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105 This chapter provides results of analyses and sensitivity studies conducted for buried structures. These studies dealt with the TGD and not PGD. The primary objectives of the TGD work were to: ⢠Identify methodologies for evaluating the ovaling response of circular conduits, as well as the racking response of rec- tangular conduits, and ⢠Conduct parametric studies and parametric evaluations for the methods being proposed. Results of analyses conducted to address these objectives are summarized in the following sections. These analyses focused on deriving a rational procedure for seismic evaluation of buried culverts and pipelines that consider the following sub- jects: (1) general properties and characteristics of culverts and pipes, (2) potential failure modes for buried culverts and pipes subject to seismic loading, (3) procedures used in current de- sign practice to evaluate seismic response of buried structures, (4) derivation of detailed rational procedures for seismic eval- uation of both rigid and ï¬exible culverts and pipes subject to TGD, taking into consideration soil-structure interaction, and (5) providing recommendations on a general methodology for seismic evaluation under the effects of PGD. These results consider both ï¬exible and rigid culverts, burial depths that range from 0.5 to 5 diameters, various cross-sectional geome- tries (for example, circular and rectangular) and wall stiff- nesses, and different properties of the surrounding soil. 9.1 Seismic Performance of Culverts and Pipelines Damage to buried culverts and pipelines during earth- quakes has been observed and documented by previous in- vestigators (NCEER, 1996; Davis and Bardet, 1999 and 2000; OâRourke, 1999; Youd and Beckman, 2003). In general, buried structures have performed better in past earthquakes than above-ground structures. Seismic performance records for culverts and pipelines have been very favorable, particu- larly when compared to reported damages to other highway/ transportation structures such as bridges. The main reason for the good performance of buried struc- tures has been that buried structures are constrained by the surrounding ground. It is unlikely that they could move to any signiï¬cant extent independent of the surrounding ground or be subjected to vibration ampliï¬cation/resonance. Compared to surface structures, which are generally unsupported above their foundations, buried structures can be considered to dis- play signiï¬cantly greater degrees of redundancy, thanks to the support from the ground. The good performance also may be partly associated with the design procedures used to construct the embankment and backï¬ll speciï¬cations for the culverts and pipes. Typical specifications require close control on backï¬ll placement to assure acceptable performance of the culvert or pipe under gravity loads and to avoid settlement of ï¬ll located above the pipe or culvert, and these strict require- ments for static design lead to good seismic performance. It is important that the ground surrounding the buried structure remains stable. If the ground is not stable and large PGD occur (for example, resulting from liquefaction, settle- ment, uplift, lateral spread, or slope instability/landslide), then signiï¬cant damage to the culvert or pipe structures can be expected. Although TGD due to shaking also can damage buried structures, compared to the effects of PGD, the damage is typically of a more limited extent. 9.2 Culvert/Pipe Characteristics Culvert/pipe products are available over a large range in terms of material properties, geometric wall sections, sizes, and shapes. Pipe sizes as small as 1 foot and as large as culverts with spans of 40 feet and larger are used in highway applica- tions. They can be composed of concrete, steel, aluminum, plastic, and other materials. Detailed information about their C H A P T E R 9 Buried Structures
106 shapes, range of sizes, and common uses for each type of cul- vert or pipe are summarized by Ballinger and Drake (1995). 9.2.1 Flexible Culverts and Pipes In general, culverts and pipes are divided into two major classes from the static design standpoints: ï¬exible and rigid. Flexible culverts and pipes typically are composed of either metal (for example, corrugated metal pipe (CMP) made of steel or aluminum) or thermoplastic materials (for example, HDPE or PVC). Flexible culverts and pipes respond to loads differently than rigid culverts and pipes. Because their oval- ing stiffness is small, relative to the adjacent soil, ï¬exible cul- verts and pipes rely on ï¬rm soil support and depend upon a large strain capacity to interact with the surrounding soil to hold their shape, while supporting the external pressures im- posed upon them. For static design, current AASHTO LRFD Bridge Design Specifications require as a minimum the following main design considerations (in addition to the seam failure) for ï¬exible cul- verts and pipes: (1) buckling (general cross sectional collapse as well as local buckling of thin-walled section), and (2) ï¬exi- bility limit for construction. Except for large box structures or other large spans with shapes other than circular [per McGrath, et al., (2002) NCHRP Report 473], the ï¬exural strength con- sideration (that is, bending moment demand) is generally not required for ï¬exible culverts and pipes. Neither current AASHTO LRFD Bridge Design Specifications nor the McGrath, et al. (2002) study has addressed seismic de- sign concerns for culvert structures. From the seismic design standpoint, there are two main factors that must be considered: 1. Bending moment and thrust evaluations: Seismic loading is in general nonsymmetric in nature and therefore may re- sult in sizable bending in the culvert structures (even for circular shape culverts). Furthermore, the behavior of thin- walled conduits (such as for the ï¬exible culverts and pipes) is vulnerable to buckling. This behavior differs somewhat from that of a rigid concrete culvert structure, for which bending moments are often the key factor in judging struc- tural performance. For buckling, thrust (that is, hoop force) is the key factor and seismically induced thrust can be signiï¬cant, particularly if the interface between the cul- vert or pipe structure and the surrounding soil is consid- ered a nonslip condition (Wang, 1993). Therefore, it is im- portant that both seismically induced bending and thrust be evaluated using published solutions for circular tube (Moore, 1989; Janson, 2003) as failure criteria for evaluating the seismic performance of CMP and polymeric conduits (for example, corrugated HDPE pipes). 2. Soil-support considerations: Implicit in the current AASHTO design assumptions for ï¬exible culverts is the existence of adequate soil support. This may be the weakness of ï¬exible culverts, in case of earthquakes, in that the soil support can be reduced or lost during liquefaction or other permanent ground failure mechanisms associated with seismic events. Signiï¬cant distortion or collapse of the cul- vert cross section is likely if soil support is reduced or lost. 9.2.2 Rigid Culverts and Pipes Rigid highway culverts and pipes consist primarily of rein- forced concreted shapes that are either precast or cast-in-place. Unreinforced concrete culverts and pipe structures are not rec- ommended for use in seismic regions. The sizes of reinforced concrete pipe (RCP) range (in diameter) from about 1 foot to 12 feet. Larger RCP can be precast on the site or constructed cast-in-place. Rectangular four-sided box culverts can be fur- nished precast in spans ranging from 3 feet to 12 feet. Larger spans can be constructed cast-in-place. Three-sided precast box culverts can be furnished in spans up to 40 feet. Unlike the ï¬exible culverts and pipes, the strain capacity of rigid culverts and pipes is much lower. Rigid culverts must develop signiï¬cant ring stiffness and strength to support ex- ternal pressures. Hence, they are not as dependent upon soil support as ï¬exible culverts. For static design, the primary design methods used for pre- cast concrete pipe, either reinforced or unreinforced, include: (1) the Indirect Design Method, based on the laboratory three- edge bearing test, known as the test; (2) a more direct design procedure that accounts for bending moment, shear, thrust/ tension, and crack width (bucking is generally not an issue with rigid converts and pipes) around the periphery of the cul- vert wall; and (3) methods employing computerized numer- ical models accounting for soil-structure interaction effects. For box culverts the static design uses the same criteria as other reinforced concrete structures (for example, beams and columns). In general, the effect of surrounding soils is ac- counted for by applying the soil pressures (active or at-rest) directly against the wall in the model, instead of fully taking advantage of the soil-structure interaction effect. Most cur- rent commercially available computer software can perform the structural analysis required for this design. For other structural shapes, consideration of soil-structure interaction becomes important and therefore is generally accounted for by using computerized numerical models. 9.3 General Effects of Earthquakes and Potential Failure Modes The general effects of earthquakes on culverts and pipe structures can be grouped into two broad categories: ground shaking and ground failure. The following sections discuss each category. As it will be demonstrated, soil-structure inter-
107 action plays a critical role in the evaluation of the effect of seis- mic loading for both ï¬exible and rigid culverts and pipes. A uniï¬ed evaluation procedure is developed in this chapter to provide a rational and reliable means for seismic evaluations as well as realistic design for buried culvert and pipe structures. 9.3.1 Ground Shaking Ground shaking refers to the vibration of the ground pro- duced by seismic waves propagating through the earthâs crust. The area experiencing this shaking may cover hundreds of square miles in the vicinity of the fault rupture. The intensity of the shaking attenuates with distance from the fault rupture. Ground shaking motions are composed of two different types of seismic waves, each with two subtypes: ⢠Body waves travel within the earthâs material. They may be either longitudinal compressional (P-) waves or transverse shear (S-) waves, and they can travel in any direction in the ground. ⢠Surface waves travel along the earthâs surface. They may be either Rayleigh waves or Love waves. As stable ground is deformed by the traveling waves, any culverts or pipelines in the ground also will be deformed. The shaking or wave traveling induced ground deformations are called transient ground deformations. When subject to transient ground deformations, the re- sponse of a buried linear culvert or pipe structure can be de- scribed in terms of three principal types of deformations: (1) axial deformations, (2) curvature deformations (refers to Figure 9-1), and (3) ovaling (for circular cross section) or racking (for rectangular cross section) deformations (refers to Figure 9-2). The axial and curvature deformations are induced by com- ponents of seismic waves that propagate along the culvert or pipeline axis. Figure 9-1 shows the idealized representations of axial and curvature deformations. The general behavior of the linear structure is similar to that of an elastic beam subject to deformations or strains imposed by the surrounding ground. Current design and analysis methodologies for pipeline systems were developed typically for long, linear structures. The principal failure modes for long, continuous pipeline struc- tures consist of (1) rupture due to axial tension (or pull out for jointed segmented pipelines), and (2) local bucking (wrinkling) due to axial compression and ï¬exural failure. If the pipelines are buried at shallow depth, continuous pipelines in com- pression also can exhibit beam-buckling behavior (that is, global bucking with upward buckling deï¬ections). If the axial stiffness of the structures is large, such as that for a large sec- tional concrete pipe, then the buckling potential in the longi- tudinal direction is small for both local buckling and global buckling. The general failure criteria for the above-mentioned potential failure modes have been documented by previous studies (OâRourke and Liu, 1996). It should be noted, however, that typical culvert structures for transportation applications are generally of limited length. For this condition, it is in general unlikely to develop signiï¬- cant transient axial/curvature deformations along the culvert structures. The potential failure modes mentioned above are not likely to take place during the earthquake. The main focus of this chapter will not be on the effects of axial/curvature de- formations. Instead, the scope of this chapter will concentrate on transverse deformations of culverts and pipes.Figure 9-1. Axial and curvature deformations. Figure 9-2. Ovaling and racking deformations.
108 The ovaling or racking deformations of a buried culvert or pipe structure may develop when waves propagate in a direc- tion perpendicular or nearly perpendicular to the longitudi- nal axis of the culvert or pipe, resulting in a distortion of the cross-sectional shape of the structure. Design considerations for this type of deformation are in the transverse direction. Figure 9-2 shows the ovaling distortion and racking deforma- tion associated with a circular culvert or pipe and a rectangu- lar culvert, respectively. The general behavior of the structure may be simulated as a buried structure subject to ground de- formations under a two-dimensional, plane-strain condition. Ovaling and racking deformations may be caused by verti- cally, horizontally, or obliquely propagating seismic waves of any type. Previous studies have suggested, however, that the vertically propagating shear wave is the predominant form of earthquake loading that governs the ovaling/racking behav- ior for the following reasons: (1) except possibly in the very near-source areas, ground motion in the vertical direction is generally considered less severe than its horizontal compo- nent, (2) vertical ground strains are generally much smaller than shearing strain because the value of constrained modu- lus is much higher than that of the shear modulus, and (3) the ampliï¬cation of vertically propagating shear wave, particu- larly in the soft or weak soils, is much higher than vertically propagating compressional wave and any other type of waves traveling in the horizontal direction. Therefore the analysis and methodology presented in this chapter addresses mainly the ef- fects of vertically propagating shear waves on ovaling/racking behavior of the buried culverts or pipes. When subject to ovaling/racking deformations, a ï¬exural type failure mode due to the combined effects of bending mo- ment and thrust force must be checked. The ï¬exural failure mode is typically the main concern for rigid culverts and pipes, such as those constructed with reinforced concrete. For ï¬ex- ible culverts and pipes (typically, thin-walled conduits con- structed with steel, aluminum, or thermoplastic such as HDPE or PVC), they are likely to be controlled by buckling, which can occur in the elastic range of stresses. For buckling, thrust is the key factor and conservative assumption must be made regarding interface condition (slip or nonslip) between the exterior surface of the conduit and the surrounding ground. An elastic buckling criterion for circular conduits in uniform soil was proposed by Moore (1989) and may be used for buck- ling potential evaluation purpose. 9.3.2 Ground Failure Ground failure broadly includes various types of ground in- stability such as faulting, landslides, liquefaction (including liquefaction-induced lateral spread, settlement, ï¬otation, etc.), and tectonic uplift and subsidence. These types of ground deformations are called permanent ground deformations. Each permanent ground deformation may be potentially cata- strophic to culvert or pipeline structures, although the dam- ages are usually localized. To avoid such damage, some sort of ground improvement is generally required, unless the design approach to this situation is to accept the displacement, local- ize the damage, and provide means to facilitate repairs. Characteristics of permanent ground deformation and its effects on culvert and pipes are extremely complex and must be dealt with on a case-by-case basis. It is unlikely that simple design procedures or solutions can be developed due to the complex nature of the problem. In this chapter, detailed study of problems associated with permanent ground deformation will not be conducted. Instead, only general guidelines and rec- ommendations on methodology for seismic evaluation under the effects of permanent ground deformation will be provided. 9.4 Current Seismic Design Practice for Culverts or Other Buried Structures Currently there is no standard seismic design methodology or guidelines for the design of culvert structures, including Section 12 within the current AASHTO LRFD Bridge Design Specifications. The NCHRP Report 473 Recommended Specifi- cations for Large-Span Culverts, (NCHRP, 2002) does not ad- dress issues related to seismic evaluation of long-span culverts. In the past, design and analysis procedures have been pro- posed by some researchers and design engineers for pipelines (for example, gas and water) or tunnel (that is, transportation or water) systems. While some of these procedures can be used for the design and analysis of culverts and pipes (for ex- ample, the transverse ovaling/racking deformation of the sec- tion, Figure 9-2), others cannot be directly applied because they are only applicable for buried structures with a long length, or with a deep burial depth. Furthermore, signiï¬cant disparity exists among engineers regarding the appropriate design philosophy and methods of analysis applicable to var- ious types of culvert structures. The following two paragraphs provide a brief description of procedures and methodologies proposed in the past for seismic evaluation of buried structures in general: ⢠OâRourke (1998) provides a general overview of lifeline earthquake engineering, including the treatment of seismic evaluation of pipelines. OâRourke and Liu (1996) present a detailed methodology for evaluating response of buried pipelines subject to earthquake effects. Pipelines responses to both transient ground deformation and permanent ground deformation were addressed in these two studies. How- ever, the focus of these studies was on pipeline behavior in the longitudinal direction which is more suitable for a long continuous buried pipeline structure. Although failure
109 criteria for axial tension and axial compression (local buckling/wrinkling and beam buckling) were developed, there were no discussions related to the procedure for eval- uating the transverse ovaling deformation of the pipeâs cross-sectional behavior. ⢠Based on the ï¬eld performance of 61 corrugated metal pipes (CMP) that were shaken by the 1994 Northridge Earthquake, Davis and Bardet (2000) provided an updated approach to evaluating the seismic performance of CMP conduits. The focus of their study was on the ovaling and buckling (of the thin metal wall) of the transverse section behavior of the CMP. This approach involves the following general steps: 1. Estimate the initial condition of compressive strain in the conduit, which is related to depth of burial. 2. Estimate the compressive ground strain induced by a ver- tically propagating shear wave, which was calculated from the closed-form solution for transient shearing strain, as 1â2 γmax = vp/2Vs, where γmax is the maximum transient shearing strain of the ground, vp is the horizontal peak particle velocity transverse to the conduit, and Vs is the average shear wave velocity of the surrounding ground. 3. Add the static and transient compressive strains. 4. Compare the strain so determined with the critical com- pressive strain that would cause dynamic buckling (due to hoop force) of the CMP pipe. The critical buckling strain (or strength) was assumed to be dependent on the stiffness of the surrounding soil (Moore, 1989). The methodology derived by Davis and Bardet, although more rational than most of the other procedures, has some drawbacks, including: ⢠The procedure is applicable for thin-walled pipes only. The failure mode considered by using this procedure is prima- rily for buckling and does not include ï¬exural (that is, bending) demand and capacity evaluation. The latter is a very important failure mode that must be considered for rigid culverts and pipes (such as those constructed with re- inforced concrete). ⢠The soil-structure interaction effect was considered in eval- uating the buckling capacity, but not in the evaluation of the demand (that is, earthquake-induced ground strains). ⢠The method assumed that the strains in the pipe coincide with those in the surrounding ground (that is, pipe de- forms in accordance with the ground deformation in the free-ï¬eld), on the basis of the assumption that there is no slippage at the soil-pipe interface. This assumption was in- correct, as Wang (1993) pointed out in his study. Wang concluded that the strains and deformations of a buried conduit can be greater, equal, or smaller than those of the surrounding ground in the free-ï¬eld, depending on the relative stiffness of the conduit to the surrounding ground. To account for the effects of transient ground deformation on tunnel structures, Wang (1993) developed closed-form and analytical solutions for the determination of seismically induced ovaling/racking deformations and the corresponding internal forces (such as moments and thrusts) for bored as well as cut-and-cover tunnel structures. The procedure pre- sented by Wang for the bored tunnels was developed from a theory that is familiar to most mining/underground engineers (Peck et al., 1972). Simple and easy-to-use seismic design charts were presented. The design charts are expressed prima- rily as a function of relative stiffness between the structure and the ground. Solutions for both full-slip and nonslip conditions at the interface between soil and the exterior surface of the tunnel lining were developed. These solutions fully account for the interaction of the tunnel lining with the surrounding ground. The results were validated through a series of ï¬nite element/difference soil-structure interaction analyses. For the cut-and-cover tunnels (with a rectangular shape), the design solutions were derived from an extensive study using dynamic finite-element, soil-structure interaction analyses. A wide range of structural, geotechnical, and ground motion parameters were considered by Wang in his study. Speciï¬cally, ï¬ve different types of cut-and-cover tunnel geometry were studied, including one-barrel, one-over-one two-barrel, and one-by-one twin-barrel conï¬gurations. To quantify the effect of relative stiffness on tunnel lining response, varying ground pro- ï¬les and soil properties were used in the parametric analyses. Based on the results of the parametric analyses, a deformation- based design chart was developed for cut-and-cover tunnels. Although these solutions were intended originally for tun- nel structures (considered a fairly rigid type of structure), the methodology is rational and comprehensive and provides a consistent and unified approach to solving the problem of buried conduits subject to ground shaking regardless of whether they are rigid or flexible structures. With some ad- justments this approach also is applicable to the culvert and pipe structures typically used for highway construction. There- fore, a more detailed discussion of Wangâs approach is given in the following section. 9.5 General Methodology and Recommended Procedures The general methodology and recommended procedures for the ovaling of circular conduits and the racking of rectan- gular conduits developed by Wang (1993) are presented in the following two sections, respectively. 9.5.1 Ovaling of Circular Conduits The seismic ovaling effect on the lining of a circular conduit is best deï¬ned in terms of change of the conduit diameter
110 (ÎDEQ) and incremental seismically induced internal forces [for example, bending moment (M) and thrust/hoop force (T)]. It should be noted that for ï¬exible types of conduits (such as thin-walled metal, corrugated or noncorrugated, and thermoplastic pipes) buckling is the most critical failure mode and therefore the thrust force, (T) is the governing quantity in the evaluation. For rigid conduits (for example, constructed with reinforced concrete), the deformation of the lining, the bending, the thrust as well as the resulting material strains are all important quantities. These quantities can be considered as seismic ovaling demands for the lining of the conduit and can be determined using the following general steps: Step 1: Estimate the expected free-ï¬eld ground strains caused by the vertically propagating shear waves of the design earthquakes using the following formula: where γmax = maximum free-ï¬eld shearing strain at the elevation of the conduit; Vs = shear (S-) wave peak particle velocity at the conduit elevation; and Cse = effective shear wave velocity of the medium sur- rounding the conduit. It should be noted that the effective shear wave velocity of the vertically propagating shear wave (Cse) should be com- patible with the level of shearing strain that may develop in the ground at the elevation of the conduit under the design earthquake shaking. An important aspect for evaluating the transient ground deformation effects on culvert and pipe structures is to ï¬rst determine the ground strain in the free-ï¬eld (in this case free- field shear strain, γmax) and then determine the response of the structures to the ground strain. For a culvert or pipe struc- ture constructed at a signiï¬cant depth below the ground sur- face, the most appropriate design ground motion parameter to characterize the ground motion effects is not PGA. Instead, PGV (in this case S-wave peak particle velocity, Vs) is a better indicator for ground deformations (strains) induced during ground shaking. This is particularly important because given the same site-adjusted PGA value, the anticipated peak ground velocity for CEUS could be much smaller than that for the WUS. The results based on the PGA versus PGV study pre- sented in Chapter 5 in this report should be used in evaluat- ing the maximum free-ï¬eld shearing strain in Equation (9-1). However, for most highway culverts and pipes, the burial depths are generally shallow (that is, within 50 feet from the ground surface). Under these conditions, it is more reason- able to estimate the maximum free-field shearing strain γ max ( )= V Cs se 9-1 using the earthquake-induced shearing stress and the strain- compatible shear modulus of the surrounding ground. In this approach, the expected free-ï¬eld ground strain caused by the vertically propagating shear waves for the design earthquake is estimated using the following equation. Ïmax = maximum earthquake-induced shearing stress; = (PGA/g) Ïv Rd; Ïv = total vertical overburden pressure at the depth cor- responding to the invert of the culvert or pipe; = γt (H + D); γt = total unit weight of soil; H = soil cover thickness measured from the ground sur- face to the crown elevation; d = diameter of the circular culvert or pipe; Rd = depth-dependent stress reduction factor; = 1.0 â 0.00233z for z <30 feet where z is the depth to the midpoint of the culvert or pipe; = 1.174 â 0.00814z for 30 feet < z <75 feet; and Gm = effective, strain-compatible shear modulus of the ground surrounding the culvert or pipe. Alternatively, the maximum free-ï¬eld shearing strain also can be estimated by a more reï¬ned free-ï¬eld site response analysis (for example, conducting SHAKE analyses). Step 2: Given γmax, the free-ï¬eld diameter change of the conduit would be: However, if the fact that there is a hole/cavity in the ground (due to the excavation of the conduit) is considered, then the diameter change in the ground with the cavity in it would be: where νm = Poissonâs ratio of the surrounding ground; and D = diameter of the conduit structure. It is to be noted that Equation (9-3) ignores the fact that there is a cavity and a conduit structure in the ground, while Equation (9-4) accounts for the presence of the cavity but ignores the stiffness of the conduit. Equation (9-4) is applica- ble for a flexible conduit in a competent ground. In this case, the lining of the conduit can be reasonably assumed to conform to the surrounding ground with the presence of a cavity in it. In the study by Davis and Bardet (2000), it was assumed that the CMP conform to the free-ï¬eld ground deformation (that is, Equation 9-3). For ï¬exible conduits such as the CMP studied by Davis and Bardet, the actual pipe deformations/ ÎD DmEQ 9-4= ± â( )2 1γ νmax ( ) ÎD DEQ-FF 9-3= 0 5. ( )maxγ γ Ïmax max ( )= Gm 9-2
111 strains should have been closer to the values predicted by Equation (9-4) rather than by Equation (9-3), suggesting that the strains in the pipes calculated in that study were probably well underestimated. This very simpliï¬ed design practice has been used frequently in the past (that is, estimate the free-ï¬eld ground deformations and then assume that the conduit structure would conform to the free-ï¬eld ground deformations). By doing this, the soil- structure interaction effect was ignored. This practice may lead to either overestimated or underestimated seismic response of the structural lining, depending on the relative stiffness be- tween the surrounding ground and the culvert. Further studies by Wang (1993) led to a more rational pro- cedure in estimating the actual lining deformation by deï¬ning the relative stiffness between a circular lining and the sur- rounding ground using two ratios designated as the compress- ibility ratio (C) and the ï¬exibility ratio (F), as follows (Peck et al., 1972): where Em = strain-compatible elastic modulus of the surrounding ground; νm = Poissonâs ratio of the surrounding ground; R = nominal radius of the conduit; El = Elastic modulus of conduit lining; νl = Poissonâs ratio of the conduit lining; Al = lining cross-sectional area per unit length along culvert axial alignment; t = lining thickness; and Il = moment of inertia of lining per unit length of tunnel (in axial direction). The ï¬exibility 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- ï¬eld. On the other hand, when F > 1, the lining is expected to deform more than the free-ï¬eld. As the ï¬exibility ratio con- tinues to increase, the lining deï¬ects more and more than the free-ï¬eld and may reach an upper limit as the ï¬exibility ratio becomes inï¬nitely large. This upper limit deï¬ection 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). F E R E Im m= â( ){ } +( ){ }1 6 112 3 1 1ν ν ( )9-6 C E R E Am m m= â( ){ } +( ) â( ){ }1 1 1 212 1 1ν ν ν ( )9-5 Step 3: The diameter change (ÎDEQ) accounting for the soil-structure interaction effects can then be estimated using the following equation: where The seismic ovaling coefï¬cient curves plotted as a function of F and νm are presented in Figure 9-3. The resulting maximum thrust (hoop) force (Tmax) and the maximum bending moment (Mmax) in the lining can be de- rived as follows: It should be noted that the solutions provided here are based on the full-slip interface assumption (which allows normal stresses, that is, without normal separation, but no tangential shear force). According to previous investigations, during an earthquake, slip at interface is a possibility only for a conduit in soft soils, or when the seismic loading intensity is very high. In most cases, the condition at the interface is be- tween 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 full-slip condition gives more conservative results in terms of maximum bending moment (Mmax) and lining deï¬ections (ÎDEQ). This conservatism is de- sirable to offset the potential underestimation (about 15 per- cent) of lining forces resulting from the use of a pseudo-static M k E R RT m mmax max max = ( ) +( )[ ]{ } = 1 6 11 2ν γ full slip 9-10( ) ( ) T k E Rm mmax max (= ( ) +( )[ ]{ }( )1 6 11 ν γ full slip 9-9) k Fm 1 2 5 = = â( ) + seismic ovaling coefficient 12 1 ν â( )6νm ( )9-8 ÎD k F DEQ full-slip 9-7= ± ( )( )1 3 1 γ max ( ) Figure 9-3. Seismic ovaling coefficient, K1.
112 model in deriving these close-form solutions in lieu of the dynamic loading condition (that is, some dynamic ampliï¬- cation effect). Therefore, the solutions derived based on the full-slip assumption should be used in evaluating the moment (Equation 9-10) and deï¬ection (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 signiï¬cantly underesti- mated under the seismic simple shear condition and may lead to unsafe results, particularly for thin-wall conduit (ï¬exible 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 modiï¬cations based on Hoegâs work (Schwartz and Einstein, 1980), is: Where the thrust/hoop force response coefï¬cient k2 is de- ï¬ned as: 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, ï¬exibility ratio, and Poissonâs Ratio. Figures 9-4 through 9-6 graphically describe their in- terrelationships. As the plots show: ⢠The seismically induced thrust/hoop force increases with decreasing compressibility ratio and decreasing ï¬exibility k F C C F m m m2 1 2 2 1 1 2 1 2 1 2 2 3 = + â( ) â â( )[ ]{ â â( ) + } â ν ν ν 2 1 2 5 2 8 6 6 8 92 ν ν ν ν ν m m m m m C C ( ) + â( )[ ]{ + â +[ ]+ â } ( -12) T k E Rm mmax max ( )= +( )[ ]{ }2 2 1 ν γ no-slip 9-11 Figure 9-4. Seismic thrust/hoop force response coefficient, k2 (no-slip interface; soil Poissonâs ratio = 0.2). Figure 9-6 Seismic thrust/hoop force response coefficient, k2 (no-slip interface; soil Poissonâs ratio = 0.5). Figure 9-5. Seismic thrust/hoop force response coefficient, k2 (no-slip interface; soil Poissonâs ratio = 0.35).
113 poses, the racking stiffness can be obtained by applying a unit lateral force at the roof level, while the base of the structure is restrained against translation, but with the joints free to rotate. The structural racking stiffness is deï¬ned as the ratio of the applied force to the resulting lateral displacement. Step 3: Derive the ï¬exibility ratio (Frec) of the rectangular structure using the following equation: where L = width of the structure; and Gm = average strain-compatible shear modulus of the sur- rounding ground. The ï¬exibility ratio is a measure of the relative racking stiff- ness of the surrounding ground to the racking stiffness of the structure. The derivation of Frec is schematically depicted in Figure 9-8. Step 4: Based on the ï¬exibility ratio obtained form Step 3 above, determine the racking ratio (Rrec) for the structure using Figure 9-5 or the following expression: The racking ratio is deï¬ned as the ratio of actual racking deformation of the structure to the free-ï¬eld racking defor- mation in the ground. The solid triangular data points in Fig- ure 9-9 were data generated by performing a series of dynamic finite element analyses on a number of cases with varying soil and structural properties, structural conï¬gurations, and ground motion characteristics. Note, however, these data were generated by using structural parameters representative of typ- ical transportation tunnels during the original development of this design methodology. The validity of this design chart was later veriï¬ed and adjusted as necessary by performing R F Frec rec rec 9-15= +( )2 1 ( ) F G K L Hm srec 9-14= ( ) ( ) ( ) ratio when the Poissonâs ratio value of the surrounding ground is less than 0.5. ⢠When the Poissonâs ratio approaches 0.5 (for example, for saturated undrained clay), the thrust response of the lining is essentially independent of the compressibility ratio. The theoretical solutions and their applicability to typical culvert and pipeline structures is further veriï¬ed for reason- ableness by numerical analysis presented in the next section. 9.5.2 Racking of Rectangular Conduits Racking deformations are deï¬ned as the differential side- ways movements between the top and bottom elevations of rectangular structures, shown as âÎsâ in Figure 9-7. The re- sulting structural internal forces or material strains in the lin- ing associated with the seismic racking deformation (Îs) can be derived by imposing the differential deformation on the structure in a simple structural frame analysis. The procedure for determining Îs and the corresponding structural internal forces [bending moment (M), thrust (T), and shear (V)], taking into account the soil-structure inter- action effects, are presented below (Wang, 1993). Step 1: Estimate the free-ï¬eld ground strains γmax (at the structure elevation) caused by the vertically propagating shear waves of the design earthquakes, refer to Equation (9-1) or Equation (9-2) and related discussions presented earlier in Section 9.4.1. Determine the differential free-ï¬eld relative dis- placements (Îfree-ï¬eld) corresponding to the top and the bottom elevations of the rectangular/box structure by: where H is height of the structure. Step 2: Determine the racking stiffness (Ks) of the structure from a simple structural frame analysis. For practical pur- Î free-field 9-13= H γ max ( ) Figure 9-7. Racking deformations of a rectangular conduit.
114 Figure 9-8. Relative stiffness of soil versus rectangular frame. Figure 9-9. Racking ratio between structure and free-field.
115 similar numerical analysis using parameters that are repre- sentative of highway culvert structures. As indicated in Figure 9-9, if Frec = 1, the structure is con- sidered to have the same racking stiffness as the surrounding ground, and therefore the racking distortion of the structure is about the same as that of the ground in the free ï¬eld. When Frec is approaching zero, representing a perfectly rigid structure, the structure does not rack regardless of the distortion of the ground in the free ï¬eld. For Frec > 1.0 the structure becomes ï¬exible relative to the ground, and the racking distortion will be magniï¬ed in comparison to the shear distortion of the ground in the free ï¬eld. This magniï¬cation effect is not caused by the effect of dynamic ampliï¬cation. Rather it is attributed to the fact that the ground has a cavity in it as opposed to the free ï¬eld condition. Step 5: Determine the racking deformation of the structure (Îs) using the following relationship: Step 6: The seismic demand in terms of internal forces (M, T, and V) as well as material strains can be calculated by im- posing Îs upon the structure in a frame analysis as depicted in Figure 9-10. It should be noted that the methodology developed above was intended to address the incremental effects due to earth- quake-induced transient ground deformation only. The seis- mic effects of transient racking/ovaling deformations on cul- verts and pipes must be considered additional to the normal load effects from surcharge, pavement, and wheel loads, and then compared to the various failure criteria considered rel- evant for the type of culvert structure in question. 9.6 Parametric and Verification Analysis Section 9.5 presents rational ovaling and racking analysis procedures robust enough to treat various types of buried conduit structures. Some simple design charts have also been developed to facilitate the design process. These design charts Î Îs R= rec free-field 9-16 ( ) have been validated through a series of parametric numerical analyses. The applications of these simple design charts to vehicular/transit tunnels also have been successfully applied in real world projects in the past, particularly for deep tunnels surrounded by relatively homogeneous ground. There are, however, differences between vehicular/transit tunnels and buried culverts and pipes. For example, tunnel structures are generally of large dimensions and typically have much greater structural stiffness than that of culverts and pipe structures. In addition, culverts and pipes are generally buried at shallow depths where the simpliï¬ed procedure developed for deep tunnels may not necessarily be directly applicable. To address the issues discussed above, numerical analysis using ï¬nite element/ï¬nite difference procedures was per- formed for a wide range of parameters representative of actual culvert properties and geometries (that is, for ï¬exible as well as rigid culverts). In addition, the parametric analysis included the construction condition in terms of burial depth. The analysis, assumptions, and results are presented in the follow- ing sections. 9.6.1 Types of Structures and Other Parameters Used in Evaluation The various parameters studied in this analysis are sum- marized in Table 9-1. 9.6.2 Model Assumptions and Results Six sets of parametric analyses were conducted. Assump- tions made and results from these analyses are summarized in the following sections. 9.6.2.1 Parametric AnalysisâSet 1 Model AssumptionsâSet 1. The parametric analysisâ Set 1 (the Reference Set) started with a 10-foot diameter cor- rugated steel pipe (or an equivalent liner plate lining) and a 10-foot diameter precast concrete pipe to represent a ï¬exible and a rigid culvert structure, respectively. Speciï¬c properties used for these two different types of culvert structures are pre- sented in Table 9-2. The soil proï¬le used for Set 1 parametric analysis was as- sumed to be a homogeneous deep (100-foot thick) soil de- posit overlying a rigid base (for example, base rock). The as- sumed Youngâs modulus and Poissonâs ratio are Em = 3,000 psi and νm = 0.3, respectively. It is recognized that this is an ideal representation of actual conditions; however, these con- ditions provide a good basis for making comparison in para- metric analysis. To account for the effects of shallow soil cover, ï¬ve cases of varying embedment depths were analyzed for each culvert Figure 9-10. Simple frame analysis of racking deformations.
116 Parameters Descriptions Structure Types FLEXIBLE CULVERTS: Corrugated Aluminum Pipe Corrugated Steel Pipe Corrugated HDPE Pipe RIGID CULVERTS: Reinforced Concrete Pipe Reinforced Concrete Box Type Burial Depths 5d, 3d, 2d, 1d, 0.5d, (âdâ represents the diameter of a circular pipe or the height of a box concrete culvert) Cross Section Geometry Types Circular Square Box Rectangular Box Square 3-sided Rectangular 3-sided Diameters of Circular Culverts 5 feet (Medium Diameter) 10 feet (Large Diameter) Wall Stiffness of Circular Culverts FLEXIBLE CULVERTS: I=0.00007256 ft 4/ft, E= 2.9E+07 psi (Steel) I=0.00001168 ft 4/ft, E= 1.0E+07 psi (Aluminum) I=0.0005787 ft 4/ft, E= 1.1E+05 psi (HDPE) Size Dimensions of Box Culverts 10 feet x 10 feet: Square Box and Square 3-sided 10 feet x 20 feet: Rectangular Box and Rectangular 3-sided Wall Stiffness of Box Culverts RIGID CULVERTS: I=0.025 ft 4/ft, t=0.67 ft, E= 4.0E+06 psi (Concrete) I=0.2 ft 4/ft, t=1.33 ft, E= 4.0E+06 psi (Concrete) Properties of Surrounding Ground* E=3,000 psi (Firm Ground) E=7,500 psi (Very Stiff Ground) Total Unit Weight = 120 psf * Note: The Youngâs Modulus values used in this study are for parametric analysis purposes only. Table 9-1. Parameters used in the parametric analysis. Culvert Properties Rigid Culvert (Concrete Pipe) Flexible Culvert (Corrugated Steel Pipe) Culvert Diameter, ft 10 10 Young's Modulus, E/(1-v2), used in 2-D Plane Strain Condition, psi 4.0E+06 2.9E+07 Moment of Inertia I, ft4/ft 0.025 ft4/ft 0.00007256 ft4/ft (=1.505 in4/ft) Sectional Area A, ft2 per ft 0.67 0.02 EI (lb-ft2 per ft) 1.44E+07 3.03E+05 AE (lb per ft) 3.86E+08 8.35E+07 Poisson's Ratio 0.3 0.3 Note: Ground condition (firm ground with Em = 3000 psi, νm = 0.3). Table 9-2. Parametric Analysis Set 1âculvert lining properties (Reference Set).
117 Cases Analyzed Soil Cover H (feet) Culvert Diameter d (feet) Embedment Depth Ratio, H/d Case 1 50 10 5 Case 2 30 10 3 Case 3 20 10 2 Case 4 10 10 1 Case 5 5 10 0.5 Case 6 2 10 0.2 Table 9-3. Analyses performed for variable embedment depths. Figure 9-11. Case 1 finite difference mesh (soil cover = 50 feet). Figure 9-12. Case 2 finite difference mesh (soil cover = 30 feet). type (that is, the ï¬exible type and the rigid type). The six cases of embedment depths are listed in Table 9-3. Figures 9-11 through 9-15 show the ï¬nite difference meshes (using computer program FLAC) used for the parametric analysis accounting for the variable culvert embedment depths. Figure 9-16 graphically deï¬nes the âEmbedment Depth Ratioâ cited in Table 9-3. Figure 9-17 shows the culvert lining modeled as continuous beam elements in the ï¬nite difference, soil-structural interaction analysis. The entire soil-structure system was subjected to an artiï¬- cially applied pseudo-horizontal acceleration of 0.3g (accelera- tion of gravity), simulating earthquake-induced vertically prop- agating shear waves. As a result, lateral shear displacement in the soil overburden will occur. A simple, uniform pseudo accelera- tion and a simple, uniform soil proï¬le (with a uniform soil stiff- ness modulus) were assumed for simplicity and are desirable in parametric analysis. Figure 9-18 presents the resulting lateral soil displacement proï¬le under lateral acceleration of 0.3g. Results of AnalysisâSet 1. Figures 9-19 and 9-20 show examples of culvert lining response in terms of lining thrust/hoop forces and bending moments, respectively. Ex- amples presented in Figures 9-19 and 9-20 are for the ï¬exi- ble culvert under the Case 1 conditions (that is, with a soil cover of 50 feet deep). As indicated, the maximum response (that is, the most vulnerable locations) occurs at the knee- and-shoulder locations around the lining, consistent with the generally observed damage/damage mechanism for buried pipes/culverts (as well as circular tunnels) during major earthquakes in the past (refer to the mechanism sketch depicted in Figure 9-2). Using the lining information presented in Table 9-2 and the soil properties of the surrounding ground (that is, Em = 3,000 psi, νm = 0.3), the compressibility ratio (C) and ï¬exibil- ity ratio (F) for the two culverts were calculated using Equa- tion (9-5) and Equation (9-6), respectively. Their values are presented in Table 9-4. The results of the analysis in terms of
118 about 15 percent to 20 percent. This result is consistent with previous studies as discussed in Section 9.5.1. The data contained in Table 9-6 is graphically presented in Figure 9-21. As seen, the ï¬exible culvert deforms signiï¬cantly more than the free ï¬eld because its ï¬exibility ratio (F = 22.6) is signiï¬cantly greater than 1.0, suggesting the ground is much stiffer than the lining. For the rigid culvert with F = 0.482 < 1.0, the lining is stiffer than the ground and therefore deforms less than the free-ï¬eld. Figure 9-22 shows the effects of culvert embedment depth on the lining deformations, expressed by the ratios of the lining to free-ï¬eld deformation. It can be seen that the ratios of the lin- ing to free-ï¬eld deformation remained almost unchanged for an embedment ratio of 1.0 or greater. When the embedment lining deformations (diameter changes) are presented in Tables 9-5 and 9-6. From these analyses the following observations were made: ⢠Flexible culverts experience greater deformation than the ground deformation in the free-ï¬eld for both full-slip and no-slip cases. ⢠Rigid culverts experience less deformation than the ground deformation in the free-ï¬eld for both full-slip and no-slip cases. ⢠The full-slip condition gives more conservative values of lining deï¬ections (ÎDEQ) than the nonslip condition by Figure 9-13. Case 3 finite difference mesh (soil cover = 20 feet). Figure 9-14. Case 4 finite difference mesh (soil cover = 10 feet). Figure 9-15. Case 5 finite difference mesh (soil cover = 5 feet). Figure 9-16. Definition of embedment depth ratio.
119 ratio is less than 1.0, the ratio of the actual culvert diameter change to the free-ï¬eld deformation decreases gradually. The culvert embedment depth, however, showed some ef- fects on the thrust/hoop force and bending response of the lining, as indicated in Figures 9-23 and 9-24. The embedment effect on the thrust response is more obvious for the rigid cul- vert than for the ï¬exible culvert. The thrust ratio presented in Figure 9-23 is deï¬ned as the maximum lining thrust obtained from the ï¬nite difference analysis normalized to that derived using the close-form solutions in Equations (9-11) and (9-12) (for the no-slip interface condition). As indicated, the theo- retical close-form solution somewhat overestimates the lining thrust/hoop forces when the culvert is buried at shallow depth. For a rigid culvert, the overestimation is no more than 15 per- cent. For a ï¬exible culvert the overestimation is negligible. The ï¬gure also shows that the effect of embedment is negligible when the embedment ratio is greater than about 3 or 4. The embedment effects on bending response are illustrated in Figure 9-24. Based on the results from the analysis, it ap- pears that the potential for overestimation of bending de- mand would occur for rigid types of culvert structures buried at shallow depths by as much as 30 to 35 percent. Figure 9-24 also suggests that the effects of embedment depth on bending response are insigniï¬cant when the embedment depth ratio is greater than about 3. It should be noted that the main reason for the overesti- mation in thrust and bending forces is that the maximum free-ï¬eld ground shearing strain used in calculating the close- form solutions (Equation 9-11 and Equation 9-12) is the maximum shearing strain that occurs at the culvert invert (instead of the average free-ï¬eld shearing strain within the culvert depth). These results suggest that the maximum free- ï¬eld ground strain is on the safe side. Figure 9-17. Culvert beam element number. Figure 9-18. Soil deformations subjected to pseudo lateral acceleration of 0.3g.
120 Figure 9-19. Culvert lining thrust/hoop force distribution (for flexible culvert in Set 1, Case 1 geometry). Figure 9-20. Culvert lining bending moment distribution (for flexible culvert in Set 1, Case 1 geometry).
121 Properties Rigid Culvert (Concrete Pipe) Flexible Culvert (Corrugated Steel Pipe) Compressibility Ratio, C 0.011 0.05 Flexibility Ratio, F 0.482 22.6 Table 9-4. Culvert lining compressibility and flexibility used in analysis. Case No. (Embedment Ratio) Free-Field Maximum Ground Shear Strain (from FLAC Analysis) γ max Diameter Change Using Eq. 9-3 Closed-Form Free-Field Ground ÎD=0.5*D* γmax (feet) Case 1 (H/d=5) 0.0129 0.065 Case 2 (H/d=3) 0.0085 0.043 Case 3 (H/d=2) 0.0064 0.032 Case 4 (H/d=1) 0.004 0.02 Case 5 (H/d=0.5) 0.003 0.015 Case 6 (H/d=0.2) 0.0022 0.011 Note: The maximum free-field ground shearing strain is the maximum shearing strain that could occur within the full depth of the culvert (that is, from the crown to the invert). In the pseudo-static FLAC analysis, the maximum ground shearing strains occur at the invert in all cases. Table 9-5. Free-field ground strain and diameter change. Case No. (Embedment Ratio) Culvert Diameter Change (ft) for Full-Slip Interface Using Eq. 9-7 Culvert Diameter Change (ft) for No-Slip Interface Using FLAC Analysis Diameter Change Ratio for No-Slip to Full-Slip For Flexible Culvert Case 1 (H/d=5) 0.169 0.129 0.77 Case 2 (H/d=3) 0.111 0.082 0.74 Case 3 (H/d=2) 0.084 0.059 0.70 Case 4 (H/d=1) 0.052 0.036 0.68 Case 5 (H/d=0.5) 0.039 0.024 0.62 Case 6 (H/d=0.2) 0.029 0.018 0.62 For Rigid Culvert Case 1 (H/d=5) 0.042 0.034 0.80 Case 2 (H/d=3) 0.028 0.021 0.77 Case 3 (H/d=2) 0.021 0.015 0.72 Case 4 (H/d=1) 0.013 0.009 0.67 Case 5 (H/d=0.5) 0.010 0.006 0.57 Case 6 (H/d=0.2) 0.007 0.004 0.51 Table 9-6. Culvert diameter changeâeffect of interface slippage condition.
122 Figure 9-21. Culvert deformations versus free-field deformations. Figure 9-22. Ratios of culvert deformations versus free-field deformations. Figure 9-23. Embedment effects on culvert maximum thrust/hoop forces. Figure 9-24. Embedment effects on culvert maximum bending moments. Additional Parametric Analysis and Results. Additional parametric analyses included (1) different circular culvert/ pipe sizes; (2) different culvert/pipe material, such as corru- gated aluminum and HDPE pipes; (3) different soil stiffness; (4) square and rectangular shape culverts (constructed with reinforced concrete; (5) 3-sided ï¬at roof rectangular concrete culverts; and (6) different culvert/pipe wall stiffness. These additional analyses were used to further verify that with some modiï¬cations, the close-form solutions developed for deep circular bored tunnels and rectangular cut-and-cover tunnels (refer to Section 9.5) also can be used for circular and rectan- gular culvert structures. 9.6.2.2 Parametric AnalysisâSet 2 Model AssumptionsâSet 2. Assumptions and parame- ters used in parametric analysis Set 2 are the same as those used in Set 1 (the Reference Case) except (1) the culvert di- ameter was reduced from 10 feet to 5 feet; (2) the total soil proï¬le depth has been reduced from 100 feet to 50 feet; and (3) the culvert embedment depth was halved in each respec- tive case to maintain the same embedment ratio (H/d). Be- cause of this reduction in culvert size the resulting compress- ibility ratio (C) was reduced from 0.011 to 0.005 and the ï¬exibility ratio (F) was reduced from 0.482 to 0.061 for the rigid culvert. Similarly for the ï¬exible culvert, C and F were reduced from 0.05 to 0.025 and from 22.6 to 2.856, respec- tively (see Table 9-7). Results of AnalysesâSet 2. Figures 9-25 through 9-27 present the results of FLAC analysis. Compared to results from Set 1 analysis (refer to Figures 9-22 through 9-24), the Set 2 results indicated that: ⢠The ratios of the actual culvert deformation to free-ï¬eld ground deformation were signiï¬cantly reduced, reï¬ecting the effect of higher culvert lining stiffness because of the re- duced culvert diameter. ⢠The bending and thrust force response of the smaller 5-foot diameter culvert, when normalized to the close- form solutions, show similar trends to that of the larger culvert (10-foot diameter). Based on results in Figure 9-26, when the burial depth is small, the close-form solutions (using the conservative maximum free-ï¬eld ground strain value at the culvert invert elevation) tend to overestimate the thrust response by up to about 20 percent for the ï¬ex- ible culvert. For the rigid culvert the overestimation is greater than about 30 percent at very shallow burial depth.
123 Culvert Properties Rigid Culvert (Concrete Pipe) Flexible Culvert (Corrugated Steel Pipe) Culvert Diameter, ft 5 5 Young's Modulus, E/(1-v2), psi 4.0E+06 2.9E+07 Moment of Inertia, ft4/ft 0.025 0.00007256 Sectional Area, ft2 per ft 0.67 0.02 EI (lb-ft2 per ft) 1.44E+07 3.03E+05 AE (lb, per ft) 3.86E+08 8.35E+07 Poisson's Ratio 0.3 0.3 Compressibility, C 0.005 0.025 Flexibility Ratio, F 0.061 2.856 Note: Ground condition (firm ground with Em = 3000 psi, νm = 0.3). Table 9-7. Parametric analysis set 2âculvert lining properties. Figure 9-25. Ratios of culvert deformations versus free-field deformations (parametric analysisâSet 2). Figure 9-26. Embedment effects on culvert maximum thrust/hoop forces (parametric analysisâSet 2). Figure 9-27. Embedment effects on culvert maximum bending moments (parametric analysisâSet 2). This suggests that the analytical methodology and procedure previously presented in Section 9.5 provide a robust ap- proach to accounting for the soil-structure interaction effect in evaluating the seismic behavior of culverts with varying characteristics. The effect of shallow embedment depth on bending shows similar trends to the thrust response (refer to Figure 9-27). 9.6.2.3 Parametric AnalysisâSet 3 Model AssumptionsâSet 3. In this set of analyses the as- sumptions and parameters are the same as those used in Set 1 (the Reference Case) except (1) the ï¬exible culvert was changed from corrugated steel pipe to corrugated aluminum pipe (with lower bending and compression stiffness com- pared to the steel pipe); and (2) the rigid concrete pipe was made even more rigid by increasing its wall thickness from 0.67 feet to 1.33 feet. The resulting compressibility ratio and ï¬exibility ratio, along with other lining properties are pre- sented in Table 9-8. Results of AnalysesâSet 3. Results from the analysis are shown in Figures 9-28 through 9-30. As indicated, the results are following the same trend as shown in results from Sets 1 and 2 analysis, even though a much more ï¬exible culvert and a much more rigid culvert were used in this set of analysis.
124 meric conduits are being used with increasing frequency, and polymers, especially high density polyethylene, are likely to be the material of choice for many drainage appli- cations in the future. The typical properties of the HDPE material are presented in Table 9-9. Youngâs modulus of 110,000 psi is appropriate for short term loading effects on HDPE pipe. Poissonâs ratio of HDPE pipe is estimated to be about 0.45. Results of AnalysesâSet 4. Figures 9-31 through 9-33 present the results of the HDPE culvert analysis. As indi- cated, the seismic behavior of the HDPE pipe also can be predicted reasonably well using the analytical procedure presented in Section 9-5. Like in other cases, if necessary, some adjustments may be made to correct the overestima- tion of thrust forces and bending moments when the pipe is buried at a very shallow depth. For conservative design pur- poses, however, it is recommended that no force reduction be made. Culvert Properties Rigid Culvert (Concrete Pipe) Flexible Culvert Aluminum CMP Culvert Diameter, ft 10 10 Young's Modulus, E/(1-v2), psi 4.0E+06 1.0E+07 Moment of Inertia, ft4/ft 0.2 0.00001168 Sectional Area, ft2 per ft 1.333 0.01125 EI (lb-ft2 per ft) 1.152E+08 1.682E+04 AE (lb, per ft) 7.678E+08 1.62E+07 Poisson's Ratio 0.3 0.3 Compressibility, C 0.005 0.256 Flexibility Ratio, F 0.060 411.7 Note: Ground condition (firm ground with Em = 3,000 psi, νm = 0.3). Table 9-8. Parametric analysis set 3âculvert lining properties. Figure 9-28. Ratios of culvert deformations versus free-field deformations (parametric analysisâSet 3). Figure 9-29. Embedment effects on culvert maximum thrust/hoop forces (parametric analysisâSet 3). Figure 9-30. Embedment effects on culvert maximum bending moments (parametric analysisâSet 3). 9.6.2.4 Parametric AnalysisâSet 4 Model AssumptionsâSet 4. Only one type of lining was analyzed in this set of analysis. The lining modeled in this analysis is a 5-foot diameter corrugated HDPE pipe. The reason for selecting HDPE in this analysis is because poly-
125 and 9-10). However, the soil stiffness has been increased from Em =3,000 psi (ï¬rm ground) to Em = 7,500 psi (very stiff ground). The entire soil proï¬le was assumed to be homogeneous. The soil overburden thickness (100-foot thick) and other condi- tions are the same as those in Set 1. Results of AnalysesâSet 5. The calculated compressibil- ity and ï¬exibility ratios also are included in Table 9-10. Be- cause of the increased ground stiffness, the ï¬exibility ratio for the rigid culvert was computed to be 1.217, slightly greater than 1.0. This suggests that the ovaling stiffness of the ground is only slightly greater than the ovaling stiffness of the rigid culvert. Based on the discussions presented in Section 9-4, when the ï¬exibility ratio is close to 1.0, the ovaling deforma- tion of the lining should be about the same as that of the sur- rounding ground. Results from the FLAC analysis in Figure 9-34 show that for the rigid culvert the ratio of the culvert deformation to the ground deformation is very close to 1.0, verifying the validity Culvert Properties Flexible Culvert (Corrugated HDPE) Culvert Diameter, ft 5 Young's Modulus, E/(1-v2), psi 1.1E+05 Moment of Inertia, ft4 per ft 0.0005787 Sectional Area, ft2 per ft 0.0448 EI (lb-ft2 per ft) 9.17E+03 AE (lb, per ft) 7.10E+05 Poisson's Ratio 0.45 Compressibility, C 2.927 Flexibility Ratio, F 94.424 Note: Ground condition (firm ground with Em = 3,000 psi, νm = 0.3). Table 9-9. Parametric analysis set 4âculvert lining properties. Figure 9-31. Ratios of culvert deformations versus free-field deformations (parametric analysisâSet 4). Figure 9-32. Embedment effects on culvert maximum thrust/hoop forces (parametric analysisâSet 4). Figure 9-33. Embedment effects on culvert maximum bending moments (parametric analysisâSet 4). 9.6.2.5 Parametric AnalysisâSet 5 Model AssumptionsâSet 5. In this set of parametric analysis, the culvert lining properties used are identical to those assumed in Set 1 (the Reference Case, refer to Tables 9-2
126 Culvert Properties Rigid Culvert (Concrete Pipe) Flexible Culvert (Corrugated Steel Pipe) Culvert Diameter, ft 10 10 Young's Modulus, E/(1-v2), psi 4.0E+06 2.9E+07 Moment of Inertia, ft4/ft 0.025 0.00007256 Sectional Area, ft2 per ft 0.67 0.02 EI (lb-ft2 per ft) 1.44E+07 3.03E+05 AE (lb, per ft) 3.86E+08 8.35E+07 Poisson's Ratio 0.3 0.3 Compressibility, C 0.027 0.127 Flexibility Ratio, F 1.217 57.122 Note: ground condition (very stiff ground with Em = 7,500 psi, νm = 0.3). Table 9-10. Parametric analysis set 5âvery stiff ground condition. Figure 9-34. Ratios of culvert deformations versus free-field deformations (parametric analysisâSet 5). Figure 9-35. Embedment effects on culvert maximum thrust/hoop forces (parametric analysisâSet 5). ⢠Youngâs Modulus, E/(1 â ν2) = 4.0E+06 psi ⢠Poissonâs Ratio, ν = 0.3 ⢠Thickness, t = 0.67 ft ⢠Moment of Inertia, I = 0.025 ft4/ft Five sets of parametric analyses have been performed con- sidering the following combinations of variables: (1) culvert sizes; (2) culvert sectional conï¬gurations; (3) soil stiffness; and (4) culvert burial depths. Table 9-11 below summarize speciï¬c parameters used in each case of analysis. The main purpose of this parametric analysis is to verify that the rectangular ï¬exibility ratio (Frec) developed in Equa- tion (9-14), Frec = (Gm / Ks) (w/h), is a proper representa- tion of the relative stiffness between the culvertâs racking stiffness and the groundâs racking stiffness. By using Frec, it is possible to accurately estimate the actual racking defor- mation of the culvert as long as the free-field ground defor- mation (Îfree-field) is known. of the analytical solutions discussed in Section 9-4. Figures 9-35 and 9-36 display similar results (normalized thrust forces and bending moments) presented in other parametric analysis cases even though the ground stiffness was signiï¬cantly changed (from Em = 3,000 psi to Em = 7,500 psi). 9.6.2.6 Parametric AnalysisâSet 6 Model AssumptionsâSet 6. The parametric analyses dis- cussed thus far focused on the ovaling behavior of culverts. In this section, a series of parametric analysis is performed for the rectangular and square shaped culverts. These culverts are as- sumed to be constructed with reinforced concrete. The sizes and geometry of these concrete box culverts are graphically presented in Figure 9-37. The concrete lining was modeled as continuous beam ele- ments in the ï¬nite difference, soil-structural interaction analysis having the following properties:
127 analysis. In this FLAC analysis, the culvert structure is in- cluded in the soil deposit model and subject to the same pseudo-horizontal acceleration used in the free-ï¬eld FLAC analysis mentioned in Step 1 above. Note that since Îs is related to Îfree-ï¬eld directly through Rrec, the racking ratio, the comparison therefore also can be made between the manually calculated Rrec = [2Frec/(1+Frec)], and Rrec computed from the FLAC analysis. 4. If the manually estimated racking deformations (or the Rrec values) are comparable to those computed by the soil- structure interaction FLAC analysis, then the simpliï¬ed procedure developed in Section 9.5.2 can be considered to be validated. Results of AnalysesâSet 6. Based on the results from the FLAC analysis (from both the free-field analysis run and the soil-structure interaction analysis run), the free- field racking deformations and the actual culvert racking deformations were obtained. Ratios of the culvert to free- field racking deformations are plotted for all five cases (for five different burial depths in each case) in Figures 9-38 through 9-42. Based on the data presented in these figures, it appears that burial depth does not have significant influ- ence on the racking deformation ratio for the rectangular type of rigid culverts. In the meantime, the structural racking stiffness (Ks) of the culvert structure in each case was determined by a simple frame analysis based on the properties of the culvert structure; the re- sults are presented in Table 9-12. Then the rectangular ï¬exibil- ity ratio (Frec) was calculated using Equation (9-14), and results also presented in Table 9-12 for each case. Figure 9-36. Embedment effects on culvert maximum bending moments (parametric analysisâSet 5). Figure 9-37. Various concrete box culvert sectional shapes and sizes used in the parametric analysisâSet 6. The veriï¬cation procedure is: 1. Determine the free-ï¬eld racking deformation of the ground (Îfree-ï¬eld). This was achieved in this analysis by ap- plying a pseudo-horizontal acceleration in the entire free- ï¬eld soil deposit in the FLAC analysis. Note that at this time the FLAC model is a free-ï¬eld soil deposit model that does not contain the culvert structure in it. The resulting free-ï¬eld racking deformations then can be directly read out from the output of the FLAC analysis. 2. Given Îfree-ï¬eld, the racking deformation of the culvert can be manually estimated by using the simple relationship presented in Equation (9-16), Îs = Rrec Îfree-ï¬eld. 3. The manually estimated racking deformation derived above then is compared to the actual racking deformation of the culvert from the soil-structure interaction FLAC
128 Structural Configurations and Soil Properties Case 1 10â x 10â Square Box, in Firm Ground (Em = 3,000 psi, ν m = 0.3) Case 2 10â x 10â Square Box, in Very Stiff Ground (Em = 7,500 psi, ν m = 0.3) Case 3 10â x 20â Rectangular Box, in Firm Ground (Em = 3,000 psi, ν m = 0.3) Case 4 10â x 10â Square 3-Sided, in Very Stiff Ground (Em = 7,500 psi, ν m = 0.3) Case 5 10â x 20â Rectangular 3-Sided, in Very Stiff Ground (Em = 7,500 psi, ν m = 0.3) Note: For each case, the effects of culvert embedment depth (of 50 feet, 30 feet, 20 feet, 10 feet, and 5 feet, measured from ground surface to top of the culvert roof) were studied. Table 9-11. Soil and structure parameters used in the analysis. Figure 9-38. Racking ratios from FLAC analysisâ Case 1. Figure 9-39. Racking ratios from FLAC analysisâ Case 2. Figure 9-40. Racking ratios from FLAC analysisâ Case 3. Figure 9-41. Racking ratios from FLAC analysisâ Case 4. The results show that for Case 1 the relative racking stiff- ness of the ground to the structure is about 1.0, suggesting that the structure would rack in conformance with the free- ï¬eld racking deformation in the ground. The results pre- sented in Figure 9-38 show clearly that the FLAC calculated racking deformations are about the same as the free-ï¬eld de- formations, validating the deï¬nition of ï¬exibility ratio (Frec) derived in Section 9-5. For Cases 2 through 5, the ï¬exibility ratios are all greater than 1.0, suggesting that the structure would deform more than the ground in the free-ï¬eld, and re- sults shown in Figures 9-39 through 9-42 support this theory. Figure 9-43 plots the racking ratio as a function of the ï¬ex- ibility ratio based on the results obtained from the FLAC analysis and then compares them with the recommended
129 ⢠For circular culverts and pipes subject to ovaling deforma- tions, the simpliï¬ed close-form solutions and procedure presented in Section 9.5.1 should provide reliable results under general conditions, with the following notes: â In selecting the design transient ground deformation pa- rameter for a culvert or pipe constructed at a signifi- cant depth below the ground surface, PGV is a better parameter in the deformation-based procedure than the site-adjusted PGA, because PGV can be used directly for estimating the shearing strain in the ground (Equa- tion 9-1). Discussions and recommendations on PGV values developed in Chapter 5 for retaining walls, slopes, and embankment should be used in evaluating the maxi- mum free-ï¬eld shearing strain in Equation (9-1). For cul- verts and pipes buried at relatively shallow depths (that is, within 50 feet of the ground surface), it is more reasonable to estimate the free-field shearing strain in the ground using the earthquake-induced shearing stress divided by the stiffness of the surrounding ground (Equation 9-2). â If a more accurate prediction of the maximum free-ï¬eld shearing strain is required, a more reï¬ned free-ï¬eld site response analysis (for example, using the SHAKE com- puter program) should be performed. â In using the simpliï¬ed approach, the no-slip interface assumption should be used in calculating the maximum thrust/hoop forces (Tmax based on Equation 9-11) in the culvert structure for conservative purposes. Results based design curve expressed by Equation (9-15), Rrec = [2Frec/ (1+Frec)]. The comparison shows reasonably good agreement between the recommended simple design solution charts and the results obtained from the numerical analyses. 9.7 Conclusions and Recommendations Simpliï¬ed seismic analysis procedures for evaluating culvert and pipe structures subjected to transient ground deforma- tions induced by ground shaking proposed in this chapter. The analysis procedures use a deformation-based methodology that can provide a more reliable prediction of culvert/pipe per- formance. The approach focuses on the deformations in the transverse section of the structure (that is, ovaling/racking de- formations) instead of the longitudinal axial/curvature defor- mations, due primarily to the general condition that typical culvert structures for transportation applications are of limited length, and as such it is in general unlikely to develop signiï¬- cant transient axial/curvature deformations along the longitu- dinal direction of the culvert structures. Based on the results of a series of parametric soil-structure interaction analysis taking various factors into considera- tion, the following conclusions and recommendations are provided: Figure 9-42. Racking ratios from FLAC analysisâ Case 5. Figure 9-43. Recommended design racking curve. Structural Racking Stiffness KS (kips/ft) Flexibility Ratio FREC Case 1 172 0.97 Case 2 172 2.4 Case 3 115 2.9 Case 4 57 7.3 Case 5 43 19.3 Table 9-12. Racking stiffness of culverts and flexibility ratios.
130 ⢠For rectangular shape culverts subject to racking deforma- tions, the simpliï¬ed procedure presented in Section 9.5.2 should provide reliable results under general conditions, with the following notes: â A series of parametric analysis was conducted verifying that the procedure can provide a reasonable estimate for the culvert racking deformations. To derive the internal forces in the structural elements, a simple frame analy- sis is all that is required (refer to Figure 9-10). â Based on the results of the parametric analysis, it ap- pears that burial depth has insignificant effects on the culvert racking deformations and therefore no further modifications to the procedure presented in Section 9.5.2 is necessary. ⢠The seismic effects of transient racking/ovaling deforma- tions on culverts and pipes must be considered additional to the normal load effects from surcharge, pavement, and wheel loads, and then compared to the various failure cri- teria considered relevant for the type of culvert structure in question. on the full-slip assumption tend to under-estimate the thrust/hoop forces. â In using the simpliï¬ed approach, the full-slip interface assumption should be used in calculating the maximum bending moments (Mmax, based on Equation 9-10) and culvert deformation (ÎDEQ, based on Equation 9-7) because it provides more conservative results than the no-slip interface assumption. A flexural type failure mode due to the combined effects of bending moment and thrust force must be checked for both rigid and flexible culverts. The flexural failure criteria may be established using the conventional capacity evaluation procedures for reinforced concrete or metals. Based on results from the soil-structure interaction analysis, the effect of shallow burial depth appears to be on the safe side, provided that the maximum free-field ground shearing strain is calculated at the most critical elevation (where the maximum ground shearing strain occurs, rather than the average ground shearing strain within the culvert depth proï¬le).