Seismic Design of Tunnels - Parsons Brinckerhoff

maximum bending moment than the no-slip interface condition. The differences are within
approximately 20 percent under seismic shear loading condition.
It should be realized, however, that these results are based on pseudo-static solutions
that do not consider the potential dynamic amplification and stress concentrations at the
tunnel excavation boundary (Mow and Pao, 1971). Previous studies suggest that a true
dynamic solution would yield results that are 10 to 15 percent greater than an equivalent
static solution, provided that the seismic wavelength is at least about 8 times greater than
the width of the excavation (cavity).
Therefore, it is prudent to adopt the more conservative full-slip assumption for the
calculation of bending moments. With this more conservative assumption, the effects of
stress amplification need not be considered.
Maximum Lining Deflection, D D lining . Figure 19 presents a plot of the maximum lining
deflections from full-slip closed form solution versus those from no-slip finite difference
analysis (noting that these lining deflections are normalized with respect to the free-field
ground deflections). Similar to the discussion presented above, lining tends to oval
(distort) more under the full-slip interface assumption. The differences, however, are very
small.
The full-slip assumption (Equation 4-10 or Equation 4-13) is recommended for
calculating the lining distortion. The effects of stress amplification need not be considered
when the conservative full-slip assumption is adopted.
It is interesting to note from the plot that almost no difference exists between the two
assumptions for Case No. 12. This can be explained by the fact that a nearly “perfectly
flexible” lining is used and little lining-ground interaction is involved in the Case No.12
analysis.
Maximum Lining Thrust, T max . For comparison, the maximum lining thrusts are
calculated using closed form solutions for both assumptions (Equations 4-8 and 4-12).
The results, along with those from the finite difference analysis, are tabulated in Table 3.
The table shows excellent agreement on the thrust response between the numerical finite
difference analysis and the closed form solution for the no-slip condition. It also verified
that the full-slip assumption will lead to significant underestimation of the lining thrust under
seismic shear condition.
Therefore, it is recommended that Equation 4-12 be used for thrust calculation. To
account for the dynamic stress amplification due to the opening, it is further recommended
that thrusts calculated from Equation 4-12 be multiplied by a factor of 1.15 for design
purpose.
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