# A Discrete Model of Rock-Soil Medium Response in the Determination of Horizontal Strain Values
## 4 Influence of Measurement Base Length on the Uncertainty of  
Horizontal Deformation  
[PERSON] [41; 53] assumed that relative horizontal elongations (hereinafter: horizontal deformations) that are caused by mining operations are a result of the discontinuous reaction of the rock mass to the influence of mining. According to [PERSON], fissures form in the rock mass, even if they are extremely small. According to this assumption, the horizontal strain that is measured at a section with a length \(l\) is a function of the fissure width and the number of fissures. Assuming that the random variable fissure width is described by the normal distribution and the random variable of the number of fissures is described by the binomial distribution and that these variables are mutually independent, [PERSON] obtained Equation (22) for the variation coefficient of the average value of the horizontal strain:  
\[M_{\epsilon}=\frac{2}{\sqrt{3}}\sqrt{\frac{1}{n}} \tag{22}\]  
where  
\(n\)--number of fissures. Assuming that the number of fissures is proportional to the length of the measured section \(l\), [PERSON] formulated the basic relationship given in Equation (2) that links the variation coefficient of the horizontal strain value with the length of the measurement section. Equation (2) shows that the greater the length of the measurement section, the lower the value of the variation coefficient of the observed/measured value of the horizontal strain. Because of this fact, [PERSON] proposed that the average maximum strain value should be determined from the average strain value for a length of the measurement section that is equal to the main influence range radius \(R\). According to [PERSON], the maximum values of horizontal strain that are determined using this method are characterized by a relatively low variation coefficient that ranges from a few to several percent. The relationship between the average maximum horizontal strain \(\overline{\epsilon}_{max}\) and the average value determined for a section with length \(l\) is given by Equation (23):  
\[\overline{\epsilon}_{max}=\sqrt{\frac{2\pi}{e}}\,\overline{\epsilon}_{max} \ \ \ (l=R) \tag{23}\]  
As a conclusion of these considerations, [PERSON] stated that the predicted values of horizontal strain should be reported with their standard deviations or variation coefficients, as this leads to a more realistic assessment of the impact of mining operations on the surface buildings and structures. Among the research in this field, the works by [PERSON] [42; 43], which still form the basis for the uncertainty assessment of forecasts and the horizontal strain values that are measured in situ, are particularly noteworthy. According to [PERSON], the variation coefficient of the horizontal tensile strain, which includes the influence of the measurement section length, can be described by Equation (24):  
\[M_{\epsilon}(l)=\frac{5}{\sqrt{l}}(0.14+0.0006\cdot H-0.0008\cdot N-0.06\cdot s _{max}) \tag{24}\]  
where  
\(H\)--depth of the mining operations (m);  
\(N\)--thickness of the loose rock overburden (m);  
\(s_{max}\)--the maximum subsidence value (m);  
\(l\)--measurement section length. [PERSON] also reported the empirically obtained findings on the average values of the variation coefficient of maximum values of tensile and compressive strain, as shown in Equations (25) and (26):  
\[M_{\varepsilon_{max}^{+}}=0.2 \tag{25}\]  
\[M_{\varepsilon_{max}^{-}}=0.3 \tag{26}\]  
The values indicated above were fully confirmed in later works, among others, by [PERSON] [48] and [PERSON] [49]. [PERSON] [42] analysed the measurement results that were obtained for changes in the length of the measurement sections along the so-called measurement lines, particularly in parts involving tensile deformations. This analysis included a total of 49 cases involving coal (35), iron ore (10) and copper ore (4) mining.