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The ring on elastic foundation (REF) model has been used as a pneumatic tire analog for many types of two-dimensional contact problems. This paper investigates the suitability of the model to reproduce hub forces during large deformation, quasi-static contact. Two types of tire contact problems are considered in this paper. The first problem is a tire deforming against a flat plate, and the second is a tire rolling over a 50 mm step. The REF model is implemented in a nonlinear, finite element program with elements that accommodate large deformation and contact. Experimental hub force measurements verify the simulation results. This paper shows that the best match with experimental data is accomplished with a model that consists entirely of beam elements and includes a softening foundation. Also, it shows that the hub force is strongly dependent on the radial stiffness of the foundation and the inflation pressure.

A brief review of tire critical speeds is given using the beam under tension as a physical model. In its most common form, this model visualizes the critical speed as that speed just sufficient to sustain a continuous sinusoidal bending wave in the tire tread band. A number of studies have been published modifying this concept by the introduction of material damping, centrifugal effects, and other characteristics, some of which aid in explaining the fact that the wave pattern observed experimentally is local and decays rapidly away from the contact patch. This paper presents a different view of the wave pattern needed for a critical speed to exist, namely that a naturally occurring local wave can arise independent of material damping and that as a practical fact, material damping may have little to do with the onset of the phenomenon. A discussion on the effect of tire design variables on critical speed is given based on the expressions derived here.

An analytical method is proposed for the prediction of load-deflection and stress-strain relations in two-dimensional elastomeric composites where the properties are highly nonlinear. Such materials are represented by various textile cords used as reinforcements in a rubber matrix. The analytical procedure involves a forward prediction process for simultaneous nonlinear equations and can readily be implemented for automated computer use. Experiments on nonlinear tubes are reported and computation of load-deflection relationships observed in those experiments are compared with experimental data. The predictions generally agree with the observed behavior.

An experimental method was developed for the measurement of air permeation leakage through tubeless automobile tires and wheels. A differential pressure measurement system was used with temperature as an independent parameter. The diffusion rates generally far exceeded the loss rates observed through either the valve core or the valve rim seal on several typical tires observed under normal static conditions. While permeation loss rates vary from tire to tire, typical losses are of the order of 0.5 to 1.0 psi (3.4 to 6.9 kPa) per month at 70° F (21° C) ambient temperature. Losses increase rapidly with increase in temperature to about 1.5 psi (10.3 kPa) per month at 90°F (32°C). An analytical model was developed which can predict pressure loss rates due to permeation under a variety of climatic conditions.

There has been considerable interest over the last twenty years in the subject of the elastic properties of cord-rubber composites. This has been due to the rather intensive study of composite material characteristics, brought about by the increased use of rigid composites in many structural applications. In addition, work on the prediction of cord rubber composite properties has also continued because of the active development of finite element programs for analysis of pneumatic tire properties. These finite element programs require a thorough knowledge of cord-rubber composite elastic characteristics, which in turn are found to be considerably more difficult to calculate than had been originally thought. The present paper introduces a simplified theory for the prediction of such properties with sufficient accuracy for engineering uses.

The degradation of adhesive strength between plies in aircraft tires was studied as a function of aging temperature, time, and environment. Adhesive strength degraded much faster than the cord tensile strength under the aging conditions used. Adhesive strength degraded faster in air than in nitrogen during the long-term aging. A 50% loss in adhesive strength is suggested as a reasonable critical value in determining tire life.

Linear relations have been observed experimentally between tire equilibrium rolling resistance, tire load, and the reciprocal of tire inflation pressure. These variables are used to formulate a simple general expression for rolling resistance, and predictions made with this expression are compared with measured data.

The question of heat generation by frictional rubbing is examined from the point of view of material elastic properties. Theory suggests and evidence is presented that low modulus materials tend to generate more internal heat during rubbing then high modulus materials of similar thermal properties.

An idealized model is proposed for heating of a pneumatic tire. A solution is obtained for the temperature rise of such a model. Using known thermal properties of rubber and known heat transfer coefficients, the time to reach thermal equilibrium is estimated.

An analytical study based on the Halpin-Tsai equations in their complex form has been conducted over a reasonably practical range of variables typical of a composite construction involving steel-cord reinforcement in a rubber matrix. Curves are presented which show the influence of several material properties on the composite characteristics of bias-angle constructions. From these curves, it is possible to deduce the influence of these constituent material properties on the overall composite response. The most significant deductions from these results are: (1). The elastic extension modulus of the composite, E′, is highly dependent on the bias angle and moderately dependent on the volume fraction of cord. On the other hand, E′ is essentially independent of the loss modulus of the matrix E″r. (2). The elastic shear modulus of the composite, G′, is also dependent on the bias angle and volume fraction of cord, but independent of the loss modulus of rubber. Also G′ is symmetric about a 45 degree bias angle. (3). The loss modulus in tension of the composite, E″, is highly dependent on the bias angle and moderately dependent on the volume fraction of cord and loss modulus of the matrix. (4). The loss modulus in shear of the composite, G″, is dependent on the bias angle and volume fraction of cord and nearly independent of the loss modulus of the matrix except at bias angles near zero and 90 degrees. (5). The loss tangent in tension of the composite, tan ΔE′ is highly dependent on the bias angle from 0 to 20 degrees and is approximately a constant for all other bias angles. The magnitude of the constant value is the same as tan δr of the matrix. (6). The loss tangent in shear of the composite, tan ΔG′ is nearly a constant except for very small or very large bias angles. The magnitude of this constant is approximately the small value of the loss modulus of the cord. Also tan ΔG is symmetric about 45 degrees. (7). For most bias angles, the loss tangent of the composite in shear is much less than the loss tangent of the composite in tension.

The basic formulation of plane stress-analysis techniques for textiles imbedded in rubber is given both from the viewpoint of a load-carrying net and of a continuous elastic material with orthotropic properties. The developments are basically dissimilar, and yet it is shown that, for material properties commonly encountered in rubber-coated textiles, both theories predict essentially the same textile loads, although the stresses carried by the rubber matrix differ in the two theories. This shows that the network approach, which is by far the simpler of the two, is perfectly valid for purposes of estimat ing cord load in cord-rubber structural members. A method is presented through the use of either theory to compute the fraction of load carried by the cord network and the fraction carried by the rubber matrix.

Results from an experimental study on rayon and nylon tire cords embedded in rubber are reported. Stress-strain curves in tension and compression on thick-walled cylinders of such materials show that much higher moduli of elasticity are exhibited by cords under tension than by cords in compression. However, the compression modulus of twisted cords is not insignificant, and such cords can apparently carry appreciable compressive loads when properly encased in rubber and when prevented from buckling. A bilinear approximation to the stress-strain curves of such materials is proposed as a means for approximate structural analysis of members utilizing these materials. Based on such a bilinear approximation, an analysis is presented for the state of stress and deformation in a laminated cord-rubber sheet under plane loads, using the principles of orthotropic materials. It is shown that four elastic constants are sufficient to describe the properties of a typical two-ply laminate when all cords are loaded in tension; a different set of four apply to the same body when it is loaded in such a way that the cords in both plies are in compression. The existence of interply stresses between lamina in built-up cord-rubber sheets is shown, and expressions are derived for the magnitude of such stresses as functions of the externally applied forces. Two different types of interply stresses are identified. Experiments were performed using laminated cord-rubber cylinders under various stress states in order to compare various moduli of elasticity with those predicted from theory. Such comparisons seem to indicate that the theories proposed adequately describe the composite materials being considered. It may generally be concluded from the work reported here that the elastic properties of any type of cord-rubber laminate may be derived from the elastic properties of a single sheet of its constituent material.

The increased use of cord- and filament-reinforced materials in structural applications during the last few years has resulted in a greater interest in their elastic properties. In part the reason for this may be found by considering the basic nature of redundant structures, in which the loads carried by individual cords are determined to some extent by the elastic characteristics of the entire system. In such situations, a knowledge of elastic characteristics becomes important to structural design practice. As a second reason for increased attention to the elastic properties of such materials, one might cite the body of work which is now developing in the general area of filamentary reinforcement of materials. A knowledge of elastic characteristics is important in obtaining optimum reinforcement properties, and such studies inevitably lead to a clearer understanding of the internal stress-states of all reinforced materials. One result of all this activity is that much of the work done in the areas of fiberglass and whisker reinforcement increases the general understanding, in a broad way, of the action of cord-reinforced rubber since in many respects the problems are similar, although major differences do exist in the structure of the reinforcement itself. The elastic properties of cord-rubber materials are understood today much better than they were even ten or fifteen years ago. A great deal of this development has paralleled, and is well represented by, internal stress analysis techniques developed for what is perhaps the primary utilization of cord-reinforced rubber, namely, the pneumatic tire. In the case of the pneumatic tire, these stress analysis techniques have essentially followed three distinct phases. In the first, the anisotropic nature of such materials was completely ignored and loads and stresses were determined on the basis of assuming the materials to be isotropic or unreinforced. In the case of shell structures, this is the equivalent of calculating the statically determinate membrane stresses. In some cases such information was valuable and in a few instances it comprised a major portion of the effects being studied so that some reliance could be placed on such an analysis. However, in general, due to the complex nature of such reinforced materials, little faith can be given to analyses based on isotropic conditions.

Plane orthotropic structures may be generated by bonding together individual layers, or plies, of material that is in itself plane orthotropic. Such orthotropic combinations usually possess reinforcement running in more than one direction when the orthotropy is introduced by imbedding wire or cordlike reinforcement in a softer matrix. The internal stress conditions are not uniform in such structures, as opposed to the usual isotropic materials. Hence, even under simple plane membrane loads they are subjected to shear stresses between the plies and to cord or reinforcement loads that are not immediately obvious. An approximation to their characteristics may be obtained by considering a structure with similar orthotropic properties uniformly distributed throughout each lamina. On the basis of exploratory tests giving ranges for the ratios of various elastic constants, it is shown that influence coefficients relating external normal stresses and interply shearing (distortion-causing) stress components as well as those relating external shearing stresses and interply normal (extension-causing) stress components may be calculated and plotted. These influence coefficients may be used to calculate the state of interply stress in multi-ply orthotropic laminates under membrane loads, provided that the external stress state is known. This external stress state may generally be obtained either by calculation or by measurement. 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 an orthotropic laminated structure are either in tension or in compression. This paper is restricted to those cases in which the stiffness of the cords is much greater than the stiffness of rubber. Thus, if the stresses in the cord directions 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 assumption, influence coefficients relating cord loads to applied external stress are presented. These indicate that the influence coefficients relating external normal stresses to cord loads are generally smaller than one. 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 90°; influence coefficients greater than 40 have been calculated. The elastic constants of a multi-plied orthotropic structure may be determined and plotted as a function of the cord half-angle α and one single numerical quantity representative of the degree of anisotropy of a single sheet of material used in the laminate. These calculations show that the cord stiffness and end count play an important part in the elastic modulus of extension Eξ at small cord angles. At larger cord angles these factors cease to be important and the over-all orthotropic modulus is determined primarily by the angle of the cords and the type of rubber used in the sheets. The cross modulus Fξη may also be determined by similar techniques, as may the orthotropic shearing modulus Gξη. Both of these are dependent upon cord stiffness and end count to some extent over the entire range of cord angles, particularly the latter term, which is extremely sensitive to this variable. Graphical and tabular values of these functions are presented and should be useful in estimating the effects of design changes on the stiffness of cord-rubber laminates.

A number of previous experimental studies on the cord load distribution in loaded and inflated bias-ply tires have revealed greatly different cord load fluctuation patterns between the crown and the side-wall regions of the tire. Typical among these is data taken from reference, where the cord load fluctuation measured at the crown center line of a typical bias-ply tire is as shown in Figure 1. The symmetry of this figure is easily explained by either bending or shell membrane considerations or by some combination of these. On the other hand, cord load fluctuations measured in the side wall of a bias-ply tire typically show fluctuations such as shown in Figures 2 or 3, also taken from the previously cited reference. Both of these latter figures show considerable asymmetry of the cord load as a function of position diagrams as the loaded tire is rolled through the contact patch. Bending of the side wall cannot account for such behavior as shown in Figures 2 or 3, since the changes in radius of curvature in the side wall region are completely symmetric about the center line through the contact patch, denoted by the zero degree point in Figures 2 and 3. Simple pressure vessel considerations are also not capable of explaining such diagrams, since changes in curvature in the side wall region are also completely symmetric and hence would result in symmetric stress distributions.