Gravity dam stresses, principal stress, FOS (2079)

Data and assumptions

• Crest width 8 m; water level 5 m below the crest; $H_w=85$ m, dam height $=90$ m; tail water $H_t=10$ m; base width $B=60$ m.
• U/s face vertical; the d/s face is taken as a straight slope from the d/s edge of the crest to the toe (horizontal run 52 m over 90 m, $\tan\phi_d=0.578$).
• $\gamma_c=24$ kN/m³ (given), $\gamma_w=9.81$ kN/m³, $\sigma_a=2500$ kN/m². The friction coefficient is not given, so $\mu=0.70$ is assumed.
• Uplift: $\gamma_wH_w=833.9$ kN/m² at the heel, $\gamma_wH_t=98.1$ kN/m² at the toe, and at the gallery (10 m from the u/s face) $98.1+\frac13(833.9-98.1)=343.4$ kN/m².

  crest 8 m
   +--+
 5 m  |   freeboard
 ~~~~~|   water 85 m
      |\
      | \  d/s slope
  G   |  \
      |   \ ~~ tail water 10 m
      +----\
      B = 60 m

Forces and moments about the toe

Results

• $\Sigma V = \Sigma W - U = 73,723.4 - 16,922.2 = 56,801.1$ kN/m
• $\Sigma H = 35,438.6 - 490.5 = 34,948.1$ kN/m
• $\Sigma M_R = 2,915,106+1,635 = 2,916,741$ kN·m/m (weights plus the tail water thrust); $\Sigma M_O = 1,658,912$ kN·m/m

Overturning: $FS_o = \Sigma M_R/\Sigma M_O = 2,916,741/1,658,912 = 1.76$ — safe (> 1.5).

Sliding (friction): $FS_s = \mu\Sigma V/\Sigma H = 0.7\times56,801.1/34,948.1 = 1.14$ — safe (> 1).

Resultant: $\bar x = (\Sigma M_R-\Sigma M_O)/\Sigma V = 1,257,829/56,801.1 = 22.14$ m from the toe; $e = B/2-\bar x = 30-22.14 = 7.86$ m; $B/6 = 10.00$ m, so the resultant is inside the middle third (no tension).

$$\sigma_{toe} = \frac{\Sigma V}{B}\left(1+\frac{6e}{B}\right) = 946.7\,(1+0.786) = 1690.4\ \text{kN/m}^2$$ $$\sigma_{heel} = 946.7\,(1-0.786) = 203.0\ \text{kN/m}^2$$

The toe stress of 1690 kN/m² is less than the allowable 2500 kN/m², so the dam is safe against crushing. The heel stress of 203 kN/m² is compressive, so there is no tension.

Shear and principal stress at the toe ($\tan\phi_d=0.578$, $p'=98.1$ kN/m²):

$$\tau_{toe} = (\sigma_{toe}-p')\tan\phi_d = (1690.4-98.1)\times0.578 = 920.0\ \text{kN/m}^2$$ $$\sigma_{1,toe} = \sigma_{toe}\sec^2\phi_d - p'\tan^2\phi_d = 1690.4\,(1+0.334) - 98.1\times0.334 = 2221.9\ \text{kN/m}^2$$

Answer: (i) maximum vertical stresses: toe = 1690 kN/m², heel = 203 kN/m²; (ii) major principal stress at the toe = 2222 kN/m² (< 2500 kN/m²); (iii) $FS$ overturning = 1.76, $FS$ sliding = 1.14.