# Application of Direct Shear Test to Analysis of the Rate of Soil Improvement with Polyester Fibres
## 2 Materials and Methods
So, e.g., a procedure to determine the uncertainties of normal stress \(u(\sigma)\) and shear stress \(u(\tau)\) is introduced. Determination of the uncertainties of the angle of internal friction and cohesion could be performed applying the same principle. Based on our experience, the standard uncertainty of the shear box was proposed to be \(u_{a}\) = 0.2% and \(u_{b}\) = 0.2% (for dimensions \(a\) and \(b\) of the shear box). Based on calibration protocols and the magnitude of force measured, the standard uncertainties \(u_{N}\) = 0.141% of normal force and \(u_{T}\)= 0.158% of shear force were applied. The normal stress is a function of normal force \(N\) and box size parameters \(a\) and \(b\). The parameters \(N\), \(a\), and \(b\) are not dependent on one another (not correlated); the standard uncertainty of the normal stress \(\sigma\) in kPa was obtained using the formula:  
\[u(\sigma)=\sqrt{\left[c_{N}.u(N)\right]^{2}+\left[c_{a}.u(a)\right]^{2}+\left[ c_{b}.u(b)\right]^{2}} \tag{4}\]  
Figure 3: Test device and tested fibres: (**a**) Polyester fibres TEXZEM PES 200 before testing; (**b**) Automatic large shear box apparatus SHEARMATIC 300; (**c**) Specimen of soil and fibre after testing (0.5% fibre, normal stress 50 kPa). where \(c_{N}\), \(c_{a}\), and \(c_{b}\) are the sensitivity coefficients relating to the particular parameters \(N\), \(a\), and \(b\), respectively, and will be obtained by the partial derivative of the model function \(f\), step by step, of \(N\), \(a\), and \(b\). So, e.g., the sensitivity coefficient for \(N\) was determined as follows:  
\[c_{N}=\frac{\partial f}{\partial N}=\frac{\partial\left(\frac{N}{a\beta} \right)}{\partial N}=\frac{1}{a.b}\Bigg{|}_{a=0.30,b=0.30} \tag{5}\]  
In the same way, \(c_{a}\) and \(c_{b}\) were determined. The absolute value of the standard uncertainty of the particular parameter, e.g., of the normal force \(u(N)\), was calculated by multiplying its estimated value by the standard uncertainty of the force transducer:  
\[u(N)=N.u_{N} \tag{6}\]  
In the same way, \(u(a)\) and \(u(b)\) and then \(u(\sigma)\) were determined based on Formula (4). Regarding the uncertainty of the shear stress \(u_{c}(\tau)\), inconformity of the specimen can give different shear stresses at the same normal stress even if the same test procedure is applied. Therefore, in this case, it was proposed that the standard uncertainty of the type A \(u_{A}(\tau)\) of shear stress existed but did not exceed 0.5%. The value \(u_{A}(\tau)\) was used to calculate the combined standard uncertainty of the shear stress, applying the following formula:  
\[u_{c}(\tau)=\sqrt{u_{A}(\tau)^{2}+u_{B}(\tau)^{2}}, \tag{7}\]  
where \(u_{B}(\tau)\) is determined in the same way as in the case of \(u(\sigma)\). In order to determine the uncertainty as precisely as possible, a further source of uncertainty was considered, namely, the covariances between correlated input quantities (normal stresses and shear stresses). In the case of DST, covariances between stresses exist since they are calculated using a common specimen area \(A\). So, e.g., covariance \(u\left(\sigma_{i},\tau_{j}\right)\) could be obtained using the formula:  
Figure 2:0  
Then, the correlation coefficient \(r\left(\sigma_{i},\tau_{j}\right)\) could be obtained using the formula:  
Figure 2:1  
Applying a similar procedure, the correlation coefficients \(r\left(\sigma_{i},\sigma_{j}\right)\) and \(r\left(\tau_{i},\tau_{j}\right)\) could be calculated. A standard uncertainty \(u(tan\varphi)\) caused by parameters \(\sigma_{i}\) and \(\tau_{i}\) and also their correlation was determined similarly as in the case of \(u(\sigma)\), applying Equation (1), considering \(tan\varphi\) as a function of eight parameters (four normal stresses and four shear stresses). A formula in [22] was used, as follows:  
\[