M M M

Shear wave front

FIGURE 28.17 Ovaling and racking deformation of tunnels: (a) Ovaling deformation of a circular cross-section and (b) racking deformation of a rectangular cross-section (© Parsons Brinckerhoff, Inc. 1993, with permission).

Tension Compression ._K.

Tunnel

Positive curvature

Positive curvature

FIGURE 28.18 Axial and curvature deformation of tunnels: (a) axial deformation along tunnel and (b) curvature deformation along tunnel (© Parsons Brinckerhoff, Inc. 1993, with permission).

FIGURE 28.18 Axial and curvature deformation of tunnels: (a) axial deformation along tunnel and (b) curvature deformation along tunnel (© Parsons Brinckerhoff, Inc. 1993, with permission).

flexible to the ground. This procedure requires minimum input, making it useful as an initial design tool and as a method of design verification.

The lining will develop axial and bending strains to accommodate the axial and curvature deformations imposed by the surrounding ground. St. John and Zahran [15] developed solutions for these strains due to compression P-waves, shear S-waves, and Rayleigh R-waves. Table 28.3 presents the solutions due to the three wave types propagating at an angle, f, in the horizontal plane with respect to the longitudinal axis of the tunnel.

The strains e due to combined axial and curvature deformations can be obtained by combining the longitudinal strains generated by axial and bending strains as follows:

For P-waves

For S-waves

For R-waves

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