# Stability Numbers for Unsupported Conical Excavations in Multi-Layered Cohesive Soils
## 1 Introduction  
The unsupported excavations are commonly encountered in civil engineering projects. Slope instability failures pose serious threat to structures, as these failures may lead to great loss of lives and property [1]. Predicting the stability of excavation is an important task for geotechnical engineers. The combined effects of geology, hydrology, and soil properties generally exist as stability issues.  
In order to design the excavation, more rigorous numerical modeling is needed, especially for large-scale complex projects. However, the performing numerical simulations are not always warranted or feasible because of time and cost constraints [2]. Therefore, stability numbers can be seen as convenient tools that provide an easier way to determine the factor of safety.  
The stability numbers (N) were used extensively as design tools, and draw the attention of many investigators. The first set of stability number was proposed by [PERSON] [3] and various stability charts have been developed [4; 5; 6; 7; 8; 9; 10; 11]. Traditionally, majority of research have studied the stability number by using the limit-equilibrium method (LEM), which is one of the most popular methods to assess the slope stability [12; 13; 14]. It is known that for the LEM not only the potential slip surface must be assumed before calculating the factor of safety, but also it cannot guarantee the best solutions [15]. It was found that the factor of safety, which is calculated by LEM, is overestimating compared with the other method [16] and using the finite-element method (FEM) can avoid this drawback, as the failure mechanism is generated automatically [17; 18]. With the finite-element method it is generally possible to model many complex conditions with a high degree of realism, including in the analyses, such asthings as nonlinear stress-strain behavior, non-homogeneous conditions, and changes in geometry during construction of an excavation [19].  
The finite element method has been widely used to simulate excavation construction steps (e.g., [20; 21; 22]). In a few research [5; 9], FEM was used to perform stability charts by using FEM, and only limited parameters such as angle of inclination, the depth ratio, width ratio, which consists of a height, and a radius at the bottom of excavation was considered. In particular, majority of studies have studied homogenous soil. Therefore, various charts based on different valuable parameters such as a strength difference between two layers, relative to the top layer thickness in multi-layer is needed.  
In this study, a series of numerical studies were conducted to assess the variation of the stability number in the multi-layer using FEM. The valuable parameters, which are (1) angle of inclination (\(\beta\)), (2) depth ratio (\(D\)/\(H\)), which is relative to the top layer thickness to excavation depth, (3) strength ratio (\(c_{u1}\)/\(c_{u2}\)), which is strength difference between two different layers on the rigid base, (4) width ratio (\(H\)/B), which is the excavation height to radius at the bottom of excavation, and (5) thickness ratio (\(T\)/\(H\)), which is the ratio of the excavation height to thickness of soil layers, are considered to confirm the stability. The stability number (\(N\)) was provided as a function of these parameters. The obtained stability numbers are compared with existing solutions published in the literature [5; 6]. The analysis results are in good agreement with published data. The stability numbers in variable conditions are presented using chart solutions. This chart shows the trend of stability number according to parameters and it can be used for engineering practice.