Seismic Design of Tunnels - Parsons Brinckerhoff

The shear (or flexural) stiffness of the soil element is taken as the ratio of the shear
stress to the corresponding angular distortion as expressed by:
t
g =
t
(Eq. 5-2)
D / H = G 109
When the rectangular frame structure is subjected to the same shear stress, t, the
stress can be converted into a concentrated force, P, by multiplying the shear stress by
the width of the structure (P= tL). The resulting expression for the angular distortion of the
structure becomes:
g= D H = P
HS 1
= tL
HS 1
(Eq. 5-3)
where S 1 = the force required to cause an unit racking
deflection of the structure
The flexural (or, racking) stiffness of the structure is, therefore, given by:
t
g =
t
D / H = S1 H
L
(Eq. 5-4)
The flexibility ratio, F, is obtained by dividing Equation 5-2 by Equation 5-4. The
resulting expression is:
F = GL
S 1 H
(Eq. 5-5)
In the expression above, the unit racking stiffness, S 1 , is simply the reciprocal of lateral
racking deflection, S 1 =1/D 1 caused by a unit concentrated force (i.e., p=1 in Figure 32A).
For a rectangular frame with arbitrary configuration, the flexibility ratio can be determined
by performing a simple frame analysis using conventional frame analysis programs such
as STAAD-III (see Figure 32A). Additional effort required to perform this type of analysis
should be minimal as most of the computer input is readily established for static design.
Special Case 1. For some of the simple one-barrel frames (Figure 32B), it is possible to
derive the flexibility ratio without resorting to computer analysis. The expression of F