for f

Vp = soil particle velocity caused by P-waves Op = soil particle acceleration caused by P-waves Cp = apparent propagation velocity of P-waves Vs = soil particle velocity caused by S-waves as = soil particle acceleration caused by S-waves Cs = apparent propagation velocity of S-waves VR = soil particle velocity caused by R-waves aR = soil particle acceleration caused by R-waves CR = propagation velocity of R-waves Hp = curvature

Source: St. John and Zahrah 1987.

It should be noted that

• S-waves generally cause the largest strains and are the governing wave type.

• The angle of wave propagation, f, should be the one that maximizes the combined axial strains.

The horizontal propagation S-wave velocity, CS, in general, reflects the seismic shear wave propagation through the deeper rocks rather than that of the shallower soils where the tunnel is located. In general, this velocity value varies from about 2 to 4km/s. Similarly, the P-wave propagation velocities, CP, generally vary between 4 and 8km/s. The designer should consult with experienced geologists and seismologists for determining CS and CP.

When the tunnel is located at a site underlain by deep deposits of soil sediments, the induced strains may be governed by the R-waves. In such deposits, detailed geological and seismological analyses should be performed to derive a reliable estimate of the apparent R-wave propagation velocity, CR.

The combined strains calculated from Equations 28.4a, 28.4b and 28.4c represent the seismic loading effect only. To evaluate the adequacy of the structure under the seismic loading condition, the seismic loading component has to be added to the static loading components using the appropriated loading combination criteria developed for the structures. The resulting combined strains are then compared against the allowable strain limits, which should be developed based on the performance goal established for the structures (e.g., the required service level and acceptable damage level). Procedure Accounting for Soil-Structure Interaction Effects

If a very stiff tunnel is embedded in a soft soil deposit, significant soil-structure interaction effects exist, and the free-field deformation procedure presented in the previous subsection may lead to an overly conservative design. In this case, a simplified BOEF procedure should be used to account for the soil-structure interaction effects. According to St. John and Zahran [15], the effects of soil-structure interaction can be accounted for by applying reduction factors to the free-field axial strains and the free-field curvature strains, as follows:

For axial strains

For bending strains

Ka L

Elll f 2n

0 0

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