The presented study deals with the development of design code for reinforced concrete and prestre... more The presented study deals with the development of design code for reinforced concrete and prestressed concrete beams, slabs and columns. The major steps in the code calibration procedure include the development of load and resistance models. It is assumed that the available load models are adequate. Therefore, the main focus of this study is the verification of resistance models. A considerable database is gathered on material strength: ordinary concrete, lightweight concrete, high-strength concrete, reinforcing steel, and prestressing steel. The test results were provided by industry and they are analyzed to determine the cumulative distribution functions and other statistical parameters. It is observed that there is a considerable improvement in quality of materials during the last 20-30 years, and this results in an increased reliability of structural components. The reliability analysis is performed to determine the reliability indices for components designed according to the current code (ACI 318-99). Based on the results, the target reliability indices are selected. New load and resistance factors are recommended based on closeness to the target reliability index.
Load and resistance parameters for bridges involve uncertainties due to natural randomness, imper... more Load and resistance parameters for bridges involve uncertainties due to natural randomness, imperfect models or human errors. Because of these uncertainties, absolute reliability is not an attainable goal. Selection of the optimum reliability level is an economical problem. Lower reliability results in frequent failures, while higher reliability requires more initial costs (materials and labor). Therefore, structural reliability is a convenient acceptability criterion in the development of a design code. The objective of this paper is to present a procedure for calculation of load and resistance factors.
The objective of this paper is to compare the reliability level of prestressed concrete bridge gi... more The objective of this paper is to compare the reliability level of prestressed concrete bridge girders designed using three codes:
Transportation Research Board 86th Annual MeetingTransportation Research Board, 2007
For a probabilistic risk assessment of an Arch bridge, an innovative prediction method for the co... more For a probabilistic risk assessment of an Arch bridge, an innovative prediction method for the combination of failure modes is proposed and compared with the conventional system reliability analysis method. The suggested method generates unexpected combinations of failure modes in significantly reduced time and efforts, compared with the previous permutation method or the conventional system reliability analysis method. In addition to that, the suggested method can be used for the verification of the system reliability with more specific predictions of failure modes. Component reliabilities of girders have been evaluated using the response surfaces of the design variables at the selected critical sections based on the maximum shear and negative moment locations. Response Surface Method (RSM) is successfully applied for reliability analyses for this relatively small probability of failure of the complex structure, which is laborious to be obtained by Monte-Carlo Simulations or by First Order Second Moment Method that can not easily calculate the derivative terms of implicit limit state functions. For the analysis of system reliability, the target bridge system is modeled as series connection system. The upper and lower probabilities of failure for the structural system have been evaluated and compared with the suggested prediction method for all the possible combination of failure modes.
Because load and resistance are subject to random vibration, reliability can be considered a rati... more Because load and resistance are subject to random vibration, reliability can be considered a rational measure of structural performance. The recently developed AASHTO 2004 bridge design code is based upon the assumption of a consistent safety level. Reliability is calculated using load and resistance models, and these models have an important effect on the results. Resistance can be considered as a function of material properties, geometry and dimensions, and the so-called professional (analysis) factor. The statistical parameters depend to a large extent on the quality of materials and workmanship. The code writers usually assume that the quality is at an average level; however, there is a considerable variation between different contractors. In particular, this applies to the properties of concrete. The aim of this paper is to establish the relationship between resistance parameters and reliability. An analysis is performed for girder bridges.
Bridge load and resistance parameters involve a considerable degree of uncertainty. Therefore, th... more Bridge load and resistance parameters involve a considerable degree of uncertainty. Therefore, the reliability analysis can be used to quantify the risk. There has been a considerable research activity in the area of structural reliability in the last two decades. There are methods available for calculation of the probability of failure for components of buildings, bridges and other structures. They are summarized in several books (e.g. Thoft-Christensen and Baker 1982; Ang and Tang 1986; Madsen, Krenk and Lind 1986; Melchers 1987). For given statistical models of loads and resistance the available methodology provides accurate results. Structural safety was evaluated for bridge components (e.g. Nowak and Grouni 1990).
The optimum safety level depends on the consequences of failure and cost of safety. Selection of ... more The optimum safety level depends on the consequences of failure and cost of safety. Selection of the target value can be based on consideration of these two parameters. Target reliability indices calculated for newly designed bridges and existing structures are different for many reasons. Reference time period is different for newly designed and existing bridges. Load model, used to calculate reliability index, depends on the reference time period. Single load path components require a different treatment than multiple load path components. The paper deals with the selection criteria for the target B for ultimate limit states (ULS) and serviceability limit states (SLS). For the covering abstract see ITRD E126028.
The development of rational design criteria requires efficient measures of structural performance... more The development of rational design criteria requires efficient measures of structural performance. Traditional methods of analysis do not reveal the actual safety reserve (degree of redundancy). Load and resistance parameters are random variables. Therefore, probabilistic methods are used to model behavior of bridges. The paper provides a review of the available methodology, including reliability measures, load models and resistance models for girder bridges.
The combination of dead load and live load is very important in design of bridge superstructure a... more The combination of dead load and live load is very important in design of bridge superstructure as, in practice, it controls the strength limit states. The basic set of load factors for the Strength I limit state is 1.25 for dead load and 1.75 for live load and dynamic load. For design cases when the dead load dominates, i.e. for Strength IV limit state, the dead load factor is 1.5. The acceptability criterion for load and resistance factors in the AASHTO LRFD Code is closeness to the target reliability index, which is assumed to be 3.5 for steel and concrete girder bridges. However, the reliability analysis performed for a full range of dead load to live load ratios indicates that when live load is about 10-20% of the total load, the reliability indices are about 3.0 which is lower than the target value of 3.5. This is an indication that the reliability level is insufficient and there is a need for increasing load factors. On the other hand, for dead load constituting about 100% of the total load (i.e. no live load), the reliability index is much higher than 3.5, which means that the load factor 1.5 can be reduced. Therefore, it is proposed to change the current Strength IV load factors to dead load of 1.4 and live load factor 1.4. The result is a more uniform reliability level for all combinations of dead load and live load. The results of reliability analysis are presented in graphs.
The objective of this paper is to present the development of the resistance factor for shear in r... more The objective of this paper is to present the development of the resistance factor for shear in reinforced concrete bridges made of lightweight concrete. The statistical model of resistance was based on the available limited experimental data. It was found that the current shear design procedures are about 15% less conservative for lightweight concrete when compared to normal weight concrete. The reliability analysis was performed and served as a basis for the selection of the resistance factor. The results indicate that a resistance factor equal to 0.8 can be applied for the shear design of lightweight concrete bridges.
Railway bridges are exposed to repetitive high stress due to the live load and constant, relative... more Railway bridges are exposed to repetitive high stress due to the live load and constant, relatively low stress due to dead load. Cyclic live load may lead to failure even when the stress level is lower than the allowable stresses. Therefore, components and connections need to be analyzed for possible damage caused by fatigue. The basic approach for estimating the remaining fatigue life of a structure element is to use S-N curves. However, those laboratory specimens tested with a constant-amplitude stress range show variability in the results. This means that fatigue resistance is a random variable. Moreover, railway bridges are subjected to variable-amplitude stress ranges during their service life. The axle load depends on many factors, not only capacity of the car but also dynamic behavior of the car and dynamic response of the bridge. If load and resistance parameters are random variables, structural performance should be measured in terms of reliability. The objective of this st...
The paper summarizes the code development procedures used for the new LRFD (load and resistance f... more The paper summarizes the code development procedures used for the new LRFD (load and resistance factor design) bridge code. The current specifications use allowable stresses and/or load factor design. The new code is based on a probability-based approach. Structural performance is measured in terms of the reliability (or probability of failure). Load and resistance factors are derived so that the reliability of bridges designed using the proposed provisions will be at the predefined target level.
Recently a considerable research effort has been devoted to bridge design and evaluation in Europ... more Recently a considerable research effort has been devoted to bridge design and evaluation in Europe and North America. This paper focuses on the review and comparison of the reliability models developed in conjunction with the development of whole life assessment methods in the UK and AASHTO LRFD code in the US. The objective of the paper is to establish time-varying reliability profiles relating reliability index to various rates of deterioration (corrosion) and develop a procedure for calculating probability of failure for different time stages. The corrosion patterns and rates are modelled for steel girder bridges on the basis of empirical relationships from laboratory tests and field observation of existing bridges. Two bridge design codes, BS 5400 in the UK and AASHTO in the UK, are compared in terms of reliability. Reliability is calculated for selected existing structures. Relationship between reliability index and rating factor is investigated. The obtained reliability spectr...
This study was part of the reliability-based calibration of the design provisions for curved gird... more This study was part of the reliability-based calibration of the design provisions for curved girder bridges. The reliability analysis for straight girder bridges for the AASHTO LRFD Code (2004) was documented by Nowak (1995, 1999). The aim of the present study is the verification of resistance factors for curved steel girder bridges. The presented research was conducted in conjunction with NCHRP Project 12-52. Three representative bridges were selected for this study. For each bridge, the analysis was performed using an advanced finite element program to determine the strains and stresses, and to compare them with the design (nominal) values. For one of the considered bridges, the analytical results were verified using field test data obtained from the University of Minnesota. The assumptions and methods used in the calibration process are also described.
Reliability analysis is performed for fatigue limit state in steel bridges. The load and resistan... more Reliability analysis is performed for fatigue limit state in steel bridges. The load and resistance parameters are treated as random values. The fatigue load model includes the magnitude and frequency of occurrence. Fatigue resistance is based on the available test data, in particular S-N curves. Reliability indices are calculated as a function of time.
The presented study deals with the development of design code for reinforced concrete and prestre... more The presented study deals with the development of design code for reinforced concrete and prestressed concrete beams, slabs and columns. The major steps in the code calibration procedure include the development of load and resistance models. It is assumed that the available load models are adequate. Therefore, the main focus of this study is the verification of resistance models. A considerable database is gathered on material strength: ordinary concrete, lightweight concrete, high-strength concrete, reinforcing steel, and prestressing steel. The test results were provided by industry and they are analyzed to determine the cumulative distribution functions and other statistical parameters. It is observed that there is a considerable improvement in quality of materials during the last 20-30 years, and this results in an increased reliability of structural components. The reliability analysis is performed to determine the reliability indices for components designed according to the current code (ACI 318-99). Based on the results, the target reliability indices are selected. New load and resistance factors are recommended based on closeness to the target reliability index.
Load and resistance parameters for bridges involve uncertainties due to natural randomness, imper... more Load and resistance parameters for bridges involve uncertainties due to natural randomness, imperfect models or human errors. Because of these uncertainties, absolute reliability is not an attainable goal. Selection of the optimum reliability level is an economical problem. Lower reliability results in frequent failures, while higher reliability requires more initial costs (materials and labor). Therefore, structural reliability is a convenient acceptability criterion in the development of a design code. The objective of this paper is to present a procedure for calculation of load and resistance factors.
The objective of this paper is to compare the reliability level of prestressed concrete bridge gi... more The objective of this paper is to compare the reliability level of prestressed concrete bridge girders designed using three codes:
Transportation Research Board 86th Annual MeetingTransportation Research Board, 2007
For a probabilistic risk assessment of an Arch bridge, an innovative prediction method for the co... more For a probabilistic risk assessment of an Arch bridge, an innovative prediction method for the combination of failure modes is proposed and compared with the conventional system reliability analysis method. The suggested method generates unexpected combinations of failure modes in significantly reduced time and efforts, compared with the previous permutation method or the conventional system reliability analysis method. In addition to that, the suggested method can be used for the verification of the system reliability with more specific predictions of failure modes. Component reliabilities of girders have been evaluated using the response surfaces of the design variables at the selected critical sections based on the maximum shear and negative moment locations. Response Surface Method (RSM) is successfully applied for reliability analyses for this relatively small probability of failure of the complex structure, which is laborious to be obtained by Monte-Carlo Simulations or by First Order Second Moment Method that can not easily calculate the derivative terms of implicit limit state functions. For the analysis of system reliability, the target bridge system is modeled as series connection system. The upper and lower probabilities of failure for the structural system have been evaluated and compared with the suggested prediction method for all the possible combination of failure modes.
Because load and resistance are subject to random vibration, reliability can be considered a rati... more Because load and resistance are subject to random vibration, reliability can be considered a rational measure of structural performance. The recently developed AASHTO 2004 bridge design code is based upon the assumption of a consistent safety level. Reliability is calculated using load and resistance models, and these models have an important effect on the results. Resistance can be considered as a function of material properties, geometry and dimensions, and the so-called professional (analysis) factor. The statistical parameters depend to a large extent on the quality of materials and workmanship. The code writers usually assume that the quality is at an average level; however, there is a considerable variation between different contractors. In particular, this applies to the properties of concrete. The aim of this paper is to establish the relationship between resistance parameters and reliability. An analysis is performed for girder bridges.
Bridge load and resistance parameters involve a considerable degree of uncertainty. Therefore, th... more Bridge load and resistance parameters involve a considerable degree of uncertainty. Therefore, the reliability analysis can be used to quantify the risk. There has been a considerable research activity in the area of structural reliability in the last two decades. There are methods available for calculation of the probability of failure for components of buildings, bridges and other structures. They are summarized in several books (e.g. Thoft-Christensen and Baker 1982; Ang and Tang 1986; Madsen, Krenk and Lind 1986; Melchers 1987). For given statistical models of loads and resistance the available methodology provides accurate results. Structural safety was evaluated for bridge components (e.g. Nowak and Grouni 1990).
The optimum safety level depends on the consequences of failure and cost of safety. Selection of ... more The optimum safety level depends on the consequences of failure and cost of safety. Selection of the target value can be based on consideration of these two parameters. Target reliability indices calculated for newly designed bridges and existing structures are different for many reasons. Reference time period is different for newly designed and existing bridges. Load model, used to calculate reliability index, depends on the reference time period. Single load path components require a different treatment than multiple load path components. The paper deals with the selection criteria for the target B for ultimate limit states (ULS) and serviceability limit states (SLS). For the covering abstract see ITRD E126028.
The development of rational design criteria requires efficient measures of structural performance... more The development of rational design criteria requires efficient measures of structural performance. Traditional methods of analysis do not reveal the actual safety reserve (degree of redundancy). Load and resistance parameters are random variables. Therefore, probabilistic methods are used to model behavior of bridges. The paper provides a review of the available methodology, including reliability measures, load models and resistance models for girder bridges.
The combination of dead load and live load is very important in design of bridge superstructure a... more The combination of dead load and live load is very important in design of bridge superstructure as, in practice, it controls the strength limit states. The basic set of load factors for the Strength I limit state is 1.25 for dead load and 1.75 for live load and dynamic load. For design cases when the dead load dominates, i.e. for Strength IV limit state, the dead load factor is 1.5. The acceptability criterion for load and resistance factors in the AASHTO LRFD Code is closeness to the target reliability index, which is assumed to be 3.5 for steel and concrete girder bridges. However, the reliability analysis performed for a full range of dead load to live load ratios indicates that when live load is about 10-20% of the total load, the reliability indices are about 3.0 which is lower than the target value of 3.5. This is an indication that the reliability level is insufficient and there is a need for increasing load factors. On the other hand, for dead load constituting about 100% of the total load (i.e. no live load), the reliability index is much higher than 3.5, which means that the load factor 1.5 can be reduced. Therefore, it is proposed to change the current Strength IV load factors to dead load of 1.4 and live load factor 1.4. The result is a more uniform reliability level for all combinations of dead load and live load. The results of reliability analysis are presented in graphs.
The objective of this paper is to present the development of the resistance factor for shear in r... more The objective of this paper is to present the development of the resistance factor for shear in reinforced concrete bridges made of lightweight concrete. The statistical model of resistance was based on the available limited experimental data. It was found that the current shear design procedures are about 15% less conservative for lightweight concrete when compared to normal weight concrete. The reliability analysis was performed and served as a basis for the selection of the resistance factor. The results indicate that a resistance factor equal to 0.8 can be applied for the shear design of lightweight concrete bridges.
Railway bridges are exposed to repetitive high stress due to the live load and constant, relative... more Railway bridges are exposed to repetitive high stress due to the live load and constant, relatively low stress due to dead load. Cyclic live load may lead to failure even when the stress level is lower than the allowable stresses. Therefore, components and connections need to be analyzed for possible damage caused by fatigue. The basic approach for estimating the remaining fatigue life of a structure element is to use S-N curves. However, those laboratory specimens tested with a constant-amplitude stress range show variability in the results. This means that fatigue resistance is a random variable. Moreover, railway bridges are subjected to variable-amplitude stress ranges during their service life. The axle load depends on many factors, not only capacity of the car but also dynamic behavior of the car and dynamic response of the bridge. If load and resistance parameters are random variables, structural performance should be measured in terms of reliability. The objective of this st...
The paper summarizes the code development procedures used for the new LRFD (load and resistance f... more The paper summarizes the code development procedures used for the new LRFD (load and resistance factor design) bridge code. The current specifications use allowable stresses and/or load factor design. The new code is based on a probability-based approach. Structural performance is measured in terms of the reliability (or probability of failure). Load and resistance factors are derived so that the reliability of bridges designed using the proposed provisions will be at the predefined target level.
Recently a considerable research effort has been devoted to bridge design and evaluation in Europ... more Recently a considerable research effort has been devoted to bridge design and evaluation in Europe and North America. This paper focuses on the review and comparison of the reliability models developed in conjunction with the development of whole life assessment methods in the UK and AASHTO LRFD code in the US. The objective of the paper is to establish time-varying reliability profiles relating reliability index to various rates of deterioration (corrosion) and develop a procedure for calculating probability of failure for different time stages. The corrosion patterns and rates are modelled for steel girder bridges on the basis of empirical relationships from laboratory tests and field observation of existing bridges. Two bridge design codes, BS 5400 in the UK and AASHTO in the UK, are compared in terms of reliability. Reliability is calculated for selected existing structures. Relationship between reliability index and rating factor is investigated. The obtained reliability spectr...
This study was part of the reliability-based calibration of the design provisions for curved gird... more This study was part of the reliability-based calibration of the design provisions for curved girder bridges. The reliability analysis for straight girder bridges for the AASHTO LRFD Code (2004) was documented by Nowak (1995, 1999). The aim of the present study is the verification of resistance factors for curved steel girder bridges. The presented research was conducted in conjunction with NCHRP Project 12-52. Three representative bridges were selected for this study. For each bridge, the analysis was performed using an advanced finite element program to determine the strains and stresses, and to compare them with the design (nominal) values. For one of the considered bridges, the analytical results were verified using field test data obtained from the University of Minnesota. The assumptions and methods used in the calibration process are also described.
Reliability analysis is performed for fatigue limit state in steel bridges. The load and resistan... more Reliability analysis is performed for fatigue limit state in steel bridges. The load and resistance parameters are treated as random values. The fatigue load model includes the magnitude and frequency of occurrence. Fatigue resistance is based on the available test data, in particular S-N curves. Reliability indices are calculated as a function of time.
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Papers by Andrzej Nowak