Gravity dam stresses with drainage gallery uplift

Data and assumptions

• Bed RL 470 m, top RL 554 m, so height $=84$ m; MWL 550 m, so $H_w=80$ m; no tail water.
• U/s face vertical. D/s face vertical from 554 to 545 m, then 0.67 : 1 to the toe, giving a horizontal run of $0.67\times75=50.25$ m.
• Crest width is not given; 8 m is assumed, so $B=8+50.25=58.25$ m.
• $\gamma_c=24$ kN/m³ (assumed), $\gamma_w=9.81$ kN/m³, $\mu=0.75$.
• Uplift: 100% of $\gamma_wH_w=784.8$ kN/m² at the heel, 50% ($392.4$ kN/m²) at the gallery (8 m from the u/s face), zero at the toe.

 554 +---+          crest
 550 |~~~|  MWL  (H_w = 80 m)
     |   |
 545 |   \          d/s slope begins
     | G  \         0.67 : 1
     |     \
 470 +------\
     heel     toe

Forces and moments about the toe

Results

• $\Sigma V = \Sigma W - U = 61,353.0 - 14,567.9 = 46,785.1$ kN/m
• $\Sigma H = P_w = 31,392.0$ kN/m
• $\Sigma M_R = 2,389,982$ kN·m/m; $\Sigma M_O = 1,424,943$ kN·m/m

Overturning: $FS_o = \Sigma M_R/\Sigma M_O = 2,389,982/1,424,943 = 1.68$ — safe (> 1.5).

Sliding (friction): $FS_s = \mu\Sigma V/\Sigma H = 0.75\times46,785.1/31,392.0 = 1.12$ — safe (> 1).

Resultant: $\bar x = (\Sigma M_R-\Sigma M_O)/\Sigma V = 965,038/46,785.1 = 20.63$ m from the toe; $e = B/2-\bar x = 29.125-20.63 = 8.50$ m; $B/6 = 9.71$ m — inside the middle third (no tension).

$$\sigma_{toe} = \frac{\Sigma V}{B}\left(1+\frac{6e}{B}\right) = 803.2\,(1+0.875) = 1506.2\ \text{kN/m}^2$$ $$\sigma_{heel} = 803.2\,(1-0.875) = 100.1\ \text{kN/m}^2$$

The toe stress of 1506 kN/m² is well within the usual allowable compressive stress of concrete (about 2500 to 3000 kN/m²; no value is given in the question). The heel stress of 100 kN/m² is compressive, so there is no tension.

Shear and principal stress at the toe ($\tan\phi_d=0.670$, $p'=0.0$ kN/m²):

$$\tau_{toe} = (\sigma_{toe}-p')\tan\phi_d = (1506.2-0.0)\times0.670 = 1009.2\ \text{kN/m}^2$$ $$\sigma_{1,toe} = \sigma_{toe}\sec^2\phi_d - p'\tan^2\phi_d = 1506.2\,(1+0.449) - 0.0\times0.449 = 2182.4\ \text{kN/m}^2$$

Answer: maximum vertical stress at the toe = 1506 kN/m²; at the heel = 100 kN/m²; factor of safety against overturning = 1.68; against sliding = 1.12. (If the actual crest width differs from 8 m, replace the crest rectangle and recompute with the same table.)