v = 110 km/h
146 m 335 m 146 m
- 25
15
5
-5
-15
-25
-35 with tuned mass damper (TMD) -45 without tuned mass damper (TMD) the truck leaves bridge Mid-point vertical displacement (mm) -55 0 10203040 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: Docent Sven Ohlsson Huvudhandledare: 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 – ______Contents ______
Abstract i
Preface iii
General Introduction and Summary 1
Part A State-of-the-art Review and a Simplified Analysis Method for Cable- 7 Stayed Bridges
1 Introduction 9 1.1 General...... 9 1.2 Review of previous research...... 15 1.2.1 Research on cable-stayed bridges...... 15 1.2.2 Research on other bridge types...... 22 1.3 General aims of the present study...... 27
2 Vehicle and Structure Modeling 29 2.1 Vehicle models...... 29 2.2 Bridge structure...... 31 2.2.1 Major assumptions...... 32 2.2.2 Differential equation of motion...... 33 2.2.3 Spring stiffness...... 34 2.3 Bridge deck surface roughness...... 38
3 Response Analysis 43 3.1 Dynamic analysis...... 43 3.1.1 Eigenmode extraction...... 43
– v – 3.1.2 Response of the bridge...... 45 3.2 Static analysis...... 49
4 Numerical Examples and Model Verifications 51 4.1 General...... 51 4.2 Simply supported bridge, moving force model...... 52 4.3 Multi-span continuous bridge with rough road surface ...... 57 4.4 Simple cable-stayed bridge...... 63 4.5 Three-span cable-stayed bridge...... 72 4.6 Discussion of the numerical results...... 80
5 Conclusions and Suggestions for Further Research 83 5.1 Conclusions of Part A...... 83 5.2 Suggestions for further research...... 85
Bibliography of Part A 87
Part B Refined Analysis Utilizing the Nonlinear Finite Element Method 97
6 Introduction 99 6.1 General ...... 99 6.2 Cable structures and cable modeling techniques ...... 101 6.3 General aims of the present study ...... 103
7 Nonlinear Finite Elements 105 7.1 General ...... 105 7.2 Modeling of cables...... 106 7.2.1 Cable element formulation...... 107 7.2.2 Analytical verification...... 111 7.3 Modeling of bridge deck and pylons...... 113
– vi – 8 Vehicle and Structure Modeling 117 8.1 Vehicle models...... 117 8.2 Vehicle load modeling and the moving load algorithm...... 121 8.3 Bridge structure...... 123 8.3.1 Modeling of damping in cable supported bridges...... 123 8.3.2 Bridge deck surface roughness...... 126 8.4 Tuned vibration absorbers...... 127
9 Response Analysis 133 9.1 Dynamic Analysis...... 133 9.1.1 Linear dynamic analysis...... 134 9.1.1.1 Eigenmode extraction and normalization of eigenvectors..... 135 9.1.1.2 Mode superposition technique ...... 136 9.1.2 Nonlinear dynamic analysis...... 138 9.2 Static analysis...... 141
10 Numerical Examples 143 10.1 Simply supported bridge ...... 144 10.2 The Great Belt suspension bridge ...... 149 10.2.1 Static response during erection and natural frequency analysis ... 151 10.2.2 Dynamic response due to moving vehicles...... 154 10.3 Medium span cable-stayed bridge...... 158 10.3.1 Static response and natural frequency analysis...... 159 10.3.2 Dynamic response due to moving vehicles – parametric study.... 162 10.3.2.1 Response due to a single moving vehicle ...... 163 10.3.2.2 Response due to a train of moving vehicles, effect of bridge- vehicle interaction and cable modeling...... 165 10.3.2.3 Speed and bridge damping effect...... 166 10.3.2.4 Effect of surface irregularities at the bridge entrance ...... 167 10.3.2.5 Effect of tuned vibration absorbers...... 168
– vii – 11 Conclusions and Suggestions for Further Research 181 11.1 Conclusions of Part B...... 181 11.1.1 Nonlinear finite element modeling technique...... 181 11.1.2 Response due to moving vehicles ...... 182 11.2 Suggestions for further research...... 184
A Maple Procedures 187 A.1 Cable element...... 187 A.2 Beam element...... 188
Bibliography of Part B 189
– viii – ______General Introduction and Summary ______
Due to their aesthetic appearance, efficient utilization of structural materials and other notable advantages, cable supported bridges, i.e. cable-stayed and suspension bridges, have gained much popularity in recent decades. Among bridge engineers the popularity of cable-stayed bridges has increased tremendously. Bridges of this type are now entering a new era with main span lengths reaching 1000 m. This fact is due, on one hand to the relatively small size of the substructures required and on the other hand to the development of efficient construction techniques and to the rapid progress in the analysis and design of this type of bridges.
Ever since the dramatic collapse of the first Tacoma Narrows Bridge in 1940, much attention has been given to the dynamic behavior of cable supported bridges. During the last fifty-eight years, great deal of theoretical and experimental research was conducted in order to gain more knowledge about the different aspects that affect the behavior of this type of structures to wind and earthquake loading. The recent developments in design technology, material qualities, and efficient construction techniques in bridge engineering enable the construction of lighter, longer, and more slender bridges. Thus nowadays, very long span cable supported bridges are being built, and the ambition is to further increase the span length and use shallower and more slender girders for future bridges. To achieve this, accurate procedures need to be developed that can lead to a thorough understanding and a realistic prediction of the structural response due to not only wind and earthquake loading but also traffic loading. It is well known that large deflections and vibrations caused by dynamic tire forces of heavy vehicles can lead to bridge deterioration and eventually increasing maintenance costs and decreasing service life of the bridge structure.
The recent developments in bridge engineering have also affected damping capacity of bridge structures. Major sources of damping in conventional bridgework have been largely eliminated in modern bridge designs reducing the damping to undesirably low levels. As an example, welded joints are extensively used nowadays in modern bridge designs. This has greatly reduced the hysteresis that was provided in riveted or bolted
– 1 – joints in earlier bridges. For cable supported bridges and in particular long span cable- stayed bridges, energy dissipation is very low and is often not enough on its own to suppress vibrations. To increase the overall damping capacity of the bridge structure, one possible option is to incorporate external dampers (discrete damping devices such as viscous dampers and tuned mass dampers) into the system. Such devices are frequently used today for cable supported bridges. However, it is not believed that this is always the most effective and the most economic solution. Therefore, a great deal of research is needed to investigate the damping capacity of modern cable supported bridges and to find new alternatives to increase the overall damping of the bridge structure.
To consider dynamic effects due to moving traffic on bridges, structural engineers worldwide rely on dynamic amplification factors specified in bridge design codes. These factors are usually a function of the bridge fundamental natural frequency or span length and states how many times the static effects must be magnified in order to cover the additional dynamic loads. This is the traditional method used today for design purpose and can yield a conservative and expensive design for some bridges but might underestimate the dynamic effects for others. In addition, design codes disagree on how this factor should be evaluated and today, when comparing different national codes, a wide range of variation is found for the dynamic amplification factor. Thus, improved analytical techniques that consider all the important parameters that influence the dynamic response, such as bridge-vehicle interaction and road surface roughness, are required in order to check the true capacity of existing bridges to heavier traffic and for proper design of new bridges.
Various studies, of the dynamic response due to moving vehicles, have been conducted on ordinary bridges. However, they cannot be directly applied to cable supported bridges, as cable supported bridges are more complex structures consisting of various structural components with different properties. Consequently, more research is required on cable supported bridges to take account of the complex structural response and to realistically 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 are seeking improvements in transport efficiency and our aging and deteriorating bridge infrastructure is being asked to carry ever increasing loads.
– 2 – The aim of this work is to study the moving load problem of cable supported bridges using different analysis methods and modeling techniques. The applicability of the implemented solution procedures is examined and guidelines for future analysis are proposed. Moreover, the influence of different parameters on the response of cable supported bridges is investigated. However, it should be noted that the aim is not to completely solve the moving load problem and develop new formulas for the dynamic amplification factors. It is to the author’s opinion that one must conduct more comprehensive parametric studies than what is done here and perform extensive testing on existing bridges before introducing new formulas for design.
This thesis contains two separate parts, Part A (Chapter 1-5) and Part B (Chapter 6- 11), where each has its own introduction, conclusions, and reference list. These two parts present two different approaches for solving the moving load problem of ordinary and cable supported bridges.
Part A, which is a slightly modified version of the licentiate thesis presented by the author in November 96, presents a state-of-the-art review and proposes a simplified analysis method for evaluating the dynamic response of cable-stayed bridges. 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 utilization of the beam on elastic bed analogy makes the presented approach also suitable for analysis of the dynamic response of railway tracks subjected to moving trains.
In Part B, a more general approach, based on the nonlinear finite element method, is adopted to study more realistic cable-stayed and suspension bridge models considering, e.g., exact cable behavior and nonlinear geometric effects. A beam element is used for modeling the girder and the pylons, and a catenary cable element, derived using “exact” analytical expressions for the elastic catenary, is used for modeling the cables. This cable element has the distinct advantage over the traditionally used elements in being able to approximate the curved catenary of the real cable with high accuracy using only one element. 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. Damping characteristics and damping ratios of cable supported bridges are discussed and a practical technique for deriving the damping
– 3 – matrix from modal damping ratios, is presented. Among other things, the effectiveness of using a tuned mass damper to suppress traffic-induced vibrations and the effect of including cables motion and modes of vibration on the dynamic response are investigated.
To study the dynamic response of the bridge-vehicle system in Part A and B, two sets of equations of motion are written one for the vehicle and one for the bridge. The two sets of equations are coupled through the interaction forces existing at the contact points of the two subsystems. To solve these two sets of equations, an iterative procedure is adopted. The implemented codes fully consider the bridge-vehicle dynamic interaction and have been verified by comparing analysis results with those found in the literature and with results obtained using a commercial finite element code.
The following basic assumptions and restrictions are made:
• elastic structural material
• two-dimensional bridge models. Consequently, the torsional behavior caused by eccentric loading of the bridge deck is disregarded
• as the damage to bridges is done mostly by heavy moving trucks rather than passenger cars, only vehicle models of heavy trucks are used
• simple one dimensional vehicle models are used consisting of masses, springs, and viscous dampers. Consequently, only vertical modes of vibration of the vehicles are considered
• it is assumed that the vehicles never loses contact with the bridge, the springs and the viscous dampers of the vehicles have linear characteristics, the bridge-vehicle interaction forces act in the vertical direction, and the contact between the bridge and each moving vehicle is assumed to be a point contact. Moreover, longitudinal forces generated by the moving vehicles are neglected.
Based on the study conducted in Part A and B, the following guidelines for future analysis and practical recommendations can be made:
• for preliminary studies using very simple cable-stayed bridge models to determine the feasibility of different design alternatives, the approach presented in Part A can
– 4 – be adopted as it is found to be simple and accurate enough for the analysis of the dynamic response. However, for analysis of more realistic bridge models where e.g. exact cable behavior, nonlinear geometric effects, or non-uniform cross- sections are to be considered, this approach becomes difficult and cumbersome. For such problems, the finite element approach presented in Part B is found to be more suitable as it can easily handle such analysis difficulties
• for cable supported bridges, nonlinear static analysis is essential to determine the dead load deformed condition. However, starting from this position and utilizing the dead load tangent stiffness matrix, linear static and linear dynamic traffic load analysis give sufficiently accurate results from the engineering point of view
• it is recommended to use the mode superposition technique for such analysis especially if large bridge models with many degrees of freedom are to be analyzed. For most cases, sufficiently accurate results are obtained including only the first 25 to 30 modes of vibration
• correct and accurate representation of the true dynamic response is obtained only if road surface roughness, bridge-vehicle interaction, bridge damping, and cables vibration are considered. For the analysis, realistic bridge damping values, e.g. based on results from tests on similar bridges, must be used
• care should be taken when the dynamic amplification factors given in the different design codes and specifications are used for cable supported bridges, as it is not believed that these can be used for such bridges. For some cases it is found that design codes underestimate the additional dynamic loads due to moving vehicles. Consequently, each bridge of this type, particularly those with long spans, should be analyzed as made in Part B of this thesis. For the final design, such analysis should be performed more accurately using a 3D bridge and vehicle models and with more realistic traffic conditions
• 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 also should consider the road surface condition.
– 5 – It is believed that Part A presents the first study of the moving load problem of cable- stayed bridges where this simple modeling and analysis technique is utilized. For Part B of this thesis, it is believed that this is the first study of the moving load problem of cable-stayed and suspension bridges where results from linear and nonlinear dynamic traffic load analysis are compared. In addition, such analyses have not been performed earlier taking into account exact cable behavior and fully considering the bridge- vehicle dynamic interaction.
Most certainly this study has not provided a complete answer to the moving load problem of cable supported bridges. However, the author hopes that the results of this study will be a help to bridge designers and researchers, and provide a basis for future work.
– 6 –
Part A
State-of-the-art Review and a Simplified Analysis Method for Cable-Stayed Bridges
– 7 –
– 8 – Chapter ______Introduction ______
1.1 General
Studies of the dynamic effects on bridges subjected to moving loads have been carried out ever since the first railway bridges were built in the early 19th century. Since that time vehicle speed and vehicle mass to the bridge mass ratio have been increased, resulting in much greater dynamic effects. In recent years, the interest in traffic induced vibrations has been increasing due to the introduction of high-speed vehicles, like the TGV train in France and the Shinkansen train in Japan with speeds exceeding 300 km/h. The increasing dynamic effects are not only imposing severe conditions upon bridge design but also upon vehicle design, in order to give an acceptable level of comfort for the passengers.
Modern cable-stayed bridges with their long spans are relatively new and have been introduced widely only since the 1950, see Table 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, see the excellent book by Gimsing [27] and also [28, 68, 75, 76, 79]. Cable supported bridges are special because they are of the geometric-hardening type, as shown in Figure 1.3 on page 16, which means that the overall stiffness of the bridge increases with the increase in the displacements as well as the forces. This is mainly due to the decrease of the cable sag and increase of the cable stiffness as the cable tension increases.
Compared to other types of bridges, the dynamic response of cable-stayed bridges subjected to moving loads is given less attention in theoretical studies. Static analysis and dynamic response analysis of cable-stayed bridges due to earthquake and wind loading, received, and have been receiving most of the attention, while only few
– 9 – studies, see section 1.2.1, have been carried out to investigate the dynamic effects of moving loads on cable-stayed bridges. However, with increasing span length and increasing slenderness of the stiffening girder, great attention must be paid not only to the behavior of such bridges under earthquake and wind loading but also under dynamic traffic loading as well.
The dynamic response of bridges subjected to moving vehicles is complicated. This is because the dynamic effects induced by moving vehicles on the bridge are greatly influenced by the interaction between vehicles and the bridge structure. The important parameters that influence the dynamic response are (according to previous research conducted in this field, see section 1.2):
• vehicle speed • road (or rail) surface roughness • characteristics of the vehicle, such as the number of axles, axle spacing, axle load, natural frequencies, and damping and stiffness of the vehicle suspension system • the number of vehicles and their travel paths • characteristics of the bridge structure, such as the bridge geometry, support conditions, bridge mass and stiffness, and natural frequencies.
For design purpose, structural engineers worldwide rely on dynamic amplification factors (DAF), which are usually related to the first vibration frequency of the bridge or to its span length. The DAF states how many times the static effects must be magnified in order to cover additional dynamic loads resulting from the moving traffic (DAF is usually defined as the ratio of the absolute maximum dynamic response to the absolute maximum static response). Because of the simplicity of the DAF expressions specified in current bridge design codes, these expressions cannot characterize the effect of all the above listed parameters. Moreover, as these expressions are originally developed for ordinary bridges, it is believed that for long span bridges like cable- stayed bridges the additional dynamic loads must be determined in more accurate way in order to guarantee the planned lifetime and economical dimensioning.
Figure 1.1 shows the variation of the DAF with respect to the fundamental frequency of the bridge, recommended by different standards [66]. For cases where the DAF was related to the span length, the fundamental frequency was approximated from the span length. It is apparent from Figure 1.1 that the national design codes disagree on the
– 10 – evaluation of the dynamic amplification factors, and although the specified traffic loads vary in these codes, this does not explain such a wide range of variation for the DAF. In the Swedish design code for new bridges, the Swedish National Road Administration (Vägverket) includes the additional dynamic loads, due to moving vehicles, in the traffic loads specified for the different types of vehicles. This gives a constant DAF that is totally independent on the characteristics of the bridge. For bridges like cable-stayed bridges that are more complex and behave differently compared to ordinary bridges, this approach can lead to incorrect traffic loads to be used for designing the bridge.
This part of the thesis presents a state-of-the-art review and a simplified analysis method for evaluating the dynamic response of cable-stayed bridges. 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 utilization of the beam on elastic bed analogy makes the presented approach also suitable for analysis of the dynamic response of railway tracks subjected to moving trains.
2.0 Canada CSA-S6-88m OHBDC Swiss SIA-88, single vehicle Swiss SIA-88, lane load 1.8 AASHTO-1989 India, IRC Germany, DIN1075 U.K. - BS5400 (1978) 1.6 France LCPC D/L=0.5 France LCPC D/L=5 D/L = Dead load / Live load 1.4
Dynamic amplification factor factor (DAF) amplification Dynamic 1.2
1.0 0 1 2 3 4 5 6 7 8 9 10 Bridge fundamental frequency (Hz) Figure 1.1 Dynamic amplification factors used in different national codes [66]
– 11 – Bridge name Country Center span Year of Girder (m) completion material Tatara Japan 890 1999 Steel Pont de Normandie France 856 1995 Steel Qingzhou Minjiang China (Fuzhou) 605 1996 Composite Yangpu China (Shanghai) 602 1993 Composite Xupu China (Shanghai) 590 1996 Composite Meiko-Chuo Japan 590 1997 Steel Skarnsund Norway 530 1991 Concrete Tsurumi Tsubasa Japan 510 1994 Steel Öresund Sweden/Denmark 490 2000 Steel Ikuchi Japan 490 1991 Steel Higashi-Kobe Japan 485 1994 Steel Ting Kau Hong Kong 475 1997 Steel Seohae South Korea 470 1998 unknown Annacis Island Canada 465 1986 Composite Yokohama Bay Japan 460 1989 Steel Second Hooghly India (Calcutta) 457 1992 Composite Second Severn England 456 1996 Composite Queen Elizabeth II England 450 1991 Composite Rama IX Thailand (Bangk.) 450 1987 Steel Chongqing Second China (Sichuan) 444 1996 Concrete Barrios de Luna Spain 440 1983 Concrete Tongling China (Anhui) 432 1995 Concrete Kap Shui Mun Hong Kong 430 1997 Composite Helgeland Norway 425 1991 Concrete Nanpu China (Shanghai) 423 1991 Composite Vasco da Gama Portugal 420 1998 unknown Hitsushijima Japan 420 1988 Steel Iwagurujima Japan 420 1988 Steel Yuanyang Hanjiang China (Hubei) 414 1993 Concrete Uddevalla Sweden 414 2000 Composite Meiko-Nishi Ohashi Japan 405 1986 Steel S:t Nazarine France 404 1975 Steel Elorn France 400 1994 Concrete Vigo-Rande Spain 400 1978 Steel
Table 1.1 Major cable-stayed bridges in the world
– 12 – Dame Point USA (Florida) 396 1989 Concrete Houston Ship Channel USA (Texas) 381 1995 Composite Luling, Mississippi USA 372 1982 Steel Duesseldorf-Flehe Germany 368 1979 Steel Tjörn (new) Sweden 366 1981 Steel Sunshine Skyway USA (Florida) 366 1987 Concrete Yamatogawa Japan 355 1982 Steel Neuenkamp Germany 350 1970 Steel Ajigawa (Tempozan) Japan 350 1990 Steel Glebe Island Australia 345 1990 Concrete ALRT Fraser Canada 340 1985 Concrete West Gate Australia 336 1974 Steel Talmadge Memorial USA (Georgia) 335 1990 Concrete Rio Parana (2 bridges) Argentina 330 1978 Steel Karnali Nepal 325 1993 Composite Köhlbrand Germany 325 1974 Steel Guadiana Portugal/Spain 324 1991 Concrete Kniebruecke Germany 320 1969 Steel Brotonne France 320 1977 Concrete Mezcala Mexico 311 1993 Composite Erskine Scotland 305 1971 Steel Bratislava Slovakia 305 1972 Steel Severin Germany 302 1959 Steel Moscovsky Ukraine (Kiev) 300 1976 Steel Faro Denmark 290 1985 Steel Dongying China (Shandong)288 1987 Steel Mannheim Germany 287 1971 Steel Wadi Kuf Libya 282 1972 Concrete Leverkusen Germany 280 1965 Steel Bonn Nord Germany 280 1967 Steel Speyer Germany 275 1974 Steel East Huntington USA 274 1985 Concrete Bayview USA 274 1990 Composite River Waal Holland 267 1974 Concrete Theodor Heuss Germany 260 1958 Steel Yonghe China (Tianjin) 260 1987 Concrete
Table 1.1 (continued)
– 13 – Oberkassel Germany 258 1975 Steel Rees-Kalkar Germany 255 1967 Steel Weirton-Steubenville USA 250 1986 Steel Chaco/Corrientes Argentina 245 1973 Concrete Papineau-Leblanc Canada 241 1971 Steel Kärkistensalmi Finland 240 1996 Composite Maracaibo Venezuela 235 1962 Concrete Pasco Kennewick USA 229 1978 Concrete Jinan Yellow River China (Shandong) 220 1983 Concrete Toyosato-Ohashi Japan 216 1970 Steel Onomichi-Ohashi Japan 215 1968 Steel Strömsund Sweden 183 1956 Steel
Table 1.1 (continued)
1000
900 Steel girder 800 Composite girder Concrete girder 700
600
500
400 Length of center span (m)
300
200
100 1950 1960 1970 1980 1990 2000 Year of completion
Figure 1.2 Span length increase of cable-stayed bridges in the last fifty years
– 14 – 1.2 Review of previous research
1.2.1 Research on cable-stayed bridges
In recent years the dynamic behavior of cable-stayed bridges has been a source of interesting research. This includes free vibration and forced vibration due to wind and earthquakes, see for example [2, 9, 47]. However, literature dealing with the dynamics of these bridges due to moving vehicles is relatively scarce.
For a cable-stayed footbridge, theoretical and experimental study on the effectiveness of tuned mass dampers, TMD’s, was carried out in [6]. In this study, tests with one and two persons jumping or running were performed, and acceleration responses with the TMD locked and unlocked were compared. In [59, 60], modal testing of the Tjörn bridge, a cable-stayed bridge in Sweden with a 366 m main span, is described. And in [11], dynamic load testing on the Riddes-Leytron bridge, a cable-stayed bridge in Switzerland with a 60 m main span, is presented.
Previous investigations on the dynamic response of cable-stayed bridges subjected to moving loads are summarised in the following:
Fleming and Egeseli (1980) [21, 22] compared linear and nonlinear dynamic analysis results for a cable-stayed bridge subjected to seismic and wind loads. The nonlinear dynamic response due to a single moving constant force was also studied. A two- dimensional (2-D) harp system cable-stayed bridge model with a main span of 260 m was adopted, and the bridge was discretized using the finite element method. The nonlinear behavior of the cables due to sag effect and the nonlinear behavior of the bending members due to the interaction of axial and bending deformations, were considered. Fleming et al. showed that although there is significant nonlinear behavior during the static application of the dead load, the structure can be assumed to behave as a linear system starting from the dead load deformed state for both static and dynamic loads, as illustrated in Figure 1.3. This means that influence lines and superposition technique can be used in the design process.
Considering only seismic loading a similar comparison was conducted in [2] and the same conclusion was made.
– 15 – Generalized force
ic m a n y d r a e n li dynamic n o c load eigenvalue n mi yna problem ar d linear dynamicline ic tat cable structures r s dead ea tic lin sta load on ar non-cable structures n line
Generalized displacement Figure 1.3 Schematic diagram showing the difference between the behavior of cable structures and non-cable structures and also the accuracy in the results from different analysis procedures
Wilson and Barbas (1980) [89] performed theoretical and experimental works on cable-stayed bridge models to determine the dynamic effects due to a moving vehicle. For the theoretical bridge model, a 2-D undamped continuous Bernoulli-Euler beam resting on discrete evenly spaced elastic supports, was adopted. The vehicle was modeled using one or two constant forces travelling at constant speeds. For the solution of the problem, mode superposition technique was used. All bridge cables were approximated by linear springs with equal stiffness, and solutions with two to five cables in the main span were presented. Only the main span was considered in this study, and the road surface roughness was neglected. The experimental models consisted of straight steel beams (cross section = 0.0492 cm × 1.97 cm and length = 2.36 m) spliced end to end at the supports (springs) so that continuous spans of up to 23.6 m could be tested. By prestressing the bridge model, an initial flatness to within ± 0.2 cm under self-weight was achieved. For the interior span supports, coil springs were used. One or two linear induction motors running in a separate track above the bridge model were used to move the point load vehicle at constant speeds in the range of 1.22 m/s to 8.85 m/s. The total vehicle model weight was about 1.2 kg. Wilson et al. presented diagrams showing, for both the theoretical and the experimental models, the influence of the speed parameter on the DAF values for displacements and bending
– 16 – moments. To show the influence of cable stiffness, diagrams with different values for the spring stiffness were also presented. The results showed good agreement between the theoretical and the experimental work. According to Wilson et al., the main reasons for the differences in the results were due to the inability of the experimental system to maintain constant speed, and the neglection of the inertia effects of the experimental transit load in the theoretical model. Wilson et al. concluded also that increasing the spring stiffness at the supports will for most cases lead to an increase in the bridge dynamic response.
Rasoul (1981) [69] used the structural impedance method1 and studied the dynamic response of bridges due to moving vehicles. The bridge flexibility functions were evaluated by using a static analysis of the bridge subjected to unit loads. A simply supported beam, a continuous beam, and very simple cable-stayed bridges were studied. For the cable-stayed bridges, two different analysis methods were used, namely an approximate method using the concept of continuous beam with intermediate elastic supports, fixed pylon heads and with the cables approximated by springs, and a more exact method taken into account the effect of the axial force in the girder and the transverse displacement of the pylons by using the reduction method. Solutions with different girder damping ratios for a simple 2-D cable-stayed bridge with only two cables were presented. The traffic load was modeled as a series of vehicles traversing along the bridge. Each vehicle was modeled with a sprung mass and an unsprung mass giving a vehicle model with two degrees of freedom (2 DOF). Different traffic conditions were studied, and the effect of vehicle speed and bridge damping on DAF was presented. Rasoul concluded that bridge damping was one of the important parameters affecting the DAF, and that the DAF was considerably higher for the cables than for other elements of the bridge. Rasoul found also that for a single vehicle travelling at constant speed, the moving force solutions are good approximations of the exact solutions. The road surface roughness was totally neglected in this study.
Alessandrini, Brancaleoni and Petrangeli (1984) [3] studied the dynamic response of railway cable-stayed bridges subjected to a moving train. The bridge was discretized using the finite element method, and geometric nonlinearities for the cables were considered by using an equivalent modulus of elasticity. The solution was carried
1 In this study, the equation of motion of the bridge was formulated in an integral form using the flexibility function (Green’s function) for the bridge.
– 17 – out using a direct time integration procedure (explicit algorithm). 2-D fan type cable- stayed bridges with steel deck and center spans of about 160, 260, and 412 m were adopted. Five different train lengths of 12-260 m and three different values for the mass per unit length of the train to the mass per unit length of the bridge were considered. The train was simulated using moving masses at three different speeds of 60, 120, and 200 km/h. DAF values for mid-span vertical displacement, axial force in the longest center span cable, and axial force in the anchor cables, were presented and compared with those obtained by the Italian Railways Steel Bridge Code. Alessandrini et al. concluded that, for most cases, the standard expression for DAF given in the Italian Railway Code were not admissible for cable-stayed bridges. It was also found that for speeds of up to about 120 km/h, the dynamic effects were small if not negligible. For speeds higher than 120 km/h the DAF values increase rapidly and for speeds of about 200 km/h, DAF values greater than those prescribed by the Italian Railway Code were observed. The rail surface roughness was neglected in this study.
Brancaleoni, Petrangeli and Villatico (1987) [8] presented solutions for the dynamic response of a railway cable-stayed bridge subjected to a single moving high-speed locomotive. The bridge was discretized using the finite element method and geometric nonlinearities were considered in the analysis. The analysis was carried out using a direct time integration procedure (explicit algorithm). A 2-D modified fan type cable- stayed bridge with concrete deck and a main span of 150 m, was adopted. The bridge deck and the pylons were modeled using beam elements, while nonlinear cable elements with parabolic shape functions were adopted for the cables. For the bridge, a Rayleigh type damping producing 2 % of the critical on the first mode has been used. Solutions for a total train weight of about 95 tons, treated as a set of moving forces, a set of moving masses, and a four axles 6 DOF sprung mass model, were presented. Three different train speeds were considered, 60, 120, and 200 km/h. Diagrams showing the variation of DAF with speed for the three different vehicle models, and time histories for the mid-span vertical displacements, were presented. The rail surface roughness was neglected in this study. Brancaleoni et al. concluded that treating the train as a set of moving forces or moving masses results in lower DAF values for the girder bending moments and the cable axial forces, and higher DAF values for the center span vertical displacements. Brancaleoni et al. showed also that bending moment amplification factors were greater than those for cable axial forces and center span vertical displacements. The rail surface roughness was neglected in this study.
– 18 – Walther (1988) [80] performed experimental study on a cable-stayed bridge model with slender deck to determine the dynamic displacements produced by the passage of a 250 kN vehicle at different speeds. The bridge model, which was equipped with rails and a launching ramp, represented a 3 span modified fan type cable-stayed bridge with a 200 m main span and about 100 m side spans. The deck and the two A-shaped pylons were made of reinforced microconcrete, while piano cord wires with a diameter of 2 to 3 mm were used for the cables. The scale adopted was 1/20 giving a total length of about 20 m for the bridge model and a model vehicle weight of 62.5 kg. Different model vehicle speeds from 0.6 to 3.8 m/s (corresponds to real vehicle speeds of about 10 to 61 km/h) were used, and tests with and without a plank in the main span were undertaken to simulate different road surface conditions. Time histories for mid- span vertical displacements were presented, for centric and eccentric vehicle movements, with or without a plank, and for fixed joint and free joint at mid-span. Based on measured data, vertical accelerations were calculated and a study of physiological effects (human sensitivity to vibrations) was undertaken. Walther concluded that from the physiological effects point of view, the structure could be considered acceptable to tolerable depending on the road surface condition. The maximum DAF value for mid-span vertical displacement was found to be 1.3. Walther found also that placing a joint at the center of the bridge deck only give very local effects and have little influence on the global dynamic behavior of the model.
Indrawan (1989) [45] studied the dynamic behavior of Rama IX cable-stayed bridge in Bangkok due to an idealized single axle vehicle travelling over the bridge at constant speeds. The 450 m main span, modified fan type, single plane, cable-stayed bridge, was modeled in 2-D. The dynamic response was analyzed using the finite element method and mode superposition technique, including only the first 10 modes of vibration. All analyses were carried out in the frequency domain and time domain responses were calculated using the fast Fourier transform (FFT) technique. The bridge deck and pylons were modeled using beam elements while truss elements were used for the cables. When evaluating the stiffness of each cable, the cable sag was considered by using an equivalent tangent modulus of elasticity. Time histories showing cable forces, mid-span vertical displacements, and pylon tops horizontal displacements, were presented for different types of vehicle models moving over a smooth surface, a rough surface, and a bumpy surface, at speeds of 36 to 540 km/h. The single axle vehicle was modeled as a constant force, an unsprung mass, and a sprung mass (1 DOF system). For the sprung mass vehicle model the assumed natural frequency and damping ratio were 1.39 Hz and 3.5 % respectively. The inertial effect
– 19 – in the vehicle due to bridge vibrations was totally neglected by the author. The road surface roughness was generated from a power spectral density function (PSD) (the same as the one used here in sec. 2.3). Since Rama IX bridge is equipped with tuned mass dampers (TMD) to suppress wind induced oscillations, a comparison was made between the dynamic response with and without the presence of a TMD. The TMD was assumed to be installed at mid-span and tuned to the first flexural mode of vibration. Indrawan found that the TMD was very effective in reducing the vibration level of cables anchored in the vicinity of the mid-span. But he suggested that, instead of using TMD’s, viscous dampers should be installed in all cables to more effectively increase the fatigue life of the cables. The analysis results showed also that the DAF increases with increasing vehicle speed and can for bumpy surface reach very high values.
Khalifa (1991) [49] carried out an analytical study on two cable-stayed bridges with main spans of 335 m and 670 m. The 3 spans cable-stayed bridges were of the double plane modified fan type, and were modeled in 3-D and discretized using the finite element method. The dynamic response was evaluated using the mode superposition technique, where each equation was solved adopting the Wilson-Θ numerical integration scheme. The linear dynamic analysis, based on geometrically nonlinear static analysis (see Figure 1.3), was conducted using the deformed dead load tangent stiffness matrix. The effect of including cable modes on the overall bridge dynamics was investigated by discretizing each cable of the longer bridge as one element and as eight equal elements. The dynamic response was evaluated for a single moving vehicle and a train of vehicles moving in one direction or in both directions. The vehicles, travelling with constant speeds of about 43 to 130 km/h over a smooth and a rough surface, were approximated using a constant moving force model and a sprung mass model. For the sprung mass vehicle model the assumed natural frequency and damping ratio were 1 or 3 Hz and 3 %, respectively. The road surface roughness was generated from a power spectral density function (PSD) (the same as the one used here in sec. 2.3). Diagrams showing the influence of bridge damping ratio, cable vibrations, vehicle model type, vehicle speed, number of vehicles, traffic direction, and deck condition, on the bridge dynamic response, were presented. A stress-life fatigue analysis was also conducted to estimate the virtual cable life under continuous moving traffic loads. Khalifa found that the fatigue life of stays cables were relatively very short if they were subjected to extreme vibrational stresses resulting from a continuous fluctuating heavy traffic. The results also showed that the magnitude of the dynamic response was influenced by the bridge damping ratio, the type of vehicle model, and
– 20 – the roughness of the bridge deck. The author recommended discretizing each cable into small elements when calculating the dynamic response due to environmental and service dynamic loads.
Wang and Huang (1992) [84] studied the dynamic response of a cable-stayed bridge due to a vehicle moving across rough bridge decks. The vehicle was simulated by a nonlinear vehicle model with 3-axles and seven degrees of freedom. A 2-D modified fan type cable-stayed bridge with concrete deck and a main span of 128 m, was adopted. The bridge deck roughness was generated using PSD functions. The dynamic response was analyzed using the finite element method and the geometric nonlinear behavior of the bridge due to dead load was considered. The equation of motion for the vehicle was solved using the fourth-order Runge-Kutta integration scheme, and an iterative procedure with mode superposition technique was used for solving the equation of motion for the bridge. Wang et al. concluded that the mode superposition procedure used was effective and involved much less computation, because accurate results of the bridge dynamic response could be obtained based on solving only 8 to 12 equations of motion of the bridge. Wang et al. noted that the DAF of all components of the bridge were generally less than 1.2 for very good road surface, but increased tremendously with increasing road surface roughness. High values of DAF were noted at the girder near the pylons and at the lower ends of the pylons and piers, but comparatively small DAF values were noted at the girder adjacent to the mid-span of the bridge.
Miyazaki et al. (1993) [55] carried out an analytical study on the dynamic response and train running quality of a prestressed concrete multicable-stayed railway bridge planned for future use on the high-speed Shinkansen line. For the analysis, the simulation program DIASTARS, developed at the Japanese Railway Technical Research Institute, was used. The railway track and the bridge structure were modeled using the finite element method. In this study, a 2-D and a 3-D bridge models of a two span cable-stayed bridge, were used. The 2-D bridge model together with a simple 12 cars train model consisting of only constant forces were used to evaluate the dynamic response of the bridge, while the 3-D bridge and the 3-D train model were used to evaluate the train running quality. The 3-D Shinkansen train model consisted of 12 cars where each car consisted of a body, two bogies, and four wheelsets giving 23 DOF. The track was assumed to be directly placed on the bridge deck surface, and the rail surface roughness was neglected. The 3-D bridge deck was modeled by 3-D beam elements connected to the cables through transversely extended rigid beams. In the
– 21 – study, a comparison was also made with the design value of DAF specified in the Japanese Design Standards for Railway Concrete Structures. Miyazaki et al. presented diagrams showing the speed, 0-400 km/h, influence on the DAF for the deck and pylons bending moments, deck and pylons shear forces, deck and pylons axial forces, and axial forces in cables. For the vehicle, diagrams were presented showing wheel load variations and vertical car body accelerations. Miyazaki et al. concluded that the examined PC cable-stayed bridge had a satisfactory train running quality (acceptable riding comfort). For the different bridge members, the authors recommended different values for the coefficient included in the DAF expression in the Japanese design standard.
Chatterjee, Datta and Surana (1994) [14] presented a continuum approach for analyzing the dynamic response of cable-stayed bridges. The effects of the pylons flexibility, coupling of the vertical and torsional motion of the bridge deck due to eccentric vehicle movement, and the roughness of the bridge surface, were considered. The vehicle was simulated using a vehicle model with 3 DOF and 3-axles. A PSD function was used to generate the road surface roughness and mode superposition technique was adopted for solving the equation of motion of the bridge. Chatterjee et al. investigated the influence of vehicle speed, eccentrically placed vehicle, spacing between first and second vehicle axles, and bridge damping ratios on the dynamic behavior of a double-plane harp type cable-stayed bridge with roller type cable-pylon connections and a main span of 335 m. Chatterjee et al. concluded that pylon rigidity and the nature of cable-pylon connection have significant effect on the natural frequencies of vertical vibration, but no effect on those of torsional vibration. Chatterjee et al. noted that idealizing the vehicle as a constant force leads to overestimation of the DAF compared to the sprung mass model. The same conclusion was found when assuming that there is no eccentricity in the vehicle path. And finely, it was noted that increasing the axle spacing of the vehicle, or not including the roughness of the bridge surface, decreases the DAF values.
1.2.2 Research on other bridge types
The dynamic effects of moving vehicles on bridges have been investigated by various researchers, using bridge and vehicle models of varying degrees of sophistication.
– 22 – A review of the early work on the dynamic response of structures under moving loads was presented in the paper by Filho [20]. For a thorough treatment of the analytical methods used for problems of moving loads with and without mass in both structures and solids, see the excellent book by Frýba [23]. In this book, analysis of sprung and unsprung mass systems moving along a beam covered with elastic layer of variable stiffness and surface irregularities, were presented. The dynamics of railway bridges and railway vehicle modeling are described in the book by Frýba [24] and the book by Garg and Dukkipati [25]. Interesting research was also presented by Olsson, see Table 1.2, where he derived a structure-vehicle finite element by eliminating the contact degrees of freedom of the vehicle. The stiffness and damping matrices thus became time-variant and non-symmetric.
Previous investigations on the dynamic response of other bridge types subjected to moving loads are summarized in Table 1.2 below.
Author(s) Bridge type Vehicle model Surface Other remarks like roughness analysis methods used etc. function Hillerborg (1951) SSB SMS-1-1-2 not considered theoretical & experimental [34] study Hirai et al. (1967) suspension MF, moving pulsating not considered theoretical & experimental [36] bridge force study Veletsos et al. 3-SB cantilever SMS-3-3-2 not considered lumped mass method (1970) [77] , SSB Yoshida et al. SSB, SS slab MF, MM not considered FEM (1971) [93] Nagaraju et al. 3-SB MF, SMS-1-1-2 not considered continuum approach, mode (1973) [57] cantilever superposition Ting et al. (1974) SSB MM not considered structural impedance [72] method
Table 1.2 Previous investigations on the dynamic response of other bridge types subjected to moving loads. SMS-x-y-z=sprung mass system with x-axles, y degrees of freedom, and in z dimensions, MF=moving force, MM=moving mass, SSB=simply supported beam, x-SB=x span beam, SS xx=simply supported xx, FEM=finite element method
– 23 – Genin et al. SSB, MF, SMS-1-1-2, harmonic structural impedance (1975) [26] 2-SB air cushion system sinusoidal method Ginsberg (1976) SSB multiple not considered structural impedance [29] SMS-1-1-2 method Filho (1978) [20] SSB SMS-1-2-2 not considered FEM Blejwas et al. SSB MM, SMS-1-2-2 harmonic Lagrange’s eqn. with (1979) [7] sinusoidal multipliers Chu et al. (1979) SS girder & SMS-4-3-3 for not considered lumped mass method [16] truss railway each railcar Gupta et al. SS orthotr. SMS-2-3-2 not considered vehicle braking, eccentric (1980) [31] plate, SSB loading Ting et al. SSB MF, MM, not considered review, different analysis (1980,1983) SMS-1-2-2 procedures and vehicle [73, 74] models Hayashikawa et SSB, 2-SB, MF not considered eigen stiffness matrix al. (1981) [32] 3-SB method Hayashikawa et suspension MF not considered continuum approach, mode al. (1982) [33] bridge superposition Mulcahy (1983) SS orthotr. SMS-2-4-3, 10 mm bump finite strip method, vehicle [56] plate SMS-3-7-3 braking Olsson (1983, SSB MF, MM, harmonic FEM, special bridge- 1985) [63, 62] SMS-1-2-2 cosine vehicle element Schneider et al. SSB MF, MM not considered used the FEM package (1983) [71] ADINA Arpe (1984) SSB SMS-2-4-2 not considered theoretical & experimental [4, 5] study Hino et al. (1984) 1-SB cantilever SMS-1-1-2 not considered FEM, direct time [35] integration Palamas et al. SSB, 2-SB SMS-1-1-2 sinusoidal, Rayleigh-Ritz method (1985) [65] pothole Chu et al. (1986) SS PC railway SMS-4-23-3 PSD lumped mass method [17] Honda et al. 2-SB, 3-SB, 4- SMS-1-2-2 PSD, bump at 1 vehicle & multiple (1986) [37] SB, 5-SB, SSB entrance groups of vehicles
Table 1.2 (continued)
Olsson (1986) SSB, 2-SB, MF, MM, SMS-1-2-2, not considered FEM, special bridge-
– 24 – [64] 6-SB SMS-2-4-2, vehicle element, vehicle SMS-2-6-2, SMS-2-7-2 braking Inbanathan et al. SSB MF, MM considered FEM, PSD for interaction (1987) [44] force Bryja et al. suspension multiple MF not considered random highway traffic (1988) [10] bridge Diana et al. suspension SMS-4-23-3 for each not considered FEM, different traffic (1988) [19] bridge railcar conditions Coussy et al. SSB SMS-2-2-2 PSD continuum approach, mode (1989) [18] superposition Wang (1990) [81] SS PC railway SMS-4-23-3 for each PSD influence of ramp/ bridge railcar track stiffness Hwang et al. SSB SMS-2-4-2, PSD traffic simulations, one and (1991) [43] SMS-3-7-2 two trucks Olsson (1991) SSB MF not considered compared analytical [61] solution with FEM Wang et al. SS truss SMS-4-23-3 for each PSD lumped mass method (1991) [82] railway railcar Huang et al. continuous SMS-3-12-3 PSD FEM, one and two trucks (1992) [39] multigirder Wang et al. SS multigirder SMS-2-7-3, PSD FEM, one and two trucks (1992) [85] SMS-3-12-3 Wang et al. SSB SMS-2-7-3, bump, PSD FEM, validation of vehicle (1992) [83] SMS-3-12-3 models Knothe et al. review of dynamic modeling of railway track and of vehicle-track interaction (1993) [50] Nielsen (1993) beam on elastic MM, SMS-1-3-2, harmonic sinus- railway structures, [58] foundation,3-D SMS-2-4-2, SMS-2-6-2 oidal for rail- compared theoretical and track model head, wheelflat experimental results Saadeghvaziri SSB, MF not considered used the FEM package (1993) [70] 3-SB ADINA Wang et al. no bridge SMS-2-7-3, bump, PSD only validation of the (1993) [86] SMS-3-12-3 vehicle models
Table 1.2 (continued)
Wang (1993) [87] SS truss SMS-4-23-3 for each PSD lumped mass method railway railcar
– 25 – Cai et al. (1994) SSB, 2-SB moving pulsating force, not considered continuum approach, mode [12] SMS-1-2-2 superposition Chatterjee et al. suspension SMS-1-1-2, PSD continuum approach, mode (1994) [15] bridge SMS-3-3-2, SMS-3-6-3 superposition Wakui et al. describes a computer program developed using FEM and mode superposition to solve the (1994) [78] dynamic interaction problem between high speed railway vehicles, each of SMS-4-31-3, and railway structures Yener et al. slab on SSB’s MF, SMS-1-3-2, not considered FEM, different traffic (1994) [92] SMS-2-6-2 conditions Chatterjee et al. arch bridge MF not considered mixed and lumped mass (1995) [13] method Green et al. 3-SB, 4-SB SMS-4-11-2 PSD, 20 mm compared leaf sprung with (1995) [30] bump air sprung vehicles Huang et al. thin walled SMS-3-12-3 PSD FEM (1995) [40] box-girder Huang et al. hor. curved SMS-3-12-3 PSD FEM, one and two trucks (1995) [41] I-girder Humar et al. SS orthotr. SMS-1-2-2 not considered FEM, different traffic (1995) [42] plate conditions Lee (1995) [51] 2-SB, 3-SB, MF not considered beams on one-sided point 4-SB constraints Lee (1995) [52] SSB rigid wheel not considered unknown wheel nominal motion, FEM Paultre et al. arch, box ambient & controlled dynamic bridge testing (1995) [67] girder traffic Yang et al. (1995) SSB, 3-SB, MF, MM, SMS-1-2-2, PSD FEM, special bridge- [90, 91] 5-SB SMS-3-6-2 vehicle element
Table 1.2 (continued)
1.3 General aims of the present study
In all the aforementioned studies on the dynamic behavior of cable-stayed bridges, authors either used very simple vehicle models, or very complicated and time-
– 26 – consuming vehicle and bridge models. In [21, 22, 89], the vehicle was modeled as a constant moving force, neglecting the vehicle inertial effects, and in [69, 3, 8, 55], the road (or rail) surface roughness was neglected and only the elastic displacements of the bridge, caused by the varying position of the vehicle, were considered. The opposite assumption was made in [45], where the bridge elastic displacements were neglected and only the excitation caused by the road surface roughness was considered. Of course, the assumptions made by those authors are acceptable, if for example the vehicle is travelling at low speed, the road surface is smooth, and the vehicle mass to the bridge mass ratio is low.
The vehicle inertial effects, the road surface roughness, and the bridge displacements were considered in [49]. However, the formulations for the coupling equations (equations (2.4a-c) in section 2.1) are, according to the author’s opinion, incorrect.
Only the models developed in [84, 14] are believed to be general and handle the bridge-vehicle contact problem correctly. On the other hand, the vehicle models used are very complicated and, as Frýba [24] pointed out, very detailed and complicated vehicle models are unnecessary, if the main purpose is to study the bridge dynamic response. In the work presented here, the most detailed vehicle model used consists of two degrees of freedom, as this is adequate for large span bridges, according to Frýba.
The main aims of this study are as follows:
• to develop a general but simple analysis tool which fully consider the bridge- vehicle interaction, including all inertial terms, in evaluating the dynamic response of bridges subjected to moving vehicles • to investigate on the applicability of the beam on elastic bed analogy and the finite difference method for dynamic analysis of cable-stayed bridges. Moreover, to show that the proposed simplified analysis method, which uses the finite difference method and the mode superposition technique for dynamic response evaluation, is very efficient and is easy to implement and understand • to analyze the dynamic response of simple cable-stayed bridge models and to study the influence of different vehicle models and the influence of different parameters, such as vehicle speed and bridge deck surface roughness, on the dynamic response.
– 27 – For this purpose a computer code has been developed using the MATLAB language [53], where the dynamic interaction between the bridge and the vehicle is included by utilizing an iterative scheme. Time histories and dynamic amplification factors are presented as functions of a limited set of parameters for quite simple but representative bridge and vehicle models. The implemented code has been verified by comparing analysis results with those obtained using the commercial finite element code ABAQUS [1]. Special emphasis is put on verification of the proposed model and on investigating the effects of local and global irregularities on the dynamic response.
Part of this work was presented earlier at the 15th Congress of IABSE, Copenhagen, 1996 [48].
– 28 – Chapter ______Vehicle and Structure Modeling ______
2.1 Vehicle models
Heavy vehicles consist of several major components, such as tractors, trailers and suspension systems, and can be modeled by a set of lumped masses, springs and dampers. As illustrated in Figure 2.1, the vehicle models used in this study include a moving force model, a moving mass model, and a sprung mass model with two degrees of freedom. The moving force model (constant force magnitude) is sufficient if the inertia forces of the vehicle are much smaller than the dead weight of the vehicle. For a vehicle moving along a straight path at a constant speed, these inertia effects are mainly caused by bridge deformations (bridge-vehicle interaction) and bridge surface irregularities. Hence factors that are believed to contribute in creating vehicle inertia effects include: high vehicle speed, flexible bridge structure, large vehicle mass, small bridge mass, stiff vehicle suspension system and large surface irregularities. In the present study, the adopted sprung mass model is a one-axle vehicle model of a real multi-axle vehicle. This model is acceptable, when the bridge span is considerably larger than the vehicle axle base [24], as the case is for cable supported bridges. The author believes that the use of simplified models may be more effective in identifying correlation between the governing bridge-vehicle interaction parameters and the bridge response. Very detailed vehicle models are unnecessary and will not bring any great advantage, when the main purpose is to study the dynamic response of bridges.
Heavy roadway vehicles generate most of their dynamic wheel loads in two distinct frequency ranges [30]: body-bounce and pitch motions at 1.5-4 Hz and wheel-hop
– 29 – motion at 8-15 Hz. This explains the increase of some of the specified DAF in Figure 1.1, for bridges with a fundamental frequency in the range of 1 to 5 Hz.
v(t) w2(t)
m2 v(t) v(t)
w1(t) w1(t) (m1+m2)g kS cS
m1 m1+m2
Moving force model Moving mass model Sprung mass model
Figure 2.1 Vehicle modeling
Considering the sprung mass model, shown in Figure 2.1, and denoting the contact force between the bridge and the vehicle by Ft( ) , defined positive when it acts downward on the bridge, the following equations of motion can be established [23, 63]:
2 d w1 d w21d w −+(mmgm12) − 1 +kwSS( 21 − w) + c − +=Ft( ) 0 (2.1) d t 2 d t d t
2 d w2 d w2 d w1 − m2 − kS ()w2 − w1 − cS − = 0 (2.2) dt 2 dt dt
Equations (2.1) and (2.2) are the dynamic equilibrium equations for the unsprung mass and the sprung mass, respectively. Referring to Figure 2.1, wt1( ) and wt2 ( ) are the displacements of the vehicle unsprung mass m1 and the vehicle sprung mass m2, respectively, k S the stiffness of the linear spring connecting the two masses, cS the damping coefficient of the viscous damper, and g the acceleration of gravity. It should be noted that wt2 ( ) is measured from the equilibrium position under the dead weight m2 g. The contact force may be expressed by use of equations (2.1) and (2.2) giving:
– 30 – 2 2 d w1 d w2 Ft( ) =+( m12 m) g + m 1 +m2 (2.3) d t 2 d t 2 where the first term on the right-hand side is the dead weight (static part) of the contact force and the other terms represent the inertia effects.
2 d w1 The contact force for the moving mass model will be Ft()=+ ( m12 m ) g + , d t 2 and for the moving force model Fmmg= ()12+ .
Assuming that the vehicle never loses contact with the bridge (that is F()t > 0), and that the deformation between the unsprung mass center and the bridge deck center line may be neglected, the following coupling equations for the point of contact,
xt()= xv () t (see Figure 2.2), must be fulfilled [20, 63]:
wt1()=+ yxt,t( ()) rxt( ()) (2.4a)
∂y ∂y ∂r wt()=++v v (2.4b) &1 ∂x ∂t ∂x
2 222 ∂ y 2 ∂ y ∂y ∂ y ∂ r 2 ∂r wt&& 1()=+v 2v ++++a v a (2.4c) ∂x2 ∂∂xt ∂x ∂t 2 ∂x2 ∂x
where wt&1() and wt&& 1() denote the unsprung mass vertical velocity and acceleration, respectively, v and a the vehicle velocity and acceleration in the longitudinal direction, respectively, y ()x , t the bridge vertical displacement, and r()x the surface irregularity function. The first term on the right-hand side of equation (2.4c) represent the influence of the bridge deck curvature (centripetal acceleration), the second term the influence of Coriolis acceleration, and the fourth term the influence of the acceleration of the point of contact in the vertical direction.
2.2 Bridge structure
For the present study, the fan-shaped self or earth anchored cable-stayed bridge scheme shown in Figure 2.2 is adopted. To make the presentation of the model more
– 31 – clear, the derivation of the equations in this section will be presented including only the main span of the bridge as shown in Figure 2.2, and assuming that the stiffening girder, having a uniform mass and flexural rigidity, is simply supported at the pylons. Of course the developed computer code is very general and capable of handling the more realistic case including side spans, suspended or not suspended, and as many supports as needed.
Figure 2.2 Idealized vehicle in contact with a cable-stayed bridge
2.2.1 Major assumptions
The following assumptions are made:
• multicable system with small stays spacing compared to the bridge length • negligible cable mass • the cables are idealized as vertical springs continuously distributed along the length of the stiffening girder • according to the usual erection procedures, the bridge in its initial configuration under dead load is free from bending moments, while only axial forces are present
– 32 – • cable forces under dead load are so adjusted that all displacements remain zero • axial girder forces have negligible effect on the frequencies and mode shapes and are therefore neglected • only in-plane flexural behavior of the bridge is considered. The torsional behavior caused by eccentric loading of the bridge deck is disregarded in this study • bridge damping is small and therefore neglected • when the vehicle enters the bridge, the vertical deflection and the vertical velocity of the moving vehicle are assumed to be zero.
2.2.2 Differential equation of motion
The governing equation of motion for vertical vibration of the bridge at any section of the stiffening girder (idealized as a Bernoulli-Euler beam on elastic supports) is given by [23]:
∂ 4 yx,t( ) ∂ 2 yx,t( ) EIgg ++kxyx,t( ) ( ) mg =−−δ ( xxFtv ) ( ) (2.5) ∂x 4 ∂t 2
where δ is the Dirac delta function, Eg the modulus of elasticity, I g the moment of inertia, mg the mass per unit length, and kx( ) the spring stiffness (to study ordinary beam type bridges kx( ) is set to zero). The effects of rotatory inertia and shear deformation are neglected as the cross-sectional dimensions of the stiffening girder are small in comparison with its length and the higher vibration modes are not significantly excited.
The boundary conditions are:
∂22yt(0, ) ∂ yLt( , ) yt(00,,) ==== 00,,, yLt( ) 0 (2.6a-d) ∂x22∂x and the initial conditions are:
∂y()x,0 y()x,0 = 0, = 0 (2.7a,b) ∂t
– 33 –
2.2.3 Spring stiffness
Using the notations of Figure 2.2, the stiffness of the spring idealizing cable i is given by [75]:
2 c EAiisin α i ki = (2.8) Li
Denoting the allowable cable stress by σa , the dead load and the live load per unit length by qqgq and , the cross-sectional area of cable i is given by [9]:
(qqsgq+ ) Ai = (2.9) σαaisin
Due to its own dead weight, a stay cable actually takes the shape of a curved line, rather than a straight one, between the two anchorage points. When the cable tension increases, the sag decreases, and the apparent axial stiffness of the inclined cable increases. In the present study, the cable geometric nonlinearity, due to the change of the sag and shape under varying stresses (forces), is approximately taken into account by introducing the following equivalent tangent modulus of elasticity [27, 9, 84]:
Ec L Ex( ) == Ei 22 0 ≤≤x (2.10) γ c x 2 1+ 3 Ec 12σo
where Ec is the modulus of elasticity for the straight cable, γ c the specific weight of the cable material, and σo the initial tensile stress in the cable. As cable forces caused by the vehicle load are small when compared to those created by dead load, the starting equlibrium configuration under dead load is used [9, 84, 14] and σo is here set equal to σ g , which is the tension stress due to dead load qg and is given by [9]:
qg σσga= (2.11) ()qqgq+
– 34 – After substituting (2.9) into (2.8), the following equation can be established for the spring stiffness per unit length due to the elongation of the cables in the main span:
c Ex( ) ( qgq+ q) 1 L kx( ) = 2 0 ≤≤x (2.12) σa H x 2 1+ H
The horizontal force on the pylon top due to the tensile force Fi in the main span cable i is:
(qqsqqsgq+ ) ( gq+ ) TFii==cosα i = x (2.13) tanαi H
Neglecting the stiffness of the pylon, the area of the anchor cable per unit length of the main span and the total area of each anchor cable can be expressed as:
qqgq+ Axo( ) = x (2.14) σαa H cos o
L/2 2 tot qqgq+ L AAxxo ==∫ o( ) d (2.15) 0 σαa H cos o 8
In equation (2.15), the dead weight of the side spans are not included because the side spans are not considered in this derivation.
The elongation of the anchor cable, the horizontal displacement of the pylon top, and the vertical displacement in the main span at joint i, due the force Fi in cable i are:
FLiio cosα ∆Lo = (2.16) EAoocosαo
∆LFLcosα o iio (2.17) b == 2 cosαo EAoocos αo
– 35 – x δ = b (2.18) i H
The internal force in cable i due to the vertical displacement δi = 1, and the spring stiffness are:
EAHcos2 α ()δi =1 oo o1 Fi = (2.19) Lxo cosαi
FEAH()δi =122sinαα cos 1 a iioo o (2.20) ki == 2 1 Lo x
After substituting (2.14) into (2.20), the following equation can be established for the spring stiffness per unit length due to the elongation of the anchor cables:
ELHqo sgq( + q) 1 L a (2.21) kx( ) = 2 0 ≤≤x σa Lxo 2
Where Eo is evaluated according to equation (2.10). Referring now to the pylons, a horizontal force at the top gives the horizontal displacement:
TH3 b = ip (2.22) 3EIpp
Thinking of the pylon as a fictitious anchor cable having the fictitious area Ap , the force Ti gives the horizontal displacement:
∆LTLo i o b == 2 (2.23) cosαo EAoop cos α
Equating both equations (2.22) and (2.23) for b, we find that the fictitious cable area is:
– 36 – 3EILppo Ap = 32 (2.24) EHoop cos α
To approximately include the effect of the pylons, equation (2.21) is modified to give:
tot ELHqo sgq( + q) AAo + p 1 L a (2.25) kx( ) = 2 tot 0 ≤≤x σa Lo Axo 2
And finally, the resulting spring stiffness that includes the effect of the elongation of the cables in the main span, the elongation of the anchor cables, and the stiffness of the pylons is:
1 L kx= 0 ≤≤x (2.26) ( ) 11 + 2 kxca( ) kx( )
The pylons shortening effect is very small and is therefore not considered in equation (2.26).
Figure 2.3 shows the spring stiffness, k(x), for a fan-shaped cable-stayed bridge with L = 150 m, and Figure 2.4 shows the fixed pylon top curve in Figure 2.3 but with the suspended side spans included.
2.0 )
2 fixed pylon top free pylon top N/m
6 1.5 free pylon top,I p =0 ) (10 x (
k 1.0
0.5 Spring stiffness, 0.0 0 25 50 75 100 125 150 x (m)
– 37 – Figure 2.3 Spring stiffness for a fan-shaped cable-stayed bridge with L=150 m, H=30 11 2 11 2 4 6 m, Hp=60 m, Ec =2.1·10 N/m , Ep=0.35·10 N/m , Ip=25 m , σa =720·10 2 4 4 4 3 N/m , qg=12·10 N/m, qq=5·10 N/m, and γ c =9·10 N/m
2.0 ) 2 N/m 6 1.5 ) (10 x (
k 1.0
0.5
Spring stiffness, 0.0 0 50 100 150 200 250 x (m)
Figure 2.4 Spring stiffness for a fan-shaped cable-stayed bridge (fixed pylon top) with the same input data as in Figure 2.3 but with the side spans, Ls=50 m, included
Equations (2.1-2.5) and (2.26), together with the boundary and initial conditions, define the analytical model of the problem.
2.3 Bridge deck surface roughness
The characteristics of road surface roughness is expressed generally by the power spectral density (PSD) function which is assumed from a stationary normal probability process with a zero mean value [38]. The PSD, S(Ω) , of road surface roughness is approximated generally by an exponential function [38, 46] as:
Sa(ΩΩ) = −n (2.27) in which Ω is the wave number, a is the spectral roughness coefficient, and n is the spectral roughness exponent.
– 38 –
In the present study, the bridge road surface roughness is simulated using the one- sided PSD function:
358. ⋅ 10−4 0 <Ω < 0.05 (2.28) S(Ω) = −−4194. 00107. ⋅ 10 ΩΩ 0.05 < < 1 presented in [38], and shown in log-log by the bold line in Figure 2.5. The exponent n and the coefficient a were calculated by the least square method using measurements on 56 bridges.
The random surface roughness profile is assumed to be the sum of series of sinusoidal waves and is generated by the following formula [49]:
M rx( ) =+∑αϕiiisin(Ω x ) (2.29) i=1
where αi is the amplitude of the sinusoidal wave, ϕi is a random phase angle with uniform distribution in the interval {02, π} , and Ωi is the ith wave number within the specified PSD interval and is given by:
ΩΩi =+−min ∆Ω(i 1) (2.30) in which the wave number increment ∆Ω is defined as:
ΩΩ− ∆Ω = max min (2.31) M where M is the total number of wave number increments in the interval between the minimum wave number Ωmin and the maximum wave number Ωmax in the defined spectra. The unit of the wave number is (m-1) and the unit of the spectrum is (m3). The amplitude αi is related to the PSD function S(Ωi ) by:
α2 S(Ω∆Ω=) i (2.32) i 2
– 39 – 10 )
3 Analytical m
-4 Simulated 1 ) (10 Ω
0.1
0.01 PSD of surface roughness, S(
0.001 0.01 0.1 1 Wave number, Ω (1/m)
Figure 2.5 Power spectral density of road surface roughness
0.02 0.01 0 -0.01 -0.02 Surface roughness (m) 0 50 100 150 Distance along the bridge (m)
Figure 2.6 Simulated road surface roughness profile for a bridge with L=150 m
– 40 – The simulated road surface roughness profiles are different depending on the random phase angles ϕi , used in equation (2.29). Figure 2.6 shows a sample of the simulated random road surface roughness profile used in this study for bridges with L = 150 m.
To compare the analytical PSD, equation (2.28), with the PSD of the sample shown in Figure 2.6, a power spectrum analysis of the sample is performed using the following relationship [83]:
1 2 S (Ω=) F (2.33) sim N
where F is the length N fast Fourier transform (FFT) of the sample, and Ssim(Ω) is the resulting PSD estimate for the sample. Using the symmetry property of real FFTs, the spectral estimates corresponding to negative wave numbers are removed and a compensation is made for them in the positive side. The resulting one-sided PSD estimate for the sample in Figure 2.6 is indicated by the fine solid line in Figure 2.5.
The bridge-vehicle interaction forces can now be calculated including the effect of road surface roughness by using the simulated profile of the bridge road.
– 41 –
– 42 – Chapter ______Response Analysis ______
3.1 Dynamic analysis
3.1.1 Eigenmode extraction
Before analyzing the response of the bridge to moving vehicles it is important to study the free vibration of the bridge model. The free vibration analysis is here an essential first step in obtaining the forced vibration response of the bridge and the parameters that affect the free vibration also affect the response to moving vehicles.
The differential equation governing the free vibration of the bridge model is obtained by setting the right-hand side in equation (2.5) equal to zero, giving:
∂ 4 yx,t( ) ∂ 2 yx,t( ) EI ++kxyx,t( ) ( ) m =0 (3.1) gg ∂x4 g ∂t 2
The displacement function yxt(), can be expressed as a product of two functions, one involving only the space coordinate x, called the mode shape function or eigenfunction zx( ) , and the other one involving the variable time and called the generalized coordinates φ(t) . Then when the bridge vibrates in its ith natural bending mode, the displacement at any location varies harmonically with time and can be expressed as [88]:
yxtiiiiiiii( ,cossin) == zx( ) φ ( t) zxa( ) ( ω t+ bω t) (3.2)
– 43 – where ωi is the ith circular frequency of the free motion of the bridge. By expressing
φi (t ) as in equation (3.2) and substituting into equation (3.1) gives:
d 4 zx( ) EI i +−kxz( ) ( x) mω2 z( x) =0 (3.3) gg d x4 igii
Dividing the bridge model into n equal segments each of length h, giving n-1 unknowns, and replacing the derivative in equation (3.3) with its finite difference approximation, the following eigenvalue problem is obtained2:
AZ= λ Z (3.4) where
c1 −41 0 −−441c2 14−−c3 41 λ1 0 A = OOOOO , λ = O (3.5a,b) 14−−c 41 0 λn−1 3 ( nn−−11x ) 14−−c2 4 0 14− c 1 ( nn−−11x )
zz11,,nL 1− 1 Zzz==[]12,,,L zn− 1 MM (3.5c) zz n,−−−11L n,n 1 1( nn−−11x )
4 mhg kx( ) kx( 23,, ) λω==+=+2 , c 561 hc4 and K h4 (3.5d,f) i i 1 23, ,K EIgg EIgg EIgg where i is the mode number and column i in Z gives the ith mode shape. The first term in equation (3.5e) for c1 must be changed from 5 to 7, if fixed end supports are assumed. If rotational springs with stiffness Km are introduced at end supports, the first term mentioned above is evaluated as: 6+(η-1) /(η+1), where η = Km h /(2Eg Ig).
2 If nothing else is mentioned, bold upper case letters are used for matrices and bold lower case letters for vectors. The transpose of a matrix or a vector is denoted with the superscript T.
– 44 –
Mode shapes and corresponding circular frequencies are now obtained numerically by solving equation (3.4). 3.1.2 Response of the bridge
Using mode superposition technique, the solution of equation (2.5) can be written as:
∞ yxt( , ) = ∑ zi ( x) φi ( t) (3.6) i=1
Substituting equation (3.6) into equation (2.5) gives:
∞ IV ∑ ( EIggi zφφφδ i++=−− kxz( ) i i mz gi&& i) ( x xv ) F (3.7) i=1
Further, both sides of the above equation are multiplied with eigenmode z j and integrated3 over the length of the bridge. Using the orthogonality properties of the eigenfunctions and setting the normalization constant4 equal to unity, the following equation is obtained (see ref. [88] for more details on derivation):
2 F &φ&i + ωi φi = − zi ()xv (3.8) mg where the contact force, F, is time depended and therefore evaluated at each time step using the expressions given in section 2.1.
When dividing the bridge model into n equal segments each of length h and setting the time step to ∆t = h / v , which corresponds to having the vehicle load applied only at the segment joints, the term zxiv( ) in equation (3.8) will be the value in the ith eigenvector zi and the row number corresponding to the point of contact (i.e. where the contact force is applied).
L 3 The properties of the Dirac function gives ∫ δ( xxzxxzx−=vi) ( ) d iv( ) 0 L 4 2 T Normalization constant is defined as ∫ zxi ( ) d x or in vector form zzii 0
– 45 –
The solution of the simple second order differential equation in (3.8) is:
2 Fz ()x (F zi ()xv + φoiωi mg ) φ& i v oi (3.9) φi = − 2 + 2 cosωi ∆t + sin ωi ∆t ωi mg ωi mg ωi
where φφoii and & o are the initial values for the step and can, when using the orthogonality and normalization relationships, be evaluated as [88]:
T φoii= zyo (3.10a)
T φ& oii= zy& o (3.10b) where yyoo and & are the bridge vertical displacement and velocity vectors at the previous time step.
Substituting into equation (3.6), the bridge displacement vector at each time step is obtained from the sum:
s 2 F z ()x (F zi ()xv + φoi ωi mg ) φ& y = z − i v + cosω ∆t + oi sin ω ∆t (3.11) ∑ i 2 2 i i i=1 ωi mg ωi mg ωi where s is the number of included modes, s ≤ n − 1. The expressions for the vertical velocity and acceleration vectors for the bridge are obtained, by differentiating equation (3.11), as:
s 2 ()F zi (xv )+ φoi ωi mg y = z − sin ω ∆t + φ& cosω ∆t (3.12) & ∑ i i oi i i=1 ωi mg
s Fz x + φ ω2 m ( i ( v ) oi i g ) &y& = zi − cosωi ∆t − φ& oi ωi sin ωi ∆t (3.13) ∑ m i=1 g
– 46 – Using the MATLAB language [53], a computer code has been developed for analysis of the dynamic response. The dynamic interaction between the bridge and the vehicle is included by utilizing the iterative scheme shown in Figure 3.1. The recalculation of bridge displacements is repeated in each step until convergence is obtained. In this study, the following convergence criterion is considered:
max( yyjj− −1 ) ≤ Tolerance (3.14) max( y j )
where y j is the jth estimate for the bridge displacement vector at the current time step. For the results presented in this study, the tolerance was set to 510⋅ −6 .
The mode superposition technique is also used to obtain the dynamic bending moment vector at each time step, giving:
s m = −Eg I g y′′ = −Eg I g ∑ z′i′φi (3.15) i=1 where zi′′ is evaluated using the finite difference approximation.
The obtained displacement, moment, velocity, and acceleration vectors are now collected in matrices for processing and plotting.
– 47 – Simulation start
Evaluate the natural frequencies and the corresponding vibration modes of the bridge
Determine the longitudinal position of the
vehicle on the bridge, xv = xv+ v ∆t
Based on previously determined quantities, set initial values for the vehicle sprung mass and for the unsprung mass predict w1, w&1,and w&&1
Solve the equations of motion of the vehicle and calculate the interaction force F(t) Move vehicle to next position
Solve equ. (3.11-3.13) and (3.15) to determine Calculate new the bridge response using mode superposition w1, w&1,and w&&1 technique
Check No convergence
Yes
No Vehicle leaves the right end of the bridge
Yes Calculate static displacements and bending moments using equ. (3.16) and (3.18)
End
Figure 3.1 Overall computational scheme
– 48 – 3.2 Static analysis
Using the same discretization technique as in section 3.1, the static displacement vector is obtained from the expression:
mmgh+ 3 st ( 12) −1 yAj =− p j (3.16) EIgg where A−1 is the inverse of the matrix given in equation (3.5a) and:
0 M 0 jth row p j = 1← (3.17) 0 M 0
st where j is the joint number where the vehicle load is applied. The obtained y j is a column vector containing the bridge static displacements corresponding to a loading case where the vehicle static load is applied at joint number j.
The static bending moment is computed using the expression given in equation (3.15). Approximating the second derivative of the displacement by its finite difference, gives st the ith element in the static moment vector m j to be equal to:
st Eg I g st st st mi, j = − ()yi+1, j − 2yi, j + yi−1, j (3.18) h2
– 49 – – 50 – Chapter ______Numerical Examples and Model Verifications ______
4.1 General
In this chapter, two beam examples and two cable-stayed bridge examples are studied and results obtained using the present model, described in chapter 2 and 3, are presented. For verification, the present model results are compared with those obtained using a commercial finite element code. The finite element code ABAQUS/Standard 5.4 [1] was used for this comparison and was run on a SUN SPARCstation 20. This code is considered as one of the most powerful finite element codes available on the market today. In ABAQUS, 2-D geometrically nonlinear analyses were performed starting from the dead load configuration and direct time integration of the dynamic response was selected with a fixed time increment. The time increment was set equal to the total time needed for the vehicle to cross the bridge divided by the number of increments. The number of increments was chosen for each numerical example so convergent results were obtained.
From ABAQUS element library, a T2D2 (2-node 2-D linear) truss element was selected to model the cables and a B23 (2-node 2-D cubic Euler-Bernoulli) beam element for the rest. The number of elements, in each numerical example, was increased until a converged solution was obtained. For each cable in the cable-stayed bridge examples, the equivalent tangent modulus of elasticity, calculated according to 3 equation (2.10), and the cable density γ c = 9000kg / m , were given in the ABAQUS input files. The vehicle was modeled using SPRING2, DASHPOT2 and MASS elements.
For the interaction problem the SLIDE LINE option was used to define the surface on which the vehicle and the bridge may interact, and the NO SEPARATION parameter
– 51 – was included in the SURFACE BEHAVIOR option. At the node where the vehicle and the bridge interact, a special purpose contact element ITT21 was used. As the position of this node is time dependent, the AMPLITUDE and the BOUNDARY options were used to modify this position for each increment.
The modeling approach used in ABAQUS for the bridge-vehicle contact problem, described above, is the simplest one, but still very complicated and requires a lot of time and experience in ABAQUS and in finite element modeling. Other modeling approaches, e.g., developing user subroutines in FORTRAN and including them in the input file for ABAQUS or using another type of contact definition, were tested for simple problems. The results obtained from these approaches agreed well with those from the above described modeling procedure but the implementation became much more complex. Therefore, results using these more complex approaches are not presented here, and all the ABAQUS curves presented in this study are those obtained using the SLIDE LINE and the ITT21 contact element approach described above.
The program developed for the present model, using the MATLAB language [53], was also run on the same SUN SPARCstation 20 machine. This made it possible to not only compare the obtained solutions but also the time consumed by the computer. In MATLAB the CPUTIME command was used to measure the CPU time needed to run the program. This value was then compared with the USER TIME value given at the end of the ABAQUS output file. For the present model solutions, the number of modes that participate in the analysis and need to be considered and the number of segments, n, which also determines the time step used for the analysis, were found using a trial and error procedure.
5 In the following examples, dynamic amplification factors for displacements (DAFd) and for bending moments (DAFm), are presented. The definition for DAFd and DAFm adopted in this study is the ratio of the absolute maximum live load dynamic response to the absolute maximum live load static response.
4.2 Simply supported bridge, moving force model
The simply supported bridge subjected to a constant force, F, moving at constant speed, v, is the easiest problem to tackle and belongs to the very few moving load problems that can be solved analytically.
5 In the literature, the impact factor defined as I=100*(DAF-1) (%) is found sometimes instead.
– 52 – Adopting the moving force model means that the influence of the inertia of the vehicle mass and the bridge-vehicle interaction are neglected. The dynamic effects are then caused only by the varying position of the force. Still this is a good approximation, see [64], for low values of the speed parameter α (defined later in equation (4.4)) and low values of the vehicle mass to bridge mass ratio.
v =68.1 m/s (α =0.25)
y(x,t) F =347000 N x
. 10 2 Eg Ig =9.92 10 Nm
xv(t) mg =11400 kg/m L =34 m
Figure 4.1 Simply supported bridge subjected to a constant moving force
Using the notations of Figure 4.1, the analytical solutions (here referred to as the exact solutions) for the displacement and bending moment are given as [23]:
FL3 96 ∞ 1 παv ixπ yxt(,)= sini t − sin(ω t) sin (4.1) 48 EI 4 ∑ 422 L i i L ggπ i=1 ii(1− α / )
FL 8 ∞ 1 παv ixπ Mxtm(,)( x,t ) = sini t − sin(ω t) sin (4.2) 4 2 ∑ 222 L i i L π i=1 ii(1− α / )
where i is the mode number, ωi the circular frequency for the ith mode of vibration, and α the non-dimensional speed parameter. ωi and α are defined as:
2 iπ EIgg ωi = (4.3) L mg
πv α = (4.4) ω1L
– 53 – The problem defined by Figure 4.1 was solved using the exact analytical model, equations (4.1-4.4), the present model, and the ABAQUS model. The first 20 natural frequencies, the normalized vertical mid-span displacement, and the normalized mid- span moment are presented in Table 4.1, Figure 4.2, and Figure 4.3, respectively. One can notice from Table 4.1 that the ABAQUS model, which uses the finite element method, always gives higher natural frequency values (stiffer solution) compared to the exact ones, whereas the present method, which uses the finite difference approximation, always gives lower values than the exact ones.
Mode Exact Present ABAQUS number 70 150 200 300 34 70 (bending) segments segments segments segments elements elements 1 4.01 4.01 4.01 4.01 4.01 4.01 4.01 2 16.03 16.02 16.03 16.03 16.03 16.03 16.03 3 36.08 36.02 36.06 36.07 36.07 36.08 36.08 4 64.13 63.96 64.10 64.11 64.12 64.13 64.13 5 100.21 99.79 100.12 100.16 100.19 100.21 100.21 6 144.30 143.43 144.11 144.19 144.25 144.31 144.30 7 196.41 194.80 196.06 196.21 196.32 196.43 196.41 8 256.53 253.79 255.93 256.20 256.38 256.59 256.54 9 324.68 320.29 323.72 324.14 324.44 324.78 324.68 10 400.83 394.15 399.37 400.01 400.47 401.03 400.85 11 485.01 475.24 482.87 483.80 484.47 485.36 485.03 12 577.20 563.39 574.17 575.50 576.44 577.79 577.24 13 677.41 658.41 673.24 675.06 676.37 678.36 677.47 14 785.64 760.13 780.02 782.48 784.23 787.11 785.72 15 901.88 868.33 894.49 897.71 900.03 904.10 902.01 16 1026.14 982.80 1016.57 1020.75 1023.74 1029.40 1026.30 17 1158.41 1103.30 1146.23 1151.55 1155.36 1163.10 1158.70 18 1298.70 1229.60 1283.40 1290.08 1294.86 1305.20 1299.10 19 1447.01 1361.43 1428.02 1436.30 1442.25 1456.00 1447.50 20 1603.34 1498.54 1580.03 1590.20 1597.49 1615.40 1604.00
Table 4.1 Comparison of the first 20 natural frequencies (Hz) for the problem defined by Figure 4.1
– 54 – A convergence study was conducted to determine the time step for the exact model, and the number of segments and elements needed for the present and the ABAQUS models. For the following numerical investigation, the 150 segments solution was chosen for the present model and the 70 elements solution with 70 increments for the
ABAQUS model. For the exact model, a time step of ∆t = L/(100 v) = 0.005 seconds was chosen and the mid-span vertical displacement and the mid-span moment were obtained from equations (4.1) and (4.2) by introducing x = L/2. For the exact model and the present model, the first 20 modes of vibration were considered.
To make the results easily understood, all moment and displacement curves presented in Figure 4.2 and Figure 4.3 are normalized by dividing with the maximum static values obtained using the three different methods. For the static response, only the results from the present model are presented, in Figure 4.2 and Figure 4.3, as all static curves from the two other solution methods coincide.
-0.2
0 Exact Present 0.2 Static (present) ABAQUS 0.4
0.6
0.8
1
1.2 Normalized vertical mid-span displacement
1.4 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.2 Vertical displacement at mid-span (normalized with respect to maximum static displacement) versus vehicle position for the problem defined by Figure 4.1
– 55 – -0.2
Exact 0 Present Static (present) 0.2 ABAQUS
0.4
0.6
0.8 Normalized mid-span moment 1
1.2 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.3 Moment at mid-span (normalized with respect to maximum static moment) versus vehicle position for the problem defined by Figure 4.1
It can be seen from the figures above that a negative dynamic mid-span moment is obtained when the vehicle enters the bridge, and that the dynamic moment curves looks similar to the dynamic displacement curves but are more irregular.
The maximum normalized displacement (corresponds to DAFd) in Figure 4.2 are 1.258, 1.258, and 1.259 for the exact model, the present model, and the ABAQUS model, respectively. The corresponding data for the bending moment (corresponds to
DAFm) are 1.093, 1.093, and 1.083, respectively. The time consumed by the computer to solve the problem using the exact model, the present model, and the ABAQUS model was about 2, 14, and 70 CPU seconds, respectively.
One should remember that in this example only the mid-span (x = L/2) vertical displacements and mid-span moments are presented, for different vehicle positions, in Figure 4.2 and Figure 4.3. However, the x-value where the maximum response occurs
– 56 – may differ from L/2. From the present model solution, values of 1.260 for x/L = 0.48 and 1.179 for x/L = 0.41 were obtained for the maximum normalized displacement and the maximum normalized moment.
For the present model, the solution of equation (3.11) converges very fast and may therefore be truncated after few modes (5 modes for the current example). However, as well known, the convergence of equation (3.15) for the bending moment is much slower. In reference [57], a better method for calculating the dynamic bending moment is described.
From Figure 4.2, Figure 4.3, and Table 4.1 one can conclude that the results from the present model are in very good agreement with those obtained using ABAQUS and the exact expressions. In addition, the results from the present model for the maximum normalized mid-span displacement can also be compared with the results 1.259, 1.258, and 1.251 given in reference [64], [20], and [93], respectively.
When comparing with the exact solution, the results also indicate that the moving load problem in ABAQUS was correctly modeled.
4.3 Multi-span continuous bridge with rough road surface
A four-span continuous bridge model with one fixed end, as shown in Figure 4.4, was adopted for this investigation. This problem was chosen to illustrate the flexibility of the proposed analysis method.
v =30 m/s y(x,t) . 4 m2 =31700 kg cs =8.6 10 Ns/m . 6 m1 =3000 kg ks =9.12 10 N/m x
. 10 2 Eg Ig =9.92 10 Nm x (t) v mg =11400 kg/m 20 m 25 m 60 m 45 m
Figure 4.4 Idealized four-span continuous bridge with one fixed end and rough road surface subjected to an idealized 2 DOF vehicle. See chapter 2 for the description of the symbols
– 57 –
In this example, the road surface roughness was considered by using the profile shown in Figure 2.6. This means that the excitation of the dynamic system was not only caused by the elastic displacements of the bridge but also by the roughness of the road surface. The 2 DOF vehicle model, consisting of one sprung mass and one unsprung mass, was adopted. The vehicle characteristics are similar to those used in reference [65] and correspond to a fundamental frequency of 2.7 Hz and a damping ratio of 8 %.
As in the simply supported bridge example, section 4.2, comparison was also made here between the present model solution and the one obtained using ABAQUS. After conducting some convergence study, the solution with 300 segments and 25 included modes was chosen for the present model and a 150 elements solution with 300 increments for the ABAQUS model. For the ABAQUS model, the computer needed approximately 570 CPU seconds to solve the problem. The corresponding time for the present model was about 240 CPU seconds which is only about 42 % of the time consumed by ABAQUS.
Considering the entire bridge, the dynamic amplification factors for displacement and bending moment, calculated using the present model, were found to be DAFd =1.396 and DAFm =1.296, respectively. These values can be compared with the DAF values 6 for the center point of the bridge DAFd =1.387 and DAFm =1.126. For the fixed end, the DAFm was as high as 1.705, which means that the dynamic moment at the fixed end is about 70 % larger than the static one. Therefore it is very important for the designer not to neglect these additional dynamic loads in the design process. Studying Figure 1.1, it is interesting to note that only the “Swiss SIA-88, single vehicle” curve considers such high DAF values.
In the following pages, mode shapes, natural frequencies, some time histories of vertical displacements, bending moments, and vertical acceleration, are presented. The presented results indicate very good agreement between the two different solutions and much less computer time consumption for the present model. One can observe that the agreement for the dynamic curves in Figures 4.6 and 4.7 is not as good as for the first numerical example, and it is not possible to infer which solution is closer to the correct one. The differences in the two solution methods are believed to cause the differences in the results, see section 4.6 for the discussion of the numerical results.
6 For a certain point P, the DAF is defined as the ratio of the absolute maximum live load dynamic response at P to the absolute maximum live load static response at the same point.
– 58 – 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147
Mode 1 (2.04 Hz) Mode 2 (4.27 Hz) Mode 3 (6.48 Hz) Mode 4 (10.00 Hz) Mode 5 (12.26 Hz)
Figure 4.5 The first 5 mode shapes calculated using the present model
Mode Present ABAQUS Mode Present ABAQUS number 300 150 number 300 150 (bending) segments elements (bending) segments elements 1 2.04 2.04 14 53.39 53.56 2 4.27 4.27 15 63.79 64.00 3 6.48 6.48 16 68.77 69.03 4 10.00 10.00 17 77.87 78.20 5 12.26 12.27 18 87.07 87.46 6 14.41 14.42 19 93.79 94.28 7 16.58 16.60 20 107.87 108.42 8 22.64 22.67 21 112.06 112.72 9 26.56 26.60 22 123.04 123.82 10 33.57 33.63 23 131.21 132.10 11 38.12 38.20 24 139.45 140.53 12 43.60 43.71 25 156.52 157.73 13 50.25 50.38
Table 4.2 Comparison of the first 25 natural frequencies (Hz) for the problem defined by Figure 4.4
– 59 – 4.0 (a) 2.0
0.0
-2.0
-4.0 Present
-6.0 ABAQUS Vertical displacement (mm) Static (present & -8.0 ABAQUS )
-10.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
4.0 (b)
2.0
0.0 Present
ABAQUS -2.0 Vertical displacement (mm) Static (present & ABAQUS )
-4.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.6 Vertical displacement histories for the problem defined by Figure
4.4: (a) at the center of the bridge (x = 75 m); (b) at the center of
the last span (x = 127.5 m)
– 60 – 1000 (a)
0
-1000
-2000 Present
Bending moment (kNm) ABAQUS -3000 Static (present & ABAQUS ) -4000 0 0.2 0.4 0.6 0.8 1 Vehicle position, x v /L
1500 (b)
500
-500
-1500 Present
Bending moment (kNm) ABAQUS -2500 Static (present & ABAQUS ) -3500 0 0.2 0.4 0.6 0.8 1 Vehicle position, x v /L
Figure 4.7 Bending moment histories for the problem defined by Figure 4.4:
(a) at the center of the bridge (x = 75 m); (b) at the center of the last
span (x = 127.5 m)
– 61 – 15.0 (a) Present
5.0 ABAQUS
-5.0
-15.0 Vertical displacement (mm)
-25.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
4.0 (b) )
2 2.0
0.0
-2.0 Vertical acceleration (m/s Present
ABAQUS
-4.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.8 Displacement and acceleration histories for the sprung mass of the vehicle: (a) sprung mass vertical displacement; (b) sprung mass vertical acceleration
– 62 – 4.4 Simple cable-stayed bridge
A simple fan-shaped cable-stayed bridge with rough road surface and with only the main span suspended, was chosen for this numerical investigation. The geometrical configuration and the mechanical properties are shown in Figure 4.9. Optimal values, see [9], were chosen for the ratios of side span to the main span length and pylon height to main span length, corresponding to 1/3 and 1/5, respectively. Rotational springs with stiffness Km were introduced at end supports of the main span girder to simulate the influence of the approach spans stiffness. The girder is also supported vertically at the pylons but is totally independent of the pylons. The chosen dead weight, qg, (including surfacing, cables, anchorages, railings, etc.) is equivalent, for example, to 0.50 m of concrete for a bridge width of 14 m. The chosen live load, qq, corresponds to a uniformly distributed load of 4.3 kN/m2, which also allows for point loads. This gives a load ratio of qq /qg = 0.35, close to that of practice. The allowable cable stress, σa , corresponds to 45 % of the chosen ultimate tensile strength for the cables which is 1670 MPa. The cross-sectional area of the cables and the corresponding cable spring stiffness were evaluated using the expressions given in section 2.2.3.
The characteristics of the 2 DOF vehicle model and the road surface roughness profile are similar to those used in the previous example. If nothing else is stated in connection with the presented figures in the following pages, this 2 DOF vehicle model and the speed of 30 m/s (108 km/h) were used when calculating the response.
y(x,t)
v =30 m/s
Km Km 30 m 40 m x 10 m
xv (t) 50 m 150 m 50 m
Figure 4.9 Idealized fan-shaped cable-stayed bridge with rough road surface 11 2 subjected to an idealized 2 DOF vehicle. Ec =2.0·10 N/m , 11 2 4 11 2 4 Eg=0.30·10 N/m , Ig=4 m , Ep=0.30·10 N/m , Ip=10 m , 6 2 4 4 4 σa =750·10 N/m , qg=17·10 N/m, qq=6·10 N/m, and γ c =9·10 N/m3. See chapter 2 for the description of the symbols
– 63 – Satisfactory results were obtained by discretizing the bridge girder into 150 elements in ABAQUS and using 300 increments, while the present model required 300 segments with 25 included modes. The computer time consumed for the two models was about 625 and 325 CPU seconds, respectively.
Mode shapes, natural frequencies, some time histories of vertical displacements and bending moments, and the influence of the rotational spring stiffness, vehicle speed, different vehicle models and road surface roughness on the dynamic response, are presented. The results are shown for two alternatives, namely, fixed and free pylon top.
Studying Table 4.3, Figures 4.11 and 4.12, one can observe good agreement between the solutions of the two different methods, specially for the fixed pylon top alternative. For the free pylon top curves, the simple expressions derived in section 2.2.3, which consider the anchor cable stiffness and the pylon stiffness, gave larger static and dynamic displacements for the present model, compared with the ABAQUS model. The bending moment responses shown in Figure 4.12 are underestimated, by the present model, at xv /L = 0.5. However, by increasing the number of modes considered to 50, a better agreement will be obtained at this vehicle position, and of course more CPU seconds will be consumed. See section 4.6 for discussion of the results. 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 Mode 1 (0.94 Hz) Mode 2 (1.56 Hz) Mode 3 (2.53 Hz) Mode 4 (3.93 Hz) Mode 5 (5.73 Hz)
Figure 4.10 The first 5 mode shapes for the girder calculated using the present
model for fixed pylon top and with Km=∞
– 64 –
Fixed pylon top Free pylon top Mode Present ABAQUS Present ABAQUS number 300 150 300 150 (bending) segments elements segments elements 1 0.94 0.94 0.76 0.76 2 1.56 1.55 1.38 1.42 3 2.53 2.53 2.41 2.50 4 3.93 3.92 3.85 3.92 5 5.73 5.71 5.68 5.71 6 7.92 7.90 7.88 7.89 7 10.49 10.46 10.47 10.46 8 13.44 13.40 13.42 13.40 9 16.77 16.72 16.75 16.72 10 20.46 20.41 20.45 20.41 11 24.52 24.47 24.51 24.47 12 28.96 28.90 28.95 28.90 13 33.76 33.72 33.75 33.72 14 38.93 38.78 38.92 38.78 15 44.47 44.52 44.46 44.52 16 50.37 50.30 50.37 50.30 17 56.64 56.62 56.64 56.62 18 63.27 63.26 63.27 63.26 19 70.27 70.29 70.27 70.29 20 77.63 77.68 77.63 77.68 21 85.35 85.44 85.35 85.44 22 93.44 93.57 93.44 93.57 23 101.88 102.07 101.88 102.07 24 110.68 110.94 110.68 110.94 25 119.85 120.18 119.85 120.18
Table 4.3 Comparison of the first 25 natural frequencies (Hz) for the problem
defined by Figure 4.9 with Km=∞. The number of ABAQUS elements includes only elements for the girder
– 65 – 2.0 (a) 0.0 Present
-2.0 ABAQUS Static (present & -4.0 ABAQUS ) -6.0
-8.0
-10.0 Vertical displacement (mm)
-12.0
-14.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
4.0 (b) Present 0.0 ABAQUS
-4.0 Static (present) Static(ABAQUS ) -8.0
-12.0 Vertical displacement (mm) -16.0
-20.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.11 Vertical displacement histories at the center of the bridge (x = 75 m)
calculated for rough road surface and with Km=∞: (a) fixed pylon top; (b) free pylon top
– 66 – 1000 (a)
0
-1000
-2000 Present
Bending moment (kNm) ABAQUS -3000 Static (present & ABAQUS ) -4000 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
1500 (b)
500
-500
-1500
-2500 Present
Bending moment (kNm) ABAQUS -3500 Static (present) Static (ABAQUS ) -4500 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.12 Bending moment histories at the center of the bridge (x = 75 m)
calculated for rough road surface and with Km=∞: (a) fixed pylon top; (b) free pylon top
– 67 – (a) 1.125 For the entire bridge
At point x=75 m 1.115
d 1.105 DAF 1.095
1.085
1.075 0 0.2 0.4 0.6 0.8 1 1.2 12 Spring stiffness, K m (10 Nm)
2.0 (b)
1.8
1.6 m DAF 1.4 For the entire bridge At point x=75 m At point x=150 m 1.2
1.0 0 0.2 0.4 0.6 0.8 1 1.2 12 Spring stiffness, K m (10 Nm)
Figure 4.13 The influence of the rotational spring stiffness, Km, on the dynamic amplification factors for displacement and bending moment (rough
road surface and fixed pylon top): (a) DAFd; (b) DAFm
– 68 – 1.40 (a) Fixed pylon top, rough road 1.35 Fixed pylon top, smooth road
=30 m/s Free pylon top, rough road 1.30 v
1.25 d 1.20 DAF 1.15
1.10
1.05
1.00 35 50 65 80 95 110 125 140 155 170 185 200 215 Vehicle speed (km/h)
2.0
(b) Fixed pylon top, rough road
1.8 =30 m/s Fixed pylon top, smooth road v Free pylon top, rough road
1.6 m DAF 1.4
1.2
1.0 35 50 65 80 95 110 125 140 155 170 185 200 215 Vehicle speed (km/h)
Figure 4.14 The vehicle speed influence on the dynamic amplification factors for
displacement and bending moment calculated for Km=∞: (a) DAFd
for the entire bridge; (b) DAFm at point x = 150 m
– 69 – 2.0 (a) 0.0 Sprung mass model Moving mass model -2.0 Moving force model Static -4.0
-6.0
-8.0
-10.0 Vertical displacement (mm)
-12.0
-14.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
1.0
Sprung mass model -1.0 (b) Moving mass model Moving force model -3.0 Static
-5.0
-7.0
-9.0 Vertical displacement (mm) -11.0
-13.0 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.15 Vertical displacement histories at the center of the bridge (x = 75 m) calculated using different vehicle models (fixed pylon top and
Km=∞): (a) rough road surface; (b) road surface with no roughness
– 70 – 1.25 g
) Sprung mass, rough road 2
m Sprung mass, smooth road +
1 1.15
m Moving force ) / ( t (
F 1.05
0.95
0.85 Normalized contact force, 0.75 0 0.2 0.4 0.6 0.8 1 Vehicle position, x v /L
Figure 4.16 Normalized contact force calculated for fixed pylon top and with
Km=∞
The presented curves, in Figure 4.13, for the influence of the rotational springs on the
DAF values, indicate fast convergence with increasing Km value. The same DAF values, as if totally fixed end supports were assumed, are obtained already for a Km 12 value of 1.2·10 Nm. This Km value corresponds to a value of 6.43 for the first term for c1 in equation (3.5e). As mentioned earlier in section 3.1.1, a value of 5 corresponds to having a free end support and 7 to a fixed end support.
From Figure 4.14, it can be seen that very high DAF values can be obtained already at normal vehicle speeds. For example, for a speed of 108 km/h, fixed pylon top, rough road surface and Km = ∞, the obtained DAF values for displacement and bending moment, considering the entire bridge, are DAFd =1.093 and DAFm =1.531, respec- tively. For the fixed end, the DAFm obtained was as high as 1.745. Figure 4.15 and 4.16 indicate that for normal vehicle speeds, most of the excitation of the dynamic system is caused by the roughness of the road surface and very little is caused by the elastic displacement of the bridge itself. It can also be concluded that, when assuming a road surface with no roughness and a normal vehicle speed, the moving force model
– 71 – is fully adequate for this example and there is no need for using complicated vehicle models. 4.5 Three-span cable-stayed bridge
For this fourth and last example, a symmetric three-span fan-shaped cable-stayed bridge was adopted with the geometrical configuration and the mechanical properties shown in Figure 4.17. The horizontal displacements of the pylon tops were neglected in this study as it was assumed that these displacements are small due to having the side spans vertically fixed at four points. The cross-sectional area of the cables and the corresponding cable spring stiffness were evaluated using the expressions given in section 2.2.3. The 2 DOF vehicle model was used with the same characteristics as for the last two examples.
The effect of local irregularities was investigated by simulating a 3 cm pot-hole
located at about x =150 m. This pot-hole represents a local defect in the road surface or an initial joint defect.
v =30 m/s y(x,t) 80 x 3 cm pot-hole xv(t) 128 m 400 m 128 m (8 x 16 m) (25 x 16 m) (8 x 16 m)
Figure 4.17 Idealized fan-shaped cable-stayed bridge subjected to an idealized 2 11 2 11 2 4 DOF vehicle. Ec =2.0·10 N/m , Eg=0.30·10 N/m , Ig=10 m , 6 2 4 4 4 σa =750·10 N/m , qg=17·10 N/m, qq=6·10 N/m, and γ c =9·10 N/m3. See chapter 2 for the description of the symbols
If nothing else is stated in connection with the presented figures in the following pages, the bridge characteristics given in Figure 4.17 and the vehicle characteristics given in Figure 4.4 were used when calculating the response.
Satisfactory results were obtained by discretizing the bridge girder into 369 elements in ABAQUS and using 738 increments, while the present model required 533 segments with 25 included modes. The computer time consumed for the two models was about
– 72 – 4000 CPU seconds for the ABAQUS model and 1000 CPU seconds for the present model. This indicates that the present method involves much less computation, and therefore is much more computationally efficient, compared to the finite element method with direct time integration.
Mode shapes, natural frequencies, some time histories of vertical displacements and bending moments, and the influence of some parameters (vehicle speed, pylon height and vehicle spring stiffness) on the dynamic amplification factors, are shown in the following pages.
It can be seen from Table 4.4 that the natural frequencies of this cable-stayed bridge are very close to one another, and that excellent agreement is obtained especially when comparing with the ABAQUS values calculated neglecting cable mass. The ABAQUS frequencies calculated neglecting cable mass ( γ c = 0) are only presented in Table 4.4 for comparison and have not been used for dynamic response calculations.
The obtained DAF values, for the problem defined by Figure 4.17, are: for the entire bridge DAFd =1.056 and DAFm =1.118; for the center point DAFd =1.046 and DAFm =1.074; for the point at x =128 m DAFm =1.179; and for the point at x = 528 m DAFm =1.264.
Mode 1 (0.50 Hz) Mode 2 (0.66 Hz) Mode 3 (0.84 Hz) Mode 4 (1.07 Hz) Mode 5 (1.39 Hz)
0 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 512 544 576 608 640
Figure 4.18 The first 5 mode shapes for the girder calculated using the present model
– 73 –
Mode Present ABAQUS ABAQUS number 533 segments 369 elements 369 elements 4 3 (bending) γ c = 9·10 N/m γ c = 0 1 0.50 0.49 0.50 2 0.66 0.65 0.66 3 0.84 0.83 0.84 4 1.07 1.06 1.07 5 1.39 1.37 1.39 6 1.80 1.78 1.80 7 2.31 2.29 2.31 8 2.91 2.88 2.91 9 3.59 3.56 3.60 10 4.36 4.32 4.36 11 5.20 5.15 5.20 12 6.08 6.04 6.09 13 6.65 6.61 6.66 14 6.73 6.68 6.74 15 7.13 7.07 7.14 16 7.90 7.85 7.91 17 8.29 8.25 8.31 18 8.69 8.64 8.71 19 9.52 9.45 9.54 20 10.34 10.29 10.37 21 10.79 10.75 10.83 22 11.34 11.29 11.38 23 12.33 12.25 12.36 24 13.10 13.06 13.16 25 13.41 13.38 13.47
Table 4.4 Comparison of the first 25 natural frequencies (Hz) for the problem defined by Figure 4.17. The number of ABAQUS elements includes only elements for the girder
– 74 – Studying Figure 4.19 and Figure 4.20, it is noted that the present solution is somewhat stiffer than the ABAQUS solution. This is probably due to the differences in the two models. In the ABAQUS model the cables are placed 16 meters apart, whereas in the present model there is always a spring7 supporting the girder beneath each vehicle position. Still, from an engineering point of view, one can say that the ABAQUS and the present model results exhibit good matches, because the important values in Figure 4.19 and Figure 4.20 are those giving the maximum response.
Figure 4.21 shows the variation of the DAF with the speed of the vehicle for three different girder moment of inertia. It can be seen that high DAF values are obtained for low vehicle speeds. Thus for this case, limitation of the speed of vehicles will not necessarily avoid damage effects on bridges.
The effect of pylon height on the DAF is shown in Figure 4.22. It is obvious that increasing pylon height results in lower DAF values.
0
-5
Present -10 ABAQUS
-15 Static (present & ABAQUS ) Vertical displacement (mm) -20
-25 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.19 Vertical displacement histories at the center of the bridge (x = 328 m)
7 Used to idealize the cables.
– 75 – 2000 (a) 1000
0
-1000
-2000 Present
-3000 ABAQUS
Bending moment (kNm) -4000 Static (present & ABAQUS ) -5000
-6000 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
4000 (b) Present 3000 ABAQUS
2000 Static (present & ABAQUS )
1000
0 Bending moment (kNm)
-1000
-2000 0 0.2 0.4 0.6 0.8 1
Vehicle position, x v /L
Figure 4.20 Bending moment histories: (a) at the center of the bridge (x = 328 m);
(b) at the first pylon (x = 128 m)
– 76 – 1.12 4 (a) Ig=5Ig=5 m4m 1.11 /s 4
m Ig=10Ig=10 m4m
1.10 30 4 = Ig=30Ig=30 m4m v 1.09
d 1.08
DAF 1.07
1.06
1.05
1.04
1.03 50 70 90 110 130 150 170 Vehicle speed (km/h)
1.80 (b) 4 1.70 Ig=5Ig=5 m4m 4 Ig=10Ig=10 m4m 4
1.60 =30 m/s Ig=30Ig=30 m4m v 1.50 m 1.40 DAF 1.30
1.20
1.10
1.00 50 70 90 110 130 150 170 Vehicle speed (km/h)
Figure 4.21 The influence of vehicle speed on the dynamic amplification factors
for displacement and bending moment calculated for different Ig
values: (a) DAFd for the entire bridge; (b) DAFm at point x = 128 m
– 77 – 1.060
(a) For the entire bridge 1.055 At point x=328 m
1.050 d 1.045 DAF
1.040 m 80 =
1.035 H
1.030 50 60 70 80 90 100 110 120 130 140 150 Pylon height, H (m)
1.30 ( At point x=128 m At point x=328 m 1.25 At point x=528 m (b) For the entire bridge =80 m
1.20 H m 1.15 DAF
1.10
1.05
1.00 50 60 70 80 90 100 110 120 130 140 150 Pylon height, H (m)
Figure 4.22 The influence of pylon height on the dynamic amplification factors
for displacement and bending moment: (a) DAFd; (b) DAFm
– 78 – 1.07 (a)
1.06
1.05 d
DAF 1.04 At point x=328 m For the entire bridge N/m 6 10 . 1.03 =9.12 S k 1.02 2.28 4.56 6.84 9.12 11.4 13.68 15.96 18.24 6 Vehicle spring stiffness, k S (10 N/m)
(1.35 Hz) (1.91 Hz) (2.34 Hz) (2.70 Hz) (3.02 Hz) (3.31 Hz) (3.57 Hz) (3.82 Hz) 1.50 (b) N/m 1.40 6 10 . =9.12
1.30 S k m 1.20 DAF
1.10 At point x=128 m At point x=328 m 1.00 At point x=528 m For the entire bridge 0.90 2.28 4.56 6.84 9.12 11.4 13.68 15.96 18.24 6 Vehicle spring stiffness, k S (10 N/m)
Figure 4.23 The influence of vehicle spring stiffness on the dynamic amplification factors for displacement and bending moment:
(a) DAFd; (b) DAFm
– 79 – Figure 4.23 shows the variation of the DAF with the vehicle spring stiffness. It is clear that increasing vehicle spring stiffness results in increasing contact forces and increasing DAF values. The reason for this is believed to be that the vehicle sprung mass will more and more become unsprung as the spring stiffness increases.
From Figure 4.22 and Figure 4.23 we can infer that the DAF can reach high values for the girder near the pylons, and that the DAF at the center of the bridge are comparatively small. This situation should be considered in the design practice of the bridge. These high DAF values, for the girder near the pylons, are believed to be caused by the higher modes (mode number 8 and 7), which have natural frequencies close to that of the vehicle (2.7 Hz). Mode number 8 has a zero point at the center of the bridge (x = 328 m) and maximum points close to the pylons.
4.6 Discussion of the numerical results
In this chapter, four problems have been studied in order to demonstrate the efficiency and the validity of the proposed model. The results obtained using the present model are generally in good agreement with those obtained using the ABAQUS model.
For the first numerical example, extremely good agreement is observed between the three different solutions, and it was not easy to produce Figures 4.2 and 4.3 so all curves can be distinguished. The reason for this extremely good agreement is the simplicity of the problem, as no surface roughness was included and the moving force model was adopted.
As the problems studied got more complicated due to having rough road surface and more complex bridge and vehicle models, the differences between the ABAQUS solutions and the present model solutions increased. As a general remark, one should bear in mind that both methods are approximate, and it is not always possible to infer which solution is closer to the correct one.
The main parameters that are believed to cause the differences in the dynamic response are:
• cable spacing. For large cable spacing compared to bridge length, the present model will give stiffer solutions due to using continuously distributed vertical spring to idealize the cables
– 80 – • location of the point studied. Differences can also be the result of studying the response of a point situated between two cables instead of a point adjacent to a cable. For the same reason as above, the present model will give stiffer solutions for points between cables • cable mass. The cable mass is neglected in the present model but not in the ABAQUS model. Using the present model, this will also lead to stiffer solutions • the surface roughness. For the present model, the surface roughness profile, r(x), and its derivatives were given at each segment joint, while for the ABAQUS model the roughness profile was considered when giving the vertical location of each node. If more increments are needed than the number of nodes, the vehicle locations will not always coincide with one of those nodes, and it is not clear how ABAQUS evaluates the vertical locations of these inter nodal points situated on the SLIDE LINE • numerical damping. The influence of the automatically introduced numerical (artificial) damping on the ABAQUS solution. For problems tested with no numerical damping, bad numerical stability was observed.
– 81 –
– 82 – Chapter ______Conclusions and Suggestions for Further Research ______
5.1 Conclusions of Part A
The present work has reviewed previous research conducted in the field of dynamic response of bridges subjected to moving vehicles, and presented a simplified analysis method for evaluating the dynamic response of cable-stayed bridges.
For the evaluation of the dynamic response due to moving vehicles, a computer code was developed which fully consider the dynamic interaction between the bridge and the vehicle. Time histories and the influence of some parameters, e.g. vehicle speed, on the dynamic amplification factors were presented. The results were compared with results obtained using the commercial finite element code ABAQUS.
Good agreement was obtained between the ABAQUS results and the results from the present model. Moreover, the presented model proves to be simple to use, computationally economical, and the finite difference method serves as an efficient tool for analysis of such problems. However, the major drawback for the present model is that it is not capable of considering the interaction between the side spans and the main span through the cables. As a conclusion, referring to the discussion in section 4.6 of the numerical results, it has been shown that the theoretical formulations used are valid and that the present model correctly handles the moving load problem.
There are several conclusions that can be drawn from the numerical results presented in the previous chapter:
– 83 – • the surface roughness has great effect on the dynamic response. Thus, when calculating the DAF, the roughness of the road surface should be considered. To reduce damage to bridges, attempts should be made to eliminate irregularities in the deck, in the approach pavements and over bearings • it was found that limitation of the speed of vehicles will not necessarily avoid damage effects on bridges • in some cases greater dynamic amplification factors can be obtained than those given by some of the current bridge design codes • as expected, if the vehicle travels at a normal speed and smooth bridge deck surface is assumed, and if the ratio of the vehicle mass to bridge mass is very small, the effect of bridge-vehicle interaction is small compared to the bridge inertia. Therefore, for this case, the moving force solution represents an acceptable approximation for the sprung mass solution.
The simplified analysis method presented in this part of the thesis can be used for dynamic analysis of conventional type of bridges, and for comparing different design alternatives of cable-stayed bridges in the feasibility design stage. For the final design stage, and specially for long span cable-stayed bridges, the author recommends carrying out a three-dimensional finite element analysis, as the nonlinearities, cable masses, axial forces, and the three-dimensional motion of the structure may no longer be ignored.
For the present model, it is clear that the modeling assumptions ignore several factors that may significantly affect the response, particularly those related to cable mass, bridge damping, and vehicle configuration. Moreover, only fan-shaped cable-stayed bridges were considered in this study. However, the developed computer code is general and flexible and can be improved to handle cable-stayed bridges with other cable configurations (harp-shaped or modified fan-shaped), more realistic traffic loads such as a sequence of moving vehicles, or more realistic bridge structures including damping and with variable mass and flexural rigidity of the stiffening girder.
The results obtained in the previous chapter indicate that the dynamic behavior of bridges due to moving vehicles is too complicated to be approximated using a simple formula for dynamic amplification factors, as adopted in many of the current bridge design codes. This study and previous studies have shown that parameters such as
– 84 – span length (or fundamental frequency of the bridge), road surface condition, bridge damping, and design vehicle speed should be considered in these DAF formulas.
Conducting this study, it was also found that very few commercial finite element codes are capable of handling the bridge-vehicle interaction problem and the moving load problem. The commercial finite element codes capable of handling such problems are very expensive and require powerful computers. The proposed model on the other hand has shown to be computationally efficient and can be simply implemented, e.g. using FORTRAN, MATLAB [53], or even Microsoft Excel, on an ordinary PC, giving a much more economical solution.
5.2 Suggestions for further research
Based on the present study, the following suggestions for further research can be given:
• referring to the discussion in section 4.6 of the numerical results, a sensitivity study need to be carried out to investigate the effects of cable spacing, cable mass neglection, variation in natural frequencies, etc., on the present model results. This is necessary to find the limits of the proposed simplified analysis method • further work is needed to develop the expressions derived in chapter 2 and the implemented code, in order to study the response of cable-stayed bridges with other cable configurations and to include the effect of bridge damping and axial forces • investigate the effect of span length, bridge damping, bridge-vehicle mass ratio, multiple vehicles, and torsional loading on the dynamic response of more realistic cable-stayed bridge models • study the effect of vehicle braking on the dynamic response of cable-stayed bridges • study the effectiveness of tuned mass dampers (TMD) on suppressing vibrations induced by moving vehicles • perform extensive instrumentation and testing on an existing cable-stayed bridge, or on a laboratory model, to further enhance the understanding of the behavior of this type of bridge structures, and to verify theoretical models.
To study the above listed theoretical topics, the author recommends using the finite element method. This is necessary if more realistic three-dimensional bridge and
– 85 – vehicle models are to be used for analysis. Axial forces, geometric nonlinearities, torsional modes of vibration, cable modes of vibration, and bridge damping can then easily be considered in the analysis.
– 86 – ______Bibliography of Part A ______
[1] ABAQUS User’s Manual, Hibbitt, Karlsson & Sorensen, Inc., Providence, Rhode Island, 1994.
[2] Abdel-Ghaffar A.M., Nazmy A.S., ‘3-D Nonlinear Seismic Behaviour of Cable- Stayed Bridges’, J. Struct. Eng., ASCE, Vol. 117, No. 11, Nov. 1991, pp. 3456-3476.
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– 95 –
– 96 –
Part B
Refined Analysis Utilizing the Nonlinear Finite Element Method
– 97 –
– 98 – Chapter ______Introduction ______
6.1 General
Although several long span cable-stayed and suspension bridges are being build or proposed for future bridges, little is known about their dynamic behavior under the action of moving traffic loads. Cable supported bridges are complex structures consisting of various structural components with different properties. Thus, to take account of the complex structural response and to more realistically predict their response due to traffic loading, a detailed structural analysis is required. Various studies of the dynamic response due to moving vehicles have been conducted on ordinary bridges, see the state-of-the-art review in Part A of this thesis. However, due to reasons stated above, they cannot be directly applied to modern cable supported bridges.
Due to the low damping, lightweight and high flexibility of modern long span cable- stayed and suspension bridges, vibrations induced by traffic can be a serious problem. Vibration effects should play a much more dominating role for the design and not be underestimated, as the ratio between traffic load and dead load can be larger for these modern bridges. It is well known that long-term vibration of the bridge deck and cables (in particular the cables in cable-stayed bridges) might enhance and accelerate the fatigue damage on the bridge. Costly repairs and modifications have been undertaken on relatively new suspension and cable-stayed bridges, because the possibility of fatigue caused by traffic-induced vibrations had not been sufficiently investigated at the design stage. In addition, dynamic forces of heavy vehicles can lead to bridge deterioration and eventually increasing maintenance costs and decreasing service life of the bridge structure. Therefore it ought to be evident that it is important for engineers to not only use design code formulas but also be able to accurately
– 99 – investigate and understand the vibration effects at the design stage. At this stage the bridge structure can readily be modified, rather than having to make costly modifications later on.
For cable supported bridges and in particular long span cable-stayed bridges, energy dissipation is very low and is often not enough on its own to suppress vibrations. To increase the overall damping capacity of the bridge structure, one possible option is to incorporate external dampers1 into the system. Such dampers can be found today on many existing cable supported bridges [33, 58]. However, it is not always easy to find a location with significant relative movements and enough space to accommodate these devices. Moreover, this is not always the most effective and economic solution. The application of such devices for bridge structures is discussed later on in Chapter 8. The damping characteristics and damping ratios of cable supported bridges are also discussed and a practical technique for deriving the damping matrix from modal damping ratios, is presented.
Since the cable supported bridge structures to be analyzed in this part of the thesis are flexible and can undergo large displacements, the nonlinear finite element method is utilized considering all sources of geometric nonlinearity. A beam element, which includes geometrically nonlinear effects and is derived using a consistent mass formulation, is adopted for modeling the girder and the pylons. Whereas, a two-node cable element derived using “exact” analytical expressions for the elastic catenary, is adopted for modeling the cables.
Two approaches for evaluating the dynamic response are adopted. In the first approach, the response is evaluated using the mode superposition technique utilizing the deformed dead load tangent stiffness matrix. This is a linear dynamic procedure based on results from a nonlinear static analysis. This approach has won considerable popularity in spite of its limitations, as one usually only need to consider the first dominant modes of vibration to obtain sufficiently accurate results. The second approach evaluates the nonlinear dynamic response using a direct time integration method combined with a nonlinear solution procedure. Here, the so-called Newton- Newmark algorithm is adopted. This is a much more CPU time-consuming approach. Still, nonlinear dynamic analysis is essential if it is believed that the bridge will not
1 Discrete damping devices such as viscous dampers and tuned mass dampers. A tuned mass damper (TMD) is a vibration absorber tuned to a particular mode of the bridge and consists of a mass, a viscous damper and a linear spring.
– 100 – behave linearly during the application of traffic loads. If this is the case, the natural frequencies will vary with the amplitude of response and linear dynamic analysis will consequently be inadequate.
6.2 Cable structures and cable modeling techniques
The increasing attention on cable structures is not only due to their inherent beauty but also to their stubborn nature in not easily revealing the secret of their nonlinear behavior. Cable structures exhibit geometrically nonlinear behavior, they are very flexible and undergo large displacements before attaining their equilibrium configuration. As an example, due to this inherently nonlinear behavior, conventional linear dead load analysis which assumes small displacements is often not applicable [49, 55], except in special cases in the feasibility design stage. In the final design stages however, a refined nonlinear dead load analysis procedure should be adopted.
A brief early history of the research on the behavior of cables has been published in [34], as well as a more recent history in [39]. Methods of static and dynamic analysis and the behavior of cable structures are thoroughly presented in [20, 27, 34, 49]. In [31, 32, 39], trial-and-error search procedures have been proposed for the nonlinear computer analysis of simple cable problems. For cable roof structures, analyses methods and several very illustrative design details, e.g. details of connections, are presented in [14].
For cable supported bridges, i.e. suspension and cable-stayed bridges, the trend today is to use more shallow and slender stiffening girders combined with increasing span lengths. For that reason, it is highly desirable in bridge engineering to develop accurate procedures that can lead to a thorough understanding and a realistic prediction of the structural response. Although several investigators [2, 6, 7, 12, 13, 22, 40, 41, 55] studied the nonlinear behavior of cable supported bridges, very few [2, 7] tackled the problem of using cable elements for modeling the cables.
Commercial finite element codes used in civil engineering today cannot be readily used for modeling and analysis of modern cable structures as they lack suitable cable elements that can accurately model the actual cable curvature. As the cable represents a flexible member with virtually no resistance to applied moments, the idea of replacing each cable by a bar element with equivalent cable stiffness or by several
– 101 – beam elements with negligible moment of inertia has found wide acceptance and has been adopted by many investigators and designers using commercial codes.
As the popularity of cable structures has increased, the search for more efficient methods has intensified and today various other cable modeling techniques, than the crude modeling with a bar element mentioned above, can be found in the literature. In [25], a 2-node curved finite element was developed, using cubic polynomial interpolation functions, and used for the static and dynamic analysis of 3-D prestressed cable nets. In [60], another 2-node curved finite element was developed using Lagrangian functions for the interpolation of element geometry. In [11, 49], derivations of isoparametric cable elements which includes the element curvature are presented, and in [7] a four-node isoparametric cable element is presented and used for modeling cables in cable-stayed bridges.
An iterative analysis procedure for cables, based on using “exact” analytical expressions for the elastic catenary, was suggested in [56, 57]. This approach was later adopted by other investigators, developed and used for the analysis of very simple cable structures [35, 63] and of power transmission lines [64]. In [34], the same approach was also suggested for the analysis of cable structures with appreciable sag and the applicability of this method was later demonstrated in [2] on numerical examples of cable supported bridges.
The cable element used in this thesis is derived using the “exact” analytical expressions for the elastic catenary given in [35]. The procedure presented later in section 7.2.1, to derive the element matrices, is similar to that described in [2, 34]. However, the analytical expressions for the elastic catenary adopted here are somewhat simpler and therefore easier to handle. The presented element can be used for modeling large sag cables such as suspension bridge main cables, cables in long span cable-stayed bridges, cables in large cable roofs, etc., where straight elements are not readily applicable. Despite the fact that the cable modeling technique based on the expressions given in [35] has been available for many years it has, at least to the author’s knowledge, very seldom been used for analysis of cable supported bridges.
The expressions of the internal force vectors and tangent stiffness matrices for the elements used were derived using the Maple software package for symbolic computations [52]. Samples of these Maple procedures are given in Appendix A.
– 102 – 6.3 General aims of the present study
The main objective of the work presented in this part is to study the response of more realistic, than those studied in Part A, two-dimensional bridge models considering bridge damping, exact cable behavior, nonlinear geometric effects, non-uniform cross- sections, and variable material properties. For this reason a more straightforward and general approach, based on the nonlinear finite element method, is developed to handle such analysis difficulties and allow a thorough study of the moving load problem of cable supported bridges.
The main aims of this study are as follows:
• to implement two approaches (linear dynamic and nonlinear dynamic) for evaluating the response and to find whether linear dynamic traffic load analysis is adequate when investigating the behavior of cable supported bridges under the action of moving traffic
• to better understand and outline the influence of different parameters on the behavior of cable-stayed bridge. The parameters that are believed to significantly influence the dynamic response and therefore considered in this study are: bridge- vehicle interaction, road surface roughness, vehicle speed, number of vehicles on the bridge, and bridge damping
• to investigate the influence of the bridge girder supporting condition on the response of the bridge. In addition, to study the effect of multi-element cable discretization2 on the dynamic response
• to study the efficiency of a so-called tuned mass damper (TMD) on suppressing traffic-induced vibrations and increasing the overall damping of the bridge.
Since, in this study, the main concern is to investigate the dynamic response of bridges and not the dynamics of the vehicle itself and since the spans of cable supported bridges are considerably larger than the vehicle axle base, a very simple vehicle model is adopted. This model is an improved version of the one used in Part A as it here includes both primary and secondary vehicle suspension systems. Since the main aim is not to develop design formulas for calculating the dynamic amplification factors,
2 Each cable is modeled with several catenary cable elements to include cable modes of vibration and the dynamic interaction between the vibrating cables and the bridge.
– 103 – only hypothetical trains of moving traffic are simulated and used for analysis. The effect of using more complex vehicle models, than the one mentioned above, or finer bridge models (more elements for discretizing the bridge girder and pylons) have not been investigated in this study.
For the purpose of this study, computer codes, fully capable of handling the above stated important factors and parameters, are implemented using the MATLAB language [53]. The developed codes have been tested by comparing results against those obtained using the commercial finite element code ABAQUS [1].
Parts of the work presented in the following chapters, concerning nonlinear finite element modeling, static dead load analysis, and frequency analysis of cable supported bridges, have been included in the paper ‘Some Modeling Aspects in the Nonlinear Finite Element Analysis of Cable Supported Bridges’. This paper has been accepted for publication in the journal Computer & Structures.
– 104 – Chapter ______Nonlinear Finite Elements ______
7.1 General
The cable supported structures considered in this study are cable-stayed and suspension bridges. Such bridges consist of cables, pylons and girders (bridge decks) and are usually modeled using beam and bar elements for the analysis of the global structural response [28, 41]. In the following, an alternative approach is presented where accurate and efficient cable and beam elements are used for the modeling. All sources of geometric nonlinearity, i.e. change of cable geometry under different tension load levels (cable sag effect), change of the bridge geometry due to large displacements, and axial force-bending moment interaction in the bridge deck and pylons (P-δ effect), are considered in the present study.
For simplicity the present study focuses on two-dimensional problems. Consequently, torsional effects and torsional modes of vibration are disregarded. As modern cable supported structures are flexible three-dimensional structures, two-dimensional models are of course not adequate when studying the response of such structures under the action of environmental loads like wind, traffic, and earthquakes [4, 28]. However, simplified two-dimensional bridge models are still very useful for bridge designers in the preliminary design stage, e.g. for investigating the feasibility of alternative structural solutions. For the interested reader, accurate three-dimensional cable and beam elements can be found in [35, 61].
The formulation of cable and beam element matrices will be described in the following sections. These are included for the sake of clarity and also for the purpose of having a thesis as self-contained as possible. The matrices will be given in the element local coordinate system. For each individual element in the model, the evaluated element
– 105 – matrices in the local coordinate system are transformed to global coordinate system by the usual coordinate transformation technique [11]. The structure matrices (i.e. the global tangent stiffness matrix K t , global mass matrix M, and global internal force vector p) are constructed from the transformed matrices of the individual elements of the structure by the general assembly procedure [11]. The superscript e, used later to denote e.g. the element length Le or the element nodal displacement vector qe, is omitted in this chapter for notational convenience.
7.2 Modeling of cables
The problem of analyzing cables under different configurations and loading conditions is very complex. This is because the behavior of cables is inherently nonlinear and also because large displacements introduce nonlinearities in the geometric sense.
For cable supported bridges, there are mainly three approaches used today to consider the nonlinear behavior of the cables. In the first approach each cable is replaced by one bar element with equivalent cable stiffness. This approach, often adopted when modeling cables in cable-stayed bridges, is referred to as the equivalent modulus approach and has been used by several investigators [6, 12, 13, 22, 42, 55, 68] and also in Part A of this thesis. Derivation of the equivalent tangent modulus of elasticity for the bar element, equation (2.10), is presented in [27]. It has been shown in [7] that the equivalent modulus approach results in softer cable response as it accounts for the sag effect but does not account for the stiffening effect due to large displacements. Still, for some cases, e.g. for short span cable-stayed bridges, linear analysis utilizing the equivalent modulus approach is often sufficient [27, 41], especially in the feasibility design stage. Whereas, long span cable-stayed bridges built today or proposed for future bridges are very flexible, they undergo large displacements, and should therefore be analyzed taking into account all sources of geometric nonlinearity.
The second approach is to divide each cable into several straight elements, as in [3], in order to adequately model the curved geometry of the cable. This would introduce many added degrees of freedom with a consequent increase in computer storage requirements and computational cost. In addition, numerical problems can occur and spurious results can be obtained if equilibrium conditions, at those nodes, are not fulfilled.
– 106 – The third approach to model cables is to use isoparametric elements. In [7, 11, 49], derivations of isoparametric cable elements, which include the element curvature, are presented. Using such elements one can model the curved geometry of a cable with fewer elements compared to using straight elements and obtain a better convergence [49]. However, those elements are stiffer and require numerical integration to formulate the element stiffness matrix [49].
The alternative approach presented in this thesis is based on “exact” analytical expressions for the elastic catenary. In contrast to other modeling techniques mentioned above, each cable may be represented by a single 2-node finite element, which accurately consider the curved geometry of the cable, making this method very attractive for static response calculations. Even if each cable must be divided into several catenary cable elements, to include cable modes of vibration in the dynamic analysis or external loads acting between cable ends, the author still believes that this approach is more efficient to adopt. This is mainly because fewer internal nodes need to be defined for each cable in the model. The main advantages of the proposed cable element are the reduction of degrees of freedom, the simplicity of finding the dead load geometry of the cable system, the exact treatment of cable sag, the exact treatment of cable weight as it is included in the equations used for element formulation, and the simplicity of including the effect of pretension of the cable by simply giving the unstressed cable length.
7.2.1 Cable element formulation
The procedure presented in this section determines the complete geometry of the cable, the cable element internal force vector, and its tangent stiffness matrix from a given unstressed cable length and given positions of the ends of the cable.
Consider an elastic cable element, stretched in the vertical plane as shown in Figure
7.1, with an unstressed length Lu , modulus of elasticity E, cross section area A, and weight per unit length w (uniformly distributed along the unstressed length). For the elastic catenary, the “exact” relations between the element projections and cable force components at the ends of the element are [35]:
– 107 –
u4 , P4
u3 , P3 node j
y
Ly Lu , E , A , w u2 , P2
u1 , P1 x node i Lx
Figure 7.1 Catenary cable element
Lu 1 P4 + T j Lx = −P1 + ln (7.1) EA w Ti − P2
1 T − T L = ()T 2 − T 2 + j i (7.2) y 2EAw j i w
where Ti and T j are the cable tension forces at the two nodes of the element. For the above expressions it is assumed that the cable is perfectly flexible and Hooke’s law is applicable to the cable material. The expressions for Lx and Ly in equations (7.1) and
(7.2) may be written, in terms of the end forces P1 and P2 only, as:
Lx = Lx ()P1, P2 ; Ly = Ly (P1, P2 ) (7.3)
because P1, P2 , P3, P4 ,Ti and T j are related by the following equations:
P4 = w Lu − P2 ; P3 = −P1 (7.4)
– 108 – 2 2 2 2 Ti = P1 + P2 ; T j = P3 + P4 (7.5)
Differentiating equation (7.3) and rewriting the results using matrix notation gives:
∂Lx ∂Lx ∂Ly ∂Ly dLx = dP1 + dP2 ; dLy = dP1 + dP2 (7.6) ∂P1 ∂P2 ∂P1 ∂P2
∂Lx ∂Lx dLx ∂P ∂P dP dP = 1 2 1 = F 1 ∂L ∂L (7.7) dLy y y dP2 dP2 ∂P1 ∂P2 where F is the flexibility matrix. The stiffness matrix K is given by the inverse of F as:
−1 k1 k2 K = F = (7.8) k3 k4
The tangent stiffness matrix K t and the corresponding internal force vector p for the cable element can now be obtained in terms of the four nodal degrees of freedom as
(noting that k2 = k3 ):
− k1 − k2 k1 k2 P1 − k k k P 4 2 4 2 K t = ; p = (7.9) − k1 − k2 P3 sym. − k4 P4
The element tangent stiffness matrix K t relates the incremental element nodal force T vector {} ∆P1, ∆P2 , ∆P3, ∆P4 to the incremental element nodal displacement vector T {} ∆u1, ∆u2 , ∆u3, ∆u4 . The Maple software package for symbolic computations [52] was used to perform the above mentioned operations and produce the necessary Fortran code. This Maple procedure is listed in Appendix A. However, if this package is not available the following expressions, obtained by derivation of equation (7.3), may be used to evaluate the matrices K, K t , and p:
1 L 1 P P k = − u + 4 + 2 (7.10a) 1 det F EA w T T j i
– 109 – 1 P 1 1 k = k = − 1 − (7.10b) 2 3 det F w T T j i
1 L 1 P P k = x + 4 + 2 (7.10c) 4 det F P w T T 1 j i
2 L 1 P P L 1 P P P 1 1 det F = − u − 4 + 2 x + 4 + 2 − 1 − (7.10d) EA w T T P w T T w T T j i 1 j i j i
To evaluate the tangent stiffness matrix K t , the end forces P1 and P2 must be determined first. Those forces are adopted as the redundant forces and are determined, from given positions of cable end nodes, using an iterative stiffness procedure. This procedure requires starting values for the redundant forces. Based on the catenary relationships the following expressions will be used for the starting values [35]:
w Lx w cosh λ P1 = − ; P2 = − Ly + Lu (7.11) 2 λ 2 sinh λ where
L2 − L2 λ = 3 u y −1 (7.12) 2 Lx
In cases where equation (7.12) cannot be used because the unstressed cable length is less than the chord length, a conservative value of 0.2 for λ is assumed [35]. Another difficulty arises in equation (7.12) for vertical cables. In that case an arbitrary large value of 106 for λ is used. Using equations (7.4) and (7.5), new cable projections corresponding to the assumed end forces P1 and P2 are now determined directly from T equations (7.1) and (7.2) and the misclosure vector {∆Lx , ∆Ly} is evaluated as the positions of the end nodes are given. Corrections to the assumed end forces can now be made using the computed misclosure vector as:
i+1 i ∆P1 ∆Lx P1 P1 ∆P1 = K ; = + (7.13) ∆P2 ∆Ly P2 P2 ∆P2
– 110 – where the stiffness matrix K is given in equation (7.8) and i is the iteration number.
For the present study, this iteration process continued until ∆Lx and ∆Ly are less than 1⋅10−5 . As will be demonstrated later, this iterative procedure converges very rapidly.
To determine the unstressed cable length, Lu , for cases where the initial cable tension is known instead, a similar iteration procedure can be adopted. A starting value for the unstressed cable length is assumed, e.g. equal to the cable chord length, and cable end forces P1 and P2 are computed using the iterative procedure described above. Using equation (7.5), cable tension can now be computed. This is then compared with the given initial tension to obtain a better approximation for Lu for the next iteration step.
If the complete geometry of the cable is to be determined, coordinates for a number of points along the cable must be computed. This is very simple because P1 and P2 are now known, so equations (7.1) and (7.2) can be used to compute the coordinates of any new point along the cable by simply replacing Lu by any fraction of Lu .
For the dynamic analysis, mass discretization is simply done by static lumping of the element mass at both ends giving the following lumped mass matrix (ρ is the mass density of the cable element):
1 0 0 0 ρAL 0 1 0 0 M = u (7.14) 2 0 0 1 0 0 0 0 1
7.2.2 Analytical verification
A cable hanging under its own weight and subjected to a tensile force at both ends along its chord, as shown in Figure 7.2, was studied to verify the cable element and the analysis procedure described in section 7.2.1. This problem was earlier studied in [7] using isoparametric cable elements and the published results can now be compared to results obtained here.
11 A cable with an unstressed length Lu = 312.7 m, modulus of elasticity E = 1.31·10 N/m2, cross section area A = 5.48·10-4 m2, and weight per unit length w = 46.11 N/m,
– 111 – was studied using two different models. For the two models: the cable was replaced by one catenary cable element, and by twenty beam elements with negligible moment of inertia. The beam element used is described in the next section.
To To
304.8 m
Figure 7.2 Cable under its own weight subjected to tensile force at both ends
To span the distance of 304.8 m, using a cable with the above given 1.5 (a) properties, a horizontal force of 4 To = 1.7794·10 N was needed at both ends. This force gave a mid- 1.0 span cable sag of 30.48 m and was 1 catenary cable element 20 beam elements, -5 adopted as the initial force when I =10 0.5 -4 20 beam elements, I =10 calculating the curves in Figure 7.3.
Using the two models, the sag and 0.0 0 5 10 15 20 the longitudinal displacement along (Displacement / Horizontal length) x100 Tension / Initial tension the chord of the cable were deter- 10.0 mined for different values of the (b) tensile force T and the results are 8.0 1 catenary cable element -5 plotted in Figure 7.3. Good agree- 20 beam elements, I =10 -4 20 beam elements, I =10 ment is observed when comparing 6.0 the curves for the adopted catenary cable element with those for the iso- 4.0 parametric cable element presented in 2.0 [7]. Figure 7.3 shows also that the
replacement of the curved cable by (Sag / Horizontal length) x100 0.0 several beam elements with negligi- 0 5 10 15 20 Tension / Initial tension ble flexural stiffness can give acceptable results. As the number of Figure 7.3 Response of the cable defined in Figure 7.2
– 112 – beam elements increase and the moment of inertia decreases, the results should converge to those of the catenary cable element model. One should only bear in mind that replacing the cables by several straight elements will give a stiffer structural model and consequently an underestimation of the displacements.
The results from this simple numerical experiment provide confidence in the application of the catenary cable element for modeling cables in cable supported bridges.
7.3 Modeling of bridge deck and pylons
The pylons and the bridge deck – girder or stiffening girder as it is also called – are modeled using beam elements able to resist bending, shear, and axial forces. For the present study, the simplest 2D beam element introduced in [61] is adopted and the treatment given there will be followed below when deriving the element matrices. This finite element is developed following the total Lagrangian approach and using a linear interpolation scheme for the displacement components. Previous studies, reported in [61], have shown that the element is efficient and accurate. This element is chosen also because it can handle large displacements and shear deformations and because it is simple to formulate the element matrices.
Referring to Figure 7.4, the current deformed configuration of the beam axis is described by a regular curve defined by the position vector:
so (x) = []x + u(x) i + w(x) j (7.15) where the abscissa x∈[0, L] is measured on the straight reference configuration of the beam, u(x), w(x) represent the axial and transversal displacement components and i and j are unit axis vectors. By introducing the angle θ (x) as the rotation of the cross sections (S') in the deformed configuration, the unit vectors orthogonal and parallel to the cross sections for each point on the deformed beam are obtained as:
a(x) = cosθ i + sinθ j; b(x) = −sinθ i + cosθ j (7.16)
Further, defining the deformation measures ε,γ ,κ according to:
– 113 –
u5 u y, w 6 u4 b(x) s a(x) o,x
deformed beam u2 θ (x) u 3 u1 S'
s (x) s(x) w(x) o u(x) j S i undeformed beam x, u
x L
Figure 7.4 Deformed and undeformed configuration of the beam element
ds dθ s = o = ()1+ ε a + γ b ; κ = = θ (7.17) o,x dx dx ,x and using equations (7.15) and (7.16), the following expressions are obtained:
ε = (1+ u,x ) cosθ + w,x sinθ −1 (7.18)
γ = w,x cosθ − (1+ u,x ) sinθ (7.19)
κ = θ,x (7.20)
If the constitutive relations are assumed as linear, the strain energy can be written as:
1 L Π (u) = () EAε 2 + GAγ 2 + EIκ 2 dx (7.21) i 2 ∫ 0
– 114 – where EA, GA and EI represent the axial, shear and flexural rigidities. For u, w and θ a linear interpolation scheme is used according to:
u = Ni (x) qi + N j (x) q j (7.22) where uT = {} u, w, θ , qT = u , u , u and qT = u , u , u contain the corresponding i {}1 2 3 j { 4 5 6} values of the displacements at the two nodes of the element, and Ni (x) = 1− x / L ; N j (x) = x / L are the interpolation functions.
Finally, the expressions for the internal force vector p and the element tangent stiffness matrix K t are obtained through successive differentiation of the expression for the strain energy according to:
2 ∂Πi ∂p ∂ Πi p = ; K t = = (7.23) ∂q ∂q ∂q2
T where q is the nodal displacement vector { u1, u2 , u3, u4 , u5, u6} . The Maple procedure which performs the above mentioned operations and produces the necessary Fortran code for p and K t , is listed in Appendix A.
The kinetic energy is expressed as the integral over the volume V :
1 T Π k = ρs&(x) s&(x)dV (7.24) 2 ∫V where ρ is the mass density and s&(x) represents the velocity in a general point of the beam. The position of this point is defined by the vector s(x) as shown in Figure 7.4. For this element, the final expression for the kinetic energy becomes [23]:
L L L ρ 2 ρ 2 ρ 2 Π k = Au&(x) dx + Aw&(x) dx + I θ&(x) dx (7.25) 2 ∫ 2 ∫ 2 ∫ 0 0 0
Using the interpolation functions, as in equation (7.22), the kinetic energy is written as a function of the velocity components in the nodal degrees of freedom of the element. From the resulting expression for the kinetic energy, the consistent element mass matrix is evaluated as:
– 115 – 2A 0 0 A 0 0 0 2A 0 0 A 0 ∂ 2Π ρL 0 0 2I 0 0 I M = k = (7.26) 2 ∂q& 6 A 0 0 2A 0 0 0 A 0 0 2A 0 0 0 I 0 0 2I
For more details concerning the formulation of the kinetic energy and mass matrix and the performance of this element in dynamic problems, the reader is referred to [23].
– 116 – Chapter ______Vehicle and Structure Modeling ______
8.1 Vehicle models
Vehicles have at least two suspension systems. Thus, in order to improve the earlier model adopted in Part A, an additional suspension system is introduced. Consequently, the improved model includes not only the body-bounce motion, as the one in Part A, but also the wheel-hop motion. The two suspension systems, each consisting of a spring and a damper element, can be seen as filters. The first suspension system (primary suspension) reduces the road input into the vehicle structure and creates isolation for frequencies higher than about 15 Hz [9]. The second suspension system (secondary or chassis suspension) produces isolation for frequencies higher than about 2 Hz for air suspended and 3 Hz for leaf spring suspended chassis. This system should also give sufficient damping to the axle and tire system to prevent the tire leaving the ground on very rough road surfaces. Other suspension systems, e.g. cab suspension which is used to reduce the acceleration levels in and the forces on the cab structure, are also present in modern heavy vehicles. However, the simple vehicle model adopted does not contain all suspension systems, detailed suspension nonlinearities and complexities of vehicle body motion that are typical of heavy vehicles. Despite that, it is believed that this model is sufficiently realistic, as was discussed in Part A, for the purpose of this study.
For the current study, it is assumed that the vehicle never loses contact with the bridge, the spring and the viscous damper have linear characteristics, and the contact between the bridge and the moving vehicle is assumed to be a point contact. Referring to Figure 8.1a, and denoting the contact force between the bridge and the vehicle by F(t), defined positive when it acts downward on the bridge, the following dynamic equilibrium equations for the three masses can be established:
– 117 – w3(t) v (t)
m3 (a) (b)
ks cs w2(t)
m2
kp cp w1(t)
m1
node i xc node j mode 1 mode 2 Le 1.5-4 Hz 8-15 Hz
Figure 8.1 (a) vehicle model on a bridge element; (b) typical modes of vibration
m3w&&3 + ks ()()w3 − w2 + cs w&3 − w&2 = 0 (8.1a)
m2w&&2 + ks ()()w2 − w3 + cs w&2 − w&3 + k p (w2 − w1) + cp (w&2 − w&1) = 0 (8.1b)
− m1w&&1 − k p ()()w1 − w2 − c p w&1 − w& 2 − (m1 + m2 + m3 )g + F(t) = 0 (8.1c)
Where w1, w2 and w3 are the vertical displacements of the masses measured from the static equilibrium position, k p and ks the stiffness of the linear springs connecting the masses, cp and cs the damping coefficients of the viscous dampers, and g the acceleration of gravity. A dot superscript denotes differentiation with respect to time. Using the equations above, the contact force may be expressed as:
F()t = (m1 + m2 + m3 )g + m1w&&1 + m2w&&2 + m3w&&3 (8.2) where the first term on the right-hand side is the dead weight (static part) of the contact force while the other terms represent the inertia effects. The moving force and the moving mass vehicle models can now be obtained, as shown in Part A, by modifying the contact force expression, equation (8.2). When modeling suspended roadway vehicles, the unsprung mass, m1, is always set to zero in this study. This mass is still included in the model to be able to model a vehicle as an unsprung moving mass (so- called moving mass model) or for modeling railway vehicles having unsprung wheels.
– 118 – With the assumption that the road profile cannot be rough enough to make the vehicle jump or leave the road surface, the displacement w1(t) is not an independent variable but can be coupled at each time step to the displacement of the contact point on the bridge deck. In this way this degree of freedom is eliminated and, as in Part A, the displacement w1()t and its derivatives are expressed in terms of the nodal degrees of freedom of the bridge as:
e w1 = Nc q + rc (8.3a)
e e w&1 = Nc,x q v +Nc q& + rc,x v (8.3b)
e 2 e e e 2 w&&1 = Nc,xx q v + Nc,x q& 2v + Nc,x q a +Nc q&& + rc,xx v + rc,x a (8.3c)
where Nc is a row vector containing linear interpolation functions for the vertical displacement of the beam element evaluated at the contact point xc. As shown in e Figure 8.1a, xc is the distance from left node of the element to the contact point. q is the nodal displacement column vector for the element on which the vehicle is positioned, v and a the vehicle velocity and acceleration in the longitudinal direction, and rc the surface irregularity evaluated at the contact point. The subscript x denotes derivation with respect to x.
Equations (8.1a) and (8.1b) may be written using matrix notation as:
m3 0 w&&3 cs − cs w& 3 ks − ks w3 0 + + = (8.4) 0 m2 w&&2 − cs c p + cs w& 2 − ks k p + ks w2 c p w&1 + k p w1
The vehicle model has consequently two degrees of freedom and its equation of motion can be written as:
Mvw&& + Cvw& + Kvw = fv (8.5)
In order to solve this second order equation of motion, it is transformed into a system of first order equations by complementing it with the equality Iw& − Iw& = 0 , where I is a 2x2 identity matrix. This will give the extended dynamic equilibrium equation:
– 119 – I 0 w& 0 − Iw 0 + = (8.6) 0 Mv w&& Kv Cv w& fv which can be rewritten as:
w& 0 I w 0 = −1 −1 + −1 or u& v = Auv + B (8.7) w&& − Mv Kv − Mv Cv w& Mv fv
For the present study, this first order differential equation is solved using the Matlab algorithm ode45 [53] which uses automatic step size 4th and 5th order pair Runge- Kutta-Fehlberg integration method.
The undamped natural frequencies for this simple sprung mass vehicle model is determined from the eigenvalue equation:
2 det( K v − ω Mv ) = 0 (8.8) giving the eigenvalues:
2 1 k k m k k m 4k k ω 2 = − p + s 1+ 3 ± p + s 1+ 3 − p s (8.9) 1,2 2 m2 m3 m2 m2 m3 m2 m2m3
For typical vehicle mass and stiffness values, the first vibration mode (body-bounce) excites mainly the vehicle body mass, m3, while the second mode (wheel-hop) almost only excites the wheel mass, m2, se Figure 8.1b.
Heavy roadway vehicles generate most of their dynamic wheel loads in two distinct frequency ranges, as mentioned earlier in Part A of this thesis: body-bounce and pitch motions at 1.5-4 Hz and wheel-hop motion at 8-15 Hz, se Figure 8.1b. The body- bounce mode of a vehicle is excited by relatively long and the wheel-hop mode by relatively short wavelengths of the road surface irregularities. As an example, the body-bounce mode for a vehicle traveling at 70 km/h is excited by irregularities with wavelengths of 4.9-13 m and the wheel-hop mode by irregularity wavelengths of 1.3- 2.4 m. Moreover, depending on the vehicle speed, a surface irregularity of a certain wavelength may be effective in both frequency ranges.
– 120 – 8.2 Vehicle load modeling and the moving load algorithm
The moving load problem is more complicated than other problems in structural dynamics, as the external force vector, containing the interaction forces existing at the contact points between the vehicles and the bridge, is time-dependent. This vector is totally unknown beforehand (except for the moving force problem which is uncoupled), as interaction forces are dependent on the motion of both the bridge structure and the vehicles.
The external force vectors for the elements where vehicles are positioned are obtained by adding the dead load and the moving traffic (live) nodal load vectors, i.e. e e e f = fdead + flive . Using, as before (see Part A), the Dirac function δ()x − xci to characterize the action of a unit force concentrated in point x = xci , the traffic part can be expressed as:
e a L a e T T flive = − ∑ Fi ()t ∫ N δ (x − xci )dx = − ∑ Fi ()t Nci (8.10) i=1 0 i=1 where a is the total number of vehicles on the element, Le the element length (see
Figure 8.1a), Fi ()t the interaction force between the bridge and the ith vehicle wheel, x is the distance measured from left node of the element, and Nci the row vector:
x x N = 0, 1− , 0, 0, , 0 (8.11) Le Le containing the element interpolation functions evaluated at the contact point of vehicle i, i.e. for x = xci . For consistency reasons, the same linear interpolation functions are adopted here as those used in section 7.3 for deriving the beam element matrices.
The moving load algorithm, illustrated in Figure 8.2, calculates the bridge-vehicle contact force, for each vehicle on the bridge, and prepares the global external force vector caused by the moving traffic. The code developed is capable of handling unlimited number of vehicles, all having the same velocity (independent on the deflected shape of the bridge deck), and fully considers the dynamic interaction between the vehicles and the bridge.
– 121 – START Call algorithm for each iteration. Input: initial vehicle Initialize global external vectors w 0 and w& 0 , traffic load vector, f = 0 new vehicle positions, live and surface roughness
parameters rc, rc,x, rc,xx Loop through all vehicles on the bridge
For vehicle i, initialize
w0i ,w& 0i , Mvi, Cvi and Kvi
Identify the loaded element. Evaluate x , the vectors N , N , N and ci ci ci,x ci,xx determine the loaded element’s dof
No Interaction problem
Yes
e e e Determine q ,q& ,q&& and calculate w1i ,w&1i ,w&&1i form equations 8.3
Solve differential equation 8.7 to Moving force model calculate wi ,w& i ,w&& i . Evaluate contact Fi = ()m1 + m2 + m3 g force, Fi (t), from equation 8.2
Accumulate nodal forces T flive (dof 1 → dof 6) = flive(dof 1 → dof 6) − Fi (t)Nci
Next vehicle
RETURN flive Figure 8.2 The moving load algorithm
– 122 – 8.3 Bridge structure
The bridge structures are discretized, for the nonlinear finite element analysis, using a catenary cable element and a simple beam element able to resist bending, shear and axial forces. As discussed earlier, this cable element is adopted in order to simplify cable modeling and to more accurately predict the response of cable supported bridge structures. The modeling technique and derivation of the elements are described in detail in Chapter 7. Only two-dimensional bridge models are considered in this study, consequently, the torsional behavior caused by eccentric loading of the bridge deck is disregarded. Furthermore, the nonlinearity considered is of the geometric type, as linear material behavior is assumed.
8.3.1 Modeling of damping in cable supported bridges
The assumption that bridges have no damping (adopted in Part A) is rather restrictive since vibration energy is dissipated in all structures even in very low damped cable- stayed bridges. There are various factors causing energy dissipation in bridge structures. Energy dissipation is generally developed by material nonlinearity, opening and closing of hair cracks (in reinforced concrete structures), structural damping such as friction at movable bearings and in the joints of the structure, loss of energy from foundation to ground, and aerodynamic damping by friction with air. The different forces that contribute to the damping of a structure may vary with vibration amplitude, velocity, acceleration, and stress intensity. Thus, theoretical evaluation of damping is extremely difficult and hence at present we have to rely upon the empirical approach. Attempts on theoretical approach to damping evaluation of cable-stayed bridges have been made in [43, 44] where a method is proposed to determine the damping ratio for the desired mode shapes by evaluating the overall energy dissipation and strain energy in the whole bridge. It was shown that the proposed method predicted the dependency of the damping ratio on oscillation amplitude and cable type with reasonable accuracy. For practical use however, damping models that represent more or less satisfactory approximations have been introduced. The most commonly employed damping model in the field of structural engineering is the viscous one, in which the damping force is proportional to the velocity. This model, also adopted for the present study, is very popular as it leads to the simplest mathematical treatment and generally gives the most satisfactory results [15].
– 123 – In the case of cable-stayed bridges, the dissipation of energy in cables is potentially a major contribution to the overall damping of the bridge. As pointed out in [65], the damping ratio depends on the type of cables and on the cable configuration used. Using spiral or locked coil strands instead of parallel wire strands (PWS), increases the total damping capacity of the bridge as such strand exhibits significant hysteresis in its longitudinal load-displacement behavior. Despite this, parallel wire strands have been favored in resent years due to economic and durability reasons.
A limited amount of information is available on damping of cable supported bridges. However, as more and more forced-excitation and ambient-vibration tests are conducted, fundamental data on the damping ratio is increasing drastically. Observations made from various tests suggest that the damping ratio for a suspension bridge decreases with increasing frequency (mode number) but for cable-stayed bridges it is found to be invariant with frequency [19, 37, 43]. In [19], this trend is discussed by interpreting many observed data and it was concluded that friction damping in the main cables of suspension bridges could be a possible explanation that distinguishes cable-stayed form suspension bridges.
To derive the damping matrix, one frequently used technique is to assume the damping matrix proportional to mass and stiffness matrices giving the so-called Rayleigh damping [8, 15], C = α K + β M . For practical problems the parameters α and β are often chosen based on the knowledge of the damping property of a similar structure. One disadvantage of the Rayleigh damping is the fact that damping behavior of the complete bridge structure is described only by the two parameters α and β , which are taken as constant values over the entire bridge model. On the other hand, this method gives damping matrices that have the same orthogonal properties with respect to the eigenvectors of the undamped system, as the mass and stiffness matrices. Thus, by premultiplying and postmultiplying with the mode shape matrix, the damping matrix can be diagonalized giving the matrix of generalized (or modal) damping values3 (see also section 9.1.1.2):
3 In the present study the eigenvectors are normalized such that the generalized (or modal) mass is set to unity.
– 124 – 2ξ1ω1 0 0 0 2ξ2ω2 0 T []2ξω = = Z CZ (8.12) O 0 0 2ξnωn
Due to the earlier mentioned limitation of the Rayleigh damping, the damping matrix derived gives a poor simulation of the real damping characteristics of the bridge structure. An improvement may be achieved by ignoring the proportionality condition and instead establish directly the diagonalized matrix, [2ξω], of equation (8.12). To derive a damping matrix with orthogonal properties, this method is much more flexible than the Rayleigh damping and is preferred if modal damping ratios are available. Another advantage, when compared to Rayleigh damping, is that only the damping ratios of the required modes can be accounted for. The actual expression for the damping matrix may be deduced from the assumed matrix [2ξω], as:
C= Z−T []2ξω Z−1 (8.13)
The damping matrix obtained is full regardless of whether the stiffness and mass matrices are banded or not, but this is of little significance. In practice, the inversion of the mode shape matrix requires a large computational effort. This can be avoided by taking advantage of the orthogonality properties of the mode shapes relative to the mass matrix. Thus, by using the expression3 ZTM Z = I to express Z−T and Z−1 in equation (8.13), one can obtain:
C= M Z[]2ξω ZTM (8.14)
For the present study, the ith mode damping ratio in equation (8.12) is assumed as:
−0.84 ωi ξi = 0.0042 (for suspension bridges [19]) (8.15) 2π
−0.645 ξ1 = ξ2 = Lξn = 0.237 L (for cable-stayed bridges [43]) (8.16) where L is the main span length in meter. The above two approximations are based on field forced-excitation tests.
– 125 – 8.3.2 Bridge deck surface roughness
Two types of roughness profiles are considered in this study. The first type is a random profile described using a power spectral density function. The generation of such a profile is described in detail in Part A, Chapter 2.
For the second roughness profile, a bump, assumed to vary harmonically as shown in Figure 8.3, is generated to simulate roughness of the local type. The profile is evaluated, for xbump ≤ x ≤ (xbump + Lbump ) , as:
1 2 π (x − x ) r(x) = h 1− cos bump (8.17) bump 2 Lbump
where xbump is the position where the bump starts, hbump the maximum height, and Lbump the length of the bump. It should be noted that here x and xbump are measured starting from the left end support of the bridge.
h r(x) bump
° x xbump Lbump
Figure 8.3 Bump dimensions
To obtain a realistic simulation of vehicles moving over a bump, the developed algorithm temporary reduce the time step as a vehicle reaches the bump. This is done automatically, if the time step is too large, to ensure that at least a specified number of contact points (10 for the results reported here) are obtained along the bump.
– 126 – 8.4 Tuned vibration absorbers
A vibration absorber is a device that reduces the vibration level of a protected structure. Usually such a device consists of an additional mass connected by means of an elastic and a damping element to the structure needed protection. The invention of the vibration absorber is usually associated with the name of Frahm, who in 1909 first patented a vibration absorber design [21].
There are innumerable examples of vibration absorbers being applied to control vibrations in various engineering structures. For civil engineering structures like chimneys, TV towers, and high-rise buildings, vibration absorbers are nowadays widely used mainly to reduce vibrations caused by wind or earthquakes, see for example [50, 54, 66]. For bridges, such devices are still not that common. However, as modern bridges become longer and more flexible and as energy dissipation in these bridges is very low (see section 8.3.1) and often not enough on its own to suppress vibrations, it is realized that additional vibration control measures are needed. The primary intention, in the few examples found where vibration absorbers are practically applied on long span bridges, was to control wind-induced vibrations. As such bridges not only are vulnerable to vibrations caused by wind and earthquakes but also to those caused by traffic, some investigations have been carried out to find if such devices are also efficient for controlling traffic-induced vibrations [10, 18, 33, 38, 45]. In [45] the efficiency of a tuned mass damper (TMD4) on suppressing vibrations induced by high- speed trains moving over a simple three-span bridge was studied and in [10] the efficiency of TMDs for footbridges was investigated theoretically and experimentally using walking, jumping and running tests. In [33] the effect of a TMD on the dynamic response of the Rama IX cable-stayed bridge5 in Bangkok, subjected to a moving vehicle, was studied. In the present study, the efficiency of a TMD is also investigated but this is done more correctly here as, among other things, the bridge-vehicle interaction was not fully considered in [33].
Other types of absorbers and devices used to control vibrations can also be found in
4 A TMD is a vibration absorber tuned to a particular mode of the bridge and usually consists of a mass, a viscous damper and a linear spring as illustrated in Figure 8.6. For convenience, the abbreviation TMD is used hereafter for such vibration absorbers.
5 The Rama IX cable-stayed bridge is equipped with TMDs for suppressing wind-induced flexural and torsional vibrations in the bridge deck as well as flexural vibrations in the pylons [33].
– 127 – bridge structures. As an example Figure 8.4 illustrate how cable vibrations in bridges can be controlled. Such design (a) measures for suppression of cable vibrations can be found on existing bridges. For example, auxiliary ropes are found on the Farø cable-stayed bridge in Denmark and the Normandie (b) bridge in France, dashpot dampers (shock absorbers) are installed between bridge deck and stay cables on the Brotonne bridge in France and the Sunshine Skyway bridge in Florida, and (c) stockbridge dampers are mounted on the long hangers of the Humber suspension bridge in England [58]. Figure 8.4 Design measures for suppres- sion of cable vibrations [58] TMDs are also used temporarily during erection of long span bridges, as free-standing pylons and cantilevered spans are very sensitive to wind and often need vibration control. For example, two TMDs each weighting about 40 tonnes, see Figure 8.5a, were employed during the erection of the 856 m main span (currently word record) Normandie cable-stayed bridge to stabilize the bridge structure during erection and limit the moment in the pylons. Wind tunnel tests and calculations showed that, by using TMDs, the dynamic response due to wind could be reduced by approximately 35 % [16]. As construction progressed, these TMDs were relocated and tuned to maximum efficiency. The TMDs were designed so their eigenfrequency could be adjusted from 0.1 Hz to 0.2 Hz and the relative internal damping from 3 % to 18 %. More information can be found in [51]. Most recently, in the two approach bridges for the Great Belt suspension bridge in Denmark, 32 TMDs have been installed, see Figure 8.5b, each with a mass of 8 tonnes (which approximately equals 0.5 % of the modal mass) [59].
There are two important roles of a TMD: firstly, it reduces the resonance response of the main structure, and secondly, the attached dashpot increases the overall damping of the structure by providing an additional source of energy dissipation. The efficiency of the TMD depends on the correct tuning of its parameters (eigenfrequency) relative to
– 128 – (a) (b)
Figure 8.5 TMD of the (a) Normandie Bridge during erection [16]; (b) Great Belt Bridge [59]
the eigenfrequancy of the mode of the structure it is designed to suppress. Thus, one disadvantage with such a passive device is that its performance significantly deteriorates when the dynamic characteristics of the structure are different from the original values assumed during the optimal design of the TMD. On the other hand, TMDs can still be advantageous as they work without requiring any connection to the ground or external power supply. In the literature, a great deal of attention is given to the optimization of parameters and evaluation of the efficiency of the TMD. For the present study, the following most often used optimum tuning parameters, derived in [21] for a structure with no damping, are adopted6:
ω ω = i (8.18a) tmd 1+ µ
6 These equations are used in the present study to determine the TMD parameters without considering available standard components and dimensions. However, in practice dimensions of available standard springs etc. often dictate the final choice of TMD parameters.
– 129 – 3µ ξtmd = (8.18b) 8 ()1+ µ 3
where ωtmd and ωi are the circular frequencies of the TMD and the dominant bridge mode to be tuned to, ξtmd the damping ratio of the TMD, and µ is the mass ratio which relates the TMD mass to the modal mass of the dominant bridge mode to be tuned to, µ = mtmd / mi . These equations are derived for undamped structures affected by a stationary harmonic load. Consequently, they may not be valid for a damped bridge structure subjected to moving traffic loads and one maybe should, using equations (8.18a-b) as starting values, perform a trail and error computer study to find the optimal TMD parameters. However, in [38] it was found that the difference is small and for practical design purpose these optimization formulas are suitable, as far as the bridge is lightly damped. But as expected, the response reduction was not as great as for the case with a stationary harmonic load.
The exact tuning of the TMD frequency is, as mentioned earlier, very important, whereas, the damping needs only be assumed approximately as its efficiency is relatively less sensitive for variations in ξtmd [10]. The mass ratio µ , which is usually small and lies between 0.01-0.04 for bridges [10, 45], is another important design parameter as it directly influences the response of the bridge and the relative movement of the TMD mass (Lower values of µ lead to large relative movement of the TMD mass). Therefore, when designing a TMD it is important to carefully select a suitable mass ratio, as it is not always easy to find a location with significant relative movements and enough space to accommodate the TMD. Another engineering problem is to provide a very low friction bearing surface for the TMD mass, so that the damper mass can also respond to the bridge movement at low levels of excitation.
ktmd ctmd wtmd mtmd
Figure 8.6 Cross section of bridge girder with a tuned mass damper, TMD
– 130 – Now referring to Figure 8.6, the following equilibrium equation can be established for the TMD mass including only the inertia effect:
e e mtmd w&&tmd + ctmd (w&tmd − Ntmdq& )+ ktmd (wtmd − Ntmdq )= 0 (8.19)
where mtmd is the mass of the TMD, ctmd the damping coefficient of the viscous dampers, ktmd the spring stiffness, wtmd the vertical displacement of the TMD mass measured from the static equilibrium position, qe the nodal displacement vector for the element on which the TMD is attached, and Ntmd the interpolation vector given in equation (8.11). This vector is evaluated for x = xtmd where xtmd is the distance from the left node of the TMD loaded element to the point where the TMD is attached. The spring stiffness and damping coefficient are evaluated using equations (8.18a-b). To include the force applied to this beam element by the TMD, an addition to the external force vector given in equation (8.10) is made as:
e e T flive = flive −mtmd w&&tmd Ntmd (8.20)
It is interesting to note that when a vehicle is modeled as a sprung mass system instead of a constant moving force (i.e. including bridge-vehicle interaction) one can for some cases observe a reduction in the bridge dynamic response as the vehicle acts as a vibration absorber [29]. Thus, for such cases the dynamic response, calculated by ignoring interaction, is conservative.
– 131 –
– 132 – Chapter ______Response Analysis ______
9.1 Dynamic Analysis
In the previous chapter, the vehicle equation of motion, equation (8.5), and an algorithm for assembling the global external force vector are presented. In the following, the general equation of motion for a bridge under the action of moving traffic loads is formulated.
Based on the element property matrices and the external nodal force vector derived in the foregoing chapters, one may obtain the following general equation of motion for the entire bridge by means of conventional FEM assemblage:
M q&& + Cq& + p()q = f (q,q&,q&&,t ) (9.1)
q, q&, q&& bridge node displacement, velocity, and acceleration vectors M bridge mass matrix C bridge damping matrix p(q) vector of internal elastic forces f(q,,q&,q&& t) external force vector resulting from the dead load, the moving traffic, and the tuned mass dampers
As indicated, the external force vector is not only time dependent but is also dependent on the bridge displacements, velocities and accelerations. This vector contains the interaction forces existing at the contact points between the vehicles and the bridge and thereby couples the bridge equation of motion with those of the vehicles. Consequently, the external force vector is dependent on the motion of both the bridge structure and the vehicles. To consider the bridge-vehicle interaction, an iterative
– 133 – procedure is adopted. First the displacements etc. of the contact points are assumed and the vehicle equations are solved to obtain the interaction forces and the external force vector. Then the bridge equation of motion, equation (9.1), is solved to obtain improved values of displacements etc. for the contact points. If the convergence criteria are not fulfilled, a new iteration is preformed recalculating the external force vector and resolving the bridge equation. The type of convergence criterion used in Part A is also adopted here to check the errors in not only the displacement vector, as in Part A equation (3.14), but also the velocity and acceleration vectors. The time step and the number of considered eigenmodes (for the mode superposition procedure) are problem-dependent. Accordingly, for each numerical example studied here, a convergence study was carried out with a view to getting reasonably converged reliable solutions with an optimum number of increments and optimum number of considered eigenmodes.
In the present study, two approaches are adopted for solving equation (9.1) in the time domain: one for evaluating the linear dynamic response and one for the nonlinear dynamic response. These are described in detail in the following sections.
9.1.1 Linear dynamic analysis
If the structure is assumed to respond linearly during the application of traffic loads, it is possible to evaluate the dynamic response using the mode superposition technique starting from the deformed dead load state. Thus, this linear dynamic procedure utilizes the dead load tangent stiffness matrix, Kt . The internal force vector in equation (9.1) is therefore evaluated as:
p(q) = K t q (9.2)
where Kt is obtained from a nonlinear static dead load analysis. It is believed that this approach is adequate for short and medium span cable supported bridges as far as traffic load to dead load ratios are small. Moreover, it is well known that this approach give sufficiently accurate results with minimum consumption of CPU time, as usually one only need to consider the first dominant modes of vibration. On the other hand, this approach requires frequency analysis and eigenmode extraction to start with, which can be expensive and time consuming for large systems. Furthermore, the
– 134 – effectiveness of mode superposition technique is reduced, as the required number of eigenmodes for satisfactory convergence is difficult to estimate beforehand.
The eigenmode extraction and the mode superposition technique are well described in many textbooks on structural dynamics [11]. Nevertheless, in order for this thesis to be fully contained and to facilitate understanding, the above mathematical concepts will be described in some depth in the following subsections.
9.1.1.1 Eigenmode extraction and normalization of eigenvectors
The first step of this approach is to determine the natural frequencies and mode shapes of the bridge structure. For this, the undamped free vibration is considered, as the effect of damping on the natural frequencies of most real structures is small. Thus equation (9.1) is reduced to:
M q&& + K t q = 0 (9.3) in which 0 is a zero vector. Assuming harmonic motion, which may be expressed as [11]:
q = z sin(ω t −ψ ) (9.4) where z is a vector with components independent of time, ω is a circular frequency, and ψ is a phase angle, and substituting into equation (9.3), we obtain the following eigenvalue problem:
2 (K t −ω M) z = 0 (9.5)
In the present study, this eigenvalue problem is solved using the Matlab algorithm eig 2 2 2 [53] giving n (= number of d.o.f.) eigensolutions (ω1 ,z1),(ω2 ,z2),L,(ωn ,zn ) , where zi is called the ith mode shape vector (eigenvector) and ωi is the corresponding 2 circular frequency of vibration (ωi is the eigenvalue). Storing the obtained eigen- solutions in two matrices Z and Ω2 , we can now write the n solutions to equation (9.5) as:
2 K t Z − M Z Ω = 0 (9.6)
– 135 – 2 where the matrix Z contains the eigenvectors zi in columns and the matrix Ω is a diagonal matrix, which stores the eigenvalues on its diagonal, i.e.:
ω 2 1 2 2 ω2 Z = []z1,z2,L,zn and Ω = (9.7) O 2 ωn
The obtained eigenvectors are then normalized such that the modal mass is set to T unity, i.e. mi = zi M zi = 1. From this M-normalization and the orthogonality properties of eigenvectors [11] it follows that:
T T 2 Z M Z = I and from equation (9.6) Z K t Z = Ω (9.8) where I is identity or unit matrix. As noted, one of the advantages of this normalization method is that the modal stiffnesses will be equal to the eigenvalues.
9.1.1.2 Mode superposition technique
When the natural frequencies and mode shapes are determined, the equation of motion can be solved. The vector of nodal displacement, q, can be approximated by a linear combination of s eigenvectors, with s much less than n, giving:
s q ≈ ∑ zi φi ()t = Z s φs (9.9) i=1
where φi in vector φs are the modal amplitudes and state the proportion of each eigenvector in the transformation. Substitution of the above expression into equation T (9.1) and postmultiplication of each term by Zs yields:
T T T T Zs M Zs &φ&s + Zs CZs φ& s + Zs K t Zs φs = Zs flive (9.10)
Note that the external force vector in the above equation only contains the external forces resulting from the moving traffic and the tuned mass dampers. From the
– 136 – properties of M-orthonormalized eigenvectors, equation (9.8), the above equation is reduced to:
T 2 T I&φ&s + Zs CZs φ& s + Ω φs = Z s flive (9.11)
The damping matrix C is introduced to approximate the overall energy dissipation of the bridge structure during vibration and cannot generally be constructed from element damping matrices, such as the mass and stiffness matrices. To obtain a system of s uncoupled differential equations the matrix C must have the same orthogonal properties, with respect to the eigenvectors of the undamped structure, as the mass and stiffness matrices. When this is the case, the damping matrix will be diagonalized by premultiplying and postmultiplying with the mode shape matrix giving:
T Zs CZs = diag{}c1,c2 ,L,cs = diag{2ξ1ω1,2ξ2ω2 ,L,2ξsω s } (9.12) where c1,c2 ,L,cs are the modal damping values and are here expressed using the 7 damping ratios ξ1,ξ2 ,L,ξ s . The obtained matrix is referred to as the matrix of generalized (or modal) damping values. One of the advantages of this method when compared to other methods like Rayleigh damping is that there is no need to assemble a damping matrix with the same dimension as the stiffness and mass matrix. With the introduction of this matrix we can obtain a system of s independent equations which can be written as:
2 T φ&&1 + 2ξ1ω1 φ&1 +ω1 φ1 = z1 flive 2 T φ&&2 + 2ξ2ω2 φ&2 +ω2 φ2 = z 2 flive ...... (9.13) ...... 2 T φ&&s + 2ξsωs φ&s +ω s φs = z s flive
Using the Maple software package for symbolic computations [52], a program code for the exact solution of the above equations was generated. This code is incorporated in the developed computer program for linear dynamic analysis. Initial values for equations (9.13) are obtained, from known initial values of q and q& , by premultiplying T both sides of equation (9.9) with zi Μ and taking note of equation (9.8) giving:
7 ξ i is the fraction of critical damping in mode i.
– 137 – T T φoi = zi M qo and φ&oi = zi M q& o (9.14)
Finally, having determined the elements in vector φs by solving the above s equations, the response due to live loads (i.e. traffic and TMD) is found from equation (9.9). By adding the dead load response, determined using nonlinear static analysis, the total response of the bridge is obtained.
9.1.2 Nonlinear dynamic analysis
A procedure for evaluating the nonlinear dynamic response using a direct time integration method combined with a nonlinear solution procedure, is described in the following. This method is significantly more expensive (i.e. CPU time-consuming) than the mode superposition method which is usually chosen for linear studies. Still, nonlinear dynamic analysis is essential for the present study if its is believed that the bridge will not behave linearly during the application of traffic loads. If this is the case, the natural frequencies and mode shapes will vary with the amplitude of response and the earlier described mode superposition technique will consequently be inadequate.
For this procedure, it is necessary to assemble not only the stiffness and mass matrices, but also the damping matrix. A method for modeling the damping in matrix form in terms of modal damping ratios is presented in Chapter 8. As the stiffness is a function of the response amplitude, frequencies and damping ratios will also change as the structure deforms, thus theoretically such damping matrix ought to be updated for each time step. In practice, however, this is usually not necessary, because the damping ratios used will in most cases be only approximate values evaluated from field tests, taken from codes of practice or the literature.
For this study, an implicit procedure based on Newmark’s modified average acceleration method combined with a full Newton-Raphson solution procedure is adopted for solving the bridge equation of motion. Among engineers and researchers in the field of structural engineering, the Newmark method presented 1959 is the most popular algorithm for numerical solution of the equation of motion and is generally the most suitable for nonlinear analysis [7, 23]. In the following, the adopted procedure will be briefly described. For the interested reader, more details concerning the derivation etc. can be found in [26].
– 138 –
Denoting the time step by ∆t, for the Newmark method the following expressions hold for the displacements and for the velocities:
2 1 2 qt+∆t = qt + ∆t q& t + ∆t − β q&&t + ∆t β q&&t+∆t (9.15a) 2
q& t+∆t = q& t + ∆t ()1− γ q&&t + ∆tγ q&&t+∆t (9.15b) where γ and β are the Newmark’s integration parameters and are evaluated for the modified average acceleration method as γ = 0.5 +α and β = 0.25(1+α)2 , with α > 0 . The parameter α introduces numerical damping to the solution but reduces the accuracy to the first order. The second order accurate average acceleration method is obtained by setting α = 0 (no numerical energy loss).
In analogy to the static nonlinear analysis, the residual (error in nodal forces or dynamical out-of-balance forces) can be specified by rewriting equation (9.1) to give (note that f also includes the dead load):
err r ()q = M q&& + Cq& + p ()q − f (q,q&,q&&,t ) (9.16)
err These should vanish at the end of each incremental time step, i.e. r ()qt+∆t = 0 , by means of an iterative solution technique. For the present study, a procedure using full8 Newton-Raphson iterations is adopted for solving the residual equation. Denoting k qt+∆t as the approximate value of qt+∆t resulting from iteration k, the residual after k+1 iterations is approximated by linearization around the previous iterated solution as:
err k+1 err k k k+1 k r (qt+∆t ) = r (qt+∆t )+ S(qt+∆t )(qt+∆t − qt+∆t ) (9.17)
The matrix S in the above equation is the iteration (jacobian) matrix and is deduced from:
8 For the full Newton-Raphson iteration procedure, the tangent stiffness and iteration matrices are updated in each iteration.
– 139 – err k ∂r ∂q&& ∂q& ∂f S()qt+∆t = = M + C + K t − (9.18) ∂q k ∂q ∂q ∂qqk qt+∆t t+∆t
The last term ∂f / ∂q reproduces the effect of displacement-dependent external loads and is, for simplicity, omitted in the present study, see [26]. Making now use of equation (9.15), the iteration matrix can be specified in detail as:
1 γ S()q = M + C + K t (9.19) β ∆t 2 β ∆t
At each time step, the iteration has to be started from a prediction of the displacements, velocities and accelerations. This is done here by using equation (9.15) and setting q&&t+∆t = q&&t . In each iteration, the displacement corrections are computed using equation (9.17) and setting the residual at iteration k+1 equals zero, giving:
k −1 err k ∆q = −S r (qt+∆t ) (9.20)
The corrected displacements, velocities, and accelerations are then computed as:
k+1 k k qt+∆t = qt+∆t + ∆q (9.21a)
k+1 k ∂q& k k γ k q& t+∆t = q& t+∆t + ∆q = q& t+∆t + ∆q (9.21b) ∂q β ∆t
k+1 k ∂q&& k k 1 k q&&t+∆t = q&&t+∆t + ∆q = q&&t+∆t + ∆q (9.21c) ∂q β ∆t 2
By iteratively updating the solution using the corrections ∆qk , the error in nodal forces, i.e. the residual vector rerr , is reduced when proceeding form iteration k to k+1. At each time step, this iteration process continues until the ratio of the Euclidian norm of the residual vector to that of the internal force vector has dropped below a certain chosen tolerance value. For the described procedure, choosing an adequate time step is a critical issue, as this not only drives the accuracy of the integration but also governs the stability of the iteration process. As mentioned earlier, since the time step is
– 140 – problem-dependent, a convergence study was carried out for each numerical example to find the optimum number of increment.
9.2 Static analysis
In this study, the dead load response is always evaluated using a nonlinear procedure, while the live load static response is evaluated either linearly based on the dead load tangent stiffness matrix or nonlinearly, i.e.:
nonlin q static,tot = qdead +live (9.22) and/or
nonlin lin nonlin −1 q static,tot = qdead + qlive = qdead + K t flive (9.23)
To evaluate the nonlinear static response, the stiffness matrix is reformulated as the bridge deforms. There are several procedures for evaluating the nonlinear static response and the most frequently used are the Newton-Raphson iteration schemes [11, 17, 55]. In the present study, an incremental-iterative procedure using full Newton-Raphson iterations is adopted. This procedure is generally expected to give quadratic convergence [17].
– 141 –
– 142 – Chapter ______Numerical Examples ______
In this chapter, static and dynamic responses of a simply supported bridge, a long span suspension bridge (the Great Belt suspension bridge), and a medium span cable-stayed bridge, under the action of moving vehicles, are presented. Frequency analysis was also conducted for all three examples. In the last two examples, the applicability of the catenary cable element, for modeling cables in cable supported bridges, is demonstrated. The nonlinear behavior of cable supported bridges during erection and application of the dead load is also presented. To verify and illustrate the efficiency of the developed finite element code, some of the obtained results are compared to those obtained using the commercial finite element code ABAQUS [1] and to results reported by other researchers. Details concerning the modeling technique used in ABAQUS to model the moving loads and the bridge structures are given in Chapter 4. The calculations were preformed on a Pentium Pro 200 MHz computer with 128 MB RAM and for each numerical example the CPU time used by the MATLAB process is given.
Throughout the dynamic investigation, the bridge initial velocity and acceleration vectors at time t = 0 were assumed to be zero. The dead load displacements obtained from the nonlinear static analysis were used as the bridge initial displacements. However, if nothing else is mentioned, the bridge deck initial dead load vertical displacements were assumed to be zero, when evaluating the bridge-vehicle contact forces. Zero initial values were also assumed for the moving vehicles vertical displacements and vertical velocities. Further, for convenience, the shear modulus of all beam elements in the models was evaluated as G = E / 2.6 .
A convergence study was carried out for each numerical example with a view to getting reasonably converged reliable solutions with optimum number of increments
– 143 – and optimum number of considered eigenmodes (for the mode superposition procedure). For the convergence study, the analysis was repeated using a smaller time step (more increments) until good agreement is shown between the last two analysis results. To find the optimum number of eigenmodes, to be considered in the mode superposition procedure, the results were also checked after repeating the analysis including more eigenmodes.
For the linear and nonlinear dynamic analysis results presented in the following study, the tolerances for the three criteria, used to control the convergence of the bridge- vehicle dynamic interaction (similar to equation (3.14) in Part A, see also section 9.1), were set to 0.01. For the nonlinear dynamic procedure, the tolerance for the residual based convergence criterion (see section 9.1.2) was set to 10-5. These tolerance values were chosen from the practical engineering point of view.
As the main aim of this study is not to develop formulas to be used in design specifications for calculating the dynamic amplification factors, not much emphasis was put on simulating realistic trains of moving traffic or finding the most critical points on the bridge with the highest dynamic amplification factors.
10.1 Simply supported bridge
The simply supported bridge studied earlier in Part A of this thesis was also adopted for this investigation. The response of this simple bridge model was mainly studied in order to verify the developed algorithms. This was done by comparing the results with those obtained using the commercial finite element code ABAQUS. The bridge was discretized, for both the ABAQUS and the present model, using 70 beam elements. For the present bridge model, the first four bending natural frequencies obtained utilizing the dead load tangent stiffness matrix are: 3.97, 15.41, 33.19, and 55.91 Hz. As expected, due to considering shear effects and the utilization of the deformed dead load tangent stiffness matrix, these are lower than the frequencies given in Table 4.1.
The properties for the bridge and the vehicle models are given in Figure 10.1a. The modeled vehicle, permitted in Sweden but not within the European community, is a 24 m long truck and trailer with a total weight of 60 ton. As seen, this vehicle was modeled as two separate sprung mass systems one for the truck and one for the trailer. Thus, as a result of this simplification, the interaction between the truck and trailer was
– 144 – disregarded and only vertical modes of vibration of the vehicle were considered. The body-bounce and wheel-hop frequencies for the truck and the trailer models were chosen as 1.89 Hz and 11.35 Hz, with the corresponding mode shapes shown in Figure 8.1b. The vehicle model was assumed to move, from the left to the right, at the constant speed of 25 m/s (90 km/h) on a rough road surface having the profile shown in Figure 10.2a.
In Figure 10.1b-d and 10.2b-d, the ABAQUS and the present mode superposition solutions, due to only traffic load, are plotted on the same diagrams and the results are found to be in very good agreement with each other. About 30 % increase in the vehicle weight is observed in Figure 10.2b as a result of having about 9 mm higher surface at the bridge entrance than the approach pavement. Only 4 modes were considered when producing the present solution curves, except for the bending moment, which required 10 modes to converge. No bridge damping was included, ξ = 0 , except the automatically introduced numerical (artificial) damping in the ABAQUS solution. This numerical damping is believed to cause the difference in Figure 10.1d for the mid-point vertical acceleration. The present linear and nonlinear static traffic load responses were found to be identical. The problem was also solved using the present direct integration method but the results are not shown as these were found to be nearly identical to the mode superposition results, except for the mid-point vertical acceleration. For the present direct integration solution, to obtain a stable mid- point vertical acceleration curve with good agreement to the curves in Figure 10.1d, it was essential to introduce numerical damping by setting the parameter α to be equal to 0.05. For the three solutions, 300 increments were chosen with a time step of about 6 ms. In Table 10.1, dynamic amplification factors (DAF) for mid-point9 vertical displacement and bending moment are given together with the corresponding CPU time and maximal iterations per increment required for solving the problem.
For the results in Figures 10.1 and 10.2 it was assumed that, as this is usually the case, no initial dead load vertical displacements in the bridge deck were present. It is normally attempted to compensate by precambering the bridge deck during construction so the dead load displacements are cancelled by the specified camber. If no compensation is made during construction, the initial displaced geometry of the
9 For a certain point P, the DAF is defined as the ratio of the absolute maximum live load dynamic response at P to the absolute maximum live load static response at the same point.
– 145 – (a) v = 25 m/s Truck and Trailer 0.0 (b) ABAQUS m1 = 0 kg m = 4400 kg Present 2 -1.0 m3 = 35600 kg Static (present) . 4 24 m cp = 1.6 10 Ns/m . 4 cs = 12.0 10 Ns/m -2.0 . 6 kp = 14.0 10 N/m /2 m3 m3 k = 8.0 .106 N/m s -3.0 ks cs ks /2 cs /2 m m /2 Bridge 2 2 . 10 2 E = 3.0 10 N/m kp /2 cp /2 kp cp 4 -4.0 I = 3.307 m m1 m1 /2
2 Vertical displacement (mm) 12 m A = 4.75 m -5.0 34 m m = 11400 kg/m 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Time (s)
0.0 0.5 (d)
ABAQUS ) 2 -1.0 Present (10 modes) ABAQUS (c) 0.3 Static (present) Present
-2.0 0.0 -3.0
-0.3 -4.0 Bending moment (MNm) Vertical acceleration (m/s
-5.0 -0.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Time (s) Time (s)
Figure 10.1 Mid-point vertical displacement (b), bending moment (c), and acceleration histories (d) for the problem defined in (a). The road surface roughness profile is shown in Figure 10.2a. The dashed vertical lines indicate when the trailer enters and the truck leaves the bridge 15.0 1.4 (a) (b) ABAQUS-truck 1.3 ABAQUS-trailer 10.0 Present-trailer 1.2 Present-truck
5.0 1.1
1.0 0.0
Normalized contact force 0.9 Surface roughness (mm)
-5.0 0.8 0246810121416182022242628303234 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Distance along the bridge (m) Time (s)
25.0 ABAQUS-truck (c) ABAQUS-truck 2.0 (d) ABAQUS-trailer )
ABAQUS-trailer 2
Present-trailer 15.0 Present-trailer Present-truck Present-truck 1.0
5.0 0.0
-5.0 -1.0 Vertical displacement (mm) Vertical acceleration (m/s -15.0 -2.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Time (s) Time (s)
Figure 10.2 Normalized bridge-vehicle contact force (b), truck and trailer body vertical displacement (c), and acceleration histories (d), when travelling over a bridge having the road surface profile shown in (a). The dashed vertical lines indicate when the trailer enters and the truck leaves the bridge 0.0 (a) No initial displacement (linear) 20.0 (b) No initial displacement (linear) With initial displacement (linear) With initial displacement (linear) With initial displacement (nonlinear) -1.0 With initial displacement (nonlinear) 10.0
-2.0 0.0
-3.0 -10.0
-4.0 -20.0 Vertical displacement (mm) Vertical displacement (mm) -5.0 -30.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.00 0.25 0.50 0.75 1.00 1.25 Time (s) Time (s)
Figure 10.3 Effect of initial dead load displacements on mid-point vertical bridge displacement (a) and on vertical displacement of the truck body (b)
0.0 (a) ξ = 0 (linear) -0.5 ξ = 0 (linear) ξ = 0.05 (linear) ξ = 0.05 (linear) ξ (b) -1.0 = 0.05 (nonlinear) ξ = 0.05 (nonlinear) -1.5 -2.0 -2.5 -3.0
-3.5 -4.0 Bending moment (MNm) Vertical displacement (mm) -5.0 -4.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Time (s) Time (s)
Figure 10.4 Effect of damping on mid-point vertical bridge displacement (a) and bending moment (b) bridge deck must be considered, as this will influence the bridge-vehicle contact force and thereby the bridge and vehicle responses. This is shown, using the two present analysis methods, in Figure 10.3 where mid-point vertical bridge displacements, due to traffic load only, and truck body displacements, are plotted. The effect of damping is shown in Figure 10.4, where the undamped solution and the solution with 5 % damping ratio, assumed constant for all modes, are plotted.
Mode Superposition (present) Direct Integration 4 modes 10 modes (present)
DAFd 1.142 1.145 1.143
DAFm 1.095 1.151 1.154 CPU time (s) 60 85 830 Max. iterations/inc. 3 3 2
Table 10.1 Dynamic amplification factors for mid-point vertical displacement and bending moment with the corresponding required CPU time and maximal iterations per increment
The comparisons, made using this simple numerical example, provide some confidence in the application of the implemented direct integration and mode superposition algorithms. Therefore, in the following sections, only analysis results obtained using these two algorithms will be presented and no comparison will be made to results from commercial codes like ABAQUS.
10.2 The Great Belt suspension bridge
In this numerical example, the behavior of the Great Belt suspension bridge during girder erection and under the action of moving vehicles is presented. A frequency analysis for the completed bridge was also conducted.
The Great Belt (Storebælt) suspension bridge in Denmark has, with its 1624 m main span, the second largest span in the world10. The bridge deck – the girder or stiffening
10 Akashi Kaikyo suspension bridge in Japan holds currently the worlds record with a main span length of 1991 m. Both the Great Belt bridge and the Akashi Kaikyo bridge were opened for traffic in 1998.
– 149 – girder as it is called – is a 31 m wide and 4 m deep steel box. This girder is continuous over the full cable supported length of 2694 m, i.e. without expansion joints at the pylons. Like many other modern suspension bridges, the girder section is streamlined (almost wing shaped) to resist strong wind action. The two main cables, each include a total of 18648 high strength galvanized steel wires 5.38 mm in diameter and have a final diameter of 0.85 m, were erected in the autumn of 1996 by the air-spinning method11 [27]. To minimize deflections under asymmetric traffic load, the main cables are fixed to the stiffening girder through rigid clamps at mid-span. The ratio of cable sag to main span length was chosen to be 1/9 as this was found economically favorable. This gives higher pylons than usual and therefore a more flexible structure. The total height of the concrete pylons, including cable and cable saddle, is approximately 258 m.
180
77.6 535 m 1624 m 535 m
Figure 10.5 Geometry of the Great Belt suspension bridge
For the results presented here, a simple bridge model consisting of 126 elements, 89 nodes and a total of 215 active degrees of freedom, was used. The bridge geometry shown in Figure 10.5 and the properties given in Table 10.2 were taken from [28, 47]. The section properties for the pylons were assumed by the author to give the first and second pylon frequencies 0.147 Hz and 0.803 Hz, which are acceptable when compared to the measured pylon frequencies reported in [46]. In the bridge model, the girder was assumed to be simply supported at the ends, and every third hanger from the original bridge was included and modeled using catenary cable elements. The main cables were assumed to be fixed at the pylon tops and the pylons were rigidly fixed to the piers. To simplify the data input process, all internal cable nodes were vertically
11 Much the same method was used for suspension bridges more than 100 years ago, e.g. for the Brooklyn bridge in New York dating from 1883.
– 150 – positioned at the same level as the girder, and the configuration of the main cable under dead load was determined accurately, during analysis, after few iterations.
E (N/m2) A (m2) I (m4) w (t/m)
Girder 2.1·1011 1.00 3.32 14.78 (31 m × 4 m) Pylons 0.4·1011 2 × 37.5 2 × 750 2 × 90 0 Æ 75.5 m Pylons 0.4·1011 2 × (32.5, 30, 25) 2 × (275, 200, 150) 2 × (78, 72, 60) 75.5 Æ 257.6 m Cable 2.1·1011 2 × 0.41 - 2 × 3.45 side spans Cable 2.1·1011 2 × 0.40 - 2 × 3.36 main span Hangers 2.1·1011 2 × 0.025 - †
† Mass of hangers and clamps are considered distributed uniformly along the main cable and included in the cable mass.
Table 10.2 Parameters for the model of the Great Belt suspension bridge
10.2.1 Static response during erection and natural frequency analysis
Erection of a suspension bridge involves many challenging problems, especially aerodynamic stability problems, which relate to the fact that the bridge structure is incomplete, thus various structural components do not receive or render the kind of support intended in the complete structure. The erection of the girder may proceed in a number of different ways [36, 47, 48]. For the analysis presented in this thesis it was assumed that the erection proceeded simultaneously from mid-span and anchor blocks towards the pylons. To study the erection procedure of the Great Belt suspension bridge and calculate the initial profile for the free hanging cables (i.e. the initial cable lengths, the cable sag at mid-span, and the pylon tops horizontal displacement), the simple 2D model described earlier is used.
To start the analysis, the initial side span and main span cable lengths were determined using a trial and error procedure. Thus the initial cable lengths were estimated and the
– 151 – final dead load profile, i.e. cable sag at mid-span and pylon top displacements, were determined and compared with the desired dead load profile shown in Figure 10.5. The estimated values were then improved and the calculation was repeated until the required dead load profile was obtained. It was found that if the initial lengths of the side span and main span cables were chosen as 564.8 m and 1672.7 m, the calculated mid-span cable sag and pylon tops horizontal displacements would be 180.09 m and 0.04 m. Those initial cable lengths were therefore considered to be good enough and were used for all the following results presented for this example.
When the initial cable lengths were known, different erection stages 185 were analyzed and the results are (a) plotted in Figure 10.6. Studying this figure one can notice that, to arrive 180 specified final cable sag under at the desired dead load profile, the dead load =180 m pylon tops must be displaced about 0.85 m outwards and the cable sag 175 should be about 173 m, for 0 % erected girder. This pylon top Cable sag at main span (m) 170 displacement is needed to counteract 0 20406080100 the displacement caused by the % of bridge deck erected elongation of the side span main 1.5 cable during the subsequent erection (b) of the girder. Consequently, to compensate for the later erection of 1.0 the girder the saddles at the pylon tops must therefore, prior to cable erection, be horizontally displaced in 0.5 relation to the vertical pylon axis. This is done either by displacing the saddles in relation to the pylon tops 0.0 or by pulling back the pylons with 0 20406080100 Pylon tops horizontal displacement (m) % of bridge deck erected so called tie-back cables or a combination of the two methods Figure 10.6 Cable sag variation (a) and [36]. It is therefore necessary to horizontal displacement of pylon specify this displacement of the tops (b) during girder erection saddles to arrive at vertical pylons
– 152 – with zero bending in the final dead load condition. During the constructing of the Great Belt suspension bridge, tie-back cables running from each pylon top to the nearest anchor block were used pulling back each pylon 1.24 m [28] prior to main cable erection. When the main cables were erected and the tie-back cables were dismantled, the pylon tops moved back about 0.20 m [28]. Thus, the remaining displacement before erecting the girder was about 1 m. Acceptable agreement is found, according to the author’s opinion, when comparing this value with the one obtained from the present analysis, i.e. 0.85 m. It is worth noting that the aims and the essential goals of this investigation were to study the bridge response using a simple model and to check the efficiency and applicability of the presented finite elements for modeling cable supported bridges. For this reason some simplifications and assumptions were made when modeling the bridge structure and not much emphasis was put on using the exact properties etc. for each bridge member. This is believed to be the major explanation for the differences in the results.
Frequency analysis was also conducted for the completed bridge and the first three natural frequencies and mode shapes are given in Figure 10.7.
mode 1: 0.099 Hz (0.100 Hz)
mode 2: 0.112 Hz (0.115 Hz)
mode 3: 0.130 Hz (0.135 Hz)
Figure 10.7 Natural frequencies and mode shapes for the lowest three vertical bending modes of vibration. Values inside brackets are reported in [48]
– 153 – In [47, 48], analytical frequency results from a 3D finite element model and results from a 1:200 scale aeroelastic bridge model, made for wind tunnel testing, are presented. These results are also given in Figure 10.7 within brackets and as can be seen the agreement is very good when comparing with the result obtained from the present analysis.
The CPU time used by the MATLAB process, to find the tangent stiffness matrix at the dead load deformed state and to solve the system eigenvalue problem determining all 215 modes of vibration, was about 200 seconds.
10.2.2 Dynamic response due to moving vehicles
The simple suspension bridge model described earlier was also adopted to study the response of the Great Belt Suspension bridge under the action of two moving trucks. Each truck, moving on a smooth road surface from the left to the right at the constant speed of 25 m/s (90 km/h), was assumed to have a total weight of 44 ton and a length of 18.75 m. The free distance between the two trucks was assumed to be 40 m. The body-bounce and wheel-hop frequencies, for each truck model, were chosen as 1.89 and 11.35 Hz. The corresponding mode shapes are shown in Figure 8.1b and the model properties are given in Figure 10.8.
The problem was solved using both the v = 25 m/s m m m = 0 kg implemented linear dynamic (mode 3 3 1 ks cs ks cs m2 = 4840 kg superposition) and nonlinear dynamic m3 = 39160 kg m2 m2 4 k c k c cp = 1.76 10 Ns/m (direct integration) procedures. For the p p p p c = 13.2 104 Ns/m m m s 1 1 k = 15.4 106 N/m two procedures, reasonably converged p 6 58.75 m ks = 8.8 10 N/m reliable solutions were obtained using
600 increments corresponding to a time Figure 10.8 Model of the two trucks step of about 0.18 s. For the mode super- position procedure, the first 25 modes were found to be sufficient for calculating the dynamic response of the bridge and the trucks, as 25- and 30-mode solutions did not differ significantly. Bridge damping ratios were evaluated according to equation (8.15) giving: ξ1 = 0.029; ξ2 = 0.026; ξ3 = 0.023Letc. No numerical damping was introduced in the direct integration procedure.
– 154 – Some interesting diagrams are shown in Figure 10.9 and 10.10 where the linear and nonlinear solutions are plotted. These figures do not show very significant difference between the two solutions. Although some differences can be observed between the linear and nonlinear dynamic moment, acceleration, and contact force curves in Figures 10.9 and 10.10, these are insignificant from the engineering point of view.
In Table 10.3, dynamic amplification factors for the horizontal displacement of the left pylon top and the bending moment at the fixed end of the left pylon are given together with the corresponding CPU time and maximal iterations per increment required for solving the problem. Even though the excitation of the dynamic system was only caused by the elastic displacement of the bridge itself, as road surface with no roughness was assumed, the maximum dynamic moment for the fixed end of the left pylon was found to be nearly 30 % larger than the static one, see Figure 10.10a.
Linear Dynamic Nonlinear Dynamic (mode superposition) (direct integration)
DAFd (pylon top) 1.127 1.123
DAFm (pylon fixed end) 1.250 1.274 CPU time (s) 845 5930 Max. iterations/inc. 4 3
Table 10.3 Dynamic amplification factors for the horizontal displacement of the left pylon top and the bending moment at the fixed end of the left pylon with the corresponding required CPU time and maximal iterations per increment
It can be concluded from the numerical results obtained in this study that, utilizing the dead load tangent stiffness matrix, linear static and linear dynamic traffic load response analysis of long span suspension bridges is adequate.
The response was also evaluated neglecting bridge damping and it was found that correct estimation of bridge damping is very important, as this will greatly affect the dynamic response of the bridge-vehicle system. As an example, the dynamic amplification factor for the deck vertical displacement at the center of the bridge will increase from about 1.10 (Figure 10.9b) to about 1.22, if bridge damping is not considered.
– 155 – 30
20 (a)
10
0
-10
-20 Dynamic (nonlinear) Dynamic (linear) -30 Static (linear & nonlinear)
Horizontal displacement (mm) -40 0 102030405060708090100110 Time (s)
100
50
0 -50 (b) -100
-150 Dynamic (nonlinear) -200 Dynamic (linear) -250 Static (linear & nonlinear) Vertical displacement (mm) -300 0 102030405060708090100110 Time (s)
0.06 (c) )
2 0.04
0.02
0.00
-0.02
-0.04 Nonlinear Linear Vertical acceleration (m/s -0.06 0 102030405060708090100110 Time (s)
Figure 10.9 Horizontal displacement of the left pylon top (a); bridge deck vertical displacement (b) and vertical acceleration (c) at the center of the bridge. The curves are not easy to distinguish, as the responses are almost identical
– 156 – 40 (a) Dynamic (nonlinear) 20 Dynamic (linear) Static (linear & nonlinear)
0
-20
-40 Bending moment (MNm)
-60 0 102030405060708090100110 Time (s)
15 (b) Dynamic (nonlinear) 10 Dynamic (linear) Static (linear & nonlinear) 5
0
-5
-10 Bending moment (MNm)
-15 0 102030405060708090100110 Time (s)
1.08 Nonlinear 1.06 (c) Linear 1.04
1.02
1.00
0.98
0.96
0.94 Normalized contact force
0.92 0 102030405060708090100110 Time (s)
Figure 10.10 Bending moment at the fixed end of the left pylon (a) and for the bridge deck at the left pylon (b); normalized bridge-vehicle contact force for the first truck (c)
– 157 – 10.3 Medium span cable-stayed bridge
A 2D model of the cable-stayed bridge described in [55] was adopted for this investigation. The bridge geometry is shown in Figure 10.11 and the properties are given in Table 10.4. This bridge is similar in configuration to an existing bridge in Japan (The Meiko-Nishi Bridge in Nagoya) with some modifications in dimension. The static and dynamic behavior of this bridge has been studied earlier by several other investigators [4, 40, 55].
E (N/m2) A (m2) I (m4) w (t/m)
Girder 2.0·1011 0.93 0.26 19.64 †
Girder 2.0·1011 1.11 1.29 19.64 † central part Pylons 2.8·1010 13.01 34.52 30.65 above deck level Pylons 2.8·1010 18.58 86.31 43.78 below deck level Links 2.0·1011 0.56 0.10 4.38 deck to pylons † Including weight of cross beams.
2 2 Cable no. E (N/m ) A (m ) Lu (m) w (t/m)
1, 24 2.0·1011 0.0362 158.13 0.398 2, 11, 14, 23 2.0·1011 0.0232 134.66 0.255 3,10, 15, 22 2.0·1011 0.0204 111.64 0.225 4, 9, 16, 21 2.0·1011 0.0176 89.43 0.194 5, 8, 17, 20 2.0·1011 0.0139 68.80 0.153 6, 7, 18, 19 2.0·1011 0.0113 51.69 0.125 12, 13 2.0·1011 0.0372 158.12 0.409
Table 10.4 Parameters for the cable-stayed bridge model defined in Figure 10.11
– 158 – node 43 . element 61 5x3 cable16 45.7 cable12 cable13
cable 6 . elem. 36 . . . element 52 30.5 node 7 node 14 node 36 element 30 node 13 node 18 . 146.3 m 335.3 m 146.3 m
Figure 10.11 Geometry of the cable-stayed bridge. The cables are numbered from the left to the right starting with cable 1
For the model, it was assumed that the girder was pinned at the ends, i.e. only rotations were allowed, and elastically connected to the pylons by vertical links. The pylons were assumed to be rigidly fixed to the piers, and all cables were assumed fixed to the pylons and to the girder at their joints of attachment. The simplest model analyzed in this example, i.e. the model with one element per cable, was composed of 66 elements and 43 nodal points.
10.3.1 Static response and natural frequency analysis
Figure 10.12 shows the nonlinear behavior of the model under static dead load, described in terms of the vertical displacement of the girder at the center of the bridge and the tension in cable 12 and 13. Examining this figure, a hardening characteristic with respect to the applied load is apparent. It is also evident that at the start there is a significant nonlinear behavior during the static application of the dead load. Thus, a nonlinear static analysis under dead load is essential to arrive at the deformed dead load tangent stiffness matrix. For this cable-stayed bridge with modest main span length, as the nonlinearity is not so strong above this dead load equilibrium point, it is believed that this bridge will behave as a linear system, when affected by live static and dynamic loads, starting from this dead load deformed state. This means that influence lines and superposition technique can be used in the design process. However, as the span length increases this nonlinearity will get more pronounced [55] and linear live load analysis might no longer be adequate.
For the frequency analysis, to include cable motions, natural frequencies were also determined replacing each stay cable by 3, 5, and 7 catenary cable elements. This was
– 159 – 5 5 (a) (b) 4 4
3 3 tangent tangent
2 eigenvalue 2 problem Dead load multiplier Dead load multiplier 1 1
0 0 -3-202356 0 10203040506070 Vertical displacement of the girder at the Tension in cable 12 and 13 (MN) center of the bridge (m)
Figure 10.12 Nonlinear behavior of the cable-stayed bridge defined in Figure 10.11: (a) vertical displacement; (b) cable tension
mode 1: 0.334 Hz (0.311 Hz)
mode 2: 0.437 Hz (0.411 Hz)
mode 3: 0.703 Hz (0.650 Hz)
Figure 10.13 Natural frequencies and mode shapes for the first three vertical bending modes of vibration. Values inside brackets are reported in [4]
– 160 – easily done as the preprocessor code developed can automatically refine the model if the user requests it. For the simplest model, i.e. the model with one element per cable, the resulting modes of vibration only include the vibrating girder and pylons. Thus, cable modes and the dynamic interaction between the vibrating cables and the bridge were disregarded. For the finer models, pure cable modes, i.e. additional new mode shapes characterized only by vibrating cables, were obtained between those basic bending modes. Moreover, it is evident that cable motions are associated with every mode of vibration, as can be noticed in Figure 10.13. For the four alternative models, Table 10.5 presents a comparative frequency study of the first ten vertical bending modes of vibration. The order in which these modes appear is given inside brackets.
Vertical Natural frequencies (Hz) and mode order bending 1 3 5 7 mode no. element/cable elements/cable elements/cable elements/cable 1 0.332 (1) 0.334 (1) 0.334 (1) 0.334 (1) 2 0.436 (2) 0.437 (2) 0.437 (2) 0.437 (2) 3 0.692 (3) 0.700 (7) 0.703 (7) 0.703 (7) 4 0.734 (4) 0.739 (8) 0.741 (8) 0.742 (8) 5 0.868 (5) 0.874 (13) 0.875 (13) 0.875 (13) 6 1.044 (6) 1.051 (18) 1.053 (18) 1.053 (18) 7 1.212 (7) 1.218 (27) 1.220 (23) 1.220 (23) 8 1.214 (8) 1.220 (28) 1.222 (24) 1.222 (24) 9 1.379 (9) 1.388 (33) 1.387 (29) 1.388 (29) 10 1.671 (10) 1.690 (42) 1.680 (38) 1.687 (38)
Table 10.5 Comparison of the first ten natural frequencies for vertical bending modes of vibration. Each cable was modeled using 1, 3, 5 and 7 catenary cable elements. Mode order is given inside brackets
Satisfactory agreement is found when comparing the results from the static and frequency analysis presented here with those reported in [4, 40, 55]. The disagreement in the frequency results shown in Figure 10.13 is believed to be due to the fact that the
– 161 – catenary cable element used in the present study is stiffer12 than the one bar element with an equivalent modulus used in [4, 55]. Moreover, the girder and pylons were modeled in [4, 55] using conventional beam elements modified by the stability functions and a diagonal lumped mass matrix was adopted for all elements.
For the simplest model with a total of 119 active degrees of freedom, the CPU time used by the MATLAB process, to find the tangent stiffness matrix at the dead load deformed state and to solve the system eigenvalue problem determining all 119 modes of vibration, was about 15 seconds. This indicates high efficiency of the presented elements.
10.3.2 Dynamic response due to moving vehicles – parametric study
For the cable-stayed bridge shown in Figure 10.11, a study was conducted to assess the importance of different factors that influence the dynamic response due to moving vehicles. The investigation was focused on the following factors that are believed to have some kind of influence on the dynamic amplification factor: the effect of bridge- vehicle interaction, number of vehicles on the bridge, surface irregularities at the bridge entrance, vehicle speed, bridge damping, cable modeling, tuned vibration absorbers, and girder supporting conditions.
Some interesting figures are presented at the end of this numerical example, where element and node numbers are referred to. The locations of these nodes and elements are shown in Figure 10.11. The corresponding dynamic amplification factors (DAF) for the different studied cases are given in Table 10.6 at the end of this numerical example. Also, the absolute maximum bridge deck vertical acceleration at the center of the bridge and the maximum normalized bridge-vehicle contact force are given for each case in this table. The plotted cable axial forces are the average of the axial forces at the two ends of the cable. Also, one should observe that the dead load response (always evaluated using a nonlinear procedure) was subtracted from the solutions, thus only the responses due to traffic loads are plotted. The linear and nonlinear static traffic load responses were found to be almost identical and are not plotted separately for the sake of making the figures more clear. However, these static solutions corresponding to the linear and nonlinear dynamic solutions were used when
12 This can be concluded when comparing Figure 7.3a with the one published in [7] showing the response for the one bar element with an equivalent modulus of elasticity.
– 162 – calculating the DAF. The following was adopted for the parametric study unless otherwise specified:
• The simplest bridge model, i.e. the model with one element per cable.
• Vehicles moving from the left to the right at the constant speed of 25 m/s (90 km/h) on a smooth road surface.
• The same 44 ton vehicle model as for the Great Belt suspension bridge example. The body-bounce and wheel-hop frequencies for the truck model were chosen as 1.89 and 11.35 Hz. See Figure 10.8 for the model properties.
• 1000 increments corresponding to a time step of 0.025 s.
• The first 30 modes were considered for the mode superposition solution.
• No numerical damping was introduced in the direct integration procedure.
• Bridge damping ratios were evaluated according to equation (8.16) giving:
ξ1 =ξ2 =Lξ30 = 0.0056.
10.3.2.1 Response due to a single moving vehicle
In this study, linear and nonlinear dynamic responses due to a 44 ton moving truck are compared. In Table 10.6 (original configuration), some results for different time steps corresponding to 500, 1000, and 1500 increments are given. A linear mode superposition analysis, including all 119 modes of vibration and using 1000 increments, was also performed for this case and the results are also given in Table 10.6. During the convergence study, for the mode superposition procedure, it was found that the 30- and 35-mode solutions did not differ significantly. However, as seen from the results in Table 10.6, there are higher modes than 35 that are affecting the response. For example, for the bending moment at the right pylon fixed end, the DAF decreases from 1.180 to 1.159 when considering all 119 modes of vibration. Comparing the 500, 1000, and 1500 increment solutions it is clear that the 500- solution is not good enough and the 1000-solution does not give perfectly converged results. However, from the practical engineering point of view, analyses with 1000 increments and 30 modes give reasonably converged reliable solutions. Consequently, unless otherwise specified, 1000 increments and 30 modes were adopted for this investigation.
– 163 – Some linear and nonlinear response curves, for this single moving vehicle case, are presented in Figures 10.14 – 10.16. By examining those figures, it is evident that the two dynamic solutions demonstrate similar behavior throughout the time of loading and the agreement between them is so good that it is even difficult to see the two curves in some of the figures. This is an important observation since the linear dynamic analysis is far less complicated and requires far less computer time than the nonlinear dynamic analysis. Although there are some minor differences at the peak regions in, for example, Figures 10.15b and 10.16a, which also lead to differences in the DAF (see Table 10.6), these are insignificant from the engineering point of view as they are comparatively small when compared to those resulting from other simplifications and assumptions made when designing a bridge. For the linear and nonlinear dynamic solutions, a maximum of 3 and 2 iterations per increment were required and the corresponding CPU time was 470 s and 1890 s, respectively.
Analysis results indicated that cable 6 has, for the case studied here, the highest DAF of all cables (1.275). The response curves for the tension in this cable, Figure 10.16a, show a fluctuation with a frequency of about 1.19 Hz. This fluctuation in cable tension, also the dominant frequency for the vertical acceleration of the corresponding cable-girder attachment point, was caused by vibration modes 7 and 8 having a frequency of about 1.21 Hz.
As mention earlier, for this bridge model it was assumed that the girder was pinned at the ends. If a simply supported girder at the ends was assumed instead, which is the traditional way of supporting such a medium long bridge girder, the natural frequencies for the first three vertical bending modes of vibration will decrease to 0.314, 0.413, and 0.676 Hz. The DAF for this simply supported girder case are also included in Table 10.6. As can be seen, the DAF for the tension in cable 6 and for the vertical displacement at node 18 are lower for this case. However, a significant increase in the DAF was obtained for the tension in cable 13, the horizontal displacement of the right pylon top, and for the fixed end moment of the same pylon.
Based upon the presented results, three general conclusions can be made. The first is that linear dynamic analysis give sufficiently accurate results as there is no apparent nonlinear behavior of the bridge during application of the moving traffic load. The second conclusion is that high DAF values can be reached (1.39 for the axial force in the deck at the left pylon) even though a road surface with no roughness was assumed.
– 164 – The third conclusion is that the supporting condition of the bridge girder has a significant influence on the dynamic response.
10.3.2.2 Response due to a train of moving vehicles, effect of bridge-vehicle interaction and cable modeling
To investigate the effect of having a number of vehicles simultaneously on the bridge, a hypothetical train of four moving 44 ton trucks was adopted. The distance between truck models and model properties are shown in Figure 10.8. According to the study conducted in Part A, one of the important factors affecting the bridge response is the dynamic interaction between the vehicles and the bridge deck. Therefore, an investigation of the bridge-vehicle interaction effect was also conducted for this case, where the moving vehicles were modeled either as constant moving forces (i.e. ignoring interaction) or as sprung masses moving on a smooth road surface. In addition, an investigation of the effect of including cable motions and modes of vibrations on the dynamic response was conducted where each cable was either modeled using 1 or 5 catenary cable elements. 1500 increments corresponding to a time step of 0.021 s was required for this analysis.
For the study of bridge-vehicle interaction effect, Figure 10.17 shows some response curves obtained from a nonlinear dynamic and nonlinear static traffic load analysis. The DAF values are given in Table 10.6. Examining this figure and also Table 10.6, it was found that the moving force model, which usually gives negligible differences in results compared to the sprung mass results when the road surface roughness is ignored, gives for this flexible bridge significant differences especially for the tension in cable 6. As also noticed the DAF decreases for the moving force model except for the tension in cable 6 where the moving force model increases the DAF tremendously. It is believed that this difference in response occurs, as mentioned earlier in section 8.4, as the result of having the vehicles acting as vibration absorbers when modeled as sprung mass systems.
Comparing the nonlinear dynamic results of the one truck case (original configuration) with the case of four trucks modeled as sprung masses, it was found that the maximal dynamic part of the contact force increases by 87 %, and the absolute maximal vertical acceleration of the girder at the bridge center by 125 %. On the other hand, no particular trend was found when comparing the DAF of the two cases, as the DAF
– 165 – increases significantly for some of the studied elements and nodes but decreases for others.
The case of four moving trucks modeled as sprung masses was also solved for the bridge model with 5 catenary cable elements per cable. Consequently, cable motions and cable vibration modes were considered in this analysis. In Figure 10.18, some response curves obtained from a nonlinear dynamic and nonlinear static traffic load analysis are presented. Mixed results were obtained when the new solution was compared with the 1 element per cable solution, see Table 10.6. As an example, the 1 element per cable case underestimated the dynamic response and therefore the DAF for the tension in cable 6 but overestimated it for the tension in cable 13. Furthermore, no particular trend could be found for the cables, as the tension in all cables were effected regardless of their length or position (side or main span). Evaluation of the DAF for the entire bridge model revealed that the largest difference in response occurred for the axial force in deck element 36 (see Figure 10.18c), where the DAF increased from 1.388 to 1.658, when the 5 element per cable model was adopted. For the cable tension, the largest difference was found for cable 16 where the DAF increased from 1.333 to 1.422. The fluctuation of tension in this cable is shown in Figure 10.18d. A significant increase in DAF (from 1.400 to 1.596), see Figure 10.18b, was also found for the bending moment in pylon element 61.
As a conclusion, the dynamic interaction between the vehicles and the bridge deck should always be taken into account even if a road surface with no roughness is assumed. Moreover, to avoid an underestimation of the dynamic response, the cables should always be modeled so their motion and modes of vibration are considered in the analysis.
10.3.2.3 Speed and bridge damping effect
To investigate the effect of vehicle speed and bridge damping on the dynamic response, a single 44 ton moving truck was adopted. This is the same case as the so- called original configuration but is solved here for the new speeds of 50, 70, 110, and 130 km/h. 1500 increments were used for the solution of the 50 and the 70 km/h cases,
and 1000 increments for the rest. The original configuration (v = 90 km/h) was also analyzed here for new bridge damping ratios, namely: 0, 0.01, 0.015, 0.02, 0.03. Only
– 166 – nonlinear dynamic analysis was performed and the results are presented in Figure 10.19 and in Table 10.6.
The vertical displacement of the girder at the center of the bridge is shown in Figures 10.19a and 10.19b for different speeds and damping ratios. The variation of the DAF, for some selected elements and nodes, with respect to different speeds and damping ratios are shown in Figures 10.19c and 10.19d. As seen, even though the DAF tends to reduce at certain speeds, it is evident that the response generally increases with the increase in vehicle speed.
As expected, damping reduces the DAF. The effect of damping on the reduction of the bridge dynamic response was found to be considerable. Among the selected elements and nodes, the amount of reduction is largest for the axial force in pylon element 52 and the vertical displacement at node 18, while the influence of damping on the axial force in deck element 30 and the tension in cable 6 are comparatively small. In addition, it is noted in Figure 10.19d that the relationship between bridge damping ratio and the DAF is not always linear.
Form this study one can conclude that the response generally increases with the increase in vehicle speed and that bridge damping have a significant effect upon the response and should be considered if accurate representation of the true dynamic response is required.
10.3.2.4 Effect of surface irregularities at the bridge entrance
To roughly study the effect of irregularities in the deck and over the bearings at the entrance to the bridge, a 3 cm high and 2.5 m long bump located at the left end of the bridge, was generated as described in section 8.3.2. Results from the nonlinear dynamic analysis are presented in Figure 10.20 and in Table 10.6. Figures 10.20a-c show response curves for a vehicle speed of 90 km/h, and in Figure 10.20d some curves are plotted to show the variation of the DAF with the vehicle speed.
From those results we can infer that the effect of this type of irregularity, on the bridge-vehicle response, is considerable. As noted from Table 10.6, a 45 % larger contact force than the static one was obtained. Also, for as little bump height as 3 cm, which is not unusual, a significant increase in the axial force in deck element 30 was
– 167 – found, increasing the DAF from 1.39 to 1.60. Of course, the response of nodes and elements that are located close to the bridge entrance were affected the most by this bump, whereas, e.g., the bridge deck vertical displacement and acceleration at the center of the bridge were not affected at all. Again the highest DAF for all cables was found for cable 6. Moreover this bump affected the response of this cable more than the response of other cables.
From this study one can conclude that not only maintenance of the bridge road surface is important to reduce damage to bridges but also the elimination of irregularities (unevenness) in the approach pavements and over bearings is important as these strongly influence the response.
10.3.2.5 Effect of tuned vibration absorbers
The effectiveness of a tuned mass damper (TMD) in suppressing vibrations due to a single 44 ton moving truck is investigated in this study. The truck was assumed to move on a smooth road surface at the constant speed of 110 km/h. The TMD was positioned at the center of the bridge (node 14) and tuned to the first bending mode of vibration (0.33 Hz). The optimum tuning parameters given in equation (8.18) were adopted, where the mass ratio was set to 0.005 giving a TMD mass of approximately 15.6 ton. Some results from the nonlinear dynamic analysis are presented in Table 10.6 and in Figures 10.21-10.23. The results for this case presented in Table 10.6 should be compared with those corresponding to v =110 km/h.
Examining the results, it was found that the two anchorage cables 1 and 24 had the largest reduction in the DAF value. The DAF for cable 1 (highest DAF among all cables) was reduced by the TMD from 1.348 to 1.306. For cable 24, the tension due to traffic load is shown in Figure 10.22c, and the DAF was reduced from 1.130 to 1.063. A considerable reduction was also found for the horizontal displacement of node 43 and the moment in pylon element 52, see Table 10.6. However, The results also revealed that the TMD not always is very effective in reducing the maximum dynamic response during the forced vibration period (i.e. when the vehicle is on the bridge). In fact, due to the interaction between the bridge-vehicle-TMD systems, the response and the DAF for certain elements and nodes can even increase due to the TMD. As an example, the DAF for cable 14 increased from 1.049 to 1.079, when the TMD was considered. However, it is evident that the TMD is very effective in reducing the
– 168 – vibration level in the free vibration period, even for the case of cable 14. This is due to the increase of the overall damping of the bridge by the TMD.
Power spectral densities (PSD) of the vertical acceleration response of the girder at the center of the bridge (node 14) and the horizontal acceleration response of the right pylon top (node 43), with and without TMD, are presented in Figures 10.23b and 10.23c. The corresponding bending modes of vibration are also indicated in those figures. It is clear from those figures that several modes were excited, but the first mode has the largest contribution to the response of the two chosen nodes. To further reduce the vibration level of the bridge one might install extra TMDs, e.g. tuned to the third mode and installed at the top of the two pylons. As expected and observed in those figures, a TMD tuned to the first mode is only effective in reducing the contribution of this mode to the dynamic response. This is one of the disadvantages of a TMD, see the discussion in section 8.4.
In addition, it is noted that the PSD for the vertical response at the center of the bridge does not have peaks corresponding to the antisymmetric modes as these mode shapes have zero displacements at this node, e.g. see mode 2 in Figure 10.13. On the other hand, as seen in Figure 10.23c and also Figure 10.13, these antisymmetric modes do contribute for the pylon tops horizontal response. This should be taken into consideration when conducting field measurements to estimate the vibration frequencies of a bridge of this type, as an accelerometer installed at the pylon top will identify not only the symmetric modes but also the antisymmetric ones.
From this investigation it is concluded that a TMD, despite the fact that it is not always effective in reducing the maximum dynamic response during the forced vibration period, increases the overall damping of the bridge by working as an additional energy dissipater.
– 169 – 0.14 20 (a) Dynamic (nonlinear) Nonlinear ) (b) Dynamic (linear) 2 0.10 Linear 10 Static (linear & nonlinear) 0.06 0 0.02 -10
-20 -0.02
-30 -0.06
-40 -0.10 Vertical acceleration (m/s Vertical displacement (mm) -50 -0.14 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
20 0.50 (c) Dynamic (nonlinear) (d) Dynamic (linear) 0.25 10 Static (linear & nonlinear) 0.00 0 -0.25 -10 -0.50 Dynamic (nonlinear) Dynamic (linear) Axial force (MN) -20 -0.75 Static (linear & nonlinear) Vertical displacement (mm) -30 -1.00 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
Figure 10.14 Vertical displacement (a) and acceleration histories (b) at node 14; vertical displacement at node 18 (c); axial force in deck element 30 at node 7 (d). The dashed vertical lines indicate when the vehicle is at the left and right pylon 15 10
(a) Dynamic (nonlinear) (b) 10 Dynamic (linear) Static (linear & nonlinear) 5 5
0 0
-5 -5 Dynamic (nonlinear) -10 Dynamic (linear)
Bending moment (MNm) Static (linear & nonlinear) Horizontal displacement (mm) -15 -10 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
0.80 0.80
(c) (d) Dynamic (nonlinear) 0.60 Dynamic (linear) 0.40 Static (linear & nonlinear) 0.40
0.20 0.00
0.00 -0.40 Axial force (MN) Dynamic (nonlinear) Shear force (MN) -0.20 Dynamic (linear) Static (linear & nonlinear) -0.40 -0.80 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
Figure 10.15 Horizontal displacement of node 43 (a); bending moment (b), axial force (c), and shear force (d) in element 52 at node 36. The dashed vertical lines indicate when the vehicle is at the left and right pylon 0.45 0.20 Dynamic (nonlinear) Dynamic (nonlinear) (a) (b) Dynamic (linear) Dynamic (linear) 0.35 0.16 Static (linear & nonlinear) Static (linear & nonlinear) 0.12 0.25
0.08 0.15 0.04 0.05 0.00 Axial force (MN) Axial force (MN) -0.05 -0.04
-0.08 -0.15 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
0.50 0 (d) Nonlinear (c) Nonlinear ) 2 Linear Linear -10 0.25
-20 0.00
-30
-0.25 -40 Vertical displacement (mm) Vertical acceleration (m/s -50 -0.50 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
Figure 10.16 Axial force in cable 6 (a); axial force in cable 13 (b); truck body vertical displacement (c), and acceleration histories (d). The dashed vertical lines indicate when the vehicle is at the left and right pylon 30 25 Dynamic (sprung mass model) 20 (a) (b) Dynamic (moving force model) 20 Static 10 15 0 10 -10 5 -20 0 -30 -5 -40 Dynamic (sprung mass model) Dynamic (moving force model) -50 Bending moment (MNm) -10 Vertical displacement (mm) Static -60 -15 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (s) Time (s)
2.5 0.30
(c) 0.25 (d) Dynamic (sprung mass model) 2.0 Dynamic (moving force model) 0.20 Static 1.5 0.15
1.0 0.10
0.05 0.5 0.00 Axial force (MN) Dynamic (sprung mass model) Axial force (MN) 0.0 Dynamic (moving force model) -0.05 Static -0.5 -0.10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (s) Time (s)
Figure 10.17 Vertical displacement at node 18 (a); bending moment (b), and axial force (c) in element 52 at node 36; axial force in cable 6 (d)
40 3.5 Dynamic (1 element/cable) (a) 2.5 (b) 20 Dynamic (5 elements/cable) Static 1.5 0 0.5 -0.5 -20 -1.5 -40 -2.5
-60 -3.5 -4.5 Dynamic (1 element/cable) -80 Dynamic (5 elements/cable) Bending moment (MNm) -5.5 Vertical displacement (mm) Static -100 -6.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (s) Time (s)
0.6 0.5 Dynamic (1 element/cable) Dynamic (1 element/cable) (c) Dynamic (5 elements/cable) 0.5 Dynamic (5 elements/cable) 0.4 Static Static 0.4 0.3 (d) 0.3 0.2
0.1 0.2
0.0 0.1 0.0 Axial force (MN) -0.1 Axial force (MN)
-0.2 -0.1
-0.3 -0.2 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (s) Time (s)
Figure 10.18 Vertical displacement at node 14 (a); bending moment in pylon element 61 at the attachment point of cable 17; (b), axial force in element 36 at node 13 (c); axial force in cable 16 (d)
25 25 (a) (b) 15 15
5 5
-5 -5
-15 -15
-25 -25 v = 50 km/h ξ = 0 -35 v = 90 km/h -35 ξ = 0.0056 v = 110 km/h -45 -45 ξ = 0.03 Vertical displacement (mm) Vertical displacement (mm) -55 -55 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 Time (s) Time (s)
1.40 1.50 (c) (d) 1.40 1.30
1.30 1.20 1.20 1.10 1.10
1.00 Dynamic amplification factor 1.00 Dynamic amplification factor 50 70 90 110 130 0 0.005 0.01 0.015 0.02 0.025 0.03 Vehicle speed (km/h) Bridge damping ratio, ξ
Vertical displacement at node 14 Axial force in element 52 at node 36 Axial force in cable 6 Axial force in deck element 30 at node 7 Bending moment in element 52 at node 36 Axial force in cable 13 Figure 10.19 Vertical displacement at node 14 calculated for different vehicle speeds (a) and damping ratios (b); the influence of vehicle speed (c) and damping ratio (d) on some dynamic amplification factors 0.20 0.40 (a) Dynamic (0 mm bump) (b) 0.16 Dynamic (30 mm bump) 0.15 Static 0.12 -0.10 0.08
0.04 -0.35
0.00 Dynamic (0 mm bump) -0.60 Dynamic (30 mm bump) Axial force (MN) Axial force (MN) -0.04 Static -0.85 -0.08
-0.12 -1.10 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)
20 1.6 (c) Dynamic (0 mm bump) (d) 10 1.5 Dynamic (30 mm bump) 0 1.4 -10 1.3 -20 1.2 -30 Axial force in deck element 30 at node 7 1.1 -40 Bending moment in element 52 at node 36
Vertical displacement (mm) Axial force in cable 6
-50 Dynamic amplification factor 1.0 0 5 10 15 20 25 50 70 90 110 130 Time (s) Vehicle speed (km/h)
Figure 10.20 Axial force in cable 6 (a); axial force in deck element 30 at node 7 (b); truck body vertical displacement (c); the influence of vehicle speed on some dynamic amplification factors (30 mm bump) (d)
Effect of tuned vibration absorbers
25
15 (a)
5
-5
-15
-25
-35 with TMD
-45 without TMD Vertical displacement (mm) -55 0 10203040506070 Time (s)
15 (b) 9
3
-3
-9
-15
-21 with TMD
-27 without TMD Vertical displacement (mm) -33 0 10203040506070 Time (s)
10 (c) 6
2
-2
-6 with TMD -10 without TMD
Horizontal displacement (mm) -14 0 10203040506070 Time (s)
Figure 10.21 Vertical displacement at node 14 (a) and node 18 (b); horizontal displacement at node 43 (c). The dashed vertical line indicate when the truck leaves the bridge
– 177 – Effect of tuned vibration absorbers
90 (a) 60
30
0
-30
-60
-90 Vertical displacement (mm) -120 0 10203040506070 Time (s)
0.40 (b) ) 2 0.20
0.00
-0.20
Vertical acceleration (m/s -0.40 0 10203040506070 Time (s)
0.60
(c) with TMD 0.40 without TMD
0.20
0.00
Axial force (MN) -0.20
-0.40 0 10203040506070 Time (s)
Figure 10.22 Vertical displacement (a) and acceleration (b) of the TMD mass; axial force in cable 24 (c). The dashed vertical line indicate when the truck leaves the bridge
– 178 – Effect of tuned vibration absorbers
12
(a) with TMD 8 without TMD
4
0
-4 Bending moment (MNm)
-8 0 10203040506070 Time (s)
st rd 1 3 1.E+0 (b) with TMD th th 9 5 without TMD 1.E-2
1.E-4 PSD
1.E-6
1.E-8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Frequency (Hz)
1.E+0 st 1 with TMD (c) rd 3 th 6 without TMD th 1.E-2 nd 2 10 th th 4 th th th 9 5 7 ,8 1.E-4 PSD
1.E-6
1.E-8 0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.0 Frequency (Hz)
Figure 10.23 Bending moment in element 52 at node 36 (a); power spectral densities (PSD) of the vertical acceleration response at node 14 (b) and the horizontal acceleration response at node 43 (c) (y axis in log scale)
– 179 – original configuration - convergence study simply supported 4 trucks 4 trucks moving number of increments 500 1000 1500 1000 500 1000 1500 girder force model linear dynamic - 30 modes all modes nonlinear dynamic linear nonlin. linear nonlin. linear nonlin. Vertical displacement at node 14 1.207 1.186 1.179 1.186 1.208 1.187 1.180 1.131 1.130 1.060 1.056 1.048 1.048 Vertical displacement at node 18 1.270 1.243 1.234 1.243 1.240 1.235 1.230 1.119 1.112 1.286 1.265 1.171 1.162 Horizontal displacement of node 43 1.086 1.065 1.057 1.065 1.094 1.060 1.053 1.250 1.247 1.279 1.274 1.202 1.199 Axial force in deck element 30 at node 7 1.404 1.393 1.391 1.394 1.374 1.390 1.387 1.360 1.364 1.237 1.246 1.205 1.214 Shear force in element 52 at node 36 1.212 1.174 1.164 1.160 1.205 1.154 1.149 1.222 1.087 1.285 1.271 1.198 1.181 Axial force in element 52 at node 36 1.284 1.270 1.268 1.268 1.279 1.301 1.275 1.270 1.290 1.251 1.202 1.229 1.173
Factors (DAF) Bending moment in element 52 at node 36 1.222 1.180 1.166 1.159 1.210 1.152 1.150 1.431 1.442 1.276 1.262 1.207 1.188 Axial force in cable 6 1.316 1.294 1.284 1.284 1.240 1.275 1.268 1.240 1.238 1.086 1.035 1.358 1.345 Dynamic Amplification Axial force in cable 13 1.157 1.133 1.125 1.133 1.164 1.135 1.126 1.232 1.234 1.239 1.242 1.211 1.213 Absolute maximum vertical accel. at node 14 (m/s2)0.1600.125 0.123 0.125 0.158 0.140 0.124 0.121 0.136 0.336 0.316 0.275 0.266 Maximum normalized bridge-vehicle contact force 1.049 1.039 1.036 1.039 1.049 1.039 1.036 1.041 1.041 1.071 1.073 1.000 1.000
4 trucks v = 50 v = 70 v = 110 v = 130 ξ = ξ = ξ = ξ = ξ = bump TMD 5 elem/cable km/h km/h km/h km/h 0 0.01 0.015 0.02 0.03 30 mm v = 110 nonlinear nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. nonlin. Vertical displacement at node 14 1.087 1.041 1.132 1.282 1.204 1.220 1.164 1.143 1.124 1.093 1.188 1.245 Vertical displacement at node 18 1.289 1.057 1.003 1.427 1.060 1.285 1.202 1.170 1.143 1.102 1.233 1.378 Horizontal displacement of node 43 1.272 1.178 1.086 1.131 1.205 1.106 1.032 1.010 0.992 0.984 1.059 1.065 Axial force in deck element 30 at node 7 1.237 1.112 1.176 1.414 1.427 1.401 1.381 1.372 1.362 1.329 1.601 1.413 Shear force in element 52 at node 36 1.272 1.195 1.100 1.331 1.230 1.211 1.127 1.102 1.097 1.087 1.172 1.266 Axial force in element 52 at node 36 1.272 1.189 1.241 1.511 1.461 1.395 1.258 1.222 1.193 1.168 1.329 1.463
Factors (DAF) Bending moment in element 52 at node 36 1.268 1.178 1.104 1.323 1.244 1.239 1.123 1.103 1.097 1.087 1.211 1.260 Axial force in cable 6 1.101 1.024 1.101 1.206 1.408 1.300 1.258 1.241 1.226 1.202 1.331 1.205 Dynamic Amplification Axial force in cable 13 1.208 1.082 1.008 1.104 1.205 1.180 1.107 1.082 1.063 1.038 1.138 1.083 Absolute maximum vertical accel. at node 14 (m/s2) 0.316 0.067 0.091 0.145 0.202 0.195 0.118 0.099 0.086 0.075 0.141 0.137 Maximum normalized bridge-vehicle contact force 1.078 1.024 1.042 1.059 1.060 1.039 1.039 1.039 1.039 1.040 1.446 1.059
Table 10.6 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 cable- stayed 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, three- dimensional 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 multi- element cable discretization. This is also necessary to avoid an underestimation of the bridge dynamic response
• bridge 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 so- called 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);
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– 194 –