Stresses in gravity dam with drain holes

Geometry and assumptions (per metre length)

• Water depth $h_1$ = 65 m, water surface 5 m below crest, so dam height = 70 m.
• Top width 12 m; downstream face vertical for the top 15 m, then sloping from 55 m height down to the toe.
• Base width $B$ = 10 + 40 = 50 m (drain holes 10 m from heel). Tailwater depth $h_2$ = 5 m.
• Downstream slope: horizontal 50 − 12 = 38 m over vertical 55 m, so $\tan\alpha = 38/55 = 0.6909$ (α measured from vertical).
• Uplift with drains (usual IS/textbook rule): heel $= \gamma_w h_1$; at drains $= \gamma_w[h_2 + \frac{1}{3}(h_1 - h_2)]$; toe $= \gamma_w h_2$.
• Silt, wave and earthquake forces are ignored. Moments are taken about the toe.

      12 m
     +----+       crest (70 m)
  ~~~|~~~~|  5 m
     |    |  15 m vertical d/s
 65m |    +     (55 m)
 wat |  W1  \
 er  |    W2  \
     |          \  ~~~ tailwater 5 m
heel +-----------\ toe
     <10>|<---40--->
       drains   B = 50 m

Uplift pressures

• Heel: $10 \times 65 = 650$ kN/m²
• Drains: $10 \times [5 + (65 - 5)/3] = 10 \times 25 = 250$ kN/m²
• Toe: $10 \times 5 = 50$ kN/m²

Forces and moments about the toe

(Tailwater wedge width on the slope $= 5 \times 0.6909 = 3.455$ m.)

$$\begin{aligned} \sum U &= 10{,}500\ \text{kN} \\ \sum V &= 20{,}160 + 25{,}080 + 86.4 - 10{,}500 = 34{,}826.4\ \text{kN} \\ \sum H &= 21{,}125 - 125 = 21{,}000\ \text{kN} \\ \sum M_R &= 1{,}522{,}708\ \text{kN·m} \\ \sum M_O &= 457{,}708 + 352{,}500 = 810{,}208\ \text{kN·m} \end{aligned}$$

Position of resultant

$$\begin{aligned} \bar{x} &= \frac{1{,}522{,}708 - 810{,}208}{34{,}826.4} = 20.459\ \text{m from toe} \\ e &= 25 - 20.459 = 4.541\ \text{m} < B/6 = 8.333\ \text{m (no tension)} \end{aligned}$$

(a) Maximum vertical stresses

$$\begin{aligned} \frac{\sum V}{B} &= \frac{34{,}826.4}{50} = 696.53\ \text{kN/m}^2, \quad \frac{6e}{B} = 0.545 \\ p_{v,toe} &= 696.53(1 + 0.545) = 1076.1\ \text{kN/m}^2 \\ p_{v,heel} &= 696.53(1 - 0.545) = 316.9\ \text{kN/m}^2 \end{aligned}$$

(b) Major principal stress at the toe

With tailwater pressure at the toe $p' = 10 \times 5 = 50$ kN/m², $\tan^2\alpha = 0.4774$, $\sec^2\alpha = 1.4774$:

$$\begin{aligned} \sigma_1 &= p_v \sec^2\alpha - p'\tan^2\alpha \\ &= 1076.1 \times 1.4774 - 50 \times 0.4774 \\ &= 1565.9\ \text{kN/m}^2 \end{aligned}$$

(c) Shear stress on a horizontal plane near the toe

$$\begin{aligned} \tau &= (p_v - p')\tan\alpha \\ &= (1076.1 - 50) \times 0.6909 = 708.9\ \text{kN/m}^2 \end{aligned}$$

Answer: Vertical stress at toe = 1076.1 kN/m², at heel = 316.9 kN/m² (both compressive); major principal stress at toe = 1565.9 kN/m²; shear stress near toe = 708.9 kN/m².