Directors: S. K. Clark and Ro Ao Dodge ORA Project 02957 administered through OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1962 Revised November 1962

The idea for this report was suggested to the authors by Dr. Harold Howe of the United States Rubber Company, who also furnished the numerical data for some of the tables in this report. His contributions are gratefully acknowledged.

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 B. F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATES RUBBER COMPANY 111

NOMENCLATURE x Ex = modulus in x direction (cord direction) Ey = modulus in y direction - Y Y (perpendicular to cord) F = cros s -modulus Gxy = shear modulus SINGLE SHEET OF FABRIC = modulus in! direction E, = modulus in ~ direction _Fe,= cross-modulus = cord half angle EVEN NUMBER OF LAMINATED FABRIC SHEETS

I. FOREWORD It has been shown in Ref. 1 that the elastic constants of orthotropic layered materials may be calculated in terms of a small number of parameters. This information is fundamental to a quantitative understanding of materials used in pneumatic tire construction, and is presented here in the hope that it will provide a convenient numerical guide to aid the ever-broadening applications of mechanics to the tire industry.

II. SUMMARY A digital computer program described in Ref. 2 is used to solve the simultaneous equations relating stress and strain in an orthotropic two-dimensional sheet. From the Solutions the elastic constants of the composite sheet may be determined. These are presented in dimensionless form over a range of variable encompassing all commonly used textile and wire fabric in rubber.

III o DETERMINATION OF ELASTIC CONSTANTS The elastic constants of layered orthotropic laminates may be expressed in terms of the elastic constants of each layer and the angle at which the layers are oriented when assembled. For those cases where a single angle of assembly is chosen, and where each structure is built up of pairs of identical plies laminated together so that an even number of them always exists, it is possible to express the elastic constants of the total structure as a function of this angle and of the elastic constants of a single sheet of the material. This is the most common case encountered in the use of coated fabrics, and merits some detailed attention. By means of certain simplifying assumptions, as discussed in Refo 1, the elastic constants of a layered structure of the type described may be expressed as a function of the angle of assembly and of a dimensionless quantity representative of the degree of anisotropy of a single sheet of the fabric. Tabular data on these elastic characteristics of layered structures are presented, similar in form to those presented in Ref. 3, but with considerably more precision and in somewhat greater detail. The primary difference between the graphical data of Ref. 3, and the tabular information presented here is that here the ratio Gxy/Ey is taken to be 0025, instead of the value 0.333 used previously. In order to demonstrate that the former value is correct, consider the process of simple shear on a typical element of coated fabric in which 5

S = cord spacing d = cord diameter GR = rubber modulus F = deflection T = shear stress Referring to Fig.l, and assuming the cords to be rigid compared to the rubber, the S RUBBER GR CORD Fig. 1 effective shear strain over several cord lengths is given by Yeff = where s = _ (S - d) GR Hence, 7eff = (1 - d GR S and the effective shear modulus of the composite is Geff = (1) eff 7eff (1 - To find a similar expression for extension perpendicular to the cords, refer to Fig. 2 in which a similar geometry is shown. Here, stresses are applied causing

the rubber between cords to extend. _ c-y Y L rF -..... s RUBBER WEB 1dA Y -AAAol(u cry Fig. 2. 8t' - Cx (S -d) EWEB But EWEB must be determined using the fact that the web is prevented from contracting freely in the x direction due to the rigidity of the cords. For this, note that from Hooke's law, the strain in the x direction vanishes, E0 = = Yx- icER ER *x = ax ER ER In the y direction, [z =x = y (1 _ 42) ER -ER ER Hence the effective modulus of the web is ET Cx ER EWEB = Ey — L 7y

The mean, or effective, modulus of the composite is = EWEB ER Eeff WEB ER 1 (~-) 1-d (( - (2 The ratio of effective moduli of the composite is Geff =Gxy GR(1 -2) Eeff Ey ER But it is well known for linearly elastic materials that G E (4) 2(1 + i) Using Eq. (4) in Eq. (3) gives Gx _ 1-p.2 - 1)G ~~~~(5) Ey 2(1 + p) 2 For an incompressible material such as rubber, p = 0.5 and = 0.25 (6) E RUBBER WEB In all of the subsequent data the ratio Ex/Fxy is taken to be 0.5. The origin of this numerical value is thoroughly discussed in Ref. 1. The dimensionless quantity Gxy/Ex of a single sheet may be thought of as characterizing the degree of anisotropy of the sheet. It is a parameter used in these computations and is allowed to range over eight separate values so that interpolation can be performed readily where it is required.

The tables are presented using a notation borrowed from digital computer practice, in which all numbers are given as decimals, followed by a capital E, and in turn followed by a number representing the exponent of ten which should be used as a multiplier for the decimal number. For example,.123861E 3 should be read as 123.861 while.996865E-2 should be read as.00996865o The symmetry properties of the elastic constants may be used to generate data for cord angles lying between 450 and 900. These properties were reviewed in Ref. 4, but are repeated here for convenience. They are: Gr (900-ca) = G%(Ca) F (900-c) = F~ e (a) E.g(450~+) = E.(45 -a ) E(45 ) (45 +C) -ca) Values of E,/Ex are given in Table I, F~U/Ex in Table II, and GOl/Ex in Table III. It should be noted that the modulus obtainable from an ordinary tensile test will correspond to the quantity Et, if the proper value of cord angle and Gxy/Ex are chosen, up to a cord angle of about 560. At this point the tensile loads applied to a specimen cause the cords to move through the zero load condition and, as the angle increases, to load the cords into compression. At this point the ratio Gxy/Ex changes since the effective modulus of the cord Ex changes. Thus, a new value of Gxy/Ex becomes applicable and, if this new value is used in reading from the tables, the resulting E~ will now correspond to the measured modulus on up to a cord angle of 900~ 9

TABLE I DIMENSIONLESS EXTENSION MODULUS Eg/Gxy Gxy/Ex Cori x 10-4 5 x 10-4 1 x 10-3 2 x 10-3 4 x 10-3 6 x 10-3 8 x 10-3 1 x 10-2 Angle (Degrees) 0 10000.00000 2000.00010 1000.00000 500.00006 250.00003 166.66668 125.00001 100.00000 1 9988.55100 1998.08340 999.06635 499.54036 249.77298 166.51655 124.88822 99.91119 2 9926.72640 1991.23840 995.99362 498.09461 249.07586 166.05935 124.54917 99.64246 3 9735.57090 1976.23750 989.98075 495.46642 247.86156 165.27505 123.97212 99.18726 4 9304.22820 1947.97440 979.74028 491.33753 246.05351 164.13104 123.13948 98.53485 5 8542.17540 1900.12740 963.60775 485.28785 243.54965 162.58366 122.02778 97.67090 6 7452.50150 1826.44750 939.74055 476.82723 240.22795 160.58051 120.60877 96.57817 7 6163.48630 1722.70900 906.44662 465.45123 235.95607 158.06382 118.85102 95.23730 8 4867.09190 1588.76730 862.60515 450.71075 230.60331 154.97485 116.72225 93.62851 9 3718.65900 1429.71250 808.08798 432.29940 224.05557 151.25901 114.19178 91.73253 10 2788.01040 1255.25350 744.06324 410.14770 216.23434 146.87332 111.23382 89.53301 11 2074.82900 1077.25070 672.94147 384.48581 207.11358> 141.79241 107.83115 87.01807 12 1544.98740 906.73134 598.02607 355.86530 196.73571 136.o1604 103.97833 84.18278 13 1156.77460 751.67414 522.88638 325.11920 185.21966 129.57406 99.68518 81.03074 14 873 ~ 17138 616.35554 450.73114 293.26131 172.75992 122.52961 94.97849 77.57594 15 665.27420 501.83225 383.98392 261.35713 159.61583 114.97809 89.90382 73.84353 16 511.79197 407.00669 324.11612 230.38510 146.08855 107.04378 84.52371 69.87002 17 397.47019 329.64273 271.72777 201.14411 132.49502 98.87133 78.91598 65.70233 18 311.48820 267.10094 226.74881 174.19615 119.13950 90.61560 73.16887 61.39582 19 246.18127 216.79748 188.67671 149.86555 106.28884 82.43048 67.37616 57.01171 20 196.10283 176.42211 156.77735 128.26784 94.15676 74.45807 61.63088 52.61357 21 157.34872 144.01422 130.23672 109.35821 82.89591 66.82066 56.02014 48.26410 22 127.09873 117.96397 108.25311 92.98161 72.59855 59.61478 50.62036 44.02147 23 103.29800 96.97393 90.09090 78.91840 63.30303 52.90969 45.49409 39.93701 24 84.43233 80.01072 75.10329 66.91927 55.00oo406 46.74741 40.68857 36.05311 25 69.37682 66.25663 62.73687 56.72976 47.66332 41.14595 36.23516 32.40215 26 57.28679 55.06627 52.52694 48.10664 41.22035 36.10312 32.15089 29.00650 27 47.52316 45.99052 44.08818 40.82641 35.60156 31.60116 28.43993 25.87894 28 39.59783 38.44740 37.10362 34.68969 30.72740 27.61116 25.09619 23.02379 29 33.13480 32.29851 31.31356 29.52192 26.51736 24.09694 22.10545 20.43816 30 27.84253 27.23113 26.50595 25.17249 22.89386 21.01843 19.44792 18.11358 31 23.49315 23.04393 22.50787 21.51291 19.78436 18.33419 17.10018 16.03738 32 19.90723 19.57574 19.17811 18.43406 17.12249 16.00319 15.03680 14.19400 33 16.94266 16.69717 16.40136 15.84392 14.84862 13.98620 13.23173 12.56615 34 14.48621 14.30387 14.08330 13.66508 12.90986 12.24653 11.65928 11.13575 35 12.44706 12.31134 12.14661 11.83255 11.25975 10.75050 10.29480 9.88463 36 10.75204 10.65089 10.52773 10.29182 9.85775 9.46764 9.11514 8.79509 37 9.34185 9.26641 9.17443 8.99719 8.66870 8.37061 8.09890 7.85023 38 8.16823 8.11198 8.04317 7.91033 7.66224 7.43516 7.22655 7.03424 39 7.19168 7.14981 7.09847 6.99905 6.81222 6.63989 6.48045 6.33251 40 6.37979 6.34869 6.31050 6.23632 6.09617 5.96600 5.84479 5.73116 41 5.70580 5.68280 5.65451 5.59941 5.49481 5.39705 5.30551 5.21962 42 5.14759 5.13067 5.10983 5.06916 4.99160 4.91873 4.85014 4.78548 43 4.68676 4.67441 4.65918 4.62940 4.57239 4.51858 4.46770 4.41952 44 4.30797 4.29905 4.28804 4.26647 4.22505 4.18578 4.14851 4.11310 10

TABLE I (Concluded) DIMENSIONLESS EXTENSION MODULUS Et/Gxy Sxy/Ex Cord 1 x 10-4 5 x 10-4 1 x 10-3 2 x 10-3 4 x 10-3 6 x 10-3 8 x 10-3 1 x 10'-2 Angle Degrees ) 45 3.99840 3.99205 3.98421 3.96881 3.93917 3.91097 3.88412 3.85852 46 3.74730 3.74287 3.73739 3.72662 3.70585 3.68603 3.66711 3.64902 47 3.54561 3.54261 3.53889 3.53157 3.51744 3.50393 3.49100 3.47863 48 3.38572 3.38376 3-.38134 3.37658 3.36738 3.35858 3.35016 3.34209 49 3.26115 3.25996 3.25850 3.25561 3.25005 3.24473 3.23965 3.23480 50 3.16645 3.16582 3.16504 3.16352 3.16059 3.15782 3.15519 3.15271 51 3.09697 3.09674 3.0o9645 3.09590 3.o09486 3.09392 3.09305 3.09228 52 3.04877 3.04881 3.04887 3.04899 3.04927 3.o4960 3.04998 3.05041 53 3.01847 3.01870 3.01898 3.01956 3.02073 3.02193 3.02316 3.02441 54 3.00319 3.00354 3.00397 3.00484 3.oo00658 3.00834 3.01012 3.01190 55 3.ooo00048 3.00090 3.00141 3.00245 3.00453 3.oo00662 3.00872 3.01082 56 3.00823 3.00867 3.00923 3.01036 3.01260 3.01485 3.01710 3.01936 57 3.02462 3.02508 3.02565 3.02679 3.02907 3.03136 3.03365 3.03594 58 3.04811 3.04855 3.04911 3.05023 3.05246 3.05471 3.05695 3.05920 59 3.07735 3.07778 3.07831 3.07937 3.08150 3.o8364 3.08579 3.08794 60 3.11121 3.11161 3.11210 3.11309 3.11508 3.11708 3.11908 3.12109 61 3.14869 3.14905 3.14950 3.15041 3.15223 3.15407 3.15591 3.15776 62 3.18893 3.18926 3.18967 3.19049 3.19214 3.19380 3.19547 3.19715 63 3.23121 3.23150 3.23187 3.23260 3.23407 3.23555 3.23705 3.23856 64 3.27489 3.27514 3.27546 3.27611 3.27741 3.27872 3.28004 3.28138 65 3.31941 3.31963 3.31991 3.32047 3.32161 3.32276 3.32392 3.32510 66 3.36431 3.36450 3-36475 3-36523 3.36621 3.36721 3.36822 3.36924 67 3.40918 3.40935 3.40955 3.40997 3.41081 3.41166 3.41253 3.41341 68 3.45367 3.45381 3.45398 3.45434 3.45505 3.45577 3.45651 3.45727 69 3.49747 3.49759 3.49773 3.49803 3.49863 3.49924 3.49986 3.50050 70 3.54033 3.54042 3.54054 3.54079 3.54128 3.54179 3.54231 3.54285 71 3.58200 3.58208 3.58218 3.58238 3.58279 2.58321 3.58364 3.58408 72 3.62231 V 3.62237 3.62245 3.62261 3.62294 3.62329 3.62364 3.62400 73 3.66108 3.66113 3.66119 3.66132 3.66159 3.66186 3.66215 3.66244 74 3.69816 3.69820 3.69825 3.69835 2.69856 3.69878 3.69901 3.69925 75 3.73344 3.73347 3.73351 3.73359 3.73375 3.73393 3.73411 3.73430 76 3.76681 3.76683 3.76686 3.76692 3.76705 3.76718 3.76732 3.76747 77 3.79817 3.79819 3.79821 3.79825 3.79835 3.79845 3.79856 3.79867 78 3.82745 3.82746 3.82748 3.82751 3.82758 3.82766 3.82774 3.82782 79 3.85458 3.85459 3.85460 3.85462 3.85467 3.85473 3.85479 3.85485 80 3.87950 3.87951 3-87952 3.87953 3.87957 3.97961 3.87965 3.87970 81 3.90218 3.90218 3.90219 3.90220 3.90222 3.90225 3.90228 3.90231 82 3.92256 3.92256 3.92256 3.92257 3.92258 3.92260 3.92262 3.92265 83 3.94061 3.94061 3.94061 3.94061 3.94062 3.94064 3.94065 3.94067 84 3.95630 3.95630 3.95630 3.95631 3.95631 3.95632 3.95633 3.95634 85 3.96962 3.96962 3.96962 3.96962 3.96962 3.96963 3.96963 3.96964 86 3.98054 3.98054 3.98054 3.98054 3.98054 3.98054 3.98055 3.98055 87 3.98904 3.98904 3.98904 3.98905 3.98905 3.98905 3.98905 3.98905 88 3.99513 3.99513 3.99513 3.99513 3.99513 3.99513 3.99513 3.99513 89 3.99878 3.99878 3.99878 3.99878 3.99878 3.99878 3.99878 3.99878 90 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 11

TABLE II DIMENSIONLESS EXTENSION MODULUS FS /Gxy NGxy/Ex Corl 1 x 10-4 5 x 10-4 1 x 10-3 2 x 10-3 4 x 10-3 6 x 10-3 8 x 10-3 1 x 10-2 Angle (Degrees ) 0 20000.00100 4000.00030 2000.00010 1000.00010 500.00005 533333556 250.00002 200.00001 1 7923.40910 3064.95620 1735.07380 928.94189 481.51305 324.98417 245.25724 196.94254 2 2814.54060 1800.05080 1240.93980 765.44054 433.34912 302.22895 232.02599 188.29040 3 1353.79120 1064.72570 840.41826 591.28888 371.21662 270.53329 212.81602 175.39799 4 782.06946 675.84984 577.76360 447.79282 308.85029 235.71924 190.59465 159.97389 5 505.41100 458.69939 411.19633 340.64636 253.62751 202.02846 167.88048 1435.61106 6 351.77019 328.41894 303.25698 262.96644 207.76940 171.73223 146.35422 127.51547 7 257.96189 245.13384 230.78905 206.61142 170.82894 145.61889 126.89873 112.44819 8 196.60442 189.053436 180.35480 165.18821 141.41317 123.62835 109.82315 98.79625 9 154.32606 149.60243 144.09032 134.20320 118.01494 105.31875 95.09471 86.68480 10 123.98290 120.90151 117.25933 110.59767 99.31935 90.13486 82.51065 76.08026 11 101.48205 99.39821 96.91128 92.29453 84.27109 77.53696 71.80449 66.86564 12 84.34389 82.89230 81.14702 77.86941 72.05374 67.05172 62.70376 58.88944 13 70.99596 69.95957 68.70625 66.33081 62.04496 58.28416 54.95748 51.99386 14 60.40223 59.64684 58.72909 56.97682 53.77195 50.91278 48.34620 46.02952 15 51.85799 51.29778 50.61459 49.30228 46.87500 44.67942 42.68388 40.86223 16 44.87043 44.44883 43.93309 42.93748 41.07866 39.37763 37.81513 36.37488 17 39.08661 38.76539 38.37142 37.60774 36.17065 34.84254 33.61145 32.46712 18 34.24842 34.00114 33.69721 33.10598 31.98596 30.94211 29.96692 29.05382 19 30.16360 29.97162 29.73523 29.27401 28.39527 27.57032 26.79439 26.06320 20 26.68662 26.53656 26.35149 25.98949 25.29636 24.64157 24.02201 23.43492 21 23.70568 23.58778 23.44217 23.15672 22.60783 22.08645 21.59054 21.11829 22 21.13378 21.04081 20.92586 20.70007 20.26428 19.84832 19.45087 19.07071 23 18.90240 18.82896 18.73805 18.55918 18.21280 17.88077 17.56221 17.25632 24 16.95705 16.89902 16.82712 16.68545 16.41028 16.14551 15.89056 15.64488 25 15.25391 15.20814 15.15140 15.03943 14.82138 14.61085 14.40745 14.21083 26 13.75749 13.72155 13.67696 13.58887 13.41691 13.25037 13.08898 12.93251 27 12.43879 12.41078 12.37600 12.30722 12.17266 12.04197 11.91496 11.79150 28 11.27398 11.25240 11.22558 11.17250 11.06844 10.96710 10.86836 10.77212 29 10.24329 10.22695 10.20664 10.16638 10.08731 lo.oloo09 9.93467 9.86098 30 9.33031 9.31826 9.30327 9.27354 9.21501 9.15771 9.10159 9.04661 31 8.52130 8.51278 8.50217 8.48109 8.43952 8.39869 8.35859 8.31919 32 7.80472 7.79910 7.79210 7.77818 7.75062 7.72346 7.69669 7.67028 33 7.17085 7.16763 7.16361 7.15560 7.13968 7.12388 7.10820 7.09264 34 6.61149 6.61025 6.60870 6.60559 6.59931 6.59296 6.58653 6.58003 35 6.11969 6.12010 6.12060 6.12154 6.12327 6.12477 6.12605 6.12712 36 5.68961 5.69137 5.69356 5.69787 5.70622 5.71424 5.72194 5.72931 37 5.31629 5.31917 5.32275 5.32983 5.34367 5. 35708 5.37009 5.38270 38 4.99559 4.99939 5.00411 5.01346 5.03179 5.04965 5.06703 5.08396 39 4.72402 4.72857 4.73422 4.74542 4.76741 4.78888 4.80983 4.83028 40 4.49875 4.50389 4.51028 4.52295 4.54788 4.57224 4.59606 4.61935 41 4.31743 4.32304 4.33001 4.34384 4.37105 4.39768 4.42374 4.44925 42 4.17824 4.18419 4.19160 4.20629 4.23522 4.26355 4.29130 4.31847 43 4.07979 4.08598 4.09369 4.10898 4.13910 4.16861 4.19753 4.22586 44 4.02110 4.02743 4.03532 4.05096 4.08178 4.11199 4.1416o 4.17061 45 4.00160 4.00798 4.01592 4.03168 4.06274 4.09318 4.12301 4.15225 12

TABLE III DIMENSIONLESS SHEAR MbDULUS G,/Gxy Cord gle 1 x 10'4 5 x 10-4 1 x 10-3 2 x 10-3 4x10-3 6x10-3 8x10-3 1x10-2 Angle (Degrees) _ 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1 4.04403 1.60807 1.3u358 1.15133 1.07521 1.04984 1.03715 1.02954 2 13.16128 3.42934 2.21284 1.60460 1.30048 1.19911 1.14842 1.11801 3 28.30736 6.45491 3.72336 2.35759 1.67470 1.44708 1.33327 1.26499 4 49.40829 10.67006 5.82777 3.40663 2.19607 1.79255 1.59080 1.46975 5 76.36158 16.05425 8.51582 4.74662 2.86202 2.23384 1.91975 1.73130 6 109.03508 22.58115 11.77439 6.37100 3.66933 2.76878 2.31852 2.04837 7 147.26868 30.21905 15.58764 8.27190 4.61405 3.39478 2.78517 2.41941 8 190.87539 38.93080 19.93703 10.44005 5.69159 4.10880 3.31742 2.84261 9 239.64838 48.67381 24.80118 12.86485 6.89670 4.90734 3.91269 3.31591 10 293.35750 59.40084 30.15667 15.53452 f8.22350 5.78652 4.56806 3.83701 11 351.71513 71.05932 35.97715 18.43605 9.66553 6.74205 5.28035 4.40336 12 414.44077 83.59296 42.23440 21.55531 11.21576 7.76929 6. 04609 5.01221 13 481.25742 96.93924 48.89803 24.87708 12.86664 8.86321 6.86155 5.66059 14 551.82156 111.03597 55.93536 28.38515 14.61012 10o.Q1851 7.72275 6.34534 15 625.79789 125.81243 63.31242 32.06262 16.43774 11.22953 8.62550 7.06313 16 702.79501 141.19447 70.99283 35.89124 18.34056 12.49040 9.56540 7.81045 17 782.47873 157.11215 78.93950 39.85266 20.30931 13.79496 10.53788 8.58368 18 864.47608 173.48461 87.11340 43.92739 22.33444 15.13687 11.53818 9.37904 19 948.23580 190.23233 95.47533 48.09575 24.40603 16.50958 12.56146 10.19265 20 1033.58100 207.27751 103.98372 52.33725 26.51403 17.90639 13.60271 11.02055 21 1120.09670 224.53039 112.59841 56.63139 28.64816 19.32054 14.65686 11.85873 22 1207.01510 241.91285 121.27665 60.95710 30.79800 20.74507 15.71876 12.70307 23 1294.28050 259.33744 129.97512 65.29337 32.95302 22.17311 16.78324 13.54947 24 1381.15820 276.71626 138.65202 69.61917 35.10287 23.59766 17.84518 14.39381 25 1467.67440 293.97239 147.26607 73.91316 37.23703 25.01177 18.89931 15.23197 26 1552.96380 311.01393 155.77488 78.15492 39.34500 26.40859 19.94057 16.05987 27 1636.79510 327.76246 164.13650 82.32309 41.41653 27.78130 20.96383 16.87348 28 1718.89230 344.13693 172.30981 86.39764 43.44169 29.12320 21.96414 17.66884 29 1798.49640 360.05285 180.25780 90.35946 43.41046 30.42774 22.93663 18.44208 30 1875.18750 375.42892 187.93601 94.18723 47.31311 31.68857 23.87648 19.18938 31 1949.39370 390.21040 195.31440 97.86526 49.14087 32.89966 24.77925 19.90718 32 2019.95710 404.30994 202.39214 101.37340 50.88440 34.05495 25.64046 20.59193 33 2086.59360 417.65511 209.01480 104.69493 52.53516 35.14889 26.45592 21.24030 34 2149.47440 430.18153 215.27088 107.81427 54.08534 36.17611 27.22165 21.84916 35 2207.70040 441.84443 221.09170 110.71573 55.52751 37.13165 27.93395 22.41550 36 2261.42010 452.57674 226.44722 113.38536 56.85426 38.01083 28.58933 22.93662 37 2310.37580 462.30802 231.31281 115.81017 58.05933 38.80936 29.18458 23.40990 38 2353.93810 471.02934 235.66119 117.97836 59.13696 39.52336 29.71684 23.83311 39 2392.05830 478.66357 239.47507 119.87935 60.08160 40.14939 30.18349 24.20414 40 2424.77140 485.19194 242.73435 121.50402 60.88899 40.68441 30.58230 24.52121 41 2451.88180 490.57844 245.42226 122.84393 61.55490 41.12568 30.91121 24.78278 42 2472.73800 494.79475 247.52659 123.89286 62.07641 41.47111 31.16875 24.98752 43 2488.11920 497.81459 249.03622 124.64584 62.45047 41.71908 31.35357 25.13451 44 2497.o06590 499.63526 249.94376 125.09898 62.67577 41.86831 31.46485 25.22300 45 2500.12500 500.24762 250.24899 125.25018 62.75093 41.91814 31.50200 25.25251 13

IV REFERENCES 1. S. K. Clark, Interply Shear Stresses in Cord Rubber Laminates, The Unirersity of Michigan, ORA Technical Report 02957-4-T, Ann Arbor, Michigan, October 1960. 2. S. K. Clark, R. N. Dodge, N. L. Field, and B. Herzog, Digital Computation of Two-Ply Elastic Characteristics, The University of Michigan, ORA Technical Report 02957-12-T, Ann Arbor, Michigan, October 1961. 3. S. K. Clark, The Elastic Constants of Cord-Rulbber Laminates, The University of Michigan, ORA Technical Report 02957-6-T, Ann Arbor, Michigan, October 1960. 4. S. K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, The University of Michigan, ORA Technical Report 02957-3-T, Ann Arbor, Michigan, October 1960. 15

V. DISTRIBUTION LIST 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. S. Attwood 1 R. A. Dodge 1 G. J o VanWylen 1 The University of Michigan ORA File 1 S. K. Clark 1 Project File 10 17

UNIVERSITY OF MICHIGAN III3 9015 02828 3805II I II 3 9015 02828 3805