1. Total immediate settlement of the foundation

It is performed by the method of layer-by-layer summation using the linear elastic half-space (Boussinesq problem).

The depth of compressible stratum H_C is reached when the following condition σ_zp = k*σ_zy is fulfilled, by means of a specified compressible depth factor k.

where

E_i - modulus of elasticity of the i-th layer of soil along the loading path;

E_ei - modulus of elasticity of the i-th layer of soil along the unloading path; by default E_ei = E_i;

σ_zp,i - stress in the i-th layer of soil from external load;

σ_zy,i - stress in the i-th layer from the dead weight of the soil excavated from the pit;

n - number of subsoil layers from the foundation base to the depth of compressible stratum H_C.

If the dead weight of the soil at the foundation base level is greater than the average pressure below the base, then the settlement S = 0.8*W3, otherwise the settlement S = 0.8*(W1 + W2).

The settlement for any time t due to consolidation is determined by the formula:

where
h - thickness of the elementary layer;
m_S - relative compressibility coefficient;
ρ - pressure acting on the top of the layer;
N - value determined by formula:

where
t - the time when the settlements are calculated;
C_v - the soil consolidation coefficient determined by formula:

where k_f - filtration coefficient; i_w - unit weight of water.

Settlement caused by creep is determined by formula:

where
β - coefficient equal to 0.8;
h_i - thickness of the i-th layer;
b_ki - soil creep parameter that depends on additional pressure (pressure from external load) in this layer and is determined after compression tests;
t - time when the settlement is calculated;
t_e - time when filtration (primary) consolidation is completed.

Soil creep is a consequence of the processes that occur in soils during primary consolidation (soil compaction due to the squeezing of water out of the soil pores (filtration)). Creep is therefore also known as secondary settlement. Soil creep develops after filtration (primary) consolidation is considered to be established (or completed).

To determine the end time te of filtration (primary) consolidation, the following algorithm is used.

In equation (3), the expression in square brackets is called the degree of compaction U:

Primary consolidation is considered to be steady when the degree of compaction U is 95% or more. If the value of U=0.95 is substituted into equation (6), then the value of N may be found. In this case, the N will be equal to 2.8. With formula (3), we can determine the time of steady consolidation:

If we substitute N = 2.8 into equation (7), then the steady consolidation time will be equal to:

4. Calculation of subgrade moduli according to EN 1997-1:2004

To calculate the subgrade moduli, the mean values (within the fixed depth of the compressible stratum H_C) of the modulus of elasticity E_gr and lateral expansion coefficient m_gr are used. These values are calculated by formulas:

Method 1. Subgrade modulus C1 is calculated according to mean values of modulus of elasticity and Poisson's ratio of soil:

Method 2. Subgrade modulus C1 is calculated according to Winkler method:

where q - the mean pressure under the base of foundation.

Method 3. To determine subgrade modulus C1, the formula of method 1 is used. The difference is that in this case correction factor u to modulus of elasticity of the i-th sublayer is introduced for calculation of mean modulus of elasticity E_gr3. This factor varies from u₁=1 at the level of foundation base up to u_n=12 at the level of calculated depth of compressible stratum. It is accepted that factor u varies according to square parabola law:

Moreover, it is also accepted that additional vertical stress is distributed uniformly along the depth. Then

Method 3 is suggested in order to remove limitations of the first two methods. Limitation of method 1 - it is impossible to take into account that modulus of elasticity increases along the depth. It causes too high values of settlements and therefore too low values of subgrade modulus C1. Limitation of method 2 - at the points where applied loads vary significantly, subgrade modulus also varies significantly, but that is not proved by theory. This limitation is also placed even when modulus of elasticity increases along the depth.

For methods 1 and 3, the subgrade modulus C2 is calculated by formula:

For method 2, the subgrade modulus C2 is not calculated.