# Application of Direct Shear Test to Analysis of the Rate of Soil Improvement with Polyester Fibres
## 2 Materials and Methods
\begin{array}{l}u(\tan\varphi)^{2}=u(\beta)=\sum\limits_{i=1}^{n}\left(c( \sigma_{i})u(\sigma_{i})\right)^{2}+\sum\limits_{i=1}^{n}\left(c(\tau_{i})u( \tau_{i})\right)^{2}+\\ \hskip 14.226378 pt+2.\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}c(\tau_{i}) c(\tau_{j})r\left(\sigma_{i},\sigma_{j}\right)u(\sigma_{i})u\left(\sigma_{j} \right)+\\ \hskip 14.226378 pt+2.\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}c(\tau_{i}) c(\tau_{j})r\left(\tau_{i},\tau_{j}\right)u(\tau_{i})u\left(\tau_{j} \right)+\\ \hskip 14.226378 pt+2.\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}c(\sigma_{i} )c\left(\tau_{j}\right)r\left(\sigma_{i},\tau_{j}\right)u(\sigma_{i})u\left( \tau_{j}\right)\end{array}  \tag{10}\]The uncertainty range of the angle of friction \(\varphi\) is asymmetric [22] since uncertainty is defined for \(tan\varphi\) (not for \(\varphi\)). According to [22], if one notes \(\beta_{0}=tan(\varphi_{0})\), then \(\beta\) (slope of strength line) belongs to the \([\beta_{0}-u(\beta);\)\(\beta_{0}+u(\beta)]\) interval. Consequently, \(\varphi\) varies in the \([atan(\beta_{0}-u(\beta));atan(\beta_{0}+u(\beta))]\) range. Thus, two components of \(u(\varphi)\) can be differentiated, i.e.,  
\[u_{-}(\varphi)=atan(\beta_{0})-atan(\beta_{0}-u(\beta)), \tag{11}\]  
which is the left-hand side of \(u(\varphi)\), and  
\[u_{+}(\varphi)=atan(\beta_{0}+u(\beta))-atan(\beta_{0}) \tag{12}\]  
which is the right-hand side of \(u(\varphi)\). The expanded uncertainty (using a coverage factor k = 2) of \(\varphi\) should be split in the right-hand interval \(U_{\varphi+95\%}=2\cdot[atan(\beta_{0}+u(\beta))-\varphi_{0}]\) and the left-hand interval \(U_{\varphi-95\%}=2\cdot[\varphi_{0}-atan(\beta_{0}-u(\beta))]\). The larger values from them will be introduced in the paper as \(U_{\varphi}\). The standard uncertainty of cohesion \(u(c)\) was determined in the same way as in the case of \(u(tan\varphi)\), applying Equation (2), and the expanded uncertainty \(U_{c}\) was determined by multiplying the standard uncertainty \(u(c)\) by the coverage factor k = 2. To examine the influence of covariance between correlated input quantities (normal stresses and shear stresses), \(U_{\varphi}\) and \(U_{c}\) were calculated without using correlation coefficients, marked as case A, and using correlation coefficients, marked as case B (see Formulae (9) and (10)). To explore the magnitude of the uncertainty in the case of the worst-case estimation, marked as case C, the uncertainty was calculated by adding linearly the uncertainty contributions \(c(\sigma_{i}).u(\sigma_{i})\) and \(c(\tau_{i}).u(\tau_{i})\) and not considering the correlation coefficients. The morphology of the fibres before and after shearing under various normal stresses was investigated by SEM. Fibres were taken from the shear surface of the center ninth of specimens of size 10 cm \(\times\) 10 cm (shear box of size 30 cm \(\times\) 30 cm) after shearing under normal stresses of 50 kPa and 300 kPa, and their morphologies were compared. A JEOL JSM 5500 LV microscope (JEOL Ltd., Tokyo, Japan) operating in backscattered electron mode was used. The observations were carried out for fibres sputtered with gold using the JEOL JFC 1200 ionic sputter coater.