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79 CHAPTER 6 Conclusions Several critical issues relative to the design of concrete box- Both single- and multi-cell concrete box-girder bridges are girder bridges were identified at the beginning of the project. covered by this project. They may be cast-in-place or precast Methods exercised in this study that were intended to address and may be constructed segmentally or on falsework. these issues consisted of a survey of the state of practice, a re- · Appropriate levels of analysis and design. Selecting the view of published literature, analytical studies of global and type of global analysis that should be used for curved con- local response, discussions of the experienced research team crete box-girder bridges is one of the most important issues that attempted to reach a consensus on critical design re- addressed by this project. Published research shows that quirements, and a review by an advisory panel with expertise these types of bridges are most accurately analyzed using in this area of study. 3-D finite element or similar techniques. Unfortunately, Several conclusions were drawn from the research conducted these analysis methods are tedious and in general not prac- in this project. In many cases, these conclusions have trans- tical for production design work. Also, in many cases, more lated into recommended AASHTO LRFD Specification pro- simplified analysis methods will produce acceptable results. visions as presented in Appendix A. Analysis guidelines were To determine the range of applicability of various analysis also developed to assist designers in performing response methods, a detailed global analysis study was undertaken. analysis. These are provided in Appendix C. Other conclu- The first step was to identify a more simplified 3-D analy- sions found that current design practice was adequate and did sis approach that would yield results comparable to the not require a change. The following paragraphs discuss con- more detailed finite element technique. This was accom- clusions relative to the critical issues defined at the beginning plished with the grillage analogy approach. In this method, of the project. the bridge is simulated as a grillage of beam elements in the longitudinal and transverse direction. Guidelines for · Applicability. Curved concrete box-girder bridges are preparing the computer model, performing the analysis, used throughout the United States. Most modern bridges and interpreting results were developed and are included of this type are prestressed. A review of the state of practice in Appendix C. From the designer's point of view, this in the United States found that both single- and multi-cell analysis method has advantages over the finite element box-girder bridges are widely used. The predominate approach. Besides being a smaller and less computationally construction type in some West Coast states is multi-cell intense analytical model, the grillage analogy produces box-girder bridges cast-in-place on falsework. This type is results in terms of the structural members commonly con- also widely used throughout the United States. Single-cell sidered by the designer. This makes it easier to design these box-girder bridges are also common, but tend to dominate elements, whereas the finite element approach would in- the type of box-girder construction used on the East Coast. volve considerable post processing of analytical results to East Coast construction also uses more precast segmental accomplish the same goal. construction than is used on the West Coast where cast- Second, the limits of applicability of three analytical ap- in-place construction is more dominant, even when seg- proaches were assessed. The three methods considered were mental methods are used. Some states do not use this type 1. Plane Frame Analysis. This allows the bridge to be ana- of bridge on a regular basis. Curved spread box beams are lyzed as if it were straight. an emerging structure type, but are not widely used at this 2. Spine Beam Analysis. This is a space frame analysis in time. which the superstructure is modeled as a series of
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80 straight, chorded beam elements located along the cen- software is not used. Fortunately, the whole-width design terline of the superstructure. approach as described in the LRFD specifications was shown 3. Full 3-D Analysis. This includes several different sophis- to yield conservative results when used in conjunction with ticated approaches that include the grillage analogy de- the plane or spine beam approaches. This will greatly simplify scribed above as well as the finite element and other the effort of the designer in determining live load response. sophisticated approaches. When using this approach, it is important to distinguish Extensive parameter studies were performed that in- between the longitudinal response along each of the webs cluded the effects of structural framing (simply supported and the effect of torsion across the whole section. When or continuous), span lengths, radius of curvature, cross torsional response is being assessed by the whole-width section (including bridge width), and bearing configura- design approach, the number of live load lanes should be tion on the response of bridges. These studies included reduced to the actual number of lanes that can fit on the both grillage analysis and spine beam analysis for which cross-section and adjusted by the multiple presence factor plane frame analysis constituted the case of a bridge with a and dynamic load factor (for truck loading only). very large radius. These studies showed that the radius-to- When a 3-D model of the bridge (either a spine beam or span length ratio as represented by the central angle be- grillage analogy model) is being used, it is important to tween two adjacent supports was the dominant parameter consider the transverse position of prestress tendons. The that determined the accuracy of the various analysis meth- length of the various tendons will have an effect on friction ods. The span length-to-width ratio (aspect ratio) of the losses and tendons will also produce a transverse response superstructure also had a minor effect. Based on these pa- in the bridge superstructure. rameter studies, the following limits for the various types Vehicular effect may be assessed using the method pre- of analysis are recommended: scribed in the LRFD Bridge Design Specifications. Other 1. For central angles less than or equal to 12 degrees, plane load conditions such as centrifugal forces, breaking or frame analysis is acceptable. acceleration forces, wind, etc. should be determined 2. For central angles between 12 and 46 degrees and an according to the LRFD Specifications and then applied to aspect ratio above 2.0, spine beam analysis is required. the spine beam model. If the plane frame approach is being 3. For central angles between 12 and 46 degrees and an used, these loads may be analyzed in the same manner as aspect ratio less than 2.0, sophisticated 3-D analysis is if the bridge were straight. The effect of bridge supereleva- required. tion can usually be ignored. 4. For central angles greater than 46 degrees, sophisticated · Torsion design. The design of concrete box sections for 3-D analysis is required. torsion is covered in the current AASHTO LRFD Specifi- 5. For all bridges with otherwise unusual plan geometry, cations. However, some clarification of these requirements sophisticated 3-D analysis is recommended. is in order. These were discussed in the review of published · Section properties and member stiffness. The section literature included in Chapter 3. properties and member stiffnesses that should be used in Torsion demands usually translate to additional lateral the spine beam analysis and the grillage analogy analysis shear demands in the webs of concrete box-girders. These are critical and are discussed in the analysis guidelines pre- may be determined from both the spine beam and grillage sented in Appendix C. For the spine beam analysis, the analogy methods. cross-sectional area and the three rotational moments of In the case of the spine beam analysis, the torsion de- inertia are important. In the case of the grillage analogy, all mands are taken directly from the torsion forces generated six section properties of each beam member are required. in the spine beam. These forces must be transformed into Special formulae for some of these section properties are shear flow around the perimeter of the box section. This used to simulate various aspects of the behavior of a curved shear flow will increase the effective shear in one web while concrete box-girder bridge. This in turn requires special decreasing it in another. Webs should be designed for the interpretation of some of the results. combined flexural and torsional shear. · Critical position of live loads. The number and position In the case of the grillage analogy, the effects of torsion of the live load lanes in the transverse direction as well as on web shear are partially accounted for because each web their position along the longitudinal axis of the bridge are is explicitly included in the analytical model. However, be- critical for curved concrete box-girder bridges. Given the cause of the way torsional stiffness of the superstructure is number of possible load positions, it will be desirable to distributed to the individual longitudinal members of the use the live load generating capabilities of sophisticated grillage model, the total effect of torsion on the entire cross commercially available software to rigorously envelope section is not completely accounted for by the longitudinal the live load response. This can be a daunting task if such member shear demands. To correct for this deficiency, it is
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81 necessary to consider the torsional forces in each of the lon- shear and torsion, regional transverse bending can re- gitudinal members at a given longitudinal location in sult in the need for more stirrup reinforcement in the the grillage model and apply the sum of these torsions to the webs. Regional transverse bending also exacerbates flex- entire cross section to obtain residual shear flow about the ural cracking of the concrete cover as described in Item 2 perimeter of the section. This is done in a manner similar above. to that used for the spine beam. When this residual shear · Consideration of stresses at critical locations. Several flow is combined with the flexural shear in the extreme critical stresses should be considered in the design of longitudinal members, the correct demands to be used for curved concrete box-girder bridges. These include web shear design are obtained. 1. Axial stresses in the top and bottom slabs and the webs. The procedures to be used for torsion design for both the These stresses result from vertical flexure of the bridge spine beam and grillage analogy analysis methods are illus- between supports and the primary and secondary effects trated in the example problem included in Appendix B. of longitudinal prestressing. Regional transverse bend- · Tendon breakout. Extensive analytical studies were per- ing of the superstructure may also occur and should be formed to investigate lateral bursting stresses in curved considered when determining these stresses. Because concrete box-girder bridges with internal prestress tendons. the web lengths vary in a curved bridge, moments and The first step in these studies was to verify that the nonlin- flexural shears in each web may also vary. This effect is ear finite element models used could accurately predict lat- best captured in the grillage analogy approach. To best eral tendon breakout behavior observed in experimental capture it with the spine beam approach, prestress ten- studies performed at the University of Texas. The results of dons should be located at their correct transverse posi- the analyses compared well with the experimental results tions with respect to the bridge centerline. so that there is confidence that both the experimental and 2. Shear stresses in the webs. These stresses result from the analytical results have yielded accurate results. flexural and torsional behavior of the superstructure. Based on parameter studies conducted using these veri- Torsion results in shear flow around the perimeter of fied nonlinear finite element techniques and the results of the cross section that should be combined with the flex- the University of Texas studies, modifications to the spec- ural shear. In continuous superstructures or between the ifications for considering in-plane force are recommended. joints in precast superstructures, these shear forces result These include in diagonal tension stresses that can combine with the 1. A method for assessing the local lateral shear resistance to flexural tensile stresses resulting from regional trans- pullout. These provisions are the recommendations verse bending. Stirrup design may be accomplished by from the University of Texas, which were further veri- combining the reinforcing requirements for each of fied by the nonlinear finite element parameter studies these actions. At the joints in precast bridges, the shear conducted as part of this study. They also include pro- is carried by a shear friction mechanism. visions for considering the effect of construction toler- 3. Transverse stresses in the cross section. These stresses can ances, which have been shown by past failures to have a generally be determined using the same methods used significant effect on web performance and are discussed for a straight bridge. They govern the design of the deck in Chapter 5. and soffit. The transverse deck and soffit reinforcing 2. A method for checking flexural cracking of the unreinforced must also participate in carrying the shear flow gener- concrete cover over the inside of the prestress tendons. ated by torsion, but because concrete is often sufficient This is a new requirement that applies only to vertically for this purpose, this is often not a significant consider- stacked tendons. It is included to prevent maintenance, ation in design. architectural, and structural problems that can arise due 4. Flexural and lateral shear stresses in the vicinity of pre- to longitudinal cracking of the web. The results are used stress tendons. Complex stresses are developed in the to determine the need for web and duct tie reinforce- webs of curved concrete box-girders due to the lateral ment. Vertical duct stacks are limited to three tendons forces developed by the curvature of prestress tendons. high and concrete cover over the inside of the ducts Simplified methods for assessing these effects have been should be maximized. Generic web and duct tie rein- developed and are included in the recommended LRFD forcement details are included in the commentary. specifications and commentary. 3. A method for calculating the regional transverse bending Because design for the above forces is often optimized, moments within a web. These moments result from the it is prudent to evaluate these forces at several longitudinal regional transverse bending of a web between the top locations along the length of the bridge. Prestress forces and bottom slab of the bridge due to lateral prestress and path location, web and slab thicknesses, and the size forces. When combined with global forces such as flexural and spacing of stirrups can be designed accordingly.
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82 · Bearing load and bearing movement considerations. Both analogy and finite element studies performed as part of this the spine beam and grillage analogy methods of analysis project demonstrated that interior diaphragms have a min- will accurately predict elastic bearing forces if used accord- imal effect on the global response of a curved concrete box- ing to the criteria outlined in the proposed AASHTO girder bridge with a 400-ft radius and 300-ft span lengths. LRFD Bridge Design Specifications and Commentary and Therefore, it is proposed that the requirement that interior the Analysis Guidelines included in Appendix C. Because diaphragms be included in bridges with a radius less than of the curvature of the structure, the bearing forces at any or equal to 800 feet be eliminated. It is recommended that longitudinal position along the bridge will vary across the end diaphragms still be used at all supports. width of the bridge. · Post-tensioning sequence. Because the curvature of tendons In addition to this, both field experience and time- can increase the transverse bending of the super structure dependent analysis show that the bearing forces will and result in tensions on the inside of the curve and com- change over time. The extent of this change is not accu- pression on the outside of the curve, it is recommended rately determined by currently available time-dependent that at least one tendon on the inside of the curve be software because of the treatment of torsion creep in these stressed first. programs, but software that takes into account axial creep With respect to varying the final distribution of prestress is thought to give conservative results. In lieu of a time- forces across the width of the bridge, there does not seem dependent analysis, elastically determined abutment dead to be any significant advantage in doing this. Although load torsions should be increased by 20%. It is recommended webs to the inside of the curve are shorter and thus theo- that bearing force capacities be designed to accommodate retically subject to less dead load and live load bending both initial and long-term conditions. forces, decreasing prestress forces for this web will be over- Methods for addressing bearing design when bearing come by the transverse bending of the bridge that will put forces are excessive (i.e., either too high or too low) may in- the inside web in tension. Thus it is thought to be impor- clude, but not be limited to, one or more of the following: tant to model tendons in their correct transverse position 1. Size individual bearings to accommodate the calculated for analysis, but a relatively even distribution of prestress range of bearing forces. forces is desirable. It is theoretically more important for the 2. Specially design bearings so that they will not be dis- designer to consider the incidental distribution of prestress placed if the applied load goes into tension or very low forces as allowed by some construction specifications. compression. · Skew effects. Analytical studies were performed to consider 3. Provide ballast in the superstructure to ensure that the the effect of skew at the abutments on the overall response envelope of bearing forces is within an acceptable range. of the bridge. It is commonly known that skew will affect 4. Reshore the structure at its bearing locations prior to the shears in the web near the obtuse corner of a skewed setting the bearings and then release the shoring after abutment support. The point of the study was to determine the bearings are set. if bridge curvature altered the relationship between the rel- 5. Use an outrigger diaphragm to increase the eccentricity ative response of a skewed and non-skewed abutment. of the individual bearings. Two skew cases were studied. One case was where the skew 6. Place the bearing group eccentric to the centerline of the occurred at only one of the abutments and the other case superstructure in order to make the individual bearing was where both abutments were skewed but in opposite forces more equal. directions. The second case is the likely orientation of a 7. Select bridge framing to better control bearing forces. curved bridge that crosses over an obstruction that is lin- Balancing the center and end span lengths can mitigate ear in orientation. In both cases, it was found that the rela- bearing problems. tionship between the response of a non-skewed support Considering the curvature of long bridges in a spine and the skewed support followed the same relationship as beam analysis can mitigate excessive design movements at for a straight bridge. Thus, it was concluded that existing the bearings due to temperature change and possibly elim- skew correction factors apply to curved concrete box-girder inate the need for interior expansion joints. Care should be bridges analyzed by the spine beam method. taken that bearing travel is through the center of movement The effect of skew on interior supports was not studied so that binding of the shear keys does not occur. Prestress nor were the effects of different abutment skew configura- shortening may occur along a slightly different orientation. tions. In all cases, a grillage analogy analysis would capture · Diaphragms. Current AASHTO LRFD provisions require any effect of skew. This method should be used to analyze that diaphragms be used for curved box-girder bridges any curved concrete box-girder bridges with large skew an- with a radius of less than 800 feet, but also allows that they gles at the interior supports or abutment skew configura- be omitted if justified by analysis or tests. Analytical grillage tions that vary significantly from those studied.
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83 · Lateral restraint issues. Horizontal curvature may result · Construction methods. The effect of construction meth- in force demands in the lateral direction at the supports if ods on the behavior of curved box-girder bridges is lateral restraint is present and is modeled as rigid elements critical. However, the analysis methods studied apply to for computer analysis. Such may be the case for supports staged construction analysis as well as cast-in-place on consisting of integrally cast abutments or piers. In these falsework construction. The same parameters can be used cases, lateral restraint should be modeled as the stiffness of to select the most appropriate analysis method except the restraining element under consideration. In the case of that time-dependent analysis should be used. Commer- bearings, however, steel or concrete shear keys usually pro- cially available software does not consider torsion creep, vide lateral restraint. These are usually provided with a but should yield generally conservative results and is small transverse gap to prevent binding. This gap is large adequate for design until more sophisticated software is enough to prevent lateral restraint and, for gravity load developed. purposes, should be modeled as a lateral force release. The · External post-tensioning deviators. The use of precast key to properly considering lateral restraint is to accurately construction results in less weight and quicker onsite model the actual condition and to use either spine beam or assembly and is thus increasing in popularity. Deviation grillage analogy analysis. blocks or saddles for external prestress tendons in curved · Thermal effects. Thermal movement and prestress shorten- precast concrete box-girder bridges may be designed in the ing will result in movements in different directions at the same manner as for straight bridges using strut-and-tie expansion joints in curved structures. This difference should methods or as recommended by an experimental study at be reflected in a properly conducted spine beam or grillage the University of Texas (Beaupre et al., 1988). For LRFD analysis. Bearings should be capable of travel through the design of deviators, a load factor of 1.7 should be used for center of movement, although normal gaps provided in the prestress deviator force and capacity reduction () factors shear keys will allow for slight variations in movement. should be 0.9 for direct tension and flexure and 0.85 for · Time-dependent effects. Because of the interaction between shear. It is recommended that reinforcing bar sizes in de- bending moment and torsion in curved bridges, consider- viation saddles be limited to #5s to ensure the proper de- ation of time-dependent effects is important. However, velopment of this reinforcement. rigorous 3-D analysis to determine the time-dependent It is recommended that deviation saddles in tightly effects of torsion is not present in commercially available curved bridges be continuous across the bottom soffit. An- software. This requires a reliance on the time-dependent other consideration for curved bridges is that straight seg- software available. Fortunately, torsion creep is expected to ments of tendons cannot rub against the interior of the mitigate the effect that has been observed at the bearings, webs. If necessary, the designer should include extra devia- and thus this software will tend to yield conservative results tion blocks or saddles to prevent this from happening. A de- for bearing force redistribution. It can be used for design viation saddle design example, which is reproduced from purposes until improved software is developed. the University of Texas report, is included in Appendix B. Vertical construction cambers can use the results from · Friction loss/wobble. The current formulae for determin- currently available time-dependent software. In fact, many ing prestress losses due to friction and wobble apply to design engineers interviewed claim to have had good re- curved bridges if the 3-D effects of angle change and tendon sults from 2-D time-dependent analysis. However the length are considered. It is necessary to explicitly consider bending of tall columns and twisting of the superstructure the difference in tendon length in individual webs and thus had to be approximated using elastic 3-D spine beam prestress tendons should be modeled in their actual trans- analysis. verse location in a 3-D spine beam or grillage analogy In the case of curved bridges, horizontal cambers may be analysis. Friction losses should be based on a tendon curved required for segmental construction. A curved concrete in space when a curved bridge is being designed using 2-D box-girder bridge with relatively tall piers in California that analysis techniques. was constructed by the segmental cantilever method re- · Web and flange thickness limits. It is recommended that quired a horizontal camber of approximately 3 inches at webs and flanges be designed based on structural and the pier. In other words, the pier had to be constructed constructability considerations. No minimum thickness 3 inches out of plumb. requirements are recommended by this study.