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113 ratio when the Poisson's ratio value of the surrounding poses, the racking stiffness can be obtained by applying a unit ground is less than 0.5. lateral force at the roof level, while the base of the structure is · When the Poisson's ratio approaches 0.5 (for example, for restrained against translation, but with the joints free to rotate. saturated undrained clay), the thrust response of the lining The structural racking stiffness is defined as the ratio of the is essentially independent of the compressibility ratio. applied force to the resulting lateral displacement. Step 3: Derive the flexibility ratio (Frec) of the rectangular The theoretical solutions and their applicability to typical structure using the following equation: culvert and pipeline structures is further verified for reason- ableness by numerical analysis presented in the next section. Frec = (Gm K s ) ( L H ) (9-14) 9.5.2 Racking of Rectangular Conduits where L = width of the structure; and Racking deformations are defined as the differential side- Gm = average strain-compatible shear modulus of the sur- ways movements between the top and bottom elevations of rounding ground. rectangular structures, shown as "s" in Figure 9-7. The re- sulting structural internal forces or material strains in the lin- ing associated with the seismic racking deformation (s) can The flexibility ratio is a measure of the relative racking stiff- be derived by imposing the differential deformation on the ness of the surrounding ground to the racking stiffness of the structure in a simple structural frame analysis. structure. The derivation of Frec is schematically depicted in The procedure for determining s and the corresponding Figure 9-8. structural internal forces [bending moment (M), thrust (T), Step 4: Based on the flexibility ratio obtained form Step 3 and shear (V )], taking into account the soil-structure inter- above, determine the racking ratio (Rrec) for the structure action effects, are presented below (Wang, 1993). using Figure 9-5 or the following expression: Rrec = 2 Frec (1 + Frec ) (9-15) Step 1: Estimate the free-field ground strains max (at the structure elevation) caused by the vertically propagating shear The racking ratio is defined as the ratio of actual racking waves of the design earthquakes, refer to Equation (9-1) or deformation of the structure to the free-field racking defor- Equation (9-2) and related discussions presented earlier in mation in the ground. The solid triangular data points in Fig- Section 9.4.1. Determine the differential free-field relative dis- ure 9-9 were data generated by performing a series of dynamic placements (free-field) corresponding to the top and the bottom finite element analyses on a number of cases with varying elevations of the rectangular/box structure by: soil and structural properties, structural configurations, and free-field = H max (9-13) ground motion characteristics. Note, however, these data were generated by using structural parameters representative of typ- where H is height of the structure. ical transportation tunnels during the original development Step 2: Determine the racking stiffness (Ks) of the structure of this design methodology. The validity of this design chart from a simple structural frame analysis. For practical pur- was later verified and adjusted as necessary by performing Figure 9-7. Racking deformations of a rectangular conduit.
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114 Figure 9-8. Relative stiffness of soil versus rectangular frame. Figure 9-9. Racking ratio between structure and free-field.