Stability of gravity dam (crest 1530 m)

Data and assumptions (per metre length)

• Dam height $= 1530 - 1470 = 60$ m; upstream water depth $h_1 = 1525 - 1470 = 55$ m; tailwater depth $h_2 = 6$ m.
• Top width 6 m, base 60 m, upstream face vertical, downstream face straight from crest to toe: $\tan\alpha = 54/60 = 0.9$.
• $\gamma_w = 9.81$, $\gamma_c = 24$ kN/m³; μ (not given) assumed 0.7.
• "Full uplift": full hydrostatic uplift at the heel; with the gallery, IS 6512 rule: head at gallery $= h_2 + \frac{1}{3}(h_1 - h_2) = 6 + 49/3 = 22.33$ m.

Uplift intensities: heel 539.6, gallery 219.1, toe 58.9 kN/m².

      6
    +--+  1530
    |  |\
 55 |  | \     60 m
 m  |W1|W2\
    |  |   \  ~~ TW 6 m
    +--+----\ 1470
    <10>|<-50->
       gallery  B = 60 m

Forces and moments about the toe

$$\begin{aligned} \sum U &= 10{,}742\ \text{kN} \\ \sum V &= 8{,}640 + 38{,}880 + 158.9 - 10{,}742 = 36{,}937\ \text{kN} \\ \sum H &= 14{,}837.6 - 176.6 = 14{,}661\ \text{kN} \\ \sum M_R &= 1{,}892{,}799, \quad \sum M_O = 690{,}420\ \text{kN·m} \end{aligned}$$

Overturning

$$FS_o = \frac{1{,}892{,}799}{690{,}420} = 2.74 > 1.5 \quad \text{(safe)}$$

Sliding

$$FS_s = \frac{0.7 \times 36{,}937}{14{,}661} = 1.76 > 1 \quad \text{(safe)}$$

Tension

$$\begin{aligned} \bar{x} &= \frac{1{,}892{,}799 - 690{,}420}{36{,}937} = 32.55\ \text{m from toe} \\ e &= 30 - 32.55 = -2.55\ \text{m} \end{aligned}$$

The resultant is 2.55 m from the centre on the heel side; $|e| < B/6 = 10$ m, so no tension.

Compression

$$\begin{aligned} \frac{\sum V}{B} &= 615.6\ \text{kN/m}^2, \quad \frac{6|e|}{B} = 0.255 \\ p_{heel} &= 615.6(1 + 0.255) = 772.7\ \text{kN/m}^2 \\ p_{toe} &= 615.6(1 - 0.255) = 458.5\ \text{kN/m}^2 \end{aligned}$$

Principal stress at toe $= 458.5(1.81) - 58.9(0.81) = 782.2$ kN/m². All stresses are well below 3000 kN/m² (safe).

Answer: FS overturning = 2.74, FS sliding = 1.76, e = 2.55 m (within middle third, no tension), maximum stress = 772.7 kN/m² at heel < 3000 kN/m². The dam is safe. (If drains are ignored and full uplift acts over the whole base, FS overturning = 1.98 and FS sliding = 1.42; still safe.)