Smith's Elements of Soil Mechanics. Ian Smith

      i.e.

      (2.30)

This equation is Laplacian and involves the two coordinate variables x_t and z. It can be solved by a flow net provided that the net is drawn to a vertical scale of z and a horizontal scale of:

      (2.31)

      2.15.2 Calculation of seepage quantities in an anisotropic soil

      This is exactly as before:

      (2.32)

      and the only problem is what value to use for k.

Using the transformed scale, a square flow net is drawn, and N_f and N_d are obtained. If we consider a ‘square’ in the transformed flow net, it will appear as shown in Fig. 2.26a. The same figure, drawn to natural scales (i.e. scale x = scale z), will appear as shown in Fig. 2.26b.

      Let k′ be the effective permeability for the anisotropic condition. Then k′ is the operative permeability in Fig. 2.26a.

      Hence, in Fig. 2.26a:

Fig. 2.26 Transformed and natural ‘squares’. (a) Transformed. (b) Natural.

and, in Fig. 2.26b:

      (2.33)

Example 2.8 Seepage loss through dam (i)

The cross‐section of an earth dam is shown in Fig. 2.27a. Assuming that the water level remains constant at 35 m, determine the seepage loss through the dam. The width of the dam is 300 m, and the soil is isotropic with k = 5.8 × 10⁻⁷ m/s.

       Solution:

      The flow net is shown in Fig. 2.27b. From it, we have N_f = 4.0 and N_d = 14.

Fig. 2.27 Example 2.8. (a) The problem. (b) Flow net.

Example 2.9 Seepage loss through dam (ii)

A dam has the same details as in Example 2.8 except that the soil is anisotropic with k_x = 5.8 × 10⁻⁷ m/s and k_z = 2.3 × 10⁻⁷ m/s.

      Determine the seepage loss through the dam.

       Solution:

      Transformed scale for x direction,

      This means that, if the vertical scale is 1 : 500, then the horizontal scale is 0.63 : 500 or 1 : 794.

The flow net is shown in Fig. 2.28. From the flow net, N_f = 5.0 and N_d = 14.

Fig. 2.28 Example 2.9.

Example 2.10 Seepage loss through dam (iii)

      A dam has the same details as in Example 2.8, except that there is no filter drain at the toe. Determine the seepage loss through the dam.

       Solution:

The flow net is shown in Fig. 2.29 and from it we see N_f = 4.0 and N_d = 18 (average). From the flow net, it is also seen that a + Δa = 22.4 m. Now α = 45°, and hence:

      Hence Δa = 7.6 m

Скачать книгу