--- title: "Overflow section stability with gates and piers" subject: "Hydropower Engineering" chapter: 4 key: "semester_7/hydropower_engineering:gravity-dam-overflow-section-stability" source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/gravity-dam-overflow-section-stability site: BCT Engineering --- # Overflow section stability with gates and piers Hydropower Engineering · Chapter 4: Headworks of Storage Plants - Asked in: 2074 Ashwin, 10 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/hydropower_engineering/gravity-dam-overflow-section-stability - Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-4 ## Question Check the stability of the overflow section of the gravity dam shown in the figure. Assume the weight of concrete, gates, piers and weight of water over the crest, $W_{total} = 3.0 \times 10^4$ kN. Moment of weight of concrete, gates, piers and water above the crest etc. about toe $M_{toe} = 10^6$ kN-m. Neglect all forces other than weight, uplift pressure and water pressure. Also check for tension. Take μ = 0.75 and q = 1400 kN/m². [Figure: overflow section with base at RL 100.0 m, crest at RL 140.0 m, reservoir water level RL 160.0 m, tail water level RL 120.0 m; drainage gallery 8 m from the upstream face; base width 47 m; 18 m dimension on the downstream side.] ## Short answer Overturning factor of safety 1.39 (below 1.5, not safe). Sliding by friction alone 0.77 (not safe); shear-friction factor 4.96 (safe). No tension occurs (heel compressive stress 88.9 kN/m²). Overall, the section needs widening or improved drainage for overturning. ## Answer #### 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. ```text 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 | Force | Magnitude (kN/m) | Arm from toe (m) | Moment about toe (kN·m/m) | |---|---|---|---| | **Vertical (resisting)** | | | | | W_total (concrete, gates, piers, water on crest) | 30,000.0 | 33.33 | 1,000,000 | | **ΣW (downward)** | **30,000.0** | | **1,000,000** | | **Horizontal / uplift (overturning)** | | | | | Water thrust $P_w=\frac12\gamma_w H_w^2$ | 17,658.0 | 20.00 | 353,160 | | Uplift part 1 (upward) | 2,616.0 | 43.00 | 112,488 | | Uplift part 2 (upward) | 1,046.4 | 44.33 | 46,390 | | Uplift part 3 (upward) | 7,651.8 | 19.50 | 149,210 | | Uplift part 4 (upward) | 2,550.6 | 26.00 | 66,316 | | **Uplift U** | **13,864.8** | | | | Tail water thrust (resisting) | 1,962.0 | 6.67 | 13,080 | | **Total overturning** | | | **ΣM_O = 727,564** | #### 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the line of action of the total vertical load (concrete, gates, piers, water over the crest) found? A. Arm = total moment about the toe / total load = 10^6 / (3 x 10^4) = 33.33 m B. Arm = B / 2 = 23.5 m C. Arm = total load / moment D. Arm = B / 3 Answer: A. Moment divided by force gives the lever arm from the toe where the load acts. (Key point: Position of the vertical load) 2. With the gallery 8 m from the u/s face, how is the uplift drawn? A. Linear from heel to toe, ignoring the gallery B. Drops from the heel value to toe + one-third of (heel - toe) at the gallery, then linearly to the toe value C. Zero everywhere beyond the gallery D. Constant at 196.2 kN/m^2 Answer: B. The gallery cuts the pressure to 327.0 kN/m^2 (toe pressure plus 1/3 of the difference), then it falls linearly to the toe value 196.2 kN/m^2. (Key point: Uplift at the gallery) 3. Why is the section free of tension even though it is unsafe in sliding? A. Because e = 5.80 m is less than B/6 = 7.83 m, so the resultant is within the middle third B. Because the sliding factor is below 1 C. Because the allowable stress q is 1400 kN/m^2 D. Because uplift is neglected Answer: A. Tension depends only on the resultant position: e < B/6 keeps both toe (598) and heel (89) stresses compressive. 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