Gravity dam stability with seismic load

Data and assumptions

• Base RL 100, crest RL 150: height 50 m. Water RL 135: $H_w=35$ m. Crest width 6 m. U/s face slope 0.1 : 1 (leaning inwards going up), so the heel A is at $x=0$ and the u/s edge of the crest at $x=5$; d/s slope 0.8 : 1, so the toe B is at $x=11+40=51$ m. Base $B=51$ m.
• The base (section A-B) is the worst location: it carries the largest load and moment.
• $\gamma_c=24$ kN/m³, $\gamma_w=9.81$ kN/m³. No tail water. Hydrostatic thrust and uplift (full, triangular, no drainage: $343.4$ kN/m² at the heel to zero at the toe) act together with the seismic loads.
• Seismic: horizontal force $0.25W$ from the upstream side (towards the downstream) and vertical force $0.15W$ upward, both at the centre of gravity (CG).

   6 m
  +----+  RL150
 /     |
/      |  RL135 ~~ water
/      |
A(heel) \___________  B(toe)
 0.1:1       0.8:1       B = 51 m

Self-weight and CG

Area $=\frac12(6+51)\times50=1425$ m². $W=24\times1425=34,200$ kN/m. CG: $\bar x=19.05$ m from the heel ($B-\bar x=31.95$ m from the toe), $\bar z=18.42$ m.

Seismic: $F_h=0.25W=8,550$ kN/m; $F_v=0.15W=5,130$ kN/m (upward), so the effective weight $=0.85W=29,070$ kN/m.

Forces

$\Sigma V=29,070+601-8,755=20,915$ kN/m; $\Sigma H=6,009+8,550=14,559$ kN/m.

Overturning

$$FS_o=\frac{\Sigma M_R}{\Sigma M_O}=\frac{958,653}{525,285}=1.83$$

This is above 1.5 (a value of 1.1-1.5 is adequate for the seismic case), so the dam is safe against overturning.

Sliding

Friction coefficient $\mu=\tan36^\circ=0.727$; shear resistance $q=700$ kN/m².

$$FS_{friction}=\frac{\mu\Sigma V}{\Sigma H}=\frac{0.727\times20,915}{14,559}=1.04$$ $$SFF=\frac{\mu\Sigma V+qB}{\Sigma H}=\frac{0.727\times20,915+700\times51}{14,559}=3.50$$

Friction alone is just above 1. Including the shear resistance of the foundation contact, $SFF=3.50$, which is well above the minimum usually required for loading with earthquake, so the section is safe in sliding.

Material failure (stresses on the base)

$\bar x=(\Sigma M_R-\Sigma M_O)/\Sigma V=20.72$ m from the toe; $e=25.5-20.72=4.78$ m $<B/6=8.5$ m (no tension).

$$\sigma_{toe}=\frac{20,915}{51}\left(1+\frac{6\times4.78}{51}\right)=641\ \text{kN/m}^2\ (<2500),\qquad \sigma_{heel}=179\ \text{kN/m}^2$$

Both are compressive and less than the allowable 2500 kN/m² (foundation).

Major principal stress at the toe (no tail water, $\tan\phi_d=0.8$):

$$\sigma_1=\sigma_{toe}(1+\tan^2\phi_d)=641\times1.64=1051\ \text{kN/m}^2<3000\ \text{kN/m}^2$$

so the concrete is also safe.

Conclusion: the dam is safe against overturning ($FS_o=1.83$), sliding ($SFF=3.50$) and material failure (maximum vertical stress 641 kN/m² < 2500 kN/m², principal stress 1051 kN/m² < 3000 kN/m², no tension).