# Primary Consolidation

Courses > Foundation Analysis and Design > Settlement of Shallow Foundations > Primary Consolidation

### Introduction

Primary compression results from changes in effective stress and is time-dependent. For general definitions, remarks and types of settlement refer to Settlement types, definitions, and general equation article.

### Concepts and Formulas

#### Virgin Compression

Virgin compression occurs when the soil is normally consolidated and the effective vertical stress increases from the beginning to the end of the stress increment being analyzed. For virgin compression, the following equation is the primary consolidation settlement:

$S_c = \int_{z=z_f}^{z=z_{f0}} \frac{C_c}{1+e_0}\Delta(log\sigma'_v)dz$

for each layer with constant parameters

$S_c = \frac{C_c}{1+e_0}\Delta(log\sigma'_v)H_0$

In a more familiar way it can be rewritten as:

$S_c = \frac{C_cH_0}{1+e_0}log\frac{\sigma'_{vf}}{\sigma'_{v0}}$

where

Cc = virgin compression index

H0 = height of the layer

e0 = void ratio

subscripts "0" and "f" for vertical stresses mean "initial" and "final", respectively.

$\Delta(log\sigma'_v) = log\sigma'_{vf}-log\sigma'_{v0} = log\frac{\sigma'_{vf}}{\sigma'_{v0}}$

#### Decompression (Rebound)

Decompression (aka rebound or heave) occurs when the soil is either normally consolidated or overconsolidated and the effective vertical stress decreases from the beginning to the end of the stress increment being analyzed

$S_c = \int_{z=z_f}^{z=z_{f0}} \frac{C_r}{1+e_0}\Delta(log\sigma'_v)dz$

for each layer with constant parameters

$S_c = \frac{C_r}{1+e_0}\Delta(log\sigma'_v)H_0$

In a more familiar way it can be rewritten as:

$S_c = \frac{C_rH_0}{1+e_0}log\frac{\sigma'_{vf}}{\sigma'_{v0}}$

where

H0 = height of the layer

e0 = void ratio

subscripts "0" and "f" for vertical stresses mean "initial" and "final", respectively.

$\Delta(log\sigma'_v) = log\sigma'_{vf}-log\sigma'_{v0} = log\frac{\sigma'_{vf}}{\sigma'_{v0}}$

#### Recompression plus Virgin Compression

Recompression plus virgin compression occurs when the soil is overconsolidated and the effective vertical stress increases from the beginning to the end of the stress increment being analyzed

$S_c = \int_{z=z_f}^{z=z_{f0}} \frac{C_r}{1+e_0}\log\frac{\sigma'_p}{\sigma'_{v0}} + \frac{C_c}{1+e_0}\log\frac{\sigma'_{vf}}{\sigma'_p}dz$

for each layer with constant parameters

$S_c = H_0(\frac{C_r}{1+e_0}\log\frac{\sigma'_p}{\sigma'_{v0}} + \frac{C_c}{1+e_0}\log\frac{\sigma'_{vf}}{\sigma'_p})$

where

Cc = virgin compression index

H0 = height of the layer

e0 = void ratio

subscripts "0", "f" and "p" for vertical stresses mean "initial", "final" and "preconsolidation", respectively.

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