**5. Response surface methodology and optimization process**

The response surfaces method is a set of mathematical techniques that use experimental design to determine the range of independent input variables [26]. This method makes it possible, thanks to empirical mathematical models, to determine an approximation relation between the output responses (Y) the swelling pressure **P kPa <sup>s</sup>** ( ) , and the input variables (dry unit weight ( / ) **<sup>3</sup>** γ **<sup>d</sup> kN m** , water content **w** (%) , plasticity index **Ip** (%) Saturation degree **Sr**(%) , the preconsolidation pressure **P kPa <sup>c</sup>** ( ) and the Plasticity Index (**Ip** ) and limite of plasticity WL) to

#### **Figure 9.**

*Response surface 3D representing the swelling pressure dependence on the plasticity index % and the limit of liquidity (%).*

#### **Figure 10.**

*Response surface 3D representing the swelling pressure dependence on the dry unit weight (kN/m3 ) and the water content (%).*

optimize process parameters to achieve desirable responses. In this method, the answer can be written in the following form:

$$\mathbf{Y} = \phi(\gamma d.w.C\mathbf{\hat{y}}...\mathbf{I}p.\mathbf{W}l.\mathbf{P}c) \tag{3}$$

Where Y is the swelling pressure as the output process and φ is the response function, the approximation of Y is proposed using a quadratic mathematical model, which helps to study the interaction effects of process parameters with geotechnical characteristics. In the present work, the second order mathematical model based on RSM is given by the following elements:

*Swelling Clay Parameters Investigation Using Design of Experiments (A Case Study) DOI: http://dx.doi.org/10.5772/intechopen.95443*

$$Y = \mathcal{X}\_o + \sum\_{i=1}^k \mathcal{Y}\_i \mathcal{X}\_i + \sum\_{\forall}^k \mathcal{Y}\_{\emptyset} \mathcal{X}\_i \mathcal{X}\_j + \sum\_{i=1}^k \mathcal{X}\_i^2 + \mathcal{e}\_{\emptyset} \tag{4}$$

$$\left(\mathcal{E}\_{\boldsymbol{\eta}} = \mathcal{Y}\_{\boldsymbol{\eta}} - \overline{\mathcal{Y}}\_{\boldsymbol{\eta}}\right) \tag{5}$$

Where x0 is the free term of the regression equation, the coefficients Y1, Y2,…, Yk and Y11, Y22,…, Ykk are the linear and quadratic terms respectively, while Y12, Y13,…, Y(k- 1) are the interactive terms and εij presents the fit error for the regression model.

#### **Figure 11.** *3D response surface of Ps (kPa) dependence on the* ( ) <sup>3</sup> γ<sup>d</sup> *kN /* m **vs IP** *(%).*

#### **Figure 12.**

*Response surface 3D representing the swelling pressure dependence on the plasticity index and the saturation degree (%).*

#### **Figure 13.**

*Response surface 3D representing the swelling pressure dependence on the Preconsolidation pressure and the plasticity index (%).*

### **Figure 14.** *3D surface of Ps vs.* ( ) **<sup>3</sup>** γ **<sup>d</sup> kN m***/ and* **w %.**

On the other hand, the coefficient of determination R2 is defined by the ratio of the dispersion of the results, given by the relationship:

$$R^2 = \frac{\sum \left(\overline{\boldsymbol{y}}\_i - \overline{\boldsymbol{y}}\right)^2}{\sum \left(\overline{\boldsymbol{y}}\_i - \overline{\boldsymbol{y}}\right)^2} \tag{6}$$

Where yi: is the calculated response to the ith experience; *<sup>i</sup> y* : is the average value of the measured responses.

*Swelling Clay Parameters Investigation Using Design of Experiments (A Case Study) DOI: http://dx.doi.org/10.5772/intechopen.95443*

Analysis of variance (ANOVA) is used to test the validity of the model, as well as to examine the significance and suitability of the model. The model is adequate within a 95% confidence interval. When the values of P are less than 0.05 (or 95% confidence), the models obtained are considered statistically significant. In other words, the closer the R2 approaches to the value 1, the model is compatible with the real (experimental) values.

3D representation on **Figure 14** clearly optimize the parameters effects on Ps value, based on RSM multifactor data, numerical optimization is possible. Including factors and propagation of error for all variables is available in the settings of Design-Expert software, and limits factor ranges to factorial levels (−1 to +1) in coded values, the area of this experimental design provides the best predictions.
