Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges (2008)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

8A large body of published literature is related to curved concrete box-girder bridges. Some of the documents most important to this project are discussed below. Codes and Design Standards Currently, there is no U.S. code specifically developed for the design of curved concrete box-girder bridges. AASHTO LRFD Bridge Design Specifications provide comprehensive specifica- tions with commentary for the design of highway bridges. Our review of Codes and Design Standards is summarized below. AASHTO, 2004, AASHTO LRFD Bridge Design Speci- fications, 3rd Edition with Interims, AASHTO, Washing- ton, D.C. A number of sections apply to the design issues associated with curved concrete box-girders. Selected specifications articles are as follows: • Article 4.6.1.2.2, “Single-Girder Torsionally Stiff Super- structures,” allows for the analysis of horizontally curved, torsionally stiff single-girder superstructures for global force effects as a curved spine beam. • Article 4.6.1.2.3, “Multicell Concrete Box-Girders,” allows for the design of horizontally curved cast-in-place multicell box-girders as single-spine beams with straight segments, for central angles up to 34°within one span, unless concerns about force effects dictate otherwise. • Article 4.6.3.4, “Cellular and Box Bridges,” allows for the refined analysis of cellular bridges by any of the methods specified in Article 4.4, “Acceptable Methods of Structural Analysis,” except the yield line method, which accounts for the two dimensions seen in plan view and for the model- ing of boundary conditions. Models intending to quantify torsional warping and/or transverse frame action should be fully three-dimensional. • Article 4.7.4.3, “Multispan Bridges,” specifies the minimum requirements for the seismic analysis of multispan bridges. Analysis requirements are based on the classification of a bridge as “regular” or “irregular.” The classification of a curved bridge includes the maximum subtended angle and whether the spans are continuous or are multiple simple-spans. • Article 5.4.6, “Ducts,” specifies the requirements for duct material and curvature. • Article 5.8, “Shear and Torsion,” specifies comprehensive design procedures for flexural shear and torsion. The mod- ified compression field theory is specified for flexural re- gions. Strut-and-tie models are specified for regions near discontinuities. Alternative design procedures are permit- ted for segmental bridges. • Article 5.9.1.6, “Tendons with Angle Points or Curves,” cross references Articles 5.4.6 and 5.10.4 for duct curvature and stress concentration considerations, respectively. • Article 5.9.5.2.2.2, “Friction,” specifies the friction loss due to curvature and includes specific requirements for deter- mining the total 3-D angle change as typically found in curved girders with draped tendons. • Article 5.10.3.3.3, “Curved Post-Tensioning Ducts,” spec- ifies the clear distance between curved ducts as required for tendon confinement as specified in Article 5.10.4.3 but not less than that required for straight ducts. • Article 5.10.4.3, “Effects of Curved Tendons,” specifies that, where tendons are placed in curved webs, additional cover and/or confinement reinforcement shall be provided. • Article 5.10.4.3.1, “In-Plane Force Effects,” defines the in- plane deviation force effects due to the change in direction of the tendon as Fu-in = Pu/R where Pu is the factored tendon force and R is the radius of curvature of the tendon. Specific requirements for local lateral shear on the unreinforced concrete cover are given and neglect any increase in lateral shear capacity for widely spaced tendons. Where the factored in-plane deviation force exceeds the lateral shear resistance C H A P T E R 3 Published Literature Review

of the concrete cover, tieback reinforcement is required. Where stacked ducts are used in curved girders, the moment resistance of the concrete cover, acting in flexure, shall be investigated, but no specific methodology or stress require- ments are provided. For curved girders, the global flexural effects of out-of-plane forces shall be investigated. • Article 5.13.2.2, “Diaphragms,” requires the use of di- aphragms at abutments, piers, and hinge joints. Interme- diate diaphragms may be used in beams in curved systems or where necessary to provide torsional resistance. Inter- mediate diaphragms shall be used in curved box-girders with an inside radius of less than 800 feet. Diaphragms may be omitted where tests or structural analysis show them to be unnecessary. • Article 14.4.1, “General,” specifies the movement require- ments for joints and bearings. It includes the requirement to consider the effects of curvature, skew, rotations, and support restraint. The commentary includes additional discussion pertinent to curved bridges. With respect to torsion design, a detailed review of the specifications was performed. The following briefly describes the design methods, outlines the basic steps of designing a box section for the combined actions of flexural shear, torsion, and moment, and includes a discussion (in italics) where fur- ther guidance is required in interpreting or applying the LRFD specification. Design Methods Two basic design methods, specified in Articles 5.8.3 and 5.8.6, depend on construction method and structure type. A sectional model using the modified compression field theory with a variable angle truss model is the basis of Article 5.8.3 and applies in most cases. Article 5.8.6 contains the flexural shear and torsion provisions specific to segmental post-tensioned concrete box-girder bridges. A conservative expression for the concrete contribution and a 45° truss model are assumed. General Comment – There is a minor conflict between Arti- cles 5.8.3 and 5.8.6. Article 5.8.3 states that Article 5.8.6 may be used for segmental post-tensioned concrete box-girder bridges while Article 5.8.6 states that Article 5.8.6 shall be used for seg- mental post-tensioned concrete box-girder bridges in lieu of Article 5.8.3. It needs to be clarified whether Article 5.8.6 is a permissible or mandatory procedure for segmental bridges. Additionally, Commentary Article C5.8.6.1 states, “For types of construction other than segmental box-girders, the provisions of Article 5.8.3 may be applied in lieu of the provisions of Arti- cle 5.8.6.” It appears that the word may should be replaced by shall, unless the intent is to permit Article 5.8.6 as an alternative design method for bridge types other than segmental. Design Steps (General Sectional Model) The following outlines the basic steps of designing the exterior web of a box section for the combined actions of flex- ural shear, torsion, and moment. It is based on the provisions of Article 5.8.3 and, therefore, does not cover the steps for a segmental post-tensioned concrete box-girder bridge. Step 1 – Determine the Controlling Load Cases Determine the controlling load cases for the applicable strength limit states. Consider the concurrent actions on the section. As a minimum, consider the following two cases: 1. Maximum flexural shear and concurrent actions 2. Maximum torsion and concurrent actions Perform Steps 3 through 7 separately for each the above cases and any additional cases that may potentially govern the design. Step 2 – Determine the Cross-Section Parameters Acp – total area enclosed by outside perimeter of concrete cross section (Article 5.8.2.1) pc – the length of the outside perimeter of concrete cross section (Article 5.8.2.1) (Comment – LRFD Article 5.8.2.1 does not address the case when the thickness of the flange of a non-segmental box sec- tion is less than the effective web width. This is addressed in Article 12.2.10 of the Segmental Guide Specification (Reference 2) and LRFD Article 5.8.6.3 for segmental bridges.) Ao – area enclosed by the shear flow path (in.2) (Article 5.8.2.1) ds – the length of the torsional shear flow path on the exte- rior web (in.) (Commentary 5.8.2.1) ph – perimeter of the centerline of the closed transverse tor- sion reinforcement (in.) (Article 5.8.2.1) bv – effective web width (in.) (Article 5.8.2.9) dv – effective shear depth (in.) (Article 5.8.2.9) Step 3 – Check the Web Width Verify that the effective web width is adequate to prevent web crushing: Vu ≤ φVn Vn 0.25f b d +V (Equation 5.8.3.3-2)c v v p≤ ′ A Pc 2A b (Equation -5)cp2 o v/ . . .≤ 5 8 2 1 9

(Comment – It may be prudent to reduce Vn by the torsional shear when web crushing governs the capacity. Provisions for considering the combined flexural shear and torsional shear are required by Article 12.3.1 of the Segmental Guide Specification and LRFD Equation 5.8.6.5-3 for segmental bridges.) Increase web width if Equation 5.8.3.3-2 is not satisfied. Step 4 – Calculate Shear Stress Check if torsion must be considered: Tu > 0.25φTcr (Equation 5.8.2.1-3) For Tu > 0.25φTcr, calculate the equivalent factored shear force, Vu, acting on the web where the flexural shear and the torsional shear are additive as follows: Vu = Vu (flexure) + Tuds/(2Ao) (Equation 5.8.2.1-7) (Comment – Vu is determined for a single girder and Tu is act- ing on the total cross section.) General Comment – The fifth paragraph of Commentary C5.8.2.1 regarding the equivalent factored shear force would benefit from additional clarification. There is a mention of a stress limit for the principal tension at the neutral axis of the sec- tion but the specific code section was not referenced. Article 5.8.5 provides limits on the principal tensile stress in the webs of seg- mental concrete bridges at the Service III limit state and during construction. As the principal stresses are checked using service load, Vu is not applicable. It appears that the intent of the equiv- alent factored shear force is to consider the increased shear force and resulting shear stress due to torsion in calculating εx, θ, β, and Vc. It appears that equivalent factored shear force should not be considered in determining the required shear capacity, φVn, or the required tensile capacity specified in Article 5.8.3.5. Using the equivalent factored shear force, Vu, acting on the exterior web where the flexural shear and the torsional shear are additive, calculate the shear stress as follows: vu = (Vu − φVp)/(φbvdv) (Equation 5.8.2.9-1) (Comment – Note that vu includes the effects of flexural shear and torsional shear.) Step 5 – Calculate and x and Find  and  Calculate using vu calculated in Step 4. This accounts for the increased shear stress due to torsion and will be used to determine θ and β from Table 5.8.3.4.2-1. v /fu c′ v / fu c′ T f A p f f cr c cp c pc c = ′ + ′ 0 125 1 0 125 2 . . (Equation 5.8.2.1-4) (Equation 5.8.3.4.2-1) using the equivalent factored shear force, Vu, determined in Step 4. (Comment – This simply adds the tensions due to flexural shear and to torsion for the exterior web where the flexural shear and torsional shear are additive and appears to be con- servative. Collins and Mitchell (1980) pages 399–400, use an equivalent longitudinal tension for combined flexural shear and torsion equal to the square root of the sum of the squares of the individually calculated tensions for flexural shear and for torsion.) If whole-width design is used, forces acting on the total sec- tion are applied. In this case, the shear force is conservatively taken as the equivalent factored shear force, Vu, determined in Step 4 multiplied by the total number of webs. (Comment – Whole-width design is not specifically addressed and would benefit from clarification in this area.) This accounts for the increased longitudinal tensile force due to torsion and will be used to determine θ and β from Table 5.8.3.4.2-1. Using the calculated values of and εx, find θ and β from Table 5.8.3.4.2-1. Step 6 – Determine Required Spacing of Stirrups The amount of transverse reinforcement required for shear is found from Vu ≤ φVn (Comment – Clarify that the factored flexural shear, not the equivalent factored shear force, is used for Vu, as the transverse reinforcement for torsion is determined separately.) Vn = Vc + Vs + Vp (Equation 5.8.3.3-1) Where Vc = 0.0316β bvdv (Equation 5.8.3.3-3) Vs = (Avfydvcot θ)/s (Equation C5.8.3.3-1) Av/s = Vs/(fydvcot θ) The amount of transverse reinforcement required for tor- sion is found from Tu ≤ φTn ′fc v /fu c′ Calculate εx u v u u p p M d N V V A = + + − − | | . . | | cot0 5 0 5 θ s po s s p ps f E A E A ⎛⎝⎜ ⎞⎠⎟ ⋅ +2 ( ) 10

Where Tn = (2AoAtfycot θ)/s (Equation C5.8.3.6-2) At/s = Tn/(2Aofycot θ) For the exterior web of a box section, the combined area of both stirrup legs in the web, Astirrups, contributes to Av and At, therefore the maximum spacing of the stirrups, Smax, is given by: Smax = Astirrups/[(Av/s)flexural shear + (Av/s)torsion] Step 7 – Check the Longitudinal Reinforcement The required tensile capacity of the longitudinal reinforce- ment on the flexural tension side of the member is found from Equation 5.8.3.5-1. (Comment – Clarify that the flexural shear, not the equivalent factored shear force, is used for Vu, as the additional longitudinal reinforcement for torsion is determined separately.) The longitudinal reinforcement required for torsion, in addition to that required for flexure, is found from Al = Tn ph/(2Aofy) (Equation 5.8.3.6.3-2) Comments: Article 5.8.3.6.3 would benefit from a clarification that Al is also in addition to the required tensile capacity from Equa- tion 5.8.3.5-1 when Equation 5.8.3.5-1 exceeds the longitudinal reinforcement required for flexure. The distribution of Al within the cross section needs to be clar- ified. Article 5.3, “Notation,” and Article 5.8.6.4, “Torsional Reinforcement,” define it as the area of longitudinal torsion re- inforcement in the exterior web of the box-girder, which appears to be incorrect. LRFD Equations 5.8.3.6.3-2 and 5.8.6.4-3 are essentially identical to the equation in Article 12.3.8 of the Seg- mental Guide Specification, which specifies that Al shall be dis- tributed around the perimeter of the closed stirrups. Prestressing steel should also be permitted to satisfy Equa- tion 5.8.3.6.3-2 similar to Article 12.3.8 of the Segmental Guide Specification and LRFD Commentary Article C5.8.6.4 for seg- mental bridges. The area of longitudinal torsion reinforcement in the flexural compression zone should be permitted to be re- duced similar to Segmental Guide Specification Article 12.3.9 and LRFD Equation 5.8.6.4-4 for segmental bridges. Design Steps (Segmental Box-Girder) The following outlines the basic steps of designing the exterior web of a box section for the combined actions of flexural shear, torsion, and moment. It is based on the provi- sions of Article 5.8.6 and, therefore, applicable to a segmental post-tensioned concrete box-girder bridge. Step 1 – Determine the Controlling Load Cases The design for flexural shear and torsion in segmental bridges shall be performed at the strength limit state per Ar- ticle 5.8.6.2. The shear component of the primary effective longitudinal prestress force, Vp, shall be added as a load effect with a load factor of 1.0. The component of inclined flexural compression or tension, in the direction of the applied shear, in variable depth members shall be considered when deter- mining the design factored shear force. In accordance with Article 5.8.5, principal stresses at the neutral axis of segmental bridges shall not exceed the tensile stress limits of Table 5.9.4.2.2-1 at the Service III limit state and Table 5.14.2.3.3-1 during construction Determine the controlling load cases for each of the three applicable limit states separately (Strength, Service III, and during construction). Consider the concurrent actions on the section. As a minimum, consider the following two cases for each of the limit states: 1. Maximum flexural shear and concurrent actions 2. Maximum torsion and concurrent actions Perform Steps 3 through 6 separately for each of the above Strength cases and any additional Strength cases that may govern the design. In Step 4, check the principal stresses separately for each of the above Service III and construction cases and any additional Service III and construction cases that may govern the design. Step 2 – Determine the Cross-Section Parameters Ao – area enclosed by the shear flow path (in.2) (Arti- cle 5.8.6.3) be – effective width of shear flow path, but not exceeding the minimum thickness of the webs or flanges compris- ing the closed box section.(in.) be shall be adjusted to account for the presence of ducts as specified in Arti- cle 5.8.6.1. (Article 5.8.6.3) be may be taken as Acp/pc (Article 5.8.6.3) Acp – area enclosed by outside perimeter of concrete cross section (in.2) (Article 5.8.6.3) pc – the outside perimeter of concrete cross section (in.) (Article 5.8.6.3) ph – perimeter of the centerline of the closed transverse tor- sion reinforcement (in.) (Article 5.8.6.4) 11

bv – effective web width (in.) (Article 5.8.6.5) de – effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) (Article 5.8.6.4) dv – effective shear depth (in.) (Article 5.8.6.5) Step 3 – Check if Torsion Must be Considered Check if torsion must be considered: Tu > 1⁄3φTcr (Equation 5.8.6.3-1) Tcr = 0.0632K 2Aobe (Equation 5.8.6.3-2) ≤ 2.0 (Equation 5.8.6.3-3) fpc = Unfactored compressive stress in concrete after prestress losses have occurred either at the centroid of the cross section resisting transient loads or at the junction of the web and flange when the centroid lies in the flange. (Article 5.8.6.3) Step 4 – Check the Web Width Verify the effective web width is adequate to prevent web crushing: Vu ≤ φVn Vn ≤ 0.379 bvdv (Equation 5.8.6.5-2) Vu/(bvdv) + Tu/(2Aobe) ≤ 0.474 (Equation 5.8.6.5-3) (Comment – It appears that Vu and Tu should be replaced by Vn and Tn, respectively, in this equation to be consistent with Article 12.3.1b of the Segmental Guide Specification.) Increase web width if either Equation 5.8.6.5-2 or Equa- tion 5.8.6.5-3 is not satisfied. Check the allowable principal tensile stress for Service Limit State III and during construction in accordance with Article 5.8.5. Consider the compressive stress due to vertical tendons in the webs. Increase the web width or the vertical prestressing force in the web if the allowable principal stresses are exceeded. Step 5 – Determine Required Spacing of Stirrups The amount of transverse reinforcement required for shear is found from Vu ≤ φVn Vn = Vc + Vs (Equation 5.8.6.5-1) ′fc 0.5 ′fc 0.5 K= 1+ f f pc c0 0632. ′ ′fc Where Vc = 0.0632K bvdv (Equation 5.8.6.5-4) Vs = Avfydv/s (Equation 5.8.6.5-5) Av/s = Vs/(fydv) The amount of transverse reinforcement required for tor- sion is found from Tu ≤ φTn Where Tn = 2AoAvfy/s (Equation C5.8.6.4-2) Av/s = Tn/(2Aofy) For the exterior web of a box section, the combined area of both stirrup legs in the web, Astirrups, contributes to the trans- verse hoop reinforcement for flexural shear and torsion, there- fore, the maximum spacing of the stirrups, Smax, is given by Smax = Astirrups/[(Av/s)flexural shear + (Av/s)torsion] When vertical tendons are provided in the web, the design yield strength for flexural shear and torsion design shall be taken in accordance with Article 5.8.2.8. Step 6 – Check the Longitudinal Reinforcement The minimum additional longitudinal reinforcement re- quired for torsion (in addition to that required for other con- current actions), shall satisfy Al = Tuph/(2φAofy) (Equation 5.8.6.4-3) Comment: The distribution of Al within the cross section needs to be clarified. Article 5.3, “Notation,” and Article 5.8.6.4, “Torsional Reinforcement,” define it as the area of longitudinal torsion reinforcement in the exterior web of the box-girder, which appears to be incorrect. Article 5.8.6.4 contains a con- flicting statement that Al shall be distributed around the perim- eter of the closed stirrups in accordance with Article 5.8.6.6 which appears to be correct. LRFD Equation 5.8.6.4-3 is essentially identical to the equation in Article 12.3.8 of the Seg- mental Guide Specification which specifies that Al shall be dis- tributed around the perimeter of the closed stirrups. In determining the required amount of longitudinal rein- forcement, the beneficial effect of longitudinal prestressing is taken into account by considering it equivalent to an area of reinforcing steel with a yield force equal to the prestressing force. (Commentary Article C5.8.6.4) Subject to the minimum reinforcement requirements of Article 5.8.6.6, the area of additional torsion reinforcement in ′fc 12

the flexural compression zone may be reduced by an amount equal to Mu/(0.9defy) (Equation 5.8.6.4-4) AASHTO (2003a) Guide Specifications for Horizontally Curved Steel Girder Highway Bridges, American Associ- ation of State Highway and Transportation Officials, Washington, D.C. This is a recently published AASHTO Guide Specification for curved steel bridges, including box-girders. It has been sug- gested that the design specification contained therein be used as a model for NCHRP Project 12-71. This specification discusses design philosophy and limit states and includes provisions for loads; structural analysis; design of flanges, webs, shear connec- tors, bearings, splices and connections; deflections; and con- structability. It also includes a construction specification and design examples for both an I girder and box-girder bridge. AASHTO (1999) Guide Specifications for Design and Con- struction of Segmental Concrete Bridges, 2nd Edition with Interims, American Association of State Highway and Transportation Officials, Washington, D.C. This second edition of the guide specifications for segmen- tal concrete bridges was prepared for use in conjunction with the Standard Specifications for Highway Bridges (non-LRFD), and subsequent interim revisions to these specifications. This publication, which was developed by a broad-based commit- tee organized by the American Segmental Bridge Institute, embodies several concepts, which are significant departures from previous design and construction provisions. It is based on recent research in the United States and abroad. The com- mittee included representatives of state DOTs, the FHWA, academicians, consulting engineers, contractors, and suppliers. Some of the details of this specification were discussed above. Design Philosophy A number of books and papers have been written about the design of concrete box-girder bridges. Many of these discuss the effect of horizontal curvature on the behavior of these bridges. A few of these publications are discussed below. Menn, C. (1990) Prestressed Concrete Bridges, ISBN 3-7643- 2414-7, Birkhauser Verlag, Basel This book provides engineers with a comprehensive overview of the fundamental principles governing the design and construction of concrete bridges. The content is based on the author’s direct experience gained from the design and construction of bridges in Switzerland. Although much of the content is based on Swiss standards and practices, the mate- rial stresses fundamental principles. This book covers straight, skewed, and curved bridges consisting of both open sections and closed box sections. The book addresses issues applicable to horizontally curved box-girder bridges. Article 4.6.4, “Detailing,” discusses the deviation forces generated by curved post-tensioning tendons. The deviation force per unit length is determined as the tendon force divided by the radius of curvature of the tendon. An example of the regional transverse bending moment in the web of a horizon- tally curved member is presented. Article 5.1.4, “Structural Models for Bridge Superstruc- tures,” provides guidance on developing analytical models that can be applied to curved box-girder bridges. For example, techniques for developing grillage models of multi-cell box- girders are presented. Article 5.3.2(b), “Web Design for Shear and Transverse Bending,” presents a rational method for the design of webs subject to combined shear and regional transverse bending, a condition that occurs in horizontally curved post-tensioned box-girder bridges. The method is based on Swiss practice and neglects the concrete contribution to the shear capacity. Shear- regional transverse bending interaction diagrams are presented. Article 5.3.4, “Diaphragms,” discusses the function, neces- sity, and design of internal diaphragms. Diaphragms are rec- ommended at abutments and piers. The use of intermediate diaphragms is usually not necessary in straight and lightly curved box-girder bridges. Article 6.1.3(b) “Influence of Girder Curvature,” discusses the qualitative difference in superstructure displacements due to temperature and shrinkage versus longitudinal prestress- ing in horizontally curved bridges. Article 7.6, “Curved Girder Bridges,” is devoted entirely to horizontally curved bridges. The article includes subsections on Conceptual Design, Analysis, Transformation of Torque into Torsional Sectional Forces, Prestressing, and Prestress- ing Concept and Tendon Layout. The discussion of the conceptual design of curved box- girders points out the role of torsion in design and how, at ultimate loads, torsion and bending moment can be redistrib- uted. The effect of torsion on bearing forces may require the bearings to be placed away from the webs. Expansion bearings must be able to accommodate both temperature and prestress shortening displacements, which will be in different directions. The book presents a simplified method for analyzing curved bridges iteratively. The method does not satisfy com- patibility equations exactly, but greatly simplifies the compu- tational effort. An example is given. Simple vector diagrams are presented to illustrate how torsional section forces are developed by a variation in the direction of longitudinal bend- ing moments due to the curvature of the superstructure and resisted by shear flow around the perimeter of the box section. 13

Prestressing can produce longitudinal and transverse bend- ing and shear forces as well as torque in curved box-girder bridges. Torsion, which can increase flexural stresses, must be considered in determining the required prestressing force. Prestressing can also be used to enhance torsional and trans- verse bending resistance, although this is often avoided for economic reasons. Sennah, K. M., and Kennedy, J. B. (2001) “State-of-the-Art in Curved Box-Girder Bridges,” Journal of Bridge Engi- neering, ASCE, Vol. 6, No. 3, pp. 159–167. The objective of this paper was to provide highlights of the most important references related to the development of cur- rent guide specifications for the design of straight and curved box-girder bridges. As such, it provided an excellent bibliog- raphy from which to identify other papers that were reviewed in detail. Subjects discussed in this review included (1) different box-girder bridge configurations, (2) construction issues, (3) deck design, (4) load distribution, (5) deflection and camber, (6) cross-bracing requirements, (7) end diaphragms, (8) thermal effects, (9) vibration characteristics, (10) impact factors, (11) seismic response, (12) ultimate load-carrying capacity, (13) buckling of individual components forming the box cross section, (14) fatigue, and (15) curvature limitations provided by the codes for treating a curved bridge as a straight one. The literature survey presented herein encompasses (1) the construction phase, (2) load distribution, (3) dynamic response, and (4) ultimate load response of box-girder bridges. ASCE Committee on Construction Equipment and Tech- niques (1989) “Concrete Bridge Design and Construction in the United Kingdom,” Journal of Construction Engineer- ing and Management, Vol. 115, No. 4, pp. 618–635. The design and construction of concrete bridges in the United Kingdom has changed rapidly during recent decades. Better analytical methods, increased mechanization, and better planning in the construction of these bridges have brought this about. However these steps have also resulted in new problems for the engineer, contractor, and supervisor. This paper shows the different approaches on several factors. The paper is divided into three parts as follows: 1. The design of bridges in classes for span and type with ref- erence to the pertinent factor for that design; 2. The contractor’s approach to construction that illustrates the need for flexibility in the construction method in order to meet contract deadlines; and 3. The views of the supervising engineer and his means of achieving a balance between the designer’s intentions and the contractor’s proposals. Schlaich, J., and Scheef, H. (1982) Concrete Box-Girder Bridges, ISBN 3 85748 031 9, International Association for Bridge and Structural Engineering, Zurich, Switzerland This publication is the outcome of a comprehensive survey of concrete box-girder bridges. The publication is divided into three main parts, “Design,” “Structural Analysis,” and “Dimensioning and Detailing.” A comprehensive reference list is included. The publication addresses straight, skew, and curved bridges. The “Design” section covers several aspects of curved bridges. Recommendations are given for when the longitudinal bending moments can be determined as for a straight bridge and then combined with the torsional effects without consider- ing the coupling of the two effects on each other. Alternative substructure configurations are discussed for curved bridges. The “Structural Analysis” section discusses the mutual in- fluence of the longitudinal bending moments and torsional moments for horizontally curved bridges. A simple table of equations based on classical curved beam theory is presented for several different loading conditions of a curved single- span bridge with fixed supports. The “Dimensioning and Detailing” section covers several aspects of curved bridges. Dimensioning and reinforcement of the web for flexural shear, torsion, and regional trans- verse bending is addressed, including a rational method of designing the web reinforcement for combined shear and regional transverse bending. The influence of horizontal curvature on the movements at bearings is also discussed. Response of Curved Concrete Box-Girder Bridges Global Analysis Most published research seems to be directed toward the global response of box-girder bridges. Several analytical tech- niques have been studied. Many of these are relatively com- plex, but many others are suitable for production design practice. Our current belief is that a properly applied grillage analogy method provides good results and may be most suit- able for analyzing bridges with significant curvature. The fol- lowing paragraphs discuss some of the papers and reports that were reviewed. Al-Rifaie, W. N., and Evans, H. R. (1979) An Approximate Method for the Analysis of Box-girder Bridges that are Curved in Plan, Proc., Int. Association of Bridges and Struc- tural Engineering, Int. Association for Bridge and Structural Engineering (IABSE), pp. 1–21. An approximate method for analyzing curved box-girder bridges using the nodal section method is described. This 14

method, originally developed for straight box-girders, has been adapted for curved box-girders and is useful for pre- liminary design of these structures. The method developed applies to simple-span, single-cell box-girders. In this procedure, the transverse nodal section is idealized as a plane frame. Nodes on the frame are assumed to be fixed against translation but free to rotate. Each frame is analyzed and the reactions at the nodes are determined. The reactions are then applied to longitudinal plates that represent the components of the box-girder. The plates can only resist in- plane forces. The final step is to apply a “sway correction” procedure that will make the displacements of the nodes in each of the transverse nodal sections compatible with the deflections of the nodes along the edges of the longitudinal plates. This approach results in substantial computational savings over the finite element method and is well suited to preliminary design studies. The method was checked against finite element results for model bridges representing both concrete and steel box-girders. Several different span-to-radius ratios and loading conditions were considered. In general, good cor- relation was found for critical stresses, although some dis- crepancy was found for non-critical stresses. The method as developed does not accurately account for shear lag in the deck. Studies have extended this method to multi-cell box-girders and have developed methods to account for shear lag in straight box-girders. These refinements are also being con- sidered for curved box-girders. Bazant, Z., and El Nimeiri, M. E. (1974) “Stiffness Method for Curved Box-girders at Initial Stress,” Journal of the Structural Division, Vol. 100, No. 10, pp. 2071–2090. A sophisticated numerical method of analysis of the global behavior of long curved or straight single-cell girders with or without initial stress is presented. It is based on thin-wall beam elements that include the modes of longitudinal warping and of transverse distortion of the cross section. Deforma- tions due to shear forces and transverse bi-moment are in- cluded, and it is found that the well-known spurious shear stiffness in very slender beams is eliminated because the inter- polation polynomials for transverse displacements and for longitudinal displacements (due to rotations and warping) are linear and quadratic, respectively, and an interior mode is used. The element is treated as a mapped image of one parent unit element and the stiffness matrix is integrated in three dimensions, which is numerical in general, but could be carried out explicitly in special cases. Numerical examples of deformation of horizontally curved bridge girders, and of lateral buckling of box arches, as well as straight girders, validate the formulation and indicate good agreement with solutions by other methods. This method is most applicable to steel box-girders and is of little use to our project. Buragohain D. N., and Agrawal, B. L. (1973) “Analysis of Curved Box-Girder Bridges,” Journal of the Structural Divi- sion, Vol. 99, No. 5, pp. 799–819. A discrete strip energy method is presented for the analy- sis of curved box-girder bridges of arbitrary cross section and various forms of curved folded plate structures simply sup- ported at the two ends and composed of elements that may, in general, be segments of conical frustra. The method de- scribed applies to orthotropic material properties, arbitrary cross sections, constant curvature, and pinned supports at both ends. The method is based on harmonic analysis in the circumferential direction. The total potential energy of the structure is discretized into energy due to extension and bending and energy due to shear and twisting. The two types of circumferential strip elements are obtained by using a modified finite difference discretization in the transverse direction. The use of minimum energy principles yields two types of element matrices assembled to form the overall stiff- ness matrix of the structure following stiffness matrix proce- dures. Results of two examples obtained by the method are compared with available solutions. The applicability of this paper to NCHRP 12-71 is limited because it only applies to simply supported bridges, and the tool (software) to imple- ment this method is not readily available. Choudhury, D., and Scordelis, A. C. (1988) “Structural Analysis and Response of Curved Prestressed Concrete Box-Girder Bridges,” Transportation Research Record 1180, Transportation Research Board, National Research Council, Washington, D.C., pp. 72–86. A numerical finite element analysis method for linear- elastic analysis and nonlinear material analysis of curved pre- stressed concrete box-girder bridges is demonstrated through two examples. A curved nonprismatic thin-walled box-beam element is used to model the bridges. The cross section of the element is a rectangular single-cell box with side cantilevers. Eight displacement degrees of freedom, includ- ing transverse distortion and longitudinal warping of the cross section, are considered at each of the three element nodes. Prestressing, consisting of post-tensioned bonded tendons in the longitudinal direction, is considered. For nonlinear material analysis, the uniaxial stress-strain curves of concrete, reinforcing steel, and prestressing steel are modeled. The shear and the transverse flexural responses of the box-beam cross section are modeled using trilinear con- stitutive relationships based on cracking, yielding, and ulti- mate stages. The first example demonstrates the versatility 15

of the numerical method in determining the linear-elastic distribution of forces in a three-span prestressed box-girder bridge of curved plan geometry and variable cross section. Dead load, live load, and prestressing load cases are analyzed. In the second example, overload behavior and ultimate strength of a three-span curved prestressed concrete box- girder bridge under increasing vehicular load are investi- gated. The different response characteristics of the bridge induced by different transverse locations of the overload vehicle are presented. Although the finite element formulation might be detailed and comprehensive and conducive to studying the ultimate behavior of concrete box-girder bridges, its applicability to the planned global elastic analysis studies is limited due to its complex nature. Chu, K. H. and Pinjarkar, S. G. (1971) “Analysis of Hori- zontally Curved Box-Girder Bridges,” Journal of the Struc- tural Division, Vol. 97, No. 10, pp. 2481–2501. A finite element method for the analysis of simply supported curved girder bridges with horizontal sector plates and vertical cylindrical shell elements is outlined. Stiffness coefficients of sector plates are presented herein whereas stiffness coefficients of shell elements are based on Hoff’s solution of Donnell’s equations. The authors claim that this analysis is much more accurate than other methods of analysis. Results of a sample bridge analysis are shown with stresses and deflections reported for a simply supported multi-cell concrete bridge. Some interesting results, particularly those with respect to the effect of radius of curvature, were ob- tained. Although a comparison is made of the results of a curved twin box-girder bridge obtained by the proposed method and another approximate analytical method (Tung, 1967), no comparisons with other (simpler) analysis methods are given. The FEM analysis tool itself is not readily available and thus is of limited use, but the results of the analysis can serve as a comparison case for measuring the accuracy of other methods, if more detail on the presented example can be obtained. Bridge Design System (BDS) (1986) A Computer Pro- gram for Analysis and Design of Multi-Cell Box-girder Bridges, ECC. The described software program is the most commonly used software for design of multi-cell box-girder bridges. Bridges are modeled as plane frames ignoring all horizontal curve effects. This modeling technique is significant for NCHRP Project 12-71 because the technique is commonly used in practice and its limits of applicability need to be investigated. Computers and Structures, Inc. (1998) “SAP2000 – Inte- grated Finite Element Analysis and Design of Structures,” CSI, Berkeley, California. This reference constitutes the concrete structure portion of the SAP2000 Manual, with emphasis on design code check analysis. SAP2000 features integrated modules for design of both steel and reinforced concrete structures. The pro- gram provides the user with options to create, modify, analyze, and design structural models. The program is struc- tured to support various design codes for the automated design and check of concrete frame members. The program currently supports several foreign and domestic design codes. Given that the design code check features of the pro- gram focus on frame analysis, these design code checks are of limited usefulness for the specialized needs of curved con- crete box-girder bridge “local” or sectional (“regional”) analysis. But the program is, of course, very useful for global analysis. Chapter II of this reference outlines various aspects of the concrete design procedures of the SAP2000 program. This chapter describes the common terminology of concrete design as implemented in SAP2000. Each of six subsequent chapters gives a detailed description of a specific code of prac- tice as interpreted by and implemented in SAP2000. Each chapter describes the design loading combination, column and beam design procedures, and other special consideration required by the code. Aside from the obvious use as a SAP user reference, this document is useful as a summary (and side-by-side compari- son) of various design codes for concrete columns and beams. Other than this, it is of limited direct utility to NCHRP Proj- ect 12-71. There is no coverage of design or analysis of prestressing in this document. Fu, C. C., and Tang, Y. (2001) “Torsional Analysis for Pre- stressed Concrete Multiple Cell Box,” ASCE Journal of Engineering Mechanics, Vol. 127, No. 1, pp. 45–51. Using the Softened Truss Model, the authors present the formulation for calculating torsional effects in a multi-cell reinforced and prestressed concrete box-girder bridge. This paper asserts that because concrete box-girder sections are not made of thin webs and flanges, the stress distribution in these components is not constant and varies through the thickness, causing the effective stiffness of the member to be less than that observed at low values of load (torque). The formulation is coded in a computer program and the results from an example problem are presented. This re- search may be of some value to NCHRP Project 12-71 if the methodology can be simplified and used as the basis of sim- plified methods for calculating torsional effects. However, 16

the presented paper, in its current form, is too complex to be used in practical design situations or for parametric studies. Lopez, A., and Aparico, A. C. (1989) “Nonlinear Behavior of Curved Prestressed Box-Girder Bridges, IABSE Periodica, Zurich, Vol. 132, No. 1, pp. 13–28. This paper describes an analytical study of the ultimate strength of horizontally curved reinforced and prestressed concrete box-girder bridges. The analysis was performed using materially nonlinear plane stress finite elements (i.e., panels) that exhibited membrane action. The material was assumed to have a variable modulus of elasticity that was strain dependent. Panel behavior was based on the evolutive truss analogy with peak stress reduction (Vecchio and Collins, 1986). Reinforcing steel and prestress strand were stressed uniaxially according to an assumed multi-linear stress strain relationship. Section warping was not considered. Classical matrix analysis techniques were used to perform the analysis. A five-span bridge was selected to study the difference be- tween linear and nonlinear response. Live loads were located at various transverse positions and the behavior was observed as the intensity of these loads was increased. Based on these studies the following conclusions were drawn: 1. The structural response was highly nonlinear at ultimate loads. 2. The form and degree of internal force redistribution at ultimate loads depended on the loading case considered. 3. Internal forces were redistributed due to progressive cracking and structural coupling between bending and torsion. 4. The type of failure depended on the loading case considered. 5. Ultimate internal force response could be evaluated accu- rately using plastic sectional analysis. 6. Transverse prestressing significantly affected post-cracking response. The following criteria are proposed for design of curved prestressed box-girder bridges. 1. The response of the bridge under service loads can be accurately predicted using elastic models. 2. Elastic models cannot accurately (and will often non- conservatively) predict the ultimate limit state. 3. When using linear analysis to determine the factor of safety against failure, cracked flexural and torsional sec- tion properties should be used to determine demands and plastic sectional analysis should be used to determine capacities. Meyer, C. (1970) “Analysis and Design of Curved Box-Girder Bridges,” Structural Engineering and Structural Mechanics Report No. UC SESM 70-22, University of California, Berkeley. The history of curved bridges and the highway geometric requirements of these structures are discussed. The report outlines the methods developed over the years for analyzing curved bridges. These include straight beam approximation, curved beam theory, refined curved beam theories, plate and grillage analysis methods, finite element analysis, and the fi- nite strip method analysis of curved folded plates. Refined curved beam theories are required to analyze thin-walled box sections that can experience warping of the cross section in the transverse direction. Because concrete box sections have relatively thick walls, warping is generally small and ordinary curved beam theory can be used successfully. Two methods of analysis are developed in the form of computer programs. The first program, FINPLA2, uses the finite element method. The second program, CURSTR, uses the finite strip method of curved folded plates. The solution methodology requires that loadings be applied in the form of Fourier series. The programs yield essentially the same results. The CURSTR program was used to study wheel load dis- tribution in 1-, 2-, 3-, and 4-cell concrete box-girder bridges. Several parameter studies were conducted with different cur- vatures, span lengths, deck widths, depth-to-span ratios, and loading configurations. With respect to single-cell boxes, the following observa- tions were made: 1. The girder on the inside of the curve is stiffer than the girder on the outside of the curve and will attract more load. 2. Load distribution improves with an increase in curvature. This behavior is independent of span, cell width, and depth-to-span ratio. 3. The girder on the outside of the curve has a larger statical moment because of its longer span. 4. The combination of items 1 and 3 results in nearly equal moments in the two girders. 5. The influence of span length on load distribution is simi- lar to straight girders. 6. The influence of depth-to-span ratio is also similar. For two-cell boxes: 1. The moments in the middle girder and the girder on the inside of the curve increase with curvature. 2. The moment in the girder on the outside of the curve de- creases with curvature up to a certain level and then starts to increase. 17

3. The influence of span on load distribution is small. 4. Cell width accelerates the curvature effects. The response of 3- and 4-cell box-girders exhibits similar characteristics to 1- and 2-cell boxes. With respect to negative moments over a “fixed” support, 1. The girders may be assigned moments proportional to their moments of inertia. 2. Load distribution is generally worse in continuous bridges. For design, approximate methods are justified and even preferred in most cases. A girder moment distribution factor is developed: Bridges with curvatures radii large than 1000 ft. may be considered straight for analysis purposes. Nakai, H., and Heins, C. P. (1977) “Analysis Criteria for Curved Bridges,” Journal of the Structural Division, ASCE, Vol. 103, No. 7, pp. 1419–1427. The paper reports on a series of stiffness equations and lim- iting angle equations developed for determining the need for analysis of a bridge as a curved structure. The equations con- sider the type of supporting element, (i.e., open girder, spread box, or single-cell box), bending and torsional stiffness and central angles, and the induced stresses and deformation. It appears that these equations are specific to steel girders and warping torsion is a part of this methodology. However, the overall approach may be used or modified to apply to concrete bridges and NCHRP Project 12-71, especially for flowchart- ing the decision path for analysis. The paper provides equations for moment, stress, and de- flection of curved and straight bridges. Design criteria for curved bridges have been formulated using these equations, along with parametric studies. The range for the param- eter ψ, which “relates the cross-sectional geometry and the spacing between the outside girders, is determined for multiple I, twin box, and monobox-girders. Data for multi- cell girders are not available from this paper. The bounds for the torsional stiffness parameter κ are derived and depend on “the central angle,” which is the total horizontal angle the girder passes through between supports, the torsional rigidity of the cross section, and EI. The deflection ratio is primarily dependent on γ, which “reflects the bending and torsional stiffness of the girders.” The relationships between γ and the central angle were also found for the three bridge types studied. α = + − ′⎡ ⎣⎢ ⎤ ⎦⎥1 2 1 B R L 600 . Conclusions made in the paper are as follows: 1. A series of stiffness parameter equations and limiting angle equations have been presented, which provide informa- tion to the designer in determining the need for a curved girder analysis. The expressions are functions of the girder types, bending and torsional stiffness, and central angles. 2. The evaluation of κ gives the following criteria: “when κ is less than 0.4, evaluation of stresses due to pure torsion may be omitted. When κ is greater or equal to 10, evalua- tion of stresses due to warping may be omitted.” Sennah, K. M., and Kennedy, J. B. (2002) “Literature Review in Analysis of Curved Box-Girder Bridges,” Journal of Bridge Engineering, ASCE, Vol. 7, No. 2, pp. 134–143. The curvilinear nature of box-girder bridges, along with their complex deformation patterns and stress fields, have led designers to adopt approximate and conservative methods for their analyses and design. Recent literature on straight and curved box-girder bridges has dealt with analytical formula- tions to better understand the behavior of these complex structural systems. Few authors have undertaken experi- mental studies to investigate the accuracy of existing methods. This paper presents highlights of references pertaining to straight and curved box-girder bridges in the form of single- cell, multiple-spine, and multi-cell cross sections. The litera- ture survey presented herein deals with (1) elastic analysis, and (2) experimental studies on the elastic response of box-girder bridges. The elastic analysis techniques discussed include 1. Orthotropic Plate Theory Method 2. Grillage Analogy Method 3. Folded Plate Method 4. Finite Strip Method 5. Finite Element Method The orthotropic plate method lumps the stiffness of the deck, webs, soffit, and diaphragms into an equivalent or- thotropic plate. The Canadian Highway Bridge Design Code (Canadian Standards Association, 1988) recommends limit- ing this method to straight bridges with multi-spine cross sections. Parameter studies indicated that acceptable results are given for bridges with three or more spines. In the grillage analogy method, the multi-cellular structure is idealized as a grillage of beams. The CHBDC does not rec- ommend this method be used for sections with less than three cells or box beams. This method requires special attention to the modeling of shear lag and the torsional stiffness of closed cells. When modeling is properly done, this method yields results that compare well with finite element techniques. 18

The folded plate method uses plates to represent the deck, webs, and soffit of box-girders. Diaphragms are not modeled. The plates are connected along their longitudinal edges and loads are applied as harmonic load functions. The method is time consuming and only applicable to restrictive support conditions. The finite strip method has been widely researched. It is essentially a special case of the finite element method but requires considerably less computational effort because a limited number of finite strips connected along their length are used. Its drawback is that it is limited to simply supported bridges with line supports and thus not applicable as a gen- eral use analysis tool for production design. With the advent of powerful personal computers and com- puter programs, the finite element method has become the method of choice for complex structural problems. Many researchers have applied this technique to the analysis of curved box-girder bridges. A problem that occurs is that a large number of flat plate elements are required to properly model the curved elements of a curved bridge. Several researchers have attempted to overcome this difficulty by developing special elements or using special substructuring techniques. The versatility of this method has allowed re- searchers to investigate several aspects of bridge behavior, in- cluding dynamics, creep, shrinkage, and temperature. Curved box-girder structures cannot be accurately analyzed using the classical curved beam theory developed by Saint- Venant because it does not account for warping, distortion, and bending deformations of the individual wall elements of the box. Vlasov first developed an adaptation of Saint Venant theory to thin-walled sections. Even this adaptation does not account for all warping and bending stresses. Considerable research effort has been expended over the years to develop computational techniques to overcome shortcomings in the present theory. Several laboratory experiments involving model box-girder bridges have been conducted over the years. In general, these experiments have shown good agreement with analytical results, particularly those obtained using the finite element method of analysis. In conclusion, the finite element method, though more dif- ficult to apply, accounts for all relevant behavior in curved box- girder bridges and yields the most reliable analysis results. Many computer programs have been developed specifically for box- girder bridges, but most of these are not commercially available. Turkstra, C. J., and Fam, A. R. M. (1978) “Behavior Study of Curved Box Bridges,” Journal of the Structural Division, ASCE, Vol. 104, No. 3, pp. 453–462. A numerical analysis of several single-cell curved box-girder sections with variable curvature, length, web spacing, number of diaphragms, and loading was performed. The effects of these parameters on longitudinal stresses are considered, based on selected numerical results. Implications for prelim- inary design are presented for both concrete and composite concrete/steel sections Reilly, R. J. (1972) “Stiffness Analysis of Grids Including Warping,” Journal of the Structural Division, ASCE, Vol. 98, No. 7, pp. 1511–1523. Two methods of including warping effects in the stiffness method of analysis are presented. Method B seems to be superior to Method A for cases where the warping constant is not large. In the limiting case where Iw = 0, the warping effects disappear and leave only the familiar GIx/L. When Iw is small relative to Ix (approximately pL > 5 for each element) com- putational errors grow, because the stiffness matrix tends to become singular as the elements on the main diagonal associated with warping approach zero. This is not a serious practical problem, as good solutions can be obtained using an ordinary grillage analysis, neglecting warping for structures where warping stiffness is small and p is large. Composite bridges seem to fall near the borderline where warping can be neglected. The bridge used in the example above was non- composite so that warping would be significant. Bimoment and warping torsion are obtained for grillage structures. Re- sults of computer programs based on these methods are shown to agree closely with published solutions for straight beams, a curved beam, and a curved highway bridge. Meyer, C., and Scordelis, A. C. (1971) “Analysis of Curved Folded Plate Structures,” Journal of the Structural Division, Vol. 97, No. 10, pp. 2459–2480. A finite strip method of analysis is presented which can be used to analyze curved folded plate structures simply sup- ported at the two ends and composed of elements that may, in general, be segments of conical frustra. The method is based on a harmonic analysis in the circumferential direction, with the loadings expressed by Fourier series, and on a finite ele- ment stiffness analysis in the transverse direction. The direct stiffness method is used to assemble the structure stiffness matrix and to determine displacements and element stresses. A description of a general computer program developed for the analysis and the results of several examples are also given. Okeil, A. M., and El-Tawil, S. (2004) “Warping Stresses in Curved Box-Girder Bridges: Case Studies,” Journal of Bridge Engineering, Vol. 9, No. 5, ASCE. This paper discusses case studies performed on 18 actual composite steel-concrete box-girder bridges. These analytical 19

studies were conducted using the computer program ABAQUS and a special 7-degree-of-freedom beam-column element that can account for warping. These studies, which were designed to investigate warping-related stresses in these bridges, found that in all cases the effect of warping stress was insignificant. The 1997 AASHTO curved girder specifications limits the span-to-radius ratio for designing the bridges as straight. These ratios were found to be conservative by a factor of 2 or more when it comes to the need to consider warping. (Given that concrete box-girders will have thicker webs and soffits, they are even less vulnerable to warping, and it is likely that the effects of warping can be ignored in almost all of these bridges.) Laboratory Experiments Most, although not all, laboratory experiments related to curved concrete box-girder bridges have been conducted on small-scale Plexiglas or metal models of these bridges. A large- scale test of a concrete structure was performed at the Univer- sity of California at Berkeley during the 1970s. In general, these tests have shown that refined analytical techniques predict structural behavior quite well. The following paragraphs discuss published papers and reports on these tests in greater detail. Aneja, I. K., and Roll, F. (1971) “A Model Analysis of Curved Box-Beam Highway Bridge,” Journal of the Structural Division, Vol. 97, No. 12, pp. 2861–2878. Fabrication, preparation, and instrumentation of a Plexiglas model of a horizontally curved box-beam highway bridge are described. The model was extensively instrumented with rosette strain gages at three cross sections. Experimental data for three lane-loading conditions were obtained. An approximate theoretical analysis of the model was obtained by using the finite element method, which showed that finite element models with curved shell elements provide better predictions than those with straight plate elements. A typical comparison between the experimental and theoretical stress distribution across the mid- span gage section for one of the loading conditions is shown graphically. The comparison shows a good agreement between the shapes but not the magnitudes of the stress plots obtained experimentally and theoretically. Experimental data at the three gage sections for each load condition is also given. Aslam, M., and Godden, W. G. (1975) “Model Studies of Multicell Curved Box-Girder Bridges,” Journal of the Engi- neering Mechanics Division, Vol. 101, No. 3, pp. 207–222. A model study on the static response of curved box-girder bridges is presented, and a close agreement is found between the test and analytical results. The prototype bridge was a four-cell reinforced concrete design that was 33 ft 10 in. (10.31 m) wide and 4 ft 10 in. (1.47 m) deep and had a radius of curvature of 282 ft (86 m). A 1⁄29 scale aluminum model was studied for spans of 60 in. (1,500 mm), 45 in. (1,140 mm), and 30 in. (760 mm), with or without a midspan radial diaphragm. The quantities measured were (1) Boundary reactions; (2) strains at a radial section close to midspan; and (3) deflections at selected points. The data were reduced by computer, and dis- tribution graphs of tangential plate forces, radial-bending moments, and deflections were plotted by Calcomp plotter. Based on the model data, some general observations are made regarding the behavior of curved box-girder bridges. Fam, A. R. M. and Turkstra C. J. (1976) “Model Study of Horizontally Curved Box-Girder,” Journal of the Structural Division, Vol. 102, No. 5, pp. 1097–1108. This paper describes an experimental study of a single- span horizontally curved Plexiglas box-girder beam with di- aphragms and flange overhangs. Static loads were applied at midspan to cause a complex pattern of membrane and bend- ing stresses with the effects of diaphragms clearly evident. Experimental results in typical cases are shown graphically and compared with the results of a special-purpose finite element program developed especially for curved box analysis. This program used the softened truss model theory applied to a prestressed concrete multiple-cell box. In this theory, the concrete torsional problem is solved by combining equilib- rium and compatibility conditions and constitutive laws of materials. Until now, the theory has been applied only to the case of pure torsion with a single-cell section. An algorithm is presented to deal with the torsional problem for reinforced concrete and prestressed concrete box-girder bridge super- structures with multiple-cell sections. Results are compared with previous theoretical and experimental work for single-cell cases. Good agreement was obtained between experimental and analytical results. Heins, C. P., Bonakdarpour, B. P., and Bell, L. C. (1972) “Multicell Curved Girder Model Studies,” Journal of the Structural Division, Vol. 98, No. 4, pp. 831–843. The behavior of a single two-span, three-cell Plexiglas model is predicted by the Slope Deflection Fourier Series Technique. This analytical technique had previously been applied to only open cross-sectional, I-type bridge systems. The model was tested under various static concentrated loads. The resulting experimental deflection, rotation, and strain data for some loadings are reported. Effects of single and multicell torsional properties are examined. Results indicate that single-cell properties can be applied in the analy- sis, and warping effects may be neglected. 20

Scordelis, A. C., Elfgren, L. G., and Larsen, P. K. (1977) “Ultimate Strength of Curved RC Box-Girder Bridge,” Journal of the Structural Division, Vol. 103, No. 8, pp. 1525–1542. Results obtained in a study of a large-scale curved two-span four-cell reinforced concrete box-girder bridge model are pre- sented. The model, which was a 1:2.82 scale replica of a proto- type, had overall plan dimensions of 72 ft (21 m) long by 12 ft (3.7 m) wide. The radius of curvature was 100 ft (30.5 m). This represents the sharpest curvature normally used for bridges in the California highway system. Experimental and theoretical results are considered for reactions, steel and concrete strains, deflections, and moments due to conditioning overloads pro- ducing stress values as high as 2.5 times the nominal design stress. The loading to failure and the ultimate strength behav- ior is examined. The excellent live-load overload capacity of the bridge is evaluated and comparisons are made with the similar behavior of an earlier tested straight bridge model. Conclusions appropriate for the design of this type of bridge are given. Design Issues Bearings Although several bearing failures consisting of uplift, over- load, or binding have been experienced in curved box-girder bridges, no published research exclusively addressing this issue was found. However, because an accurate 3-D analysis will account for differences in bearing forces and displace- ments, several references that deal with global analysis and laboratory experimentation deal with this issue (Aslam and Godden, 1975; Scordelis et al., 1977; Choudhury and Scordelis, 1988; Sennah and Kennedy, 2002). This issue is also discussed in some textbooks (Menn, 1990). Diaphragms Diaphragms help prevent excessive distortions of the cross section, facilitate wheel load distribution, and distribute transverse load. The following two papers discuss research on determining the number and spacing of interior diaphragms in box-girder bridges. Oleinik, J. C. and Heins, C. P. (1975) “Diaphragms for Curved Box-Girder Bridges,” Journal of Structural Engi- neering, Vol. 101, No. 10, pp. 2161–2178. A finite difference procedure is used to determine the re- sponse of a single-span curved single box-beam bridge with any number of interior diaphragms. The bending and torsional distortions as well as cross-sectional distortions can then be determined throughout the curved box-girder. The forces that are determined include bending moment and flexural shear, pure torsion, warping torsion, and bi-moment. These forces, in addition to distortional functions, yield resulting normal bending, normal warping, and normal distortional stresses. The technique is then used to determine the dead load and live load response of a series of typical curved box beams. A study of the data has resulted in a series of empirical design equations. Abendroth, R. E., Klaiber, F. W., and Shafer, M. W. (1995) “Diaphragm Effectiveness in Prestressed-Concrete Girder Bridges,” Journal of Structural Engineering, Vol. 121, No. 9, pp. 1362–1369. Each year many prestressed-concrete (P/C) girder bridges are damaged by overheight vehicles or vehicles transporting overheight loads. The effects of this type of loading on P/C bridge behavior were investigated for various types and loca- tions of intermediate diaphragms. The research included a comprehensive literature review; a survey of design agencies; the testing of a full-scale, simple-span, P/C girder-bridge model with eight intermediate diaphragm configurations, as well as a model without diaphragms; and the finite element analyses of the bridge model assuming both pinned- and fixed- end conditions. The vertical load distribution was deter- mined to be essentially independent of the type and location of the intermediate diaphragms, while the horizontal load distribution was a function of the intermediate diaphragm type and location. Construction details at the girder sup- ports produced significant rotational-end restraint for both vertical and horizontal loading. Both the vertical and hori- zontal load distributions were affected by the girder-end restraint. A fabricated intermediate structural steel diaphragm was determined to provide essentially the same type of re- sponse to lateral and vertical loads that was provided by the reinforced-concrete intermediate diaphragms currently used by the Iowa DOT. Flexure and Flexural Shear Beyond the issue of global analysis, the mechanism for re- sisting flexural and shear stresses in box-girders is important. The mechanisms of shear resistance and its interaction with flexural stresses in reinforced and prestressed concrete have been well researched (Marti, 1999, and Vecchio and Collins, 1986). Also, the effectiveness of the deck and soffit slabs in resisting flexural compressive forces has been studied. This includes the phenomenon commonly known as shear lag. Several published papers and reports have dealt with these issues. Some of these are discussed below. 21

Chang, S. T., and Zheng, F. Z. (1987) “Negative Shear Lag in Cantilever Box-Girder with Constant Depth,” Journal of Struct. Eng., Vol. 113, No. 1, pp. 20–35. This paper addresses the classical phenomenon of shear lag in box-girders and draws attention to distinguishing between positive and negative shear lag. The effects of shear lag and negative shear lag in cantilever box-girders are analyzed through a variation approach and finite element techniques. Expressions are derived to determine the region of negative shear lag effect with the interrelation of span/width parame- ters involved. The theoretical results obtained are compared with a Plexiglas model test. Finally, conclusions are drawn with regard to further study and research. Positive shear lag is the phenomenon in which, near the support of a cantilever, flange longitudinal stresses near the web are larger than away from the web. But for a cantilever box-girder with constant depth under a uniform load, away from the support, the bending stress in the deck near the webs is smaller than the stress away from the webs. This is a result of negative shear lag. Using the principle of minimum potential energy, following Reissner’s procedure with slight modifications, shows that the additional moment created by flange shear deformation plays an important role in both positive and negative shear lag. For a single point load at the free end of the cantilever, only positive shear lag is created. When there is a uniformly distributed load along the full span of the cantilever box-girder however, negative shear lag occurs. The region of the cantilever affected by negative shear lag is from the free end to more than 3⁄4 of the cantilever length from the free end. Negative shear lag affects a larger region than positive shear lag. With a finite element model analysis, three load cases were considered; a distributed load, a point load, and a combina- tion of a downward point load and an upward distributed force. This analysis showed that negative shear lag occurred only with the first load case of a distributed load. This model was consistent with the results from the minimum potential energy method. Negative shear lag depends on not only the load case but also the boundary conditions. The ratio of the length of the cantilever to the width of the box-girder affects the amount of moment caused by shear. As the ratio increases, both positive and negative shear lag decrease. Actual testing using a Plexiglas model confirmed the the- oretical results. When a uniform load is applied, not only is positive shear lag more severe compared with a point load, but negative shear lag is also present. A cross-sectional analy- sis of shear stress in the flange is taken at several locations. Near the fixed end where shear lag is greatest, the bending stress near the web is much larger than the stress away from the webs. At a cross section where negative shear lag is significant, the bending stress away from the webs is greater than the stress near the webs. This paper is only indirectly applicable to this project be- cause the paper does not deal specifically with curved girders. However, given that shear lag effects are an important con- sideration in developing analysis and design strategies, the conclusions in this paper, and the theoretical solutions are noteworthy. In short, the relevant conclusions are 1. Positive shear lag may occur under both point and uni- form load, but negative shear lag occurs only under uni- form load. 2. Negative shear lag also depends on the ratio of L/b, where b is the net width of the box section. The smaller the ratio, the more severe are the effects of positive and negative shear lag. 3. Negative shear lag depends on the boundary condition of displacement as well as on the external force applied to the girder. 4. In cantilever box-girders, although the negative shear lag yield in the region of the bending stress is small, the relative additional stress induced by this effect is often considerably greater. It cannot be neglected. It should never be believed that in all cases only positive shear lag is produced. Chang, S. T., and Gang, J. Z. (1990) “Analysis of Cantilever Decks of Thin-Walled Box-Girder Bridges,” Journal of Structural Eng., Vol. 116, No. 9, pp. 2410–2418. This paper, which addresses the cantilever decks (“wings”) of single-cell box-girder bridges, does not make any distinc- tion about the effects of horizontal curves, but it does present some useful qualitative information about cantilever deck evaluation, in general. The paper reports on a spline finite strip approach used to analyze the cantilever decks. Effects of distortion of thin- walled box sections are taken into account by treating the cantilever deck as a slab with horizontally distributed spring supports along the cantilever root. Perspex model tests were conducted in the model structural laboratory at Tong Ji Uni- versity. The results based on the spline finite strip method are compared with those of the model test. Simplified solutions are also given for the distribution of transverse moment along the cantilever root. A Plexiglas model of a single-cell box-beam was evaluated. As a point load moved transversely across the box-girder, the bending stress and membrane stress at the root of the over- hang of the deck were obtained. From this analysis, it was observed that it is reasonable to treat the cantilever decks as 22

cantilever slabs with horizontally distributed spring support along the cantilever root with the spring constant K depend- ing on dimensions and material properties of the deck. The spline finite strip analysis was shown to be in close agreement with actual model test results. Tests showed that the moment along the cantilever root approaches zero as the longitudinal distance from the point load increases. The maximum moment at the root is when the point load is at the end of the cantilever and approximately zero when the load is not on the cantilever. There is also sagging in the can- tilever around the point load. Although sagging moment is only local, many point loads at the same longitudinal loca- tion can cause significant sagging moment, so this should not be ignored. Conclusions made in this paper are as follows: 1. Cantilever decks of thin-walled box-girder bridges can be treated as cantilever slabs with horizontally distributed spring support along the cantilever root, taking into ac- count the influence of local distortion of the box section. 2. Spline finite strip results, based on the simplified idealiza- tion of the cantilever decks of thin-walled box-girder bridges, are in close agreement with test results. The sagging moment at the cantilever root can be obtained in conjunc- tion with information tabulated in the article. 3. In cantilever slabs of infinite length and large cantilever length, sagging moment cannot be ignored. The sagging moment may be taken from information also provided in the article. Other than the general information provided for evaluat- ing cantilever decks of straight box-girder bridges, there is no specific information pertaining to curved box-girders. Hasebe, K., Usuki, S., and Horie, Y. (1985) “Shear Lag Analysis and Effective Width of Curved Girder Bridges,” J. Eng. Mech., Vol. 111, No. 1, pp. 87–92. This paper develops guidelines and graphs for estimating the effective width of curved girder bridges. The methodol- ogy is formulated by substituting the flange stress derived from present theory into the equation of effective width def- inition for the curved girders. The required information in formulating the effective width rule for design of curved girder bridges is provided. The actual longitudinal stress distributions for the curved girders are evolved from the present theory for shear lag in order to determine the effective width. The thin-walled curved girders used in this investi- gation are based on box and channel cross sections and are analyzed for a uniform lateral load and for a concentrated load. Numerical examples are shown for several problems to in- vestigate the effect on effective width of curved girder bridges. The values of the effective width obtained by the present theory are compared with those of the straight girder bridges. Accord- ing to the results, it is acceptable to say the values of effective width of curved girder bridges are the same (approximately) as the values of the straight girder bridges. This is a very important conclusion for NCHRP Project 12-71. The inner and outer effective widths are denoted as λ1 and λ2, respectively, for a curved girder. λ0 is half of the effective width for a straight beam and 2b = width of the flange. Effec- tive width ratio is defined as the ratio of an effective width to the actual breadth of the flange. The parametric study in- volves calculating the effective width ratio for simply sup- ported girders and comparing results of (1) point load at mid span versus uniformly distributed load, (2) present theory versus folded plate theory, (3) inner and outer effective width ratio for curved girders versus effective width ratio for straight girders, and (4) box cross section versus channel cross sec- tion. In the test, the curved beam has a radius of curvature R = 4L and b/L = 0.1. In the case of a concentrated loading, the values of the effective width ratio are at minimum at the center of the span length and increase rapidly toward the supports. However, in the case of a uniformly distributed loading, the values are at maximum at the mid span and de- crease toward the supports. The values of effective width ratio of the present theory are lower than the folded plate theory curves, and the inside and outside effective width ratio agree with the values of the straight beams, irrespective of cross- sectional shapes or types of load distributions. The authors analyze the inner and outer effective widths at mid span relative to the curvature R/L. For a box-girder under a distributed load, the effective widths remain fairly constant for b/L = 0.1 when R/L > 2 and only decreases slightly for b/L = 0.2. With a point load and b/L = 0.1, the effective width also remains fairly constant for R/L > 4. When b/L = 0.2 with a point load however, the effective widths deviate significantly for small R/L. For a curved girder, as b/L increases, the difference between the inner and outer effective widths becomes significant, especially for box-girders. For small values of b/L however, those two values are almost identical. One needs to be careful when R/L is small, b/h is small, or b/L is large, especially under concentrated loads. Otherwise, it is reasonable to assume the effective width of a curved girder is equal to that of a straight girder for practical applications. Torsion Torsion design is currently addressed by the AASHTO LRFD code and, in the case of box-girders, is integrated 23

with shear design. Recent studies (Rahal and Collins, 2003) indicate that this approach yields good results when the correct value of θ is used. Several other papers on the sub- ject of torsional resistance have been reviewed and it is con- cluded that the current design methods are acceptable. Only minor clarifications of the current specifications and guidelines for applying them to box-girders are required. Following is a summary of the papers that were reviewed on torsion design. Collins, M. P., and Mitchell, D. (1980) “Shear and Torsion Design of Prestressed and Non-Prestressed Concrete Beams,” Journal of the Prestressed Concrete Institute, Vol. 25, No. 5, pp. 32–1000. This document presents design proposals for flexural shear and torsion of prestressed and non-prestressed con- crete beams based on the compression field theory. A rational method is proposed which addresses members subject to flexural shear, torsion, combined flexural shear and flexure, and combined torsion, flexural shear and flexure. Early de- sign procedures using truss models are also presented. The compression field theory, a development of the traditional truss model for flexural shear and torsion, considers in addition to the truss equilibrium conditions, geometric compatibility conditions and material stress-strain rela- tionships. The compression field theory can predict the fail- ure load as well as the complete load-deformation response. Measured and predicted response of numerous members is presented. The proposed design recommendations are pro- vided in code format. Comparisons with the provisions of the ACI 318-77 and CEB codes are provided. Numerous worked examples are provided that demonstrate the pro- posed method. Rahal, K. N., and Collins, M. P. (1996) “Simple Model for Predicting Torsional Strength of Reinforced and Prestressed Concrete Sections,” ACI Structural Journal, Vol. 93, No. 6, pp. 658–666. A noniterative method for calculating the ultimate tor- sional strength and the corresponding deformations of re- inforced and concentrically prestressed concrete sections is presented. This method, based on the truss model, avoids the need for iterations by making simplifying assumptions about the thickness of the concrete diagonal, the softening of the concrete due to diagonal cracking, and the principal compressive strains at ultimate conditions. A simple check on the spalling of the concrete cover is implemented. The calculated torsional capacities of 86 beams are compared with the experimental results and very good agreement is obtained. Rahal, K. N., and Collins, M. P. (2003) “Experimental Eval- uation of ACI and AASHTO-LRFD Design Provisions for Combined Shear and Torsion,” ACI Structural Journal, Vol. 100, No. 3, pp. 277–282. The experimental results from four large nonprestressed specimens loaded in combined flexural shear and torsion are used to evaluate the torsion design procedures of ACI 318-02 and AASHTO-LRFD. Both sets of procedures calculate the required amounts of hoop reinforcement for torsion based on a space truss model with compression diagonals inclined at an angle of θ to the longitudinal axis. It is shown that the ACI provisions give very conservative results if the recom- mended value of 45 degrees is used for θ. If the lower limit of 30 degrees is used, however, some unconservative results are possible. The AASHTO-LRFD provisions predicted values of θ of approximately 36 degrees for these specimens and gave accurate estimates of the strengths. Fu, Chung C, and Yang, Dailli (1996) “Designs of Concrete Bridges with Multiple Box Cells due to Torsion Using Soft- ened Truss Model” ACI Structural Journal, Vol. 93, No. 6, pp. 696–702. Using a softened truss model, this paper presents a method for torsional design of multicell concrete bridges. Earlier researchers have successfully shown the development for solving single-cell torsion by combining the equalibrium, compatibility, and the softened constitutive laws of con- crete. By solving the simultaneous equations based on the membrane analogy, multicontinuous or separate cells can be solved. Hsu, T. T. C. (1997) “ACI Shear and Torsion Provisions for Prestressed Hollow Girders,” ACI Structural Journal, Vol. 94, No. 6, pp. 787–799. New torsion design provisions have been proposed for the 1995 ACI Building Code. As compared with the 1989 provi- sions, these generalized 1995 provisions have three advan- tages: First, they are applicable to closed cross sections of arbitrary shapes. Second, they are applicable to prestressed concrete. Third, they are considerably simplified by deleting the “torsional concrete contribution” and its interaction with flexural shear. These new provisions are suitable for applica- tion to concrete guideways and bridges because these large structures are always prestressed and are often chosen to have hollow box sections of various shapes. This paper discusses the background of the new code provisions, suggests modifi- cations to code formulas, and illustrates the application of the code provisions to prestressed hollow girders by way of a guideway example. 24

Wheel Load Distribution Wheel load distribution has been the subject of many of the analytical studies cited earlier. Other studies are discussed in the following paragraphs. Song, S. T., Y. H. Chai, and S. E. Hida (2001) “Live Load Dis- tribution in Multi-Cell Box-Girder Bridges and its Compar- ison with the AASHTO LRFD Bridge Design Specifications,” Final Report to Caltrans for Contract Number 59A0148. This report presents the results of a series of analyses done on box-girder bridges with normal and skewed supports and straight and curved geometry. The analysis models included 3-D finite element as well as grillage and single line of elements for superstructure. The goal of the project was to evaluate the LRFD live load distribution factors for box-girder bridges. However, the modeling techniques used and verified for this project can be used as guidelines for NCHRP Project 12-71. Zokaie, T., K. D. Mish, and R. A. Imbsen (1993) “Distribu- tion of Wheel Loads on Highway Bridges,” Phase 3, Final Report to NCHRP 12-26 (2). This report presents a computer program (LDFac) devel- oped for modeling bridge superstructure with straight or skewed supports and obtaining live load distribution factors. Although this computer program did not consider curved geometry specifically, the modeling process and load place- ment guidelines may be used for analysis of curved bridges as well. One of the key issues discussed in this report is the mod- eling of distortion of box-girders in a grillage analysis via an equivalent shear deformation parameter. Zokaie, T., T. A. Osterkamp, and R. A. Imbsen (1991) “Dis- tribution of Wheel Loads on Highway Bridges,” Final Report to NCHRP 12-26 (1). This report presents a series of guidelines for analysis of various bridge types. The guidelines include calculation of equivalent section property parameters to be used in plate and grillage analyses, as well as guidelines for setting the boundary conditions. Although the research did not specifi- cally consider curved bridge geometry, many of the guide- lines for modeling and analysis using common analysis tools are applicable to modeling that will be needed in NCHRP Project 12-71 global analysis studies. Tendon Breakout and Deviation Saddles Prestress tendon breakout in curved bridges has occurred on bridges over the years. It is evident from observing the reasons for these failures that they can be prevented through close attention to details such as tendon spacing and tie back reinforcement. The following paragraphs summarize the ref- erences reviewed on the subject. Beaupre, R. J., et al. (1988) Deviation Saddle Behavior and Design for Externally Post-Tensioned Bridges, Research Re- port 365-2. Center for Transportation Research, University of Texas at Austin, Austin, Texas. This report is the second in a series outlining a major study of the behavior of post-tensioned concrete box-girder bridges with post-tensioning tendons external to the con- crete section. It presents the results of an experimental pro- gram in which ten very accurately sealed reinforced concrete models of typical tendon deviators were tested. Detailed instrumentation led to a very good understanding of the behavior of the various patterns of reinforcement in the deviators. The models included two very different patterns of detailing, several arrangements of tendon inclinations, and both normal and epoxy-coated reinforcement. The report evaluates the results with respect to both simplified conventional analysis methods and strut-and- tie models. The results provide the basis for deviator de- sign recommendations and several examples are presented to illustrate the practical use of these recommendations. Caltrans (1996) Bridge Memo to Designers Manual, Memo 11-31 Curved Post-Tensioned Bridges, California Depart- ment of Transportation, Sacramento, California Memo 11-31 addresses the design of curved post- tensioned concrete box-girders for lateral prestress forces. The force effects considered are tendon confinement and web regional transverse bending. The lateral prestress force, F, is determined by dividing the jacking force (Pj) per girder by the horizontal radius (R) of the web. A standard detail for tendon confinement (see Figure 3-1) is required for all webs with a Pj/R > 100 kN per m or a horizontal radius (R) of 250 m or less. The regional transverse bending moment in the web is taken as Mu = 0.20Fhc where hc is the clear dis- tance between the top and bottom slabs. This assumes the web to act as a simple beam spanning the top and bottom slabs with a concentrated load, F, acting at mid-height of the web. The resulting simple beam moment is reduced 20% for continuity. The load factor is taken as 1.0. The design of stirrup reinforcement does not combine regional transverse bending and shear requirements. Graphs are provided to check webs for containment of tendons and adequate stir- rup reinforcement to resist regional transverse bending. A review of the 405/55 failure has led to the identification of several issues related to the Caltrans Memo to Designers 11-31. These are discussed below. 25

26 Figure 3-1. Caltrans detail A.

Cordtz, K. (2004) Design of Curved Post-Tensioned Bridges for Lateral Prestress Forces, David Evans and Associates, Inc., Roseville, California. This document presents the internal guidelines of David Evans and Associates, Inc., for the design of horizontally curved post-tensioned concrete box-girders for lateral prestress forces. The primary focus is on the regional beam action of the webs and local slab action of the cover concrete over the tendons. The document provides a discussion of the actions on curved post-tensioned girders, identifies those actions not completely addressed in current design codes and guidelines, and recom- mends design procedures that reflect current best practice. Local slab action of the concrete cover over the tendons has been identified as the major cause of failure in several curved post-tensioned bridges that did not have duct or web ties. For a web without duct or web ties, the cover concrete is the only el- ement restraining the lateral prestress force. The cover concrete acts as a plain concrete beam to restrain the lateral prestress force. The local slab is subject to lateral shear and bending from the lateral prestress force. Specific requirements for local lateral shear are given in AASHTO LRFD 5.10.4.3.1 “In-Plane Force Effects.” No specific design methodology for local flexure is given by AASHTO. The document provides interim recom- mendations for the tensile stresses in the cover concrete. Where duct ties are required, the document recommends the use of a rational method for design such as a strut and tie model. The vertical reinforcement in the web is subjected to combined global shear and regional transverse bending due to regional beam action. No specific design methodology for these combined actions is given by AASHTO. The docu- ment provides interim recommendations for the design of the stirrups. A flowchart is presented that outlines the recommended procedures. Numerous worked examples are also provided. Podolny, W., Jr. (1985), “The Cause of Cracking in Post- Tensioned Concrete Box-girder Bridges and Retrofit Pro- cedures,” Journal of the Prestressed Concrete Institute, March-April 1985. This article discusses the types of problems that lead to cracking in post-tensioned concrete box-girder bridges and have been encountered in both Europe and the United States. Cracking is attributed in a broad sense to the following fac- tors: inadequate flexural and shear capacity, non-consideration of thermal stresses, insufficient attention to stresses devel- oped by curvature of tendons, improper or inappropriate construction techniques, lack of quality workmanship to meet the tolerances necessary for problem free structures, and understrength materials. It is noted that in general the cause of cracking can be attributed to the superposition of stresses of multiple effects. The article discusses the pullout of horizontal curved ten- dons that occurred on several cast-in-place post-tensioned concrete box-girder bridges. In two of these structures, there was a combination of relatively sharp horizontal curvature, thin concrete cover over the tendons, and the bundling of large-sized tendons close together. Podolny divides the analysis of the failures of these bridges in three separate actions: 1. The global or overall girder action of the bridge together with its supporting piers and abutments. 2. Regional beam action of each web supported at the top and bottom flanges as a beam. 3. Local slab action of the concrete cover over the tendons. It appears that for both of these bridges, local slab action of the concrete cover over the tendons was the primary cause of the failure, but the regional beam action could have been a contributory cause and could, by itself, have overstressed some of the stirrups. The global action had a very small effect on these failures. Seible, F., Dameron, R., and Hansen, B. (2003), Structural Evaluation of the 405-55 HOV Connector and the Curved Girder Cracking/Spalling Problems. StD&A, San Diego, California. This document is a detailed (70 pages, including illustra- tions) project report on results of structural evaluation of the 405-55 HOV Connector’s curved girder cracking/spalling problems. The report provides background on the observed cracking and spalling (caused by horizontal breakout of web 27 Figure 3-1. (continued).

tendons in the multicell boxes), summarizes occurrences of similar problems at Las Lomas (California) and Kapiolani (Hawaii), and then documents a step-by-step analysis of the 405/55 bridge. The steps followed include global, regional, and local analysis, as suggested by Podolny. The regional and local analysis use detailed finite element modeling with crack- ing concrete constitutive models, but various hand calcula- tion methods for evaluating regional and local effects are also described and demonstrated as a check on the detailed analysis. The report also provides an in depth summary of Caltrans’s Memo to Designers (MTD) 11-31 and points to some potential shortcomings (under the right set of unusual circumstances) of the Memo. The report then lists eight con- tributory causes for the damage that occurred, but concludes that the one cause which, if corrected would have prevented the damage, was the omission of duct ties from the design. Although this report is very structure-specific and contains detailed project information that should not be directly ref- erenced in a set of design specifications, it provides a useful reference for summarizing the various analyses (both hand calculation and finite element) appropriate to this class of problems, especially at the “regional” and “local” level for girder cross-section and web evaluation. Strasky, J., 2001, Influence of Prestressing in Curved Mem- bers, Betonve Mosty, Report TK21, Prauge, Czech Republic. The influence of prestressing in curved members is discussed. To mitigate the effect of radial prestressing forces in the webs of box-girders, it is recommended that prestress tendons be sepa- rated vertically where they pass near the middle of the web. This will spread the radial forces and result in lower stresses tending to rip the strand out of the side of the web. Care is also required for tendons in the soffit of segmentally constructed bridges (straight or curved) where tendons are anchored in blisters along the length of the bridge and deviated across the width of the soffit, resulting in transverse tensile stresses that can fail the soffit. Such a failure occurred in a bridge constructed with a gantry in Austria. The vertical curvature of these tendons in a haunched bridge can also present a problem. Van Landuyt, D., and Breen, J. E. (1997) Tendon Breakout Failures in Bridges, Concrete International, American Con- crete Institute, Farmington Hills, MI, November 1997. The article discusses the pullout failure of horizontal curved tendons that occurred on several cast-in-place post- tensioned concrete box-girder bridges. In both of these struc- tures, there was a combination of relatively sharp horizontal curvature, thin concrete cover over the tendons, and the bundling of large-sized tendons close together. It appears that for both of these bridges local slab action of the concrete cover over the tendons was the primary cause of the failure. The article discusses the theory of transverse stresses in curved box-girder cross sections due to post-tensioning in- cluding “distributed radial force arch action.” Horizontal curved post-tensioned bridges are subjected to the following three separate actions: 1. The global or overall girder action of the bridge together with its supporting piers and abutments. 2. Regional beam action of each web supported at the top and bottom flanges as a beam. 3. Local slab action of the concrete cover over the tendons. The article discusses the current design philosophy of the California DOT, the Texas DOT, the AASHTO Guide Speci- fications for the Design and Construction of Segmental Concrete Bridges, and the AASHTO Load and Resistance Factor Design Bridge Design Specifications for tendon confinement. A test program was carried out by the authors on webs without tendon confinement reinforcement. Both closely and widely spaced ducts (duct spacing less than or greater than one duct diameter, respectively) were tested, and different lateral shear failure modes were observed. Recommendations for lateral shear capacity are proposed that are more conser- vative than the AASHTO LRFD Specifications and yielded a consistently narrow range of factors of safety for the four test specimens (1.99 to 2.34). A recommendation for a design methodology for the flexure of the concrete cover is not pro- posed because of the lack of understanding of its behavior. The test specimens did not fail in regional transverse bending of the web due to the formation of a second load path after formation of flexural cracks in the web. The load path is en- visioned primarily as a vertical one with the lateral prestress load carried through flexural bending of the web until crack- ing. Once cracking occurred, the stiffness was reduced and the load was carried primarily through longitudinal arching until a local lateral shearing failure occurred. A University of Texas thesis (Van Landuyt, 1991) that discusses this research in greater detail was also reviewed. Time Dependency The redistribution of stresses due to creep and shrinkage may be important in curved concrete bridges. This issue is discussed in at least one of the references previously described (Menn, 1990). A paper further exploring this issue is de- scribed below. Zhang, L., Liu, M., and Huang, L. (1993) Time-Dependent Analysis of Nonprismatic Curved PC Box-Girder Bridges, 28

Conference Proceeding Paper, Computing in Civil and Building Engineering, pp. 1703–1710. This proceeding consists of papers presented at the Fifth International Conference on Computing in Civil and Build- ing Engineering held in Anaheim, California, June 7–9, 1993. The proceedings cover five major areas of concern: (1) com- puting in construction, (2) geographic information systems, (3) expert systems and artificial intelligence, (4) computing in structures, and (5) computing in transportation. Within these broad topics are subjects such as (1) computer analysis of cable-stayed bridges; (2) artificial intelligence in highway CAD systems; (3) automated systems for construction bidding; (4) effect of automation on construction changes; (5) optimal seismic design of structures; and (6) CAD instruction for civil engineering students. The book also presents several papers discussing different aspects of multimedia information sys- tems and geographic information systems. Vehicular Impact Vehicular impact in curved bridges is different than in straight bridges. A paper addressing this subject is discussed below. Rabizadeh, R. O., and Shore, S. (1975) “Dynamic Analysis of Curved Box-Girder Bridges,” Journal of the Structural Division, ASCE, Vol. 101, No. 9, pp. 1899–1912. The finite element technique is used for the forced vibration analysis of horizontally curved box-girder bridges. Annular plates and cylindrical shell elements are used to discretize the slab, bottom flanges, and webs. Rectangular plate elements and pin-jointed bar elements are used for diaphragm discretization. The applied time varying forcing function used in this analysis represents a vehicle that is simulated by two sets of concentrated forces with components in both the radial and transverse direc- tion. The position of these concentrated forces is moved at a constant radial velocity in circumferential paths on the bridge. The effect of centrifugal forces is considered and the effect of damping of the bridge is neglected in the analysis. The mass condensation technique is used to reduce the number of cou- pled differential equations obtained from the finite element method. The resulting differential equations are solved by the linear acceleration method. Several bridges with practical geometries are analyzed and impact factors are calculated. Seismic Response Several papers and reports have addressed the seismic re- sponse of curved box-girder bridges. Most of these deal with substructure response and are beyond the scope of this study. However, at least one presentation (no paper available) has implications for superstructure design. This study is de- scribed below. Ibrahim, A. M. M., et al. (2005) Torsional Analysis and De- sign of Curved Bridges with Single Columns - LFD vs. LRFD Approach, Paper presented at the Western Bridge Engi- neers Conference, Portland, OR. This unpublished study compared torsional superstruc- ture design of a curved concrete box-girder bridge subjected to seismic loading using the Load Factor Design approach currently used by Caltrans and the AASHTO LRFD method. Torsion is induced in the superstructure not only by curva- ture, but also by column plastic hinging of the single column bent during an earthquake. Caltrans practice is to design the superstructure in bridges with monolithic columns to remain elastic. In this case, the limitation on seismically induced tor- sional forces resulting from column yielding causes a redis- tribution of superstructure forces in the box-girder. Design Optimization Several combinations of slab and web width can be selected to resist the applied loads. Although design optimization is generally not the subject of design specifications, at least one paper reviewed addressed this issue. Ozakea, M., and Tavsi, N. (2003) “Analysis and Shape Optimization of Variable Thickness Box-Girder Bridges in Curved Platform,” Electronic Journal of Structural Engi- neering International, Vol. 3, Queensland, Australia. This paper deals with the development of reliable and effi- cient computational tools to analyze and find optimum shapes of box-girder bridges in curved planforms in which the strain energy or the weight of the structure is minimized subject to certain constraints. The finite strip method is used to determine the stresses and displacements based on Mindlin-Reissner shell theory. An automated analysis and optimization proce- dure is adopted which integrates finite strip analysis, para- metric cubic spline geometry definition, automatic mesh generation, sensitivity analysis, and mathematical program- ming methods. It is concluded that the finite strip method offers an accurate and inexpensive tool for the optimization of box-girder bridges having regular prismatic-type geome- try with diaphragm ends and in curved planform. Detailing The detailing of prestressed concrete in bridges is addressed by several agencies that commonly use this structural form. 29

A general reference published by VSL International, a large prestresser with international experience, is discussed below. Rogowsky, D. M. and Marti, P. (1991) Detailing for Post- Tensioning, VSL International Ltd., Bern, Switzerland. Detailing for Post-Tensioning includes discussions and examples demonstrating the forces that are produced by post-tensioning, in particular, those in anchorage zones and regions of tendon curvature. Emphasis is placed on the use of strut-and-tie models to determine the tensile reinforcement requirements. Article 4.4, “Tendon Curvature Effects,” deals with special issues associated with curved tendons, including in-plane deviation forces, out-of-plane bundle flattening forces, minimum radius requirements, and minimum tangent length requirements. The radial force generated by a curved tendon is given as P/R where P is the tendon force and R is the radius of curvature of the tendon. Methods for preventing tendon breakout in thin curved webs include adequate lateral shear capacity of the concrete cover (adequate cover) or pro- viding tieback reinforcement. Summary A considerable body of research has been conducted on box-girder bridges. Much of this is useful to this project. With respect to global response analysis, research can be broadly divided between steel and concrete bridges. Concrete bridges have been found to be stiff enough so that torsion warping can generally be ignored. Sophisticated elastic analysis techniques such as finite element methods have been shown to produce excellent results that compare well with physical testing. It is therefore not necessary to do any more sophisticated research on this subject. It is necessary for our project to explore the accuracy of less sophisticated methods such as grillage analysis. If grillage analysis methods can be shown to produce reliable results, than they can be used both in design and as a verification tool for even less sophisticated analysis methods. The goal is to identify the simplest meth- ods that can be used safely. It also seems that several potential configurations of curved box-girder bridges need further study from the designer’s point of view. Although some research work has been per- formed on skewed bridges, bearings, and interior diaphragms, most of it has not found its way into design specifications. Part of our goal is to develop design procedures to handle these issues. Conventional reinforced and prestressed concrete design methods can be used for curved concrete box-girder design, provided accurate global demands can be established. Con- siderable work has been performed over the years in these areas. Torsion design, particularly as it applies to box-girders is well established, and further refinement of these methods is beyond the scope of this project. The local behavior of prestressed tendons in curved con- crete box-girder bridges is an issue to be addressed by this project. Although excellent research has been conducted at the University of Texas (Van Landuyt, 1991) this needs to be studied further using available analytical techniques. 30