Seismic Design of Tunnels - Parsons Brinckerhoff

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Figure 18. Influence of Interface Condition on Bending Moment 78
maximum bending moment than the no-slip interface condition. The differences are within approximately 20 percent under seismic shear loading condition. It should be realized, however, that these results are based on pseudo-static solutions that do not consider the potential dynamic amplification and stress concentrations at the tunnel excavation boundary (Mow and Pao, 1971). Previous studies suggest that a true dynamic solution would yield results that are 10 to 15 percent greater than an equivalent static solution, provided that the seismic wavelength is at least about 8 times greater than the width of the excavation (cavity). Therefore, it is prudent to adopt the more conservative full-slip assumption for the calculation of bending moments. With this more conservative assumption, the effects of stress amplification need not be considered. Maximum Lining Deflection, D D lining . Figure 19 presents a plot of the maximum lining deflections from full-slip closed form solution versus those from no-slip finite difference analysis (noting that these lining deflections are normalized with respect to the free-field ground deflections). Similar to the discussion presented above, lining tends to oval (distort) more under the full-slip interface assumption. The differences, however, are very small. The full-slip assumption (Equation 4-10 or Equation 4-13) is recommended for calculating the lining distortion. The effects of stress amplification need not be considered when the conservative full-slip assumption is adopted. It is interesting to note from the plot that almost no difference exists between the two assumptions for Case No. 12. This can be explained by the fact that a nearly “perfectly flexible” lining is used and little lining-ground interaction is involved in the Case No.12 analysis. Maximum Lining Thrust, T max . For comparison, the maximum lining thrusts are calculated using closed form solutions for both assumptions (Equations 4-8 and 4-12). The results, along with those from the finite difference analysis, are tabulated in Table 3. The table shows excellent agreement on the thrust response between the numerical finite difference analysis and the closed form solution for the no-slip condition. It also verified that the full-slip assumption will lead to significant underestimation of the lining thrust under seismic shear condition. Therefore, it is recommended that Equation 4-12 be used for thrust calculation. To account for the dynamic stress amplification due to the opening, it is further recommended that thrusts calculated from Equation 4-12 be multiplied by a factor of 1.15 for design purpose. 79
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1991 William Barclay Parsons Fellow
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CONTENTS Foreword ix 1.0 Introducti
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Mononobe-Okabe Method 87 Wood Metho
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Figure Title Page 20 Typical Free-F
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LIST OF TABLES Table Title Page 1 F
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FOREWORD For more than a century, P
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1.0 INTRODUCTION 1
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1.2 Scope of this Study The work pe
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Figure 1. Ground Response to Seismi
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Ground Failure Ground failure broad
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- Formation of plastic hinges at th
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• The effects of overburden depth
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2.0 SEISMIC DESIGN PHILOSOPHY FOR T
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2.3 Seismic Design Philosophies for
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2.4 Proposed Seismic Design Philoso
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expressed in terms of internal mome
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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 and 52: Figure 6. Sectional Forces Due to C
Page 53 and 54: - In the JSCE (Japanese Society of
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: Table 2. Cases Analyzed by Finite D
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
Page 106 and 107: • 254 ft/sec for case I • 415 f
Page 108 and 109: locations of roof and invert are ap
Page 110 and 111: Figure 25. Structure Deformations v
Page 112 and 113: Factors Contributing to the Soil-St
Page 114 and 115: As Figures 28A and 28B show, earthq
Page 116 and 117: Figure 28A. West Coast Earthquake A
Page 118 and 119: Figure 29. Design Response Spectra
Page 120 and 121: Figure 31. Relative Stiffness Betwe
Page 122 and 123: developed for a one-barrel frame wi
Page 124 and 125: • For flexibility ratios less tha
Page 126 and 127: where g s = angular distortion of t
Page 128 and 129: Structure Deformation Free-Field De
Page 130 and 131: Because the Poisson’s Ratios of t
Page 132 and 133: Table 5. Cases Analyzed to Study th
Page 134 and 135: In comparison with the results show
Page 136 and 137: Structure Deformation Free-Field De
Page 138 and 139: • 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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