Stability checks of 50 m trapezoidal dam

Data (per metre length)

• Dam height 50 m, freeboard 5 m, so water depth $h = 45$ m (no tailwater).
• Top width 4.5 m, base 42 m, upstream face vertical; downstream slope $\tan\alpha = 37.5/50 = 0.75$.
• $\gamma_w = 10$, $\gamma_c = 25$ kN/m³; $\mu = \tan 37° = 0.754$; uplift coefficient C = 0.8.
• Crushing strength $= 10$ kg/cm² $= 10 \times 9.81 \times 10^4$ N/m² $= 981$ kN/m².

     4.5
    +--+   ---
    |  |\   5 m FB
 45 |W1| \
 m  |  |W2\    50 m
 -> |  |   \
heel+--+----\ toe
    <-- 42 m -->

Forces and moments about the toe

$$\begin{aligned} \sum V &= 5{,}625 + 23{,}437.5 - 7{,}560 = 21{,}502.5\ \text{kN} \\ \sum M_R &= 809{,}531, \quad \sum M_O = 363{,}555\ \text{kN·m} \end{aligned}$$

(i) Overturning

$$FS_o = \frac{809{,}531}{363{,}555} = 2.23 > 1.5 \quad \text{(safe)}$$

(ii) Sliding

$$FS_s = \frac{0.754 \times 21{,}502.5}{10{,}125} = 1.60 > 1 \quad \text{(safe)}$$

(iii) Tension

$$\begin{aligned} \bar{x} &= \frac{809{,}531 - 363{,}555}{21{,}502.5} = 20.74\ \text{m from toe} \\ e &= 21 - 20.74 = 0.26\ \text{m} < B/6 = 7\ \text{m} \end{aligned}$$

No tension (safe).

(iv) Crushing

$$\begin{aligned} p_{toe} &= \frac{21{,}502.5}{42}\left(1 + \frac{6\times0.26}{42}\right) = 511.97 \times 1.037 = 530.9\ \text{kN/m}^2 \\ p_{heel} &= 511.97 \times 0.963 = 493.0\ \text{kN/m}^2 \\ \sigma_1 &= p_{toe}\sec^2\alpha = 530.9 \times 1.5625 = 829.6\ \text{kN/m}^2 \end{aligned}$$

Both are below 981 kN/m² (safe).

Recommendation: The dam is safe in all four checks. The resultant is almost at the centre of the base, so stresses are nearly uniform; the section is on the heavy side and the downstream slope could be steepened to save concrete, after re-checking with earthquake and silt loads.