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Sj'?"‘*-$v. -.3 v ‘ K“ ABSTRACT A SIMULATION STUDY OF FATIGUE LIFE OF HIGHWAY BRIDGES by Egbert Hsi-Ting Chang A study of the fatigue life of simple span highway bridges is made using computer simulation. The study begins with a deterministic analysis of a bridge traversed by a vehicle. The set-up of the simulation procedure follows. Numerical results were obtained to illustrate the procedure as well as to investigate the fatigue life of an existing bridge in Michigan. Finally. the effects of small varia- tions of certain parameters that enter the problem are considered. Five parameters are considered as random variables: (i) annual vehicle volume.(ii) vehicle type.(iii) vehicle speed.(iv) vehicle axle load level, and (v) interarrival time of vehicles. The fatigue damage is considered at three “critical” points: the quarter span. mid-span. and three-quarter span. the last of which is taken to be the most critical section. The damage has been calculated on the basis of both the static and dynamic stresses. Egbert Hsi-Ting Chang For the real bridge studied, the fatigue life (referred to the three-quarter point) ranges from 12 years to 9,135 years, depending upon three factors: (1) dynamic or static stress. (ii) random or constant annual vehicle volume, and (iii) fatigue models. But it is reasonable to consider that value corresponding to the case of dynamic stress, random annual vehicle volume, and a certain model D as the best estimate. It is 45 years, which can reasonably be regarded as being within the service life of a structure of this type. Therefore, it seems that fatigue damage should be a major factor to be considered in the design of such bridges. A SIMULATION STUDY OF FATIGUE LIFE OF HIGHWAY BRIDGES by Egbert Kai-Ting Chang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1971 ACKNOWLEDGMENTS The author would like to express his deep gratitude and appreciation to his advisor,Dr. R.K. Wen. whose guidance and assistance were invaluable throughout the author's graduate program and especially during the course of this investigation. Thanks are also due the other members of the author's guidance committee, Dr.C. E. Cutts. Dr.W.A. Bradley. and Dr.J.S. Frame. for their inspiration. Special thanks are extended to ur.0ehler. Mr.Cudney. and lr.Copple of the Research Laboratory Section of the Michigan Department of State Highways for their suggestions and supply of field data. The author will forever be indebted to his wife. Rosa, whose help and encouragement have made his work immeasurably easier. ii TABLE OF CONTENTS ACKNOWLEGWNTS IIOOOOOOOOIOOOIOI0.0.0.0... LIST OF TABIES OOOOOOCOOOOOOOOOOO0.0.0.0... LIST OF FIGURES 000.000.000.00.IOOOOOOIOOOO Chapter I. INTRODUCTION ...................... 1.1 Objective and Scope ........ 1.2 Literature Review .......... 1.3 Notation ................... II. COMPUTATION OF STRESSES ........... N CO'QmUt'FWNH Idealization of Bridge ..... Idealization of Vehicles ... BridgeAVehicle System ...... Expressions of Energy ...... Equations of Motion ........ Dynamic Moments and Stresses. Minimum Dynamic Stresses ... Numerical Solution of the Equations of Motion ........ 2.8.1 Modified 8 0 Method . 2.8.2 Choice of Time Incre- ment in Numerical Integration .......... 2.8.3 Computer Program DYNA-MIC OOOOOOOOOOOOO. 2.9 Static Stresses ............ III. FATIGUE STRENGTH .................. 3.1 Critical Sections .......... 3.2 Fatigue Models ............. 3.3 Cumulative Damage Hypothesis. Iv. COMPUTER SIMULATION .OOOOOOOOQOOCOO iii Page ii vii \I-er H 13 1 14 15 20 21 22 23 23 Chapter 4.1 Generation of Random Numbers and Random Observations ..... . Random Variables ............ h.2 Simulation Process .......... 4 Choice of Sample Size ....... v. N‘lmerical Results COOOOOOOOOOOOOOOCO 501 Bridge Data OOOOOOOOIOOOOO... 5.2 Traffic Data ......OOOOCOOOOO 5.2.1 Annual Vehicle Volume . 2. 2 Interarrival Time ..... 2.3 Vehicle Type .......... 2 4 Maximum Axle Load and Axle Load Level ....... 2.5 Speed Distribution .... 2 6 2 7 2 BS Axle Spacings ......... Polar Moment of Inertia. Spring Constants and Damping Coefficients ... 5.2.9 Sample Size ........... 5.3 Relative Frequency Distribus tion from Simulation ........ .4 Stress Ranges ............... 5 Fatigue Life oooooooooooooooo 5.5.1 Dynamic Stress-Random Vehicle Volume ........ 5.5.2 Dynamic Stress-Constant Vehicle Volume ........ 5.5.3 Static Stress ......... \nU‘anI mum 5.6 Effect of RFD of Axle Loading Level .OOOOOOIOOOOOOOOOOOIOII 5.7 Effect of Vehicle Speed ..... 5.8 Effect of Bridge Approach ... 5.9 Effect of Section Modulus ... 5.10 Effect of Magnitude of Maximum Axle Load ........... VI. SUMMARY AND CONCLUSIONS ............ LIST OF REFERENCES ......................... TABLES ..................................... FIGURES .................................... APPENDIX COMPUTER PROGRAMS ................ iv LIST OF TABLES Table Page 2-1 Comparison of Numerical Solution for Different Values of n ............... 63 5-1 Relative Frequency Distribution of vehicle Types OOOOOOOOOCOOOOOOOOOOOOO 6“ 5-2 Maximum Axle Loads .................. 65 5-3 Relative Frequency Distribution of Axle Load Levels IOOOOOOOOOOOOOOOOOOO 66 5-4 Relative Frequency Distribution of Vehicle Speed ....................... 66 5-5 AXIe sPaCingS eeeeeeeeeeeeeeeeeeeeoee 6? 5-6 Mean Value. Variance. and Standard Deviation of Interarrival Time ...... 66 5-7(a) Simulated Relative Frequency Dis- tribution of Dynamic Stress Ranges .. 68 5-7(b) Simulated Relative Frequency Dis- tribution of Static Stress Ranges ... 68 5-8 Estimated Fatigue Life (Dynamic Case). 69 5-9 Fatigue Life as Affected by Dynamic and Static Computations and Vehicle valume medals OOOOOOOOOOOOOOOOOOIOOOO 70 5-10 Effect of RFD of Axle Load Levels on Fatigue Life OOOOOOOOOOOOOOOOOOOOOOOI 71 5-11 Effect of RFD of Vehicle Speed on Fatigue Life ...OOOOOOOOOOOOOOOOOOOOO 72 5-12 Effect of Section.lodulus on Fatigue Life OOOOOOOOOCOIOOOOOOOOOIO000.00COO 73 Table Page 5-13 Comparison of Effects of Section Modulus and Axle Load Level on Fatigue Life OOOOOOOOOOOIOOOOOOOCOOO '71+ vi Figure 2-1 2-2 5-1 5-2 5-3(a) 5-3(b) 5-4 LIST OF FIGURES Idealization of Bridge ............. Idealization of BridgeAVehicle SyStem ......OOOOOOO0.0.00.00.00.00. Actions and Reactions between Axles andBri-dge OOOOOOOOOOOOOOOOOOOO0.00. Typical History Curve for Dynamic Bending Moment(2-axle vehicle) ..... Typical History Curve for Dynamic Bending Moment(5-axle vehicle) ..... Typical History Curve for Dynamic Bending Moment(6-axle vehicle) ..... History Curve for Dynamic Bending Stress at a Critical Section ....... Illustration of Procedure for Obtaining a Random Observation from a Given Cumulative Distribution Function.(a) Discrete Random Variable. (b) Continuous Random Variable ..... Idealized Section of the Tested Bridge 0.00.0000....OOOOOOOOOOOOO... veh1c19TypeB ......OOOOOOOIOOOOOOOO Effect of Vehicle Speed on Stress Range at Critical Sections(Vehicle Type 281) eeeeeeeeeeeeeeoeeeeeoeeeoe Effect of Vehicle Speed on Stress Range at critical Sections(Vehicle Type 381-2) eeeeeoeeeeeeeeeeeeeeeeee Effect of Section Modulus .......... vii Page 75 76 77 78 79 80 81 82 83 an 8“ 86 37 CHAPTER I INTRODUCTION 1-1 W222. The purpose of this thesis is to study by computer simulation the fatigue damage and fatigue life of simple span highway bridges as caused by heavy vehicles. Because highway bridges are designed to have substantial reserve strength beyond their ordinary working loads. it is very rare that a bridge failure would be caused by a single passage of the vehicle or train of vehicles that governed the design. But in recent years, fatigue cracks have been observed in highway bridges (1.h.17)‘. These cracks were caused and propagated by the repeated passage of heavy vehicles extending over a period of time. If the cracks are allowed to increase in size, failure will occur. To make appropriate provisions in the design of highway bridges to prevent fatigue failure economically presents a challenging task. The 1965 AASHO specifications for bridges (16) explicitly consider fatigue as a possible mode of failure * Numbers in parentheses refer to items in the List of References. 2 for the first time. Currently. the specifications (the 1969 version) call for allowable fatigue stress as a function of "fatigue life“ which is expressed in cycles of a fixed ratio of the design minimum stress to the design maximum stress and the type and location of material. It has been pointed out (7) that the specifications do not account for the fact that fatigue damage can be done to a bridge by numerous loading situations different from that considered in design and.much research is needed to improve the present practice. The present thesis is an effort toward that end. There are two main features of the present study. Firstly. the stresses in the bridge are computed more accurately by considering the bridgedvehicles as a dynamic system. Secondly. the random nature of the vehicle loads (their types. weights. and speeds) is taken into account. Therefore. the study involves the following two major parts: (1) the development of a mathematical analysis of a dynamic bridgedvehicle system appropriate for the purpose of this study. (2) the development of a computer simulation procedure to represent the random aspects of vehicle loading history. and the estimation of the corresponding fatigue damage and fatigue life of the bridge. 3 In order to simplify the bridge-vehicle system. the bridge is idealized as a single T-beam. simply supported at the ends. And the vehicle is represented by a set of sprung load units supported by linearly elastic springs and viscous dampers. The number of sprung load units of each vehicle depends on its type. Lagrangian equation is used in deriving the equa- tions of motion for the bridge-vehicle system. An unevenness of the bridge approach is considered in the analysis. Thus a given vehicle entering the bridge will be. in general. in a state of vibration. This analysis is described in Chapter II. Both the dynamic and static stresses approaches are calculated at three sections ~-- the quarter span.mid-span. and three-quarter span. The simple influence line method is used to calculate the maximum static stresses. A fatigue model specifies a relationship between certain stress vectors and the number of cycles that the material can sustain without fatigue failure. Seven different fatigue models are used to estimate the fatigue life. In order to account for the cumulative fatigue damage suffered through the different stress levels corresponding to different vehicle loadings. the Miner's hypothesis is used. The preceding is described in detail in Chapter III. z. The computer simulation procedure is described in Chapter IV. The procedure requires that the relative frequency distribution (denoted by RFD) or probability density function (denoted by PDF) of the following five random.variablesa vehicle type. vehicle speed. vehicle axle loading level. interarrival time. and annual vehicle volume. The numerical results of the study are presented in Chapter V. They include a study of the fatigue life of an existing highway bridge in Michigan and the effects of small variations of certain parameters such as the RFD of axle loading level and the static strength of the bridge. The numerical results are obtained by use of three computer programs: STATIC. DYNAMIC. AND SIMUi written in Fortran IV for use on the CDC 6500 Computer System at Michigan State University. 1.2 Wigs. AASHD.Road Test Report (1) studied bridge fatigue damage using two approachessi) Accelerated Fatigue Tests -- bridges were excited by mechanical oscillators which worked with a constant amplitude to replace actual heavy vehicle's dynamic loadings and. ii) Increasing Load Tests -- bridges were tested with two or three heavier truck types. for each truck. after 30 trips the axle loadings were increased and another 30 trips were made across the 5 bridge. until the bridges were considered failed or further loading of the test vehicle was considered un- desirable(unsafe). Applying a fatigue model (Model G in chapter IV). it was found that the actual lengths of fatigue lives of the bridges tested were in good agreement with those computed based on Miner's hypothesis of cumulative damage. Either approach. however. deviated substantially from the actual conditions of bridgedvehicle system in service. The first approach was actually a steady motion. i.e.. with no random factors involved. The second was a very special kinds of field test. because overloaded vehicles with speed less than 25 MPH were used. which produced a maximum static bending moment at mid-span up to 2.3 times greater than the design moment corresponding to a design stress of 18 ksi. Lower stress cycles were not considered in this approach. although such cycles could also produce fatigue damage. In 1968. Cudney (3) reported field data on dynamic stress ranges. rebound stresses. RFD of vehicle types. etc. in time periods ranging from 2“ hrs. to 96 hrs, for eight highway bridges in Michigan. In order to simplify the calculation. he grouped the stress ranges into a few levels. Munse and Stallmeyer's fatigue damage data were used together with certain assumptions in deriving a fatigue model (Model F in chapter III). 6 As in the preceding case and the cases to follow. Miner's hypothesis was used to calculate the fatigue damage.D. for one year's projected traffic. Then the fatigue life in years was taken to be inverse of that quantity.i.e. . l/D. This study demonstrated the variability of such factors as vehicle types and annual vehicle volume. And hence. the dynamic stress history and its relation to fatigue life in years. Furthermore. the data. though “ necessarily incomplete because of the high cost of collecting them in the field. provide a great deal of valuable information much of which was used in this study. However. the very long fatigue lives(of the order of thousands of years) cast some doubt regarding the validity of the fatigue model used. ' Werner.Heins.and Looney(22) discussed the fatigue damage based on static stresses in simple spans (continuous and cantilever spans were assumed to be cut L into and act as several simple spans). Annual vehicle volume and the distributions of vehicle types and vehicle weights were estimated statistically from data collected within a ten-year period. In 1969. Tung (21) adopted a more analytical approach to consider the fatigue damage problem. The simple Poisson process was used as the traffic model. 7 and the response of bridge was treated as a filtered Poisson process. Vehicles can be considered to be of different type and weight but must travel at the same constant speed. He gave a numerical example by considering that all vehicles were replaced by two constant forces having the same axle spacing. From the practical point of view. his method seems too complex. when applied to real systems. 1-3 m The symbols « listed below have been adopted in this thesis. a - length of bridge approach: aij - horizontal distance between the jth axle and the centroid of the ith load unit. It is positive. when the 1th axle is in front of centroid.otherwise negatives b - depth of a sine curve bridge approach at center: °ij - damping coefficient of the 3th axle in the ith load unit: C . - critical damping of vehicle axle: D - energy dissipation function: D - fatigue damage based on one sample year: d‘iJ 8 initial static compression in the 3th axle of the ith load unit: 8 the ith level of fatigue damages simulated fatigue damage experiment: flexural rigidity of the idealized bridges natural frequency of the idealized bridge: relative frequency of the ith vehicle speed level: relative frequency of event 31: medal amplitude function (which varies with time): cumulative distribution function of the random variable X. i.e.. F(x) = P(X < x): natural frequency of vehicle axle; gravitational acceleration: number of load units of one idealised vehicle: polar moment of inertia of the ith load unit: spring stiffness of vehicle axle: stiffness of the 3th axle in the ith load unit; length of bridge span: length of the ith load unit: a sample spaces of vehicle 3390: SPOOCi and axle loading level. respectively: N(i) N(t) 5| 9 the ith element of the sample space Mi: the jth element of the sample space Mg; the kth element of the sample space M3: sprung mass of the ith load unit: dynamic bending moment at a idealized bridge section having a distance i from the left supports mass of idealized bridge per unit length: sample size: number of cycles of stress or strain of a specified character that a given specimen sustains before failure of a specified nature occurs; number of axles in the ith load unit; Poisson arrival process: number of steps of one axle to pass through the bridge: number of vehicles passed a given point within a time interval (0.t)z number of independent outcomes of E"; number of cycles at stress range level is number of cycles at stress range level i which would cause a fatigue failure: theoretical probability of event 31' 10 P13 8 instantaneous reaction between the jth axle of the ith load unit and the idealized bridge: 0 8 defined on page 89: r - number of load unit of a given vehicle type; qn I the nth generalized coordinate: Ra - reaction at left support of bridge: RN I random number: 3i I number of springs in the ith load unit: S - vehicle speed; 8 - section modulus of the idealized bridge: Sr'smax’smin'smins'Srs'smaxs'smind’Srd'smaxd quantities defined on page 30; t 3 time: T 8 kinetic energy of the whole bridgedvehicle systems T I interarrival time : Uh - total strain energy in the idealized bridges Uv 8 total strain energy in the vehicle: - total potential energy of the bridge- vehicle system; V1 8 the ith vehicle speed levels w I bridge approach curve: '13 - w value at the horizontal position of the jth axle of the ith load unit; MI 11 random variable: distance of a given section from the left end of bridge: horizontal position between the left support and the jth axle of the ith load unit: dynamic deflection of the idealized bridge measured from y. bridge deflection due to dead load: dynamic deflection of the idealized bridge corresponding to the horizontal position of the 1th axle in the ith load unit: dead load deflection of the idealized bridge corresponding to the horizontal position of the jth axle in the ith load unit: vertical displacement of the ith load unit measured from its static equilibrium position. it is positive when downward: angular displacement of the ith load unit about its centroid axis: horizontal distance between the left support and the critical section: 1. if idealized bridge is in vibration. i.e.. y f'O. otherwise. zero: °2i “315 Eula 12 v 1. if at least one axle of the ith load unit is on the idealized bridge or the bridge approach. otherwise.zero: 1. if the jth axle of the ith load unit is on the bridge. otherwise. zero: if the jth axle of the ith load unit is on the approach. otherwise. zero: time increment: defined on page 40, average interarrival time: static deflection at the centroid of the sprung mass of the ith load unite CHAPTER II COMPUTATION OF STRESSES In this chapter are presented the derivation of the differential equations of motion of simple span bridge- vohicle system. the numerical solution of these equations. and the computation of the dynamic stresses in the bridge. In addition. the much simpler case of static stress analysis is explained in the last section of the chapter. 2.1 W In this study. a typical girder or I-beam and its tributary slab area are considered as a representative unit of the bridge. The effective width of the slab follows the AASHO Specifications. section 1.7.99 (16). The resulting T-beam is simply supported at its two ends. as ,shown in Fig.2-1. The flexural rigidity. EI. and the mass.. m. of the beam are considered to be uniformly distributed along the length of beam. Internal damping and surface unevenness of the bridge are ignored. The usual.beam theory is assumed applicable for the analysis. The dynamic deflection configuration of the bridge at any instant is taken to be : y(x.t) I f(t)sin if .....(2-1) 13 14 The form was first used by Timoshenko(19). Other investi- gatbrs of bridge dynamics have also considered it and found it to be reasonably accurate. In the equation above. x I distance measured from the entry point of the bridge: L I length of the bridge span: f(t) I modal amplitude function (which varies with time): y(x.t) I dynamic deflection of the bridge measured from its static equilibrium position. Thus. the idealized bridge is a single-degree-of-freedom system. 2.2 Idgglizgtiog of Vehicleg. Because of the preceding approach. only one (longi- tudinal) line of wheels of the vehicles is considered. such wheel loads are assumed to act directly above the beam. For simplicity. the beam and the wheel loads will be referred to as the bridge-vehicle system. Each vehicle is idealized and represented by a set of load units. and each load unit consists of a point mass or a uniformly distri- buted mass connected to a linearly elastic spring or several springs. Viscous dampers are also placed in parallel with the springs. Fig.2-2 shows two idealized vehicles. 2-3 W The system considered is shown in Fig.2-2. It consists of three parts: (i) a simply supported beam 15 (idealized bridge) spanned between two rigid supports. (ii) idealized vehicles. and (iii) the approach to the bridge. The following assumptions are made in regard to the passage of the vehicles over the approach and bridge. (1) The analysis starts for each load unit when its front axle reaches the beginning of the bridge approach. (2) Prior to that time. each load unit is in its static equilibrium condition in the vertical and angular coordinates. (The bridge approach imparts an initial vibration to the load units as they enter the bridge.) (3) When the first axle of a vehicle enters the span. the bridge is either at rest having a deflection due to its own weight or in a state of vibration (caused by the passage of an earlier vehicle). (A) If there are more than one load unit in the system. the speeds of these units are the same and they remain the same during the passage over the approach and bridge. 2.4 W As indicated in Fig.2-2. two generalized coordinates are-used to specify the configuration of the sprung mass of each load unit. One for the vertical displacement. ’i' and another for the rotational displacement. 01. of the sprung mass. where i refers to the ith load unit. Thus. the total number of generalized coordinates or degrees of 16 freedom for the bridgeavehicle system is 2h+1. where h is the number of load units. By considering the energy in the whole system. the equations of motion may be derived in the following manner. The initial energy level of this system is taken to be correspondent to the conditions that the beam is in an _unstrassed horizontal position and that the springs of the load units are undeformed. (1) Totgl Strain Engggy in the Bridge 0,, - éEIg"(§+'c'1y)§xdx where beridge deflection due to dead load cnly:and 81- 1. if the bridge is in motion. i.e.. Y.f 0. otherwise. zero. Each subscript x indicates a differentiation with respect to x. (2) T035; Strain Energy Stgggg in thg ngd Units r 81 ~ - - - Uv " f ’3 *r-(d’kf"l*alj°l’°2l“ylj*yla“31.1 - 2 -wijchijJ kid eeeee(2b) in which the sum of terms in square brackets represents the total deflection of the 3th spring in the ith load unit. and aij I horizontal distance between the jth axle and the centroid of the ith load unit: it is positive. when the jth axle is in front of °21 °313 5&13‘ 17 centroid. otherwise.negative: 1. if at least one axle of the ith load unit is on the span or the bridge approach. other- wise. zero: 1. if the Jth axle of the ith load unit is on the bridge. otherwise. zero: 1. if the jth axle of the ith load unit is on the bridge approach. otherwise. zero: initial static compression in the 1th axle of the ith load unit: stiffness of the 3th axle spring in the ith load unit: number of load units: number of springs in the ith lead unit: w value measured at the position of the 1th axle in the ith load unit: dynamic deflection of the bridge at the location of the jth axle in the ith load unit: i value measured at the position of the jth axle in the ith load unit: vertical displacement of the ith load unit measured from its static equilibrium position. it is positive when downward: angular displacement of the ith load unit about its centroid axis. it is positive when clockwise. 18 (3) Totg; Pgtentigl Energy of the Bridge-Vehicle System The change of the potential energy of the beam is given by the expression - mg£L§dx - 31mg£Lydx .....(26) The change of the potential energy of the load units is given by the expression r - f E -Mig(AV)1 - '1gz1c21 J eeese(2d) where M1 I sprung mass of the ith load unit. and (4v)i I initial static deflection of the centroid of the sprung mass M1. Adding expressions (2a) to (2d). one obtains the following expression for the total potential energy of the V I Ub'+ Uv - Smggl‘ ydx - 31mg£L ydx r .- + E E -‘1g(Av)i . ligziczi J eeeee(ze) (A) W The kinetic energy of the system is given by systems: the expression - .2 - r e 2 - r .2 T I §c1£Lmy dx + c21§ iMizi + c21§ iJiOi 000.0(2f) The first term on the right-hand side of this equation represents the kinetic energy of the beam. the second and third terms represent the kinetic energy of the sprung 19 masses due to vertical and angular motion. respectively. Where the superscript dots denote derivatives with respect to time. and J1 is the polar moment of inertia of the sprung mass about its centroidal axis. (5) Energy Digsipation anction The energy dissipation function of the system is given by the expression 81 , . . a . .. D ' *§ § cij[(zi+aij°i)°21 ' (Y15*Yij)°313 . - 2 -wijcliij" ‘ .....(28) where cij I viscous damping coefficient in the jth spring of the ith load unit. 0n substituting eq.(2-1) into the expression for V. T. and D. one obtains . .I.“‘ r s . . - a . - . - z . r + constant .....(Zh) The constant term reflects the choice of the initial energy level at that corresponding to a "true” zero. - - 1‘ .2 -2 1' I thzcl + ieazixmiz1 4- J10.) .....(21) r 8 . ; . "x D I if )31 (21+a1181)821 - (yij'l'fsin-TJ'J + «x . {firms-filfifij - *155h15-‘2 ......(ZJ) 20 where s I vehicle speed: and x13 I horizontal distance between the 3th axle of the ith load unit and the beginning of the bridge approach. 2.5 Equations of Mgtign. The Lagrangian form of the equation of motion is shown in the following 1.... -21. ..ZY... +12... 0 (2-2) dt n an an aqn The symbols T. V. and D represent the quantities in the preceding section. and qn represents.the nth generalized coordinate. In this analysis. the generalized coordinates are zi. 01. and f. The governing differential equations of motion of the bridgedvohicle system are obtained by substituting expressions (2h) through (25) into Lagrange's equation for each of the generalized coordinates of the system. The resulting equations are 8 - u 1 - - - °21E "1‘1 ” ’j (“1"‘1591H3’1fy13)°3lj"lj°ul:”Fl: 8 . .3 , - . - + 21((t1..1jqp-(yij+yij)eBijdeijc.11>cij]- o ...(2-3) 21 s 521[ Jiéi + §1((ds)1j+(zi+aijei)-(§ij+yij)53id s . . , x -w115u13)k1ja ij + §1(('zi+aijei)-(l71j+fsinflL + %—fcos—-i1)caijdwijc:1:)cij ’aij ] I 0 .....(2-4) 51E§mLf + §s1L(go“: - I §1((ds)1j+(zi+aijei)- «x r s . (y13+y15))0313k11sin-ril - :1: filminuep- -._ . "11 fit "X - fix (yij+fsinjE-1~+ L cosjiil)c31jcijsin—:11] a 0 .....(2-5) 2.6 Qynggic lemon}; and Stggssgg, The dynamic bending moments are found by treating the instantaneous reactions between the axles and the beam as statically applied forces (D'Alembert's principle). Fig.2-3 shows the relations of these forces. By taking moments about the right-hand support. one obtains the reac- tion at the left-hand support. 1' 81 - .. Ra -%.[§ E pij(L+a'xij)°3ij +£L amy(L-x)dx J Ti 2 z (L+a-xij)3313[((ds)ij+(zi+a1101)- (5713*?13’”‘15”(*1+31361)'(§11*5'13)”131 ..ka oeoee(2"6) where p13 is the sum of the spring and damping forces corresponding to the 1th spring and damping device of the 22 ith load unit: and “a” is the length of the bridge approach. And then. the bending moment of any section having a dis- tance i from the left support may be expressed as follows. 'i ' Rafi -fgr £11 3315(i+a'xii)[((d8)13+(”1+aii°1) xij nT+t1. where n is some prescribed integer. then the bridge is assumed to be at rest when the current vehicle enters it. (ii) if t < nT+t1. the vibration caused by the previous vehicle is taken into account. In this case. the deterministic system involving two vehic1es is analyzed. (3) After a train of two vehicles for which a deter- ministic solution is obtained. the bridge's initial conditions for the third vehicle is always taken to be zero. (k) For each passage the fatigue damage to the bridge is calculated and accumulated. The total number of passages is equal to the sample size. which is discussed in the next section. (5) Calculate the first year's fatigue damage by the following equation. V 1 1 I D88( gs) eeeee(l"'2) D where D1 8 estimated fatigue damage at the end of the first year: Dss' cumulative fatigue damage based on the sample size: V1 = the first year's annual vehicle volume; 39 Se 8 sample size. (6) Estimate the fatigue life: (i) if D1 ;.1. the estimated fatigue life is equal to or less than one year. (ii) if D1 < 1. make an observation of the 2nd year's annual vehicle volume. the fatigue damage at the 2nd year is V2_ D2 8 D107?) .....(4-3) The general form for the ith year's fatigue damage is V1 Di as D1(Tl-) .....(u'u') where D1 = the fatigue damage caused at the ith year. and V1 8 the observation of the ith year's annual vehicle volume. Simulation experiment stOps when k fDizl .....(u-S) but k-I ZD1<1 eeeee(u-6) l Eqs.(u-5) and (“-6) imply that the estimated fatigue life is k years. h.4 Choice of §ample Size, As mentioned previously. it is important to deter- mine the sample size or the number of simulated runs. Let E denote the simulated fatigue damage experiment. and (11 be the fatigue damage caused by the passage of the ith #0 vehicle. For all of those di values. i= 1.2.3. .....N. one may group them into few levels. say 31.32.83......dn. Consider N independent outcomes of E. Let ni be the number of times that fatigue damage levels di occurs among the N outcomes. Note that n1 is a binomially distributed random variable. then the expected value and variance of ni are E(n1) = Np .....(4-7) Var(n1) 8 Np(1 - p) .....(4-8) where p is the theoretical probability of the event 31. Now the relative frequency of 31 is fai 8 ni/N. hence the expected value and the variance of £31 can be calculated as following Ema-i) . 3011/10 = E(ni)/N = p .....(4-9) Var(fai) 8 Var(ni/N) 8 Var(ni)/N2 8 p(1 - p)/N eeeee(u-1o) Applying Chebyshev's inequality to the random variable f- . one obtains d1 Prob [Ifa-1 - p| m eeeee(u-1u) where N can be treated as the sample size for this simulated experiment. CHAPTER V NUMERICAL RESULTS A simulation method for the study of the fatigue life of highway bridges has been described in the preceding chapter. In this chapter numerical results are presented. These results are obtained by use of programs written in Fortran IV for use on the Michigan State University CDC 6500 System. The results pertain to an existing bridge in Michigan subjected to heavy vehicular traffic that is reasonably representatfve'of that of the state. Additional numerical results have also been obtained to consider the effects of small variations of certain parameters that enter in the modelling of the bridge-vehicle system. Such parameters include the annual vehicle volume. vehicle axle load level. vehicle speed. the geometry of the bridge approach. and the bending strength of the bridge. 5-1 22mm The bridge considered is a composite simple span rolled I-beam bridge with welded tapered end cover plate. It has span length 78.5 ft.. angle of skew 1u°. and is located on US 23 SB over Huron River and NYC RR. 42 43 The idealized section and its properties are shown in Fig.5-1. The bridge approach is assumed to be a half sine curve with a length of 50 ft. and an amplitude of 2 inches(dip). As mentioned in chapter II. the approach would cause the vehicle to vibrate before it enters the span. 5.2 W Certain parts of the necessary data for the traffic characteristic needed for this simulation study are assumed to be probabilistic.and others are taken to be constants for simplicity. The latter are related essen- tially to the physical characteristics of the vehicles. 5.2.1 Annual Vehiglg Vglnmg --- To estimate the annual vehicle volume for the future is difficult.Generally speaking. it depends on the location of bridge.national economic growth rate.development of other transportation systems. etc.. For the years from 1962 to 1969. the annual vehicle volumes for a certain bridge have been estimated (3) to be ##4940.488099.552040.615525.635222.688580.711992. and 732640. respectively. The data are used in two ways to model future annual vehicle volumes: (1) Assume that the annual vehicle volume is a constant.equal to 444940. (2) Use the annual vehicle volume from 1962 to 1969 as given. After that. the volume is taken to be a random 4h variable. It is assumed to have an uniform distribution between two reasonably chosen limits. The upper limit should be within.a maximum acceptable number consistent with the class of highway and safety requirement. The choice of lower limit is even more a matter of judgement. In.any case. for this study the limits used are 1.200.000 and 732.640. respectively. Note that. as defined above. the constant annual vehicle volume case represents a lower density traffic than the random one. 5.2.2 Intezgggivgl Time --- For the simulation study. it is necessary to specify the times of vehicle arrivals. The simple Poisson process N(t) is used. That is. the probability that n heavy vehicles arrive within a time interval (0.t) is given by -at P[ n(t)=n]-e (13:193— .....(5-1) ml where.2 8 average number of vehicle arrivals per unit time. which is also equal to the annual vehicle volume divided by the number of time units in a year. Eq.5-1 implies that the interarrival time.T. has an exponential distribution. fT(t) 8fie'fit for all t > 0 .....(5-2) The cumulative distribution function of T is FT(t)-I-°-At for 111t>0 eeeee(5-3) ”5 The mean and variance of T is 1/,: and 1/,1‘ . respectively. Applying the procedure outlined in section h.1. one may find that the relation between a random observation. t. and random number. RN . is given by ts-l-i-l-LAE-m sees-(5-1+) 5.2.3 Vehicle Types --- Forty three vehicle types varying from a 2-ax1e truck to a 12-axle tractor-semi- trailer-full trailer type are considered in this study. Fig.5-2 shows the figures of common truck types. In Ref.(Z) are reported data of the distribution of vehicle types among a total of 2000 vehicles. The relative frequency distribution is listed in column three of Table 5-1. (It should be noted that in Ref.(2). only 21 ”groups” were considered: in each group one or more of the forty three types are lumped together in the group. For a group that has more than one type. the assumption is made that the total number of vehicles in the group is equally divided among the types.) .5.2.4 flaximgg Axle Load and Axle Load Levgl --- The maximum loads for the axles of the vehicles are listed in Table 5-2. They have been computed from empirical rules given in Ref.(20) as the maximum allowable loads depending on the axle type and spacing. In Table 5-2 are also show vertical bars between certain axles. The axles between a given pair of such bars 46 are modelled by a load unit in the analysis described in chapter II. For each vehicle type. five levels of static axle loading are considered: they are 50%.70%.80%.90%. and 100% of the maximum axle loads. The cases of 100% and 50% are assumed to correspond to a fully loaded and empty vehicle. respectively. A RFD based on certain field data (18) of these levels is given in Table 5-3. 5.2.5 Speed Dlgtrlbutlog --- Table 5-# shows the RFD of heavy vehicle speed used herein. It is based on data from Ref.(15). 5.2.6 Axle Spaglngg --- Field data show that axle spacings of vehicles of a given type are not always the same. However. for simplicity the axle spacings for each vehicle type is considered to be fixed. They are listed in Table 5-5. 5.2.? W --- The sprung mass of each load unit is assumed to be uniformly distributed. Hence the center of gravity coincides with the geometric center. And the polar moment of inertia can be calculated from the expression Ji 8 uiLi/lz eieeee(5-5) where Ji 8 polar moment of inertia of the ith load unit: Mi 8 sprung mass of the ith load unit: and #7 Li 8 length of the ith load unit. 5.2.8 Sprlng Conspanps and Damping Coefflclents --- Measurements made of heavy vehicles (23) indicated that vehicle axles have natural frequency.f. varying from 1.6 to “.1 cps. For simplicity. an average value of f 8 2.8 cps has been used to calculate the spring stiffness k as follows: 1: . 4112:2111 .....(5-6) where M is the mass of the wheel load. In each axle. damping is assumed to be viscous and "critical”. The coefficient of damping is therefore equal to c 8 #wffl .....(5-7) 5.2.9 §éfl2l24§i£2.--- For this study a sample size of N 8 10.000 vehicles has been used. Following the dis- cussions in section 4.“. this corresponds to a value of é . 0.900 and 5 . 0.015. 5.3 321W Using a sample of 10.000. the distributions of vehicle type.speed.and axle load level from simulation are listed in the appropriate columns in Tables 5-1. 5-3. and 5-4 along with the postulated distributions. The agreement is seen to be excellent. which indicates that the sample size is sufficiently large. Furthermore. the mean. variance. and standard deviation of the interarrival time are also calculated #8 and shown in Table 5-6. These again are in excellent agreement with the theoretical values for the Poisson process. 5-4 W Fatigue models. as listed in section 3.2. indicate that stress range is a dominant variable in considering fatigue damage. Hence. the RFD of both the dynamic and static stress ranges at the three critical sections are tabulated in Tables 5-7(a) and 5-7(b) respectively. An examination of Table 5-7(a) shows the following: (1) The mid-span has the largest stress ranges. which vary from 1 ksi to 11 ksi. Stress ranges at the quarter span and the three-quarter span vary from less than 1ksi to 8 ksi. In spite of the difference. the situation at the center may not be as serious as at the other sections. This is because of the consideration that the quarter span and the three-quarter span are presumed to coincide with the ends of the cover plate. (2) The RFD of the stress ranges at the quarter span and three-quarter span are roughly symmetrical. but the latter has a larger density for those stress ranges that exceed the fatigue limit. One might thus conclude that the three-quarter span is the most critical section so far as fatigue life is considered. “9 (3) At this critical section. approximately 20% of the stress ranges are greater than the fatigue limit of 3.0 ksi for models A.B.C.D.E. and G. From the probabilistic point of view. this means that only one of every five vehicles will cause fatigue damage. For model F. only four of every 100 vehicles will cause fatigue damage. this is due to the high fatigue limit 3.8 ksi for this par- ticular model. Table 5-7(b) is similar to Table 5-7(a). But two differences are worth noting. Firstly. the static stress ranges are lower than the dynamic ones. Secondly. the quarter span has a larger amount of stress ranges that are beyond the fatigue limit. This would imply that for the static approach the quarter span is more critical. 5.5 M For this numerical example. the fatigue lives at the quarter span. mid-span. and three-quarter span are computed in four ways. They are combinations from the following two factors: (1) dynamic or static stress: and (ii) constant or random annual vehicle volume. Table 5-8 lists the values of the fatigue life corresponding to the random vehicle volume as explained in section 5.2.1 for four simulation runs. An inspection of the table indicates that the values for the different simulation run do not differ from one another appreciably. 50 This means that. for the purposes of this study. the size of sampling of the postulated vehicle volume distribution is sufficiently large. It Should be remembered that the range of the vehicle volume. being from 732.600 to 1.200.000. is not very large. 5.5.1 Qypgpic Stregg - Randon Veplglg Volgle --- Table 5-9(a) shows that the mid-span has the lowest fatigue life for all seven fatigue models. And the fatigue life at the three-quarter span is less than that at the quarter span. These results are compatible with the observations made in the preceding sections. For purposes of discussion. consider the most critical section - ’the three-quarter span --'one will find that the fatigue life varies from 12 years for model B to 570 years for model F. In general. the values for the seven models can be grouped into three categories. They are (1) Semi-Log Models --- models A.B.and G yield values of the same order of magnitude with an average of in years. (2) Log-Log Models --models C.D.and E yield higher values. with an average of #9 years. (3) Log-Log Model -- model F gives the highest value. 570 years. 51 5.5.2 Qypamic Stress - Constant Vehicle Volume.--- Estimated fatigue lives for this case are listed in Table 5-9(b). The general pattern of variation of the fatigues in this table is similar to that of Table 5-9(a). But overall it has higher values of fatigue life. For example. the value for model D is twice as much as for the case considered f previously. The differences between these two cases are ' expected.since the random vehicle volume corresponds to a denser traffic (see section 5.2.1). 5.5.3 §tatl§ Stresg,--- Fatigue lives estimated by the static stress approach for both types of vehicle volumes are listed in Tables 5-9(c) and 5-9(d). They show that the static stress approach produces higher fatigue lives than the dynamic stress approach. One also notes an interesting difference in that while in the dynamic case the fatigue life for the three-quarter span is more critical(smaller) than that for the quarter span. the opposite is true for the static case. This. of course. is consistent with the observations made in section 5.4 on the stress ranges for the two cases. It may be pointed out that for the four combina- tions. the case of dynamic stress with random vehicle volume would seem.most realistic and should be given more weight in the estimation of fatigue life. 52 5.6 Effegt pf RFD of Axle Loadlng Level. The preceding results are based on one RFD of axle loading level. In this section the influence of changes in RFD of axle loading level will be considered. As it is done in the following sections on the influences of small variations of parameters. three sets of data will be considered. The first set is the same as used in ob- taining the preceding data. The others correspond to variations. The second and third sets of the RFD of axle loading level are obtained by increasing the relative frequency of the largest axle loading level by 5% and 10%. respectively. with a corresponding decrease of the lightest axle loading. The results for these three cases are listed in Table 5-10. For the dynamic stress approach. it is seen that at the critical section of the three-quarter span. the variations of fatigue life is within a negligibly small percentage. For the static stress approach. some- what larger differences are noted at its critical section. the quarter span. 5.7 W The second and third sets of the RFD of this parameter are formed as follows: (1) If the relative frequency of speed V1 MPH is 53 fi for the first set of RFD. a relative frequency equal to fi is assigned to (V1+5) MPH and (Vi+10) MPH to the second and third sets. respectively. (2) When the speed is greater than 75 MPH. it is treated as 75 MPH. Therefore. the relative frequency for 75 MPH is larger for the second and third set than the first set. The results of the estimated fatigue life are shown in Table 5-11. It is seen that the changes in RFD of vehicle speed have little influence on the fatigue life. In passing. it may be of some interest to note the effects of vehicle speed on stress range. Some represen- tative results are shown in Figs:5-3(a) and 5-3(b) for vehicle types 281 and 381-2. It is noted that these curves are oscillatory in nature. and that both the amplitude and period increase with increasing value of the speed parameter. Higher speeds do not necessarily cause higher stress ranges. .5-8 MW For a given bridge. the shape and size of its approach are not constant. they may change with weather. soil conditions. etc.. In the study based on dynamic stress. the bridge approach controls the rotational and vertical vibrations of the vehicle as it enters the span. Assume that the approach is a half sine curve with length 54 "a" and amplitude “b” (dip). Recall that for the first set a1 8 600' and b 8 2”. The second and third sets of 1 a and b values considered in this section are: for the 2nd. set : a2 8600' and b2 8 #': =1”. and the 3rd. 861: 8 a 8300” and b The results of the siudy pertainizg to the effect of bridge approach based on all vehicle types and speeds are presented below. where the values of Ti/Tj represent the maximum dynamic stress range at the critical section. three-quarter span. under the influence of the approach with dimensions a1 and bi to that of bridge approach with dimensions aj and bj' : 1.00 '- 1.12 rap-s N 1 $3- : 0.98 - 1.02 1 It is seen that for small changes in the geometry of the bridge approach. the values of dynamic stress ranges do not change a great deal. The effect on fatigue life would thus seem to be not serious. 5.9 WWO... Because the stress vector is inversely proportional to the section modulus of the idealized bridge, it is clear that to increase the section modulus is an effective way to decrease the fatigue damage. Let the I- beam used 55 in the preceding be replaced by two larger I-beams with the following properties: I- Beam 36WF230 36WF280 Moment of inertia 18935 in? 23025 in? (steel only) Moment of inertia 39768 in? #5005 in? (composite ) Distance from N.A. 25.91 in. 25.60 in. to bottom (at center ) The estimated fatigue lives are listed in Table 5-12. The fatigue lives (dynamic approach) at the three-quarter span are shown in Fig.5-# as functions of section moduli and weight. The data do not include model F. because the corresponding values are too large to be plotted. 5.10 W The case of 36WF230 considered in the preceding section dealt with the effect of an increase of the section modulus by 11.07%. It is of interest to consider the effect corresponding to a decreaser of the static load level uniformly by the same percentage. This is done by a reduction of the magnitude of the maximum axle load by this percentage. The results are shown in Table 5-13. Not unexpect- edly. the values of fatigue life are very close to those presented in the preceding section. For statical con- sideration of fatigue life. these two sets of values 56 should be the same. This comparison shows that small variations in properties of the dynamic system ( changes in the frequencies of the bridge and vehicles ) are not important. although the considerations of the dynamic nature (versus static) are important as indicated in section 5.5. CHAPTER VI SUMMARY AND CONCLUSIONS In this thesis a study is made of the fatigue life of simple span highway bridges using simulation by computer. The study begins with a deterministic analysis of a bridge traversed by a vehicle. The set-up of the simulation procedure follows. Numerical results were obtained to illustrate the procedure as well as to inves- tigate the fatigue life of an existing bridge in Michigan. Finally. the effects of small variations of certain parameters that enter the problem are considered. In this study. five parameters are considered as random variables: (1) annual vehicle volume.(ii) vehicle type; (iii) vehicle speed.(iv) vehicle axle load level. and (v) interarrival time of vehicles. The fatigue damage is considered at three "critical” points: the quarter span. mid-span. and three-quarter span. The damage has been calculated on the bases of both the static and dynamic stresses. Under the assumption that cover plates were welded to the I-beams from the quarter to the three quarter span. the latter point is considered to be the most critical section. .57 58 For the real bridge studied. the fatigue life (referred to the three-quarter point) ranges from 12 years to 9.135 years. depending upon three factors: (1) dynamic or static stress. (ii) random or constant annual vehicle volume. and (iii) fatigue models(A through G). But it is reasonable to consider that value corresponding to the case of dynamic stress. random annual vehicle volume. and model D as the best estimate. It is 45 years which is reasonably within the service life of a structure of this type. Therefore. it seems that fatigue damage should be a major factor to be considered in the design of such bridges. Although the numerical results obtained are limited in scope. they appear to warrant the following observations that bear on the directions of future efforts of research in this area. (1) In the estimation of fatigue life. the differ- ences resulting from using the dynamic stresses or the static stresses seem to be sufficiently large as to warrant the use of the former approach. It may be mentioned that the simulation procedure presented here may be applied also to other types of bridges than simple spans. (2) The traffic characteristics of heavy vehicles need to be specified more accurately. A probable weakness 59 of the present study is the assumption of the simple Poisson process to govern vehicle arrivals. For heavy vehicles. it seems likely that they would often travel in platoons. A study involving mathematical modeling from field observations would be useful. In this connec- tion. it may be noted that not only the vehicle type but the axle load level is important. This study has indicated that no empty vehicle. regardless of its type.saused a stress range which exceeded 3.5 ksi. Besides. only three of the forty three vehicle types gave rise to dynamic stresses that surpassed the assumed fatigue limit of 3.0 ksi. (3) Inasmuch as the values of the fatigue life are very sensitive to the fatigue models used. more research should be done to establish a fatigue model of more general validity. Such research probably would call for experimental work on fatigue strength involving random stress vectors including stress levels that are substan- tially lower than the lowest that has been used up to now. (4) Before the necessary items of research are done and specific rules of method of design against fatigue failure are developed. it is worthwhile to note that the limited amount of numerical results obtained herein would indicate that a moderate increase of the section modulus would greatly increase the fatigue life of the structure. 1. 3. 7. LIST OF REFERENCES AASHO Road Test: Report 4 -- Bridge Research. Highway Research Board Special Report 61 D. 1962. “Annual Study of Truck Volume on Michigan Trucklines.“ Transportation Division. Michigan Department of State Highways. Cudney. G. R. ' The Effects of Loadings on Bridge Life.” Michigan Department of State Highways. Research Report No. R-638. January 196 ”Fatigue Fractures in Welded Connections." Reports prepared by Commission XIII. International Institute of Welding. 1967. Fisher. J.W.. Frank. K.H.. Hirt. M.A.. and McNamee. B.M.. “Effects of‘Weldments on The Fatigue Strength of Steel Beams.“ National Cooperative Highway Research Program. Report No. 102. 1970. Fisher. J.W.. and Viest. I.M.. “Fatigue Tests of Bridge Materials of The AASHO Road Test.” AASHO Road Test Technical Staff Papers. Highway Research Board M Special Report 66. 1961 Fountain. R.S.. Munse. W.H.. and Sunbury. R. D.. “Spe- cifications and Design Relations ' Journal of the Structural Division. ASCE. Vol. 95. No. ST12. Proc. Paper 6291. Dec.. 1968. Hall. L.R.. and Stallmeyer. J.E.. “The Fatigue Strength of Flexural Members.” Univ. of Illinois. Dept. of Civil Eng.. Status Report to Fatigue Committee. Welding Research Council. 1959. 60 9. 10. 11. 12. 13. 10. 15. 16. 17. 61 Hillier. F.S.. and Lieberman. G.J.. “Introduction to Operations Research.“ HoldeneDay. Inc.. San Fran- c sco. California. 1967. Iverson. J.K.. “Dynamic Analysis of Nonlinear Elas- tic Frames.‘ Ph.D Thesis. Michigan State Univ.. Dept. of Civil En5.: 1968. Maurer. W.D.. "Programming: An Introduction to Compu- ter Languages and Techniques.” HoldenéDay. Inc.. San Francisco. California. 1968. I'MaximumDesirable Dimensions and Weights of Vehicles Operated on The Federal Aid Systems.“ House Docu- ment No. 358. 88th Congress. 2nd Session. Aug. 19. 196‘. pp. 143-109. Munse. W.H.. and Stallmeyer. J.E.. “Fatigue in Welded Beams and Cirders.“ Highway Research Board Bulletin 315. 1962. pp. 45-62. Newmark. N.M.. “A Method of Computation of Structural Dynamics.“ Journal of Engineering Mechanics Division. Proceedings. ASCE. July 1959. pp.67-9#. 'Spegd Report." Michigan Department of State Highways 9 9- 'Standard Specifications for Highway Bridges." Amggican Association of State Highway Officials. 19 . ”The Minutes of The January Meeting." National Research Council. Highway Research Board of The Division of’Engineering. January 18-22. 1971. 18. 19. 20. 21. 22. 23. 62 “The Study of Vehicle Axle Loadings.’ Unpublished Report. Research Laboratory Division. Michigan Department of State Highways. Timoshenko. S.P.. “Vibration Problems in Engineering.“ Van Nestrand Co.. New York. 1955. “Truck Weight and Characteristics Study." Michigan Department of State Highways. Report No. 63. Tran. Ple D1Ve. 1961-1969. Tung. C.C.. "Life Expectancy of Highway Bridges to Vehicle Loads." Journal of the Engineering Mechanics Division. Proceedings. ASCE. No.8M6. December. 1969. Werner. J.D.. Heine. C.P.. and Looney. C.T.G.. "Induce ed Bridge Leads and Moments During A Ten Year Period.“ Civil EngineeringDept.. University of Mary- Land. College Park. Md. June 1968. Whittemore. A.P.. Wiley. J.R.. Schultz. P.C.. and Pollock. D.E.. “Dynamic Pavement Loads of Heavy Highway Vehicles." National Cooperative Highway Research Program. Report No.105. 1970. 63 .cmam popmMSUIoomnp . o .cmamucwa . n .mmam popumsu » m * mm.m mm.a ~m.m ao.m sw.afi mm.m ma.m mm.s no.n snnmm com mm.m mm.a mm.m so.w so.aa mm.m mH.m m~.s so.m s-mmm co mm.m mm.a mm.m ao.m so.HH mm.m ms.m m~.s so.m Summm om ma.m am.a ma.m oo.m mm.aa ~o.a ~m.~ No.m aa.~ aunmm on ms.m mm.a as.m wa.a mm.os mm.a om.m om.~ oo.m N-Hmm com ma.m mm.e as.m ms.a mm.oH sm.a am.~ Ho.~ mo.m Nufimm om as.m Hm.a as.m oa.a m~.oa :m.s a~.~ mm.~ mo.~ mnamm om sfl.m mm.a aa.m Hm.a mm.oa ma.a ma.~ Hm.~ mm.H ~-Hmm as oa.m ma.a ca.m am.w om.m os.w ea.a su.a Hm.H am com os.m ma.a os.m ~m.o am.m Ha.o mH.H ma.H mm.a am om os.m ma.a oa.m mm.m mm.m He.m mH.H ma.a Hm.H om ow mo.m mama so.m mm.m smhm mm.w SH.H oA.H m~.H am as o 9 .NW 0 n m o 9 ed mmoapm owsmczo mmohpm ofismCSm omcmm mmehpm camp m 535w2az ESwamz Esswxms oaowso> A.me a“ mommoapmv m Mo modam> Psopommwm mom soapsaom Hmoamoszz mo ConwamQSoo . Hum magma 64 Table 5-1 Relative Frequency Distribution of Vehicle Types Vehicle Vehicle Volume Relative Freq. Relative Freq. Type (Simulation) (Field Data) (Simulation) 2D 1878 0.18990 0.18780 3 217 0.01890 0.02170 231 1568 0.15480 0.15680 331 1191 0.12030 0.11910 232 1172 0.12030 0.11720 232L 1270 0.12030 0.12700 332L 219 0.02373 0.02190 332 252 0.02373 0.02520 283L1 268 0.02373 0.02680 233 222 0.02373 0.02220 233L 227 0.02373 0.02270 ZSBLZ 250 0.02373 0.02500 233LL 230 0.02373 0.02300 3S3LL 18 0.00162 0.00180 3S3L 23 0.00162 0.00230 333L2 18 0.00162 0.00180 333 18 0.00162 0.00180 333L1 17 0.00162 0.00170 38 8 0.00110 0.00080 335 17 0.00110 0.00170 231-2 10? 0.01080 0.01070 232-2 92 0.00937 0.00920 232L-2 83 0.00937 0.00830 331-2 89 0.00937 0.00890 332-2 46 0.00472 0.00460 232- 43 0.00472 0.00430 232L-3 40 0.00472 0.00400 332L-2 50 0.00472 0.00500 352-3 14 0.00150 0.00140 233-3 14 0.00150 0.00140 333-2 9 0.00150 0.00090 232-4 12 0.00150 0.00120 332-4 13 0.00190 0.00130 233-4 20 0.00190 0.00200 333-4 26 0.00405 0.00260 333-4 43 0.00405 0.00430 332-3 9 0.00460 0.00 90 33 - 9 0.00460 0.00 90 334-5 16 0.00220 0.00160 -2 67 0.00600 0.00670 3-2 30 0.00380 0.00300 3’5 7 0.00110 0.00070 65 Table 5-2 Maximum Axle Loads 2 mm 1 e 3 1 eIJaIJ3 1 e I e 333 m mala333 3 43333 i clan/03333 k9 eeeeeee 8 3343333 “.8 ml; .elumflmu: 3333333 all m 3 33443333333 3 e 7 .IJu3 “Jewel/953350 3335333 .443 n 33 4.33433443314333 33 A Hafizlzalunlv 33 flflfl5335353335xflxfiefl33 33 ..... C C C C C C C C C C C C C C C C C C C C C C m 4343433 44443343433314.33333 33 .m5 500aflnluflmuu53330335555355flxflafl333333333 533 ................... C C C C C C C C C C C C C C u 444344343334334444344433333333333 433 M 5N¢l/.50 0 30 0 355335355nlv firlz30| 55333333333335533 . C C C C C C ...... C C C C C C C C C C C C C C C .......... C C C huuuuuzufiifl.“BauzunuuzuuBBQJQ/BBBQJQJBR.“33 3 050050053505000030050500050030303300305003 “nun-4.4.3.“aunuuuuzuunufiuuuukRRBanunua 2 5050.55005555500003005550055005050330030500.3 C C C C CCCCCCCCCCCCCC C C C C C C C C C C C C C C C C CCCCCCC 1 5555555555555555555555555555555555555555555 C C C C CCCCCCCCCCCC C C C C C C C C C C C C C C C C CCCCCCCC C C C “nan-4.414.444.4444.hubhuuubnuuuhuhuhuunuhhuuhuuu ee . . . . . . . . . . .1 llZflfiZfi3fifi33fifi3fi45lZfl122nfi2332233334422 5 hm w SSSSSSSSSSSSSSSSSTSSSSSSSSfifiSSSfiSSfiSfiS.. . W. 323223222222333333 2223322 232 23 3 32333 66 Table 5-3 Relative Frequency Distribution of Vehicle Speed Speed Relative Frequency Field Simulated (IFH) Data Data 75 0.0015 0.0013 70 0.007 0.0081 65 0.091 0.0930 60 0.2£21 0.2477 55 0.3 5 0.3469 0 0.1770 0.1737 5 0.0801 0.0843 40 0.0297 0.0301 35 0.0123 0.0126 30 0.0020 0.0017 25 0.0005 0.0006 Table 5-0 Relative Frequency Distribution of Axle Load Levels Axle Load Relative Frequency Field Simulated Levels Data Data 100% of’M.A.Lfi 0.1203 0.1167 90% of’M.A.L 0.1659 0.1687 80% of M.A.L 0.1659 0.1670 70% of M.A.L 0.1659 0.1606 50% of‘l.A.L 0.3820 0.3870 M.A.L 8 maximum axle loads Table 5-6 Mean Value.Variance. and Standard Deviation of Interarrival Time Mean Variance Value Simulated Value 70.14 Theoretical Value 70'87 5028.15 5023.61 Standard Deviation 70.91 70.87 67 Table 5-5 Axle Spacings Axle Spacings. ft. 11 10 . h. 3 863 #33 67729 C C C C C 33394 8008366 29233 01530 56 1.795 5 eeeeeeeseee e 3 0 ha 44947439355 89 1951108720.“?!“6‘ 55 33 133909814159633 33 1 1 1 1 001.2081. #2062109167656606.“ 1.9 10100202013921039060026611066267 363 9102500192054977301167118367485568024246 (In.9822039-47086019.4992999322010569998 1.122121. 1.11. :491:4015-4205221‘2023510213220008358~829uu22 “1.4.3. 9.4.3. 518 SICCCCChhlo’ZCz 2.4. 22:48:230uh.“ 3 21 11111 1 727712350170109907387609250952u°20592898172 Vehicle TYPO C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C 621.49101.42102099000911390099090010009991.2111 1111 111111111 111 111 11 111111 . 1111 LL L LLuuLu L on.»- - .LL . c .44444 n . 3m1mmzzmmmamaaaamusmmmmzmmmzsomemammasas 8 68 Table 5-7(a) Simulated Relative Frequency Distribution of Dynamic Stress Ranges Stress Range Relative Frequency of Stress Range 0.0001 Quarter Mid-span Three-quarter (psi.) Span Span 0 - 1000 0.0021 0.0000 0.0015 1001 - 2000 0.4511 0.1650 0.4249 2001 - 000 0.3712 0.4025 0.3711 001 - 000 0.1416 0.2318 0.1611 001 - 5000 0.0280 0.1291 0.0336 5001 - 6000 0.0046 0.0437 0.0055 6001 - 7000 0.0012 0.01 9 0.0021 8001 - 9000 0.0000 0.0022 0.0000 9001 -10000 0.0000 0.0008 0.0000 10001 -11000 0.0000 0.0000 Table 5-7(b) Simulated Relative Frequency Distribution of Static Stress Ranges Stress Range Relative Frequency of’Stress Range Quarter Mid-span Three-quarter (psi.) Span Span 0 - 1000 0.0732 0.0000 0.0732 1001 - 2000 0.5066 0.3437 0.4974 2001 - 000 0.3074 0.341 0.3323 001 - 000 0.0964 0.203 0.080 001 - 5000 0.0127 0.0784 0.012 _5001 - 6000 0.003 0.0231 0.0038 6001 - 7000 0.000 0.0058 0.0000 7001 - 8000 0.0000 0.0027 0.0000 8001 - 9000 0.0000 0.0010 0.0000 9001 -10000 0.0000 0.0000 0.0000 10001 -11000 0.0000 0.0000 0.0000 69 sumo uevuusdneonnv I Am>.o .cdnnnuus I Aom.o .nsnn uevuuuu I AmN.o e NH NH mH mH HH mH 0H HH NH 6H HH NH 0 can ONH :Nm mom ONH mNm cum NHH NNm mmm oHH on m N: NN om a: NN ow Ne ON am 63 ON Nm m m: NN mm 0: ON on s: 6H mm d: 6H mm 9 mm MN mm mm NN we Nm HN mm om HN me 0 NH 6 NH 0H m NH 0H m NH oH m HH m mH m 3H NH m mH HH m mH HH N NH e HmN.o Hom.o HmN.o HmN.o Hom.o HmN.o Hmu.o Hom.o HmN.orMmN.oumom.ouumN.o Hove: Ampmmzv .wwwg msmwpmm vmpwswpmm msmflvmm 398 £5.58 33 333m 63306.0. 7m 3009 70 Table 5-9 Fatigue Life as Affected by Dynamic and Static Computations and Vehicle Volume Models Fatigue Estimated Fatigue Life. years. Quarter lid-span Three-quarter Model Span Span A 14 9 13 B 13 9 12 Dynamic Stress & C 65 23 33 Random Annual D 56 22 5 Vehicle Volume E 60 22 48 P 924 120 570 G 19 12 18 A 22 12 19 B 20 11 17 Dynamic Stress e C 133 41 107 Constant Annual D 113 37 92 Vehicle Volume E 122 39 98 » F 2011 251 1234 G 33 18 29 A 43 17 0 B 40 16 is Static Stress & C 250 63 28 Random Annual D 215 57 ”2 Vehicle Volume E 229 59 261 F 3923 510 4418 G 23 74 A 87 28 101 B 79 26 92 Static Stress a C 35 128 614 Constant Annual D 55 114 522 Vehicle Volume E 489 119 561 P 8126 1102 9135 G 130 41 151 sage mLo manned on 36 on Ea Home H.235 0: $5 394.55 a»: mud on >8: bomb H.040: 3.3.9.3 maiden Hangman She. «as. 95843. 3.5.569: 33:35.33. 71 3093. was: mpg . b 3 o 5 9003 on 3.».3 I 0.53 m C o H» was on 3.?H. I 93% a 3 8 mm mg“ on 3.»? I 0.33 U mm m» m mg“ on 3.?3 I Pummo a. mm: 50 mac n 3 um am > 3 m H» mom 0H 3.».H. I 6.33 o mu mu w” «8“ on 3.».5 I 93$ 3 mm Nu. 5 m8“ on 3.».6 I obumo 3 mac Ham mum a 3 H» 3 > 3 m pm $03 on 35L. I 0%me a mo NH 5 mg“ on. 3.».3 I 0.33 U mu 5 kn .33 on 3.?H. I 0.3mm 3 mm no km mom on 3.?H. I 983 a 3m H8 HNN a pm 3. um 72 Table 5-11 Effect of RFD of Vehicle Speed on Fatigue Life RFD of Vehicle Fatigue Estimated Fatigue Life, years. Quarter Mid-span Three-quarter Speed (MPH) Model Span Span 75 - 0.0015 65 - 0.091 A 14 9 13 60 - 0.25 1 B 13 9 12 55 - 0.3 5 C 65 23 23 go - 0.1770 D 56 22 5 5 - 0.0801 E 60 22 48 40 - 0.0297 F 924 120 570 35 - 0.0123 G 19 12 18 30 - 0.0020 25 A 0.0005 75 ' 0a0094 70 - O. 091“ 65 - 0.25 1 A 13 9 12 60 - 0.3 5 B 12 9 11 55 - 0.1770 C 6 21 0 0 - 0.0801 D 5 20 3 5 - 0.0297 E 58 20 47 40 - 0.0123 F 853 116 525 35 - 0.0020 G 18 11 16 30 - 0.0005 25 - 0.0000 70 - 0.2 31 65 - 0.3 5 A 13 9 12 60 - 0.1770 B 12 9 11 55 - 0.0801 C 22 22 it 0 - 0.0297 D 20 5 - 0.0123 R 58 21 47 40 - 0.0020 F 811 114 497 35 - 0.0005 G 18 12 17 30 - 0.0000 25 - 0.0000 73 Table 5-12 Effect of Section.nodulus on Fatigue Life I-Beam. Section Fatigue Estimated Fat e Life ears modulus Model _fiiarter iid-span Three-quarter (1n? ) Span Span A 14 9 13 B 13 9 12 C 65 23 23 36WF194 1376.88 D 56 22 5 E 60 22 48 r 924 120 570 G 19 12 18 A 1 1 45 B 24 1 39 C 309 59 26? 36WF230 1534.85 D 249 52 21 s 268 55 23 3 5509 476 4090 G 80 21 71 A 114 19 109 B 98 17 95 C 768 95 701 36WF280 1755.29 D 612 82 562 E 663 87 610 F x 1153 x G 178 27 172 x.é values greater than 10000. 74 3 mm mm a HmNN nun mNam m mnN Nb «pm u “234“ gun on nnm n 33 H253” coon ed? New 8 3m 0 0.3um no 033er Ne mH mm m we 3 H 4. 2. 3 cm a cue: was comm m «N mm mwN m HN Nn aaN a aaS.HH an ueHseoa pan on mom 0 souvoem no somehocH an H as m we H an 4 cam am Hove: 82366329 53ng Mega .usuoa .ouHH oamHvum cops-Hens osmeum oqu osteam no Hosea econ oHu< ecu osHuuozecoHpoom No upooeem.uo noanuaaoo NHIm oHnua . J T T T\5T; Idealized Bridge Section First Quarter S I l I m \i F" : i l i 1 T i J '_—‘=_-——_—___—-f__———_—— : .... . : a" i" F 0.25L 0.25L 0.25L 0.25L 4— Effective Width-4 hé Section A-A % 1"“ Cover Plate Fig. 2-1 Idealization of Bridge 76 I326 e do Hmo>1owuuum conuudov H NI N 6.: @234 E Pan: UUOA SFH ? H.“ m\ a N.“ . La 03.78 and :3 c0053 23.30.83 2.3 5.833 nINéfim 7 7. 3:: 39H ,3 one \ v.25 Gwen avatmv 2.? 78 Aeaongo> oHHdIuv .pmoaoz.ws«vsom awesome you o>hso hhopmwm Hmoamhe cum .me Anvsooomv osaa mN.H oo.H ma.o ammo mN.o o P H _ .o L _ q — 4 W Ho>aq econ on< ssaqus .mmz ow soanm oHngo> nN ooze oHoHeo> H.0I .o.o 11 ”0° pueg o 4918 mutt!!! guano“ 98 a 79 AoHoHn¢> onuunv .vsesoz_w:Hucom oaamshn you o>hso ascend: Hmoanhe mIN .mah Amusooomv cede omua mm.n coma upmo Ho>oH anon on< aschHz .362 mm coonm 6HoHno> Nmm ooze oHoHea> 0.0 «.01 -o.o Puafl 0 4948 “n 19“ auemou 0 Acaoanob saonwV .vcoaos waausom oaasshn you o>hso hhoPmnm Hooanha mum .mHm Anusooomv cede mN.H om.H mNnH ap.H no.o ammo mm.o one _ _ 1 H 1 4 H 4‘.“ .01 Io.o -.N.o [130° 80 Ho>0q anon oHa< ssstsz .mmz ow uoonw oHoHna> .:w.o NIHmm some oHoHnos pueq o 4318 In men 98 queue“ 81 Static Bending Stress at e span length Pig.3-1(a) History Curve for Static Bending Stress at a Critical Section .p II II 0 ‘fi .____ _____ __ m5 q-l g”; I O -§aa srd and 0 Oct-i ”up 0"! g8 _ S a '_.9; f nind-— 1E: . WW span length Pig.3-1(b) History Curve for Dynanic Bending Stress at a Critical Section 82 P(x) 1. o ————————————— :——. Random} z Decimal ........... .---_--.: Number E (a) i I r——J I O : Random ' Var able 1 x 3 Random 1 2 3 Observation x” P(x) 1.0 ____________ Random Decimal +---—- _ --.____ Number : (b) I | 0 4L Random Random Variable Observation Fig.h-1 Illustration of Procedure for Obtaining a Random.0bservation from a given Cumulative Distribution Function. (a) Discrete Random Variable. (b) Continuous Random Variable. 83 r" ,4: L * . s l L. A 7C; 0.251. * 0.591. * 0.25LHi 1:78.52 % 4 17.5" Section A-A 36WF194 E;;;£;;3 lfl'xi' Fig. 5-1 Idealized Section of the Tested Bridge 2-P 253— 6 59-!- sh 2—5 3sn-2 ni- H #1, —3' _ 2-D 35l-3 L ‘gilllll!__l!!_ ‘B‘lUU’ 00 3. 351-4 {A Her—7m 25' 3s__2_- 2 252 352- 3 00 400— 00 -mwmnw”w“h ZSZL 352- 4 ail—o o o slim 00 00 --- J— ...i 253— 2$l-3 352- 5 Ila-c 000 (fir—3m lies—7.3m 253Ll 2a-4 352-6 _‘!‘Li; __ <3 00 'E‘i-rr o c0» 00 ‘U‘l65"'7§77iiffifi _ 253L2_ 252 2 353-2 ‘1‘LTKW w QQ_ o 'E‘i-O 03<3 o oap??"73 253fffi 232-3 353- 3 i}6‘1451"(3 ‘3 C3 ‘U‘lHD (XDET-WET ‘U‘lOU' 000—0 00 3Sl -_ _ 252—:- 353-4 352 252-5 353—5 #39 “0ng __'3 00m #00 000-53368 353“ 252-6 353-6“ _"I'7—_:09_0_5:___ .4 #0 com 000% 353Ll 253-3 453 ”16‘100 o 00 Qliqu <1x>ET'7§3 16‘i1f7-7if7 353L2 253-4 454 »;E‘ioo 00 0 1651-0 QJDTET7§T ‘5‘L§3‘7§ifif 353LL 253- 5 453- 6 ‘U‘loo o o C) ‘U‘l-U' C00 oocxx> ‘U‘lxxr coocmxxro Fig. 5-2 Vehicle Types m5 «soaaoom auoaaaso an omens anoupm so oooom oaa«:o>.uo scones Auvnrm.m«m .mmx . ooonm ouodso> on no om mm on me 33 mm on nu . _ — p u _ 0 some nevussuuoousa p< Hmw aha oaodgo> sinus sseais stasis/eiusu sseaas otasufiq ma usoapoom aeoapauo as «mean amoupm so ooosm oaodso>_no «conga Anvnum.m«a .mm: . soonm oaoago» pa mo no mm on ma 9: mm on mm canons“: p< nuns nopndsd v suns novhdsduoonnp p «lawn onhH oaoano> ..n.H 1&2" sauna sseais stasis/eiueu sseais otusufiq 8? 700” o' 600«* Model C a Model E ’1 Model D 2. m 500" u C E g #00“ 2 E! .p “ odel G 5 300- Model A '4 Model B O a. 'l'. ‘3 an 20on -o 0 E .p a III 100" 0 f 4 1376.88 153“.” 1755.29 (36WF19fl) (BGWPZBO) (36WF280) Section Modulus of the Idealized Bridge, in.3 Fig. 5-4 Effect of Section Modulus APPENDIX COMPUTER PROGRAMS For completeness the three main computer programs written for this study are given here. They are programs (1) DYNAMIC. (ii) STATIC. and (iii) SIMUl. As mentioned in sections 2.8.3 and 2.9. the first two programs are prepared to calculate the dynamic and static stress vectors: minimum stresses. maximum stresses. and stress ranges for three critical sectioned quarter . mid-. and three-quarter spans. The program.SIMUi is written to estimate the fatigue life; it is explained in the next section in detail. A-1 2:2£:!I_§I!!l The procedure described in the section h.) is imple- mented by this program. If one considers the vehicle type. vehicle speed. and vehicle axle load level having sample spaces R1. M2. and R3. respectively. where fli-[m11.m12.m13... ...mlpj. RZ-[m21.m22.m23.....mij. and fl3-Lm31.m32.m33..... mar]. The stress vectors are obtained by use of programs STATIC and DYNAMIC for all possible combinations of sample points from M1. M2. and M3. The data are punched out and 88 89 serve» as the input deck for SIMUl. They are loaded into 12 one-dimension arrays: SR1, SR2. SR3. SMAX1. SMAXZ, SMAXB, SMIN1. SMINZ. SMINB. SSRi. 5532. and SSRB. The first nine arrays are data for the dynamic case --- dynamic stress range. maximum stress. and minimum stress at the three critical sections. and the last three arrays are data for the static case --- static stress ranges at the three critical sections. The static minimum stress is a constant for each section: it is simply the dead load stress. The maximum static stress is simply the minimum stress plus the stress range. Thus there is no need to store these stresses. These data loaded in the 12 one-dimension arrays are needed to calculate the fatigue damage value. when the vehicles crossing the span are not in train. The sequence of input data stored in the arrays can be expressed by the following relationships. For each array. the quantity associated with the event of vehicle type “11’ vehicle speed "25' and axle load level mak. (where i.j.and k run to p,q,and r. respectively). is stored in the ch element of that array where, for the dynamic case. Q - paqe(m3k - 1) + qe(m11 - 1) + '21 .....(A-1) and 1 < 0 < peqer , for the static case. 0 - P*(‘3k - 1) + I11 .....(A-2) and 1 < O < per When the simulation calls for a platoon of two 90 vehicles. program DYNAMIC is executed in program SIMUI. The two successive vehicles are treated as one special vehicle by SIMUl. While the type and axle load level of each vehicle are completely independent. the speeds for both are the same. The axle spacing between the last axle of the first vehicle and the first axle of the second vehicle is equal to the vehicle speed times the interarrival time. Note that such platooning is not considered in the static case. Program SIMUl also includes three subroutines: The subroutine SELECTl is used to determine random observations from random.numbers. The Binary Search technique (12) is used for this purpose. The subroutine RPD100 counts the number of occurrences of each stress level. The subroutine RATIG calculates the cumulative fatigue damage of the bridge. A flow chart of the program.SIMUl is shown in Pig.A-1. A.2.1 angzgl - The main variable names used in the programs are listed below in the alphabetical order. The following:nine names are applicable to all three major programs. These applicable to specific programs are described under the various program names. COMI 8 composite moment of inertia of the idealized bridge section: INDEXWT(I) - axle load level: L a span length of bridge: 91 NOA I number of axles of vehicle: NOT I number of load units representing a vehicle: NOCS I number of critical sections: NOVT I number of vehicle types: NOVW I number of axle load levels: VT(I) I the ith vehicle type. A-Z-Z W - BM(I) I bending moment at a critical section. when the ith axle is at the section: DB(I) I distance between the neutral axis and the lower extreme fiber at the ith section: INDEX(I). INDEX1(I). INDEX2(I) I coefficients for bending moment influence line at the ith section: MBM I maximum bending moment at a section: SSR(I.J) I maximum static stress range. caused by the ith vehicle type. at the jth section: WT(I) I maximum allowable load of the ith axle: XX(I) I distance between the first and the ith axle. :.2.3 W - A(I.J) I horizontal distance between the jth axle and the centroid of the ith load unit: BMASS I mass per unit length of the idealized bridge: C(I.J) I damping coefficient of the 1th axle in the ith load unit: CPS I natural frequency of the idealized bridge: D DB1. DB2 DLD DLDD DELTAT DS (I . J) D31. D82 92 amplitude of the sine curve represent- ing the bridge approach: distance between the neutral axis and the lower extreme fiber at the quarter spans and mid-span. dead load deflection of the idealized bridge: the first derivative of DLD: time increment: initial static compression in the ith axle of the jth load unit: static bending stresses. which are caused by the.dead load of bridge. at the end of cover plate and the ‘mid-span. respectively: DYNM1(I). DYNM2(I). DYNM3(I) 3 dynamic bending moment at the critical sections. corresponding to the ith time increment: 32(1). 33(I.J). Eh(I.J) I quantities defined on E1 P PD. PDD P1 FDI. PDDl K(I.J) LL p. they are denoted by c c and chij: respectively: 21' 313' flexural rigidity of the idealized bridge: amplitude of dynamic deflection. see Eq. 2-1: the first and the second derivative of P: P. when time is equal to t+ t: FD. PDD respectively. when time is equal to t+ t: spring stiffness of the jth axle in the ith load unit: length of bridge approach: 93 LLL I total length of bridge and bridge approach: NOAOE(I) I number of axles of the ith load unit: NOVV I number of vehicle speed levels: P(I) I length of the ith load unit: PERIOD I natural period of vibration of the idealized bridge: POLARM(I) I polar moment of inertia of the ith load unit: SMOE I Young's modulus: SMOI I moment of inertia of I-beam(excluding slab): SITA(I) I angular displacement of the ith load unit at time t: SITAD(I). SITADD(I) I the first and second deriva- tive of SITA(I). respectively: SITA1(I) I angular displacement of the ith load un t at tile t+ t: SITAD1(I). SITADD1(I)I the first and second deriva- tive of SITA1(I): SPEED(I) I the ith vehicle speed level: STORE I t time! SPIID(I): SR1. SR2. SR3 I maximum dynamic stress ranges at the critical sections: STRESSl. STRESSZ. STRE883 I maximum dynamic stresses at critical sections: STRESSb. STRESSS I minimum dynamic stresses at critical sections -- quarter span (or three-quarter span) and mid-span. respectively: TLL I length of bridge approach/vehicle speed: 9h TOTALWT(I) I weight of the ith load unit: TXXLLL I time between the entry of the front axle of a vehicle to the bridge approach to the departure of the last axle from the bridge: VMASS(I) I mass of the ith load unit: WT(I.J) I the 3th axle load of the ith load unit: XXX(I.J) I distance between the first axle of the first load unit and the Jth axle of the ith load unit: 2(1) I vertical displacement of the ith load unit at time t: ZD(I). ZDD(I)Z(I)the first and second derivative of I 3 21(1) I vertical displacement of the ith load unit at the end of time t+ t: ZD1(I). ZDD1(%)? the first and second derivative of 1 I : YB(I.J) I vertical deviation of the bridge approach from the horizontal at the position of the ith axle in the jth load unit: YBD(I.J) I the first derivative of YB(I.J). $2.4 W AA(I.J.K) I horizontal distance between the kth axle and the centroid of the jth load unit for the ith vehicle type:‘ AAT I average interarrival time: AB I index of fatigue model: ADD(I) I number of vehicle type 1 observed: AVV(I) I the ith annual vehicle volume: 95 SMAX1(I). SMAXZ(I). SMAX3(I) I maximum dynamic stresses at the critical sections. corres ending to each index I(I INDEXR§I SMIN1(I). SMIN2(I). SMIN3(I) I minimum dynamic stresses at the critical sections. corres ending to each index I(I INDEXRgt SSR1(I). SSR2(I). SSR3(I) I static stress ranges at the critical sections corresponding to each index I(I INDEXSS: STAND? I standard deviation of interarrival tho. IAT’ STORESP(I) I number of vehicle speed 1 observed: STOREWT(I) I number of axle load level 1 observed: STORE1(I) I vehicle type i: SRFD1(I). SRFD2(I). SRPD3(I) I simulated relative frequency of the ith dynamic stress range level at the critical sections: SSRFD1(I). SSRPD2(I). SSRPD3(I) I simulated rela- tive frequency of the ith static stress ranges level at the critical sections: TOTAL I sample size: UVT(I) I cumulative relative frequency of the ith vehicle type: UVV(I) I cumulative relative frequency of the ith vehicle speed: UVW(I) I cumulative relative frequency of the ith axle load level: UPPERL I upper limit of the simulated annual vehicle volume: VARIANS I sample variance of the interarrival time. IAT: 96 BLIFE(I.J) I estimated fatigue life at section 1 using fatigue model i: CAR99(I.J) I cumulative fatigue damage at section 3 using the ith fatigue model: CAR88(I.J) I the first year's fatigue damage at the 1th section. estimated by using the ith fatigue model: DLSl. DLSZ I dead load bending stresses at the critical sections: IAT I interarrival time: INDEXR. INDEXS I quantities defined by qu.(A-1) and (A-Z). respectively: J1 I random decimal number: LITTLEN(INDEXR) I number of cycles. corresponding to the dynamic stress range with index INDEXR: LOWERL I lower limit of the simulated annual vehicle volume: NOA1(I) I number of axles of vehicle type i: NOLU(I) I number of load units of vehicle type i: NOAOELU(I.J) I number of axles in the jth load unit of vehicle typei: NOFM I number of fatigue models: RFI(I) I relative frequency of vehicle type i: RFW(I) I relative frequency of axle load level i: RPS(I) I relative frequency of vehicle speed i: RN I random.decimal number: SMALLN(INDEXS) I number of cycles. corresponding to the static stress range with index IND3188 97 WT1(I.J.K) I the kth axle load of the jth load unit in the ith vehicle type: XBAR I sample mean of IAT: XXX1(I.J.K)I distance between the first axle of the the first load unit and the 3th axle of the ith load unit: YOGD I number of years for which the annual vehicle volumes are prescribed: A.2.5 W K I index: L1. L2. L3 I parameters used in binary search: NN I total number of sample points of a random variable: UL(I) I cumulative relative frequency of the ith sample point. A.2.6 Sgbrggfiine BFDlgo SRL I stress range value at quarter span: SRM I stress range value at mid-span: SRR I stress range value at three-quarter span: SSSSS(I) I the ith stress range level. A.2.7 W N1.N2.N3.N4.N5.N6.N7 I for a given stress vector. the number of cycles which would cause a fatigue failure as defined by fatigue models.A. B. C. D. E. P. and C. respectively: SR11. SR22. SR33 I stress range value at the critical sections: SMAX11. SMAXZZ. Slaxll I maximum stress values at the critical sections: 98 SMINlI. SMINZZ. SMIN33 I minimum stress values at the critical sections: NNNNN I number of cycles at a given stress range: STORE(I.J) I cumulative fatigue damage at the jth segtion estimated by the ith fatigue n09. 99 C MT 3 ‘ CALCULATE AND ACCUMULATE] IT E CHOOSE 3 RANDOM NUMBERS RNl. RNZ. AND RN}. RNl I>VEHICLE TYPE.m11 RNZ =>VEHICLE SPEED.m21 RN3 I>VEHICLE AXLE LOAD LEVEL.m3k HE STATIC FATIGUE DAMAG CALCULATE THE DYNAMIC STRESS VECTOR FOR THE PLATOON CASE FIND THE STATIC STRESS VECTOR.BY USING EQ.(A—2) CALCULATE AND ACCUMULATE THE STATIC FATIGUE DAMAGE CHOOSE A RANDOM NUMBER RN“ I>INTERARRIVAL TIME * DOES THE PLATOON CASE OCCUR? SEE SECTION h.3 CHOOSE 2 RANDOM NUMBERS RNS AND RN6. RNS I>VEHICLE TYPE.II18 RN6 I>VEHICLE AXLE LOAD LEVEL.m3t l FIND THE STATIC STRESS VECTOR.BY USING EQ.(A-2) Pig. A-l Flow Chart for Program.SIMU1 HE DYNAMIC FATIGUE AMAGE ICALCULATE AND ACCUMULATE T D IS SAMPLING SET A PARAMETER T0 COMPU- TE DYNAMIC FATIGUE DAMAGE I—.—® CALCULATE THE CUMULATIVE FATIGUE DAMAGE AT THE END OF THE FIRST YEAR. D1 IS Dl > 1.? 0 CD CHOOSE A RANDOM NUMBER RN? I>NEXT YEAR'S ANNUAL VEHICLE VOLUME CALCULATE THE CUMULATIVE FATIGUE DAMAGE.2 D1. BY USING EQ.(h-#) 100 FATI GUE DAMAGE (STATIC) BEEN “BLIFE” CONSIDERED? 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