A dam is a hydraulic structure built primarily to impound water upstream. Gravity dams rely mainly on their own weight for stability rather than arch action.
Common forces considered in board problems include gravity force, water pressure, silt pressure, uplift, seismic force, and other horizontal thrusts.
Dam problems commonly check overturning, sliding, and location of the resultant at the base.
For no tension at the base, the resultant should fall within the middle third.
If the resultant vertical load is eccentric, base pressure varies linearly.
A trapezoidal concrete dam has top thickness 0.60 m, bottom thickness 4.2 m, height 7 m, and water depth 6 m. Unit weight of concrete is $24 \text{ kN/m}^3$. Compute hydrostatic force, resisting moment, and factor of safety against overturning.
Answer: $P = 176.58 \text{ kN}$, $M_R = 1118.90 \text{ kN}\cdot\text{m}$, and $FS = 3.17$.
A dam has $W_1 = 216 \text{ kN}$, $W_2 = 356.4 \text{ kN}$, water force $P = 177 \text{ kN}$, and coefficient of friction $\mu = 0.50$. Determine the factor of safety against sliding.
Answer: The factor of safety against sliding is 1.62.
A concrete gravity dam is triangular in cross-section with base width B = 5.0 m and height H = 8.0 m. The dam retains water to its full height. Unit weight of concrete is 24 kN/m³. Uplift pressure acts on the base, varying from full hydrostatic ($\gamma_w H$) at the heel to zero at the toe. Find the factor of safety against overturning about the toe. Consider a 1 m strip.
Forces (per 1 m strip, moments about toe):
Answer: FS against overturning = 0.54 — the dam is unsafe! This demonstrates why a purely triangular gravity dam with only its own weight is insufficient; heel reinforcement, counterweights, or anchor tie-backs are required when uplift is significant.
A concrete gravity dam (1 m strip) has the following forces at the base: total vertical resultant $R_y = 850 \text{ kN}$, sum of righting moments about the toe $\sum M_R = 2400 \text{ kN·m}$, and sum of overturning moments $\sum M_O = 1100 \text{ kN·m}$. The base width is 6.0 m. Determine the resultant eccentricity $e$, verify no-tension condition, and compute maximum and minimum base pressures.
Since $e = 1.471 \text{ m} > B/6 = 1.0 \text{ m}$, tension develops at the toe — middle third criterion is violated.
Answer: $e = 1.471$ m exceeds $B/6 = 1.0$ m, so tension exists at the toe (−66.7 kPa). Maximum base pressure is 350.2 kPa at the heel. This dam cross-section needs redesign to bring the resultant within the middle third.
A concrete dam retains water to a depth of 6.0 m. In addition to water pressure, saturated silt has accumulated on the upstream face to a depth of 1.5 m (measured from the base). The silt has an equivalent unit weight of 18 kN/m³ (buoyant weight already deducted) and acts as a horizontal fluid pressure. The dam weighs 540 kN per meter width and its center of gravity is 1.8 m from the toe. The base is 4.5 m wide. Compute the factor of safety against overturning about the toe with friction coefficient $\mu = 0.60$.
Answer: FS against overturning = 2.68 (safe). FS against sliding = 1.64 (marginally safe — typically requires ≥ 1.5). Silt accumulation significantly increases the overturning and sliding loads on a dam.
A concrete dam has a rectangular cross-section 3.0 m wide and 6.0 m tall. Unit weight of concrete is 23.5 kN/m³ and water stands at full height (6.0 m) on the upstream face. No silt, no uplift. The toe is downstream and the heel is upstream. For a 1 m strip: (a) compute FS against overturning and sliding ($\mu = 0.55$), (b) locate the resultant at the base, and (c) determine if tension develops.
Since $e = 0.835 > B/6 = 0.50$, tension develops at the toe.
Answer: FS overturning = 1.80 (OK). FS sliding = 1.32 (borderline). The resultant falls outside the middle third ($e > B/6$), so tension develops at the toe. A rectangular dam section alone is generally not efficient — trapezoidal profiles are preferred.