Seismic Design of Tunnels - Parsons Brinckerhoff

where
g s = angular distortion of the structure
D s = lateral racking deformation of the structure
g free-field = shear distortion/strain of the free-field
D free-field = lateral shear deformation of the free-field
The racking coefficients, R, obtained from the analyses are presented in the last
column of Table 4 for all 25 cases.
Note that the total structural deformation obtained from the finite element analyses
contains a rigid body rotational movement, which causes no distortion to the cross-section
of the structure. Therefore, this portion of the movement is excluded in the calculation of
the structure racking deformation.
Effect of Relative Stiffness. As expected, results of the analyses indicate that the relative
stiffness between the soil medium and the structure has the most significant influence on
the structure response. This is demonstrated in Figure 33, where the structure racking
coefficients, R, are plotted against the flexibility ratios, F.
• When the flexibility ratio approaches zero, representing a perfectly rigid structure, the
structure does not rack regardless of the distortion of the ground in the free-field. The
normalized structure distortion (i.e., R) increases with the increasing flexibility ratio. At
F=1, the structure is considered to have the same stiffness as the ground and
therefore is subjected to a racking distortion that is comparable in magnitude to the
ground distortion in the free field (i.e., Rª1).
• With a flexibility ratio greater than 1.0, the structure becomes flexible relative to the
ground and the racking distortion will be magnified in comparison to the shear
distortion experienced by the ground in the free field. This latter phenomenon is not
caused by the effect of dynamic amplification. Rather, it is primarily attributable to the
fact that the ground surrounding the structure has a cavity in it (i.e., a perforated
ground). A perforated ground, compared to the non-perforated ground in the free
field, has a lower stiffness in resisting shear distortion and thus will distort more than
will the non-perforated ground.
An interesting presentation of these data for rectangular structures is shown in Figures
34 and 35, where the closed-form solutions obtained for the normalized circular lining
deflections (Figure 15 in Chapter 4) are superimposed. Note that the definitions of
flexibility ratio, F, are different.
• For circular tunnels, Equation 4-6 is used.
• For rectangular tunnels, Equation 5-5, 5-6 or 5-7, as appropriate, is used.
114