Overturning and sliding check with shear strength

Geometry and assumptions (per metre length)

• Dam height $= 13 + 20 = 33$ m; water depth $h = 30$ m; no tailwater.
• Top width 6 m; downstream face vertical for 13 m, then slopes 15 m horizontally over 20 m height; base $B = 21$ m.
• $\gamma_w = 9.81$ kN/m³, $\gamma_c = 24$ kN/m³, μ = 0.7, shear strength $q = 1400$ kN/m² (taken per m² of base).
• Uplift is taken as full (C = 1), varying from $\gamma_w h$ at heel to zero at toe.

      6 m
    +----+
    |    |  13 m
 30 | W1 +
 m  |     \  W2   20 m
 -> |      \
heel+-------\ toe
    <6>|<-15->
       B = 21 m

Forces and moments about the toe

$$\begin{aligned} \sum V &= 4{,}752 + 3{,}600 - 3{,}090.2 = 5{,}261.9\ \text{kN} \\ \sum M_R &= 121{,}536, \quad \sum M_O = 44{,}145 + 43{,}262 = 87{,}407\ \text{kN·m} \end{aligned}$$

Overturning

$$FS_o = \frac{121{,}536}{87{,}407} = 1.39$$

Sliding

$$\begin{aligned} FS_s &= \frac{\mu\sum V}{\sum H} = \frac{0.7 \times 5{,}261.9}{4{,}414.5} = 0.83 \\ SFF &= \frac{\mu\sum V + Bq}{\sum H} = \frac{3{,}683.3 + 21 \times 1400}{4{,}414.5} = 7.49 \end{aligned}$$

Results

(Without uplift: $\sum V = 8{,}352$ kN, $\sum M_O = 44{,}145$ kN·m.)

Answer: With full uplift, FS overturning = 1.39 (slightly below 1.5) and friction FS = 0.83, but the shear friction factor = 7.49 > 5, so the dam is safe against sliding when the shear strength of the joint is counted. Without uplift, FS overturning = 2.75 and SFF = 7.98 (safe). To improve overturning safety, drainage holes should be provided to reduce uplift.