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57 12" Typ Inside of Curve, @ all sections Typ @ all # 4 @ 12", Alternative sections Sides for 135 hooks 3" Clr 3" Clr To Duct To Duct Typ 2" clr Typ 2" clr 3" Clr 3" Clr To Duct To Duct Duct Typ Case B Case A-1.5" Only Case B-1 Duct Dia Case A-1.5" #4 12" 1.5" Stirrups Typ Type 1 Type 2 Type 3 Type 4 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). The maximum principal strain contours illustrate the general The reason for this is a tendency toward lateral shear failure level of damage to the concrete surrounding the tendon ducts of the overall web. When the ducts are located at the mid- (maximum tensile strain regardless of orientation). height, the lateral shear is divided equally between the top and The following strain thresholds are important quantities to bottom of the web. But when the ducts move down, the bot- compare to when viewing these results. tom of the web carries most of the lateral shear, and this is a different mechanism than the failure modes observed for ten- 11 = 1.6 10-4 first visible cracking (micro-cracking don ducts at mid-height. So the "quarter-height" cases can be occurs at about half of this strain) compared to each other (and the "bottom" cases), but should 2.0 10-3 first rebar yield not be compared directly with the "midheight" cases. "Pc" 1.0 10-2 1% strain in rebar; typically wide open only applies to the "midheight" cases. cracks/sometimes spalling concrete For purposes of interpreting and comparing the results, the following damage criteria should be considered: In Table 5-4, delaminations ("distortion") information is provided at "Duct 1," "Duct 3," and "Duct 5." This refers to Stirrup rebar strain exceeding yield (i.e., 0.2% strain for a measure across the bottom edge, top edge, and centerline of Grade 60 steel); note that for Load Factor Design, concrete the duct assembly. reinforcement is designed to yield at the ultimate member forces. Visible concrete cracking occurs at strains of approxi- Discussion of Results mately 0.016%, but this is not necessarily web failure; The analysis results have been used to compare the web concrete with maximum principal strains of 0.3% can be design parameters. The first general observation is the com- considered to be heavily cracked. Concrete with strains in parison of the midheight cases with the "1/4 height" and "bot- excess of 1.0% will generally show wide-open cracks and tom" cases. It was generally observed that when the ducts potential spalling from the section. occur near the bottom of the web (either "quarter-height" or Significant distortion or delamination (change of width "bottom" as was tested in Configurations 4M and 14M) the of the webs) would also represent an upper bound on force at "failure" is substantially lower than when the ducts serviceable capacity for webs; the delamination is evidence are placed at the midheight, i.e., on average as much as 25% of a local splitting or lateral shear failure within the web. to 40% lower when comparing these cases with similar cases. It was arbitrarily assumed that a crack width of 1/16" is an

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

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59 Figure 5-20. Deformed shape/strain contour when the same force is applied to each web. Model 11M included web ties. The results compared with failure mode. Moving the ducts toward the curve outside face their respective baselines are shown in Table 5-7. within the webs also contributed to resistance against delam- These results demonstrate significant influence on resist- ination and local lateral shear damage. ance to lateral bending and tendon pullout caused by web thickness. Stirrups yielded sooner, and concrete damage Web Slope and web delamination was more extensive with the thinner webs. As described earlier, the sloped webs in this analysis series For stirrup yield, capacity formulae based on regional flex- were found to be significantly weaker (3040%) than the ure considerations appear to be appropriate for design. Dif- vertical webs, but part of this difference was caused by being ferences in stirrup yield and especially web delamination were exterior webs rather than interior. Exterior webs have more also significantly influenced when the web ties were added be- flexible end conditions at their connection with the top and cause the web ties tended to eliminate the web delamination 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 Figure 5-21. Tendon forces applied to a beam model comparisons because these have the baseline values for all of typical cross-section. properties, but they investigate duct-tie configurations 2a and

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60 Table 5-2. Deflections. 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

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61 Table 5-2. (Continued). 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 4a/b. Direct comparisons of Force versus Deflection are bundles farther apart (4.5" versus 1.5" separation). So in gen- shown in Figures 5-22a and 5-22b below. eral, a prudent recommendation is to require a maximum of These figures show the vertical webs to be stiffer and 3 ducts per bundle. When the individual ducts were separated stronger than the sloping webs, but show that the differences (i.e., Config. 3A) and moved toward the curve's outside edge in force capacity (strength) are negligibly small. A compari- of the web, performance further improved. In fact, as meas- son of the occurrence of rebar yield and web delamination is ured by the delamination criteria, Configuration 3A exceeded shown in Table 5-8. 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 Cover Thickness impractical for designers to spread individual ducts apart due Cover thickness was varied in Models 7M, 8M, and 13M. to lack of space in the web and due to requirements on loca- Table 5-9 summarizes sample strength comparisons. tion of C.G. of the tendon group. The conclusion reached is that cover thickness influences lateral pullout resistance, but is not the only driver of pullout Number and Configuration of Duct Ties resistance. The results of the parameter study were influenced by the fact that when the cover is reduced, for the same overall This was evaluated by comparing Configurations 4A and web thickness, the moment arm for the stirrups is increased, 4B, to Configurations 1, 2, and 3. This is covered by compar- and this is an off-setting influence on pullout resistance. As will ing Model/Webs 11M-D to 10M-D, 12M-B to 10M-B, 13M- be discussed further in the conclusions, it appears appropriate A-D to 10M-A-D, 12M-C to 12M-B, and 14M-B, D to 9M-B. to check cover concrete thickness for resistance to initial crack- This comparison is shown in Table 5-11. ing, but not to include cover concrete tensile strength in the The conclusions from these comparisons are that web/duct calculation of regional transverse bending strength. ties make a significant contribution to the resistance to lateral tendon breakout. Number and Configuration of Tendon Ducts Stirrups This was evaluated by comparing Configurations 1, 2, and 3 (from Figure 5-17), which involve comparing Models 1M vs. Stirrup spacing was evaluated by comparing Model-Webs 2M, 3M-D vs. 2M-D, 5M vs. 2M, and 10M-C vs. 7M-C. Re- 6M-A, B, and D to 2M-A, B, and D, comparing 7M-A, B, D sults are shown in Table 5-10. to 2M-A, B, D, and comparing 13M-A, B, D to 12M-A, B, D. Clearly, when the ducts are spread apart, the performance These comparisons are shown in Table 5-12. significantly improves. Roughly 20% resistance force improve- This indicates that web section strength is significantly in- ment was demonstrated by separating the 5-duct bundle into fluenced by the stirrup spacing only when web/duct tie rein- two bundles (Config. 2A versus Config. 1), and an addi- forcement is NOT used or when the web-splitting/lateral tional 4% improvement was demonstrated by spreading the shear-failure does not occur. In other words, if the failure

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62 Table 5-3. Stirrup strain (%) adjacent to mid-height of ducts (Duct 1 bottom, Duct 3 middle, Duct 5 top). 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