91Ó°ÊÓ

It is given in the Research publishedin the Journal Tunnelling and Underground Space Technology (July 2014), which evaluated the safety of underground tunnels with built-in rigid soils. A factor of safety (FS), measured as the ratio of capacity over demand,was determined for three different areas of tunnels

For Tunnel Face,

Assume that the tunnel face is an area of tunnels for the factor of safety (FS), i.e., x follows a normal distribution with a mean of 1.2 and standard deviation of 0.16

i.e.$\mathrm{μ}=1.2  and  \mathrm{σ}=0.16$

x=1

The z-score is,

$\begin{array}{rcl}z& =& \frac{x-\mathrm{μ}}{\mathrm{σ}}\\ & =& \frac{1-1.2}{0.16}\\ & =& -1.25\\ & & \end{array}$

The probability of tunnel failure is lower than or equal to 1 is,

$\begin{array}{rcl}P\left(xâ\copyright ½1\right)& =& P\left(zâ\copyright ½-1.25\right)\\ & =& 1-P\left(zâ\copyright ½1.25\right)\\ & =& 1-0.8944\\ & =& 0.1056\\ & & \end{array}$

$P\left(xâ\copyright ½1\right)=0.1056$

Therefore, the probability of tunnel face likely to result in failure FS is lower than or equal to 1 is 0.1056.

For Tunnel Walls,

Assume that the tunnel walls are an area of tunnels for the factor of safety (FS), i.e., x follows a normal distribution with a mean of 1.4 and standard deviation of 0.20

i.e.$\mathrm{μ}=1.4  and  \mathrm{σ}=0.20$

x=1

The z-score is,

$$\begin{gathered} z=\frac{x-\mathrm{μ}}{\mathrm{σ}} \\ =\frac{1-1.4}{0.20} \\ =-2 \end{gathered}$$

The probability of tunnel failure is lower than or equal to 1 is,

$\begin{array}{rcl}P\left(xâ\copyright ½1\right)& =& P\left(zâ\copyright ½-2\right)\\ & =& 1-P\left(zâ\copyright ½2\right)\\ & =& 1-0.9772\\ & =& 0.0228\\ & & \end{array}$

$P\left(xâ\copyright ½1\right)=0.0228$

Therefore, the probability of tunnel walls likely to result infailure when FS is lower than or equal to 1 is 0.0228.

For Tunnel Crown,

Assume that the tunnel crown is an area of tunnels for the factor of safety (FS), i.e., x follows a normal distribution with a mean of 2.1 and standard deviation of 0.70

i.e.$\mathrm{μ}=2.1  and  \mathrm{σ}=0.70$

x=1

The z-score is,

$\begin{array}{rcl}z& =& \frac{x-\mathrm{μ}}{\mathrm{σ}}\\ & =& \frac{1-2.1}{0.70}\\ & =& -1.5714\\ & ≈& -1.57\\ & & \end{array}$

The probability of tunnel failure is lower than or equal to 1 is,

$\begin{array}{rcl}P\left(xâ\copyright ½1\right)& =& P\left(zâ\copyright ½-1.57\right)\\ & =& 1-P\left(zâ\copyright ½1.57\right)\\ & =& 1-0.9418\\ & =& 0.0582\\ & & \end{array}$

$P\left(xâ\copyright ½1\right)=0.0582$

Therefore, the probability of tunnel crown likely to result in failure when FS is lower than or equal to 1 is 0.0582.

So, Tunnel Face is more likely to result in failure.