Nonlinear structural design optimization of cable stayed bridges

Contents

Title Page

Acknowledgements

Abstract

Contents

List of Tables

List of Figures

Chapter 1
1.1 Introduction
1.2 Literature Review
1.3 Objective
1.4 Optimal Design of Cable-Stayed Bridges
1.5 Structural Design Optimization
1.6 Optimization Techniques
1.7 Load Factor Optimization
1.8 Cable Stayed Bridge Structural Concept

Chapter 2
2.1 Cable Stayed Bridge Components
2.1.1 Deck
2.1.2 Pylon
2.1.3 Stay Cables
2.2 Stay cables types
2.2.1 Locked coil stays
2.2.2 Helical or spiral strand stays
2.2.3 Bar bundles
2.2.4 Parallel wire strand stays
2.2.5 New parallel wire strand stays
2.2.6 Parallel strand stays
2.2.7 Advanced composite stays
2.3 Stay cable arrangement
2.3.1 Fan arrangement
2.3.2 Semi-harp arrangement
2.3.3 Harp arrangement
2.4 Modeling the Stay Cables
2.5 Modeling the Pylon and Deck
2.5.1 Pylon
2.5.2 Deck
2.5.3 Dampers

Chapter 3
3.1 Finite Element Model s of the Cable Stayed Bridge
3.1.1 General Static Analysis
3.1.2 Boundary Conditions of the model
3.1.3 Load Assignation
3.1.4 Seeding and Meshing Step

Chapter 4
4.1 Analysis of the Results
4.1.1 Pylons Support Analysis
4.1.2 Pylons Support Reactions Analysis
4.1.3 Pylons Support S.Mises Stresses Analysis
4.2 Deck support Analysis
4.2.1 Deck support Reaction Analysis
4.2.2 Deck supports S.Mises Stress Analysis
4.3 Pylons Analysis
4.3.1 Pylons Deflection Analysis
4.3.2 Pylons S.Mises Analysis
4.4 Deck Analysis
4.4.1 Deck Deflection Analysis
4.4.2 Deck S.Mises Analysis
4.5 Stay Cables Analysis
4.5.1 Stay Cables Deflection Analysis
4.5.2 Stay Cables S.Mises Stresses Analysis
4.6 Deck Support Analysis (Pylon Shape Effect)
4.6.1 Deck Support Reaction Analysis (Pylon Shape Effect)
4.6.2 Deck Support S.Mises Analysis (Pylon Shape Effect)
4.6.3 Deck Deflection Analysis (Pylon Shape Effect) for the Left Deck
4.6.4 S.Mises Analysis (Pylon Shape Effect) for the Left Deck
4.6.5 Deck Deflection Analysis (Pylon Shape Effect) for the Deck Center
4.6.6 Deck S.Mises Analysis (Pylon Shape Effect) for the Deck Center
4.6.7 Deck Deflection Analysis (Pylon Shape Effect) for the Right Deck
4.6.8 Deck S.Mises Analysis (Pylon Shape Effect) for the Right Deck

Chapter 5
5.1 Discussion
5.2 Conclusions
5.3 Recommendations for Future Researches
5.4 References

List of Tables

[illustration not visible in this excerpt]

List of Figures

[illustration not visible in this excerpt]

Chapter 1

1.1 Introduction

The history of cable stayed bridges dates back to 1595, and have been constructed all over the world. The Swedish Stromsund Bridge, designed in 1955, is known as the first modern cable-stayed bridge (Wilson and Gravelle). The total length of the bridge is 332 m, and its main span length is 182 m. It was opened in 1956, and it was the largest cable-stayed bridge of the world at that time. This bridge was constructed by Franz Dischinger, from Germany, who was a pioneer in construction of cable-stayed bridges. The designers realized that cable stayed style requires less material for cables and deck and can be erected much easier than other bridges. This is mainly due to advances in design and construction method and the availability of high strength steel cables. The Theodor Heuss Bridge was the second true cable-stayed bridge and was erected in 1957 across the Rhine River at Dusseldorf. It had a main span of 260 m and side spans of 108 m which was larger than the Stromsund. It has a harp cable arrangement with parallel stays and a pylon composed of two free-standing posts fixed to the deck. The reason for choosing the harp style was aesthetics appearance. The Severins Bridge in Köln designed in 1961 was the first fan shape cable stayed bridge, which has a A-shape pylon. In this bridge, the cross section of the deck was similar to the one used in Theodor Heuss bridge. The Flehe Bridge was the first semi-harp type which was erected in 1979 in Dusseldrof, Germany over the Rhine River. There markable feature of this bridge was the reinforced concrete tower, which has the shape of an inverted Y.

Cable stayed bridges have good stability, optimum use of structural materials, aesthetic, relatively low design and maintenance costs, and efficient structural characteristics. Therefore, this type of bridges are becoming more and more popular and are usually preferred for long span crossings compared to suspension bridges. Figures 1,2,3 and 4 are samples of cables stayed bridges. A cable stayed bridge consists of one or more towers with cables supporting the bridge deck. In terms of cable arrangements, the most common types of cable stayed bridges are fan, Semi harp, and harp bridges. Because of their large size and nonlinear structural behavior, the analysis of these types of bridges is more complicated than conventional bridges. In these bridges, the cables are the main source of nonlinearity. Obtaining the optimum distribution of post-tensioning cable forces is an important task and plays a major role in optimizing the design of cable stayed bridges, and optimum design of a cable-stayed bridge with minimum cost while achieving strength and serviceability requirements is a challenging task.

illustration not visible in this excerpt

Figure 1 - Russky Island Bridge

http://en.skmost.ru/objects/auto/bosfor/

illustration not visible in this excerpt

Figure 2 - Royal Haskoning Bridge in Dubai

http://www.building.co.uk/royal-haskoning-wins-design-contest-for-dubai-bridge/3119479.article

illustration not visible in this excerpt

Figure 3 - Le Viaduc de Millau Bridge

illustration not visible in this excerpt

http://www.funonthenet.in/articles/millau-viaduct-worlds-tallest-bridge.html

Figure 4 - Yokogawa Bridge

http://kids.britannica.com/comptons/art-115965/The-Yokohama-Bay-Bridge-in-Japan-a-steel-cable-stayed

1.2 Literature Review

Long (1999), has employed an efficient algorithm based on internal penalty algorithm to optimize the cost of cable-stayed bridges. Similar to the scheme presented by Simoes and Negrao (1994), the pylon height, span length, and the number of stay cables received a pre-assigned value.

A convex scalar function has been employed to optimize the cost of the deck in cable-stayed bridges by Simoes and Negrao (2000). In this scheme, post-tensioning forces of stay cables are also considered. The convex scalar function combines the dimensions of the bridge and the post-tensioning cable forces. In this scheme pylon height and span length are not considered as design variables in this optimization technique.

Jajnic (2003), presented an efficient scheme to find optimal tensioning strategy for the construction of cable-stayed bridges in which three main sources of non- linearity are considered. Also, bending moment distribution at the deck connections is used to find the optimum cable forces.

Sung (2006), studied optimum post-tensioning cable forces for cable-stayed bridges. It is shown that the upper limits on cable forces for the cable-stayed bridges are due to a dead load. In this work, post-tensioning forces of stay cables are taken into account. The Mau-Lo Hsi cable-stayed bridge located in Taiwan is considered as a case study.

Lee (2008), proposed an optimization of tensioning strategy for asymmetric cable-stayed bridge and its effect on the construction process. Unit load method (ULM) is considered as it can take into account the actual construction process while other approaches are based on configuration of the final structure only. The numerical results are given to show the validity of the proposed approach.

A scheme based on genetic algorithm is incorporated for optimizing the cable stayed bridges by Lute (2009). The computation time of the genetic algorithm has been reduced employing a support vector machine. In this method, the number of stay cables is assumed to be constant. Moreover, the effect of post-tensioning cable forces has not been considered in this optimization scheme.

Most recently, Baldomir (2010) investigated cable optimization of long span cable-stayed bridges. The cross section of stay cables as well as constraint of cable stress and deck displacement is considered.

Zhang and Wu (2011) used an optimization method of unknown load factor to determine the cable forces to achieve an ideal state. Then, the ideal cable forces are established and a construction stage analysis is performed. A study based on the recent work conducted by Hassan (2012), proposed a new technique for optimal design of cable-stayed bridges with semi-harp arrangements based on a finite element model, B-spline curves, and real coded genetic algorithm.

1.3 Objective

The objective of this research work is to:

1- Find a suitable certain type of between multiple cable stayed Bridge models with three arrangements of stay cables (Fan, Semi harp and Harp) which is stiffer and the most robust model by using structural design optimization in the topology aspect for each model of the cable stayed bridges.
2- Optimize the shape aspect between three types of cable stayed bridges models for Pylons shape (Double out, single center and single inclined out) and their efficiency in the design of the cable stayed bridges in relation with stresses, reactions and deflection of the main components of the cable stayed bridge models which are the stay cables, deck and the pylons.
3- Obtain the optimum model of the cable stayed bridge that is a combination of the structural design optimization for the shape and the topological design together that is the most appropriate case as a safe and serviceable structure without oscillation, un-stability and failure due to the loads of the structure itself and other existing static and dynamic loads.
4- Recognize the sources of nonlinearity in the structural components of the cable stayed bridge and to discover the reasons beyond these nonlinearities associated with the design optimization process.

1.4 Optimal Design of Cable-Stayed Bridges

In the recent decade, considerable research has been conducted on the optimum design of cable-stayed bridges, which are considered as the most suitable structure system for medium to long span bridges with span length ranging from 200 to about 1000 meter. This is due to their aesthetic, economic, and the ease of erection.

Achieving the optimum design for cable-stayed bridges is a challenging task. This is due to the fact that the design is influenced by a large number of variables, including geometrical configurations, number of stay cables, types of pylons, arrangement of the stay cables, and the types of main girder. As mentioned in the previous chapter, stay cables of a cable-stayed bridge are typically post-tensioned to counter balance the effect of dead load on the deck and pylon. These cable forces affect directly the performance and the economic efficiency of cable-stayed bridges. An entropy-base optimization algorithm to optimize the cost of cable-stayed bridges was proposed by Simoes and Negrao (1994). The locations of stay cable along the main girder and pylon, and the cross-sectional sizes of the deck, pylons, and stay cables were considered as the design variables. In their work, the number of stay cables and the mid-span length were assumed as pre assigned constant parameters. Long (1999) used an internal penalty function algorithm optimize the cost of cable-stayed bridges with composite superstructure. The bridge is modeled as a 2-D structure while including the geometric non-linear effect. The design variables included the parameters that describe the cross sectional dimensions of the bridge elements. The height of pylon, the mid-span length, and the number of stay cables are kept constant with pre assigned values. Simoes and Negrao (2000) proposed a function called convex scalar function, which is used to optimize the cost of the deck in cable-stayed bridges. Convex scalar function combines dimensions of the cross-sections of the bridge and post-tensioning cable forces. The design variables included maximum allowable stresses, minimum stresses in the stay cables, and deflections of the deck. The pylon height and the mid-span length were not considered by Simoes and Negrao (2000). Recently, Lute (2009) has proposed a genetic algorithm (GA) which was employed to reduce the computational time of optimizing cable-stayed bridges. It was indicated that the genetic algorithm is an efficient tool for solving cable-stayed bridge optimization problems. The number of stay cables was treated as a pre-set design variable, and the effect of post-tensioning cable forces was not considered in their study.

In the design cable-stayed bridge, an important step is determining the tensioning forces of stay cables to achieve a desired geometry of the bridge after construction, especially under the reaction of dead load. Many structural analysis techniques were proposed to solve this problem. The different models of cable have been investigated.

The elastic cable is assumed to be perfectly flexible and possesses only tension stiffness; it is incapable of resisting compression, shear and bending forces. When the weight of the cable is neglected, the cable element can be considered as a straight member. But under action of its own dead load and axial tensile force, a cable supported at its end will sag into a catenary shape. The axial stiffness of a cable will change with changing sag. When a straight cable element for a whole inclined cable stay is used in the analysis, the sag effect has to be taken into account. On the consideration of the sag nonlinearity in the inclined cable stays, it is convenient to use an equivalent straight cable element with an equivalent modulus of elasticity, which can well describe the catenary action of the cable. The concept of a cable equivalent modulus of elasticity was first introduced by Ernst. This cable element has been used popularly in cable – stayed bridge design.

More accurate than cable element with an equivalent modulus of elasticity, some cable elements have been proposed such as nonlinear geometry cable element proposed by the last of authors, nonlinear cable element by R. Karoumi, elastic catenary cable element with an unknown initial length.

Pareto’s economic principle (Pareto 1893) is gaining increasing acceptance to multi-objective optimization problems. In minimization problems a solution vector is said to be Pareto optimal if no other feasible vector exists that could decrease one objective function without increasing at least another one. The optimum vector usually exists in practical problems and is not unique. In regards to reliability-based design, several alternative formulations exist. A comprehensive review can be found in Thoft-Christensen (1991). The material cost together with maximum probability of failure and measures of the structural performance and imposed by manufacturing and technical considerations are the objectives to be minimized. Size, shape, material configuration and loading parameters may be allowed to vary during the optimization process. Bounds must be set for average cross-sectional and geometric design variables in order to achieve executable solutions and required aesthetic characteristics.

The overall objective of cable-stayed bridge design is to achieve an economic and yet safe solution. One of the factors conventionally adopted is the cost of material used. A second set of goals arises from the requirement that the stresses should be as small as possible. The optimization method requires that all these goals should be cast in a normalized form. Another set of goals arises from the imposition of lower and upper limits on the sizing variables, namely minimum cable cross sections to prevent topology changes and executable dimensions for the stiffness girder and pylons cross sections. Similar bounds must be considered for the geometric design variables. Additional bounds are set when geometric design variables are considered, to ensure that no geometry violation occurs when these design variables are updated.

Additional goals may be established in order to ensure the desired geometric requirements during the optimization process (mesh discretization, ratios of variation of cable spacing on deck and pylons, etc). For these, the chosen approach was to initially supply all the necessary information, by means of a geometry coefficients set describing such conditions.

1.5 Structural Design Optimization

The optimization problem is classified on the basis of nature of equations with respect to design variables. If the objective function and the constraints involving the design variable are linear then the optimization is termed as linear optimization problem. If even one of them is nonlinear it is classified as the non-linear optimization problem. In general the design variables are real but some time they could be integers for example, number of layers orientation angle, etc. The behavior constraints could be equality constraints or inequality constraints depending on the nature of the problem. The goal of optimization is to find the best solution among a set of candidate solutions using efficient quantitative methods. In this framework, decision variables represent the quantities to be determined, and a set of decision variable values constitutes a candidate solution. An objective function, which is either maximized or minimized, expresses the goal, or performance criterion, in terms of the decision variables. The set of allowable solutions, and hence, the objective function value, is restricted by constraints that govern the system. .

Structural optimization is seeking the best set of design parameters defining a structural system structural design in a rational manner and has been proved to be much more efficient than the conventional trial-and-error design process. Due to the developments of faster digital computers, more sophisticated computing techniques and more frequent use of finite element method. Structural optimization techniques have found their way into many facets of engineering practice for the sake of design improvement during the past decades. To some extent, design optimization has become a standard tool in many industrial fields and covers applications in civil engineering mechanical engineering, vehicle engineering and more. .

1.6 Optimization Techniques

There are 4 optimization techniques that can be applied to case of the cable stayed bridge and combined together so that to design more efficiently and to refine the results with respect to the design parameters involving in the structural design optimization. These techniques are:

1 -Topology optimization
Optimize material distribution in a 3d domain finding the best design that meets your requirements. Narrow the field of solutions to a set of feasible solutions including manufacturing constraints.
2 -Topography optimization
Enhance structural performances such as stiffness and maximum local tresses reduction to find the best load path to place structural ribs and reinforcement plates.
3 - Size optimization
Optimize the thicknesses and more geometric features of your model while maintaining the same overall shape.
4 -Shape optimization
Optimize your model by changing the shape of your design space to find the best profile of your section. Find the best curvature of your beam section while staying the given boundary conditions.

1.7 Load Factor Optimization

The permanent state of stress in a cable-stayed bridge subject to its dead load is determined by the tension forces in the cable stays. They are introduced to reduce the bending moments in the main girder and to support the reactions in the bridge structure. The cable tension should be chosen in a way that bending moments in the girders and the pylons are eliminated or at least reduced as much as possible. Hence, the deck and pylon would be mainly under compression under the dead load. Initial pre-stressing forces can be calculated through optimizing the equilibrium state. The calculation of the ideal cable pre stressing forces by the optimization is restricted to the linear analysis as the different loadings are superposed. The initial cable pre-stressing forces are obtained by the unknown load factor function and the initial equilibrium state analysis of a completed cable-stayed bridge.

1.8 Cable Stayed Bridge Structural Concept

The concept of a cable stayed bridge is simple although the loading mechanism is not so easy to predict. A bridge carries mainly vertical loads acting on the girder. The stay cables provide intermediate supports for the girder so that it can span a long distance. The basic structural form a cable stayed bridge is a series of overlapping triangles that connect the deck to the pylons. The deck, the cables and the pylons are under predominant axial forces, with the cables under tension and both the pylon and the deck under compression. Figures 5 and 6 are showing the stay cables, deck and the pylons that are connected with each other to form the cable stayed bridge. Axially loaded members are generally more efficient than flexural members. This contributes to economy of a cable stayed bridge. They also have less steel consumption but on the other hand larger stress variations can occur and their structural behavior is complex. Nowadays, cable stayed bridges are the most common bridge type for long span bridges and can span up to 1000 m and come in various forms because of economy and aesthetics. They are beautiful structures that appeal to most people. The pylons are most visible elements of a cable stayed bridge and therefore contribute the most from an aesthetic point of view. A clean and simple configuration is preferable with free standing pylons. Under special circumstances they can also serve as tourist attractions, for example when lighting is a part of the design which enhances the beauty and visibility of the bridge at night.

illustration not visible in this excerpt

Figure 5 - River Suir Bridge

http://www.mageba-group.com/fr/804/References.htm?Reference=36340

illustration not visible in this excerpt

Figure 6 - Yeosu Dolsan Bridge

http://www.rjkoehler.com/travelog/2012/06/yeosus-dolsan-bridge-and-expo-venue/

Typical cable stayed bridge is a deck with one or two pylons erected above the piers in the middle of the span. The cables are attached diagonally to the girder to provide additional supports. A typical cable-stayed bridge is depicted in Figure 7. The pylons form the primary load-bearing structure in these types of bridges. Large amounts of compression forces are transferred from the deck to the cables to the pylons and into the foundation as shown in Figure 7. The design of the bridge is conducted such that the static horizontal forces resulting from dead load are almost balanced to minimize the height of the pylon. Cable stayed-bridges have a low center of gravity, which makes them efficient in resisting earthquakes. Cable stayed bridges provide outstanding architectural appearance due to their small diameter cables and unique overhead structure.

illustration not visible in this excerpt

Figure 7 - A simple illustration of typical cable-stayed Bridge

Chapter 2

2.1 Cable Stayed Bridge Components

A cable stayed bridge is composed of three main components (Deck, Pylon and Stay Cables) which are more detailed in identifying their roles in the following sections.

2.1.1 Deck

The deck or road bed is the roadway surface of a cable-stayed bridge. The deck can be made of different materials such as steel, concrete or composite steel-concrete. The choice of material for the bridge deck determines the overall cost of the construction of cable stayed bridges. The weight of the deck has significant impact on the required stay cables, pylons, and foundations (Bernard 1988). As one can see in Figure 8, the composite steel-concrete deck is composed of two structural edge girders. These girders are attached by transverse steel floor beams. The precast reinforced concrete

illustration not visible in this excerpt

Figure 8 - Composite deck

deck is supported by these two main girders. This type of composite steel-concrete deck has more advantages as follow (Hassan 2012):

1- The own weight of a composite deck is less than a concrete deck.
2- The light steel girders can be erected before applying the heavy concrete slab.
3- The stay cables have more resistance against rotation anchoring to the outside main
steel girders.
4- The redistribution of compression forces due to shrinkage and creep onto the steel girders are minimized by using the precast slab.

2.1.2 Pylon

Pylons of cable stayed bridges are aimed to support the weight and live load acting on the structure. There are several different shapes of pylons for cable stayed bridges such as a Trapezoidal pylon, Twin pylon, A-frame pylon, and Single pylon. They are chosen based on the structure of the cable stayed bridge (for different cable arrangements), aesthetics, length, and other environmental parameters.

2.1.3 Stay Cables

Cables are one of the main parts of a cable-stayed bridge. They transfer the dead weight of the deck to the pylons. These cables are usually post-tensioned based on the weight of the deck. The cables post-tensioned forces are selected in a way to minimize both the vertical deflection of the deck and lateral deflection of the pylons. There are four major types of stay cables including, parallel-bar, parallel-wire, standard, and locked-coil cables. The choice of these cables depends mainly on the mechanical properties, structural properties and Economical criteria.

2.2 Stay cables types

Problems arose with the stays of early cable stay bridges as a result of deficiencies with the anchorage design, steel material problems and inadequate corrosion resistance. The development of modern stay systems has largely overcome these problems providing designs that minimize bending of the stay at the anchorage face and incorporate a double corrosion protection system throughout. Available stay systems include:

1- Locked coil (prefabricated)
2- Helical or spiral strand (prefabricated)
3- Bar bundles
4- Parallel wire strand (PWS)
5- New PWS (prefabricated)
6- Parallel strand
7- Advanced composites

2.2.1 Locked coil stays

Locked coil stays have been incorporated into many of the earliest cable-stay bridges. The stays are factory produced on planetary stranding machines, each layer being applied in a single pass through the machine and contra-laid between each layer. The core of the stay is composed of conventional round steel wires while the final layers comprise Z-shaped steel wires which lock together creating an extremely compact stay cross-section. A typical example of a locked coil stay is illustrated in Figure 9. Modern locked coil stays provide all the wires in a finally galvanized condition and will achieve a tensile strength of up to 1770N/mm2. The stays are commonly anchored by zinc filled sockets although sometimes stays that are sheathed with a polyethylene protection have their sockets filled with epoxy resin. The largest locked coil stays manufactured to date are the 167mm diameter stays supplied for the Rama IX Bridge over the Chao Phraya River, Bangkok.

illustration not visible in this excerpt

Figure 9 - Locked coil cable

2.2.2 Helical or spiral strand stays

Helical or spiral strand stays, which are illustrated in Figure 10, are also factory fabricated on a planetary stranding machine similar to the locked coil stay but are entirely manufactured from finally galvanized round steel wires. The wires are usually of 5mm diameter with a tensile strength of either 1570N/mm2 or 1770N/mm2. The largest spiral strand stays manufactured to date are 164mm diameter, as supplied for the Queen Elizabeth II Bridge over the River Thames at Dartford.

illustration not visible in this excerpt

Figure 10 - Spiral strand cable

[...]