Bridge System Safety and Redundancy (2014)

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39 4.1 Redundancy of Bridge Systems under Lateral Load Traditionally, structural design codes have defined a structure’s capacity in terms of the ability of its individual members to sustain the applied loads using a linear-elastic analysis. Given that the failures of individual members do not necessarily lead to the collapse of the system, structural redundancy is defined as the ability of a structural system to continue to carry load after one critical member reaches its load carrying capacity. Based on the system behavior explained in Section 3.1 of this report as described in Fig- ure 3.1 and to remain consistent with the definition of bridge redundancy established for systems under vertical loads as explained in NCHRP Report 406 and Chapter 2, quantifiable measures of system redundancy for bridges subjected to a distributed lateral load are proposed as fol- lows for force-based designs: (4.1) 1 R P P fu u p = where Rfu gives the redundancy ratio or the system reserve ratio expressed in terms of the forces, Pp1 gives the force capacity of a bridge system under lateral load assuming linear-elastic behavior and assuming that failure takes place when the most critical member reaches its plastic limit as typically done when using a force-based analysis, and Pu gives the ultimate capacity of the system accounting for the entire system’s nonlinear behavior. The analyses of the results of hundreds of bridge systems and substructure bents have demonstrated that a simple empirical model can be used to describe the relationship between the ultimate capacity of a multi-column bridge substructure system represented by Pu and the lateral load carrying capacity of one column represented by Pp1 as func- tions of the number of columns in the bent and the ultimate curvature capacity of the bent columns. This relationship is expressed by an equation of the form (4.2)1P P F Cu p mc u tunc tconf tunc = + ϕ − ϕ ϕ − ϕ    ϕ where Pp1 gives the capacity of a bridge system under lateral load assuming that the analysis is performed using linear- elastic behavior and failure is defined when one column reaches its maximum load carrying capacity as typically done when using a force-based analysis, Fmc is a multi-column fac- tor, Cj is a curvature factor, ju is the ultimate curvature of the weakest column in the bent, jtunc is the average curvature for a typical unconfined column, jtconf is the average curvature for a typical confined column. The typical curvature values for the confined and unconfined columns are extracted from the results of the survey conducted in NCHRP Report 458. For a particular bridge system, Pp1 is calculated using a lin- ear structural analysis of the system under the effect of the applied lateral load. To find Pp1, failure is defined as the load at which one column reaches its ultimate capacity. The value for the ultimate curvature at failure ju is calculated from the ultimate plastic analysis of the column’s cross section. Values for Fmc, Cj, jtunc, and jtconf have been extracted from the analysis of a large number of bridges with two-column, three-column, and four-column bents. The bents analyzed included a range of column sizes, vertical reinforcement ratios, and confinement ratios. The analyses also considered the effect of different foundation stiffnesses. The recommended values for these parameters are provided in Table 4.1. The values for jtunc and jtconf are the average curvatures obtained from the analysis of the column sizes used in NCHRP Report 458. The columns analyzed in NCHRP Report 458 represent typical column sizes and reinforcement ratios collected from a national survey conducted as part of the study. The val- ues for jtunc and jtconf are used in Equation 4.2 to compare C H A P T E R 4 Force-Based System Safety and Redundancy of Bridges

40 the confinement ratio of the column being evaluated to the average confinement ratios observed in typical confined and unconfined columns. This chapter summarizes the results of the analyses con- ducted during this project and those extracted from NCHRP Report 458. The results of the analyses of typical bridge sys- tem configurations considered different bridge and column dimensions, confinement and reinforcement ratios, founda- tion stiffnesses, and deficiencies in the columns and their connecting elements. The results are used to calibrate a sys- tem factor equation for use during the force-based design and safety evaluation of columns of bridges subjected to dis- tributed lateral load. Specifically, this chapter consists of the following: • Section 4.2 summarizes the results obtained in this study and in NCHRP Report 458 from all the analyses of bridge systems subjected to lateral load at the superstructure level. Tables listing all the results are provided for bridges con- sisting of two-column, three-column, and four-column bents with various column heights and cross-section dimensions, vertical and transverse reinforcement ratios, foundation stiffnesses, as well as different connections between columns and superstructures. • Section 4.3 summarizes the validation of the proposed model for the cases previously analyzed. • Section 4.4 contains a few additional analyses to study if the proposed model remains essentially valid when con- sidering P-delta effects. Also, additional analyses are per- formed to consider different foundation stiffnesses for bridge systems having three-column bents. Additionally, this section describes how the model can be adjusted to account for the effect of inadequate cap beams on sub- structure redundancy and how to account for columns weak in shear. Examples describing how an engineer can use the proposed model along with the necessary adjust- ments are also provided. • Section 4.5 gives the conclusions. 4.2 Summary of Bridge Analyses and Results This summarizes the bridge models analyzed during the course of this study along with a summary of the results obtained during the course of this study and in NCHRP Report 458. The bridges analyzed in this study consist of a continuous three-span I-girder steel bridge with two bents supported by three columns each, a three-span bridge carrying a multi-cell prestressed concrete bridge superstruc- ture where each bent is formed by two columns, and a three- span bridge carrying two prestressed concrete girder boxes where each bent has two columns. The results of the analyses performed in this study are supplemented by the results of the two-column and four-column bents analyzed in NCHRP Report 458. Two-Column Bents Supporting a Prestressed Concrete Twin Box-Girder Bridge Bridge Description A three-span (80-ft, 120-ft, 80-ft) continuous bridge with two precast prestressed concrete box sections is selected for analysis. Figure 4.1 gives the frame element model used in this set of analyses. Prestressed Concrete Box Figure 4.2 shows the dimensions of the box cross sec- tion. The material properties of concrete and steel are listed in Tables 4.2 and 4.3, respectively. The correspond- ing stress-strain curves for steel and concrete are obtained from the library of the program xtract, which are based on the Mander model for concrete, parabolic strain hard- ening steel model for reinforcing steel and low-relaxation strands model in Collins and Mitchell’s book for prestressed strands. Variable Applicability Recommended Value mcF , multi-column factor Two-column bents 1.10 Three-column bents 1.16 All other multi-column bents 1.18 C , curvature factor All systems 0.24 tunc , typical unconfined column ultimate curvature All systems 3.64 x 10-4 (1/in) tconf , typical confined column ultimate curvature All systems 1.55 x 10-3 (1/in) Table 4.1. Recommended values for redundancy parameters.

41 Figure 4.1. The 3-D, profile, and elevation views of the twin box-girder bridge. (a) Configuration of prestressing steel (b) Dimensions of one box (c) Spacing of two boxes (d) Grillage model for box-girder bridge Figure 4.2. Detailed dimensions of cross section of the twin box bridge.

42 Bridge Columns The bridge columns are the most important contributors for the resistance of the bridge to lateral load. The lateral load is applied on the top of the bents, in particular on the middle point of the cap beams. The cross section of the four typical columns used for the substructure is shown in Figure 4.3. The column height used in the base case is 20 feet. In this study, three cases with different column confine- ment ratios, rs, are investigated. The original confinement ratio is 0.01. Two other cases are analyzed where the con- finement ratios meet the LRFD seismic criteria for bridges of category B and C. In summary, the three confinement ratios are • rs = 0.01 (base case), • rs = 0.003 (category B), and • rs = 0.005 (category C). The pushover analysis of the frame element model is per- formed using SAP2000. Moment-curvature relationships are used to model the nonlinear behavior of the columns. This nonlinear behavior is related to the axial load acting on it. In fact, different axial loads lead to different moment-curvature curves. Therefore, an important input for the pushover analy- sis is the axial force versus moment interaction (P-M) curve. As an example, Figure 4.4 shows the moment interaction diagram for the column in the base case (rs = 0.01) that is valid for all bending axes because the section is axisymmetric. In the base case, the axial load value due to dead load is about P = 512 kip and the axial load value due to dead load plus live load is about P = 523 kip assuming a load combination factor of 0.20. Figure 4.5 gives the different moment- Parameters Box Slab Column Unconfined Confined 28 Day Strength (ksi) 7.250 4.350 4.000 4.000 Crushing Strain 4.000E 3 4.000E 3 4.000E 3 18.740E 3 Elasc Modulus (ksi) 4853 3759 3605 3605 Table 4.2. Concrete properties. Table 4.3. Steel properties. Parameters Rebar Strand Yield stress (ksi) 60 Fracture stress (ksi) 90 270 Failure strain 90.00E 3 4.20E 2 Figure 4.3. Column cross section for base cases. Figure 4.4. Column P-M interaction curve for base case.

43 Figure 4.5. Column M-phi curve for different values of axial load. curvature (M-phi) curves constructed using the software XTRACT to illustrate the differences due to the axial load. SAP2000 updates the M-phi curve based on the axial load calculated at each step of the pushover analysis by updating the ultimate moment and curvature capacities using the P-M curve and the equivalent energy principle. Figure 4.5 also shows that the difference in the M-phi curve between the two cases when P = 512 kips and P = 523 kips is negligible where the blue line practically coincides with the purple line with a maximum difference smaller than 0.13%. It is observed that even a doubling of the axial load will result in a relatively small change in the M-phi curve. In this par- ticular case, because the loading on the column lies below the balanced point of the P-M interaction curve, the moment capacity of the section increases by about 4% when the axial load is increased to 1000 kips. Cap Beam The dimensions of the cap beam are provided in Figure 4.6. Bearings The bearings are assumed to be placed at the abutments below each box’s web. The bearings at the abutments are modeled as linear-elastic springs. In the base case model, the Figure 4.6. Dimensions of cap beam.

44 bearings are assumed to allow for longitudinal expansion and for transverse displacement. The same stiffness values are assumed for the bearings in the longitudinal and lateral directions. The displacement in the perpendicular direction to the slab is assumed to be locked. Table 4.4 summarizes the spring values assumed for the base case. Summary of Results The three-span box-girder bridge system analyzed in the base case consists of two prestressed concrete boxes sup- ported on two columns at each bent. The columns have integral connections to the cap beam. A sensitivity analy- sis is performed to study how variations in column dimen- sions and other parameters affect the results of the pushover analysis. Specifically, the Nonlinear Static Pushover Analy- sis (NSPA) is used to study the sensitivity of the results to the following parameters: (1) column height, (2) column confinement ratio, and (3) column reinforcement ratio. The numerical results from the analyses are summarized in Table 4.5, which gives the diameter of the column, its height, the longitudinal reinforcement ratio in percent, the force Pp1 which gives the failure load of the first column assuming elasto-plastic behavior and using the force-based approach, the force Pu, which gives the ultimate curvature for the entire system assuming nonlinear behavior, the ultimate curvature for the columns, the lateral displacement of the system at failure du, and the maximum displacement when one col- umn fails d1. The analysis included columns designed with lateral con- finement reinforcement ratios rs = 1%, 0.3% (detail cate- gory B) and 0.5% (detail category C). The columns’ height was varied between 4.6-ft, 8.3-ft, 15-ft, 20-ft, 26.7-ft, and 33.3-ft. Although some of the low column heights may not be typical, they are used in order to study the effect of large changes in bridge configurations. Different longitudinal reinforcement ratios for the 6-ft diameter columns varying between 1.66% (original design with steel area As = 67.5 in2), 1.44%, 1.22%, 0.99%, and 0.83% are also investigated. The validation of Equation 4.2 for the ultimate capacity of the system is verified in Figure 4.7, which plots Pu from Equation 4.2 versus that obtained from the SAP2000 analy- sis. Figure 4.7 shows that the data points are clearly aligned along the equal force line indicating that Equation 4.2 is rea- sonably accurate with a regression coefficient R2 = 0.99. The COV of the ratio between the value from Equation 4.2 and the SAP2000 is less than 4%. The worst cases are those for which the bearing stiffness is assumed to be over 10 times the actual stiffness, which is extremely high for this type of bridge. Two-Column Bents Supporting a Multi-Cell P/S Concrete Box Bridge Bridge Description A multi-cell prestressed concrete bridge system is inves- tigated to understand the behavior of such systems when subjected to lateral loads applied on top of the bents. The configuration of the multi-cell prestressed concrete box- girder bridge system analyzed is a variation on an actual bridge configuration that had been designed to sustain high levels of seismic motions. The superstructure is a three-span continuous prestressed multi-cellular box girder with dia- phragms located over the abutments and over the bents. The bridge is assumed to have three spans with a middle span of 150 feet and two end spans 110 feet in length each, as shown in Figure 4.8. Each bent is formed by two columns connected integrally to the diaphragms. For the base case, all the col- umns are 20 feet high and their diameters are 72 inches. The superstructure is supported on elastomeric bearings at the abutments. Figure 4.9 shows detailed dimensions of the box cross section. The rigid connections of the columns to the dia- phragms would allow for the transfer of forces and moments between the two subsystems. This design should provide a higher capacity to sustain lateral loads as compared to tradi- tional bearing on bent bridges. The basic bridge configura- tion assumes that the foundation is very stiff, approaching fixed conditions. Summary of Results The three-span box-girder bridge system analyzed in the base case consists of a multi-cell prestressed concrete box supported on two columns at each bent. Following the analysis of the base case bridge, NSPA is used to study the sensitivity of the structure to (1) different bearing stiffness values at the abutments, (2) different column diameters and reinforcement ratios, (3) changes in the rigidity of the con- nections between the columns and diaphragm elements, and (4) different foundation stiffness. Numerical results are summarized in Table 4.6, which gives the column diameter and height, with the longitudinal reinforcement ratio and the ultimate bending moment curvature along with the force, Pp1, that causes the failure of the first column assum- ing elastic behavior and the ultimate capacity of the non- Table 4.4. Bearing stiffness values. Left Abutment (kip/in.) Right Abutment (kip/in.) kx 4 4 ky 4 4 kz

45 linear system represented by the force Pu and the maximum displacement at failure du. The displacement d1 is the maxi- mum displacement if the pushover analysis is performed on a single column. The sensitivity analysis assumed column heights equal to 20-ft, 25-ft, and 30-ft with lateral confinement reinforcement ratios rs = 1%, 0.3% (detail category B) and 0.5% (detail cat- egory C), in addition to cases where the maximum curva- ture is reduced by 50% and 75% to account for deficiencies in design. The columns’ diameters were varied between 6-ft, 7-ft, and 8-ft. Different foundation stiffnesses are used to represent pile and spread footing foundation as well as rigid foundation. The possibility of integral column/superstructure connections is compared to the cases where the load is trans- ferred between the superstructure and the columns through bearing supports. Changes in the abutment bearing stiffness also are considered, including the case where the bearings have negligible stiffness. Original case (column lateral confinement ratio=1.0%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) 50% rebar 72 240 0.83 2364 3624 0.00168 13.94 14.10 60% rebar 72 240 0.99 2708 4072 0.00166 13.39 13.72 73% rebar 72 240 1.22 3126 4552 0.00154 12.14 12.53 87% rebar 72 240 1.44 3526 5052 0.00140 11.26 11.53 100% rebar 72 240 1.66 4150 5535 0.00132 10.55 10.86 H=55 in. 72 55 1.66 14969 22662 0.00132 2.20 2.21 H=100 in. 72 100 1.66 9028 12618 0.00132 4.42 4.53 H=180 in. 72 180 1.66 5386 7188 0.00132 7.93 7.98 H=280 in. 72 280 1.66 3602 4854 0.00132 10.55 10.86 H=320 in. 72 320 1.66 3184 4334 0.00132 14.05 14.15 H=400 in. 72 400 1.66 2585 3679 0.00132 17.63 17.93 H=450 in. 72 450 1.66 2315 3408 0.00132 -- -- H=500 in. 72 500 1.66 2097 3189 0.00132 -- -- Category B (column lateral confinement ratio=0.3%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) 50% rebar 72 240 0.83 2351 2989 0.000847 6.88 6.96 60% rebar 72 240 0.99 2676 3341 0.000783 6.41 6.41 73% rebar 72 240 1.22 3039 3808 0.000735 5.73 5.99 87% rebar 72 240 1.44 3437 4250 0.000688 5.49 5.58 100% rebar 72 240 1.66 3808 4692 0.000652 5.23 5.27 H=55 in. 72 55 1.66 13768 19758 0.000652 1.22 1.21 H=100 in. 72 100 1.66 8297 10939 0.000652 2.18 2.19 H=180 in. 72 180 1.66 4949 6158 0.000652 3.93 3.95 H=280 in. 72 280 1.66 3304 4068 0.000652 5.23 5.27 H=320 in. 72 320 1.66 2920 3602 0.000652 7.00 7.04 H=400 in. 72 400 1.66 2370 2982 0.000652 8.75 8.81 H=450 in. 72 450 1.66 2122 2700 0.000652 -- -- H=500 in. 72 500 1.66 1922 2497 0.000652 -- -- Category C (column lateral confinement ratio=0.5%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) 50% rebar 72 240 0.83 2430 3218 0.00113 8.65 9.30 60% rebar 72 240 0.99 2764 3576 0.00104 7.87 8.61 73% rebar 72 240 1.22 3160 4064 0.000936 7.80 8.12 87% rebar 72 240 1.44 3536 4500 0.000864 7.14 7.50 100% rebar 72 240 1.66 3906 4986 0.000857 6.98 7.04 H=55 in. 72 55 1.66 14088 20811 0.000857 1.62 1.61 H=100 in. 72 100 1.66 8496 11542 0.000857 2.89 2.93 H=180 in. 72 180 1.66 5069 6524 0.000857 5.13 5.27 H=280 in. 72 280 1.66 3390 4338 0.000857 6.98 7.04 H=320 in. 72 320 1.66 2996 3854 0.000857 8.84 9.39 H=400 in. 72 400 1.66 2433 3214 0.000857 10.88 11.76 H=450 in. 72 450 1.66 2179 2928 0.000857 -- -- H=500 in. 72 500 1.66 1973 2730 0.000857 -- -- Table 4.5. Results summary of two box-girder bridge with two-column bent.

46 Figure 4.7. Lateral capacity by Equation 4.2 vs. lateral capacity from SAP2000 for two box-girder bridges. y = 0.874x R² = 0.99 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu by Eq .4 .2 (k ip s) Pu from SAP2000 (kips) P/s concrete twin box Linear (P/s concrete twin box) Figure 4.8. The 3-D isometric view and span dimensions. Figure 4.9. Typical cross section of bridge system. (a) (b)

47 Table 4.6. Results summary of multi-cell box-girder bridge with two-column bent. Category B (column lateral confinement ratio=0.3%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Base Case 72 240 1.66 4907 5527 0.00061 4.48 4.49 50% Phi 72 240 1.66 4907 5018 0.00031 2.16 2.10 75% Phi 72 240 1.66 4907 5312 0.00046 3.32 3.30 Spring Top Column 72 240 1.66 2460 2704 0.00061 5.39 5.40 25 ft 72 300 1.66 4011 4584 0.00061 5.67 5.62 30 ft 72 360 1.66 3406 4004 0.00061 6.89 6.78 Pile Foundation 72 240 1.66 3649 5448 0.00061 6.98 7.01 Spread Foundation 72 240 1.66 5451 5823 0.00061 7.51 8.33 No bearing 72 240 1.66 4561 5152 0.00061 4.45 4.58 2X bearing 72 240 1.66 4940 5844 0.00061 4.47 4.48 4X bearing 72 240 1.66 4982 6317 0.00061 4.47 4.46 10X bearing 72 240 1.66 5053 7160 0.00061 4.45 4.48 Diameter–7 ft 84 240 1.66 7582 8287 0.00053 4.75 4.62 Diameter–8 ft 96 240 1.66 10984 11925 0.00046 4.88 5.02 Category C (column lateral confinement ratio=0.5%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Base Case 72 240 1.66 5047 5865 0.00082 6.03 5.97 50% Phi 72 240 1.66 5047 5300 0.00041 2.94 3.01 75% Phi 72 240 1.66 4907 5592 0.00061 4.49 4.52 Spring Top Column 72 240 1.66 2530 2942 0.00082 7.16 7.16 25 ft 72 300 1.66 4125 4977 0.00082 7.61 7.43 30 ft 72 360 1.66 3503 4332 0.00082 9.23 9.00 Pile Foundation 72 240 1.66 3753 5858 0.00082 8.65 8.80 Spread Foundation 72 240 1.66 5604 6128 0.00082 9.19 9.98 No bearing 72 240 1.66 4693 5385 0.00082 6.02 6.10 2X bearing 72 240 1.66 5081 6286 0.00082 6.02 5.95 4X bearing 72 240 1.66 5124 6905 0.00082 6.02 5.97 10X bearing 72 240 1.66 5197 7984 0.00082 6.02 5.98 Diameter–7 ft 84 240 1.66 7806 8837 0.00071 6.29 6.32 Diameter–8 ft 96 240 1.66 11334 12650 0.00062 6.54 6.61 (continued on next page)

48 The validation of Equation 4.2 for the ultimate capacity of the system is verified in Figure 4.10, which plots Pu from Equation 4.2 versus the one obtained from the SAP2000 analysis. Figure 4.10 shows that the data points are almost aligned along the equal force line, indicating that Equa- tion 4.2 is reasonably accurate with a regression coefficient R2 = 0.86. The COV of the ratio between the value from Equation 4.2 and the SAP2000 is less than 10%. The maxi- mum differences are those corresponding to soft foun- dations and for the cases where the bearing stiffness is extremely high. Three-Column Bents Supporting an I-Girder Bridge Bridge Description A combined 3-D space frame model is used to analyze the behavior of a combined superstructure-substructure multi-girder bridge system under lateral loads. Figure 4.11 gives the profile and elevation views of the frame element model used in this set of analyses. The bridge has three spans with a middle span that is 80-ft long and two end spans of 50-ft length each. Each bent is formed by three 27.58-ft columns connected by a cap beam. The six girders are connected to the cap beams through bearing supports. The analysis accounts for material nonlinearity of the col- umns, cap beams, and superstructure under the effect of lateral loads while a reduced level of traffic load is applied on the bridge. Figure 4.12 illustrates the 3-D space frame model of the combined superstructure-substructure system. The longitu- dinal members labeled “A” represent the contribution of the composite I-girders to the longitudinal bending. The trans- verse elements labeled “B” model the bending of the slabs in the transverse direction. The vertical elements labeled “C” represent the columns. The elements labeled “D” represent the cap beams. The elements labeled E are link elements rep- resenting the connection between the superstructure and substructure provided by the bearings. The elements labeled F are rigid links used to connect the centers of the longi- Original Design (column lateral confinement ratio=1.0%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Base Case 72 240 1.66 5326 6643 0.00123 9.15 8.99 50% Phi 72 240 1.66 5326 5723 0.00061 4.49 4.39 75% Phi 72 240 1.66 5326 6210 0.00092 6.82 6.77 Spring Top Column 72 240 1.66 2669 3474 0.00123 10.71 10.72 25 ft 72 300 1.66 4353 5702 0.00123 11.51 11.66 30 ft 72 360 1.66 3697 5108 0.00123 13.93 13.94 Pile Foundation 72 240 1.66 3961 6603 0.00123 12.00 12.00 Spread Foundation 72 240 1.66 5912 6947 0.00123 12.60 13.72 No bearing 72 240 1.66 4956 5848 0.00123 9.09 9.10 2X bearing 72 240 1.66 5362 7277 0.00123 9.09 8.97 4X bearing 72 240 1.66 5407 8231 0.00123 9.09 8.98 10X bearing 72 240 1.66 5485 9933 0.00123 9.09 8.98 Table 4.6. (Continued). Figure 4.10. Lateral capacity by Equation 4.2 vs. lateral capacity from SAP2000 for multi-cell box bridges. y = 0.9606x R² = 0.8601 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu by Eq .4 .2 (k ip s) Pu from SAP2000 (kips) Multi cell P/s concrete box

49 tudinal girders to the center of the cap beams through the bearings. Figure 4.13 shows cross sections of the column, cap beam, and girder. The 20 rebars in the column have 1.25-in. diam- eters. The 28 rebars in the cap beam have 5⁄8 in. (0.625-in.) diameters. The reinforcement is assumed to have a yielding stress Fy = 60 ksi. The girders are assumed to be Grade 36. The unconfined concrete strength is assumed to be 4 ksi. A confinement ratio of 2.4 × 10-3 is assumed in the columns. Summary of Results An extensive sensitivity analysis is performed to study how variations in the loading condition and the structure’s Figure 4.11. Profile and elevation view of the bridge. Figure 4.12. The 3-D space frame model of three-span I-girder bridge.

50 properties affect the response of the bridge system. Specifically, the sensitivity analysis performed in this section describes the effect of changes in the following parameters: (1) load- ing condition, (2) column height, (3) foundation flexibility, (4) concrete confinement, and (5) superstructure-substructure connection type. Numerical results of the nonlinear analyses are summarized in Table 4.7. The analysis included 27.5-ft, 32.5-ft, and 37.5-ft columns designed with lateral confinement reinforcement ratios rs = 0.24%, 0.3% (detail category B) and 0.5% (detail category C) in addition to cases where the maximum curvature is reduced by 50% and 75% to account for deficiencies in design. Differ- ent foundation stiffnesses are used to represent pile founda- tions and spread footing foundation as well as rigid and pinned foundations. The possibility of integral girder/cap-beam/ column connections is compared to girder bearings on cap beam designs and to cases where the top of the column is pinned to the cap beam. Also, the analysis compares the behavior when a lateral load is applied to the case when the load is in the lon- gitudinal direction. Table 4.7 gives the results for the ultimate load capacity and the ultimate displacements in comparison to the load at which the first column reaches its capacity, assuming linear analysis and the maximum displacement of one column. The validation of Equation 4.2 for the ultimate capacity of the system is verified in Figure 4.14, which plots Pu from Equation 4.2 versus that obtained from the SAP2000 analysis. (a) Cross section of bridge columns 14.4'' 36'' (b) Cross section of cap beam 11 ' 44 '' 44'' 6'' 2. 5''8.8''8.3'' 4.4'' (c) Cross section of I-girder 96'' 8' ' 12'' 1. 5'' 24 '' 0.5'' 12'' 0. 87 5'' Figure 4.13. Cross sections of bridge members. Table 4.7. Results summary of three-column bent supporting an I-girder bridge. Category B (column lateral confinement ratio=0.3%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Base Case 36 331 2.41 765 1077 0.00106 8.0 8.3 32.5-ft 36 391 2.41 723 1070 0.00106 10.2 10.4 37.5-ft 36 451 2.41 709 1065 0.00106 12.0 12.7 Pile Foundation 36 331 2.41 788 1125 0.00106 9.0 8.6 Spread Foundation 36 331 2.41 798 1113 0.00106 8.7 8.7 Base Pinned 36 331 2.41 572 871 0.00106 11.4 11.0 Longitudinal Load 36 331 2.41 622 962 0.00106 13.5 8.3 Integral Top/Fixed Base 36 331 2.41 772 1080 0.00106 8.0 8.3 Integral Top/Pinned Base 36 331 2.41 567 839 0.00106 10.6 11.0 Integral Longitudinal 36 331 2.41 756 1131 0.00106 8.6 8.3 Column Top Pinned 36 331 2.41 582 870 0.00106 11.3 11.0 50% Phi 36 331 2.41 765 943 0.00053 5.3 5.1 75% Phi 36 331 2.41 765 1032 0.00079 7.0 6.8

Base Case 36 331 2.41 761 1060 0.00096 7.5 7.8 32.5-ft 36 391 2.41 719 1044 0.00096 9.6 9.8 37.5-ft 36 451 2.41 706 1047 0.00096 11.5 11.9 Pile Foundation 36 331 2.41 784 1095 0.00096 8.3 8.1 Spread Foundation 36 331 2.41 794 1071 0.00096 7.7 8.1 Base Pinned 36 331 2.41 569 839 0.00096 10.6 10.5 Longitudinal Load 36 331 2.41 619 922 0.00096 12.6 7.8 Integral Top/Fixed Base 36 331 2.41 768 1063 0.00096 7.5 7.8 Integral Top/Pinned Base 36 331 2.41 564 834 0.00096 10.4 10.4 Integral Longitudinal 36 331 2.41 752 1096 0.00096 7.8 7.8 Column Top Pinned 36 331 2.41 579 831 0.00096 10.4 10.4 50% Phi 36 331 2.41 761 929 0.00048 5.0 5.1 75% Phi 36 331 2.41 761 1011 0.00072 6.4 6.4 Category B (column lateral confinement ratio=0.3%) New data using M-phi curve from multi-girder bridge in Category B, Category B below is different from the above one Category B Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) Diameter–20 ft 72 240 1.66 3512.1 3767 0.000717 Diameter–32.5 ft 72 391 1.66 2359.9 2676.3 0.000717 Diameter–37.5 ft 72 451 1.66 2252.8 2480.7 0.000717 Diameter–7 ft 84 391 1.66 5365.9 5865.9 0.000574 Diameter–7 ft 84 240 1.66 8079.5 9116.3 0.000574 Diameter–8 ft 96 391 1.66 7733.2 8541.4 0.000516 Diameter–8 ft 96 240 1.66 11603.3 13508.8 0.000516 Pile Foundation 96 240 1.66 8870.5 11528.4 0.000516 Spread Foundation 96 240 1.66 11881.2 13016.1 0.000516 Category C (column lateral confinement ratio=0.5%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Base Case 36 331 2.41 773 1212 0.00138 10.4 10.3 32.5-ft 36 391 2.41 730 1219 0.00138 13.1 12.9 37.5-ft 36 451 2.41 717 1246 0.00138 15.6 15.5 Pile Foundation 36 331 2.41 796 1232 0.00138 10.9 10.7 Spread Foundation 36 331 2.41 806 1226 0.00138 10.8 10.8 Base Pinned 36 331 2.41 578 970 0.00138 13.3 13.1 Longitudinal Load 36 331 2.41 628 1080 0.00138 15.7 10.3 Integral Top/Fixed Base 36 331 2.41 780 1214 0.00138 10.4 10.4 Integral Top/Pinned Base 36 331 2.41 573 966 0.00138 13.2 13.1 Integral Longitudinal 36 331 2.41 764 1237 0.00138 10.5 10.4 Column Top Pinned 36 331 2.41 588 967 0.00138 13.2 13.1 50% Phi 36 331 2.41 773 989 0.00069 6.3 6.1 75% Phi 36 331 2.41 773 1107 0.00104 8.5 8.2 Original Design (column lateral confinement ratio=0.24%) Diameter (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) u (in-1) u of system (in.) 1 for column (in.) Table 4.7. (Continued).

52 Figure 4.14 shows that the data points are clearly aligned along the equal force line indicating that Equation 4.2 is rea- sonably accurate with a regression coefficient R2 = 0.99. The COV of the ratio between the value from Equation 4.2 and the SAP2000 is less than 9%. The maximum differences are those corresponding to a soft foundation. Two-Column and Four-Column Bents in NCHRP Report 458 Bridge Description NCHRP Report 458 also analyzed many bridge bents sub- jected to lateral load, and analyzed individual bents rather than entire bridge systems. However, an overview of the results of analyses performed in this study has shown that the results from single bents are very similar to those of entire bridge systems. Accordingly, the results from NCHRP Report 458 can be used to supplement the database assembled in this study. In NCHRP Report 458, the gravity loads applied on the substructure include both the dead load and the entire AASHTO HL-93 vehicular live load. The analysis process incremented the lateral load until system failure occurred. In the NCHRP Report 458 analyses it was assumed that the vertical loads (dead load and vehicular live load) are set at their maximum design values. This approach is conservative as it is generally unlikely that the vehicular live load will be at its expected maximum value when the maximum lateral (wind, seismic, etc.) load is applied on the structure. The pier configurations used in the NCHRP Report 458 analysis are illustrated in Figures 4.15 and 4.16 for the two-column and four-column bents, respectively. The material properties (concrete strength, yielding stress of steel) and geometric properties (section size and amount and location of reinforcement) combine to produce the moment curvature and the capacity of the column section. For Figure 4.14. Lateral capacity by Equation 4.2 vs. lateral capacity from SAP2000 for I-girder bridges. y = 1.03x R² = 0.99 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu b y Eq . 4 .2 (k ip s) Pu from SAP2000 (kips) I girder bridges Linear (I girder bridges) Figure 4.15. Configuration of two- column bent in NCHRP Report 458. 2.0m 10.0m 2.0m Figure 4.16. Configuration of four-column bent in NCHRP Report 458.

53 Table 4.8. Parameters for two-column bent. Variation # Variation Low Average High 1 Height [m] 4 11 18 2 Width [m] 0.8 1.2 1.6 3 Concrete Strength [MN/m2] 22 27 32 4 Steel Strength [MN/m2] 400 450 500 5 long [%] 1.10 2.30 3.50 6 trans [%] 0.18 0.32 0.45 Table 4.9. Parameters for four-column bent. Variation # Variation Low Average High 1 Height [m] 3.5 6.5 9.5 2 Width [m] 0.5 1.0 1.5 3 Concrete Strength [MN/m2] 22 27 32 4 Steel Strength [MN/m2] 400 450 500 5 long [%] 0.60 1.85 3.10 6 trans [%] 0.18 0.32 0.45 the corresponding concrete crushing strain is eu = 0.004. The second value is for the confined columns, which are assumed to crush when the concrete strain reaches the value ec = 0.015. NCHRP Report 458 used an in-house program to perform the nonlinear pushover analysis. The model used for the analysis accounts for the P-delta effect produced when large values of lateral displacement interact with gravity loads to increase the moments in the columns. Foundation stiffness coefficients for the eight categories are summarized in Table 4.10 for the two-column bents and in Table 4.11 for four-column bents. These foundation stiffness coefficients are obtained using the standard procedure devel- oped in a FHWA funded study (Lam and Martin, 1986). Also, different column heights and column dimensions are assumed as listed in the second and third columns of Table 4.12. The lon- gitudinal reinforcement ratio for each column configuration is provided in the fourth column of Table 4.12. Summary of Results The numerical results for all the two-column and four- column bents analyzed in NCHRP Report 458 are summa- rized in Tables 4.12 and Table 4.13, respectively. For each a. Four-Column Bent—Average Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 77800 58300 1870000 2 spread\stiff\ 118000 88300 2830000 3 extension\soft\ 369000 8030 54195 4 extension\normal\ 923100 36500 109932 5 extension\stiff\ 1661000 127200 188483 6 pile\soft\ 450000 12580 94170 7 pile\normal\ 1126000 57200 235000 8 pile\stiff\ 2026000 199000 424000 b. Four-Column Bent—Low Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 38900 29200 234000 2 spread\stiff\ 58900 44200 354000 3 extension\soft\ 184615 2647 7339 4 extension\normal\ 461500 12050 14000 5 extension\stiff\ 830700 42000 23000 6 pile\soft\ 450300 12580 94170 7 pile\normal\ 1126000 57240 235400 8 pile\stiff\ 2026000 199300 423800 c. Four-Column Bent—High Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 117000 87500 6310000 2 spread\stiff\ 177000 133000 9560000 3 extension\soft\ 553900 15350 168600 4 extension\normal\ 1380000 69900 356000 5 extension\stiff\ 2490000 243300 628600 6 pile\soft\ 1013000 28300 565000 7 pile\normal\ 2533000 128800 1413000 8 pile\stiff\ 4560000 448500 2543000 Table 4.11. Four-column bent, foundation stiffness. the two-column bent and four-column bent cases, material parameters and column details are summarized in Tables 4.8 and 4.9, respectively. Two limiting values for the strain that produce concrete crushing are given in NCHRP Report 458. The first value assumes that the columns are unconfined and Table 4.10. Two-column bent, foundation stiffness. a. Two-Column Bent—Average Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 97200 72900 3650000 2 spread\stiff\ 147000 110000 5530000 3 extension\soft\ 443077 5226 113726 4 extension\normal\ 1107000 17784 220882 5 extension\stiff\ 1994000 46628 367348 6 pile\soft\ 675400 18870 376700 7 pile\normal\ 1689000 85870 941700 8 pile\stiff\ 3039000 299000 1695000 b. Two-Column Bent—Low Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 61500 46100 999000 2 spread\stiff\ 93100 69800 1510000 3 extension\soft\ 295358 2283 34614 4 extension\normal\ 738462 7474 65038 5 extension\stiff\ 1329231 19067 105915 6 pile\soft\ 450300 12580 94170 7 pile\normal\ 1126000 57240 235400 8 pile\stiff\ 2026000 199300 423800 c. Two-Column Bent—High Column Width Kvertical (kN/m) Ktransverse (kN/m) Krotation (kNm) 1 spread\normal\ 120000 89900 7120000 2 spread\stiff\ 182000 136000 10800000 3 extension\soft\ 590769 9259 260623 4 extension\normal\ 1476923 32421 519329 5 extension\stiff\ 2658462 86849 879287 6 pile\soft\ 1351000 37730 1413000 7 pile\normal\ 3377000 171700 3531000 8 pile\stiff\ 6079000 598000 6357000

54 Table 4.12. Results summary of two-column bent in NCHRP Report 458. (Column lateral confinement ratio=0.6%) Confined core concrete ultimate strain is 0.015 First Member Unconfined Confined Unconfined Confined Foundation Type b×d Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) Pu (kips) u (in-1) u (in-1) Spread_Normal 47.2 ×47.2 433 2.3 2481 2821 2973 3.01E-04 1.217E-03 Spread_Stiff 47.2 ×47.2 433 2.3 2479 2857 2986 3.01E-04 1.217E-03 Extension_Soft 47.2 ×47.2 433 2.3 1606 1754 1893 3.01E-04 1.217E-03 Extension_Normal 47.2 ×47.2 433 2.3 1692 1956 2298 3.01E-04 1.217E-03 Extension_Stiff 47.2 ×47.2 433 2.3 1825 2181 2727 3.01E-04 1.217E-03 Piles_Soft 47.2 ×47.2 433 2.3 1902 2228 2772 3.01E-04 1.217E-03 Piles_Normal 47.2 ×47.2 433 2.3 2215 2728 2987 3.01E-04 1.217E-03 Piles_Stiff 47.2 ×47.2 433 2.3 2427 2967 3011 3.01E-04 1.217E-03 Spread_Normal 47.2 ×47.2 157.5 2.3 5072 5877 7595 3.01E-04 1.217E-03 Spread_Stiff 47.2 ×47.2 157.5 2.3 5258 6237 7960 3.01E-04 1.217E-03 Extension_Soft 47.2 ×47.2 157.5 2.3 4667 4745 4966 3.01E-04 1.217E-03 Extension_Normal 47.2 ×47.2 157.5 2.3 4714 4924 5448 3.01E-04 1.217E-03 Extension_Stiff 47.2 ×47.2 157.5 2.3 4834 5186 6077 3.01E-04 1.217E-03 Piles_Soft 47.2 ×47.2 157.5 2.3 5142 5451 6270 3.01E-04 1.217E-03 Piles_Normal 47.2 ×47.2 157.5 2.3 5883 6510 8160 3.01E-04 1.217E-03 Piles_Stiff 47.2 ×47.2 157.5 2.3 6673 7505 8382 3.01E-04 1.217E-03 Spread_Normal 47.2 ×47.2 708.7 2.3 1557 1739 1758 3.01E-04 1.217E-03 Spread_Stiff 47.2 ×47.2 708.7 2.3 1573 1761 1771 3.01E-04 1.217E-03 Extension_Soft 47.2 ×47.2 708.7 2.3 956 1062 1110 3.01E-04 1.217E-03 Extension_Normal 47.2 ×47.2 708.7 2.3 1071 1248 1446 3.01E-04 1.217E-03 Extension_Stiff 47.2 ×47.2 708.7 2.3 1187 1424 1641 3.01E-04 1.217E-03 Piles_Soft 47.2 ×47.2 708.7 2.3 1210 1439 1647 3.01E-04 1.217E-03 Piles_Normal 47.2 ×47.2 708.7 2.3 1423 1752 1768 3.01E-04 1.217E-03 Piles_Stiff 47.2 ×47.2 708.7 2.3 1528 1796 1796 3.01E-04 1.217E-03 Spread_Normal 31.5 ×31.5 433 2.3 753 849 849 3.50E-04 1.543E-03 Spread_Stiff 31.5 ×31.5 433 2.3 747 856 856 3.50E-04 1.543E-03 Extension_Soft 31.5 ×31.5 433 2.3 447 486 481 3.50E-04 1.543E-03 Extension_Normal 31.5 ×31.5 433 2.3 480 560 638 3.50E-04 1.543E-03 Extension_Stiff 31.5 ×31.5 433 2.3 516 630 746 3.50E-04 1.543E-03 Piles_Soft 31.5 ×31.5 433 2.3 511 614 728 3.50E-04 1.543E-03 Piles_Normal 31.5 ×31.5 433 2.3 593 755 824 3.50E-04 1.543E-03 (in.)

55 bent, the results show the ultimate capacity Pu assuming that the columns are confined and also assuming that the columns are unconfined. These results are compared to those obtained when the first column reaches its limiting capacity, assuming linear behavior. The validity of Equation 4.2 for the ultimate capacity of the unconfined and confined two-column bent systems analyzed in NCHRP Report 458 is verified in Figure 4.17 (a) and Figure 4.17 (b), respectively. The figures plot Pu from Equation 4.2 versus that obtained from the NCHRP Report 458 analysis and show that the data points are clearly aligned along the equal force line, indicating that Equation 4.2 is reasonably accurate with a regression coefficient R2 = 0.99 and R2 = 0.97 for the unconfined and confined two-column bents. The COV of Table 4.12. (Continued). Piles_Stiff 31.5 ×31.5 433 2.3 638 835 852 3.50E-04 1.543E-03 Spread_Normal 63 ×63 433 2.3 4984 5649 7085 2.55E-04 1.013E-03 Spread_Stiff 63 ×63 433 2.3 5135 5826 7116 2.55E-04 1.013E-03 Extension_Soft 63 ×63 433 2.3 4099 4358 4849 2.55E-04 1.013E-03 Extension_Normal 63 ×63 433 2.3 4516 4963 6074 2.55E-04 1.013E-03 Extension_Stiff 63 ×63 433 2.3 5084 5700 7030 2.55E-04 1.013E-03 Piles_Soft 63 ×63 433 2.3 5948 6697 7136 2.55E-04 1.013E-03 Piles_Normal 63 ×63 433 2.3 6279 7020 7176 2.55E-04 1.013E-03 Piles_Stiff 63 ×63 433 2.3 6007 6874 7175 2.55E-04 1.013E-03 Spread_Normal 47.2 ×47.2 433 1.1 1519 1821 1851 3.56E-04 1.585E-03 Spread_Stiff 47.2 ×47.2 433 1.1 1510 1852 1860 3.56E-04 1.585E-03 Extension_Soft 47.2 ×47.2 433 1.1 963 1127 1267 3.56E-04 1.585E-03 Extension_Normal 47.2 ×47.2 433 1.1 966 1293 1625 3.56E-04 1.585E-03 Extension_Stiff 47.2 ×47.2 433 1.1 1018 1469 1789 3.56E-04 1.585E-03 Piles_Soft 47.2 ×47.2 433 1.1 1087 1497 1791 3.56E-04 1.585E-03 Piles_Normal 47.2 ×47.2 433 1.1 1226 1849 1868 3.56E-04 1.585E-03 Piles_Stiff 47.2 ×47.2 433 1.1 1324 1877 1877 3.56E-04 1.585E-03 Spread_Normal 47.2 ×47.2 433 3.5 3420 3756 4075 2.66E-04 1.107E-03 Spread_Stiff 47.2 ×47.2 433 3.5 3422 3823 4092 2.66E-04 1.107E-03 Extension_Soft 47.2 ×47.2 433 3.5 2223 2339 2507 2.66E-04 1.107E-03 Extension_Normal 47.2 ×47.2 433 3.5 2389 2604 2964 2.66E-04 1.107E-03 Extension_Stiff 47.2 ×47.2 433 3.5 2604 2889 3448 2.66E-04 1.107E-03 Piles_Soft 47.2 ×47.2 433 3.5 2678 2944 3496 2.66E-04 1.107E-03 Piles_Normal 47.2 ×47.2 433 3.5 3176 3591 4088 2.66E-04 1.107E-03 Piles_Stiff 47.2 ×47.2 433 3.5 3481 3990 4124 2.66E-04 1.107E-03 (Column lateral confinement ratio=0.6%) Confined core concrete ultimate strain is 0.015 First Member Unconfined Confined Unconfined Confined Foundation Type b×d Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) Pu (kips) u (in-1) u (in-1) (in.)

56 Table 4.13. Results summary of four-column bent in NCHRP Report 458. (Column lateral confinement ratio=0.6%) Confined core concrete ultimate strain is 0.015 First Member Unconfined Confined Unconfined Confined Foundation Type b×d (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) Pu (kips) u (in-1) u (in-1) Spread_Normal 39.4 ×39.4 255.9 1.85 3787 4651 4857 3.68E-04 1.494E-03 Spread_Stiff 39.4 ×39.4 255.9 1.85 3923 4707 4876 3.68E-04 1.494E-03 Extension_Soft 39.4 ×39.4 255.9 1.85 2468 2692 3011 3.68E-04 1.494E-03 Extension_Normal 39.4 ×39.4 255.9 1.85 2570 3016 3622 3.68E-04 1.494E-03 Extension_Stiff 39.4 ×39.4 255.9 1.85 2749 3377 4346 3.68E-04 1.494E-03 Piles_Soft 39.4 ×39.4 255.9 1.85 2581 2915 3466 3.68E-04 1.494E-03 Piles_Normal 39.4 ×39.4 255.9 1.85 2884 3550 4679 3.68E-04 1.494E-03 Piles_Stiff 39.4 ×39.4 255.9 1.85 3175 4080 4848 3.68E-04 1.494E-03 Spread_Normal 39.4 ×39.4 137.8 1.85 6388 7520 8841 3.68E-04 1.494E-03 Spread_Stiff 39.4 ×39.4 137.8 1.85 6587 7992 8984 3.68E-04 1.494E-03 Extension_Soft 39.4 ×39.4 137.8 1.85 4494 4731 5210 3.68E-04 1.494E-03 Extension_Normal 39.4 ×39.4 137.8 1.85 4695 5079 5949 3.68E-04 1.494E-03 Extension_Stiff 39.4 ×39.4 137.8 1.85 4775 5457 6802 3.68E-04 1.494E-03 Piles_Soft 39.4 ×39.4 137.8 1.85 4747 5040 5757 3.68E-04 1.494E-03 Piles_Normal 39.4 ×39.4 137.8 1.85 5102 5744 7306 3.68E-04 1.494E-03 Piles_Stiff 39.4 ×39.4 137.8 1.85 5323 6442 8808 3.68E-04 1.494E-03 Spread_Normal 39.4 ×39.4 708.7 1.85 2697 3224 3290 3.68E-04 1.494E-03 Spread_Stiff 39.4 ×39.4 708.7 1.85 2773 3262 3308 3.68E-04 1.494E-03 Extension_Soft 39.4 ×39.4 708.7 1.85 1687 1918 2109 3.68E-04 1.494E-03 Extension_Normal 39.4 ×39.4 708.7 1.85 1825 2180 2664 3.68E-04 1.494E-03 Extension_Stiff 39.4 ×39.4 708.7 1.85 2006 2470 3145 3.68E-04 1.494E-03 Piles_Soft 39.4 ×39.4 708.7 1.85 1798 2116 2523 3.68E-04 1.494E-03 Piles_Normal 39.4 ×39.4 708.7 1.85 2088 2616 3197 3.68E-04 1.494E-03 Piles_Stiff 39.4 ×39.4 708.7 1.85 2349 3018 3280 3.68E-04 1.494E-03 Spread_Normal 19.7 ×19.7 255.9 1.85 500 581 581 3.96E-04 1.692E-03 Spread_Stiff 19.7 ×19.7 255.9 1.85 514 586 586 3.96E-04 1.692E-03 Extension_Soft 19.7 ×19.7 255.9 1.85 292 318 295 3.96E-04 1.692E-03 Extension_Normal 19.7 ×19.7 255.9 1.85 326 366 403 3.96E-04 1.692E-03 Extension_Stiff 19.7 ×19.7 255.9 1.85 360 414 481 3.96E-04 1.692E-03 Piles_Soft 19.7 ×19.7 255.9 1.85 468 567 573 3.96E-04 1.692E-03 Piles_Normal 19.7 ×19.7 255.9 1.85 514 592 592 3.96E-04 1.692E-03

57 Piles_Stiff 19.7 ×19.7 255.9 1.85 535 597 597 3.96E-04 1.692E-03 Spread_Normal 59 ×59 255.9 1.85 10340 12261 15674 2.86E-04 1.211E-03 Spread_Stiff 59 ×59 255.9 1.85 10710 13034 15850 2.86E-04 1.211E-03 Extension_Soft 59 ×59 255.9 1.85 8166 8735 10110 2.86E-04 1.211E-03 Extension_Normal 59 ×59 255.9 1.85 8307 9452 12110 2.86E-04 1.211E-03 Extension_Stiff 59 ×59 255.9 1.85 8813 10505 14496 2.86E-04 1.211E-03 Piles_Soft 59 ×59 255.9 1.85 9402 10662 14219 2.86E-04 1.211E-03 Piles_Normal 59 ×59 255.9 1.85 10793 13248 16508 2.86E-04 1.211E-03 Piles_Stiff 59 ×59 255.9 1.85 12242 15286 16541 2.86E-04 1.211E-03 Spread_Normal 39.4 ×39.4 255.9 0.60 1739 2484 2529 4.46E-04 1.981E-03 Spread_Stiff 39.4 ×39.4 255.9 0.60 1795 2541 2542 4.46E-04 1.981E-03 Extension_Soft 39.4 ×39.4 255.9 0.60 1189 1527 1846 4.46E-04 1.981E-03 Extension_Normal 39.4 ×39.4 255.9 0.60 1305 1797 2347 4.46E-04 1.981E-03 Extension_Stiff 39.4 ×39.4 255.9 0.60 1417 2118 2463 4.46E-04 1.981E-03 Piles_Soft 39.4 ×39.4 255.9 0.60 1277 1727 2285 4.46E-04 1.981E-03 Piles_Normal 39.4 ×39.4 255.9 0.60 1476 2280 2488 4.46E-04 1.981E-03 Piles_Stiff 39.4 ×39.4 255.9 0.60 1628 2527 2536 4.46E-04 1.981E-03 Spread_Normal 39.4 ×39.4 255.9 3.10 5791 6595 7107 3.19E-04 1.344E-03 Spread_Stiff 39.4 ×39.4 255.9 3.10 5988 6749 7131 3.19E-04 1.344E-03 Extension_Soft 39.4 ×39.4 255.9 3.10 3652 3864 4190 3.19E-04 1.344E-03 Extension_Normal 39.4 ×39.4 255.9 3.10 3792 4206 4827 3.19E-04 1.344E-03 Extension_Stiff 39.4 ×39.4 255.9 3.10 4023 4600 5575 3.19E-04 1.344E-03 Piles_Soft 39.4 ×39.4 255.9 3.10 3809 4126 4667 3.19E-04 1.344E-03 Piles_Normal 39.4 ×39.4 255.9 3.10 4241 4837 5980 3.19E-04 1.344E-03 Piles_Stiff 39.4 ×39.4 255.9 3.10 4650 5459 7042 3.19E-04 1.344E-03 (Column lateral confinement ratio=0.6%) Confined core concrete ultimate strain is 0.015 First Member Unconfined Confined Unconfined Confined Foundation Type b×d (in.) Height (in.) Long. rebar ratio (%) Pp1 (kips) Pu (kips) Pu (kips) u (in-1) u (in-1) Table 4.13. (Continued). the ratio between the value from Equation 4.2 and NCHRP Report 458 is less than 7% and 11% for the unconfined and confined two-column bents, respectively. The maximum dif- ferences are those corresponding to a soft foundation. Figure 4.18 (a) and (b) plot Pu from Equation 4.2 versus that obtained from the NCHRP Report 458 analysis for the unconfined and confined four-column bent systems, respec- tively. The figures show that the data points are clearly aligned along the equal force line indicating that Equation 4.2 is rea- sonably accurate with a regression coefficient R2 = 0.99 and R2 = 0.97 for the unconfined and confined four-column systems. The COV of the ratio between the value from Equation 4.2 and NCHRP Report 458 is less than 8% and 14% for the unconfined and confined four-column system, respectively. The maximum differences are those corresponding to a soft foundation. The larger differences and wider spread of data points around the equal force line observed in Figures 4.17 and 4.18 for the confined cases are due to the larger P-delta effects in NCHRP Report 458 related to the high applied vertical live load. When the columns are highly confined and the vertical load is very high, the P-delta effects produce a softening in the

58 force versus lateral deformation curve. This has caused some difficulty in defining the exact failure point in the NCHRP Report 458 model. Model Verification The empirical model proposed in this study to establish the relationship between the ultimate capacity of a multi- column bridge substructure system and the lateral load carry- ing capacity of one column in the system is a function of the number of columns in the bent and the ultimate curvature capacity of the bent columns. This relationship is expressed by an equation of the form: (4.2)1P P F Cu p mc u tunc tconf tunc = + ϕ − ϕ ϕ − ϕ    ϕ where Pp1 gives the capacity of a bridge system under lateral load assuming that the analysis is performed using linear- elastic behavior and failure is defined when one column reaches its maximum load carrying capacity as typically done when using a force-based analysis, Fmc is a multi-column fac- tor, Cj is a curvature factor, ju is the ultimate curvature of the weakest column in the bent, jtunc is the average curvature for a typical unconfined column, jtconf is the average curvature for a typical confined column. The typical curvature values for the confined and unconfined columns are extracted from the results of the survey conducted in NCHRP Report 458. For a particular bridge system, Pp1 is calculated using a linear structural analysis of the system under the effect of the applied lateral load. To find Pp1, failure is defined as the load at which one column reaches its ultimate capac- ity. The value for the ultimate curvature at failure ju is Figure 4.18. Lateral capacity by Equation 4.2 vs. lateral capacity from NCHRP Report 458 for four-column bents. y = 0.96x R² = 0.99 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 Pu by Eq .4 .2 (k ip s) Pu from NCHRP Report 458 (kips) two column_unconfined (a) Unconfined (b) Confined y = 0.97x R² = 0.97 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 2000 4000 6000 8000 10000 Pu by Eq .4 .2 (k ip s) Pu from NCHRP Report 458 (kips) two column_confined Figure 4.17. Lateral capacity by Equation 4.2 vs. lateral capacity from NCHRP Report 458 for two-column bents. y = 0.99x R² = 0.99 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu by Eq .4 .2 (k ip s) Pu from NCHRP Report 458 (kips) Unconfined y = 0.96x R² = 0.97 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu by Eq .4 .2 (k ip s) Pu from NCHRP Report 458 (kips) Confined (a) Unconfined (b) Confined

59 calculated from the ultimate plastic analysis of the column’s cross section. Values for Fmc, Cj, jtunc, and jtconf have been extracted from the analysis of a large number of bridges with two-column, three-column and four-column bents. The bents analyzed included a range of column sizes, vertical reinforcement ratios, and confinement ratios. The analyses also considered the effect of different foundation stiffnesses. The recom- mended values for these parameters are provided in Table 4.1, shown previously. The values for jtunc and jtconf are the aver- age curvatures obtained from the analysis of the column sizes used in NCHRP Report 458. The columns analyzed in NCHRP Report 458 represent typical column sizes and rein- forcement ratios collected from a national survey conducted as part of the study. The values for jtunc and jtconf are used in Equation 4.2 to compare the confinement ratio of the column being evaluated to the average confinement ratios observed in typical confined and unconfined columns. Figures 4.19, 4.20, and 4.21 summarize the results pre- sented in the previous section. These figures are provided to visualize how well the proposed model of Equation 4.2 matches the results obtained from the nonlinear pushover analysis of hundreds of bridge systems subjected to lateral load applied at the top of the bent. The plots show the values of the system capacity Pu obtained from Equation 4.2 versus the values obtained from the pushover analysis. Figure 4.19 is for systems with two-column bents analyzed during the course of this study in green as well as those ana- lyzed during NCHRP Report 458 in blue for unconfined col- umns and in red for confined columns. Different levels of confinement ratios were considered during the analyses per- formed in NCHRP Report 458. The green data points labeled NCHRP 12-86 were obtained during the course of this study using a large range of sensitivity analyses. The green data points also represent a combination of different confine- ment ratios. The solid lines give the trend lines obtained from a regression analysis of the data. The equations in the figure give the equations of the trend lines that describe the relationship between the estimated Pu obtained from Equa- tion 4.2 and the calculated Pu obtained from the nonlinear pushover analysis of actual bridge systems. A perfect match would lead to a trend line having an equation of the form y = 1.0x with a coefficient of regression R2 = 1.0. The results in Figure 4.20 are for the bridge systems with three-column bents analyzed in this study including all the sensitivity analy- ses. Figure 4.21 shows the results of the four-column bents analyzed in NCHRP Report 458 including those with con- fined columns (in red) and unconfined columns (in blue). All the trend lines in Figures 4.19, 4.20, and 4.21 have slopes close to 1.0 and coefficients of regression R2 also close to 1.0. This serves to confirm that Equation 4.2 provides a good model for estimating the ultimate capacity of bridge systems Figure 4.19. Two-column model verification. y = 0.92x R² = 0.97 0 5000 10000 15000 20000 25000 5000 10000 15000 200000 25000 Pu by Eq .4 .2 (k ip s) Pu from SAP2000 and NCHRP Report 458 (kips) Unconfined NCHRP 458 Confined NCHRP 458 NCHRP 12-86 COV=9.0% Figure 4.20. Three-column model verification. y = 1.03x R² = 0.99 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Pu by Eq .4 .2 (k ip s) Pu from SAP2000 (kips) NCHRP 12 86 COV=8.9% Figure 4.21. Four-column model verification. y = 0.97x R² = 0.98 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 2000 4000 6000 8000 10000 12000 14000 160000 18000 Pu by Eq .4 .2 (k ip s) Pu from NCHRP Report 458 (kips) Unconfined NCHRP 458 Confined NCHRP 458 COV=11.5%

60 subjected to lateral load. Another parameter that identifies the variation between Equation 4.2 and the analysis results is the COV of the ratio between the equation and the analysis results. It is observed that the COV is about 9% for the two- and three-column systems and about 11.5% for the four- column system. The higher COV in the latter case is primar- ily due to the difficulty of determining the failure point when P-delta effects are considered for soft foundations as observed in the NCHRP Report 458 data set. The analyses performed in this study and in NCHRP Report 458 assumed that the hinges form in the columns and that any weakness in the cap beams, shear capacity, member detailing, and connections is represented by reduction in the ultimate curvatures of the columns. Chapter 5 of this report describes how the weaknesses change the ultimate curvature that should be used in Equation 3.1, which is also given above as Equation 1.3. 4.3 Calibration of System Factors Calibration Approach The calibration of system factors for the force-based approach can be executed following the procedure out- lined in Chapter 2. The process is based on Figure 3.1 that describes the behavior of bridges under lateral load assum- ing that the performance is being evaluated using the force levels on the vertical axis. The procedure as presented in this chapter assumes that the lognormal reliability model is valid for the force-based analysis process. Accordingly, the evalu- ation of the safety of a bridge under lateral load also can be expressed in terms of the probability of failure, which for the force-based approach represents the probability that the load carrying capacity, P, is smaller than the applied extreme load, LE. = ≤  Pr 1 (4.3)P P LE f Assuming that both the capacity and the load can be expressed by lognormal probability distributions, the prob- ability of failure can be expanded as ln * ln * ln * * (4.4) 2 2 2 2 P P LE P LE f P LE P LE ( )[ ]( ) ( ) = Φ − −ξ + ξ     = Φ − ξ + ξ       where P* is the median of the capacity P, xP is the dispersion of the lognormal distribution of the capacity. The variables with the subscript “LE” are the statistics for the load. F is the cumulative normal distribution function. In this case, the reliability index for the system defined as, bsystem, can be calculated as ln * * (4.5) 2 2 P LE system u P LE ( )β = ξ + ξ The reliability index for the failure of the first column defined as, bcolumn, can be calculated as ln * * (4.6) 1 2 2 P LE column p P LE ( )β = ξ + ξ In the program Hazus (2003) developed by FEMA for eval- uating the seismic risk of structures, the combined dispersion for capacity and demand for typical bridge structures and structural members is set at 0.602 2P LEξ + ξ = for all damage types and all bridge members. As explained in Chapter 2, a probabilistic measure of sys- tem redundancy can be expressed in terms of the additional reliability provided by the system compared to that of the member and can be defined as the reliability index margin (4.7)u system column∆β = β − β Substituting Equations 4.5 and 4.6 into Equation 4.7, the reliability index margin is obtained as ln * * ln * * (4.8) 2 2 1 2 2 P LE P LE u u P LE p P LE ( )( ) ∆β = ξ + ξ − ξ + ξ Or ln * * (4.9) 1 2 2 P P u u p P LE ( ) ∆β = ξ + ξ This reliability margin can be used as a probabilistic mea- sure of redundancy. Bridges that meet a reliability margin criterion can be defined as being sufficiently redundant. Those that do not meet the criterion are non-redundant. For example, NCHRP Report 406 observed that redundant super- structure systems subjected to vertical loads have been tradi- tionally associated with bridges that meet or exceed a target system reliability margin Dbu target = 0.85. The 0.85 target mar- gin was selected to match the average reliability margin of all four-girder bridges because four-girder bridges have been widely accepted as being redundant. Following the approach of NCHRP Report 406, bridge sys- tems that do not meet the minimum target reliability margin should be designed to higher standards by applying a system factor fs. The system factor should be calibrated to offset the

61 difference between the target reliability margin and the reli- ability margin that a system designed using current methods provides. Following the same rationale, it can be assumed that bridges under lateral loads should produce a target reliabil- ity margin Dbu target to be classified as sufficiently redundant. Those that do not meet the target reliability margin must be designed to meet higher system reliability levels. The higher reliability levels should offset the difference between the cal- culated Dbu and the target value Dbu target. This can be for- mulated by first evaluating the deficit between the reliability index margin provided by a particular bridge system, Dbu, and the target reliability index margin Dbu target. (4.10)deficit targetu u u∆β = ∆β − ∆β A negative Dbu deficit means that the system provides a higher level of redundancy than the minimum required. A positive deficit indicates that the bridge system configuration does not provide a sufficient level of redundancy. To compensate for the lack of redundancy, the members of the bridge should be designed to higher standards to ensure that the system pro- vides a reliability index higher than the one obtained accord- ing to current standards to offset the reliability deficit. This can be formulated as (4.11)u deficit u targetsystemN system system u( )β = β + ∆β = β + ∆β − ∆β where bNsystem is the reliability index that a more conservatively designed system should achieve to compensate for the lack of adequate levels of redundancy while bsystem is the reliability index obtained for the current design. bNsystem can be expressed as ln * * (4.12) N 2 2 LP LE system N u P LE ( )β = ξ + ξ where Pu*N is the required ultimate load capacity needed to reach the system reliability bNsystem. Substituting Equations 4.5, 4.9, and 4.12 into Equa- tion 4.11 leads to ln * * ln * * ln * * (4.13) N 2 2 2 2 u target 1 2 2 P LE P LE P P u P LE u P LE u p P LE ( )( ) ( ) ξ + ξ = ξ + ξ + ∆β − ξ + ξ       The above expression is simplified to give ln * * (4.14) N 1 2 2 u target P P u p P LE ( ) ξ + ξ = ∆β which can be written as ln * * * * (4.15) N 1 N 1 N 1 2 2 u target P P P P u p p p P LE     ξ + ξ = ∆β where Pp1*N is the required column capacity obtained by adjust- ing the currently required member capacity Pp1* by a system factor fs such that * *1N 1P Pp p s = φ so that ln * * 1 (4.16) N 1 N 2 2 u target P LF u p s P LE φ     ξ + ξ = ∆β Or ln 1 * * (4.17) N 1 N 2 2 u target P Ps u p P LEφ     = ξ + ξ ∆β Finally * * exp (4.18) N 1 N 2 2 u targetP Ps u p P LE( )φ = ( )− ξ +ξ ∆β The evaluation of the system factor can then be executed if the target margin reliability Dbu target is set, and if the dis- persion coefficient 2 2P LEξ + ξ as well as the redundancy ratio * * N 1 N P P u p are known. In NCHRP Report 458, it was assumed that the ratio * * N 1 N P P u p remains constant, independent of the value of P*N1p such that * * * * N 1 N 1 R P P P P u u p u p = = . This is a reasonable assumption as long as the redundancy ratio Ru is evaluated for bridges having similar configurations including the same number of col- umns, similar column cross sections and column heights. This led to the development of very complex tables of sys- tem factors that attempt to cover a large array of substruc- ture systems with different combinations of properties and dimensions. Instead, a review of the NCHRP Report 458 data, comple- mented with the results of the analyses performed during the course of this study, have shown that the relationship between the ultimate capacity of a multi-column bridge substructure system and the lateral load carrying capac- ity of one column can be reasonably well represented as a function of the number of columns in the bent and the ultimate curvature capacity of the bent columns as dem- onstrated in Section 4.2 of this chapter. A system’s ultimate

62 capacity under lateral load can be represented by an equa- tion of the form: = + ϕ − ϕ ϕ − ϕ    φ (4.2)1P P F Cu p mc u tunc tconf tunc where Pp1 gives the capacity of a bridge system under lateral load assuming linear-elastic behavior as typically done when using a force-based analysis, Fmc is a multi-column factor, Cj is a curvature factor, ju is the ultimate curvature of the weak- est column in the bent, jtunc is the average curvature for a typical unconfined column, jtconf is the average curvature for a typical confined column. For a particular bridge system, Pp1 is calculated using a lin- ear structural analysis of the system under the effect of the applied lateral load. To find Pp1, failure is defined as the load at which one column reaches its ultimate capacity assuming elasto-plastic behavior. The value for the ultimate curvature at failure ju is calculated from the ultimate plastic analysis of the column’s cross section. Values for Fmc, Cj, jtunc, and jtconf have been extracted from the analysis of a large number of two-column, three-column, and four-column bents. The bents analyzed included a range of column sizes, vertical reinforcement ratios, and confine- ment ratios. The analyses also considered the effect of dif- ferent foundation stiffnesses. The recommended values for these parameters are provided in Table 4.1. The values for jtunc and jtconf are the average curvatures obtained from the analysis of the column sizes used in NCHRP Report 458. The columns analyzed in NCHRP Report 458 represent typi- cal column sizes and reinforcement ratios collected from a national survey conducted as part of the study. The values for jtunc and jtconf are used in Equation 4.2 to compare the con- finement ratio of the column being evaluated to the average confinement ratios observed in typical confined and uncon- fined columns. The implementation of Equation 4.2 in combination with the recommended factors provided in Table 4.1 into Equa- tion 4.18 will lead to the system factors to be used when evaluating the force capacity of bridge systems subjected to lateral loads. Specifically, the system factor is a function of the target reliability level as well as the combined dispersion coefficient. Substituting Equation 4.2 into Equation 4.18, the system factor is exp (4.19) 2 2 F Cs mc u tunc tconf tunc LF LE u targetφ = + ϕ − ϕ ϕ − ϕ     ( )− ξ +ξ ∆β ϕ The target reliability index margin, Dbu target, must be set by the code writing authorities to match those of acceptable values obtained from systems recognized as being adequately redundant. The dispersion coefficient 2 2P LEξ + ξ must reflect the uncertainties associated with estimating the strength and the applied loads. As an example, a dispersion coefficient equal to 0.60 is used in the program Hazus (2003) prepared by FEMA for the analysis of seismic damage risk. The exponential term in Equation 4.18 is defined as the system risk factor as it reflects the acceptable level of risk for bridge collapse that can be tolerated such that exp (4.20) 2 2 s LF LE u targetℜ = ( )− ξ +ξ ∆β Table 4.14 gives the risk coefficient, ℜs for different values of Dbu target and dispersion coefficients. As an example, Figure 4.22 shows how the system factor, fs, varies as a function of the curvature of the bridge column. The figure shows the system factor for different numbers of columns in a bent. The system factors in Figure 4.22 are cal- culated assuming a target reliability margin Dbu target = 0.50 and a dispersion coefficient 0.602 2P LEξ + ξ = such that the risk factor is ℜs = 074. The results in Figure 4.22 show a moderate increase in the system factor as the number of columns in the bent increases from two to four columns by 0.06. However, the more signifi- cant increase in the system factor is due to the increase in the confinement ratio, which is reflected by the increase in the ultimate curvature of the column. In fact, from Equation 4.19, it is observed that an increase in the ultimate curvature of the section from 3.0 × 10-4 (1/in) to 2.0 × 10-3 (1/in) results in an increase in the system factor by about 0.25. A curvature of 3.0 × 10-4 (1/in) is obtained for this column when the column is unconfined. A confinement ratio of 0.003, which according to the AASHTO LRFD Seismic Design Guide corresponds to a structural detail category B, produces an ultimate curvature equal to 0.974 × 10-3 (1/in). A confinement ratio of 0.005, which corresponds to structural detail category C, produces a curvature equal to 1.30 × 10-3 (1/in). A confinement ratio of 0.008 produces a curvature equal to 1.76 × 10-3 (1/in). A confinement ratio of 0.01 produces a curvature equal to 2.05 × 10-3 (1/in). Implementation Example A simple example is presented to illustrate the procedure that an engineer would follow to include redundancy during the safety evaluation of a bridge system subjected to lateral load. In this example, the researchers assume that the seis- mic design of a bridge system with two-column bents in a low seismic region calls for each column to have a moment capacity Mp = 3.85 × 104 kip-in. To meet the preliminary design requirements, the engineer selects a section that has a diameter D = 3.6-ft. The column is reinforced by verti- cal steel bars such that the vertical reinforcement ratio is 1.85%. Furthermore, the engineer uses a confinement ratio

63 rs = 0.003. The analysis of the section determines that the moment capacity of the column meets the requirement and is equal to Mp = 3.85 × 10-4 kip-in. and the ultimate curvature is ju = 0.974 × 10-3 (1/in). A linear-elastic analysis of the system under lateral load indicates that one column reaches its moment capacity Mp = 3.85 × 10-4 kip-in. when a lateral force P1p = 714 kips is applied on the two-column bent. To reduce the risk of collapse under an extreme event, the engineer would like to verify that the system will have significantly additional capacity in case one of the columns reaches its design moment capacity. Using Equation 4.2, Target reliability margin targetu Dispersion coefficient 2 2 P LE Risk factor s 0.3 0.2 0.94 0.3 0.91 0.4 0.89 0.5 0.86 0.6 0.84 0.4 0.2 0.92 0.3 0.89 0.4 0.85 0.5 0.82 0.6 0.79 0.5 0.2 0.90 0.3 0.86 0.4 0.82 0.5 0.78 0.6 0.74 0.6 0.2 0.89 0.3 0.84 0.4 0.79 0.5 0.74 0.6 0.70 0.7 0.2 0.87 0.3 0.81 0.4 0.76 0.5 0.70 0.6 0.66 0.8 0.2 0.85 0.3 0.79 0.4 0.73 0.5 0.67 0.6 0.62 0.9 0.2 0.84 0.3 0.76 0.4 0.70 0.5 0.64 0.6 0.58 1.0 0.2 0.82 0.3 0.74 0.4 0.67 0.5 0.61 0.6 0.55 Table 4.14. Risk coefficient for different target reliability margins.

64 the engineer calculates the ultimate load capacity of the sys­ tem to be 714 1.10 0.24 0.974 10 3.64 10 1.55 10 3.64 10 874 1 3 4 3 4 P P F C kips u p mc u tunc tconf tunc = + ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = φ − − − − The redundancy ratio is obtained as 874 7141 R P Pu u p = = = 1.22 . This indicates that this bridge with two­column bents does provide some level of redundancy. However, this level must be checked against the specified requirements. It is assumed that the specifications applicable to the jurisdiction where the bridge is to be built require a risk factor ℜs = 0.74 corresponding to a reliability margin Dbu target = 0.50. To verify whether the bridge columns meet this requirement, a system factor is calculated from Equation 4.19 as 0.74 1.22 0.90F Cs s mc u tunc tconf tunc φ = ℜ + ϕ − ϕ ϕ − ϕ     = × =ϕ Because the system factor is less than 1.0, the bridge is not sufficiently redundant to meet the risk requirements. Thus, the bridge columns must be designed to a higher moment capac­ ity such that fs Mp = 3.85 × 104 kip­in. That is, the moment capacity must be increased by (1/0.90) so that Mp = 4.28 × 104 kip­in. A higher moment capacity will not turn a non­redundant system into a redundant one but it will help compensate for the relatively low level of system reliability by increasing the reliability of the columns to compensate for the low level of redundancy. Alternatively, the engineer may decide to improve the over­ all reliability of the system by increasing the confinement ratio from the original rs = 0.003 to a higher ratio rs = 0.008. In this case, the new system factor is calculated from Equa­ tion 4.19 to be fs = 1.02 which is higher than 1.0. Thus, the moment capacity Mp = 3.85 × 104 kip­in will be acceptable as it will produce the required redundancy level as long as the lateral reinforcement is improved to produce a confinement ratio equal to rs = 0.008. Reliability Check Using Equation 4.9 with Pu/Pp1 = 1.22, it is found that the original reliability index margin is Dbu = 0.33, which is lower than the target value Dbu target = 0.50. The deficit in the reliabil­ ity index margin is Dbu deficit = Dbu target - Dbu = 0.17. Increas­ ing the moment capacity of the column by a factor 1/0.9 will increase the overall reliability of the system to compensate for the low level of redundancy. 4.4 Additional Verifications of Model P-Delta Effect P­delta effect, also known as geometric nonlinearity, involves the equilibrium and compatibility relationships of a struc­ tural system loaded about its deflected configuration. Of par­ ticular concern is the application of gravity load on bridge columns with large lateral deformations. If the deformations become sufficiently large as to break from linear compatibility relationships, then using SAP2000’s large­displacement and large­deformation analyses becomes necessary. This condition magnifies the bending moment in the column and reduces the deformation capacity. As explained in the SAP2000 manual, the two sources of P­delta effect are illustrated in Figure 4.23, and described as follows: • P-d effect, or P­“small­delta,” is associated with local defor­ mation relative to the element chord between end nodes. Figure 4.22. System factor vs. ultimate curvature for example column with risk factor ℜs 5 0.74. 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.0005 1E 17 0.0005 0.001 0.0015 0.002 0.0025 0.003 sy st em Curvature/ in 1 Four column Three column Two column Figure 4.23. P-delta about column.

65 Typically, P-d only becomes significant at unreasonably large displacement values, or in especially slender columns. • P-D effect, or P-“big-delta,” is associated with displace- ments relative to member ends. Unlike P-d, this type of P-delta effect is critical to nonlinear modeling and analysis. Gravity loading will influence structural response under significant lateral displacement. To verify that Equation 4.2 is still valid for predicting bridge system capacity when P-delta effect is taken into account, the analysis of the three-column multi-girder bridge described in Section 4.2 is performed for the following three cases: (1) without geometric nonlinearity, (2) with P-d effect, and (3) P-delta including large displacement effects. Twenty-four bridge-bent models with various diameters ranging from 6-ft to 8-ft and different column heights ranging from 20-ft to 40-ft are investigated. The results of the analysis for first column failure Pp1 are compared to the system’s ultimate capacity Pu obtained from the SAP2000 analysis and the pre- dicted Pu by Equation 4.2 are listed in Table 4.15. The results with P-delta effect (in red) and those without the effect (in blue) also are compared in Figure 4.24. The last column in Table 4.15 shows that the errors remain within the same range whether P-delta effects are consid- ered or not, with the largest differences observed for the very slender long columns when the error in Equation 4.2 may increase by less than 4%. The trend lines in Figure 4.24 show that R2 remains essentially similar whether the P-delta effects are considered or not. Effect of Foundation Stiffness on Bridge Systems with Three-Column Bents The analyses performed in NCHRP Report 458 consid- ered different foundation types and soil conditions to reflect Table 4.15. Comparison of results with and without P-delta effect. Category B Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature System Capacity by Eq. 4.2 Error D (in.) H (in.) Pu (kips) Pp1 (kips) u (in-1) Pu (kips) None 451-inch 72 451 2480.7 2252.8 0.000717 2774.22 11.83% P 451-inch 72 451 2279.5 2121.2 0.000717 2612.16 14.59% P 451-inch 72 451 2259.5 2119.9 0.000717 2610.56 15.54% None Diameter–7 ft 84 240 9116.3 8079.5 0.000574 9715.24 6.57% P Diameter–7 ft 84 240 9074.1 8076.9 0.000574 9712.11 7.03% P Diameter–7 ft 84 240 9018.6 8074.5 0.000574 9709.23 7.66% None Pile 96 240 11528.4 8870.5 0.000516 10562.1 -8.38% P Pile 96 240 11464 8863.1 0.000516 10553.3 -7.94% P Pile 96 240 11347.6 8863.9 0.000516 10554.2 -6.99% None Pile 96 360 8087.1 6210.4 0.000516 7394.71 -8.56% P Pile 96 360 8002.6 6201.6 0.000516 7384.23 -7.73% P Pile 96 360 7972.3 6202.5 0.000516 7385.31 -7.36% None Pile 96 480 6401 4936.2 0.000516 5877.52 -8.18% P Pile 96 480 6290.4 4926.1 0.000516 5865.5 -6.75% P Pile 96 480 6252.1 4927 0.000516 5866.57 -6.17% None Spread 96 240 13016.1 11881.2 0.000516 14146.9 8.69% P Spread 96 240 13010 11820.5 0.000516 14074.6 8.18% P Spread 96 240 12866.1 11796.6 0.000516 14046.2 9.17% None Spread 96 360 9169.2 8459.6 0.000516 10072.8 9.86% P Spread 96 360 9125.3 8405.8 0.000516 10008.8 9.68% P Spread 96 360 9089.3 8383.6 0.000516 9982.34 9.83% None Spread 96 480 7225.7 6696 0.000516 7972.91 10.34% P Spread 96 480 7157.2 6651.4 0.000516 7919.81 10.66% P Spread 96 480 7091.9 6671.4 0.000516 7943.62 12.01% NOTES: “P ” (i.e., P-“small-delta”) is associated with local deformation relative to the element chord between end nodes. The includes P-“small-delta” plus large displacement. equilibrium equations take into partial account the deformed configuration of the structure. “P- ” “None” means neither P-“small-delta” nor P-“large-delta” effect is considered.

66 typical substructure design cases. To investigate situations where the foundations may be overdesigned or under- designed, this section analyzes the multi-girder bridge with three-column bents assuming different foundation stiff- nesses. The analysis is then performed for the different foun- dation stiffness values shown in Table 4.16, which has been extracted from NCHRP Report 458, where foundations and supporting soils were grouped into eight categories. In this section, 40 bridge system models with eight types of foundations are studied. The analysis results are sum- marized in Table 4.17. The results also are presented in Fig- ure 4.25, which plots the estimated ultimate system capacity Pu obtained from Equation 4.2 versus the value of Pu obtained from the pushover analysis for the different column dimen- sions and foundations analyzed. The analysis included 3-ft diameter columns designed with lateral confinement rein- forcement ratios rs = 0.24% (original detail), 0.3% (detail category B) and 0.5% (detail category C), in addition to cases where the diameter is increased up to 8-ft with lateral con- finement reinforcement ratios rs = 0.3% to obtain a much larger range of the ultimate system capacity. Although the errors in Table 4.17 are found to vary between -14.68% and +12.78%, the trend line in Figure 4.25 shows a regression slope of 1.02 with a regression coefficient R2 of 0.99. This demonstrates that, generally speaking, the ultimate system capacity estimated by Equation 4.2 gives a good approxima- tion to the actual ultimate capacity of the bridge system and the bridge foundation stiffness has no overall effect on the proposed equation. Reduction in Column Curvature The analyses performed in the previous sections of this report and in NCHRP Report 458 assumed that hinges form in the columns. Accounting for weaknesses in the cap beams, shear capacity, member detailing, and connections can be accommodated by reducing the ultimate curvature of the columns. FHWA’s Seismic Retrofitting Manual for Highway Structures: Part 1—Bridges provides a set of models that can be used to estimate the system’s ductility for different types of inadequacies in the design of the bridge system. This section describes how the models in the FHWA report can be adopted during the application of Equation 4.2. For that purpose, it is proposed to use a ductility reduction factor that can be applied on the ultimate curvature ju in Equation 4.2 to account for the reduction in ductility due to weaknesses in the system design. Also, in some cases, a mod- ification of the force Pp1 may need to be applied when the capacity of the first member to fail is significantly reduced. The proposed approach is specifically described for the cases when the cap beam is weaker than the bridge columns and when the column shear capacity leads to shearing failures before the ultimate moment capacity of a section is reached. Weaknesses in lap splicing and other connection details can follow the same approach using the models described in the FHWA report. y = 1.03x R² = 0.95 y = 1.02x R² = 0.95 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Es ti m at ed Pu by Eq .4 .2 (k ip s) Pushover analysis Pu (kips) Data w/ P Delta Data w/o P Delta Linear (Data w/ P Delta) Linear (Data w/o P Delta) Figure 4.24. Comparison of results with and without P-delta effect. Table 4.16. Foundation types and stiffness. Foundaon types K_vercal(kN/m) K_transverse (kN/m) K_rotaon (kN m) 1 spread\normal\ 97200 72900 3650000 2 spread\sff\ 147000 110000 5530000 3 extension\so\ 443077 5226 113726 4 extension\normal\ 1107000 17784 220882 5 extension\sff\ 1994000 46628 367348 6 pile\so\ 675400 18870 376700 7 pile\normal\ 1689000 85870 941700 8 pile\sff\ 3039000 299000 1695000

Table 4.17. Summary of results for three-column bridge bents with various foundations. Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature System Capacity by Eq.4.2 Error Category B 72'' Column D (in.) H (in.) Pu (kips) Pp1 (kips) u (in-1) Pu (kips) 1 spread\normal\ 96 240 13016.1 11881.2 0.000516 14146.9 8.69% 2 spread\stiff\ 96 240 13206.3 11846 0.000516 14105 6.81% 2 spread\stiff\ 96 180 17012.6 15037.2 0.000516 17904.8 5.24% 2 spread\stiff\ 96 150 19847.4 17460.4 0.000516 20790.1 4.75% 2 spread\stiff\ 96 480 7195.2 6636.9 0.000516 7902.54 9.83% 3 extension\soft\ 96 700 4115.9 3858.4 0.000516 4594.19 11.62% 4 extension\normal\ 96 400 5518.6 5200.4 0.000516 6192.11 12.20% 5 extension\stiff\ 96 400 5869 5099 0.000516 6071.37 3.45% 6 pile\soft\ 96 240 7198.4 6818.1 0.000516 8118.3 12.78% 7 pile\normal\ 96 700 4922.4 3828.4 0.000516 4558.47 -7.39% 7 pile\normal\ 96 240 11528.4 8870.5 0.000516 10562.1 -8.38% 8 pile\stiff\ 96 240 12904.5 9419.7 0.000516 11216 -13.08% 8 pile\stiff\ 96 480 6955.3 5349.7 0.000516 6369.88 -8.42% Category B 36'' Column D (in.) H (in.) Pu (kips) Pp1 (kips) u (in-1) Pu (kips) Error 1 spread\normal\ 36 331 1113 798.349 0.00106 1038.53 -6.69% 2 spread\stiff\ 36 331 1112.4 784.9 0.00106 1021.03 -8.21% 2 spread\stiff\ 36 451 922.8 727.1 0.00106 945.843 2.50% 2 spread\stiff\ 36 150 1752.2 1385.4 0.00106 1802.19 2.85% 3 extension\soft\ 36 150 1650.8 1386.4 0.00106 1803.49 9.25% 4 extension\normal\ 36 451 883.6 759.7 0.00106 988.251 11.84% 4 extension\normal\ 36 150 1866.1 1290.4 0.00106 1678.61 -10.05% 5 extension\stiff\ 36 451 912.2 738.7 0.00106 960.933 5.34% 6 pile\soft\ 36 331 1122.2 809.8 0.00106 1053.42 -6.13% 7 pile\normal\ 36 331 870 581.638 0.00106 756.62 -13.03% 8 pile\stiff\ 36 331 1126.6 785.6 0.00106 1021.94 -9.29% Category C 36'' Column D (in.) H (in.) Pu (kips) Pp1 (kips) u (in-1) Pu (kips) Error 1 spread\normal\ 36 331 1226 806.364 0.00138 1101.17 -10.18% 2 spread\stiff\ 36 331 1235.9 792.7 0.00138 1082.51 -12.41% 3 extension\soft\ 36 331 1388.6 867.6 0.00138 1184.79 -14.68% 4 extension\normal\ 36 331 1268.4 794.2 0.00138 1084.56 -14.49% 5 extension\stiff\ 36 331 1237.8 781.7 0.00138 1067.49 -13.76% 6 pile\soft\ 36 331 1270.4 818 0.00138 1117.06 -12.07% 7 pile\normal\ 36 331 1232 796.269 0.00138 1087.38 -11.74% 8 pile\stiff\ 36 331 1225.8 793.5 0.00138 1083.6 -11.60% Original 36'' Column D (in.) H (in.) Pu (kips) Pp1 (kips) u (in-1) Pu (kips) Error 1 spread\normal\ 36 331 1071.186 794.341 0.00096 1017.24 -5.04% 2 spread\stiff\ 36 180 1509.2 1178.8 0.00096 1509.58 0.03% 3 extension\soft\ 36 180 1722.9 1200.7 0.00096 1537.62 -10.75% 4 extension\normal\ 36 180 1609.8 1107 0.00096 1417.63 -11.94% 5 extension\stiff\ 36 180 1554.3 1111.7 0.00096 1423.65 -8.41% 6 pile\soft\ 36 180 1610.2 1197.6 0.00096 1533.66 -4.75% 7 pile\normal\ 36 331 1095.164 784.356 0.00096 1004.45 -8.28% 8 pile\stiff\ 36 180 1512 1183.8 0.00096 1515.98 0.26%

68 Effect of Inadequate Cap Beams In a properly designed bridge system, the cap beam should be at least as strong as the columns. Thus, no plastic hinges are expected to form in the cap beam during a pushover analysis. However, reinforced or prestressed concrete cap beams may yield and form plastic hinges due to inadequacies in their design or to reductions in their capacities due to deteriora- tion or other phenomena. Plastic hinges form at locations of peak bending moment and are therefore likely to occur in the end regions of the columns or cap beams. The interac- tion between columns and cap beams and their strengths will determine the likely mode of failure. Equation 4.2 gives the relationship between the ultimate capacity of a multi-column bridge system and the lateral load carrying capacity of one column as a function of the number of columns in the bent and the ultimate curvature capacity of the bent columns. The equation has been developed based on the assumption of having strong cap beams. FHWA’s Seismic Retrofitting Manual for Highway Structures: Part 1—Bridges uses a capacity/demand (C/D) ratio to account for the effect of weak cap beams of bridge bents. The same concept can be used to reflect the weakness of the cap beam when using Equation 4.2. However, in the context of this study, the defini- tion of demand is not the “demand of the design earthquake” as defined in the FHWA report, but the minimum demand on the cap beam strength to make it at least as strong as the column. Two parameters need to be checked to verify that the cap beam is at least as strong as the column: the moment capacity of the beam and the ultimate curvature of the cap beam. The moment capacity of the cap beam divided by the moment capacity of the column is defined as the moment capacity of demand ratio C/Dmoment. The ultimate curvature of the cap beam divided by the ultimate curvature of the column is defined as curvature capacity over demand ratio C/Dcurvature. In a properly designed system, capacity over demand ratios, C/D, for both the moment and the curvature should always be greater than or equal to 1.0. If C/D is less than 1.0, the sys- tem’s ability to carry lateral load is reduced. Three cases can be considered, as shown in Figure 4.26. For Case A, the moment capacity of the cap beam is larger than that of the column, but the ultimate curvature of the beam is lower than that of the column. For Case B, the moment capacity of the beam is Figure 4.26. Model for cap beam-column capacities. Figure 4.25. Effect of foundation on Equation 4.2. y = 1.02x R² = 0.99 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000 Es ti m at ed Pu by Eq .4 .2 (k ip s) Pushover analysis Pu (kips) Bridges w/ various foundations Linear (Bridges w/ various foundations) (a) Beam-column connection for a three-column bent showing potential top bent hinge locations (b) Idealized moment-curvature relationships for cap beam (in red) and column (in black)

69 lower than that of the column but higher than the column’s plastic moment. In Case C, the cap beam’s moment capacity is lower than the column’s plastic moment. The adjustments to be made for Equation 4.2 in each of these cases are studied in this section. Cap Beam Capacity Higher Than Column Capacity Case A: This is the situation when C/Dcurvature of the cap beam leads to a reduction in the curvature capacity of the beam- column connection, which can be modeled by an equivalent reduction in the column’s curvature. The reduction can be rep- resented by a factor gjc = C/Dcurvature if less than 1.0, otherwise no reduction is needed and one can use gjc = 1.0. The modi- fied Equation 4.2 is then presented as (4.21)c beam column γ = ϕ ϕϕ (4.22)1P P F Cu p mc c u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    ϕ ϕ Cap Beam Capacity Lower Than Column Capacity When the cap beam strength is weaker than the column strength and C/Dmoment is less than 1.0, then two cases can be considered. Case B considers the situation where the cap beam moment capacity is higher than the plastic moment capacity of the column but lower than the ultimate capac- ity of the column. In this case, the ultimate curvature ju in Equation 4.2 needs to be reduced to reflect the lower moment capacity of the system as well as the lower ultimate curvature. Case C considers the situation where the cap beam moment capacity is even lower than the plastic moment capacity of the column. In this case, both the load at failure assuming linear-elastic behavior, Pp1, as well as the ultimate curvature would need to be reduced. Changing the force Pp1 reflects the insufficient moment capacity of the connection between the cap beam and the bridge columns. In this case, Pp1 is calculated as the load at which the cap beam reaches its moment capacity. Case B: If the cap beam strength is higher than the plastic moment of the column, but less than the ultimate moment capacity of the column, [Mcol.ultimate > Mbeam.plastic > Mcol.plastic], then gjc for the beam-column connection will depend on the cap beam capacity including moment and curvature capac- ity, which will cause a weak beam-column connection. The column curvature reduction factor gjc is used to reduce ju using the following equations: (4.23) . . . . M M M M c M beam plastic col plastic col ultimate col plastic beam column γ = γ γ = − − × ϕ ϕϕ ϕ (4.24)1P P F Cu p mc c u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    φ ϕ Such that gjcju ≈ effective curvature for beam-column connection. Case C: If the cap beam strength is weaker than the plastic moment of the column, [Mbeam.plastic < Mcol.plastic], then Pp1 is the load at which a plastic hinge forms in the cap beam assuming linear-elastic analysis and gjc is used to reduce the column curvature. (4.25)c beam column γ = ϕ ϕϕ (4.26)1P P F Cu p mc c u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    ϕ ϕ Forty bridge-bent models are analyzed using SAP2000 and compared to Equations 4.21 through 4.26 as appropri- ate for Cases A, B, and C to demonstrate the validity of the proposed approach for treating bridges with weak cap beams. For simplicity, Equations 4.21 through 4.26 will be referred to as modified Equation 4.2. Weak Cap Beam Model Verification To verify if Equations 4.21 through 4.26 are valid, 40 bridge models with 7-ft diameter columns having different column heights in the range of 16.7-ft up to 32.6-ft are investigated. All column sections have a transverse confinement ratio of 0.3%. The result of the linear-elastic analysis at which the first member reaches its plastic capacity is defined as Pp1. The values for Pp1, as well as those of the system ultimate capacity Pu obtained from the nonlinear SAP2000 pushover analysis, are presented in Table 4.18. The table also gives the column curvatures and the capacity over demand ratios C/Dcurvature column curvature and C/Dmoment. The SAP2000 results for Case A are compared to Pu predicted by Equation 4.22 show- ing a maximum difference on the order of 7.64%. The maxi- mum differences for Case B and Case C are 8.26% and 8.64%, respectively. Figure 4.27 plots all the data of the predicted Pu from the modified empirical model versus the pushover analysis val- ues. All the trend lines in Figure 4.27 have slopes close to 1.0 and the coefficients of regression are R2 = 0.97, R2 = 0.95, and R2 = 0.94 for Cases A, B, and C, respectively. These results show that the proposed equations can be used in engineering practice to estimate the ultimate capacity of bridge systems subjected to lateral load for cases involving weaknesses in the moment and curvature capacities of the cap beams. Several examples are presented to illustrate how an engineer can use

70 Case A Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature C/D Ratio Moment C/D Ratio Curvature System Capacity by Eq.4.22 Error D (in.) H (in.) Pu (kips) Pp1 (kips) Phi_ult. (in-1) M * * Pu (kips) 84 391 5859 5244.8 0.000574 1.26 1.00 6306.6 7.64% 84 391 5859 5244.8 0.000574 1.26 0.90 6245.7 6.60% 84 391 5811.6 5244.8 0.000574 1.26 0.80 6184.8 6.42% 84 391 5811.6 5244.8 0.000574 1.26 0.70 6123.9 5.37% 84 391 5785.2 5244.8 0.000574 1.26 0.60 6063.0 4.80% 84 391 5766 5244.8 0.000574 1.26 0.50 6002.1 4.10% 84 391 5742.4 5244.8 0.000574 1.26 0.40 5941.2 3.46% 84 391 5714.4 5244.8 0.000574 1.26 0.30 5880.3 2.90% 84 360 6291.4 5609.4 0.000574 1.26 0.80 6614.8 5.14% 84 330 6775.4 6018.3 0.000574 1.26 0.80 7097.0 4.75% 84 300 7384.1 6495.5 0.000574 1.26 0.80 7659.7 3.73% 84 270 8205.3 7058.2 0.000574 1.26 0.80 8323.3 1.44% 84 240 9103.1 7729.2 0.000574 1.26 0.80 9114.5 0.13% 84 200 10801.7 8848.9 0.000574 1.26 0.80 10434.9 -3.40% Case B Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature C/D Ratio Moment C/D Ratio Curvature System Capacity by Eq.4.24 Error D (in.) H (in.) Pu (kips) Pp1 (kips) Phi_ult. (in-1) M * * Pu (kips) 84 391 5713.3 5244.8 0.000574 0.00 1.25 5697.6 -0.27% 84 391 5725.8 5244.8 0.000574 0.21 1.25 5827.1 1.77% 84 391 5740.8 5244.8 0.000574 0.30 1.25 5880.3 2.43% 84 391 5740.2 5244.8 0.000574 0.40 1.25 5941.2 3.50% 84 391 5746.6 5244.8 0.000574 0.50 1.25 6002.1 4.45% 84 391 5747.6 5244.8 0.000574 0.60 1.25 6063.0 5.49% 84 391 5762.6 5244.8 0.000574 0.74 1.25 6150.6 6.73% 84 391 5769 5244.8 0.000574 0.90 1.25 6245.7 8.26% 84 360 6161.1 5609.4 0.000574 0.21 1.25 6232.1 1.15% 84 330 6619.7 6018.3 0.000574 0.21 1.25 6686.4 1.01% 84 300 7214.6 6495.5 0.000574 0.21 1.25 7216.6 0.03% 84 270 7932.9 7058.2 0.000574 0.21 1.25 7841.8 -1.15% 84 240 8782.4 7729.2 0.000574 0.21 1.25 8587.3 -2.22% 84 200 10424.4 8848.9 0.000574 0.21 1.25 9831.3 -5.69% Table 4.18. Comparison of system capacities from SAP2000 and Equations 4.22, 4.24, and 4.26.

71 the proposed approach when evaluating the ultimate load carrying capacity of a bridge system subjected to distributed lateral load. Implementation Example: Weak Cap Beam Case A Case A is the situation when C/Dcurvature of the cap beam leads to a reduction in the curvature capacity of the beam- column connection, which can be modeled by an equivalent reduction in the column’s curvature. The reduction can be represented by a factor gjc and the modified Equation 4.22 is then presented as 1P P F Cc beam column u p mc c u tunc tconf tunc γ = ϕ ϕ = + γ ϕ − ϕ ϕ − ϕ    ϕ ϕ ϕ y = 1.03x R² = 0.97 y = 1.00x R² = 0.95 y = 1.02x R² = 0.94 0 2000 4000 6000 8000 10000 12000 0 2000 4000 6000 8000 10000 12000 Pu b y m od ifi ed E q. 4. 2 (k ip s) Pu from SAP2000 (kips) Case A Case B Case C Figure 4.27. Predicted Pu vs. Pu from SAP2000 for weak cap beams. Case C Diameter Column Height System Capacity Plastic Capacity Ultimate Column Curvature C/D Ratio Moment C/D Ratio Curvature System Capacity by Eq.4.26 Error D (in.) H (in.) Pu (kips) Pp1 (kips) Phi_ult. (in-1) M * * Pu (kips) 84 391 5084.7 4314.3 0.000574 N/A 1.00 5187.8 2.03% 84 391 4896 4314.3 0.000574 N/A 0.50 4937.3 0.84% 84 391 4535.2 4314.3 0.000574 N/A 0.20 4787.0 5.55% 84 391 3730.1 3294.0 0.000574 N/A 0.05 3597.5 -3.55% 84 391 3955.1 3294.5 0.000574 N/A 0.09 3613.4 -8.64% 84 391 3191 3020.4 0.000574 N/A 0.05 3298.7 3.38% 84 391 3199.7 3020.6 0.000574 N/A 0.09 3313.0 3.54% 84 391 3180.6 3020.6 0.000574 N/A 0.05 3298.9 3.72% 84 391 3180.6 3020.6 0.000574 N/A 0.09 3313.0 4.16% 84 360 3359.5 3233.4 0.000574 N/A 0.09 3546.4 5.56% 84 330 3548.1 3480.7 0.000574 N/A 0.09 3817.7 7.60% 84 300 3768.8 3600.7 0.000574 N/A 0.09 3949.3 4.79% NOTES: * When M or is larger than 1.0, the maximum value of 1.0 is used to calculate the reduction factor c M which is applied for the ultimate curvature of column, otherwise use the actual values. M is not applicable (N/A) for Case C. Table 4.18. (Continued).

72 This example is for a three-span continuous bridge with two three-column bents where the lateral confinement rein- forcement ratio of each column is rs = 0.3% (detail cate- gory B). Each column’s height is 32.6-ft with a 7-ft diameter. The columns of the bridge system are based on stiff founda- tions that are assumed to be fixed. A cross-section analysis shows that the plastic moment capacity Mp for the cap beam without axial loads is 250,000 kip-in. The ultimate curva- tures for the cap beam and column without axial load are 3.6 × 10-4 in-1 and 7.2 × 10-4 in-1. The ultimate curvature of the middle column subjected to the axial load of 214 kips including dead load and 20% of live load is 5.74 × 10-4 in-1. The pushover analysis shows that the first column reaches its plastic capacity when the lateral force is Pp1 = 5,244.8 kip. The steps necessary to obtain the maximum ultimate capacity of the system and the system factor fs are as follows: 1. Determine the curvature capacity/demand (C/Dcurvature) ratio gjc 3.60 10 7.20 10 0.50 1.0 4 4 c beam column γ = ϕ ϕ = × × = ≤ϕ − − 2. Estimate the ultimate system capacity. According to Equation 4.22, the lateral load capacity of the entire bridge system is 5244.8 1.16 0.24 0.50 5.74 10 3.64 10 1.55 10 3.64 10 6,002 1 4 4 3 4 P P F C kips u p mc c u tunc tconf tunc ( ) = + γ ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ ϕ − − − − The pushover analysis would give Pu = 5,766 kips. The error in Pu observed when Equation 4.22 is used as com- pared to the pushover analysis is 6002 5766 5766 100% 4%ε = − × = 3. Find the system factor. The bridge columns with the weak cap beam in Case A should be evaluated using a system factor equal to exp exp 6,002 5,244.8 0.850.6 0.5 F Cs mc c u tunc tconf tunc u targetφ = + γ ϕ − ϕ ϕ − ϕ     =   = −ξ×∆β ϕ ϕ − × Implementation Example: Weak Cap Beam Case B If the cap beam strength is higher than the plastic moment of the column, but less than the ultimate moment capacity of the column, [Mcol.ultimate > Mbeam.plastic > Mcol.plastic], then gjc for the beam-column connection will depend on the cap beam capacity accounting for both the moment and curvature capacities, which will cause a weak beam-column connection. The column curvature reduction factor gjc is used to reduce ju using the following equations: . . . . M M M M c M beam plastic col plastic col ultimate col plastic beam column γ = γ γ = − − × ϕ ϕϕ ϕ 1P P F Cu p mc c u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    φ ϕ such that gjc ju ≈ effective curvature for beam-column connection. In this example, the same multi-girder three-span bridge with two three-column bents is used where the lateral confine- ment reinforcement ratio is rs = 0.3% (detail category B). The column height is 32.6-ft with a 7-ft diameter. The columns are fixed to the stiff foundation. A cross-section analysis shows that the plastic moment capacity Mp for cap beam without axial loads is 202,000 kip-in. The ultimate curvatures for the cap beam and column without axial load are 9.03 × 10-4 in-1 and 7.2 × 10-4 in-1. The ultimate moment, plastic moment and ultimate curvature of the middle column subjected to the axial load of 214 kips including dead load and 20% of live load is 214,600 kip-in., 198,600 kip-in., and 5.74 × 10-4 in.-1, respec- tively. The pushover analysis shows that the first column reaches its plastic capacity when the lateral force is Pp1 = 5,244.8 kip. The steps necessary to obtain the maximum ultimate capacity of the system and the system factor fs are as follows: 1. Determine the moment capacity/demand (C/Dmoment) ratio gM 202,000 198,600 214,600 198,600 0.21 . . . . M M M M M beam plastic col plastic col ultimate col plastic γ = − − = − − = 2. Determine the curvature capacity/demand (C/Dcurvature) ratio gj 9.03 10 7.20 10 1.25 1.0 therefore use 1.0 4 4 beam column γ = ϕ ϕ = × × = ≥ γ = ϕ − − ϕ 3. Determine the reduction factor gjc 0.21 1.0 0.21c Mγ = γ γ = × =ϕ ϕ

73 4. Estimate the ultimate system capacity. According to Equa- tion 4.24, the lateral load capacity of the entire bridge system is 5,244.8 1.16 0.24 0.21 5.74 10 3.64 10 1.55 10 3.64 10 5,827.1 1 4 4 3 4 P P F C kips u p mc c u tunc tconf tunc ( ) = + γ ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ ϕ − − − − The pushover analysis would give Pu = 5,725.8 kips. The error in Pu observed when Equation 4.24 is used as compared to the pushover analysis is 5827.1 5725.8 5725.8 ε = − × 100% 1.77%= 5. System factor. The bridge columns with the weak cap beam in Case B should be evaluated using a system factor equal to exp exp 5,827.1 5,244.8 0.820.6 0.5 F Cs mc c u tunc tconf tunc u targetφ = + γ ϕ − ϕ ϕ − ϕ     =   = −ξ×∆β ϕ ϕ − × Implementation Example: Weak Cap Beam Case C If the cap beam strength is weaker than the plastic moment of the column [Mbeam.plastic < Mcol.plastic], then gjc is used to reduce column curvature using the following equation: 1P P F Cc beam column u p mc c u tunc tconf tunc γ = ϕ ϕ = + γ ϕ − ϕ ϕ − ϕ    ϕ ϕ ϕ Note: In the linear-elastic analysis used in Case C, Pp1 is the load at which the moment in the cap beam reaches its plastic moment capacity. The same multi-girder three-span bridge with two three- column bents where the lateral confinement reinforcement ratio is rs = 0.3% (detail category B) is used in this example. Each column’s height is 30.0-ft with a 7-ft diameter. The columns are fixed to the stiff foundation. A cross-section analysis shows that the plastic moment capacity Mp for the cap beam without axial loads is 30,000 kip-in. The ultimate curvatures for the cap beam and column without axial load are 6.49 × 10-5 in-1 and 7.2 × 10-4 in-1. The ultimate curvature of the middle column subjected to the axial load of 214 kips including dead load and 20% of live load is 5.74 × 10-4 in-1. The pushover analysis shows that the cap beam reaches its plastic capacity when the lateral force is Pp1 = 3,233.4 kips. The steps necessary to obtain the maximum ultimate capacity of the system and the system factor fs are as follows: 1. Determine the curvature capacity/demand (C/Dcurvature) ratio gjc 6.49 10 7.20 10 0.09 1.0 5 4 c beam column γ = ϕ ϕ = × × = ≤ϕ − − 2. Estimate ultimate system capacity. According to Equa- tion 4.26, the lateral load capacity of the entire bridge system is 3233.4 1.16 0.24 0.09 5.74 10 3.64 10 1.55 10 3.64 10 3,546.4 1 4 4 3 4 P P F C kips u p mc c u tunc tconf tunc ( ) = + γ ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ ϕ − − − − The pushover analysis would give Pu = 3,359.5 kips. The error in Pu observed when Equation 4.26 is used as compared to the pushover analysis is 3546.4 3359.5 3359.5 100% 5.56%ε = − × = 3. System factor. The bridge columns with the weak cap beam in Case C should be evaluated using a system factor equal to exp exp 3546.4 3233.4 0.810.6 0.5 F Cs mc c u tunc tconf tunc u target ( ) φ = + γ ϕ − ϕ ϕ − ϕ     = = −ξ×∆β ϕ ϕ − × Effect of Column Shear Weak shear model: Equation 4.2 and the modified ver- sions in Equations 4.22, 4.24, and 4.26 are derived based on the assumption that the shear strength of a column is suf- ficiently large so that no column shear failure occurs when the bridge system is subjected to incremental lateral loads. However, in inadequately designed bridges or deteriorated bridges, column shear failure may occur prior to flexural yielding or after flexural yielding but before bending failures. Using the same approach followed by the FHWA Seismic Retrofitting Manual (2006), a weakness in the shear capac- ity can be expressed in terms of the ratio of shear capacity over shear demand where in this context shear demand is the shear capacity corresponding to the load at which the column

74 reaches its bending moment capacity. This ratio will be repre- sented by the term C/Dshear. Figure 4.28 describes the model for the shear capacity evaluation process where Vu(d) is defined as the maximum column shear force observed in the column during the push- over analysis assuming that the column will fail in bending moment. As an example, in idealistic elasto-plastic condi- tions, Vu(d) is the shear resulting from plastic hinging at both the top and bottom of the column if both ends are fixed or the maximum shear force in the column when the fixed end reaches its maximum moment capacity if the other end is pinned. The shear demand in idealistic conditions can be cal- culated as Vu(d) = Mu/Lceff where Lceff is the effective column length, which depends on boundary conditions at the two ends of the column. For fixed-fixed columns Lceff = Height of column/2. For pinned-fixed columns Lceff = Height of column. The actual boundary conditions of the columns are usually neither fixed-fixed nor pinned-fixed because the actual stiff- nesses of the cap beam and foundations are not infinitely stiff, which may cause difficulties in determining the effective length. Therefore, it is recommended that Vu(d) be obtained from the pushover analysis. Following the FHWA Seismic Retrofitting Manual (2006) and Figure 4.28, Vi(c) is defined as the initial shear resistance of the undamaged column, which includes the resistance of the gross concrete section and the transverse steel. Also, Vf(c) is defined as the final shear resistance of the damaged column, which considers only the transverse steel that is effectively anchored. The overall shear capacity of the column can then be modeled by a trilinear curve as shown in Figure 4.28. Three possible cases mentioned in the FHWA manual are considered for evaluating the capacity over demand, C/Dshear, ratio for column shear. Case A represents the situation where the initial shear capacity Vi(c) is lower than the shear demand Vu(d). Case B reflects the situation where the shear demand is lower than Vi(c) but higher than Vf(c). Case C is when the shear demand is lower than Vi(c) and Vf(c). The three cases are treated as described next. Case A: If the initial shear resistance of the undamaged column is insufficient to withstand the maximum shear force due to plastic hinging, [Vi(c) < Vu(d)], a brittle shear failure may occur prior to the formation of a plastic hinge. In this case, the ultimate system capacity Pu is estimated by multi- plying Pp1, which is the capacity of the system when one col- umn reaches its moment capacity by C/Dshear ratio, which is defined as :C D V c V dshear i u ( ) ( )= (4.27)1P P V c V d u p i u ( ) ( )= Case B: If the initial shear resistance of the column is suf- ficient to withstand the maximum shear force due to plastic hinging, but the final shear resistance of the column is not, [Vi(c) > Vu(d) > Vf(c)], then gv = C/Dshear ratio for column shear will depend on the amount of flexural yielding, which will cause a degradation in shear capacity from Vi(c) to Vu(d). The shear reduction factor gv is used to reduce ju using the following equations: (4.28) V c V d V c V c V i u i f ( ) ( ) ( ) ( )γ = − − (4.29)1P P F Cu p mc V u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    φ Figure 4.28. Resolution of shear demand and shear capacity.

75 such that gVju ≈ effective curvature for columns weak in shear. Case C: If the final shear resistance of the column is suf- ficient to withstand the maximum shear force due to plastic hinging, [Vf(c) > Vu(d)], then no modification shall be made to Equation 4.2. As described in the FHWA manual, the method proposed for evaluating the effect of column shear failure on bridge system redundancy is based on engineering judgment and assumes an idealized model of shear column behavior. This method may be visualized by examining the assumed rela- tionships between shear capacity and shear demand as shown in Figure 4.28. Case A occurs when the column cannot achieve flexural yielding because of a low initial shear capacity. In this case, the column’s C/Dshear ratio is calculated by dividing the initial shear capacity of the column by the shear demand. Case B will result when a shear failure is expected to occur due to shear capacity degradation resulting from plastic hinging of the column. Case C is assumed when the degradation in col- umn shear capacity is not expected to result in a shear failure. Weak shear model verification: The implementation of Equation 4.27 for Case A for brittle shear failures is straight- forward. For Case C, shear failure will not take place because the shear capacity of the columns is sufficient to withstand the maximum shear force due to plastic hinging. Therefore, Equation 4.2 is valid without any modification. On the other hand, Equations 4.28 and 4.29 need to be validated for col- umns that meet the Case B criteria. Following the FHWA manual’s model of Figure 4.28, the shear capacity of a Case B column starts to decrease when the ductility is larger than 2.0 as the concrete damages during the pushover analysis. As shown in Figure 4.28, the reduc- tion in the shear capacity will not allow the curvature of the column section to reach its ultimate value and failure takes place at a lower ductility. Based on Figure 4.28, a measure of the ductility reduction can be calculated as V c V d V c V c V i u i f ( ) ( ) ( ) ( )γ = − − where Vi(c) = initial shear resistance of the undamaged col- umn including the resistance of the gross concrete section and the transverse steel, Vf(c) = final shear resistance of the damaged column accounting for the transverse steel that is effectively anchored, and Vu(d) = demand shear force. When Vu(d) = Vf(c) the shear reduction factor is gv = 1.0. This means that the shear resistance of the column is suf- ficient to withstand the maximum shear force due to plastic hinging and no reduction in the column’s ductility capacity is observed. When Vu(d) = Vi(c) the shear reduction is gv = 0, which means that the initial shear resistance is only capable of withstanding the linear-elastic loading stage after which point the column fails in shear with very little ductility. In this case, we assume that the column will fail at or close to the initiation of plastic hinging. Based on the logic described above and assuming a lin- ear relationship between the moment and the shear, as well as a linear relationship between the moment and the curva- ture, the shear reduction factor can be approximately used to express the reduction in the curvature capacity of the col- umn using the schematic of Figure 4.29. Therefore, the shear reduction factor gv can be applied in Equation 4.29 to obtain the ultimate capacity of a multi-column bridge system whose column may be weak in shear. In the following parts of this section, the validity of the model is verified by comparing the results obtained using Equation 4.29 and those obtained from SAP2000 for columns with different levels of shear capacity. To demonstrate the validity of the proposed approach for treating bridges with potential weaknesses in shear strength, several bridge systems with different levels of shear capacity are analyzed. The three-span multi-girder steel bridge with three columns per bent is used as the base case for the analy- ses described in this section. Two analyses are performed: the first analysis is performed assuming that the M-phi curve is elasto-plastic in order to find Pp1 and the results are used to find an approximation to the shear demand force Vu(d), which is assumed to be the shear force when one column reaches its plastic moment capacity Mp. The second analysis accounts for the nonlinear M-phi curve as well as the duc- tile shear hinge to find Pu. As shown in Figure 4.28, the shear hinge model requires as input the initial shear capacity Vi(c) and the displacement Dy, which is the displacement at which Mp is reached and that can be taken from the first linear-elastic analysis. The model assumes that the shear force capacity begins to degrade when the displacement is 2Dy. Also, the shear hinge model requires the final shear capacity Vf(c), Figure 4.29. Relationship between insufficient shear capacity, moment, and curvature. p Mp Mu Ultimate curvature of column in bending Reduced curvature due to shear weakness Moment

76 which is assumed to be reached when the displacement is at 5 Dy. A linear interpolation is used to find the shear capacity when the displacement is between 2 Dy and 5 Dy. Following the shear-strength models of Ang et al. (1989) and Wong et al. (1993) that have been used in the FHWA manual, the initial shear force capacity is defined by V c V V (4.30)i si ci( ) = + where Vsi is the shear strength due to the reinforcing steel and Vci is the shear strength due to the concrete. The steel’s con- tribution to the shear capacity is = pi ′ 2 (4.31)V A f D s si sh yh and the concrete shearing strength is 0.37 1 3 (in megapascals)(4.32a)V P f A f Aci c g c e= α + ′     ′ 4.45 1 3 (in pounds per square inch)(4.32b) V P f A f Aci c g c e= α + ′     ′ where the column aspect ratio a is equal to 1.0, and D and D′ = column diameter and core diameter measured to the centerline of the transverse hoop or spiral, which has a cross- sectional area Ash and yield strength fyh. The effective shear area is Ae = 0.8Ag. According to the FHWA manual, the final shearing force Vf(c) is shear capacity of the transverse steel that is effectively anchored represented by Vsi and residual concrete capacity is not considered. To verify the validity of Equation 4.29, 16 bridge models with columns having various diameters ranging from 3-ft to 8-ft and different column heights in the range of 16.7-ft up to 40-ft are investigated. All column sections have a transverse confinement ratio of 0.3%. The results of shear capacities, shear demand, shear reduction factors, first column yielding displacement, plastic capacity Pp1, and system ultimate capacity Pu obtained from the SAP2000 analysis are listed in Table 4.19. The SAP2000 results are compared to Pu predicted by Equa- tion 4.29 showing a maximum difference on the order of 13%. Figure 4.30 plots all the data of the predicted Pu versus the pushover analysis values. The trend line in Figure 4.30 shows a trend line with a slope equal to 1.02 and regression coeffi- cient R2 = 0.99. A perfect model would produce a slope equal to 1.00 and R2 = 1.0. The results in Figure 4.30 demonstrate that the proposed shear reduction factor approach can be used to estimate the reduction in the column’s curvature due to weaknesses in the shear capacity of bridge columns. Implementation Example: Shear Example for Case A—If the initial shear resistance of the undamaged column is insuf- ficient to withstand the maximum shear force due to plastic Table 4.19. Results summary for Case B. Dia. In. Col. Length In. Vu(d) Kips Vi(c)=Vci +Vf Kips Vci kips Vf(c) kips V y In. Pu by SAP kips Pp1 kips u in 1x10 3 Es. Pu kips Error 36 391 101 343 251 92 0.96 3.79 1078 723 1.06 934 13% 36 300 131 343 251 92 0.84 2.28 1125 802 1.06 1017 10% 36 200 193 343 251 92 0.60 1.07 1392 1042 1.06 1265 9% 72 480 436 1305 939 366 0.93 3.57 3152 2314 0.717 2825 10% 72 400 507 1305 939 366 0.85 2.61 3563 2621 0.717 3171 11% 72 300 639 1305 939 366 0.71 1.62 4447 3234 0.717 3847 13% 84 480 783 1768 1270 499 0.78 2.52 4956 4518 0.574 5315 7% 84 440 850 1768 1270 499 0.72 2.15 5299 4858 0.574 5686 7% 84 400 930 1768 1270 499 0.66 1.81 5753 5264 0.574 6122 6% 84 360 1026 1768 1270 499 0.58 1.50 6284 5754 0.574 6641 6% 84 320 1145 1768 1270 499 0.49 1.22 7004 6355 0.574 7266 4% 84 280 1294 1768 1270 499 0.37 0.97 7914 7109 0.574 8031 1% 84 240 1487 1768 1270 499 0.22 0.75 9095 8080 0.574 8985 1% 84 200 1749 1768 1270 499 0.02 0.57 9887 9375 0.574 10201 3% 96 280 1836 2303 1652 651 0.28 0.92 11483 10228 0.516 11413 1% 96 240 2101 2303 1652 651 0.12 0.72 12186 11603 0.516 12753 5%

77 hinging, [Vi(c) < Vu(d)], a brittle shear failure may occur prior to the formation of a plastic hinge and the C/Dshear ratio may be calculated and multiplied by the capacity when one column reaches the plastic moment of the weakest column in the bent, such that 1P P V c V d u p i u ( ) ( )= This example assumes that the bridge has three-column bents where the lateral confinement reinforcement ratio in the columns is rs = 0.3% (detail category B) and the longitu- dinal reinforcement ratio is r = 1.66%. Each column’s height is 16.67-ft with a 7-ft diameter. The columns are based on stiff foundations that are assumed to be fixed and the cap beam is also assumed to be very rigid. The reinforcement is assumed to have a yielding stress Fy = 60 ksi. The unconfined concrete strength is assumed to be 4 ksi. The axial force on a single column is equal to 130 kip. A cross-section analysis shows that the plastic moment capacity Mp is 197,400 kip-in. The pushover analysis shows that the first column reaches its plastic capacity when the lateral force is Pp1 = 6,355 kip. The steps necessary to obtain the maximum ultimate capac- ity of the system and the system factor fs are as follows: 1. Determine the shear capacity contributed by concrete Vci. 4.45 1 3 4.45 1.0 1 3 130,000 4000 84 4 4000 0.8 84 4 1000 1270 2 2 V P f A f A kips ci c g c e ( ) ( ) ( )( ) = α + ′     ′ = + × × pi ×     pi = 2. Determine the shear capacity contributed by reinforce- ment Vsi. For a circular column with hoop reinforcement, the lateral confinement ratio is expressed as 4 4 2 V V D A s D A sD s s c sh shρ = = pi ′ pi ′ = ′ Where, D′ = column core diameter measured to the cen- terline of the transverse hoop, which has a cross-sectional area Ash and yield strength fyh. S is spacing of hoops. The effective shear area is Ae = 0.8Ag. The shear capacity due to the steel is 2 where 4 V A f D s s A D si sh yh sh s = pi ′ = ρ ′ which leads to 2 4 8 3.1416 60 0.003 84 8 499 2 2 V A f D D A f D kips si sh yh s sh yh s ( )( )( ) = pi ′ ρ ′ = pi ρ ′ = = 3. Determine initial shear resistance of the undamaged col- umn Vi(c). 1,270 499 1,769V c V V kipsi ci si( ) = + = + = 4. Determine the maximum column shear force result- ing from plastic hinging Vu(d). For this simple exam- ple, an approximation to the shear demand can be obtained as Vu(d) = Mu/Lceff. For the fixed-fixed column Vu(d) = Mu/Lceff = 2Mu/L = 2 × 197,400/200 = 1,974 kips. 5. Estimate the ultimate system capacity. Since Vi(c) = 1,769 kips < Vu(d) = 1,974 kips, a brittle shear failure will occur prior to the formation of a plastic Figure 4.30. Estimated Pu by Equation 4.29 vs. Pu from SAP2000. y = 1.02x R² = 0.99 0 2000 4000 6000 8000 10000 12000 14000 0 5000 10000 15000 Pr ed ic te d Pu b y Eq .4 .2 9 (k ip s) Pu by Pushover analysis (kips)

78 hinge during the pushover analysis. Therefore, according to Equation 4.27, the lateral load capacity of the entire bridge system is 6,355 1,769 1,974 5,6951P P V c V d kipu p i d ( ) ( )= = × = 6. Find the system factor. This bridge system will fail in brittle shear and is non-redundant. Therefore, the bridge columns should be evaluated using a system factor equal to exp exp 5,695 6355 0.660.6 0.5 F Cs mc u tunc tconf tunc u target ( ) φ = + ϕ − ϕ ϕ − ϕ     = = −ξ×∆β ϕ − × Implementation Example: Shear Example for Case B If the initial shear resistance of the column is sufficient to withstand the maximum shear force due to plastic hing- ing, but the final shear resistance of the column is not, [Vi(c) > Vu(d) > Vf (c)], then the C/Dshear ratio for the column will depend on the extent of flexural yielding, which will cause a degradation in the shear capacity from Vi(c) to Vu(d). The ultimate curvature capacity of the column ju obtained from the ultimate bending capacity is multiplied by the shear reduction factor gv to obtain the actual curvature at failure and the system capacity is calculated from 1P P F Cu p mc v u tunc tconf tunc = + γ ϕ − ϕ ϕ − ϕ    ϕ where effective curvature V c V d V c V c V i u i f V u ( ) ( ) ( ) ( )γ = − − γ ϕ ≈ This example assumes the same multi-girder three-span bridge with two three-column bents where the lateral con- finement reinforcement ratio is rs = 0.3% (detail category B) and the longitudinal reinforcement ratio is r = 1.66%. The column height is 26.7-ft with a 7-ft diameter. The columns are fixed to the stiff foundation and the rigid cap beam. The reinforcement is assumed to have a yielding stress Fy = 60 ksi. The unconfined concrete strength is assumed to be 4 ksi. The axial load is 130 kips and the plastic moment capacity of the column Mp is 197,400 kip-in. The pushover analysis assum- ing elastic behavior indicates that the first column reaches its plastic moment capacity at Pp1 = 6,355 kip. The nonlinear pushover analysis shows that the ultimate lateral load capac- ity is Pu is 7003.8 kip. The steps necessary to obtain the maximum ultimate capac- ity of the system and the system factor fs are as follows: 1. Determine the shear capacity contributed by concrete, Vci. 4.45 1 3 4.45 1.0 1 3 130,000 4000 84 4 4000 0.8 84 4 1000 1270 2 2 V P f A f A kip ci c g c e ( ) ( ) ( )( ) = α + ′     ′ = + × × pi ×     pi = 2. Determine the shear capacity contributed by the rein- forcement Vsi. For a circular column with hoop reinforce- ment, the lateral confinement ratio is expressed as 4 4 2 V V D A s D A sD s s c sh shρ = = pi ′ pi ′ = ′ where, D′ = column core diameter measured to the center- line of the transverse hoop, which has a cross-sectional area Ash and yield strength fyh. S is spacing of hoops. The effective shear area is Ae = 0.8Ag. The steel shear capacity is 2 where 4 V A f D s s A D si sh yh sh s = pi ′ = ρ ′ By combining the two equations, the contribution of the steel reinforcement to the shear capacity is 2 4 8 3.1416 60 0.003 84 8 449 2 2 V A f D D A f D kips si sh yh s sh yh s ( )( )( ) = pi ′ ρ ′ = pi ρ ′ = = The final shear resistance of the damaged column, Vf(c), includes only that transverse steel which is effectively anchored, giving Vf(c) = Vsi = 499 kip. 3. Determine the initial shear resistance of the undamaged column Vi(c). 1,270 499 1,769V c V V kipi ci si( ) = + = + = 4. Determine the maximum column shear force resulting from plastic hinging Vu(d). Assuming elastic behavior, the pushover analysis indi- cates that the first column reaches its plastic moment

79 capacity when the load is Pp = 6355 kip. At that load, the shear force in the column is assumed to be equal to the demand shear force Vu(d), which is equal to 1,145 kip. Another way to estimate this lateral force Vu(d) is to use the equation Vu(d) = Mu/Lceff. For a fixed-fixed column, Vu(d) = Mu/Lceff = 2Mu/L = 2 × 197,400/320 = 1,234 kips. The 8% difference between the two approaches is partially due to the flexibility of the cap beam but also due to the axial forces that are obtained in the col- umn when the pushover analysis is performed as com- pared to the Mu/Lceff approach that ignores the axial forces. 5. Determine the shear reduction factor and find load factor Pu. According to Equation 4.28, the reduction factor for the ultimate curvature is found from 1769 1145 1769 449 0.49 V c V d V c V c V i u i f ( ) ( ) ( ) ( )γ = − − = − − = and the ultimate system capacity for the bridge is 6,355 1.16 0.24 0.49 5.74 10 3.64 10 1.55 10 3.64 10 7,266 1 4 4 3 4 P P F C kip u p mc V u tunc tconf tunc ( ) = + γ ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ − − − − The pushover analysis would give Pu = 7003.8 kip. The error in Pu observed when Equation 4.29 is used as com- pared to the pushover analysis is 7266 7003.8 7003.8 100% 4%ε = − × = Using the more approximate Vu(d) = Mu/Lceff, the shear reduction factor is calculated as 1769 1234 1769 449 0.405 V c V d V c V c V i u i f ( ) ( ) ( ) ( )γ = − − = − − = and the ultimate system capacity is 6,355 1.16 0.24 0.405 5.74 10 3.64 10 1.55 10 3.64 10 7,203 1 4 4 3 4 P P F C kip u p mc V u tunc tconf tunc ( ) = + γ ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ − − − − The error in Pu is 7203 7003.8 7003.8 100% 3%ε = − × = 6. System factor. This bridge system will fail in shear due to shear capac- ity degradation resulting from plastic hinging of the column and is non-redundant. Therefore, the bridge col- umns should be evaluated using a system factor equal to exp exp 7,203 6,355 0.840.6 0.5 F Cs mc V u tunc tconf tunc u targetφ = + γ ϕ − ϕ ϕ − ϕ     =   = −ξ×∆β ϕ − × The system factor is calculated to be, fs = 0.84. Implementation Example: Shear Example for Case C If the final shear resistance of the column is sufficient to withstand the maximum shear force due to plastic hing- ing, [Vf(c) > Vu(d)], then no modification shall be made to Equation 4.2. In this example, the researchers assume that the bridge system and column properties are the same as those given in the previous example for Case B except for the column height, which is taken to be 33.33-ft, and the boundary con- dition, which is assumed to be pinned at the bottom, and the column is assumed to be fixed to the cap beam. The pushover analysis performed assuming elastic behavior shows that the lateral force Pp1 at which the first column reaches its plastic moment capacity is Pp1 = 5,263.8 kip. The nonlinear pushover analysis gives an ultimate capacity Pu = 6,122 kip. Following the same steps outlined in the Case B example, the researchers find Vf(c) = Vsi = 499 kip. Also, for a fixed- pinned column, Vu(d) = Mu/Lceff = Mu/L = 197,400/400 = 493.5 kips < 499 kips = Vf(c). Therefore, no modification shall be made to Equation 4.2, and the ultimate system capacity is 5,263.8 1.16 0.24 5.74 10 3.64 10 1.55 10 3.64 10 6,330 1 4 4 3 4 P P F C kip u p mc u tunc tconf tunc = + ϕ − ϕ ϕ − ϕ     = + × − × × − ×     = ϕ − − − − If comparing results of the pushover analysis, the researchers find that the error in Pu is 6,330 6,122 6,122 100% 3.4%ε = − × =

80 This bridge system will fail in ductile bending and the col- umns should be evaluated using a system factor equal to exp exp 6,330 5,263.8 0.890.6 0.5 F Cs mc u tunc tconf tunc u targetφ = + ϕ − ϕ ϕ − ϕ     =   = −ξ×∆β ϕ − × The system factor is calculated to be, fs = 0.89 which is higher than that needed when the failure is due to shear. 4.5 Conclusions This chapter presents the proposed model for estimating the system capacity of bridge systems subjected to uniform lateral load at the superstructure level and how the model can be used to define system factors that can be applied during the safety evaluation of new and existing bridge sys- tems. A summary of the results of the analyses conducted during the course of the project and those assembled from NCHRP Report 458 to validate the proposed model is given. A method to adjust the model to account for weaknesses in the pier cap beams and weaknesses in the shear capacity of the columns also is described. Several examples of how the equation can be used for particular types of bridges are provided. References Ang, B. G., Priestley, M. J. N., and Paulay, T. (1989) “Seismic Shear Strength of Circular Reinforced Concrete Columns,” ACI Structural Journal 86(1) 45–59. Buckle, I., et al. (2006) Seismic Retrofitting Manual for Highway Struc- tures: Part 1—Bridges (No. FHWA-HRT-06-032). Collins, M. P. and D. Mitchell (1991) Prestressed Concrete Structures. Prentice-Hall, Englewood Cliffs, NJ. Liu, D., et al. (2001) NCHRP Report 458: Redundancy in Highway Bridge Substructures. Transportation Research Board, National Research Council, Washington, D.C. Mander, J. B. and Priestly, M. J. N. (1988) “Observed Stress-Strain Behavior of Confined Concrete,” Journal of Structural Engineering, ASCE, 114(8) Aug, 1827–1849. Wong, Y. L., Paulay, T., and Priestley, M. N. (1993) “Response of Cir- cular Reinforced Concrete Columns to Multi-Directional Seismic Attack,” ACI Structural Journal 90(2) 180–191.