THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Engineering Mechanics Department of Mechanical Engineering Tire and Suspension Systems Sesearch Group Technical Report No. 27 FORE-AND-AFT STIFFNESS CHARACTERISTICS OF PNEUMATIC TIRES R. N. Dpdge David'rne S. K. Clark ORA Project 02957 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1966

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 iii

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS vii I. INTRODUCTION 1 II. SUMMARY 3 III. ANALYSIS 4 IV. COMPARISON OF THEORY WITH EXPERIMENT 11 V. ACKNOWLEDGMENTS 21 VI. REFERENCES 22 VII. DISTRIBUTION LIST 23 v

LIST OF ILLUSTRATIONS Table Page I. Summary of Geometric, Elastic, and Structural Properties of Five Automotive Tires 13 II. Summary of Experimental and Calculated Values of Kf 13 Figure 1. Idealized model of pneumatic tire for analyzing fore-and-aft spring rates. 4 2. Loaded element of the elastic bar portion of the model. 5 3. Meridional cross-section of the tire. 8 4. Experimental apparatus for checking Ks. 9 5. Load-deflection data for double tube experiment used to confirm the expression for Ks. 10 6. Idealized mid-line profiles of five pneumatic tires. 12 7. Photograph of experimental apparatus for Kf. 14 8. Schematic of fore-and-aft spring-rate test. 14 9. Fore-and-aft load-deflection curves. Tire No. 1. 15 10. Fore-and-aft load-deflection curves. Tire No. 2. 16 11. Fore-and-aft load-deflection curves. Tire No. 3. 17 12. Fore-and-aft load-deflection curves. Tire No. 4. 18 13. Fore-and-aft load-deflection curves. Tire No. 5. 19 vii

I. INTRODUCTION Pneumatic tires are functional parts of many operating mechanisms. In order to effectively design and engineer such mechanisms, it is often necessary to know the mechanical properties of the component parts3 including the tires. One of the major roles of the Tire and Suspension Systems Research Group has been to study and analyze some of the important mechanical properties of pneumatic tires and to present rational methods for predicting them. For the past several years an extensive structural analysis of the tire treated as a nonisotropic toroidal shell has been pursued. This has been a large undertaking and, although not complete, is slowly becoming usable. The analysis for the axisymmetric inflation problem currently provides good numerical results, and the analysis for the static loaded tire shows considerable promise of being equally useful. However, these analyses require a great deal of understanding and practice on the part of the user before practical results can be gotten from them. Thus, even though such analyses will eventually be available for more complete understanding of the pneumatic tire's role as an elastic body, less complicated methods for estimating important mechanical properties of tires are highly desirable. Several pat.hs have been followed by this research group in developing methods for predicting mechanical properties of tires. One technique involved modeling of the pneumatic tire as a cylindrical shell supported by an elastic foundation.5-7 This model provides relations for predicting several properties involving deformation in the plane of the wheel, such as patch length 1

vs. vertical deflection, vertical load vs. vertical deflection, plane vibration characteristics, transmissability characteristics, and dynamic response to a point load. 4 A more recent attempt4 involved analyzing the pneumatic tire as a string on an elastic foundation. This model is primarily used to predict lateral stiffness characteristics, vertical stiffness characteristics and twisting moments. However, this model has been found to be useful only when the inflation pressure is high, such as in the case of aircraft tires. This report presents a method for predicting the fore-and-aft stiffness characteristics of pneumatic tires. Fore-and-aft properties are important in the overall analysis of a tire since they represent the contributions of the carcass and tread to braking and tractive elasticity. A different model is required here since the cylindrical shell and the string on the elastic foundation do not provide means for transmitting such loads. It is hoped that this model will prove satisfactory for predicting fore-and-aft characteristics 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 serves as 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 model. A set of static tests was performed to establish an experimental value for the fore-and-aft spring-rate for the various tires. 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 characteristics. 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. F a EatcBrRIM Elastic Bar F ~ ASE E RIM Elastic Shear Foundation a a Figure 1. Idealized model of pneumatic tire for analyzing fore-and-aft spring rates. If it is assumed that the restraining force of the 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 ad

vantage is taken of the symmetry present in the model. (Ks'u)dx F/2 S " — S + a' dx d////// ra Figure 2. Loaded element of the elastic bar 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 Ks is provided only by the shear resistance of the sidewall. From Figure 2, it is seen that one may approximate K 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, a-Ks u = O S = TAs = EeAs = AsE a 5 ^8

where T is the stress, e the resulting strain, As the cross-sectional area of the bar at any location, and E the effective extension modulus of the tread region in the circumferential direction, Thus,;2u 2 U -q2u = o (1) ax2 where 2 Ks AsE The general solution of this equation is u = C1 cosh qx + C2 sinh qx (2) The boundary conditions for this problem are at x = 0, S = 2 (3) at x = ta, S = 0 Substituting (3) into (2) gives -F F C C = (4) C1 = 2AsE tanh 7taq 2 = 2AsEq Thus, u(x) = F ~ sinh qx - cosh (5) 2AsEq L tanh qr 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 load. Thus, 6

Kf = u(o) - 2AEq (tanh qua) (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 foundation. 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 of the tread region in the circumferential direction ASE, the effective shear modulus of the sidewall 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, A. 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 averaging criteria must be established in order to compute ASE and G for a given tire section. Such a criteria can be established by referring to Figure 3 and defining the effective ASE and G by: 4i.. (AsE)d ( (A E) e = (A5E)d f. [E (h[ +ht Et)] d s eff. 4) i o o (7) 4( r G G124d eff. d4 hi 7

ETh hC \ Shear Modulus G2 Extension Modulus A Normal to Plane of Paper E8 Figure 3. Meridional cross-section of the tire. In Eqs. (7), $ is the location angle in the meridional plane, Eq is the extension modulus in the circumferential direction, hc is the effective thickness of carcass at any location, ht is the effective tread thickness at any section, Et is the Young's modulus of the tread stock, and G12 is the effective shear modulus of the carcass at any location. Using References 1 and 2 as a guide, EG and G12 can be calculated for any meridional location knowing only certain elastic and geometric parameters required by most tire designers. 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 in8

D Dial Indicator Figure 4. Experimental apparatus for checking K5. dicator. 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, dA H Figure 4. Experimental apparatus for checking Ks, dicator. The slope of the experimental load-deflection curve., related to KsY K 2GH s A A summary of this experiment is presented below: B = 10.0 in. d2 = 1.252 in. dl = 0.986 in. H = 0.133 in. 9

el = 0.10 in. A = 1.72 in. e2 = 0.15 in. G = 200/3 lb/in.2 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 K 5 = 3) = 20.6 lb/in./in. A 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 8 10" TUBE LENGTH 6 - ~ /X, ^SLOPE 9 o - - Z LENGTH KS=041 /10- 22.0 #/IN. /IN. 4 - 2 0 I I ~1I I.01.02.03.04.05 DEFLECTION- IN. Figure 5. Load-deflection data for double tube experiment used to confirm the expression for Ks. 10

IV. COMPARISON OF THEORY WITH EXPERIMENT In an attempt to investigate the applicability of Eq. (6), a series of static fore-and-aft stiffness tests were run on representative tires. Before reporting these tests and their results, the five tires used are described in detail. The idealized centerline profiles of the tires are shown in Figure 6. Tire No. 1 is a domestic 4-ply, 8.00 x 14 bias-ply tire with standard nylon cord. Tire No. 2 is a 2-ply, 7.50 x 14 bias-ply tire with standard nylon cord. Tire No. 3 is an imported 4-ply, 5.90 x 15 bias-ply tire with nylon cord. Tire No. 4 is an imported 7.50 x 14 radial-ply tire with overheads reinforced with wire cord. Tire No. 5 is a European made 155 mm x 15 in. radial-ply tire with overheads reinforced with nylon cord. Table I is a summary of the pertinent elastic and geometric parameters required from the five tires. Using the results in this table, Figure 6, and Eqs. (6) and (7), 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 8. In this testing procedure the tires were loaded vertically to a fixed deflection. Then a varying fore-and-aft load was applied and the resulting deflection recorded. The slope of these load-deflection curves represents the experimental foreand-aft spring-rates. These tests were run for different vertical deflections and inflation pressures. The results of these tests are summarized in Figures 9 through 13. 11

6 6 5 5Tire No. 1 5 Tire No.2 Tire No. 3 4 - 4 4 3 3 IN. \ IN.3 \ IN. 2 2 2 I" I 0 1 2 3 4 0 2 3 ~__________ IN. 0 1 2 3 4 IN. IN. 6 5 s5 T Tire No.4 Tire No.5 4 4 IN. \ IN. 3 3 2 2 0 1 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. 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 0.220 0.160 0.250 0.280 G - effective shear modulus for Ks 28440. 47164. 43876. 269. 144. 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 0.610 0.0.3142 0.256 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. k^~~N 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 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

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

o 1/2 in. vertical deflection x 3/4 in. vertical deflection A I in. vertical deflection o 1 1/4 in. vertical deflection 300 15 psi 25psi 35psi xa x~~~~~~~~~~ (n)2 00~ ^a ^ co 0X 0 X 0 ~~,2OO A~~~0A AX 00 S0 Cr a 0~~~~~~~~~~~~0L i x o a w A X S a a 2~~~~~~~~ ^ 0 X L~ x > 1002 0 I I I I 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 I in. vertical deflection o 1 1/4 in. vertical deflection 18 psi 28 psi 38psi 0 200 Cn x x CD ~ ~ ~ ~ ~ ~ ~ x0 -J g x x o I ^ ^ A 0 0 0 x 0 X w ~00 ~ ^ ^ ^ x 0 X o X o B ^ L2 A 0 0 0 ~~~9loo~igr 10 Xoeac-f oddfeto uvs ieN.2 U U A o 0 0 0 x 0 I I I I I I I I f I I I 1 O.01.02.03 04.05.06.07 08 0.01.02.03 04.05.06.07.08 0.01.02.03 0, 0 0 0 0 HORIZONTAL DEFLECTION Figure 10. Fore-andy-aft load1-deflection curves. Tire No. 2.

o 1 /2 in. vertical deflection x 3/4 in. vertical deflection I in. vertical deflection o 1 1/4 in. vertical deflection 9 psi 19 psi 29 psi e200 U ~^x o Cr A~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 a x 0 U. A A w A 2 x S 0 0 ~lOO 20 X-?- L. ___ o_ 0 )( 0 Al U ~~~o0 2o cI X X 0 x B B 0 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 loadc-deflection curves. Tire No. 3.

o 1/2 in. vertical deflection x 3/4 in. vertical deflection A I in. vertical deflection o 1 1/4 in. vertical deflection 18 psi 28 psi 38 psi 200 U) -J H S S 2 o: rr a x Il 000 X- X-o QE x X XO B o 0 x o o x LI_ 0 0 0 c x O x O Ii Ro D x o eP tf o 8 o F ~ a o S o a o.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-deflection curves. Tire No. 4.

o 1/2 in. vertical deflection x 34 in. vertical deflection A I in. vertical deflection o 1 1/4 in. vertical deflection 10 psi 20 psi 30 psi 200 CO )0 00 1 Ldl ^ o a " "100~ ~0 ac W 0 K ^O,r ^ ^ 5 0 K - o 0 0 0 S S 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. 5.

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 pressure. Since Eq. (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 fashion. However, since the experimental values are nearly the same for all conditions examined, any results used as a comparison with the calculated values will serve as a meaningful check. 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 formulated above gives a method for approximating the fore-and-aft spring-rate of pneumatic tires using only the geometric, elastic, and structural properties required by most tire designers. 20

V. ACKNOWLEDGMENTS The authors wish to thank Mr. B. Bourland, Mr. P. A. Schultz, and Mr. B. Bowman for their assistance in obtaining the experimental data presented in this report. 21

VI. REFERENCES i. Clark, S. K., "The Plane Elastic Characteristics of Cord-Rubber Laminates," The University of Michigan, ORA Technical Report 02957-3-T, October 1960. 2. Clark, S. K., R. N. Dodge, N. L. Field, and B. Herzog, "Inflation of a Pneumatic Tire," The University of Michigan, ORA Technical Report 0295714-T, February, 1962. 3. Smiley, R. F., and W. B. Home, "Mechanical Properties of Pneumatic Tires with Special Reference to Modern Aircraft Tires," NACA Technical Note 41i10 National Advisory Committee for Aeronautics, Washington, DoC., January, 1958. 4. Clark, So K., "Simple Approximations for Force-Deflection Characteristics of Aircraft Tires," The University of Michigan, ORA Technical. Report 05608-8-T, December, 1965. 5o Dodge, Ro N., "Prediction of Pneumatic Tire Characteristics from a Cylindrical Shell Model," The University of Michigan, ORA Technical Report 02957-25-T, March, 1966. 6. Clark, S. K., "An Analog for the Static Loading of a Pneumatic Tire," The University of Michigan, ORA Technical Report 02957-19-T, March, 1964. 7. Tielking, Jo T,, "Plane Vibration Characteristics of a Pneummatic Tire Model," The University of Michigan, ORA Technical Report 02957-22-T, June, 1965. 22

VII. 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 The University of Michigan ORA File 1 S. K. Clark 1 Project File 10 23

UNIVERSITY OF MICHIGAN 3 9111111111 11 1111102539 7079 3 9015 02539 7079