## 1 Introduction  
Air springs are composed of an air bag, steel beads, a bead ring plate, an upper plate, a lower plate, and a rubber spring as the auxiliary spring. The air bag is made of multilayered, reinforced fiber, rubber laminated matrix ply, inner and outer rubber shells. For railways, the diaphragm type in the form of a round tire is mainly used, and for buses and passenger cars, the rolling lobe type in the form of a sleeve is mainly used. Air springs have a low natural frequency, which could minimize transmission of vibration. Air springs can also change the spring stiffness by adjusting the air volume and internal pressure. Research has been conducted to predict the stiffness of air springs using theoretical models because the stiffness of the air spring is the most important among various specifications. It is mainly affected by the shape and cord angles of the bag, but the properties of the composite material are not considered in theory. [PERSON] [1] constructed a vertical stiffness analytical model of a rolling lobe air-spring based on hydrodynamics and thermodynamics and verified by comparing with the experimental data. The influence of the geometrical parameters such as diaphragm volume, auxiliary chamber volume, and the shape coefficient were analyzed and discussed. [PERSON] et al. [2] established an analytical formula to predict the vertical stiffness of rolling lobe spring and verified by comparisons to an axisymmetric finite element simulations and experimental data. Precise prediction of the vertical stiffness was obtained using the formula in cases where the reservoir volume was relatively small. [PERSON] et al. [3] constructed a model covering both the vertical and horizontal directions. In the vertical direction, they obtained the air spring's reaction force by using the derivation of the pneumatics equations. In the horizontal direction, they used a quadratic function to fit the trends of the variation in the air spring's horizontal stiffness, which was simulated by the finite element method. Since the cord angle affects the spring constant, it has been used as a variable in the study of stiffness. [PERSON] et al. [4] studied the aging mechanism of air springs. They analyzed the effect of aging on the vertical static stiffness of an air spring through an accelerated aging test and finite element simulation. They also examined the vertical static stiffness of air springs with different cord angles and the existence of emergency auxiliary spring during aging. It was found that the vertical stiffness increases with aging, as the cord angle is small, and there is no auxiliary spring. [PERSON] and [PERSON] [5] investigated the effect of design variables with the variation of the cord angle of an air bag. The orthotropic material and axisymmetric model were adopted. It was revealed that an air bag had different modes of deformation according to the cord angle and had different modes of change in the outer diameter and the fold height with deformation. [PERSON] and [PERSON] [6] also developed finite element code considering the orthotropic material properties and degenerated shell element to analyze a diaphragm-type air spring. They simulated the inflation and lateral deflection with a half symmetric model. In the analysis of air springs, angles are often modeled and analyzed as one value; thus, it is difficult to accurately input the angle and thickness due to the complex shape in which the values vary at all points along the latitude. The analysis of the fiber reinforced to support the load in the rubber base material is often used by the rebar element in the ABAQUS software. If the rebar element is used, the angle of the cord can be input differently for each element, and the change of the angle can also be tracked. [PERSON] et al. [7] developed a simulation technology that accurately predicts the static and dynamic characteristics of air springs.