# Behaviour of Horseshoe-Shaped Tunnel Subjected to Different In Situ Stress Fields
## 2 Methods of In Situ Stress Measurement
### In Situ Stresses-Depth Relationship  
It is common practice to conduct two-dimensional stress analysis using a vertical field stress of \(\sigma_{\rm v}\) and a lateral (horizontal) stress of K \(\sigma_{\rm v}\) (e.g., \"k\" is the horizontal-to-vertical stress ratio) as depicted in Figure 1. The overburden stress (e.g., vertical) is the product of unit weight of overlying rock and the depth below surface. However, in complex tectonic environments, this stress may be lower [75; 76; 77] or higher than the calculated overburden stress [78; 79; 80]. The vertical stress can be calculated using simple relationship as per Equation (1) [81]:  
\[\sigma_{\rm v}=0.027\;\rm Z \tag{1}\]  
where 0.027: is the average unit weight of rock mass, \(\rm MN/m^{3}\). Z: is the depth below surface, m.  
The lateral (e.g., horizontal) stress is affected by topographic features and plate tectonics [26; 27; 28; 69; 82; 83; 84; 85; 86; 87]. The in situ stress ratio, k, is defined as the horizontal-to-vertical stress ratio, as per Equation (2) [88; 27] and its limits is defined as given in Equation (3) [89]:  
\[\rm K=\left(\frac{\sigma_{h}}{\sigma_{v}}\right) \tag{2}\]  
\[\rm\frac{100}{Z}+0.3\;\leq K\leq\frac{1500}{Z}+0.5 \tag{3}\]  
Thus, by substituting from Equations (1) and (2) into Equation (3), the limits of horizontal in situ stress (e.g., \(\sigma_{h}=k\;\sigma_{v}\)) are given as per Equation (4) [90]:  
\[\rm 2.7+0.0081\;Z\leq\sigma_{h}\leq 40.5+0.0135\;Z \tag{4}\]  
where, depth (Z) is in meters and stress is in MPa units. Therefore, if the depth, Z, is 500 m, then \"\(K\)\" ranges from 0.50 to 3.50 (e.g., by substitution in Equation (3)) and horizontal in situ stress (\(\sigma_{h}\)) ranges according from 6.75 MPa to 47.25 MPa (e.g., by substitution in Equation (4)). However, if depth, Z, is 2000 m, then \"K\" ranges from 0.35 to 1.25 and horizontal in situ stress (\(\sigma_{h}\)) ranges according from 18.90 MPa to 67.50 MPa. Thus, the value of \"K\" tends to be smaller at great depths and vice versa at shallow depths ([PERSON], 2015). However, these relationships between \"K\", depth (Z) and in situ vertical stress (\(\sigma_{v}\)) are useful in feasibility design studies, but it is noteworthy that they vary from the shield of one country to another [27; 69; 83; 84; 91; 92; 93; 94; 95; 96; 97; 98; 99].  
Figure 1: In situ horizontal and vertical stresses imposed on rock mass before creating tunnel opening.  
In this study, sensitivity stability analysis of a tunnel opening has been investigated at different values of in situ stress ratio, K, of 0.5, 1, 1.5, 1.65, 1.80, 2.0, 2.13 and 2.50 and various depths, Z, of 50-, 100-, 150-, 200- and 250-ms below surface. Four reference points (e.g., #1, #2, #3 and #4) have been assigned in the perimeter of the tunnel as shown in Figure 1, to assess stress-displacement around them. The stability of the tunnel opening is evaluated in terms of major induced-stress, stress concentration, strength of rock mass around tunnel opening, ratios of wall convergence, roof sag and floor have and extent of yielding zones into rock mass surrounding the tunnel. For explanations/mathematical relationships of these evaluation criteria [100; 101; 102]. The model geometry; dimensions and boundary conditions are presented in the following section (e.g., Section 3. Modelling set up).