In this thesis the reliability analysis of reinforced concrete bridge decks subjected to high cycle fatigue was undertaken. A simply supported (single span) reinforced concrete bridge deck was specifically used for the investigation. The statistical models of capacity loss were derived. The uncertainties in structural resistance and the applied loadings were fully accommodated using probabilistic method. The limit state function(Failure Modes) for the flexural capacity of the deck slab, shear capacity of the deck beam and flexural capacity of the deck beam girder was developed and evaluated using the First Order Reliability Method(FORM). The entire process was implemented through a developed MATLAB program “dc_fatigue.m”. Analysis was carried out on all the failure modes considered by varying the geometrical and material properties of the system and their respective safety indices were determined. Failure due to shear in the deck beam gave the least safety index range of 6.06 to -2.11 at 3cycles/min for 10years and 40years respectively indicating the most critical section, when compared to the flexural failure with a safety index range value of 7.59 to 0.25 at 3cycles/min for 10years and 40years respectively. It was generally observed that the reliability index increased as the depth of section and concrete grade increased. Also, the coefficient of variation due to the concrete strength and depth of section decreased with increase in reliability index, from a safety index range of 7.61 to -0.47 at 5% covariance and from 3.47 to -0.23 at 20% covariance under stress load of 3cycles/min. This trend is expected as quality control plays a very important role in achieving this level of safety in the system. Thus, for a single span bridge of 15m and less under high level of traffic load, a deck beam depth of 1200mm, deck slab of 250mm and concrete strength class of 30-35 are adequate for design and construction.
TABLE OF CONTENTS
Title page ii Declaration iii Certification iv Dedication v Acknowledgement vi Abstract vii Table of Contents viii List of Figures xi List of Tables xv List of Appendices xv List of Symbols/Abbreviation xvii CHAPTER ONE 1.0 INTRODUCTION 1.1 Preamble 1 1.2 Problem Statement 3 1.3 Justification of Study 4 1.4 Aim and Objectives 6 1.4.1 Aim 6 1.4.2 Objectives 6 1.5 Scope and Limitation 7 1.5.1 Scope 7 1.5.2 Limitation 7
CHAPTER TWO 2.0 LITERATURE REVIEW 2.1 Preamble 8 2.2 Basis of Fatigue 10 2.3 Classification of details in concrete structural component 11 2.4 Fatigue 12 2.4.1 Fatigue of concrete 12 2.4.2 Fatigue of reinforcement 13 2.4.3 Fatigue of reinforced concrete 14 2.5 Fatigue failure of reinforced concrete structures 14 2.5.1 Shear and Bond failures 15 2.6 Structural Reliability 16 2.7 Limit State Principles 16 2.7.1 Failure Events and Basic Random Variables 18 2.7.2 Reliability Index 20 2.7.3 Hasofer-Lind Reliability Index 21 CHAPTER THREE 3.0 RESEARCH METHODOLOGY 3.1 Estimation of parameters for reliability of bridge deck 23 3.2 Bridge Model 23 3.3 Structural Material 24 3.4 Load Combination 24 3.4.1 Dead Load Effect 25 3.4.2 Live Load Effect 25 3.5 Structural Resistance Models 28 3.6 Generation of Limit State Function 29 3.6.1 Failure of the Slab 30
188.8.131.52 Applied Dead Load 30 3.6.2 Failure Model for Beams 31 184.108.40.206 Interior Beams 32 220.127.116.11.1 Limit State Function for Shear in Interior Beam 33 18.104.22.168 Limit State Function for Moment Flexure in Interior Beam 34 3.7 Computer Analysis Procedure 36 3.7.1 MATLAB Program 37 CHAPTER FOUR 4.0 RESULT DISCUSSION 4.1 Failure due to moment flexure in Slab (Failure Mode One) 39 4.1.1 Effect of deck slab depth variation on the Safety Index 39 4.1.2 Effect of Coefficient of Variation due to concrete strength under bending on the Safety Index with respect to the stress cycle 40 4.1.3 Effect of Coefficient of Variation due to slab depth under bending on the Safety Index with respect to the stress cycle 42 4.2 Failure due to shear in deck beam(Failure Mode Two) 44 4.2.1 Effect of beam depth variation on the Safety Index with respect shear stress under stress cycle 44 4.2.2 Effect of concrete strength variation on the Safety Index with respect to stress cycle46 4.2.3 Effect of coefficient of variation due to concrete strength under shear on the safety index with respect to the stress cycle. 47 4.2.4 Effect of coefficient of variation due to beam depth under shear on the safety index with respect to the stress cycle 49 4.3 Failure due to Moment Flexure (Failure Mode Three) 50
4.3.1. Effect of beam depth variation on the Safety Index with respect
to flexure under stress Cycle 51 4.3.2 Effect of concrete strength variation under bending on the Safety Index with respect to stress cycle 52 4.3.3 Effect of coefficient of variation due to concrete strength under flexure on the Safety Index with respect to the stress cycle 54 4.3.4 Effect of coefficient of variation due to beam depth under flexure on the Safety Index with respect to the stress cycle 56 CHAPTER FIVE 5.0 CONCLUSION AND RECOMMENDATION 5.1 Conclusion 59 5.2 Recommendation 59
INTRODUCTION 1.1 Preamble Researches on fatigue behaviour of concrete materials started at the end of the 19th century, due to failure in many concrete structures caused by fatigue rupture of the concrete (Rteil et.al, 2011). Results of experimental and theoretical studies of the fatigue properties of plain concrete, reinforcing bars, prestressing tendons, and also of structural concrete members, have been accumulating steadily over the past thirty years. The assessment of existing structures will become a more frequent task for engineers in the near future due to the increasing age of existing infrastructure. These may be due to reasons such as(Jansen, 1996); Change in intended use of the structure, new regulations with higher load requirements for the structure, indications of ongoing deterioration in the structure, unusual incidents during use (e.g. vehicle impact, fire, earthquakes), inadequate serviceability, discovery of design or construction errors. The purpose of reliability analysis in this research is to verify the overall stability and establishment of action effects, i.e. the distribution of internal forces and moments. In turn, this will enable the calculation of stresses, strains, curvature, rotation and displacements. In certain complex structures, the type of analysis used (e.g. finite-element analysis) will yield internal stresses and strains and displacements directly.
To carry out the analysis, both the geometry and the behaviour of the structure will need to be idealized. Commonly, the structure is idealized by considering it as made up of elements depicted.
Some of the causes in variation of uncertainty in structural analysis and design by Svensson (1997) include: the material properties, structural properties of components, load variation, parameter estimations and model errors. Even if Svensson’s analysis is sufficiently accurate, it is necessary to note that there is a fundamental difference among the causes described. The first three items are actual random variables and they constitute the casualty of the investigated phenomenon. On the contrary, the fourth and fifth items are caused by a lack of knowledge of the true parameters involved (usually due the limited amount of experimental data) or of the fatigue damage mechanism under variable-amplitude loading or of the structural analysis of the component investigated. These last items cause an additional uncertainty margin to evaluation of the actual variability of the fatigue phenomena. According to this judgement, the first group implies a problem in estimating a random variable, while the second group relates to the evaluation of variance and proper confidence interval on the obtained estimation. Thus, an accurate assessment of fatigue strength under service loading should include both correct estimation of each statistical parameter involved and sound evaluation of the confidence interval of each estimated quantity, particularly when the overall probability of failure is concerned.
Design of reinforced concrete bridges is normally done on the basis of structural analysis. The purpose of the analysis is to find a distribution of sectional forces which fulfils limit state criteria and is suitable for design. In the past, structural analyses were often done with simplified models, for example two-dimensional (2D) equivalent beam or frame models. Such a model is not able to describe the distribution of forces in transverse direction.
Therefore a design according to a 2D equivalent model will not give a true linear elastic distribution, even though the design might fulfil requirements in ultimate limit state (ULS) after sufficient plastic redistribution(Karin et.al,2010). Current practice for the design of reinforced concrete bridges is based on a linear elastic structural analysis in which a suitable distribution of sectional forces is sought. Such an analysis is today required to account for all the structural response, implying a demand for three-dimensional (3D) models capable of describing the force distribution in longitudinal and transverse directions. 1.2 Problem Statement Deterioration of reinforced concrete bridge decks has been a major problem in the country. Both Federal Ministry of Works and Federal Roads Maintenance Agency records show that as at the year 2008, there were 1,696 Bridges that had been completed from year to year and were in use on the Federal Highways nationwide. However,as at date some of these bridges are requiring emergency intervention and some have even been shut to traffic. According to the data of the “national bridge inventory” obtained from the U.S. Department of Transportation, it is estimated that deficiencies occur mostly in the decks in more than half of the bridges in United States (Mabsout et.al 2004).
It is not only an economic problem but also risk to commuters who traverse these bridges. Some of the deficiencies may include; slight damages to the deck surfaces, spalling of top concrete surface and the decrease in serviceability of the deck which may lead to danger to the public. As such there is the need to understand the behaviour of bridge deck under service conditions and develop a reliable way to check the serviceability of the deck.
In the Nigeria, most of the bridge decks are constructed as reinforced concrete slabs supported by precast prestressed beam girders. Such decks have traditionally been designed using the “empirical strip method”, based on a conventional beam theory, which assumed that the slab is continuous over fixed supports. As a result, the top part of the slab is reinforced with steel bars to resist the negative moments, and the bottom part of the slab is reinforced with steel bars to resist the positive moments. It is believed that the decks designed by empirical method are more resistant to deterioration due to fewer sources of corrosion (fewer steel rebars). This empirical method has been adopted in the current AASHTO LRFD code (2010). In bridges, there are usually a large number of cycles of significant live load, and fatigue will almost always precede fracture. Therefore, controlling fatigue is practically more important than controlling fracture because this can result in increased displacement of the reinforcing bar relative to the concrete, known as slip (Rteil et.al, 2011). Therefore, in this thesis, a reliability based analysis with a probabilistic view would be carried out to check the uncertainties in structural resistance and applied loads. Limit state function for flexural capacity of deck slab and beam as well as shear capacity in deck beam were developed and evaluated using FORM. The entire process was implemented through a developed MATLAB program. 1.3 Justification of the Study
In the past, fatigue failure has been primarily regarded as a design problem associated with civil engineering structures such as highways bridges, car parks and rail bridges which are subjected to reasonably well defined man-induced repeated loads. However, serious
fatigue problems can also arise as the result of complex, naturally occurring, ill-defined patterns of repeated loads. For example, fatigue problems in concrete structures have arisen recently in relation to off-shore and marine construction(Karin et.al, 2010). In RC slab damage process, microcracks occur and gradually develop, and at the final stage, large cracks occur and slab displacement increases sharply. Mid-span deflection caused by fatigue is thought to be a possible indicator for the degree of slab damage under the influence of overall progress of slab cracking.The fatigue lives of flexural members are typically a function of the stress range induced in the reinforcement (Braimah et.al,2006). It is practically hard to exactly interpret the loading histories of bridge deck. The failure can occur even if the maximum stress is below the ordinary strength of the material. Some materials have a certain stress limit e.g. steel, which means that the stress variation below a certain level can be repeated infinitely many times without fatigue failure. Fatigue failure is characterized by fracture in a localized area of a structure which is exposed to cyclic loading (Karin et.al, 2010).
A single lane of a Class A highway bridge experiences an average daily truck traffic of over 1000, or 27 million trucks over a 75 year design life (CSA A23.3, 2010).Bridge decks directly sustain these repeated moving wheel loads and may be susceptible to fatigue damage (El-Ragaby et al. 2007). Fatigue life and fatigue strength of reinforced concrete elements areinfluenced by many factors including the material properties of the concrete and reinforcement, reinforcement ratio, transverse reinforcement, minimum and maximum values of repeated loading, range and rate of loading as well as environmental factors such as temperature and humidity. The response of a member subjected to fatigue loading is
affected by both the strength of the materials and the interaction between the concrete and reinforcement (Higgins et al. 2006). It is generally agreed that the most important factor influencing fatigue behaviour is the applied load range,or rather the induced stress range in each component. The results of this research can be usefully employed in practice only when appropriate fatigue design formats have been developed which provide adequate, but not excessive, margins of safety against fatigue failure. The safety margins and the associated design safety coefficients would allow realistically for inherent variability in loads, in the fatigue properties of the materials, and in the performance of members and systems under high cyclic loads. 1.4 Aim and Objectives 1.4.1 Aim The aim of this research is to assess the fatigue reliability of a typical RC bridge deck under limit state condition. 1.4.2 Objectives The objectives of the study include;
i. Identify the possible fatigue modes of failure of the bridge deck.
ii. Develop a performance function for each mode of failure.
iii. Establish the statistical models for each random variables.
iv. Develop a MATLAB program to evaluate the reliability functions, and compute probabilities of failure and safety indices.
v. Carry out sensitivity analysis using the developed program.
1.5 Scope and Limitation 1.5.1 Scope The research project was to treat fatigue failure in reinforced concrete bridge deck. It is devoted to the testing of the time-dependent safety of RC bridge deck using statistical toolsMATLAB and FORM.In order to simplify and emphasize the fatigue analysis, resistance models and load values together with information on their stochastic parameters have been developed, evaluated and implemented in accordance with the provisions of the Eurocode system. 1.5.2 Limitations This work is limited to numerical analysis and not laboratory work was carried out.
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