Updating the AASHTO LRFD Movable Highway Bridge Design Specifications (2021)

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29 C H A P T E R 3 Update of the AASHTO LRFD MHBDS and Development of RBM Development of Reliability-Based Methodology Auburn University took the lead in developing an RBM that is consistent with approach used the AASHTO LRFD for Highway Bridge Design Specifications for fixed bridges. For the MHBDS, structural design primarily references back to the fixed bridge design specification with a few exceptions to address load conditions that are unique to movable bridges. From work done in Phase 1 of the project, it was determined that further development and application of RBM was only practical for mechanical design. Furthermore, from data collected from owners, it was found that the overall outcomes that have been historically realized by movable bridge owners were acceptable and similar outcomes were desired with any updates and modifications. This allowed the Auburn team, led by Andrzej (Andy) Nowak, to execute a meaningful calibration of load and resistance factors. The complete Auburn report including design examples is included in Appendix C. Following are important portions of the report that provide details on development of the RBM, Reliability Indices, Recommended Factors, and a summary of results. Overview The objective is to develop and incorporate a consistent RBM in the Mechanical Design of the AASHTO LRFD MHBDS. Firstly, the limit states that are in the existing MHBDS are summarized, and the design format of the mechanical design in existing MHBDS is shown. Afterward, service limit state, overload limit state, and fatigue limit state are considered for LRFD calibration purposes for operating machinery. An overview of the calibration procedure and recommended load and resistance factors are developed. The selection of the target reliability index (βT) and check of the β in the current MHBDS is presented. Lastly, design examples, based on current MHBDS and LRFD format of Mechanical design, are summarized. Limit States in Mechanical Design The limit states in mechanical design must be clearly defined. A Limit State distinguishes what is acceptable and what is not on the basis of the design criterion with a certain safety margin. Service, Overload and the Fatigue Limit States are defined in this chapter. Following are abbreviations that are utilized and are consistent with the existing MHBDS (see MHBDS Article 5.2 “Definitions” for more detail): • FLT: Full Load Torque • ST: Starting Torque

30 • BDT: Breakdown Torque • AT: Acceleration Torque • PT: Peak Torque • E-Stop: Emergency Stop of Bridge (cuts all power and sets all brakes) Service Limit State The Service Limit State imposes boundaries on stresses and deformations in mechanical components. Elastic deformations in machinery components should be limited. Loading effects on operating mechanical components come from the prime mover or braking. The load is defined as torque produced by the prime mover or braking under regular service conditions. The resistance of the mechanical components, in this case operating machinery, is based on the yield strength of material. The yield criterion is used to define the behavior of material. Resistance is represented by the maximum shear criterion. Loading needs to be less than the factored resistance of the mechanical component. The load is limited in case of: • Electric motor – 1.5 Full Load Torque produced by the motor (150% FLT of the electric motor, MHBDS Ref: 5.7.1). • Hydraulic motor or cylinder – Normal working pressure – adjustable value (relief valve setting of the pump or circuit, MHBDS Ref: 7.4.2). • Dynamic Braking – During Normal Operation - from motor brakes or machinery brakes, regenerative braking, or hydraulic braking (MHBDS Ref: 5.7.3 and 7.4.2). Definition: Service Limit State is exceeded when the load caused by the prime mover or braking exceeds loads under normal service specified by the code to limit elastic deformations. Mechanical Design in the Current MHBDS Service and Overload Limit State The design formula in the current MHBDS is as follows: 𝑄 = 𝜑 ∙ 1𝑛 ∙ 𝑅 (1) where: 𝑅 - nominal resistance 𝑄 - nominal load 𝜑 – resistance factor 𝑛 – safety factor In the current specification, the load factor is not defined, it is implicitly assumed to be equal to 1.0.

31 Resistance factors listed in MHBDS section 6.4.1.1 are shown in Figure 24. Figure 24. Resistance Factors (current MHBDS 6.4.1.1) Safety factors listed in MHBDS section 6.6.1 are shown in Figure 25. Safety Factors (current MHBDS 6.6.1) 5. Figure 25. Safety Factors (current MHBDS 6.6.1) The design formula in the current specification for Service Limit State for cast steel is specified in Eq. 3.2. On the load side there is no load factor, only nominal load is taken into account. On the resistance side, the safety factor and resistance factor reduce the resistance to 25%. 𝑄 ≤ 14 ∙ 𝑅 (2) where: 𝑅 - nominal resistance 𝑄 - nominal load under overload case

32 The design formula in the current specification for Service Limit State for forged, drawn, rolled and wrought steel is specified in Eq. 3.3. On the load side, there is no load factor, the only nominal load is considered. On the resistance side, the safety factor and resistance factor reduce the resistance to 33%. 𝑄 ≤ 13 ∙ 𝑅 (3) The design formula in the current specification for Overload Limit State for cast steel and for forged, drawn, rolled and wrought steel is specified in Eq. 3.4. On the load side there is no load factor, only nominal load is taken into account. On the resistance side, the safety factor and resistance factor reduce the resistance to 75%. 𝑄 ≤ 34 ∙ 𝑅 (4) Fatigue Limit State – Shaft (infinite life) Design for infinite fatigue life of a shaft in the current MHBDS is as follows: 32𝜋𝑑 𝐾 𝑀𝜎 + √3𝐾 𝑇2𝜎 ≤ 0.80 (5) where: 𝑑 – diameter 𝜎 – endurance limit 𝜎 – Yield strength of steel 𝑀 – Moment 𝑇 – Torque 𝐾 – Fatigue stress concentration factor (bending) 𝐾 – Fatigue stress concentration factor (bending) The endurance limit, 𝜎 in Eq.3.4 is defined as: 𝜎 = 𝛼𝜎 (𝐶 𝐶 𝐶 𝐶 𝐶 ) (6) where: 𝛼 - material property parameter 𝜎 – specified ultimate tensile strength 𝐶 – size factor 𝐶 – surface factor 𝐶 – reliability factor 𝐶 – temperature factor 𝐶 – miscellaneous factor The above parameters in the interaction equation are discussed in Mechanical Engineering Design by Shigley.

33 Overload Limit State The Overload Limit State imposes boundaries on stresses from high magnitude loading. This limit state is exceeded less frequently than the Service Limit State. It should be considered to preserve mechanical components from localized yielding that may permanently deform machinery components and their supports while allowing higher stresses for load cases of limited frequency. Loading effects on operating mechanical components come from the prime mover or braking. The load is defined as loading produced by the prime mover or braking under overload condition specified in MHBDS. The resistance of the mechanical components is based on the yield strength of the material. The yield criterion is used to define the behavior of material. Resistance is represented by the maximum shear stress criterion. Loading needs to be less than the factored resistance of the mechanical component. Load cases for Overload Limit State are as follows: • Case 1a: 1.5 ST or 1.5 BDT for uncontrolled AC motors (MHBDS Ref: 5.7.1) • Case 1b: 1.0 ST or 1.5 AT for controlled AC motors (MHBDS Ref: 5.7.1) • Case 1c: 3.0 FLT for controlled DC motors (MHBDS Ref: 5.7.1) • Case 1d: Hydraulic Pressure at Maximum Working Pressure (the larger of system pressure relief or overriding pressure relief valve setting on the pump) • Case 1e: 1.0 PT at Full Throttle for Internal Combustion Engine (MHBDS Ref: 5.7.1) • Case 2a: Uncontrolled dynamic braking (E-Stop) with a modified resistance factor of 1.5 (AASHTO MHBDS: 5.7.3) • Case 2b: Static (Holding) Braking Combined Motor and Machinery Brake Torque (MHBDS Ref: 5.7.3) Definition: Overload Limit State is exceeded when the loading caused by the prime mover or braking exceeds high and extreme loading specified by the code to avoid permanent deformations. In Overload Limit State, normal operating loads are exceeded. Loadings covered under this limit state include uncontrolled stopping or actuation of the movable span. Loadings include abrupt uncontrolled stopping of the span such as can occur with a loss of power; an emergency stop or a limit switch failure. The frequency of these loads is such that they will not impair serviceability. Fatigue Limit State The fatigue limit state applies restrictions on stress range. Mechanical components subjected to cyclical loading need to be checked. For more than one million cycles, the design is based on infinite life (endurance limit as defined in the specifications). Extreme Event Limit State The Extreme Event Limit State assures structural survival of a bridge during a major earthquake, flood, or collision by a vessel, vehicle, or ice flow, possibly under scoured conditions. For movable bridge components, this also ensures survivability with possible plastic deformation but avoids catastrophic failure. The operability of the movable bridge may be compromised.

34 Reliability-based Calibration Procedure In the new generation of design specifications, safety is provided in terms of load and resistance factors determined in the reliability-based calibration procedure. The code calibration requires the knowledge of statistical parameters of load and resistance. Reliability can be measured in terms of the reliability index. The load and resistance factors are selected so that the designed structures or machinery will have at least the minimum acceptable reliability, i.e., β will be at least equal to the target reliability index, βT. The target reliability index depends on the consequences of failure and relative cost (cost of a unit of safety). The code calibration procedure [3 and 4] can be summarized as follows: 1. Formulate the limit state function and identify variables – For Service, Overload and the Fatigue Limit States, definitions and acceptability criteria were established. 2. Identify and select representative machinery types and design cases. 3. Determine load and resistance parameters for the selected design cases. 4. Gather statistical parameters for load and resistance – The information from existing literature and engineering judgement was used to develop statistical parameters. 5. Develop a reliability analysis procedure - Reliability can be calculated using either a closed-form formula or simulation techniques like the Monte Carlo method. The reliability index for each case can be calculated using closed formulas available for particular types of probability distribution functions in the literature. In this report, the reliability calculations were based on a closed-form solution. 6. Calculate the reliability index for the current design code – The reliability index β for the limit states in the current specification is calculated. 7. Review the results and select the target reliability index – Based on the calculated reliability index in the current MHBDS and experience from the current engineering practice, the 𝛽 , was selected. 8. Select potential load and resistance factors – The optimum values of load and resistance factors that correspond to the “design point” were selected. 9. Calculate the reliability index for selected load and resistance factors – The reliability index corresponding to potential load and resistance factors is calculated for verification. Limit State Function The general form of a limit state function can be presented as follows: 𝑔 = 𝑅 − 𝑄 = 0 (3.7) where: 𝑅 – resistance (random variable) Q – load (random variable) and the probability of failure is: 𝑃 = 𝑃𝑟𝑜𝑏(𝑔 < 0) (3.8)

35 Limit State functions considered in the mechanical design are as follow: Service Limit State – Electrical Motor It is assumed that the Service Limit State is exceeded when the torque caused by the motor exceeds a selected torque, which is 1.50 FLT in the current specifications. Therefore, in the limit state function (Eq. 3.7), R is the maximum allowed torque for Service Limit State and Q represents applied torque. Overload Limit State – Electrical Motor It is assumed that the Overload Limit State is exceeded when the torque caused by the motor exceeds a selected torque related to FLT, ST, BDT, or AT depending on the type of the prime mover and governing case. Therefore, in the limit state function (Eq. 3.7), R is the maximum allowed torque for Overload Limit State and Q represents applied torque. Service Limit State – Hydraulic Motor It is assumed that the Service Limit State is exceeded when the torque caused by the motor exceeds an acceptable torque. Therefore, in the limit state function (Eq. 3.7), R is the maximum allowed torque for Service Limit State and Q represents normal working pressure. Overload Limit State – Hydraulic Motor It is assumed that the Overload Limit State is exceeded when the torque caused by the motor exceeds an acceptable torque. Therefore, in the limit state function (Eq. 3.7), R is the maximum allowed torque for Overload Limit State and Q represents maximum working pressure. Fatigue Limit State – Shaft (Infinite Life) It is assumed that the fatigue limit state of a shaft (infinite life) is exceeded when the reversed bending moment due to rotation in combination with a steady torsional moment exceeds 0.80. Therefore, fatigue check is based on the interaction equation where Q is represented by moment and torque and R is based on material fatigue strength. Statistical Parameters of Load and Resistance The statistical parameters needed for the reliability analysis include: λ - bias factor, which is the ratio of the mean maximum value and the nominal value (Eq. 3.9) λ = 𝜇𝑄 (3.9) V- coefficient of variation, which is the ratio of standard deviation and mean maximum value (Eq. 3.10) 𝑉 = 𝜎𝜇 (3.10) Electrical Motor – Service Limit State: In the limit state function, Eq. 3.7, where R represents the maximum allowable torque, the resistance is assumed to be 1.50 FLT by using reverse engineering. In the current specification, the load has to be less

36 than or equal to the factored resistance. In this case, it is assumed that the nominal value of R is 1.50 FLT and the coefficient of variation is 0.15 The value of the coefficient of variation is selected based on experience, the failure matrix shown in Table 6 and engineering judgement. Table 6. Failure Matrix – Service Limit State Movable Bridge System Component Redundancy Average Time Inoperable (days) Average Time to Correct (days) Traffic Impacts Average Cost Average Repair / Replacement Interval (years) SERVICE LIMIT STATE Operating Machinery Drive Shaft Yes or No 5 90 vehicular or marine traffic $ 20,000 25 Support Machinery Trunnion Shaft No 15 180 vehicular or marine traffic $ 200,000 50 Locking Devices Span Lock Bar No 2 120 vehicular traffic delays $ 30,000 15 Note: Average Cost values were estimated by RT members in 2018 dollars. . The failure matrix for Service Limit State shown in Table 6 indicates the acceptable average time to correct (repair or replace) for: operating machinery, support machinery and locking devices. Having all that information based on the current practice, it is decided what is an acceptable percentage of the bridges that will exceed the limit state before the average time to repair. In other words what variation is tolerable? In this case, the following question needs to be answered: What percentage of movable bridge systems can fail before assumed acceptable average time to repair? From experts’ opinions, 10% of bridges exceeding the limit state is acceptable. As an example, let’s consider that 10% of operating machinery can fail or need repair after 20 years of operation. The period of 20 years is an assumed number. Considering this example, where average time to repair is 25 years, and 10% of the operating machinery can fail after 20 years of operation. From the reliability point of view, having an acceptable average value and probability of exceedance, the standard deviation and coefficient of variation can be found. It is assumed that the distribution is normal and then the COV of load is equal to 15%. Based on observations and expert opinions, an electric motor produces 1.25 FLT as the average maximum loading during normal operation. Therefore, the mean maximum value of load 𝑄 is assumed as 1.25 FLT with the coefficient of variation of 0.15. Furthermore, strain gage data collected on a variety of movable bridges (typically for Strain Gage Balancing) has been assembled and is used in the analysis. Those measurements can be used to justify statistical information regarding the performance of the movable bridge components and the whole structure under ideal operating conditions (typically with no snow, rain, or ice and wind speeds less than 5 mph). The received data recorded by strain gages to the drive shaft component for opening and closing torque in terms of % of FLT were recorded for various movable bridges in the US. The data is presented in Table 7.

37 Table 7. Opening and Closing Torques for Sample of Movable Bridges in US ALL Values are maximums and are %FLT Opening Closing Bridge Name State Bridge Type Motor HP Prime Mover TS-O TA-O TCV-O TS-C TA-C TCV-C Isabel Holmes NC B-TR 150 AC-C 70.3 44.9 34.4 45.7 23.5 20.7 Jax Bridge FL B-ST 450 AC-C 79 61.4 39.1 46.5 11 27 Lapalco LA B-TR 60 160.5 140 47 90 47.7 20.8 North Draw LA B-RL 100 79 65.8 53 31.9 30.6 48.2 Casco Bay ME B-TR 150 AC-C 79.6 41.2 31.5 66.8 28.8 8.5 Metropolitan NY B-TR 30 AC-C 73.4 53.7 63.4 20.2 38.1 34.2 Charles Berry OH B-TR 300 AC-C 34 20.6 16.7 41.4 16.8 17.7 Seabrook LA B-ST 150 AC-U 137.6 208.4 88.8 108.9 131.4 71.2 Charlevoix MI B-RL 30 63.4 45.2 25.4 81.9 42.8 36.3 Clarksville TN SW-CB 22.4 IC 220.4 131.5 82.1 135.9 112.8 81.7 Macombs Dam NY SW-RB 60 AC-C 213.5 96.7 77 18.5 95 85.3 Memorial NC VL-TD 100 AC-C 131.6 114.4 86 42.6 73.4 53.3 Route 1 & 9 NJ VL-TD 200 47.2 56.6 49.5 43.5 45 33.5 Route 44 NJ VL-SD 30 156.7 83.6 77.6 100.7 50.6 19 West 3rd Street OH VL-SD 125 58.8 49.2 33.9 10 21.9 20.7 Willow Ave OH VL-TD 37.5 84.6 65.5 35 18.3 35.2 42.3 Columbus Rd OH VL-SD 125 18.8 18 9.4 8 8.4 4.5 Houghton MI VL-TD 150 119.9 58.2 29.7 106.7 50.3 19.9 Florida Ave LA VL-TD 100 AC-C 103.5 65.3 45.8 80.7 59 17.8 MOPAC LA VL-SD 40 AC-C 40.5 15.8 63.5 92.5 112.3 108.3 An additional explanation of the terminology used in Table 7 is presented below. The bridge type description can be defined as: B-TR = Bascule-Trunnion B-RL = Bascule-Rolling Lift B-ST = Bascule-Strauss B-DU = Bascule-Dutch SW-CB = Swing-Cent Bearing SW-RB = Swing-Rim Bearing SW-BT = Swing-Bobtail VL-TD = Vert Lift-Tower Dr VL-SD = Vert Lift-Span Dr The prime movers can be defined as: AC-U = AC Induction-Uncontrolled AC-C = AC Induction-Controlled DC-C = Direct Current – Controlled HYDR = Hydraulic IC = Internal Combustion Engine MAN = Manual Operation

38 Torque values indicated in Table 7 can be specified as: TS = Starting Torque TA = Accelerating or Decelerating Torque from Prime Mover (not mechanical brakes) TCV = Constant Velocity Torque Distribution of starting torque for opening and closing is shown on Figure 2424. Figure 26. Starting Torque for Opening and Closing (data from two bridges were excluded, as they were considered “outliers”) As noted previously, the values in Table 7 represent actual data taken under ideal conditions. As part of this project, the team has analyzed comprehensive design loadings for dozens of movable bridges. From those analyses, average baseline loadings for wind snow and ice loadings that should be considered in addition to Table 7 values are: • “Missing Loads” from Table 7 Values - Bascule Bridges: Wind 33% FLT, Ice 33% FLT - Swing Bridges: Wind 40% FLT, Ice 0% FLT (negligible) - Vertical Lift Bridges: Wind 35% FLT, Ice 35% FLT Even if the “Missing Loads” are added to Table 7 values, the existing bridges strain gage torque data indicates that for most cases, assumption for the mean maximum value of load 1.25 FLT is conservative and representative for all types of movable bridges. The nominal value of the load was calculated from the design formula Eq. 3.2 and Eq. 3.3 taking into account that nominal resistance is 1.50 FLT. Based on observations and expert opinions, an electric motor produces 1.25 FLT as the average maximum during normal operation. Therefore, the mean maximum value of the load is assumed as 1.25 FLT with the coefficient of variation of 0.15.

39 The nominal load 𝑄 for the cast, steel is equal to 0.38 FLT and for forged, drawn, rolled, wrought steel it is 0.50 FLT. So, the bias factors are: λ , = 𝜇𝑄 = 1.25 𝐹𝐿𝑇0.38 𝐹𝐿𝑇 = 3.33 (3.11) λ , , , = 𝜇𝑄 = 1.25 𝐹𝐿𝑇0.50 𝐹𝐿𝑇 = 2.50 (3.12) The statistical parameters of load and resistance for Service Limit State for Electric Motor are summarized as follows: λ , = 3.33 λ , , , = 2.50 V = 0.15 λ = 1.00 V = 0.15 Electrical Motor – Overload Limit State: It is assumed that the Overload Limit State is exceeded when the torque caused by the motor exceeds a selected torque that is related to: FLT, ST, BDT, AT, depending on the type of the prime mover and governing case. Therefore, in the limit state function (Eq.3.7), R is the maximum allowed torque for Overload Limit State and Q represents applied torque. For different prime movers, statistical parameters for Overload Limit State can be established. The nominal value of load is calculated from Eq. 3.4, taking into consideration safety factors for Overload Limit State. The mean maximum value of load (μ) for various types of prime movers is presented in 8. Table 8. Overload Limit State – Machinery Design for Prime Movers Prime Mover Overload Limit State Rn Qn μ λ AC (Uncontrolled) greater of 1.5ST or 1.5BDT 1.5ST or 1.5BDT 1.50 greater of 1.5ST or 1.5BDT 1.00 AC (Controlled) greater of 1.0ST or 1.5AT 1.0ST or 1.5AT 1.00 greater of 1.0ST or 1.5AT 1.00 DC (Controlled) 3.0 FLT 3.0 FLT 3.00 3.0 FLT 1.00 I.C. Engines 1.0PT at Full Throttle 1.0 PT 1.00 1.0 PT 1.00 The coefficient of variation of 0.10 is selected, and it is less than the COV for Service Limit State due to the fact that the frequency of occurrence Overload Limit State is very rare. Statistical parameters of load and resistance for Overload Limit State in mechanical design are summarized as follow: λ = 1.00 V = 0.10 λ = 1.00

40 V = 0.10 Hydraulic Motor – Service Limit State: The nominal value of the load was calculated from design formula Eq. 3.2 and Eq. 3.3 considering that nominal resistance is 1.0 Effective Pressure Torque. Based on observations and expert opinions, a hydraulic motor produces 0.80·Effective Pressure Torque as the average maximum during normal operation. Therefore, the mean maximum value of the load is assumed as 0.80·Effective Pressure Torque with the coefficient of variation of 0.15. The nominal load 𝑄 for cast steel is equal to 0.25 Effective Pressure Torque and for forged, drawn, rolled, wrought steel it is 0.33 Effective Pressure Torque. So, the bias factors are: λ , = 𝜇𝑄 = 0.80 0.25 = 3.20 (3.13) λ , , , = 𝜇𝑄 = 0.80 0.33 = 2.40 (3.14) The statistical parameters of load and resistance for Service Limit State in Hydraulic Design are summarized as follows: λ , = 3.20 λ , , , = 2.40 V = 0.15 λ = 1.00 V = 0.15 Hydraulic Motor – Overload Limit State: The nominal value of the load was calculated from design formula Eq. 3.2 and Eq. 3.3 considering that nominal resistance is 1.0 Effective Pressure Torque. Based on observations and expert opinions a hydraulic motor produces 1.00·Effective Pressure Torque as the average maximum for an overload event. Therefore, the mean maximum value of load Q is assumed as 1.00·Effective Pressure Torque with the coefficient of variation of 0.10. The nominal value of applied torque is calculated from Eq.3.4 substituting values of nominal resistance of 1.40 Effective Pressure Torque, nominal load 𝑄 is 1.05 Effective Pressure Torque. So, the bias factor is: λ = 𝜇𝑄 = 1.00 · 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑇𝑜𝑟𝑞𝑢𝑒1.05 · 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑇𝑜𝑟𝑞𝑢𝑒 = 0.95 (3.15) The statistical parameters of load and resistance for Overload Limit State in Hydraulic Design are summarized as follows: λ = 0.95 V = 0.10 λ = 1.00 V = 0.10 The summary of mean and nominal values of load are shown in Table 9 followed by statistical parameters in shown in Table 10.

41 Table 9. Summary of Mean and Nominal Values of Load Limit State Forge Steel Cast Steel Rn in terms of Forged Steel Cast Steel Mean Maximum load Rn Rn Qn Qn μ Electric Motor Design Service 1.5 1.5 1.5 FLT 0.50 0.38 1.25 Overload - - - - - - AC (Uncontrolled) 1.5 1.5 1.5 ST or 1.5 BDT 1.13 1.13 1.13 AC (Controlled) 1.0 1.0 1.0 ST or 1.5 AT 0.75 0.75 0.75 DC (Controlled) 3.0 3.0 3.0 FLT 2.25 2.25 2.25 I.C. Engines 1.0 1.0 1.0 PT 0.75 0.75 0.75 Hydraulic Motor Design Service 1.0 1.0 Effective Pressure Torque 0.33 0.25 0.8 Overload 1.4 1.4 Effective Pressure Torque 1.05 1.05 1.0 Table 10. Summary of Statistical Parameters Statistical parameters Resistance Load (Forged Steel) Load (Cast Steel) 𝜆 𝑉 𝜆 𝑉 𝜆 𝑉 Electric Motors Service Limit State 1.00 0.15 2.50 0.15 3.33 0.15 Overload Limit State 1.00 0.10 1.00 0.10 1.00 0.10 Hydraulic Motors Service Limit State 1.10 0.15 2.40 0.15 3.20 0.15 Overload Limit State 1.00 0.10 0.95 0.10 0.95 0.10 Reliability Index Calculation Procedure The reliability index, β, is related to the probability of failure, 𝑃 , by β = −ɸ (P ) (3.16) where: ɸ – inverse standard normal distribution 𝑃 – probability of failure

42 The reliability index, β, can be calculated using the following formula (Eq. 3.17). It is adequate when R and Q are normal random variables. If both R and Q are normal random variables, the reliability index can be calculated using Eq. 3.17 [from “Reliability of Structures” by Nowak and Collins]. 𝛽 = 𝜇 − 𝜇𝜎 + 𝜎 (3.17) where: 𝜇 - mean value of resistance, 𝜇 - mean value of load, 𝜎 - standard deviation of resistance, 𝜎 - standard deviation of load. Mean values of load and resistance can be calculated using Eq. 3.9 as follows. 𝜇 = 𝜆 𝑅 (3.18) By substituting Eq. 3.10 and 3.18 to the reliability index equation (Eq.7.2), β is: 𝛽 = 𝜆 𝑅 − 𝜆 𝑄𝜆 𝑉 𝑅 + 𝜆 𝑉 𝑄 (3.19) where: 𝜆 - bias factor of resistance, 𝑉 - coefficient of variation of resistance, 𝜆 - bias factor of load, 𝑉 - coefficient of variation of load, 𝑄 - nominal load, 𝑅 - nominal resistance. In the design using current MHBDS, the nominal resistance from the design formula (Eq.3.1) is: 𝑅 = 𝑛 𝑄𝜑 (3.20) By substituting Eq. 3.20 into the reliability index equation (Eq.3.19), is:

43 𝛽 = 𝜆 𝑛𝜑 − 𝜆𝜆 𝑉 𝑛𝜑 + 𝜆 𝑉 (3.21) In design using LRFD, the nominal resistance from the design formula (Eq. 3.1) is: 𝑅 = 𝛾𝑄𝜙 (3.22) By substituting Eq. 3.22 into the reliability index equation (Eq.3.19) is: 𝛽 = 𝜆 𝛾𝜙 − 𝜆𝜆 𝑉 𝛾𝜙 + 𝜆 𝑉 (3.23) Reliability Index β – Current MHBDS Specifications Resistance factor (𝜑) and safety factor (𝑛 ) from the current specifications, and statistical parameters from are summarized in Table 7 along with the result from substitution into the reliability index equation (Eq.3.21). Table 11. Reliability Index Calculated for the Current Specification Forged Steel Cast Steel Limit State β current MHBDS β current MHBDS Electric Motor Design Service 0.85 0.85 Overload - - AC (Uncontrolled) 2.00 2.00 AC (Controlled) 2.00 2.00 DC (Controlled) 2.00 2.00 I.E. Engines 2.00 2.00 Hydraulic Design Service 1.00 1.00 Overload 2.35 2.35

44 Selection of the Target Reliability Index (𝛽 ) Based on the current practice, the relative frequencies of exceedance for Service Limit State and Overload Limit State can be defined as follows: In normal operation, Service Limit State occurs with a frequency of 98%, and the probability of reaching the upper bound of Service Limit State is 15%. The probability of exceedance can be related to reliability index β, by the Eq. 3.16. By substituting P = 0.15 in Eq. 3.16 corresponds to the target reliability index of about 1.0. Overload Limit State occurs with the frequency of 1.99% and the probability of reaching the upper bound of Overload Limit State being 0.49%. By substituting P = 0.0049 in Eq. 3.16 corresponds to the target reliability of about 2.75. Summarizing values of reliability index from the definition using relative frequencies and the current specification Table 12 presents a selection of target reliability. Table 12. Selection of Target Reliability Forged Steel Cast Steel β - relative frequencies βt Limit State β current MHBDS β current MHBDS Mechanical Design Service 0.85 0.85 1.00 1.00 Overload - - - - AC (Uncontrolled) 2.00 2.00 2.75 2.00 AC (Controlled) 2.00 2.00 2.75 2.00 DC (Controlled) 2.00 2.00 2.75 2.00 I.E. Engines 2.00 2.00 2.75 2.00 Hydraulic Design Service 1.00 1.00 1.00 1.00 Overload 2.35 2.35 2.75 2.50 Recommended Load and Resistance Factors and the Resulting Reliability Index The general design formula in LRFD is defined as: 𝛾 · 𝑄 ≤ 𝜙 ∙ 𝑅 (3.24) where: 𝛾 −load factor 𝜙 −resistance factor 𝑄 − nominal load 𝑅 − nominal resistance

45 The recommended load and resistance factors are calculated based on design point procedure as shown in the publication “Revised load and resistance factors for the AASHTO LRFD Bridge Design Specifications” (by Nowak and Iatsko). Eq. 3.25 and Eq.10.5 are used to calculate load and resistance factors for established statistical parameters. The coordinates of the designed point for the load can be determined from Eq. 3.25 𝑄∗ = 𝜇 + 𝛽𝜎𝜎 + 𝜎 (3.25) where: 𝜇 − mean value of Q 𝛽 − reliability index 𝜎 − standard deviation of Q 𝜎 − standard deviation of R The optimum load factor can be computed using following equation: 𝛾 = 𝜆 · 𝑄∗𝜇 (3.26) where: 𝜆 − bias factor of Q 𝑄∗ − coordinate of the design point for Q defined by Eq. 3.25 𝜇 − mean value of Q The coordinates of the designed point for resistance can be determined from Eq. 3.27 𝑅∗ = 𝜇 − 𝛽𝜎𝜎 + 𝜎 (3.27) Optimum resistance factor can be calculated from the Eq. 3.28. 𝜙 = 𝜆 · 𝑅∗𝜇 (3.28) 𝜆 − bias factor of R 𝑅∗ − coordinate of the design point for R defined by Eq. 3.27 𝜇 − mean value of R Since Service Limit State for Electric and Hydraulic Motors have the same target reliability index (also true for the Overload Limit State), load and resistance factors will be consistent for electric and hydraulic motors. The load and resistance factors obtained from Eq. 3.25 and Eq. 3.26 were further factored by the same value to obtain the recommended load and resistance factors as shown below in Table 13. By substituting recommend load and resistance factors in the Eq. 3.23, reliability index β was checked for closeness to target reliability index 𝛽 .

46 Table 13. Recommended Load and Resistance Factors for Service and Overload Limit State Limit State Forged Steel Cast Steel Forged Cast βT Load factor (γ) Resistance factor (ϕ) Load factor (γ) Resistance factor (ϕ) β β Electric Motor Service 2.75 0.90 2.75 0.65 0.94 1.11 1.00 Overload 1.25 0.90 1.25 0.90 2.27 2.27 2.00 Hydraulic Motor Service 2.75 0.90 2.75 0.70 1.12 0.96 1.00 Overload 1.25 0.90 1.25 0.90 2.59 2.59 2.50 Summary of Examples As part of the complete Auburn University report (see Appendix C), several sample problems were performed with excellent correlation. Summary Table 14 below includes results for the Service and Overload Limit States. The two columns furthest to the right reveal the correlated values of d (required minimum shaft diameter) and Rn (nominal resistance). In all cases, the values obtained utilizing the proposed new methodology and factors is very slightly more conservative. The comparison between the existing code and proposed LRFD format indicates a small difference, but shows that calibrated load and resistance factors are coherent with current practice. Table 14. Summary of Calculations for Existing vs. New RBM – Service and Overload Limit States Factored Load ns ϕ ϕ γ ϕ d Rn = Factored Resistance Service Limit State Electric Motor Current Spec 3.25 3.00 1.00 N/A N/A N/A 1.40 9.75 Proposed LRFD 9.10 N/A N/A 0.90 2.80 0.90 1.41 9.93 Overload Limit State Electric Motor Current Spec 6.50 3.00 2.25 N/A N/A N/A 1.35 8.66 Proposed LRFD 8.12 N/A N/A 0.90 1.25 0.90 1.37 9.03 Service Limit State Hydraulic Motor Current Spec 3.25 3.00 1 N/A N/A N/A 1.40 9.75 Proposed LRFD 8.76 N/A N/A 0.90 2.70 0.90 1.41 9.93 Overload Limit State – Hydraulic Motor Current Spec 4.37 3.00 2.25 N/A N/A N/A 1.18 5.82 Proposed LRFD 5.46 N/A N/A 0.90 1.25 0.90 1.20 6.06

47 Furthermore, Summary Table 15 for the fatigue limit state is shown below. The proportion of both sides of the interaction equation remains consistent with the current specification. Table 15. Summary of Calculations for Existing vs. New RBM – Fatigue Limit State γ ϕ Interaction Equation Right side of interaction equation Fatigue Limit State Current Spec - - 0.235 0.80 Proposed LRFD 1.25 1.00 0.293 1.00 Updates to the MHBDS The primary deliverable for Phase 3 work was to provide a draft version of the specifications. Section 5 (Mechanical Design Loads and Power Requirements) had the most comprehensive updates. Section 6 (Mechanical Design) is the only other part that had substantial changes due to incorporation of the Reliability-Based Design approach. Other than Sections 5 and 6, changes were primarily limited to general updates, corrections, and coordination with new parts of Sections 5 and 6. Summaries of updates for each section follow. SECTION 1 – GENERAL PROVISIONS Section 1 updates were primarily to coordinate the extensive changes in Sections 5 and 6. There was some older “legacy” wording that was removed, as it is no longer applicable. Other changes include clarifications to items that were known to be open to misinterpretation. SECTION 2 – STRUCTURAL DESIGN Section 2 had relatively minor updating. The Service Limit State had previously been undefined for movables and is now included in 2.4.2.4. Much of the other updating deals with clarifications and corrections relating to rolling lift bascules (2.5.1.1.3). SECTION 3 – SEISMIC DESIGN No changes were made to this section. SECTION 4 – VESSEL COLLISION CONSIDERATIONS No changes were made to this section. SECTION 5 – MECHANICAL DESIGN LOADS AND POWER REQUIREMENTS Section 5 primarily deals with “external loading” such as wind, ice, friction, inertia, and similar. These are loads that must be overcome and controlled by the prime mover (e.g., and electric motor) and braking system. Once these loads are determined, a standard prime mover is selected to operate the bridge along with a braking system to safely control (stop) motion of the movable bridge. One of the largest variables that is utilized in determining the sizing of prime movers and braking systems is the wind load. Wind loading was separated and made its own Article 5.4 (Wind Loading During Operation). Included was an entirely new sub-article 5.4.3 (Site-Specific Procedure) which allows the designer (at the owner’s discretion) flexibility to utilize site-specific parameters to determine alternate wind loading values.

48 SECTION 6 – MECHANICAL DESIGN Section 6 is where the reliability-based design methodology is primarily applied. Load and resistance factors that were derived from the Auburn work are presented for application in mechanical design (6.4.1.1). Where practicable, machinery resistance employs the RBDM (6.6), but there are specific components that still employ industry-standard design philosophies such as allowable stress and pressure / velocity design criteria for bronze bearings. For this and similar cases (e.g., anti-friction bearing, gearing, and wire ropes) load cases may be unfactored or have a load modifier applied. Other changes include general updates, clarifications, and corrections. SECTION 7 – HYDRAULIC DESIGN There was coordination with the vastly changed Sections 5 and 6. There was also a substantial amount of updating outdated references. Required containment of for hydraulic fluid was added in 7.5.1. Use of fixed displacement pumps with variable speed motors was included in 7.5.2.1 with parallel updates in 7.5.5.1 and 7.6.5.1. Commentary regarding dynamic braking of hydraulic motors was added in C7.5.11. Cylinder buckling was updated to include Johnson’s formula (7.5.12.3). SECTION 8 – ELECTRICAL DESIGN Section 8 had relatively minor updating, primarily for outdated references. APPENDIX A – SI VERSIONS OF EQUATIONS, TABLES AND FIGURES Updated to reflect updates in other sections.