F[(3 - 2Vm)-(l - 2Vm)C]~ 1(1 - 2Vm)2 + 2 F[(3 - 2Vm) + (l - 2Vm)C] + C[2 - 8Vm + 6vm]+6 - 8vn

where C is the compressibility ratio.

The seismically induced strains due to the ovaling effect need to be combined with the strains resulting from nonseismic loading and then checked against the allowable strain limits consistent with the performance goal established for the design of the tunnel lining. Evaluation of Racking Deformations of Rectangular Tunnels

Racking deformations are defined as the differential sideways movements between the top and bottom elevations of the rectangular structures, shown as As in Figure 28.20. The resulting material strains in the lining associated with the seismic racking deformation, As, can be derived by imposing the differential deformation on the structure in a structural frame analysis. The procedure for determining, As taking into account the soil-structure interaction effects, is presented below [12].

Estimate the free-field ground strains gmax (at the structure elevation) caused by the vertically propagating shear waves of the design earthquakes (see Section Determine Afree-field, the differential free-field relative displacements corresponding to the top and the bottom elevations of the rectangular structure (see Figure 28.20), by

Afree-field = hgmax (28.17)

where h is the height of the structure.

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