where Ei is Young's modulus of tunnel lining, Al is the cross-sectional area of the lining, Kh is the transverse soil spring constant, Ka is the longitudinal soil spring constant, L is the wave length of the P-, S-, or R-waves, and Il is the moment of inertia of the lining cross-section.

It should be noted that the axial strain calculated from Equation 28.5a should not exceed the value that could be developed using the maximum frictional forces, Qmax, between the lining and the surrounding soils. Qmax can be estimated using the following expression:

where f is the maximum frictional force per unit length of the tunnel. Evaluation of Ovaling Deformations of Bored or Mined Circular Tunnels

The seismic ovaling effect on the lining of bored or mined circular tunnels is best defined in terms of change of tunnel diameter, DDeq. For practical purposes, the ovaling deformations can be assumed to be caused primarily by the vertically propagating shear waves. DDeq can be considered as seismic ovaling deformation demand for the lining. The procedure for determining DDeq and the corresponding lining strains is outlined as follows [12].

Estimate the expected free-field ground strains caused by the vertically propagating shear waves of the design earthquakes. The free-field ground strains can be estimated using the following formula:

where gmax is the maximum free-field shear strain at the elevation of the tunnel, VS is the S-wave peak particle velocity at the tunnel elevation, and CSE is the effective shear wave velocity of the medium surrounding the tunnel.

Alternatively, the maximum free-field shear strain can be estimated by a more refined free-field site response analysis (such as in Ref. [16]).

The effective shear wave velocity of the vertically propagating shear wave, CSE, must be compatible with the level of the shear strain that may develop in the ground at the elevation of the tunnel under the design earthquake shaking. A rough estimate of the ratio CSE/CSS (where CSS is the low-strain shear wave velocity of the surrounding medium) can be made as follows:

• For stiff to very stiff soil, CSE/CSS may range from 0.7 to 0.9 for low to moderate earthquake shaking and from 0.5 to 0.7 for strong shaking.

Alternatively, site-specific response analyses can be performed for estimating CSE. Site-specific response analyses should be performed on soft soil sites.

The values of the low-strain shear wave velocity, CSS, can be determined using geophysical testing techniques in the field such as P-S logger, cross-hole, and seismic cone penetration methods or estimated from empirical correlation.

By ignoring the stiffness of the tunnel, which is applicable for tunnels in rock or in stiff or dense soils, the lining can be reasonably assumed to conform to the surrounding ground with the presence of a cavity due to the excavation of the tunnel (but without the presence of the lining). The resulting diameter change of the tunnel is

where vm is Poisson's ratio of the surrounding ground and D is the diameter of the tunnel.

If the structure is stiff relative to the surrounding soil, then the effects of soil-structure interaction should be taken into consideration. The relative stiffness of the lining is measured by the flexibility ratio, F, defined as follows:

where Em is the strain-compatible elastic modulus of the surrounding ground, _i is the nominal radius of the tunnel lining, V is Poisson's ratio of the tunnel lining, and Il,1 is the moment of inertia of the lining per unit width of tunnel along the tunnel axis.

The strain-compatible elastic modulus of the surrounding ground, Em, should be derived using the strain-compatible shear modulus, Gm, corresponding to the effective shear wave propagating velocity, CSE.

The moment of inertia of the tunnel lining per unit width, Il,1, should be determined based on the expected behavior of the selected lining under the combined seismic and static loads, accounting for cracking and joints between segments and between rings as appropriate.

The diameter change, Adeq, accounting for the soil-structure interaction effects can then be estimated using the following equations:

where k1 is the Seismic ovaling coefficient. The seismic ovaling coefficient curves plotted as a function of F and vm are presented in Figure 28.19. The resulting bending moment induced maximum fiber strain, em, and the axial force (i.e., thrust) induced strain, eT, in the lining can be derived as follows:

where tl is the thickness of the lining.

The solutions presented in Equations 28.10 through 28.13 assume that a full-slippage condition exists along the soil-lining interface, which allows normal stresses (without normal separation) but no tangential shear force. The full-slippage assumption yields slightly more conservative results in estimating

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