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45 Girder Action 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 GLOBAL the local FE analysis. Local Analysis Validation/Demonstration REGIONAL Case (UT Test Case) Bean Action LOCAL Slab Action 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 Figure 5-1. Types of actions considered. a 1/3 scale representation of the configuration of Las Lomas, a well-known bridge that failed in lateral tendon breakout. (so has the limitations of) the constitutive formulations Test Model and Test Conduct within the ABAQUS program, a widely distributed, well- respected commercial product for solving problems with a The following test model and test conduct description high degree of material nonlinearity. come from the test report and thesis: "The Effect of Duct This program was chosen partly because of the authors' de- Arrangement on Breakout of Internal Post-Tensioning gree of experience using it, but also to demonstrate the use of a Tendons in Horizontally Curved Concrete Box Girder Webs," widely available tool that other interested engineers or re- by D.W. Van Landuyt, 1991. searchers could use to examine their own special design cases. The box cross-section was a scaled version of Las Lomas ABAQUS is certainly not the only product available for this type with changes made for simplifying construction (Figure 5-3). of analysis. To avoid having to build cantilever forms, the girder was built Despite the limitations noted, the authors believe the FE and tested in an inverted position. This did not significantly work herein has demonstrated an approach to modeling box- affect results (it was assumed by the researchers that gravity girder cross-sections, especially a method for applying the loads are unimportant to breakout). The model top slab loads and boundary conditions, and this has satisfied one of therefore represents the bottom slab at Las Lomas, etc. The the goals for this project. top slab thickness at Las Lomas varied transversely and the Mn jd V LOCAL MOMENT C T ML LOCAL SLAB W = F/L dmin L F dmin CT V Mn V ~ F/2 T ~ Mn /(jd) ML ~ FL/12 Figure 5-2. Regional and local actions on a web.
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46 16'-3.75" 1'-4" 6'-2.5" 4" R=22' PT 3'-8" 7'-0" C L PC 9'-0.25" 6'-0" 4" 1'-4" R=18' 5'-0" = 16.07° 1'-11.38" 14'-4.38" "Typical" Segment Modeled Top slab 4" 1.5" chamfer 2'-5" 3'-0" Bottom slab 3" 1'-4" 3'-8" 1'-4" 4" 4" 7'-0" Figure 5-3. Plan and end view of U.T. girder test specimen (from Van Landuyt, 1991). bottom varied longitudinally. Average thicknesses were calcu- other part of the girder or testing apparatus. If the curve were lated, scaled and rounded off to an integral number of inches. not sharp enough, anchorage zone failure might have resulted At Las Lomas, the centerline distance between interior and or the strands would have been loaded to an unsafe level. exterior webs was 11 feet. This scaled to 3 ft 8 in. The model Duct arrangement controlled the design of the curve radius. was constructed with a centerline web-to-web distance of The capacity to resist lateral shear failure was calculated for 4 feet so that the radius of each web was a whole number. To each tendon, assuming two failure planes would form and save on materials and labor, the cantilevers were shortened that the maximum concrete strength would be 5000 psi. 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 Fr = 2 5000 (2)12 (1.125 ) = 3.8k / ft dimensions of the web were considered important. The exact scaled values of 4 in for the thickness and 3 ft for the overall where Fr is the lateral (radially oriented relative to the curve) height were maintained (Figure 5-4). prestress force. The web radii were chosen to be small enough so that the Therefore a total Fr of 15.2 k/ft was required for all four tendon breakout in the web would occur before failure of any tendons. A jacking force of 372 kips could be delivered from
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47 104 Figure 5-4. Girder #1 cross section in curved region (from Van Landuyt, 1991). the loading apparatus. An 18-ft radius for the inside of web scaling of stirrup sizes. Equivalent spacing of #6's at 21.3 inches was selected, as it would permit a total Fr of 20.6 k/ft. This was was scaled to 6-mm bars at 7 inches. The spacing was not more than a third larger than the anticipated failure load. The increased to account for the greater yield strength of the previously determined cell width mandated that the com- Swedish bar. Stirrup spacing was reduced in the anchorage panion web radius be 22 feet. The curve length of 5 feet was zones to 51/2 inches. chosen for the 18-ft radius. The curved region was the most A four-tendon bundle and three other promising arrange- difficult part of the model to construct and was kept as short ments were tested. Only duct positioning varied from web to as possible. A 5-ft curve was more than twice the clear height web; all other details remained the same. Duct size at Las of the web and was thought to be sufficient to allow regional Lomas was not given, although based on the maximum num- transverse bending. A 5-ft straight transition zone on each end ber of strands (28), ducts of approximately 41/2-inch O.D. insulated the curve from the complex stresses at anchorages. should have been used. Scaling required 11/2-inch ducts for Las Lomas was reinforced with GR40 #5 stirrups spaced at the model, but the major manufacturers of post-tensioning 15 inches. No standard bars match this on a 1/3 scale. The duct apparently do not make this size. The nearest available closest match was 6 mm, 75 ksi bars from Sweden already duct size (1.75 inches) was used. available in the lab. This is nearly equivalent to a #2 bar. The Specimen BC is the duct arrangement similar to the one at stirrup spacing needed to be adjusted to reflect the imprecise Las Lomas. A slight modification was made for the model
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48 (Figure 5-4). A straight vertical formation was used in lieu of 1.75" the zigzag because it was considered more universal. The 1.67" 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 Figure 5-5. on the web vertical axis. Strand Specimen 1.0DC follows the Texas State Department of positions in Highway and Public Transportation design of the San Antonio curve (from "Y" project with an arrangement that maintained a clear Van Landuyt, spacing between ducts equal to the diameter of the duct. It is 1991). believed that this allows for better consolidation and, more importantly, eliminates the single large discontinuity found at with the strand showed an actual yield of 276 ksi and ultimate Las Lomas. This arrangement is conservative and would be of 289.5 ksi. considered an upper limit beyond which further spacing of A loading system was developed to apply gradually in- ducts would provide no benefit. The scaled vertical spacing was creasing load simultaneously to all tendons with an equal 1.75 inches. All ducts were centered on the web vertical axis. force in each tendon. It was necessary to consider how the All test concrete strengths were much greater than the strand or strands that would constitute a tendon would bear 28-day design strength of 3500 psi. The slab and web concrete on a duct. The ducts at Las Lomas were nearly filled to capac- had higher overall strengths and faster strength gains as is ity with strands. This meant that almost the entire 180 degrees typical of concrete containing super-plasticizers. The web of the duct on the inside of the curve was in contact with the concrete strength was 5300 psi. strands. That same type of load distribution could be approx- 6-mm-diameter hot-rolled bars were used for all rein- imated with a minimum of three 1/2-inch-diameter strands forcement. Tensile tests on bars conducted at the lab showed per duct (Figure 5-5). Given that there were four ducts, a total average yield strength of 75 ksi. of twelve strands could be safely stressed to 0.75fpu to develop Galvanized, corrugated, folded metal ducts were used in all a maximum force of 372 kips. instances. The outside ridge-to-outside ridge dimension was Regional beam behavior was monitored by deflection of 1.75 inches. The inside diameter was 1.60 inches and the the web relative to the top and bottom slabs. U-shaped frames gauge was 0.035 inches. were mounted to the web face on the outside of the curve Post-tensioning was applied to the specimens with 7-wire, (Figure 5-6). The actual attachment points were about 1 /2-inch Ø, 270 ksi, low-relaxation strands. Test data provided 2 inches below the top slab and 2 inches above the bottom PC Potentiometer WS2 WS4 WS6 WS8 PT 14.5" NOMENCLATURE: WS# Location with respect to stirrup # (±)1") Web deflection with respect to slabs Figure 5-6. Web potentiometer configuration (from Van Landuyt, 1991).
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49 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 INSIDE INSIDE deflection anticipated in the first 2 inches is negligible. A sin- FACE FACE 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 R=22' R=18' which the potentiometer stem could bear. Mounting the potentiometers on the outside face of the curve meant that deflections should not be influenced by BC 1.0DC 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- INSIDE INSIDE crete. The web-slab potentiometer nomenclature is given in FACE FACE 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 R=22' R=18' by wires/potentiometers placed in tubes cast through the webs above and below the duct group. Sudden movements BO 0.2DC in these measurements were good indicators of imminent failure. Figure 5-7. Specimen failure plans Figure 5-7 is a sketch of how the concrete cracked and (from Van Landuyt, 1991). failed during the test. Figure 5-8 compares tendon horizontal force versus deflections for the four different webs tested. 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 Finite Element (FE) Model and Analysis boundary conditions are illustrated in Figure 5-9. This An FE model of Girder BC and 1.0DC was developed for shows how a 3-D slice was modeled with horizontal tendon a typical slice of the test model in the curved region, as was loads applied directly to the inner surfaces of the ducts. The 18 16 14 1.0DC 12 10 BO Ft (k/ft) 8 PC 6 BC PT 4 0.2DC WS2 WS4 WS6 WS8 2 ELEV SECTION 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 DEFLECTION (inches) Figure 5-8. Comparison of web deflections relative to slabs at Stirrup #6 (from Van Landuyt, 1991).
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50 Figure 5-9. Finite element model and boundary conditions. wedge slice model configuration and "symmetry" boundary A typical deformed shape from the analysis (at the 1.0 DC conditions allowed horizontal (radial) displacement to occur web displacement of 0.07 inches) is shown in Figure 5-10. naturally in the FE model, and this in turn, causes "longi- The amount of deflection and the sharpness of the flexural tudinal" compression prestress in the concrete in a manner curvature is significantly more severe in Girder-BC than in similar to the actual test specimen. What is precisely math- Girder-1.0DC. This agrees with test observations. In the ematically represented with such boundary conditions is a illustration, displacements are magnified by 50. wedge-slice of a complete circle, but the boundary condi- Figure 5-11 shows the same deformed shape, but with con- tions are also reasonably accurate for a section of a curve. tours of maximum principal strain. In reinforced concrete The tendon forces were applied in incremental fashion with analysis, these are one of the most effective ways to show enough equilibrium iterations and small load increments to damage and deformation distributions in the concrete and achieve solution convergence at every increment. This solution rebar. Concrete stress contours are generally not helpful, be- procedure is often referred to as Full-Newton, where upon the cause after concrete cracks, the stress reduces to nearly zero, next load increment; the structure stiffness is updated, based on so a zero or small tensile stress may be displayed in a zone that the material damage, which has occurred (concrete cracking/ is already highly damaged. But maximum principal strains crushing and steel yielding). The analysis was run out to large (generally, the maximum tension found in any orientation web displacements and significant damage (failure), i.e., well for a given point in the continuum) can indicate concrete beyond the displacements plotted in Figure 5-8. cracking, which occurs at strain of approximately 1.50E-4 to