Gravity dam section: shear stress near toe (2082)

Assumptions (from the figure description)

• Dam height $=5+40+20=65$ m. Crest width 10 m. U/s face vertical down to 20 m above the base, then flared with a horizontal run of 15 m to the heel. Water level 5 m below the crest, so $H_w=60$ m. D/s face vertical for the top 10 m, then sloping (35 m horizontal run over 55 m height) to the toe. Base width $B=15+10+35=60$ m.
• Tail water depth $H_t=8$ m. $\gamma_c=24$ kN/m³, $\gamma_w=10$ kN/m³, $\mu=0.75$.
• Uplift: full hydrostatic head at the heel ($\gamma_wH_w=600$ kN/m²) and at the toe ($\gamma_wH_t=80$ kN/m²). The drainage gallery (assumed with its centre 5 m from the heel) reduces the uplift at the gallery to $80+\frac13(600-80)=253.3$ kN/m². Earthquake is neglected.

 crest 10 m
  +--------+
 ~|~~5 m~~~|  water level
  |        |  d/s vertical 10 m
  |  40 m  \
  |         \  slope: 35 m run in 55 m
  |  20 m    \
   \__________\_ tail water 8 m
  15     10     35   B = 60 m

Forces and moments about the toe

(The tail water also acts on the sloping d/s face; its vertical component is included as a resisting load, and its horizontal thrust acts at $H_t/3$ above the base.)

Stability

• $\Sigma V=50,003.6-11,300.0=38,703.6$ kN/m; $\Sigma H=18,000.0-320.0=17,680.0$ kN/m
• $\Sigma M_R=1,741,699$ kN·m/m (including tail water); $\Sigma M_O=779,167$ kN·m/m

Overturning: $FS_o=1,741,699/779,167=2.24$ — safe (> 1.5).

Sliding (friction only): $FS_s=\mu\Sigma V/\Sigma H = 0.75\times38,703.6/17,680.0=1.64$ — safe (> 1).

Shear-friction factor: $SFF=\dfrac{\mu\Sigma V+\tau B}{\Sigma H}=\dfrac{0.75\times38,703.6+2200\times60}{17,680.0}=9.1$ — very safe (> 3 to 5).

Resultant and stresses: $\bar x=962,532/38,703.6=24.87$ m from the toe; $e=30-24.87=5.13$ m; $B/6=10$ m (inside the middle third).

$$\sigma_{toe}=645.1\,(1+0.513)=976.0\ \text{kN/m}^2,\quad \sigma_{heel}=645.1\,(1-0.513)=314.1\ \text{kN/m}^2$$

Both are compressive and below the allowable 2500 kN/m².

Shear stress on a horizontal plane near the toe and heel

Face slopes: d/s $\tan\phi_d=35/55=0.636$; u/s (flare) $\tan\phi_u=15/20=0.75$. Water pressures at the faces: $p'=\gamma_wH_t=80$ kN/m² at the toe and $p=\gamma_wH_w=600$ kN/m² at the heel.

$$\tau_{toe}=(\sigma_{toe}-p')\tan\phi_d=(976.0-80)\times0.636=570.2\ \text{kN/m}^2$$ $$\tau_{heel}=(p-\sigma_{heel})\tan\phi_u=(600-314.1)\times0.75=214.4\ \text{kN/m}^2$$

Both are much smaller than the average shear strength of 2200 kN/m². Principal stresses: $\sigma_{1,toe}=\sigma_{toe}\sec^2\phi_d-p'\tan^2\phi_d=1338.9$ kN/m², $\sigma_{1,heel}=\sigma_{heel}\sec^2\phi_u-p\tan^2\phi_u=153.3$ kN/m².

Conclusion: the dam is safe against overturning (2.24), sliding (1.64; shear-friction factor 9.1), tension (none: the minimum stress of 314 kN/m² is compressive) and crushing (maximum principal stress 1339 kN/m² < 2500 kN/m²).