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（2.35）
Where = height of pressure arching
= height at any point of the pressure arching
= physical index of rock mass, and
for loose rock mass without cohesion
for clayey rock mass, where
for intact rock mass
Where = uniaxial compression strength.
= cohesion of rock mass
= internal friction angle
= half width of an underground structure
● Horizontal pressure
Generally, vertical pressure is the leading load, however, horizontal pressure only needs to be considered for soft rock ( 2). Based on Rankine’s theory, the horizontal pressure can be expressed as
（2.36）
Where = unit weight of rock mass
= depth of an underground structure
= internal friction angle of rock mass
Note that the horizontal pressure appears to be a triangular distribution at the direction of depth and if the rock masses are composed of multi-layer rocks, horizontal pressure should be determined, respectively.
● Bottom pressure
In some extreme conditions, underground structures are constructed in weak and swelling rock mass. The upheaving of surrounding rock can produce bottom reaction pressure and its value is usually smaller than that of horizontal pressure. Details for the determination of bottom pressure refer to the references.
2.3.2 Rational methods
Rational methods of design are based on theories of elasticity and plasticity and are approached through the concepts of strain and stress. They include the consideration of in situ stresses and the loss of inherent strength of rock due to removal of confining pressure because of excavation.
Figure 2.8 shows rock mass with a circular opening, and a rock mass subjected to a radial pressure “ ” at boundaries at great distances from the center of the circular opening of radius “ ”. The stresses and strains due to creation of an opening in such a homogeneous and isotropic rock mass under biaxial pressure loading and plane strain conditions can be found out as follows:
For elastic zone, we have
Figure 2.8 Rock mass with a circular opening (2.37)
However, for non-elastic zone, we have
(2.38)
Where = reaction pressure caused by support
= in situ stress and , where and are unit weight of rock mass and depth, respectively
= radius of a circular opening
R= radius of inelastic zone
On the interface of elastic and inelastic zone, stresses and should satisfy the Eqs.（2.37）and 2.38).
Hence
（2.39）
And
（2.40）
Note that r=R， Eq.（2.39）is equal to Eq.(2.40)
From which
(2.41)