The influence of soil specimens shape and dimensions on consolidation parameters used in foundation settlement prediction

The theory of elasticity is used in geotechnical engineering for the calculation of foundation settlement. The equation used in the evaluation of consolidation settlement uses the soil compressibility parameters which usually are determined on cylindrical soil specimens with ϕ 71.4 mm. Laboratory tests performed on soil specimens showed that the parameters that are used in foundation settlement analysis are influenced by the dimensions of the tested soil specimens and the shape of that specimen. The paper presents the influence of specimen size, shape and method used for the coefficient of consolidation evaluation on the final value of the foundation settlement. The analysis was performed using the Settle3 software from Rocsience, considering the case of an elastic foundation flexible square foundation (1.5 m × 1.5 m) which transmits to the ground a pressure of 100 kPa.


Introduction
The structure of a building can be divided in two parts, the superstructure which is the part of the building that develops above the ground and the substructure which is the part of the construction located under the ground surface [1].The structural engineers in general deal with the superstructure which is made from man-made materials like concrete, steel or wood, materials for which the mechanical properties are well known.The substructure refers to foundation and the soil located inside the bulb of pressure influence beneath the foundations.In general, the substructure design is done by engineers with knowledge in soil behavior, also known as geotechnical engineers which can adopt a proper foundation based on the ground conditions encountered in that site.Geotechnical design requires to check if the soil under foundations does not reach a limit state which can be a serviceability limit state (SLS) or an ultimate limit state (ULS).The adequate behavior of the foundation to avoid the serviceability limit state implies the calculation of the total settlement and limit this value to an accepted one.The values of the settlement can induce supplementary stress in superstructure elements; therefore, it is important that the final values of the settlement to be known from the design stage in order to avoid structural damage of the building.
The most used methods for the evaluation of the shallow foundation settlements are based on the theory of elasticity.The Romanian design standard NP112/2014 presents such a method for the computation of the total settlement using the method of superposition [2], method that is based on the theory of elasticity.
The change in the stress state of the foundation soil due to construction load will lead to a rearrangement of the soil particles which in turn leads to the decrease in soil volume which results in the settlement of the structure.The rearrangement of the solid particles can be almost immediate or time dependent according to the permeability of the soil layer underneath the foundation.
In the case of the cohesive soils the rate of the settlement is time dependent.The total settlement (  ) is the sum of the immediate settlement (  ), from the primary consolidation (  ) and from the secondary consolidation or creep (  ).
For the cohesive soils the immediate settlements (  ) represents 0.15% from the total settlement [3], [4] and the settlement from the secondary consolidation is usually small compared to the one from the primary consolidation [5].The primary consolidation settlement can be computed using the consolidation theory developed by Terzaghi using the laboratory oedometer testing results [3]: Equation 2 can be expressed in a different form such as: where:   -compression index;  0 -initial void ratio;   -coefficient of volumetric compressibility; ′ 0 -effective overburden pressure corresponding to the layer; ∆′ -increase of effective pressure due to the foundation loading; H -layer thickness.
The rate of settlement can be calculated using the coefficient of consolidation (  ).This parameter can be established in based on the laboratory testing results on undisturbed soil specimens using the onedimensional consolidation theory.There are two methods that are the most used ones for the estimation of the consolidation coefficient value which are known as Casagrande's logarithm of time fitting method and the Taylor's square root of time method [3], [6], [7].
Both methods use the Terzaghi's consolidation theory, but the resulting values of the consolidation coefficient are different, usually the Taylor's method yields higher values compared to the values from Casagrande's method [8].Therefore, it is very important to know which method is suitable to be used in the evaluation of the consolidation coefficient, parameter which significantly influences the consolidation settlement of a soil under the foundation.
There is limited scientific literature focused on the influence of the specimen dimension and shape on the final values of the consolidation coefficient [9].The objective of this research is to present the influence of the soil specimen shape, dimensions and method used for evaluation of the consolidation coefficient in the prediction of the consolidation settlement of shallow foundations.

Site investigation and geotechnical findings
The soil investigated in this research was collected from the city of Iasi, Romania.Sampling and laboratory testing were performed on undisturbed soil samples.To minimize the specimen disturbance for the laboratory consolidation testing, monolith samples were collected from the lateral sides of an open excavation.

Geotechnical parameters of the investigated soil
Grain size distribution tests performed on the collected samples classify the soil as silty clay and according to the Atterberg limits, the soil exhibits a medium plasticity (  = (16 ÷ 25.1)%).
Oedometer consolidation testing was performed in accordance to the European standards.The tested specimens had different shapes and dimensions.Each specimen was incrementally loaded till 500 kPa and each load was maintained 24 h, time needed for the completion of the consolidation phase.The stress-strain curves of the oedometer test results obtained on saturated specimens, as corresponding values established on three identical specimens for each category of shape and dimensions, are presented in Figure 1.The parameters used in the settlement evaluation and obtained from the oedometer testing as average values (based on S1, S2, S3 recordings) for specimens with circular and square cross-sections with different dimensions are summarized in table 1.  [10]:   = 0.003 • (/100 + 7)

Settlement prediction
For the consolidation settlement calculation, a flexible square footing foundation (1.5 m x 1.5 m) was considered to be placed on the investigated silty clay transmitting a pressure of 100 kPa to the ground (figure 2).Settle3 software from Rocsience was used for the settlement evaluations.The loading induced stress distribution can be computed using Boussinesq method.The quarry point for the settlement was chosen in the centre of the foundation footing.
The soil behaviour can be considered as being linear or non-linear.For the linear behaviour the software requires the   parameter for the settlement evaluation due to consolidation; for the non-linear behaviour, the modulus is not constant, being a function of the soil stress and thus, the software requires other parameters like compression index (  ) and recompression index (  ) resulting from the oedometer testing.The pre-consolidation pressure is equal to pc = 50 kPa.

The influence of the specimen dimension
In the geotechnical laboratories the consolidation tests can be performed on specimens with different dimensions.In this study the dimensions of specimens were 54.7 mm, 71.4 mm and 112 mm as diameters for cylindrical specimens with circular cross-section and two different dimensions for prismatic specimens with square cross-section (60 mm and 100 mm).This paper refers to the influence of the used specimens' dimensions on the laboratory test values and on the final results of the primary consolidation settlement.Figure 3 shows that the specimen dimensions have an important influence on the final values of the compression index (  ) and on the coefficient of volume compressibility (  ).For smaller specimen dimensions these two parameters used in the settlement prediction as input parameters lead to lower settlements compared with the ones based on larger specimens' dimensions.An increase of the soil diameter from 54.7 mm to 112 mm induces an increase of approximately 40% of both soil parameters.In using the Settle3 software for the consolidation settlement evaluation, the following soil parameters are required from laboratory testing: the natural unit weight, saturated unit weight and depending on the analysis type, the soil behaviour can be defined as being linear, in which case it needs the coefficient of volumetric compressibility (  ), or non-linear, for which the consolidation parameters need to be established (  ,   ,   ) (Figure 4).
For the time of final consolidation, the software requires the value of the coefficient of consolidation which can be evaluated using Taylor's method or Casagrande's method.
The results of total consolidation settlement with depth after 100 years from placing the foundation loading (time at which one can be sure that the final settlement is completed) is given in Figure 5.The first conclusion is that when increasing the specimen size, larger values for the final consolidation settlement will result.By approximately doubling the specimen size from 54.7 mm to 112 mm and considering the linear soil behaviour, the difference in consolidation settlement is approximately 37 mm and in the case of non-linear soil behaviour approximately 32 mm (Figure 5).

The influence of specimen shape
In this research two types of specimen's shapes were used, cylindrical specimens with three different diameters (ϕ 54.7 mm, ϕ 71.4 mm and ϕ 112 mm) and prismatic specimens with square cross-sections of two different dimensions (60 x 60 mm and 100 x 100 mm). Figure 6 presents the influence of the specimen size and shape on the final consolidation settlement under the foundation.The parameter obtained from tested prismatic specimens with square crosssections with the dimensions of 60 x 60 mm used in the consolidation settlement evaluation led to larger value (67.31 mm) compared to the specimen that is of cylindrical shape with the diameter of ϕ 71.4 mm for which the resulting value is 64.05 mm (for linear soil behaviour, Figure 6 a).If the soil behaviour is changed to non-linear (Figure 6 b) there is the same trend in soil settlement for prismatic specimens with dimensions of 60 x 60 mm compared with cylindrical specimen with the diameter of ϕ71.4 mm; in case of non-linear soil behaviour, for the larger specimens (both the cylindrical and prismatic ones) the final value of total consolidation under the foundation's centre is approximately the same (approx.88 mm).

The influence of the method used for the consolidation coefficient evaluation on the time of 90% degree of consolidation settlement
The time needed for the final consolidation settlement can be evaluated using the coefficient of consolidation (  ).The value of this parameter can be obtain using the graphical methods: Casagrande's method known as the logarithm of time fitting method and the Taylor's method also known as the square root of time method.From Figure 7 one can see that the method used for the evaluation of the coefficient of consolidation (  ) has a great influence on the parameter magnitude.For the tested soil, the Taylor's method yields higher value compared to the Casagrande's method.The evolution of the consolidation settlement in time was studied for time periods of: 0-year, 0.05-year, 0.5-year, 1.0 year, 5.0 years, 10.0 years, 50.0 years and 100.0 years.Figure 8 presents the variation of the total consolidation settlement with the log of time.The log of time scale was used to capture the time when the consolidation settlement can be considered as 90% from the final one.Using Taylor's method, the tine needed for 90% of the total consolidation settlement is 1.0 year.The specimen size and shape does not have significant influence over the time of consolidation (Figure 8).
Using the value of the coefficient of consolidation evaluated based on Casagrande's method in the timesettlement analysis, the results indicated that the specimen dimensions influence the time for 90% of the total consolidation settlement.The consolidation parameters obtained after testing the cylindrical soil specimens with diameters of 54.7 mm and 71.4 mm led to one year time needed for reaching 90% of the total consolidation settlement.The tested specimens of 112 mm diameter indicate a time of 5 years.

Conclusions
The current research is focused on the analysis of the influence of the specimen dimension and shape used in consolidation tests with the application in the prediction of the final value of the foundation settlement.The final value of the foundation soil consolidation settlements is important in the geotechnical design.Usually, in geotechnical laboratories the tested specimens are cylindrical and in most of the cases with a diameter of 71.4 mm.This research presents the consolidation results obtained on three different cylindrical specimens' diameters (ϕ 54.7 mm; ϕ 71.4 mm and ϕ 112 mm) and two prismatic specimens with square cross-section (60 mm and 100 mm the side).
The consolidation settlement was computed using the Settle3 software, where the soil behavior can be defined as being linear, in which case it needs the coefficient of volumetric compressibility (  ), or non-linear, for which the consolidation parameters need to be defined (  ,   ,   ).
The specimen dimensions have an important influence of the final values of the compression index (  ) and on the coefficient of volumetric compressibility (  ).For the smaller soil specimen dimensions these two parameters used in the geotechnical design as input parameters lead to lower values of the settlement compared with the larger soil specimens.An increase of the soil specimen diameter from 54.7 mm to 112 mm led to an increase of both parameters (  ,   ) of approximately 40%.
Another conclusion is that the final value of the consolidation settlement is influenced by the specimen dimensions.Larger specimens lead to larger values of the final consolidation settlement.
The time needed for 90% of the consolidation settlement depends on the method used in the evaluation of the coefficient of consolidation (  ).Using the Taylor's method, the total consolidation settlement is 1.0 year and is not influenced by the specimen size.Using the value of the coefficient of consolidation evaluated based on Casagrande's method in the time-settlement analysis, the results indicated that the specimen dimensions influence the time for 90% of the total consolidation settlement.

Figure 3 .
Figure 3. Variation of the consolidation parameters with the specimen dimensions.

Figure 4 .
Defining soil properties in Settle3 software for: a) linear and b) non-linear soil behaviour.

Figure 5 .
Variation of the maximum computed value of total settlement with depth for: a) linear and b) non-linear soil behaviour.

Figure 6 .
Figure 6.Variation of the maximum computed value of the total settlement with specimen dimension for: a) linear soil behaviour and b) non-linear soil behaviour.

Figure 7 .
Figure 7. Variation of the coefficient of consolidation depending on the used method.

Figure 8 .
Figure 8.Time depending consolidation settlement (for 90% consolidation) based on the consolidation parameters evaluated using Casagrande's method (C) and Taylor's method (T)

Table 1 .
Consolidation parameters for the tested soil.Values computed using the equation proposed by Azzous