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Analysis method for the design of reinforced concrete bridge barrier and cantilever deck under railing… Wong , Joe 2005

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ANALYSIS M E T H O D FOR T H E DESIGN OF REINFORCED C O N C R E T E BRIDGE BARRIER AND C A N T I L E V E R D E C K UNDER RAILING LOADS AS SPECIFIED IN CAN/CSA-S6-00 (CANADIAN HIGHWAY BRIDGE DESIGN CODE) by J O E W O N G B.A.Sc, University of British Columbia, Canada, 2002 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Civil Engineering THE UNIVERSITY OF BRITISH COLUMBIA August 2005 © Joe Wong, 2005 Abstract The objective of this thesis is to develop a rational and effective method for designing reinforced concrete bridge parapets and cantilevered decks so that such a method could be easily applied in practice against railing loads, as specified in the C A N / C S A - S 6 - 0 0 Canadian Highway Bridge Design Code. The maximum moment dispersal angle ( M M D A ) is the most promising overall o f the methods being considered for this task, including yield line analysis ( Y L A ) , finite element analysis (FEA) and the dispersal angle method. The M M D A provides a means of approximating maximum moments, which were evaluated using the linear elastic F E A at locations of interest on both the traffic barriers and the deck overhang using dispersal angles, which are provided in the form of tables. The M M D A is an improved version of the maximum moment envelope ( M M E ) method, which had been initially developed based on concepts from the dispersal angle method as well as the F E A . The improved M M D A method takes advantage of the accuracy o f F E A and the simplicity of the classic concept of load dispersion, while eliminating some of the issues of unconventionality found in the dispersal angle method. Hence, M M D A is an improvement on the dispersal angle method, as was suggested in the Commentary o f S6-00. It minimizes possible inconsistencies between the code design methods and the F E A results. This thesis summarizes the design criteria, methods of analysis, and load applications for bridge traffic barriers and deck overhang design that has been suggested by various jurisdictions, including AASHTO LRFD Bridge Design Specifications 2004, Washington State D O T Bridge Design Manual LRFD, C A N / C S A - S 6 - 8 8 Design of Highway Bridges and C A N / C S A - S 6 - 0 0 Canadian Highway Bridge Design Code. i i Table of Contents Abstract i i List of Tables v i i List o f Figures ix Acknowledgment x i i Chapter 1: Introduction 1 Chapter 2: Background Research and Literature Review 3 2.1 A A S H T O L R F D Bridge Design Specifications 3 2.1.1 Limi t States and Resistance Factors 4 2.1.2 Test Levels 5 2.1.3 Traffic Barrier 6 2.1.4 Deck Overhang 11 2.1.5 Comments 12 2.2 Washington State D O T Bridge Design Manual L R F D 13 2.2.1 Traffic Barrier 13 2.2.2 Deck Overhang 14 2.2.3 Design Criteria 15 2.2.4 Comments 15 2.3 C A N / C S A - S 6 - 0 0 and C A N / C S A - S 6 - 8 8 16 2.3.1 Traffic Barrier 16 2.3.2 Performance Level 17 2.3.3 Design Criteria 17 2.3.4 Deck Overhang 19 i i i 2.3.4.1 Refined Methods 19 2.3.4.2 Simplified Methods 19 2.3.4.3 Dispersal Angle Method 20 2.3.5 Comments 22 Chapter 3: Methods o f Analysis 23 3.1 Y i e l d Line Analysis ( Y L A ) 23 3.1.1 Design Philosophy 24 3.1.2 Sample Application 24 3.1.3 Comments 26 3.2 Finite Element Analysis ( F E A ) 27 3.2.1 Basic Principle 27 3.2.2 Finite Element Model 28 3.2.2.1 Element Properties 28 3.2.2.2 Deck Overhang 29 3.2.2.3 Traffic Barrier 29 3.2.2.4 Connection/Anchorages 30 3.2.2.5 Overall Dimensions 30 3.2.2.6 Material 31 3.2.2.7 Mesh and Geometry 31 3.2.2.8 Loads 34 3.2.3 Stress Averaging. 35 3.2.4 Comments 36 3.3 Dispersal Angle Method 37 3.3.1 Mechanic of Behavior 37 iv 3.3.2 Development of Dispersal Angle Method 40 3.3.3 Comments 46 3.4 Maximum Moment Envelope ( M M E ) 46 3.4.1 Development 47 3.4.2 Applications 49 3.4.3 Comments 52 3.5 Maximum Moment Dispersal Angle ( M M D A ) 53 3.5.1 Method 1 - Modification factors 54 3.5.1.1 Development and Application 54 3.5.1.2 Comments 61 3.5.2 Method 2 - New Dispersal Angles 61 3.5.2.1 Development and Application 62 3.5.2.2 Comments 67 3.5.3 Results 68 3.5.4 Examples 75 Chapter 4: Conclusion 80 Chapter 5: Recommendations and Future Developments 84 References 86 Appendix A : Simple Cantilever F E M for Determination of Mesh Size 87 Appendix B : Load Applications in PL-2 F E M for Inner Portion 89 Appendix C: Load Applications in PL-2 F E M for End Portion 93 Appendix D : Load Applications in PL-3 F E M for Inner Portion 97 Appendix E : Load Applications in PL-3 F E M for End Portion 101 Appendix F: F E A Results and M M D A Spreadsheet for PL-2 Inner Portion 105 v Appendix G : F E A Results and M M D A Spreadsheet for PL-2 End Portion 116 Appendix H : F E A Results and M M D A Spreadsheet for PL-3 Inner Portion 127 Appendix I: F E A Results and M M D A Spreadsheet for PL-3 End Portion 138 Appendix J: Sample Transverse Moment Due to Design Loads Plots 149 Appendix K : Spreadsheet for Moment Calculations Using Dispersal Angle Method ... 154 Appendix L : Constants vs. Cantilever Length Relationship for M M E 158 v i List of Tables Table 1 - Design Forces for Traffic Railings 4 Table 2 - Bridge Railing Test Levels and Crash Test Criteria 6 Table 3 - Impact Design Forces for Traffic Barriers and Deck Overhang 15 Table 4 - Unfactored Loads on Traffic Barriers 18 Table 5 - Transverse Moments in Cantilever Slabs Due to Horizontal Railing Loads 21 Table 6 - Element Properties 29 Table 7 - Material Properties 31 Table 8 - Simple Cantilever Testing Results for Increasing Mesh Density in Y-Direct ion 32 Table 9 - Simple Cantilever Testing Results for Increasing Mesh Density in X-Direct ion 32 Table 10 - Deck Test Results for Increasing Mesh Density in X-Direction 33 Table 11 - Deck Test Results for Increasing Mesh Density in Y-Direction 33 Table 12 - Results of New Dispersal Angle Method for PL-3 Inner Portion 69 Table 13 - Results of New Dispersal Angle Method for PL-3 End Portion 69 Table 14 - Results of New Dispersal Angle Method for PL-2 Inner Portion 69 Table 15 - Results of New Dispersal Angle Method for PL-2 End Portion 70 Table 16 - Simplified Dispersal Angles 75 Table 17 - Linear Interpolation for Dispersal Angles 75 Table 18 - Example 1 76 Table 19- Example II 76 Table 20 -Example III 77 vi i Table 21 - M M E Approximation for PL-2 Internal Portion (thk250-oh600-0.87) 106 Table 22 - M M E Approximation for PL-2 Internal Portion (fhk250-oh900-0.87) 108 Table 23 - M M E Approximation for PL-2 Internal Portion (thk250-ohl200-0.87) 110 Table 24 - M M E Approximation for PL-2 Internal Portion (thk250-ohl 500-0.87) 112 Table 25 - M M E Approximation for PL-2 Internal Portion (thk250-ohl 500-0.87) 114 Table 26 - M M E Approximation for PL-2 End Portion (thk250-oh600-0.87) 117 Table 27 - M M E Approximation for PL-2 End Portion (thk250-oh900-0.87) 119 Table 28 - M M E Approximation for P L - 2 End Portion (thk250-ohl200-0.87) 121 Table 29 - M M E Approximation for PL-2 End Portion (thk250-ohl500-0.87) 123 Table 30 - M M E Approximation for PL-2 End Portion (thk250-oh 1800-0.87) 125 Table 31 - M M E Approximation for PL-3 Internal Portion (thk275-oh600-1.07) 128 Table 32 - M M E Approximation for PL-3 Internal Portion (thk275-oh900-1.07) 130 Table 33 - M M E Approximation for PL-3 Internal Portion (thk275-oh 1200-1.07) 132 Table 34 - M M E Approximation for PL-3 Internal Portion (thk275-ohl 500-1.07) 134 Table 35 - M M E Approximation for PL-3 Internal Portion (thk275-ohl 800-1.07) 136 Table 36 - M M E Approximation for PL-3 End Portion (thk275-oh600-1.07) 139 Table 37 - M M E Approximation for PL-3 End Portion (thk275-oh900-1.07) 141 Table 38 - M M E Approximation for PL-3 End Portion (thk275-ohl200-1.07) 143 Table 39 - M M E Approximation for PL-3 End Portion (thk.275-oh1500-1.07) 145 Table 40 - M M E Approximation for PL-3 End Portion (thk275-ohl 800-1.07) 147 Table 41 - Spreadsheet for Moment Calculations Using Dispersal Angle Method 155 Vlll List of Figures Figure 1 - Bridge Railing Design Forces, Locations and Distribution Length 5 Figure 2 - Y L A of Barrier for Inner Portion 8 Figure 3 - Y L A of Barrier for End Portion 9 Figure 4 - Sample Standard Concrete Barriers Commonly Used in Washington State.... 14 Figure 5 - Application of Rail ing Loads 18 Figure 6 - Y L A Model for Deck 25 Figure 7 - Load Dispersion for Cantilever Deck Inner Portion 38 Figure 8 - Load Dispersion for Cantilever Deck End Portion 40 Figure 9 - Dispersal Angle Concept 41 Figure 10 - Moment Intensity vs. Deck Distances Graph fro PL-1 44 Figure 11 - Moment Intensity vs. Deck Distance Graph for PL-2 44 Figure 12 - Moment Intensity vs. Deck Distance Graph for PL-3 45 Figure 13 - M M E Concept 47 Figure 14 - Transverse Load on PL-3 Inner Portion Barrier and Deck with 1200mm Overhang 49 Figure 15 - Transverse Moment Distribution due to PL-3 Transverse load 50 Figure 16 - M M E Spreadsheet for Barrier 50 Figure 17 - Plots for Transverse Moment on Barrier 51 Figure 18 - M M E Spreadsheet for Deck 51 Figure 19 - Plot for Deck Transverse Moment 52 Figure 20 - Spreadsheet Used to Determine Modification Factor " A " 55 Figure 21 - Plot of F E A , Dispersal Angle, and Modification Factor " A " Results 56 ix Figure 22 - Spreadsheet Used to Determine Modification Factor " B " 57 Figure 23 - Plot of F E A , Dispersal Angle, and Modification Factor " B " Results 57 Figure 24 - Spreadsheet Used to Determine Modification Factor " C " 58 Figure 25 - Plot of F E A , Dispersal Angle, and Modification Factor " C " Results 59 Figure 26 - Spreadsheet Used to Present the Results of Various Methods 60 Figure 27 - Plot of F E A , Dispersal Angle, and Modification Factors Results 60 Figure 28 - M M D A Concept by Changing Dispersal Angles 62 Figure 29 - Loading Condition and System Configuration for PL-2 Inner Portion 71 Figure 30 - Transverse Moment Intensity for PL-2 Inner Portion 72 Figure 31 - Loading Condition and System Configuration for PL-2 End Portion 73 Figure 32 - Transverse Moment Intensity for PL-2 End Portion 74 Figure 33 - Typical PL-3 Cast-In-Place Barrier 78 Figure 34 - Section Capacity Calculation Performed by "Response-2000" 78 Figure 35 - Simple Cantilever F E M for Determination of Mesh Size 88 Figure 36 - PT in PL-2 F E M for Inner Portion 90 Figure 37 - P L in PL-2 F E M for Inner Portion 91 Figure 38 - P V in PL-2 F E M for Inner Portion 92 Figure 39 - PT in PL-2 F E M for End Portion 94 Figure 40 - P L in PL-2 F E M for End Portion 95 Figure 41 - P V in PL-2 F E M for End Portion 96 Figure 42 - PT in PL-3 F E M for Inner Portion 98 Figure 43 - P L in PL-3 F E M for Inner Portion 99 Figure 44 - P V in PL-3 F E M for Inner Portion 100 Figure 45 - PT in PL-3 F E M for End Portion 102 x Figure 46 - P L in PL-3 F E M for End Portion 103 Figure 47 - P V in PL-3 F E M for End Portion 104 Figure 48 - Sample Plot of PT for PL-3 Barrier Internal Portion with 1800mm Overhang 150 Figure 49 - Sample Plot of PT for PL-3 Deck Internal Portion with 1800mm Overhang 151 Figure 50 - Sample Plot of P V for PL-3 Deck Internal Portion with 1800mm Overhang .' 152 Figure 51 - Sample Plot of Combined Loads for PL-3 Deck Internal Portion with 1800mm Overhang 153 Figure 52 - Constant A vs. Cantilever Length Plot for M M E 159 Figure 53 - Spreadsheet M M E Calculations for Constant A 159 Figure 54 - Constant B vs. Cantilever Length Plot for M M E 160 Figure 55 - Spreadsheet M M E Calculations for Constant B 160 Figure 56 - Constant C vs. Cantilever Length Plot for M M E 161 Figure 57 - Spreadsheet M M E Calculations for Constant C 161 x i Acknowledgment I would like to express my deepest gratitude to M r . Kev in Basin (Chief Bridge Engineer) and M s . Sharlie Huffman (Bridge Seismic Engineer) from the Ministry of Transportation of British Columbia. Their support and contributions to this research have been invaluable to my work. I would also like to acknowledge my research supervisor, Dr. S. F. Stiemer, for his continual guidance and for the immense effort he has put into making this research an interesting and enlightening experience for me. His wisdom and sense of humor have helped me to overcome challenges, and delighted me in many ways. Further, I would like to express my appreciation for my fellow students: Nathan Loewen and Chun Hai (Sean) Xiao, both o f whom participated in this project. Last, but not least then, I sincerely thank my family and friends for the time and effort they have put into encouraging me and supporting me throughout my journey. x i i Chapter 1: Introduction The objective of this research is to develop an effective tool that is practical and which can be easily used to design reinforced concrete parapets, as well as the cantilever portion of bridge decks, against railing loads as specified in the C A N / C S A - S 6 - 0 0 Canadian Highway Bridge Design Code. This design tool should allow designers to calculate the design forces o f both the cantilevered deck and the traffic barrier/parapet system with minimum effort. It should also seek a compromise with the current code S6-00, and should effectively improve, or upgrade, the current code's proposed methods. The method developed in this research combines the concepts of finite element analysis and the dispersal angle method, called the maximum moment dispersal angle ( M M D A ) method. The scope of this research focuses on the design criteria for performance level 2 and 3. These two performance levels have the highest load magnitude and the widest area of affect, based on the standards of the S6-00. This report w i l l include a general discussion o f the methods suggested by various jurisdictions, such as AASHTO LRFD Bridge Design Specifications 2004, Washington State D O T Bridge Design Manual LRFD, C A N / C S A - S 6 - 8 8 Design of Highway Bridges and C A N / C S A - S 6 - 0 0 Canadian Highway Bridge Design Code. Background information such as site condition ranking systems, specified design loads and their applications, design criteria, and analysis methods are topics included in that review. A discussion of the strengths and weaknesses found in each jurisdiction is also essential for contributing towards the improvement of current methods. For the purposes of this research, several methods of analysis have been used for study, including yield line analysis ( Y L A ) , finite element analysis (FEA) , and dispersal 1 angle methods. The theory behind each method is reviewed in detail, then following by a discussion and an investigation of their advantages and disadvantages. In order to come up with a method that takes advantages of the above methods while minimizing their drawbacks, a new method called "maximum moment envelope" ( M M E ) w i l l be introduced. However, this new concept deviates from a possible ideal outcome of this research because of its complexity and difficulty when applying to practice. A s a result, a further improved version of this method, called the "maximum moment dispersal angle" ( M M D A ) wi l l be established. This one has proven to be more useful and effective than other methods. The results of the M M D A wi l l then be presented in the form of a table for easy and efficient application. Finally, examples usage of this new method as well as examples of some of the traditional methods wi l l be shown in a correlation study. 2 Chapter 2: Background Research and Literature Review A s part of the literature review, relevant jurisdictions w i l l be studied and discussed in detail in the following section in order to provide background information on this topic. The jurisdictions being reviewed are the AASHTO LRFD Bridge Design Specifications, Washington State D O T Bridge Design Manual LRFD, CAN/CSA-S6-88 Design of Highway Bridges and CAN/CSA-S6-00 Canadian Highway Bridge Design Code. Methods of analysis, loading criteria, traffic barriers and bridge deck overhang designs are all issues included in this discussion. Although the design methods suggested by each jurisdiction have their own distinct advantages, only a few of them have been selected for the following discussion as the scope o f this research. Ultimately one method is proposed for further studies as far as the scope of this research allows. The review of various codes provides solid background information in making such a decision. Details of the recommended design methods can be found in Chapter 3 "Methods of Analysis." 2.1 AASHTO LRFD Bridge Design Specifications The third edition of the American Association of State and Highway and Transportation Officials {AASHTO) published in 2004 is considered in this study. In general, AASHTO codes are well recognized and are used as the basis of many other official codes in different municipalities. For example, the Canadian Highway Bridge Design Code C A N / C S A - S 6 - 0 0 , which is discussed in a later section, makes many references to the AASHTO as a supporting document. Reviewing this code is needed in order to understand the philosophy behind various design methods in other codes. 3 2.1.1 Limit States and Resistance Factors ASSTHO Clause 13.6.1 states that the strength limit should be applied according to load combinations specified in ASSTHO Table 3.4.1-1, using resistance factors as specified in ASSTHO Articles 5.5.4, 6.5.4, 7.5.4, and 8.5.2. For an extreme event limit state, the bridge deck design forces may be determined by an ultimate strength analysis using the loads specified in Table 1 and applied as illustrated in Figure 1 as factored loads. Table 1 - Design Forces for Traffic Railings (Data source: AASHTO Table Al3.2-1) Design Forces and Designations Railing Test Levels TL-1 T L - 2 TL-3 T L - 4 T L - 5 A TL-5 T L - 6 F t , Transverse (kip) 13.5 27 54 54 116 124 175 F L , Longitudinal (kip) 4.5 9.0 18.0 18 39 41 58 F v , Vertical (kip) Down 4.5 4.5 4.5 18 50 80 80 L t and L L (ft) 4.0 4.0 4.0 3.5 8.0 8.0 8.0 L v (ft) 18.0 18.0 18.0 18.0 40.0 40.0 40.0 H e (min)(in) 18 20 24 32 40 42 56 M i n . H Height of Rai l (in) 27 27 27 32 40 54 90 4 Figure 1 - Bridge Railing Design Forces, Locations and Distribution Length (Data source: AASHTO Figure A13.2-1) 2.1.2 Test Levels AASHTO has introduced seven test levels in correspondence with the NCHRP Report 350, Recommended Procedures for the Safety Performance Evaluation of Highway Features. The seven test levels in AASHTO Clause 13.7.2 are specified so as to evaluate the performance factors of bridge railings, which include structural adequacy, occupant risk, and post-impact behavior of the test vehicles. The seven test levels being considered are T L - 1 , T L - 2 , T L - 3 , T L - 4 , T L - 5 A , T L - 5 , and T L - 6 as ranked from the lowest level of TL-1 to the highest level of T L - 6 . TL-1 is generally acceptable for a work zone with low posted speed limits, and for very low-volume, low-speed local streets. T L - 6 on the other hand, is generally acceptable for applications on freeways with high speed, high traffic volume, a higher ratio of heavy vehicles, and otherwise unfavorable site conditions. AASHTO Commentary Clause C13.7.2 also states that the T L - 4 railing is expected to satisfy the majority of interstate design requirements. Table 2 provides corresponding 5 vehicle weights, speeds, and angles of impact as the testing criteria for the different test levels. Table 2 - Bridge Railing Test Levels and Crash Test Criteria (Data source: AASHTO Table 13.7.2.1) Vehicle Characteristics Small Automobiles Pickup Truck Single-Unit Van Truck Van-Type 1 Tractor-Tanker Tractor-Trailers | Trailers W(kip) 1.55 1.8 4.5 18.0 50.0 80.0 80.0 B(ft) 5.5 5.5 6.5 7.5 8.0 8.0 8.0 G ( i n ) 22 22 27 49 64 73 81 Crash angle, 0 20° 20° 25o 15° 15° 15° 15° Test Level Test Speeds (mph) TL-1 30 30 30 N / A N / A N / A N / A T L - 2 45 45 45 N / A N / A N / A N / A TL-3 60 60 60 N / A N / A N / A N / A T L - 4 60 60 60 50 N / A N / A N / A T L - 5 A 60 60 60 N / A 50 N / A N / A TL-5 60 60 60 N / A N / A 50 N / A T L - 6 60 60 60 N / A N / A N / A 50 2.1.3 Traffic Barrier AASHTO Clause 13.7.3 gives detailed design criteria for traffic railings. A n existing railing system can be used without further analysis or testing i f it has been previously tested and proven crashworthy. However, a new system can only be used i f it is first approved by full-scale crash tests. The specifications on minimal edge thickness for concrete deck overhangs supporting deck-mounted post systems or concrete barriers are 8 inches, and 12 inches for a side-mounted post system. The height of concrete railings with slopping surfaces should be greater than, or equal to, 27 inches for TL-3 and 32 6 inches for T L - 4 . Furthermore, concrete railing with a vertical surface should have a minimum height of 27 inches. Appendix A in AASHTO Chapter 13 provides detail information on the specifications of railing geometry and anchorages, railing design forces, and barrier and deck overhang designs. In short, anchorages and reinforcement in concrete barriers should be properly designed, using, amongst other things, bond, honks, embedded plates, and sufficient embedment length, so that the yield strength o f the connections can be fully developed. Pull-out failure is an example of immature failure where the yield strength o f the connections is not fully developed. The design loads for barriers under the impact of different test levels should be taken as specified in Table 1, and should be applied as illustrated in Figure 1. The transverse and longitudinal loads are applied at the effective height of the vehicle rollover force, H e such that: 2Ft Where: G = height of vehicle center o f gravity above bridge deck, as specified in Table 2 (ft) W = weight of vehicle corresponding to the required test level, as specified in Table 2 (kip) B = out-to-out wheel spacing on an axle, as specified in Table 2 (kip) F t = transverse force corresponding to the required test level as specified in Table 1 (kip) The vertical loads are applied on top of the railing but need not be applied in conjunction with the transverse and longitudinal loads. 7 AASHTO makes use of yield line analysis ( Y L A ) in the capacity design of reinforced concrete and prestressed concrete barriers, or parapets, as described in AASHTO Clause A13.3.1. The concept of yield line analysis is explained later under Section 3.1 " Y i e l d Line Analysis." According to the yield line patterns predefined by ASSTHO as shown in Figure 2 and 3, though, a unique parameter called the critical length of failure pattern, Lc , is introduced so as to determine the total transverse resistance o f the railing, R w , as described by Equations (2) - (5). Figure 2 - Y L A of Barrier for Inner Portion (Data source: AASHTO Figure CA13.3.1-1) 8 For inner portion analysis: R. r 2 V 8 M , + SMW + M c L c 2 ^ H V The critical wall length L c is taken as: c 2 V ^  J SH(Mb+Mw) H Figure 3 - Y L A of Barrier for End Portion (Data source: AASHTO Figure CA13.3.1-2) For end portion analysis: R... = f 2 V M,+M... + M ^ V H J c 2 fr \ v 2 y H M c J Where: H height of wall (ft) L c = critical length of yield line failure pattern (ft) L t = longitudinal length of distribution of impact force F t (ft) R w = total transverse resistance of the railing (kip) Mb = additional flexural resistance of beam in addition to M w , i f any, at top of wall (kip-ft) M c = flexural resistance o f cantilevered walls about an axis parallel to the longitudinal axis of the bridge (kip-ft/ft) M w = flexural resistance of the wall about its vertical axis (kip-ft/ft) F t = transverse force specified in Table 1 (kip) It is important to note then, that this Y L A has assumed that the failure mechanism occurs within the barrier and does not extend to the deck. In other words, the deck must be designed to properly resist the impact loads so that it does not fail before the barrier does, otherwise Equation (2) - (5) are not valid. Also , the longitudinal length of the barrier should be sufficient, meaning that it should be greater than or equal to the critical length, L c , for the assumed yield line pattern to take place within the barrier. A s such, some other means of analysis should be used for capacity calculations i f the barrier is considered to be short. For example, a precasted discontinuous barrier may be considered short based on its flexural resistances. Lastly, it is assumed that the negative and positive resisting moments of barriers are equal. The resisting moment of a barrier depends on its geometry, its material, and on the layout of reinforcement. Ideally, a perfectly symmetrical barrier, with respect to its vertical axis, would yield the same negative and positive resisting moments. However, barriers used in practice are usually perfectly vertical on the outer surface while having the inner surface slightly sloped, meaning that the negative and positive resisting 10 moments may not in fact be equal. In this case, adjustment of Equation (2) - (5) may be necessary to ensure the accuracy of the barrier capacity calculations. When the total transverse resistance of the railing, Rw, is determined, it is to be subsequently compared with the specified loads in Table 1 for structural adequacy. 2.1.4 Deck Overhang Deck overhang design should be performed as suggested under ASSTHO Article A13.4, with considerations for three separate design cases. The first case is an ultimate limit state design, with an application of the transverse and longitudinal loads as specified in Table 1 and Figure 1. The second case also involves an ultimate limit state design, but this one takes only the vertical loads specified in Table 1 and Figure 1. The last case is a strength limit state design, with applied loads as specified in ASSTHO Article 3.6.1. For the first design case, the deck may be expected to provide flexural resistance, acting coincidently with the tensile force, T, which can be calculated using the following equation: R. w T = (6) LC+2H Where: Rw barrier resistance specified in Equation (2) or (4) L c critical length of yield line failure pattern (ft) H height of barrier (ft) T tensile force per unit of deck length (kip/ft) 11 2.1.5 Comments The design philosophy of AASHTO Y L A can be well applied to the predefined yield line pattern i f we assume that a failure occurred only within the barrier and that it did not extend to the deck. This assumption, while it is valid, would take full advantage of the power of Y L A . However, design procedures must be clearly defined to ensure such assumptions continue to work towards the safety of designs. In addition, this method becomes invalid i f the barrier is considered short, or at least shorter than the critical length, Lc . In some cases, precasted barriers may be considered short barriers and should therefore be checked carefully before the application of the Y L A method, as suggested by Equation (2) - (5). Furthermore, the drawback of this design method lies in the limitations of Y L A , which are discussed later in Section 3.1 " Y i e l d Line Analysis." Also, AASHTO makes use of rather conservative design criteria in their designs for the deck overhang. They state that the deck overhang capacity, which acting coincident with the tensile force specified in Equation (6), may be designed to exceed the transverse flexural resistance of the barrier. Since the crash test system is oriented towards improving the likelihood of survival in an accident and not necessary the ultimate limit state design, the barrier is l ikely to be over-designed, which would lead to an over-design of the deck. On the other hand, localized failure at the edge of the deck is a mode of brittle failure which may lead to casualties, so conservative design procedures may, in fact, be necessary to ensure the deck is capable of resisting the forces transferred down from the barriers. Overall, the design methods specified in this code work well i f all o f its assumptions prove accurate, although that is a far-fetched possibility in some 12 circumstances. The instructions are easy to follow though, and are easy to carry out in practice as well . 2.2 Washington State DOT Bridge Design Manual LRFD The Washington State Department of Transportation (WSDOT) Bridge Design Manual (BDM) 2005 is a guide for those who design bridges in Washington State. In association with that guide, the ASSTHO LRFD Bridge Design Specifications is the second of the two basic documents for highway bridge and structure design in Washington State. BDM supplements ASSTHO by providing additional directions, design aides, and examples. It also takes precedence when conflicts arise between the two documents' standards. 2.2.1 Traffic Barrier Design guidelines for the bridge traffic barriers, which can be found in Chapter 10.2 of the B D M , are developed in accordance with Chapter 13 of the AASHTO LRFD Bridge Design Specifications. BDM Clause 10.2.1 gives general guidelines for the types and heights o f the barriers to be used on highway bridges depending on the different traffic conditions involved. T L - 4 is the test level adopted by BDM from AASHTO, and so, standard bridge traffic barriers in Washington State are commonly designed to accommodate it. Similar to ASSTHO, BDM also carries over the railing test level grading system according to site conditions. Two samples of the standard T L - 4 concrete barriers, which have been previously crash tested and are now commonly used in Washington State, are shown in Figure 4. The "Shape F " traffic barrier has been crash tested twice, 13 first in late 1996 under NCHRP 230, and then again more recently under NCHRP 350. It was proven to have met the safety requirements. A newer traffic barrier, "Single Slope," was introduced later in the '90s to speed up construction by using the 'slip forming' method. However, the drawback of that 'slip forming' method was that it resulted in an increase of the concrete cover required by W S D O T on the traffic side of the barrier. 5HAFE F SINGLE SLOPE Figure 4 - Sample Standard Concrete Barriers Commonly Used in Washington State (Data source: WSDOT Bridge Design Manual Figure 10.2.3.2) 2.2.2 Deck Overhang BDM Clause 10.2.4 "Design Criteria" states that the W S D O T barriers are reinforced as per the crash test results in NCHRP Report 350. This may lead to an over-design of the traffic barrier though, and hence an over-design of deck overhangs. Therefore, the nominal transverse barrier resistance, Rw, that is transferred from the barrier to the deck should be equal to 120% of the transverse loads, F t , as specified in Table 1. This prevents 14 the over-design o f deck overhangs. BDM also requires, as does ASSTHO, that the flexural resistance of the deck exceed that of the barrier at its base, so that the yield line pattern remains valid by having failures occur within the barrier. 2.2.3 Design Criteria B y inheriting the method of analysis used in ASSTHO, BDM produces its own table of impact design forces for traffic barriers and deck overhangs based on the few types o f barriers that are most commonly used in Washington State. This can be found in Table 3. Table 3 - Impact Design Forces for Traffic Barriers and Deck Overhang (Data source: WSDOTBridge Design Manual Section 10.2.4, pg. 10-18) Traffic Barrier/Cantitevered Slab Design Yield Line Theory Parameters Shape F Interior End Single Slope Interior End Shape F 42" Interior End Single Slope 42" Interior • End Traffic Barrier Design Mc (ft-kips/ft) 20.62 20.62 19.39 19.39 29.18 29.18 25.22 25.22 MwH (ft-kips) 42.48 45.98 44.72 42.17 97.83 96.91 81.06 77.50 Lc(ft) 8.61 4.75 9.19 4.79 14.48 9.26 14.30 9.17 Rw (kips) 133.09 73.48 125.79 65.53 241.47 154.33 205.99 132.17 Ft (kips) 54.00 54.00 54.00 54.00 124.00 124.00 124.00 124.00 Deck Overhand Design Ms (ft-kips/ft) 12.40 17.13 12.36 17.56 24.24 32.04 24.46 28.60 T (kips/ft) 4.65 6.43 4.36 6.20 6.93 9.15 6.99 8.17 1.2*Ft(kips) 64.80 64.80 64.80 64.80 148.80 148.80 148.80 148.80 Deck to Barrier Reinforcement As required (ir^ a/ft) 0.29 0.41 0.29 0.42 0.44 0.59 0.50 0.59 As provided 0.41 0.41 0.41 0.41 0.66 0.66 0.66 0.66 S1 Bars #5@ 9in #5(g 8 9in #6@ Sin #6@8in S2 Bars #4@ 18in #4@ 18in #5 @ 16in #5 @ 16in Weight Area W (lbs/ft) 472.5 505.3 729.0 692.2 A (in A2) 425.2 454.8 656.1 623.0 2.2.4 Comments The W S D O T Bridge Design Manual, using ASSTHO Bridge Design Specifications as its backbone, is a useful resource for traffic barrier and deck overhang design. It provides 15 extensive background information and clear design criteria. It also went one step further in solidifying the impact design forces, by applying some of the design methods proposed in ASSTHO for the standard traffic barriers used in Washington State. These design forces are presented in Table 3, making them easy to follow and to use in practice. However, this table is only valid for certain types of barriers, making it less useful for those who are outside of Washington State. Nonetheless, BDM is a good example of making full use of the ASSTHO Bridge Design Specifications. 2.3 CAN/CSA-S6-00 and CAN/CSA-S6-88 The Canadian Highway Bridge Design code S6-00 and the Design of Highway Bridges S6-88 are both prepared by the Canadian Standards Association (CSA) for highway bridge design in Canada. CAN/CSA-S6-00 combines with and replaces both CAN/CSA-S6-88 Design of Highway Bridges, and OHBDC-91-01 Ontario Highway Bridge Design Code, becoming the standard national code used in Canada. This section reviews mainly the design methods and criteria suggested in S6-00, while also including some materials from S6-88 for reference. 2.3.1 Traffic Barrier S6-00 Clause 12.5.1 states that traffic barriers shall be crash tested to determine their effectiveness in reducing the consequences of vehicles leaving the highway upon the occurrence of an accident. S6-00 Clause 12.5.2.3 provides detailed information on crash test requirements. The adequacy of a barrier that has the same details as those of an 16 existing barrier can be determined from an evaluation of the performance of the existing barrier. 2.3.2 Performance Level Performance levels (PL) 1, 2, and 3 are the ranking systems used in S6-00 to determine the site condition for a bridge. Alongside that, the barrier exposure index is used in accordance with S6-00 Clauses 12.5.2.1.2 and 12.5.2.1.3 to determine the performance level of a bridge site. This exposure index is evaluated based on the estimated average annual daily traffic for the first year after construction, as well as on the highway type, curvature, and grade, and superstructure height factors which are shown in S6-00 Tables 12.5.2.1.2 (a) - (d). Wi th that, then, the barrier exposure index is used, along with barrier clearance, design speed, and ratio of trucks as shown in S6-00 Tables 12.5.2.1.3 (a) - (c), to determine the performance level suitable for the bridge site. The minimum barrier heights for P L - 1 , 2, and 3 traffic barriers are 0.68 m, 0.80 m, and 1.05 m respectively, as laid out in S6-00 Table 12.5.2.2. 2.3.3 Design Criteria Under S6-00 Clause 12.5.2.4, traffic barrier anchorages should be designed to resist the unfactored loads as specified in Table 4, and applied as illustrated in Figure 5. These are also used in the calculation of design forces for deck overhang. Unlike AASHTO, where the vertical load is applied independently of the other two loads, the transverse and longitudinal loads are to be applied in tandem with the vertical load on the barrier, as 17 specified in S6-00. For reference, the transverse load used in S6-88 was much lower; 35 k N and 45 k N for roadway widths of 9m or less, and greater than 9m, respectively. Also , the transverse load should be distributed over a longitudinal length of 1.5 m. Table 4 - Unfactored Loads on Traffic Barriers (Data source: S6-00 Table 3.8.8.1) Performance Level Transverse Load - PT (kN) Longitudinal Load -P L (kN) Vertical Load - P V (kN) PL-1 50 20 10 PL-2 100 30 30 PL-3 210 70 90 Note: . . . . . . . (a) Traffic barrier typos are illustrative only and other types may be used. • (b) Transverse load "P," shall be applied over a barrier length of 1200 mm for PL-1 barriers, 1050 mm lor PL-2 barriers, and 2400 mm for PL-3 barriers. > • ; : i . ; . (c) Longitudinal load "P," shall be applied at the same locations'arid over the same barrier lengths as Pt FSf-poM, and railing barriers, the longitudinal load shaft not be distributed to more than 3 posts. (d) Vertical load "Pf shall be applied over a bamer length <rf5500 mm for PL-1 and PL-2 barriers and 12000|rnm ••: for PL-3 barriers. - ••• • ., > .... • (e) • These loadsshallbe used forthe design oftraffic barrier anchorages anddecks only. Figure 5 - Application of Railing Loads (Data source: S6-00 Figure 12.5.2.4) 18 2.3.4 Deck Overhang According to S6-00 Clause 5.4.7, analysis of deck overhang under railing loads should be carried out using the methods given in S6-00 Clause 5.7.1.6.3. Those include refined methods in accordance with S6-00 Clause 5.9, and yield line analysis. 2.3.4.1 Refined Methods The refined methods suggested by Clause 5.9 are divided into four categories. The first category includes methods for general applications, such as grillage analogy, orthotropic plate theory, finite element analysis, finite strip, folded plate, and semi-continuum method, in accordance with S6-00 Clause 5.9.1. The second category is a specific application of influence surfaces in accordance with S6-00 Clause 5.9.2. Model Analysis is the third category specified under S6-00 Clause 5.9.3, and it involves the testing of a physical model. Lastly, S6-00 suggests that other methods may be used upon approval. The limitations of applicability for the above methods can be found in S6-00 Table 5.9.4. 2.3.4.2 Simplified Methods S6-00 Clause C5.7.1.6.3 in the Commentary recalls a simplified method from the OHBDC and S6-88 which was used for determining moments in the deck due to concentrated horizontal railing loads. This method assumes that the horizontal loads transferred down from the barrier to the deck are distributed over a constant length of 1.5m at the outer edge of the deck. A s the moments progress into the deck due to the horizontal loads, this length is increased by 0.8 times the distance between the outer edge 19 of the deck and the section of the deck being analyzed. That results in a dispersal angle of 21°. Since the design loads in the previous code were much lower than those specified in this code, it is necessary to perform the more rational, refined methods that were suggested in S6-00 Clause 5.9 2.3.4.3 Dispersal Angle Method The S6-00 Commentary also mentions the use of dispersal angles, resulting from F E A , in some Canadian provinces. This is shown in Table 5. Details of this dispersal angle method are presented in Section 3.3 "Dispersal Angle Method." A point worth noting here is that although load dispersal is an old concept from the previous code, when it is used in combination with the F E A results, it becomes an effective tool for approximating the dispersion of forces on the deck. 20 Table 5 - Transverse Moments in Cantilever Slabs Due to Horizontal Railing Loads (Data source: S6-00 Commentary Table C5.7.1.6.3) Horizontal Load or Moment Performance Level 3 Performance Level 2 Dispersion at Inner Portion of Deck Barrier Barrier with Rail Factored Horizontal Load Pt (Clause 3.8.8.1) 357 kN 170 kN Length of Load Application (Clause 12.5.2.4) 2400 mm 1050 mm Height of Load Application Above Deck (clause 12.5.2.4) 900 mm 700 mm Moment in Inner Portions of Deck per metre at Face of Barrier 83 kN-m/m 38 kN-m/m Dispersal Angle for Barrier Dispersal Angle for Deck ' 0 = 42° 0 = 47° e = 56° 0 = 55° Tensile Force in Inner Portion of Deck at Deck Edge 144 kN/m 100 kN/m Dispersal Angle for Barrier Dispersal Angle for Deck 0 = 3° e = 10° 0 = 25° 0 =20° Moment in End Portion of Deck per metre at Face of Barrier , 102 kN-m/m 52 kN-m/m Dispersal Angle for Barrier Dispersal Angle for Deck 0 =48° e = 45° 0 = 55° 0 = 55° Tensile Force in End Portion of Deck at Deck Edge 161 kN/m 142 kN/m Dispersal Angle for Barrier Dispersal Angle for Deck 0 = 0° a = 0° 0 = 8° 9 = 8° 21 2.3.5 Comments In general, S6-00 and its Commentary provide users with great flexibility in the design of traffic barriers, anchorages, and deck overhangs by offering a varied range of suggested methods. However, the list o f available analysis methods may be too broad for most users, and so may lead to confusion when being applied in practice. Without a standardized method of analysis, the resulting design loads may vary significantly from one designer to another simply because they have been calculated using very different methods. While flexibility is valuable, one standard design method should be created as tool for the majority o f designers to use in common. O f course, in order for a tool to be used commonly by people, it must also be effective and easy to apply. Hence, finding such a design method becomes the objective o f this research. Certainly is is also sensible that users who demand more sophisticated methods of analysis, such as those suggested by the code, are still given the authority to do so. 22 Chapter 3: Methods of Analysis There are a number of methods available for carrying out the traffic barrier and deck overhang design calculations, as we saw in the pervious section. Some are better than others. While studying every method in detail is not possible here, considering the scope of this thesis, a few of the better, more well-known methods have been selected for review in this section. Those methods include yield line analysis, finite element analysis, and the dispersal angle method. In addition, two new and improved concepts w i l l be introduced as having evolved from some of the existing methods through this research. They are referred to as the maximum moment envelope ( M M E ) method and the maximum moment dispersal angle ( M M D A ) method. As suggested by their names, these two new methods act as a means of approximating "maximum" moment intensities for the design forces, which would in effect, give the design an additional level of safety. The M M D A method is recognized as the most useful of the available methods for the purposes of this research, because o f its overall strength. Then finally, some design examples w i l l be presented and the results w i l l be compared using the different methods mentioned before. 3.1 Yield Line Analysis (YLA) According to S6-00 Clause 5.4.7, Y L A may be used in the ultimate limit state design along with elastic analysis for the analysis of decks. Clause 5.11.3.2 also briefly explains the theory of Y L A and puts forward the criteria used by Y L A to determine inelastic-dynamic responses. 23 3.1.1 Design Philosophy In short, Y L A is basically another form o f plastic analysis that uses the principle o f virtual work. With an assumed yield-line pattern for the reinforced concrete slab, the ultimate plastic capacity of the slab can be determined by equating external work to internal work without knowing the actual load. This is illustrated as follows: Y,PS = \Miais + YdMt ( 7 ) where P is the concentrated external load, 8 is the deformation, M is any distributed moments that satisfy equilibrium with P, K is any distributed curvature that is compatible with hinge rotation, cp, and s is the length of each member. This analysis gives an upper bound solution such that the collapsed load obtained is either higher or equal to the true failure load of the system. Therefore, i f an incorrect set of yield-line patterns is assumed, the collapsed load obtained could potentially be overestimated for the given slab reinforcements. 3.1.2 Sample Application The model can be set up using some variables to represent the geometry of the yield-line pattern, instead of requiring the use of some assumed actual values. Hence, the true yield-line pattern can be determined by optimizing the load with respect to these variables. A model of the deck has been set up as shown in the following figure, with two variables (a and b) to represent its yield-line pattern. 24 Yie ld - l i ne S t r e s s distr ibut ion Figure 6 - Y L A M o d e l for Deck The model in Figure 6 is developed using many assumptions. First, the deck fails in flexural, not in shear, so that Y L A is applicable. Second, only the force in x-direction contributes to the external work applied to the system, while the force in y-direction is not considered. Also , both the barrier and the connection between the barrier and the deck must be properly designed so that they survive the impact; hence, the impact force can be successfully transferred down to the deck. Lastly, both the barrier and the connection suffer minor damage, such that little or no energy is lost there. Based on these assumptions, the following equation is derived: Where: 25 d = the assumed width of the deck subjected to the impact load transferred from the barrier attached to it (m) w = the distance from the edge of the deck to the nearest girder underneath the deck, where negative bending moment is max (m) m x l = height of impact location to the deck (m) m y l = positive plastic moment of the deck in x-direction (kN*m/m) m x 2 = negative plastic moment of the deck in x-direction (kN*m/m) m y 2 = negative plastic moment of the deck in y-direction (kN*m/m) m x 3 = negative plastic moment of the deck in x-direction at the location above the girder (kN*m/m) a = variable representing the yield-line pattern b = variable representing the yield-line pattern F = failure load of the deck; a function of a and b (kN) A = virtual displacement (m) 3.1.3 Comments Note, however, that this method is only valid i f the corrected set of yield line patterns can be predicted in advance. Therefore, it should be used in conjunction with other methods to ensure that the assumptions are correct. For example, it could be used with F E A . In that case, the result of a simple F E A shows an undesirable localize failure mechanism that Y L A is not capable of dealing with. However, i f the outer portion of the deck is reinforced, the next failure mechanism should take place at the girder and hence Y L A can be used to predict such a failure load. Again, the accuracy of the Y L A solution depends heavily on the accuracy o f the predicted yield line pattern. But for an element as large as a bridge deck, with its irregular loadings and configuration, the yield line pattern can be 26 very complicated and difficult to predict. Also , the fact that Y L A gives an upper bound solution makes the design unconservative. Therefore, it may not be the best tool to use for a method of analysis in this case, as compared to the F E A . 3.2 Finite Element Analysis (FEA) F E A is listed as one of the most feasible methods of analysis in S6-00 because of its ability to simulate very complicated, three-dimensional structures. The concept of F E A was first introduced in the 1950s, and has been thoroughly used and improved over the last five decades. Although the technique may appear complex to begin with, the basic theory is actually rather straightforward. 3.2.1 Basic Principle The basic principle of the F E A is to divide the actual geometry of a structure into discrete pieces called finite elements. These elements are then joined together by nodes to form a mesh. The number and the type of elements should be chosen in a way that allows the best approximation o f the overall geometry to be achieved through the combined elemental representations. A s a structure is divided into its discrete pieces, the governing equation of each element is calculated and then they are combined to determine the system equations. Those system equations describe the behavior o f the whole structure. A s such, variables such as temperature, displacement, and stress can be evaluated. In this case, the force and stress distribution of a system containing two planes (the barrier and deck overhang) that intersect at a 90 degree angle with specified loads in 27 three directions, can best be evaluated using F E A . For this reason, a linear elastic finite element model ( F E M ) of the traffic barrier and bridge deck system was developed. That was done in preparation for this research by me and my fellow student, Sean Xiao, using SAP2000. Besides the following, further information on the development of the F E M can also be found in Xiao ' s paper, listed in the References. 3.2.2 Finite Element Model This section describes the various modeling details involved in the development of the F E M . Note that the detailing of the F E M plays a big part in the accuracy of the F E A results. Although two F E M s may be set up to simulate the same structures, the difference in their detailing may yield significantly different results. Therefore, one needs to take precautions when making decisions regarding each detail o f a F E M , so as to ensure the accuracy of the F E A . 3.2.2.1 Element Properties Shell elements are chosen for the F E M for their ability to carry forces and moments lying both in their plane as well as transverse to their plane. These shell objects are defined as thick plates which can also demonstrate transverse shear deformation. Compared to solid elements, shell elements achieve greater accuracy with greater comfort, and are therefore more commonly used. A s for the geometry o f these, quadrilateral elements are used because they are more reliable than triangular elements. However, the shell elements are linear, instead of quadratic, meaning that the mesh density must actually be much higher to ensure the accuracy o f results. 28 Table 6 - Element Propert ies PL-2 PL3 Element type Thick shell (linear) Material Reinforced concrete Actual deck thickness 250mm 275mm Modeled deck thickness 170mm 195mm Modeled barrier thickness - top 140mm Modeled barrier thickness - middle 195mm Modeled barrier thickness - bottom 260mm 3.2.2.2 Deck Overhang The variation between actual and modeled deck thickness is a result of minimum cover, rebar size, and the deck's neutral axis. Since the difference in deck thickness for different performance levels may lead to inconsistent results, a series of tests were run in order to determine the sensitivity of deck thickness on stress distribution. The results show that the effects are minimal, as compared to other factors such as cantilever length. This is because the range of possible deck thickness is very narrow. 3.2.2.3 Traffic Barrier The barrier is modeled in three sections with various thicknesses, in an attempt to reflect the actual varying thickness that arises from a sloped surface. In terms of the barrier types, precasted reinforced concrete barriers are used for PL-2 F E M and Cast-In-Place reinforced concrete barriers are used for PL-3 F E M . Note that it is possible to use other types of barriers on each performance level. For example, Cast-In-Place reinforced concrete barriers can be used for both PL-2 and 3, or a post type barrier could be used for PL-2 . However, taking into consideration all possible barrier-PLs, combinations of 29 different ones would result in too much data being generated by the F E A and would therefore make them unmanageable. In doing so, it would also violate the simplicity objective of this research. Consequently, for simplicity's sake as well as for convenience, just one barrier type w i l l be specified for each performance level. 3.2.2.4 Connection/Anchorages As for anchorages^ connections between barriers and decks are assumed to be continuous, instead o f in a series o f discrete joints. This assumption may be valid for PL-3 because rebar connections resemble continuous connections under certain circumstances. For P L -2, in the meantime, the effect of discrete joints on force distribution may or may not be significant, but further investigation is suggested for future studies. 3.2.2.5 Overall Dimensions The F E M that was constructed is 14m in length (the direction parallel to traffic), and varies from 2.6m to 3.8m in depth (the direction perpendicular to the traffic). This variance depends upon the cantilever length. A s mentioned above, the barriers are attached to the deck as continuous connections. Their heights, for P L - 1 , 2, and 3, are 0.68m, 0.80m, and 1.05m respectively, as specified in S6-00 Table 12.5.2.2. The deck then, is rested on two bridge girders, which are located 2m apart. The outer girder is represented by a series of roller supports that allow all rotations and horizontal translations in both x and y directions, although translations in z directions are restrained. The inner girder is modeled by a series of fixed supports, which serve to prevent all rotations and translations. 30 3.2.2.6 Material The material property table below provides detailed information about all o f the default materials. Regular, reinforced concrete is used as the F E M material for the purpose of load calculation only. Capacity designs should then be performed using the design loads calculated without regard for this default material. However, it is essential to check the two materials at the end to ensure the discrepancy between their properties is within a reasonable range. Table 7 - M a t e r i a l Proper t ies Material type Reinforced concrete - Isotropic Mass per volume 2.4 kN-s2 4 m Weight per volume 23.6 kN ~n7 Modulus of elasticity 24800000 kN 2 m Passion's ratio 0.2 Unitless Shear modulus 10300000 kN m2 Specified compressive strength, fc 27600 kN m2 Bending reinforced yield strength, fy 414000 kN m2 Shear reinforced yield strength, fys 275000 kN m2 3.2.2.7 Mesh and Geometry Mesh is a collection of elements joined together by nodes. A s two of the basic properties of a F E M , mesh size and geometry have great effects on the F E A results. Shell elements usually have an aspect ratio of 1:1 to 1:2 as is the convention. For this reason, the basic 31 framework of the F E M is built using 0.35mx0.35m shell elements. However, a much denser mesh is commonly applied in an area of interest to ensure the accuracy of the results produced by the F E A . In order to determine the mesh size then, a series of tests were conducted for convergence. The testing contained two parts: increasing mesh density in the x-direction, and then increasing it in the y-direction independently. A point of reference at the base o f the barrier (node 16 in the F E M ) was selected to be used for comparing the results. The results of PL-2 test are presented in the following tables. Table 8 - Simple Cantilever Testing Results for Increasing Mesh Density in Y-Direction Mesh size (mxm) Max moment intensity(kN*m/m) Vertical displacement (mm) 0.35X0.35 145 -0.835 0.35x0.18 168 -0.842 0.35X0.088 194 -0.847 0.35x0.044 217 -0.849 0.35x0.022 233 -0.85 0.35x0.011 242 -0.85 0.35x0.0055 245 -0.85 0.35x0.0028 246 -0.85 Table 9 - Simple Cantilever Testing Results for Increasing Mesh Density in X-Direction Mesh size (mxm) Max moment intensity(kN*m/m) Vertical displacement (mm) 0.18x0.011 345 -0.85 . 0.088x0.011 452 -0.85 0.044x0.011 536 -0.85 0.022x0.011 590 -0.85 0.011x0.011 614 -0.85 For clarification, x-direction in this F E M is parallel to the direction of traffic, while y-direction is perpendicular to the direction of traffic. A s shown in the above tables, both 32 the force and displacement yield converge at higher mesh densities in the y-direction. In the case of the x-direction, only displacement converged at higher mesh densities and the forces kept increasing as mesh density increased. In order to explain this phenomenon and to ensure the F E M is error-free, a much simpler F E M was built so that results could be hand-calculated for comparison with the F E A results under the same testing conditions as the original F E M . This mini-model was a simple cantilever model built using shell elements, as shown in Appendix A . The cantilever is 1.4m in length and 0.35m in width and has 2 point loads of 10 k N applied to the end. Hence, the moment at the fixed support can be calculated easily by hand. Again, this F E A is tested in two parts as before and the results are shown in the following tables. Table 10 - Deck Test Results for Increasing Mesh Density in X-Direction Mesh size (mxm) M a x moment intensity(kN*m/m) 0.35x0.35 -78.95 0.18x0.35 -78.96 0.088x0.35 -78.97 0.044x0.35 -79 0.022x0.35 -79.06 Hand calculation -80 Table 11 - Deck Test Results for Increasing Mesh Density in Y-Direction Mesh size (mxm) Max moment intensity(kN*m/m) 0.22x0.35 -78.95 0.22x0.18 -81.1 0.22x0.088 -82.4 0.22x0.044 -84.6 0.22x0.022 -86.9 Hand Calculation -80 33 Similar to the original F E A results, this simple cantilever F E M yields results (which are forces, in this case) that converge when meshing in the x-direction, but not when in the y-direction. This simple test shows that by increasing the mesh density in one direction, which is the direction parallel to the cantilever, it is enough to ensure that the force converges on the real solution. Over-meshing in the other direction only produces results that deviate from the real solution. The existence of this phenomenon is probably due to the F E A coding of SAP2000. The details of the connectivity between elements in SAP2000 are hidden factors that might have significant effects on the results. For P L - 3 , the F E A results converged even when very coarse mesh was used, because o f the simplicity of the model. The mesh size of 0.35m by 0.15m at the area of interest seemed to yield a convergence, while also using minimal computational effort. For P L - 2 , however, the use of a much denser mesh size of 0.35m by 0.01 l m was necessary for convergence. This was due to the more complex loading mechanism and structural configuration (Discontinuity o f barriers). 3.2.2.8 Loads According to S6-00, the anchorages and cantilever decks are to be designed for the possible forces that could be transmitted from the barrier subject to the impact loads, as specified in Table 4 and applied in Figure 5. The loads given in Table 4, are then distributed evenly amongst the nodes along the length of the barrier depending on the performance level, as is again specified in Figure 5. The safety factor is the 1.7 live load factor, as specified in Table 3.5.1(a) per S6-00. 34 Various loading combinations are set up while keeping under consideration the effect of cantilever length, the location of impact, and the performance levels. There are five different cantilever lengths being tested that range from 600mm to 1800mm, and there are two possible locations of impact: the inner portion and the end portion. In total, 80 various loading cases are being considered. The application of railing loads on PL-2 ' s inner and end portions, and PL-3 ' s inner and end portions of the bridges are presented in Appendices B , C, D , and E respectively. 3.2.3 Stress Averaging Each element has its own forces or stresses, and typically they w i l l be different at the common points of different elements. Thus, abrupt changes may be seen across those common points from element to element. The finer the mesh, the closer these common values wi l l become, and vice-versa. SAP2000 has a function that averages the stresses at any given point by averaging the stresses from all of the surrounding shell elements that are both connected to the point and are visible in the active window, or on the plane at which the user is looking. Then, when SAP2000 plots the stresses for a particular shell element on screen, it plots those averaged stresses at the points that are under consideration instead of using the actual stresses calculated from the F E A . If the model has some kind of discontinuity, for example when two planes meet at an angle, one wi l l need to perform this stress averaging function on each plane independently, by enabling this option during the plotting process. This w i l l avoid the problem of averaging across the two planes, which would give incorrect results because the stress along the two planes is not continued relative to the element's local axes. 35 In addition, the stress averaging function provides an alternative way of checking the F E M for errors, i f any. B y turning this function off, the user can carefully observe and decide i f the abrupt changes in stresses among elements are too great. If this change is much more apparent than the ones found with stress averaging, it is likely that the F E M contains errors such as inappropriate mesh sizes or connectivity problems. This "test" may be performed until the user is satisfied with the changes in the stresses across the elements. In this case, the model has discontinuity because the barriers are intersecting the deck at a right angle. Therefore, it is necessary to check for any possible errors that may exist in the F E M . That would be done through the use of the stress averaging function in each of the barrier planes and deck planes independently. 3.2.4 Comments The data generated by the F E A are immense, although only part of them is actually useful for the purpose o f this research. Therefore, data from only the areas of interest are being collected and stored in a spreadsheet. The spreadsheet itself is then completed by further data processing later in Section 3.5 "Maximum Moment Dispersal Angle ." These data can be found in Appendices F, G , H , and I under the F E A results column. Sample graphs in which the forces vs. the distances have been plotted for a cantilever length of 1800mm, can be found in Appendix J. In terms of the method o f analysis, F E A is by far the most reliable way for determining load calculations. The results generated using F E A are usually more realistic than those generated through other methods, because of its ability to allow detailed specifications of each component, of realistic load assignments, and of 36 simultaneous load combinations. However, the drawback of its power is that the results are valid for only the very specific cases in which they are modeled. These results may or may not be applicable to other cases with slightly different specifications. That makes it a rather weak tool for general usage. In order to cover a reasonable scope, this method requires many individual remodelings and analyses for every possible case within the scope of interest. Hence, F E A requires too much time and effort, making it impractical to use in the course of regular design. 3.3 Dispersal Angle Method The method of dispersal angle, which has been used in some Canadian provinces, is introduced in the S6-00 Commentary Clause C.5.7.1.6.3. Dispersal angles can be determined by a finite element analysis, and can be used to calculate barrier and deck overhang moment intensity based on the railing loads. This is a classic concept that tries to approximate the nature of load dispersion. To understand the strengths and weaknesses of this method, then, one must understand the concepts behind it. In the following sections, the behavior of true load dispersion is discussed in detail and compared to the assumed load dispersion described using the dispersal angle method. 3.3.1 Mechanic of Behavior Figure 7 shows an inner portion cantilever bridge deck subjected to a concentrated load, P, as well as the actual moment intensity distribution along its two sections. The pattern of distribution of moment intensity comes out in the shape of a bell-curve, with a peak, or 37 maximum moment intensity. It can be seen that the moment intensity drops rapidly as it moves away from the peak and then drops more gradually as it reaches zero. Just as well , the maximum moment intensity increases as it moves towards the first support and away from the point load, P. On the other hand, the distance in the longitudinal direction, or x-direction, between the two end points where moment intensity diminishes to zero becomes greater as it progresses into the first support from the load P. The spread-out of the zone defined by the points where moment intensity reaches zero is know as load dispersion. This dispersion of load, however, is nonlinear and is dependent on the stiffness of the element. Also , note that the total area under each bell curve is equal to the total moment, which is determined by overall static equilibrium, at any given section. In this case, the total moment is the concentrated load, P, times the distance between the load and the section of interest. Figure 7 - Load Dispersion for Cantilever Deck Inner Portion 38 A cantilever deck at the end portion, which is shown in Figure 8(a), would have a slightly different moment intensity distribution than that of the inner portion. A s mentioned above, the total area under the curve of any given section is equal to the total moment of that section, which equals to P * D , where P is the concentrated load and D is the distance from the load to the section being analyzed. In order to satisfy equilibrium then, the moment beyond the free edge represented by A L must be redistributed within the cantilever deck. To do that, the area A L is "reflected back," forming a mirror image that uses the free edge as the plane of reference, as shown in Figure 8(b), and that redistributes itself back into the moment represented by AR , as shown in Figure 8(c). A s a result, the peak, or maximum moment intensity, as well as the moment in the region closest to the free edge, w i l l be affected significantly. Therefore, a separate analysis for the inner and end portions should be carried out to ensure such effects are taken into account. 39 Figure 8 - Load Dispersion for Cantilever Deck End Portion. 3.3.2 Development of Dispersal Angle Method Dispersal angle method is basically a simple method that was developed while "trying" to represent true load dispersions through the assignment of parameters called dispersal angles. It also assumes that the dispersion is linear, instead of nonlinear. In Figure 9, the dispersal angle, 0, which can be defined by F E A , is used to calculate the moment intensity of the barrier and deck overhang based on the concentrated load, P. Note that this method is applicable for either the point load, or the distributed load. The zone of dispersion is represented by two lines that are determined by the given dispersal angle, 0. Since the actual load dispersal is nonlinear and cannot be captured using one single dispersal angle, this method may produce results that are either overestimated or 40 underestimated depending on the location of the section making the design unconservative in some cases. Figure 9 - Dispersal Angle Concept The pattern of moment intensity distribution is assumed to be uniform instead o f to be the bell-curve distribution that it actually is. In order to satisfy equilibrium, the total area under the uniform distribution is equal to the total moment of that particular section. The total moment is simply P*D, where P is the load and D is the distance from the load to the section being analyzed. However, this total moment is now evenly distributed along the dispersal length and is bounded by the two lines that are defined in the dispersal angle. A t this point, it is clear that the uniform distributions are not good representations of the original bell curve distributions, because they do not capture the peak, or maximum moment intensity. The uniform distribution only serves as an approximation of the average moment intensity of the original bell curve distribution. Hence, it underestimates 41 the moment at the center and overestimates the moment near the two sides, yielding unconservative design loads. A spreadsheet is set up to calculate the moment distribution at various sections of the cantilever slab based on the railing loads. The dispersal angles are taken from the S6-00 Commentary Clauses C5.7.1.6.3 (which was determined by F E A ) and are then applied in the spreadsheet to different performance levels with the use of Equations (9) - (12) using the parameters specified in Table 4 and Figure 5. Since there was no specific information regarding P L - 1 , it was assumed to have dispersal angles similar to those of PL-2 . The dispersal angle for vertical loads is assumed to be zero at this point, and that is verified later by the F E A . The torsion on the deck, due to longitudinal loads, is not considered significant. Figures 10 to 12 show some examples of moment intensity calculations for the inner portions of decks with overhang lengths of 1000mm using the spreadsheet developed. Transverse moment on barrier due to transverse load: M = P T x H (9) D L + H x T A N ( f 9 i / J r ) Transverse moment on deck due to transverse load: MT = FT x H B (10) ( D L + H B x TAN(0bPT )) + (D x T A N ( 0 d P T )) Transverse moment on deck due to vertical load: MV = P F x D (11) D L + D x T A N ( 0 r f W , ) Total transverse moment on deck due to transverse and vertical loads: MC = (MT + MV) (12) 42 Where: M transverse moment intensity on barrier due to transverse load (kN*m/m) M T = transverse moment intensity on deck due to transverse load (kN*m/m) M V = transverse moment intensity on deck due to vertical load (kN*m/m) M C = transverse moment (kN*m/m) due to combined loads: PT, P V , and P L PT transverse load (kN) * safety factor (1.7) P V vertical load (kN) * safety factor (1.7) H height from the point of interest on the barrier to point of impact (m) H B height of barrier base to point of impact (m) D L = dispersal length (m) D distance of deck from the point of interest to the barrier (m) GbPT = dispersal angle of barrier (degree) GdPT = dispersal angle of deck (degree) ©dPV = dispersal angle of deck (dgree) 43 Moment vs. Distance (PL-1) 1200.00 1000.00 800.00 600.00 400.00 200.00 200 400 600 Distance (mm) 800 1000 " Moment due to transverse load™1*™ Moment due to vertical load Total Figure 10 - Moment Intensity vs. Deck Distances Graph fro PL -1 Moment vs. Distance (PL-2) 35.00 30.00 c 20.00 e o 2 15.00 10.00 5.00 0.00 200 400 600 Distance (mm) 1000 " Moment due to Transverse Loads Moment due to Vertical Loads Total Moment Figure 11 - Moment Intensity vs. Deck Distance Graph for PL -2 44 Moment vs. Distance (PL-3) 100.00 200 600 Distance (mm) 800 1000 " Moment due to Transverse L o a d s " * - Moment due to Vertical Loads * Total Moment Figure 12 - Moment Intensity vs. Deck Distance Graph for P L - 3 Since it appears as though the patterns associated with the change of moment intensity depend on the performance level, the following conclusion can be made. Figure 10 shows that the moment distribution for PL-1 (post-type barriers) is highly concentrated at the base of the posts but dissipates quickly as it extends outward onto the deck. Transverse loads dominate in this case. Local slab reinforcement should be designed properly to resist the high moment intensities here. Figure 11 shows that the moment distribution for PL-2 (precasted type barriers) is high at both the base of the barriers and the girder. In this case, both transverse loads and vertical loads dominate. Both areas also require additional reinforcement to deal with the railing loads, however. For P L - 3 , (Cast-In-Place barriers), the moment distribution is higher at the girder areas because of the much greater vertical loads which are being applied at the railings, causing 45 the vertical loads to dominate as shown in Figure 12. Meanwhile, the reinforcement of the whole cantilever slab may be designed around the highest moment in that area. Although the dispersal angle method is a quick and efficient one for use in preliminary design processes where design moments are being estimated, it should still be checked for accuracy using other methods, such as the F E A . 3.3.3 Comments The results generated using the dispersal angle method can be found in Appendix K . Also, Appendices F to I show results as calculated using the dispersal angle method, while further comparing that to the F E A . Note that the discrepancy between the results calculated using these two methods can be as great as 20%. The various assumptions inherent in the dispersal angle method are the main reason for these great differences in results. Also , because this method is based on F E A findings, it also inherits the advantages and disadvantages from the F E A method. This was discusses previously in Section 3.2 "Finite Element Analysis". However, the biggest strength o f the dispersal angle method is that its concept is relatively simple and takes minimal effort to apply in practice. 3.4 Maximum Moment Envelope (MME) A new method called the maximum moment envelope ( M M E ) wi l l be introduced here in as a means of capturing the true load dispersion on traffic barriers and deck overhangs. Unlike with the dispersal angle, the maximum moment intensity of each section can be 46 evaluated using this method, as shown in Figure 13. These maximum moment intensities can then be used as design loads in order to yield a safer design than would be had using the average moment intensities generated with uniform distribution in the dispersal angle method. The process of developing this method as well as the uses of the method, are presented in the following sections. Figure 13 - M M E Concept 3.4.1 Development A s mentioned before, the pattern of moment intensity distribution is a bell curve with a well-defined peak, or maximum moment intensity. The purpose of the M M E is to capture these peaks using the F E A . A s such, these peaks, which are collected and stored in a spreadsheet, form a distribution of maximum moment intensity all the way from the location of impact to the first support. This distribution is thus called the M M E , and it is 47 represented by a chosen simple function. Different scenarios have different distributions and so are represented by different functions. For example, the transverse moment distribution of the PL-3 inner portion deck under transverse load is represented by a function different from that of the PL-2 end portion barrier under vertical load. These scenarios include combinations of various cantilever lengths (600mm, 900mm, 1200mm, 1500mm, or 1800mm), of different locations of impact (inner or end portion), of elements of interest (barrier or deck), of different loads (transverse, longitudinal, or vertical), and of various performance levels (1,2, or 3). They may be represented by either the same, or different, functions depending on the combinations. For example, varying cantilever length has a minor effect on the shape of the selected function, while a variance in the location of impact results in a change of the function itself. The functions to be selected should be simple and should be represented by no more than a one or two constants, such as A*sqrt(x), s in(x A A), or cos(A*x), where A is the constant that can be adjusted to fit the shape of the distribution, and x is the distance from the location of impact to the section being considered. Although the function type may or may not change depending on the situation, it can be predicted that the constants that define the functions wi l l change from case to case. In particular, the constant o f cantilever length is l ikely to undergo changes. In other words, both the function type and the constants are the basic parameters for determining the moment distribution of any scenario. The value of the constants, determined by matching the results of the F E A , is then represented by another set of functions that varying depending on the cantilever length, alone. Examples of this process are shown below in detail. 48 3.4.2 Applications Figures 14 and 15 show a PL-3 barrier and the inner portion o f a deck with a 1200mm overhang under a transverse load, and a corresponding transverse moment distribution. For the barrier, two functions (linear and square root) are proposed in Figure 16 in order to match the F E A results through the varying o f constants A and B . It can be seen from Figure 17 that the linear approximation is a better representation o f the F E A results. For the deck, only an exponential function is applicable, as seen in Figure 18. Figure 19 shows that the exponential function with one constant matches well with the F E A results. Examples of the relationship between the constants and the cantilever lengths can be found in Appendix L . Figure 14 - Transverse Load on PL-3 Inner Portion Barrier and Deck with 1200mm Overhang 49 Figure 15 - Transverse Moment Distribution due to PL-3 Transverse load } Microsoft Excel - MME S] 9* 6* 3«w insert Format loots Data BJndow D * i r i f l ' # a y . J <fe • ' • H Ik i H27 ' • « •Jaljsi A B C ' 5 E F ! 6 H I 2 3 Barrier Transverse Moment due to PT 4 Trial Function 1 Trial Function 2 5 D i s t a n c e f r o m B a r r i e r T o p ( m m ) F E A M a x N e g a t i v e A p p r o x i m a t e d M o m e n t M a x N e g a t i v e | k N * m / m ) M o m e n t ( k N " m / m ) F i n i t e D i f f e r e n c e s A p p r o x i m a t e d M a x N e g a t i v e M o m e n t ! k N * m / m ) F i n i t e D i f f e r e n c e s 6 0 0 0 0 0 0 8 . . 1 8 0 10 1 10 4 8 0 9 5 2 3 8 0 3 8 0 9 5 2 3 8 2 0 3 1 9 7 5 7 7 6 10 2 1 9 7 5 7 8 3 6 0 2 2 1 2 0 9 6 1 9 0 4 7 6 -1 1 3 8 0 9 5 2 2 8 7 3 6 4 7 7 0 1 6 6 3 6 4 7 7 0 1 5 4 0 3 3 3 3 1 4 4 2 8 5 7 1 4 -1 8 5 7 1 4 2 9 3 5 . 1 9 4 8 5 2 8 4 1 8 9 4 8 5 2 8 4 7 2 0 4 2 9 4 1 9 2 3 8 C 9 5 2 - 0 9 7 6 1 9 0 5 4 0 . 6 3 9 5 1 5 5 3 - 2 2 6 0 4 8 4 5 9 0 0 5 1 2 5 2 4 0 4 7 6 1 9 1 2 0 4 7 6 1 9 4 5 4 3 6 3 5 9 6 4 - 5 . 7 6 3 6 4 0 4 1 0 8 0 6 0 5 6 2 8 8 5 7 1 4 2 9 2 3 8 5 7 1 4 2 9 4 9 7 7 3 0 3 8 2 2 - 1 0 7 2 6 9 6 2 . 14 15 s u m = 1 9 5 4 E - 1 4 1 E - 0 6 16 17 Approximation 18 T r i a l F u n c t i o n l 19 A 0 0 6 20 Y A ' x 21 T n a l F u n c t i o n 2 22 B 1 5 1 4 5 4 5 3 2 1 23 < < Y B ' s q r t ( x ) • M \ PL3-Bamer-PT / Pi 3-Oett-PT \sheet 1 / Sheets/Sheets/ |<] Ready •ir Figure 16 - M M E Spreadsheet for Barrier 50 |3 ivtrrosoft 1 xirt profleptot . 5 X 5*J Be Edit » w insert Format loots Chart SMndo* Help - - • x ' ** • &, • * * Transverse Moment on Barrier duo to Transverse Load (PL-3) z 70 "| 50 i | z 1 40 I I 30 f I S 20-1 n i • -f 0 200 400 GOO 800 1000 12 50 Distance from Barrier Top (mm) [ —*— FEA ResiJts --»-Trial Function 1 Trtal funclion 2 ] H 4 * H \PL3-Barrier-VT/Pi3~Deck-vT /Sheetl /Sheet2 /Sheet3/ \i u r Ready .• • f • start] 4^FuncdonS"fro... | ) _jf*>T ;i.Qproffleptot Mcrosoft Power... | £ g « 9 < J f l 4:29PM Figure 17 - Plots for Transverse Moment on Barrier } Ntcrosoft Excel - proftteptot i ] S i B t Sew Insert Format loots Qata ffiftdow He*) - * I'D - a, . »J . •M 3 : Anal 1 10 | I ./ U ; M M Wi ® "i* : • G36 - « =SUM(G28G34) A 9 C D E F H 1 24 25 Dock Transversa Moment due to VT 26 ; 27 Distance from Barrier Base (mm) FEA Max Negative Moment (kN*m/m) Approximated Max Negative MomentfkN'm/m) Finite Differences 0 60 5 62 89 2 39 • 150 52 6 54 99 2 39 300 46 9 48 09 1 19 450 42 1 42 06 -0 04 600 37 9 36 78 -1 12 • 750 34 2 32 16 -2 04 930 30 9 28 13 -2 77 i 5 sum = 1 -3 164E-07 38 Approximation 39 A = 0 0009 40 N 62 89 (obtained from barner MME approximation) 41 Trial Funcbon 42 Y N'e^-A'x) 43 44 45 AS !< < _ » M \ PL 3-Barrier-VT / PL3-Deck-VT \sheet 1 / SheeK / Sheets / l«l 1 • Read 1 ) Functions--fro.., MSAP200O-PL3O... ; : «Jp ro«ep lo t i^WaosoftPower... j (S • « "jf ©H 4:30PM Figure 18 - M M E Spreadsheet for Deck 51 II 1 II — — S] Be B * view Insert F a r m * loots ffa . . a - O 3 ? ': • » D « c k T r a n s v t r M o m e n t due to T r a n s v e r s a L o a d ;o GO J 1 I E 30 . " 2 J 20 Plot A rea j • J o 100 200 300 400 500 S00 700 800 900 10 Distance from Barrier Base (mm) 3D |~+--FEA *• Trial Function | J K « < I H \ Pt-3-Barner-VT \ P L 3 - D e c k - V T / S h e e t l / S h e e t s / s h e e t 3 / \ i i Raadg }* start] ^ F u r x O o n s - f r o . . . | 'x'.SAFZXO-PL3Q...] ^ j M o T ] j % ] p r o f l e p l o t ^ M c r o s o f t P w * . . . | - 4 : 3 2 P M Figure 19 - Plot for Deck Transverse Moment The results of this method ought to provide a table containing various constant functions that can be used to calculate the relevant constants. Depending on the scenario, those can then be used together with the distribution function to determine the M M E needed for any particular design. 3.4.3 Comments The M M E is realistic in its representations o f the true moment distribution along barriers and deck overhangs. This method also gives the maximum moments that designers should design for, instead o f just the average values calculated using the dispersal angle method. A s such, this design is a conservative one. Keep in mind what came up before, where the moment distribution yielded by the dispersal angle method was assumed to be 52 uniform and so it may have underestimated the design moments by as much as 20%. The application of this M M E method, however, is much more involved and does require much more time and effort than the dispersal angle method, making it less practical. The various functions and constants that define the moment distribution curve may appear to be confusing for designers at first, as well . That might make people unwilling to adopt this new method. In general, M M E provides an alternative way for mapping the distribution of the maximum moment intensity of the barrier and deck, although it does deviate slightly from one of the objectives of this research, which is to compromise with the current code design methods. 3.5 Maximum Moment Dispersal Angle (MMDA) The dispersal angle method and the M M E each have their own strengths and weaknesses. In order to counter the weaknesses o f these two methods while preserving their strengths, a combined method called the maximum moment dispersal angle ( M M D A ) was developed during this research. M M D A takes advantages of the use of maximum moment intensity, while also using the dispersal angle theory as its backbone. In other words, the M M D A makes use of the basic concept of the dispersal angle while improving the results of that system, but without making it too much more complicated, as the M M E did. There are two ways of achieving such an affect. The first way is by adding modification factors to the results generated by the current dispersal angles to scale or improve those results until they suit the F E A results. The second way is by determining new dispersal angles altogether, so that the results produced by these new angles would give the maximum moment intensity distribution as calculated by the F E A . Both 53 methods are presented in details in the following sections although only one method (termed method 2) is recommended for further studies after consideration of their advantages and disadvantages, and after consultation with the Ministry of Transportation ( M O T ) o f British Columbia. 3.5.1 Method 1 - Modification factors The application of this method requires a calculation o f the average moment intensity using the current dispersal angles given by the Commentary o f the S6-00 code. A s mentioned before, a spreadsheet was developed so that by simply entering a few basic input parameters such as performance level, live load factors, overhang distances, and properties of the barriers, as shown in Appendix K , this calculation could be carried out. Then, the results can be transferred into another spreadsheet for the purposes of further developing this method. 3.5.1.1 Development and Application Figure 20 shows the spreadsheet used to determine modification factor A by adjusting the results that were calculated in the previous spreadsheet in order to better represent the F E A results. The Cast-In-Place barrier located in the inner portion is subjected to performance level 3 transverse loads. The modification factor A is in linear relationship with the results calculated using current the dispersal angle and it is to be scaled by the modification factor A of 1.17 in order to yield the maximum moment intensity results produced by the F E A at the base of the barrier, where maximum moment usually occurs. Figure 21 illustrates the concept best, as it shows the results plotted using the current 54 dispersal angle, F E A , along with the modification factor A . Although the modification factor method does not capture the maximum moment intensity on every point along the barrier, it does provide a better approximation than would the code dispersal angle at the base of the barrier, where maximum moment occurs. jMkrosoii • K t i '%] Be £dtJ.Vew"Insert "Format' Iocs Cjsti Window•* Help' < " B D MME Approximation for Performance Level 3 (thk275-oh600-1.0 Internal Poition Barrier Transverse Moment _ _ Due ID P T Distance Nogativo fromBamer Moment5. Top (mm)''" (kN*m/m)° FEA Max Negative Moment Finite r. . •• ii stance \ . om Barrier Moment(kN"m.'mi Differences ' Top (mm) Due to PT FEAMax Negative Moment (kN^m/mL Due to FEA Max Negative x Moment (kN'm/m) •*/," 0"-', 180 ""540 ' 720 '..-.900 >s 1070 ' ' . -0-17 1 -SRiillt 72 9-'Sl.f 103 4 sum --ooo - r. i/1 Oi 0 jljjjfl 180! 2 2 4 ! -IBIlliii 360! 46"6[-56 95 -i >•,! 540! 68.5'!-JitlRlIII lipll 720 \ 87']-900! 107]-lltlillll 0 00 1070! m2j-*' v i I ApproximerxM Trial Funcfconi Deck Transverse Moment due to PT Modification Factor I ; FEA Max Irijctanpft M a n a h i B i ™ Imnjxiuad.Mav r. i + 4 »' N / Constant B-Int / Comtent C-lrtt \ I M M £ - P L 3 / M i * . ^ i>sr^ al*n3e / FEA Max Modificati Improved Max:;^  . :Monahi«a jMpnahwo •»••• ? Figure 20 - Spreadsheet Used to Determine Modification Factor " A " 55 120 PL3-Barrier-275-600-1.07-PT-lnt 0 200 400 600 800 1000 1200 Distance from Barrier Base (mm) FEA « - Dispersal Angle * Modification Figure 21 - Plot of F E A , Dispersal Angle, and Modification Factor " A " Results However, not all modification factors are in a linear relationship with the results. The relationship depends on how closely the new curve is able to simulate the original one, which is described by the F E A results. Looking at the deck of the same system under the same transverse loads, the moment calculated using the code dispersal angle exponentially related to its modification factor B of 1.1E-3 as shown in Figure 22. It can be seen from Figure 23 that the use of the exponential relationship in this case could yield a good approximation of the F E A results. Note that, the use of a linear relationship here would yield a bad approximation though, especially at the deck supports. 56 t Mure .-lift I xn-l MM1 V ] F)e Edt view Insert Fermal g 10 i j 8 B i ™ 5 i n | s | • . 8 X • A. - »' ;20 B D * A*N Deck Transverse Moment due to PT , 2 1. ' 24 IS lis . i ^29 ' 30, L1L g l l (B 38 : 33 • F / , V . » Distance Negative fromBarner Moment Base (mm) (kN^ itvm) 300 103 4 9^1 4 sum Approximation ?tlllBllllii!l|^! v • • Modificati F E A Max Improved Max. improved Max J i stance Negative Negative Negative Finite "om Barrier Moment Moment(kN"m." Moment(kN*rn/m! Differe ices Base (mm) (kN*m/m) m) 0 00 0 12872 128.2C . .—2 7 9 ' 8 3 ' v ' " * ' i l l l l f "75 123 126.0:-Mill l l l l l l lSft l i i l l i i 150 120 123.9; 72 70 l i i i l 225 120 12l'.9r 1 20 300 ZZZ3I? 20.0C 0.00 ::: •^^^••wBIIBII A 1.2( l l l l B zzzz: • "5E-0! Y A 'e"(-Bx)'N : Deck Transverse Moment due to PV Distance from Barner No-Dispersion Modification Factor FEA Max Negative Moment Approximated Max Negative j Finite Base (mm) :(kN*m/m) iMoment(kN*m/m) iDifferences OJ _ _0 : _ 0 00: ^ ^ 0.00: > ~NY Constant B-lnt / ConstarrtCIrrt V^tE-PL3 /^JME^2' Approximated Max Negative j Finite Moment(kN*m/m) iDifferences 0.00! 0.00 Distance from Barrier FEA Max Negative Moment No-Qis, [Approximated/ i Max Negative -|Moment(kN*m Base (mm) ;(kN*m/m) ir 0! 01 0 Q U Figure 22 - Spreadsheet Used to Determine Modification Factor "B" PL3-Deck-275-600-1.07-PT-lnt 20 0 4 , , , , , , 1 0 50 100 150 200 250 300 350 Distance from Barrier Base (mm) [—•— FEA -a— Dispersal Angle a.— Modification"] Figure 23 - Plot of F E A , Dispersal Angle, and Modification Factor "B" Results 57 The spreadsheet used to calculate the modification factor C in the case of the deck under the specified vertical loads, as well as the plotting of various analytical methods appear as shown in Figures 24 and 25, respectively. The modification factor C is in linear relationship with the results, and has been found to equal 0.96. 3&] fie gdjt yjew Insert^ Fc/mat loote data ffiixlow j - Hdp I '" A38 'StllllllS ' f l i p B Deck Transverse Moment due to PV ffi! fl i ! 4 ; ;_4-, t i l . i 4 Distance Negative . s i . ,t , ' " I . » 1 T . - " 1 /io-Disparsion H4omorit(kN*m/m) Difference! Approximated Max N&gatwa Finite Morrient(kN"rn/m) Differences wBmm -oco I l i l l P HEP | p i f l § i P C 0 96 • • 1 1 1 " 2 8 i l l jpf l lllpllit 37 lljjljjg llllllllilpS^ liilllllff !it!ill|ll§i|ll • ^ B • S M I -> 0 00 }>0 92 -',1 84 l l p l t l fnal^unoicfp CM L O O S J - : « , " O O I -0 04j 0 00 Oi 75\ 50|" 2 5 ! " DOT :(kN*m/m) 11 0 I T i r e ] " 2.08}" '319!' 4.3]"' No-Dis/ig Approximated v«lj Max Negative M iMoment(kN*mp b'BiS . , 1 9 j "2."8?;»1 53 Deck Transverse Moment due to Combined loads Distance from Barrier Base (mm) 0 Cede Dispersal Angle_ FEA Max Negative Moment (kN"m/m) 103.6 Approximated Max Negative Moment(kN*m/m) 88.28 Combined Modifications Ready Finite Differences j J 5 . 3 2 ! u'/ CcnsantB~-lnt / Const«c' :w"VlME-PL3/"MNE-PL2 Approximated Max Negative IFinite Moment( kN'm/mJiDifferences _ 103 40 _ -0_20 / •'«'•- : / . -*--.vnrv ,/ FEA Max Distance JNegative from Barrier j Moment Base (mm) i(kN*m/m) 0! 127.9 iiiiililllBP ' Sum=779.M95S»t . -r _CodeDispi:m Approximated Max Negative ^ Moment) kN'nVjKi m) ^ 106 4 '» i Figure 24 - Spreadsheet Used to Determine Modification Factor " C " 58 PL3-Deck-275-600-1.07-PV-lnt E 1 3 g 2.5 E o s « n s 1 150 200 D i s tance f rom Barr ier Base (mm) - F E A No Dispersion Modification j Figure 25 - Plot of F E A , Dispersal Angle, and Modification Factor " C " Results Figure 26 is a table presenting, for comparison, the code dispersal angle results, the modification factor results, and the F E A results of the deck moment intensity due to combined loads. It is clear that the code dispersal angles yield results significantly different from those of the F E A , especially at the two ends of the deck overhang. Interestingly, the code dispersal angles have underestimated the results by 15.32 kN*m/m (14.8%) at the location of the barrier and have overestimated the results by 13.18 kN*m/m (19.5%) at the first support. However, the contribution of the modification factors did manage to greatly reduce these errors to 0.20 kN*m/m (0.002%) and 0.98 kN*m/m (0.01%) at the location of the barrier and the support, respectively. Figure 27 shows the plotted results as calculated using the three methods. It shows the improvement that the modification factors have brought upon the current code dispersal angles. 59 Microsoft Excel W l ] Be Edt' View insert1 c cjmat * lools Data window, Hetb I:; *i U _i <a c * '1 v J ^ ' >< E - } » l i t - ' J " Aral "™"**A54 ' ""W^^MM - - X . f > - r? x | sum Approximation: c ;= Tnal Funct ion: Y != 0.3625: 0 9 6 ; C * N I - "3 Deck Transverse Moment due to Combined loads [56 |~57 §§ ,'eo IS : 62 -< if Distance from Eamsr • t V '> .-103.6 HEPS® • » 67 5 ApL'-'-mr- .:-).: /V ' . : ' "~!- ' :o3 [Distance Max Negative ~ Finite Max Negative Finite I f r o m B a m e r Momenl(kN"m/m! Dirferences Mornent(kN*m/m) Oiffe e (mm) F E A Max Negative Moment (kN*m/m) ^5 "S608 ~ 84 OS lp^ fl|ll|§| -6 22 Ijjjjljl 6 09 IBS 68,4$ / _ _-0 20 Oi 127.9 0 45| 75: " 123.4 -1 11)1 150 122,9 - '*i 225: 126 300! 129 > "Jl ... C o d e Dispc,< Approximated Max Negative >. Moment(kN*m.>i m) 106 4 ' 105 2 104 0< 1 0 3 O 5 -102 Of MME Approximation for Performance Level 3 (thk275-oh900-1.0 Barrier Transverse Moment 67 Due to P T Due to Combined Modification Factor : F E A Max Distance : Negative " H - " " « >,">./ CorstantB-lrit.-/ - j • . •• > M n i r ; ( stf r- . / •; ,«.*«,>>• ' Due to PT_ "i F E A Max Due to Of-J F E A Max " " Negative 5um=3012 454867 Figure 26 - Spreadsheet Used to Present the Results of Various Methods PL3-Deck-275-600-1.07-Combined-lnt 150 200 Distance from Barrier Base (mm) - Code Dispersal Angle Figure 27 - Plot of F E A , Dispersal Angle, and Modification Factors Results 60 3.5.1.2 Comments Although, the results can be improved by adding modification factors while leaving the current dispersal angles unchanged, that does add complications to the method and in so doing, defeats the main objective of this research. The introduction of various relationship and modification factors could potentially lead to confusion similar to that found in the M M E method. Therefore, another method is proposed below as a better alternative. 3.5.2 Method 2 - New Dispersal Angles The second method functions to generate results that match those of the F E A at critical locations such as base o f the barrier and the support of the deck overhang. This is done by changing the dispersal angle itself. The dispersal angle is adjusted so that the uniform moment intensity distribution ends up equal to the maximum moment intensity calculated by the F E A , as shown in Figure 28. The black line there represents the true bell-curve moment intensity distribution that has been calculated by the F E A . Additionally, the red line represents the results calculated using the code dispersal angle, and the blue line represents the new dispersal angle and the ways it has been adjusted to fit the F E A results, or the black line, at certain location. Keep in mind that the new dispersal angle is defined differently than the current one. Instead o f describing the dispersion of forces, the new angle is used as a tool for finding the maximum moment intensity. 61 Figure 28 - M M D A Concept by Changing Dispersal Angles 3.5.2.1 Development and Application Another spreadsheet was set up for the development of this method, and it can be found in Appendices F to I. The new dispersal angle is to be optimized so that the moment intensity at the location o f interest comes out equal to the maximum moment intensity calculated using the F E A . The equations used in calculating moment intensities with the code dispersal angles are slightly modified for the application of this method. A few new parameters have even been added so that the new moment intensities can better approximate the F E A results. Equation (13) is used to calculate the transverse moment, M , of the barrier resulting from the specified transverse loads, PT, in the S6-00 code. 0 d P T is the new dispersal angle of the barrier, which is ultimately to be determined by the spreadsheet. H and D L have the same definitions as before, where H is the height measured from the location of impact to 62 the section being analyzed and D L is the distributed length of the transverse load. The only new parameter in this case is N , which depends upon the dispersion of forces: single-sided or double-sided. For the case of a PL-3 inner portion, the forces are freely dispersed onto both sides of the barrier because the Cast-In-Place barrier is a continuous element. Therefore, the dispersion must be multiplied by two to take those effects into account (N = 2). However, for other cases such as a PL-3 end portion, where one side is discontinuous, or a PL-2 inner and end portion, where the precasted barriers are discontinuous at the edge, the dispersion is limited to one side only. Recall that the worst loading condition is when the load is applied at the end of these elements, as shown in Appendices B to E . A s a result, N equals unity. Note that the effects of vertical loads and longitudinal loads have very little or no contribution to the transverse moment of the barrier. Therefore, the transverse moment intensity, M , caused by a transverse load, is a good approximation for the total design moment intensity of a barrier. M= ™ (13) D L + H x TAN(0bPT) x N l Where: M = transverse moment intensity (kN*m/m) PT = transverse load (kN) * safety factor (1.7) H = height from the point of interest on the barrier to point of impact (m) D L = dispersal length (m) ©bPT = dispersal angle of barrier (degree) N l = 2 for PL3 inner portion, 1 for other cases 63 Equation (14) calculates the transverse moment intensities, M , of the bridge deck that result from a transverse load, PT, as specified in the S6-00 code. The fact that impact force transfers from the barrier down to the deck, caused the denominator of this equation to be divided into two parts. The first part describes the dispersion of loads on the barrier in ways similar to Equation (13) above, while the second part describes the load dispersion on the bridge deck itself. N l and N2 are two parameters that determine whether the load dispersion is single or double-sided in the barrier and the deck, respectively, as was mentioned before. However, the dispersion across the connection between the barrier and the deck is not necessarily a smooth transition, meaning that the dispersal length on the barrier may not be equivalent to the dispersal length on the deck at the connection point. In fact, only in the case of a PL-3 does the connection between the barrier and the deck share the same dispersal length as would be predicted. However, in the case o f PL-2s, the discontinuity of the precasted barrier forces the dispersions at the joint to behave quite differently. Similar to a case of fluid flowing from a narrow channel into an open water, as the fluid leaves the narrow channel it immediately spreads out and travels in all directions to fi l l the area. The impact load behaves like the fluid in this case. A s the load transfers across the connection from the discontinuous barrier (narrow channel) into the continuous bridge deck (open area), the load immediately spreads out and disperses in all directions, causing an abrupt change in the dispersal length at the joint. After conducting a series of observations, this change in dispersal length is about two times the original length. In other words, the load disperses over a certain length as it approaches the connection, and this length doubles itself as the load subsequently moves across the connection, causing a great reduction in moment intensity on the deck. This phenomenon is defined by the new parameter, N3 . 64 MT- (14) ( D L + H B x TAN(0bPT ) x N l ) x N3 + (D x T A N ( 0 d P T ) x N2) Where: M T = transverse moment intensity on deck due to transverse load (kN*m/m) PT = transverse load (kN) * safety factor (1.7) H B = height of barrier base to point of impact (m) D L = dispersal length (m) D = distance of deck from the point of interest to the barrier (m) © b P T = dispersal angle of barrier (degree) 0dPT = dispersal angle of deck (degree) N l = 2 for PL3 inner portion, 1 for other cases N 2 = 1 for end portion, 2 for inner portion N3 = 1 f o r P L 3 , 2 f o r P L 2 Equation (15) is provided for the purpose o f calculating moment intensities o f decks based on vertical loads per the S6-00 code. Unlike the transverse load, a vertical load transfers directly down the barrier onto the deck and creates a transverse moment in only the deck. B y intuition, an axial load distributed over a long length would yield very little or no dispersion among an element. The distributed length is considered long when compared to the depth o f the element. In this case, the length to depth ratio is as high as 8 and 13 for PL-2 and P L - 3 , respectively. Therefore, the barrier acts more as a medium for transmitting the load rather than means of dispersion. With this assumption, the dispersal length, D L , on the deck is set as equivalent to that of the barrier, which is specified in the code. N2, again, takes into consideration the dispersal affect; a single-sided dispersion for end portions and a double-sided dispersion for inner portions. 65 MV = P K x D (15) D L + D x T A N ( ( 9 r f / 3 K ) x N 2 Where M V transverse moment intensity on deck due to vertical load (kN*m/m) P V vertical load (kN) * safety factor (1.7) D distance of deck from the point of interest to the barrier (m) D L dispersal length (m) © d P V dispersal angle of deck (degree) N 2 1 for end portion, 2 for inner portion The effect of longitudinal loads is minimal in some cases, although still significant enough to be considered in others. For P L - 3 , the longitudinal load was neglected for the inner portion cases because o f its insignificant contribution, yet it was still considered for the end portion cases because o f its affect was notable. The contribution of longitudinal loads is more significant for PL-2 scenarios, though, and in those cases it must be considered. However, the longitudinal load creates a twisting moment in the deck which is difficult to capture using the concept of a dispersal angle. Through observation, the proportion of transverse moment generated by the longitudinal load to the total transverse moment of the deck seems to be fairly constant. For the case o f PL-3 with cantilever length equal to or greater than 900mm, this moment is approximately 7% of the total transverse moment. For cantilever length less than 900mm, this moment is insignificant and can be ignored for simplicity's sake. In terms of PL-2 , the transverse moment of the deck resulting from longitudinal load at both the inner and end portions, with a cantilever length of 900mm or greater, is about 12% of the 66 total transverse moment. Just as well , for a cantilever length of less than 900mm, this ratio is reduced to approximately 5% for both inner and end potion. These proportions have been incorporated into the Equation (16) during the calculations for the total moment intensity that results from the combination of a transverse load, a vertical load, and a longitudinal load. The new parameter, N L , acts like a scale factor, and applies to the sum of moment intensities resulting from transverse and vertical loads. At this point, the total moment intensity, M C , takes into consideration the effect of all loads. MC = (MT + MV)NL (16) Where: M C = transverse moment (kN*m/m) due to combined loads: PT, P V , and P L M T = transverse moment (kN*m/m) due to PT M V = transverse moment (kN*m/m) due to P V N L = longitudinal load factor = 1 for PL3 inner portion = 1 for PL3 outer portion with cantilever length less than 900mm = 1.07 for PL3 outer portion with cantilever length equal or greater than 900mm = 1.05 for PL2 with cantilever length less than 900mm = 1.12 for P L 2 with cantilever length equal or greater than 900mm 3.5.2.2 Comments A s mentioned before, the second method of finding new dispersal angles is preferred over the first, which involves the addition of modification factors. This is because of the simplicity and effectiveness of the second method, two characteristics which coincide 67 with the primary objective of this research. B y directly applying the new dispersal angles using the equations presented above, one can easily determine the maximum moment intensities in any particular area of interest, unlike with the modification factor method, where one must calculate the moment intensities first, using the code dispersal angles, and then add scale factors into various relationships later on, so as to improve the results. Therefore, the new dispersal angle method is recommended for further studies. The overall superior results of this method are presented below. 3.5.3 Results Loading conditions depend upon performance levels, overhang distances, and the properties of barriers and decks. Therefore, since load dispersion, or the dispersal angle, depends upon various parameters, including the loading conditions, many different combinations of these parameters must be tested for analysis to be thorough enough to ensure accuracy. For this reason, F E A has been performed repeatedly so that the necessary scenarios could all be completed. The data that have been used to carry out this research were collected in the spreadsheet available in Appendices F to I. To be precise, 80 different scenarios were performed in consideration of all the possible combinations of the important parameters. The resulting dispersal angles of the different scenarios are presented in the following tables. 68 Table 12 - Results of New Dispersal Angle Method for PL-3 Inner Portion Overhang Distance (mm) Angle of Barrier due to PT Angle of Deck due to PT Angle of Deck due to PV 600 31.2 75.5 34.1 900 30.8 77.2 32.0 1200 31.6 77.3 25.8 1500 32.8 77.6 24.9 1800 34.1 77.0 26.6 *AII angles in degrees Table 13 - Results of New Dispersal Angle Method for PL-3 End Portion Overhang Distance (mm) Angle of Barrier due to PT Angle of Deck due to PT Angle of Deck due to PV 600 28.4 34.2 -77.2 900 31.6 46.5 -65.4 1200 31.0 50.9 -57.4 1500 31.5 55.1 -51.5 1800 32.5 57.1 -43.5 *AII angles in degrees Table 14 - Results of New Dispersal Angle Method for PL-2 Inner Portion Overhang Distance (mm) Angle of Barrier due to PT Angle of Deck due to PT Angle of Deck due to PV 600 -25.1 70.9 62.5 900 -25.1 70.2 71.1 1200 -23.6 69.5 69.9 1500 -24.1 66.0 65.4 1800 -24.6 65.2 61.8 *AII angles in degrees 69 Table 15 - Results of New Dispersal Angle Method for PL-2 End Portion Overhang Distance (mm) Angle of Barrier due to PT Angle of Deck due to PT Angle of Deck due to PV 600 7.5 -10.2 -36.7 900 -7.1 48.0 -20.5 1200 -14.1 63.1 -70.3 1500 -19.4 70.1 -77.6 1800 -23.0 74.5 -79.7 *AII angles in degrees Notice that the dispersal angles obtained in some cases are negative. This phenomenon is inevitable, for reasons given below. A s mentioned before, the new dispersal angle has a slightly different definition than that of the current dispersal angle. While the current angles try to capture the load dispersion to a certain extent, the new angle is a means of calculating maximum moment intensity at specific locations of interest. Instead of providing the average moment intensity calculated using the current angles, the new angles provide the designer with a tool for approximating the critical moment intensity that should be designed for. To achieve such a goal, the new angles are adjusted so that the resulting dispersal length within which the load is distributed would produce a moment intensity that matches that maximum intensity obtained by the F E A as shown in Figure 28. For this reason, negative dispersal angles do seem to be possible based on the F E A results. For example, i f the F E A moment intensity at a location of interest is so large that the dispersal length, calculated using a positive dispersal angle which cannot be produced i f the total load is distributed evenly throughout, was needing to be shortened, then the dispersal angle would have to go in a negative direction to cause the resulting moment intensity to increase until it reached the F E A results. In fact, cases where the dispersal angles became negative occurred for the same reasons as given 70 above. It can be seen from the Tables 12-15 that the two scenarios involving negative dispersal angles took place at the deck overhangs located at the end portion for both PL-3 and PL-2 , and at the barrier for PL-2 at both the inner and end portions. The results of the F E A , which can be found in Appendices F to I, indicate that extremely high moment concentrations exist in both of the above scenarios. For deck end portion systems, this high moment intensity is concentrated at the end of the first support, meaning that the force flows toward this location as it travels across the deck. That is possibly due to the discontinuity of the deck. For the case of PL-2 barriers at the inner portions, which are shown in Figures 29 and 30, the extremely high moment concentration occurs at the bottom corners of the barrier, meaning they are most likely due to barrier discontinuity. 71 Figure 30 - Transverse Moment Intensity for PL-2 Inner Portion For the last case o f PL-2 with the barrier at the end portion, the moment concentration is found at the bottom corner opposite where the transverse load is applied, unlike what happened with the inner portion. The phenomenon shown in Figures 31 and 32 may seem surprising at first, but become sensible when one understands clearly the loading conditions and configurations o f the system. Since the end portion is discontinuous, the outer most corner o f the barrier would be left unsupported. Based on the laws of physics, a force would always take the path with the highest stiffness, which in this case is the interior side of the barrier where it is still connected firmly to the body of the deck. In other words, the part of the barrier closest to the connection between the barrier and the deck, but farthest from the discontinuous edge, is the area that attracts the most forces, or moment intensity, as it is in this case. Hence, the high moment 72 concentration ofthe PL-2 end portion occurs on the interior side ofthe barrier. The PL-2 inner portion, on the other hand, has both sides of the barrier evenly supported by the deck. S SAPTOuo thkxjfj ohsoo (Pl .2-erid o.37)_«ttert .meshed Be Eo* yiew Celine Dew Select assgn Analyze Dtspjay Design actions Help Jai.xj Plight CSick on any Shell ElemenHor detailed diagram Figure 32 - Transverse Moment Intensity for PL-2 End Portion Although the new dispersal angles are found and presented in Tables 12 - 15, it is necessary to further reduce and simplify this large amount of data into something more useful and practical to apply. It can be easily seen that some dispersal angles are more sensitive to changes in overhang distance than others. For cases where the angles act more independently of the overhang distances and have values within a reasonable range, the angles can be simplified and represented by one single dispersal angle; the mean of all relevant angles. However this method of simplification cannot be applied in those cases where the differences between angles are too great. Note that in some cases the differences between angles can be as much as two times the amount, depending on the variation caused by the overhang distance. To take into account this large change in angles, two angles for each o f the 600mm and 1800mm overhang distances were given. 74 The angles between those two cantilever lengths could then be calculated using linear interpolation. Although this method cannot provide exact angles, as shown in Tables 12 -15, it can sufficiently approximate the change of angles depending on the overhang distance. The resulting simplified angles are presented in the following tables. Table 16 - Simplified Dispersal Angles Performance Level 3 - Cast-in-place Reinforced Concrete Barriers Performance Level 2 - Precasted Reinforced Concrete Barriers Inner Portion End Portion Inner Portion End Portion Barrier due to PT 31 31 -24 LI Deck due to PT 77 50 67 L I Deck due to P V 25 LI 65 LI *A11 angles in degrees *For " L I " , refer to Linear Interpolation Table # Table 17 - Linear Interpolation for Dispersal Angles PL-3 Deck End Portion due to P V PL-2 Barrier End Portion due to PT PL-2 Deck End Portion due to PT PL-2 Deck End Portion due to P V 600mm -77 8 -10 -37 1800mm -44 -23 75 -80 *A11 angles in degrees 3.5.4 Examples Tables 18 and 19 depict the results calculated using the new dispersal angles, the current code dispersal angles, and the F E A results. The new dispersal angle results were determined using Equations (13) - (16) in the previous section. 75 Table 18 - Example I Example I - PL3 with 1800mm Cantilever at Inner Portion F E A Results New Dispersal Angles Current Dispersal Angles Transverse moment at base of barrier due to PT (kN*m/m) 99.3 99.3 88.2 Transverse moment at deck support due to PT (kN*m/m) 22.7 22.7 50.6 Transverse moment at deck support due to P V (kN*m/m) 17 17 19.1 Transverse moment at deck support due to combined loads (kN*m/m) 39.8 39.7 69.8 Table 19 - Example II Example II - PL3 with 1200mm Cantilever at End Portion F E A Results New Dispersal Angles Current Dispersal Angles Transverse moment at base of barrier due to PT (kN*m/m) 125 125 106 Transverse moment at deck support due to PT (kN*m/m) 103.5 103.5 91.2 Transverse moment at deck support due to P V (kN*m/m) 8.6 8.6 7.7 Transverse moment at deck support due to combined loads (kN*m/m) 118.8 119.9 98.9 76 Table 20 - Example III Example III - PL2 with 1500mm Cantilever at Inner Portion F E A Results New Dispersal Angles Current Dispersal Angles Transverse moment at base of barrier due to PT (kN*m/m) 224 224 63.2 Transverse moment at deck support due to PT (kN*m/m) 22 22 21 Transverse moment at deck support due to P V (kN*m/m) 5.7 5.7 11.1 Transverse moment at deck support due to combined loads (kN*m/m) 31 31 32 In general, the new dispersal angles tend to generate more accurate results (closer to the F E A results) than those generated by the existing ones. The new dispersal angles also capture areas of high moment concentration, due to the more complicated structural configurations and loading mechanisms. A free piece of software called "Response-2000" is used to carry out the section capacity calculations on a typical PL-3 Cast-In-Place barrier provided by M O T as shown in Figure 33. "Response-2000" is a very user-friendly analysis program that calculates the strength and ductility of a reinforced concrete cross-section subjected to shear, moment, and axial load. Details of this software can be found in the website provided in the "References" section of this thesis. 77 CONCRETE BRIDGE RAILING F SHAPE (SBC04c) Figure 33 - Typical PL-3 Cast-In-Place Barrier FSeDefine Loads Sc*ve, vfew/'OptK"-. *•«+_' Cmss Section Longitudinal Strain Shrinkage & Thermal Strain Nine Graphs IU II Auto Ftanqp Control. M-ex Control: M-Phl Currpnl I u.'iffs V |09 \ 45.46 pot top bot Crack Diagram Long. Reinforcement Stress Long. Reinf Stress at Crack top top 562.6 Longitudinal Concrete Stress - = JODJ Internal Forces C: 859 kN 203 mm 62 mm T: 859 kN N+M M: 227 kNm N " OOkN M 7712 kNn Figure 34 - Section Capacity Calculation Performed by "Response-2000" 78 "Response-2000" has determined the flexural capacity of the typical PL-3 Cast-In-Place Barrier, which is shown in Figure 34 to be 227kN*m/m at its base. For comparison, the design loads calculated by the M M D A method can be found in Tables 18 and 19 and come out to 99.3kN*m/m and 125kN*m/m for the 1800mm inner portion and 1200mm for the end portion. Therefore, the typical PL-3 barrier that is being considered satisfies the safety requirements for both of these two examples. The drawback of this method, however, is that it is only effective for calculating moments at critical locations, such as the bases of barriers, and deck supports. The fact that this method assumes linear load dispersions, instead of more realistic non-linear dispersions, makes it unsuitable for moment calculation done in between the end points. Again, this new dispersal angle concept is significantly different from that of the existing dispersal angle, and should be used with a strict adherence to the guidelines listed in the previous section. Overall, this method w i l l probably be able to take designers one step closer to a safer design while requiring a minimum of effort. 79 C h a p t e r 4: C o n c l u s i o n The jurisdictions reviewed in this report include AASHTO LRFD Bridge Design Specifications, Washington State D O T Bridge Design Manual LRFD, CAN/CSA-S6-88 Design of Highway Bridges and CAN/CSA-S6-00 Canadian Highway Bridge Design Code. Each of them has its own strengths and limitations. AASHTO, for one, is rich in technical information about the topics of traffic barriers and deck overhang designs, although the accuracy of the design method it suggests, the Y L A , relies heavily on assumptions. For example, the deck overhang capacity has to be designed to exceed the flexural resistance of the barrier, or else the failure mechanism may not develop in the barrier, making the Y L A invalid in this case. However, since the barrier is designed for survival instead of for the ultimate strength of the barrier itself, it is l ikely that the barrier is significantly over-designed and therefore wi l l result in the deck overhang being over-designed as well . This over-design issue is minimized by the W S D O T Bridge Design Manual. Seeing itself as a supplement to AASHTO, it offers an alternative design criterion for the deck overhang, replacing AASHTO's. B y suggesting that the nominal transverse barrier resistance, Rw, transferred from the barrier to the deck should be equal to 120% of the transverse loads, F t , as specified in AASHTO, the over-design of the deck overhang can be prevented here. W S D O T Bridge Design Manual also provides a useful table of design forces for several standard traffic barriers used in Washington State, making the design procedures themselves, more consistent. S6-00 offers a list o f available design methods without specifying a standard. This may lead to confusion and cause inconsistency in designs. While the flexibility o f using 80 various refined methods is something that should be kept, it would also be useful to introduce a standardized method so that there is some consistency amongst most applications for the majority of designers. The product offered by this research, the M M D A , is an ideal tool that manages to suits such purposes. After the literature review of various jurisdictions, a discussion about the various methods of analysis that are available is also beneficial and is necessary i f a better method is to be developed. AASHTO made use of the Y L A for it power, but at the same time inherited its weaknesses. The Y L A is an ultimate limit design method that approximates the ultimate capacity of a system by predicting its failure mechanism. Many assumptions must be made to ensure that the failure mechanism predicted by the Y L A actually occurs and is therefore valid. While this method is powerful for systems with simple configurations and load applications, it begins reaching its limitations as soon as the system and its load applications become more complex, for example, when deck overhangs are put together with traffic barriers. Many failure mechanisms are possible for the deck overhang and the number of assumptions needed to ensure their occurrence may easily become too great to remain practical. Furthermore, Y L A produces an upper bound solution, which makes the design unconservative since it becomes possible for the system to fail sooner than predicted. F E A is a much better method for analyzing complex systems. B y correctly entering the necessary parameters, system configurations, and load applications, the F E A program can carry out an analysis using whichever method the user specifies, including any o f a static, dynamic, linear or non-linear analysis. B y knowing the stress distribution within the deck overhang and barriers, their capacities can be designed accordingly. However, the time and human resource needed to develop an F E A is usually impractical 81 for the industry. Hence, the dispersal angle becomes a better tool in this case because of its efficiency. Dispersal angles, which have been developed using F E A , allow for simple calculations to approximate the load dispersion on the deck overhang so that it can be properly designed using minimal time and effort. Although this method is convenient, the underlying theory makes it unconservative. The fact that this method assumes the load dispersion to be linear when in fact it is non-linear, as well as assuming the load distribution is uniform when it is actually a bell curve distribution, causes an overestimation of results in one area and an underestimation in another. In order to prevent this inconsistency in results, a new method called the M M E was introduced. B y simulating the actual non-linearity of load dispersion with different functions for different scenarios, the M M E was developed in order to determine the "maximum" moment intensities for traffic barriers and deck overhangs, ultimately yielding a safer design. However, seeing that the functions may actually be far too complicated and unfamiliar for most people, an improved version of the M M E called the M M D A was developed to address those concerns. The M M D A uses dispersal angles. These are obtained by the F E A , which is a system developed specifically for the purposes of this research. Those dispersal angles are then used to evaluate the maximum moment intensities in the barrier and deck overhang, instead of just finding the average moment intensities calculated through the use of the dispersal angle method. Using the M M E as its backbone, the M M D A works in conjunction with the dispersal angle method, making it more user-friendly and easier for industries to adopt, while also eliminating any lingering weaknesses of the dispersal 82 angle method. Tables 16 and 17 present the results of the M M D A , which should be used in accordance with Equations (13) - (16). 83 C h a p t e r 5: R e c o m m e n d a t i o n s a n d F u t u r e D e v e l o p m e n t s Due to the scope of this thesis, performance level 1 analysis is not included and it is recommended as one of the possible future research topics. Since PL-1 involves mostly post type railings, which yield significantly different load dispersions on the deck overhang than they do on the concrete parapets, it is a necessary area for further analysis. The connections between the barriers and the decks are assumed to be continuous in the F E M , and are not seen as real, discrete connections that should be located at a particular distance apart from one another. A s an extension of this research, the F E M can be modified by adding discrete connections/joints in between the barrier and the deck to reflect the true nature of the existence of anchorages/rebar. This may create a significant difference in the load dispersions generated by F E A . However, it is recommended that this be used to compare the differences between load dispersion for the F E M with and without these discrete connections. Another recommendation for future research is a plastic analysis of the traffic barriers and deck systems in the development of an ultimate limit state design. Similar to the M M D A method, the method to be developed in evaluation of barrier and deck moment intensities for P L - 1 , 2, and 3 in accordance with the plastic analysis o f F E A , should also require minimal time and effort. It should also be easy for designers to apply in practice. The dispersal angle method is suggested as a place to begin, but other methods are fine i f they happen to satisfy the objectives. However, i f other methods are proposed, the results calculated using the proposed methods and conventional dispersal angle methods should still be provided for comparison nonetheless. To then compare the 84 design forces calculated by the plastic analysis with those from the design examples provide by the M O T , examples should be supplied. Reliability is another important recommendation in developing a better design method. Recall that the new dispersal angle varies with different cantilever lengths as well as with the magnitude of variation from case to case, as shown in Table 1 2 - 1 5 . Instead of representing a range of dispersal angles that is within reason and which has a mean value, reliability can be applied here to determine a more representative dispersal angle. The concept of reliability is also applicable for replacing the linear interpolation method, in cases where the range of dispersal angles is too broad to be accurately represented by the mean value. 85 R e f e r e n c e s A A S H T O L R F D Bridge Design Specifications, American Association of State Highway and Transportation Officials, 2004 Bentz, Evan Response-2000 [Computer software] University of Toronto http://www.ecf.utoronto.ca/~bentz/r2k.htm Bridge Design Manual L R F D , Washington State Department of Transportation, 2005 http://www.wsdot.wa.gov/fasc/EngineeringPublications/BDMSections.htm Canadian Highway Bridge Design Code - C A N / C S A - S 6 - 0 0 , CSA International, 2000 Chun Hai Xiao "Analysis and Design of Bridge Deck and Barrier" University of British Columbia, 2004 Design of Highway Bridge - C A N / C S A - S 6 - 8 8 , Canadian Standards Association 1988 Dilger, W . H . , Tadros, G.S. and Chebib, J. "Bending moments in cantilever slabs -Developments in Short and Medium Span Bridge Engineering '90" V o l 1, pp. 256-276. Canadian Society for Civil Engineering, Montreal, 1990 Jategaonkar, R., M . S . Cheung "Bridge analysis using finite elements" Montreal, Canadian Society for Civil Engineering, c l985 Microsoft Office 2002 [Computer software], Microsoft Corporation, © 1983-2001 Troitsky, M . S. "Orthotropic bridges theory and design [by] M . S. Troitsky" Cleveland, James F. Lincoln Arc Welding Foundation, 1967 Mosley, W . H . (Wil l iam Henry), Bungey, J. H . "Reinforced concrete design" Basingstoke, Macmillan Education, 1987 Ontario Highway Bridge Design Code, Ministry of Transportation of Ontario, Downsview, Ontario, 1992. S A P 2000 Nonlinear 8.08 [Computer software], Computers and Structures, Inc. © 1984 -2002 Xanthakos, Petros P. "Theory and design of bridges" New York, Wiley, c l994 Yi lmaz , Cetin., Wasti, Syed Tanvir, North Atlantic Treaty Organization. Scientific Affairs Division. "Analysis and design of bridges" Boston, NATO Advanced Study Institute on Analysis and Design of Bridges (1982 Turkey), 1984 86 A p p e n d i x A : S i m p l e C a n t i l e v e r F E M f o r D e t e r m i n a t i o n o f M e s h S i z e 87 i SAP2000 siinplfffiritilpwilcst 0 0 OO e i n w • 5T o < n - i w re S S3 CZ) re f File Edit £ View. DefHeTcVaw Select :fesign Analyze- : Dismay Design" Options Help", mm B l i t 4 l< | |S jr'» / J © & HI 3-d xy a ia r . 'CroV * • , - ; ? H . "- . 1 ' . rf £ i loint L O . K I S (pv) (As Dchiicl) '•M • it . ,.lklZ.. T.; * „• XO.OOYp.pO HOOtf^. J GLOBAL 3l K N- m- c "z. A p p e n d i x B : L o a d A p p l i c a t i o n s i n P L - 2 F E M f o r I n n e r P o r t i o n 89 A p p e n d i x C : L o a d A p p l i c a t i o n s i n P L - 2 F E M f o r E n d P o r t i o n 93 Figure 39 - PT in PL-2 F E M for End Portion 94 A p p e n d i x D : L o a d A p p l i c a t i o n s i n P L - 3 F E M f o r I n n e r P o r t i o n 97 Figure 42 - P T in PL-3 F E M for Inner Portion 98 A p p e n d i x E : L o a d A p p l i c a t i o n s i n P L - 3 F E M f o r E n d P o r t i o n 101 p ^ g ^ ° a i M ^ a i M M • 'File Edit" View sbeftne^Draw Select-- Assign" Analyze' Diselay- Design Of • • - : 8 o ? i ' V e i V . . XD GI! Ti) Oil 03 j' 5LOBAL § | j | KM rrtC__ ~ | | Figure 47 - PV in PL-3 F E M for End Portion 104 A p p e n d i x F : F E A R e s u l t s a n d M M D A S p r e a d s h e e t f o r P L - 2 I n n e r P o r t i o n 105 Table 21 - M M E Approximation for PL-2 Internal Portion (thk250-oh600-0.87) MME Approximation for Performance Level 2 (thk250-oh600-0.87) Internal Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 24.7 - 31.68 6.98 360 41.7 - 69.42 27.72 540 73.5 - 115.12 41.62 720 98.4 - 171.62 73.22 870 230 - 230.00 0.00 sum/number of section - 21.36 Approximation: Dispersal Angle -25.1 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 120 115.00 -5.00 75 72 86.03 14.03 150 63 68.72 5.72 225 54 57.21 3.21 300 49 49.00 0.00 sum/number of section = 3.59 Approximation: Dispersal Angle 70.9 Degree 106 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 75 0.9 0.70 -0.20 0.66 -0.24 150 1.4 1.39 -0.01 1.26 -0.14 225 1.9 2.09 0.19 1.80 -0.10 300 2.3 2.78 0.48 2.30 0.00 sum/number of section = 0.09 -0.10 Approximation: Dispersal Angle — 62.5 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 121 40.75 -80.25 115.00 -6.00 75 73 39.17 -33.83 86.69 13.69 150 64 37.84 -26.16 69.98 5.98 225 58 36.70 -21.30 59.01 1.01 300 53 35.75 -17.25 51.30 -1.70 sum/number of section — -35.76 2.60 107 Table 22 - M M E Approximation for PL-2 Internal Portion (thk250-oh900-0.87) MME Approximation for Performance Level 2 (thk250-oh900-0.87) Internal Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 23 - 31.68 8.68 360 42 - 69.42 27.42 540 73 - 115.12 42.12 720 101 - 171.62 70.62 870 230 - 230.00 0.00 sum/number of section = - 21.26 Approximation Dispersal Angle — -25.1 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 120 115.00 -5.00 150 61 69.76 8.76 300 46 50.07 4.07 450 36 39.05 3.05 600 32 32.00 0.00 sum/number of section 2.18 Approximation Dispersal Angle 70.2 Degree 108 Deck Transverse Moment due to PV No-Dispcrsion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.7 1.39 -0.31 1.20 -0.50 300 2 2.78 0.78 2.11 0.11 450 2.7 4.17 1.47 2.82 0.12 600 3.4 5.56 2.16 3.40 0.00 sum/number of section = 0.82 -0.05 Approximation: Dispersal Angle — 71.1 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 121 40.75 -80.25 115.00 -6.00 150 61 37.84 -23.16 70.96 9.96 300 48 35.75 -12.25 52.18 4.18 450 41 34.26 -6.74 41.87 0.87 600 38 33.24 -4.76 35.40 -2.60 sum/number of section -25.43 1.28 109 Table 23 - M M E Approximation for PL-2 Internal Portion (thk250-ohl200-0.87) MME Approximation for Performance Level 2 (thk250-oh1200-0.87) Internal Portion „ Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 22 - 31.51 9.51 360 41 - 68.58 27.58 540 71 - 112.82 41.82 720 98 - 166.56 68.56 870 221 - 221.00 0.00 sum/number of section - 21.07 Approximation: Dispersal Angle -23.6 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 121 110.50 -10.50 225 54 58.13 4.13 450 37 39.43 2.43 675 28 29.84 1.84 900 24 24.00 0.00 sum/number of section -0.42 Approximation Dispersal Angle 69.5 Degree 110 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 225 2.1 2.09 -0.01 1.70 -0.40 450 2.6 4.17 1.57 2.88 0.28 675 3.6 6.26 2.66 3.74 0.14 900 4.4 8.35 3.95 4.40 0.00 sum/number of section — 1.63 0.01 Approximation: Dispersal Angle 69.9 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 122 40.75 -81.25 110.50 -11.50 225 55 36.70 -18.30 59.83 4.83 450 40 34.26 -5.74 42.32 2.32 675 33 32.87 -0.13 33.58 0.58 900 33 32.20 -0.80 28.40 -4.60 sum/number of section -21.24 -1.67 111 Table 24 - M M E Approximation for PL-2 Internal Portion (thk250-ohl500-0.87) MME Approximation for Performance Level 2 (thk250-oh1500-0.87) Internal Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 24 - 31.57 7.57 360 41 - 68.86 27.86 540 71 - 113.60 42.60 720 98 - 168.26 70.26 870 224 - 224.00 0.00 sum/number of section — - 21.18 Approximation: Dispersal Angle — -24.1 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 120 112.00 -8.00 300 54 55.37 1.37 600 35 36.78 1.78 900 26 27.53 1.53 1200 22 22.00 0.00 sum/number of section = -0.66 Approximation: Dispersal Angle = 66.0 Degree 112 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 300 2.3 2.78 0.48 2.25 -0.05 600 3.5 5.56 2.06 3.77 0.27 900 4.7 8.35 3.65 4.87 0.17 1200 5.7 11.13 5.43 5.70 0.00 sum/number of section = 2.32 0.08 Approximation: Dispersal Angle 65.4 Degree Deck Transverse Moment due to Combined toads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 120 40.75 -79.25 112.00 -8.00 300 55 35.75 -19.25 57.62 2.62 600 40 33.24 -6.76 40.55 0.55 900 33 32.20 -0.80 32.40 -0.60 1200 31 32.08 1.08 27.70 -3.30 sum/number of section — -21.00 -1.75 113 Table 25 - M M E Approximation for PL-2 Internal Portion (thk250-ohl500-0.87) MME Approximation for Performance Level 2 (thk250-oh1800-0.87) Internal Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 24 - 31.63 7.63 360 41 - 69.14 28.14 540 71 - 114.37 43.37 720 98 - 169.94 71.94 870 227 - 227.00 0.00 sum/number of section — - 21.58 Approximation Dispersal Angle — -24.6 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 119 113.50 -5.50 375 49 50.59 1.59 750 31 32.55 1.55 1125 23 23.99 0.99 1500 19 19.00 0.00 sum/number of section — -0.27 Approximation Dispersal Angle 65.2 Degree 114 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 375 2.9 3.48 0.58 2.77 -0.13 750 4.4 6.95 2.55 4.61 0.21 1125 5.8 10.43 4.63 5.92 0.12 1500 6.9 13.91 7.01 6.90 0.00 sum/number of section = 2.95 0.04 Approximation: Dispersal Angle — 61.8 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 119 40.75 -78.25 113.50 -5.50 375 55 34.94 -20.06 53.37 -1.63 750 40 32.58 -7.42 37.16 -2.84 1125 33 32.05 -0.95 29.91 -3.09 1500 29 32.60 3.60 25.90 -3.10 sum/number of section = -20.62 -3.23 115 A p p e n d i x G : F E A R e s u l t s a n d M M D A S p r e a d s h e e t f o r P L - 2 E n d P o r t i o n 116 Table 26 - M M E Approximation for PL-2 End Portion (thk250-oh600-0.87) MME Approximation for Performance Level 2 (thk250-oh600-0.87) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max from Negative FEA Max Improved Max Barrier Moment Negative Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 28 - 28.50 0.50 360 46 - 55.77 9.77 540 59 - 81.88 22.88 720 59 - 106.92 47.92 870 127 127.00 0.00 - 11.58 Dispersal Angle 7.5 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 66 63.50 -2.50 75 67 63.87 -3.13 150 65 64.24 -0.76 225 63 64.62 1.62 300 65 65.00 0.00 -0.95 Dispersal Angle -10.2 Degree Deck Transverse Moment due to PV 117 No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 75 0.4 0.70 0.30 0.70 0.30 150 1 1.39 0.39 1.42 0.42 225 1.9 2.09 0.19 2.15 0.25 300 2.9 2.78 -0.12 2.90 0.00 0.15 0.19 Dispersal Angle -36.7 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 68 64.52 -3.48 63.50 -4.50 75 67 62.33 -4.67 64.57 -2.43 150 66 60.39 -5.61 65.66 -0.34 225 66 58.67 -7.33 66.77 0.77 300 71 57.14 -13.86 67.90 -3.10 -6.99 -1.92 118 Table 27 - M M E Approximation for PL-2 End Portion (thk250-oh900-0.87) MME Approximation for Performance Level 2 (thk250-oh900-0.87) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 25 - 29.78 4.78 360 43 - 60.88 17.88 540 55 - 93.39 38.39 720 61 - 127.41 66.41 870 157 157.00 0.00 18.21 Dispersal Angle -7.1 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 75 78.50 3.50 150 65 72.13 7.13 300 63 66.71 3.71 450 59 62.05 3.05 600 58 58.00 0.00 3.48 Dispersal Angle — 48.0 Degree 119 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.2 1.39 0.19 1.41 0.21 300 2.7 2.78 0.08 2.84 0.14 450 4.2 4.17 -0.03 4.30 0.10 600 5.8 5.56 -0.24 5.80 0.00 0.00 0.09 Dispersal Angle = -20.5 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 83 64.52 -18.48 78.50 -4.50 150 64 60.39 -3.61 73.53 9.53 300 65 57.14 -7.86 69.55 4.55 450 67 54.56 -12.44 66.36 -0.64 600 68 52.53 -15.47 63.80 -4.20 120 Table 28 - M M E Approximation for PL-2 End Portion (thk250-ohl200-0.87) MME Approximation for Performance Level 2 (thk250-oh1200-0.87) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 24 - 30.46 6.46 360 40 - 63.79 23.79 540 59 - 100.44 41.44 720 56 - 140.90 84.90 870 178 178.00 0.00 22.37 Dispersal Angle -14.1 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 84 89.00 5.00 150 57 75.56 18.56 300 55 65.65 10.65 450 53 58.03 5.03 600 52 52.00 0.00 7.85 Dispersal Angle 63.1 Degree 121 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.3 1.39 0.09 1.51 0.21 300 3.1 2.78 -0.32 3.28 0.18 450 5.4 4.17 -1.23 5.41 0.01 600 8 5.56 -2.44 8.00 0.00 -0.78 0.08 Dispersal Angle = -70.3 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 97 64.52 -32.48 89.00 -8.00 150 62 60.39 -1.61 77.06 15.06 300 63 57.14 -5.86 68.93 5.93 450 66 54.56 -11.44 63.44 -2.56 600 68 52.53 -15.47 60.00 -8.00 122 Table 29 - M M E Approximation for PL-2 End Portion (thk250-ohl500-0.87) MME Approximation for Performance Level 2 (thk250-oh1500-0.87) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 22 - 31.02 9.02 360 37 - 66.30 29.30 540 59 - 106.80 47.80 720 55 - 153.75 98.75 870 199 199.00 0.00 26.41 Dispersal Angle -19.4 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 88 99.50 11.50 150 61 77.78 16.78 300 55 63.84 8.84 450 49 54.14 5.14 600 47 47.00 0.00 8.45 Dispersal Angle 70.1 Degree 123 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 2 1.39 -0.61 1.59 -0.41 300 3.4 2.78 -0.62 3.69 0.29 450 6.7 4.17 -2.53 6.63 -0.07 600 11 5.56 -5.44 11.00 0.00 -1.84 -0.04 Dispersal Angle — -77.6 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 106 64.52 -41.48 99.50 -6.50 150 51 60.39 9.39 79.37 28.37 300 52 57.14 5.14 67.54 15.54 450 55 54.56 -0.44 60.77 5.77 600 66 52.53 -13.47 58.00 -8.00 -8.17 7.04 124 Table 30 - M M E A p p r o x i m a t i o n for P L - 2 E n d Po r t i on (thk250-oh!800-0.87) MME Approximation for Performance Level 2 (thk250-oh1800-0.87) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 _ 0.00 0.00 180 26 - 31.42 5.42 360 41 - 68.19 27.19 540 57 - 111.77 54.77 720 71 - 164.28 93.28 870 217 217.00 0.00 25.81 Dispersal Angle — -23.0 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 91 108.50 17.50 150 45 77.73 32.73 300 44 60.56 16.56 450 44 49.60 5.60 600 42 42.00 0.00 14.48 Dispersal Angle 74.5 Degree 125 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.5 1.39 -0.11 1.64 0.14 300 3.7 2.78 -0.92 3.98 0.28 450 7.5 4.17 -3.33 7.61 0.11 600 14 5.56 -8.44 14.00 0.00 -2.56 0.11 Dispersal Angle = -79.7 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 112 64.52 -47.48 108.50 -3.50 150 65 60.39 -4.61 79.37 14.37 300 63 57.14 -5.86 64.54 1.54 450 62 54.56 -7.44 57.21 -4.79 600 62 52.53 -9.47 56.00 -6.00 126 A p p e n d i x H : F E A R e s u l t s a n d M M D A S p r e a d s h e e t f o r P L - 3 I n n e r P o r t i o n Table 31 - M M E Approximation for PL-3 Internal Portion (thk275-oh600-1.07) MME Approximation for Performance Level 3 (thk275-oh600-1.07) Internal Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 17,1 - 24.55 7.45 360 37.5 - 45.33 7.83 540 56.7 - 63.14 6.44 720 72.9 - 78.58 5.68 900 87.1 - 92.10 5.00 1070 103.4 - 103.40 0.00 sum/number of section - 4.63 Approximation Dispersal Angle — 31.2 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 103.4 103.40 0.00 75 91.4 89.41 -1.99 150 82.7 78.76 -3.94 225 73.3 70.37 -2.93 300 63.6 63.60 0.00 sum/number of section = -1.77 Approximation: Dispersal Angle 75.5 Degree 128 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 75 0.87 0.96 0.09 0.95 0.08 150 1.83 1.91 0.08 1.88 0.05 225 2.8 2.87 0.07 2.80 0.00 300 3.7 3.83 0.13 3.70 0.00 sum/number of section 0.07 0.03 Approximation Dispersal Angle 34.1 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 103.6 88.28 -15.32 103.40 -0.20 75 92.3 86.08 -6.22 90.36 -1.94 150 84.6 84.09 -0.51 80.64 -3.96 225 76.2 82.29 6.09 73.17 -3.03 300 67.5 80.68 13.18 67.30 -0.20 sum/number of section — -0.56 -1.87 129 Table 32 - M M E Approximation for PL-3 Internal Portion (thk275-oh900-1.07) MME Approximation for Performance Level 3 (thk275-oh900-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 17.2 - 24.58 7.38 360 37.6 - 45.42 7.82 540 56.9 - 63.33 6.43 720 73.4 - 78.87 5.47 900 87.7 - 92.49 4.79 1070 103.9 - 103.90 0.00 sum/number of section - 4.56 Approximation: Dispersal Angle 30.8 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 103.9 103.90 0.00 150 85.6 76.49 -9.11 300 70.5 60.53 -9.97 450 56.3 50.07 -6.23 600 42.7 42.70 0.00 sum/number of section -5.06 Approximation: Dispersal Angle = 77.2 Degree 130 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from Barrier Base (mm) 0 FEA Max Negative Moment (kN*m/m) 0 Approximated Max Negative Moment(kN*m/m) 0.00 Finite Differences 0.00 Approximated Max Negative Moment(kN*m/m) 0.00 Finite Differences 0.00 150 1.68 1.91 0.23 1.88 0.20 300 3.5 3.83 0.33 3.71 0.21 450 5.4 5.74 0.34 5.48 0.08 600 7.2 7.65 0.45 7.20 0.00 sum/number of section = 0.27 0.10 Approximation: Dispersal Angle = 32.0 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from Barrier Base (mm) 0 FEA Max Negative Moment (kN*m/m) 103.9 Approximated Max Negative Moment(kN*m/m) 88.28 Finite Differences -15.62 Approximated Max Negative Moment(kN*m/m) 103.90 Finite Differences 0.00 150 87.3 84.09 -3.21 78.38 -8.92 300 74 80.68 6.68 64.23 -9.77 450 61.8 77.92 16.12 55.55 -6.25 600 50.1 75.70 25.60 49.90 -0.20 sum/number of section = 5.91 -5.03 131 Table 33 - M M E Approximation for PL-3 Internal Portion (thk275-oh!200-1.07) MME Approximation for Performance Level 3 (thk275-oh1200-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle FEA Max FEA Max Distance Negative Negative Improved Max from Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0 0 0.00 0.00 180 17.2 17.1 -0.1 24.51 7.31 360 37.5 37.4 -0.1 45.21 7.71 540 56.7 56.6 -0.1 62.92 6.22 720 73 72.9 -0.1 78.24 5.24 900 87 86.9 -0.1 91.62 4.62 1070 102.8 102.7 -0.1 102.80 0.00 sum/number of section = -0.09 4.44 Approximation: Dispersal Angle — 31.6 Degree Deck Transverse Moment due to PT New Dispersal Angle FEA Max Distance Negative Improved Max from Barrier Moment Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences 0 102.8 102.80 0.00 150 86.8 75.65 -11.15 300 74.3 59.84 -14.46 450 62.9 49.50 -13.40 600 52.4 42.21 -10.19 750 42.4 36.79 -5.61 900 32.6 32.60 0.00 sum/number of section — -7.83 Approximation: Dispersal Angle 77.3 Degree 132 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.6 1.91 0.31 1.89 0.29 300 3.4 3.83 0.43 3.73 0.33 450 5.2 5.74 0.54 5.54 0.34 600 7 7.65 0.65 7.30 0.30 750 8.9 9.56 0.66 9.02 0.12 900 10.7 11.48 0.78 10.70 0.00 sum/number of section = 0.48 0.20 Approximation: Dispersal Angle 25.8 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 102.7 88.28 -14.42 102.80 0.10 150 88.5 84.09 -4.41 77.54 -10.96 300 77.7 80.68 2.98 63.58 -14.12 450 68 77.92 9.92 55.04 -12.96 600 59.4 75.70 16.30 49.51 -9!89 750 51.2 73.92 22.72 45.81 -5.39 900 43.2 72.52 29.32 43.30 0.10 sum/number of section = 8.92 -7.59 133 Table 34 - M M E Approximation for PL-3 Internal Portion (thk275-ohl500-1.07) MME Approximation for Performance Level 3 (thk275-oh1500-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance from Barrier Top (mm) FEA Max Negative Moment (kN*m/m) FEA Max Negative Moment (kN*m/m) Improved Max Finite Negative Finite Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 17 - 24.42 7.42 360 37.3 - 44.88 7.58 540 56.3 - 62.28 5.98 720 72.3 - 77.25 4.95 900 85.8 - 90.27 4.47 1070 101.1 - 101.10 0.00 sum/number of section = 4.34 Approximation: Dispersal Angle = 32.8 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from Barrier Base (mm) 0 FEA Max Negative Moment (kN*m/m) 101.1 Improved Max Negative Moment(kN*m/m) 101.10 Finite Differences 0.00 150 86.5 74.28 -12.22 300 75.5 58.71 -16.79 450 65.6 48.53 -17.07 600 56.8 41.36 -15.44 750 48.6 36.04 -12.56 900 41 31.93 -9.07 1050 33.5 28.66 -4.84 1200 26 26.00 0.00 sum/number of section -9.78 Approximation: Dispersal Angle = 77.6 Degree 134 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.5 1.91 0.41 1.89 0.39 300 3.3 3.83 0.53 3.74 0.44 450 5.1 5.74 0.64 5.54 0.44 600 6.9 7.65 0.75 7.31 0.41 750 8.6 9.56 0.96 9.04 0.44 900 10.4 11.48 1.08 10.73 0.33 1050 12.2 13.39 1.19 12.38 0.18 1200 14 15.30 1.30 14.00 0.00 sum/number of section — 0.48 0.29 Approximation: Dispersal Angle — 24.9 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 100.9 88.28 -12.62 101.10 0.20 150 88.1 84.09 -4.01 76.17 11.93 300 78.8 80.68 1.88 62.45 16.35 450 70.7 77.92 7.22 54.08 16.62 600 63.6 75.70 12.10 48.67 14.93 750 57.2 73.92 16.72 45.08 12.12 900 51.3 72.52 21.22 42.66 -8.64 1050 45.6 71.45 25.85 41.04 -4.56 1200 40 70.66 30.66 40.00 0.00 sum/number of section = 4.72 -9.44 135 Table 35 - M M E A p p r o x i m a t i o n for P L - 3 In ternal Po r t ion (thk275-oh!800-1.07) MME Approximation for Performance Level 3 (thk275-oh 1800-1.07) Barrier Transverse Moment Due toPT Due to Combined New Dispersal Angle Distance from Barrier Top (mm) FEA Max Negative Moment (kN*m/m) FEA Max Negative Moment (kN*m/m) Improved Max Finite Negative Finite Differences Moment(kN*m/m) Differences 0 0 - 0.00 0.00 180 17.1 - 24.31 7.21 360 37.2 - 44.52 7.32 540 56 - 61.59 5.59 720 72.2 - 76.19 3.99 900 85.1 - 88.83 3.73 1070 99.3 - 99.30 0.00 sum/number of section = 3.98 Approximation: Dispersal Angle = 34.1 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from Barrier Base (mm) 0 FEA Max Negative Moment (kN*m/m) 99.3 Improved Max Negative Moment(kN*m/m) 99.30 Finite Differences 0.00 300 76.1 59.29 -16.81 600 59.6 42.26 -17.34 900 45.9 32.83 -13.07 1200 33.9 26.84 -7.06 1500 22.7 22.70 0.00 sum/number of section = -9.05 Approximation: Dispersal Angle = 77.0 Degree 136 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 300 3.3 3.83 0.53 3.73 0.43 600 6.7 7.65 0.95 7.29 0.59 900 10.1 11.48 1.38 10.67 0.57 1200 13.6 15.30 1.70 13.91 0.31 1500 17 19.13 2.13 17.00 0.00 sum/number of section = 1.1125 0.32 Approximation: Dispersal Angle 26.6 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications FEA Max Distance Negative Approximated Approximated from Barrier Moment Max Negative Finite Max Negative Finite Base (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 99.1 88.28 -10.82 99.30 0.20 300 79.4 80.68 1.28 63.02 16.38 600 66.2 75.70 9.50 49.55 16.65 900 56 72.52 16.52 43.51 12.49 1200 47.5 70.66 23.16 40.75 -6.75 1500 39.8 69.76 29.96 39.70 -0.10 sum/number of section = 11.60 -8.70 137 Appendix I: FEA Results and MMDA Spreadsheet for PL-3 End Portion 138 Table 36 - M M E Approximation for PL-3 End Portion (thk275-oh600-1.07) MME Approximation for Performance Level 3 (thk275-oh600-1.07) End Portion Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 _ 0.00 0.00 180 22.4 - 25.73 3.33 360 46.6 - 49.53 2.93 540 68.5 - 71.60 3.10 720 87 - 92.13 5.13 900 107 - 111.27 4.27 1070 128.2 - 128.20 0.00 - 2.68 Dispersal Angle — 28.4 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 128.2 128.20 0.00 75 123 126.05 3.05 150 120 123.96 3.96 225 120 121.95 1.95 300 120 120.00 0.00 1.79 Dispersal Angle 34.2 Degree 139 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 75 1.05 0.96 -0.09 0.98 -0.07 150 2.08 1.91 -0.17 2.02 -0.06 225 3.19 2.87 -0.32 3.13 -0.06 300 4.3 3.83 -0.48 4.30 0.00 -0.21 -0.04 Dispersal Angle — -77.2 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 127.9 106.45 -21.45 128.20 0.30 75 123.4 105.23 -18.17 127.03 3.63 150 122.9 104.09 -18.81 125.99 3.09 225 126 103.04 -22.96 125.08 -0.92 300 129 102.06 -26.94 124.30 -4.70 -21.66 0.28 140 Table 37 - M M E A p p r o x i m a t i o n for P L - 3 E n d Po r t i on (thk275-oh900-1.07) MME Approximation for Performance Level 3 (thk275-oh900-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 22 - 25.59 3.59 360 45.9 - 49.03 3.13 540 67.3 - 70.56 3.26 720 85.8 - 90.41 4.61 900 104.5 - 108.78 4.28 1070 124.9 - 124.90 0.00 - 2.69 Dispersal Angle — 31.6 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 124.9 124.90 0.00 150 106.77 118.76 11.99 300 106.66 113.20 6.54 450 104.55 108.13 3.58 600 103.5 103.50 0.00 4.42 Dispersal Angle — 46.5 Degree 141 Deck Transverse Moment due to PV Distance No-Dispersion New Dispersal Angle from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 2.05 1.91 -0.14 1.97 -0.08 300 4.11 3.83 -0.29 4.05 -0.06 450 6.27 5.74 -0.53 6.25 -0.02 600 8.59 7.65 -0.94 8.59 0.00 -0.38 -0.03 Dispersal Angle -65.44 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 123.7 106.45 -17.25 124.90 1.20 150 115.8 104.09 -11.71 120.73 4.93 300 111.86 102.06 -9.80 117.24 5.38 450 114.7 100.33 -14.37 114.38 -0.32 600 118.8 98.85 -19.95 112.09 -6.71 -14.6 0.90 142 Table 38 - M M E A p p r o x i m a t i o n for P L - 3 E n d Por t ion (thk275-ohl200-1.07) MME Approximation for Performance Level 3 (thk275-oh1200-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0 0 0.00 0.00 180 21.5 21.2 -0.3 25.62 4.12 360 45.7 45 -0.7 49.12 3.42 540 67.5 65.7 -1.8 70.75 3.25 720 85.5 84.7 -0.8 90.73 5.23 900 104 103 -1 109.23 5.23 1070 125.5 123 -2.5 125.50 0.00 -1.01 3.03 Dispersal Angle — 31.0 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 125.5 125.50 0.00 150 114 118.32 4.32 300 106 111.92 5.92 450 105 106.17 1.17 600 101 100.99 -0.01 750 95 96.28 1.28 900 92 92.00 0.00 1.81 Dispersal Angle — 50.9 Degree 143 Deck Transverse Moment due to PV No-Dispersion New Dispersal Angle Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 2 1.91 -0.09 1.95 -0.05 300 4.03 3.83 -0.21 3.98 -0.05 450 6.13 5.74 -0.39 6.09 -0.04 600 8.32 7.65 -0.67 8.30 -0.02 750 10.57 9.56 -1.01 10.60 0.03 900 13 11.48 -1.53 13.00 0.00 -0.56 -0.02 Dispersal Angle -57.4 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 123 106.45 -16.55 125.50 2.50 150 115 104.09 -10.91 120.27 5.27 300 110 102.06 -7.94 115.90 5.90 450 112 100.33 -11.67 112.26 0.26 600 114 98.85 -15.15 109.28 -4.72 750 111 97.61 -13.39 106.88 -4.12 900 113 96.58 -16.42 105.00 -8.00 -13.14 -0.41 144 Table 39 - M M E A p p r o x i m a t i o n for P L - 3 E n d Po r t i on (thk275-ohl500-1.07) MME Approximation for Performance Level 3 (thk275-oh1500-1.07) Barrier Transverse Moment Due to PT Due to Combined New Dispersal Angle Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 180 21.7 - 25.60 3.90 360 45.3 - 49.04 3.74 540 67.1 - 70.59 3.49 720 84.7 - 90.46 5.76 900 103.5 - 108.85 5.35 1070 125 - 125.00 0.00 3.18 Dispersal Angle 31.5 Degree Deck Transverse Moment due to PT New Dispersal Angle Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 125 125.00 0.00 150 112.4 116.79 4.39 300 103.7 109.59 5.89 450 101.3 103.23 1.93 600 98 97.56 -0.44 750 93.8 92.49 -1.31 900 89.1 87.91 -1.19 1050 82.8 83.77 0.97 1200 80 80.00 0.00 1.14 Dispersal Angle 55.1 Degree 145 Deck Transverse Moment due to PV No-Dispersion Modification Factor Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 150 1.8 1.91 0.11 1.94 0.14 300 3.9 3.83 -0.07 3.95 0.05 450 6 5.74 -0.26 6.02 0.02 600 8.3 7.65 -0.65 8.16 -0.14 750 10.5 9.56 -0.94 10.38 -0.12 900 12.6 11.48 -1.13 12.67 0.07 1050 14.8 13.39 -1.41 15.04 0.24 1200 17.5 15.30 -2.20 17.50 0.00 -0.33 0.03 Dispersal Angle -51.5 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 121.8 106.45 -15.35 125.00 3.20 150 112.6 104.09 -8.51 118.73 6.13 300 106.9 102.06 -4.84 113.54 6.64 450 107.3 100.33 -6.97 109.25 1.95 600 108.1 98.85 -9.25 105.72 -2.38 750 107.9 97.61 -10.29 102.86 -5.04 900 106.9 96.58 -10.32 100.58 -6.32 1050 103.8 95.74 -8.06 98.81 -4.99 1200 104.8 95.07 -9.73 97.50 -7.30 -7.28 -0.90 146 Table 40 - M M E A p p r o x i m a t i o n for P L - 3 E n d Po r t i on (thk275-oh!800-1.07) MME Approximation for Performance Level 3 (thk275-oh1800-1.07) Barrier Transverse Moment Due to PT Due to Combined Modification Factor Distance FEA Max FEA Max from Negative Negative Improved Max Barrier Moment Moment Finite Negative Finite Top (mm) (kN*m/m) (kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0 180 21.8 - 25.56 3.76 360 45.5 - 48.89 3.39 540 66.6 - 70.27 3.67 720 84.4 - 89.94 5.54 900 102.9 - 108.09 5.19 1070 124 - 124.00 0.00 3.08 Dispersal Angle 32.5 Degree Deck Transverse Moment due to PT Modification Factor Distance from FEA Max Barrier Negative Improved Max Base Moment Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences 0 124 124.00 0.00 300 103.3 107.75 4.45 600 96.8 95.27 -1.53 900 89 85.38 -3.62 1200 80 77.35 -2.65 1500 70.7 70.70 0.00 -0.56 Dispersal Angle 57.14 Degree 147 Deck Transverse Moment due to PV No-Dispersion Modification Factor Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 0 0.00 0.00 0.00 0.00 300 3.8 3.83 0.03 3.92 0.12 600 8 7.65 -0.35 8.03 0.03 900 12.7 11.48 -1.23 12.35 -0.35 1200 16.9 15.30 -1.60 16.90 0.00 1500 21.7 19.13 -2.58 21.70 0.00 -0.95 -0.03 Dispersal Angle — -43.5 Degree Deck Transverse Moment due to Combined loads Code Dispersal Angle Combined Modifications Distance from FEA Max Barrier Negative Approximated Approximated Base Moment Max Negative Finite Max Negative Finite (mm) (kN*m/m) Moment(kN*m/m) Differences Moment(kN*m/m) Differences 0 120 106.45 -13.55 124.00 4.00 300 105.7 102.06 -3.64 111.67 5.97 600 105.7 98.85 -6.85 103.30 -2.40 900 105 96.58 -8.42 97.73 -7.27 1200 102.8 95.07 -7.73 94.25 -8.55 1500 99.6 94.20 -5.40 92.40 -7.20 -7.60 -2.57 148 Appendix J: Sample Transverse Moment Due to Design Loads Plots 149 PL3-Barrier-oh1800-PT-lnner 120 0 200 400 600 800 1000 1200 Distance from Barrier Top (mm) • FEA —•— Code Disperal Angle New Dispersal Angle PL3-Deck-oh1800-PV-lnner w" s 25 i - -- i C/5 3 1600 O Distance from Barrier Base (mm) FEA »- No Dispesal Angle * New Dispersal Angle OTQ C 1 re i 3 •a_ re" 2 ST o o 3 t r 5' CL o • f l i O re re w o o s 3! oo o PL3-Deck-oh1800-Combined-lnner 120 200 400 600 800 1000 Distance from Barrier Base (mm) 1200 1400 1600 O < re •FEA -Code Disperal Angle New Dispersal Angle ore Appendix K: Spreadsheet for Moment Calculations Using Dispersal Angle Method 154 Table 41 - Spreadsheet for Moment Calculations Using Dispersal Angle Method PROJECT MoT - Joe Wong SECTION 1 TITLE Transverse Moments on Bridge Barrier and Deck DATE 12/06/2005 FILE MMEv3.xls TIME 6:01 P M INPUT PARAMETERS L o a d s Target performance level P L = 3 Live load factor fl = = 1.7 Overhang Distance oh = = 1800 Barrier Type i.)Sleel Bridge Railings (PL-1) Post spacing d = 1200 [mm] Width of Post Base wp = = 300 [mm] CALCULATIONS Depth of cantilever portion dd oh-DemoAppl icat ion!G19/2 = 1500 [mm] Loads Transverse load Pt = lf(PL=1,50,if(PL=2,100,if(PL=3,210,"Error"))) 210 [kN] Longitudinal load PI lf(PL=1,20,if(PL=2 130,if(PL=3,70,"Error"))) 70 [kN] Vertical load Py lf(PL=1,10,if(PL=2,30,if(PL=3,90,"Error"))) 90 [kN] Height of impact h = lf(PL=1,750,if(PL=2,870,if(PL=3,1070,"Error"))) 1070 [mm] Barrier length for transverse and longituduenal loads = l f(PL=1,1200,if(PL=2,1050,if(PL=3,2400,"Error")» 2400 [mm] Barrier length for vertical loads Lv = lf(PL=1,5500,if(PL=2,5500,if(PL=3,12000,"Error"))) 12000 [mm] Factored transverse load FPt P f f l 357 [kN] Factored longitudinal load FPI = 119 [kN] Factored vertical load F P y Py*fl 153 [kN] Factored moment due to transverse load ( N A . for PL-1) Mt = F P f h 381990 [kN'mm] Dispersal Angle of Barrier Moment @ inner portion thetami = lf(PL=1,"N.A.",if(PL=2,56,if(PL=3,42,"Error"))) 42 [degree] Tensile force @ inner portion thetati = lf(PL=1,"N.A.",if(PL=2,25,if(PL=3,3,"Error"))) 3 [degree] Moment @ end portion thetame = lf(PL=1,"N.A.",if(PL=2,55,if(PL=3,48,"Errof'))) 48 [degree] Tensile force @ end portion thetate = lf(PL=1,"N.A.",if(PL=2,8,if(PL=3,0,"Error"))) 0 [degree] Dispersal Length @ Barrier Base Moment @ inner portion Ltmi = Lt+2*h*tan(radians(thetami)) = 4326.86 [mm] Tensile force @ inner portion Ltti = Lt+2"h*tan(radians(thetati)) = 2512.15 [mm] Moment @ end portion Ltme = Lt+h*tan(radians(thetame)) = 3588.36 [mm] Tensile force @ end portion Ltte = Lt+h"tan(radians(thetate)) = 2400.00 [mm] Dispersal Angle of Deck Moment @ inner portion thetaid = lf(PL=1,55,if(PL=2,55,if(PL=3,47,"Error"))) 47 [degree] Tensile force @ inner portion thetatd = l f(PL=1,20,if(PL=2,20,if(PL=3,10,"Error))) 10 [degree] Moment @ end portion thetaed lf(PL=1,55,if(PL=2,55,if(PL=3,45,"Error"))) 45 [degree] Tensile force @ end portion thetafe = lf(PL=1,8,if(PL=2,8,if(PL=3,0,"Erroi J '))) 0 [degree] Vertical for both portions thetavd = 0 = 0.00 [degree] INTERNAL PORTION Barrier ii.) Cast-in-place Concrete Barrier (PL-3) or Precast Concrete Barrier (PL-2) Distance from barrier top @ section: 0 hO = 0*h 0.00 [mm] 1 h i = 0.2 'h 214.00 [mm] 2 h2 0.4*h 428.00 [mm] 3 h3 = 0.6*h 642.00 [mm] 4 h4 0.8*h 856.00 [mm] 5 h5 h 1070.00 [mm] Transverse moment due to transverse load ( § section: 0 mhO = F P f h O 0 1 mh1 FPt*h1 76398 2 mh2 F P f h 2 152796 3 mh3 = FPt*h3 229194 4 mh4 = FPt*h4 305592 5 mh5 = F P f h 5 381990 Dispersal length @ section: 0 dO = Lt+2*h0*tan(radians(thetami)) = 2400.00 1 d1 = Lt+2*h1*tan(radians(thetami)) = 2785.37 2 d2 = Lt+2*h2*tan(radians(thetami)) = 3170.75 3 d3 = Lt+2*h3"tan(radians(thetami)) = 3556.12 4 d4 = Lt+2*h4*tan(radians(thetami)) = 3941.49 5 d5 = Lt+2*h5*tan(radians(thetami)) = 4326.86 Transverse moment distribution @ section: 0 mdO = mhO/dO 0.00 1 md1 = mh1/d1 27.43 2 md2 = mh2/d2 48.19 3 md3 mh3/d3 64.45 4 md4 mh4/d4 77.53 5 md5 mh5/d5 88.28 155 Deck i.) Steel Bridge Railings (PL-1) Traverse load per post p Vertical load per post Pyp Moment due to transverse toad Mtp Moment due to vertical load Myp Distance @ sections 0 ddO 1 dd1 2 dd2 3 dd3 4 dd4 5 dd5 Dispersal length of transverse moment At 0*dd DO At 0.2*dd D1 At 0.4'dd D2 At 0.6'dd D3 At 0.8'dd D4 At dd D5 Transverse moment distribution At 0*dd MdtpO At 0.2*dd Mdtpl At 0.4'dd Mdtp2 At 0.6*dd Mdtp3 At 0.8*dd Mdtp4 At dd Mdtp5 Dispersal length of vertical load At O'dd DvO At 0.2*dd Dv1 At 0.4'dd Dv2 At 0.6*dd Dv3 At 0.8*dd Dv4 At dd Dv5 Moment due to vertical load At O'dd MvO At0.2*dd Mv1 At 0.4'dd Mv2 At 0.6*dd Mv3 At 0.8*dd Mv4 At dd Mv5 Moment distribution due to vertical load At O'dd At 0.2*dd At 0.4*dd At 0.6*dd At 0.8*dd Atdd Total Moment At O'dd At 0.2*dd At 0.4'dd At 0.6*dd At 0.8*dd Atdd MdvO Mdv1 Mdv2 Mdv3 Mdv4 Mdv5 MttO Mtt1 Mtt2 Mtt3 Mtt4 Mtt5 if(Lt<d,FPt,if(Lt<2'd,FPt/2,if(Lt<3'd,FPt/3,"Error))) = 119.00 [kN] if(Lv<d,FPy,if(Lv<2*d,FPy/2,if(Lv<3*d,FPy/3,if(Lv<4*d, FPy/4,if(Lv<5*d.FPy/5,if(Lv<6*d,FPy/6,if(Lv<7*d,FPy/7, if(Lv<8*d,FPy/8,"Error")))))))) = Error [kN] Ptp'h = 1.27E+05 [kN'mm) Pyp'O = #VALUE! [kN'mm/mm] O'dd 0.2*dd 0.4'dd 0.6*dd 0.8*dd dd 0 [mm] 300 [mm] 600 [mm] 900 [mm] 1200 [mm] 1500 [mm] wp = 300.00 [mm] 0.2'dd*tan(radians(thetaid))*2+wp = 943.42 [mm] 0.4*dd'tan(radians(thetaid))*2+wp = 1586.84 [mm] 0.6*dd*tan(radians(thetaid))*2+wp = 2230.26 [mm] 0.8*dd'tan(radians(thetaid))*2+wp = 2873.68 [mm] dd*tan(radians(thetaid))*2+wp = 3517.11 [mm] Mtp/wp = 424.43 [kN'mm/mm] Mtp/D1 = 134.97 [kN'mm/mm] Mtp/D2 = 80.24 [kN'mm/mm] Mtp/D3 = 57.09 [kN'mm/mm] Mtp/D4 = 44.31 [kN'mm/mm] Mtp/D5 = 36.20 [kN'mm/mm] wp = 300.00 [mm] 0.2*dd*tan(radians(thetavd))*2+wp = 300.00 [mm] 0.4'dd*tan(radians(thetavd))*2+wp = 300.00 [mm] 0.6*dd*tan(radians(thetavd))*2+wp = 300.00 [mm] 0.8*dd*tan(radians(thetavd))*2+wp = 300.00 [mm] dd*tan(radians(thetavd))'2+wp = 300.00 [mm] Pyp'O'dd = #VALUE! [kN'mm] Pyp'0.2*dd = #VALUE! [kN'mm] Pyp*0.4'dd = #VALUE! [kN'mm] Pyp'0.6*dd = #VALUE! [kN'mm] Pyp'0.8*dd = #VALUE! [kN'mm] Pyp'dd = #VALUE! [kN'mm] MvO/DvO Mv1/Dv1 Mv2/Dv2 Mv3/Dv3 Mv4/Dv4 Mv5/Dv5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! [kN'mm/mm] [kN'mm/mm] [kN'mm/mm] [kN'mm/mm] [kN'mm/mm] [kN'mm/mm] MdvO+MdtpO = #VALUE! [kN'mm/mm; Mdv1+Mdtp1 = #VALUE! [kN'mm/mm; Mdv2+Mdtp2 = #VALUE! [kN'mm/mm; Mdv3+Mdtp3 = #VALUE! [kN'mm/mm; Mdv4+Mdtp4 = #VALUE! [kN'mm/mm; Mdv5+Mdtp5 = #VALUE! [kN'mm/mm; 156 \ii.) Cast-in-place Concrete Barrier (PL-3) or Precast Concrete Barrier (PL-2) Distance @ sections 0 ddOii = 0*dd = 0 [mm] 1 ddlii = 0.2*dd = 300 [mm] 2 dd2ii = 0.4*dd = 600 [mm] 3 dd3ii = 0.6*dd = 900 [mm] 4 dd4ii = 0.8*dd = 1200 [mm] 5 dd5ii = dd = 1500 [mm] Dispersal length of transverse moment AtO'dd DOii = Ltmi = 4326.86 [mm] At0.2*dd D1ii = 0.2*dd*tan(radians(thetaid))*2+Ltmi = 4970.29 [mm] At0.4*dd D2ii = 0.4*dd*tan(radians(thetaid))*2+Ltmi = 5613.71 [mm] At0.6*dd D3ii = 0.6*dd*tan(radians(thetaid))*2+Ltmi = 6257.13 [mm] At0.8*dd D4ii = 0.8*dd*tan(radians(thetaid))*2+Ltmi = 6900.55 [mm] Atdd D5ii = dd*tan(radians(thetaid))*2+Umi = 7543.97 [mm] Transverse moment distribution AtO'dd MdtpOii = Mt/DOii = 88.28 [kN*mm/mm] At0.2*dd Mdtplii = Mt/D1ii = 76.85 [kN'mm/mm] At0.4*dd Mdtp2ii = Mt/D2ii = 68.05 [kN'mm/mm] At0.6*dd Mdtp3ii = Mt/D3ii = 61.05 [kN*mm/mmj At0.8*dd Mdtp4ii = Mt/D4ii = 55.36 [kN*mm/mm] Atdd Mdtp5ii = Mt/D5ii = 50.64 [kN*mm/mm] Moment due to vertical load At 0*dd At0.2*dd At 0.4*dd At 0.6*dd At0.8*dd At dd MvOii Mvlii Mv2ii Mv3ii Mv4ii Mv5ii FPy'O'dd FPy*0.2*dd FPy*0.4*dd FPy*0.6*dd FPy*0.8*dd FPy'dd 0 [kN'mm] 45900 [kN'mm] 91800 [kN'mm] 137700 [kN'mm] 183600 [kN'mm] 229500 [kN'mm] Dispersal length of vertical load AtO'dd DvOii = Lv = 12000.00 [mm] At0.2*dd Dvlii = 0.2*dd*tan(radians(thetavd))*2+Lv = 12000.00 [mm] At0.4*dd Dv2ii = 0.4*dd*tan(radians(thetavd))*2+Lv = 12000.00 [mm] At0.6*dd Dv3ii = 0.6*dd*tan(radians(thetavd))*2+Lv = 12000.00 [mm] At0.8*dd Dv4ii = 0.8*dd*tan(radians(thetavd))*2+Lv = 12000.00 [mm] Atdd Dv5ii = dd*tan(radians(thetavd))*2+Lv = 12000.00 [mm] Moment distribution due to vertical load At O'dd MdvOii = MvOii/DvOii 0.00 [kN'mm/mm] At0.2*dd Mdv1 ii = Mv1ii/Dv1ii = 3.83 [kN'mm/mm] At0.4*dd Mdv2ii = Mv2ii/Dv2ii = 7.65 [kN'mm/mm] At0.6*dd Mdv3ii = Mv3ii/Dv3ii = 11.48 [kN'mm/mm] At0.8*dd Mdv4ii = Mv4ii/Dv4ii = • 15.30 [kN'mm/mm] At dd Mdv5ii = Mv5ii/Dv5ii = 19.13 [kN'mm/mm] Total Moment At O'dd MttOii = MdvOii+MdtpOii 88.28 [kN'mm/mm] At0.2*dd Mttlii = Mdv1ii+Mdtplii 80.68 [kN'mm/mm] At0.4*dd Mtt2ii Mdv2ii+Mdtp2ii 75.70 [kN'mm/mm] At0.6*dd Mtt3ii = Mdv3ii+Mdtp3ii 72.52 [kN'mm/mm] At0.8*dd Mtt4ii = Mdv4ii+Mdtp4ii 70.66 [kN'mm/mm] At dd Mtt5ii = Mdv5ii+Mdtp5ii 69.76 [kN'mm/mm] 157 Appendix L: Constants vs. Cantilever Length Relationship for MME 158 V ] Be ,6*t View '-insert cp/mat.jloc* Chart ,'SMndow Help - l a lx l Constant A vs. Cantilever Length MOT ' • . ^ JliiNlllllllS • • t i l l l i M M l l i i s i i i BIIIll IB Mi ' 400 GOO 00 1000 1200 1400 1600 1800 Cantilever Length (mm) - F E A Approximation | !i< < • M \ Constant A - I n t / ' e j ^ ^ ^ - y , ^ ^ ^ | ^ - ^ n ^ j y Kre-R2/.MME-PL1 7 DspenwtAnpje ",|'< ;' ?Bilirjliiiiiili!i^ Figure 52 - Constant A vs. Cantilever Length Plot for M M E 3 )^ Bte 6c»t yraw Insert Fa mat lools Qsta Bfndow tfet> "A351 f» , fij Length (mm) ;•- - it .• u ' D 349! 350] ill 1353 §•>' 3 - . i i 356 ;357 •358 [359" 36 -5§* 36: Approximation of Various Parameters for Different Cantilever Li Constant A • FC A 1200', 1500 ' 1800 Approximation Differences' Equation - I 1 18 -.1'16 • • 13 y=constent ooo • ~-5 79E-05 ' fllilllltlffilll Length (mm) iFEA 900 i 1200! 1500! '"i860!" 1.20! T l 7 [ T.15}" .....„.J.. I Approximation A " ' 1 1 i'.'i'l i 'i 'fS 1.14* 1.1JJB -5 39E-0!'":; 12: "3 :onstant B i364! {•365! |366! j387, 368 369 [370 !371' !372 Length (mm) ; F E A ^Approximation iDifferences ; Equation 600: 1.10E-03! 1.07E-03! -2.92E-05J 900i 9 20E-04: 9.14E-04: -6.30E-06I 1200: 7.18E-04: 7.62E-04 : 4.42E-05: y=mx+b 1500: 5.90E-04: 6.10E-04: 2.05E-05: 1800: 4.87E-04: 4.58E-04! -2.92E-05: Length (mm) !FEA 600: -4.73E-05: 5 55E-05: 1200: 1.00E+00I 1500! 5.90E-04! 1800: 4.87E-04: Approximation;!^ 1.07E-OfS;5 9.14E-Sg| 7.62E-(K$S 6.10E-6SM 4.58E-SH rConstant«-lnt / Constant B-iht / ConstantC-Int \ M M E PL3/"MME-P12 /MME-ftl / aspersalAnde ,l < I " ; Figure 53 - Spreadsheet M M E Calculations for Constant A 159 3 Microsoft excel - MME FJe Bit View insert Format loots Chart yvjndow tlep * ' J j * U *^ ^ " 1 4 ^ "* " ' ^ * ' Constant B vs. Cantilever Length | FEA ••••••• Approximation j • « / ConstantA-lnt\^onstant"B-IntCoretantOjnt;../^^-PIO.^€^F/'wSTf-?'OBpersiflrigfe',j 1 200 400 600 800 1000 1200 WOO 1600 1800 2000 Cantilever Length (mm) Figure 54 - Constant B vs. Cantilever Length Plot for M M E J Microwft I »• '•! M' I! S ] Fjle Edit aew ... LI 5 . . . . . . ~ ^ g ^ . . . Insert ,Foj mat-"'loots s . L ngth (mm 357 359 360 3 6 l 362: 0.00! -5 .79E-05 : 1.23! -5 3 9 E - 0 ' 1 2. 363^ Constant B l r 366 367 §£ S : 370 H §Z? 37 374 37" Length (mm) ilS^^H'ffli^'S'SiB L, -[<i-»r..-«.= _-.j->r'.~ 600 - 900 1200 1500 1800 -10E-03 , 9 20E-04 7 18E-04 5 90E-04 4 97E-04 .' • . -- 7 8 2 E - W 6 10E-04 4 58E-04 - 5 07E-07 1.37E-03 -2 02E -05 Ilg§fi|j iSiMlPI I ISjLt.lMUfI*:. 600 -4 .73E-05 Approximation 1.07E-0 -y=mx+b* 900: 5.55E-05J " ' l200| I .OOE+Ooi 1500: 5 .90E-04! """"i" SOOT4^87E-04T 9.14E-CK "J62E-CV '6.'10E-CV "'4™58E-0' 4.46E-0>: 1.37E-0S: 376 Length 37$ If! 3 7 9 is 3 8 ! < « • • . / [Finite (mm) [ F E A [Approximation ^Differences [Equation 600i 9 .62E-01 ! 9 \ 5 8 E - 0 1 j - 4 . 4 3 E - 0 3 ! 900[ 9 .30E-01 ! / 9 . 3 8 E - 0 1 ] 8 . 3 8 E - 0 3 ! 12001 9.16E-01 9 " i 8 E - b l l 2 . 1 6 E - 0 3 ] y=mx+b 1500: 9.01E-01 i 8 .99E-01! -1.67E-03J Constant A lint 7 OcirisrJiritB-Int"/ CcjnsWitC~lnT\MME P13 / MME-pi.2~/ Length (mm) : F E A 600 i 9001 120bT 1500! tm / :> «"i*rigk.jl±l. 1.11E+00I 1.10E+00! 9.16E-01I 9 .01E-01 ! ^Approximations 9 58E-0" 9 . 3 8 E - 0 J 9 . 1 8 E - 0 * 8 . 9 9 E - 0 ^ ».m;6pm0089SI2_ Figure 55 - Spreadsheet M M E Calculations for Constant B 160 E v i t r i o l ! =xc&-JMME :&p):8eiSEo)t £ew Insert Format loos flurt fflrWow belp •>U.:.s*iU 5i a ^ ; ^ ' -j i -* : • l | 4 " Constant C vs. Cantilever Length 200 400 Rrm„„ •Value (X) axia | OO 1000 1200 1400 1600 1800 Cantilever Length (mm) — FEA Approximation J iRie^ x^|ij||^ ii?J|ji Figure 56 - Constant C vs. Cantilever Length Plot for M M E ] Microvolt Exf i. M'll .5§] B e - B i t * yjew 'insert Format- loots Data" &jndow y i^p A 3 7 7 iliiSisili^l Length (mm) * it O f2- 9 *j • • ! i377_ [378 380 BB 38 <384 Ijjp s i J390J '3911 3^92 393 39* ' ' 3961 397 Length (mm) iBillfRl , , 900 1200 ^HfiSo* • - A a , , -'•' 9 30E-01 9 16E-01 ' 9 01E-01 ''58 84E-0I illilwll lllISilisllllllllii^^^^M Approximation Differences Equation ? ' 9 58E-01 -4.43E-03 . ' 8 38E-01 8 38E-03 9 18E-0! 2 16E-03 8.99E-01 -1 67C-03 8*79E-01 ,-4 43E-03 stant C angth (mm)jFEA ysfrw-D 5 600! 9o6T 1200!'' 1500;" 180o]" 1.11E+00 T i 61+00 •••g~i6E-bi _ _ _ _ _ 8 84 :r Vi Approximation^ I 9.58E-0~il 9 38E-0 {f 9 18E-0 8 99E-0 : ; 8.79E-0;-S 2 22E-16 -6.S4E-05, 9" 9*76-01 Length (mm) IFEA 600! 9bbT "1260T 1500 j 1800! 9.62E-01 ' '93OE-61 ' gi 'SE-Oi 8.84E-di ^Approximation 1 9 T 6 2 E - 6 7 _______ 9 .03 | ;0 i f 8 . 8 4 E - 0 1 Finite Differences I Equation [745E-67] l" 22E-02J 576E-03! y=A*exp(-B*x) :2T14E-63I I T I T E - 0 8 1 2.01E-02! 1.00E+00! ••••y;pgg^5;" Length (mm) iFEA 600: 900! J200T '"i5bbT l l b b T 1.11E+00 J J O E + O O ' 9?16E-01 "9'blE-oT "8.84E-6T -1.89E-0-J3 '9"97!-b!S| Approximation;;:^ : 9 . 6 2 E - 0 ' - : ' 1 9.42E-0 9.22E-0 9.03E-0-ft^ 8.84E-o'*":< 1 00E+0C'-;-7,09E-0R,,( _ . lHf_.a«Hs#(^tarit:A-1nt / Constante-ific / b_nstantC- lnt^M.^-P-3 /w Ready . ... Sum=«JlD,181229 Figure 57 - Spreadsheet M M E Calculations for Constant C 161 DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF BRITISH C O L U M B I A CHECKLIST FOR GRADUATING GRADUATE STUDENTS o Apply for Graduation either on-line at http://students.ubcxa/current/graduation.crn'?page=applv (Deadline dates are found in the Calendar). Inform the Graduate Secretary that you have applied for graduation. o Submission of Theses For PhD dissertations, the requirement is as follows. The department does not require a hard copy for archiving in the CEME Reading Room. PhD dissertations will be available to UBC users through the ProQuest distribution service via the UBC Library. Any copies required by you and your supervisor should be agreed upon by both of you. It is the responsibility of the student to produce al! of the copies required. For MASc theses, the requirements are as follows. The department does not require a hard copy for archiving in the CEME Reading Room. The Reading Room is rapidly filling with MASc theses and many of these have become badly worn through use by subsequent students. The new requirement is for a CD version of your thesis in PDF format. The department will be making MASc theses available to UBC users through our web site. Note that MASc theses are not readily available via any other means, as the UBC Library stores copies only in microfiche form and MASc theses are not distributed through ProQuest. Any copies required by you and your supervisor should be agreed upon by both of you. It is the responsibility of the student to produce all of the copies required. If you do not have the Adobe Acrobat software required to convert your word processor document to PDF, it is available in the graduate computer lab downstairs in CEME. A word processor document is not adequate because it is not a read-only file. o Desk and office area must be cleaned, and the Department's Graduate Secretary advised of your departure, o Lockers must be cleared out and advise the CEGSS. o Lab work area must be cleaned and left tidy, including the disposal of all samples, and the return of lab keys, equipment, tools and instrumentation to the Machine Shop or Lab Manager. o Computer accounts: "Home" directory should be cleaned out and the Computer Laboratory Manager advised of your departure. o Advise the Department office of your forwarding address and have someone check your mailbox after your leave. If you request the Department office to forward your mail this will be done for 3 months only. o Return all copy cards to the CEGSS copy card representative for refund. o Return all books and periodicals to the Library and Reading Room. o Return all keys to Parking & Security Access Control Centre and present receipt to the Department office for a refund. \\Office\Home\GRADl\WrNWORD\2004-2005 Grad Forms-orientation\CHECKLST for Graduating.DOC 9/30/2004 

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