Principal stresses at toe and heel with silt and earthquake

Assumptions (from the figure)

• Crest width 10 m, height 110 m, base width 100 m, u/s face vertical, d/s face straight from the crest edge to the toe (run 90 m, $\tan\phi_d=90/110=0.818\approx0.8:1$). Water depth $H_w=94$ m, silt depth $h_s=20$ m. No tail water.
• $\gamma_c=25$, $\gamma_w=10$ kN/m³ (given). Uplift: $\gamma_wH_w=940$ kN/m² at the heel, one-third of it at the gallery (8 m from the heel), zero at the toe.
• Silt: submerged unit weight $10$ kN/m³, $\phi=30^\circ$ so $K_a=\frac13$: $P_s=\frac12K_a\gamma_sh_s^2$ acting at $h_s/3$.
• Earthquake: $\alpha_h=0.1$ acting towards the downstream (worst case). Inertia force $=\alpha_hW$ at the centre of gravity of the dam. Hydrodynamic pressure by Zangar: $p_e=C_m\alpha_h\gamma_wH_w$ with $C_m=0.735$, resultant $P_e=0.726\,p_eH_w$ at $0.412H_w$ above the base.

Calculations

Dam weights: $W_1=25\times10\times110=27{,}500$ kN, $W_2=25\times\frac12\times90\times110=123{,}750$ kN, total $W=151,250$ kN/m. Centre of gravity height $\bar z=40.00$ m.

$P_s=\frac12\times\frac13\times10\times20^2=666.7$ kN/m; $p_e=0.735\times0.1\times10\times94=69.1$ kN/m²; $P_e=0.726\times69.1\times94=4,715$ kN/m; $F_i=0.1\times151,250=15,125$ kN/m.

$$\Sigma V = W-U = 151,250-19,427 = 131,823\ \text{kN/m}$$ $$\bar x=\frac{\Sigma M_R-\Sigma M_O}{\Sigma V}=\frac{10,037,500-3,544,993}{131,823}=49.25\ \text{m from the toe},\quad e=50-49.25=0.75\ \text{m}$$

($B/6=16.67$ m: the resultant is inside the middle third.)

$$\sigma_{toe}=\frac{131,823}{100}\left(1+\frac{6\times0.75}{100}\right)=1,377.4\ \text{kN/m}^2,\quad \sigma_{heel}=1,318.2\left(1-0.045\right)=1,259.0\ \text{kN/m}^2$$

Principal stresses

Toe (no tail water, $p'=0$):

$$\sigma_1=\sigma_{toe}(1+\tan^2\phi_d)=1,377.4\,(1+0.669)=2,299.5\ \text{kN/m}^2,\quad \sigma_3=0$$

Shear stress $\tau_{toe}=\sigma_{toe}\tan\phi_d=1,127.0$ kN/m².

Heel (u/s face vertical, so no shear on the face and the face is a principal plane). The normal stress on the face is the total water, hydrodynamic and silt pressure $p=940+69.1+66.7=1075.8$ kN/m².

• Vertical principal stress: $\sigma_z=1,259.0$ kN/m²
• Horizontal principal stress: $\sigma_h=p=1,075.8$ kN/m²
• Major principal stress at the heel $=1,259.0$ kN/m², minor $=1,075.8$ kN/m².

Answer: toe — σ_z = 1,377 kN/m², major principal stress = 2,300 kN/m², τ = 1,127 kN/m²; heel — σ_z = 1,259 kN/m² (face normal stress 1,076 kN/m²).