# Modified Equation of Shear Strength with Respect to Saturation
## 1 Introduction  
Shear strength is regarded as one of the important mechanical properties in the forecast of the stability of slopes, subgrade bearing capacity, and pressure against a soil-retaining structure. Taking slope failure as an example, after reservoir impounding, reservoir water level fluctuation and rainfall infiltration have become main factors that affect slope stability [1, 2, 3, 4]. There are several reasons for this phenomenon. First, infiltrated water increases the sliding area's weight and the pore water pressure, leading to slope failure [5, 6]. Second, infiltrated water softens the soil's shear strength and leads to slope failure [7, 8, 9]. Therefore, shear strength has a significant impact on the analysis of stability for slope in geotechnical engineering [10]. Many theories or empirical equations have been developed to predict or estimate the shear strength of unsaturated soil. [PERSON] pointed out an effective stress shear strength equation to explain the mechanical behavior of unsaturated soil [11, 12, 13]. Later, two independent stress sources including net normal stress \((\sigma-\mu_{a})\) and matric suction \((\mu_{a}-\mu_{w})\) were used to interpret the behavior of soil [14]. These two equations claimed by [PERSON] and [PERSON] were of great importance for shear strength in an unsaturated soil area, however,-matrix suction \((\mu_{a}-\mu_{w})\), which is difficult to test, was still included in these two models [15; 16]. In order to develop an empirical model of shear strength for unsaturated soil, many scholars such as [PERSON] [17] and [PERSON] [18] have pointed out an empirical model with fitting parameters. Many of these equations also adopted the soil-water retention curve (SWCC) as the controlling parameter to project shear strength [10; 19]. [PERSON] [10] showed a procedure to predict the shear strength of an unsaturated soil by using the soil-water characteristic curve and the saturated shear strength parameters. [PERSON] [20] proposed an effective stress equation that was able to predict the soil strength and deformation behavior. [PERSON] [21] proposed a closed-form equation for effective stress in unsaturated soil that only required two controlling parameters including the inverse of the air entry pressure and the pore size spectrum number. Other scholars have also discussed empirical models that considered the air-entry value [22] or the distinct element method [23]. However, these models focused on the relationship between the shear strength and matric suction rather than directly considering the water content. In addition, experiments for suction are time-consuming and need an expensive and sophisticated laboratory test system [24; 25]. This resulted in a limited application of shear strength empirical equations in engineering and scientific areas. Up to now, there has only been a limited practical application of the unsaturated soil shear strength theory in engineering practice. Nonetheless, engineers can measure the degree of saturation with a simple experimental method at each location. As summarized above, for various engineering fields, it is of great importance to propose a simpler method to directly estimate the shear strength of unsaturated soil including saturation. This research, based on the unsaturated soil shear strength theory proposed by [PERSON]'s soil-water characteristic curve model, a new multiple parameter shear strength model was developed to describe the influence of saturation. In this model, the matric suction was replaced by the degree of saturation. In the end, the equation was a function of saturation degree, residual saturation degree, effective cohesion, effective angle of internal friction, and net normal stress. The proposed model was validated against experimental points from experiments such as the triaxial shear test and the SWCC test as well as the literature data reported for clayed-silty sand [26] and Ankara clay [27]. Section 2 introduces the development of the new shear strength model.