Gravity dam stability and shear stress at toe (2081)

Assumptions and data

• RL of base $=305$ m; top $=390$ m (the printed 1390 m is read as 390 m); dam height $=85$ m. Maximum water level 385 m, so $H_w=385-305=80$ m. Tail water 311 m, so $H_t=6$ m.
• Crest width 5 m; u/s face vertical; d/s face vertical from 390 to 380 m, then sloping to the toe. Base width $B=56$ m, so the d/s slope runs 51 m horizontally over 75 m (about 2 : 3 as drawn), $\tan\phi_d=51/75=0.68$.
• $\gamma_w=10$, $\gamma_c=24$ kN/m³, $\mu=0.70$, allowable stress 2500 kN/m², shear strength 2000 kN/m².
• Uplift: $\gamma_wH_w=800$ kN/m² at the heel, $\gamma_wH_t=60$ kN/m² at the toe; the gallery (8 m from the u/s face) reduces the pressure there to $60+\frac13(800-60)=306.7$ kN/m². Earthquake is neglected.

 390 +----+
     |    | crest 5 m
 385~|~~~~|  MWL (H_w = 80 m)
     |    |
     |    |  vertical to 380
     |    \
     |     \  slope 2 : 3
     |      \
     | G     \
 311 |        \~~ tail water 6 m
 305 +---------\
     |<-- B = 56 m -->|

Forces and moments about the toe

Stability checks

• $\Sigma V = \Sigma W - U = 56,222.4 - 13,226.7 = 42,995.7$ kN/m
• $\Sigma H = 32,000.0 - 180.0 = 31,820.0$ kN/m
• $\Sigma M_R = 2,106,826$ kN·m/m (tail water thrust moment included); $\Sigma M_O = 1,344,711$ kN·m/m

Overturning: $FS_o = \Sigma M_R/\Sigma M_O = 2,106,826/1,344,711 = 1.57$ — safe (> 1.5).

Sliding (friction): $FS_s = \mu\Sigma V/\Sigma H = 0.7\times42,995.7/31,820.0 = 0.95$ — less than 1: not safe by friction alone.

Shear-friction factor: $SFF = (\mu\Sigma V + \tau B)/\Sigma H = (0.7\times42,995.7 + 2000\times56)/31,820.0 = 4.47$ — safe (above the usual minimum of 3 to 5 for normal loading, at the lower end).

Resultant: $\bar x = (\Sigma M_R-\Sigma M_O)/\Sigma V = 762,115/42,995.7 = 17.73$ m from the toe; $e = B/2-\bar x = 28-17.73 = 10.27$ m; $B/6 = 9.33$ m — outside the middle third (tension at the heel).

$$\sigma_{toe} = \frac{\Sigma V}{B}\left(1+\frac{6e}{B}\right) = 767.8\,(1+1.101) = 1613.0\ \text{kN/m}^2$$ $$\sigma_{heel} = 767.8\,(1-1.101) = -77.4\ \text{kN/m}^2$$

The toe stress of 1613 kN/m² is below the allowable 2500 kN/m², so the dam is safe against crushing. The heel stress is -77 kN/m² (tensile), so tension occurs at the heel.

Shear and principal stress at the toe ($\tan\phi_d=0.680$, $p'=60.0$ kN/m²):

$$\tau_{toe} = (\sigma_{toe}-p')\tan\phi_d = (1613.0-60.0)\times0.680 = 1056.0\ \text{kN/m}^2$$ $$\sigma_{1,toe} = \sigma_{toe}\sec^2\phi_d - p'\tan^2\phi_d = 1613.0\,(1+0.462) - 60.0\times0.462 = 2331.1\ \text{kN/m}^2$$

Shear stress near the toe: $\tau_{toe}=1056$ kN/m², which is about half of the average shear strength of 2000 kN/m², so it is safe. The principal stress of 2331 kN/m² is also below the allowable 2500 kN/m².

Conclusion: overturning and the shear-friction factor are satisfactory; the friction-only sliding factor (0.95) is below 1, but this is covered by the shear strength of the concrete-rock contact (SFF above). The resultant lies slightly outside the middle third, so a small tension of about 77 kN/m² appears at the heel; better drainage or a slightly wider base would remove it.