Bridge System Safety and Redundancy (2014)

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31 3.1 Behavior of Bridge Systems under Distributed Lateral Load When a bridge system is subjected to an increasing distrib- uted lateral load, its response is often represented in terms of the maximum lateral displacement at a critical point as a function of the applied lateral load’s intensity. A typical bridge behavior can be represented using the green plot shown in Figure 3.1. This behavior is usually simulated using a pushover analysis whereby a load is applied laterally on the bridge sys- tem and an incremental load analysis is performed taking into consideration the nonlinear behavior of each bridge member. The load is continuously incremented until the bridge fails. The load that causes the bridge to fail is designated as Pu in Fig- ure 3.1. This load defines the maximum load carrying capacity of the system. The displacement at bridge failure is designated as duc, which represents the ultimate displacement capacity of the system. Traditionally, to simplify the analysis process, especially in previous decades when computational tools to perform non- linear incremental pushover analyses were not widely avail- able, bridge engineers used the results of linear-elastic analyses to check the safety of bridge systems subjected to lateral loads. During the bridge design process, bridge members are pro- portioned based on their plastic member capacities even when a linear-elastic analysis is performed to find the load effects. This method, known as the force-based approach, has been the standard approach for checking the safety of bridges under all types of lateral loading conditions. The forced-based approach can be represented by the brown curve in Figure 3.1 where a linear-elastic analysis behavior is assumed until one member reaches its plastic capacity. The load that causes the one member to reach its plastic capacity is designated as Pp1 in Figure 3.1. This load defines the “nominal” load carrying capacity of the system. Because earthquakes impose displacements on structural systems rather than forces, the traditional force-based approach, which is sufficiently valid for other types of loadings, had to be modified in order to take into consideration the effect of mem- ber ductility on the seismic response of bridges. Accordingly, the results of a linear-elastic seismic analysis are adjusted by apply- ing a response modification factor on the linear-elastic response of bridge members to reduce the calculated elastic forces and moments to “equivalent plastic” load effects. Bridge members are designed and detailed to meet these “equivalent plastic” load effects. In principle, the response modification factor would account for the plastic deformation that may be sustained until a theoretical failure point marked by “x” on the brown curve of Figure 3.1 is reached. The ratio of the plastic deformation limit to the linear-elastic limit represents the ductility of the member. The ductility of the system also is assumed to be represented by the maximum response of the most critical member. Although this traditional force-based approach has been the standard method for years, it does not provide an adequate representation of the safety of the entire bridge system. This is because a linear-elastic analysis does not properly model the redistribution of the loads to the members of the bridge as the members undergo nonlinear deformations and because neither the bridge members and certainly nor the system exhibit pure elasto-plastic (bilinear) behaviors. For this reason, in recent years, the seismic assessment of bridge systems has shifted from the force-based approach to a response-based approach. Spe- cifically, the recently implemented AASHTO Guide Specifica- tions for LRFD Seismic Bridge Design (2011) are based on a displacement-based approach whereby the system capacity is defined in terms of the maximum displacement that can be sus- tained by the system before system collapse. This is represented by duc in Figure 3.1. By ensuring that the displacement capacity is higher than the imposed seismic displacement demand, dd, the bridge system will be safe. According to the AASHTO LRFD seismic guide provisions, the ultimate displacement capacity of a bridge system can be directly obtained using a pushover analysis of the entire bridge system. This will directly define the ultimate system C H A P T E R 3 Displacement-Based System Safety and Redundancy of Bridges

32 response capacity duc. Alternatively, for bridges in low seis- mic regions, the AASHTO seismic guide provides equations that give the maximum displacement capacity of bridge col- umns in function of the column height, cross-section size, end constraints, and lateral confinement reinforcement ratio. These equations were presumably derived to provide a con- servative estimate of the maximum displacement that indi- vidual bridge columns can sustain, which also is assumed to be equivalent to the maximum displacement that the system can sustain. The response of an individual column can be modeled as shown by the purple curve in Figure 3.1 and the column response capacity is represented by d1c. 3.2 Redundancy of Bridge Systems under Lateral Load Measures of Redundancy Traditionally, structural design codes define 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 carry- ing capacity. Based on the system behavior explained in Sec- tion 3.1 as described in Figure 3.1, and to remain consistent with the definition of bridge redundancy established for sys- tems 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 follows: For force-based designs: (3.1) 1 R P P fu u p = For displacement-based designs: (3.2) 1 Rdu uc c = δ δ Analysis of Typical Bridge Configurations To obtain estimates of the force-based and displacement- based redundancy of typical bridge system configurations, this study performed the pushover analysis of several bridges and compared their system response capacities, duc, to their individual column response capacities, d1c. Also, the study compared the systems’ ultimate load capacities, Pu, to the first member plastic capacities, Pp1. The analyses were performed on continuous three-span I-girder steel bridges with two bents supported by three columns each, three-span bridges carrying a multi-cell prestressed concrete bridge superstructure where each bent consisted of two columns and three-span bridges carrying two prestressed concrete girder boxes where each bent consisted of two columns. The load transfer mechanism between the I-girder superstructure and the substructure was through bearings on a cap beam, although the possibility of having an integral connection between cap beam, I-girders, and columns was also analyzed. For the box-girder bridges, the response of bridges with integral connections between the Figure 3.1. Representation of typical behavior of bridge systems under distributed lateral load.

33 columns and the superstructure is compared to the effects of pinned connections. For all bridge configurations, the effect of changes in column height and size, lateral confinement and longitudinal reinforcement ratios, reduction in member curvatures due to deficiencies in column/connection detailing and changes in foundation stiffness are investigated. Evaluation of Displacement-Based Redundancy The results of the pushover analyses performed on the entire bridge system are compared to the results performed on one column for each of the bridge configurations and variations on the base case configurations analyzed in this study. The ultimate system response capacity, duc, is compared to the column response capacity, d1c, for the I-girder bridges and the box-girder bridges. The results for the I-girder bridges are presented in Fig- ure 3.2, which plots duc versus d1c for each column detailing and size analyzed. The values of the plotted displacements are provided in the last two columns of 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. Different foundation stiffnesses are used to represent pile foundations and spread footing foundations as well as rigid foundations and pinned foundations. The possibility of integral girder/cap-beam/ column connections is compared to girder-bearings on cap beam designs. Figure 3.2 shows the plot of the displacement capacity of the system versus the displacement obtained when one isolated column is analyzed. The data points are clearly aligned along the equal displacement line indicating that duc is very close to d1c or Rdu = 1.0 for all of the cases analyzed. This demonstrates that the displacement capacity of one column reflects the displacement capacity of the entire bridge struc- tural system very accurately and the redundancy of the system is directly accounted for when using the displacement-based approach for evaluating bridges under lateral loads. These results are expected due to the large stiffness of the deck under the effect of lateral load, which will ensure the compatibility of the displacements of all of the columns at the deck level. The results for the multi-cell box-girder bridge are presented in Figure 3.3. The values of the displacements are provided on the last two columns of Table 4.6. The analysis included 20-ft, 25-ft, and 30-ft columns designed with lateral confinement reinforcement ratios rs = 1%, 0.3% (detail category B) and 0.5% (detail category C) in addition to cases where the maxi- mum curvature 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 foundations as well as rigid foundations. The possibility of integral column/ superstructure connections is compared to the cases where the load is transferred between the superstructure and the 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Sy st em di sp la ce m en t ( in ) One column displacement (in) Orig. Conf. Category C Category B Equalline Figure 3.2. System displacement versus one-column displacement for I-girder bridge.

34 columns through bearing supports. Changes in the abutment bearing stiffnesses also are considered, including the case where the bearings have negligible stiffness. Figure 3.3 shows the plot of the displacement capacity of the system versus the displacement obtained when one isolated column is analyzed. Here again, the data points are clearly aligned along the equal displacement line indicating that duc is very close to d1c or Rdu = 1.0 for all the cases analyzed. This also demonstrates that the displacement capacity of one column reflects the displacement capacity of the entire bridge structural system very accurately and the redundancy of the system is directly accounted for when using the displacement-based approach for evaluating bridges under lateral loads. The results for the two-box-girder bridge are presented in Figure 3.4. The displacements are provided on the last two col- umns of Table 4.5. The analysis included columns designed with lateral confinement reinforcement ratios rs = 1%, 0.3% (detail category 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 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Sy st em di sp la ce m en t ( in ) One column displacement (in) Orig. Conf. Cat. C Cat. B Equalline Figure 3.3. System displacement versus one-column displacement for multi-cell box-girder bridge. 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Sy st em di sp la ce m en t ( in ) One column displacement (in) Cat. B Cat. C Orig. Equalline Figure 3.4. System displacement versus one-column displacement for two-box-girder bridge.

35 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% also are investigated. Here again, the data points are clearly aligned along the equal displacement line indi- cating that duc is very close to d1c or Rdu = 1.0 for all the cases analyzed. This confirms that the displacement capacity of one column reflects the displacement capacity of the entire bridge structural system very accurately and the redundancy of the sys- tem is directly accounted for when using the displacement-based approach for evaluating bridges under lateral loads. These results are due to the large stiffness of the superstructure, which ensures the compatibility of the lateral displacement of the top of the bridge columns of symmetric bridge configurations. Comments on Displacement-Based Redundancy The results shown in Section 3.2 demonstrate very clearly that the displacement capacity of a bridge system is equal to the displacement capacity of its most critical column. This fact has been recognized in the AASHTO Guide Specifications for LRFD Seismic Bridge Design, which provides empirical equa- tions to calculate the displacement capacity of bridge columns with structural detailing categories SDC B and SDC C. The equations are given as : 0.12 1.27 ln 0.32 0.12 (3.3) 1 0 0 0 0 i i iδ∗ = × − Λ  −  ≥ × For category B H H H B c : 0.12 2.32 ln 1.22 0.12 (3.4) 1 0 0 0 0 For category C H H H B c i i iδ∗ = × − Λ  −  ≥ × where d1c is the code-specified displacement capacity of the column in inches, L = 2 for columns fixed at the top and the bottom, L = 1 for fixed-pinned columns, B0 is the width of the column in feet, and H0 is the height in feet. According to the AASHTO LRFD Seismic Design Specifica- tions, column-based Equations 3.3 and 3.4 should be used to determine the seismic displacement capacity of a bridge sys- tem. Tables 3.1 and 3.2 compare the displacement capacities obtained from Equations 3.3 and 3.4 to those obtained from the nonlinear analysis of the systems for the I-girder bridges and the multi-cell box-girder bridges analyzed. The results in Tables 3.1 and 3.2 show the large differences between the displacements obtained from the pushover analy- sis and those of Equation 3.3 and 3.4. The ratio between the two displacements ranges from 1.01 up to 2.75, demonstrating an inconsistent level of conservatism in the AASHTO equations. The sources of the AASHTO empirical equations are not known, and it is clear from the sensitivity analysis performed in this study that they generally provide a safe lower bound estimate of the displacement capacity of bridge columns under lateral load. However, the level of safety was found to be incon- sistent as the AASHTO equations ignore important parameters that affect the displacement capacity of bridge columns. The most notable omission is that of the longitudinal reinforce- ment ratio and the material properties. Therefore, it may be worthwhile to direct future research to developing improved models for evaluating the displacement capacity of bridge System Analysis System Analysis Column Height Load Capacity Displacement Capacity Code Displacement Ratio of Actual Displacement to Code Detail Category B H (inch) Pult. (kip) uc (in) *1c (in) Base Case 331 1077 7.99 5.35 1.49 Height = 32.6 ft 391 1070 10.24 7.15 1.43 Height = 37.6 ft 451 1065 11.99 9.07 1.32 Base Pinned 331 871 11.38 8.27 1.38 Integral Top – Fixed Base 331 1080 7.97 5.35 1.49 Integral Top – Pinned Base 331 839 10.58 8.27 1.28 Pinned Top – Fixed Base 331 870 11.29 8.27 1.37 Detail Category C H Pult. (kip) uc (in) *1c (in) δ δ δ δ Base Case 331 1212 10.44 7.68 1.36 Height = 32.6 ft 391 1219 13.12 10.58 1.24 Height = 37.6 ft 451 1246 15.62 13.70 1.14 Base Pinned 331 970 13.33 13.00 1.03 Integral Top – Fixed Base 331 1214 10.40 7.68 1.35 Integral Top – Pinned Base 331 966 13.16 13.00 1.01 Pinned Top – Fixed Base 331 967 13.15 13.00 1.01 Table 3.1. Comparison of code displacement to analysis results for I-girder bridge.

36 columns that take into consideration the columns’ structural properties in addition to their dimensions. 3.3 Calibration of System Factors for Displacement-Based Approach Procedure The evaluation of the safety of a bridge under lateral load can be expressed in terms of the probability of failure, which for the displacement-based approach can be presented in terms of the probability that the ultimate system displacement capacity, duc, is smaller than the displacement demand dd: Pr 1 (3.5)Pf uc d = δ δ ≤     It is common in probabilistic seismic hazard analysis (Hazus, 2003) to describe both the seismic demand and capacity by log- normal probability distributions. Accordingly, the probability of failure can be expanded as ln ln ln ln 1 1 ln 1 1 (3.6) 2 2 2 2 2 2 2 2 P V V V V f uc d c d uc d c d uc d d c c d    ( ) ( ) [ ]( )( ) = Φ − δ − δ  ξ + ξ       = Φ − δ δ     ξ + ξ       = Φ − δ δ + +     + +       where duc is the median of the displacement capacity duc, __ duc is its mean value, xc is the dispersion of the lognormal distribu- tion of the capacity and Vc is the COV of the capacity. The variables with the subscript “d” are the statistics for the dis- placement demand. F is the cumulative normal distribution function. Using the lognormal model, the reliability index for the system defined as, bsystem, can be calculated as ln ln 1 1 ln 1 1 (3.7) 2 2 2 2 2 2 V V V V system uc d c d uc d d c c d   [ ]( )( )β = δ δ     ξ + ξ = δ δ + +     + + The reliability index for one column defined as, bcolumn, can be calculated as ln ln 1 1 ln 1 1 (3.8) 1 2 2 1 2 2 2 2 V V V V column c d c d c d d c c d   [ ]( )( )β = δ δ     ξ + ξ = δ δ + +     + + 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 2 2c dξ + ξ = 0.60 for all damage types and all bridge members. If the demand on the system is set in terms of the basic seismic input, which could be related to the peak ground acceleration and the overall properties of the entire system such as its natural period and soil conditions, and given that as shown in Figures 3.2, 3.3, and 3.4, the system displacement and the one-column displacement capacities are equal, then all the variables in bsystem and bcolumn have the same values. System Analysis System Analysis Column Height Load Capacity Displacement Capacity Code Displacement Ratio of Actual Displacement to Code Detail Category B H (inch) Pult. (kip) uc (in) *1c (in) Base Case 240 5527 4.49 2.40 1.87 Spring on Top 240 2704 5.40 2.90 1.86 Height =25 ft 300 4584 5.62 3.00 1.87 Height =30 ft 360 4004 6.78 3.60 1.88 Diameter =7ft 240 8287 4.62 2.40 1.93 Diameter = 8ft 240 11925 5.02 2.40 2.09 Detail Category C H (inch) Pult. (kip) uc (in) δ δ δ δ*1c (in) Base Case 240 5865 5.97 2.40 2.49 Spring on Top 240 2942 7.17 3.78 1.90 Height =25 ft 300 4977 7.43 3.00 2.48 Height =30 ft 360 4332 9.00 3.60 2.50 Diameter =7ft 240 8837 6.33 2.40 2.64 Diameter = 8ft 240 12650 6.61 2.40 2.75 Table 3.2. Comparison of code displacement to analysis results for multi-cell box-girder bridge.

37 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 defined by the reliability index margin as (3.9)u system column∆β = β − β Substituting Equations 3.7 and 3.8 into Equation 3.9, the reliability index margin is ln ln ln (3.10) 2 2 1 2 2 1 2 2 u uc d c d c d c d uc c c d       ∆β = δ δ     ξ + ξ − δ δ     ξ + ξ = δ δ   ξ + ξ As demonstrated in Figures 3.2, 3.3, and 3.4, the one column displacement capacity is essentially equal to the system dis- placement capacity such as d1c = duc leading to Dbu = 0. Since the system does not provide any additional reliability compared to that of the most critical member, then bridge systems designed to meet the displacement-based requirements of the AASHTO Guide Specifications for LRFD Seismic Bridge Design are not redundant. Therefore, they must be designed to higher reliability index levels than equivalent redundant systems. As explained in Chapter 2 and in NCHRP Report 406, redun- dant superstructure systems subjected to vertical loads have been defined as those that meet a target system reliability mar- gin Dbu target = 0.85. This target margin was selected to match the average reliability margin of all four-girder bridges assum- ing that the overall COV for the safety margin is approximately equal to 0.25, which reflects the uncertainties in estimating the superstructure capacity and live load. Bridge systems that do not meet this 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 difference between the target reliability margin and the reliability mar- gin that the system provides. In NCHRP Report 458, the target reliability margin was set at Dbu target = 0.50 based on a COV for the safety margin equal to 0.35 reflecting the uncertainty in estimating the substructure capacity and lateral wind load. For example, assume that the same target margin Dbu target = 0.85 set for superstructures under vertical loads also is used for systems subjected to lateral load being evaluated using the displacement-based approach. As shown in this report, the displacement-based approach shows that the provided reli- ability margin is Dbu = 0. Therefore, following the same calibra- tion process outlined in Section 2.5, all bridges designed using the displacement-based approach should include a system factor fs such that ln (3.11) 1 2 2 column c s d c d column u target u   ( )β∗ = δ φ δ     ξ + ξ = β + ∆β − ∆β where bcolumn is the reliability index that has been specified based on the member reliability criteria as shown in Equation 3.5, bcolumn is the reliability index for the column after applying the system factor, and Dbu deficit = (Dbu target - Dbu) is the deficit in the reliability index margin that the new columns’ design should offset. Substituting Equation 3.8 into Equation 3.11 and expanding, provides    ln ln ln (3.12) 1 2 2 2 2 1 2 2 target column c d c d s c d c d c d u u( ) ( )β∗ = δ δ     ξ + ξ − φ ξ + ξ = δ δ     ξ + ξ + ∆β − ∆β (3.13)2 2 target 2 2 targete es c d u u c d uφ = =( ) ( )− ξ +ξ ∆β −∆β − ξ +ξ ∆β Equation 3.13 can be used for any dispersion value and any target reliability index margin. For example, if (Dbu target - Dbu) = (0.85 - 0) = 0.85 and 2 2c dξ + ξ = 0.60 Equation 3.13 can be solved for the system factor fs = 0.60. This means that the calculated capacity should be reduced by a factor of 0.60 to meet the new target reliability level. Alternatively, this indicates that the calculated capacity can only sustain 0.60 times the required demand with a sufficient level of reliability. This system factor fs = 0.60 would be applied to reflect the lack of system redundancy when evaluating the safety of bridges using the displacement-based approach to give an additional reliability index equal to 0.85 on top of the reliability index and safety factors already embedded in the displacement-based design methodologies. Such embedded reliability is included by using the 1,000-year design earthquake, which has about 5% probability of exceedance within 50 years or 7.2% prob- ability of exceedance within 75 years, which is the design life stipulated in the AASHTO LRFD. If no other safety factors are applied, Equation 2.7 shows that a 7.2% probability would lead to a reliability index b = 1.45. In actuality, for detail categories B and C, the use of Equa- tions 3.3 and 3.4 proposed in the AASHTO LRFD seismic provisions include some additional safety. As demonstrated in Tables 3.1 and 3.2, the safety factor seems to be in the range of 1.0 to 2.75 for different column dimensions, end conditions, and confinement detailing for bridges in categories C and B. For detail category D, the AASHTO LRFD seismic provisions recommend that the displacement capacity be evaluated using a pushover analysis that would give very good approximations of the member as well as the system capacity. Since no rigorous evaluation of the safety factors implied in the AASHTO LRFD seismic provisions was performed, the reliability index implied in the AASHTO seismic provisions is not exactly known but is higher than b = 1.45. Applying an additional system factor of 1/0.60 will lead to a lower bound reliability index of b = 2.30.

38 Sensitivity Analysis and Recommendation A sensitivity analysis is presented to study the effect of the dispersion coefficient and the target reliability index margin on the system factor for the displacement-based approach as calculated from Equation 3.13. The results are presented in Table 3.3 and in Figure 3.5, which show that the system factor is sensitive to the target reliability index margin and the dispersion coefficient. Although the target margin for superstructures under vehicular live loads can be established based on configura- tions known to provide sufficient levels of redundancy, it is much more difficult to decide on the appropriate target reli- ability margin that a system subjected to seismic displace- ment demand should be able to achieve. It is suggested that the target reliability index margin should at a minimum be set at Dbu target = 0.50. This recommendation is based on the fact that the 1,000-year design earthquake may be providing a system reliability level on the order of b = 1.45. Providing an additional reliability equal to 0.50 would raise that avail- able system reliability to a value close to 2.0, which is slightly lower than the target reliability set for bridge members being evaluated under vertical load for operating rating. According to Table 3.3, a target reliability margin Dbu target = 0.50 when the dispersion coefficient is equal to 0.60 will require a system fac- tor fs = 0.74 or an additional safety factor = 1.35 (=1/ fs). This value is approximately equal to the overstrength factor tradi- tionally used when detailing bridge columns whose capacity was set based on ultimate bending moment criteria. References AASHTO (2011) Guide Specifications for LRFD Seismic Bridge Design, 2nd ed, Washington, D.C. Buckle, I., et al. (2006) Seismic Retrofitting Manual for Highway Struc- tures, Part 1 Bridges, FHWA-HRT-06-032, Turner-Fairbank Highway Research Center, McLean, VA. HAZUS - MH MR4 (2003) Multi-Hazard Loss Estimation Methodology Earthquake Model, Technical Manual, Department of Homeland Security, Emergency Preparedness and Response Directorate, FEMA Mitigation Division, Washington, D.C. Target Reliability Margin Dispersion Coefficient 22 dc System Factor s 0.30 0.2 0.94 0.3 0.91 0.4 0.89 0.5 0.86 0.6 0.84 0.40 0.2 0.92 0.3 0.89 0.4 0.85 0.5 0.82 0.6 0.79 0.50 0.2 0.90 0.3 0.86 0.4 0.82 0.5 0.78 0.6 0.74 0.60 0.2 0.89 0.3 0.84 0.4 0.79 0.5 0.74 0.6 0.70 0.70 0.2 0.87 0.3 0.81 0.4 0.76 0.5 0.70 0.6 0.66 0.80 0.2 0.85 0.3 0.79 0.4 0.73 0.5 0.67 0.6 0.62 0.90 0.2 0.84 0.3 0.76 0.4 0.70 0.5 0.64 0.6 0.58 1.00 0.2 0.82 0.3 0.74 0.4 0.67 0.5 0.61 0.6 0.55 targetu Table 3.3. Variation of system factor with target margin and dispersion. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.40.2 0.6 0.8 1 1.2 1.4 1.6 sy st em fa ct or , s target u dispersion=0.20 dispersion=0.30 dispersion=0.40 dispersion=0.50 dispersion=0.60 Figure 3.5. System factor for different target reliability and dispersion coefficient.