THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Research Group Technical Report No 6 FORE-AND-AFT STIFFNESS CHARACTERISTICS OF PNEUMATIC TIRES Ro N. Dodge DavidO-ne. - S, K, Clark ORA Project 05608 supported by: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NsG-344 WASHINGTON, Do Co administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1966 (Revised June 1967)

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TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v I. INTRODUCTION 1 II. SUMMARY 3 III. ANALYSIS 4 IV. COMPARISON OF THEORY WITH EXPERIMENT 11 Vo APPENDIX I 21 VIo APPENDIX II 31 VII. ACKNOWLEDGMENTS 33 VIII. REFERENCES 34 iii

LIST OF ILLUSTRATIONS Table Page Io Summary of Geometric, Elastic, and Structural Properties of Five Automotive Tires 13 II, Summary of Experimental and Calculated Values of Kf 13 III. Comparison of Results Using Approximate Moduli 26 Figure 1, Idealized model of pneumatic tire for analyzing fore-and-aft spring rates. 4 2, Loaded element of portion of the model 5 3o Meridional cross-section of the tire showing symbols used. 8 4. Experimental apparatus for checking K. 9 Load-deflection data for double tube experiment used to confirm the expression for Kso 10 6. Idealized mid-line profiles of five pneumatic tires. 12 7. Photograph of experimental apparatus for Kfo 14 8o Schematic of fore-and-aft spring-rate test, 14 9, Fore-and-aft load-deflection curves. Tire No, 1. 16 10o Fore-and-aft load-deflection curves, Tire No, 2. 17 11, Fore-and-aft load-deflection curves. Tire No, 3, 18 12. Fore-and-aft load-deflection curves. Tire No, 4, 19 13. Fore-and-aft load-deflection curves. Tire Noo 5. 20 14, Comparison of exact and approximate circumferential modulus Eg. 22 15. Comparison of exact and approximate shear modulus G. 24 v

LIST OF ILLUSTRATIONS (Continued) Figure Page 16o Sidewall location for effective G. 26 17. Description of hypothetical tire for example problemo 28 vi

Io INTRODUCTION Pneumatic tires are functional parts of many dynamic systems. In order to effectively design and engineer such systems, it is often necessary to know the mechanical properties of the component parts, including the tires One of the major roles of this research group has been to study and analyze some of the important mechanical properties of pneumatic tires and t;o present rational methods for predicting themo Several paths have been followed by this research group in developing approximate met;hods for predicting mechanical, properties of tires. One technique involved modeling of the penumatic tire as a cylindrical shell 5-7 supported by an elastic foundation This model gives relations for predicting several properties involving deformation in the plane of the wheel, such as contact patch length vs. vertical deflection, vertical load vso vertical deflection, plane vibration characteristics, transmissabil.Lit y characteristics, and dynamic response to a point loado 4 A more recent attempt involved analyzing the pneumatic tire as a string on an elastic foundationo This model, is primarily used to predict lateral stiffness characteristics, vertical stiffness characteristics and twisting momentso However, it has been found to be useful only when the i.nflation pressure is high, such as in the case of aircraft tires. This report presents a method for predicting the fore-and-aft sti.ffness characteristics of pneumatic tireso Fore-and-aft properties are important in the overall analysis of a tire since they represent the contributions of 1

the carcass and tread to braking and tractive elasticity. A different model is required here since neither the cylindrical shell nor the string on the elastic foundation provide means for transmitting such loadso It is hoped that; this model will prove satisfactory for predicting fore-and-aft charac teristics of various tire designs. 2

II. SUMMARY An elastic bar supported by a foundation exhibiting elasticity in shear serves as a model for determining the fore-and-aft stiffness properties of a pneumatic tire. The differential equation representing the deformation is derived and solved, and the resulting solution gives a means for calculating a fore-and-aft spring rate for the model. A series of five tires of various sizes and structures was used for testing the validity of the proposed modelo A set of static tests was performed to establish an experimental value for the fore-and-aft spring-rate for the various tires0 Additional structural data, required by the analytical solution of the model, were also obtained from the tires, A comparison of the calculated and experimental results was reasonably satisfactory, indicating that the proposed model can be used to roughly approximate fore-and-aft stiffness characteristicso A complete tabular summary of the geometry and composition of the five tires is included for easy reference. All experimental and analytical results are summarized and compared in graphical form,

III. ANALYSIS To represent fore-and-aft stiffness characteristics, the pneumatic tire is idealized as an elastic bar supported by an elastic shear foundation (see Figure 1). The elastic bar portion of the model represents the tread region of the tire which is loaded by the fore-and-aft load F. In addition to the restraint offered by the stiffness of the tread region itself, resistance to deformation by the load F is provided by the tires' ability to withstand shearing forces in the sidewall regions. This portion of the tire is represented in the model by the elastic shear foundation. ~ —~~ ~Elastic Bar RIM F AsE RIM Elastic Shear F Foundation Figure 1. Idealized model of pneumatic tire for analyzing fore-and-aft spring rates. If it is assumed that the restraining force of that e elastic shear foundation is directly proportional to the displacement, an elemental segment of the elastic bar can be set in equilibrium as shown in Figure 2. Note that adlb

vantage is taken of the symmetry present in thn mA X (Ks'u)dx F/2 _ - S+'dx / dx Figure 2. Loaded element of a portion of the model. In Figure 2, S is the force acting on the bar, u is the displacement, and Ks is the spring rate per unit length of the shear foundation. It is assumed that K is provided only by the shear resistance of the sidewall. s From Figure 2, it is seen that one may approximate 2GH Ks = A where G is the effective shear modulus of the sidewall, H is the sidewall thickness, and A is the length along the sidewall from the rim to the point of intersection of the tread and carcass. From equilibrium of the element, - K u = 0 S = TAs = EeAs = AsE 5x

where T is the stress, e the resulting strain, A. the cross-sectiona:l area of the bar at any location, and E the effective extension modulus of the tread region in the circumferential directiono Thus, 2u 2. -- u u=O (1) ax" where 2 Kc = -- AsE The general solution of this equation is u = C1 cosh qx + C2 sinh qx (2) The boundary conditions for this problem are determined by assuming t~hat each half of the tire (fore-and-aft of the contact patch) is equally Loaded, so that at x = O, S at x = -a, S (3) Substituting (3) into (2) gives C -F F (4) Cl = 2AsE tanh Trag 2 2A Eq Thus, u(x) = F Fsinh qx cosh (q) 6

The fore-and-aft stiffness is determined by finding the ratio of the applied load to the displacement at the point of application of the loado Thus, Kf = - - 2AsEq (tanh qta) (6) Equation (6) now represents a relationship for the fore-and-aft spring-rate of a pneumatic tire idealized as an elastic bar supported by a shear foundationo As can be seen From Eqs. (1) and (6), the application of Eq. (6) to a real tire requires a knowledge of the effective stiffness (AsE) of the tread region in the circumferential direction, the effective shear modulus of the side-wall region G,the effective sidewall thickness H, and the length along the mean meridional section from the rim to the intersection of the tread and carcass, Ao The extension modulus in the circumferential direction and the shear modulus of the carcass usually vary from one location to another in the meridional direction because of the orthotropic nature of the tire carcass, so some criteria must be established to compute ASE and G for a given tire sectiono Both the extension and shear modulus used for the results listed in this report were obtained by averaging the actual values of these properties throughout the cross-section0 This technique has been successful because the variation in these properties has not been too nonlinear. However, it has a great disadvantage in a simplified analysis such as this because it requires lengthy calculations which cannot be done efficiently without, the aid of a digital computer0 For this reason an effort has been made to obtain some simplified,^ but reasonably accurate, approximations for the A E 7

and G necessary for calculating the fore-and-aft spring constant. The results of this effort are included in Appendix I and give a satisfactory approximate technique for calculating these properties. ET Shear Modulus G Extension Modulus Normal to Plane of Paper E8 r X Figure 3. Meridional cross-section of the tire showing symbols used. In order to establish some validity for the assumption that the shear foundation modulus Ks can be estimated by considering shear effects only, a simple experiment was performed by gluing a metal strip along the line of contact of two rubber cylinders placed side by side (see Figure 4). A load was attached to the bar and the resulting deflection was measured by the dial indicator. The slope of the experimental load-deflection curve, related to Ks, was then compared with the value of Ks obtained from the relation given above, 2GH Ks = At A summary of this experiment is presented below: 8

,=, /- Dial Indicator H d, = o.986 in. H = 0.133 in. 1 e = 0.10 in. A = 1.72 in. e2 = 0.15 in. G = 200/3 lb/in. From the test data (Figure 5), the slope of the load-deflection curve yields a Ks = 22.0 lb/in./in. The calculated value for the double tube is 9

4GH 3 Ks = 4G = 20.6 lb/in./in. 1.72 (A factor of 4 appears in this computation because of the double tube arrangement.) The close comparison between the experimental Ks and the calculated one, assuming that the foundation is flat rather than curved, indicates that any curvature effects are minor. 10 810" TUBE LENGTH 6 SLOPE 9 LENGTH Ks = O 22.0 #/IN./IN. 4 - 2 -.01.02.03.04.05 DEFLECTION- IN. Figure 5. Load-deflection data for double tube experiment used to confirm the expression for Ks. 10

IVo COMPARISON OF THEORY WITH EXPERIMENT In order to investigate the validity of Eqo (6), a series of static fore-and-aft stiffness tests were run on representative tireso Before reporting these tests and their results, the five tires used are described in detailo The idealized centerline profiles of the tires are shown in Figure 6o Tire No 1 is a domestic 4-ply, 8o00;x 14 bias-ply ti.re with standard nylon cord. Tire No. 2 is a 2-ply, 7o50 x 14 bias-ply tire witlh standard nylon cordo Tire No 3 is an imported 4-ply, 5090 x 15 bias-ply tire with nylon cordo Tire No. 4 is an imported 7 50 x 14 radiaL-ply tire with overheads reinforced with wire cordo Tire No 5 is a European made 155 mm x 15 ino radial-ply tire with overheads reinforced with nylon cordo Table I is a summary of the pertinent elastic and geometric parameters required from the five tireso Using the results in this table: Figure 6, Eq. (6) and the proper elastic properties, it is possible to calculate the fore-and-aft stiffness of the five tires, Carrying out these computations gives the calculated values presented in Table II. To check the accuracy of the calculated values, the five tires were tested in the apparatus illustrated in Figures 7 and 8o In this testing procedure the tires were loaded vertically to a fixed deflectiono Then a varying fore-and-aft load was applied and the resulting deflection recordedo The slope of these load-deflection curves represents the experimental folreand-aft spring-rateso These tests were run for different* verrtical defl.ections and inflation pressureso The results of these tests are summarized i.n Fig11

6 5 5 Tire No. 5No. 2 Tire No. 3 4 4- 44 IN. IN. \ IN. 2 2 2 I I 0 I 2 3 4 0 I 2 3 IN. 0 1 2 3 4 IN. IN. 6 5 - Tire No.4 ~ Tire No. 5 4 4 IN. IN. \ 3 y20 2 3 I 0 I 2 3 0 I 2 3 4 IN. IN. Figure 6. Idealized mid-line profiles of five pneumatic tires.

TABLE I SUMMARY OF GEOMETRIC, ELASTIC, AND STRUCTURAL PROPERTIES OF FIVE AUTOMOTIVE TIRES Tire 1 Tire 2 Tire 3 Tire 4 Tire 5 Item (Ref. 5, Table I) Bias-Ply Bias-Ply Bias-Ply Radial-Ply Radial-Ply 8.00x14 7.50x14 5.90x15 7.50x14 155mm x15 AO - outside radius of tire 12.525 13.94 12.875 13.44 12.25 L - half circumference 39.35 43.79 40.45 42.22 38.48 ET - extension modulus, tread rubber 670. 560. 481. 690. 490. A - length, mean meridional section 4.6408 5.8404 4.9217 5.7238 4.5529 H - effective thickness for Ks 0.164 0.110 0.160 0.250 0.280 G - effective shear modulus for Ks 28440. 47164. 43876. 269. (Gxy) 44. (Gxy) AsE - effective spring rate, circumferential 779. 747. 760. 11906. 23027. Ks - spring rate, shear foundation 2010. 1747. 2852. 18.98 23.50 BETAC - cord half angle, crown 0.6458 0.6283 0.6109 0.3142 0.2356 ROC - radial location, crown 12.120 13.38 12.355 12.92 11.70 ROB - radial location, rim 7.078 7.20 7.515 7.06 7.59 R - idealized radius, sidewall 3.22 3.03 2.84 3.13 2.17 Yc - y-coordinate, center for R 1.935 2.490 2.092 2.454 1.939 Xc - x-coordinate, center for R -0.057 0.465 -0.146 0.405 0.704 R1 - idealized radius, crown region 3.40 2.62 2.76 3.54 4.88 ERUB - extension modulus, carcass rubber 438. 310. 370. 625. 300. GRUB - shear modulus, carcass rubber 146. 103. 123. 208. 100. MURUB - Poisson ratio, carcass rubber 0.500 0.500 0.500 0.500 0.500 GCORD - shear modulus, cord 705. 705. 705. 705. 705. AESUBC - spring rate, cord 200. 623. 317. 350. 307. DIAMC - effective diameter, cord 0.025 0.040 0.026 0.025 0.023 MUC - Poisson ratio, cord 0.700 0.700 0.700 0.700 0.700 TPLY - effective ply thickness 0.041 0.055 0.040 0.040 0.040 NCORD - cord count, crown 26. 19. 24. 18. 20. ALPSTR - normal angle, intersection 0.7746 0.6085 0.6665 0.6427 0.5775 ALPHR - normal angle, rim 2.2148 2.5331 2.3973 2.4714 2.6792 BW - tread width 4.36 4.40 3.20 4.60 3.68 TABLE II SUMMARY OF EXPERIMENTAL AND CALCULATED VALUES OF Kf Kf - lb/in. Tire Experimental Calculated 1 2530 2503 2 2300 2286 3 2780 2944 4 1340 1469 5 1265 950 13

Figure 7. Photograph of experimental apparatus for Kf. Figure 8. Schematic of fore-and-aft spring-rate test. 14

ures 9 through 13. In general it can be seen from these curves that the fore-and-aft springrate increases only slightly with increasing vertical load and with increasing internal pressureo Since Eqo (6) does not account for the slight increase due to these factors, a comparison between the experimental and calculated results must be made in a somewhat arbitrary fashiono However, since the experimental values are nearly the same for all conditions examined, any resu:ltrs used as a comparison with the calculated values will serve as a meaningful checko The comparisons -shown in Table II are based on experimental values obtained. from vertical tire deflections of one inch and by use of manufacturers rated inflation pressure, The experimental values were determined by measuring the slopes in the linear portions of the load-deflection curves~ These comparisons indicate that the simple model formulat-ed above gives a method for approximating the fore-and-aft spring-rate of pneumatic tires using only T',he geometric, elastic, and structural properties required by most tire designersO At the end of the Appendix II an example problem is worked out, illustrating how an approximate fore-and-aft spring rate can be calculated if the correcrt input data is availableo 15

o 1/2 in. vertical deflection x 3/4 in. vertical deflection A I in. vertical deflection o I 1/4 in. vertical deflection 300 15 psi 25 psi 35 psi X 0 A 0(200~|X 0 O o a x L \ x o a o oo Aa o o x 0 x O.01.02.03.04.05.06.07.08 0.01 02.03.04.05.06 0.01.02 - HORI 0 Figure 9. Fore-and-aft load-deflection curves. Tire No. 1.x 0.01.02.03.04.05.06.07.08 0.01 02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 HORIZONTAL DEFLECTION-IN. Figure 9. Fore-and-aft load-deflection curves. Tire No. i.

o 1/2 in. vertical deflection x 3/2 in. vertical deflection ~A I in. vertical deflection o 1 1/4 in. vertical deflection 18 psi 28 psi 38psi 200 (n x I x 0x _J 0 X X 0 I, o A 0 0 0 X 0 LL. A o U 0 c1,, o I o 0>o x o (a o 0 0 H o -x 0 0.01.02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 HORIZONTAL DEFLECTION Figure 10. Fore-and-aft load-deflection curves. Tire No. 2.

o I /2 in. vertical deflection x 3/4 in. vertical deflection A I in. vertical deflection o I 1/4 in. vertical deflection 9 psi 19 psi 29 psi u~ 200 z 0 A~ x Um~~~~~~0 x o 0 X 0. AO x 0 O0 D A A A 0 W A X x O 100 x x x 0 x 0 0 o x o x 0I- 0 X B x a o ~x 0 0.01.02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 HORIZONTAL DEFLECTION Figure 11. Fore-and-aft load-deflection curves. Tire No. 35.

o 1/2 in. vertical deflectioi. x 3/4 in. vertical deflection I in. vertical deflection o 1 1/4 in. vertical deflection 18 psi 28 psi 38 psi 200 (n CD -j I W 2100~2 x o ~ ~ ~ ~ ~~~Q a x ax \O loQ x-o o0~~~~~~~~~~~~~~~ 00x0 w-B O 0 x 0~~~~~~ a 0 o 0.01.02.03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 0 01.02.03.04.05.06.07.08 HORIZONTAL DEFLECTION - IN. Figure 12. Fore-and-aft load-leflection curves. Tire No. 4.

o 1/2 in. vertical deflection x 3/4 in. vertical deflection A I in. vertical deflection o 1 1/4 in. vertical deflection 10 psi 20 psi 30 psi 200 0) bJ 000 O 0 0L W 0 0[ 0.01.02.03.04.05.06.07.08 0.01.02 03.04.05.06.07.08 0.01.02.030 B t a a o 0.01.02.03.04.05.06.07.08 0.01.02 03.04.05.06.07.08 0.01.02.03.04.05.06.07.08 HORIZONTAL DEFLECTION-IN Figure 13. Fore-and-aft load-deflection curves. Tire No. ~.

V. APPENDIX I This Appendix presents the results of an effort to simplify the computations involved in obtaining the extension modulus of the carcass in the circumferential direction, Eg, and the shear modulus of the carcass, G. Reference 1 gives exact expressions for the moduli of laminated orthotropic two-dimensional sheets, and these are good representations for the elastic constants of a tire carcass. However the expressions derived in Ref. 1 are quite lengthy to evaluate. They can be simplified considerably by considering the structure to be made of inextensible cords, so that the modulus of elasticity Ex, parallel to the cords in a single ply, becomes indefinitely large. By simplifying the expressions of Ref. 1 in this way, one gets 4G 2 C2 E ( si ) cos2 + E(c _ x (7) sin a where a is the local cord half-angle at any meridional location, G is the xy shear modulus of an individual ply and E is the extension modulus of an iny dividual ply in the direction normal to the cords. This is a very good approximation for EG as long as the cord half-angle is not less than 30~. This is shown in Figure 14 where Eq. (7) is compared with the exact formulation of Ref. 1. Thus, Eq. (7) is a good approximation for an ordinary bias-construction tire, where the angle is almost always greater than 30~. An approximation for the shear modulus can be obtained from a strength of materials analysis. This is reproduced in Appendix II in detail, and gives 21

106 Theoretical (Ref. 1) \ ~- Approximate (Eq. 7) ~\ ~~~~Ex = 1.2683 x 105 "105 \' \ Ey = 1370.7 10 ~'5 *, \ \ Fxy= 2.2557 x 10 CD ~\ \ \Gxy= 272.92 L,,I 4 I ^\ I \\ coF 103 I 0 15 30 45 60 75 90 Cord Half- Angle a~ Figure 14. Comparison of exact and approximate circumferential modulus E9. 22

2 2 G E sin a ccs a (8) x where E is the extension modulus in the direction of the cords of an x individual ply of the tire carcass material. This is a very good approximation for the shear modulus except at cord half-angles near 0 and 90" as shown in Figure 15 where the exact shear modulus expression from Ref. 1 is compared with Eqo (8)o Again, this is valid approximation for almost all tire constructions since cord angles usually lie between 50~ and 60"o Reference 2 presents a concise method for calculating Ex, E y and G as a function of the geometric and elastic properties of a single xy ply of the carcass material. The expressions for these moduli are listed below: Ex = (AESUBC) (NCORD)/TPLY E = ERUB l + 2o9 xy Gx =705 K %H\s + GRUB(1-,s sH) where: K(DIAMC) kH PLYXs = K(DIAMC)(NCORD) The constant k is an area coefficient equal to 4 /4o The other parameters are defined in Table I of this report. Unfortunately both the expression for Eg and G are still functions of the cord half-angle, C, which varies from position to position around the cross-section. However, 23

105 Theoretical (Ref. 1) -- Approximate (Fig. 8) 104 Eo= 1371 I -I 0 15 30 45 60 75 90 Cord Half-Angle - a0 Figure 15. Comparison of exact and approximate shear modulus G. 10O~^ E x 1.268xl05

it has been found possible to select average locations at which the values of the extension modulus and shear modulus represent useful values fo:r the entire cross-section. The extensional stiffness of the "bar" is represented by AsE and primarily depends on the extensional stiffness of the carcass in the tread shoulder region of the tireO The cord half-angle in the tread and shoulder region of the tire ordinarily varies from about 55~ to 45~ and since the computation for EB is very simple at 45~, an effective Eg is selected as the value corresponding to the value of E at 45~. The total AsE is then calculated by multiplying the EO at 450 by the total carcass thickness and the width of the tread. Thus, AsE = Eg(450) o H * BW (1O) The shear modulus of the supporting foundation, G, is the shear modulus of the sidewall portion of the tire. Therefore G is approximated by its value at a location mid-way between the rim and the shoulder. In Figure 16 this position is noted by the dimension RS. The cord-half-angle, as, is approximated at this position by the "cosine law": RS co = cos = co (II) Where 3 is the cord half-angle at the crown. The relative accuracy of these approximate moduli is examined by comparing their values with those used previously for Tires 1, 2, and 3 in Table I. These comparisons and the resulting effects on the fore-andaft spring constant, Kf, are shown in Table III 25

TABLE III COMPARISON OF RESULTS USING APPROXIMATE MODULI Tire 1 Tire 2 Tire 3 G-From Table I 28440 47164 43876 G-Approx. from Eqs. (8) and (11) 29000 48500 45400 AsE —From Table I 779 747 760 AsE-Approx. from Eq. (7) - a = 45~ 780 580 533 Kf-From Table II 2503 2286 2944 Kf-Using Approx. A E and G 2530 2070 2520 Kf —Experimental 2530 2300 2780 r I ~ ROC RI ROB Axle __ Figure 16. Sidewall location for effective G. 26

As can be seen, the comparisons are relatively good and since this model for calculating Kf is simple, it seems justifiable to use these simpler expressions for the moduli in approximate calculations of the spring rate Kf. To summarize the method for calculating Kf and to illustrate the relative ease with which it can be done if one uses the approximate moduli outlined above, an example problem is given below. PROBLEM STATEMENT Calculate an approximate fore-and-aft spring rate for the tire described in Figure 17. This tire roughly corresponds to a standard 9.50 by 14 four-ply tire. The elastic properties of the individual ply are calculated first by referring to Eqs. (9): (.028) H (.033)= 0.7 xs =/ (.028)(20) = 0.500 Ex = (300)(20)/(.033) = 1.82 x 105 psi = 530 L 1+ 2.9 0l758 x 0.500~ = 1760 psi I C = 1z- o.500 j:o G = 705(0.7854)(0.758)(0.500) + 180 (1-0.758 x 0.500) 325 psi The extension modulus EO is then computed from Eq. (7), using ca = 45~: 27

5.35 14.25 13.30 RS 7.30 Axle, _ ___ AESUBC =300 lb/ in. NCORD=20 ends/in. TPLY =0.033 in. ERUB = 550 psi GRUB = 180 psi H = 0.132 in. BW = 5.35 in. P = 37.5~ ROB = 7.30 in. ROC = 14.25 in. RI =13.30 in. A = 9.70 in. DIAMC = 0.028 in. a = 14.83 in. Figure 17. Description of hypothetical tire for example problem.

E = = 4(325) = 1300 psi Thus AsE can be found from Eq. (10): AsE = 1300(0.132)(5.35) = 915 lb The next property to calculate is G, whose mean value is found at a location approximately half way between the rim and the shoulder. The radial location RS is determined by locating the point half way between ROB and RI (see Figure 16), which for this example is: RS RI + ROB 730 + 1330= 10.0 in 2 2 The cord half-angle at this location is found from Eq. (11): cos a - 10 cos 3750 = 0.573 and sin as = 0.819 Thus from Eqo (8): G = 1.82 x 105(0.573)2(0.819)2 = 4.00 x 104 psi Now referring to Eq. (6), the fore-and-aft spring rate is Kf = 2(AsE)(q)tanh qra) The value for q is found from Eq. (1) and the definition of Ks: 2GH 2(4.00 x 10 )(0.132) 10 Ks = -~ = 1070 lb/ino/in. 9.90 q - 1070 - l.08/in. 29

Therefore, the spring rate is Kf = 2(915)(1.08)(1) = 1980 lb/ino 30

VI. APPENDIX II Consider a plane element made up of a series of parallel cords, with end count n, each carrying a tension load To as shown below. T To The number of cords per unit length / on the upper and lower faces is n cos a. The tension component per cord tangential to the upper or lower faces is To sin a. Hence, / the shear force per unit length T is To sin a * n cos a. If the thickness of the lamina is h, then the shear stress T, which by equilibrium acts on all edges equally, is T = To T sin a cos a (12) Now consider deformation of the element shown above in the shear direction, based on the concept that the cord will elongate under tension To. From the geometry, x sin a = e~o where c is cord strain, Lo the original cord length. x = eQo/sin a. 31

/0/ a 7 But shear strain is defined as x = o (13) = cos a ao sin a cos a sin a cos a Also, cord tension and strain are related by the cord spring rate To = (AE)c. e Hence, shear modulus G is n T n sin a cos a G T T = o2h / (AE)c sin a cos a r TO n 2 2 (AE )c sin2 a cos2 C (14) But the extension modulus parallel to the cords is (AE)c, = E, so that G = Ex sin2 a cos2 (15) 32

VII. ACKNOWLEDGMENTS The authors wish to thank Mr. Bo Bourland, Mr. P. A. Schultz, and Mr. B. Bowman for their assistance in obtaining the experimental data presented in this report, 33

VIII. REFERENCES 1. Clark, S. K., "Internal Characteristics of Orthotropic Laminates," Textile Research Journal, Vol. 33, No. 11, Nov. 19630 2o Clark, S. K., Dodge, R. N., Field, N. L., "Calculation of Elastic Constants of a Single Sheet of Rubber-Coated Fabric,r The Univo of Michigan, ORA. Technical Report 02957-14-T, Febo 1962. 3o Smiley, R. F., and Home, Wo Bo, "Mechanical Properties of Pneumatic Tires with Special Reference to Modern Aircraft Tires," NACA Technical Note 4110, National Advisory Committee for Aeronautics, Washington, D. C, January, 1958. 4 Clark, S. K., "Simple Approximations for Force-Deflection Characteristics of Aircraft Tires," NASA Contractor Report, NASA CR-439, Washington, D.C., May, 1966. 5o Dodge, Ro N., "Prediction of Pneumatic Tire Characteristics from a Cylindrical Shell Model " Paper presented at SAE Mid-Year Meeting, Chicago, May 1965o 6 Clark, S. K., "An Analog for the Static Loading of a Pneumatic Tire " Paper presented at SAE Mid-Year Meeting, Chicago, May 1965o 7. Tielking, J. To, "Plane Vibration Characteristics of a Pneumatic Tire Model)" Paper presented at SAE Mid-Year Meeting, Chicagc, May 1965o 34

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