Seismic Design of Tunnels - Parsons Brinckerhoff

Comments on Closed Form Solutions
According to previous investigations, during an earthquake slip at interface is a
possibility only for tunnels in soft soils, or when seismic loading intensity is severe. For
most tunnels, the condition at the interface is between full-slip and no-slip. In computing
the forces and deformations in the lining, it is prudent to investigate both cases and the
more critical one should be used in design. The full-slip condition gives more conservative
results in terms of maximum bending moment, M max , and lining deflections DD.
This conservatism is desirable to offset the potential underestimation (10 to 15
percent) of lining forces resulting from the use of equivalent static model in lieu of the
dynamic loading condition (Mow and Pao, 1971). Therefore, the full-slip model is adopted
for the present study in evaluating the moment and deflection response of a circular tunnel
lining.
The maximum thrust, T max , calculated by Equation 4-8, however, may be significantly
underestimated under the seismic simple shear condition. The full-slip assumption along
the interface is the cause. Therefore, it is recommended that the no-slip interface
assumption be used in assessing the lining thrust response. The resulting expressions,
after modifications based on Hoeg’s work (Schwartz and Einstein, 1980), are:
T max =±K 2 t max R
E m
=±K 2
2(1+n m )
Rg max
(Eq. 4-12)
where the lining thrust response coefficient, K 2 , is defined as:
F[ ( 1- 2n m )- ( 1 - 2n m )C ]- 1 (
K 2 =1 +
2 1- 2n m) 2 + 2
F[ ( 3 - 2n m )+ ( 1 - 2n m )C ]+ C È5 2
- 8nm +
˘
6nm
Î2 ˚+ 6 - 8nm
F = flexibility ratio as defined in Eq. 4-6
C = Compressibility ratio as defined in Eq. 4-5
E m , n m = modulus of elasticity and Poisson’s Ratio of medium
R = radius of the tunnel lining
t max = maximum free-field shear stress
g max = maximum free-field shear strain
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