Question

Safety of underground tunnels. Research published in the journal Tunnelling and Underground Space Technology (July 2014) evaluated the safety of underground tunnels 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 made from shotcrete: tunnel face, tunnel walls, and tunnel crown. FS was determined to be normally distributed in each area, with means and standard deviations shown in the table. Tunnel failure is considered to occur when FS is lower than or equal to 1. Which tunnel area is more likely to result in failure? Why?

	Mean	Standard Deviation
Tunnel Face	1.2	0.16
Tunnel Walls	1.4	0.2
Tunnel Crown	2.1	0.7

Short answer

Tunnel Face is more likely to result in failure.

Step-by-step solution

Given information

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

Calculatethe probability

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.$\mu =1.2 and \sigma =0.16$

x=1

The z-score is,

$\begin{array}{rcl}z& =& \frac{x-\mu }{\sigma }\\ & =& \frac{1-1.2}{0.16}\\ & =& -1.25\\ & & \end{array}$

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

$\begin{array}{rcl}P\left(x⩽1\right)& =& P\left(z⩽-1.25\right)\\ & =& 1-P\left(z⩽1.25\right)\\ & =& 1-0.8944\\ & =& 0.1056\\ & & \end{array}$

$P\left(x⩽1\right)=0.1056$

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.$\mu =1.4 and \sigma =0.20$

x=1

The z-score is,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{1-1.4}{0.20} \\ =-2 \end{gathered}$$

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

$\begin{array}{rcl}P\left(x⩽1\right)& =& P\left(z⩽-2\right)\\ & =& 1-P\left(z⩽2\right)\\ & =& 1-0.9772\\ & =& 0.0228\\ & & \end{array}$

$P\left(x⩽1\right)=0.0228$

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.$\mu =2.1 and \sigma =0.70$

x=1

The z-score is,

$\begin{array}{rcl}z& =& \frac{x-\mu }{\sigma }\\ & =& \frac{1-2.1}{0.70}\\ & =& -1.5714\\ & \approx & -1.57\\ & & \end{array}$

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

$\begin{array}{rcl}P\left(x⩽1\right)& =& P\left(z⩽-1.57\right)\\ & =& 1-P\left(z⩽1.57\right)\\ & =& 1-0.9418\\ & =& 0.0582\\ & & \end{array}$

$P\left(x⩽1\right)=0.0582$

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.