Overflow section stability with gates and piers

Data

Base RL 100 m and reservoir RL 160 m, so $H_w=60$ m; tail water RL 120 m, so $H_t=20$ m; $B=47$ m; $\gamma_w=9.81$ kN/m³; $\mu=0.75$; unit shear strength of the dam-foundation contact $q=1400$ kN/m². Total vertical load of the concrete, gates, piers and water over the crest: $W=3.0\times10^4$ kN with a moment about the toe of $10^6$ kN·m, so it acts at $10^6/(3\times10^4)=33.33$ m from the toe. Other loads (tail water weight, earthquake, silt) are neglected, as stated.

 160 ~~~~~~~\  reservoir RL 160
 140 -------+==crest (overflow) RL 140
            |  \
            | G \
            |    \
 120 ........|     \~~~ tail water RL 120
 100 --------+------\
       heel         toe   B = 47 m

Uplift: with the gallery 8 m from the u/s face, the pressure drops from $\gamma_wH_w=588.6$ kN/m² at the heel to $\gamma_wH_t+\frac13(\gamma_wH_w-\gamma_wH_t)=327.0$ kN/m² at the gallery, and to $\gamma_wH_t=196.2$ kN/m² at the toe.

Forces and moments about the toe

Checks

• $\Sigma V=W-U=30,000.0-13,864.8=16,135.2$ kN/m
• $\Sigma H=P_w-P_{tail}=17,658.0-1,962.0=15,696.0$ kN/m
• $\Sigma M_R=10^6+13,080=1,013,080$ kN·m/m; $\Sigma M_O=727,564$ kN·m/m

Overturning: $FS_o=1,013,080/727,564=1.39$ — below the usual 1.5, so not safe.

Sliding (friction only): $FS_s=\mu\Sigma V/\Sigma H=0.75\times16,135.2/15,696.0=0.77$ — not safe (< 1).

Sliding (shear-friction factor, using $q$):

$$SFF=\frac{\mu\Sigma V+qB}{\Sigma H}=\frac{0.75\times16,135.2+1400\times47}{15,696.0}=\frac{12,101.4+65,800}{15,696.0}=4.96$$

This is above the usual 3-5, so the section is safe in sliding when the shear strength of the joint is counted.

Resultant: $\bar x=285,516/16,135.2=17.70$ m from the toe; $e=23.5-17.70=5.80$ m; $B/6=7.83$ m, so the resultant is inside the middle third: no tension.

$$\sigma_{toe}=343.3\,(1+0.741)=597.7\ \text{kN/m}^2,\qquad \sigma_{heel}=343.3\,(1-0.741)=88.9\ \text{kN/m}^2$$

The heel stress is compressive, so there is no tension, and the toe stress (598 kN/m²) is moderate.

Conclusion: the section has no tension and is safe in sliding by the shear-friction factor (SFF = 4.96), although friction alone (0.77) would not be enough. The factor of safety against overturning (1.39) is below the usual 1.5; since the resultant lies inside the middle third, the section is acceptable against overturning only if 1.39 is allowed, otherwise the base should be widened or the uplift reduced by better drainage.