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Technical Manual for Design and Construction of Road Tunnels - Civil Elements
Chapter 13 - Seismic Considerations
13.5 Seismic Evaluation Procedures - Ground Shaking Effects
Underground tunnel structures undergo three primary modes of deformation during seismic shaking: ovaling/racking, axial and curvature deformations. The ovaling/racking deformation is caused primarily by seismic waves propagating perpendicular to the tunnel longitudinal axis, causing deformations in the plane of the tunnel cross section (Refer to Figure 13-3 , Wang, 1993; Owen and Scholl, 1981). Vertically propagating shear waves are generally considered the most critical type of waves for this mode of deformation. The axial and curvature deformations are induced by components of seismic waves that propagate along the longitudinal axis (Refer to Figure 13-14 , Wang, 1993; Owen and Scholl, 1981).
Figure 13-13 Tunnel Transverse Ovaling and Racking Response to Vertically Propagating Shear Waves
Figure 13-14 Tunnel Longitudinal Axial and Curvature Response to Traveling Waves
13.5.1 Evaluation of Transverse Ovaling/Racking Response of Tunnel Structures
The evaluation procedures for transverse response of tunnel structures can be based on either (1) simplified analytical method, or (2) more complex numerical modeling approach, depending on the degree of complexity of the soil-structure system, subsurface conditions, the seismic hazard level, and the importance of the structures. The numerical modeling approach should be considered in cases where simplified analysis methods are less applicable, more uncertain, or inconclusive, or where a very important structure is located in a severe seismic environment or where case history data indicate relatively higher seismic vulnerability for the type of tunnel, such as rectangular cut-and-cover tunnels in seismically active areas. The numerical modeling approach is further discussed in Section 188.8.131.52.
184.108.40.206 Simplified Procedure for Ovaling Response of Circular Tunnels
This section provides methods for quantifying the seismic ovaling effect on circular tunnel linings. The conventionally used simplified free-field deformation method, discussed first, ignores the soil-structure interaction effects. Therefore its use is limited to conditions where the tunnel structures can be reasonably assumed to deform according to the free-field displacements during earthquakes.
A refined method is then presented in Section 220.127.116.11 that is equally simple but capable of eliminating the drawbacks associated with the free-field deformation method. This refined method - built from a theory that is familiar to most mining/underground engineers - considers the soil-structure interaction effects. Based on this method, a series of design charts are developed to facilitate the design process.
Ovaling Effect: As mentioned earlier, ovaling of a circular tunnel lining is primarily caused by seismic waves propagating in planes perpendicular to the tunnel axis. The results are cycles of additional stress concentrations with alternating compressive and tensile stresses in the tunnel lining. These dynamic stresses are superimposed on the existing static state of stress in the lining. Several critical modes may result (Owen and Scholl, 1981):
Free-Field Shear Deformations: As mentioned previously, the shear distortion of ground caused by vertically propagating shear waves is probably the most critical and predominant mode of seismic motions. It causes a circular tunnel to oval and a rectangular underground structure to rack (sideways motion), as shown in Figure 13-13 . Analytical procedures by numerical methods are often required to arrive at a reasonable estimate of the free-field shear distortion, particularly for a soil site with variable stratigraphy. Many computer codes with variable degree of sophistication are available (e.g., SHAKE, FLUSH, FLAC, PLAXIS, et al.). The most widely used approach is to simplify the site geology into a horizontally layered system and to derive a solution using one-dimensional wave propagation theory (Schnabel, Lysmer, and Seed, 1972). The resulting free-field shear distortion of the ground from this type of analysis can be expressed as a shear strain distribution or shear deformation profile versus depth.
For a deep tunnel located in relatively homogeneous soil or rock and in the absence of detailed site response analyses, the simplified procedure by Newmark (1968) and Hendron (1985) may provide a reasonable estimate, noting, however, that this method tends to produce more conservative results particularly when the effect of ground motion attenuation with depth (refer to Table 13-1) is ignored. Here, the maximum free-field shear strain, γmax, can be expressed as
VS = Peak particle velocity
The effective shear wave velocity of the vertically propagating shear wave, Cse, should be compatible with the level of the shear strain that may develop in the ground at the elevation of the tunnel under the design earthquake shaking. The values of Cse can be estimated by making proper reduction (to account for the strain-level dependent effect) from the small-strain shear wave velocity, Cs, obtained from in-situ testing (such as using the cross-hole, down-hole, and P-S logging techniques). For rock, the ratio of Cse/Cs can be assumed equal to 1.0. For stiff to very stiff soil, Cse/Cs may range from 0.6 to 0.9. Alternatively, site specific response analyses can be performed for estimating Cse. Site specific response analyses should be performed for estimating Cse for tunnels embedded in soft soils
An equation relating the effective propagation velocity of shear waves to effective shear modulus, Gm, is expressed as:
ρ = Mass density of the ground
An alternative simplified method for calculating the free-field ground shear strain, γmax, is by dividing the earthquake-induced shear stresses (τmax) by the shear stiffness (i.e., the strain-compatible effective shear modulus, Gm). This method is especially suitable for tunnels with shallow burial depths.
In this simplified method the maximum free-field ground shear strain is calculated using the following equation:
Gm = Effective strain-compatible shear modulus of ground surrounding tunnel (ksf)
τmax = Maximum earthquake-induced shear stress (ksf)
σv = Total vertical soil overburden pressure at invert elevation of tunnel (ksf)
γt = Total soil unit weight (kcf)
H = Soil cover thickness measured from ground surface to tunnel crown (ft)
D = Height of tunnel (or diameter of circular tunnel) (ft)
Rd = Depth dependent stress reduction factor; can be estimated using the following relationships:
Rd = 1.0 - 0.00233z for z < 30 ft
Rd = 1.174 - 0.00814z for 30 ft < z < 75 ft
Rd = 0.744 - 0.00244z for 75 ft < z < 100 ft
Rd = 0.5 for z > 100 ft
z = the depth (ft) from ground surface to the invert elevation of the tunnel and is represented by z = (H+D).
Lining Conforming to Free-Field Shear Deformations: When a circular lining is assumed to oval in accordance with the deformations imposed by the surrounding ground (e.g., shear), the lining's transverse sectional stiffness is completely ignored. This assumption is probably reasonable for most circular tunnels in rock and in stiff soils, because the lining stiffness against distortion is low compared with that of the surrounding medium. Depending on the definition of "ground deformation of surrounding medium," however, a design based on this assumption may be overly conservative in some cases and non-conservative in others. This will be discussed further below.
Shear distortion of the surrounding ground, for this discussion, can be defined in two ways. If the non-perforated ground in the free-field is used to derive the shear distortion surrounding the tunnel lining, the lining is to be designed to conform to the maximum diameter change, ΔDfree-field, shown in the top of Figure 13-15.
Figure 13-15 Shear Distortion of Ground - Free-Field Condition vs Cavity In-Place Condition
The maximum diametric change of the lining for this case can be derived as:
D = the diameter of the tunnel
On the other hand, if the ground deformation is derived by assuming the presence of a cavity due to tunnel excavation (bottom of Figure 13-15, for perforated ground), then the lining is to be designed according to the diametric strain expressed as:
νm = the Poisson's Ratio of the medium
Equations 13-8 and 13-9 both assume the absence of the lining. In other words, tunnel-ground interaction is ignored.
Comparison between Equations 13-8 and 13-9 shows that the perforated ground deformation would yield a much greater distortion than the free-field case (non-perforated ground). For a typical ground medium, the difference could be as much as three times. Based on the assumptions made, some preliminary conclusions can be drawn as follows:
Based on the discussions above, it can be further suggested that a lining with a greater distortion stiffness than the surrounding medium should experience a lining distortion even less than the free-field deformation. This latest case may occur when a tunnel is built in soft to very soft soils. It is therefore clear that the relative stiffness between the tunnel and the surrounding ground (i.e., soil-structure interaction effect) plays an important role in quantifying tunnel response during the seismic loading condition. This effect will be discussed next.
Importance of Lining Stiffness- Compressibility and Flexibility Ratios: To quantify the relative stiffness between a circular lining and the medium, two ratios designated as the compressibility ratio, C, and the flexibility ratio, F (Hoeg, 1968, and Peck et al., 1972) are defined by the following equations:
Em = Strain-compatible elastic modulus of the surrounding ground
υm = Poisson's ratio of the surrounding ground
Rl = Nominal radius of the tunnel lining
υl = Poisson's ratio of the tunnel Lining
Il,1 = Moment of inertia of lining per unit width of tunnel along the tunnel axis.
tl = The thickness of the lining
Of these two ratios, it often has been suggested that the flexibility ratio is the more important because it is related to the ability of the lining to resist distortion imposed by the ground. As will be discussed later, the compressibility ratio also has a significant effect on the lining thrust response.
For most circular tunnels encountered in practice, the flexibility ratio, F, is likely to be large enough (say, F>20) so that the tunnel-ground interaction effect can be ignored (Peck, 1972). It is to be noted that F > 20 suggests that the ground is about 20 times stiffer than the lining. In these cases, the distortions to be experienced by the lining can be reasonably assumed to be equal to those of the perforated ground (i.e., ΔDcavity).
This rule of thumb procedure may present some design problems when a very stiff structure is surrounded by a very soft soil. A typical example would be to construct a very stiff immersed tube in a soft lake or river bed deposit. In this case the flexibility ratio is very low, and the stiff tunnel lining could not be realistically designed to conform to the deformations imposed by the soft ground. The tunnel-ground interaction effect must be considered in this case to achieve a more efficient design.
In the following section a refined procedure taking into account the tunnel-ground interaction effect is presented to provide a more accurate assessment of the seismic ovaling effect on a circular lining.
18.104.22.168 Analytical Lining-Ground Interaction Solutions for Ovaling Response of Circular Tunnels
Closed form analytical solutions have been proposed (Wang, 1993) for estimating ground-structure interaction for circular tunnels under the seismic loading conditions. These solutions are generally based on the assumptions that:
The expressions of these lining responses are functions of flexibility ratio and compressibility ratio as presented previously in Equations 13-10 and 13-11 . The expressions for maximum thrust, Tmax, bending moment, Mmax, and diametric strain, ΔD/D, can be presented in the following forms:
K1 and K2 are defined herein as lining response coefficients. The earthquake loading parameter is represented by the maximum shear strain induced in the ground (free-field), γmax, which may be obtained through a simplified approach (such as Equation 13-15 or 13-16 ), or by performing a site-response analysis.
The resulting bending moment induced maximum fiber strain, εm, and the axial force (i.e., thrust) induced strain, εT, can be derived as follows:
To ease the design process, Figure 13-16 shows the lining response coefficient, K1, as a function of flexibility ratio and Poisson's Ratio of the ground. The design charts showing the lining coefficient K2, primarily used for the thrust response evaluation, are presented in Figure 13-17 , Figure 13-18 , and Figure Figure 13-19 for Poisson's Ratio values of 0.2, 0.35 and 0.5, respectively.
Figure 13-16 Lining Response Coefficient, K1 (Full-Slip Interface Condition)
Figure 13-17 Lining Response Coefficient, K2, for Poisson's Ratio = 0.2 (No-Slip Interface Condition)
Figure 13-18 Lining Response Coefficient, K2, for Poisson's Ratio = 0.35 (No-Slip Interface Condition)
Figure 13-19 Lining Response Coefficient, K2, for Poisson's Ratio = 0.5 (No-Slip Interface Condition)
It should be noted that the solutions in terms of Mmax, ΔDmax, and εm provided herein are based on the full-slip interface assumption. For the maximum thrust response Tmax the interface conditions is assumed to be no-slip. These assumptions were adopted because full-slip condition produces more conservative results for Mmax and ΔDmax, while no-slip condition is more conservative for Tmax. During an earthquake, in general, slip at interface is a possibility only for tunnels in soft soils, or when seismic loading intensity is severe. For most tunnels, the condition at the interface is between full-slip and no-slip. In computing the forces and deformations in the lining, it is prudent to investigate both cases and the more critical one should be used in design.
The conservatism described above is desirable to offset the potential underestimation of lining forces resulting from the use of equivalent static model in lieu of the dynamic loading condition. 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, the full-slip model is recommended in evaluating the moment and deflection response (i.e., Figure 13-16 and Equation 13-15 ) of a circular tunnel lining.
Using the full-slip condition, however, would significantly underestimate the maximum thrust, Tmax, under the seismic simple shear condition. Therefore, it is recommended that the no-slip interface assumption be used in assessing the lining thrust response (Equation13-16 ).
Effective Lining Stiffness: The results presented above are based on the assumption that the lining is a monolithic and continuous circular ring with intact, elastic properties. Many circular tunnels are constructed with bolted or unbolted segmental lining. Besides, a concrete lining subjected to bending and thrust often cracks and behaves in a nonlinear fashion. Therefore, in applying the results presented herewith, the effective (or, equivalent) stiffness of the lining should be used. Some simple and approximate methods accounting for the effect of joints on lining stiffness can be found in the literature.
Ie < I and n > 4
22.214.171.124 Analytical Lining-Ground Interaction Solutions for Racking Response of Rectangular Tunnels
General: Shallow depth transportation tunnels are often of rectangular shape and are often built using the cut-and-cover method. Usually the tunnel is designed as a rigid frame box structure. From the seismic design standpoint, these box structures have some characteristics that are different from those of the bored circular tunnels, besides the geometrical aspects. The implications of three of these characteristics for seismic design are discussed below.
First, cut-and-cover tunnels are generally built at shallow depths in soils where seismic ground deformations and the shaking intensity tend to be greater than at deeper locations, due to the lower stiffness of the soils and the site amplification effect. As discussed earlier, past tunnel performance data suggest that tunnels built with shallow soil overburden cover tend to be more vulnerable to earthquakes than deep ones.
Second, a box frame usually does not transmit the static loads as efficiently as a circular lining, resulting in much thicker walls and slabs for the box frame. As a result, a rectangular tunnel structure is usually stiffer than a circular tunnel lining in the transverse direction and less tolerant to distortion. This characteristic, along with the potential large seismic ground deformations that are typical for shallow soil deposits, makes the soil-structure interaction effect particularly important for the seismic design of cut-and-cover rectangular tunnels, including those built with the sunken/immersed tube method.
Third, typically soil is backfilled above the structure and possibly between the in-situ medium and the structure. Often, the backfill soil may consist of compacted material having different properties than the in-situ soil. The properties of the backfill soil as well as the in-situ medium should be properly accounted for in the design and analysis. The effect of backfill, however, cannot be accounted for using analytical closed-form solutions. Instead, more complex numerical analysis is required for solving this problem if the effect of backfill is considered significant in evaluating seismic response of a cut-and-cover tunnel.
The evaluation procedures presented in this section are based on simplified analytical method. The more refined numerical modeling approach is discussed in Section 126.96.36.199.
Racking Effect: During earthquakes a rectangular box structure in soil or in rock will experience transverse racking deformations (sideways motion) due to the shear distortions of the ground, in a manner similar to the ovaling of a circular tunnel discussed in Section 188.8.131.52. The racking effect on the structure is similar to that of an unbalanced loading condition.
The external forces the structure is subjected to are in the form of shear stresses and normal pressures all around the exterior surfaces of the box. The magnitude and distribution of these external earth forces are complex and difficult to assess. The end results, however, are cycles of additional internal forces and stresses with alternating direction in the structure members. These dynamic forces and stresses are superimposed on the existing static state of stress in the structure members. For rigid frame box structures, the most critical mode of potential damage due to the racking effect is the distress at the top and bottom joints (refer to Figure 13-1 , Figure 13-11 , Figure 13-12 and Figure 13-13 ).
Realizing that the overall effect of the seismically induced external earth loading is to cause the structure to rack, it is more reasonable to approach the problem by specifying the loading in terms of deformations. The structure design goal, therefore, is to ensure that the structure can adequately absorb the imposed racking deformation (i.e., the deformation method), rather than using a criterion of resisting a specified dynamic earth pressure (i.e., the force method). The focus of the remaining sections of this chapter, therefore, is on the method based on seismic racking deformations.
Free-Field Racking Deformation Method It has been proposed in the past that a rectangular tunnel structure be designed by assuming that the amount of racking imposed on the structure is equal to the "free-field" shear distortions of the surrounding medium, as illustrated in Figure 13-20 (i.e., Δfree-field = Δs). The racking stiffness of the structure is ignored with this assumption.
Figure 13-20 Soil Deformation Profile and Racking Deformation of a Box Structure
The free-field deformation method serves as a simple and effective design tool when the seismically induced ground distortion is small, for example when the shaking intensity is low or the ground is very stiff. Given these conditions, most practical structural configurations can easily absorb the ground distortion without being distressed. The method is also a realistic one when the racking stiffness of the structure is comparable to that of its surrounding medium.
It has been reported (Wang, 1993), however, that this simple procedure could lead to overly conservative design (i.e., when Δfree-field > Δs) or un-conservative design (i.e., when Δfree-field < Δs), depending on the relative stiffness between the ground and the structure. The overly conservative cases generally occur in soft soils. Seismically induced free-field ground distortions are generally large in soft soils, particularly when they are subjected to amplification effects. Ironically, rectangular box structures in soft soils are generally designed with stiff configurations to resist the static loads, making them less tolerant to racking distortions. Imposing free-field deformations on a structure in this situation is likely to result in unnecessary conservatism, as the stiff structure may deform less than the soft ground.
On the other hand, the un-conservative cases arise when the shear stiffness of the ground is greater than the racking stiffness of the structures - a behavior similar to that described for the ovaling of circular tunnel (Section 184.108.40.206). To more accurately quantify the racking response of rectangular tunnel structures a rational procedure accounting for the tunnel-ground interaction effect is presented in the following section.
Tunnel-Ground Interaction Analysis: Although closed-form solutions accounting for soil-structure interaction, such as those presented in Section 220.127.116.11, are available for deep circular lined tunnels, they are not readily available for rectangular tunnels due primarily to the highly variable geometrical characteristics typically associated with rectangular tunnels. Complex earthquake induced stress-strain conditions is another reason as most of the rectangular tunnels are built using the cut-and-cover method at shallow depths, where seismically induced ground distortions and stresses change significantly with depth.
To develop a simple and practical design procedure, Wang (1993) performed a series of dynamic soil-structure interaction finite element analyses. In this study, the main factors that may potentially affect the dynamic racking response of rectangular tunnel structures were investigated. These factors include:
A total number of 36 dynamic finite element analyses were carried out to account for the variables discussed above. Based on the results of the analyses, a simplified procedure incorporating soil-structure interaction for the racking analysis of rectangular tunnels was developed. The step-by-step procedure is outlined below (Wang, 1993).
Step 1: Estimate the free-field ground strains γmax (at the structure elevation) caused by the vertically propagating shear waves of the design earthquakes, see Section 18.104.22.168 in deriving the free-field ground strain using various methods. Determine Δfree-field, the differential free-field relative displacements corresponding to the top and the bottom elevations of the box structure (see Figure 13-20 ) by using the following expression:
H = height of the box structure
Alternatively site-specific site response analysis may be performed to provide a more accurate assessment of Δfree-field. Site-specific site response analysis is recommended for tunnels embedded in soft soils.
Step 2: Determine the racking stiffness, Ks, of the box structure from a structural frame analysis. The racking stiffness should be computed using the displacement of the roof subjected to a unit lateral force applied at the roof level, while the base of the structure is restrained against translation, but with the joints free to rotate. The ratio of the applied force to the resulting lateral displacement yields Ks. In performing the structural frame analysis, appropriate moment of inertia values, taking into account the potential development of cracked section, should be used.
Step 3: Determine the flexibility ratio, Fr, of the box structure using the following equation:
W = Width of the box structure
H = Height of the box structure
Gm = Average strain-compatible shear modulus of the surrounding ground between the top and bottom elevation of the structure
Ks = Racking Stiffness of the box structure
The strain-compatible shear modulus can be derived from the strain-compatible effective shear wave velocity, Cse, see Equation 13-4 ).
Detailed derivation of the flexibility ratio, Fr, is given by Wang (1993).
Step 4: Based on the flexibility ratio obtained from Step 3 above, determine the racking coefficient, Rr, for the proposed structure. The racking coefficient, Rr, is the ratio of the racking distortion of the structure embedded in the soil, Δs, to that of the free-field soil, Δfree-field, over the height of the structure (see Figure 13-20 ):
From a series of dynamic finite element analyses, Wang (1993) presented results showing the relationship between the structure racking and the flexibility ratio, Fr. The values of Rr vs. Fr obtained from the dynamic finite element analyses are shown in Figure 13-21 (a) and Figure 13-21 (b). Also shown in these figures are curves from closed-form static solutions for circular tunnels (refer to Section 22.214.171.124). The solutions shown in the figures are from the full-slip solution presented by Wang (1993) and Penzien (2000) and the no-slip solution presented by Penzien (2000). As can be seen in the figures, the curves from the closed-form solutions provide a good approximation of the finite element analysis results. These curves can therefore be used to provide a good estimate of the racking of a rectangular tunnel as a function of the flexibility ratio defined by Equation 13-21 . The analytical expressions for the curves in Figure 13-21 are:
For no-slip interface condition:
For full-slip interface condition:
Several observations can be made from Figure 13-21 . When Fr is equal to zero, the structure is perfectly rigid, no racking distortion is induced, and the structure moves as a rigid body during earthquake loading. When Fr is equal to 1, the racking distortion of the structure is approximately the same as that of the soil (exactly equal to that of the soil for the no-slip interface condition). For a structure that is flexible relative to the surrounding ground, (Fr > 1), racking distortion of the structure is greater than that of the free-field. As noted by Penzien (2000), if the structure has no stiffness (i.e., Fr → ∞), Rr is approximately equal to 4(1- νm ), which is the case of an unlined cavity.
Figure 13-21 Racking Coefficient Rr for Rectangular Tunnels (MCEER-06-SP11, Modified from Wang, 1993, and Penzien, 2000)
Step 5: Determine the racking deformation of the structure, Δs, using the following relationship:
Step 6: The seismic demand in terms of internal forces as well as material strains are calculated by imposing Δs upon the structure in a frame analysis as depicted in Figure 13-22 (MCEER-06-SP11). Results of the analysis can also be used to determine the detailing requirements.
As indicated in Figure 13-22 , two pseudo-static lateral force models are recommended. The more critical responses from the two models should be used for design. If the displacements are large enough to cause inelastic deformation of the structure, inelastic soil-structure interaction analyses should be performed to assess structural behavior and ensure adequate strength and displacement capacity of the tunnel structure.
Under the loading from the design earthquake, inelastic deformation in the structure may be allowed depending on the performance criteria and provided that overall stability of the tunnel is maintained. Detailing of the structural members and joints should provide for adequate internal strength, and ductility and energy absorption capability if inelastic deformation is anticipated.
Figure 13-22 Simplified Racking Frame Analysis of a Rectangular Tunnel
(MCEER-06-SP11, Modified from Wang, 1993)
Step 7: The effects of vertical seismic motions can be accounted for by applying a vertical pseudo-static loading, equivalent to the product of the vertical seismic coefficient and the combined dead and design overburden loads used in static design. The vertical seismic coefficient can be reasonably assumed to be two-thirds of the design peak horizontal acceleration divided by the gravity. This vertical pseudo-static loading should be applied by considering both up and down direction of motions, whichever results in a more critical load case should govern.
Step 8: Seismic demands due to racking deformations and vertical seismic motions are then combined with non-seismic loads using appropriate load combinations. A load factor of 1.0 is recommended in the load combination criteria.
126.96.36.199 Numerical Modeling Approach
The analytical solutions presented in Sections 188.8.131.52 and 184.108.40.206 for transverse response of tunnel structures (i.e., ovaling for circular tunnels and racking for rectangular tunnels) have been developed based on ideal conditions and assumptions as follows:
The actual soil-structure system encountered in the field for underground structures are more complex than the ideal conditions described above and may require the use of numerical methods. This is particularly true in cases where a very important tunnel structure is located in a severe seismic environment.
For transverse ovaling/racking analysis, two-dimensional finite element or finite difference continuum method of analysis is generally considered adequate numerical modeling approach. The model needs to be developed with the capability of capturing SSI effects as well as appropriate depth-variable representations of the earth medium and the associated free-field motions (or ground deformations) obtained from site-response analyses of representative soil profiles.
There are three types of two-dimensional continuum method of analysis that have been used in engineering practice and they are described in the following sections.
Pseudo-Static Seismic Coefficient Deformation Method: In pseudo-static seismic coefficient deformation method, the ground deformations are generated (induced) by seismic coefficients and distributed in the finite element/finite difference domain that is being analyzed. The seismic coefficients can be derived from a separate one-dimensional, free-field site response analysis.
The pseudo-static seismic coefficient deformation method is suitable for underground structures buried at shallow depths. The general procedure in using this method is outlined below:
Pseudo-Dynamic Time-History Analysis The procedure employed in pseudo-dynamic analysis is similar to that for the pseudo-static seismic coefficient deformation method, except that the derivation of the ground displacements and the manner in which the displacements are imposed to the two dimension continuum system are different. The pseudo-dynamic analysis consists of stepping the soil-structure system statically through displacement time-history simulations of free-field displacements obtained by a site response analysis performed using vertically propagating shear waves (e.g., SHAKE analyses). Under the pseudo-dynamic loading, the transverse section of a tunnel structure will be subject to these induced ground distortions. Figure 13-23 shows an example of a two-dimensional continuum finite element analysis performed for an immersed tube tunnel structure subject to static stepping of a pseudo-dynamic displacement time history. In this model both the geologic medium (e.g., soil) and the tunnel structure were modeled as continuum solid elements. As indicated in the figure, in addition to the natural in-situ soils, the model can also consider the effect of the backfill material (within the dredged trench) on the ovaling/racking response of the tunnel structure. If warranted, the inelastic behavior of the tunnel structure can also be accounted for and incorporated into the model.
Figure 13-23 Example of Two-dimensional Continuum Finite Element Model in Pseudo-Dynamic Displacement Time-History Analysis
The model shown in Figure 13-23 includes both the geologic medium and the structure in one model. Alternatively, the analysis can also be performed in a de-coupled manner, where the tunnel structure is analyzed separately from the surrounding geologic medium. This de-coupled analysis involves the following two general steps:
Dynamic Time History Analysis: Generally, the inertia of a tunnel is small compared to that of the surrounding geologic medium. Therefore, it is reasonable to perform the tunnel deformation analysis using pseudo-static or pseudo-dynamic analysis in which displacements or displacement time histories are statically applied to the soil-structure system. The dynamic time history analysis can be used to further refine the analysis when necessary, particularly when some portion(s) of the tunnel structure can respond dynamically under earthquake loading, i.e., in the case where the inertial effect of the tunnel structure is considered to be significant.
In a dynamic time history analysis, the entire soil-structure system is subject to dynamic excitations using ground motion time histories as input at the base of the soil-structure system. The ground motion time histories used for this purpose should be developed to match the target design response spectra and have characteristics that are representative of the seismic environment of the site and the site conditions (refer to Section 13.2.3).
Figure 13-24 shows a sample dynamic time history analysis using a two-dimensional continuum finite difference model for a cut-and-cover box structure. It should be noted in the figure that, the side boundary conditions in a dynamic time history analysis should be in such a manner that out-going seismic waves be allowed to pass through instead of being trapped within the soil-structure system being analyzed. Special energy absorbing boundaries should be incorporated into the model to allow radiation of the seismic energy rather than trapping it.
Figure 13-24 Sample Dynamic Time History Analysis Model
13.5.2 Evaluation of Longitudinal Response of Tunnel Structures
Similar to the procedures discussed for the evaluation of transverse response of tunnel structures, the evaluation procedures for the longitudinal response of tunnel structures can also be based on either simplified analytical method or more complex numerical modeling approach, depending on the degree of complexity of the soil-structure system, the seismic hazard level, and the importance of the structures. Section 220.127.116.11 discusses the simplified free-field deformation method, which ignores the soil-structure interaction effects. A refined method is then presented in Section 18.104.22.168 that considers the soil-structure interaction effects based on analytical beam-on-elastic-foundation theory. The more comprehensive and complex method using numerical modeling approach is discussed in Section 22.214.171.124.
126.96.36.199 Free-field Deformation Procedure
This procedure assumes that the tunnel lining conforms to the axial and curvature deformations of the ground in the free-field (i.e., without the presence of the tunnel). While conservative, this assumption provides a reasonable evaluation because, in most cases, the tunnel lining stiffness is considered relatively flexible to that of the ground. This procedure requires minimum input, making it useful as an initial design tool and as a method of design verification.
The lining will develop axial and bending strains to accommodate the axial and curvature deformations imposed by the surrounding ground. St. John and Zahran (1987) developed solutions for these strains due to compression P-waves, shear S-waves, and Rayleigh R-waves.
The strains ε due to combined axial and curvature deformations can be obtained by combining the longitudinal strains generated by axial and bending strains as follows:
It should be noted that:
The horizontal propagation S-wave velocity, CS, in general, reflects the seismic shear wave propagation through the deeper rocks rather than that of the shallower soils where the tunnel is located. In general, this velocity value varies from about 2 to 4 km/sec. Similarly, the P-wave propagation velocities, CP, generally vary between 4 and 8 km/sec. The designer should consult with experienced geologists/seismologists for determining CS and CP. In the absence of site-specific data, the horizontal propagation S-wave and P-wave velocities can be assumed to be 2.5 km/sec and 5 km/sec, respectively.
When the tunnel is located at a site underlain by deep deposits of soil sediments, the induced strains may be governed by the R-waves. In such deposits, detailed geological/seismological analyses should be performed to derive a reliable estimate of the apparent R-wave propagation velocity, CR.
The combined strains calculated from Equations 13-26 , 13-27 , and 13-28 represent the seismic loading effect only. To evaluate the adequacy of the structure under the seismic loading condition, the seismic loading component has to be added to the static loading components using appropriated loading combination criteria developed for the structures. The resulting combined strains are then compared against the allowable strain limits, which should be developed based on the performance goal established for the structures (e.g., the required service level and acceptable damage level).
188.8.131.52 Procedure Accounting for Soil-Structure Interaction Effects
If a very stiff tunnel is embedded in a soft soil deposit, significant soil-structure interaction effects exist, and the free-field deformation procedure presented above may lead to an overly conservative design. In this case, a simplified beam-on-elastic-foundation procedure should be used to account for the soil-structure interaction effects. According to St. John and Zahran (1987), the effects of soil-structure interaction can be accounted for by applying reduction factors to the free-field axial strains and the free-field curvature strains, as follows:
For axial strains:
For bending strains:
It should be noted that the axial strain calculated using the procedure presented above should not exceed the value that could be developed using the maximum frictional forces, Qmax, between the lining and the surrounding soils. Qmax can be estimated using the following expression:
f = Maximum frictional force per unit length of the tunnel
184.108.40.206 Numerical Modeling Approach
Numerical modeling approach for the evaluation of longitudinal response of a tunnel structure is desirable for cases where tunnels encounter abrupt changes in structural stiffness or run through highly variable subsurface conditions (where the effect of spatially varying ground motions due to local site effect becomes significant). These conditions include, but are not limited to, the following:
Numerical analysis for the evaluation of longitudinal response of a tunnel structure is typically performed by a three-dimensional pseudo-dynamic time history analysis in order to capture the two primary modes of deformation: axial compression/extension and curvature deformations. As discussed previously, since the inertia of a tunnel is small compared to that of the surrounding geologic medium, the analysis is generally performed by using the pseudo-dynamic approach in which free-field displacement time histories are statically applied to soil springs connected to the model of the tunnel (to account for the soil-structure interaction effect). The general procedure for the pseudo-dynamic time history analysis in the longitudinal direction involves the following steps.
13.6 Seismic Evaluation Procedures - Ground Failure Effects
As mentioned earlier, the greatest risk to tunnel structures is the potential for large ground movements as a result of unstable ground conditions (e.g., liquefaction and landslides) or fault displacements. In general, it is not feasible to design a tunnel structure to withstand large ground displacements. The proper design measures in dealing with the unstable ground conditions may consist of:
With regard to the fault displacements, the best strategy is to avoid any potential crossing of active faults. If this is not possible, then the general design philosophy is to accept and accommodate the displacements by either employing an oversized excavation, perhaps backfilled with compressible/collapsible material, or using ductile lining to minimize the instability potential of the lining. In cases where the magnitude of the fault displacement is limited or the width of the sheared fault zone is considerable such that the displacement is dissipated gradually over a distance, design of a strong lining to resist the displacement may be technically feasible. The structures, however, may be subject to large axial, shear and bending forces. Many factors need to be considered in the evaluation, including the stiffness of the lining and the ground, the angle of the fault plane intersecting the tunnel, the width of the fault, the magnitude as well as orientation of the fault movement. Analytical procedures are generally used for evaluating the effects of fault displacement on lining response. Some of these procedures were originally developed for buried pipelines (ASCE Committee on Gas and Liquid Fuel Lifelines, 1984). Continuum finite-element or finite-difference methods have also been used effectively for evaluating the tunnel-ground-faulting interaction effects.
The following sections will discuss briefly the general considerations and methodology used in dealing with various types of ground failure effects.
13.6.1 Evaluation for Fault Rupture
General: Assessing the behavior of a tunnel that may be subject to the direct shear displacements along a fault includes, first, characterizing the free-field fault displacement (i.e., displacements in the absence of the tunnel) where the fault zone crosses the tunnel and, second, evaluating the effects of the characterized displacements on the tunnel.
Figure 13-25 is an example of such a relationship, which shows that the amount of displacement is strongly dependent on earthquake magnitude and can reach maximum values of several feet or even tens of feet for large-magnitude earthquakes.
Analyzing Tunnels for Fault Displacement: When subjected to fault differential displacements, a buried structure with shear and bending stiffness tends to resist the deformed configuration of the fault offset, which induces axial and shear forces and bending moments in the structure. The axial deformation is resisted by the frictional forces that develop at the soil-tunnel interface in the axial direction, while shear and curvature deformations are caused by the soil resistance normal to the tunnel lining or walls.
Figure 13-25 Maximum Surface Fault Displacement vs. Earthquake Moment Magnitude, Mw (Wells and Coppersmith, 1994)
In general, analytical procedures for evaluating tunnels subjected to fault displacements can follow those used for buried pipelines. Three analytical methods have been utilized in the evaluation and design of linear buried structures (ASCE Committee on Gas and Liquid Fuel Lifelines, 1984). They are: (1) Newmark-Hall procedure, (2) Kennedy et al. procedure, and (3) Finite element approach. For detailed evaluation of transportation tunnels at fault crossing, however, it is generally believed that finite element method is more appropriate than other methods. The finite element method is preferred because it can incorporate realistic models of the tunnel and surrounding geologic media. The tunnel is modeled using finite elements, which may incorporate nonlinear behavior (Figure 13-26).
Figure 13-26 Analytical Model of Tunnel at Fault Crossing (ASCE, 1984)
Transverse and axial springs connected to the tunnel model soil normal pressures on the tunnel lining or walls and axial frictional resistance (Figure 13-27); these springs may also incorporate nonlinear behavior if applicable (Figure 13-28). Many commercially available finite element codes may be considered for analyzing the response of tunnels to fault displacement.
Figure 13-27 Tunnel-Ground Interaction Model at Fault Crossing (ASCE Committee on Gas and Liquid Fuel Lifelines, 1984)
Figure 13-28 Analytical Model of Ground Restraint for Tunnel at Fault Crossing (ASCE Committee on Gas and Liquid Fuel Lifelines, 1984)
13.6.2 Evaluation for Landsliding or Liquefaction
If liquefiable soil deposits or unstable soil masses susceptible to landsliding are identified along the tunnel alignment, then more detailed evaluations may be required to assess whether liquefaction or landsliding would be expected to occur during the design earthquake and to assess impacts on the tunnel.
If slope movements due to landsliding or lateral spreading movements due to liquefaction intersect a tunnel, the potential effects of these movements on the tunnel are similar to those of fault displacement. As is the case for fault displacements, tunnels generally would not be able to resist landsliding or lateral spreading concentrated displacements larger than a few inches without experiencing locally severe damage.
If liquefaction were predicted to occur adjacent to a tunnel lining or wall, a potential consequence could be yielding of the lining or wall due to the increased lateral earth pressure in the liquefied zone. The pressure exerted by a liquefied soil may be as large as the total overburden pressure. The potential for liquefaction to cause uplift of a tunnel embedded in liquefied soil, or for the tunnel to settle into the soil, should also be checked.
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