# Role of Bimodal Water Retention Curve on the Unsaturated Shear Strength
## 3 Mathematical Equation
### Proposed Equation  
Many studies showed that pore size distribution (PSD) of the soil can be determined from the differentiation of water retention curve [43; 45]. In this study the PSD of the soil mixtures was differentiated from its corresponding WRC which has been modelled using [PERSON] et al. [17], equation. Therefore, the equation from [PERSON] et al. [17], was differentiated to generate a PSD of the soil mixtures. The PSD equation used in this study is presented in Equation (7). Figure 1: Variables of bimodal water retention curve. \[\frac{d\theta}{d\psi}=\left[1-\frac{ln\left(1+\frac{\psi}{\psi^{\prime}}\right)}{ln \left(1+\frac{10^{6}}{\psi^{\prime}}\right)}\right]\left\{ \begin{array}{l} \left[\frac{\theta_{41}-\theta_{41}}{\sqrt{2\pi}s_{1}\psi}exp\left(-\frac{ \left(ln\left(\frac{\theta_{41}-\psi}{\theta_{41}-\theta_{41}}\right)\right)^{2 }}{2s_{1}{}^{2}}\right)\right]+\\ \left[\frac{\theta_{22}-\theta_{22}}{\sqrt{2\pi}s_{2}\psi}exp\left(-\frac{ \left(ln\left(\frac{\theta_{42}-\psi}{\theta_{42}-\theta_{42}}\right)\right)^{2 }}{2s_{2}{}^{2}}\right)\right]\end{array} \right\}+ \tag{7}\]  
where all parameters were taken from Equation (1) after [PERSON] et al. [17]  
In this study, a new equation that can be used for a best fitting of the unsaturated shear strength with bimodal WRC is proposed. The shear strength equation for suction lower than \(AEV_{1}\) is the same as [PERSON] et al. [44] equation. The new equation (Equation (8)) was proposed since the experimental data from the unsaturated shear strength of soil with bimodal characteristics has unique behavior for suctions higher than \(AEV_{1}\). The rationale behind the development of the new equation is simply to incorporate the characteristics of the first sub-curve of the bimodal WRC into the equation. The parameters related to standard deviation in the bimodal WRC (s\({}_{1}\) and s\({}_{2}\)) have been studied by [PERSON] et al. [17; 23], [PERSON] et al. [12]. They concluded that the parameters related to standard deviation can be used to estimate bimodal WRC based on grain-size distribution. Therefore, the parameters related to standard deviation were used in the development of the new bimodal unsaturated shear strength equation. The parameter s\({}_{1}\) and s\({}_{2}\) from Equation (1) are incorporated in the new shear strength equation (Equation (10)) to replace parameter \"\(n\)\" from [PERSON] [43] model because shear strength of bimodal soil is affected by dual-porosity characteristics of PSD within the soil. The parameters S\({}_{1}\) and S\({}_{2}\) are discrete values representing the width of sub-curve 1 and sub-curve 2 in bimodal WRC (Figure 1). The reader is referred [PERSON] et al. [17; 23], and [PERSON] et al. [12] for details of these parameters. In the proposed equations, the parameter s\({}_{1}\) is used to represent the effect of sub-curve 1 of bimodal WRC on the unsaturated shear strength for suctions between \(AEV_{1}\) and \(AEV_{2}\). Additionally, the parameter s\({}_{2}\) is used to represent the effect of sub-curve 2 of bimodal WRC on the unsaturated shear strength for suctions beyond \(AEV_{2}\). Void ratio and fine content are selected as the parameters in the Equation (11) since these properties play important role in clay materials. Soil with bimodal WRC are commonly associated with the presence of clay materials [12]. In addition, the equation was developed to take into account the characteristics of the second sub-curve of the bimodal WRC. For AEV\({}_{1}\) < matric suction < \(AEV_{2}\):  
\[\tau=ct+[(\sigma-u_{a})+AEV_{1}]tan\ \phi\ \prime+[(u_{a}-u_{w})-AEV_{1}]b_{1} \Theta^{k_{1}}tan\ \phi\ \prime \tag{8}\]  
\[k_{1}=[log\ (\ u_{a}-u_{w})-log\ AEV_{1}]^{y} \tag{9}\]  
\[b_{1}=-0.245\{ln\ [\ s_{1}\big{(}I_{p}+4.4\big{)}\big{]}\}^{2}+2.114\{ln\ [\ s_{1} \big{(}I_{p}+4.4\big{)}\big{]}\}-3.522 \tag{10}\]  
\(s_{1}\) = parameter related to geometric standard deviation 1.