Design elementary profile of masonry dam

Data

$H=228-198=30$ m, $S_c=2.4$, $K=1$ (full uplift), $\tan\phi=\mu=0.70$, permissible stress $1200$ kN/m², $\gamma_w=9.81$ kN/m³. Reservoir full (HFL at the apex).

Base width from stability conditions

No tension:

$$B_1=\frac{H}{\sqrt{S_c-K}}=\frac{30}{\sqrt{2.4-1}}=25.35\ \text{m}$$

Sliding ($FS=1$):

$$B_2=\frac{H}{\mu(S_c-K)}=\frac{30}{0.70\times(2.4-1)}=30.61\ \text{m}$$

The larger value governs: adopt $B=31$ m, with the vertical u/s face and the d/s slope $B/H=1.03:1$.

Check at $B=31$ m (per metre)

• $W=\frac12\times31\times30\times2.4\times9.81=10,948$ kN; $U=\frac12\times1\times9.81\times30\times31=4,562$ kN; $P=\frac12\times9.81\times30^2=4,414$ kN
• $\Sigma V=W-U=6,386$ kN; $\Sigma M_R=W\cdot\frac23B=226,258$; $\Sigma M_O=P\frac{H}{3}+U\frac23B=138,419$ kN·m
• $\bar x=\dfrac{87,839}{6,386}=13.75$ m from the toe, $e=B/2-\bar x=1.75$ m $<B/6=5.17$ m: no tension.
• Sliding: $\mu\Sigma V/P=0.7\times6,386/4,414=1.01\ge1$ — safe.
• Overturning: $\Sigma M_R/\Sigma M_O=1.63$.

$$\sigma_{toe}=206.0\left(1+\frac{6\times1.75}{31}\right)=275.6\ \text{kN/m}^2,\qquad \sigma_{heel}=136.4\ \text{kN/m}^2$$

Principal stress at the toe $\sigma_1=\sigma_{toe}(1+\tan^2\phi_d)=275.6\,(1+1.068)=569.9$ kN/m² $<1200$ kN/m² — safe.

  HFL 228 ^   apex
          |\
          | \
     30 m |  \   d/s slope
          |   \
  198 ----+----\
         B = 31 m

Result: elementary profile with base width $B=31$ m, height 30 m, vertical u/s face. For the practical section add a crest width (about 3-4 m) and a freeboard above HFL (2-3 m).