Development and Calibration of AASHTO LRFD Specifications for Structural Supports for Highway Signs, Luminaires, and Traffic Signals (2014)

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12 C H A P T E R 3 Load Models and Calibration LRFD Limit-State Format The LRFD format is widely used for structural design of buildings, bridges, and other structures. In 1994, the AASHTO LRFD Bridge Design Specifications (LRFD-BDS) was published in its first edition for bridge design and is now in its sixth edition. The limit-state format is: Q R Ri i n r∑γ ≤ ϕ = where: gi = load factors, Qi = load effects, j = resistance factors, Rn = resistance, and Rr = factored resistance. The researchers considered the loads for design that are presented in Table 10. Dead Load Parameters Dead load is the weight of structural and permanently attached nonstructural components. Variation in the dead load, which affects statistical parameters of resistance, is caused by variation of gravity weight of materials (concrete and steel), variation of dimensions (tolerances in design dimensions), and idealization of analytical models. The bias factor (ratio of mean to nominal) value of dead load is l = 1.05, with a coef- ficient of variation (Cov) = 0.10 for cast-in-place elements, and l = 1.03 and Cov = 0.08 for factory-made members. The assumed statistical parameters for dead load are based on the data available in the literature (e.g., Ellingwood et al., 1980; Nowak, 1999). Wind Load Model Note that wind is now an extreme limit state. In ASCE/ SEI 7-10, the wind speeds are increased significantly in the new wind hazard maps. The load factor, however, is decreased from 1.6 to 1.0, which is the same as seismic events in the document. Because a seismic event is consid- ered an extreme event within the LRFD-BDS, within the LRFD-LTS specifications, so is wind. The increases in wind speeds are nominally balanced in most locations of the coun- try with the decreased load factors that result in nominally the same wind pressures. Figure 1 illustrates a typical wind hazard map for the west- ern half of the United States for the most common struc- tures. These winds have a mean recurrence interval (MRI) of 700 years, with a 7% exceedance probability of 50 years. Figure 2 illustrates a typical wind hazard map for the eastern half of the United States. For this level of wind, ASCE assigns an importance level of II; the number of people considered at risk (for buildings) is between two and 200 people (see the small figure inserts along the right side). This level of risk was aligned with LTS structures of a typical nature where they could fall on a roadway. Note that in the Midwest, the wind speed was 90 mph in ASCE/SEI 7-05 and was 90 mph in STD-LTS-6. In this region, it is now 115 mph. Because the wind pressure is proportional to the square of speed, the increase in pressure is (115/90)2 = 1.632, which is close to the value of the wind load factor of 1.6 in ASCE/ SEI 7-05. The wind load factor is now 1.0; therefore, for much of the United States, the wind pressures did not change. How- ever, in coastal regions, the new maps incorporate better data, and the wind maps in some areas have changed. This is auto- matically included in LRFD-LTS specifications as the ASCE/ SEI 7-10 maps are used directly. Table 11 is the load combination table for the LRFD-LTS specifications. The abbreviations provided in Table 10 are used in this table. Interpretation, Appraisal, and Application

13 The Strength I limit state for dead load only (Comb. 1) was calibrated. The Strength I limit state for dead load and live load was considered a minor case and may control only for components that support personnel servicing the traffic devices (Comb. 2). A live load factor based on ASCE/SEI 7-10 was used directly. The Extreme I limit state combines dead loads with wind loads (Comb. 4). This is an important limit state. This combi- nation was a strength limit in the allowable stress design (ASD) LTS specification. The combination is termed “extreme” because ASCE/SEI 7-10 uses new wind speed maps that are associated with a unit load factor. (Note that a unit load factor is also used for seismic events, which are definitely considered extreme events.) Therefore, in the LRFD-LTS specifications, the term “extreme” is used. The Extreme I limit state that combines dead load, wind, and ice (Comb. 5) was studied in detail, and it was determined that it will not be critical in the vast majority of cases, and in Load Abbrev. Description Limit State Dead load components DC Gravity Strength Live load LL Gravity (typically service personnel) Strength Wind W Lateral load Extreme Ice IC Gravity Strength Wind on ice WI Lateral Extreme Truck gust TrG Vibration Fatigue Natural wind gust NWG Vibration Fatigue Vortex-induced vibration VIV Vibration Fatigue Combined wind on high- mast towers HMT Vibration Fatigue Galloping-induced vibration GIV Vibration Fatigue Table 10. LTS loads. Notes: 1. Values are nominal design 3-s gust wind speeds in m/s (mph) at 10 m (33 ft) above ground for Exposure C category, 2. Linear interpolation between wind contours is permitted. 3. Islands and coastal areas outside the last contour shall use the last wind speed contour of the coastal area. 4. Mountainous terrain, gorges, ocean promontories, and special wind regions shall be examined for unusual wind conditions. 5. Wind speeds correspond to approximately a 7% probability of exceedance in 50 years (Annual Exceedance Probability = 0.00143, MRI = 700 Years) Figure 1. Typical ASCE/SEI wind hazard map for the western United States.

14 Category II: 7% probability of exceedance in 50 years (Annual Exceedance Probability = 0.00143, MRI = 700 Years) 90 mph (’05) Figure 2. Typical ASCE/SEI wind hazard map for the eastern United States. Comb. No. Limit State Calibrated? Permanent Transient Fatigue (loads applied separately) DC LL W IC TrG NWG VIV HMT GIV 1 Strength I Yes 1.25 2 Strength I No 1.25 1.6 3 Strength I Yes 1.1/0.9 4 Extreme I Yes 1.1/0.9 1.0 5 Extreme I Studied in detail X X X 6 Service I No 1.0 1.0 7 Service III No 1.0 1.0 8 Fatigue I No, except for HMT 1.0 1.0 1.0 1.0 1.0 9 Fatigue II No 1.0 1.0 1.0 1.0 1.0 1.0 Table 11. Limit states considered in the LRFD-LTS specifications.

15 the few cases where it will be critical, it is close to the dead load combined with wind (Comb. 4). Details are presented in Appendix A. The Service I and III limit states were not calibrated, and the same factors that were used in the previous ASD-based specifications were used. The Fatigue I limit is often critical, depending on the con- nection details. Significant work has been conducted on the fatigue performance of LTS connections (Connor et al., 2012, and Roy et al., 2011). The recommendations of the researchers of those projects were used without further calibration. The Fatigue II limit is for the finite-life approach used to determine remaining service life for an in-service structure. Wind Load Information from ASCE/SEI 7-10 and Available Literature According to ASCE/SEI 7-10, the basic wind speed, V, used in the determination of design wind load on buildings and other structures should be determined from maps included in ASCE/SEI 7-10 (Fig. 26.5-1), depending on the risk category, with exceptions as provided in Section 26.5.2 (special wind regions) and 26.5.3 (estimation of basic speeds from regional climatic data). For Risk Category II, it is required to use the map of wind speed V700 (Fig. 26.5-1A), corresponding to an approximately 7% probability of exceedance in 50 years (annual exceedance probability = 0.00143, MRI = 700 years) (see Figure 1 and Figure 2). For Risk Categories III and IV, it is required to use the map of wind speed V1700 (Fig. 26.5-1B), corresponding to an approxi- mately 3% probability of exceedance in 50 years (annual exceed- ance probability = 0.000588, MRI = 1,700 years) (not shown in this report; reference ASCE/SEI 7-10). For Risk Category I, it is required to use the map of wind speed V300 (Fig. 26.5-1C), corresponding to an approximately 15% probability of exceedance in 50 years (annual exceed- ance probability = 0.00333, MRI = 300 years) (not shown in this report; see ASCE/SEI 7-10). The basic wind speeds in ASCE/SEI 7-10 (Fig. 26.5-1) are based on the 3-s gust wind speed map. The non-hurricane wind speed is based on peak gust data collected at 485 weather stations where at least 5 years of data were available (Peterka, 1992; Peterka and Shahid, 1998). For non-hurricane regions, measured gust data were assembled from a number of stations in state-sized areas to decrease sampling error, and the assem- bled data were fit using a Fisher-Tippett Type I extreme value distribution. The hurricane wind speeds on the United States Gulf and Atlantic coasts are based on the results of a Monte Carlo simulation model described in Vickery and Waldhera (2008) and Vickery et al. (2009a, 2009b, and 2010). Statistical Parameters for Wind Load Variables The wind pressure is computed using the following formula: 0.0256 2i i i i i ( )=P K K G V C psfz z d d where: V = basic wind speed (mph), Kz = height and exposure factor, Kd = directionality factor, G = gust effect factor, and Cd = drag coefficient. The parameters V, Kz, Kd, G, and Cd are random vari- ables, and the distribution function of wind pressure and the wind load statistics are required to determine appropri- ate probability-based load and load combination factors. The cumulative distribution function (CDF) of wind speed is particularly significant because V is squared. However, the uncertainties in the other variables also contribute to the uncertainty in Pz. The CDFs for the random variables used to derive the wind load criteria that appear in ASCE/SEI 7-10 are summarized in Table 12 (Ellingwood, 1981). Statistical Parameters of Resistance Load-carrying capacity is a function of the nominal value of resistance (Rn) and three factors: material factor (m), rep- resenting material properties, fabrication factor ( f), repre- senting the dimensions and geometry, and professional factor (p), representing uncertainty in the analytical model: i i i=R R m f pn Parameter Mean/Nominal Cov CDF Exposure factor, Kz 1.0 0.16 Normal Gust factor, G 1.0 0.11 Normal Pressure coefficient, Cp 1.0 0.12 Normal Table 12. Wind load statistics (Ellingwood, 1981).

16 The statistical parameters for m, f, and p were considered by various researchers, and the results were summarized by Ellingwood et al. (1980) based on material test data available in the 1970s. The actual strength in the structure can differ from structure to structure, but these differences are included in the fabrication and professional bias factors (lf and lp). Material parameters for steel were established based on the yield strength data. The typical parameters are listed in Table 13 to Table 16. The resistance (load-carrying capacity) is formulated for each of the considered limit states and structural components: Bending resistance, elastic state: M f Sy i= Bending resistance, plastic state: i=M f Zy Shear resistance: 0.57i i=V A fshear y Torsion capacity: 0.5 0.57 i i i=T J d fy Axial capacity: i=P A fy Generally, the limit state that controls the design of luminaires is calculated using an interaction equation for load combination that produces torsion, shear, flexure, and axial force [Sec- tion C-H3-8, AISC Steel Construction Manual (AISC, 2010)]. 1.0 2 +  + +  ≤ P P M M V V T T r c r c r c r c where: P = axial force, M = bending moment, V = shear, and T = torsion. The terms with the subscript r represent the required strength (load effect), and those with the subscript c represent the cor- responding available strengths (load-carrying capacity). Parameters λ Cov Static yield strength, flanges 1.05 0.10 Static yield strength, webs 1.10 0.11 Young’s modulus 1.00 0.06 Static yield strength in shear 1.11 0.10 Tensile strength of steel 1.10 0.11 Dimensions, f 1.00 0.05 Table 13. Statistical parameters for material and dimensions (Ellingwood et al., 1980). Limit State Professional Material Fabrication Resistance Cov Cov Cov Cov Tension member, yield 1.00 0 1.05 0.10 1.00 0.05 1.05 0.11 Tension member, ultimate 1.00 0 1.10 0.10 1.00 0.05 1.10 0.11 Elastic beam, LTB 1.03 0.09 1.00 0.06 1.00 0.05 1.03 0.12 Inelastic beam, LTB 1.06 0.09 1.05 0.10 1.00 0.05 1.11 0.14 Plate girders in flexure 1.03 0.05 1.05 0.10 1.00 0.05 1.08 0.12 Plate girders in shear 1.03 0.11 1.11 0.10 1.00 0.05 1.14 0.16 Beam – columns 1.02 0.10 1.05 0.10 1.00 0.05 1.07 0.15 Note: LTB = lateral-torsional buckling. Table 14. Resistance statistics for hot-rolled steel elements (Ellingwood et al., 1980). Limit State Resistance Cov Tension member 1.10 0.11 Braced beams in flexure, flange stiffened 1.17 0.17 Braced beams in flexure, flange unstiffened 1.60 0.28 Laterally unbraced beams 1.15 0.17 Columns, flexural buckling, elastic 0.97 0.09 Columns, flexural buckling, inelastic, compact 1.20 0.13 Columns, flexural buckling, inelastic, stiffened 1.07 0.20 Columns, flexural buckling, inelastic, unstiffened 1.68 0.26 Columns, flexural buckling, inelastic, cold work 1.21 0.14 Columns, torsional–flexural buckling, elastic 1.11 0.13 Columns, torsional–flexural buckling, inelastic 1.32 0.18 Table 15. Resistance statistics for cold-formed steel members (Ellingwood et al., 1980).

17 The limit-state function can be written: , 1.0 1 1 2 2 3 3 4 4 2 ( ) = − +  − + g Q R Q R Q R Q R Q R i i The interaction equation is a nonlinear function; therefore, to calculate combined load-carrying capacity, Monte Carlo simulation was used for each random variable. This proce- dure allows for finding function g and calculating reliability index b. For calibration purposes, using a first-order second- moment approach, the resistance parameters were assumed to have a bias factor of 1.05 and a coefficient of variation of 10%. The details are provided in Appendix A. LRFD Reliability Analysis The calibration between ASD and LRFD is based on the cali- bration of ASCE/SEI 7-05 50-year V50 wind speed and ASCE/ SEI 7-10 700-year V700 wind speed. The ASCE/SEI 7-10 wind speed maps for a 700-year wind are calibrated to the ASCE/ SEI 7-05 50-year wind speed, where the difference between LRFD design wind load factors (ASCE/SEI 7-05 gW = 1.6 vs. ASCE/SEI 7-10 gW = 1.0) is (V700/V50)2 = 1.6. Thus, the LRFD ASCE/SEI 7-05 V50 wind speed is equivalent (for pressures that are proportional to V2) to the ASCE/SEI 7-10 V700 wind speed. Likewise, the ASCE/SEI 7-10 V300 and V1700 winds speeds are equivalent to ASCE/SEI 7-05 V50 wind speeds adjusted for importance [low Ilow = 0.87 = (V300/V700)2 and high Ihigh = 1.15 = (V1700/V700)2]. The ASCE/SEI 7-05 V50 wind speed is used as the mean wind speed (adjusted for design map values compared to statistical means) for the reliability analyses. Flexural Resistance The flexural resistance is discussed here, and other actions and combinations of actions are provided in detail in Appendix A. The LRFD design requirement for a structure at the opti- mal design limit is: max 2 1 φ = γ γ + γ  R M M M n D D D D W W where: Rn = nominal resistance, MD = nominal dead load (DL) moment, MW = nominal wind load (WL) moment, gD1 = dead load design load factor (used in conjunction with dead + wind case), gD2 = deal load design load factor (dead load only case), gW = wind load design load factor, and f = resistance factor. To meet the design limit, the nominal resistance is: max 1 1 2 1[ ] = φ γ φ γ + γ      R M M M n D D D D W W The mean resistance is: = λR RR n where: lR = bias factor for strength variable R, and R _ = statistical mean of variable R. At the optimal design limit, the mean of R becomes: max 2 1[ ] = λ φ γ λ φ γ + γ      R M M M R D D R D D W W The coefficient of variation for the strength is CovR. Load The total applied nominal moment at the ASCE/SEI 7-10 700-year wind speed is: 7001M M MT D= + where: MT1 = total nominal moment at ASCE/SEI 7-10 700-year wind speed, MD = dead load moment, and M700 = nominal moment from wind at ASCE/SEI 7-10 700-year wind speed. Limit State Resistance Cov Tension member, limit-state yield 1.10 0.08 Tension member, limit-state ultimate 1.10 0.08 Beams, limit-state yield 1.10 0.08 Beams, limit-state lateral buckling 1.03 0.13 Beams, limit-state inelastic local buckling 1.00 0.09 Columns, limit-state yield 1.10 0.08 Columns, limit-state local buckling 1.00 0.09 Table 16. Resistance statistics for aluminum structures (Ellingwood et al., 1980).

18 To standardize the comparisons between ASD and LRFD, and for any specified-year wind, all analyses and comparisons are based on the total nominal moment for the LRFD 700-year total applied moment equal to 1.0: 1.07001M M MT D= + = and, the dead load moment can be represented by: 1 700M MD = − The calibration and comparisons vary the M700 wind load effect from 1.0 to 0.0, while MD varies from 0.0 to 1.0 so that the total applied nominal moment at the ASCE/SEI 7-10 700-year load remains 1.0. The total applied nominal moments for ASD and other LRFD year wind speeds is adjusted to be equivalent to the ASCE/SEI 7-10 700-year wind speed load case. Given that the nominal moment from wind for any year wind can be determined by: 700 2 700M V V MWT T =   where: VT = wind speed for any year T wind speed, and MWT = nominal wind moment at any year T. and the total applied nominal moment becomes: 1 700 700 2 7002M M M M V V MT D WT T( )= + = − +   where: MT2 = total applied nominal moment at any year T wind speed. To determine the mean wind moment for the reliability analyses, the mean moment at the 50-year wind speed is determined from the ASCE/SEI 7-10 wind speed relation: 0.36 0.10ln 12 50V T VT [ ]( )= + or: )(= + = λ0.36 0.10ln50V V T V T V T where: lV = bias factor for wind speed at year T, and the nominal wind moment at the 50-year wind speed becomes: 50 2 700M MV= λ The nominal moment at the 50-year wind speed is propor- tional to V2 by: 50 50 2 ∝M K K GC Vd z d where: Kd = directionality coefficient, Kz = elevation coefficient, G = gust effect factor, and Cd = drag coefficient. The mean wind moment for the reliability analyses is: 50 50 2 ∝M K K GC Vd z d where the variables are the means. Assuming that Kd does not vary, the other non–wind-speed variables’ nominal values are related to the means by the bias factors. Combining them into a single bias factor lP gives: = λ λ λ = λK GC K GC K GCz d K G C z d p z dz d where: p K G Cz dλ = λ λ λ and: Kd does not vary. Considering that the map design values may differ from the statistical mean of the 50-year wind speed, the mean 50-year wind speed can be represented by: 50 50 700V V VX X V= λ = λ λ where: 50 50V Xλ = µ = bias for the 50-year wind speed, m50 = mean 50-year wind speed, and V50 = map design 50-year wind speed. The mean wind moment for the reliability analyses becomes: 50 2 2 700M MP V X= λ λ λ where: 700 700 2 ∝M K K GC Vd z d Referring back to the basis that all comparisons are equated with a total ASCE/SEI 7-10 applied nominal moment of: 1700M MD + =

19 and using that the nominal dead load moment and mean dead load moment are: 1 1 700 700 M M M M D D D ( ) = − = λ − where: lD = bias factor for dead load moment, The mean load effect on the structure becomes: 150 700 2 2 700Q M M M MD D P V X( )= + = λ − + λ λ λ where Q = the mean moment. To find the coefficient of variation for Q, first the coeffi- cient of variation for the mean wind moment is determined from: 2 2 2 2 250Cov Cov Cov Cov CovM V K G Cz d( )= + + + noting that V in the V2 term is 100% correlated, and the coef- ficient of V2 (CovV2) is two times the coefficient of variation of V (CovV). Combining the statistical properties for the dead and wind moments to determine the coefficient of variation for the total mean moment Q results in 1 700 2 2 2 700 2 50Cov Q Cov M Cov M Q Q Q D D M P V X[ ] [ ]( ) = σ = λ − + λ λ λ Reliability Indices Q and R are assumed to be lognormal and independent: ln ln 1 ln ln 1 ln 1 2 ln 2 ln 2 2 ln 1 2 ln 2 ln 2 2 R Cov Q Cov R R R R Q Q Q Q( ) ( ) µ = − σ σ = + µ = − σ σ = + where s is the standard deviation of the variable indicated. The reliability index b is: ln ln ln ln 2 ln 2 1 2 ln 2 ln 2 ln 2 ln 2 R QR Q R Q R Q R Q ( ) β = µ − µ σ + σ =   − σ + σ σ + σ Implementation The LRFD reliability analysis was coded into a spreadsheet to study four regions in the United States: • Florida Coastal Region, • Midwest and Western Region, • Western Coastal Region, and • Southern Alaska Region. Inputs for LRFD reliability analyses spreadsheet: , , per ASCE/SEI 7-10 design wind speeds , , per ASCE/SEI 7-05 design wind speeds 300 700 1700 50 50 50 V V V V Vµ µ Regional wind statistics are provided in Table 17. Global inputs (same for all regions) are: , , Cov , Cov , , , Cov , Cov , Cov , , ,1 2 λ λ λ λ λ φ γ γ γ D R D R Kz G Cd Kz G Cd D D W Table 18 provides global inputs (inputs are highlighted). V50 50 COV 50 V300 V700 V1700 Florida Coastal 150 130 0.14 170 180 200 Midwest &West 90 75 0.1 105 115 120 West Coast 85 67 0.095 100 110 115 Southern Alaska 130 110 0.105 150 160 165 Table 17. Regional wind statistics. COV BIAS BIASD 1.03 COVkz 0.16 1.00 COVD 0.08 COVG 0.11 1.00 BIASR 1.05 COVCd 0.12 1.00 COVR 0.10 Total BiasP 1.00 D+W DOnly 0.90 D 1.10 1.25 W 1.00 Table 18. Input to reliability calibration (all regions).

20 The results for the Midwest and Western Region ASCE/ SEI 7-10 700-year wind speed are shown in Table 19 (other regions are similar). For the 300-year wind speed, the results are shown in Table 20. Notice that the total nominal moment, MT2, is less than 1.0 because the wind moment, M300, is less than M700. Likewise, for the 1,700-year wind speed, MT2 is larger than 1.0 since M1700 is greater than M700, as shown in Table 21. Using the 300-year wind speed requires less nominal resis- tance; conversely, using the 1,700-year wind speed increases the required nominal resistance. Because the mean load Q and its variation do not change, this difference in required nominal resistance changes the reliability indices b accordingly. ASD Reliability Analysis Because the LRFD reliability analyses are based on the total nominal moment MD + M700 = 1.0, the ASD analyses must adjust the moments for a consistent comparison. 700 YearWind V700 115 T 700 V50 91.00991 Theory BIASX 0.8241 V700/V700 1.00 (V700/V700) 2 COVV 0.100 (V300/V700) 2 1.00 1.00 Equiv BIASV 0.79 M700 MT2 M700/MT2 Rn R lnR Q COVM50 COVQ lnQ LRFD 1.00 1.00 1.00 1.11 1.17 0.10 0.43 0.30 0.30 0.30 3.35 0.90 1.00 0.90 1.12 1.18 0.10 0.49 0.30 0.24 0.24 3.54 0.80 1.00 0.80 1.13 1.19 0.10 0.55 0.30 0.19 0.19 3.69 0.70 1.00 0.70 1.14 1.20 0.10 0.61 0.30 0.15 0.15 3.77 0.60 1.00 0.60 1.16 1.21 0.10 0.67 0.30 0.13 0.13 3.75 0.50 1.00 0.50 1.17 1.23 0.10 0.73 0.30 0.11 0.10 3.60 0.40 1.00 0.40 1.18 1.24 0.10 0.79 0.30 0.09 0.09 3.34 0.30 1.00 0.30 1.19 1.25 0.10 0.85 0.30 0.08 0.08 2.98 0.20 1.00 0.20 1.20 1.26 0.10 0.91 0.30 0.08 0.08 2.57 0.10 1.00 0.10 1.25 1.31 0.10 0.97 0.30 0.08 0.08 2.38 0.00 1.00 0.00 1.39 1.46 0.10 1.03 0.30 0.08 0.08 2.71 Table 19. Reliability indices for Midwest and Western Region (MRI  700 yrs). 300 Year Wind V300 105 T 300 Theory V300/V700 0.91 (V300/V700) 2 (V300/V700) 2 0.83 0.87 Equiv M300 MT2 M300/MT2 Rn R lnR Q COVM50 COVQ lnQ LRFD 0.83 0.83 1.00 0.93 0.97 0.10 0.43 0.30 0.30 0.30 2.77 0.75 0.85 0.88 0.96 1.00 0.10 0.49 0.30 0.24 0.24 2.92 0.67 0.87 0.77 0.99 1.03 0.10 0.55 0.30 0.19 0.19 3.04 0.58 0.88 0.66 1.02 1.07 0.10 0.61 0.30 0.15 0.15 3.11 0.50 0.90 0.56 1.04 1.10 0.10 0.67 0.30 0.13 0.13 3.12 0.42 0.92 0.45 1.07 1.13 0.10 0.73 0.30 0.11 0.10 3.03 0.33 0.93 0.36 1.10 1.16 0.10 0.79 0.30 0.09 0.09 2.86 0.25 0.95 0.26 1.13 1.19 0.10 0.85 0.30 0.08 0.08 2.61 0.17 0.97 0.17 1.16 1.22 0.10 0.91 0.30 0.08 0.08 2.32 0.08 0.98 0.08 1.25 1.31 0.10 0.97 0.30 0.08 0.08 2.38 0.00 1.00 0.00 1.39 1.46 0.10 1.03 0.30 0.08 0.08 2.71 Table 20. Reliability indices for Midwest and Western Region (MRI  300 yrs).

21 Using the ASCE/SEI 7-05 criteria for the ASD design, the wind moment for a 50-year wind speed is: 50 700 50 700 2 Design M M Design V V =   Considering that the Design V50 may differ from V50 = (lV)2V700, a bias factor, lDesign, is introduced, and: 50 2 50 700 2 700 2 2 700 50 50 Design M V V M M Design V V Design Design V Design = λ   = λ λ λ = The total ASD design moment, MT3, consistent with MD + M700 = 1.0, becomes: 150 700 2 2 7003M M Design M M MT D Design V( )= + = − + λ λ Resistance The LRFD nominal resistance is assumed to be the plastic moment capacity. To directly compare resistances between LRFD and ASD sections, the nominal resistance for the ASD design is increased by the section shape factor for a compact section: R SF Mn y= where SF is the shape factor. The allowable stress for a compact section using the allowed overstress factor (OSF) of 4/3 for wind loads is: 4 3 0.66 0.66F F OSF Fallow y y( ) ( )( )= = Using moments instead of stresses, the allowable moment is OSF (0.66) My, and the design requirement for an optimal design is: 0.66 50OSF M M Design M Iy D( )( ) = + where: I = Ilow = 0.87 (low importance) comparable to ASCE/SEI 7-10 300-year wind speed, I = Imed = 1.00 (medium importance) comparable to ASCE/ SEI 7-10 700-year wind speed, and I = Ihigh = 1.15 (high importance) comparable to ASCE/SEI 7-10 1,700-year wind speed. The nominal resistance (to directly compare to the LRFD design) is determined by increasing the design strength by the shape factor as: 1 0.66 1 700 2 2 700R SF M SF OSF M M In y Design V[ ]( )= = − + λ λ For the ASD reliability analyses, the statistical properties are: R RR n= λ 1700 YearWind V1700 120 T 1700 Theory V1700/V700 1.04 (V1700/V700) 2 (V1700/V700) 2 1.09 1.15 Equiv M1700 MT2 M1700/MT2 Rn R lnR Q COVM50 COVQ lnQ LRFD 1.09 1.09 1.00 1.21 1.27 0.10 0.43 0.30 0.30 0.30 3.62 0.98 1.08 0.91 1.21 1.27 0.10 0.49 0.30 0.24 0.24 3.84 0.87 1.07 0.81 1.21 1.27 0.10 0.55 0.30 0.19 0.19 4.01 0.76 1.06 0.72 1.21 1.27 0.10 0.61 0.30 0.15 0.15 4.09 0.65 1.05 0.62 1.21 1.28 0.10 0.67 0.30 0.13 0.13 4.06 0.54 1.04 0.52 1.22 1.28 0.10 0.73 0.30 0.11 0.10 3.89 0.44 1.04 0.42 1.22 1.28 0.10 0.79 0.30 0.09 0.09 3.58 0.33 1.03 0.32 1.22 1.28 0.10 0.85 0.30 0.08 0.08 3.17 0.22 1.02 0.21 1.22 1.28 0.10 0.91 0.30 0.08 0.08 2.69 0.11 1.01 0.11 1.25 1.31 0.10 0.97 0.30 0.08 0.08 2.38 0.00 1.00 0.00 1.39 1.46 0.10 1.03 0.30 0.08 0.08 2.71 Table 21. Reliability indices for Midwest and Western Region (MRI  1,700 yrs).

22 and: , Cov , and are unchanged.lnQ Q Qσ The coefficient of variation for the strength (resistance) is CovR. The equations for determining the reliability indices are identical to those used for the LRFD cases. Implementation For the four regions, the ASD reliability analyses require additional inputs. Inputs for ASD are: • Importance factors Ilow = 0.87, Imed = 1.00, and Ihigh = 1.15; • Shape factor SF = Zx/Sx = 1.30 for a circular section; and • Wind overstress factor OSF = 4/3 = 1.333. The results for the Midwest and Western Region ASCE/SEI 7-05, medium importance Imed = 1.00 are shown in Table 22. The LRFD required nominal strength is shown for direct comparison. For the Midwest and Western Region for a low importance Ilow = 0.87, the results are shown in Table 23. Table 24 provides results for a high importance Ihigh = 1.15. LRFD ASD Strength Total Design Ratio Equiv Moment RnLRFD RnLRFD M50 MT3 M50/MT3 MD+M50I RnASD RnASD ASD 1.11 0.61 0.61 1.00 0.61 0.90 1.23 2.69 1.12 0.55 0.65 0.85 0.65 0.96 1.17 2.94 1.13 0.49 0.69 0.71 0.69 1.02 1.11 3.20 1.14 0.43 0.73 0.59 0.73 1.08 1.06 3.44 1.16 0.37 0.77 0.48 0.77 1.13 1.02 3.63 1.17 0.31 0.81 0.38 0.81 1.19 0.98 3.74 1.18 0.24 0.84 0.29 0.84 1.25 0.94 3.77 1.19 0.18 0.88 0.21 0.88 1.31 0.91 3.71 1.20 0.12 0.92 0.13 0.92 1.36 0.88 3.57 1.25 0.06 0.96 0.06 0.96 1.42 0.88 3.39 1.39 0.00 1.00 0.00 1.00 1.48 0.94 3.19 Table 22. Midwest and Western Region reliability indices (I  1.00). BiasDes= 0.988903 LRFD ASD Strength Total Design Ratio Equiv Moment RnLRFD RnLRFD M50 MT3 M50/MT3 MD+M50I RnASD RnASD ASD 0.93 0.61 0.61 1.00 0.53 0.79 1.18 2.25 0.96 0.55 0.65 0.85 0.58 0.86 1.12 2.49 0.99 0.49 0.69 0.71 0.63 0.93 1.07 2.75 1.02 0.43 0.73 0.59 0.67 0.99 1.02 3.00 1.04 0.37 0.77 0.48 0.72 1.06 0.98 3.23 1.07 0.31 0.81 0.38 0.77 1.13 0.95 3.39 1.10 0.24 0.84 0.29 0.81 1.20 0.92 3.48 1.13 0.18 0.88 0.21 0.86 1.27 0.89 3.49 1.16 0.12 0.92 0.13 0.91 1.34 0.87 3.43 1.25 0.06 0.96 0.06 0.95 1.41 0.89 3.33 1.39 0.00 1.00 0.00 1.00 1.48 0.94 3.19 Table 23. Midwest and Western Region reliability indices (I  0.87).

23 Notice that the total nominal moment, MT3, does not change, but the total design moment MD + M50I changes with the importance factor, resulting in different required nomi- nal strength Rn. Similarly, for high importance, the required nominal strength Rn increases as shown in the following for the Midwest and Western Region. The importance factors directly change the required nomi- nal resistances. Because the mean load Q and its variation do not change (not shown in these tables but the same as in the LRFD tables), this difference in required nominal resistances changes the reliability indices b accordingly. Calibration and Comparison Using the proposed flexure load and resistance factors, and with the statistical properties incorporated into the reliability analyses, the plots in Table 25 compare the reliability indices for the four regions between current ASD design procedures and the proposed LRFD procedures. The Minimum Beta plots represent the minimum indices over the four regions. Similarly, the Average Beta plots show the averages over the four regions. For the LRFD 300-year, 700-year, and 1,700-year wind speed cases, the equivalent ASD designs use Ilow = 0.87, Imed = 1.00, and Ihigh = 1.15 importance factors, respectively. The proposed LRFD procedures result in comparable but more consistent reliability over the range of designs. For low-importance structures (using 300-year wind speeds), the reliability indices are lower, as intended. Likewise, for higher-importance structures (1,700-year wind speeds), the reliability indices are higher. This is shown in Figure 3 for the LRFD procedures. The ratios are the averages over the four regions. At low wind moments (gD2MD controls the design), there is no difference. However, for higher wind moments, the required strength increases for high-importance structures and decreases for lower-importance structures. As expected, the LRFD-required strength at a higher per- centage of wind load (MWind/MTotal high) is greater than that required for ASD. This behavior is demonstrated in Figure 4, where the ratios are the average for the four regions. At a total moment where the wind is responsible for approx- imately 60% or more of the total, the proposed LRFD-LTS procedures will require more section capacity than the current ASD procedures. Below 60%, the LRFD-LTS procedures will require less section capacity than ASD. Implementation Setting Target Reliability Indices The statistical characterization of the limit-state equation and the associated inputs are presented in the preceding sec- tions. The reliability indices are computed based on the current ASD practice and the LRFD-LTS specifications. Comparisons made and presented previously are based on the recommended load and resistance factors. These factors are illustrated for the 700-year wind speeds (MRI = 700 yrs). This MRI is for the typical structure; however, some consideration is warranted for structures that are located on routes with low average daily traffic (ADT) or that are located away from the travelway, whereby failure is unlikely to be a safety issue. Similarly, con- sideration is also warranted for structures located on heavily traveled roads, where a failure has a significant chance of harm- ing travelers or suddenly stopping traffic, possibly creating a LRFD ASD Strength Total Design Ratio Equiv Moment RnLRFD RnLRFD M50 MT3 M50/MT3 MD+M50I RnASD RnASD ASD 1.21 0.61 0.61 1.00 0.70 1.04 1.16 3.14 1.21 0.55 0.65 0.85 0.73 1.08 1.12 3.41 1.21 0.49 0.69 0.71 0.76 1.13 1.07 3.67 1.21 0.43 0.73 0.59 0.79 1.17 1.04 3.90 1.21 0.37 0.77 0.48 0.82 1.22 1.00 4.06 1.22 0.31 0.81 0.38 0.85 1.26 0.97 4.13 1.22 0.24 0.84 0.29 0.88 1.30 0.93 4.09 1.22 0.18 0.88 0.21 0.91 1.35 0.91 3.94 1.22 0.12 0.92 0.13 0.94 1.39 0.88 3.73 1.25 0.06 0.96 0.06 0.97 1.43 0.87 3.47 1.39 0.00 1.00 0.00 1.00 1.48 0.94 3.19 Table 24. Midwest and Western Region reliability indices (I  1.15).

24 0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal Average Beta 700 Year LRFD ASD0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal Minimum Beta 700 Year LRFD ASD 0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal MinimumBeta 300 Year LRFD ASD 0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal Average Beta 300 Year LRFD ASD 0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal Average Beta 1700 Year LRFD ASD 0.00 1.00 2.00 3.00 4.00 00.511.5 Be ta MWind/MTotal MinimumBeta 1700 Year LRFD ASD Table 25. Minimum and average reliability indices (all regions). Figure 3. Resistance ratios for different return periods.

25 situation conducive to a traffic collision with the structure or a chain-reaction impact of vehicles. Ultimately, judgment is used to set the target reliability indices for the different applications. The target reliability index (b) is often based on typical average performance under the previous design specifications (i.e., ASD). However, even in the ASD methods, an importance factor was considered: 0.87 and 1.15 for less important and more important applica- tions, respectively. Some variations were also considered for hurricane versus non-hurricane regions. There were similar concerns for the LRFD-LTS specifi- cations’ assignment of the MRI considered for design. Less important structures are assigned an MRI of 300 years, while an important structure uses an MRI of 1,700 years. Typical structures are assigned an MRI of 700 years. The description of this implementation is provided next with the resulting reliability indices for each region. Implementation into Specifications The possible structure locations were divided into two primary categories: 1. Failures where a structure is likely to cross the travelway and, within those structures, those that are located on a typical travelway versus a lifeline travelway, which are those that are critical for emergency use/egress; and 2. Failures where a structure cannot cross the travelway and that, consequently, are of lesser importance. Within these categories, the ADT is used to further distin- guish the consequence of failure. The traffic speed was initially considered in the research but was not used in the final report based on simplicity and judgment. Table 26 summarizes this approach. From this design approach, Table 26 establishes the MRI and directs the user to the appropriate wind hazard map, which provides the design wind speed. Computed Reliability Indices Based on the load and resistance statistical characteristics, the reliability indices b are computed for the four regions for a wind-to-total-load ratio of 0.5 and 1.0. The 0.5 ratio is typi- cal of a traffic signal pole, and the 1.0 ratio is typical of a high- mast pole. Other ratios were computed; however, for brevity, only these two are illustrated. Table 27 illustrates the relationship between Table 26 and the computed values. For example, assume that a structure is located on a travelway with ADT of between 1,000 and 10,000, and a failure could result in a structure crossing the roadway. From Table 26, the MRI is 700. The statistical properties for the 700-year wind in the region of interest (Midwest and Western in this case) are then used to compute b. The computed value of b = 3.60 for WL/(DL + WL) = 0.5 is shown in Table 27. Similarly, b = 3.35 for the WL/(DL + WL) = 1.0. Figure 4. Resistance ratios LRFD versus ASD. Risk Category Typical High Low Traffic Volume <35 N/A N/A ADT < 100 300 1,700 300 100 < ADT 1,000 700 1,700 300 1,000 < ADT 10,000 700 1,700 300 ADT > 10,000 1,700 1,700 300 Typical: Support failure could cross travelway. High: Support failure could stop a lifeline travelway. Low: Support failure could not cross travelway. Roadside sign supports: use 10-year MRI. Table 26. MRI related to structure location and consequence of failure.

26 Other indices were computed for load ratios in each region. The results are illustrated in Appendix A. Note that for the same region and location, the load ratio of 0.5 has a higher b than does the ratio of 1.0. This is because the wind-dominated structure will experience a higher load vari- ability (all wind) than one that is 50% dead load. Comparing the same application (cell) across regions, the region with the lower wind variability will have a higher b. The resulting indices are reasonable for the various appli- cations, and the load and resistance factors were accordingly set. The load factors are summarized in Table 11. Sensitivities The previous discussion outlined the results of assign- ment of load and resistance factors and the resulting reliabil- ity indices. It is useful to illustrate the sensitivities of these assignments to the resulting reliability indices. The minimum and average values for all regions are used as a demonstration by varying the dead load, wind load, and resistance factors for steel flexure strength and extreme limit states (see Table 28). Note that an increase in resistance factor f decreases the reliability index b. An increase in load factor g increases b. (Midwest and West)Load Ratio [WL/(DL+WL) = 0.5] Importance Traffic Volume Typical High Low ADT<100 3.03 3.89 3.03 100<ADT≤1000 3.60 3.89 3.03 1000<ADT≤ 10000 3.60 3.89 3.03 ADT>10000 3.89 3.89 3.03 Typical: Failure could cross travelway High: Support failure could stop a life line travelway Low: Support failure could not cross travelway Roadway sign supports: use 300 years (Low) (Midwest and West)Load Ratio [WL/(DL+WL) = 1.0] Importance Traffic Volume Typical High Low ADT<100 2.77 3.62 2.77 100<ADT≤100 0 3.35 3.62 2.77 1000<ADT≤ 10000 3.35 3.62 2.77 ADT>10000 3.62 3.62 2.77 Typical: Failure could cross travelway High: Support failure could stop a life line travelway Low: Support failure could not cross travelway Roadway sign supports: use 300 years (Low) Importance Typical High Low Traffic Volume <35 n/a n/a ADT<100 300 1700 300 100<ADT1000 700 1700 300 1000<ADT10000 700 1700 300 ADT>10000 1700 1700 300 Typical: Failure could cross travelway High: Support failure could stop a life line travelway Low: Support failure could not cross travelway Roadway sign supports: use 300 years Table 27. Relationship between MRI and computed reliability indices (Midwest and Western Region load ratio  0.5 and 1.0). Typical traffic signal structures have load ratios in the region of one-half, while the high-mast poles have very little dead load effect, and ratios are nearer to unity. In Table 28, the area contained within the dotted line indicates the region that is of typical interest. Scope of Appendix A Appendix A outlines the complex calibration process and includes more detail than the brief description in the main body of the report. Appendix A includes: • Wind statistic quantification, • Resistance quantifications, • Calibration for different actions and interactions, and • Monte Carlo simulations. Calibration Summary Judgment must be employed in the calibration regard- ing the performance of existing structures under the current specifications and setting the target reliability index b for the LRFD-LTS specifications.

27 Table 28. Sensitivity of the reliability index to load and resistance factors. Parameters Minimum β Average β Resistance Ratio Baseline = 0.9 dead-only = 1.25 dead = 1.1 wind = 1.0 = 0.9 dead-only = 1.35 dead = 1.1 wind = 1.0 = 0.9 dead-only = 1.25 dead = 1.2 wind = 1.0 = 0.95 dead-only = 1.25 dead = 1.1 wind = 1.0 = 1.0 dead-only = 1.25 dead = 1.1 wind = 1.0 = 0.85 dead-only = 1.25 dead = 1.1 wind = 1.0

28 The LRFD-LTS specifications were calibrated using the standard ASD-based specifications as a baseline. The variabili- ties of the loads and resistances were considered in a rigorous manner. The wind loads have higher variabilities than the dead loads. Therefore, a structure with a high wind-to-total-load ratio will require higher resistance and associated resistances compared to ASD. This increase is on the order of 10% for high-mast structures. For structures with a wind-to-total-load ratio of approximately 0.5 to 0.6 (e.g., cantilever structures), the required resistance will not change significantly. It is important to note that resistance is proportional to sec- tion thickness and proportional to the square of the diameter [i.e., a 10% resistance increase may be associated with a 10% increase in thickness (area) or a 5% increase in diameter or area]. The reliability index for the LRFD-LTS specifications is more uniform over the range of load ratios of practical interest than are the current ASD-based specifications. Examples Table 29 illustrates 15 example designs that were used to demonstrate the LRFD-LTS specifications. These problems were solved with the support of Mathcad (Version 15), a pop- ular computer utility program for engineering computations, and are available as a PDF in Appendix C. Figure 5 provides a typical first page as an example. The problem description is followed by a table of contents show- ing that section of the report. Example No. Title Graphic 1 Cantilevered Overhead Sign Support – Truss with Post 2 Traffic Sign Support Structure 3 High Mast 4 Monotube Overhead Traffic Signal and Sign Support Bridge Table 29. Example designs.

29 (continued on next page) Example No. Title Graphic 7 Street Lighting Pole with 10-ft Mast Arm 8 Steel Roadside Sign Support 9 Cantilevered Monotube Support for a Dynamic Message Sign 5 Overhead Truss Span- Type Support (Steel) 6 Span Wire and Poles Table 29. (Continued).

30 Example No. Title Graphic 12 Prestressed Concrete Light Pole 13 Luminaire Support – Fiber-Reinforced Polymer (FRP) 14 Street Light Pole – Timber 15 Road Sign – Timber 10 Aluminum Pole Design 11 Span Wire with Prestressed Concrete Poles Table 29. (Continued).

31 Figure 5. Typical problem statement.