Stability assessment of 3D slopes is commonly presented as dimensionless charts. Even though the chart provides quick stability assessments, it offers estimates due to the linear interpolation caused by its absence of specific slope conditions. Moreover, it is challenging to be incorporated into the modern digital design workflow. This issue can be mitigated by obtaining closed-form solutions through regression analysis. The charts typically fall into two forms: c/γHF _s versus tanϕ/F _s and F _s/tanϕ versus c/γHtanϕ (c = soil cohesion; γ = soil unit weight; H = slope height; F _s = factor of safety, ϕ = angle of internal friction). Chien and Tsai (2017) and Huang and Ji (2022) adopted the former and latter to obtain closed-form solutions of the 2D slope, respectively. However, the factor of safety in the stability equation of Chien and Tsai (2017) is implicitly expressed and obtained by solving the equation. Consequently, this paper employs the latter form to obtain the closed-form solutions of the 3D slope. The straight finite slope is used as the baseline to conduct regression analysis in this paper. This regression analysis method is innovatively used to obtain the closed-form solutions of reinforced slopes.

Numerous stability charts show the relationship between F _s/tanϕ and c/γHtanϕ is typically a power function when c/γHtanϕ is small and a linear function when c/γHtanϕ is large. Huang and Ji (2022) used a power function (Eq. 1) to address this function for the 2D slope. Both A and B in Eq. 1 are functions of the slope angle β (as shown in Eqs 1, 2). To be consistent with the form of Eqs 2, 3, the expression for “A” is rewritten as a quadratic function. The parameter B is split at c/γHtanϕ = 1 to improve the accuracy while meeting the continuity condition of the equation. For a cohesionless slope, the factor of safety is precisely calculated as tanϕ/tanβ according to the 2D slope model. This conclusion is similarly applied to the 3D slope model. Note that the slope angles β in stability equations in this paper are inputted as degrees (°). F s _ 2 D tan ⁡ ϕ = A c γ H ⁡ tan ⁡ ϕ B + 1 tan ⁡ β (1) A = 10.5 + 2.9 × 10 − 4 β 2 − 0.091 β (2) B = 0.72 − 3.5 × 10 − 5 β 2 + 0.0032 β 0.83 − 2.2 × 10 − 5 β 2 + 0.0026 β (3)

The solutions of the 2D stability analysis methods are conservative, as they do not account for the 3D end effects. The 3D effects can be interpreted as the difference in F _s/tanϕ for the same c/γHtanϕ and the same slope angle in the stability charts. For a cohesionless slope (i.e., c/γHtanϕ = 0), the factor of safety obtained by both the infinite and 3D slope model is equal to tanϕ/tanβ, resulting in zero 3D effects. On the other hand, the 3D effects increase approximately linearly with c/γHtanϕ. Consequently, the 3D effects can be expressed as a proportional function of c/γHtanϕ with a coefficient of A _3D. The coefficient A _3D is a function of the width-to-height ratio L/H (L = slope width) and the slope angle. The regression analysis revealed that A _3D can be concisely written as a power function of (L/H)sinβ, which captures the geometric feature. A 3 D = 2.29 L H sin ⁡ β − 1.12 (4)

Based on the stability equation for the 2D slope obtained by Huang and Ji (2022) and the 3D effects defined by Eq. 4, the stability equation for 3D slopes is expressed as F s tan ⁡ ϕ = A 3 D c γ H ⁡ tan ⁡ ϕ + A c γ H ⁡ tan ⁡ ϕ B + 1 tan ⁡ β (5)

In this study, three assumptions are made for the reinforced slope: 1) reinforcements are required to be long enough to avoid compound failure; 2) the filled soil is cohesionless; 3) the vertical spacing between reinforcement layers is constant throughout the structure. Hence, the tensile strength of reinforcement layers can be distributed evenly along the height and width of the slope. If a step-wise layout of reinorcements is adopted, the tensile strength is divide into different evenly distributed sctions (Michalowski, 1997). The dimensionless form, k _u/γHF _s (k _u = nT _ult/H, n = the number of reinforcement layers, T _ult = the ultimate tensile strength of reinforcements), is used to assess the required strength of reinforced slopes. In the limit equilibrium framework, the factor of safety of the reinforced slope is applied to the soil strength and reinforcement strength: ϕ _m = tan⁻¹[tan(ϕ)/F _s], c _m = c/F _s and k _t = k _u/F _s (ϕ _m and c _m are mobilized soil strength, k _t = required reinforcement strength at limit state). Therefore, the stability charts for reinforced slopes can be represented in the form of F _s/tanϕ versus k _u/γHtanϕ. The same regression analysis procedure as the unreinforced slope is conducted to develop the closed-form solutions for reinforced slopes, and the results are as follows: F s tan ⁡ ϕ = A 3 D _ r k u γ H ⁡ tan ⁡ ϕ + A r k u γ H ⁡ tan ⁡ ϕ B r + 1 tan ⁡ β (6) A r = 7.48 − 1.63 × 10 − 5 β 3 + 0.00316 β 2 − 0.2 β (7) B r = 1.65 − 3.04 × 10 − 6 β 3 + 6.45 × 10 − 4 β 2 − 0.0461 β 2.21 − 2.66 × 10 − 6 β 3 + 6.77 × 10 − 4 β 2 − 0.0567 β (8) A 3 D _ r = 0.65 L H sin ⁡ β − 1.33 (9)

Both Eqs 5, 6 are concise closed-form solutions consisting of the 2D slope solution (Eqs 8, 9) and the 3D effects (Eq. 7). They can degrade into 2D slope solutions when L/H tends to be infinite. To verify the accuracy, five hundred sets of calculated parameters are randomly generated, and then the factors of safety are calculated with the LE method and closed-form solutions. As shown in Figure 2, the coefficients of determination (R ²) approach 1 and the root mean square error (RMSE) is nearly 0, indicating a high level of agreement between the LE methods and closed-form solutions. Though Eqs 6, 7 are developed for straight slopes, the kinds of closed forms can be extended to other geometry slopes (e.g., convex slopes and V-shaped slopes) by simply modifying the coefficient of 3D effects.

Convex filled slopes are common in mountainous areas. Zhang et al. (2019) presented a series of design charts of the corner reinforced slope based on the internal stability. The charts provided the required reinforcement length for each layer. This paper presents the design procedure of internal stability for convex slopes with a turning arc. Due to space limitations, the stability charts of reinforced convex slopes with a turning arc are not presented, and the analysis method refers to Yang et al. (2020) and Zhang et al. (2023). An example is illustrated here to illustrate the design procedure. The parameters of the example slope are selected as: soil cohesion c = 0 kPa, soil friction ϕ = 40°, unit weight γ = 20 kN/m³, slope height H = 6 m, slope angle β = 45°, turning angle θ = 90°, bottom curvature radius R = 12 m, reinforcement spacing S _v = 0.6 m, reinforcement layers n = 10. The design safety of factor F _s = 1.5 is used for internal stability. Both the static and dynamic cases (seismic acceleration k _h = 0.3) are assessed. For the given slope, the following steps are needed for internal stability design:

(1) For a required factor of safety of internal stability, F _s, the design angle of internal friction is ϕ _m = tan⁻¹(tanϕ/F _s);

(4) Select the maximum total embedment length for each layer, l = l _e + l _s, as the design embedment length.

Figure 3 illustrates the calculated total embedment length of the reinforcement at each layer for both 2D and 3D analysis. On the one hand, the 3D surface is deeper than the 2D surface, requiring a longer reinforcement length in the sliding zone. On the other hand, the 3D analysis results in a shorter embedment length in most layers compared to the 2D analysis due to the smaller required strength, T _max. Generally, in terms of the final result, the 3D analysis yields greater values of the required total length than the 2D analysis. That is, it would be unconservative if using the 2D method to deal with a 3D reinforced slope problem. However, the 2D method is typically conservative for the unreinforced slope.

A 67-m-tall and 1H:1V reinforced soil slope (RSS) was constructed at Yeager Airport in Charleston to support an extension runway. This RSS structure is a typical example of a convex slope with a turning arc. The design specified a vertical spacing of reinforcement of 46 cm in the lower third of the RSS, and it was increased to 91 cm at higher elevations. It resulted in a total of 85 layers of reinforcements. On 12 March 2015, after a service period of about 8 years, the RSS structure catastrophically collapsed in a compound mode with a distinct 3D failure characteristic. The failure surface sheared through more than 30 layers of reinforcements throughout the exposed head scarp. Only the primary uniaxial geogrids of 20G were used for the failure cross section of the RSS. The broken layers had a manufacturer published minimum average roll value, ultimate strength T _u = 187.8 kN/m and allowable strength T _a = 66.1 kN/m (Berg et al., 2020). However, the tests made by VandenBerge et al. (2021) suggested that their effective strength T _e at the time of the RSS collapse was estimated to be 117.8 kN/m, due to the installation damage and 8-year creep effect. The immediate cause of the failure was the decrease in the shear strength of the soil-rock interface below the reinforced soil. The shear strength decreased from the peak strength toward the fully softened strength during the service period. The strength for the fully softened strength is approximateϕ = 20°, which is much less than the design strength ϕ = 36°.

A forensic analysis of the Yeager Airport slope is conducted using the 3D variational LE method. According to the RSS plan view proposed by Triad Engineering, the RSS is simplified as a convex slope with a 45° turning corner and R/H = 1. Firstly, the stability required reinforcement strength at the limit state is investigated using the design soil strength ϕ = 36° and c = 0 kPa. The result shows that the required strength T _max = 33.1 kN/m is much less than the allowable strength T _a = 66.1 kN/m, which means that the RSS is stable enough if the weak layer does not exist. Then, the fully softened strength (ϕ = 20°) is taken into account for the stability analysis. To simplify the calculation, the fully softened strength is adopted for the entire slip surface. Actually, the simplification will yield a larger value of T _max than the true value, because the soil strength at the upper slip surface is replaced by the fully softened strength. However, it helps to determine the effects of the weak layer on slope stability. Figure 4 illustrates the comparison between the 3D slip surface obtained from the 3D analysis and the observed field, showing a good agreement. The calculated slip volume is approximately 120 thousand cubic meters. According to the results of the 3D analysis, only approximately 60 layers of reinforcements are mobilized. In this situation, the required strength of the upper 60 reinforcement layers is 139.5 kN/m, exceeding the effective strength T _e = 117.8 kN/m. These results verify that the weak layer is the cause of the RSS failure.