Seismic Design of Tunnels - Parsons Brinckerhoff

Numerical Analysis
A series of computer analyses using finite difference code (FLAC, 1989) is performed
to verify the proposed procedure in the previous section. The mesh and the lining-ground
system used in these analyses are shown in Figure 17. The assumptions made for these
analyses include the following:
• Plane strain conditions are assumed.
• Seismic shear wave loading is simulated by pure shear conditions with shear stresses
applied at far external boundaries.
• Taking advantage of the anti-symmetric loading conditions, only one quarter of the
entire lining/ground system is analyzed. Rollers are provided at planes of antisymmetry.
• Lining is modeled by a series of continuous flexural beam elements of linear elasticity.
• Ground (medium) is modeled as linear elastic material.
• No-slip condition along the lining-ground interface is assumed.
A total of 13 analyses are performed. In order to cover a wide range of possible
effects of lining-ground interaction, the parameters for lining and ground are varied.
Following is a list of the range of the variations:
Range of E m = from 325 ksf to 72000 ksf
n m = 0.25 and 0.333
E l /(1-n l
2) = 518400 ksf and 662400 ksf
Range of t = from 0.5 feet to 2.0 feet
The resulting flexibility ratios, F, and compressibility ratios, C, are tabulated in Table 2.
To make the level of seismic loading within a reasonable range, the boundary shear
stresses (t max ) are made to result in the maximum free-field shear strains (g max ) in the
range between 0.001 and 0.008.
Results and Recommendations
Maximum Bending Moment, M max . The resulting maximum bending moments are first
calculated for each of the 13 cases by using the full-slip closed form solution, Equation 4-
9. These values are then compared to those obtained from the no-slip finite difference
analysis. A plot of comparison in terms of dimensionless bending moment between the
two is shown in Figure 18. As expected, the full-slip interface assumption results in higher
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