Seismic Design of Tunnels - Parsons Brinckerhoff

developed for a one-barrel frame with equal moment of inertia, I L , for roof and invert slabs
and equal moment of inertia, I H , for side walls is given by:
F = G ÊH 2 L
24 Ë EI H
+ HL2
EI L
ˆ
¯
(Eq. 5-6)
where
E = plane strain elastic modulus of frame
G = shear modulus of soil
I L , I H = moments of inertia per unit width for slabs and walls, respectively
Note that the expressions by Equation 5-6 and Equation 5-7 that follow are valid only
for homogeneous, continuous frames with rigid connections. Reinforced framed concrete
structures are examples of this type of construction.
Special Case 2. The flexibility ratio derived for a one-barrel frame with roof slab moment
of inertia, I R , invert slab moment of inertia, I I , and side wall moment of inertia, I W , is
expressed as:
F = G ÊHL 2
12 Ë EI R
Y
ˆ
¯
(Eq. 5-7)
where
( 1+ a2)a1 + 3a2
Y= ( )2 + ( a 1+ a 2 )3a ( 2 +1) 2
( 1 + a 1 + 6a 2 ) 2
Ê
a 1 =
IR
Ë ˆ¯ and a Ê
2 = IR
Ë
ˆ¯
I I
I W
H
L
E = plane strain elastic modulus of frame
G = shear modulus of soil
I R , I I , I W = moments of inertia per unit width
Implications of Flexibility Ratios. The derivation of the flexibility ratio presented in this
section is consistent with that for the circular tunnels. The theoretical implications are:
• A flexibility ratio of 1.0 implies equal stiffness between the structure and the ground.
Thus, the structure should theoretically distort the same magnitude as estimated for
the ground in the free-field.
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