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

Chapter: Chapter 5 - Regional and Local Response Analysis Studies

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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
×
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Suggested Citation:"Chapter 5 - Regional and Local Response Analysis Studies." National Academies of Sciences, Engineering, and Medicine. 2008. Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC: The National Academies Press. doi: 10.17226/14186.
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44 Three types of actions, as shown in Figure 5-1, have been considered in the analysis work for this project. 1. 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 action of the concrete cover over the tendons, and/or local lateral shear/breakout failure adjacent to the ducts. This is sometimes referred to as Lateral Tendon Breakout (LTB) Global or overall girder action of the bridge together with its supporting piers and abutments is covered in Chapter 4. This chapter focuses on “regional” and “local” action. Regional beam action considers each web as a beam sup- ported at the top and bottom flanges. The regional moments can be determined from a 2-D frame analysis of the cross sec- tion. The prestress lateral force is determined individually for each web with due consideration for the allowable variation in prestress force between webs. The compressive reactive forces on the concrete are applied as distributed loads on the webs and as concentrated loads at the centerlines of the slabs. The system is in static equilibrium and the support reactions will be zero. Local slab action of the concrete cover over the tendons has been identified as a 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 element restraining the lateral prestress force. The cover concrete acts as a plain concrete beam to restrain the lateral prestress force, F, as shown in Figure 5-2. The “local slab” is subject to lateral shear and bending from the lateral prestress force. The web is subject to regional transverse bending which results in tensile stresses on the local slab. Detailed local analysis models were used to evaluate the local stresses resulting from longitudinal tendons generating transverse forces on curved webs. Finite element (FE) models were developed to perform parameter studies, investigate capacities and damage modes, and prescribe a methodology to prevent such damage. Detailed 3-D, nonlinear analyses were performed using ABAQUS Version 6.5 “damaged plas- ticity” concrete cracking model. Such models provided tendon horizontal force plots versus deformation, and these, in com- bination with strain contour and crack pattern plots show the evolution of damage with increasing force. Comparison of such plots among different geometries and reinforcing schemes provides quantitative and qualitative parameter sensitivity evaluation. The FE analyses were used to provide insights into where damage first accumulates and develops, but these parameter sensitivity comparisons are not exhaustive, and there are limitations to what can and cannot be reliably predicted by FE analysis. A limitation of the parameter studies, for exam- ple, is that, for cases with reduced cover, the web thickness was held constant, and this tends to increase the moment arm to the web stirrups, which partially offsets the reduction in strength associated with reduced cover. Limitations on the FE simulations, for example, include the fact that ultimate failures caused by discrete crack propa- gation are difficult to predict. FE analysis practitioners (e.g., members of ACI/ASCE Committee 447 “Finite Element Analysis of Reinforced Concrete”) have a range of opinions on how best to predict the propagation of individual cracks in a structure component with a lot of rebar. Some advocate the use of fracture-mechanics-based algorithms shown to predict propagation of single cracks in plain concrete reason- ably well. But for practical FE analysis of concrete with a lot of rebar and a lot of cracks, the industry standard approach is to use smeared crack models, as was used here. What these models do reasonably well is predict “zones of likely crack formation” and strain distributions in concrete and rebar, which provide insight into causes and triggers for failure. Experience and judgment is required for the interpretation of the results. The work herein is based on C H A P T E R 5 Regional and Local Response Analysis Studies

(so has the limitations of) the constitutive formulations within the ABAQUS program, a widely distributed, well- respected commercial product for solving problems with a high degree of material nonlinearity. This program was chosen partly because of the authors’ de- gree of experience using it, but also to demonstrate the use of a widely available tool that other interested engineers or re- searchers could use to examine their own special design cases. ABAQUS is certainly not the only product available for this type of analysis. Despite the limitations noted, the authors believe the FE work herein has demonstrated an approach to modeling box- girder cross-sections, especially a method for applying the loads and boundary conditions, and this has satisfied one of the goals for this project. Ultimately, the design code recommendations in this re- port were developed by our team’s years of collective design and construction observation experience, research of litera- ture and existing codes, extensive hand calculations (similar to what we think designers should use), global analysis, and the local FE analysis. Local Analysis Validation/Demonstration Case (UT Test Case) The local analyses were begun with a validation case simu- lating Specimen “BC” from prior research conducted at the University of Texas (Van Landuyt, 1991). The case studied is a 1⁄3 scale representation of the configuration of Las Lomas, a well-known bridge that failed in lateral tendon breakout. Test Model and Test Conduct The following test model and test conduct description come from the test report and thesis: “The Effect of Duct Arrangement on Breakout of Internal Post-Tensioning Tendons in Horizontally Curved Concrete Box Girder Webs,” by D.W. Van Landuyt, 1991. The box cross-section was a scaled version of Las Lomas with changes made for simplifying construction (Figure 5-3). To avoid having to build cantilever forms, the girder was built and tested in an inverted position. This did not significantly affect results (it was assumed by the researchers that gravity loads are unimportant to breakout). The model top slab therefore represents the bottom slab at Las Lomas, etc. The top slab thickness at Las Lomas varied transversely and the 45 GLOBAL Girder Action LOCAL Slab Action REGIONAL Bean Action Figure 5-1. Types of actions considered. dmin ML dmin TC CT LOCAL SLAB W = F/L LOCAL MOMENT L F V V M Mn n V ~ F/2 T ~ Mn /(jd) ML ~ FL/12 jd Figure 5-2. Regional and local actions on a web.

bottom varied longitudinally. Average thicknesses were calcu- lated, scaled and rounded off to an integral number of inches. At Las Lomas, the centerline distance between interior and exterior webs was 11 feet. This scaled to 3 ft 8 in. The model was constructed with a centerline web-to-web distance of 4 feet so that the radius of each web was a whole number. To save on materials and labor, the cantilevers were shortened slightly from a scaled value of 1 foot 10 inches to 1 ft 4 in. (The actual cantilever length has little effect on web behavior.) The dimensions of the web were considered important. The exact scaled values of 4 in for the thickness and 3 ft for the overall height were maintained (Figure 5-4). The web radii were chosen to be small enough so that the tendon breakout in the web would occur before failure of any other part of the girder or testing apparatus. If the curve were not sharp enough, anchorage zone failure might have resulted or the strands would have been loaded to an unsafe level. Duct arrangement controlled the design of the curve radius. The capacity to resist lateral shear failure was calculated for each tendon, assuming two failure planes would form and that the maximum concrete strength would be 5000 psi. where Fr is the lateral (radially oriented relative to the curve) prestress force. Therefore a total Fr of 15.2 k/ft was required for all four tendons. A jacking force of 372 kips could be delivered from F k ftr = ′′ ′′ =2 5000 2 12 1 125 3 8( ) ( . ) . / 46 14'-4.38" 9'-0.25" 1'-11.38" 16'-3.75" 4" 3'-8" 4" 1'-4" 7'-0" 5'-0" = 16.07° R=18' PC PTR=22'CL 1'-4" 6'-2.5" 6'-0" "Typical" Segment Modeled Top slab Bottom slab 1.5" chamfer 1'-4" 4" 3'-8" 4" 1'-4" 7'-0" 4" 2'-5" 3" 3'-0" Figure 5-3. Plan and end view of U.T. girder test specimen (from Van Landuyt, 1991).

the loading apparatus. An 18-ft radius for the inside of web was selected, as it would permit a total Fr of 20.6 k/ft. This was more than a third larger than the anticipated failure load. The previously determined cell width mandated that the com- panion web radius be 22 feet. The curve length of 5 feet was chosen for the 18-ft radius. The curved region was the most difficult part of the model to construct and was kept as short as possible. A 5-ft curve was more than twice the clear height of the web and was thought to be sufficient to allow regional transverse bending. A 5-ft straight transition zone on each end insulated the curve from the complex stresses at anchorages. Las Lomas was reinforced with GR40 #5 stirrups spaced at 15 inches. No standard bars match this on a 1⁄3 scale. The closest match was 6 mm, 75 ksi bars from Sweden already available in the lab. This is nearly equivalent to a #2 bar. The stirrup spacing needed to be adjusted to reflect the imprecise scaling of stirrup sizes. Equivalent spacing of #6’s at 21.3 inches was scaled to 6-mm bars at 7 inches. The spacing was not increased to account for the greater yield strength of the Swedish bar. Stirrup spacing was reduced in the anchorage zones to 51⁄2 inches. A four-tendon bundle and three other promising arrange- ments were tested. Only duct positioning varied from web to web; all other details remained the same. Duct size at Las Lomas was not given, although based on the maximum num- ber of strands (28), ducts of approximately 41⁄2-inch O.D. should have been used. Scaling required 11⁄2-inch ducts for the model, but the major manufacturers of post-tensioning duct apparently do not make this size. The nearest available duct size (1.75 inches) was used. Specimen BC is the duct arrangement similar to the one at Las Lomas. A slight modification was made for the model 47 104 Figure 5-4. Girder #1 cross section in curved region (from Van Landuyt, 1991).

(Figure 5-4). A straight vertical formation was used in lieu of the zigzag because it was considered more universal. The relative horizontal offsets between ducts in a zigzag pattern can change from bridge to bridge, depending on the clear distance between the stirrup legs and the diameter of the ducts. No large difference in behavior between the two arrangements was anticipated. The vertical bundle height at Las Lomas was approximately 16 inches; a vertical stacked bundle would have been 171⁄2 inches. All ducts were centered on the web vertical axis. Specimen 1.0DC follows the Texas State Department of Highway and Public Transportation design of the San Antonio “Y” project with an arrangement that maintained a clear spacing between ducts equal to the diameter of the duct. It is believed that this allows for better consolidation and, more importantly, eliminates the single large discontinuity found at Las Lomas. This arrangement is conservative and would be considered an upper limit beyond which further spacing of ducts would provide no benefit. The scaled vertical spacing was 1.75 inches. All ducts were centered on the web vertical axis. All test concrete strengths were much greater than the 28-day design strength of 3500 psi. The slab and web concrete had higher overall strengths and faster strength gains as is typical of concrete containing super-plasticizers. The web concrete strength was 5300 psi. 6-mm-diameter hot-rolled bars were used for all rein- forcement. Tensile tests on bars conducted at the lab showed average yield strength of 75 ksi. Galvanized, corrugated, folded metal ducts were used in all instances. The outside ridge-to-outside ridge dimension was 1.75 inches. The inside diameter was 1.60 inches and the gauge was 0.035 inches. Post-tensioning was applied to the specimens with 7-wire, 1⁄2-inch Ø, 270 ksi, low-relaxation strands. Test data provided with the strand showed an actual yield of 276 ksi and ultimate of 289.5 ksi. A loading system was developed to apply gradually in- creasing load simultaneously to all tendons with an equal force in each tendon. It was necessary to consider how the strand or strands that would constitute a tendon would bear on a duct. The ducts at Las Lomas were nearly filled to capac- ity with strands. This meant that almost the entire 180 degrees of the duct on the inside of the curve was in contact with the strands. That same type of load distribution could be approx- imated with a minimum of three 1⁄2-inch-diameter strands per duct (Figure 5-5). Given that there were four ducts, a total of twelve strands could be safely stressed to 0.75fpu to develop a maximum force of 372 kips. Regional beam behavior was monitored by deflection of the web relative to the top and bottom slabs. U-shaped frames were mounted to the web face on the outside of the curve (Figure 5-6). The actual attachment points were about 2 inches below the top slab and 2 inches above the bottom 48 1.75" 1.67" Figure 5-5. Strand positions in curve (from Van Landuyt, 1991). WS2 WS4 WS6 WS8 14.5" PC PT Potentiometer NOMENCLATURE: WS# Web deflection with respect to slabs Location with respect to stirrup # (±)1") Figure 5-6. Web potentiometer configuration (from Van Landuyt, 1991).

slab. This permitted the construction of only one type of frame, which could be mounted on either web. Ideally, the frame should have been mounted on the slabs. However the deflection anticipated in the first 2 inches is negligible. A sin- gle potentiometer was mounted at the mid-height of the frame (which is also the c.g. of the tendon group). A small mirror glued to the specimen provided a smooth surface on which the potentiometer stem could bear. Mounting the potentiometers on the outside face of the curve meant that deflections should not be influenced by local beam action; regional beam behavior should be solely responsible for measured deflections. Also potentiometers attached to the back face were protected from exploding con- crete. The web-slab potentiometer nomenclature is given in Figure 5-6. The description begins with the letters WS to signify that deflections of the web relative to the slab are being measured. The number of the stirrup nearest the potentiome- ter follows these letters. Web delaminations were measured by wires/potentiometers placed in tubes cast through the webs above and below the duct group. Sudden movements in these measurements were good indicators of imminent failure. Figure 5-7 is a sketch of how the concrete cracked and failed during the test. Figure 5-8 compares tendon horizontal force versus deflections for the four different webs tested. Finite Element (FE) Model and Analysis An FE model of Girder BC and 1.0DC was developed for a typical slice of the test model in the curved region, as was shown in Figure 5-3. The cross-section and duct geometry (as was shown in Figures 5-4 and 5-5) were modeled per the test configuration and dimensions. The FE model and boundary conditions are illustrated in Figure 5-9. This shows how a 3-D slice was modeled with horizontal tendon loads applied directly to the inner surfaces of the ducts. The 49 BC 1.0DC BO 0.2DC INSIDE FACE INSIDE FACE INSIDE FACE INSIDE FACE R=22' R=18' R=22' R=18' Figure 5-7. Specimen failure plans (from Van Landuyt, 1991). 0 2 4 6 8 10 12 14 16 18 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 DEFLECTION (inches) Ft (k/ft) BC BO 1.0DC 0.2DC PC WS8WS6WS4WS2 PT ELEV SECTION Figure 5-8. Comparison of web deflections relative to slabs at Stirrup #6 (from Van Landuyt, 1991).

wedge slice model configuration and “symmetry” boundary conditions allowed horizontal (radial) displacement to occur naturally in the FE model, and this in turn, causes “longi- tudinal” compression prestress in the concrete in a manner similar to the actual test specimen. What is precisely math- ematically represented with such boundary conditions is a wedge-slice of a complete circle, but the boundary condi- tions are also reasonably accurate for a section of a curve. The tendon forces were applied in incremental fashion with enough equilibrium iterations and small load increments to achieve solution convergence at every increment. This solution procedure is often referred to as Full-Newton, where upon the next load increment; the structure stiffness is updated, based on the material damage, which has occurred (concrete cracking/ crushing and steel yielding). The analysis was run out to large web displacements and significant damage (failure), i.e., well beyond the displacements plotted in Figure 5-8. A typical deformed shape from the analysis (at the 1.0 DC web displacement of 0.07 inches) is shown in Figure 5-10. The amount of deflection and the sharpness of the flexural curvature is significantly more severe in Girder-BC than in Girder-1.0DC. This agrees with test observations. In the illustration, displacements are magnified by 50. Figure 5-11 shows the same deformed shape, but with con- tours of maximum principal strain. In reinforced concrete analysis, these are one of the most effective ways to show damage and deformation distributions in the concrete and rebar. Concrete stress contours are generally not helpful, be- cause after concrete cracks, the stress reduces to nearly zero, so a zero or small tensile stress may be displayed in a zone that is already highly damaged. But maximum principal strains (generally, the maximum tension found in any orientation for a given point in the continuum) can indicate concrete cracking, which occurs at strain of approximately 1.50E-4 to 50 Figure 5-9. Finite element model and boundary conditions.

2.00E-4. Figure 5-11 shows widespread cracking in the vicinity of the ducts and much more severe cracking at the BC configuration than at the 1.0DC. Figure 5-12 shows similar strain contours, but in an end view at a lower displacement. Here the crack zones are also fully formed, but are easier to see in cross section. Figure 5-13 shows the same deformed shape as Figure 5-12, but viewed in conjunction with principal strains, is used to estimate the ex- tent of cracks. Using this information, and judgment from experience with many similar analyses, actual cracks implied by the analysis are drawn onto the figure. Comparing to Fig- ure 5-7 shows similar trends in the location of the cracking as compared to the test. In making this conclusion, it is assumed that the test also showed other small cracks in the vicinity of the ducts, but the cracks shown in Figure 5-7 were the primary failure planes. Another significant analysis versus test comparison is shown in Figure 5-14: the lateral force versus deflection at the mid-height of the duct bank relative to the slab. The compar- ison shows that for both the BC and 1.0DC girder webs, the analysis is simulating the response (initial stiffness and stiff- ness degradation with accumulating damage) and the lateral force capacity reasonably well. These comparisons and experience with many similar re- inforced concrete analyses indicate that the FE representation and modeling strategy are appropriate for moving on to the prototype models. Local Analysis of Multicell Box-Girders For the NCHRP 12-71 project study, there are two basic local model types: • Multi-cell box (three-cell box with super-elevation/ vertical interior webs, and inclined exterior webs; also ran with vertical exterior webs) • Single-cell box (prototypical pre-cast box with super- elevation and inclined webs) The prototype geometry for the multicell box girder was super-elevated since this is how curved girders often occur in practice. In the multicell studies conducted, the only effect of super-elevation was a small difference in the end conditions of 51 Figure 5-10. Deformed shape (displacements  50) at 1.0 DC deflection of 0.07 inches. Figure 5-11. Contours of max principle strains at 1.0DC deflections of 0.07 inches.

the regional moment calculation in the sloping exterior webs. Though not studied in depth, there was no apparent effect on the behavior of vertical webs caused by super-elevation so it was not included in the single cell section that was studied. The variables that may significantly influence local behav- ior are as follows: • Web depth, • Web thickness, • Web slope, • Cover thickness, • Number and configuration of tendon ducts, • Number and configuration of duct ties, • Stirrups, and • Concrete material properties, especially assumed tensile strength. A last variable, Pt/R, is evaluated by the analyses producing Pt/R versus deformation curves. So each analysis includes the full range of Pt/R up to failure. The analysis matrix to study the parameters for the multi-cell series is shown in Table 5-1. Model Prototype: Three-Cell Cast-In-Place Box Girder A multi-cell, cast-in-place box configuration was used as shown in Figure 5-15. The basic model (shown in Figure 5-16) uses 3-D elements and a slice, with out-of-page thickness equal to one stirrup spacing. The stirrups (and other rebar) are modeled explicitly (unlike in two dimensions, where the stir- rup is smeared). This allows introduction of the out-of-page compression due to prestress. Using this model framework, geometry variabilities were introduced directly into the models—e.g., one web can have one thickness, and another have a different thickness. Also, by using this model prototype, the effects of web slope are included and can be compared. Webs A and D have different slopes and can be compared with B and C, which are vertical. Webs A and D demonstrate the differences related to web sloping away from the radius of curvature versus sloping toward the radius of curvature. (Two additional cases were later added with vertical-web exterior webs to provide additional comparisons.) The tendon duct arrangements and local reinforcements are shown in Figure 5-17. In configuration Type 4, analyses were run with (4b) and without the center web ties (4a). This refers to the two rebar ties in the middle of the group of ducts. In configuration Type 3, separation of the ducts by 11⁄2 inches was found to provide increased resistance to lateral tendon breakout. Further separation may provide even higher resist- ance, but it was the opinion of the project team that enforc- ing even larger separations between the ducts begins to pose a very significant limitation on the effectiveness of the pre- stressing. Designers need to be allowed some flexibility in duct placement in order to achieve a range of vertical posi- tions of prestressing within the webs, for typical designs. 52 Figure 5-12. Contours for max principle strains at BC deflection of 0.03 inches (displ  25).

The boundary conditions for the models were the same as for the UT Test simulation, which produced reasonable cor- relation between analysis and test. The model is a sector slice taken from a curve. The dimension longitudinally varies between inside and outside edge, but on average is equal to 1.5 ft. This is also the stirrup spacing for the baseline model. This bridge example is assumed to be on an 800-foot curve radius, so the sector width varies slightly from the inside of the curve (Web D) to the outside (Web A). The concrete properties were = 5,000 psi, and Young’s Modulus = 4,030.5 ksi. The rebar was Grade 60. Plots of ma- terial stress-strain curves for the concrete and steel are shown in Figure 5-18. Tensile strengths (ft) for concrete when tested in direct uniaxial tension can show large variations, but most results fall within the range to ; is consid- ered a reasonable average. Multicell Models—Analysis Results The 16 different multicell girder local models were devel- oped, and analyses were completed. The results are shown in this chapter through the following plots and tables. These allow for qualitative and quantitative assessment and com- parison of the cases analyzed. • Lateral Force vs. Deflection of Web Midheight • Lateral Force vs. Deflection of Web Quarter-height • Maximum Principal Strain Contours in Concrete at 75%, 100%, 125%, 150% Pc 5 ′fc6 ′fc4 ′fc ′fc • Strains in stirrup rebar at 3 locations along duct bank at 75%, 100%, 125%, 150% Pc • Distortions (change in web width) at three locations along duct bank at 75%, 100%, 125%, 150% Pc Pc refers to a lateral force applied to the web that will cause theoretical web failure calculated using conventional means and removing various safety factors. This creates a frame- work for comparing the results of the detailed FE analysis to a baseline capacity. The displacements were measured at the “outside curve” edge of the webs. Although the four individual webs in each multi-cell model tended to act independently, the local analyses re- quired a decision as to loading of the individual webs. In planning this loading it was found that the interaction of the webs with their end conditions (i.e., the stiffness character- istics of the top and bottom slabs) was important to how the webs behaved. An initial study using the baseline geometry (Model 1M) and all-elastic material properties showed that web mid-height deflections varied as shown when equal loads (1,000 lbs per web) were applied to the webs as indi- cated in Figures 5-19 and 5-20. These figures show how the exterior web ends are freer to rotate than the interior webs. For the two extremes of fixed-fixed versus pinned-pinned, the ratios of mid- height moment to applied tendon force (P) would be h/8 versus h/4 or a ratio of one-half. But the web end conditions are neither fixed-fixed nor pinned-pinned. One way to quantify these differences associated with end effects was to apply the tendon forces to a beam model, as shown in the deformed shape plot of Figure 5-21. This exercise produced the following ratios of web mid- height moments to applied force: Web A B C D 0.186h 0.145h 0.147h 0.171h h = 92.75 in. (7.73 ft.) Normalizing to the pinned-pinned condition (M = P × h/4) gives ratios of: 0.744 0.580 0.588 0.684 Considered as coefficients, these can be compared with the Caltrans Memo-to-Designers Formula: Mu = 0.8(1⁄4)(Pj/R)hc Where Pj is the tendon force (j is for “jacking force”) Thus, for this case, Caltrans uses a continuity factor of 0.8 for design. Based on the work performed for this project, factors of 0.6 for interior webs and 0.7 for exterior webs are proposed. 53 Figure 5-13. Estimated cracking in the webs based on strains (displ.  25).

The overall distribution of these moments to webs agrees well with damage trends observed in the FE analysis, so one finding from the Local Analysis study may be that design- ers should account for this effect in more detail than to just apply a single formula to calculate mid-height moment demand. Further study of this using fully nonlinear properties showed that once concrete cracking begins to occur, the dif- ferences between webs become even larger. So it was eventu- ally decided to choose a baseline prestress force divided by the four webs, then increase this force for the interior webs, and reduce this force for the exterior webs. Using this as a basis and choosing a prestress force large enough to cause signifi- cant damage in all of the parametric models led to the fol- lowing total applied forces in kips/ft. This is analogous to Pj/R. Of course in some cases, the webs failed prior to reach- ing this total load. Web A B C D 13 k/ft 17 k/ft 17 k/ft 15 k/ft In order to establish a baseline for comparison with de- sign calculations, as previously mentioned, Pc is defined as a “Capacity” calculated using conventional means, but removing safety factors, so as to make direct comparison to finite element analyses. For the interior (B or C webs) of the multicell geometry prototype, Pc was calculated as follows. φMn = 8.7 k-ft/ft Removing the resistance factor φ = 0.9, Mn = 9.7 k-ft/ft Applying an over-strength factor for rebar strain harden- ing (which is included in the FE analysis), Mo = Mn × 1.125 = 10.9 k-ft/ft The moment-fixity effect is rounded off at 0.6. It is also rec- ognized that the duct forces are not applied at one point in midheight, but are instead, applied at five points distributed along 18 inches of the height. This decreases the moment to 54 18.0 1.0DC UT Test BC UT Test 1.0 DC FE Model BC FE Model 16.0 14.0 12.0 10.0 8.0F r ( K/ ft) 6.0 4.0 2.0 0.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Deflection (inches) 0.040 0.045 0.050 0.055 0.060 0.065 0.070 Figure 5-14. Force vs. deflection (FE model compared with U.T. Test).

88.4% of that caused by a single point load at midheight. So the baseline Pc becomes Pc = [10.9/(h/4)]/0.6/0.884 = 10.6 k/ft This is used as “100%” in the results tables and plots. But for the exterior webs, the “100%” force is less, because the load is applied in the proportions previously listed. The result of all this is to proportion loads so that webs will be “failing” more-or-less simultaneously, therefore maintaining proper flexural stiffnesses in webs and flanges at all loading stages. The results of all the multi-cell analyses are shown in Ta- bles 5-1, 5-2, 5-3, and 5-4 and figures. The figures are shown in Appendix E-b and are numbered as Figures E-b-1, E-b-2, etc. 55 Analysis # Model Type Web # Web Thickness Duct/Tie Config. Bundle Vert. Pos. Stirrup Spacing (in.) Cover Thickness Concr. Tens. Str. (x√fc`) 1M "baseline" Multi-cell A, D B C 12 12 12 1 1 1 midheight midheight midheight 18 18 18 3 3 3 4 4 4 2M Multi-cell A, D B C 12 12 12 2a 2a 2a midheight midheight midheight 18 18 18 3 3 3 4 4 4 3M Multi-cell A, D B C A-10, D-12 10 14 2b 2a 2a midheight midheight midheight 18 18 18 3 3 3 4 4 4 4M Multi-cell A, D B C 12 12 12 2a 2a 2a 1/4 height 1/4 height bottom 18 18 18 3 3 3 4 4 4 5M Multi-cell A, D B C 12 12 12 2b 2b 2a midheight midheight midheight 18 18 18 3 3 3 4 4 6 6M Multi-cell A, D B C 12 12 12 2a 2a 2a midheight midheight midheight 12 12 18 3 3 3 4 4 2 7M Multi-cell A, D B C 12 12 12 2a 2a 2a midheight midheight midheight 27 27 18 3 3 2 4 4 4 8M Multi-cell A, D B C 12 12 12 2a 2b 2a midheight midheight midheight 18 18 18 2 2 4 4 4 4 9M Multi-cell A, D B C 12 12 12 3a 2a 2a 1/4 height 1/4 height bottom 18 18 18 2 2 2 4 4 4 10M Multi-cell A, D B C 12 12 12 3a 3a 3a midheight midheight midheight 27 27 18 3 3 2 4 4 4 11M Multi-cell A, D B C A-10, D-12 10 14 4a 4a 4b midheight midheight midheight 18 18 18 3 3 3 4 4 4 12M Multi-cell A, D B C 12 12 12 4a 4a 4b Midheight midheight midheight 18 18 18 3 3 3 2 4 6 13M Multi-cell A, D B C 12 12 12 4a 4a 4a midheight midheight midheight 27 27 18 3 3 2 4 4 4 14M Multi-cell A, D B C 12 12 12 4b 4b 4b 1/4 height 1/4 height bottom 18 18 18 3 3 3 4 4 4 2m-Vert Multi-cell A, D B C 12 12 12 2a 2a 2a midheight midheight midheight 18 18 18 3 3 3 4 4 4 11M-Vert Multi-cell A, D B C A-10, D-12 10 14 4a 4a 4b midheight midheight midheight 18 18 18 3 3 3 4 4 4 Table 5-1. Parameters for multi-cell girder local response analysis.

56 275.75” 275.75” 70.86” 47.25” 102” 72.83” 72.83” 102” 47.25” 70.86” 11 0. 25 ” 9” 8.5” & Var 12” & Var 13” & Var 10% & Var I-405 - 55 HOV Connector O/C General Cross-Section (inches) Figure 5-15. Multicell girder cross-section geometry. Figure 5-16. Example finite element mesh for multicell local analysis prototype. All duct diameters are 41⁄2”.

The maximum principal strain contours illustrate the general level of damage to the concrete surrounding the tendon ducts (maximum tensile strain regardless of orientation). The following strain thresholds are important quantities to compare to when viewing these results. ε11 = 1.6 × 10−4 first visible cracking (micro-cracking occurs at about half of this strain) 2.0 × 10−3 first rebar yield 1.0 × 10−2 1% strain in rebar; typically wide open cracks/sometimes spalling concrete In Table 5-4, delaminations (“distortion”) information is provided at “Duct 1,” “Duct 3,” and “Duct 5.” This refers to a measure across the bottom edge, top edge, and centerline of the duct assembly. Discussion of Results The analysis results have been used to compare the web design parameters. The first general observation is the com- parison of the midheight cases with the “1⁄4 height” and “bot- tom” cases. It was generally observed that when the ducts occur near the bottom of the web (either “quarter-height” or “bottom” as was tested in Configurations 4M and 14M) the force at “failure” is substantially lower than when the ducts are placed at the midheight, i.e., on average as much as 25% to 40% lower when comparing these cases with similar cases. The reason for this is a tendency toward lateral shear failure of the overall web. When the ducts are located at the mid- height, the lateral shear is divided equally between the top and bottom of the web. But when the ducts move down, the bot- tom of the web carries most of the lateral shear, and this is a different mechanism than the failure modes observed for ten- don ducts at mid-height. So the “quarter-height” cases can be compared to each other (and the “bottom” cases), but should not be compared directly with the “midheight” cases. “Pc” only applies to the “midheight” cases. For purposes of interpreting and comparing the results, the following damage criteria should be considered: • Stirrup rebar strain exceeding yield (i.e., 0.2% strain for Grade 60 steel); note that for Load Factor Design, concrete reinforcement is designed to yield at the ultimate member forces. • Visible concrete cracking occurs at strains of approxi- mately 0.016%, but this is not necessarily web failure; concrete with maximum principal strains of 0.3% can be considered to be heavily cracked. Concrete with strains in excess of 1.0% will generally show wide-open cracks and potential spalling from the section. • Significant distortion or delamination (change of width of the webs) would also represent an upper bound on serviceable capacity for webs; the delamination is evidence of a local splitting or lateral shear failure within the web. It was arbitrarily assumed that a crack width of 1⁄16” is an 57 Type 1 Type 2 Type 3 Type 4 12" Typ @ all sections Inside of Curve, Typ @ all sections 3" Clr To Duct Typ Duct Typ Stirrups Typ 3" Clr To Duct Typ 3" Clr To Duct 3" Clr To Duct 2" clr 2" clr Case B Only 1. 5" # 4 12" # 4 @ 12", Alternative Sides for 135° hooks Ca se A -1 .5 " Ca se B -1 D uc t D ia Ca se A -1 .5 " Figure 5-17. Tendon duct and local reinforcement for multicell box local analysis prototypes (note that this figure is an idealization of bar placements that are implemented in the FE analysis).

total forces (sum of all tendon ducts in the web) applied when any part of the stirrup reaches yield, and when the web dis- tortion reaches 0.06 inch. Using these criteria and the results tables and plots has re- sulted in the following observations. Web Depth This parameter was varied indirectly by subjecting the webs to wide ranges of moments and horizontal shears. Based on ob- servations of the analysis results, web depth can be adequately accounted for by considering and designing for web moments. Web Thickness Web thickness was varied in Model 3M (A – 10 inches, B – 10 inches, C – 14 inches), and similarly in Model 11M. 58 Figure 5-18. Stress vs. strain curves concrete and steel in local FE analysis. Figure 5-19. Deformed shape of typical cross-section with the same force applied to each web. indicator of such a failure. For 12” webs, this represents a distortion of 0.06 inches, and a distortion ratio (average strain through the section) of 0.5%. For sections with web ties, this means the web ties have yielded; for sections without web ties, the section is at a web splitting or a web lateral shear failure condition. Two of the criteria, Stirrup Yield and Web Delamination, have been summarized in Tables 5-5 and 5-6. These are the

Model 11M included web ties. The results compared with their respective baselines are shown in Table 5-7. These results demonstrate significant influence on resist- ance to lateral bending and tendon pullout caused by web thickness. Stirrups yielded sooner, and concrete damage and web delamination was more extensive with the thinner webs. For stirrup yield, capacity formulae based on regional flex- ure considerations appear to be appropriate for design. Dif- ferences in stirrup yield and especially web delamination were also significantly influenced when the web ties were added be- cause the web ties tended to eliminate the web delamination failure mode. Moving the ducts toward the curve outside face within the webs also contributed to resistance against delam- ination and local lateral shear damage. Web Slope As described earlier, the sloped webs in this analysis series were found to be significantly weaker (30–40%) than the vertical webs, but part of this difference was caused by being exterior webs rather than interior. Exterior webs have more flexible end conditions at their connection with the top and bottom slab, and this produces larger mid-height moments. Comparison of Webs A to D for the inclined webs show that Web A is generally weaker than D by about 10%, due to ori- entation of slope relative to the direction of the tendon force. In order to examine the differences between sloped webs and vertical webs more directly, two additional analyses were performed with exterior webs converted to vertical webs. The strain contour and Force versus Deflection plots are included in Appendix F. Models 2M and 11M were chosen for these comparisons because these have the baseline values for all properties, but they investigate duct-tie configurations 2a and 59 Figure 5-20. Deformed shape/strain contour when the same force is applied to each web. Figure 5-21. Tendon forces applied to a beam model of typical cross-section.

60 Web A Web B Web C Web D Percent Mid Quarter Mid Quarter Mid Quarter Mid Quarter Model # Capacity 1m 75% 0.0290 0.0287 0.0360 0.0262 0.0271 0.0210 0.0199 0.0139 100% 0.0830 0.0686 0.0962 0.0555 0.0797 0.0420 0.0554 0.0225 125% 0.1851 0.1553 0.2249 0.1371 0.1981 0.1115 0.1390 0.0625 150% 0.4067 0.3567 0.5017 0.3382 0.4859 0.3083 0.3379 0.1867 2m 75% 0.0309 0.0340 0.0242 0.0262 0.0237 0.0259 0.0254 0.0216 100% 0.0872 0.0891 0.0702 0.0690 0.0673 0.0672 0.0673 0.0573 125% 0.1966 0.2026 0.1702 0.1629 0.1676 0.1623 0.1566 0.1366 150% 0.4455 0.4509 0.3821 0.3756 0.4080 0.3897 0.3477 0.3067 3m 75% 0.0554 0.0520 0.0411 0.0403 0.0225 0.0293 0.0309 0.0266 100% 0.1524 0.1395 0.1323 0.1191 0.0615 0.0792 0.0863 0.0751 125% 0.3711 0.3367 0.3314 0.2972 0.1571 0.1974 0.2103 0.1841 150% 2.2610 2.2670 1.6490 1.5490 0.9640 1.2280 0.9860 0.6550 4m 75% 0.2148 0.3214 0.2289 0.3171 0.1890 0.2885 0.2141 0.2803 100% 0.8136 1.2218 0.9112 1.2538 0.7589 1.1392 0.8323 1.0922 125% 2.3248 3.5099 2.7380 3.7348 2.2520 3.3725 2.4692 3.2460 150% 11.1900 16.3600 11.3300 15.5600 9.9500 15.1500 12.4100 14.2900 5m 75% 0.0251 0.0269 0.0190 0.0202 0.0172 0.0195 0.0179 0.0142 100% 0.0665 0.0687 0.0518 0.0539 0.0409 0.0481 0.0511 0.0415 125% 0.1385 0.1408 0.1103 0.1114 0.0938 0.0986 0.1082 0.0870 150% 0.2707 0.2731 0.2248 0.2240 0.1954 0.1983 0.2109 0.1726 6m 75% 0.0352 0.0386 0.0278 0.0304 0.0506 0.0385 0.0257 0.0231 100% 0.1240 0.1290 0.1051 0.1048 0.1610 0.1313 0.0887 0.0840 125% 0.3047 0.3182 0.2684 0.2603 0.3685 0.3123 0.2182 0.2071 150% 0.7176 0.7433 0.6550 0.6354 0.8475 0.7286 0.5231 0.4981 7m 75% 0.0279 0.0298 0.0220 0.0230 0.0207 0.0225 0.0206 0.0170 100% 0.0789 0.0799 0.0583 0.0606 0.0548 0.0587 0.0596 0.0486 125% 0.1674 0.1702 0.1295 0.1325 0.1283 0.1305 0.1332 0.1074 150% 0.2980 0.3033 0.2425 0.2393 0.2493 0.2427 0.2385 0.1935 8m 75% 0.0313 0.0337 0.0217 0.0253 0.0214 0.0253 0.0235 0.0202 100% 0.0922 0.0916 0.0545 0.0660 0.0621 0.0680 0.0671 0.0558 125% 0.2020 0.2030 0.1338 0.1560 0.1639 0.1630 0.1563 0.1285 150% 0.4180 0.4180 0.2815 0.3252 0.3478 0.3461 0.3340 0.2714 9m 75% 0.1442 0.1983 0.1270 0.1825 0.1067 0.1682 0.1481 0.1654 100% 0.5575 0.7819 0.5310 0.7502 0.4394 0.6840 0.5756 0.6430 125% 1.6680 2.3691 1.6878 2.3781 1.3931 2.1697 1.7823 1.9963 150% 9.7190 13.7340 8.8200 12.3520 7.6240 11.8020 11.0610 11.3050 10m 75% 0.0230 0.0254 0.0178 0.0197 0.0178 0.0195 0.0156 0.0122 100% 0.0591 0.0627 0.0397 0.0480 0.0390 0.0463 0.0427 0.0363 125% 0.1350 0.1428 0.0892 0.1096 0.0924 0.1098 0.0990 0.0847 150% 0.2458 0.2636 0.1778 0.2091 0.1755 0.2090 0.1939 0.1626 11m 75% 0.0388 0.0390 0.0253 0.0273 0.0183 0.0213 0.0195 0.0154 100% 0.1024 0.1019 0.0683 0.0765 0.0453 0.0566 0.0575 0.0491 125% 0.2601 0.2544 0.1764 0.1961 0.1160 0.1450 0.1534 0.1270 150% 1.1300 1.2390 0.7370 0.7760 0.6300 0.6890 0.5420 0.3130 12m 75% 0.0212 0.0236 0.0168 0.0181 0.0161 0.0180 0.0153 0.0113 100% 0.0493 0.0537 0.0355 0.0414 0.0343 0.0412 0.0407 0.0328 125% 0.1051 0.1118 0.0761 0.0864 0.0705 0.0860 0.0991 0.0748 150% 0.1949 0.2089 0.1515 0.1688 0.1332 0.1642 0.1939 0.1454 13m 75% 0.0190 0.0202 0.0161 0.0169 0.0162 0.0175 0.0154 0.0117 100% 0.0372 0.0392 0.0285 0.0319 0.0288 0.0327 0.0324 0.0256 125% 0.0750 0.0780 0.0533 0.0610 0.0541 0.0627 0.0630 0.0516 150% 0.1201 0.1258 0.0986 0.1089 0.1059 0.1164 0.1122 0.0903 Table 5-2. Deflections.

4a/b. Direct comparisons of Force versus Deflection are shown in Figures 5-22a and 5-22b below. These figures show the vertical webs to be stiffer and stronger than the sloping webs, but show that the differences in force capacity (strength) are negligibly small. A compari- son of the occurrence of rebar yield and web delamination is shown in Table 5-8. Cover Thickness Cover thickness was varied in Models 7M, 8M, and 13M. Table 5-9 summarizes sample strength comparisons. The conclusion reached is that cover thickness influences lateral pullout resistance, but is not the only driver of pullout resistance. The results of the parameter study were influenced by the fact that when the cover is reduced, for the same overall web thickness, the moment arm for the stirrups is increased, and this is an off-setting influence on pullout resistance. As will be discussed further in the conclusions, it appears appropriate to check cover concrete thickness for resistance to initial crack- ing, but not to include cover concrete tensile strength in the calculation of regional transverse bending strength. Number and Configuration of Tendon Ducts This was evaluated by comparing Configurations 1, 2, and 3 (from Figure 5-17), which involve comparing Models 1M vs. 2M, 3M-D vs. 2M-D, 5M vs. 2M, and 10M-C vs. 7M-C. Re- sults are shown in Table 5-10. Clearly, when the ducts are spread apart, the performance significantly improves. Roughly 20% resistance force improve- ment was demonstrated by separating the 5-duct bundle into two bundles (Config. 2A versus Config. 1), and an addi- tional 4% improvement was demonstrated by spreading the bundles farther apart (4.5” versus 1.5” separation). So in gen- eral, a prudent recommendation is to require a maximum of 3 ducts per bundle. When the individual ducts were separated (i.e., Config. 3A) and moved toward the curve’s outside edge of the web, performance further improved. In fact, as meas- ured by the delamination criteria, Configuration 3A exceeded 200% Pc, so the improvement in delamination performance was very large. The influence on stirrup yield performance by spreading individual ducts apart was only 5%, and it is often impractical for designers to spread individual ducts apart due to lack of space in the web and due to requirements on loca- tion of C.G. of the tendon group. Number and Configuration of Duct Ties This was evaluated by comparing Configurations 4A and 4B, to Configurations 1, 2, and 3. This is covered by compar- ing Model/Webs 11M-D to 10M-D, 12M-B to 10M-B, 13M- A-D to 10M-A-D, 12M-C to 12M-B, and 14M-B, D to 9M-B. This comparison is shown in Table 5-11. The conclusions from these comparisons are that web/duct ties make a significant contribution to the resistance to lateral tendon breakout. Stirrups Stirrup spacing was evaluated by comparing Model-Webs 6M-A, B, and D to 2M-A, B, and D, comparing 7M-A, B, D to 2M-A, B, D, and comparing 13M-A, B, D to 12M-A, B, D. These comparisons are shown in Table 5-12. This indicates that web section strength is significantly in- fluenced by the stirrup spacing only when web/duct tie rein- forcement is NOT used or when the web-splitting/lateral shear-failure does not occur. In other words, if the failure 61 Web A Web B Web C Web D Percent Mid Quarter Mid Quarter Mid Quarter Mid Quarter Model # Capacity 14m 75% 0.1016 0.1346 0.0841 0.1233 0.0842 0.1228 0.1207 0.1239 100% 0.3100 0.4193 0.2772 0.4126 0.2657 0.3889 0.3656 0.3779 125% 0.8000 1.0955 0.7350 1.0775 0.7158 1.0229 0.9539 0.9897 150% 2.1159 2.8840 1.8586 2.7535 1.9289 2.7251 2.5100 2.6583 2MVert 75% 0.0153 0.0160 0.0170 0.0172 0.0162 0.0171 0.0154 0.0162 100% 0.0338 0.0318 0.0400 0.0343 0.0398 0.0348 0.0319 0.0302 125% 0.0913 0.0748 0.0896 0.0731 0.1033 0.0796 0.0806 0.0665 150% 0.1824 0.1475 0.1917 0.1481 0.2198 0.1664 0.1831 0.1401 11MVert 75% 0.0314 0.0295 0.0245 0.0254 0.0146 0.0182 0.0191 0.0206 100% 0.0686 0.0594 0.0474 0.0479 0.0255 0.0320 0.0325 0.0355 125% 0.1449 0.1190 0.1026 0.0977 0.0489 0.0614 0.0677 0.0697 150% 0.2706 0.2221 0.2076 0.1937 0.0935 0.1189 0.1365 0.1337 Table 5-2. (Continued).

62 Web A Web B Web C Web D Percent Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Model # Capacity 1m 75% 0.0002 0.0003 0.0002 0.0002 0.0005 0.0004 0.0002 0.0004 0.0002 0.0002 0.0003 0.0002 100% 0.0007 0.0011 0.0005 0.0006 0.0017 0.0010 0.0005 0.0015 0.0006 0.0006 0.0011 0.0005 125% 0.0015 0.0023 0.0013 0.0012 0.0045 0.0016 0.0012 0.0033 0.0013 0.0013 0.0026 0.0013 150% 0.0026 0.0057 0.0018 0.0017 0.0104 0.0026 0.0020 0.0085 0.0026 0.0024 0.0071 0.0019 2m 75% 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 100% 0.0007 0.0003 0.0003 0.0003 0.0005 0.0008 0.0003 0.0012 0.0003 0.0006 0.0004 0.0003 125% 0.0013 0.0008 0.0006 0.0007 0.0013 0.0018 0.0008 0.0018 0.0008 0.0013 0.0019 0.0007 150% 0.0026 0.0015 0.0012 0.0018 0.0021 0.0039 0.0019 0.0083 0.0014 0.0024 0.0019 0.0012 3m 75% 0.0003 0.0003 0.0002 0.0003 0.0002 0.0003 0.0002 0.0001 0.0001 0.0003 0.0005 0.0001 100% 0.0009 0.0008 0.0005 0.0012 0.0008 0.0008 0.0004 0.0004 0.0003 0.0010 0.0012 0.0003 125% 0.0023 0.0019 0.0013 0.0027 0.0019 0.0018 0.0010 0.0009 0.0006 0.0018 0.0032 0.0009 150% 0.0209 0.0186 0.0207 0.0225 0.0153 0.0218 0.0178 0.0170 0.0172 0.0242 0.0244 0.0164 4m 75% 0.0011 0.0019 0.0007 0.0008 0.0013 0.0014 0.0009 0.0015 0.0010 0.0016 0.0013 0.0003 100% 0.0025 0.0029 0.0025 0.0019 0.0045 0.0080 0.0027 0.0068 0.0046 0.0056 0.0021 0.0011 125% 0.0093 0.0235 0.0090 0.0065 0.0153 0.0246 0.0107 0.0171 0.0158 0.0197 0.0048 0.0021 150% 0.0789 0.0978 0.0787 0.0793 0.0756 0.0994 0.0870 0.0796 0.0876 0.1241 0.0923 0.0653 5m 75% 0.0002 0.0002 0.0001 0.0002 0.0001 0.0002 0.0001 0.0001 0.0001 0.0003 0.0003 0.0001 100% 0.0007 0.0003 0.0003 0.0007 0.0004 0.0003 0.0003 0.0003 0.0002 0.0007 0.0012 0.0002 125% 0.0015 0.0009 0.0006 0.0016 0.0009 0.0006 0.0009 0.0008 0.0003 0.0014 0.0020 0.0005 150% 0.0029 0.0015 0.0012 0.0033 0.0015 0.0012 0.0019 0.0017 0.0007 0.0025 0.0040 0.0010 6m 75% 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0007 0.0006 0.0004 0.0004 0.0004 0.0001 100% 0.0016 0.0009 0.0004 0.0007 0.0010 0.0004 0.0018 0.0016 0.0011 0.0014 0.0004 0.0003 125% 0.0041 0.0022 0.0015 0.0014 0.0027 0.0012 0.0042 0.0036 0.0023 0.0028 0.0026 0.0008 150% 0.0099 0.0065 0.0037 0.0027 0.0026 0.0031 0.0110 0.0029 0.0054 0.0081 0.0020 0.0017 7m 75% 0.0002 0.0002 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 100% 0.0006 0.0006 0.0003 0.0005 0.0005 0.0003 0.0005 0.0005 0.0003 0.0006 0.0006 0.0003 125% 0.0013 0.0012 0.0005 0.0010 0.0010 0.0007 0.0010 0.0010 0.0007 0.0013 0.0013 0.0007 150% 0.0026 0.0017 0.0010 0.0017 0.0019 0.0013 0.0017 0.0019 0.0013 0.0023 0.0019 0.0014 8m 75% 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0001 0.0001 0.0001 0.0003 0.0003 0.0001 100% 0.0009 0.0008 0.0003 0.0003 0.0004 0.0003 0.0004 0.0004 0.0004 0.0008 0.0009 0.0003 125% 0.0019 0.0016 0.0005 0.0008 0.0010 0.0007 0.0012 0.0010 0.0008 0.0015 0.0019 0.0008 150% 0.0040 0.0035 0.0012 0.0015 0.0022 0.0017 0.0024 0.0020 0.0017 0.0024 0.0046 0.0018 9m 75% 0.0010 0.0005 0.0002 0.0007 0.0009 0.0006 0.0008 0.0008 0.0004 0.0013 0.0006 0.0001 100% 0.0042 0.0022 0.0010 0.0029 0.0042 0.0025 0.0027 0.0036 0.0022 0.0062 0.0013 0.0003 125% 0.0147 0.0083 0.0019 0.0101 0.0133 0.0105 0.0100 0.0118 0.0070 0.0203 0.0038 0.0012 150% 0.0747 0.0664 0.0559 0.0708 0.0742 0.0753 0.0721 0.0713 0.0688 0.1066 0.0592 0.0514 10m 75% 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 100% 0.0004 0.0004 0.0003 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0008 0.0003 0.0002 125% 0.0009 0.0009 0.0007 0.0003 0.0003 0.0005 0.0004 0.0003 0.0007 0.0016 0.0006 0.0004 150% 0.0016 0.0016 0.0011 0.0008 0.0007 0.0012 0.0008 0.0007 0.0013 0.0027 0.0014 0.0008 11m 75% 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0001 0.0001 0.0001 0.0003 0.0001 0.0001 100% 0.0006 0.0005 0.0004 0.0003 0.0002 0.0005 0.0002 0.0001 0.0002 0.0008 0.0002 0.0002 125% 0.0016 0.0013 0.0011 0.0008 0.0005 0.0012 0.0004 0.0002 0.0005 0.0021 0.0006 0.0005 150% 0.0103 0.0091 0.0109 0.0108 0.0048 0.0117 0.0098 0.0042 0.0096 0.0154 0.0070 0.0086 Table 5-3. Stirrup strain (%) adjacent to mid-height of ducts (Duct 1 – bottom, Duct 3 – middle, Duct 5 – top).

mode is tending toward local duct pullout, stirrups are not a very effective deterrent against this failure mode. But if the duct layout and duct ties are properly detailed to eliminate the local pullout failure mode, the stirrup spacing does define the web “regional” beam strength. Concrete Material Properties, Especially Assumed Tensile Strength This was evaluated by comparing Model-Webs 5M-C to 2M-C, 6M-C to 2M-C, 12M-D to 11M-D, and 12M-C to 12M-B. This comparison is shown in Table 5-13. So the web section strength tends to be significantly influ- enced by the concrete strength only when the section is prone to web-splitting/local-lateral shear-failures, i.e., when vul- nerable duct placement is used or web/duct tie reinforcement is NOT used. When web/duct tie reinforcement is used, con- crete tensile strength has little effect on the section strength. This tends to further underscore the importance of providing web/duct tie reinforcement; because of the various parame- ters involved in reinforced concrete design, concrete tensile strength has wide variability, and low reliability. Thus de- signers should be directed toward design rules that will ensure good performance regardless of variabilities in concrete tensile strength. Local Analysis of Single-Cell Box Girders Model Prototype: Single-Cell CIP Box Girder A single-cell box configuration was used as shown in Fig- ure 5-23, with the tendon duct arrangements and local rein- forcements shown in Figure 5-16 and further illustrated in Figure 5-23. The duct and tie configurations are referred to as #6 and #6a, to differentiate from the configurations of the multi-cell. The only difference between these is that #6 has no duct ties, and #6a has the two rebar ties as shown. The basic model uses 3D elements and a slice, with out-of-page thick- ness equal to one stirrup spacing. The stirrups (and other rebar) are modeled explicitly (unlike in 2D, where the stirrup is smeared). This allows introduction of the out-of-page compression due to prestress. Using this model framework, geometry variabilities were introduced directly into the mod- els—e.g., one web can have one thickness and another have a different thickness. Webs 1 and 2 demonstrate the differences related to web sloping away from the radius of curvature ver- sus sloping toward the radius of curvature. Web 1 - left, Web 2 - right. (Duct/Tie configuration “6”. “6a” would include horizontal crossties.) Similar to the multi-cell analysis series, in order to estab- lish a baseline for comparison with design calculations, Pc was 63 12m 75% 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 100% 0.0002 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0008 0.0002 0.0002 125% 0.0005 0.0008 0.0007 0.0004 0.0004 0.0005 0.0004 0.0003 0.0005 0.0017 0.0007 0.0006 150% 0.0009 0.0013 0.0014 0.0009 0.0008 0.0010 0.0008 0.0005 0.0009 0.0034 0.0013 0.0011 13m 75% 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 100% 0.0002 0.0002 0.0003 0.0002 0.0001 0.0002 0.0001 0.0001 0.0002 0.0006 0.0002 0.0002 125% 0.0006 0.0005 0.0006 0.0003 0.0002 0.0004 0.0002 0.0002 0.0003 0.0011 0.0004 0.0003 150% 0.0009 0.0008 0.0009 0.0006 0.0005 0.0007 0.0005 0.0005 0.0007 0.0017 0.0007 0.0005 14m 75% 0.0005 0.0005 0.0002 0.0001 0.0001 0.0004 0.0002 0.0001 0.0003 0.0011 0.0002 0.0001 100% 0.0017 0.0015 0.0009 0.0004 0.0007 0.0021 0.0010 0.0004 0.0013 0.0034 0.0008 0.0002 125% 0.0045 0.0048 0.0019 0.0011 0.0013 0.0063 0.0032 0.0014 0.0032 0.0111 0.0011 0.0004 150% 0.0125 0.0143 0.0046 0.0017 0.0022 0.0182 0.0113 0.0036 0.0081 0.0273 0.0018 0.0012 2MVert 75% 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 100% 0.0003 0.0003 0.0002 0.0005 0.0004 0.0003 0.0004 0.0004 0.0003 0.0003 0.0003 0.0002 125% 0.0010 0.0009 0.0006 0.0009 0.0010 0.0008 0.0011 0.0010 0.0006 0.0008 0.0010 0.0007 150% 0.0019 0.0017 0.0011 0.0019 0.0021 0.0016 0.0023 0.0020 0.0015 0.0020 0.0021 0.0016 11MVert 75% 0.0002 0.0001 0.0002 0.0002 0.0001 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 100% 0.0006 0.0002 0.0007 0.0002 0.0002 0.0006 0.0002 0.0002 0.0002 0.0002 0.0002 0.0003 125% 0.0013 0.0007 0.0014 0.0006 0.0005 0.0011 0.0003 0.0003 0.0004 0.0005 0.0005 0.0006 150% 0.0024 0.0015 0.0026 0.0013 0.0011 0.0019 0.0006 0.0006 0.0008 0.0011 0.0011 0.0012 Web A Web B Web C Web D Percent Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Duct 1 Duct 3 Duct 5 Model # Capacity Table 5-3. (Continued).

64 Web A Web B Web C Web D Percent Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Model # Capacity 1m 75% 0.0020 0.0029 0.0016 0.0029 0.0088 0.0020 0.0020 0.0078 0.0010 0.0020 0.0116 0.0016 100% 0.0033 0.0092 0.0041 0.0368 0.0283 0.0039 0.0039 0.0303 0.0029 0.0041 0.0251 0.0035 125% 0.0075 0.0209 0.0078 0.0254 0.0693 0.0107 0.0078 0.0732 0.0078 0.0062 0.0251 0.0073 150% 0.0101 0.0409 0.0150 0.0508 0.1387 0.0234 0.0156 0.1621 0.0244 0.0089 0.0591 0.0133 2m 75% 0.0016 0.0012 0.0014 0.0010 0.0049 0.0020 0.0010 0.0039 0.0020 0.0034 0.0032 0.0014 100% 0.0026 0.0035 0.0018 0.0020 0.0146 0.0029 0.0020 0.0137 0.0039 0.0067 0.0098 0.0019 125% 0.0051 0.0943 0.0044 0.0010 0.0342 0.0059 0.0068 0.0332 0.0059 0.0158 0.0221 0.0053 150% 0.0113 0.1010 0.0087 0.0029 0.0742 0.0146 0.0293 0.0791 0.0117 0.0443 0.0488 0.0089 3m 75% 0.0019 0.0013 0.0010 0.0029 0.0029 0.0020 0.0020 0.0049 0.0020 0.0024 0.0033 0.0011 100% 0.0023 0.0031 0.0032 0.0098 0.0117 0.0020 0.0039 0.0107 0.0039 0.0064 0.0087 0.0017 125% 0.0052 0.0055 0.0087 0.0244 0.0342 0.0059 0.0078 0.0264 0.0107 0.0112 0.0242 0.0052 150% 0.1803 0.1789 0.1843 0.1748 0.3154 0.1875 0.2803 0.4268 0.2910 0.2535 0.3563 0.2105 4m 75% 0.0019 0.0170 0.0021 0.0078 0.0127 0.0049 0.0078 0.0137 0.0059 0.0104 0.0110 0.0022 100% 0.0048 0.0133 0.0042 0.0293 0.0596 0.0039 0.0264 0.0459 0.0068 0.0380 0.0437 0.0083 125% 0.0249 0.0081 0.0074 0.0947 0.2266 0.0010 0.0752 0.1211 0.0117 0.0918 0.1244 0.0292 150% 0.9973 0.8594 0.8428 1.1494 1.7051 0.9990 1.1553 1.5244 1.1211 1.0413 1.2327 0.9574 5m 75% 0.0014 0.0013 0.0021 0.0020 0.0029 0.0020 0.0010 0.0029 0.0020 0.0034 0.0024 0.0006 100% 0.0013 0.0025 0.0026 0.0068 0.0059 0.0029 0.0020 0.0059 0.0029 0.0065 0.0037 0.0016 125% 0.0042 0.0045 0.0056 0.0186 0.0146 0.0049 0.0049 0.0176 0.0049 0.0133 0.0059 0.0029 150% 0.0065 0.0088 0.0093 0.0361 0.0312 0.0107 0.0117 0.0371 0.0098 0.0253 0.0130 0.0046 6m 75% 0.0016 0.0019 0.0013 0.0029 0.0039 0.0020 0.0059 0.0117 0.0039 0.0026 0.0032 0.0014 100% 0.0044 0.0149 0.0020 0.0068 0.0186 0.0029 0.0225 0.0381 0.0107 0.0075 0.0117 0.0028 125% 0.0080 0.0174 0.0046 0.0146 0.0479 0.0098 0.0547 0.0801 0.0234 0.1099 0.0321 0.0105 150% 0.0193 0.0282 0.0110 0.0391 0.1055 0.0186 0.1113 0.1660 0.0537 0.0447 0.0764 0.0140 7m 75% 0.0021 0.0007 0.0014 0.0020 0.0039 0.0010 0.0029 0.0049 0.0020 0.0027 0.0040 0.0014 100% 0.0021 0.0036 0.0034 0.0059 0.0088 0.0029 0.0049 0.0098 0.0049 0.0082 0.0073 0.0025 125% 0.0035 0.0064 0.0046 0.0117 0.0195 0.0068 0.0156 0.0244 0.0088 0.0184 0.0169 0.0052 150% 0.0068 0.0130 0.0093 0.0283 0.0391 0.0146 0.0303 0.0498 0.0195 0.0358 0.0369 0.0109 8m 75% 0.0014 0.0014 0.0011 0.0020 0.0020 0.0020 0.0029 0.0010 0.0010 0.0040 0.0037 0.0012 100% 0.0010 0.0049 0.0029 0.0029 0.0049 0.0029 0.0029 0.0098 0.0029 0.0083 0.0057 0.0022 125% 0.0039 0.0077 0.0056 0.0078 0.0137 0.0049 0.0244 0.0303 0.0068 0.0197 0.0174 0.0047 150% 0.0075 0.0167 0.0103 0.0156 0.0321 0.0107 0.0527 0.0625 0.0176 0.0478 0.0438 0.0084 9m 75% -0.0001 0.0014 0.0013 0.0049 0.0117 0.0039 0.0029 0.0098 0.0029 0.0031 0.0032 0.0011 100% 0.0012 0.0015 0.0031 0.0215 0.0361 0.0078 0.0078 0.0352 0.0059 -0.0040 0.0089 0.0035 125% -0.0140 -0.0261 0.0055 0.0801 0.1162 0.0059 0.0352 0.1143 0.0146 -0.0002 0.0296 0.0111 150% 0.7518 0.6562 0.6789 0.9424 1.1504 0.7197 0.7256 1.1426 0.8164 0.7270 0.9038 1.8598 Table 5-4. Distortion (web thickness change) across the mid-height of ducts (Duct 1 – bottom, Duct 3 – middle, Duct 5 – top).

defined as a “Capacity” calculated using conventional means, but removing resistance factors. This provided the best direct comparison to finite element analyses. For the webs of the single-cell geometry prototype, Pc was calculated as follows: h = 9.67 feet φMn = 42.1 k-ft/ft Removing the safety factor φ = 0.9, Mn = 46.8 k-ft/ft Applying an over-strength factor for rebar strain harden- ing (which is included in the FE analysis), Mo = Mn × 1.125 = 52.6 k-ft/ft As described earlier, moment-fixity effects were approxi- mately 0.6 for interior webs and 0.7 for exterior webs. Since there were only three ducts distributed vertically, the increase to capacity caused by load distribution was small, so no ca- pacity increase was applied for this. So the baseline Pc became Pc = [52.6/(h/4)]/0.7 = 31.1 k/ft 65 10m 75% 0.0014 0.0019 0.0013 0.0010 0.0020 0.0020 0.0010 0.0029 0.0010 0.0021 0.0013 0.0016 100% 0.0026 0.0035 0.0013 0.0020 0.0039 0.0020 0.0010 0.0039 0.0029 0.0035 0.0025 0.0024 125% 0.0041 0.0046 8.0000 0.0049 0.0088 0.0049 0.0020 0.0117 0.0049 0.0061 0.0062 0.0031 150% 0.0060 0.0095 9.0000 0.0127 0.0176 0.0078 0.0059 0.0225 0.0088 0.0082 0.0146 0.0064 11m 75% 0.0015 0.0010 0.0015 0.0020 0.0020 0.0010 0.0010 0.0020 0.0010 0.0019 0.0016 0.0008 100% 0.0560 0.0010 0.0021 0.0029 0.0029 0.0010 0.0010 0.0039 0.0020 0.0033 0.0021 0.0024 125% 0.0036 0.0033 0.0032 0.0049 0.0078 0.0029 0.0029 0.0107 0.0049 -0.0090 0.0086 0.0049 150% 0.0809 0.0963 0.0978 0.0977 0.1387 0.0781 0.1211 0.1963 0.1328 0.0939 0.1518 0.1160 12m 75% 0.0014 0.0014 0.0014 0.0008 0.0008 0.0008 0.0020 0.0020 0.0020 0.0018 0.0018 0.0018 100% 0.0020 0.0020 0.0020 0.0016 0.0016 0.0016 0.0039 0.0039 0.0039 0.0030 0.0030 0.0030 125% 0.0041 0.0041 0.0041 0.0041 0.0041 0.0041 0.0078 0.0078 0.0078 0.0038 0.0038 0.0038 150% 0.0078 0.0078 0.0078 0.0093 0.0093 0.0093 0.0127 0.0127 0.0127 0.0071 0.0071 0.0071 13m 75% 0.0015 0.0017 0.0008 0.0010 0.0020 0.0020 0.0000 0.0020 0.0020 0.0014 0.0014 0.0017 100% 0.0023 0.0026 0.0021 0.0020 0.0029 0.0020 0.0000 0.0029 0.0020 0.0026 0.0027 0.0017 125% 0.0022 0.0027 0.0021 0.0039 0.0068 0.0029 0.0000 0.0068 0.0039 0.0041 0.0058 0.0028 150% 0.0047 0.0045 0.0021 0.0078 0.0137 0.0059 0.0000 0.0117 0.0059 0.0059 0.0117 0.0017 14m 75% 0.0018 0.0018 0.0012 0.0010 0.0022 0.0004 0.0010 0.0020 0.0020 0.0016 0.0022 0.0011 100% 0.0030 0.0029 0.0015 0.0042 0.0043 0.0065 0.0010 0.0049 0.0059 0.0013 0.0053 0.0019 125% 0.0072 -0.0008 0.0017 0.0097 0.0109 0.0086 -0.0010 0.0059 0.0117 0.0060 0.0104 0.0047 150% 0.0228 -0.0302 0.0064 0.0268 0.0279 -0.0193 -0.0195 0.0146 0.0098 -0.0168 0.0364 0.0140 2MVert 75% 0.0015 0.0028 0.0020 0.0020 0.0029 0.0010 0.0010 0.0029 0.0020 0.0021 0.0029 0.0020 100% 0.0021 0.0057 0.0020 0.0059 0.0088 0.0020 0.0010 0.0088 0.0029 -0.0017 0.0050 -0.0233 125% 0.0065 0.0204 0.0059 0.0000 0.0000 0.0039 0.0049 0.0264 0.0000 0.0046 0.0175 0.0074 150% 0.0132 0.0397 0.0109 0.0332 0.0488 0.0068 0.0117 0.0557 0.0117 0.0074 0.0371 0.0163 11MVert 75% 0.0010 0.0010 0.0010 0.0010 0.0020 0.0010 0.0020 0.0020 0.0010 0.0022 0.0010 0.0019 100% 0.0000 0.0020 0.0000 0.0020 0.0029 0.0010 0.0039 0.0039 0.0029 0.0024 0.0030 0.0019 125% 0.0000 0.0039 0.0010 0.0039 0.0049 -0.0013 0.0059 0.0068 0.0049 0.0050 0.0069 0.0038 150% 0.0000 0.0107 0.0020 0.0068 0.0146 0.0000 0.0127 0.0107 0.0088 0.0091 0.0148 0.0068 Web A Web B Web C Web D Percent Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Duct1 Duct 3 Duct 5 Model # Capacity Table 5-4. (Continued).

66 Total Force (in % of Pc and in K/ft) Model # Web AWeb A Web B Web B Web C Web C Web D Web D 1M 119.55% 9.93 107.36% 11.38 111.04% 11.77 122.41% 11.18 2M 141.66% 11.77 128.85% 13.66 128.67% 13.64 129.38% 11.82 3M 121.11% 10.06 118.77% 12.59 144.72% 15.34 134.24% 12.26 4M - 6.24 - 8.59 - 8.57 - 7.42 5M 134.96% 11.21 133.11% 14.11 150.47% 15.95 126.25% 11.53 6M 105.54% 8.77 118.54% 12.57 105.75% 11.21 117.92% 10.77 7M 140.98% 11.71 154.06% 16.33 153.87% 16.31 147.93% 13.51 8M 124.72% 10.36 148.49% 15.74 145.38% 15.41 129.86% 11.86 9M - 7.28 - 9.59 - 9.58 - 7.75 10M 157.23% 13.06 162.45% 17.22 161.60% 17.13 142.01% 12.97 11M 131.45% 10.92 141.51% 15.00 148.15% 15.70 126.77% 11.58 12M 160.55% 13.34 165.81% 17.58 165.58% 17.55 135.53% 12.38 13M 180.74% 15.01 184.97% 19.61 184.00% 19.50 166.09% 15.17 14M - 8.65 - 10.47 - 11.81 - 8.48 2MVert 151.05% 12.55 148.79% 15.77 145.06% 15.38 151.69% 13.85 11MVert 140.59% 11.68 152.49% 16.16 165.41% 17.53 167.19% 15.27 - Percentages not shown for cases other than ducts placed at midheight * Never reached delamination limit, so the delamination at 200% of Pc is shown Model # Web A Web B Web C Web D % Pc Total Force Deflection % Pc Total Force Deflection % Pc Total Force Deflection % Pc Total Force Deflection (K/ft) (in) (K/ft) (in) (K/ft) (in) (K/ft) (in) 1M 165.03% 13.708 3.440 121.05% 12.831 0.193 118.90% 12.604 0.156 155.08% 14.163 0.425 2M 130.38% 10.830 0.249 143.12% 15.170 0.301 142.06% 15.059 0.305 159.45% 14.563 0.446 3M 136.29% 11.321 0.568 120.84% 12.809 0.278 123.99% 13.143 0.148 124.87% 11.405 0.190 4M - 9.458 1.573 - 10.620 0.914 - 10.677 0.787 - 9.337 0.853 5M 200% * 16.613 0.830 184.92% 19.601 0.513 172.64% 18.300 0.384 200% * 18.266 0.688 6M 162.10% 13.464 3.103 131.33% 13.921 0.332 114.65% 12.153 0.257 142.79% 13.041 0.365 7M 200% * 16.613 0.753 173.24% 18.364 0.407 161.42% 17.111 0.346 179.26% 16.372 0.429 8M 200% * 16.613 14.363 178.17% 18.886 1.810 142.28% 15.082 0.274 167.97% 15.341 1.032 9M - 10.058 1.504 - 10.939 0.602 - 11.042 0.528 - 10.767 1.181 10M 200% * 16.613 10.122 200% * 21.200 7.139 200% * 21.200 7.533 200% * 18.266 3.868 11M 145.14% 12.056 0.519 137.96% 14.624 0.259 138.95% 14.729 0.183 140.55% 12.836 0.235 12M 200% * 16.613 7.883 200% * 21.200 7.160 200% * 21.200 5.334 200% * 18.266 3.725 13M 200% * 16.613 2.717 200% * 21.200 2.575 200% * 21.200 2.697 200% * 18.266 2.230 14M - 16.830 9.277 - 20.186 6.473 - 27.218 15.109 - 16.288 6.762 2MVert 166.40% 13.822 0.289 158.62% 16.814 0.250 153.77% 16.299 0.248 171.58% 15.670 0.315 11MVert 200% * 16.613 10.559 200% * 21.200 8.047 200% * 21.200 6.917 200% * 18.266 7.757 Table 5-5. Lateral prestress force at stirrup yield (0.2% strain) for stirrups on inside of curve. Table 5-6. Lateral prestress force at web delamination of 0.5% (0.06” for 12” web). This was used as “100%” in the results tables and plots. The results of the single-cell analyses are shown in Tables 5-14, 5-15, 5-16, and 5-17, and the figures in the remainder of this chapter. The figures are included in Appendix Fa and are numbered (Fig. B-1, B-2, etc.). The maximum principal strain contours illustrate the general level of damage to the concrete surrounding the tendon ducts (maximum tensile strain regardless of orientation). Single-Cell Models – Analysis Results The results of the 10 different single-cell box-girder local models allow for qualitative and quantitative assessment and comparison of the cases analyzed. • Lateral Force vs. Deflection of Web Mid-height • Lateral Force vs. Deflection of Web Quarter-height

67 Model-Web Force at stirrup yield (kips/ft) Difference 2M-A vs. 3M-A 11.77 vs. 10.06 -15% change between 12” vs. 10” 2M-B vs. 3M-B 13.66 vs. 12.59 -8% change between 12” vs. 10” 2M-C vs. 3M-C 13.64 vs. 15.34 12% change between 12” vs. 14” 12M-A vs. 11M-A 13.34 vs. 10.92 -18% change between 12” vs. 10” 12M-B vs. 11M-B 17.58 vs. 15.00 -15% change between 12” vs. 10” Model-Web Force at web delam. (kips/ft) Difference 2M-B vs. 3M-B 15.17 vs. 12.81 -16% change between 12” vs. 10” 12M-A* vs. 11M-A 16.61 vs. 12.06 -27% change between 12” vs. 10” 12M-B* vs. 11M-B 21.20 vs. 14.62 -31% change between 12” vs. 10” * Never reached delamination limit Table 5-7. Effect of web thickness – thin webs. 0 2 4 6 8 10 12 14 16 Fr (K ips /Ft ) 0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 3.00E-01 3.50E-01 4.00E-01 Defl (in) Model 2M Web A Model 2M Web D Model 2MVert Web A Model 2MVert Web D Figure 5-22a. Model 2M and 2MVert force vs. deflection comparison for Webs A and D (quarter height). • Maximum Principal Strain Contours in Concrete at 75%, 100%, 125%, and 150% Pc • Strains in stirrup rebar at 3 locations along duct bank at 75%, 100%, 125%, and 150% Pc • Distortions (change in web width) 3 locations along duct bank 75%, 100%, 125%, and 150% Pc Detailed results are included in Appendix F-a. Discussion of Results Analyses of these models showed similar trends as the multi- cell model analyses, and there were no surprises as to the per- formance of the sections. As expected, the sections with duct ties performed better than those without. Having the double row of tendons was found to concentrate the local damage area within the web, but having the 20-inch web thickness with local reinforcement was quite adequate to accommodate this.

68 0 2 4 6 8 10 12 14 16 18 20 0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 3.00E-01 3.50E-01 4.00E-01 Defl (in) Fr (K ips /Ft ) Model 11M Web A Model 11M Web D Model 11MVert Web A Model 11MVert Web D Figure 5-22b. Model 11M and 11MVert force vs. deflection comparison for Webs A and D (mid height). Model-Web Force at stirrup yield (kips/ft) Difference 2M-A vs. 2Mvert-A 11.77 vs. 12.55 7% change with vertical web 2M-D vs. 2Mvert-D 11.82 vs. 13.85 17% change with vertical web 11M-A vs. 11Mvert-A 10.92 vs. 11.68 7% change with vertical web 11M-D vs. 11Mvert-D 11.58 vs. 15.27 32% change with vertical web Model-Web Force at web delam. (kips/ft) Difference 2M-A vs. 2Mvert-A 10.83 vs. 13.82 28% change with vertical web 2M-D vs. 2Mvert-D 14.56 vs. 15.67 8% change with vertical web 11M-A vs. 11Mvert-A* 12.06 vs. 16.61 38% change with vertical web 11M-D vs. 11Mvert-D* 12.84 vs. 18.27 42% change with vertical web * Never reached delamination limit Table 5-8. Effect of web slope – thin webs. Model-Web Force at stirrup yield (kips/ft) Difference 8M-A vs. 2M-A 10.36 vs. 11.77 14% change between 2" vs. 3" 8M-C vs. 2M-C 15.41 vs. 13.64 -11% change between 4" vs. 3" 8M-D vs. 2M-D 11.86 vs. 11.82 -0.3% change between 2" vs. 3" Model-Web Force at web delam. (kips/ft) Difference 8M-C vs. 2M-C 15.08 vs. 15.06 -0.2% change between 4" vs. 3" 8M-D vs. 2M-D 15.34 vs. 14.56 -5% change between 2" vs. 3" Qualitative evidence can also be obtained by examining the strain plots in the Appendixes for the models where cover thickness was varied. Table 5-9. Effect of cover thickness – thin webs.

Similar to the multi-cell studies, the analysis results can be used to compare the web design parameters. For purposes of interpreting the 3D finite element analysis results, the follow- ing damage limit criteria are suggested: • Stirrup rebar strain exceeding yield (i.e., 0.2% strain for Grade 60 steel); for Load Factor Design, concrete rein- forcement is designed to yield; yield should be considered an upper bound criteria for unfactored loads. • Visible concrete cracking occurs at strains of approxi- mately 0.016%, but this is not necessarily web failure; concrete with maximum principal strains of 0.3% can be considered to be heavily cracked. Concrete with strains in excess of 1.0% will generally show wide-open cracks and potential spalling from the section. • Significant distortion or delamination (change of width of the webs) also represents an upper limit on capacity for webs; the delamination is evidence of a local split- ting or lateral shear failure within the web; it was again assumed that an upper bound on crack width of 1/16” is an indicator of such a failure. For 20” webs, this repre- sents a distortion ratio (average strain through the section) of 0.3%. For sections with web ties, this means the web ties have yielded; for sections without web ties, the section is at a web splitting or a cover concrete spalling condition. One of the criteria, Stirrup Yield, has been summarized in Table 5-18. These are the total forces (sum of all tendon ducts in the web) applied when any part of the stirrup reaches yield. Using these criteria, and examining the results tables and plots resulted in the following observations. Web Slope Similar to the multi-cell series, the inside radius web, slop- ing toward the center of the curve, was found to be roughly 10% stronger than the outside radius web, sloping away from the center of the curve. Further study indicated a possible rea- son for this was that loading the inside radius web (Web 2) created positive transverse moment in the top slab adjacent to the web (tension on the bottom of the slab), whereas load- ing Web 1 created negative moment. The slab resistance to 69 Model-Web Force at stirrup yield (kips/ft) Difference 1M-A vs. 2M-A 9.93 vs. 11.77 19% change - Config. 1 vs. 2A 1M-B vs. 2M-B 11.38 vs. 13.66 20% change - Config. 1 vs. 2A 1M-C vs. 2M-C 11.77 vs. 13.64 16% change - Config. 1 vs. 2A 3M-D vs. 2M-D 12.26 vs. 11.82 -4% change - Config. 2B vs. 2A 5M-B vs. 2M-B 14.11 vs. 13.66 -3% change - Config. 2B vs. 2A 10M-C vs. 7M-C 17.13 vs. 16.31 -5% change - Config. 3A vs. 2A Model-Web Force at web delam. (kips/ft) Difference 1M-B vs. 2M-B 12.83 vs. 15.17 18% change - Config. 1 vs. 2A 1M-C vs. 2M-C 12.60 vs. 15.06 19% change - Config. 1 vs. 2A 5M-B vs. 2M-B 19.60 vs. 15.17 -23% change - Config. 2B vs. 2A 10M-C* vs. 7M-C 21.20 vs. 17.11 * change - Config. 3A vs. 2A * Never reached delamination limit Table 5-10. Effect of different duct configurations – thin webs. Model-Web Force at stirrup yield (kips/ft) Difference 10M-B vs. 12M-B 17.22 vs. 17.58 2% change - Config. 3A vs. 4A 10M-C vs. 12M-C 17.13 vs. 17.55 2% change - Config. 3A vs. 4B 10M-A vs. 13M-A 13.06 vs. 15.01 15% change - Config. 3A vs. 4A 10M-B vs. 13M-B 17.22 vs. 19.61 14% change - Config. 3A vs. 4A 10M-C vs. 13M-C 17.13 vs. 19.50 14% change - Config. 3A vs. 4A 10M-D vs. 13M-D 12.97 vs. 15.17 17% change - Config. 3A vs. 4A 12M-B vs. 12M-C 17.58 vs. 17.55 -0.2% change - Config. 4A vs. 4B 9M-B vs. 14M-B 9.59 vs. 10.47 9% change - Config. 2A vs. 4B Model-Web Force at web delam. (kips/ft) Difference 9M-B vs. 14M-B 10.94 vs. 20.19 85% change - Config. 2A vs. 4B Table 5-11. Effect of duct tie arrangements – thin webs.

70 Model-Web Force at stirrup yield (kips/ft) Difference 6M-A vs. 2M-A 8.77 vs. 11.77 34% change with 50% more stirrup steel 6M-B vs. 2M-B 12.57 vs. 13.66 9% change with 50% more stirrup steel 6M-D vs. 2M-D 10.77 vs. 11.82 10% change with 50% more stirrup steel 7M-B vs. 2M-B 16.33 vs. 13.66 -16% change with 33% less stirrup steel 7M-D vs. 2M-D 13.51 vs. 11.82 -13% change with 33% less stirrup steel 13M-A vs. 12M-A 15.01 vs. 13.34 -11% change with 33% less stirrup steel 13M-B vs. 12M-B 19.61 vs. 17.58 -10% change with 33% less stirrup steel 13M-D vs. 12M-D 15.17 vs. 12.38 -18% change with 33% less stirrup steel Model-Web Force at web delam.(kips/ft) Difference 6M-B vs. 2M-B 13.92 vs. 15.17 9% change with 50% more stirrup steel 6M-D vs. 2M-D 13.04 vs. 14.56 12% change with 50% more stirrup steel 7M-B vs. 2M-B 18.36 vs. 15.17 -17% change with 33% less stirrup steel 7M-D vs. 2M-D 16.37 vs. 14.56 -11% change with 33% less stirrup steel Table 5-12. Effect of stirrup spacing – thin webs. Model-Web Force at stirrup yield (kips/ft) Difference 2M-C vs. 5M-C 13.64 vs. 15.95 17% change with 50% larger concr. strength 2M-C vs. 6M-C 13.64 vs. 11.21 -18% change with 50% smaller concr. strength 12M-B vs. 12M-C 17.58 vs. 17.55 -0.2% change with 50% larger concr. strength Model-Web Force at web delam. (kips/ft) Difference 2M-C vs. 5M-C 15.06 vs. 18.30 22% change with 50% larger concr. strength 2M-C vs. 6M-C 15.06 vs. 12.15 -19% change with 50% smaller concr. strength Table 5-13. Effect of concrete strengths – thin webs. these moments was stronger (about 2 times stronger, based on typical deck reinforcing) in positive moment than in negative moment, and this translated to more strength in the associated web. Cover Thickness Cover thickness was varied in Model 6S, where increases for Webs 1 and 2 were by 50% and 75%, respectively. Table 5-19 summarizes the relevant strength comparisons. The local concrete damage was less severe with thicker cover, but the comparisons for stirrup yield were incon- clusive because (1) for 2-inch cover and above, cover fail- ure did not control the failure mode, and (2) when the cover was less, the “moment arm” between the stirrups was more, and this increased capacity rather than decreas- ing it. Number and Configuration of Tendon Ducts The only variation studied in the single-cell case was the positioning of the duct group (studied in Configurations 2S and 10S). The number and relative position of the ducts to each other was held constant. But it was again observed that, when the ducts occurred near the bottom of the web (either “quarter-height” or “bottom” as tested in Configurations 4M and 14M), the force at “failure” was substantially lower than when the ducts were placed at the mid-height, i.e., on average as much as 25% to 40% lower when comparing these cases to similar cases. The reason for this was a tendency toward lateral shear failure of the overall web, and a tendency toward flexural damage in the top slab, thus weakening the whole system. When the ducts were located at the mid-height, the lateral shear was divided equally between the top and bottom of the web. But when the ducts moved down, the bottom of the web carried most of the lateral shear. This is a different mechanism than the failure modes observed for tendon ducts at mid-height, but one that still warrants consideration in design. Number and Configuration of Duct Ties This was evaluated by comparing duct tie Configuration 6a, which had duct ties, to Configuration 6, which had no duct

71 21’- 6”21’- 6” 10’- 0” 11’- 6” 5’-0” 9” 18 ” 20” 16 ” 9’- 0” 9’- 0” 3’-6” 12 ’ - 6” 10 ” 9” Ty p 24 ” @ Pi er ~3.5” 6” 6” 2” Cl 1.5” Cl Tendons assumed 6-31 F 0.6” 1 web Ducts 5”00 20” # 4 at 12” Typ # 7 at 12” Typ # 6 at 12” Figure 5-23. Tendon duct and local reinforcement for the local analysis prototype for a single-cell box. Analysis # Model Type Web # Web- ties Web Thickness Duct/Tie Config.* Bundle Vert. Pos. Stirrup Spacing(in.) Cover Thickness Concr. Tens. Str. (x√fc`) 1S "baseline" Single-cell 1 2 N N 20 20 6 6 midheight midheight 12 12 1.5"/2" 1.5"/2" 44 2S Single-cell 1 2 N N 20 20 6 6 1/4 height bottom 12 12 1.5"/2" 1.5"/2" 4 4 3S Single-cell 1 2 N N 20 20 6 6 midheight midheight 12 12 1.5"/2" 1.5"/2" 4 6 4S Single-cell 1 2 N N 20 20 6 6 midheight midheight 8 12 1.5"/2" 1.5"/2" 4 2 5S Single-cell 1 2 N N 20 20 6 6 midheight midheight 18 12 1.5"/2" 1.5"/2" 44 6S Single-cell 1 2 N N 20 20 6 6 midheight midheight 12 12 2.5"/3" 3:/3.5" 44 7S Single-cell 1 2 Y Y 20 20 6a 6a midheight midheight 12 12 1.5"/2" 1.5"/2" 44 8S Single-cell 1 2 Y Y 20 20 6a 6a midheight midheight 12 12 1.5"/2" 1.5"/2" 26 9S Single-cell 1 2 Y Y 20 20 6a 6a midheight midheight 18 8 1.5"/2" 1.5"/2" 44 10S Single-cell 1 2 Y N 20 20 6a 6a 1/4 height bottom 12 12 1.5"/2" 1.5"/2" 44 Table 5-14. Single-cell box variations/parameter study. ties. Table 5-20 compares Model/Webs 7S to 1S, 8S to 3S, and 9S-1 to 5S-1. These comparisons show that, for the wider web (20 inches) and double row of tendon ducts, the ties do not make a significant difference in the force to cause stirrup yield, but they make a large difference in the delamination-damage that can occur within the web. Delaminations (width changes in the web) are reduced by 24 to 31% with duct ties as compared to without duct ties. Stirrups Stirrup spacing was evaluated by comparing Model-Webs 4S-1 to 1S-1, 5S-1 to 1S-1, and 9S-1, 2 to 7S-1, 2. The results are shown in Table 5-21. The web section strength tends to be significantly influ- enced by the stirrup spacing for this geometry also, perhaps even more so than for the multi-cell geometry. Again, stirrup spacing is a driver of web “regional” beam strength.

Concrete Material Properties, Especially Assumed Tensile Strength The effect of concrete strength was evaluated by compar- ing Model-Webs 3S-2 to 1S-2, 4S-2 to 1S-2, and 8S-1, 2 to 7S-1, 2. The results are shown in Table 5-22. So, repeating the trend observed in the multi-cell analysis, the web section strength tends to be only marginally influenced by the concrete strength, and mostly this influence occurs when web/duct tie reinforcement is NOT used. When web/ duct tie reinforcement is used, concrete tensile strength has less effect on the section strength. Of the various parameters in- volved in reinforced concrete design, concrete tensile strength has wide variability, and low reliability, so designers should use design rules that will ensure good performance, regardless of variabilities in concrete tensile strength. Conclusions from Local Analyses General Observations on Capacity Using the capacity definitions described in this section (Pc, developed based on regional transverse bending consid- erations), it was found that all of the multi-cell box-girders achieved this target capacity. The baseline (Model 1M) inte- rior webs achieved it marginally (i.e., stirrup yield was reached at 107% of Pc), while stronger details that use spreading apart the ducts, adding duct ties, or moving the ducts toward the curve-outside-face of the web reached as high as 185% of Pc. The variations in force to cause local duct bank breakout (either local shearing or web delamination) were even larger, depending on the detailing used, so the detailing significantly influences resistance to lateral pullout. For the single-cell example, with the 20-inch webs and double row of ducts, the finite element analysis showed capacities that were mostly lower than the hand-calculated regional transverse bending capacity (i.e., stirrup yield was reached at a range from 52% Pc up to 100% Pc), but this is explained by the fact that, for the thicker web, failures were dominated by local lateral shearing. Summary of Influences from Detailing Parameters • Web Depth can be adequately accounted for by consider- ing and designing for web moments. • Web Thickness significantly influences resistance to re- gional transverse bending and tendon pullout. For stirrup yield, capacity formulae based on regional flexure consid- erations appear to be appropriate for design. • Web Slope. Sloped webs were found to be significantly weaker (roughly 30%) than the vertical webs, but much of this difference is caused because these are exterior webs rather than interior ones. Exterior webs have more flexible end conditions at their connection with the top and bottom slab, and this produces larger mid-height moments. Comparison of Webs A to D for the inclined webs show that Web A is generally weaker than D by about 10%. It is believed this is due to the difference in positive bending versus negative bending strength of the top slab. Lateral force for Web D applies positive moment 72 Web 1 Web 2 Percent Mid Quarter Mid Quarter Model # Capacity 1S 75% 0.176 0.174 0.179 0.158 100% 0.404 0.402 0.418 0.360 125% 1.076 1.090 1.208 0.978 150% 2.496 2.499 2.903 2.224 2S 75% 0.888 1.251 0.724 1.068 100% 2.340 3.365 1.924 2.852 125% 4.521 6.484 3.647 5.410 150% 26.702 36.174 22.345 31.707 3S 75% 0.156 0.152 0.135 0.131 100% 0.366 0.357 0.338 0.307 125% 0.918 0.906 0.871 0.778 150% 2.239 2.161 2.246 1.868 4S 75% 0.272 0.274 0.314 0.256 100% 0.678 0.681 0.764 0.609 125% 1.805 1.775 2.050 1.570 150% 4.030 3.904 4.542 3.445 5S 75% 0.140 0.140 0.159 0.137 100% 0.295 0.298 0.351 0.293 125% 0.685 0.701 0.961 0.722 150% 1.728 1.815 2.526 1.835 6S 75% 0.195 0.190 0.192 0.170 100% 0.462 0.457 0.463 0.405 125% 1.212 1.204 1.288 1.066 150% 2.805 2.691 2.974 2.362 7S 75% 0.168 0.168 0.174 0.154 100% 0.372 0.377 0.400 0.355 125% 0.953 0.987 1.056 0.923 150% 2.286 2.335 2.600 2.169 8S 75% 0.205 0.192 0.155 0.154 100% 0.435 0.420 0.357 0.341 125% 1.095 1.046 0.866 0.846 150% 2.588 2.447 2.127 1.996 9S 75% 0.146 0.148 0.182 0.151 100% 0.320 0.335 0.472 0.365 125% 0.849 0.915 1.409 1.011 150% 2.102 2.292 3.413 2.463 10S 75% 0.808 1.131 0.659 0.972 100% 2.301 3.242 1.877 2.785 125% 4.462 6.274 3.493 5.187 150% 23.358 31.548 19.232 27.371 Table 5-15. Deflections (inches) measured at mid-height of webs on backside.

73 Web 1 Web 2 Percent Duct 2 Duct 4 Duct 6 Duct 2 Duct 4 Duct 6 Model # Capacity 1S 75% 0.00166 0.00137 00097 00150 00150 00150 100% 0.00421 0.00222 00173 00310 00310 00310 125% 0.01851 0.01023 00275 01506 01506 01506 150% 0.04107 0.03204 01021 04032 04032 04032 2S 75% 0.01429 0.00345 00187 00680 00311 00188 100% 0.03765 0.01701 00450 02543 01638 00724 125% 0.07417 0.04288 01750 04969 03558 02096 150% 0.24986 0.15104 06873 21465 10079 05699 3S 75% 0.00152 0.00127 00090 00158 00097 00049 100% 0.00376 0.00218 00176 00326 00236 00164 125% 0.01529 0.00902 00279 01341 00712 00421 150% 0.04083 0.03034 00937 03334 02640 01838 4S 75% 0.00332 0.00190 00134 00165 00172 00175 100% 0.01269 0.00563 00244 00361 00580 00520 125% 0.03219 0.02447 00713 01271 02075 02036 150% 0.06683 0.05776 01953 03541 04853 04659 5S 75% 0.00126 0.00112 00076 00140 00141 00110 100% 0.00189 0.00177 00137 00244 00332 00187 125% 0.00942 0.00452 00200 01017 01499 00759 150% 0.02689 0.01755 00329 03406 03425 03028 6S 75% 0.00178 0.00137 00109 00150 00132 00113 100% 0.00642 0.00204 00190 00415 00284 00215 125% 0.02163 0.00915 00323 01628 01319 00910 150% 0.04799 0.02955 00911 03942 03223 02922 7S 75% 0.00159 0.00135 00092 00160 00140 00124 100% 0.00329 0.00215 00160 00525 00335 00259 125% 0.01451 0.01098 00257 01872 01541 00807 150% 0.03987 0.03238 00851 04636 04096 02690 8S 75% 0.00162 0.00149 00134 00154 00106 00059 100% 0.00387 0.00339 00277 00407 00243 00158 125% 0.01690 0.01528 01066 01612 00684 00332 150% 0.04019 0.03653 02766 04088 02560 01352 9S 75% 0.00129 0.00110 00072 00207 00174 00151 100% 0.00207 0.00178 00129 00841 00630 00387 125% 0.01140 0.00698 00196 02662 02184 01958 150% 0.03106 0.02321 00373 06473 05138 04844 10S 75% 0.01253 0.00783 00210 00703 00380 00184 100% 0.03730 0.02768 00920 02789 01708 00750 125% 0.07189 0.05973 02640 05278 03636 01851 150% 0.23078 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.19175 09182 19842 09676 04984 Table 5-16. Stirrup strain (%) on curve inside - face.

74 Web A Web B Percent Duct 2 Duct 4 Duct 6 Duct 2 Duct 4 Duct 6 Model # Capacity 1S 75% 0.0126 0.0104 0.0026 0.0128 0.0247 0.0095 100% 0.0351 0.0297 0.0053 0.0442 0.0630 0.0316 125% 0.0732 0.0666 0.0179 0.0987 0.1464 0.0876 150% 0.1398 0.1446 0.0673 0.1702 0.3469 0.2093 2S 75% 0.0169 0.0278 0.0064 0.0472 0.0849 0.0226 100% 0.0635 0.0779 0.0303 0.1207 0.2037 0.0671 125% 0.1959 0.2289 0.1114 0.2546 0.4035 0.1122 150% 0.4625 0.4472 0.2166 0.4946 0.7850 0.2093 3S 75% 0.0165 0.0084 0.0031 0.0051 0.0108 0.0034 100% 0.0348 0.0277 0.0092 0.0176 0.0455 0.0137 125% 0.0658 0.0512 0.0196 0.0374 0.0875 0.0412 150% 0.1347 0.1356 0.0813 0.1102 0.2152 0.1084 4S 75% 0.0245 0.0141 0.0039 0.0339 0.0496 0.0332 100% 0.0569 0.0392 0.0132 0.0928 0.1219 0.0903 125% 0.1079 0.0971 0.0463 0.1995 0.2876 0.2064 150% 0.2377 0.2341 0.1441 0.4056 0.5930 0.4169 5S 75% 0.0077 0.0094 0.0031 0.0112 0.0223 0.0087 100% 0.0247 0.0212 0.0058 0.0356 0.0521 0.0250 125% 0.0459 0.0462 0.0133 0.0862 0.1145 0.0600 150% 0.0734 0.0898 0.0346 0.1594 0.3204 0.1824 6S 75% 0.0158 0.0129 0.0021 0.0106 0.0235 0.0048 100% 0.0366 0.0328 0.0077 0.0406 0.0632 0.0175 125% 0.0935 0.0748 0.0302 0.0965 0.1555 0.0424 150% 0.1985 0.2021 0.1147 0.2057 0.3781 0.1055 7S 75% 0.0106 0.0093 0.0026 0.0104 0.0217 0.0075 100% 0.0213 0.0237 0.0053 0.0274 0.0479 0.0153 125% 0.0425 0.0405 0.0085 0.0573 0.0939 0.0261 150% 0.0567 0.0824 0.0253 0.1254 0.2166 0.0563 8S 75% 0.0936 0.0157 0.0069 0.0106 0.0173 0.0052 100% 0.0207 0.0273 0.0164 0.0246 0.0503 0.0127 125% 0.0413 0.0574 0.0461 0.0442 0.0872 0.0250 150% 0.0735 0.1541 0.0933 0.0922 0.1443 0.2584 9S 75% 0.0057 0.0095 0.0032 0.0390 0.0227 -0.0218 100% 0.0189 0.0190 0.0062 0.0298 0.0480 0.0126 125% 0.0345 0.0361 0.0139 0.0630 0.1241 0.0217 150% 0.0386 0.0645 0.0274 0.1379 0.3115 0.0543 10S 75% 0.0132 0.0250 0.0064 0.0399 0.0661 0.0072 100% 0.0124 0.0600 0.0154 0.0800 0.1449 0.0163 125% 0.0751 0.1321 0.0279 0.1604 0.2918 0.0202 150% 0.1939 0.2563 0.0306 0.2885 0.5176 0.0376 Table 5-17. Distortion (web thickness changes – inches) at mid-height of ducts.

to the top slab, and the positive moment reinforcement is approximately 2 times that of the negative moment reinforcement. • Cover Thickness. Inside face duct cover influences lateral pullout resistance, but is not the only driver of pullout resistance. The results of the parameter studies are influ- enced by the fact that when the cover is reduced, for the same overall web thickness, the moment arm for the stir- rups is increased, which is an offsetting influence on pullout resistance. It appears appropriate to check cover concrete thickness for resistance to initial cracking, but not to include cover concrete tensile strength in calculating regional transverse bending strength. • Number and Configuration of Tendon Ducts. When ducts are spread apart, the performance significantly improves. Roughly 20% resistance force improvement was demonstrated by separating the 5-duct bundle into two bundles, and an additional 4% improvement was demonstrated by spreading the bundles farther apart (4.5 inches versus 1.5 inches of separation). It is believed prudent to require a maximum of 3 ducts per bundle. When individual ducts were separated and moved toward the curve’s outside face of the web, performance further improves. When measured by the delamination/local- lateral shear criteria, Duct Configuration 3A exceeded 200% Pc, so the improvement in delamination perform- ance was very large. However, it is often impractical for designers to spread individual ducts apart due to lack of space in the web and due to requirements on location of the C.G. of the tendon group. • Number and Configuration of Duct Ties contribute sig- nificantly to resistance to lateral tendon breakout. • Stirrups. Regional transverse bending strength is directly tied to stirrup area, but it controls the design only when web/duct tie reinforcement is NOT used or when the web- splitting/lateral shear-failure does not occur. In other words, if the failure mode is tending toward local duct breakout, stirrups are not a very effective deterrent against this failure mode. But if the duct layout and duct ties are properly de- tailed to eliminate the local pullout failure mode, the stirrup spacing does define the web “regional” beam strength. • Concrete Material Properties, Especially Assumed Ten- sile Strength. Web section strength can be significantly in- fluenced by concrete tensile strength only when the section is prone to web-splitting/local-lateral shear-failures, i.e., when vulnerable duct placement is used or web/duct tie re- inforcement is NOT used. When web/duct tie reinforce- ment is used, concrete tensile strength has little effect on the section strength. Thus designers should be directed toward design rules that will ensure good performance, regardless of variabilities in concrete tensile strength. Recommendations for Web Capacity Design Web capacity design for lateral tendon force resistance should be a three-step calculation: Regional flexure check, local- lateral shear/breakout check, and cover concrete cracking check Regional Transverse Bending The regional mechanism is the web acting as a vertical beam loaded laterally near its center. Fundamentally, the cal- culation follows the equation: Mu = (Load Factor)(Moment Fixity Factor)(1⁄4)(Pj/R) hc This equation (a modified version of the Caltrans Equa- tion) and the corresponding stirrup spacing should be evalu- ated for each web of a box-girder separately—not for the total box divided by the number of webs. The radius is different for each web, and it was found that the moment fixity factor is also different. AASHTO LRFD currently applies a load factor of 1.2 to the Pjack tendon force, which is judged to be rea- sonable. Appropriate moment fixity factors are 0.6 for inte- rior webs and 0.7 for exterior webs. The stirrup sizing and spacing should then be calculated using Ultimate Strength design such that φMn ≥ Mu 75 Total Force (K/ft) Model # Web 1 Web 2 1S 25.83 28.18 2S 16.05 17.88 3S 26.21 28.6 4S 20.1 26.17 5S 31.18 28.77 6S 24.16 28.04 7S 26.21 26.64 8S 26.12 27.95 9S 29.87 23.78 10S 16.09 18.38 Table 5-18. Force at stirrup yield (0.2% strain). Model-Web Force at Stirrup Yield (kips) Difference 6S-1 vs. 1S-1 24.16 vs. 25.83 7% increase with 3” vs. 2” 6S-2 vs. 1S-2 28.04 vs. 28.18 0% increase with 3.5” vs. 2” Table 5-19. Effect of cover thickness – thick webs.

However, the Vs stress in the stirrups due to vertical shear in the web should be added to the stress due to flexure in the sizing and spacing of the stirrups. At the midheight of the web, on the inside-curve side of the web, these stresses are di- rectly additive. Local Lateral Shear Check The local lateral shear mechanism involves the complex behavior that develops in the concrete and stirrup region immediately adjacent to the duct bank. This may be checked by the following equations developed by the University of Texas (Van Landuyt, 1991): For a strip of web 1 foot long, the applied lateral shear de- mand along a plane deff long is Vd = Pj/R  2 Vc capacity of the cover-beam along this plane may be taken as Where φ = 0.75 (reduced due to uncertainties in concrete quality within the cover-beam) When the spacing between ducts is greater than or equal to the duct diameter deff = dc + (Duct Diam.)/4 + ∑s/2 V dc eff= ′φ24 fc or deff = tw – (Duct Diam)/2 whichever is least. where s = space between ducts (assume 0 if s < 1.5” or for single ducts) tw = thickness of web When the spacing between ducts is less than the duct diameter or for single ducts deff = dc + (Duct Diam)/4 where dc = cover over the ducts Figure 5-24 shows what is intended by the above equations for deff. There has been discussion within the industry as to selecting deff (some refer to this as the “lateral shearing plane depth”). Some say this should be no greater than dc (the cover concrete depth) due to uncertainties in the con- crete interaction with the ducts, but the local analyses conducted here allow for the extra width of 1⁄4 of a duct diameter. If this lateral shear is exceeded, the most effective design remedy is the addition of duct-tie reinforcement. 76 Model-Web Force at Stirrup Yield (kips) (anddelamination at 100%Pc) Difference 26.21 vs. 25.83 2% incr. with Duct ties7S-1 vs. 1S-1 (0.024” vs. 0.035”) (31% less delamination) 26.64 vs. 28.18 -5% change with Duct ties7S-2 vs. 1S-2 (0.048” vs. 0.063”) (24% less delamination) 27.95 vs. 28.60 -2% change with Duct ties8S-2 vs. 3S-2 (0.050” vs. 0.046”) (little change in delamination) 29.87 vs. 31.18 -4% change with Duct ties9S-1 vs. 5S-1 (0.019” vs. 0.025”) (24% less delamination) Table 5-20. Effect of duct ties – thick webs. Model-Web Force at stirrup yield (kips) Difference 4S-1 vs. 1S-1 20.1 vs. 25.83 29% increase with 50% more stirrup steel 5S-1 vs. 1S-1 31.18 vs. 25.83 21% decrease with 33% less stirrup steel 9S-1 vs. 7S-1 29.87 vs. 26.21 14% decrease with 50% less stirrup steel 9S-2 vs. 7S-2 23.78 vs. 26.64 21% increase with 50% more stirrup steel Model-Web Force at stirrup yield (kips) Difference 3S-2 vs. 1S-2 28.60 vs. 28.18 2% increase with 50% larger concrete tensile strength 4S-2 vs. 1S-2 26.17 vs. 28.18 7% decrease with 50% smaller concrete. strength 8S-1 vs. 7S-1 26.12 vs. 26.21 0% change with 50% smaller concrete. strength 8S-2 vs. 7S-2 27.95 vs. 26.64 5% increase with 50% larger concrete. strength Table 5-21. Effect of stirrup spacing – thick webs. Table 5-22. Effect of material strength – thick webs.

Cover Concrete Cracking Check Evaluating the cracking of the cover concrete is a check that is made to ensure serviceability because it is recommend that the lateral tendon forces be completely carried by the strength elements of the above two checks. But this serviceability check remains critical to achieving a good design, because sig- nificant cover cracks running along the tendons should be avoided for long-term structure durability. The flexure on the cover beam involves a complex mecha- nism because it is uncertain what the level of adhesion is of the cover concrete to the duct bank and to the concrete surrounding the duct bank. Assuming there is no adhesion between the metal duct and the web concrete in the radial direction of the duct, the flexure calculation proceeds as fol- lows. The cover-beam acts as a vertical beam “built-in” or fixed at top and bottom. Thus the following moments are produced: L is the height of the duct bank and φMn ≥ Mu I = bdc3 12 M wL center = 2 24 M wL L Lends = = 2 2 12 12( /Pj/R/ ) Where Mn is defined by an allowable tensile stress for concrete of , and φ = 0.55. The allowable tensile stress should also be reduced by the tensile stress in the concrete at the crit- ical point due to regional transverse bending. Although this may appear quite conservative in terms of choice of tensile strength and choice of φ, once cracking begins within the in- terior of the cover concrete near the top and bottom of the duct bank, the moment at the center of the duct banks quickly becomes So these factors and conservative tensile strengths are judged appropriate to prevent this progressive cracking mechanism from occurring. Other Local Detailing Guidelines A further guideline, which has come out of the local analysis work and from examination of some local breakout failures in various bridges and test structures, is to limit the number of ducts of a sub-bundle to no more than three. Sub-bundles should then be separated by either a duct-tie rebar or by a minimum of 1⁄3 of one duct diameter (for ex- ample, 11⁄2 inches for the analyses performed here). Duct ties should be well anchored with hooks around stirrup reinforcement. A generic duct tie detail is shown in Figure 5-25. M wl center = 2 8 5 ′fc 77 dc dc R inside face tw deff = lesser of: 2 24 4 R inside face deff = tw - deff = dc + deff = dc + For Single Ducts or for “s” <For “s” ≥ Figure 5-24. Definition of deff (after Van Landuyt, 1991).

Construction Tolerances Designers should consider the practical aspects of con- struction tolerances when checking and implementing their designs. Construction tolerances should be held to industry standards—it is not the point of this design recommendation to modify these, but designers may wish to consider conser- vatively allowing for field variations in web width and in rebar placement of up to ±0.5 inch when evaluating issues of web regional transverse bending strength, local breakout resist- ance, and, particularly, cover-beam strength. Dimensional changes of 0.5 inch can make considerable difference in the stresses in the web concrete and reinforcing steel. The following is an example of how design and construc- tion issues can affect conditions for lateral tendon breakout. As a box-girder gets deeper, the stirrup cage gets deeper. As the stirrup cage gets deeper, it becomes more flexible laterally, especially in areas of low lateral shear demand where designers often specify stirrup spacing as large as 24 inches. During the web and soffit pour, the stirrup cage has been shown to deflect laterally within the web form due to unbalanced concrete placement, vibration process, and duct float. Duct float, in combination with sloped exterior webs, can often lead to a substantial reduction in concrete cover between the stirrup cage and the interior face of the web. This may be mitigated somewhat by rebar spacer requirements at midheight of webs to help control stirrup movement during the pour, but the designer should be aware of possible variations in the actually constructed dimensions. Several conditions can aggravate the chances for lateral tendon breakout, including 1. Reduction of cover over the duct or rebar, which can af- fect resistance to breakout. 2. Excessive wobble of the ducts, which can result in either reduced resistance to breakout or locally elevated lateral forces. 3. Out-of-plane forces in a vertically curved tendon due to wedging of the stand. 4. Pressure from grout leakage due to poor quality duct (ex- cessive flexibility), damaged duct, or improperly sealed duct. 5. Distortion of empty ducts acted on by adjacent stressed ducts. 6. Local curvature in ducts near anchorage zones or blisters The specified load and resistance factors (1.2 and 0.75) re- flect the assumption that construction tolerances are reason- ably well controlled. If this may not be the case, the designer may wish to consider one of the following three options. 1. Use higher load factors and/or lower resistance factors. Some engineers familiar with the potential problems have recommended φ factors be reduced from 0.75 to 0.55 for local lateral shear failure. Load factors could also be raised above 1.2 to say 1.5. 2. Use dimensions that include an allowance for misplace- ment of the duct, rebar, or forms. As suggested above, crit- ical dimensions could be reduced by 0.5 inch or even 1.0 inch 3. When in doubt, provide web and duct tie reinforcement Tendon breakout failures can be expensive to repair. Al- though the recommended design specifications should pro- vide an adequate factor of safety in most cases, the designer is ultimately responsible for assessing the likely conditions in the field. 78 inside of Curve #4 #4 #5 Stirrups 12" Web Duct Tie Web Tie, Typ 3" clr to Duct 2" clr to Stirrup Figure 5-25. Generic web and duct tie detail.

Next: Chapter 6 - Conclusions »
Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges Get This Book
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TRB's National Cooperative Highway Research Program (NCHRP) Report 620: Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-Girder Bridges explores proposed specifications and examples for the design of horizontally curved concrete box-girder highway bridges.

Potential LRFD specifications and design examples illustrating the application of the design methods and specifications are available online as appendixes to NCHRP Report 620.

Appendix A - Proposed LRFD Specifications and Commentary

Appendix B - Example Problems

Appendix C - Global Analysis Guidelines

Appendix D - State of Practice Summary for the United States

Appendix E - Detailed Global Analysis Results

Appendix F - Detailed Local Analysis Results

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