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

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132 A P P E N D I X C Auburn University Report of Reliability- Based Methodology for Mechanical Design with Design Examples PREPARED BY Andrzej S. Nowak J. Michael Stallings Sylwia Stawska Anjan Ramesh Babu Auburn University Auburn, AL SUBMITTED TO Jeffrey Newman Kevin Johns Thomas Murphy Maria Lopez Zolan Prucz Lance Borden Modjeski and Masters, Inc. Mechanicsburg, PA Andrzej Nowak Auburn University Auburn, AL Jim Phillips Alec Noble Paul Skelton Hardesty and Hanover, LLC New York, NY Ian Buckle Independent Consultant Reno, NV July 2020

133 Table of Contents 1. Overview of the Report ............................................................................... 134 2. Limit States in Mechanical Design ................................................................. 134 3. Mechanical Design in the current MHBDS specifications ................................. 137 4. Reliability-based Calibration Procedure ......................................................... 140 5. Limit state function ..................................................................................... 141 6. Statistical parameters for Load and Resistance .............................................. 143 7. Reliability index calculation procedure .......................................................... 153 8. Reliability index β – current MHBDS specifications ......................................... 155 9. Selection of the Target Reliability Index (𝛽𝑇) ................................................ 156 10. Recommended Load and Resistance Factors and the resulting Reliability Index 157 11. Fatigue Limit State – Shaft (infinite life) ........................................................ 159 12. Electric Motor Example – Service Limit State, current specification .................. 162 13. Electric Motor Example – Service Limit State, LRFD format ............................. 165 14. Electric Motor Example – Overload Limit State, current specification ............... 167 15. Electric Motor Example – Overload Limit State, LRFD format .......................... 169 16. Hydraulic Motor Example – Service Limit State, current specification ............... 170 17. Hydraulic Motor Example – Service Limit State, LRFD format .......................... 172 18. Hydraulic Motor Example – Overload Limit State, current specification ............ 174 19. Hydraulic Motor Example – Overload Limit State, LRFD format ....................... 175 20. Fatigue Limit State, current specification ....................................................... 177 21. Fatigue Limit State, LRFD format ................................................................. 182 22. Summary of Examples ................................................................................. 186 23. References ................................................................................................. 188

134 1. Overview of the Report The objective of this report is to develop and incorporate a consistent Reliability-Based Methodology in the Mechanical Design of the AASHTO Load and Resistance Factor Design (LRFD) Movable Highway Bridge Design Specifications (MHBDS) [1]. 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 shown. 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. 2. Limit States in Mechanical Design The Limit States in Mechanical Design have to be clearly defined. 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. Abbreviations that are used throughout the report (see AASHTO Movable Ref. 5.2 “Definitions” for more detail): • FLT: Full Load Torque • ST: Starting Torque • BDT: Breakdown Torque • AT: Acceleration Torque • PT: Peak Torque • E-Stop: Emergency Stop of Bridge (cuts all power and sets all brakes)

135 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, AASHTO Movable Ref: 5.7.1). • Hydraulic motor or cylinder – Normal working pressure – adjustable value (relief valve setting of the pump or circuit, AASHTO Movable Ref: 7.4.2). • Dynamic Braking – During Normal Operation - from motor brakes or machinery brakes, regenerative braking, or hydraulic braking (AASHTO Movable 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. 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.

136 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 (AASHTO Movable Ref: 5.7.1) • Case 1b: 1.0 ST or 1.5 AT for controlled AC motors (AASHTO Movable Ref: 5.7.1) • Case 1c: 3.0 FLT for controlled DC motors (AASHTO Movable 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 (AASHTO Movable Ref: 5.7.1) • Case 2a: Uncontrolled dynamic braking (E-Stop) with a modified resistance factor of 1.5 (AASHTO Movable Ref: 5.7.3) • Case 2b: Static (Holding) Braking Combined Motor and Machinery Brake Torque (AASHTO Movable 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.

137 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. 3. Mechanical Design in the Current MHBDS Specifications Service and Overload Limit State Design formula in the current specification is as follow: 𝑄 = 𝜑 ∙ 1𝑛 ∙ 𝑅 (3.1) 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. Resistance factors listed in MHBDS Section 6.4.1.1 are shown in Figure 1.

138 Figure 1. Resistance factors [Ref. 1]. Safety factors listed in MHBDS Section 6.6.1 are shown in Figure 2. Figure 2. Safety factors [Ref. 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 (3.2) where: 𝑅 - nominal resistance

139 𝑄 - nominal load under overload case 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 (3.3) 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 (3.4) Fatigue Limit State – Shaft (infinite life) Design formula in the current specification for infinite fatigue life of a shaft is as follows: 32𝜋𝑑 𝐾 𝑀𝜎 + √3𝐾 𝑇2𝜎 ≤ 0.80 (3.5) where: 𝑑 – diameter 𝜎 – endurance limit 𝜎 – Yield strength of steel 𝑀 – Moment 𝑇 – Torque 𝐾 – Fatigue stress concentration factor (bending)

140 𝐾 – Fatigue stress concentration factor (bending) The endurance limit, 𝜎 in Eq.3.4 is defined as: 𝜎 = 𝛼𝜎 (𝐶 𝐶 𝐶 𝐶 𝐶 ) (3.6) 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 All the above parameters in the interaction equation are discussed in Mechanical Engineering Design book [9]. 4. 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 [4]. 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:

141 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 judgment 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. 5. Limit State Function The general form of a limit state function can be presented as follows: 𝑔 = 𝑅 − 𝑄 = 0 (5.1) where:

142 𝑅 – resistance (random variable) Q – load (random variable) and the probability of failure is 𝑃 = 𝑃𝑟𝑜𝑏(𝑔 < 0) (5.2) 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. 5.1), 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. 5.1), 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. 5.1), R is the maximum allowed torque for Service Limit State and Q represents normal working pressure.

143 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. 5.1), 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. 6. Statistical Parameters for 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. 6.1) λ = 𝜇𝑄 (6.1) V- coefficient of variation, which is the ratio of standard deviation and mean maximum value (Eq. 6.2) 𝑉 = 𝜎𝜇 (6.2) Electrical Motor – Service Limit State: In the limit state function, Eq. 5.1, 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 than or equal to the factored resistance. In this case,

144 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 1 [Ref. 8] and engineering judgment. Table 1. 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 1 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,

145 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 2.

146 Table 2 Opening and closing torque for movable bridges in the 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 2 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

147 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 Torque values indicated in Table 2 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 3.

148 Figure 3. Starting Torque for opening and closing (data from 2 bridges were excluded, as they were considered “outliers”). As noted previously, the values in Table 2 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 1 values are: • “Missing Loads” from Table 2 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

149 Even if the “Missing Loads” are added to Table 2 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. 3.3taking 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. 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 (6.3) λ , , , = 𝜇𝑄 = 1.25 𝐹𝐿𝑇0.50 𝐹𝐿𝑇 = 2.50 (6.4) 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

150 (Eq.5.1), 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 Table 3. Table 3. 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 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

151 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 (6.5) λ , , , = 𝜇𝑄 = 0.80 0.33 = 2.40 (6.6) 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 the design formula Eq. 3.4 considering that nominal resistance is 1.00 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:

152 λ = 𝜇𝑄 = 1.00 · 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑇𝑜𝑟𝑞𝑢𝑒1.05 · 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑇𝑜𝑟𝑞𝑢𝑒 = 0.95 (6.7) 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 4A followed by statistical parameters in shown in Table 4B. Table 4A. Summary of mean and nominal values of load Limit State Forge Steel Cast Steel Rn in terms of Forge d 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 4B5. Summary of statistical parameters Statistical parameters Resistance Load Load

153 (Forged Steel) (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 7. Reliability Index Calculation Procedure The reliability index, β, is related to the probability of failure, 𝑃 , by β = −ɸ (P ) (7.1) where: ɸ – inverse standard normal distribution 𝑃 – probability of failure The reliability index, β, can be calculated using the following formula (Eq. 7.2). 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 the Eq. 7.2 [Ref. 4]. 𝛽 = 𝜇 − 𝜇𝜎 + 𝜎 (7.2) where: 𝜇 - mean value of resistance, 𝜇 - mean value of load, 𝜎 - standard deviation of resistance, 𝜎 - standard deviation of load.

154 Mean values of load and resistance can be calculated using Eq. 6.1 as follows. 𝜇 = 𝜆 𝑅 (7.3) By substituting Eq. 6.2, 7.2 and 7.3 to the reliability index equation (Eq.7.2), β is: 𝛽 = 𝜆 𝑅 − 𝜆 𝑄𝜆 𝑉 𝑅 + 𝜆 𝑉 𝑄 (7.4) 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 (7.5) By substituting Eq. 7.5 into the reliability index equation (Eq.7.4), is: 𝛽 = 𝜆 𝑛𝜑 − 𝜆𝜆 𝑉 𝑛𝜑 + 𝜆 𝑉 (7.6) In design using LRFD, the nominal resistance from the design formula (Eq. 3.1) is:

155 𝑅 = 𝛾𝑄𝜙 (7.7) By substituting Eq. 7.7 into the reliability index equation (Eq.7.4) is: 𝛽 = 𝜆 𝛾𝜙 − 𝜆𝜆 𝑉 𝛾𝜙 + 𝜆 𝑉 (7.8) 8. Reliability Index β – Current MHBDS Specifications Resistance factor (𝜑) and safety factor (𝑛 ) from the current specifications, and statistical parameters from Section 6 are summarized in Table 2 along with the result from substitution into the reliability index equation (Eq.7.6).

156 Table 6. 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 9. Selection of the Target Reliability Index (𝜷𝑻) 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. 7.1. By substituting P = 0.15 in Eq. 7.1 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. 7.1 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 7presents a selection of target reliability.

157 Table 7. 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 10. Recommended Load and Resistance Factors and the Resulting Reliability Index Design formula in LRFD is defined as: 𝛾 · 𝑄 ≤ 𝜙 ∙ 𝑅 (10.1) where: 𝛾 −load factor 𝜙 −resistance factor 𝑄 − nominal load 𝑅 − nominal resistance The recommended load and resistance factors are calculated based on design point procedure as shown in Ref. 6. Eq. 10.2 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. 10.2 𝑄∗ = 𝜇 + 𝛽𝜎𝜎 + 𝜎 (10.2) where: 𝜇 − mean value of Q

158 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 (10.3) where: 𝜆 − bias factor of Q 𝑄∗ − coordinate of the design point for Q defined by Eq. 10.2 𝜇 − mean value of Q The coordinates of the designed point for resistance can be determined from Eq. 10.4 𝑅∗ = 𝜇 − 𝛽𝜎𝜎 + 𝜎 (10.4) Optimum resistance factor can be calculated from the Eq. 10.5. 𝜙 = 𝜆 · 𝑅∗𝜇 (10.5) 𝜆 − bias factor of R 𝑅∗ − coordinate of the design point for R defined by Eq. 10.4 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

159 from Eq. 10.2 and Eq. 10.3 were further factored by the same value to obtain the recommended load and resistance factors as shown below in Table 8. By substituting recommend load and resistance factors in the Eq. 7.8, reliability index β was checked for closeness to target reliability index 𝛽 . Table 8. 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 11. Fatigue Limit State – Shaft (Infinite Life) In the current specification [1] it is assumed that the Fatigue Limit State for infinite life of a shaft is e xceeded when the reversed bending moment due to rotation in combination with a steady torsional moment exceeds value of 0.8. Therefore, fatigue check interaction equation where Q is represented by moment and torque and R is based on fatigue strength of material other factors are taken into consideration in the fatigue check. Fatigue Limit State Design formula in the current specification for infinite fatigue life of a shaft is specified in Eq. 3.4. The endurance limit as part of Eq. 3.4 of current MHBDS was shown in Eq 3.5 and it is a function of many factors. Substituting all the variables in the interaction equation (Eq. 3.4) it is shown as follows:

160 32𝜋𝑑 ⎣⎢⎢ ⎢⎢⎢ ⎢⎢⎡⎝⎛1 + 11 + √𝑎√𝑟 (𝐾 − 1)⎠⎞𝑀𝛼𝜎 (𝐶 𝐶 𝐶 𝐶 𝐶 ) + √3⎝⎛1 + 11 + √𝑎√𝑟 (𝐾 − 1)⎠⎞𝑇2𝜎 ⎦⎥⎥ ⎥⎥⎥ ⎥⎥⎤ ≤ 0.80 (11.1) where: 𝜎 - Tensile strength of steel 𝜎 - Yield strength of steel 𝛼 - material property parameter 𝑑 - diameter 𝐶 - size factor 𝐶 - surface factor 𝐶 - reliability factor 𝐶 - temperature factor 𝐶 - miscellaneous factor 𝑟 – radius, notch or fillet √𝑎 - Neuber constant 𝑀 - Moment 𝑇 - Torque 𝐾 - stress concentration factor for bending 𝐾 - stress concentration factor for torque Analysis of Fatigue Limit State represented by interaction equation and thorough investigation of all the variables was conducted. The safety margin in the current specification is sufficient and is confirmed from the bridge owner survey and the calibration is based on that conclusion. The interaction equation in the current specification was adjusted to LRFD format. The right side of the equation was changed to 1.00 instead of 0.80, to be consistent with the existing LRFD code format. Unity of the

161 right side of the equation indicates that there is no additional safety margin included in the equation. The safety in LRFD codes is represented by load and resistance factors. In this case, the value of 0.80 is eliminated from the equation. Furthermore, load and resistance factors need to be calculated. In order to be consistent with Fatigue Limit State defined in AASHTO LRFD code, where resistance factor (𝜙) is equal to 1.00. In the new LRFD MHBDS, resistance factors (𝜙) is assumed as 1.00. The load factor (𝛾) is assumed as 1.25 to satisfy the current format. The new LRFD format of the fatigue interaction equation check is specified in Eq. 11.2. 32𝜋𝑑 𝛾𝐾 𝑀𝜙𝜎 + 𝛾√3𝐾 𝑇𝜙2𝜎 ≤ 1.00 (11.2) where: 𝜙 = 1.0 – resistance factor 𝛾 = 1.25 – load factor In the commentary of the current specification, the interaction equation can be defined as a one-way bending cycle for each complete bridge operation (for trunnions of bascule bridges). In that case, the interaction equation on the right side is equal to 1.00. The LRFD format for one-way bending can be represented as: 32𝜋𝑑 𝛾𝐾 𝑀𝜙𝜎 + 𝛾√3𝐾 𝑇𝜙2𝜎 ≤ 1.00 (11.3) where: 𝜙 = 1.0 – resistance factor 𝛾 = 1.00 – load factor Mechanical elements prone to fatigue, are those subjected to cyclical loading. Typical mechanical components include:  Shafts and Trunnions: Article 6.7.4.1  Open Spur Gearing: Article 6.7.5.2.2

162  Sheaves: Article 6.8.3.4.1  Span Lock / Tail Lock Bars:  Machinery Supports: Article 6.7.3.3.2 Those components are to be checked for Fatigue Limit Sate. 12. Electric Motor Example – Service Limit State, Current Specification On the basis of the Lance V. Borden paper “Torque Characteristics of Wound Rotor Motors Revisited”, it is shown that the calculated theoretical FLT values have very little variance with actual test data. Theoretical FLT values of 600 RPM motors (assuming 3% slip): For 20 HP: FLT = 20 HP * 5252 / (600 rpm * 0.97) = 180.5 lb-ft For 50 HP: FLT = 50 HP * 5252 / (600 rpm * 0.97) = 451.2 lb-ft Our example will utilize a 20 HP motor. Also, note that the tested Breakdown Torques do indeed represent significant potential loading, which must be resisted (in the Overload Limit State).

163 Table 9. Wound Rotor Motors properties (600 RPM nominal) [Ref. 2]. Factored Load: In the paper [2] there is no information about the type of Control System. However, most movable bridge installations with wound rotor motors are “uncontrolled”. Therefore, it is assumed as AC uncontrolled. Drive shaft is made of forged steel.

164 Table 10 Prime mover loads for machinery design for AC (uncontrolled) prime mover [Ref. 1]. Because in current MHBDS there is no load factor, so that means that factored load is equal to nominal load. 𝑄 = 1.5 · 𝐹𝐿𝑇 𝑄 = 1.5 · (180.5) 𝑙𝑏. 𝑓𝑡 𝑄 = 270.75 𝑙𝑏 · 𝑓𝑡 𝑄 = 3.25 𝑘𝑖𝑝 · 𝑖𝑛 Factored Resistance: In the current MHBDS there is resistance and safety factor, so that mean that factored resistance is: 𝜑 ∙ 1𝑛 ∙ 𝑅 𝑅 = 𝐹2 𝜋𝑑16 𝜑 = 1.0 𝑛 = 3.0

165 Factored resistance: 𝜑 ∙ 1𝑛 ∙ 𝐹2 𝜋𝑑16 Design Formula: Assuming factored load is equal to factored resistance, which represents boundary of limit state 𝑅 = 𝑄 = 3.25 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.40 𝑖𝑛. The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.40)16 = 9.75 𝑘𝑖𝑝 · 𝑖𝑛 13. Electric Motor Example – Service Limit State, LRFD Format AC Uncontrolled The following example is based on the Lance V. Borden paper “Torque Characteristics of Wound Rotor Motors Revisited” as was the example Service Limit State – current specification. Factored Load: γ · 𝑄 γ = 2.75 𝑄 = 1.5 · FLT

166 𝑄 = 1.5 · (180.5) lb · ft 𝑄 = 270.75 lb · ft γ · 𝑄 = 2.75 · 3.25 kip · in = 8.96 kip · in Factored Resistance 𝜙 ∙ 𝑅 𝜙 = 0.90 𝑅 = F2 πd16 R = 𝜙 ∙ F2 πd16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 8.96 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (1/𝜙) = 1.41 𝑖𝑛 The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.420)16 = 9.93 𝑘𝑖𝑝 · 𝑖𝑛

167 Comparison between existing methodology and proposed: Current Specification Proposed LRFD Method dmin 1.40 in 1.41 in Rn 9.75 kip-in 9.93 kip-in This simple example provides similar results when utilizing the proposed LRFD Methodology when compared to existing methodology. The required minimum diameter and corresponding resistance are slightly more conservative in results in LRFD format example. 14. Electric Motor Example – Overload Limit State, Current Specification Assume DC controlled prime mover Factored Load In the current MHBDS there is no load factor so that the factored load is equal to the nominal load. 𝑄 = 3.0 · 𝐹𝐿𝑇 𝑄 = 3.0 · (180.40) 𝑙𝑏. 𝑓𝑡 𝑄 = 541.5 𝑙𝑏 · 𝑓𝑡 𝑄 = 6.50 𝑘𝑖𝑝 · 𝑖𝑛

168 Factored Resistance In the current MHBDS there is resistance and safety factor so that the factored resistance is: 𝜑 ∙ 1𝑛 ∙ 𝑅 𝑅 = 𝐹2 𝜋𝑑16 𝜑 = 2.25 𝑛 = 3.00 Factored resistance: 𝜑 ∙ 1𝑛 ∙ 𝐹2 𝜋𝑑16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 6.50 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.35 𝑖𝑛. The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.1.35)16 = 8.66 𝑘𝑖𝑝 · 𝑖𝑛

169 15. Electric Motor Example – Overload Limit State, LRFD Format The following example is based on the Lance V. Borden paper “Torque Characteristics of Wound Rotor Motors Revisited” as was the example Service Limit State – current specification. Factored Load: γ · 𝑄 γ = 1.25 𝑄 = 3.0 · FLT 𝑄 = 3.0 · (180.5) lb · ft 𝑄 = 541.2 lb · ft γ · 𝑄 = 1.25 · 6.50 kip · in = 8.12 kip · in Factored Resistance: 𝜙 ∙ 𝑅 𝜙 = 0.90 𝑅 = 𝜙 = 0.90 Factored load: 𝜙 ∙ Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 8.12 𝑘𝑖𝑝 · 𝑖𝑛

170 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 ((𝑛𝜑 )) = 1.37 𝑖𝑛 The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.37)16 = 9.03 𝑘𝑖𝑝 · 𝑖𝑛 Comparison between existing methodology and proposed: Current Specification Proposed LRFD Method dmin 1.35 in 1.37 in Rn 8.66 kip-in 9.03 kip-in This simple example provides similar results when utilizing the proposed LRFD Methodology when compared to existing methodology. The required minimum diameter and corresponding resistance are slightly more conservative in results in LRFD format example. 16. Hydraulic Motor Example – Service Limit State, Current Specification Design Assumptions: Normal Working Pressure 𝑃 = 2.230 𝑘𝑠𝑖 Normal working pressure would be determined based on the selection of the hydraulic motor and physically established by a field adjustable relief valve.

171 Hydraulic Motor displacement 𝑑 = 9.154 𝑖𝑛 Factored Load: In the current MHBDS there is no load factor so that the factored load is equal to the nominal load. 𝑄 = 𝑑 𝑃 2𝜋 𝑄 = 270.74 𝑙𝑏. 𝑓𝑡 𝑄 = 3.25 𝑘𝑖𝑝 · 𝑖𝑛 Factored Resistance: 𝜑 ∙ 1𝑛 ∙ 𝑅 𝑅 = 𝐹2 𝜋𝑑16 𝜑 = 1.00 𝑛 = 3.00 Factored Resistance: 𝜑 ∙ 1𝑛 ∙ 𝐹2 𝜋𝑑16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 3.25 𝑘𝑖𝑝 · 𝑖𝑛

172 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.40 𝑖𝑛. The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋 (1.070)16 = 9.75 𝑘𝑖𝑝 · 𝑖𝑛 17. Hydraulic Motor Example – Service Limit State, LRFD Format Design Assumptions: Normal working Pressure 𝑃 = 2.230 𝑘𝑠𝑖 Normal working pressure would be determined based on the selection of the hydraulic motor and physically established by a field adjustable relief valve. Hydraulic Motor displacement 𝑑 = 9.154 𝑖𝑛 Factored Load: 𝛾 · 𝑄 𝑄 = 𝑑 𝑃 2𝜋 𝛾 = 2.75 𝑄 = 270.471 𝑙𝑏.𝑓𝑡 2.75 · 3.25 𝑘𝑖𝑝 · 𝑖𝑛 = 8.93 𝑘𝑖𝑝 · 𝑖𝑛

173 Factored Resistance: 𝜙 ∙ 𝑅 𝜙 = 0.90 𝑅 = F2 πd16 Factored Resistance: 𝜙 ∙ F2 πd16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 8.93 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.41 𝑖𝑛 The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.40)16 = 9.93 𝑘𝑖𝑝 · 𝑖𝑛 Comparison between existing methodology and proposed: Current Specification Proposed LRFD Method dmin 1.40 in 1.41 in Rn 9.75 kip-in 9.93 kip-in

174 This simple example provides the same results when utilizing the proposed LRFD Methodology when compared to existing methodology. The required minimum diameter and corresponding resistance gives the same results in LRFD format example. 18. Hydraulic Motor Example – Overload Limit State, Current Specification Design Assumptions Maximum working Pressure 𝑃 = 3.00 𝑘𝑠𝑖 The maximum working pressure is set at 3.0 ksi, a common setting for a non-adjustable relief valve. Hydraulic Motor displacement 𝑑 = 9.154 𝑖𝑛 Factored Load: In current MHBDS there is no load factor, so that factored load is equal to nominal load. 𝑄 = 𝑑 𝑃 2𝜋 𝑄 = 363.87 𝑘𝑖𝑝 · 𝑓𝑡 𝑄 = 4.37 𝑘𝑖𝑝 · 𝑖𝑛 Factored Resistance: 𝜑 ∙ 1𝑛 ∙ 𝑅 𝑅 = 𝐹2 𝜋𝑑16 𝜑 = 2.25

175 𝑛 = 3.00 Factored Resistance: 𝜑 ∙ 1𝑛 ∙ 𝐹2 𝜋𝑑16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 4.37 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.18 𝑖𝑛. The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.18)16 = 5.82 𝑘𝑖𝑝 · 𝑖𝑛 19. Hydraulic Motor Example – Overload Limit State, LRFD Format Design Assumptions: Normal working Pressure 𝑃 = 3.0 𝑘𝑠𝑖 Normal working pressure would be determined based on the selection of the hydraulic motor and physically established by a field adjustable relief valve.

176 Hydraulic Motor displacement 𝑑 = 9.154 𝑖𝑛 Factored Load: 𝛾 𝑄 𝛾 = 1.25 𝑄 = 𝑑 𝑃 2𝜋 𝑄 = 2363.87 𝑙𝑏.𝑓𝑡 Factored Load 1.25 · 4.37 𝑘𝑖𝑝 · 𝑖𝑛 = 5.46 𝑘𝑖𝑝 · 𝑖𝑛 Factored Resistance: 𝜙 ∙ 𝑅 𝜙 = 0.90 𝑅 = F2 πd16 Factored resistance: 𝜙 ∙ F2 πd16 Design Formula: Assuming load is equal to resistance: 𝑅 = 𝑄 = 5.46 𝑘𝑖𝑝 · 𝑖𝑛 Finding shaft diameter for A36 steel

177 𝑑 = 𝑄 · 2𝐹 16𝜋 (𝑛𝜑 ) = 1.20 𝑖𝑛 The nominal resistance for this calculated shaft diameter (d) is: 𝑅 = 𝐹2 𝜋(1.20)16 = 6.06 𝑘𝑖𝑝 · 𝑖𝑛 Comparison between existing methodology and proposed: Current Specification Method Proposed LRFD Method dmin 1.18 in 1.20 in Rn 5.82 kip-in 6.06 kip-in This simple example provides similar results when utilizing the proposed LRFD Methodology when compared to existing methodology. The required minimum diameter and corresponding resistance are slightly more conservative in LRFD format example. 20. Fatigue Limit State, Current Specification Fatigue Analysis of shaft assuming reverse bending and steady torsion based on ASHTO LRFD Movable Highway Bridge Design Specifications 2nd Edition 2007. Pinion integral with shaft:

178 Pinion shaft material is specified as 3588HN, according to ASTM A291, Class 7 with tensile strength of 165 ksi and yield strength of 135 ksi for range of 10 – 20 in. Input Data: Input diameters 𝑑 = 10 𝑖𝑛 𝐷 = 13 𝑖𝑛 Tensile strength 𝜎 = 165,000 𝑝𝑠𝑖 Yield Strength 𝜎 = 135,000 𝑝𝑠𝑖 Fillet radius 𝑟 = 0.375 𝑖𝑛 Bearing load 𝑅 = 76,600 𝑙𝑏 Moment arm to shoulder 𝑎 = 5.5 𝑖𝑛 Bending moment at shoulder 𝑀 = 𝑅 𝑎 = 421,300 𝑙𝑏 − 𝑖𝑛

179 Bearing effective friction radius 𝑑 = 5 𝑖𝑛 Friction coefficient 𝑣 = 0.12 Torsional moment due to friction 𝑇 = 𝑅 𝑑 𝑣 = 45960 𝑙𝑏 − 𝑖𝑛 Ratios 𝐷𝑑 = 1.300 𝑟𝑑 = 0.038 Specify Stress Concentration Factors Stress concentration factor for bending [10] 𝐾 = 2.16

180 Stress concentration factor for torsion [10] 𝐾 = 1.8 Neuber’s Constant, from the Table 6.7.3.2-1[1] √𝑎 = 0.032

181 Fatigue notch sensitivity factor 𝑞 = 11 + √𝑎√𝑟 = 0.95 Modified stress concentration factor for bending 𝐾 = 1 + 𝑞(𝐾 − 1) = 2.102 Modified stress concentration factor for torsion 𝐾 = 1 + 𝑞(𝐾 − 1) = 1.800 Endurance limit factors: Size factor 𝐶 = 0.673 Surface factor 𝐶 = 0.699 Reliability factor 𝐶 = 1.0 Temperature factor 𝐶 = 1.0 Miscellaneous factor 𝐶 = 1.0 Material property parameter 𝛼 = 0.4 Endurance limit: 𝜎 = 𝛼𝜎 (𝐶 𝐶 𝐶 𝐶 𝐶 ) = 38777 𝑝𝑠𝑖

182 Fatigue Check 32𝜋𝑑 𝐾 𝑀𝜎 + √3𝐾 𝑇2𝜎 ≤ 0.80 0.235 ≤ 0.80 Fatigue check for reversed bending is ok. 21. Fatigue Limit State, LRFD Format Fatigue Analysis of shaft assuming reverse bending steady torsion for proposed LRFD format. Pinion integral with shaft: Pinion shaft material is specified as 3588HN, according to ASTM A291, Class 7 with the tensile strength 165 ksi and yield strength 135 ksi for diameter range 10 – 20 in. Input Data: Input diameter 𝑑 = 10 𝑖𝑛 𝐷 = 13 𝑖𝑛

183 Tensile strength 𝜎 = 165,000 𝑝𝑠𝑖 Yield Strength 𝜎 = 135,000 𝑝𝑠𝑖 Fillet radius 𝑟 = 0.375 𝑖𝑛 Bearing load 𝑅 = 76,600 𝑙𝑏 Moment arm to shoulder 𝑎 = 5.5 𝑖𝑛 Bending moment at shoulder 𝑀 = 𝑅 𝑎 = 421,300 𝑙𝑏 − 𝑖𝑛 Bearing effective friction radius 𝑑 = 5 𝑖𝑛 Friction coefficient 𝑣 = 0.12 Torsional moment due to friction 𝑇 = 𝑅 𝑑 𝑣 = 45960 𝑙𝑏 − 𝑖𝑛 Ratios 𝐷𝑑 = 1.300 𝑟𝑑 = 0.038 Specify stress concentration factors Stress concentration factor for bending [10]

184 𝐾 = 2.16 Stress concentration factor for torsion [10] 𝐾 = 1.8 Neuber’s Constant, from the Table 6.7.3.2-1[1]

185 √𝑎 = 0.032 Fatigue notch sensitivity factor 𝑞 = 11 + √𝑎√𝑟 = 0.95 Modified stress concentration factor for bending 𝐾 = 1 + 𝑞(𝐾 − 1) = 2.102 Modified stress concentration factor for torsion 𝐾 = 1 + 𝑞(𝐾 − 1) = 1.800 Endurance limit factors: Size factor 𝐶 = 0.673 Surface factor 𝐶 = 0.699

186 Reliability factor 𝐶 = 1.0 Temperature factor 𝐶 = 1.0 Miscellaneous factor 𝐶 = 1.0 Material property parameter 𝛼 = 0.4 Endurance limit: 𝜎 = 𝛼𝜎 (𝐶 𝐶 𝐶 𝐶 𝐶 ) = 38777 𝑝𝑠𝑖 Fatigue Check Load factor: 𝛾 = 1.25 Resistance factor 𝜙 = 1.00 32𝜋𝑑 𝛾𝐾 𝑀𝜙𝜎 + 𝛾√3𝐾 𝑇𝜙2𝜎 ≤ 1.00 0.293 ≤ 1.00 Fatigue check for reversed bending is ok. 22. Summary of Examples Summary table for the examples concerning Service and Overload Limit State is created. The comparison between the existing code and proposed LRFD format indicates a small difference. It means that calibrated load and resistance factors are coherent with current practice.

187 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 Furthermore, the summary table for the Fatigue Limit State is shown below. The proportion of both sides of the interaction equation remains consistent with the current specification. 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

188 23. References 1. AASHTO LRFD Movable Highway Bridge Design Specifications 2nd Edition 2007 2. Lance V. Borden, P.E. Senior Associate Modjeski and Masters. Inc., “Torque characteristics of Wound-Rotor Motor Revisited”. 3. Modjeski and Masters Inc., University of Nebraska, Lincoln, University of Delaware, and NCS Consultants, LLC. SHRP 2 Report S2-R19B-RW-1: Bridges for Service Life Beyond 100 Years: Service Limit State Design, Transportation Research Board, 2015. 4. Nowak A. S., Kevin R. Collins, Reliability of Structures”, 2000 5. Phillips, J., and J. Newman. NCHRP Project 12-112, Limit State Definition, Transportation Research Board, 2019. 6. Nowak, A. S., and Iatsko, O. (2017). “Revised load and resistance factors for the AASHTO LRFD Bridge Design Specifications.” PCI Journal, 13, May-June 2017. 7. Hydraulic Motor Design Example by Jim Philips Hardesty & Hanover, May 29th 2019 8. Limit State – Failure Matrix, J. Phillips Hardesty & Hanover and Jeff Newman Modjeski and Masters. Inc, July 23rd 2019. 9. Shigley J.E., Mitchell L.D., “Mechanical Engineering Design”, McGraw-Hill Book Company 10. Norton, R. L. 1998. “Machine Design: An Integrated Approach”, Revised Printing. Prentice-Hall Inc., Upper Saddle River, NJ.