Response of cable stayed and susspension bridge to moviing vehicles

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v = 110 km/h 335 m 146 m 25 15 -5 the truck leaves the bridge Mid-point vertical displacement (mm) - 146 m -15 -25 -35 -45 with tuned mass damper (TMD) without tuned mass damper (TMD) -55 10 20 30 40 Time (s) Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Royal Institute of Technology Department of Structural Engineering TRITA-BKN Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B 44 SE Doctoral Thesis Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden Akademisk avhandling Som med tillstånd av Kungl Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari 1999 kl 10.00 i Kollegiesalen, Valhallavägen 79, Stockholm Avhandlingen försvaras på svenska Fakultetsopponent: Huvudhandledare: Docent Sven Ohlsson Professor Håkan Sundquist TRITA-BKN Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B 44 SE Stockholm 1999 Response of Cable-Stayed and Suspension Bridges to Moving Vehicles Analysis methods and practical modeling techniques Raid Karoumi Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden _ TRITA-BKN Bulletin 44, 1998 ISSN 1103-4270 ISRN KTH/BKN/B 44 SE Doctoral Thesis To my wife, Lena, to my daughter and son, Maria and Marcus, and to my parents, Faiza and Sabah Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12 februari 1999  Raid Karoumi 1999 KTH, TS- Tryck & Kopiering, Stockholm 1999 Abstract This thesis presents a state-of-the-art-review and two different approaches for solving the moving load problem of cable-stayed and suspension bridges The first approach uses a simplified analysis method to study the dynamic response of simple cable-stayed bridge models The bridge is idealized as a Bernoulli-Euler beam on elastic supports with varying support stiffness To solve the equation of motion of the bridge, the finite difference method and the mode superposition technique are used The second approach is based on the nonlinear finite element method and is used to study the response of more realistic cable-stayed and suspension bridge models considering exact cable behavior and nonlinear geometric effects The cables are modeled using a two-node catenary cable element derived using “exact” analytical expressions for the elastic catenary Two methods for evaluating the dynamic response are presented The first for evaluating the linear traffic load response using the mode superposition technique and the deformed dead load tangent stiffness matrix, and the second for the nonlinear traffic load response using the Newton-Newmark algorithm The implemented programs have been verified by comparing analysis results with those found in the literature and with results obtained using a commercial finite element code Several numerical examples are presented including one for the Great Belt suspension bridge in Denmark Parametric studies have been conducted to investigate the effect of, among others, bridge damping, bridge-vehicle interaction, cables vibration, road surface roughness, vehicle speed, and tuned mass dampers From the numerical study, it was concluded that road surface roughness has great influence on the dynamic response and should always be considered It was also found that utilizing the dead load tangent stiffness matrix, linear dynamic traffic load analysis give sufficiently accurate results from the engineering point of view Key words: cable-stayed bridge, suspension bridge, Great Belt suspension bridge, bridge, moving loads, traffic-induced vibrations, bridge-vehicle interaction, dynamic analysis, cable element, finite element analysis, finite difference method, tuned mass damper –i– – ii – Preface The research presented in this thesis was carried out at the Department of Structural Engineering, Structural Design and Bridges group, at the Royal Institute of Technology (KTH) in Stockholm The project has been financed by KTH and the Axel and Margaret Ax:son Johnson Foundation The work was conducted under the supervision of Professor Håkan Sundquist to whom I want to express my sincere appreciation and gratitude for his encouragement, valuable advice and for always having time for discussions I also wish to thank Dr Costin Pacoste for reviewing the manuscript of this report and providing valuable comments for improvement Finally, I would like to thank my wife Lena Karoumi, my daughter and son, and my parents for their love, understanding, support and encouragement Stockholm, January 1999 Raid Karoumi – iii – – iv – original configuration - convergence study number of increments Dynamic Amplification Factors (DAF) Vertical displacement at node 14 Vertical displacement at node 18 Horizontal displacement of node 43 Axial force in deck element 30 at node Shear force in element 52 at node 36 Axial force in element 52 at node 36 Bending moment in element 52 at node 36 Axial force in cable Axial force in cable 13 Absolute maximum vertical accel at node 14 (m/s2) Maximum normalized bridge-vehicle contact force 1000 500 1500 1000 linear dynamic - 30 modes all modes 1.186 1.179 1.207 1.186 1.243 1.234 1.270 1.243 1.086 1.065 1.057 1.065 1.404 1.393 1.391 1.394 1.212 1.174 1.164 1.160 1.270 1.268 1.284 1.268 1.180 1.166 1.222 1.159 1.316 1.294 1.284 1.284 1.157 1.133 1.125 1.133 0.125 0.123 0.160 0.125 1.049 1.039 1.036 1.039 Dynamic Amplification Factors (DAF) trucks Vertical displacement at node 14 Vertical displacement at node 18 Horizontal displacement of node 43 Axial force in deck element 30 at node Shear force in element 52 at node 36 Axial force in element 52 at node 36 Bending moment in element 52 at node 36 Axial force in cable Axial force in cable 13 Absolute maximum vertical accel at node 14 (m/s2) Maximum normalized bridge-vehicle contact force Table 10.6 elem/cable nonlinear 1.087 1.289 1.272 1.237 1.272 1.272 1.268 1.101 1.208 0.316 1.078 v = 50 v = 70 simply supported 1000 1500 500 nonlinear dynamic 1.208 1.187 1.180 1.240 1.235 1.230 1.094 1.060 1.053 1.374 1.390 1.387 1.205 1.154 1.149 1.279 1.301 1.275 1.210 1.152 1.150 1.240 1.275 1.268 1.164 1.135 1.126 0.158 0.140 0.124 1.049 1.039 1.036 v = 110 v = 130 = girder linear nonlin 1.131 1.130 1.119 1.112 1.250 1.247 1.360 1.364 1.222 1.087 1.270 1.290 1.431 1.442 1.240 1.238 1.232 1.234 0.121 0.136 1.041 1.041 = = trucks trucks moving linear nonlin 1.060 1.056 1.286 1.265 1.279 1.274 1.237 1.246 1.285 1.271 1.251 1.202 1.276 1.262 1.086 1.035 1.239 1.242 0.336 0.316 1.071 1.073 = = force model linear nonlin 1.048 1.048 1.171 1.162 1.202 1.199 1.205 1.214 1.198 1.181 1.229 1.173 1.207 1.188 1.358 1.345 1.211 1.213 0.275 0.266 1.000 1.000 bump TMD km/h km/h km/h km/h 0.01 0.015 0.02 0.03 30 mm v = 110 nonlin nonlin nonlin nonlin nonlin nonlin nonlin nonlin nonlin nonlin nonlin 1.041 1.132 1.282 1.204 1.220 1.164 1.143 1.124 1.093 1.188 1.245 1.057 1.003 1.427 1.060 1.285 1.202 1.170 1.143 1.102 1.233 1.378 1.178 1.086 1.131 1.205 1.106 1.032 1.010 0.992 0.984 1.059 1.065 1.112 1.176 1.414 1.427 1.401 1.381 1.372 1.362 1.329 1.601 1.413 1.195 1.100 1.331 1.230 1.211 1.127 1.102 1.097 1.087 1.172 1.266 1.189 1.241 1.511 1.461 1.395 1.258 1.222 1.193 1.168 1.329 1.463 1.178 1.104 1.323 1.244 1.239 1.123 1.103 1.097 1.087 1.211 1.260 1.024 1.101 1.206 1.408 1.300 1.258 1.241 1.226 1.202 1.331 1.205 1.082 1.008 1.104 1.205 1.180 1.107 1.082 1.063 1.038 1.138 1.083 0.067 1.024 0.091 1.042 0.145 1.059 0.202 1.060 0.195 1.039 0.118 1.039 0.099 1.039 0.086 1.039 0.075 1.040 0.141 1.446 0.137 1.059 Dynamic amplification factors (DAF), absolute maximum vertical acceleration at node 14, and maximum normalized bridge-vehicle contact force Note that even the linear dynamic analysis referred to in this table is based on the dead load tangent stiffness matrix obtained from a nonlinear static analysis Chapter Conclusions and Suggestions for Further Research 11.1 Conclusions of Part B The conclusions from the study conducted in Part B of this thesis are presented in the following two subsections In the first subsection, conclusions concerning the nonlinear finite element modeling of cable supported bridges are presented, and in the second subsection, conclusions are presented concerning the response due to moving vehicles 11.1.1 Nonlinear finite element modeling technique The present work has presented a method for modeling cable supported bridges for the nonlinear finite element analysis A two-node catenary cable element was adopted for modeling the cables and a beam element for modeling the girder and the pylons This study has shown that the adopted elements are accurate and efficient for nonlinear analysis of cable-stayed and suspension bridges It has been confirmed that the main advantages of the cable element are the simplicity of including the effect of pretension of the cable and the exact treatment of cable sag and cable weight Moreover, the iterative process adopted, to find the internal force vector and tangent stiffness matrix for the cable element, was found to converge very rapidly According to the author’s opinion, linear analysis utilizing the traditional equivalent modulus approach, is not satisfactory for modern cable-stayed bridges Modern cablestayed bridges built today or proposed for future bridges are, as they are highly flexible, subjected to large displacements The equivalent modulus approach however – 181 – accounts only for the sag effect but not for the stiffening effect due to large displacements [7] It was found that the catenary cable element is simple to formulate, accurate, and can correctly model the geometric change of the cable at any tension level This makes the element very attractive, especially for static response calculations, and the author strongly recommends the use of this element However, one drawback is when using commercial finite element codes for analysis, as only few commercial codes, e.g ABAQUS, enable the users to define their own elements This disadvantage applies also to the one bar element equivalent modulus approach It has been demonstrated that cable supported bridges have a hardening characteristic with respect to the applied load Furthermore, due to the highly nonlinear behavior during the static application of the dead load, a nonlinear static analysis is required to arrive at the deformed dead load tangent stiffness matrix Replacing each cable by several catenary cable elements has demonstrated that, in addition to obtaining new pure cable modes of vibration, cable motions are also associated with every bending mode of vibration To simplify the data input process when utilizing the multi-element cable discretization, one can start from a straight cable configuration and during analysis the cable configuration under its own weight is determined accurately after few iterations Finally, this work has only focused on two-dimensional modeling of cable supported bridges However, the catenary cable element used in this study is also applicable for modeling cables in other types of cable structures [35, 63, 64], such as: suspended roofs, guyed masts, electric transmission lines, moored floating bridges, etc Moreover, with some minor modifications of the cable element matrices this element can also be used for modeling cables for three-dimensional analysis For such analysis, threedimensional catenary cable and beam elements can be found in [35, 61] 11.1.2 Response due to moving vehicles An investigation was conducted to analyze the response of realistic two-dimensional cable-stayed and suspension bridge models under the action of moving vehicles For the analysis of the dynamic response, two approaches were implemented: one for evaluating the linear dynamic response and one for the nonlinear dynamic response Further, nonlinear geometric effects, “exact” cable behavior, and realistically – 182 – estimated bridge damping, were considered This investigation has mainly focused on comparing linear and nonlinear traffic load dynamic responses and also on the effect of bridge-vehicle interaction, road surface roughness, vehicle speed, bridge damping, cable modeling, and tuned vibration absorbers Based on this investigation of the traffic load response of cable-stayed and suspension bridges, the following conclusions can be made: • utilizing the tangent stiffness matrix (obtained from a nonlinear static analysis under dead load), linear static and linear dynamic traffic load analysis of cable supported bridges give sufficiently accurate results from the engineering point of view Moreover, the mode superposition technique was found to be very efficient as accurate results could be obtained based on only 25 to 30 modes of vibration Thus, this linear dynamic procedure is especially appropriate for analyzing bridge models with many degrees of freedom • bridge deck surface roughness and irregularities in the approach pavements and over bearings have a tremendous effect on the dynamic response To reduce damage to bridges not only maintenance of the bridge deck surface is important but also the elimination of irregularities (unevenness) in the approach pavements and over bearings It is also suggested that the formulas for dynamic amplification factors specified in bridge design codes should not only be a function of the fundamental natural frequency or span length (as in many present design codes) but should also consider the road surface condition • for more detailed and accurate studies where the most accurate representation of the true dynamic response is required, it is recommended to consider the cables motion and modes of vibration in the dynamic analysis by utilizing the multielement cable discretization This is also necessary to avoid an underestimation of the bridge dynamic responsebridge damping has a significant effect upon the response and should always be considered in such analysis Some dynamic amplification factors are very sensitive to bridge damping ratio and the relationship is not always linear Bridge damping ratios should be carefully estimated to insure more correct and accurate representation of the true dynamic response To obtain realistic damping ratios, such estimation should be based on results from tests on similar bridges Unfortunately, results from many studies of the dynamic response of cable-stayed – 183 – bridges found in the literature are not useful, as they have been conducted using either unrealistically high damping ratios for such bridges or no damping at all • a tuned mass damper is not very effective in reducing the maximum dynamic response during the forced vibration period (i.e when the vehicle is on the bridge) In fact, such a device can even increase some of the dynamic amplification factors However, the reduction of the vibration level in the free vibration period is significant as the tuned mass damper increases the overall damping of the bridge by working as an additional energy dissipater • the moving force model (constant force idealization of the vehicle load) can lead to unnecessary overestimation of the dynamic amplification factors compared to the sprung mass model It is believed that the sprung mass vehicle models are causing this by acting as vibration absorbers • the dynamic amplification factors of cable supported bridges can reach high values, higher than 1.30, even if maintenance of the road surface is made regularly This situation should be considered in the design practice of such bridges For the studied cable-stayed bridge, high dynamic amplification factors were obtained for the axial force in the girder near the pylons and for the tension in the shortest cables in the side spans For this bridge, the designer should consider installation of cable dampers especially for the shortest cables to increase the fatigue life of the cables 11.2 Suggestions for further research Based on the performed investigation, the following suggestions for further research can be given: • The effect of cable modeling and tuned mass dampers should be more thoroughly investigated using realistic trains of moving vehicles and considering road surface roughness and different vehicle speeds, as this could not be accomplished in this study due to time limitation Moreover, for future research it is suggested to use simulated trains of moving traffic based on collected statistical traffic data • The dependency of bridge response and dynamic amplification factors on the way in which the girder is connected to the pylons and on other modern girder supporting conditions, should be investigated – 184 – • Further work is needed to study the effect of using finer models (i.e more elements for discretizing the bridge girder and pylons) of the two studied cable supported bridges and also three-dimensional models to include torsional effects and torsional modes of vibration in the analysis • Extensive testing on a cable supported bridge should be performed to assess the validity of the analysis methods and the theoretical findings • Research is needed to thoroughly study active structural control of cable supported bridges and examine the effectiveness of active devices on suppressing bridge vibrations due to moving vehicles As discussed earlier in section 8.4, the performance of a tuned mass damper (TMD, passive device) significantly deteriorates when the dynamic characteristics of the bridge changes (i.e are different from the original characteristics assumed during the optimal design of the TMD) Thus, a superior solution can be obtained by using a socalled active tuned mass damper (ATMD) Such a damper comprises computer controlled servo-hydraulic actuators that can, when needed, modify the TMD properties to improve its efficiency The computer continuously monitors the dynamic characteristics of the bridge using e.g sensors attached to the bridge deck Active control of structures using cables was proposed by Freyssinet as early as 1960 [67] Today, active control is applied in advanced airplanes for suppression of aerodynamic instability, in high-speed trains like the Swedish train X2000 to improve riding comfort, and in modern cars like the Mercedes A-class to improve stability Active controls, e.g active modification of bridge deck edge shape to enhance resistance to aerodynamic instabilities like flutter, are also considered for new cable supported bridges with very long spans such as for the Messina crossing and the Gibraltar crossing It is believed that, as the cost of such active systems is high, they can only be economical for long span bridges where they can induce big saving in construction material Furthermore, since some people are perhaps not ready to rely on computers when crossing a bridge, active control should as a first step only be used to improve serviceability aspects such as riding comfort, whereas e.g the stability of the bridge have to rely entirely on the bridge structure itself For cable-stayed bridges real time vibration control can be achieved by e.g computer controlling the tension in some cables, so-called active cables, in order to counteract – 185 – traffic loads at any time Such control system is based on the idea of constantly monitoring movements of the bridge using attached sensors and via computer controlled tensioning jacks, the pretensioning force in the cables is changed Such bridges can be referred to as smart bridges as they have the capability of modifying their behavior during the dynamic loading The author believes that active vibration control of long span cable-stayed and suspension bridges will be an area of significant interest in the future Till now analysis and application of active vibration control of structures excited by moving loads have attracted limited research efforts For the interested reader, excellent literature review and state-of-the-art review on control and monitoring of civil engineering structures are found in [30, 67] Recent studies describing active control of bridges are presented in [5, 62, 69] – 186 – Appendix Maple Procedures The Maple procedures, used to generate the Fortran code for the elements presented in Chapter 7, are given below Each comment line starts with the symbol # A.1 Cable element # Tangent stiffness matrix Kt for; # the catenary cable element; readlib(fortran); with(linalg); Ly:=1/(2*E*A*w)*(Tj^2-Ti^2)+(Tj-Ti)/w; Lx:=-P1*(Lu/E/A+1/w*ln((P4+Tj)/(Ti-P2))); P3:=-P1; P4:=w*Lu-P2; Ti:=sqrt(P1^2+P2^2); Tj:=sqrt(P3^2+P4^2); f11:=diff(Lx,P1): f12:=diff(Lx,P2): f21:=diff(Ly,P1): f22:=diff(Ly,P2): f:=matrix(2,2,[f11,f12,f21,f22]): k:=inverse(f): k1:=k[1,1]: k2:=k[1,2]: k4:=k[2,2]: – 187 – Kt:=matrix(4,4,[-k1,-k2,k1,k2,-k2,-k4,k2,k4,k1,k2,-k1,-k2,k2,k4,-k2,-k4]): fortran(Kt,optimized): A.2 Beam element # Internal force vector p and; # tangent stiffness matrix Kt for; # the beam element; readlib(fortran); with(linalg); ux:=(u4-u1)/L; wx:=(u5-u2)/L; t:=(u3+u6)/2; tx:=(u6-u3)/L; e:=(1+ux)*cos(t)+wx*sin(t)-1; g:=wx*cos(t)-(1+ux)*sin(t); k:=tx; PIe:=1/2*L*E*A*e^2; PIg:=1/2*L*G*A*g^2; PIk:=1/2*L*E*I*k^2; PI:=PIe+PIg+PIk; p:=grad(PI,[u1,u2,u3,u4,u5,u6]); Kt:=hessian(PI,[u1,u2,u3,u4,u5,u6]); fortran(p,optimized); fortran(Kt,optimized); – 188 – Bibliography of Part B [1] ABAQUS User’s Manual, Hibbitt, Karlsson & Sorensen, Inc., Providence, Rhode Island, 1994 [2] Abbas S., Scordelis A., ‘Nonlinear Analysis of Cable-Stayed Bridges’, Proc Int Conference on Cable-Stayed and Suspension Bridges, Vol 2, Deauville, France, Oct 1994, pp 195-210 [3] Abdel-Ghaffar A.M., Khalifa M.A., ‘Importance of Cable Vibration in Dynamics of Cable-Stayed Bridges’, J Eng 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London Docklands’, J Constructional Steel Research, 46:1-3, 1998, paper no 59 – 194 – ... moving load problem of cable- stayed and suspension bridges The first approach uses a simplified analysis method to study the dynamic response of simple cable- stayed bridge models The bridge is idealized... predict their response due to moving vehicles Not only the dynamic behavior of new bridges need to be studied and understood but also the response of existing bridges, as governments and the industry... 1.1 and Figure 1.2 The first modern cable- stayed bridge was the Strömsund Bridge in Sweden opened to traffic in 1956 For the study of the concept, design and construction of cable- stayed bridges,
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