Seismic Design of Tunnels - Parsons Brinckerhoff

amplitude is relatively small. Using the given wavelength and the corresponding
displacement amplitude, the calculated free-field ground strains would be
significantly smaller than those calculated using the simplified equations shown in
Table 1. This suggests that it may be overly conservative to use the simplified
equations to estimate the axial and curvature strains caused by seismic waves
travelling in soils for tunnel design.
• With regard to the derivations of spring coefficients K a and K t , there is no consensus
among design engineers. The derivations of these spring coefficients differ from those
for the conventional beam on elastic foundation problems in that:
-The spring coefficients should be representative of the dynamic modulus of the
ground under seismic loads.
-The derivations should consider the fact that loading felt by the surrounding soil
(medium) is alternately positive and negative due to the assumed sinusoidal
seismic wave.
Limited information on this problem is available in the literature (SFBARTD 1960, St.
John and Zahrah, 1987 and Owen and Scholl, 1981). For preliminary design, it
appears that the expressions suggested by St. John and Zahrah (1987) should serve
the purpose:
K t = K a = 16pG m(1 - v m )
(3 - 4v m )
d
L
(Eq. 3-6)
where G m = shear modulus of the medium (see Section 4.2 in Chapter 4)
n m = Poisson’s radio of the medium
d
L
= diameter (or equivalent diameter) of the tunnel
= wavelength
• A review of Equations 3-1, 3-3 and 3-4 reveals that increasing the stiffness of the
structure (i.e., E c A c and E c I c ), although it may increase the strength capacity of the
structure, will not result in reduced forces. In fact, the structure may attract more
forces as a result. Therefore, the designer should realize that strengthening of an
overstressed section by increasing its sectional dimensions (e.g., lining thickness)
may not always provide an efficient solution for seismic design of tunnel structures.
Sometimes, a more flexible configuration with adequate reinforcements to provide
sufficient ductility is a more desirable measure.
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