# Immediate settlement computations

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## Introduction on Immediate settlement computations :

Immediate settlement results from distortion of soil mass caused by a change in total stress. For general definitions, remarks and types of settlement refer to Settlement types, definitions, and general equation article.

## Concepts and Formulas of Immediate settlement computations:

The settlement of the corner of a rectangular base of dimensions B' x L' on the surface of an elastic half-space can be computed from an equation from Theory of Elasticity as follows:

$S_{i}=q_{0}B'\frac{1-\mu^{2}}{E_{s}}(I_{1}+\frac{1-2\mu}{1-\mu}I_{2})I_{F}$

where

q0 = intensity of contact pressure in units of Es

B' = least lateral dimension of contributing base area in units of Si

I1 and I2 = influence factors, which depend on L'/B', thickness of stratum H. Poisson's ratio, and base embedment depth D

Es = average modulus of elasticity of soil layer

Influence factors can be computed using:

$I_{1}=\frac{1}{\pi}[Mln\frac{(1+\sqrt{M^{2}+1})\sqrt{M^{2}+N^{2}}}{M(1+\sqrt{M^{2}+N^2+1})} + ln\frac{(M+\sqrt{M^2+1})\sqrt{1+N^2}}{M+\sqrt{M^2+N^2+1}}]$$I_{2}=\frac{N}{2\pi}tan^{-1}(\frac{M}{N\sqrt{M^2+N^2+1}})$

where

$M = \frac{L'}{B'}$

$N = \frac{H}{B'}$

IF = embedment (or depth) factor which is equal to or less than 1.0 (for footings on the ground IF = 1.0)

### Remarks

1. For circular footings, convert to an equivalent square.
2. Determine the point where the settlement is to be computed and divide the base so the point is at the corner or common corner of one or up to 4 contributing rectangles.
3. The zone of influence is either of the following:
1. Depth z = 5B where B is the least total lateral dimension of base
2. Depth to where a hard stratum is encountered.
4. Compute zone of influence based on the actual footing dimension.

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