# Mechanical Properties of the Functionally Graded Lining for a Deep Buried Subway Tunnel
## 1 Introduction  
Underground spaces in Shanghai are congested with subways and vital urban facilities [1]. The utilization of deep underground spaces solves the problem regarding the shortage of shallow underground space [2, 3, 4]. Nevertheless, challenges facing deep buried tunnels are still considerable. For example, soil and water pressure in the stratum deeper than 50 m exceeds 1 MPa [5], which challenges the bearing capacity and waterproofness of deep buried tunnels [6]. In order to improve the supporting performance and waterproofnessof subway lining, two methods are usually applied: either to increase the thickness of the lining or to enhance the concrete strength [7]. However, both are extravagant in that concrete strength and thickness are the same in each position, which leads to a waste of concrete in corresponding positions. In this paper, the functionally graded materials (FCMs) were introduced to the design of the lining of the subway tunnel, and concrete with different elastic moduli was arranged at different positions to reduce its cost on the premise of ensuring structural safety. The concept of FGMs was proposed in 1984 by materials scientists in the Sendai area as a means of preparing thermal barrier materials [8]. The composition of the functional graded materials changed continuously from one direction to another, resulting in a continuous change in the material properties (such as elastic modulus) [9]. The variations of composition and properties in conventional composite materials and FGMs are illustrated in Figure 1.  
Extensive applications of FGMs have extended to many fields where the operating conditions are severe, including aerospace, chemical plants, nuclear energy reactors and so on [10]. Efforts were also made to study the performance of construction materials, such as metal and cement-based materials [11]. [PERSON] et al. [12] developed the sustainable structure constructed with functionally graded concretes using fibers and recycled aggregates. [PERSON] et al. [13] studied the mechanical properties of the graded concrete specimens composed of recycled aggregates and steel wires recycled from waste tires. [PERSON] et al. [14] established the concept of functionally graded fiber cement, and the use of statistical mixture designs was discussed to choose formulations and present ideas for the production of functionally graded fiber cement components. [PERSON] et al. [15] employed a functionally graded material system to make fiber more efficient in a fiber reinforced cement composite with four layers, each with a different fiber volume ratio. Differently, the study introduced in this paper aims to develop a functionally graded cement-based lining for the subway tunnel. To achieve this, the elastic modulus function is adopted, which is a power law function with respect to angle \(\alpha\). There have been many analytical results about functionally graded hollow cylinders subjected to mechanical stresses. [PERSON] et al. [16] defined the elastic modulus as a linear function varying with the radius, and Poisson's ratio is set as a constant, then, the exact solutions of the hollow cylinder with continuously graded properties are obtained. [PERSON] et al. [17] assumed the elastic modulus as a simple power law function varying through the wall thickness, and the exact solution for displacement and stress is determined when Poisson's ratio is assumed constant. Similarly, [PERSON] et al. [18; 19] analyzed the deformation of functionally graded cylinders composed of incompressible isotropic linear elastic materials, with the variations of the shear modulus in the radial direction given by a power law relation and constant Poisson's ratio. In the other case, the exact elasticity solution of a radially nonhomogeneous hollow cylinder was derived, whose elastic modulus varied in an exponential and power law function [20]. Beyond those above, the radial elastic  
Figure 1: Variations of composition and properties in conventional composite materials and FGMs.