Seismic Design of Tunnels - Parsons Brinckerhoff

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Maximum Bending Moment, Mmax. The bending moment resulting from curvature deformations is maximized when a shear wave is traveling parallel to the tunnel axis (i.e., with an angle of incidence equal to zero). The mathematical expression of the maximum bending moment is: M max = Ê L ˆ2 K Á ˜ t Á ˜ Ë2p¯ Ê ˆÊ L ˆ ËE c I c ¯Ë2p ¯ 1+ Kt 4 D (Eq. 3-3) where L, E c and D are as defined in Equation 3-1 I c = moment of inertia of the tunnel section K t = transverse spring coefficient of medium (in force per unit deformation per unit length of tunnel). Maximum Shear Force, Vmax. The maximum shear force corresponding to the maximum bending moment is derived as: V max = K t L 2p Ê ˆÊ L ˆ D = M 2p 4 max L ËE c I c ¯Ë2p ¯ 1+ Kt (Eq. 3-4) where L, E c , I c , K t and D are as defined in Equation 3-3. Comments on the Interaction Equations • The tunnel-ground interaction effect is explicitly accounted for in these formulations. The ground stiffness and the tunnel stiffness are represented by spring coefficients (K a or K t ) and sectional modulus (E c A c or E c I c ), respectively. • The application of these equations is necessary only when tunnel structures are built in soft ground. For structures in rock or stiff soils, the evaluation based on the freefield ground deformation approach presented in Section 3.3 will, in general, be satisfactory. • Equations 3-1, 3-3 and 3-4 are general mathematical forms. Other expressions of the maximum sectional forces exist in the literature. The differences are primarily due to the further maximization of the sectional forces with respect to the wavelength, L. For instance: 40
- In the JSCE (Japanese Society of Civil Engineers) Specifications for Earthquake Resistant Design of Submerged Tunnels, the values of wavelength that will maximize Equations 3-1, 3-3 and 3-4 are determined and substituted back into each respective equation to yield the maximum sectional forces. - St. John and Zahran (1987) suggested a maximization scheme that is similar to the Japanese approach except that the spring coefficients (K a or K t ) are assumed to be functions of wavelength, L, in the maximization process. Both of these approaches assume that the free-field ground displacement response amplitude, D, is independent of the wavelength. This assumption sometimes may lead to very conservative results, as the ground displacement response amplitude generally decreases with the wavelength. It is, therefore, the author’s view that Equations 3-1 through 3-4 presented in this section will provide a practical and adequate assessment, provided that the values (or the ranges of the values) of L, D, and K t (or K a ) can be reasonably estimated. A reasonable estimate of the wavelength can be obtained by L = T C s (Eq. 3-5) where T is the predominant natural period of the shear wave traveling in the soil deposit in which the tunnel is built, and C s is the shear wave propagation velocity within the soil deposit. Often, T can also be represented by the natural period of the site. Dobry, Oweis and Urzua (1976) presented some procedures for estimating the natural period of a linear or equivalent linear model of a soil site. • The ground displacement response amplitude, D, should be derived based on sitespecific subsurface conditions by earthquake engineers. The displacement amplitude represents the spatial variations of ground motions along a horizontal alignment. Generally, the displacement amplitude increases as the wavelength, L, increases. For example, the displacement spectrum chart prepared by Housner (SFBARTD, 1960) for the SF BART project was expressed by D = 4.9 x 10 -6 L 1.4 , where the units of D and L are in feet. This spectrum is intended for tunnel tubes in soft San Francisco Bay muds and was derived for a magnitude 8.2 earthquake on the San Andreas fault. The equation shows clearly that: - The displacement amplitude increases with the wavelength. - For any reasonably given wavelength, the corresponding ground displacement 41
Page 1 and 2: 1991 William Barclay Parsons Fellow
Page 3 and 4: CONTENTS Foreword ix 1.0 Introducti
Page 5 and 6: Mononobe-Okabe Method 87 Wood Metho
Page 7 and 8: Figure Title Page 20 Typical Free-F
Page 9 and 10: LIST OF TABLES Table Title Page 1 F
Page 11 and 12: FOREWORD For more than a century, P
Page 13: 1.0 INTRODUCTION 1
Page 16 and 17: 1.2 Scope of this Study The work pe
Page 18 and 19: Figure 1. Ground Response to Seismi
Page 20 and 21: Ground Failure Ground failure broad
Page 22 and 23: - Formation of plastic hinges at th
Page 24 and 25: • The effects of overburden depth
Page 27 and 28: 2.0 SEISMIC DESIGN PHILOSOPHY FOR T
Page 29 and 30: 2.3 Seismic Design Philosophies for
Page 31 and 32: 2.4 Proposed Seismic Design Philoso
Page 33 and 34: expressed in terms of internal mome
Page 35: Comments on Loading Combinations fo
Page 39 and 40: 3.0 RUNNING LINE TUNNEL DESIGN 3.1
Page 41 and 42: Ovaling or Racking Deformations The
Page 43 and 44: 3.3 Free-Field Axial and Curvature
Page 45 and 46: Simplified Equations for Axial Stra
Page 47 and 48: 3.4 Design Conforming to Free-Field
Page 49 and 50: Applicability of the Free-Field Def
Page 51: Figure 6. Sectional Forces Due to C
Page 55 and 56: Design Example 2: A Linear Tunnel i
Page 57 and 58: 5. Derive the ground displacement a
Page 59 and 60: 10. Calculate the allowable shear s
Page 61 and 62: It is believed that the only transp
Page 63: Based on Equations 3-7 and 3-8, the
Page 67 and 68: 4.0 OVALING EFFECT ON CIRCULAR TUNN
Page 69 and 70: Figure 7. Free-Field Shear Distorti
Page 71 and 72: Figure 8. Free-Field Shear Distorti
Page 73 and 74: R = radius of the tunnel lining t =
Page 75 and 76: Lining Properties Soil Properties R
Page 77 and 78: The expressions of these lining res
Page 79 and 80: Figure 11. Lining Response Coeffici
Page 81 and 82: Thrust Response Coefficient, K 2 Fi
Page 83 and 84: Thrust Response Coefficient, K 2 Fi
Page 85 and 86: Figure 15. Normalized Lining Deflec
Page 87 and 88: Figure 17. Finite Difference Mesh (
Page 89 and 90: Table 2. Cases Analyzed by Finite D
Page 91 and 92: maximum bending moment than the no-
Page 93 and 94: Table 3. Influence of Interface Con
Page 95: 5.0 RACKING EFFECT ON RECTANGULAR T
Page 98 and 99: Third, typically soil is backfilled
Page 100 and 101: thrust that is approximately 1.5 to
Page 102 and 103:
San Francisco BART In his pioneerin
Page 104 and 105:
general, flexibility can be achieve
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• 254 ft/sec for case I • 415 f
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locations of roof and invert are ap
Page 110 and 111:
Figure 25. Structure Deformations v
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Factors Contributing to the Soil-St
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As Figures 28A and 28B show, earthq
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Figure 28A. West Coast Earthquake A
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Figure 29. Design Response Spectra
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Figure 31. Relative Stiffness Betwe
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developed for a one-barrel frame wi
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• For flexibility ratios less tha
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where g s = angular distortion of t
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Structure Deformation Free-Field De
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Because the Poisson’s Ratios of t
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Table 5. Cases Analyzed to Study th
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In comparison with the results show
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Structure Deformation Free-Field De
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• Pseudo-Concentrated Force Model
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deformations result. Section 2.4 in
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Figure 40. Moments at Invert-Wall C
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Figure 42. Moments at Invert-Wall C
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Table 7. Seismic Racking Design App
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136
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• The more frequently occurring e
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Use of this method will lead to a c
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142
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Duddeck, H. and Erdman, J., “Stru
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Tunnels, Report No. UMTA-MA-06-0100
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Seismic Design of Tunnels - Parsons Brinckerhoff

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