, . . . , V , . A T ,. . , V. . ‘ . ‘ . . ‘ . n ‘ y ‘ mess NIVERSITY LIBRARIES llllllllllllllllllllllllll l lllll l ll 31293 00880 I ll This is to certify that the thesis entitled A Seismic Design Study of Two Deck—Type Arch Bridges presented by Dan Ping Xu has been accepted towards fulfillment of the requirements for Master of Science degree in Civil Engineering HEW/JIM Major professor Date November 9, 1992 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State 1 University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE wr—ll l EAL—Qt ______l__ ___l ___l—7V— MSU Is An Affirmative Action/Equal Opportunity institution remnant A SEISMIC DESIGN STUDY OF TWO DECK-TYPE ARCH BRIDGES By Dan Ping Xu A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Environmental Engineering 1992 ABSTRACT A SEISMIC DESIGN STUDY OF TWO DECK-TYPE ARCH BRIDGES By Dan Ping Xu Two bridges were studied to provide information that could aid the designers of deck-type arch bridges, when considering seismic effects, regarding two parameters: 1) column diagonal bracing and 2) the depth to width ratio of the rib box section. It was based on a computer modelling of the bridge and load system. Nonlinear elastic curved beam elements were used for the ribs, and straight beams and truss mem- bers for other members of the bridge. The seismic load was represented by the AASHTO design response spectrum. It was noted that the column diagonal lateral bracing with small cross-sectional area when used at all panels is very effective in reducing the maximum stresses in the ribs. It appears that the depth to width ratio of the rib section need not be increased for bridgs with longer spans if the cross-section area of the rib remains the same. This seems contrary to a general tendency in practice. The results indicate that the ratio need not go beyond two. ACKNOWLEDGMENTS The author wishes to express her deepest appreciation to Dr. Robert K. L. Wen of the Department of Civil and Environmental Engineering, Michigan State University, for his guidance in the preparation of this thesis and for his thought given throughout the course of this study. Thanks are also extended to the other members of the author’s Master Degree Committee: Professors Frank Hatfield and Ron Harichandran for their guidance. iii TABLE OF CONTENTS LIST OF TABLES vi LIST OF FIGURES vii LIST OF SYMBOLS x CHAPTER I INTRODUCTION 1 CHAPTER II MODELLING AND ANALYSIS METHOD 7 2.1 General 7 2.2 Bridge Model 7 2.3 Dynamic Analysis 8 2.4 Ground Motion and Damping 9 2.5 Lateral Tower Stiffness 9 2.6 Maximum Stresses 10 2.7 Maximum Response 11 2.8 Parameters 12 2.9 Computer Modelling 13 CHAPTER III DESIGN STUDIES 20 3.1 General 20 3.2 Preliminary Parametric Studies 20 3.2.1 Rotational restraint of supports 21 iv TABLE OF CONTENTS (continued) 3.2.2 Cross-sectional area of deck stringers A 5 3.2.3 Analysis method 3.2.4 Torsional constant 3.3 Natural Modes of Vibration for Out-of-plane Response 3.4 Comparison of Responses to Different Loadings 3.5 Effect of Cross-sectional Area of Column Diagonal Bracing Ad 3.6 Effect of Rib Depth to Width Ratio D/B 3.6.1 Sectional properties and natural frequencies 3.6.2 Effect on seismic responses 3.6.3 Effect on displacements and tower forces 3.6.4 Effect on responses to dead load and wind load CHAPTER IV SUMMARY AND CONCLUSION 4.1 Summary 4.2 Concluding Remarks LIST OF REFERENCES 21 22 22 23 23 24 26 27 27 29 29 66 66 68 69 LIST OF TABLES Table 2-1. Parameters of bridge system Table 2-2. Parameters for in-plane and additional parameters for out-of-plane studies Table 3.2- 1. Effects of rotational restraint at supports on natural frequencies (cps) Table 3.2-2. Effects of rotational restraint at supports on rib member forces and stresses (MCSCB) Table 3.2-3. Effects of A s on rib member forces and stresses (MCSCB) Table 3.24. Effects of A s on natural frequencies (cps) Table 3.2-5. Natural frequencies (cps) of MCSCB from different analysis Table 3.2-6. Fundamental frequencies (cps) of MCSCB for different AK'I'I‘ values Table 3.2-7. Dynamic stresses (ksi) in MCSCB for different AK'IT values Table 3.6—1. Sectional properties of MCSCB for different D/B ratios and tf/ tW ratios Table 3.6-2. Sectional properties of MSSB for different D/B ratios and tf/tW ratios Table 363. Displacements of MCSCB for different D/B ratios Table 3.6-4. Tower lateral forces (kips) of MCSCB for different D/B ratios vi Figure 1-1. Figure 1-2. Figure 2-1. Figure 2-2. Figure 3-1. Figure 3-2. Figure 3-3. Figure 3-4. Figure 3-5. Figure 3-6. Figure 3-7. Figure 3-8. Figure 3-9. Figure 3-10. Figure 3-11. Figure 3-12. LIST OF FIGURES A deck-type arch bridge Bridge model with global coordinates X, Y, Z and rib local coordinates x, y, 2, system is symmetric AASHTO acceleration spectra Box shape cross-section of arch rib First out-of-plain mode shape (2nd overall mode number) Second out-of-plain mode shape (4th overall mode number) Third out-of-plain mode shape (5th overall mode number) Fourth out-of-plain mode shape (7th overall mode number) Stress distributions under various loadings (MCSCB) Fundamental natural frequencies due to varying Ad (MCSCB) Dynamic stresses due to varying Ad (Z-motion, MCSCB) Dynamic stresses due to varying Ad (Z-motion, MSSB) Fundamental natural frequencies of MCSCB due to varying D/B ratio Stress and member forces at support due to varying D/B ratio (MCSCB) Stress and member forces at the left node of panel 2 due to varying D/B ratio (MCSCB), Z-motion seismic loading Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), Z-motion seismic loading vii 18 19 39 41 42 43 45 46 47 48 49 50 LIST OF FIGURES (continued) Figure 3-13. Figure 3-14. Figure 3-15. Figure 3-16. Figure 3-17. Figure 3-18. Figure 3-19. Figure 3-20. Figure 3-21. Figure 3-22. Figure 3-23. Figure 3-24. Figure 3-25. Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), Z-motion seismic loading Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), Z-motion seismic loading Dynamic stress distributions under lateral seismic loading (Z-motion) due to varying D/B ratio (MCSCB) Dynamic stress distributions under in-plane seismic loading (X-, Y-motion) due to varying D/B ratio (MCSCB) Dynamic stress distributions under three dimensional loading (X-, Y-, Z-motion) due to varying D/B ratio (MCSCB) Dynamic stress distributions under lateral seismic loading (Z-motion) due to varying D/B ratio (MSSB) Dynamic stress distributions under three dimensional loading (X-, Y-, Z-motion) due to varying D/B ratio (MSSB) Stress and member forces at the left node of panel 2 due to varying D/B ratio (MCSCB), dead load Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), dead load Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), dead load Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), dead load Stress and member forces at the left node of panel 2 due to varying D/B ratio (MCSCB), wind load Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), wind load viii 56 57 58 59 61 62 63 LIST OF FIGURES (continued) Figure 3-26. Figure 3-27. Suess and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), wind load Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), wind load ix 65 LIST OF SYMBOLS acceleration coefficient cross-sectional area of arch rib cross-sectional area of column bracing cross-sectional area of deck stringer width of arch rib cross section depth of arch rib cross section Young’s modulus acceleration of gravity dead load factor rise or height of bridge arch moment of inertia of arch rib about local x-axis (resisting out-of-plane bending) moment of inertia of arch rib about local y-axis (resisting in-plane bending) stiffness matrix the torsional constants to be used for one rib the local torsional constant of one rib based on cross-section shape : rib lateral stiffness : end tower lateral stiffness length of the bridge span Mass per foot of span length local bending moment about x-axis : local bending moment about y—axis number of panels LIST OF SYMBOLS (continued) [n1]: first order incremental stiffness matrices [n2]: second order incremental stiffness matrices Pz: member axial force {q}: nodal displacement vector r: radius of gyration of arch rib cross-section S: site coefficient Sx section modulus about rib local x-axis (out-of-plane bending) Sy . section modulus about rib local y-axis (in-plane bending) S A: acceleration spectrum Sm estimated maximum response (stress, member forces or displacements) T: period of vibration W: width of bridge X, Y, Z: global coordinates x, y, z rib local curvilinear coordinates (I). l 5 a: o ith circular frequency critical damping coefficient for first two modes ratio of end tower lateral stiffness to rib system lateral stiffness total stress xi CHAPTER I INTRODUCTION As lifeline structures, bridges in general should be sufficiently sound to continue functioning in an emergency situation such as that resulting from a major earthquake. Since the San Fernando earthquake in 1971, the engineering profession has given much added time and effort to the study of earthquakes in order to build stronger, safer structures. In the case of arch bridges, considerable amount of research had been done in recent years. In particular at MSU, several studies on the behavior of deck-type arch bridges (Fig. 1-1) had been made. Research on the deck-type arch bridges was reported by Dusseau and Wen [3]. The importance of seismic effects on arch bridges was assessed. The seismic responses of three actual deck-type arch bridges: SSB (193ft); CSCB (700ft) and NRGB (1400ft) were computed and compared with those of wind (combined with dead load effects). It was concluded that seismic effects are important and can govern the design. However, the computations were based on linear straight beam elements to represent the arch rib. Artificial ground motions were applied. C. M. Lee [6] developed a method of analysis and computer program that incorporated nonlinear curved beam elements for the arch rib. The geometric nonlinear analysis included was based on a model reported by Jose Lange [5]. The treatment of the elasto-plastic properties of the model was reported by Wen, Lee and Alahamd [15]. In applying Lee's analysis it was found when the duration of time of solution was sufficiently long, in the case of geometric nonlinearity, the equilibrium position would drift because of the use of the tangent stiffness matrix (the first incremental stiffness) in the solution to calculate the element resistance. In response to this challenge, a more accurate secant stiffness matrix was developed for the nonlinear elastic curved beam element by Wen and Sunhedro [13]. With that the “drift” phenomenon would be removed. The nonlinear element was subsequently incorporated by Wen [14] into a general program and used to obtain certain design aids for the in-plane response of deck type arch bridges. The design aids give response values in terms of stress and displacement amplification factors as functions of span length, L, and two major dimensionless parameters: the slendemess ratio, L/r, r is the radius of gyration of the rib cross-section, and the dead load factor G = M gL3/EIyr , where M is the total mass per unit length, E is the Young's modulus of elasticity, and Iy r is the moment of inertia of the rib section. An exploratory study of three dimensional models concentrating on certain modelling aspects and the role of the end towers had been carried out by R. Millies [9]. Attention was turned to the development of information that could be helpful to the designers considering responses in the three dimensional space. In the latter _case the number of variables of the problem was increased greatly over that for the two dimensional or in-plane case. It does not seem realistic to develop charts giving seismic responses covering all combinations of the numerous variables. However, it was recognized that not all variables are of equal significance. In his parametric variation study, by fixing the cross bracing truss elements between the ribs and the cross bars (straight beam elements) between the ribs, a study was made on the effects of the lateral stiffness of the end towers of the bridge system. Letting or be the ratio of the lateral stiffness of a tower to the lateral stiffness of the braced ribs, it was found that a value of or equal to two would be appropriate for seismic resistance. The study was based on time history solutions. The preceding works all deal with the deck-type bridge. More recently A. Bellamine has completed a study of the seismic responses of tied-type of arch bridges [1]. It was limited to in-planc responses only. The practical effects of seismic loading in relation to design live load is assessed and an “optimal” distribution of material between the deck and the rib was investigated. Also it was found that, apart from the obvious fact that the deck would be subjected to additional axial stress on account of its function as a tension tie, the seismic behavior of tied-type bridges seemed to be quite similar to that of deck-type bridges. A study of a single existing tied-bridge was also conducted by Lee and Torkamani [7]. It included considerations of soil structure interaction and unequal support motions. The objective of this study is to get a better understanding of the general three dimensional seismic responses of deck type arch bridges. Thus the general purpose of the work reported herein is similar to that reported by B. Millies. Hence, the work may be considered as a continuation of same. That is: to develop information that may aid the designers in their decision making process. . The structural designer does not always get to decide on all the major parameters of the bridge, e.g., the span length and/or width of the bridge. However, the design of the cross-section of the arch rib generally falls within his/her domain. For most deck-type : steel arch bridges, the rib cross-section usually has a box shape. A key parameter of that shape is the depth to width ratio. Its consideration is the main object of this study. Another major parameter for the bridge design is the use, or lack of it, of diagonal bracing (perpendicular to traffic) between the columns supporting the deck (see Fig. 1-2). Although the qualitative action of these bracings is known, their effectiveness is not clear. Consequently such bracing had been used in some cases and omitted in others. A study of the effects of such bracings is the second objective of this report The study is based on the theory of elastic design. Hence, stress is regarded as the main response parameter. However, the nonlinear behavior of the ribs resulting from the dead load compression is considered. The seismic analysis is based on the design response spectrum approach. Because of the large number of variables involved in the system considered, before data for investigating the previously mentioned two variables can be collected, it was necessary to conduct some preliminary studies leading to the holding of some parameters constant. They included the representation of the rotation restraint at the rib support, the cross-sectional area of the stringers of the deck and the representation of the torsional stiffness of the ribs. In the following, Chapter 2 describes the modelling of the bridge system, the method of analysis, parameters of the bridge system and computer program used. The results obtained for the study are presented in Chapter 3. A summary and conclusion are given in Chapter 4. 6qu :08 REE—one < .2 same”. HI. 539 new .oEoEEQ mm ESE? .N i .x 835908 :82 m: 98 N S .x 8&5208 Boo—m .23 $88 owetm N; Bzwi wEoEnamEo at 582-390 mead—:UHOOO 92:53 .82 a: x ”no: a: N in 2308 pt wEoEn _wcomfle 5:38 5:38 \ wcmofinémob x8e wEowE goon \ fcomcEm x8e 9:85 .032 can CHAPTER H MODELLING AND ANALYSIS METHOD Zchrrcral This chapter discusses the modelling of the bridge system and the method of analysis. The parameters used, ground acceleration input, computer method of responses and the computer program are also described herein. 2.2 W In an arch bridge, the arch ribs are the main components of the structure. Therefore, the subsystem of the ribs is given greater precision than the other components such as the deck, column and the end towers. The bridge model in the present study is a three dimensional finite element model (Fig.1-2). It contains two ribs (modelled by curved beams), which are parabolic in shape. The ribs are braced by cross-beam (modelled by straight beams) and cross-bracing (modelled by truss elements). Eight panels of equal length are used throughout the analysis. The deck system, which includes the “deck stringers”, “deck cross-bracing” and “deck bracings”, is entirely represented by truss elements. The deck and ribs are connected by “columns” modelled as truss elements which have larger stiffness. The cross—bracings between the columns at a given panel point are referred to as “column diagonal bracings.” The initial static load and the corresponding mass are assumed to be uniformly distributed on the horizontal projection and lumped at the panel points. 23 Minimum The equation of motion of the bridge subjected to earthquake excitations may expressed as [6]. [m] {ii} + [C] {u} + {r} = -lm] {fig} (1) where, [m] is the lumped mass matrix, [c] is the damping matrix of the Rayleigh type, {r} is the resistance vector, {u} is the displacement vector with respect to the ground, and the dot superscripts denote derivatives with respect to time, {ii g} is the ground acceleration vector. If linear elastic behavior is presumed, {r} = [R] {q} (2) in which [k] is the element stiffness matrix and {q} is the displacement vector. The resistance of a nonlinear elastic element may be written as [n1] [n2] . 2 + {r} = <[k1+ 3 ){q} (3) in which [n1] and [n2] are the first and second order incremental stiffness matrices [13]. 2.4 QmundeicundDamping For dynamic analysis, the AASHTO [11] design response spectrum is used as the ground motion excitations for all three directions with the vertical acceleration scaled by a factor of 3/4. Using the multimode spectral method, the acceleration spectrum, S A , is given by 1.2AS S = __ ' (4) A Tm m for Tm < 4.0 second, and 3A8 SA " $73 (5) m for Tm > 4.0 second. The value of S A need not exceed 2.5A, as indicated in Fig. 2-1. where A = acceleration coefficient, S = site soil coefficient and Tm = the period of the mth mode of vibration. The value of A = 0.4 and S = 1.0 correspond to the “strongest ground motion and soil” in the study. Damping is assumed to be of the Rayleigh type with critical damping ratios of 0.02 used for the first two modes. 2.5 LateraLIchLStifmess The stiffness of the rib system, k rib , is calculated using F = k rib z. A uniformly distributed load (F) in the z -direction is applied to the deck and ribs (with zero lateral end tower stiffness), and the corresponding crown displacement (z) is obtained. The stiffness is equal to the load applied divided by the displacement, i.e., k ib = F/z. The 1' 10 end tower lateral stiffness kt ow is set to be proportional to krib , ktow = a X krib (6) where a, is the ratio of the lateral stiffness of an end tower to that of the rib system. Following a suggestion in Millies’ study [9], a value of two is used for a in this study. Tower bracing area is then calculated as k x13 x tow tb (7) A = raw2 1 tb E where E = Young’s Modulus W = width of the bridge ltb = length of tow-bracing member Once the tower bracing area Atb is determined, the structure with the tower and tower bracings is then subjected to static load (dead load and wind load) and seismic load. \Vrnd load is also applied to the structure in this study. The magnitude of wind load used corresponds to an horizontal acceleration of 0.1 g of the bridge mass. 2.6 Maximum The maximum stress at a given cross-section of a rib is calculated as P l A r M _Y s y M x X + + (8) 0’: where I l denotes the absolute value 11 P2 = axial force Mx , M = the local bending moment about the x-axis and the y axis, respectively 8 , S y = the section moduli. x A r = cross-sectional area of arch rib. 2.7 Maximum Resmnse For an estimated maximum dynamic response based on the response spectrum, the computer program used for the study can specify either by the complete quadratic combination method (CQC) or the square root of sum of squares (SRSS) method. In a preliminary study, no major difference was found among the results by using these two methods. Therefore CQC method, considered to be the more accurate of the two, was used throughout this study. That is, = EJZSiSanPjIQanIRij (9) 8(Ci (oi +Cj (oj )coi or. J[Ciijimj .. = (10) U (7402- Mo?)2 +4C. iCjcoiJ a). J(m.2+or2) +4(C2+C2) (020312 where Sm = estimated maximum response (stress, member forces or displacements) S i = any modal response of the ith mode 1 = 1, 2, 3 corresponding to x,y,z ground motion, respectively Pi1 = participation factor (bi = ith circular frequency Qi1 = maximum response of SDOF system due to response spectrum 12 C i = damping ratio of ith mode The magnitude of p depends on the number of modes considered. It is observed in a preliminary study that the effect of those modes greater than 15 are negligible. Therefore, the number of modes used obtaining the results reported herein were set to be 20. 2.8 Parmgters The dimensional parameters that specify the bridge system being studied here are listed in the first two columns of Table 2-1. In a previous study [9], certain dimensionless parameters were defined from these basic dimensional ones and they were reproduced in Table 2-2. For the present study that deals with the cross-sectional design parameters, initially those dimensionless parameters such as Ix 1_/ Iy r (ratio of moment of inertia about x-axis to that about y-axis for the rib), Cx/rx (ratio of half width to the radius of gyration about x-axis of rib) and G = M gL3/ EIyr (the dead load displacement factor) were employed. That approach was found to be unwieldy. Sometimes after translating them into dimensional form, impractical cross-sections resulted. For example, it was attempted to investigate the effect of G on the response of the bridge with different cross-sectional properties while keeping the cross-sectional area constant. When G is increased, in other words, Iyr must be decreased. Since all the other ratios are fixed at this point, hence Ixr has to be decreased. On the other hand if G is decreased, Ixr and Iyr are b0th in increasing. Meanwhile the cross sectional area A r of the rib is fixed. One cannot expect such a cross-section. Thus, assuming that the cross-sectional area has a box shape with two axes of symmetry (Fig. 2-2), the cross-sectional properties are defined by the depth D, width B, flange thickness If and web thickness tW . For the study, the computer program computes first these from values from the data input cross-sectional area A r , D/B ratio, tw/ D ratio, and tf/tW ratio. Srnce 13 Ar = 2th-I-2(D--2tf)tW (11) the width B can be determined by Ar B = 2 (12) 4mm + <9>2l:“-"l-2(3)l1“1l a) tW D B B D tW D B Then D, tw , and tf follow. The properties such that Ixr and Iyr can then be easily calculated. 2.9 We The numerical results obtained for this study was obtained from a modified version of the program used in Millies [9]. The modifications include the following additions: (1). an analysis by use of response spectrum and model superposition. (2). an eigen analysis subroutine (Subroutine RSG)[4]. (3). input and section properties computation associated with the rib box section (as described in the preceding section). 14 Table 2-1. Parameters of bridge system. Parameter Description Values used MCSCB MSSB L length of the bridge span (ft.) 700.00 193.00 H rise of the bridge arch (ft.) 121.75 29.00 W width of the bridge (ft.) 26.00 22.00 Ar cross-sectional area of rib ( {12) 2.59 0.96 Ix r moment of inertia of rib about local x-axis ( {14) 1.84 - 13.47 0.16 - 0.95 I r moment of inertia of rib y about local y«axis( a“) 20.91 - 34.90 1.37 - 2.13 Ktr local torsional constant of one rib (see eq. 13) 6.51 - 21.46 0.61 - 1.61 er one half of the width of the rib cross-section (ft) 1.00 - 3.21 0.50 - 1.38 C r one half of the depth of the y rib cross-section (ft) 3.21 - 5.01 1.39 - 2.05 Ab cross-sectional area of rib bracing beam ( 112) 0.259 0.096 be moment of inertia of rib bracing beam about X-axis ( n“) 1.45 - 1.75 0.60 - 1.07 I yb moment of inertia of rib bracing beam about Y-axis ( ft“) 1.45 - 1.75 0.60 - 1.07 Ktb local torsional constant of bracing beam 1.45 - 1.75 0.60 - 1.07 15 Table 2-1. (continued) Parameter Description Values used MCSCB MSSB A s cross-sectional area of stringer ( {12) 0.40 - 2.00 0.80 A d cross-sectional area of column diagonal bracing ( {8) 0.00 - 0.24 0.00 - 0.08 A x cross-sectional area of rib cross- bracing ( f t2) 0.00 0.00 A db cross-sectional area of deck bracing ( ft2) 4.00 4.00 A dx cross-sectional area of deck cross-bracing ( ft2) 4.00 4.00 A c cross-sectional area of column ( {12) 25.91 9.58 E Young’s modulus (ksf) 4176000 4176000 Mr mass of the rib total mass (HR) 1 .274 1.274 M total mass per unit length of span length (k/ft) 4.80 4.80 or ratio of end tower lateral stiffness to rib system lateral stiffness 2.00 2.00 § critical damping coefficient for first two modes 0.02 0.02 N number of panels 8 8 16 Table 2-1. (continued) Parameter Description Values used MCSCB MSSB D depth of arch rib cross section (in.) 76 - 120 33.4 - 49.2 B width of arch rib cross section (in.) 24 - 77 12 - 33 tf flange thickness of arch rib cross section (in.) 1.7 - 2.8 1.48 - 2.2 tw web thickness of arch rib cross section (in.) 0.67 - 1.03 0.6 - 0.94 AK'I'I‘ torsional constant ratio (see section ) 2 - 6 2 17 Table 2-2. Parameters for in-plane and additional parameters for out-of-plane studies [9] Parameters for Range Value Used In-Plane Behavior ~ H/L 0.125 — 0.225 0.175 L/r, 100, 300 200 G 2.63 - 10.5 10.5 run 0.344 - 0.760 0.265 N 6 - 24 8 2%, 5% 2% cylry 1.00 — 1.55 1.27 L 200 - 1000 ft. 200, 600, 1000 ‘ 0 - 0.509 0.319 Y: 0 - 0.509 0.239 Additional Parameters for Out-of-Plane Behavior W 30 - 60 30 In/Iy, 0.32 - 0.11 0.32 - 0.11 cx/r, 1.00 - 1.55 1.307 AJA, Not Available 0.04 Ada, 0.10 - 0.25 0.10 IW/Ifi 0.0015 - 0.014 0.05 #EELEW Not Available 1.0 _EEng Not Available 1.0 a 0.0 - 10.0 0 - 10 A,/A, Not Available 0.183 - 0.91 z, 0 - 0.509 0.319 18 538% conga—08a OEm<< .TN 8239 mecooom E EH boron VS 19 .7. § \\\\\\\\\\\\\\\\\\k tw of arch rib. l N \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ .\\ ll D \\\\\Ns\\\\\\\\\ \ Fig. 2-2. Box shape cross-section 20 CHAPTER III DESIGN STUDIES 3.1mm This Chapter presents and discusses the results obtained for the study. The study focuses on the MCSCB bridge, which is a medium span (700 ft.) and relatively slender bridge. Some results for MSSB, which is shorter (193 ft.) and less slender, are also presented. The values of other parameters of the two bridges are listed in Table 2-1. Since much data is available on the in-plane response, the emphasis of the study is on the lateral response. Thus most data gathered corresponds to responses to ground motion in the lateral (Z) direction. For the purposes of comparison, dead load (DL) and static wind load (WL) responses are also presented as well as in-plane responses and the general case of responses to seismic excitations in all three directions in space. Two major parameters are considered herein: A d , the column diagonal bracing and D/B, the depth to width ratio of the rib section. 3.2 mlirninm Pmmem’g Studies Before numerical data on the effects of A d and D/B were collected, a preliminary investigation was carried out to fixed certain parameters that enter into the analysis. This includes the type of restraint of supports, cross-sectional area of the deck stringers and the torsional constant used for the ribs. 21 3.2.1 Rotational restraint of supports Previously in Millies’ study [9], the out-of-plane rotations of the ribs at supports were restrained, i.e., rotation about the global X and Y axis are set to be zero at supports; the supports have moment release about the Z-axis only. In practice, it is possible that the bridge is not completely laterally fixed at supports, a model with free out-of-plane rotation at the rib supports was considered. The bridge investigated was MCSCB (with A d = 0, A s = 0.4 and AK’IT = 2.0 to be defined later by using linear analysis method). In Table 3.2-1 are listed the first five natural frequencies corresponding to the two cases of the rotational degrees of freedom about the global X - and Y-axis. It is seen that compared to fixed support model, the in-plane motion frequencies of the hinge support model remain the same, but the out-of-plane frequencies are decreased somewhat. This is due to the fact that releasing the moments about the X- and Y- axis does not change the in- plane behavior of the structure but reduces the out-of—plane stiffness of the structure. In Table 3.2-2 are listed the stress resultants and stresses for the two cases. In general, releasing the restraint of out-of-plane rotation led to a reduction of the rib member forces and stresses at all nodes. As far as displacements are concerned, there was approximately a ten percent increase in the Z-direction due to the rotational releases. Since bridge supports are not completely restraint in real cases, the “free” rotation model would provide more conservative results. Therefore, in the following studies, the supports are rotationally unrestrained. 3.2.2 Cross-sectional area of deck stringers AS Because of the deck support conditions, A s has virtually no effect on the in-plane stiffness of the structure. Under out-of-plane loading, i.e., Z-load, however, the deck responds as a horizontal beam with the 2 edge stringers with cross-sectional area As acting as flanges. They do affect the lateral response. 22 In Table 3.2-3 are shown, for As = 0.4, 0.8 and 2.0 {12, the rib forces at the support, quart point and the crown for MCSCB (with A d = 0, no rotational restraint at supports and AK'I'I‘ = 2.0 (to be defined in the next section), MCSCB). It is seen that the internal forces of the rib decrease with increasing values of A s (as the deck carries more load). In Table 3.2-4 are listed for first five natural frequencies for the same bridge considered above. As expected, the frequencies corresponding to the in-plane modes . (noted “1”) are net affected by A s' The out-of-plane frequencies increase with an increase in A s . For this study the intermediate value of As equal to 0.8 ft2 was used. 3.2.3 Analysis method The program has the capability of using a linear, geometrically nonlinear or “linearized” model for dynamic analysis. The “linearized” model employs the nonlinear model for the initial dead load solution, and the subsequent response to seismic motion would be calculated based on a linear analysis using the tangent stiffness of the structure under dead load as the linear stiffness. The natural frequencies of MCSCB corresponding to the linearized and linear models are shown in Table 3.2-5 (with A s = 0.8 ftz, A d = 0.08 ft2, no rotational restraint at supports and AKTT = 2.0, MCSCB). It can be seen that the fundamental natural frequencies of the “linearized” structure are substantially lower than those of the linear model. The method of linearized analysis was chosen for use in this study. 3.2.4 Torsional constant In an actual arch bridge construction, the two ribs are often braced together between the top flanges and the bottom ones. The two ribs and bracings would act as a single box section with a torsional stiffness substantially larger than the sum of the 23 torsional stiffness of the individual rib sections. To account for this in the analysis model, the torsional parameter AK‘I’I‘ is introduced. The parameter AK'I'I‘ is the ratio of Kt to Ktr’ where Kt is the torsional constants to be used for one rib, and Kt r is the torsional constant of the box section of one rib acting alone. With an increase in AK'IT, the structure becomes stiffer in out-of-plane response, while the in-plane stifl‘ness would remain unchanged. In Table 3.2-6 (with A s = 0.8 ftz, A d = 0.08 f t2, no rotational restraint at supports, linearized analysis method, MCSCB), it is shown that AK'I'I‘ does not affect the in-plane fundamental natural frequency but increase the out-of-plane fundamental frequency. However, the effect is quite small. Similarly, increasing the value of AK'I'I‘ does not significantly influence the dynamic stresses as illustrated in Table 3.2-7. Therefore, AK'IT equals 2.0 was used for the subsequent studies. 3.3 W The vibration mode shapes of the first four modes for out-of-plane motion of the bridge are obtained for MCSCB and shown in Fig. 3-1, 3-2, 3-3 and 3—4. These four modes corresponding to 2nd, 4th, 5th and 7th overall mode number. For odd number out- of-plane motions (Fig. 3-1 and 3-3), the displacements of deck and ribs in the y direction are symmetric with respect to the crown plane of the bridge. Even number ones (Fig. 3-2 and 3-4) are anti-symmetric to the crown plane. 3.4 WWW The maximum stresses in MCSCB at the various panel points under static vertical dead load, static lateral wind load, lateral seismic loading (Z—motion), two dimensional in- plane seismic loading (X, Y motion) and three dimensional seismic loading (X, Y, Z motion) are plotted in Fig. 3-5. Due to the symmetry respect to the crown (Y—Z plane), 24 results are presented for one half of the bridge. The dynamic response of the bridge by using the response spectrum method is symmetric. It is not so if time history analysis is used [9] For combination of seismic loading, the amplification factors 1.0, 0.75, and 1.0 were applied to the ground acceleration in the X, Y and 2 direction, respectively. For the different combinations of seismic loading, stresses were initially calculated under individual ground motion, then the total stresses were calculated based on those components as the square root of sum of squares. As can be seen from the figure the stresses due to in-plane (X and Y direction) ground motion are the main components of the total stresses due to the three dimensional ground motion. It is interesting to note that the seismic stresses from the lateral seismic loading have similar distribution as the stresses from the static wind loading. When the lateral seismic acceleration S A is given by Equations (4) and (5), and the wind load corresponds to 0.1g, the values of the ratio of the seismic stress to the wind load stress at the various points are in the range of 2.4 to 2.9. This is also true for individual member forces. In other words, the behavior of the structure under wind load can be related to the one under seismic loading. If one likes to estimate the lateral seismic structure behavior, it is feasible to use a static lateral loading and apply a certain factor. The former seems much easier to deal with. 3.5 Effggt Qf Cruss-secu'unal Areu uf Column Diugunul Braging A d In deck-arch bridge construction, column diagonal bracing has been used in some cases not in others. The role of the bracing is to help to tie the deck and the ribs together so as to act more as a unit in resisting lateral loads. Ultimately all lateral loads are carried to the foundation through the end towers and through the arch rib supports. The net effects of Ad on the structural response (i.e. stresses and displacements) are not clear. In this section, data on the responses with and without the bracings are presented. 25 Comparisons are made among the following cases: 1. no column bracing 2. column bracing at 1/4 points of the bridge with area of A d equals to 0.38 percent of the rib area Ar 3. column bracing at each panel with A d equals to 0.38 percent of A r 4. column bracing at each panel with A (1 equals to 1.5 percent of Ar 5. column bracing at each panel with A d equals to 3 percent of A r 6. column bracing at each panel with A d equals to 9 percent of A r . Fundamental in-plane and out-of-plane frequencies of MCSCB due to various A d values are illustrated in Fig. 3-6. As can be expected that, statically, the efl'ect of the bracings is to increase the out-of-plane stiffness; and dynamically, the fundamental out-of- plane natural frequency is increased. The various cases with column diagonal bracings virtually have no effect on bridge in-plane frequencies. The behavior of the structure under in-plane loading is not affected by varying A d . Fig. 3-7 shows the maximum stress under lateral seismic load (Z-motion only) for MCSCB. It corresponds to those at the left end node of each curved beam member between the panel points of the rib. With column bracing, the stress at the crown decreased dramatically. The values of A d ( f t2) used in the plot, presented as percentage of the rib area, are as follows, MCSCB MSSB 0.38% A r 0.01 0.0036 1.5% Ar 0.04 0.0144 3% A r 0.08 0.0288 9% A r 0.24 0.0864 26 Similar comparisons are p10tted in Fig. 3-8 for MSSB. The stress responses have the same pattern as for the MCSCB except that stresses at the 1/81h points did not decreases as drastically. It can be seen that the case of A d equal to 0.38 percent rib area is most eflicient in terms of decreasing rib member forces and stresses, especially at the crown node. In the case of no bracing, the maximum moment occurs at the crown. The column diagonal bracings reduce the moment at the crown by providing more points of lateral load transfer (in addition to the connection at the crown). The load redistribution is enhanced by increasing stiffness of column diagonal bracing. The results obtained also indicated that column diagonal bracings can effectively reduce the dynamic stresses with relatively small members. It seems unnecessary and not economical to use too strong or heavy members. It is noticed that the effects of A d leveled off after A (I reached approximately 3 percent of A r . Therefore, an area of three percent of rib area A r seemed desirable and was used in later studies. That is, A d = 0.08 ftz for MCSCB; and Ad =0.03 £12 for MSSB. 3.6 W The ribs of steel arch bridges often have cross-sections with a box shape (Fig. 2-2). Even assuming double symmetry, there are still a number of parameters that define the proportion of the cross-section. A key parameter is the depth to width ratio. The purpose of this section is to investigate its effect on the response of the bridge. In Section 2.8, it was shown that the dimensions of a box section can be determined from the four parameters: the cross-section area Ar , the web thickness to depth ratio tw/ D , flange to web thickness ratio tf/tW and the depth to the width ratio D/B. Both the MCSCB and the MSSB were used in this study. For each bridge, A r , tw/ D , tf/tW were held constant while D/B were varied from 1.0 to 5.0 for MCSCB, and 1.0 to 4.0 for MSSB respectively. 27 3.6.1 Sectional properties and natural frequencies In order to aid the interpretation of the data to be presented, the various sectional properties of the cross-section, such as the section moduli, are presented in Table 3.6-1 for MCSCB and Table 3.62 for MSSB. In general, with an increase in the value of DB (as the section narrows) there is large decrease in Ixr (moment of inertia for out-of-plane bending) and a relatively modest increase in Iyr (moment of inertia for in-plane bending). The effects of D/B on the in-plane and out-of-plane natural frequencies of MCSCB are illustrated in Fig. 3-9.It is seen that with an increase in D/B ratio, there is a slight decrease in the out-of-plane frequency, although the increase in Ix r is large. The reason is thought to lie in the fact that the major source of the lateral stiffness of the bridge system still comes from the cross-sectional areas of the rib and the deck stringers and the “local” nature of Ix r has only a secondary effect. An increase in the D/B values resulted in increases in the in-plane frequency almost in proportion to the increase in Iy r which is the main source of the in-planc stiffness. 3.6.2 Effect on seismic responses x , My and the combined stress at the rib support due to lateral (Z) seismic motion are plotted in Fig. 3- The effects of varying D/B on the stress resultants: Pz , M 10. There is very little effect of D/B on P2 at supports. The local moments Mx . and My at supports are null. The total stress is caused by the axial force.Pz Thus, varying D/B ratio does not affect the stress 0 at the support. Member forces Pz , M , M and stress 0 at the left end node of panel 2 are X Y shown in Fig. 3-11. Again, P2 is not affected by D/B. At this node or the 1/8th point, My decreases about 26 percent when D/B increase from 1 to 3, and levels off approximately at D/B=3. The Mx distribution has a convex shape with the maximum occurring at D/B = 2. 28 Table 3.6—1 shows that an increase in D/B results in a drastic decrease in Ix r and significant decrease in the section modulus Sx while the changes in Sy are small. However, with an increase in D/B, the total dynamic stress did not increase as much as one might first think. This is because of the decrease in both Mx and My for D/B > 2.0, as illustrated in Fig. 3-11. The reader is reminded that these results were obtained with the cross-sectional area of the rib kept constant. Dynamic rib member forces for the left end of members at the other panel points are shown in Fig. 3- 12, 3-13 and 3-14 respectively. As previously, the total stress does nor increase significantly. At the crown node, Mx decreases from 3245 ft.-k to 655 ft.-k, My increases from 474 to 1647 ft.-k, and P2 decrease from 548 kips to 395 kips. Consequently, the total dynamic stress decreases from 6.9 ksi to 5.1 ksi. In Fig. 3-15 are shown the dynamic stresses due to the lateral ground motion as functions of the panel points with D/B as parameters. It is not surprising to note that the case D/B = 1 seems to yield the best “design”. For D/B = 1, the stresses are generally lower except at the crown. But note that this stress is lower than that at panel point 2. The general nature of the results is expected because the load is in the lateral direction and a smaller D/B indicates a larger lateral stiffness. In Fig. 3-16 are plotted, for in-plane seismic excitations (X and Y direction), similar data to those in Fig. 3-15. Here also the design D/B = 1.0 yields the lowest response (except at the support where the stresses are relatively small and the differences minor). This is not expected. The reasons may be (1) the in-plane section modulus Sy is not sensitive to DIR (Table 3.6-1) and (2) smaller D/B lowers the in-plane natural frequencies and also the acceleration response spectrum values (eq. (4) and (5)). In Fig. 3-17 are shown similar plots to Fig. 3-16 for the case of seismic inputs in all three dimensions. Similar observations to those regarding Fig. 3-16 can be made. In Fig. 3-18 and 3-19 are shown, for MSSB, the dynamic stresses due to lateral ground motion only and to all three dimensional ground motions, respectively. In these cases, the trends are less clear. However, larger values of D/B (3 or 4) are seen to be undesirable. The value of 1.65 (the actual value for SSB) seems the best. .29 3.6.3 Effect on displacements and tower forces The effects of D/B on maximum displacements under lateral seismic motion are shown in Table 3.6-3. The largest vertical displacement scaled by the span length, Uy, occurred at the crown. It is seen that there is a moderate increase in Uy with an increase in DB. It is believed that such an increase was due to the larger lateral torsional motion of the bridge resulting from a decrease in the lateral stiffness. The lateral displacement, scaled by span length, Uz, at the crown is an order of magnitude larger than Uy. As expected, it increased with increase in the D/B ratio. The tower lateral force, F 2 , represents the maximum horizontal reaction transferred at the support of an end tower. The effects of D/B on the tower lateral force are shown in Table 3.6-4. It is seen that they are relatively miner, of the order of one to two percent. 3.6.4 Effect on responses to dead load and wind load The effects of D/B on the MCSCB rib internal forces due to dead load are illustrated in Figs. 3-20 to 3-23 for the various panel points. It is seen that, in general, such effects are quite small, even the in-plane bending moment, My’ is seen to increase rather mildly with D/B. The total stress is practically independent of the D/B ratio. The effects of D/B on the MCSCB rib internal forces due to (statically applied) wind load are illustrated in Figs 3-24 to 3-27. for the various panel points. The effects are seen to be more significant than those due to dead load. The considerable increase in out- of-plane bending, M , as D/B increases from 1.0 to 2.0, is particularly noteworthy. For x larger values of D/B, the increase in Mx leveled off. The out-of—plane bending, My , however, decreased, and the axial force P2 was not sensitive to the changes in D/B. The net effect on the maximum stress due to these stress resultants is a significant increase (of the order of 30%) from D/B = 1.0 to D/B = 2.0. For larger values of D/B the increase levelled off. 30 dance 05:95 38:20 L: can .532: 0:39.350 885.0 :0: a. 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I m . t m . l m NHMCNI. < NCWCI < NHMVCI < 808.5 moaoba .05 aces 2565 a: :6 a< co 28:: .93 are. .5585 0:595 88:0: :5 :§ .5585 05298.80 88:0: :0: a. 33 5V 3:3; 5v newmmd 5v 3:36 m 8 202.6 5 2:25 3 Sound : 8: 588 6: $89: 85 22:: m 5v 3036 e who—m6 e muffle N 5 mhflmd av 59%.: 4on Sound _ «a QNu m< a: wdn m< N: You m< 8: 0:2: A83 ”08:030.: 3.88: :o m< h8 $00.5: .v-~.m 03m... 34 Table 3.2-5. Natural frequences (cps) of MCSCB from different analysis Mode N o. Linearized Linear 1 0.27101 0.31673 1* 2 0.31028 0.32247 0 3 0.72896 0.77617 I 4 0.78125 0.79034 0 5 1.25722 1.26190 0 6 1.39736 1.44537 1 7 1.80682 1.81774 0 8 1.82029 1.85972 0 9 1.89594 1.93431 0 10 2.05909 2.09128 I 1 1 2.43652 2.45350 I 12 2.45858 2.49048 0 13 2.65667 2.66017 0 14 3.01787 3.05821 I 15 3.03041 3.08334 0 16 3.43868 3.50449 0 17 3.86869 3.90187 0 18 3.90001 3.93363 I 19 4.23250 4.23362 I 20 4.34983 4.37691 0 at: "1": denotes in-plane (X-Y plane) and "0": denotes out-of-plane (Y-Z plane). 35 Table 3.2-6. Fundamental Frequencies (cps) of MCSCB for different AK'I'I‘ values AKTT=2 AKTI‘=3 AKTT=4 AKTT=6 in-plane 0.27101 0.27 101 0.27101 0.27101 out—of-plane 0.31028 0.31 143 0.31223 0.31325 Table 3.2-7. 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Displacements of MCSCB for different D/B ratios Uz( x104) Uy( x104) D/B tower top l/4pt crown 1/8 pt 1/4 pt. crown 1 0.0412 0.224 0.329 0.323 0.831 2.06 2 0.041 1 0.225 0.331 0.298 0.792 2.10 3 0.0414 0.229 0.335 0.297 0.764 2.21 4 0.0417 0.232 0.340 0.306 0.741 2.32 5 0.0420 0.235 0.344 0.318 0.722 2.42 Table 3.6—4. Tower lateral force (kips) of MCSCB for different D/B ratios D/B F2 1 308.78 2 308.82 3 307.26 4 305.52 5 303.93 39 A8282. 268 :80; 6:3 0927. 308 5396-80 ii at .66.. xomo .62 a: :3: x x \ (Ix. xoob “to: x 9\OIJ, ‘KOIIJC x .2 23mm a: 60.. 26 “co: x xomn 6m: 96 t6: 40 .9982. £68 =Eo>o 53 093m 0608 £396.50 968m at 68 xoon .60.. x a: to: xomn t6: x x .3 26E at 6m: 26 t6: xomo .68 26 “co: 41 .CBES. £68 =9B>o 50 0923 £68 :39.~o-.=o “SE. n... .60.. xoov .66.. x a. .8... xoob t6: ‘1; .KJ \ / x /i \ y I 4 /.9\ x x aaeam at .66.. 96 “co... xooo .66. 6:6 “cot 42 .3982. owe:— =fio>o ED 23% once £296.26 5:6”. 9.. .66. 9.. 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Stress and member forces at the left node of panel 2 due to varying D/B ratio (MCSCB), Z-motion seismic loading. Total Stresses 0 of panel 2 (ksi) 50 3000 14 9—0 Pz G-D Mx ” A—A My A 0—0 a "12 t] . x - 8“- ~10 '2 2000~ F v "2 —8 (D C . . O I’D-~‘\ O. ’ ‘ / Us ‘4— U, \\\ b6 0 \‘EL‘ g; 1000i “ .0 - -4 x 2 - IA . N . ”Ian” D. A—-———&-—"‘""'¢” I--2 31W - O I I I I I 0 0 1 2 3 4 5 6 D/ B Figure 3- 12. Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), Z—motion seismic loading. Total Stresses a of panel 3 (ksi) 51 3000 14 0—0 P2 G-CJ Mx * A—A My 1? . 0—0 a ’12 T x . m“ —10 E? 20004 _ <1~ ~8 ‘03 c . . o o. q.— "'6 o 5; 1000— ' ~ —4 X 2 I- n? . O. -2 r- O l l I l l O O ‘l 2 3 4 5 6 Figure 3- 13. Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), Z-motion seismic loading. Total Stresses o of panel 4 (ksi) 52 4000 , 14 9—0 Pz G-D Mx - . 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Total Stresses a of panel 2 (ksi) P2, Mx, My of panel 3 (kips, k—ft) 59 3500 15 .. —-————a-—--6 3000— ,«”*”' -* r14 r ~12 2500— G $ 4 __3 a .. ~10 °‘\. .. ¢ a: 20004 ' ' _ be 1500— L —6 1000~ ' —4 500““ ' H 0' -2 A—A My D'L' B-Cl Mx - o—e P2 0 ?* *2?” 2%? 4v 0 0 1 2 3 It 5 6 [L/B Figure 3 - 21. Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), dead load. Total Stresses 0 of panel 3 (ksi) Pz, Mx, My of panel 4 (kips, k—ft) 4500 14 ‘ ___.._——¢—-'—"-1 :— _-A h 5 ~12 \ e 3 ’4" -10 30001 . . —8 % e e 4* 4: _ ‘ ~6 1500- ‘ -4' 0—0 a' -2 ‘ A—A My Di“ Bun Mx ‘ e—e P2 0 1r .7. er 4 + 0 0 1 2 3 ‘1 5 6 D/B Figure 3 - 22. Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), dead load. Total Stresses a of panel 4 (ksi) P2, Mx, My of panel 5 (kips, k—ft) 61 5000 14 k”___a_“__.*m——uahu——m¢ _12 4000- _ \r e e 4'9 - ~10 3000— —8 G G 3 C o ~6 2000* ‘ ~4 10004 ’ 0—0 a -2 ~ a—a My Di" awn Mx ” are P2 0 411‘ 11‘ 1r 0 0 1 2 3 ‘4 5 6 D/B Figure 3 - 23. Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), dead load. Total Stresses a of panel 5 (ksi) P2, Mx, My of panel 2 (kips, k—ft) 62 04 . ~400— @— 3 # fie ~A) —800— - .ma—fl-m'“ // ’— * WL. /" // r —1200 I l I 1 l O 1 2 3 4 5 6 D/B Figure 3 - 24. Stress and member forces at the left node of panel 2 due to varying D/B ratio (MCSCB), wind load. Total Stresses a of panel 2 (ksi) Pz, Mx, My of panel 3 (kips, k—ft) 63 Total Stresses a of panel 3 (ksi) 1000 0—0 P2 B-U Mx A—A My 0—0 a 600— a—--- ,’ ‘3.~ 1” “fl“ ‘ a “‘a 200— . ""r” .——-"*.’ M —200- —600— W1. -1000 l 1 1 r r O 1 2 3 4 5 D/B Figure 3 - 25. Stress and member forces at the left node of panel 3 due to varying D/B ratio (MCSCB), wind load. P2, Mx, My of panel 4 (kips, k—ft) 600 6 (9—0 P2 B-CJ Mx A—A My H a . —4 200— ’--°----e~--- - ‘0 -2 —200 :3r 0 0 5 6 Figure 3 - 26. Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), wind load. Total Stresses a of panel 4 (ksi) P2, Mx, My of panel 5 (kips, k—ft) 65 400 6 9—0 Pz 134:1 Mx . b—A M 0—0 GPO—\N . 0-4 A.\\ \ X\ .4 \ '* \ \\ \ I _400... I’l’si 1- [If 6\;{7\.\.\‘ P2 I I —800- X ’l ’ I- I I '1 I I W.L. ,1 I “-1200 4% T I I I O O 1 2 3 4 5 6 Figure 3 - 27. Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), wind load. Total Stresses a of panel 5 (ksi) 66 CHAPTER IV SUMMARY AND CONCLUSION 4.IS_ummaaI The main purpose of this study is to provide information that could aid the designers of arch bridges in their decision regarding the two parameters: 1. column diagonal bracing and 2. the depth to width ratio of the rib box section. The designer basis is presumed to be elastic. Hence the chief measure of response is the maximum stress in the rib which is a combination of the effects of the axial force and the in-plane and out-of- bending moments. The study was based on a computer modelling of the bridge and load system. For the bridge, nonlinear elastic curved beam elements were used for the ribs (see Fig. 1-2), and straight beam elements were used for the cross bars between the ribs and the stringers of the deck. Truss elements were used for the cross-bracings between the ribs, for the column between the deck and the ribs, and for the deck system. The seismic load was represented by the design response spectrum of AASH'PO. For better perspectives, dead load and wind load were also considered. The dead load analysis was based on a nonlinear elastic analysis, the main feature of which was the consideration of the effect of the compression due to dead load on the stiffness of the ribs. That stiffness at the end of the dead load application was used as the linear stiffness for a linear dynamic analysis of the structure by use of the response spectrum and method of superposition. The CQC method of modal responses combination was adopted employing twenty normal modes. Because of the large number of parameters involved, two real bridges were used for the study: the Cold Spring Canyon Bridge (CSCB) in California which has a span 67 length of 700 ft. and is relatively slender, and the South Street Bridge (SSB) in Connecticut which has a span of 193 ft and is relatively stiff laterally. For the bridge models, a number of simplifications were made such as the number of panels and the exact geometry of the rib as well as the representation of the deck system. Hence they were referred to as modified versions or MCSCB and MSSB. The bulk of the data obtained pertains to MCSCB. Before collecting the data for the major parameters for the study, a preliminary parametric study was made. It resulted in the decision to release the r0tational constraints at the supports of the ribs, the choice of the deck stringer cross-sectional area and a multiplier for the torsional constant of the ribs. For the column diagonal bracing, the cross-sectional area Ad was varied from 0.38 percent to 9 percent of that of the rib, and for the lower limit, bracing at the quarter point only was also considered. It was found that such lateral bracing was effective in reducing the maximum stress and the most effective schemes seems to use the smallest area, i.e., 0.38 percent, but use them at all panels. The design parameter of the rib section depth to width D/B was varied from 1.0 to 5.0 with a fixed area of the rib cross-section and depth and web thickness ratio. For the MCSCB, it seems that the ratio of 1.0 is most effective. For the MSSB, the ratio of 1.65 seems to give the best results. Contrary to a general tendency in practice that such ratio would increase with longer spans, the results seem to indicate that the ratio need not go beyond two. The reason seems to lie in the fact that larger values of D/B do not increase the value of the section modulus for in-plane bending. That is, it does not strengthen the Structure in the vertical direction while weakens it in the lateral direction. For the same reason, an variation of D/B did not have a significant effect on the maximum stress due to dead load. Its effect on the wind load response is larger. The response to statically applied wind load is noted to be quite similar to the dynamic response to a lateral seismic load (horizontal motion normal to the bridge longitudinal axis). For the magnitude of the loading considered herein, the ratio between the stresses for the lateral seismic load and the wind load falls within the range of 2.4 to 2.9. This is 68 also true for member forces. Thus for consideration of the lateral seismic loading, a reasonably good estimation may be obtained from a static wind load analysis. 4.2 n1 in R marks The design parameters: the size of the cross-sectional area of the column diagonal bracing, and the depth to width ratio of the rib section, are studied for an effective seismic design of the deck type of arch bridges. Responses to dead load and wind load are also considered. The results provide considerable insight into the behavior of such bridges and guidance to their seismic design. Because of the large number of parameters involved in the system, it is infeasible to produce general formulas or even tables or charts as design aids. However, the information presented here should be useful in providing guidance for an initial design. A final design still need be done with the aid of a computer program using a more precise modelling of the structure proposed. For an elastic design, it is appropriate to use the tangent stiffness of ribs as the stiffness for a linear dynamic analysis. LIST OF REFERENCES 69 LIST OF REFERENCES . Bellamine, A. 1992. Seismic response of tied arch bridges. MS. Thesis, Department of Civil Engineering, Michigan State University, E. Lansing, MI, .1992. . Clough, R. W. and J. Penzien, Dynamics of Structure. McGraw-Hill Book Company, New York, 1975. . Dusseau, R. A. and R. K. Wen, “Seismic Responses of Deck-Type Arch Bridges”, Earthquake Engineering and Structural Dynamics, Vol.18, pp.701-715, 1989. . Garbow, B.S. “Subroutine RSG”. Mathematics and Computer Science Div., Argonne National Laboratory, 1983. . Lange, J ., “Elastic Buckling of Arches by Finite Element Method”, Ph.D. Dissertation, Department of Civil Engineering, Michigan State University, E. Lansing, MI, 1980. . Lee, C. M., “Nonlinear Seismic Analysis of Steel Arch Bridges”, Ph.D. Dissertation, Department of Civil Engineering, Michigan State University, E. Lansing, MI. 1990. . Lee, H. E. and M. A. M. Torkamani, 1989. Dynamic response of tied arch bridges to earthquake excitations, Department of Civil Engineering, University of Pittsburgh. ‘ Pittsburgh, Pennsylvania. . Merrit, F. 8. Editor, Structural steel designers’ handbook. Section 13, McGraw-Hill Book Company, New York, 1972. . Millies, R. 1., “Three Dimensional Elastic Seismic Response of Deck Type Arch Bridges”, M.S.Thesis, Department of Civil Engineering, Michigan State University, E.Lansing, MI 1992. 10. Raithel, A., and Franciosi.C., “Dynamic Response of Arches Using Lagrangian Approach”, Journal of Structural Engineering, April 1984, pp. 847-858. 70 LIST OF REFERENCES(continued) 11. Standard Specifications For Seismic Design of Highway Bridges, 1983. American Association of State Highway and Transportation Officials. Washington, DC. 12. U.S. Department of Transportation, “Arch Bridges”, Series No.2, Washington DC, 1977. 13. Wen, R. K., and B. Suhendro “Nonlinear Curved-Beam Element for Arch Structures”, Journal of Su'uctural Engineering, ASCE, Vol. 117, No.11, Nov. 1991, pp. 3496-3515. 14. Wen, R. K., “Seismic Behavior and Design of Arch Bridges”, Proceedings of the 4th U.S. National Conference On Earthquake Engineering, Palm Springs, California, May 1990, Vol.1. 91’. 1027-1036. 15. Wen, R. 'K., C. M. Lee and Alahamd, “Incremental Resistance and Deformations of Elasto-plastic Beams, Journal of Structural Engineering, ASCE, Vol. 115, No.5, Nov. 1989. PP. 1267-1271. Pz, Mx, My of panel 4 (kips, k—ft) Total Stresses a of panel 4 (ksi) 600 9—0 P2 G-U Mx A—A My o—o a 2004 ,I“G‘---e--‘ ‘~t:1 -200 ‘Vf. 0 5 Figure 3 - 26. Stress and member forces at the left node of panel 4 due to varying D/B ratio (MCSCB), wind load. P2, Mx, My of panel 5 (kips, k—ft) 65 400 6 0—0 Pz G-Cl Mx q A—A M H (fl/N _ 0~ A.\ \ \‘A \ ~4 \ ‘ \ \\ ’4: \ ” _400_l ,’/\E\\\ . It \A N __2 I I -800‘ I ’I ’ p I I ., I I w.L.; I -1200 41L r l l l 0 O 1 2 3 4 5 6 Figure 3 - 27. Stress and member forces at the left node of panel 5 due to varying D/B ratio (MCSCB), wind load. Total Stresses a of panel 5 (ksi) 66 CHAPTER IV SUMMARY AND CONCLUSION “Summary The main purpose of this study is to provide information that could aid the designers of arch bridges in their decision regarding the two parameters: 1. column diagonal bracing and 2. the depth to width ratio of the rib box section. The designer basis is presumed to be elastic. Hence the chief measure of response is the maximum stress in the rib which is a combination of the effects of the axial force and the in-plane and out-of- bending moments. The study was based on a computer modelling of the bridge and load system. For the bridge, nonlinear elastic curved beam elements were used for the ribs (see Fig. 1-2), and straight beam elements were used for the cross bars between the ribs and the stringers of the deck. Truss elements were used for the cross-bracings between the ribs, for the column between the deck and the ribs, and for the deck system. The seismic load was represented by the design response spectrum of AASHTO. For better perspectives, dead load and wind load were also considered. The dead load analysis was based on a nonlinear elastic analysis, the main feature of which was the consideration of the effect of the compression due to dead load on the stiffness of the ribs. That stiffness at the end of the dead load application was used as the linear stiffness for a linear dynamic analysis of the structure by use of the response spectrum and method of superposition. The CQC method of modal responses combination was adopted employing twenty normal modes. Because of the large number of parameters involved, two real bridges were used for the study: the Cold Spring Canyon Bridge (CSCB) in California which has a span 67 length of 700 ft. and is relatively slender, and the South Street Bridge (SSB) in Connecticut which has a span of 193 ft. and is relatively stiff laterally. For the bridge models, a number of simplifications were made such as the number of panels and the exact geometry of the rib as well as the representation of the deck system. Hence they were referred to as modified versions or MCSCB and MSSB. The bulk of the data obtained pertains to MCSCB. Before collecting the data for the major parameters for the study, a preliminary parametric study was made. It resulted in the decision to release the rotational constraints at the supports of the ribs, the choice of the deck stringer cross-sectional area and a multiplier for the torsional constant of the ribs. For the column diagonal bracing, the cross-sectional area Ad was varied from 0.38 percent to 9 percent of that of the rib, and for the lower limit, bracing at the quarter point only was also considered. It was found that such lateral bracing was efi‘ective in reducing the maximum stress and the most effective schemes seems to use the smallest area, i.e., 0.38 percent, but use them at all panels. The design parameter of the rib section depth to width D/B was varied from 1.0 to 5.0 with a fixed area of the rib cross-section and depth and web thickness ratio. For the MCSCB, it seems that the ratio of 1.0 is most effective. For the MSSB, the ratio of 1.65 seems to give the best results. Contrary to a general tendency in practice that such ratio would increase with longer spans, the results seem to indicate that the ratio need not go beyond two. The reason seems to lie in the fact that larger values of D/B do not increase the value of the section modulus for in-plane bending. That is, it does not strengthen the structure in the vertical direction while weakens it in the lateral direction. For the same reason, an variation of D/B did not have a significant effect on the maximum stress due to dead load. Its effect on the wind load response is larger. The response to statically applied wind load is noted to be quite similar to the dynamic response to a lateral seismic load (horizontal motion normal to the bridge longitudinal axis). For the magnitude of the loading considered herein, the ratio between the stresses for the lateral seismic load and the wind load falls within the range of 2.4 to 2.9. This is 68 also true for member forces. Thus for consideration of the lateral seismic loading, a reasonably good estimation may be obtained from a static wind load analysis. 4.2 anclgding Remarks The design parameters: the size of the cross-sectional area of the column diagonal bracing, and the depth to width ratio of the rib section, are studied for an efl‘ective seismic design of the deck type of arch bridges. Responses to dead load and wind load are also considered. The results provide considerable insight into the behavior of such bridges and guidance to their seismic design. Because of the large number of parameters involved in the system, it is infeasible to produce general formulas or even tables or charts as design aids. However, the information presented here should be useful in providing guidance for an initial design. A final design still need be done with the aid of a computer program using a more precise modelling of the structure proposed. For an elastic design, it is appropriate to use the tangent stiffness of ribs as the stiffness for a linear dynamic analysis. LIST OF REFERENCES 69 LIST OF REFERENCES . Bellamine, A. 1992. Seismic response of tied arch bridges. MS. Thesis, Department of Civil Engineering, Michigan State University, E. Lansing, MI, .1992. . Clough, R. W. and J. Penzien, Dynamics of Structure. McGraw-Hill Book Company, New York, 1975. . Dusseau, R. A. and R. K. Wen, “Seismic Responses of Deck-Type Arch Bridges”, Earthquake Engineering and Structural Dynamics, Vol.18, pp.701-715, 1989. . Garbow, B.S. “Subroutine RSG”. Mathematics and Computer Science Div., Argonne National Laboratory, 1983. . Lange, J ., “Elastic Buckling of Arches by Finite Element Method”, Ph.D. Dissertation, Department of Civil Engineering, Michigan State University, E. Lansing, MI, 1980. . Lee, C. M., “Nonlinear Seismic Analysis of Steel Arch Bridges”, Ph.D. Dissertation, Department of Civil Engineering, Michigan State University, E. Lansing, MI. 1990. . Lee, H. E. and M. A. M. Torkamani, 1989. Dynamic response of tied arch bridges to earthquake excitations, Department of Civil Engineering, University of Pittsburgh. ' Pittsburgh, Pennsylvania. . Menit, F. 8. Editor, Structural steel designers’ handbook. Section 13, McGraw-Hill Book Company, New York, 1972. . Millies, R. J ., “Three Dimensional Elastic Seismic Response of Deck 'Iype Arch Bridges”, M.S.Thesis, Department of Civil Engineering, Michigan State University, E.Lansing, MI 1992. 10. Raithel, A., and Franciosi.C., “Dynamic Response of Arches Using Lagrangian Approach”, Journal of Structural Engineering, April 1984, pp. 847-858. 70 LIST OF REFERENCES(continued) 11. Standard Specifications For Seismic Design of Highway Bridges, 1983. American Association of State Highway and Transportation Officials. Washington, DC. 12. U.S. Department of Transportation, “Arch Bridges”, Series No.2, Washington DC, 1977. 13. Wen, R. K., and B. Suhendro “Nonlinear Curved-Beam Element for Arch Structures”, Journal of Structural Engineering, ASCE, Vol. 117, No.11, Nov. 1991, pp. 3496-3515. 14. Wen, R. K., “Seismic Behavior and Design of Arch Bridges”, Proceedings of the 4th U.S. National Conference On Earthquake Engineering, Palm Springs, California, May 1990, Vol.1. PP. 1027-1036. 15. Wen, R. K., C. M. Lee and Alahamd, “Incremental Resistance and Deformations of Elasto-plastic Beams, Journal of Structural Engineering, ASCE, Vol. 115, No.5, Nov. 1989. PP. 1267-1271. nICHIGnN STATE UNIV. LIBRRRIES llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 31293008803664