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53 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- Figure 5-13. Estimated cracking in the webs based cated in Figures 5-19 and 5-20. on strains (displ. 25). These figures show how the exterior web ends are freer to rotate than the interior webs. For the two extremes The boundary conditions for the models were the same as of fixed-fixed versus pinned-pinned, the ratios of mid- for the UT Test simulation, which produced reasonable cor- height moment to applied tendon force (P) would be h/8 relation between analysis and test. The model is a sector slice versus h/4 or a ratio of one-half. But the web end conditions taken from a curve. The dimension longitudinally varies are neither fixed-fixed nor pinned-pinned. One way to between inside and outside edge, but on average is equal to quantify these differences associated with end effects was to 1.5 ft. This is also the stirrup spacing for the baseline model. apply the tendon forces to a beam model, as shown in the This bridge example is assumed to be on an 800-foot curve deformed shape plot of Figure 5-21. radius, so the sector width varies slightly from the inside of This exercise produced the following ratios of web mid- the curve (Web D) to the outside (Web A). height moments to applied force: The concrete properties were fc = 5,000 psi, and Young's Modulus = 4,030.5 ksi. The rebar was Grade 60. Plots of ma- Web terial stress-strain curves for the concrete and steel are shown in Figure 5-18. Tensile strengths (ft) for concrete when tested A B C D in direct uniaxial tension can show large variations, but most 0.186h 0.145h 0.147h 0.171h results fall within the range 4 fc to 6 fc ; 5 fc is consid- h = 92.75 in. (7.73 ft.) ered a reasonable average. Normalizing to the pinned-pinned condition (M = P h/4) gives ratios of: Multicell Models--Analysis Results 0.744 0.580 0.588 0.684 The 16 different multicell girder local models were devel- oped, and analyses were completed. The results are shown in Considered as coefficients, these can be compared with the this chapter through the following plots and tables. These Caltrans Memo-to-Designers Formula: allow for qualitative and quantitative assessment and com- parison of the cases analyzed. Mu = 0.8(1/4)(Pj/R)hc Lateral Force vs. Deflection of Web Midheight Where Pj is the tendon force (j is for "jacking force") Lateral Force vs. Deflection of Web Quarter-height Thus, for this case, Caltrans uses a continuity factor of 0.8 for Maximum Principal Strain Contours in Concrete at 75%, design. Based on the work performed for this project, factors of 100%, 125%, 150% Pc 0.6 for interior webs and 0.7 for exterior webs are proposed.

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54 1.0DC UT Test BC UT Test 1.0 DC FE Model BC FE Model 18.0 16.0 14.0 12.0 10.0 Fr (K/ft) 8.0 6.0 4.0 2.0 0.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 Deflection (inches) Figure 5-14. Force vs. deflection (FE model compared with U.T. Test). The overall distribution of these moments to webs agrees In order to establish a baseline for comparison with de- well with damage trends observed in the FE analysis, so one sign calculations, as previously mentioned, Pc is defined finding from the Local Analysis study may be that design- as a "Capacity" calculated using conventional means, but ers should account for this effect in more detail than to just removing safety factors, so as to make direct comparison apply a single formula to calculate mid-height moment to finite element analyses. For the interior (B or C webs) of demand. the multicell geometry prototype, Pc was calculated as Further study of this using fully nonlinear properties follows. showed that once concrete cracking begins to occur, the dif- ferences between webs become even larger. So it was eventu- Mn = 8.7 k-ft/ft ally decided to choose a baseline prestress force divided by the Removing the resistance factor = 0.9, four webs, then increase this force for the interior webs, and reduce this force for the exterior webs. Using this as a basis Mn = 9.7 k-ft/ft and choosing a prestress force large enough to cause signifi- cant damage in all of the parametric models led to the fol- Applying an over-strength factor for rebar strain harden- lowing total applied forces in kips/ft. This is analogous to ing (which is included in the FE analysis), Pj/R. Of course in some cases, the webs failed prior to reach- ing this total load. Mo = Mn 1.125 = 10.9 k-ft/ft Web 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 A B C D midheight, but are instead, applied at five points distributed 13 k/ft 17 k/ft 17 k/ft 15 k/ft along 18 inches of the height. This decreases the moment to

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

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56 275.75" 275.75" 10% & Var 9" 13" & Var 110.25" 12" & Var 8.5" & Var 70.86" 47.25" 102" 72.83" 72.83" 102" 47.25" 70.86" 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".