*T HE U NI VE R SI TY OF M IC HI G AN COLLEGE OF ENGINEERING Department of~ Engineering Mechanics Department of' Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No, 4i CORD LOADS IN CORD~-RUBBER LAMINATES S-4 K* Clark.Project Directors: SI* K*,Clark and R. A. Dodge UMRI Project 02957 administered by:

I) I )

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 F6 GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATES RUBBER COMPANY

TABLE OF CONThENTS Page LIST OF FIGURES i I. STATEMENT1 Il, SUMMARY 2 IIlL CALCULATION OF CORD LOADS 3 IV4 EXAMPLE 13 V.* ACKNOWLEDGMENTS 15'VIo, REFERENCES 16 ViII DISTRIBUTION LIST 17

LIST OF FIGURES Figu~re Page 1 Typical lamina with both interply anid external stresses acting. 2,. L vs. half-angle a~ for a range of values of Gxy/EX*, 9 15 M vs. half-angle Cl for a range of values of G~y/EX*. 10 4 N vs. half-angle cy for a range of values of GXYEX~ 11 5 Cylindrical tube of two-ply constru~ction. 13

I, STATEM\EN~T The calculation of cord loads in cord~-rubber laminates must be preceded by a knowledge of the interply stresses acting in these laminates0 This is be~-.cause the total stress state acting on any particular lamina is made up of that portion of the total external stresses carried by this lammna plus the interply stresses acting on, it* Thus only a partial answer to cord loads is given by considering only the effects of external stresses. Reference 1 presented equations for the values of the interply stresses as functions of the elastic constants of the laminates and as functions of the cord half-angleo Having these expressions, it should be possible to combine them with the external stresses in such a way as to predict the cord loads for any external loading condition'.

IT1 -SUMMARY By proper combination of the -interply and external, stresses, expressions for the stress in the direction of the cords may be obtained for those cases where the cords in all plies of a laminated structure are either in tension. or in compression, This report will be restricted to those cases in which the stiffness of the cords is much greater than the stiffness of the rubber. Thus,'if the stresses in the cord direction are known,,'it may be assumed that these stresses are equally distributed among the cords only, i.e., that the cords carry all the tension load or compression load in their direction. Under this assumptions influence coefficients relating cord loads to ap~plied external1 stress are presented0, These indicate that the influence coef~ficients relating external normal. stresses to cord loads are generally smaller than unity, However) the application of external shearing stresses can result in extremely high cord loads in those cases where the cord half-angles are either close to zero or close to 900, since influence coefficients greater than 4-0 have been calculated,

IIIL CALCULATION OF CORD LOADS As discussed in Ref. 1, the expressions for interply stresses as functions of the externally applied stre~sses may be written as' (,a-12a13 -aii4a2M) (a, 22-a~2 T) L\~~/(a11a2_- al.a2ja~ = - alla (all2, 32aI r, a33 a-33 1 Assuming a two-ply laminate as before, it may be seen at once that the total stress state of one lamina is determined by the sum of the external stresses and the interply stresses just given.0 In constructing this sum, it, will be assumed that both plies of the -two-ply laminate have the same elastic properties, that is, they may both have cords in tension or they may both have cords in compression. The case of one ply being in tension and the other being in comnpression is specifically omitted from consideration here. Figure 1 shows one typical lamina of a laminated sheet with the st resses acting on it, as assumed in this report,,, From the general transformation expressions used in Ref,. 1, the expression for the stress'in the cord direction may be written at once as ax (a~ + a~)42 + 2(ar~l + ar')Ixox~ + (ay~ + r 2 ay = aX+a~e + (a a)yfy (a e (2) a. A N 1 Nfi2I5

where l Cosa Ily - sinca 1SI f si aeyTr1 = Cos a (a) O'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "1, T~~~~~~~~~ Fig. 1. Typical lamina with both interply and external~~~~'7 Thecor stes Tpcalno bedeeminedit bythaitdirectn examinatinolhesrs in th xsdrectios scincetecrga.culybefre noasaeo

compressive strain by means of stresses acting at right angles to it. Thus to determine the sign of stress in' a cord'it is necessary to examine the expression for the cord strain, x E~ F() xy Reference 2 shows that the modulus Ex may be approximated as Ex AE)e,(n) t where n = end count t=thickness of a single ply = cord cross~-sectional area EC = cord modulus ( =E cord spring constant per unit length of cord. Equation (3) can now be written as (~x)(Ex) ax (ay -(ex) (AE) cn) Fxy (t) ~~~~~(5) From this'it is seen at once that the cord load may be- obtained., since it is the product of the strain in the cord direction and the (A)d This gives = (AE)c(Cx) = x (cy()a6 where P is the load in a single cord,

.be -necessary to specify, factors t and no since these are variables of construction.- The quantity in brackets may be calculated and plotted as a function of the other elastic variables of the systera* This may be seen most easily by inserting the terms of Eqs4 (2) into Eqs' 6) giving the final expression for cord load as (AE) C~tjJ Fa2aia3-aa2N al1+- Cosx'EX C_ nL\al/\ ll2a a~2 / allj T + 2 [r - a\ Cy sin Ccosa + I + (a2a1_3 alla;23' 2 TI T 1a2 ~/~Ij Ex ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a2ai2a:3 a1123,~ +1 2 i.F ~al aI~a22 ali2 / alIi ~2E F~ K?2k, CKY~ sina01coscet Ex - ~~aiia23j1csJ Fxy {a + a2a3 2 Jr IO2 Equation (7) may be simplified considerably by rewriting it in terms of symbols in the form

where L,, Ml and N are dimensionless'Influence coefficients., since the dimensions of hetem o=,the left side, the cord load, are identical to the dimension of the terms on the right,, namely', the product of stress and thickness over end count. From Eq.: (8) it may be seen that the f actors L. M., and N are the only ones which need to be calculated and plotted, Their values are given by Eqs. 9 L =coscaicos a { sin sin a(2x Lsin a + a3Cos (l M=sin aYsin.a -2( Cos a] -Cos a(~ Cos (a + 2 sin a] (2-E)x i2a a12 /a12a,13 a112' L \Fxy/ i~~~- Lkai) ajj1a22 a a12 + 2sin a cos ae l + E + sin 2 a (~~cos2ale 12alg3 - a1:1a23\ L K~xy) jK a11a22. a%)(9 In regard to the nunerical calculation of the terms L, M. and N,'it may be see-t-n t.hat.+ the o)nl elast P.icP(- cnstntqn which e-nteizrs dietyanavral is

depend only upon one variable, namely, the ratio which has been found to Ex span a range running from 10 4 to 10 l1 for all observed cases for both cord tension and compression~ It should be possible, then, to calculate the factors L, M, and N as functions of the variables Go, and the cord half-angle a. This Ex has been done and the results are presented in Figs. 2, 35, and 4. Trial calculations showed that nearly identical curves resulted if Ex was chosen to Fxy. be 1/3 instead of 1/2. The cord loads may be determined from the information presented in these figures along with the use of Eqo (8), which requires a knowledge of the end count and the ply thickness. The numerical value of L may be found by the Ex method described in Ref, 2,* Examination of the values of L, M, and N indicates that cord loads due to external normal stresses approach a maximum in the vicinity 30-355 of cord angle, that they approach zero between 55 and 60~ of cord angle, and become negative for larger cord angles. Similar conclusions apply to M, which is the mirror image about the 45~ line of L. In both of these curves~ it is seen that a distinct possibility exists that cords may go into compression, as evidenced by the values of L in excess of 55~,. These calculations indicate a cord halfangle somewhat larger than 55~ as being that associated with zero cord tension due to the presence of external stressesg partly because the interply stresses, discussed in Ref. 2. become very small in this vicinity and contribute little to the cord loads here. If it were not for this, it might be expected that this zero load angle would be considerably differenat from that of 54j750, derived from the usual condition of an ine~xtsensible cord imbedded in a ru~bber

0e, -GxyE//FExI= or01/ 1.2 -.2 -A2

1.4 Gx /E 1- a 10- / 1.2 1.0.8 / Ex /Fxy 1/2 or I/3 G'xy/Ey =1/3.6C II1.2 Q 15~ 30 450~ 600 75~ 9 O0 CORD HALF-ANGLE- a w.'~~~l -2_ Fig. 3. M vs. half-angle (X for a range of values of Gxy/Ex. 10

45 40 Ex /Fxy -1/2 or 1/3 35 3025 -- -4_ _ __ _ _ 20 - Gxy/ExO- 10 Gxy/Ex = 0151A 10 0 1~~50 300 450 6007590

block whose Poisson's ratio is one-half..Figure 4i shows that the presence of external shear stresses can result in extremely high cord loads umder certain conditions, Generally,, these condi' tions require that the cord half-angles be very small or else be very close to 900,, which is the equivalent., and also that the cords be quite stiff compared with the rubber matrix in which they are imbedded, This means that such structures as braided wire cords imbedded in rubber would be par'ticularly susceptible to large cord loads if these laminates were used at small angles. It is quite surprising that cord loads of the magnitudes, indicated can be generated, and it is hoped that some experimental evidence can be obtained on this point in the future;

IV. EXAMPTLE The example used in IRef. 2 will again be chosen here since it is a fairly simple geometry and can be used to illustrate all the principles involved. Consider a circular tube of 5~-in. diameter such as shown in Fig. 5. We desire to determine the cord loads in each of the two plies of this tube due to the application of the same external stresses as previously used,, these beiIng =r 62,5 psi a = 125. 0 Psi Crh = 6.58 psi It will be necessary here to assume that the end count and the ply thickness are known. The ply thickness,,.050 in. was previously defined for purposes of calculating the stress magnitudes. The end count will be assumed to be 20 cords per inch. 15~ ~ ~~~01

Utiizig te crve inFig~ 2 5~and 4i~ along with the values of ply thickness and end coutnt just given, the cord load in the tension ply may be obtained directly from the positive values of L, M, and N as given in Eq. (8) This gives, again assuming as in the previou~s example that 1 0., pte[(l,20 375) (62w5) + (~04125) (125*0) + (25) (658)] ply.22lb per cord (10) In regard to the compression ply,'it is seen that the fu~nctions L,, M, and N must be evaluated for negative angles a. Examination of Eqs., (9) *indicates that both L and N are unchanged when evaluated for negative c1 in place of positive oe. That means that the influence on cord load of normal stresses will be just the same in both plies of the tube inm question~ However., examination of EqIs, (9) further indicates that the term N does change sign when evaluated for negative angles in place of positive angleso In that case, the correct form of Eqs* (8) applicable to the other-ply of this tube is-, = O2.2 [(15y5)(6245) + (-0l25) (125*0) (2,3) (6.38) 20 - 1l9 lbs per cord (1 From these calculations), it is seen t~hat the cord loads are different in the two plies but that both sets of cords are still'in a state of tension,, A.similar example could be presented utilizing all compression data but would

V, ACKNOWLEDGMENTS The calculations necessary for presenting this infornation were performed by Mr, Richard N. Dodge with assistance from Mr,. D. Ho Robbins, Mr,. De E. Zimmer, and Miss Gwendolynne Change Thanks are due to them for their care and patience in this lengthy task. 15

ITT, REFE~RENCES 1~ SK. CarkThe Plane Elastic Characteristics of,Cord-Ru~bber Laminates, The University of Michigan Research Instituite, Teclinical Report 02957-3-T. Ann Arbor., Michigan, 2, SK, CarkInteplyShear Stresses in Cord-Ru~bber Laminates, The Uniiversity of Michigan Research Instituite., Technical Report 02957-4i-T, Ann. Arbor, Michigan.

DISTRIBUTION LIST Name No. of Copies The General Tire and Rubber Ca. 6 Akron., Ohio The Firestone Tire and Rubber C~o 6 Akron, Ohio BF.. Goodrich Tire Co. Akron,, Ohio Goodyear Tire and Rubber Co. Akron,4 Ohio United States Rubber Co* 6 DetroitI Michigan'Sw So Attwood. R. A. Dodge G-t J. Van Wylen The University of Michigan Research Institute File 1 5, K. Clark 1 Project File 10

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