Abstract
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 E ξ 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 F ξη may also be determined by similar techniques, as may the orthotropic shearing modulus G ξη. 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.
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