## 1 Introduction  
Expansion joints can be eliminated by exploiting the concept of a jointless bridge to essentially avoid noise while driving, reduced durability of the bridge, erosion of the substructure, and high maintenance costs resulting from damage to the expansion devices [1; 2; 3; 4; 5; 6]. Due to its outstanding service performance, low life-cycle cost, and favorable economic and social benefits, jointless bridges have been widely applied [7; 8; 9; 10; 11]. For bridges with expansion joints, the longitudinal expansion and contraction deformation of the girder induced by temperature variations (\(u_{\mathrm{t}}\)) can be absorbed by the gap at the expansion joint (\(\Delta u_{0}\)) installed between the girder end and the abutment backwall, as depicted in Figure 1. Therefore, no \(u_{\mathrm{t}}\) will be transferred to the abutment backwall and approach slab. For jointless bridges, the approach slab is longitudinally connected to the girder to transfer the \(u_{\text{t}}\) to the soil of the connecting road [12; 13; 14; 15; 16; 17], as depicted in Figure 2. The translation and rotation of the approach slabs due to the \(u_{\text{t}}\) can cause horizontal cracks and vertical settlement on the pavement of the connecting road at the junctions between the approach slab and the girder and at the end of the approach slab, as well as the settlement and cracking of the approach slab [18; 19; 20]. According to an investigation conducted by the Federal Highway Administration of the United States in 2004, the settlement and cracking of the approach slabs are two of the most prevalent diseases of jointless bridges in the United States, accounting for 46% and 28% of problems, respectively [21; 22]. It can be pointed out that the approach slab must be considered in the design of jointless bridges [23; 24; 25]. The typical approach slab is a reinforced concrete (RC) solid slab. When the pavement of the connecting road is concrete pavement, a grade flat approach slab (GFAS) is commonly employed. The GFAS can be considered part of the pavement structure because its elevation is usually the same as the pavement of the connecting road [23]. In many states of the United States, the GFAS is adopted in the standard design drawing [26; 27], of which the typical structure is shown in Figure 2a. For a jointless bridge with a GFAS, \(u_{\text{t}}\) is mainly absorbed through soil deformation (\(\Delta u_{23}\)) within the range from \(x_{2}\) to \(x_{3}\) at the end of the GFAS, as shown in Figure 2a. Nevertheless, since the deformation is concentrated in one small area at the end of the GFAS, cracks and settlement of the pavement are prone to occur in that small area, and the corresponding maintenance expense increases [28; 29]. To address this shortcoming, the buried flat approach slab (BFAS) was proposed and is now widely used in Europe, the United States, and Australia [14; 20; 30; 31]. The BFAS is horizontally embedded at a depth (\(H\)), which can be calculated as the vertical distance from the pavement surface to the lower surface of the approach slab, of which the typical structure is shown in Figure 2b. The  
Figure 1: Bridge with expansion joint. Figure 2: Absorption and transfer of \(u_{\text{t}}\) by approach slab: (**a**) FAS; (**b**) FBAS. BFAS is particularly suitable for connecting roads with asphalt concrete pavement [2]. For a jointless bridge with a BFAS, cracks and settlement at the end of the approaching slab can be effectively reduced because \(u_{\text{t}}\) is mainly absorbed through soil deformation between \(x_{1}\) and \(x_{2}\) and between \(x_{2}\) and \(x_{3}\) above the BFAS, which can be denoted as \(\Delta u_{12}\) and \(\Delta u_{23}\), respectively, as shown in Figure 2b. GFASs and BFASs can be referred to as flat approach slabs (FASs) because the elevations at both ends are the same. Under the influence of \(u_{\text{t}}\), an interaction between the approach slab and the soil can be found, namely the approach slab-soil interaction (ASSI).