THE UNIVERSITY OF M I C H I G A N COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No. 3 INTERPLY SHEAR STRESSES IN CORD-RUBBER LAMINATES S. K. Clark Project Directors: S. Ko Clark and R. A. Dodge UMRI Project 02957 administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR October 1960

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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 B. F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATES RUBBER COMPNAY ii

TABLE OF CONTENTS Page LIST OF FIGURES iv I. STATEMENT 1 II. SUMMARY 2 III. PHYSICAL CONSIDERATIONS 3 IV. INFLUENCE COEFFICIENTS FOR INTERPLY STRESSES 6 V. EXAMPLE 25 VI. ACKNOWLEDGJMENTS 27 VII. REFERENCES 28 iii

LIST OF FIGURES Figure Page 1 Idealized structure of cord-rubber sheet. 10 2 Assumed modes of deformation. 10 3 Loading process by means of which relations between Ex and Fxy may be visualized. 12 4 vs. cord half-angle a for various values of Gxy/Ex. 17 5 a~y/a~/, vs. cord half-angle a for various values of Gxy/Ex. 18 6 -al3/a33 vs. cord half-angle a for various values of Gxy/Ex. 19 7 -a23/a33 vs. cord half-angle a for various values of Gxy/Ex. 20 8 Two-ply cylinder. 25 iv

I. STATEMENT Previous analytical work reported in Ref. 1 showed that interply stresses* could be obtained as a function of the applied external stresses by means of exact solutions to the equilibrium equations for those cases where the cord in both plies of a two-ply structure was in either tension or compression. These relationships between the applied external stresses and the resulting interply stresses might be thought of as dimensionless influence coefficients. The purpose of the work reported here was to investigate the nature and range of these influence coefficients in terms of the variables normally present in the construction of cord-rubber laminates. *All interply stresses are transmitted by shear between plies. However, this interply stress may have shearing (distortion-causing) or normal (extensioncausing) effects on each separate ply. The total stress transmitted between plies will henceforth be referred to as the interply stress, and its components will be called the shearing and normal components.

II. SUMMARY It is shown that, based on exploratory tests giving ranges for the ratios of various elastic constants, influence coefficients relating external normal stresses and interply shearing (distortion-causing) stress components as well as those relating external shearing stresses and interply normal (extensioncausing) stress components may be calculated and plotted. These influence coefficients may be used to calculate the state of interply stress in multi-ply laminates, with certain restrictions, provided that the external stress state is known. This external stress state may generally be obtained either by calculation or by measurement.

III. PHYSICAL CONSIDERATIONS Reference 1, previously mentioned, contains equations relating interply stresses to external stresses in a two-ply structure, These equations are valid only for those cases where the cords in both plies of a two-ply structure are either in tension or in compression and specifically do not apply to that case where the cords of one ply are in tension while the cords of the other ply are in compression. Restricting ourselves to the former cases, these influence coefficients, or dimensionless relationships, may be calculated directly if the elastic constants of each ply constituting the two-ply laminate are known, and if the angles between the cords in each of the two plies are known. The interply stresses inherent in a two-ply structure may be visualized as identical with those in a four-ply, six-ply, eight-ply or any even-plied structure subject to the restriction that each of the pairs of plies be equally loaded. This will be the case if one considers only the effect of membrane forces in structures composed of such laminates. These membrane forces are commonly those due to external loads applied in such a way that little bending occurs, and also due to internal pressure. Therefore, these dimensionless relationships, or influence coefficients, are applicable to a rather wide range of laminates. However, under these restrictions the interply stresses will be generated between alternate laminates only. For example, in a four-ply structure of this type, interply stresses would only exist between the first and second, and third and fourth plies. One might very well ask what has happened to the interply stresses between the second and third plies. It is not difficult 3

to visualize, based on the analysis of the Ref. 1, that each of the outer plies in this four-ply structure, to wit, the first and fourth plies, requires exactly the interply stresses derived in this reference to distort in a manner compatible with the applied forces and with the known orthotropic nature of the material. Thus it becomes necessary for the full interply stresses to be generated between the first and second and between third and fourth plies. In this case, it may be seen that there is no need for any interply stresses between the second and third plies, and as a matter of fact the existence of such interply stresses would not be physically possible. This conclusion is based on the assumption that all four plies distort or strain exactly the same amount. To the extent that this is not correct, interply stresses may be generated between the second and third plies. At this time there is no direct experimental evidence on this point. The presence or absence of interply stresses on laminations composed of any number of plies in which alternate plies are laid at the same angle could be discussed similarly. It would be visualized that in this case interply stresses would exist only between pairs of plies, the first pair being composed of the first and second ply, the second pair being composed of third and fourth ply, and so forth. Examination of the pertinent equations for these influence coefficients indicates that they are dimensionless numbers depending on the cord half-angle a and also depending on the elastic characteristics of a single ply of the laminate taken in the directions parallel and perpendicular to the cords. Since only ratios of elastic constants enter into these calculations or expressions, 4

it should be possible to calculate these over a wide enough range so that one might have values of these influence coefficients available for any type of structure commonly used. One of the difficulties inherent in this scheme is that four elastic constants enter into the calculations of the influence coefficients, for those cases where the cords are all in tension, while four different elastic constants of a single sheet enter into the calculations when all the cords of a laminate are in compression. Some thought has been given to methods for reducing the number of independent elastic constants since it would be extremely difficult to present these influence coefficients as functions of four or eight independent variables. The next section of this report deals with the methods for reducing the number of variables here to one. For the present it will only be noted that this can be done from theoretical considerations and from the results of some fairly simple tests on cord-rubber sheets. 5

IV. INFLUENCE COEFFICIENTS FOR INTERPLY STRESSES The equations relating the interply stresses to the externally applied stresses were given in Ref. 1 and are repeated here: = a2al3 - al la23 Cyrq \ allaa22 a+12 L r(aala12a13 - a11ay + a1l Era l /\alla22 - a2) a,1 trl al_33 a33 Ot (1) where a' represent interply stresses and the a represent externally applied stresses. As previously mentioned, it may be seen that each of these relations between an interply stress and an externally applied stress is a dimensionless quantity. It may also be noted that the distortion-causing, or shearing, component ai' of the interply stress is caused only by the application of normal external stresses a~ and an. The normal, or extension-causing, components a' and a' of the total interply stress can be caused only by the application of external shear stresses ai.- For example, in the case of the inflation stresses or centrifugal stresses in a symmetrical pneumatic tire, no external shear stresses are present due to the symmetry. Thus, the only type of interply stress generated here is the distortion-causing component a' i. The existence of these interply stresses oa, %, and aG depends on the rigidity of the elastic material between the two plies. This elastic material 6

must be stiff enough so that very small strain differences, local in nature, are sufficient to generate the interply stresses of Eqs. (1). It has been tacitly assumed in this discussion that this stiffness is present. Results of experiments performed on the cylindrical tubes described in Ref. 1 are in good agreement with predictions based on this assumption. In other words, all experimental evidence collected to date seems to indicate that in cord-rubber laminates of the class commonly used in pneumatic tires it is possible to generate the full value of the interply stresses given by Eqs. (1). The consequence of failing to generate these interply stresses is incompatibility of the strain state of the different plies in a multi-ply laminate, and it is believed that the situation is not physically realizable for the types of laminates in question here. Thus, Eqs. (1) represent the state of interply stresses to the best of present knowledge under those conditions where all plies of the multi-ply laminate are either in tension or in compression. In the case when all plies have their cords in tension, then the constants aij appearing in Eqs. (1) should be evaluated utilizing properties of a single ply of cord-rubber material based on those conditions where the cord is in a state of tension. In those cases where all plies of a multi-ply laminate are in such a stress state that the cords in each ply are in compression, it will be necessary to calculate the aij terms appearing in Eqs. (1) on the basis of the elastic constants of a single sheet taken in such a way that the textile cords are in a state of compression. A notation previously adopted will be continued here, namely, that such terms as the latter will be given the symbols ail, the prime denoting the use of compression characteristics. 7

Once the extension, or normal, components and distortion, or shearing, components of the interply stress are determined, it is possible to determine the maximum and minimum values of the total interply stress by the usual stress transformation usually associated with Mohr's circle. This expression gives for the maximum or minimum interply stress the value o' +o CY C2 1/2 CY' + T, + ~+l (2) max 2 / min. It is most convenient to consider the characteristics of the ratios of the aij terms appearing in Eqs. (1). Notice that, since these ratios are dimensionless, it should be possible to divide each one of the aij terms appearing there by the term Gxy, which physically represents the shearing modulus of a single sheet of parallel cords encased in rubber. One may, for example, choose a all as a typical aij term for further study. Notice that each of the aij terms depends on the same four basic elastic constants, these being Ex, Ey_ Fxy and Gxy. The expression for all, taken from Ref. 1, is given in Eq. (3): cos4tc + sin4U + ssi2a cos2a G - -2 (3) Ex Ey xy The product all times Gxy may now be written as Eq. (4): (all)(Gxy) = G cos4a + xy sin4a + sin2cos1 - (4) Ex Ey Fxy Equation (4) shows that the term (all)(Gxy) is dependent upon the angle of inclination a of the cords and upon three separate ratios of elastic constants. These ratios of elastic constants will now be discussed in detail. The quantity Ey used here is nothing more than the extension of a single cord-rubber sheet in the direction perpendicular to the cords, expressed as 8

the ratio stress over strain. This process of extension may be synthesized by imagining the structure of the cord-rubber sheet to be idealized into that shown in Fig. 1, in which a series of rectangular cords equal in thickness to that of the sheet are joined by elastic rubber webs. The cords are presumed to completely rigid in comparison with the webso One may imagine that all the extension in the y direction takes place due to extension of the rubber between the rigid cords. This holds true no matter how close the cords are spaced together. On the other hand, the term Gxy may be thought of as the shear modulus of such a sheet taken in a direction parallel and perpendicular to the cord direction. It may be visualized that the process of shearing deformation takes place only in those regions of the sheet lying between the cords, provided that one assumes that the cords are again indefinitely rigid in comparison with the rubber and do not themselves participate in the shearing deformation. This line of reasoning leads one to believe that the process of shearing deformation and the process of deformation at right angles to the cord directions take place in exactly the same rubber elements. This is illustrated in Fig. 2, in which the assumed form of each type of deformation is shown. Now it should be true that some relationship exists between this shearing modulus and this extensional modulus perpendicular to the cords, since these processes are taking place in the same pieces of isotropic material. Thus, these two processes of shearing and extension take place on exactly the same kind of basis as if the material were isotropic, and the relationship which would exist between them is of the form derived for an isotropic elastic body. 2(1 + ~) 9

y 7 7-7 K. 77" 7 $ 7 CORD RUBBER WEB Fig. 1. Idealized structure of cord-rubber sheet. ~xy 0,x I I II II r'xy ~'xy (a) EXTENSION IN y (b) SHEAR DISTORTION DIRECTION Fig. 2. Assumed modes of deformation. 10

where Ci is Poisson's ratio. It has been shown many times that for rubber compounds Poisson's ratio is approximately one-half. Thus, the relationship between Gxy and Ey should ideally be E 3 Another ratio which may be determined from similar considerations is the ratio Ex/Fxy. To visualize this situation, imagine that it is desired to extend a cord-rubber sheet in the direction parallel to the cords by means of normal stresses acting as shown in Fig. 3. If no transverse restraints are present, there should be no reason why the lateral contraction of this sheet should not be in exactly the same ratio to the longitudinal extension as if no cords were present in the sheet. This presumes that the cords act as lines or extremely stiff wires of negligible cross-sectional area, so that they do not influence in any way the tendency toward lateral contraction evidenced by the surrounding Ex rubber matrix. It could thus be imagined that the ratio, which is the wellxy known Poisson's ratio, is exactly equal to the Poisson's ratio for rubber. This Poisson's ratio, as previously noted, is one-half. It is thus seen that two of these ratios are known, Gxy 1 Ex 1 Ey 3 Fy 2 Thus, two of the three ratios appearing in Eq. (4) can be estimated numerically from theoretical considerations. No similar method has been discovered for Gx port. 11

~t'~*O CORDS RUBBER Fig. 3. Loading process by means of which relations between Ex and Fxy may be visualized. 12

A physical interpretation of Gxy/Ex is that this represents the ratio of two elastic constants which might be thought of as expressing for a single ply, the relative stiffness in shear of the rubber interstices between the cords compared with the longitudinal stiffness of the cords themselves. It is thus a rather basic parameter defining, if one wishes to state it in this fashion, a quantitative measure of the anisotropy of a single ply of cord-rubber combination.* Since no method is available for readily predicting the range of values of this term, an experimental program to obtain this information and to check some of the previous conclusions was performed. A number of single sheets of various types of cords imbedded in rubber were obtained. Samples from each of these sheets were subjected to various elastic tests which enabled the quantities Gxy, Ex, Ey) and Fxy to be determinted. In many respects these tests were rather crude in instrumentation and were exploratory in the sense that they were a necessary prerequisite to more thorough experiments of this same nature which are to be run shortly for other purposes. They did, however, serve to define these elastic constants sufficiently over a wide range of cords and rubbers, including rayon, nylon, and steel. Having these elastic constants, it was possible to check the conclusions G Ex concerning the values of the ratios xy and E Allowing for some small exEy Fxy perimental error, these tests confirmed predictions except for one case, to be discussed in some detail in a subsequent paragraph. An additional benefit of this exploratory set of experiments was that the range of the ratio Gxy/Ex was found experimentally. The limits of this ratio were chosen to be from 10-4 to *A value of the ratio Gxy/Ex = 533 indicates an isotropic, incompressible body. 13

10-2 for subsequent computations, allowing some margin of safety on either side of the experimental results. Knowing the range of this ratio, it is now possible to calculate numerically the product (all)(Gxy) using the value of oneE third for Gxy/Ey and using the value of one-half for F, and in addition G Fxy taking various values of the ratio xy, these values lying at even increments Ex throughout the chosen range of this term. In other words, the product (all) (Gxy) can be reduced to a function of two variables, these being the half-angle a and the ratio xy. This should erable easy graphical presentation of this Ex product, as well as of all other products of terms of the form (all)(Gxy). This means that the influence coefficients relating externally applied stresses and interply stresses may be calculated and plotted. A completely different set of products (aij)(Gxy) may be obtained if one uses the values of the constants Ex, E F and Gx y all representing elastic constants of a single sheet for those conditions where the cords are in a state G Ex of compression. The arguments concerning the ratio XY and the ratio - are Ey Fxy believed to hold equally well for compression as for tension. Therefore, these ratios for compression should have the values previously determined for tension. There will be, however, a considerable change in the range of values of the Gxy ratio -, since the shear modulus of the sheet will still be approximately Ex the same due to the fact that it is primarily governed by the stiffness of the rubber matrix. The quantity Ex will, however, be reduced since Ref. 2 showed that the compression modulus of a typical textile cord is considerably less in compression than in tension. This means that this ratio will generally take on somewhat larger values for all plies in compression than if all plies are in 14

tension. Examination of this point by means of tests on samples from various sheets indicated that these conclusions were correct, and that it was necessary G only to extend the range of the ratio G-x by a factor of ten once more to cover Ex all those cases involving both compression arid tension. It is thus seen that the factor (all)(Gxy), and consequently all other products of the forms (aij) (Gxy), can be written as functions of the half-angle a and of the ratio Gxy where this latter ratio will always be somewhere between 10-4 and 10-1, with the cords in compression generally giving rise to larger values of the ratio than if the cords are in tension. There is some question in i:terpreting the test results concerning the valEv ue of the ratio -. It may be seen by reference to Fig. 3 that the only way xy in which completely free inward contraction, of such a sheet can accompany longitudinal extension is if the specimen is indefinitely long. This is because practically all the tensile load is carried by the cords. If the specimen is gripped in the usual fashion at the ends, it is not possible for the cords to move closer together at these points. Due to the normal process of contraction at the center of the specimen, it will be seen that each of the cords at the outer edges of the specimen will actually be deflected in the center and, due to their tension loads, will exert an outward force on the soft rubber matrix. Then they will be, in effect, slightly deflected cables under large tension loads. Due to the low stiffness of the encasing rubber, these tensile components in the direction opposite that of lateral contraction may prevent the free lateral contraction of the sides of the specimen. It is believed that this phenomenon was observed ir the tests performed on samples cut from various 15

kinds of sheets, as previously described~ It was not possible to obtain specimens longer than 12 inches for these tests, and for this reason it is believed E that the ratio is really one-half even though the test results indicated an xy average value for five differe.at materials, i.ncludinng wire, of approximately one-thirdo This point will be resolved by future work but a present answer to it may be obtained by calculating these influence coefficients described here for two different values of the ratio, these values being one-half and onexy third. These calculations indicated orly negligible differences between the influence coefficients using the two values of this ratio, and it may be concluded that the actual determination of the true value of this ratio is not important to the present problem0 In the subsequent calculations, the value of this ratio is taken to be one halfo The calculations giving the numerical values of these influence coefficients were performed and checked on desk calculators. They are presented in Figs. 4-7. The interply stresses discussed here are obtained without consideration of the stress-concentration. factor which arises due to the proximity of cords as they cross over one another in adjacent plies. Thus the values given here may be considered as local, or point by point, average interply stress valueso These will reflect all the actual effects except this stress concentration effect, and hence will be useful for predicting the relative severities of interply stresses, for comparing various designs in terms of their susceptability to high stresses at ply interfaces, and as a basis for further, more detailed studies on the local stress distribution.. To summarize, it might be said that interply stress magnitudes may be determi.ned as average values, taken locally 16

100 Ex /Fxy = I/2 or 1/3 Gxy/Ex = 1/3 Gxy/Ex = 4 a7- /o-7p = 0 at 00 in all Cases /Gxy/Ex = I 0 1.0~~ ~17 1.0 ~.~"0 CORD HALF-ANGLE aQ I I I 17

100 _.. Ex /Fxy - 1/2 or 1/3; Gxy/Ey- 1/3'c /cX = O at a =90~ in all Cases Gxy/Ex= 10'4 10 ty/Ex = 0-3 Gx Ex= 10-2 ~- I /oi/ 10 — 0 150 300 450 600 7 50 90 CORD HALF - ANGLE a 0.1II I I Fig. 5. o~ /e vs. cord half-angle Lx for various values of GIy/E 18

0.7, l l Ex /Fxy= 1/2 or 1/3; Gxy/Ey = 1/3 0.6 Gxy/Ex = 104 and 10-3 o. 6 — Gxy/Ex =' I 0.5 / 0.4 0.3 Gxy/Ex = 00.2 0.1I 150 30~ 450 60 750 ~90 CORD HALF ANGLE-a -0, 1...... Fig. 6. -al3/a33 vs. cord half-angle u for various values of Gxy/Ex.

0.7 Ex / Fxy 1/2 or 1/3 Gxy/Ex = I Olnd 10-3 Gxy/Ey = 1/3 ~~~~~~0.6 x I I ~-G Gy/Ex = 152 0.5 0.4 Gxy/Ex= IO' o 0.3 0.2 0.1 150 /~300 450 600 750 900 ORD HALF - ANGLE a -I.v Ii 0 9 Fig. 7. -a23/a33 vs. cord half-angle a for various values of Gxy/Ex. 20

at any point on a body whose membrane stress state is knowno These average values are obtained by smoothing off the peaks and valleys of the actual interply stress distribution, caused by the proximity of the cords as they cross over one another, over dimensions approximately equal to the cord spacing. Examination of Figso 4 through 7 indicates several interesting conclusions of a general nature. These will be discussed below: (a) In regard to the relationship between external normal stresses and the resultant shearing compo.rnent of the i.terply stress, Figs. 4 and 5, it may be seen that the value of the interply stress ge:nerated can, never exceed the individual value of the normal stress, ioe., the influence coefficient is smaller than one under all conditions. For most values of the ratio Gxy/Ex, including those values commonly associated with tension in the cords (namely, 10-4, 10-3, 10-2), there is very little difference in the interply stress influence coefficient. However, for Gxy/Ex = 10-1, the interply shearing stress reduces sharply. This indicates that, under conditions where the cords go into compression in both plies, the resulting interply stresses are less than for external stresses of equal magnitude but applied in such a way that the cords are in tension. An alternate way of looking at this set of results is to consider that the two most extreme curves, those pertaining to Gxy/Ex = 10-4 and 10-3, are indistinguishable. It is suspected that the meaning of this is that these curves pertain to a limiting case of extreme anisotropy as far as this type of interply stress is concerned, and that they thus form an upper bound for it. The lower bound is formed by taking Gxy/Ex = 1/3, correspondirng to an isotropic body, in which case all inrterp:ly stresses valnisho 21

The maximum value of interply shearing components are generated. at a cord angle of about 35~, while zero values of this function occur at 0t, approximately 600 and at 90~~ As an example of the use of these curves, it may be observed that the major component of stress in a normal passenger car tire, namely, the meridional component, has a cord angle of approximately 55' associated with it at the crown. This indicates a small interply stress caiused from this source. On the other hand, the circumferential compone.nt of tire stresses have a cord angle of approximately 35~ associated with them at the crown, and here shearing components of the total interply stress of the order of 0o6 times the applied circumferential stress are possible. One could thus say that, in. normal automobile tire construction, the interply stresses i:: the vicinity of the crown are quite dependent on the presence or absence of circumferential stresses. (b) Figures 6 and 7 show the relation, between externally applied shear stresses and the resulting normal, or extension-causing, components of the interply stress. For the most part, this if'luene coefficient is less than one and decreases regularly both with increasing cord angle a and increasing size of the ratio Gxy/Ex. However, these calculations indicate that extremely high values of the normal component of the ninterply stress are possible under conditions of low cord angle, say less than 150 or 20~, and particularly under conditions of simultaneous small value of the:ratio Gxy/Ex. The exploratory tests conducted on sheet materials, described earlier in this report, indicated that the smallest values of the ratio Gxy/Ex were obtained from wire cord samples~ Thus, the use of wire-ord laminates at very 22

small angles, say 50 or less, would. appear to generate interply stresses of the most severe type~ It should be observed that the curves of Figs. 6 and 7 are distinctly different Cfor each of the values of Gxy/Ex) ard that no limit has apparertly been reached here on the interply stresses generated by bodies of extreme anisotropy. (c) It may be seen that - al evaluated at some angle 0, is equal to a33 a23 evaluated at (90~ - G). This is as expected., although it is not at all a33 obvious from the algebraic form of Eqs. (l)u A similar relatio:n holds between the quantities ~/a, and o/a, in that C,(G) C'(90 - 9) oOn(D) = Cn(90 o- G) Detailed use of Figso 4-7 requires,of course, that the ratio Gxy/Ex be known. This quantity can vary over quite a wide range, depending on the geometry and materials of constructionr of the single cord-rubber sheet. Approximate methods for estimating this ratio have been developed and will be presented here, pending more thorough work on. this subject. For laminates in which the modulus of the cord is much greater than the modulus of the rubber, it will be possible to approximate Ex by a simple expression which presumes all load in the cord direction to be taken by the cords, so that t 25

where n = end count t = thickness of a single ply Ac = cord cross-sectional area EC = cord modulus (AE)c = cord spring constant per ur.it length The approximation of Gxy can be made in a similar fashion~ Figure 2 shows that this quantity may be idealized as the shear distortion ir. the rubber interstices between cords, and that the distortionr of these regions is solely responsible for the over-all distortion in shear of the body. This means that the single cord-rubber sheet is stiffer than a solid rubber sheet of the same dimensions by a factor of s - d' where s is cord spacing and d is cord diameter. Hence, one may write an expression for Gxy as Gxy G= (6) where GR and ER are the shear and extensional moduli of rubber, respectively. Using Eqso (5) and (6), the quantities Gxy and, Ex may be readily estimated, thus allowing quantitative use of the curves and graphs of this report.

V. EXAMPLE An example problem is given below in which the use of the information of Figs. 4-7 appears. Given: A two-ply, laminated pressure vessel such as shown in Fig. 8, with end plugs so that an internal pressure of 5.0 psi can be carried. In addition, an external torque of 25.0 inch pounds is to be carried..100 Fig. 8. Two-ply cylinder. To find: The state of interply stress. Solution: It will be assumed that Ref. 3 has been consulted and that it has been established that the cords in both plies are in a state of tension. Following this, the external stresses may be calculated. The i axis is taken arbitrarily to be the longitudinal direction while the T axis is the circumferential direction. Thus, from simple pressure vessel expressions, Y 5ox2 =.5 62.5 psi 2x.1 = 2.5 125.0 psi o1.1 250 = 6.35 psi j tx. 1x5x2.5

It will also be assumed that the properties of each sheet used to laminate the tube are known and that Gxy/Ex is equal to 10-3 From Figso. 4 and 5 and Eqo (1), a~a = (~57)(62~ 5) + (0)(250) = 356 psi T = (1,7)(6-38) = lo 85 psi ao'~ = ( 58)(6 38) = 5 7 psi Finally, from Eqo (2), the resultant value of the total interply stress is given by its maximum or minimum values as 212 l10-8 + 3~\) + |7 0+3 7-28 + 3558 min m ax = 43.1 psi al. = -28 5 psi max mn26

VI. ACKNOWLEDGMENTS The calculations were performed primarily by Mr. Richard N. Dodge, with assistance from Mr. D. H. Robbins, Mr. D. E. Zimmer and Miss Gwendolynne Chang. Thanks are due to them for their care in completion of this rather lengthy task. 27

VII. REFERENCES 1. S.K. Clark, The Plane Elastic Characteristics of Cord-Rubber Laminates, The University of Michigan Research Institute, Technical Report 02957-3-T, Ann Arbor, Michigan. 2. S.K. Clark and M.D. Coon, The Elastic Characteristics of Textile Cords in Compression, The University of Michigan Research Institute, Technical Report 02957-2-T, Ann Arbor, Michigan. 3. S.K. Clark, Cord Loads in Cord Rubber Laminates, The University of Michigan Research Institute, Technical Report 02957-5-T, Ann Arbor, Michigan. 28

DISTRIBUTION LIST Name No. of Copies The General Tire and Rubber Co. 6 Akron, Ohio The Firestone Tire and Rubber Co. 6 Akron, Ohio B.F. Goodrich Tire Co. 6 Akron, Ohio Goodyear Tire and Rubber Co. 6 Akron, Ohio United States Rubber Co. 6 Detroit, Michigan S. S. Attwood 1 Ra. A. Dodge 1 G. J. Van Wylen 1 The University of Michigan Research Institute File 1 S. K. Clark 1 Project File 10 29

UNIVERSITY OF MICHIGAN 3III III;1151 21i112 13811 3 9015 02828 3854