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109 criteria for axial tension and axial compression (local To account for the effects of transient ground deformation buckling/wrinkling and beam buckling) were developed, on tunnel structures, Wang (1993) developed closed-form there were no discussions related to the procedure for eval- and analytical solutions for the determination of seismically uating the transverse ovaling deformation of the pipe's induced ovaling/racking deformations and the corresponding cross-sectional behavior. internal forces (such as moments and thrusts) for bored as Based on the field performance of 61 corrugated metal pipes well as cut-and-cover tunnel structures. The procedure pre- (CMP) that were shaken by the 1994 Northridge Earthquake, sented by Wang for the bored tunnels was developed from a Davis and Bardet (2000) provided an updated approach to theory that is familiar to most mining/underground engineers evaluating the seismic performance of CMP conduits. The (Peck et al., 1972). Simple and easy-to-use seismic design focus of their study was on the ovaling and buckling (of charts were presented. The design charts are expressed prima- the thin metal wall) of the transverse section behavior of the rily as a function of relative stiffness between the structure and CMP. This approach involves the following general steps: the ground. Solutions for both full-slip and nonslip conditions 1. Estimate the initial condition of compressive strain in at the interface between soil and the exterior surface of the the conduit, which is related to depth of burial. tunnel lining were developed. These solutions fully account 2. Estimate the compressive ground strain induced by a ver- for the interaction of the tunnel lining with the surrounding tically propagating shear wave, which was calculated from ground. The results were validated through a series of finite the closed-form solution for transient shearing strain, as element/difference soil-structure interaction analyses. 1 /2 max = vp/2Vs, where max is the maximum transient For the cut-and-cover tunnels (with a rectangular shape), the shearing strain of the ground, vp is the horizontal peak design solutions were derived from an extensive study using particle velocity transverse to the conduit, and Vs is the dynamic finite-element, soil-structure interaction analyses. average shear wave velocity of the surrounding ground. A wide range of structural, geotechnical, and ground motion 3. Add the static and transient compressive strains. parameters were considered by Wang in his study. Specifically, 4. Compare the strain so determined with the critical com- five different types of cut-and-cover tunnel geometry were pressive strain that would cause dynamic buckling (due studied, including one-barrel, one-over-one two-barrel, and to hoop force) of the CMP pipe. The critical buckling one-by-one twin-barrel configurations. To quantify the effect of strain (or strength) was assumed to be dependent on the relative stiffness on tunnel lining response, varying ground pro- stiffness of the surrounding soil (Moore, 1989). files and soil properties were used in the parametric analyses. Based on the results of the parametric analyses, a deformation- The methodology derived by Davis and Bardet, although based design chart was developed for cut-and-cover tunnels. more rational than most of the other procedures, has some Although these solutions were intended originally for tun- drawbacks, including: nel structures (considered a fairly rigid type of structure), the methodology is rational and comprehensive and provides a The procedure is applicable for thin-walled pipes only. The consistent and unified approach to solving the problem of failure mode considered by using this procedure is prima- buried conduits subject to ground shaking regardless of rily for buckling and does not include flexural (that is, whether they are rigid or flexible structures. With some ad- bending) demand and capacity evaluation. The latter is a justments this approach also is applicable to the culvert and very important failure mode that must be considered for pipe structures typically used for highway construction. There- rigid culverts and pipes (such as those constructed with re- fore, a more detailed discussion of Wang's approach is given inforced concrete). in the following section. The soil-structure interaction effect was considered in eval- uating the buckling capacity, but not in the evaluation of 9.5 General Methodology and the demand (that is, earthquake-induced ground strains). Recommended Procedures The method assumed that the strains in the pipe coincide with those in the surrounding ground (that is, pipe de- The general methodology and recommended procedures forms in accordance with the ground deformation in the for the ovaling of circular conduits and the racking of rectan- free-field), on the basis of the assumption that there is no gular conduits developed by Wang (1993) are presented in slippage at the soil-pipe interface. This assumption was in- the following two sections, respectively. correct, as Wang (1993) pointed out in his study. Wang concluded that the strains and deformations of a buried 9.5.1 Ovaling of Circular Conduits conduit can be greater, equal, or smaller than those of the surrounding ground in the free-field, depending on the The seismic ovaling effect on the lining of a circular conduit relative stiffness of the conduit to the surrounding ground. is best defined in terms of change of the conduit diameter

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110 (DEQ) and incremental seismically induced internal forces using the earthquake-induced shearing stress and the strain- [for example, bending moment (M) and thrust/hoop force compatible shear modulus of the surrounding ground. In this (T)]. It should be noted that for flexible types of conduits approach, the expected free-field ground strain caused by the (such as thin-walled metal, corrugated or noncorrugated, and vertically propagating shear waves for the design earthquake thermoplastic pipes) buckling is the most critical failure mode is estimated using the following equation. and therefore the thrust force, (T) is the governing quantity in the evaluation. For rigid conduits (for example, constructed max = max Gm (9-2) with reinforced concrete), the deformation of the lining, the max = maximum earthquake-induced shearing stress; bending, the thrust as well as the resulting material strains are = (PGA/g) v Rd; all important quantities. These quantities can be considered as v = total vertical overburden pressure at the depth cor- seismic ovaling demands for the lining of the conduit and can responding to the invert of the culvert or pipe; be determined using the following general steps: = t (H + D); t = total unit weight of soil; Step 1: Estimate the expected free-field ground strains H = soil cover thickness measured from the ground sur- caused by the vertically propagating shear waves of the design face to the crown elevation; earthquakes using the following formula: d = diameter of the circular culvert or pipe; Rd = depth-dependent stress reduction factor; max = Vs Cse (9-1) = 1.0 - 0.00233z for z <30 feet where z is the depth to where the midpoint of the culvert or pipe; max = maximum free-field shearing strain at the elevation = 1.174 - 0.00814z for 30 feet < z <75 feet; and of the conduit; Gm = effective, strain-compatible shear modulus of the Vs = shear (S-) wave peak particle velocity at the conduit ground surrounding the culvert or pipe. elevation; and Cse = effective shear wave velocity of the medium sur- Alternatively, the maximum free-field shearing strain also rounding the conduit. can be estimated by a more refined free-field site response analysis (for example, conducting SHAKE analyses). It should be noted that the effective shear wave velocity of Step 2: Given max, the free-field diameter change of the the vertically propagating shear wave (Cse) should be com- conduit would be: patible with the level of shearing strain that may develop in DEQ-FF = 0.5 max D (9-3) the ground at the elevation of the conduit under the design earthquake shaking. However, if the fact that there is a hole/cavity in the ground An important aspect for evaluating the transient ground (due to the excavation of the conduit) is considered, then the deformation effects on culvert and pipe structures is to first diameter change in the ground with the cavity in it would be: determine the ground strain in the free-field (in this case free- DEQ = 2 max (1 - m ) D (9-4) field shear strain, max) and then determine the response of the structures to the ground strain. For a culvert or pipe struc- where ture constructed at a significant depth below the ground sur- m = Poisson's ratio of the surrounding ground; and face, the most appropriate design ground motion parameter D = diameter of the conduit structure. to characterize the ground motion effects is not PGA. Instead, PGV (in this case S-wave peak particle velocity, Vs) is a better It is to be noted that Equation (9-3) ignores the fact that indicator for ground deformations (strains) induced during there is a cavity and a conduit structure in the ground, while ground shaking. This is particularly important because given Equation (9-4) accounts for the presence of the cavity but the same site-adjusted PGA value, the anticipated peak ground ignores the stiffness of the conduit. Equation (9-4) is applica- velocity for CEUS could be much smaller than that for the ble for a flexible conduit in a competent ground. In this WUS. The results based on the PGA versus PGV study pre- case, the lining of the conduit can be reasonably assumed to sented in Chapter 5 in this report should be used in evaluat- conform to the surrounding ground with the presence of a ing the maximum free-field shearing strain in Equation (9-1). cavity in it. However, for most highway culverts and pipes, the burial In the study by Davis and Bardet (2000), it was assumed depths are generally shallow (that is, within 50 feet from the that the CMP conform to the free-field ground deformation ground surface). Under these conditions, it is more reason- (that is, Equation 9-3). For flexible conduits such as the CMP able to estimate the maximum free-field shearing strain studied by Davis and Bardet, the actual pipe deformations/

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111 strains should have been closer to the values predicted by Step 3: The diameter change (DEQ) accounting for the Equation (9-4) rather than by Equation (9-3), suggesting that soil-structure interaction effects can then be estimated using the strains in the pipes calculated in that study were probably the following equation: well underestimated. This very simplified design practice has been used frequently DEQ = 1 3( k1 F max D )( full-slip ) (9-7) in the past (that is, estimate the free-field ground deformations where and then assume that the conduit structure would conform to k1 = seismic ovaling coefficient the free-field ground deformations). By doing this, the soil- structure interaction effect was ignored. This practice may lead = 12 (1 - m ) ( 2 F + 5 - 6 m ) (9-8) to either overestimated or underestimated seismic response of The seismic ovaling coefficient curves plotted as a function the structural lining, depending on the relative stiffness be- of F and m are presented in Figure 9-3. tween the surrounding ground and the culvert. The resulting maximum thrust (hoop) force (Tmax) and the Further studies by Wang (1993) led to a more rational pro- maximum bending moment (Mmax) in the lining can be de- cedure in estimating the actual lining deformation by defining rived as follows: the relative stiffness between a circular lining and the sur- rounding ground using two ratios designated as the compress- Tmax = {(1 6 ) k1 [ Em (1 + m )] R max }( full slip ) (9-9) ) ibility ratio (C) and the flexibility ratio (F), as follows (Peck et al., 1972): M max = {(1 6 ) k1 [ Em (1 + m )] R 2 max } C = { Em (1 - 1 ) } {E1 A1 (1 + m )(1 - 2m )} 2 R (9-5) p) = RTmax ( full slip (9-10) F = { Em (1 - 1 ) } {6E1 I1 (1 + m )} 2 R3 (9-6) It should be noted that the solutions provided here are based on the full-slip interface assumption (which allows where normal stresses, that is, without normal separation, but no Em = strain-compatible elastic modulus of the surrounding tangential shear force). According to previous investigations, ground; during an earthquake, slip at interface is a possibility only for m = Poisson's ratio of the surrounding ground; a conduit in soft soils, or when the seismic loading intensity R = nominal radius of the conduit; is very high. In most cases, the condition at the interface is be- El = Elastic modulus of conduit lining; tween full-slip and no-slip. l = Poisson's ratio of the conduit lining; In computing the forces and deformations in the lining, it Al = lining cross-sectional area per unit length along culvert is prudent to investigate both cases, and the more critical one axial alignment; should be used in design. The full-slip condition gives more t = lining thickness; and conservative results in terms of maximum bending moment Il = moment of inertia of lining per unit length of tunnel (Mmax) and lining deflections (DEQ). This conservatism is de- (in axial direction). sirable to offset the potential underestimation (about 15 per- cent) of lining forces resulting from the use of a pseudo-static The flexibility ratio (F) tends to be the governing factor for the bending response of the lining (distortion) while the compressibility ratio (C) tends to dominate the thrust/hoop forces/strains of the lining. When F < 1.0, the lining is consid- ered stiffer than the ground, and it tends to resist the ground and therefore deforms less than that occurring in the free- field. On the other hand, when F > 1, the lining is expected to deform more than the free-field. As the flexibility ratio con- tinues to increase, the lining deflects more and more than the free-field and may reach an upper limit as the flexibility ratio becomes infinitely large. This upper limit deflection is equal to the deformations displayed by a perforated ground, calcu- lated by the Equation (9-4) presented above. The strain-compatible elastic modulus of the surrounding ground (Em) should be derived using the strain-compatible shear modulus (Gm) corresponding to the effective shear wave propagating velocity (Cse). Figure 9-3. Seismic ovaling coefficient, K1.

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112 model in deriving these close-form solutions in lieu of the dynamic loading condition (that is, some dynamic amplifi- cation effect). Therefore, the solutions derived based on the full-slip assumption should be used in evaluating the moment (Equation 9-10) and deflection (Equation 9-7) response of a circular conduit (that is, culvert/pipe in this study). The maximum thrust/hoop force (Tmax) calculated by Equation (9-9), however, may be significantly underesti- mated under the seismic simple shear condition and may lead to unsafe results, particularly for thin-wall conduit (flexible culverts and pipes) where buckling potential is the key poten- tial failure mode. It is recommended that the no-slip interface assumption be used in assessing the lining thrust response. The resulting expression, after modifications based on Hoeg's Figure 9-4. Seismic thrust/hoop force response work (Schwartz and Einstein, 1980), is: coefficient, k2 (no-slip interface; soil Poisson's ratio = 0.2). Tmax = {k2 [ Em 2 (1 + m )] R max } no-slip (9-11) Where the thrust/hoop force response coefficient k2 is de- fined as: k2 = 1 + { F [(1 - 2 m ) - (1 - 2 m )C ] - 1 2 (1 - 2 m ) C + 2} 2 {F [(3 - 2m ) + (1 - 2m )C ] + C [5 2 - 8 m + 6 m ] 2 + 6 - 8 m} (9-12) A review of Equation (9-11) and the expression of k2 sug- gests that the maximum lining thrust/hoop force response is a function of compressibility ratio, flexibility ratio, and Poisson's Ratio. Figures 9-4 through 9-6 graphically describe their in- terrelationships. As the plots show: Figure 9-5. Seismic thrust/hoop force response The seismically induced thrust/hoop force increases with coefficient, k2 (no-slip interface; soil Poisson's decreasing compressibility ratio and decreasing flexibility ratio = 0.35). Figure 9-6 Seismic thrust/hoop force response coefficient, k2 (no-slip interface; soil Poisson's ratio = 0.5).