^{1}

^{2}

^{1}

^{2}

^{3}

^{1}

^{2}

^{2}

^{4}

^{1}

^{2}

^{3}

^{4}

Taylor’s

The

Cojean and Fleurisson [

In this study, the original Taylor’s

Figure _{1}C_{1} represents the potential sliding mass; _{1} is the slope height; _{1} is the central angle of the failure surface arc AB_{1}; _{1} is the inclination angle of the chord AB_{1}; and nH_{1} is the distance between the slope toe and the outlet of potential failure surface. When

A simple slope model (adapted from Taylor [

From the geometry of Figure

According to Taylor’s derivation, the gravity of the potential sliding mass AB_{1}_{1} is_{1} is the unit weight of the soil above the water level.

Let

Equation (

The gravity moment of potential sliding mass AB_{1}_{1} is_{1} is the arm of gravity to the center of the

Taylor has derived the equations in two cases, namely, (i) the outlet of the failure surface is at the slope toe (i.e., _{1}_{1}

In Figure _{1}_{1}. The arm of AEC_{1}

The gravity of parallelogram AEC_{1}F is _{1}

Replacing the last term of equation (_{1}_{1} is

Let

Then, equation (

With modified equation (

The submerged slope is shown in Figure _{2} to _{3} as shown in Figure _{2} to _{3}. Supposing there is no water recharge at the inner side of the slope, the pore water pressure on the failure surface can be expressed by the following equation:_{2}) and water level after drawdown (_{3}), _{u} is the pore water pressure coefficient, and

A slope model showing water level and the geometric denomination.

A slope model showing water drawdown and the geometric denomination.

Two extreme situations are considered. One is that the sliding mass is completely impermeable, so the soil between the water level before drawdown (_{2}) and after drawdown (_{3}) should be saturated. The pore water pressure on the failure surface can be calculated by the difference of _{2} and _{3}. The other case is that the sliding mass is completely permeable, and the pore water pressure on the failure surface is approximately zero. The general case is between the two extreme conditions, so the average unit weight of the soil between the water levels before and after drawdown is estimated by interpolating between the saturated unit weight and the unsaturated unit weight as_{a} is the average unit weight of soil between the water levels before and after drawdown, _{u} is the pore water pressure coefficient.

When the pore water pressure has not yet dissipated, _{a} equals saturated unit weight. When the pore water pressure has completely dissipated, _{a} equals the unsaturated unit weight, which means that the water level in the sliding mass is consistent with the reservoir.

The factor of safety can be obtained by superimposing three independent slope models with the same slope angle _{1}C_{1} with unsaturated unit weight as the original Taylor’s model shown in Figure _{2}C_{2}, where _{2} is the slope height, 2_{2} is the central angle of the failure surface arc AB_{2}, and _{2} is the inclination angle of the chord AB_{2}. The unit weight of AB_{2}C_{2} equals the difference between _{a} and _{1}, as shown in Figure _{3}C_{3}, where _{3} is the slope height, 2_{3} is the central angle of the failure surface arc AB_{3}, and _{3} is the inclination angle of the chord AB_{3}. The unit weight of AB_{3}C_{3} equals the difference of _{a}, as shown in Figure _{d}. Based on the derivation of Taylor, the analytical expression for the factor of safety in water drawdown slopes can be established.

Deduction of the gravity and gravity moment for the latter two models is listed in Appendix A. The polygon of force equilibrium in the case of water declining is shown in Figure

Polygon of force balance in water drawdown slopes.

The general expression for the factor of safety in the case of water drawdown is shown in equation (

Based on the general form, the factor of safety for six cases can be simplified as below.

For a simple homogenous slope as Taylor’s model, under the conditions of _{2} = _{3} = 0, _{2} = _{3} = 0,

For the completely submerged slope, under the conditions of _{2} = _{3} = _{1}, _{2} = _{3}=1, _{u} = 0, _{2} = 0, _{3} = _{1}, _{1} is replaced with

For the partially submerged slope, under the conditions of _{2} = _{3}, _{2} = _{3}, _{u} = 0, _{2} = 0, _{3} = _{1},

For the slope where water level declines very slowly or the soil is highly permeable, the water level in the slope is synchronous with the reservoir water level. In this case, the expression for the factor of safety is the same as equation (

For the slope that experiences water drawdown with a decrease of water level, the soil in the zone between the water levels before and after the water drawdown is regarded to be saturated due to capillarity. In this case, _{2} = _{sat}−_{1},

For the slope having water sudden drawdown, the soil between the water levels before and after drawdown is saturated and the pore water pressure on the failure arc has not dissipated. In this case, _{u} = _{2} = _{sat} − _{1}, and _{3} = −

The failure surface arc has three degrees of freedom, namely, _{1}, and _{1}. Given reasonable domains for the three degrees of freedom, the minimum factor of safety corresponding to the critical failure surface can be searched with any optimum method such as the golden-section method. In this study, the failure surface is determined using the computer program accomplished by the authors.

Considering the simple homogenous slope first, the factor of safety has a linear relationship with _{1} as shown in Equation (_{1} = 0.01 and 0.10, respectively, the minimum factors of safety against a range of slope inclinations and internal friction angles are shown as charts in Figure

Factor of safety charts for homogenous slopes. (a) _{1} = 0.01. (b) _{1} = 0. 1.

An example is described to explain how to use the charts. The slope height H1 is 50 m; the slope inclination angle ^{3}. With the given _{1} = 0.01 and is 1.89 when _{1} = 0.10 based on Figure _{1} is 0.034 based on the given _{1} values.

In Taylor’s chart, the factor of safety is defined as the ratio of actual cohesion to critical cohesion. To find the reliable factor of safety, it needs to adjust

For a submerged slope with the static water level, _{1} as shown in equation (_{1} = 1.0 and 2.0, _{2} is a series of values between 0 −1.0, the minimum

Factor of safety charts for different static water levels: (a) _{1} = 1.0, _{1} = 1.0, _{1} = 1.0, _{1} = 2.0, _{1} = 2.0, _{1} = 2.0,

For the case of water sudden drawdown, assume _{1} = 50 m, _{u} = 0, _{2} and the terminal water levels _{3} between 10 m and 45 m. Calculations are carried out with the orthogonal values of _{2} and _{3}. The results are presented in Figure _{2} = _{3}. The curves that intersect with the top curve show the factor of safety against the water drawdown from _{2} to _{3}. It suggests in this example that the factor of safety increases with the rise of the static water level. However, it decreases with the water sudden drawdown for all the original water levels. The larger the water level drawdown is, the lower the factor of safety is. The results are in agreement with the results by Cojean and Fleurisson [

Factor of safety charts for water sudden drawdown case.

The results in Figures

Based on Taylor’s

For simple homogenous slopes, the charts are designed for determining the factor of safety with aid of interpolation between two values of _{1} that has a linear relation with

For the slopes with static water level or water drawdown, the factor of safety is a linear function with _{1}. It is difficult to give a set of general charts due to the additional free variables such as unit weights above and below the water level. However, it is easy to obtain using the proposed computer programs. Some typical cases have been calculated and shown as charts in the paper.

For the simple slope model shown in Figure

For the model shown in Figure

Let

From Figure

Then,

For the model shown in Figure

Similarly, we have

Let

Similar to equations (

Referring to equations (

Superimposing the gravity and gravity moment of the three models, we can obtain the practical gravity and gravity moment.

Substituting equation (

Based on equation (_{2} (_{3}) is the arc length of the failure surface from point _{1} to the corresponding water depth _{2} (_{3}), 2_{p} is the center angle corresponding to the point of the resultant force

From Figure

D2 and D3 can be calculated with equation (_{2} or _{3}.

The acting position of _{2}−_{3})/3 above the water level after drawdown or (2_{3}−_{2})/3 below the top ground of the slope as shown in Figure _{3}−_{2})/3 can be expressed as

So,

Substituting equations (

Referring to Figure

So, the angle

By the cosine law in triangle ABD in Figure

By the sinusoidal law in triangle ADE in Figure

Substituting equation (B

By the cosine law and sinusoidal law in triangle BDE in Figure

So, equation (B

Substituting equations (B

A common definition of the factor of safety is the ratio of the shear strength of the soil to the shear strength mobilized to retain the limit equilibrium:_{d} is the mobilized soil cohesion, _{d} is the mobilized internal friction angle.

The mobilized internal angle is

Referring to Taylor [

Using sinusoidal law and referring to Figure

Substituting equation (C

The angle

The data used to support the findings of this study are included in the article.

The authors declare that they have no conflicts of interest.

The authors acknowledge the funding received from the National Natural Science Foundation of China (program nos. 41877242 and 41772278) and the National Key R&D Program of China (2017YFC1501302) which supported this study.