# Bias Truck Tire Deformation Analysis with Finite Element Modeling
## 1 Introduction  
Nowadays, the vast majority of vehicles are equipped with radial tires, but they have not pushed bias tires out of the market. The latter are used mainly in all-terrain vehicles as well as agricultural and heavy-duty machines. Often, these are tires of large sizes. They are still utilized since they provide several benefits, including high sidewall resistance to mechanical damage, good performance on uneven ground surfaces, silent-running operation, good tread, self-cleaning properties, single-phase tire building connected with a simple design, and prices lower than radial tires. High demand for bias tires for vehicles is present in countries with a high percentage of unpaved roads [1]. The majority of leading tire producers design and manufacture bias tires (Goodyear: bias ply tire, Continental: crossply tire). It must be noted that many experienced bias tire designers are leaving their occupation and retiring, and their successors need tools to aid their design work. Thus, actions aimed at improving the already existing tools and creating new tools facilitating bias tire design are well-founded. Tire designing is an extremely difficult and complex task, arising from the fact that a pneumatic tire is a complex, heterogeneous, and anisotropic structure made of rubber, rubberized cord fabric (cord), and wire. The most important tire load-bearing members are bead wires made of wire coils, ensuringcontact connection between the tire and the wheel rim, and body ply made of one or several cord layers, whose ends (edges) are fixed in beads through wrapping around bead wires. In a non-inflated state, a tire is quite a flaccid/flexible structure, and when inflated, it stiffens and may carry external loads. Thus, pneumatic tire mechanics is fully non-linear, i.e., physically, geometrically, and with respect to boundary conditions. Deformation analysis of tire and strength of its elements is always a major computational challenge. The subject of pneumatic tire deformation modeling was discussed as early as the 1950s. Back then, [PERSON] [2] and [PERSON] [3] proposed a mathematical description of a tire's netting model in which rubber influence on tire deformation is disregarded. Even the simplest task of determining an inflated tire inner profile with the use of a simple netting model cannot be solved with an analytical approach. Differential equation describing this profile features no solution in elementary functions set [4]. Thus, in the past, in the tire designing process, there was the need to use atlases containing profiles for tires being in equilibrium [5]. Due to high labor consumption and the low accuracy of such actions, along with an increase in the availability of electronic digital machines, various algorithms were created to support this process with computers. In [6], based on the tire netting model, an automation method of determining bias and radial tire profiles was presented, and the entire set of computational tools, verified in the industry and successfully used in tire designing process, was described in [7]. Nowadays, calculation software based on the tire netting model is used in tire industry only for the initial tire profile design. Exact and reliable tire design verification is conducted with the use of the pneumatic tire finite element model. Since [PERSON] and [PERSON] [8], being some of the pioneers, applied the finite element method (FEM) for pneumatic bias aircraft tire deformation modeling, many tire models were worked out based on this method. The complexity of the problem of aircraft tire deformation modeling is well presented in the paper [9]. Due to the radial tire application universality, many publications describing tire computational models dwell upon radial constructions, and papers on bias tires are scarce, e.g., [10; 11; 12; 13; 14]. In the paper [15], proprietary FEM software is presented for the analysis of pneumatic tire axisymmetric models aimed at supporting the design of all-steel radial tires.