Analytical Solution for the Stresses Arising in +/− Angle Ply Belts of Radial Tires
Stresses arising in the belts of radial ply tires, particularly those at the belt edge, are known to be critical to tire durability. Belt edge stresses are commonly calculated using finite element (FE) methods that provide estimates of the levels but do not necessarily give significant insight into the underlying mechanics. In contrast, analytical models can provide physical insight into the mechanisms affecting tire durability but are currently incomplete due to the challenges faced in obtaining closed-form mathematical solutions. Nevertheless, analytical solutions remain important to tire design and development because they can expose the entire design space, show the mathematical relationships between the variables, and allow rapid parameter studies. This work develops an analytical description of the belt deformations and stresses, particularly at the belt edge. The formulation captures all the first-order mechanics pertinent to finite width, antisymmetric +/− angle belt packages present in radial tires. It incorporates interply shear stresses already recognized in the literature and adds to that a new mechanism controlling the interaction of the plies via a Poisson effect. The analytical model is validated by comparison to FE simulations and is also contrasted with a classical analytical model in the literature. The design space for the belt composite is then explored by parameter variation. Finally, since all these solutions depend on homogenization of the belt layers, the analytical solution is compared to a FE model of discrete cables embedded in rubber to explore the accuracy of the homogenization step.Abstract
Lamina description. Note that the lamina is described by five parameters, the rubber modulus, the cable modulus, the cable pace, the cable diameter, and the cable angle relative to the body coordinate X. The lamina thickness is equal to the cable diameter. The material coordinate (1) follows the cable direction.
Lamina modulus versus the belt cable angle. The analytical solution, Eq (3), is compared with a finite element solution. The belt parameters are typical of those used in modern radial tire belts. The agreement is very good over the full range of possible belt cable angles.
Laminate description. Note that the laminate description adds only one additional parameter to the problem, the rubber layer thickness. The rubber in the lamina and laminate are assumed to have the same properties. The lamina, however, are laid at plus and minus angles relative to the X axis. In the tire, the X axis represents the circumferential direction, the Y axis represents the belt width direction, and the Z axis represents the radial direction.
Laminate centerline modulus versus the belt cable angle. The analytical solution, Eq (30), is compared with a finite element solution. The belt parameters are typical of those used in modern radial tire belts. The agreement is very good for belt cable angles greater than about 15 degrees. Below that angle the infinite cable modulus assumption becomes increasingly invalid. Most radial tire belts use angles from about 15 to 30 degrees. For belt cable angles greater than about 55 degrees the belt plys become effectively uncoupled and act like individual lamina. The lamina modulus, Eq (3), is also plotted for comparison purposes.
Interply shear strain representation. This diagram is looking along the Y axis at the edge of the belts. The angle describing the interply shear strain in the rubber layer is shown.
Interply shear strain versus belt width. The analytical solution, Eq (27), is compared with a finite element solution. The belt parameters are typical of those used in modern radial tire belts. The agreement is very good over the full width of the belts. The finite element solution underpredicts the peak strain at the belt edges by a few percent relative to the analytical solution. Note that the interply strain is an edge effect that effectively disappears 10 mm from the edge. Thus, to study belt edge shear strains the belt models do not need to be very wide. The decay of shear strain at the belt edges is effectively independent of the belt width.
Comparison of laminate centerline modulus versus the belt cable angle for the published solution of Puppo and Evensen, the present theory, and finite element model. The published solution lacks the stiffening mechanics due to lateral stress in the laminate.
Comparison of laminate interply shear strain versus belt width for the published solution of Puppo and Evensen, the present theory, and FE analysis. Typical base case laminate parameters are used. The published solution underpredicts the peak value at the belt edge by about 20% in this case. Smaller belt angles produce larger differences.
Parameter variation example, interply shear strain at the belt edge versus laminate parameters. Each laminate parameter is varied about a base case. The imposed strain case is represented by the darker curve. The imposed stress case is represented by the lighter curve. The variation of Poisson’s ratio and imposed lateral stress gives the same result regardless of the deformation mode of the laminate. The other parameters show complicated nonlinear behavior dependent on the deformation mode. The case of imposed strain energy density is not shown but falls between the limiting curves in this figure.
Interply shear strain versus cable diameter, lines of constant belt strength. Strain is imposed at 1%. Note that for each belt strength there is an optimum cable diameter that will minimize the interply shear strain. The imposed stress case does not show defined minimums but indicates that small cables and high belt strength is the direction to lower interply shear strain.
Complete view of the half million degree of freedom discrete finite element model of a two-ply cable reinforced laminate.
Zoomed view of internal construction of discrete finite element model of a two-ply cable reinforced laminate.
Zoomed view of internal element arrangement. The rubber layer separating the reinforced layers is meshed with quadratic elements that are aligned with the transverse and longitudinal axes of the complete ply. The geometry of these three layers are arranged such that all of the nodes lying in a plane that adjoins adjacent layers are coincident.
Zoomed view of the discrete model boundary conditions. Axial strain is introduced into the ply by applying displacements onto the end faces of the rubber layer separating the reinforced layers. The tendency of the ply to twist when subjected to the axial load is resisted by requiring the outer face of one of the reinforced layers to remain flat.
View of the locations, A and B, at the midplane of the rubber layer where the interply shear strain will be evaluated as a function of lateral position.
Comparison of discrete model and homogeneous theory interply shear strain values for the nominal case where the cable diameter is 0.56 mm, cable pace is 1.6 mm, rubber layer thickness is 0.9 mm, and belt angle is 25°. The centerline axial strain is prescribed at 0.1% (edge strain is therefore 0.08%).
Comparison of corrected discrete model and homogeneous theory interply shear strain values for the nominal case where the cable diameter is 0.56 mm, cable pace is 1.6 mm, rubber layer thickness is 0.9 mm, and belt angle is 25°. The centerline axial strain is prescribed at 0.1%.
Interply shear strain variation at the ply edge between cables and along the midplane.
Sensitivity of interply shear strain to variation in cable diameter. All other parameters are held equal to the nominal case. Points labeled “max” correspond to position “B.” Points labeled “min” correspond to position “A.” Also note that all of the discrete model interply strains have been corrected as previously described.
Contour plots of the interply shear strain at one of the ply edges for three values of cable diameter. The centerline axial strain is prescribed at 0.1%. Deflections in the load direction have been amplified by 100. The minimum and maximum points are labeled “A” and “B,” respectively.
Sensitivity of interply shear strain to variation in cable pace. All other parameters are held equal to the nominal case. Points labeled “max” correspond to position “B.” Points labeled “min” correspond to position “A.” Also note that all of the discrete model interply strains have been corrected as previously described.
Contour plots of the interply shear strain at one of the ply edges for three values of cable pace. The centerline axial strain is prescribed at 0.1%. Deflections in the load direction have been amplified by 100. The minimum and maximum points are labeled “A” and “B,” respectively.
Sensitivity of interply shear strain to variation in rubber layer thickness. All other parameters are held equal to the nominal case. Points labeled “max” correspond to position “B.” Points labeled “min” correspond to position “A.” Also note that all of the discrete model interply strains have been corrected as previously described.
Contour plots of the interply shear strain at one of the ply edges for three values of rubber layer thickness. The centerline axial strain is prescribed at 0.1%. Deflections in the load direction have been amplified by 100. The minimum and maximum points are labeled “A” and “B,” respectively.