Derive 1D primary consolidation settlement

Consider a clay layer of thickness $H$ and initial void ratio $e_0$ under a stress increase $\Delta\sigma'$, so that the void ratio decreases by $\Delta e$ (one-dimensional compression, no lateral strain).

  before: Vs + Vv0 = 1 + e0     after: Vs + Vv = 1 + e0 - Δe
  |  H   |                      | H - ΔH |

For a unit volume of solids, the initial total volume is $1 + e_0$ and the reduction in volume is $\Delta e$. Since the area is constant, the vertical strain equals the volumetric strain:

$$\frac{\Delta H}{H} = \frac{\Delta V}{V} = \frac{\Delta e}{1+e_0}$$ $$\boxed{S_c = \Delta H = \frac{\Delta e}{1+e_0}\,H}$$

In terms of the compressibility:

• With $a_v = \dfrac{\Delta e}{\Delta\sigma'}$: $\;S_c = \dfrac{a_v}{1+e_0}\Delta\sigma'\,H = m_v\,\Delta\sigma'\,H$
• With the compression index $C_c$ (NC clay): $\Delta e = C_c\log_{10}\dfrac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}$, so

$$S_c = \frac{C_c\,H}{1+e_0}\log_{10}\frac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}$$

where $\sigma'_0$ is the effective overburden pressure at the middle of the layer.