# Shear Strength and Durability of Expansive Soil Treated with Recycled Gypsum and Rice Husk Ash
## 3 Results and Discussion
### Modified Hyperbolic Model  
All tested specimens' progressive increase in shear strength indicates strain hardening in each soil. Strain increase is coupled with added stiffness, which yields higher strength values. The non-linearity of the stress-strain curve of fine-grained soil and its treated variants can be characterized as asymptotic, highlighted by the general tendency to reach a relatively constant shear strength. This curve behavior is akin to hyperbolic functions \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multicolumn{3}{c}{**Shear Strength at Failure (kPa)**} \\ \cline{2-4}  & \(\sigma_{\
u}\) = 27.25 kPa & \(\sigma_{\
u}\) = 40.88 kPa & \(\sigma_{\
u}\) = 54.50 kPa \\ \hline Untreated Soil & 23.551 & 35.813 & 41.863 \\ Soil + 5\% Gypsum & 18.849 & 36.021 & 45.141 \\ Soil + 10\% Gypsum & 24.324 & 38.285 & 47.447 \\ Soil + 15\% Gypsum & 25.769 & 40.920 & 49.201 \\ Soil + 5\% Gypsum + 10\% RHA & 24.226 & 39.801 & 48.483 \\ Soil + 10\% Gypsum + 10\% RHA & 24.018 & 38.732 & 43.685 \\ Soil + 15\% Gypsum + 10\% RHA & 23.082 & 38.357 & 42.333 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Shear strength at failure for all specimens at saturated condition. Figure 6: Experimental and model-predicted shear stress vs. shear strain and volumetric strain vs. shear strain behavior for fine-grained soil samples with gypsum and rice husk ash as complementary additives. modeled by [PERSON] and [PERSON] [51] and [PERSON] [55], where approximations are made for the non-linear elastic movement of isotropic soils using finite element methods. Modifications were applied to the model based on volumetric strain parameters introduced by [PERSON] [38] to fit results from direct shear tests. Predicted stress-strain and volumetric-strain-shear-strain plots of all specimen proportions are also illustrated as solid lines in Figures 4-6. The stress-strain and volumetric-strain-shear-strain plots generated from treated samples formed the basis for calculating the modified hyperbolic model. Transformed plots of shear strain/shear stress vs. shear strain, as shown in Figure 7, display a straight-line function that confirms the hyperbolic stress-strain behavior of the samples. The modified hyperbolic model determines the initial shear modulus of a given sample through a transformed plot. These graphs define parameters K (shear modulus number) and n (shear modulus exponent). The failure ratio characterizing the difference of shear strength at failure and asymptotic shear strength is also identified. Parameters \(\phi_{0}\) and \(\Delta\phi\) defining the relationship between the friction angle and effective stress are also known. The introduction of volumetric strain parameters highlights modifications in the initial hyperbolic model \(\alpha\) and \(\beta\). A corresponding volumetric strain number and exponent describe the variation of these parameters with the effective stress. For \(\alpha\), these are hyperbolic parameters K\({}_{\text{a}}\) and m. For \(\beta\), these are hyperbolic parameters K\({}_{\text{b}}\) and r. Hyperbolic parameters, which define the calculated shear modulus and volumetric strain parameters, are enumerated in Tables 9 and 10. After determining the hyperbolic parameters, the soil's stress-strain and volume change response can be predicted using Equations (9) and (10) for any normal effective stress and shear strain value. The modified hyperbolic model provides a satisfactory approximation of the specimen's actual stress-strain response and volumetric change. The disparity between actual and predicted stress-strain values is minimal due to the non-linear behavior of the sheared samples, which closely resembled hyperbolas due to the progressive increase in shear strength. Good estimation was also shown by the model on the volumetric response of the sample despite several slight deviations demonstrated in a few samples. A desirable outcome was attained with the modeling due to the ductile and compressive behavior of the test specimen.