Seismic Design of Tunnels - Parsons Brinckerhoff

• Equation 4-4, the perforated ground deformation, should serve well for a lining that
has little stiffness (against distortion) in comparison to that of the medium.
• Equation 4-3, on the other hand, should provide a reasonable distortion criterion for a
lining with a distortion stiffness equal to the surrounding medium.
It is logical to speculate further that a lining with a greater distortion stiffness than the
surrounding medium should experience a lining distortion even less than that calculated
by Equation 4-3. This latest case may occur when a tunnel is built in soft to very soft soils.
The questions that may be raised are:
• How important is the lining stiffness as it pertains to the engineering design?
• How should the lining stiffness be quantified relative to the ground?
• What solutions should an engineer use when the lining and ground conditions differ
from those where Equations 4-3 and 4-4 are applicable?
In the following sections (4.4 and 4.5), answers to these questions are presented.
4.4 Importance of Lining Stiffness
Compressibility and Flexibility Ratios
To quantify the relative stiffness between a circular lining and the medium, two ratios
designated as the compressibility ratio, C, and the flexibility ratio, F (Hoeg, 1968, and Peck
et al., 1972) are defined by the following equations:
Em (1 - v
Compressibility Ratio, C =
12 ) R
E 1 t (1+ v m )( 1- 2v m )
(Eq. 4-5)
Flexibility Ratio, F = Em (1- v 1 2 ) R 3
6E 1 I (1+ v m )
(Eq. 4-6)
where
E m = modulus of elasticity of the medium
n m = Poisson’s Ratio of the medium
E l = the modulus of elasticity of the tunnel lining
n l = Poisson’s Ratio of the tunnel lining
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