RELIABILITY ANALYSIS OF STRUCTURAL CONCRETE ELEMENTS by Gisli Jonsson B. Sc. The Technical College of Iceland, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1992 © Gisli Jonsson, 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. it is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of( z--/ / Z //ç/-2c-(’V4 c The University of British Columbia Vancouver, Canada Date /‘J/ /49? Abstract The reliability of reinforced concrete elements and design iceberg impact loads for offshore structures were studied. The reliability program RELAN was used to perform FORM reliability analysis for reinforced beams subjected to bending, and for a reinforced concrete wall from the Hibernia offshore structure subjected to complex loading. RELAN was also used to establish the probabilistic distribution of ice impact loads. To study the reliability of concrete beams accounting for the variability of the intervening variables, and in order to determine the theoretical flexural capacity of concrete beam, computer program TIN was developed. TIN uses a strain compatibility approach accounting for the non-linear stress-strain relationships of concrete and reinforcing steel. As a pilot study on the reliability of concrete elements, a beam designed according to the Canadian concrete code was analyzed with the objective of evaluating the effect of different spans and reinforcing steel ratios on the reliability of the beam. To study the reliability of more complex elements, an element from the icewall of the Hibernia offshore structure was used. The theoretical strength of the wall element was evaluated with program SHELL474. In order to link SHELL474 to RELAN for the reliability study, the main subroutine in SJIELL474 was modified. Since one of the major factors in reliability studies of concrete offshore structures are the uncertainties associated with extreme environmental load conditions, the statistics for ice impact loads for the Hibemia structure were derived using RELAN and applications of energy conservation principles. 11 Abstract For the purpose of deriving the ice load distributions to evaluate the reliability of the Hibemia icewall element, the program PROB, which is a product of the reliability program RELAN and the energy conservation principles, was developed. 111 Table of Contents Abstract . Table of Contents . ii iv List of Tables viii List of Figures ix Acknowledgements xi CHAPTER 1 Introduction 1 1.1. Introduction 1 1.2. Reliability Concepts 2 1.3. Concrete Elements and Variability 4 1.4. Ice Impact and Offshore Structures 5 1.5. Outline of Thesis 6 CHAPTER 2 Reliability Concepts 8 2.1. Introduction 8 2.2. Reliability Based Design 10 2.3. Calculation of the Reliability Index using the Program RELAN 13 CHAPTER 3 Review of Statistical Definitions 16 3.1. Introduction 16 3.2. Compressive Strength of Concrete 17 3.2.1. Variability in the Compressive Strength iv 18 Table of Contents 3.2.1.1. In-situ Versus Cylinders Tests 19 3.2.1.2. Size Effects 20 3.2.1.3. Influence of Rate of Loading 20 3.3. Initial Tangent Modulus 22 3.4. Reinforcing Steel 24 3.4.1. Stress-Strain Response of Reinforcement 24 3.4.2. Variability of Mechanical Properties of Reinforcement 26 3.4.2.1. The Yield Strength of Steel 26 3.4.2.2. The Modulus of Elasticity 27 3.4.2.3. Variations in Steel Area of Cross Section 28 3.5. Geometric Properties 28 CHAPTER 4 Reliability of Concrete Beams in Bending 30 4.1. Introduction 30 4.2. Flexural Strength of Reinforced Concrete Beams 31 4.2.1. Beam Program 34 4.2.2. Stress Block Factor Method 35 4.3. Design of Concrete Beams in Bending 38 4.3.1. Design According to the CSA Code 38 4.3.2. Development of Subroutines for RELAN 40 4.3.3. Variability of Intervening Variables 43 4.3.3.1. Material Statistics 43 4.3.3.2. Loads and Fitted Distributions 44 4.3.4. Example Runs 47 4.4. Effect of Span on Beam Reliability V 50 Table of Contents 4.5. Effect of Steel Ratio on Beam Reliability CHAPTER 5 Offshore Structure Ice Impact 51 59 5.1. Introduction 59 5.2. Icebergs and Multi-year floes 62 5.3. Dynamic Impact from Icebergs and Floes 63 5.4. Evaluation of the Ice Impact with Energy Principles 64 5.4.1. Calculation of the Ice Contact Area 66 5.4.2. The Ice Compressive Strength 68 5.5. Ice Impact Force for the Hibernia Structure 70 5.6. Reliability Based Formulation of Ice Impact 71 CHAPTER 6 Reliability of Concrete Offshore Structures 73 6.1. Introduction 73 6.2. Sectional Strength of Concrete Wall Elements 74 6.3. Design of Hibernia Offshore Wall Element 75 6.3.1. Modification to SHELL474 and Subroutines for RELAN 77 6.3.2. Comparison of Beam Program and Modified SHELL474 79 6.3.3. Variability of Intervening Variables 80 6.3.4. Ice Impact Prediction Using the Program PROB 82 6.3.5. Example Runs 84 6.4. Development of PDF-Functions for Load and Resistance 87 6.5. System Performance Using the Joint PDF-Functions 92 CHAPTER 7 Concluding Remarks and Further Study vi 94 Table of Contents Bibliography .97 Appendix A: FORTRAN Subroutines 102 1. Beam Program Subroutines 102 2. Modified Main Subroutine in SHELL474 113 3. Subroutines to Link RELAN and SHELL474 120 4. Subroutines for Probabilistic Evaluation of Ice Impact 126 Appendix B: Design of Concrete Beam According to CSA A23.3 134 Appendix C: Results from SHELL474 140 vii List of Tables Table 4.1 Material Factors and Nominal Values .40 Table 4.2 Intervening Material Random Variables 43 Table 4.3 Extreme Parameters 45 Table 4.4 Intervening Load Random Variables 47 Table 4.5 RELAN Results: Code Equation 47 Table 4.6 RELAN Results: Exact Equation 48 Table 4.7 Sensitivity Factors: Beam Case 49 Table 4.8 RELAN Results: Different Steel Ratios 53 Table 4.9 Sensitivity Factors: Different Steel Ratios 56 Table 6.1 Intervening Material Random Variables 81 Table 6.2 Extreme Parameters for Ice-Impact Loads 83 Table 6.3 Resultant Load Components from Linear Finite Element Analysis 84 Table 6.4 RELAN Results: Wall Element 85 Table 6.5 Sensitivity Factors: Wall Element 86 viii List of Figures Figure 1.1: Definition of the Safety Index /5 .3 Figure 2.1: Geometric Representation of FORM/SORM Reliability Calculation 12 Figure 2.2: Flow Chart for System Reliability Calculation 14 Figure 3.1: Stress-Strain Response of Concrete 18 Figure 3.2: Influence of Loading Rate on Concrete Strength 21 Figure 3.3: Stress-Strain Response of Non-Prestressed Reinforcement 25 Figure 3.4: Modified Ramberg-Osgood function 26 Figure 4.1: Definition of Sectional Parameters 32 Figure 4.2: Layer by Layer Approach 35 Figure 4.3: Stress-Block Factor Method 36 Figure 4.4: Design of Concrete Beam in Bending 39 Figure 4.5: Changes in Safety for Different Spans 50 Figure 4.6: Changes in Safety for Different Steel Ratios 54 Figure 4.7: Cumulative Distribution Curve for Different Steel Ratios 55 Figure 4.8: Changes in Material Sensitivity 57 Figure 4.9: Changes in Load Sensitivity 58 Figure 5.1: Failure Modes of Ice 60 Figure 5.2: Flow Chart for Probabilistic Approach for Ice Load 61 Figure 5.3: Icebergs and Multi-Year Floes 62 Figure 5.4: Different Structures and Ice Features 67 Figure 5.5: Indentation Pressure at Peak Load 69 Figure 6.1: Sectional Forces at Complex Concrete Structure 74 ix List of Figures Figure 6.2: Details of Hibernia GBS Icewall Design 76 Figure 6.3: Performance Function for Wall Elements 78 Figure 6.4: Comparison between TIN and Modified SHELL474 79 Figure 6.5: CDF-Curves for 100 Year Ice-Impact 82 Figure 6.6: PDF-Curves for Annual Concentric Ice-Impact 87 Figure 6.7: PDF-Curves for Annual Eccentric Ice-Impact 88 Figure 6.8: PDF-Curves for 100 Year Concentric Ice-Impact 89 Figure 6.9: PDF-Curves for 100 Year Eccentric Ice-Impact 90 Figure 6.10: Element Resistance and 100 Year Uncorrel. Eccentric Ice-Impact 91 Figure 6.11: Element Resistance and 100 Year Correlated Eccentric Ice-Impact 92 Figure B. 1: Code Requirements for Reinforcement Placing 136 x Acknowledgements I would like to express my appreciation to several people for their help during the course of my research. I am most grateful to Jim Greig and Thomas Wong in the Computer Lab for their constructive comments and assistance with development of the FORTRAN programs for microcomputers and SUN system. I am also thankful to the staff in the Civil Engineering Office for help with going through the every day life. Most importantly, I would like to thank my supervisors, Dr. Ricardo 0. Foschi and Dr. Perry E. Adebar, for their help and guidance throughout my research. I would also want to thank Dr. Ricardo 0. Foschi for providing the source codes from the reliability program RELAN, and Dr. Perry B. Adebar for providing the source codes from SHELL474, helping to modify the main subroutines in order to link SHELL474 to RELAN, and for financial support. xi CHAPTER 1 Introduction 1.1. Introduction Engineering analysis and design normally requires resolution of uncertainties. These imply that an absolute assurance of safety and performance of a design is not practically obtainable, which basically means that some risk is invariably involved. In order to deal with the uncertainties in design, building codes introduce factors of safety, or load factors. However, the variability of the intervening variables and their significance for structural safety and performance can be analyzed systematically through methods of probability theory. Reliability methods have been developed to assist engineers in making design decisions in the presence of uncertainties, where the conceptual basis is the probability of failure, P. In general, failure denotes the exceedence of a limit state but does not necessarily imply collapse. The probability Pf may be calculated from the probability distributions of the random resistance and load variables, if these distributions are known. 1 CHAPTER 1 Introduction 1.2. Reliability Concepts The main problem in general design is the determination of a structural capacity to assure adequate safety and performance of the system. Failure of a system, which generally means the realization of a specified limit state, including collapse, can be defined as the event R applied load. D, where R is the resistance and D is the demand or the Thus, as shown in Figure 1.1, the probability Pf corresponds to the probability of the event [G = R — D] <0. The application of analytical methods to compute Pf is mainly limited to rather simple systems which involve few random variables. Because usually many random variables are involved, approximate methods like the standard Monte Carlo simulation [8], and the variance reduction technique, often known as the Iterative Fast Monte Carlo (IFM) [48] procedure, have been developed to estimate the probability of failure. These methods have proven effective for the analysis of complex systems. However, in order to make the probability calculation more accessible, the First and Second Order Methods (FORM/SORM) [26] have been developed. The FORM/SORM procedures are based on the calculation of the reliability index ,B, which enables engineers to evaluate the probability of failure by using the standard normal probability distribution function (see Figure 1.1). The estimation of the probability of failure in this manner will only be exact if all the intervening variables are normally distributed and uncorrelated, and if they combine linearly in the performance function G. 2 CHAPTER 1 Introduction Ps = P1(R-D) >0)-Probability of Survival G =0 G =R-D Pt = P[(R-D) <01-Probability of Failure Figure 1.1. Definition of the Safety Index fi Since generally the variables are non-Normal and correlated, the FORMISORM procedures introduce an appropriate transformation to convert all variables to normals and to eliminate correlations if present. A geometric interpretation of the reliability index ,8 permits the development of iterative computer algorithms [26] as will be discussed in Chapter 2. Using FORM/SORM procedures, a program entitled RELAN, or RELiability ANalysis, was developed by Foschi, Folz and Yao [19]. RELAN, has been applied extensively to study the reliability of wood structures and the calibration of corresponding reliabilitybased design guidelines in Canada. 3 CHAPTER 1 Introduction 1.3. Concrete Elements and Variability The basic information required to study the reliability of reinforced concrete elements is the probability distribution of each load and resistance variable, including as a minimum, estimates of their means and standard deviations or coefficients of variation. In order to express the ultimate strength for specific values of the random variables, a theoretical deterministic calculation procedure needs to be established. This procedure can then be linked to RELAN and called for each realization of the random variables. To study the moment capacity of a simple beam, program TIN was developed. TIN calculates the ultimate flexural capacity using a general strain compatibility solution and a layer approach, using assumed non-linear stress-strain relationships for the concrete and the reinforcing steel. As a pilot study, and in order to gain a better understanding of the reliability of more complex concrete elements, the influence of different spans and different reinforcing steel ratios were studied. In case of reliability of offshore structure elements, program SHELL474, developed by Adebar and Collins [1], was used to perform the capacity calculation for an element from the Hibernia offshore structure. Because SHELL474 was written to analyze only one set of variables for each run, the main subroutine in SHELL474 had to be modified so the variables could be brought in through RELAN as random variables. A reliability calculation was then performed for one-hundred-year eccentric and concentric ice impact, as well as to determine the probability density function (PDF) for the element strength. 4 CHAPTER 1 Introduction By performing reliability calculations, the designer can gain a better understanding of the behavior of the element and the influence of the many random variables involved, and is therefore more capable of economical design and preventing drastic failures. 1.4. Ice Impact and Offshore Structures Icebergs or floes present one of the most severe threats to offshore installations in sub-polar regions, particularly off the coast of Newfoundland. Various methods are available for calculating the maximum impact load experienced during a collision. For icebergs impacting a large structure, the limit momentum approach can be used to equate the kinetic energy of the impacting icebergs with the energy dissipated during the collision and to predict the maximum impact load (see Chapter 5). Because of the nature of these loads, they are normally predicted with probabilistic procedures such as Monte Carlo simulation or FORM/SORM procedures. However, obtaining the required statistical data can sometimes be difficult, mainly because quite often the data do not exist or are not available, and therefore it is hard and sometimes even not possible to develop unambiguous characterizations of uncertainties. A major problem in establishing the impacting force is due to the difficulty of relating small-scale ice properties to a prediction of how ice behaves on a large scale. Because of this, no generally accepted methods for predicting ice loads on offshore structures exist. 5 CHAPTER 1 Introduction When the probability of failure for the offshore elements was calculated in this thesis, the intervening ice parameters, i.e. the thickness, the diameter, the velocity, and the ice compressive strength, where used as random variables. But since the impacting force would change with time due to increasing contact area, so will the stresses in the element. It is apparent that an updating finite element analysis is also needed to reevaluate the stresses in the element, when RELAN would change the ice parameters in order to find the gradient to the failure surface. However, if the joint probability function for the ice impact would be known, the randomness can be represented by only one random variable with its mean and standard deviation. Based on the theoretical model derived in Chapter 5, a program was develop in order to derive the PDF function for one hundred year eccentric and concentric iceberg. 1.5. Outline of Thesis Chapter 2 discusses the reliability concepts in more detail and introduces different methods available, including the basic principles of First and Second Order Methods and their implementation in the reliability program RELAN. The variability of the intervening variables, such as the material strength and geometric properties, are reviewed in Chapter 3, including discussion on stress-strain relationships for concrete and reinforcing steel. Chapter 4 presents a reliability calculation for simple beams, of different spans and reinforcing ratios, and explains how the reliability program RELAN was linked to the beam design program TIN, which was developed for this study. 6 CHAPTER 1 Introduction The reliability of offshore structure elements and the extreme ice loads are discussed in Chapter 5, including the energy conservation methods used in the analysis of the impacting icebergs. Chapter 6 presents the study of the offshore structure element from the Hibernia offshore structure, making use of program SHELL474. Chapter 7 includes conclusions and discusses recommendation for further study. 7 Finally CHAPTER 2 Reliability Concepts 2.1. Introduction One of the principal aims of engineering analysis and design is the assurance of system performance within the constraint of economy. This means quite often that satisfactory failure rates for different limit states corresponds to a “trade-off’ between human safety or serviceability on the one hand, and economy, including expected losses due to failures, on the other hand. In practice, satisfactory failure rates are achieved through competent structural engineering, manufacturing, and erection, and by the use of safety and serviceability criteria. Most planning and design of engineering systems must be done on a basis which requires the resolution of uncertainties because of incomplete information, e.g. the actual lifetime maximum load and the actual capacity of the structure. In view of these uncertainties, risk is unavoidable and the way it is dealt with traditionally is to apply factors or margins of safety and adopt conservative assumptions in the process of design. This is done by ascertaining that a “worst,” or minimum, resistance conditions will remain adequate under a “worst,” or maximum demand conditions. These conditions are often defined on the basis of subjective judgment and similarly the adequacy or inadequacy of the applied “margins” may be evaluated or calibrated only in terms of past experience with similar systems. 8 CHAPTER 2 Reliability Concepts The proper way to deal with uncertainties and their significance on structural safety and performance is to systematically analyze the structure through methods of probability. As before, the available resistance and actual demand cannot be determined precisely, but in order to represent or reflect the significance of uncertainty they may be modeled as “random variables”. performance may be evaluated by using The probability of system non- either Monte Carlo simulation or FORM/S ORM procedures. Monte Carlo simulation is a powerful engineering tool which enables one to perform a statistical analysis of the uncertainty in structural engineering problems, being particularly useful for complex problems where numerous random variables are related through nonlinear equations. The fundamental step in a Monte Carlo analysis is the development of a set of random numbers by simulating the samples randomly over the entire range of each variable. The only disadvantage in using Monte Carlo simulation is the cost of execution where estimating low probability of failure may take a long time. A variance reduction technique, often known as the Iterative Fast Monte Carlo (IFM) procedure, can be most advantageously applied for such cases. Instead of simulating the samples randomly over the entire range of each variable, it is concentrated at the important regions, i.e. at those regions where most of the contributions to the failure probability is expected. The significant role of probability in structural engineering lies in providing a logical framework for uncertainty analysis and a quantitative basis for risk and safety assessment. Therefore it is important that probability of failure as a measurement for uncertainty will be accepted, even if only as a relative measure of safety and performance. 9 CHAPTER 2 Reliability Concepts 2.2. Reliability Based Design In the design of most engineering systems multiple variables are involved and they may influence either the resistance or the demand. It is necessary to formulate the performance of the system in terms of basic design variables, to be able to predict the probability of non-performance. For the purpose of a generalized formulation the performance function is described mathematically as follows [19]: ,x 1 G(X)=G(x 2 where X = (x 1 , , XN) ,x) (2.1) is a N-dimensional vector of design variables such as the concrete compressive strength and the steel yield strength. Because most of these variables, some of which represent the mechanical and geometric properties of the system, while others characterize the load demands, are uncertain or random they need to be described statistically. In some cases this cannot be done without tests or surveys which would provide statistical information on each variable. The performance function, G, can be written in terms of the resistance, R, and the demand, D, as following: G=R—D (2.2) where the failure of the system will occur when the demand exceeds the resistance, i.e. G <0 and the system will survive when G >0. 10 CHAPTER 2 Reliability Concepts The situation when G = 0 is usually known as the limit state between survival and failure, where all variable combinations satisfying G = 0 are said to belong to this limit state. For simple cases it is possible to obtain the probability of failure relatively easily with analytical methods, but since there are usually many random variables involved, the calculation requires the PDF-function or the joint probability density function and multiple integration over the failure region. Since the required joint probability is rarely known and difficult to obtain, approximate methods such as Monte Carlo simulation or the FORM/SORM procedures have been developed to estimate the probability of failure. FORM/SORM procedures enable engineers to evaluate the probability of failure by using the standard normal probability distribution function J [191: (2.3) PfT’(fl) In order to make use of the standard normal distribution, the FORM/SORM calculation procedure defines a new set of variables x by transforming the original X according to: 1 1 1=1 (2.4) where X is the mean of X 1 and o 1 its standard deviation, the origin x =0 corresponds to the mean value of X. 11 CHAPTER 2 Reliability Concepts The reliability index /3 is the minimum distance between the origin and the limit state surface G(x) = 0, and the corresponding point on the limit state surface is known as the “Design Point”, as illustrated in Figure 2.1, for the case of two variables x 1 and x . 2 x 2 Tangent Plane (FORM) Failure Region G<O Point Limit State G =0 Quadratic Surface (SORM) x 1 Figure 2.1: Geometric Representation of FORM/SORM Reliability Calculation [19] If FORM/SORM procedures are used to calculate the probability of failure the estimation will be exact if all the intervening variables are normally distributed and uncorrelated, and if they combine linearly in the performance function G. Generally the variables are not normal and uncorrelated, and the performance function is non linear. Since the FORM/SORM procedures introduce an appropriate transformation to convert all variables to normals and to eliminate correlations if present, the approximation of the probability of failure, F , is influenced solely by the non-linearity 1 in 0. 12 CHAPTER 2 Reliability Concepts The difference between FORM and SOR.M methods is that FORM assumes the limit state surface G(x) can be approximated by the tangent plane at the design point, where SORM on the other hand, assumes that the true limit state can be approximated by a quadratic surface (see Figure 2.1). 2.3. Calculation of the Reliability Index using the Program RELAN As mentioned earlier the geometric interpretation of /5 permits the development of iterative computer algorithms as illustrated in Figure 2.2. RELAN, which is a general FORM/SORM FORTRAN program, must be supplemented by a description of the performance function and its gradient with respect to the intervening random variables [20]. When the performance function is linear the FORM method gives a good estimate but in case of non-linearity a more approximate estimate can be made by the SORM method. For SORM calculations, the matrix of second order derivatives of G is also needed. It is sufficient to describe the function G for each specific problem, since first and second order derivatives can be obtained numerically by RELAN. The algorithm in RELAN adjusts for the case where the random variables are not normally distributed, and also where there are correlations between the variables. 13 CHAPTER 2 Reliability Concepts INPUT: STATISTICAL PROPERTIES OF VARIABLES CONSTANTS AND NOMINAL VALUES DEFINED THROUGH INPUT FILE SELECT A RANDOM VALUE FOR EACH VARIABLE CALCULATION OF THE THEORETICAL STRENGTH UPDATE THE RANDOM VALUES EACH RANDOM VARIABLE CALCULATION OF VALUE FOR SYSTEM PERFORMANCE {GXP) rCALCULATI0N OF THE GRADIENT OF GXP WITH RESPECT TO EACH RANDOM VARIABLE fl CALCULATION OF THE SAFETY INDEX J3 COMPUTE Pf =(-f3) j OUTPUT: SUMMARIZE RESULTS Figure 2. 2. Flow Chartfor System Reliability Calculation 14 CHAPTER 2 Reliability Concepts In order to estimate the importance of each variable to the system reliability, RELAN calculates the sensitivity coefficients which indicate the relative influence of each variables uncertainty in the reliability index fi. This can be very useful for designers in cases where the system behavior is too complex and many different modes of failure might influence the system performance. To perform calculations by RELAN the user has to provide four FORTRAN subroutines, i.e. DETERM, GFUN, DFUN and D2FUN. The subroutine DETERM defines all the deterministic variables which then are shared with GFUN, DFUN and D2FUN through a common block . When the deterministic values have been defined, 1 and the random variables brought through with an array’, GFUN calculates the value of the performance function and returns that value to RELAN main subroutine. As mentioned before, the first and second order derivatives can be obtained numerically by RELAN. The user can also provide DFUN and D2FUN, which return respectively the gradient and the second derivatives of the performance function. RELAN can be used to determine the probability of failure for problems in virtually any field of study in civil engineering, i.e. structures, soil mechanics, seismic risk, construction, transportation and hydraulics. Communication Option in FORTRAN. 1 15 CHAPTER 3 Review of Statistical Definitions 3.1. Introduction A theoretical deterministic calculation procedure needs to be established to express the ultimate strength for specific values of the random variables. The values must be selected from the corresponding statistical distributions. Data on both structural resistance, i.e. strength and geometric properties, and load variables are required in order to conduct probability-based design calculations. The basic information required is the probability distribution of each load and resistance variable and estimates of its mean and standard deviation or coefficient of variation. The mean and coefficient of variations of these basic variables should be representative of values that would be expected in actual structures in situ. statistics for the material properties is through testing. The normal way to develop While frequently there are sufficient data to obtain a reasonable estimate of the probability distribution, in many other cases this must be assumed on the basis of physical argument or for convenience. This is because different techniques, instruments and the human factor, can create different results for the same specimen. The variables which most affect the strength variability of reinforced elements are the compressive strength of concrete and the yield strength of steel as well as the geometric properties. 16 CHAPTER 3 Review of Statistical Definitions 3.2. Compressive Strength of Concrete The compressive strength of concrete is one of its most important properties. Other important properties, which influence the strength and stiffness, can be approximately correlated to the compressive strength. Thus, in probabilistic calculations for concrete elements the compressive strength needs to be studied carefully. The response of concrete to uniaxial compression is usually determined by loading a standard cylinder, 150 mm in diameter and 300 mm long. loading rate is such that the maximum stress, f, The standard is reached in 2 to 3 minutes. Even though the actual shape of the stress-strain relation for concrete is not unique and depends on several factors, such as the cylinder strength, density, rate and duration of loading, the relationship between axial stress, stress, cf 8 J, and the axial strain caused by this is reasonably accurately represented by the equation [13]: 4=2_1-’ which is a parabola shown in Figure 3.1. (3.1) This parabola, which is widely used, describes the rising portion and the immediate post-peak response reasonably well but somewhat over estimates the rate of which the stress drops off at larger strains especially in cases where elements have a high degree of confinement. To capture this effect Kent and Park [29] proposed another curve for the stress-strain relationship to capture this over estimation, which consists of a second order parabola up to the maximum stress f’ at a strain s and then a linear falling branch. 17 CHAPTER 3 Review of Statistical Definitions By having a linear falling branch, this curve does not have the same problem in over estimating the compression strength at least not after it reaches the peak. fc Compression Zone Assumed Parabolic Response / Actual Response / S SC cf Figure 3.1: Stress-Strain Response of Concrete 3.2.1. Variability in the Compressive Strength An essential component in the development of probabilistic-based design for concrete elements is to introduce how various factors effect the concrete compression strength. 18 CHAPTER 3 Review of Statistical Definitions In-situ versus Cylinders Tests 3.2.1.1. Tests have shown that the strength of concrete in a structure tends to be lower than its specified design strength and may not be uniform throughout the structure. The major sources of variations in concrete strength are due to one or all of the following: i. Variations in material properties and proportions of the concrete mix. ii. Variations in mixing. iii. Transporting. iv. Placing and curing methods. v. Variations in testing procedures and the rate of loading. vi. Size effects. The reduction in the in-situ strength of concrete is partially offset by the requirement that the average cylinder strength must be about 700-900 psi (4.8-6.2 MPa) greater than the specified strength to meet the existing design codes [311. Based on this observation and on equations and data, it has been suggested [31] that the 28-day strength of concrete in a structure for minimum acceptable curing can be expressed as: fc.ctr35 = where 35 fctr 0. 675f’ is less than or equal to 1.15ff’. 19 + 1,100 (psi) (3.2) CHAPTER 3 Review of Statistical Definitions 3.2.1.2. Size Effects The phenomenon of “size effects”, which is a change in indicated unit strength due to a change in specimen size, has been noted by many researchers while investigating the properties of concrete and other materials. The effect of size on properties of concrete is particularly important if small scale models are to be used to predict the behavior of prototype structures. It can be concluded that since there are a smaller number of flaws in a smaller specimen the strength of the small specimens is on the average larger than that of the larger specimens. Despite the fact that the mean strengths are significantly affected by volume, the influence of size on the minimum strength seems to be quite small [38]. Since in reliability study of concrete elements the lower strength tail is most important it might be acceptable to neglect the effect of volume in probabilistic studies involving strength. 3.2.1.3. Influence of Rate of Loading It has been observed by testing cylinders with different loading rates, that fast loading increases the strength by about 20% while slow loading reduces it by about 20% (see Figure 3.2) [13]. However, in design the decrease in strength caused by long term loading is usually neglected because of the fact that the concrete will typically gain 20 to 40% in strength due to the hydration that occurs after 28-days. It is usually assumed that these two effects will compensate each other, resulting in a conservative assumption. 20 CHAPTER 3 Review of Statistical Definitions Few Seconds Few Minutes f; Few Months S Figure 3.2: Influence ofLoading Rate on Concrete Strength [42] In the work done by Mirza, Hatzinikolas and MacGregor [38] the mean value for the in-situ compressive strength of concrete at a given rate of loading R (psi/sec) was given by: fcstr = where 35 fcgjr [O.89(O.O81ogR)] (psi) 35 fc (3.3) is given by Equation 3.2 and the normal rate of loading for standard cylinder test is approximately 35 psi/sec. The majority of researchers have represented the distribution of concrete compressive strengths with a normal distribution. 21 CHAPTER 3 Review of Statistical Definitions A review of literature indicates that the coefficient of variation of field-cast laboratorycured specimens is in many cases between 15% and 20%, which suggests that 20% is a reasonable maximum value for average controls. However the standard deviation and the coefficient of variation are not constant for different strength levels so it appears that the average coefficient of variation can be taken as roughly constant at 10%, 15% and 20% for strength levels below 4,000 psi (27.6 MPa) for excellent, average, and poor control, respectively [38]. Based on this result, MacGregor [311 suggested that the coefficient of variation for concrete in a structure should be taken as 0.18. 3.3. Initial Tangent Modulus Although several equations are available in the literature to estimate the static modulus of elasticity of concrete, the available data on the variability of this parameter is limited. If the parabolic stress-strain relationship is used then the initial slope, is given by: (3.4) Sc If only the cylinder crushing strength of the concrete is known, then the initial tangent modulus, E , can be estimated from following approximate expression [13]: 1 = 5500f (MPa) 22 (3.5) CHAPTER 3 Review of Statistical Definitions After determining 8 can be found and the parabolic equation can then be used. In the same manner the direct cracking strength can be found by [101: fcr = 0.332/ (MPa) (3.6) where 2 is a factor accounting for the density of the concrete. An analysis of test data from the University of Illinois for 139 tests of standard cylinders of normal weight concrete indicated that a high degree of correlation, or approximately 0.9, existed between initial tangent modulus and compressive strength. In the same study, a statistical analysis of the ratios of observed to calculated modulus showed the distribution of the initial tangent modulus of concrete relative to its calculated value can be approximated by a normal distribution. Based on the same data the following relationship for initial tangent modulus was obtained [38]: = 6040O[ (psi) (3.7) and when the mean value of the initial tangent modulus is estimated from Equation 3.7 the variability relative to the calculated value should be taken as 0.08. Like the compressive strength, the mean value and dispersion of the modulus of elasticity of concrete are subject to a rate of loading effect. In order to express this effect the following equation can be used [38]: 35 ECR =(l.16—O.081ogt)E where t is the load duration in seconds. 23 (3.8) CHAPTER 3 Review of Statistical Definitions 3.4. Reinforcing Steel Since concrete is very weak in tension it has to be “reinforced” with material which is stronger in tension, like steel bars, wires or welded wire fabric. Because steel is much stronger than concrete even in compression, it can also be used to carry compressive stresses if it is desired to reduce the dimensions of the concrete section. The reinforcing steel generally in use are hot-rolled and deformed bars and colddrawn wires. Deformed bars are classified into three grades based on minimum specified yield strength: 300, 350 and 400 MPa where grade 400 bars are the most frequently used type of reinforcement in Canada. 3.4.1. Stress-Strain Response of Reinforcement Typical stress-strain curves for steel show that the initial stiffness is essentially the same even though the strength will differ a lot especially in case of prestressing steel. The stress-strain relationship for steel bars are normally assumed to be bilinear (see Figure 3.3) where the stress, and the strain caused by this stress, s, can be expressed by following: fSES8Sf where J MPa. This relationship is assumed to be valid for both tension and compression as is less or equal to f, (3.9) and the modulus of elasticity E is equal to 200000 illustrated in Figure 3.3 [13]. 24 CHAPTER 3 Review of Statistical Definitions Assumed Response / f sf Figure 3.3: Stress-Strain Response of Non-Prestressed Reinforcement [13] For strands and wire the response can be approximated by a bilinear relationship, but a more accurate representation of the stress-strain response of prestressing strands can be obtained by using the modified Ramberg-Osgood function [13]. Most frequently used strand is the low-relaxation strand. The response of this type of strand, which has a peak stress f = 1860 MPa can be described as following [13]: 0.975 f=20OO006fO.025+ [1+(ll8sf)’°] 25 1860 (MPa) 10 (3.10) CHAPTER 3 Review of Statistical Definitions f fpu= 1860 MPa 200000 MPa pf 8 Figure 3.4: Modified Ramberg-Osgood function [13] 3.4.2. Variability of Mechanical Properties of Reinforcement To understand the effects of the variability of the strength and geometrical properties of reinforcing steel on the strength of reinforced concrete member, the variability of reinforcing steel needs to be studied. 3.4.2.1. The Yield Strength of Steel The variability of yield strength depends on the source and nature of the data. The variation in strength within a single bar or strand is relatively small, while for the in-batch variation in a given heat is slightly larger. 26 CHAPTER 3 Review of Statistical Definitions When the samples are derived from different batches, from one mill or especially from different mills, there will be significantly more variation [36]. This is expected since roffing practices and quality measures vary for different manufacturers and different bar and strand sizes. Different values can be obtained for the yield strength depending on how it is defined. The most common definition for yield strength is the static yield strength which is based on the nominal area. The static yield strength seems to be desirable because the strain rate in tests is similar to what is expected in a structure and because designers use the nominal areas in their calculation [36]. The statistics for bars, stirrups and prestressing strands are mostly documented in References 31, 37 or 39. The mean values were found by calculating the ratios between the tabulated specified and the tabulated actual values, which then are used to scale the current design values. The probability distribution for the yield strength of steel bars and stirrups is assumed to be log-normal [31] but a number of investigators have recommended the use of a normal distribution for higher strength steel such as prestressing strands [39]. 3.4.2.2. The Modulus of Elasticity The modulus of elasticity of steel has been found to have a small dispersion and to be more or less insensitive to the rate of loading or the bar size. The probabilistic distribution of the modulus of elasticity for reinforcing bars or strands can be considered normal with the actual mean value equal to the specified value and a coefficient of variation 3.3% [39]. 27 CHAPTER 3 Review of Statistical Definitions 3.4.2.3. Variations in Steel Area of Cross Section The actual areas of reinforcing bars or strands tend to deviate form the nominal areas due to the manufacturing process. Most researchers have indicated that the probability distribution of steel area should be taken as normal. The mean value for the ratio between the measured and nominal areas should be 0.99 with a coefficient of variations 2.4% [36]. However, if the effect of variability in the steel area is considered negligible, Mirza and MacGregor have suggested [36] that a single value of 0.97 could be used. According to MacGregor [31] the ratio between the measured and nominal areas could also be taken to have a mean of 1.0 with a coefficient of variations 6.0%. It should be noted, that the statistics for yield strength are assumed here to reflect the nominal steel area, although in the calculations the actual area was included as a random variable. This was mainly because in the references used it was not clear whether the actual or the nominal area was applied. 3.5. Geometric Properties Variations in dimensions or geometric imperfections can significantly affect the size and hence the strength of concrete members. These variations and imperfections are caused by deviations from the specified values of the cross-sectional shape and dimensions, the position of reinforcing bars and strands, ties and stirrups, and the grades and surfaces of the constructed structures [35]. There are many reasons for these variations but two of the most important ones are the construction process, e.g. size, shape, and quality of the used forms, and curing of the concrete. 28 CHAPTER 3 Review of Statistical Definitions The process of collecting data for statistical purposes has not been standardized yet [35], mainly because it is difficult to compare the results of measurements reported by various researchers, when the quality of construction technique and equipment are different between countries. Most researchers have indicated that the probability distributions for the geometric properties should be taken as normal [39]. One should keep in mind that all suggestions for geometric properties are based on interpretation of available data and as such they should be considered preliminary [35]. 29 CHAPTER 4 Reliability of Concrete Beams in Bending 4.1. Introduction To predict the resistance of a concrete beam, the strains in the concrete and reinforcement are assumed to vary linearly such that plane sections remain plane. The compressive stresses in the concrete can be calculated using an appropriate stress-strain relationship, often assumed parabolic as was discussed in Chapter 3. If now the concrete stresses are integrated over the section, an equilibrium can be used to obtain the sectional moment and the axial force. To evaluate the moment-curvature response two methods are most often used: i. The Layer-by-Layer Approach ii. The Stress Block Factor Method were the first one is designed for microcomputers because it involves numerical integration of the stress-strain curves, while the second is more appropriate for programmable calculators or hand calculations. 30 CHAPTER 4 Reliability of Concrete Beams in Bending A beam program, which was developed, uses the Layer-by-Layer approach and a general strain compatibility with the assumed stress-strain relationships, to find the corresponding moment for each curvature step until a decrease in moment with curvature is obtained. The program uses the values before the drop but decreases the curvature step and tries to find a higher solution than the current maximum value by changing the sign of the curvature. The program keeps iterating until it drops again, then it changes the step and the sign of the curvature again. This process is repeated until the program cannot find a higher solution. In the reliability calculation, the accuracy in evaluating the moment curvature response must be sufficient to permit an accurate calculation of gradient of the performance function. When the program RELAN was linked to the moment-curvature subroutines and the concrete reliability program TIN was created, a major problem was detected in a few runs. This problem indicated that the accuracy in evaluating the moment-curvature response must be sufficient for the program to detect changes in capacity when RELAN was changing the values in order to find the gradient to the failure surface for each variable. 4.2. Flexural Strength of Reinforced Concrete Beams The assumption that plane section remains plane makes it possible to define the concrete strain with only two variables, i.e. the strain at the top and the bottom face. To define the linear strain distribution for the section the strain at the centroid, 8cefl and the curvature, q5, are used (see Figure 4.1). The curvature is equal to the change in slope per unit length along the section and also the strain gradient over the depth. 31 CHAPTER 4 Reliability of Concrete Beams in Bending If the strain distribution across the section is known, then the assumed stressstrain relationship can be used to find the distribution of stresses across the section and the moment acting at the section can be determined form the equilibrium equations [13]. b V cen 6 4:i h Centroid of Section ii...: Concrete Strains Cross Section Figure 4.1: Definition of Sectional Parameters [13] Compatibility conditions: The concrete strain at any level y can be found by: 8c=6cent;fy The strain in the bars at any level y = (4.1) is equal to the strain in the surrounding concrete: (4.2) — 32 CHAPTER 4 Reliability of Concrete Beams in Bending The strain in the prestressing tendons at any level y is equal to the strain in the surrounding concrete plus the strain difference, A, at this level: (4.3) 8€Y+1 ep Equilibrium condition: At any section the stresses, when integrated over the section, must add up to the required sectional moment M and the sectional force N: f ffdA + fdA 8 JICYdAC + ffdA + f fydA + = N JfydA (4.4) = -M (4.5) In the equilibrium equations, it has been assumed that tensile strains and stresses are positive and compressive strains and stresses are negative. The axial load, N, reacting at the section is taken positive in tension and negative in compression. The curvature, b, like the moment, M, is positive if the section develops tensile stresses at the bottom [13]. 33 CHAPTER 4 Reliability of Concrete Beams in Bending 4.2.1. Beam Program What makes it difficult to evaluate the response of a flexural member, is the varying of stresses and strains over the depth of the section. To perform the integrals of Equations 4.4 and 4.5, they can be simplified by assuming that the reinforcing bars and the prestressing tendons consist of a number of discrete elements and their contributions can be replaced by summations: yA +fyA 8 +f f fdA 3 = -M (4.6) The force in each bar or tendon is assumed to be equal to the stress at its center times the area. In order to evaluate stresses in the concrete it is also convenient to idealize the cross-section as a series of rectangular layers (see Figure 4.2) and assume that the strain in each layer is uniform and equal to the actual strain at the center of the layer. If the strain is uniform over the layer then the concrete stress will also be uniform over the layer. The force in each layer can now be found by multiplying the stress in the layer by the area of the layer, while the moment contribution can be found by multiplying the layer force by the distance between the middle of the layer and the reference axis [13]. 34 CHAPTER 4 Reliability of Concrete Beams in Bending b I Cross Section Concrete Strains Steel Strains Concrete Stresses and forces Steel Forces Figure 4.2: Layer by Layer Approach 4.2.2. The Stress Block Factor Method The layer-by-layer approach is a good procedure as an algorithm for microcomputers. However, in cases where cross sections have essentially constant widths, the concrete stress integrals can be efficiently evaluated by using stress-block factors. Instead of using the nonlinear stress distributions, equivalent uniform stress distributions are applied (see Figure 4.3). 1 and For a given compressive stress distribution, the stress-block factors a fl are determined so that the magnitude and location of the resultant force are the same in equivalent uniform stress distribution as in the actual distribution. The requirement that the magnitude of the resultant force remains the same can be described as following [13]: ffbdy = 1 fI3 a c b 35 (4.7) CHAPTER 4 Reliability of Concrete Beams in Bending Even though the stress block may be imagined to have any convenient shape, the requirement that the location of the resultant force remains the same [13]: fbydy c 1 =c—O.5/3 (4.8) fbdy where y in this case is measured from the neutral axis (see Figure 4.3). 1 ‘c cL b Compression 0.5f3CI , :pC Effective Embedment Zone h 7 O.5dzI H — T O•5cr Cross Section Concrete Strains Actual Concrete Stress Equivalent Uniform Stresses Concrete Forces Figure 4.3: Stress-Block Faetor Method [13] Many researchers have tried to develop an expression to represent the compressive stress-strain response of concrete and a simple parabola has been found to describe it reasonably well. 36 CHAPTER 4 Reliability of Concrete Beams in Bending For such case, i.e. a parabolic stress-strain curve and a constant width, b, Equations 4.7 and 4.8 can be reduced to simple expressions listed in Reference [13] by Collins and Mitchell. The Simplified Stress Block Factor Method or the Code Method, recommends that the actual concrete stress distribution should be taken as equivalent to rectangular concrete stress distribution and the strains in the steel and the concrete are assumed to vary linearly with distance from neutral axis with the maximum compressive strain in the concrete limited to 0.003. The steel stress is taken as = q5E8 q5f and tensile strength of concrete is neglected. The uniform stress and the depth of stress block recommended is essentially the 1 is taken as 0.85. same as those determined experimentally, where a Therefore the c (see Figure 4.3), where 1 uniform stress is taken as 0.85 cbf’ over a depth a = /3 taken as 0.85 for concrete strengths f fl is up to and including 30 MPa and, beyond this, it is reduced continuously at a rate of 0.08 for each additional 10 MPa of strength, but with a minimum value for fl of 0.65. In order to calculate the moment capacity for concrete beam by using the simplified stress block factor method the equilibrium for the section is expressed in terms of two forces i.e. the concrete force, C, from the compression zone and the steel force, T, in the bars (see Figure 4.3). Those two forces have to be equal so the equilibrium for the section can easily be written as C = T or in terms intervening variables [43]: q5fb/3 = a c 1 37 (4.9) CHAPTER 4 Reliability of Concrete Beams in Bending Based on this the critical moment can be written as: M. = q5fd 3 A (4.10) — where d is the effective depth, a = ,8 c and c can be found by the formula: 1 AØf = (4.11) /3 a q 1 fb 4.3. Design of Concrete Beams in Bending As a first example, a study of a simple beam is used to explore the general problem and gain an understanding of reliability based design of concrete elements. 4.3.1. Design According to the CSA Code Let us assume that we are to design a roof of a department store in down town Ottawa with a parking lot on top. The roof structure, which has a 17 m span, is carried by 500 x 1300 mm simply supported concrete beams with 5 m spacing, and 210 mm thick slab (see Figure 4.4). 38 CHAPTER 4 Reliability of Concrete Beams in Bending Dead Load 42.5 kN/m 2 LiveLoad5.OkN/m2 . Snow Load 2.5 kN/m 2 l7mc/c5m 5m I I 210mm I 1300mm 500 mm Figure 4.4: Design of Concrete Beam in Bending In addition to the dead load, the building has to be designed, for 5.0 service load and 2.5 Im2 Im2 snow load. If we now design the beam in the roof structure, the applied load can be calculated with the values listed in Table 4.1, where the factored load, qf , is given by: qf = l.25D +l.5(L +S) (4.12) and the maximum moment, Mmax is: M=,4qfL2 39 (4.13) CHAPTER 4 Reliability of Concrete Beams in Bending Table 4.1 Material Factors and Nominal Values Definition of Variables Values Ø-Material Factor for Concrete 0.6 ø-Material Factor for Steel 0.85 1 -Concrete Stress Block Factor a 0.85 -Concrete Stress Block Factor 0.8 16 f’ -Concrete Compressive Strength (MPa) 35.0 f -Steel Yield Strength (MPa) 400.0 b -Beam width (mm) 500.0 h -Beam height (mm) 1300.0 L-Beam Span (m) 17.0 D-Dead Load (m) 42.5 La-Service Load (m) 25.0 Sn-Snow Load for Ottawa (1Im) 12.5 By following the design procedure step by step, and the preceding equations and design values used, the amount of reinforcement needed for the beam is 12 No. 35 bars (see Appendix B). 4.3.2. Development of the User Subroutines for RELAN As mentioned earlier, to perform a reliability calculation by RELAN, it must be supplemented by a description of the performance function and its gradient with respect to the intervening random variables. 40 CHAPTER 4 Reliability of Concrete Beams in Bending In order to do so, the user has to provide four subroutines, i.e. DETERM, GFUN, DFUN and D2FUN. Despite the importance of all the subroutines, GFUN can be accounted to be the most important, as GFUN contains the performance function which is the core of the reliability calculation. If the first and the second order derivatives are computed numerically by RELAN, DFUN and D2FUN are not needed. Let us now look at the designed beam in Section 4.3.1 and Appendix B, to establish the performance function. The actual applied moment, Macj which the simply supported beam has to support can be written: Mact = ,3’(Dact act +Lact)L2 (4.14) which can also be written in terms of ratios between the nominal and the actual loads: Mact =2 SL 8 D (Sn Ln E1Pi S Ln S Dn + [ ) + } + ) + S } 1Li1 S Ln } }] (4.15) or in a simplified way: Mact = SnL (Do r (4.16) + Lr8) where: D= r Dn S=r Sn = L=L r D Sn 41 + Sn+Ln 8 Ln ‘ CHAPTER 4 Reliability of Concrete Beams in Bending If we now recall the fundamental formulation in Chapter 2 of the performance function, i.e. G R — D, we can write the performance function for the beam case as following: G = Mcap — SL (DrYS+ Sr + (4.17) LrS) where Mcap is the calculated theoretical strength of the beam. According to the CSA code, the maximum moment which the beam can sustain is: Mm = 5L 2 (1.2578+1.56) (4.18) and the critical moment, Me,., in Equation 4.10 can also be written as: Mcr = alq5fb/3lc(d_a/) (4.19) If the critical and the maximum moment are set equal, the performance function becomes: aiq5fb/3ic(d a) — G = Mcap — (1.25y8+ 1.5s) 42 r + Lr6) (4.20) CHAPTER 4 Reliability of Concrete Beams in Bending 4.3.3. Variability of Intervening Variables One of the most important factor in the reliability calculation, is the variability of the intervening variables. The statistical data, which will be reviewed in following sections, is more or less based on definitions in Chapter 3. 4.3.3.1. Material Statistics The statistical data used for the beam example are listed in Table 4.2. It should be noted that because of lack of information, the mean value of the concrete compression strength is only adjusted for the rate of loading and the gain in strength that occurs with time has been ignored (see Table 4.2). Table 4.2 Intervening Material Random Variables Definition of Variables / Units Mean COV Distribution/Ref. .t-Compression StrengLh (MIi) 27.77 0.18 Normal / [38] E-Concrete Stiffness (MPa) 24529.78 0.08 Normal / [38] f-Steel Yield Strength (MPa) 445.34 0.093 Log-Normal / [37] 200000.0 0.033 Normal I [37] 11.19 0.024 Normal / [36] 35.34 0.024 Normal / [36] 501.52 0.013 Normal / [37] 1301.52 0.005 Normal I [37] c-Concrete Cover (mm) 48.38 0.087 Normal I [37] Se-Spacing between Layers (mm) 25.0 0.050 Normal E -Steel Modulus of Elasticity (MPa) -Diameter of Stirrups (mm) barameter of Steel Bars b -Beam Width h -Beam Height (mm) (mm) (mm) 43 CHAPTER 4 Reliability of Concrete Beams in Bending 4.3.3.2. Load and Fitted Distributions All statistical data for the loads are listed in Table 4.4, where the dead load, D, is described with normal distribution, while the maximum annual snow load, S, and the maximum annual service load, L, is described with Gumbel extreme type I distribution. According to the CSA code, the 30 years return snow load is used in design, but because the probability of getting 30 years snow and service load at the same time is rather low, the max annual service load is used with the 30 years snow load. The basic Gumbel extreme type I distribution is written as following: F(x) = exp{—exp[—A(x—B)j} (4.21) and by rearranging the formula and solve for x we get: {ln(— ln(F(x)))} x=B+ (4.22) A where: A= 2r (0.577 B=x—i — and To be able to describe the distribution of maximum load in N-years, we need to expand Equation 4.22: {ln N ln(_ln(F(x)))} — xN =B+ A 44 (4.23) CHAPTER 4 Reliability of Concrete Beams in Bending The 30 years return snow load, will correspond to a probability of nonperformance of F(x) = 29/30 in Equation 4.22. The coefficients A and B are given in Table 4.3. Using Equation 4.23 for N = 30, the ratio Sr can then be expressed as: Sr=B*+ {i(_ ln(F(x)))} (4.24) where: A*=AB+3.3843 B*= AB+lnN and AB +3.3843 If we use this now, to find the distribution of maximum load for Ottawa and Vancouver we get following values: Table 4.3 Extreme Parameters B* Mean kNI 2 COV A B Ottawa 1.255 0.452 2.260 1.00 5.644 1.003 Vancouver 0.523 1.202 2.039 0.24 3.874 1.004 This can now be used to find the mean and the corresponding standard deviation for Sr which is the ratio between the actual and nominal value, where in case of the distribution of maximum load in N-years, the standard deviation can be found by: (4.25) 45 CHAPTER 4 Reliability of Concrete Beams in Bending By using corresponding values for Equation 4.24 and Equation 4.25, and the relationship u = xCOV, we get the mean and the covariance values listed in Table 4.4. The parameters A and B for the maximum annual service load, L, can be found by using the following relationship from Equation 4.22, assuming that the design load L is also a 30 year return value: {in(_ i(2%))} 5.0=B+ A Also, assuming that the COV of L is 0.25 (see Table 4.4), we can write: COVxB+ COVxO.5772 A = IA = A = 0.2196B and therefore the extreme parameters A and B become: B = 2.868 , A = 1.588 which gives us the equation for maximum annual service load, i.e.: L=2.868+ (— in(.—. ln(F(x)))) 1.588 46 CHAPTER 4 Reliability of Concrete Beams in Bending This equation can then be expressed in terms of the ratio between the actual and nominal load as: (_ Lr = 0.5736+ in(_- ln(F(x)))) 7.936 Table 4.4 Intervening Load Random Variables Mean COV Distribution/Ref D-Actual-Nominal Dead load Ratio 1 .0 0. 1 Normal / [23] Sr-Actual-Nominal Snow load Ratio 1.105 0.206 Extreme type I 0.647 0.25 Extreme type I Definition of Variables / Units L-ACtUa1-Noffliflal Live load Ratio 4.3.4. Example Runs By performing a FORM calculation with TIN, which is a product of the theoretical Beam subroutines and RELAN, and by formulating the performance function as in Equation 4.20 with 0.9 correlation between the concrete compression strength and the initial tangent modulus, we get following results from RELAN: Table 4.5 RELAN Results. Code Equation Product Result fl-Reliability Index (FORM) 4.252 Ps-Probability of Failure (FORM) 47 0. 106E-4 CHAPTER 4 Reliability of Concrete Beams in Bending Instead of using Equation 4.26 to describe the performance function, we can write it as function of the span or as in Equation 4.23: G = Mcap — SL (DrY8+ Sr + Lr8) which should give us a slightly higher reliability index, because the maximum moment, Mmax is now the exact value where in the other case we use the design values, which are always conservative. Now by running TIN, where the performance is a function of the exact maximum moment, we get following results from RELAN: Table 4.6 RELAN Results: Exact Equation Product Result ,8-Reliability Index (FORM) 4.270 P-Probability of Failure (FORM) 0.977E-5 One very important product of RELAN are the sensitivity factors. They can indicate what mode of failure might be expected at each time, and for that reason they play a major roll in the reliability design. 48 CHAPTER 4 Reliability of Concrete Beams in Bending Table 4.7 Sensitivity Factors: Beam Case Definition of Variables / Units Mean Sensitivity factors t-Compression Strength (MPa) 27.77 0.5 15 E-Concrete Stiffness (MPa) 24529.78 0.111 fr-Steel Yield Strength (MPa) 445.34 0.641E5 200000.0 0.0 11.19 0.366E2 35.34 0.974E-1 b-Beam Width (mm) 501.52 0.410E-1 h -Beam Height (mm) 1301.52 0. 863E- 1 48.38 0.595E1 Dr-Actual-Nominal Dead load Ratio 1.0 0.309 S-Actua1-Nomina1 Snow load Ratio 1.105 0.286 L-Actual-Nominal Live load Ratio 0.647 0.706 -Steel Modulus of Elasticity (MPa) 5 E -Diameter of Stirrups (mm) barDiameter of Steel Bars (mm) c-Concrete Cover (mm) The two preceding examples give nearly the same sensitivity factors, which was expected, because there was only a slight difference in the reliability index. If now the sensitivity factors in Table 4.7, which are good representatives of both the runs, are studied and the load sensitivity factors are excluded, it can be seen that the compression strength of the concrete influences the reliability of the beam the most. This means that if the actual load exceeds the design load we might expect a compression failure. 49 CHAPTER 4 Reliability of Concrete Beams in Bending 4.4. Effect of Span on Beam Reliability The program RELAN can be used to analyze the same cross-section, under the same loads, for different spans. Let us now use Equation 4.17 as performance function and run the program for different spans. CHANGES IN SAFETY FOR DIFFERENT SPAN THERE IS A CORRELATION OF 0.9 BETWEEN Fc AND Ect 6 5 4 f33 2 1 0 -1 12 14 16 18 20 22 24 26 SPAN (m) Figure 4.5: Changes in Safety for Different Spans On the left side of the original design span we get concrete compression failure, but by increasing the span the sensitivity of the concrete becomes less and less important until we get a combination of tension and compression failure after 18 m. 50 CHAPTER 4 Reliability of Concrete Beams in Bending The shift in the failure mode could be for the reason that the section is close to the balance point. For 14 m span, the sensitivity to compression strength was 0.965 while for steel yield strength is was 0.0. On the other hand, at 20 m span, the corresponding sensitivities were 0.402 and 0.504. It can be concluded from Figure 4.5, that by adding steel to the tension part of the section, when the span is less than 18 m, the reliability of the beam is not going to be substantially affected. By performing this type of calculation, it is possible to make an economical design, and also prevent certain failures, which can be drastic like the compression failure, and danger to human lives. The way this is normally dealt with in design, as explained earlier, is to have criteria for maximum and minimum reinforcement. From this it can be seen, that reliability calculation cannot only be used for risk assessments, but also as a tool for engineers to understand how different conditions can affect the overall behavior of the section. 4.5. Effect of Steel Ratio on Beam Reliability Most concrete codes have criteria for minimum and maximum reinforcement in concrete members in order to obtain a ductile failure. The reason for minimum reinforcement ratio is to avoid a sudden tension failure of an element. The specified minimum ratio, Pm,ll according to CSA A23.3-M84 (10.5) for a member subjected to bending is given by following ratio [43]: p= 51 (4.26) CHAPTER 4 Reliability of Concrete Beams in Bending This gives roughly the steel area required to have a strength equal to the cracking moment of an identical plain concrete section. The maximum reinforcement ratio on the other hand is to ensure that the beam reinforcement will yield prior to the concrete crushing. When the ratio is close to the upper limit the sections tend to have to small effective depth and therefore a problem with deflections. Too high reinforcement ratio may also result in a compression failure. The specified maximum ratio according to CSA A23.3-M84 (10.3.3) is given by following [43]: d 600 600+f (4.27) Limitations of deflections, convenience in placement of reinforcement and economy in design generally dictate larger overall beam dimensions with correspondingly lower reinforcement ratios, usually in the range of 30-40% of the maximum limit. These lower reinforcement ratios result in further improvement in the ductility of beams. Let us now look at the same beam but change the reinforcement ratio, in order to study the different failure modes which occur, and also to see how effective the code limits are. 52 CHAPTER 4 Reliability of Concrete Beams in Bending By running TIN, with both Vancouver and Ottawa snow load, for different steel ratios, we get the following: Table 4.8 RELAN Results: Different Steel Ratios Layer/ As Dc?17 No.bars 1/3 mm ) 2 3000 (mm) 1230.85 3.129 0.878E-3 3.034 O.121E-2 1/6 6000 1230.85 3.831 O.637E-4 3.750 0.885E-4 2/1 1000 1222.18 4.074 0.231E-4 3.926 0.432E-4 2/2 2000 1215.67 4.168 0.154E-4 4.060 0.245E-4 2/3 3000 4000 1210.62 4.276 0.949E-5 4.172 0.151E-4 1206.57 4.375 0.608E-5 5000 1203.26 1200.5 4.369 4.247 0.624E-5 0.108E-4 4.287 4.247 0.905E-5 0.108E-4 4.164 0.156E-4 1193.5 3.943 0.402E-4 3.928 0.429E-4 2/4 2/5 2/6 Ottawa Vancouver pf 3/1 6000 1000 3/2 2000 1187.49 3.798 0.731E-4 3.779 0.787E-4 3/3 3000 1182.29 3.672 0.120E-3 3.653 0.130E-3 3/4 4000 1177.74 3.547 0.195E-3 3.518 0.218E-3 3/5 5000 1173.72 3.442 0.288E-3 3.427 0.305E-3 3/6 6000 1170.15 3.330 0.434E-3 4/3 3000 6000 3.100 2.940 0.966E-3 0.164E-2 0.448E-3 0.102E-2 4/6 1152.81 1139.8 3.321 3.084 2.900 0.187E-2 The results listed in Table 4.8, which are also illustrated in Figure 4.6, show that there is a drastic change in the reliability at certain points, which basically means that we are observing different failure modes. 53 CHAPTER 4 Reliability of Concrete Beams in Bending However, because the variability of intervening variables such as the compression strength affects the failure pattern, and the code does not involve the probability directly, the criteria for maximum and minimum reinforcement may not serve their purpose. L SAFETY INDEX FOR DIFFERENT STEEL/CONCRETE RATIOS THERE IS A CORRELATION OF 0.9 BETWEEN Fc AND Ect I 6 5 4 2 0 0.005 0.01 0.015 0.02 0.025 0.03 STEEL I CONCRETE RATIO E OTTAWA SNOW LOAD 2.5 kN/m 2 1 VANCOUVER SNOW LOAD 1.9 kN/m 2 Figure 4. 6. Changes in Safety for Different Steel Ratios The cumulative distribution curve can also be plotted from the data in Table 4.8 (see Figure 4.7). 54 CHAPTER 4 Reliability of Concrete Beams in Bending By fitting some known distribution through the data in Figure 4.7 and finding the derivative of the function, the probability density curve can be established. CDF-CURVE FOR DIFFERENT STEEL / CONCRETE RATIO THERE IS A CORRELATION OF 0.9 BETWEEN Fc AND Ect 0.002 OTTAWA SNOW LOAD 2.5 kN/m 2 VANCOUVER SNOW LOAD 1.9 kN/m 2 0.00 15 Pf 0.001 0.0005 0 - 0.005 0.01 I- 0.015 - 0.02 0.025 0.03 STEEL I CONCRETE RATIO Figure 4.7: Cumulative Distribution Curve for Different Steel Ratios By looking at the sensitivity factors, which are listed in Table 4.9 and illustrated in Figures 4.8 and Figure 4.9, it can be seen that there is a shift from tension failure at point, f, to combination of both tension and compression failure at point, g, to pure compression failure at point, i. The limit for the maximum reinforcement in the code (see Figure 4.6), is to the right of point, i, which means that the code does not prevent compression failure in this case. 55 CHAPTER 4 Reliability of Concrete Beams in Bending Table 4.9 Sensitivity Factors: Different Steel Ratios Defimtion of Variables / Umts Points on Curve (see figure 4 6) f g 1 f-Compression Strength (MPa) 0.309E2 0.498 0.962 -Concrete Stiffness (MPa) 0.495E3 0.109 0.399E1 0.616 0.775E5 0.0 E -Steel Modulus of Elasticity (MPa) 0.225E4 0.0 0.0 q5 -Diameter of Stirrups (mm) 0. 166E2 0.36 1E2 0.1 19E2 0.194 0.108 0. 806E2 bar-D1ameter of Steel Bars 2.nd layer (mm) 0.124 0.765E4 0.391E2 bar1amet of Steel Bars 3.rd layer (mm) 0.0 0.0 0.769E3 b-Beam Width (mm) 0.126E-3 0.391E-l 0.331E-l h-Beam Height (mm) 0.331E-l 0.850E-1 0.285E-l C -Concrete Cover (mm) 0.269E1 0.585E1 0. 194E4 D-Actual-Nominal Dead load Ratio 0.283 0.30 1 0.159 5rcttT0mi Snow load Ratio 0.256 0.277 0.113 Lr-Actu2d-Nominai Live load Ratio 0.648 0.725 0.178 fr-Steel Yield Strength (MPa) øbar1amet of Steel Bars 1. St layer (mm) One of the explanations why the code gives an unconservative estimation for the limit of tension-compression failure in this case, could be the code assumes that the actual compression strength which we normally get is higher than the specified value, while according to Mirza, Hatzinikolas and MacGregor [38] the actual mean value for compression strength (see Table 4.2) is assumed to be lower than the specified value (see Table 4.1). 56 CHAPTER 4 Reliability of Concrete Beams in Bending It is interesting to see in Figure 4.8 and Figure 4.9, how sudden the shift is in the variable sensitivity, and also how clear difference is between different modes, i.e. tension and compression failure. L CHANGES IN SENSITIVITY WITH DIFFERENT RATIO 1 CONCRE COMPRESSIVE STRENGTH YIELD STRENGTH OF STEEL 0.8 0.: STEEL DIAMETER P’J THE 1 ST LAYER I I 0.005 0.01 0.015 0.02 0.025 0.03 STEEL I CONCRETE RATIO Figure 4.8: Changes in Material Sensitivity The cause of the relatively high sensitivity of the concrete compression strength and this sudden shift between failure modes could be, as was mentioned earlier, because by using Equation 3.3 and ignoring the increase due to hydration we might get a conservative estimation of the mean value for concrete compression strength. 57 CHAPTER 4 Reliability of Concrete Beams in Bending Further study on concrete compression strength seems to be needed in order to develop appropriate reliability based limits for maximum reinforcement and prevent this premature failure mode. CHANGES IN SENSITIVITY WITH DIFFERENT RATIO 0.8 Pct 0.6 0.4 0.2 0 0.005 0.01 0.015 0.02 STEEL / CONCRETE RATIO Figure 4.9: Changes in Load Sensitivity 58 0.025 0.03 CHAPTER 5 Offshore Structure Ice Impact 5.1. Introduction One of the major factors in reliability study of concrete offshore structures, are the uncertainties associated with extreme environmental load conditions, such as iceberg impact, imposed on the offshore structure. Because ice is not really an isotropic material, even though it could appear so, the issue of predicting ice forces on structures essentially reduces to what at first sight appears to be a relatively simple problem, that of understanding how ice deforms and fails under stress. The deformation and strength properties of ice are affected by two major characteristics i.e. temperature and brittleness. For the reason ice is an extremely brittle material it cannot resist tension very well which makes it ease for cracks to propagate. At the same time ice is a solid close to its melting point, and therefore it exhibits creep and its compressive strength is temperature dependent. This explains the various shapes of offshore structures because what designers are interested in is reducing the impact load as much as possible by failing the ice where it is weakest. The failure which the ice undergoes during an impact is crushing-, tension- and flexural failure (see Figure 5.1). 59 CHAPTER 5 Offshore Structure Ice Impact P Flexural Failure Tension Failure P Crushing Failure Figure 5.1: Failure Modes of Ice During the past two decades our understanding of these aspects of ice mechanics has advanced considerably. But despite extensive research and well quantified laboratory tests of response of ice to stress, problems remain in extrapolating from the knowledge of small-scale ice properties to a prediction of how ice behaves on a large scale. Because of this difficulty and lacking full scale tests there is no generally accepted methods for predicting ice loads on structures. One way to predict ice impacts on offshore structures is to use probabilistic programs. Most of them approach the ice impact by using either simulation process like Monte Carlo (see Figure 5.2) or FORM/SORM procedures. 60 CHAPTER 5 Offshore Structure Ice Impact ICE ENVIRONMENT FOR THE VELOPMENT LOCATION STRUCTURAL DESIGN CONCEPT ICE LOADING SCENARIOS i=1 n [ELECT SCENARIO I LOAD PROCESS CONSIDERING UNCERTAINTY IN CONCENTRATION I Hi ICE FEATURE SIZE MORPHOLOGY ICE ENVIRONMENTAL DRIVING FORCES MECHANICAL PROPERTIES OF ICE PROBABILITY PROBABILITY e.g. LOADING EVENTS TIME THICKNESS e.g. ICE VELOCITY PROBABILITY e.g. AVE. ICE PRESSURE i<n EMPLOY EXTREMAL ANALYSIS TO ASSIGN PROBABILITY DISTRIBUTIONS FOR ICE LOADINGS ON THE STRUCTURE PROBABILITY / ANNUAL MAXIMUM ICE LOAD PROVIDE ICE LOADS FOR DESIGN TO COMPARE WITH STRUCTURAL RESISTANCE Figure 5.2. Flow Chart for Probabilistic Approach for Ice Load 61 CHAPTER 5 Offshore Structure Ice Impact 5.2. Icebergs and Multi-Year Floes Icebergs are not frozen sea water as one might think but are composed of freshwater ice from land-based glaciers flowing off the land into the sea. Glacier ice develops from successive snowfalls of pure freshwater snow which compress under their own weight until they become solid ice (see Figure 5.3). The term “Multi-year ice”, which stands basically for frozen sea water, is normally defined to be ice which has survived at least two summer seasons and is formed from second-year ice by continuing dynamic action and by melting and refreezing (see Figure 5.3). Snowfall Blocky lce Shell Tabular Iceberg Ice Cap generating Icebergs. Second-Year Ice, of variable thickness, in the late summer, surface and bottom melting has occured. Second-Year Ice during its second winter, undergoing refreezing, accretion and further ice action. Mature Multi-Year Ice, 4-6 m thick, 5-10 years old, containing old smoothed ridges, hummocks and b ummocks. Figure 5.3: Icebergs and Multi-Year Floes [47] 62 CHAPTER 5 Offshore Structure Ice Impact 5.3. Dynamic Impact from Icebergs and Floes No full-scale measurements have ever been made of forces exerted during impact of an iceberg with a structure. However, the problem would seem not to be very substantially different from that of impact of very thick multi-year floe with a structure. In both cases we expect progressive failure of ice over a steadily growing contact zone, and we expect the ice mass to come to rest when all its energy has been dissipated by the failure process. The principal differences between iceberg impact and ice floe impact are following: i. An iceberg is generally free to rotate about all three axes during impact, though depending on the size of the iceberg, while an ice floe is typically able to rotate significantly only in horizontal plane, about the vertical axis. Because there are so few data on which to base analysis Sanderson [47] proposed that all rotational components of motion should be neglected. ii. The contact zone of an iceberg with a structure typically grows progressively in two dimensions rather than just one, i.e. the contact width and the contact depth both grow as penetration proceeds. iii. The failure modes of the ice may be rather different. In most cases full-thickness flexural failure is unlikely to occur during iceberg impact and we might also expect a higher degree triaxial confinement during crushing of an impacting iceberg. 63 CHAPTER 5 Offshore Structure Ice Impact 5.4. Evaluation of the Ice Impact with Energy Principles When a multi-year floe or iceberg impacts with a structure it continues moving until all its kinetic energy is dissipated. If the floe or the iceberg has a mass, m, and initial velocity, V, then it will come to rest at total penetration, x, when [11]: ,4m(1+Cm)V2 fp(x)dx (5.1) This simple energy model, based on the formulation given by Johnson and Nevel (1985), assumes that the total kinetic energy of the ice feature is observed in the progressive crushing of the ice contact zone. The kinetic energy, Ek, of the ice feature is given by the equation: (l+Cm)mV /2 Ek = 2 (5.2) where Cm is added mass factor obtained by Croasdale and Marcellus (1981): Cm =O.9h/(2z—O.9h) and (5.3) z is the water depth. Also given by Rothrock and Thorndike (1984) the area of typical floe is related to mean caliper diameter, D, by the approximate formula [47]: 2 A=O.66D 64 (5.4) CHAPTER 5 Offshore Structure ice impact From this we can say that the mass, m, is: p 1 h 2 m=O.66D where h is the ice thickness and a the density. (5.5) Now we can write following expression: , 2 0. 66Dhp 1 (1+ Cm )V fp(x)dx (5.6) The impact load, p(x), is defined as: p(x)=A(x)ci (5.7) where A(x) is the contact area and ç is the unconfined compressive strength of ice. Now we can say that the energy dissipated duriig crushing based on above will then become [11]: E=fp(x)dx 65 (5.8) CHAPTER 5 Offshore Structure Ice Impact Through energy principles we know that internal work is the same as the external work i.e.: (5.9) and from that we can get the maximum penetration and the maximum impact load. 5.4.1. Calculation of the Ice Contact Area To calculate the ice impact and the contact area for different velocity, diameter, thickness or compressive strength we need to come up with a equation where all these variables are introduced. The only complications are how we formulate the changes in the contact area for a particular ice feature and for a particular structure because as stated earlier the contact width and the contact depth both grow as penetration proceeds. Lets now look at two different structures (see Figure 5.4), i.e. a cylindrical structure and a structure with multiple wedge-shaped indentors assuming a wedge shaped ice feature, as an example to see how we can establish the impact force. The mathematical expressions for those two cases, i.e. the cylindrical and the multiple wedge-shaped structure, can now be established easily from Figure 5.4. 66 CHAPTER 5 Offshore Structure Ice Impact Plan Elevation Cylindrical Structure Wedge-Shaped Structure Figure 5.4: Different Structures and Ice Features Cylindrical structure: 4(x) = 2 2LR sin +(tan a+ tanflj(Ø— sin Øcosq)R (5.10) and: cosq5=(R—x)/R (5.11) where R is the radius of the structure, L the thickness of the ice and x the penetration (see Figure 5.4). 67 CHAPTER 5 Offshore Structure Ice Impact Multiple wedge-shaped indentors: A(x) = (2L + xtan a+ xtan,13)(x/cosy) (5.12) where L the thickness of the ice and x the penetration (see Figure 5.4). It should be noted that the angles at the top and the bottom of the ice feature could be kept as variables but in order to simplify calculation later they will be assumed to be constants. 5.4.2. The Ice Compressive Strength Observation by Sanderson [47] concluded that the upper bound of data collected at Tarsiut P-45 in 1984-1985 appeared to depend on inverse square root of area, and the fact that theory would lead us to suppose that indeed it should do so, suggests that a normalization of these data can be carried out. This means that we can normalize all pressure measurements by the inverse square root of contact area, A(x), and express . 0 then relative to a single “reference contact area”, A This means that for any measurement of stress, u, over an area, A, a normalized stress, cr*, can be calculated over the reference area using the assumption of inverse square root dependence on area. The expression for the normalized stress is as following: * /A(x) 0 A 68 C (5.13) CHAPTER 5 Offshore Structure Ice Impact Based on the same data it was concluded that a mean normalized indentation stress, Cim* is equal to 0.92 MPa with a standard deviation of, ci;’, equal to 0.45 MPa (see Figure 5.5). INDENTATION PRESSURE AT PEAK LOAD 10 a (MPa) 0.1 0.01 10 100 1000 CONTACT AREA 10000 ) 2 (m Figure 5.5: Indentation Pressure at Peak Load [47] As mentioned before the observation was based on upper bound data and because normally the mean stress represents the actual compressive strength better Sanderson suggested that factor, 2m would be used in order to express the average ratio of mean load to peak load. Based on this the compressive strength becomes: = Urnj/O2m 69 (5.14) CHAPTER 5 Offshore Structure Ice Impact 5.5. Ice Impact Force for the Hibernia Structure As an example lets look at the case which was investigated for the Hibernia Development Project in offshore Newfoundland by applying the energy theory [11]. By assuming that the unconfined compressive strength of the ice is constant the energy dissipation formula becomes: E = 2mjm* fJ(2L + x tana+ xtanfl)(x/ cosy)dx (5.15) This leads then to the final equation by using energy principles: Ek = mjm* ff .J(2L + x tan a + x tanfl)(x / cos y)dx (5.16) where: Ek __j40.66D2hR(l+Cm)V2 (5.17) and the by carrying out the integration we can find the Xm which is the maximum penetration and then find the contact area, A(x), which leads us to the maximum impact load, p(x) [47]: p(x) = %JA(X)AOJm*2m 70 (5.18) CHAPTER 5 Offshore Structure Ice Impact For the Hibernia structure the angle, y, is 450 so the energy formula can be written as following: 0. 1 hp (1 + Cm)V 2 66D 2 = 2mJm fJ(2LX +(tan a + tanfl)x )d 2 (5.19) 5.6. Reliability Based Formulation of Ice Impact The next step, in order to estimate the probability distribution of ice load, is to evaluate the integral in the energy equation by using numerical methods. The objective is to fix the basic variables, i.e. the diameter, the thickness, the velocity and the compressive strength of the ice feature, to compute the corresponding maximum ice load and use RELAN to establish the corresponding Cumulative Distribution Function. By using Gauss integration to solve X,,, we need to change the coordinate system from x to 71: 2 (5.20) 2 if we introduce this now to the original equation and solve for y = 0: = mUm Xm 1(L(l + 71)+(tan a+tan 71 x/(l + )d Ek (5.21) 2 ) — 71 CHAPTER 5 Offshore Structure Ice Impact In terms of programming procedure: 2 mm NG (5.22) 2m where NG is the number of Gauss points used, ij location of point, and w the weight at the point. To evaluate the integral we can write a simple FORTRAN Do-loop i.e.: sum = 0.0 Do 101 = 1, NG 1 sum = sum + (L(l+j)+(tana+tanfl)Xm/(1+)2)w 10 and then iterate Xm continue for solution which gives us then the contact area, A(x). After the penetration has been found for one set of random variables we can find the impacting force and express it in terms of probability. Now in order to construct the CDF-Curve for the ice impact we can write following: Pf=P(Fffl<FO) which is the probability of that the maximum impact force, (5.23) will be less than certain impact force F. The performance function in RELAN will therefore become: G= 2mJm*JJA(X)_F 72 (5.24) CHAPTER 6 Reliability of Concrete Offshore Structures 6.1. Introduction The design of a complex concrete offshore structure, which is exposed to extreme environmental loads such as icebergs and waves, involves determining the sectional forces at various locations of the structure by using a linear elastic analysis. The response due to the eight sectional forces, i.e. two normal forces N and N, a membrane shear force N, two flexural bending moments M and M, a torsional bending moment M and two transverse shear forces V and ‘, (see Figure 6.1), can be predicted using a generalization of the strain compatibility approach used for beams (see Chapter 4). While the case of a beam subjected to bending involves uniaxial strains and stresses, the case of eight sectional forces involves triaxial strains and triaxial stresses. The program SHELL474, which is based on a 3-D strain compatibility approach, was used to account for the influence of the intervening variables, and to evaluate the theoretical capacity for reliability calculation of offshore structure wall elements. SHELL474 was developed by Adebar and Collins [1] as verification of the • • new Canadian concrete offshore structure code (CSA S474). SHELL474 calculates the factored sectional resistance of an element for given concrete and reinforcement dimensions, material grades and loading ratios. 73 CHAPTER 6 Reliability of Concrete Offshore Structures 6.2. Sectional Strength of Concrete Wall Elements The following is a brief summary of the theoretical procedure used by SHELL474, but a more detailed description is given by Adebar and Collins [1]. An introduction into the strain compatibility approach for reinforced concrete in bending can also be found in Chapter 4. The three membrane forces and the three bending moments (see Figure 6.1), which a wall element is subjected to, is predicted by assuming that the three biaxial strains e, 6 and y, vary linearly over the thickness of the element. 1 1 Mxy My x Mxy Figure 6.1: Sectional Forces at Complex Concrete Structure Thus, the complete biaxial strain state can be described by six variables, i.e. three strains at the top surface and three strains at the bottom surface. For a given set of the six strain variables, the stresses in the concrete and the reinforcement can be determined from biaxial stress-strain relationships. 74 CHAPTER 6 Reliability of Concrete Offshore Structures By integrating the stresses over the thickness of the element, the six corresponding stress resultants N, N, N, M, M and M can be found. When a wall element is subjected to transverse shears V and T’, (see Figure 6.1), the out-of-plane strains , y and cannot be ignored, hence the problem involves triaxial strains and stresses. While the biaxial strains are considered over the thickness of the section, the triaxial strains are only evaluated at one location in SHELL474, e.g. at the mid-plane of the section or at the centroid of the flexural tension reinforcement. 6.3. Design of Hibernia Offshore Wall Element The concrete offshore structure chosen for the reliability study was the Hibernia Gravity Base Structure, which will stand in 80 meters of water on the Grand Banks of Newfoundland. The Hibemia GBS (1986 update design) structure has a 1.4 meters thick icewall with 30 gear teeth in order to reduce the ice impact forces. The overall diameter of the structure, from tip to tip of the teeth, is 104 meters. In the design for the ultimate limit states of Hibernia GBS the sectional strength, system ductility, and the fatigue were considered, while in the case of serviceability limit states, crack control and control of local damage were considered. According to a recent study [5], the most critical load case of all the various limit states for the design of the reinforcement in the icewall is the 100 year eccentric ice impact. The wall element used for the reliability study (see Figure 6.2), was designed for local damage with 100 year ice impact. 75 CHAPTER 6 Reliability of Concrete Offshore Structures 10-130 Grade 1860 Strands in 69 ID. Sheeting EA. Side @ 470 0/c Exterior Face 1400 mm 35M @ 235 1 5M Headed Transverse Reinforcing Bars @ 235 H x 125 V 0/c Vertical EA. Side 35M Reinforcing Bars in Bundles of 2 @ 1 25 olc Plan Note: In Accordance with Clause 5.4.1 of S474-M1989 / / 1 5M Headed Transverse Reinforcing Bars @ 125 o/c Vertical Plane CL 1400mm Elevation Section E) Figure 6.2: Details of Hibernia GBS icewall Design Adapted from Reference [5] 76 CHAPTER 6 Reliability of Concrete Offshore Structures 6.3.1. Modified SHELL474 and Subroutines for RELAN To perform reliability calculation for offshore structure wall element, the main subroutine in SHELL474, entitled Sl.FOR, was modified. Instead of fmding a solution for only one set of variables, SHELL474 is now able to calculate the capacity for many sets of random variables coming from RELAN. As mentioned earlier, the performance function is defined in the subroutine GFUN. In order to describe the performance function for the offshore structure wall element, an eight-dimensional ultimate capacity vector { N, ,, 3 N, N, M, M, M V, iç } = { ç, } needs to be evaluated. Since the applied forces and the corresponding ultimate capacity vector change with time, a linear finite element program is needed to re-evaluate the applied resultant vector at each time during the impact. Here, as an approximation, it is assumed that the applied resultant components Aj’ corresponding to the random applied load F, are constant over time and given by: Ai = where Aj (6.1) are the resultant components obtained from the finite element analysis for the load P . The performance function used for the reliability calculation of offshore 0 structure wall elements can be written in terms of vectors or as following: G=OQI—IOP where the vectors OQ and OFI are illustrated in Figure 6.3. 77 (6.2) CHAPTER 6 Reliability of Concrete Offshore Structures In detail, OP is the norm of the applied resultants, and OQI represents the norm of the capacity vector for a loading path in the direction OP. Failure, i.e. G <0, occurs when lOP! > l°Ql• Failure: G<O Figure 6.3: Performance Function for Wall Elements In terms of the eight dimensional vectors the performance function can be written as follows: G= 2 _p/j2++j2 8 lucl2+....+uc (6.3) where p is the ratio between the random applied load P and the applied load I used for the finite element analysis. The performance function can also be written in terms of the ratio between the norms of the resistance and the applied load vectors, as follows: 1 — /ci A1 2 +....+J.g ••• 78 — (6.4) CHAPTER 6 Reliability of Concrete Offshore Structures It should be noted that the components, which are combined in the resultant vector, have different units. To make the influence of each component equal to its real effect on the resultant direction, a scale factor was used. A scale factor is applied to the performance function by dividing the two bending moments and the torsional moment by the thickness of the wall. 6.3.2. Comparison of Beam Program and Modified SHELL474 To compare and test the modified version of SHELL474, both the beam program TIN and the modified SHELL474 were used to perform reliability calculation for the beam designed in Appendix B (see Figure 6.4). L COMPARISON BETWEEN BEAM VS. SHELL PROGRAM 8 BEAM PROGRAM SHELL_PROGRAM 4 6 134 2 0 0 I I 2 4 6 RATIO OF NOMINAL MEMBER CAPACITY TO AN APPLIED LOAD Figure 6.4: Comparison between TIN anti modified SHELL474 79 8 CHAPTER 6 Reliability of Concrete Offshore Structures The performance function was written in terms of the theoretical capacity for the resistance and for a constant applied load. However, it should be noted that the results presented in Figure 6.4 are only for comparison and to illustrate the reliability of the modification done to SHELL474. The variability of the load is ignored and therefore the results should not be taken out of context. 6.3.3. Variability of Intervening Variables The material statistics listed in Table 6.1 and used for the concrete offshore structure wall elements, are explained in Chapter 3. It should be noted, that the rate of loading was ignored for high strength concrete compression strength unlike what was done for the beam case. The reason was simply, that the actual concrete compression strength is normally expected to be 10-15 % higher than the specified value, and the effect from the rate of loading and the increase in strength due to additional hydration are expected to approximately cancel each other out. However, due to other factors discussed in Chapter 3, Equation 3.2 was used for high strength concrete. 80 CHAPTER 6 Reliability of Concrete Offshore Structures Table 6.1 Intervening Material Random Variables Definition of Variables / Units Mean COV Distribution/Ref. f -Compression Strength (MPa) 41.33 0.10 Normal/[38] h -Sectional Thickness (mm) 1401.52 0.0045 Normal I [37] f -Yield Strength of Stirrups (MPa) 445.34 0.093 Log-Normal I [37] 2 31 of Steel in X-dir. mm A -Area 702.0 0.06 Normal / [31] 2 32 of Steel in X-dir. mm A -Area 2000.0 0.06 Normal / [31] -Spacing of Bars in X-dir. mm 1 S 125.0 0.05 Normal -Spacing of Bars in X-dir. mm 2 S 125.0 0.05 Normal -Location of X-Bars in Z-dir. mm 1 Z 621.52 0.01 Normal I [37] -Location of X-Bars in Z-dir. mm 2 Z -616.52 0.01 Normal / [37] f -Steel Yield Strength (MPa) 1 445.34 0.093 Log-Normal / [37] -Steel Yield Strength (MPa) 2 f 445.34 0.093 Log-Normal / [37] 2 -Area of Steel in Y-dir. mm 1 A 1000.0 0.06 Normal / [31] 2 -Area of Steel in Y-dir. mm 2 A 1000.0 0.06 Normal I [31] 2 3 -Area of Steel in Y-dir. mm A 990.0 0.02 Normal / [31] -Area of Steel in Y-dir. mm 4 2 990.0 0.02 Normal / [31] 1 -Spacing of Bars in Y-dir. mm S 325.0 0.05 Normal -Spacing of Bars in Y-dir. mm 2 S 235.0 0.05 Normal 3 -Spacing of Bars in Y-dir. mm S 470.0 0.05 Normal -Spacing of Bars in Y-dir. mm 4 S 470.0 0.05 Normal 1 -Location of Y-Bars in Z-dir. mm Z 586.52 0.011 Normal / [37] -Location of Y-Bars in Z-dir. mm 2 Z -581.52 0.011 Normal! [37] 3 -Location of Y-Bars in Z-dir. mm Z 561.52 0.011 Normal/[37] -Location of Y-Bars in Z-dir. mm 4 Z -561.52 0.011 Normal / [37] f -Steel Yield Strength (MPa) 2 -Steel Yield Strength (MPa) f 3 -Strand Yield Strength (MPa) f 4 -Strand Yield Strength (MPa) f, 445.34 0.093 Log-Normal I [37] 445.34 0.093 Log-Normal I [37] 1742.54 0.025 Normal! [39] 1742.54 0.025 Normal I [39] 81 I CHAPTER 6 Reliability of Concrete Offshore Structures 6.3.4. Ice Impact Prediction Using the PROB Program By running the program PROB, which is based on the theoretical model derived through energy principles in Chapter 5, the CDF-Curves for concentric and eccentric ice impact can be derived (see Figure 6.5). To find the CDF-Function for the eccentric load, the concentric load was multiplied by the angle of the eccentricity or i/I. The nominal concentric ice impact used for the finite element analysis was 555 MN and therefore the nominal eccentric ice impact became 392.4 MN. To fit a distribution to the program results, the Gumbel distribution was used. All the statistical data for the CDF-Curves can be found in Table 6.2. CDF-CURVES FOR 100 YEAR ICE-IMPACT 1.2 1 / / / 0.8 Pf ‘I 0.6 UN CO RRELATED C ONC ENTRIC 0.4 CORRELATED CONCENTRIC 1, UNCORRELATED ECCENTRIC -.- 0.2 0 CORRELATED ECCENTRIC 7 0 500 1000 1500 LOAD (MN) Figure 6.5: CDF-Curves for 100 Year Ice-Impact 82 2000 2500 CHAPTER 6 Reliability of Concrete Offshore Structures Even though no data are available, confirming a correlation between the intervening parameters effecting the ice impact, i.e. the velocity, the thickness, the diameter and the ice compression strength, a correlation factor of 0.6 was used between the velocity and the thickness for the sake of comparison. Figure 6.5 shows results for no correlation and for the correlated case. Table 6.2 Extreme Parameters for Ice-Impact Loads Type of Ice-Impact A B 555 MN Concentnc/uncorrelated 0.0104 -8.4326 555 MN Concentric/correlated 0.0079 -15.4286 392.4 MN Eccentric/uncorrelated 0.0146 -5.9628 392.4 MN Eccentric/correlated 0.0112 -10.9097 While the CDF-Function for ice impact can be easily derived by using the procedure in Chapter 5 if enough available data exist, the effect on the applied load resultant components remains unknown. Instead of re-evaluating the changes at every time due to the increasing contact area with a finite element analysis, the 8-dimensional applied load resultant vector was scaled accordingly (see Equation 6.1) and assumed to be constant over time. 83 CHAPTER 6 Reliability of Concrete Offshore Structures 6.3.5. Example Runs The applied load resultant components used for the example runs and listed in Table 6.3, as part of preliminary study by Allyn, Yee and Adebar [5], are results from the finite element analysis program COSMOS. It should be noted that the gravity and the ballast loads have been subtracted from the resultants listed in Table 6.3. The element used for reliability study, was originally designed for load combination No. 34 according to CSA S474. Load combination No. 34, which is used for an evaluation of local damage, is a combination of gravity loads, solid ballast load and 100 year eccentric ice impact where the load factors are set to 1.0. Table 6.3 Resultant Load C’omponents from Linear Finite Element Analysis Load Vectors for Wall Elements Concentric Eccentric 1 -Normal Force in X-dir. (kN/m) N 446.0 170.0 Ny-Normal Force in Y-dir. (kNIm) -165.0 -1835.0 N -Membrane Shear Force (kNIm) 250.0 -384.0 Mi-Bending Moment in X-dir. (kNm/m) -559.0 4760.0 My-Bending Moment in Y-dir. (kNmlm) -33.0 1690.0 M),-Torsional Bending Moment (kNmlm) 12.0 -12.0 Vs-Transverse Shear in X-dir. (kNIm) -78.0 2640.0 l’ -Transverse Shear in Y-dir. (kNIm) -7.0 6.0 84 CHAPTER 6 Reliability of Concrete Offshore Structures By running the SHELL474 program for the element, the load factor, which is the ratio between the element strength resultant vector and the applied load resultant vector, was only 1.027 (see Appendix C). Since the load factor is so low for this load case, the safety index /3 was expected to be low. While the statistics for the ice impact parameters are quiet often not available and therefore assumed, and also because of the approximation for the direction of the resultant load vector, the probability of failure will not be realistic. However, because the eccentric load case is rather extreme, and for the sake of comparison, components from concentric load were also applied. The results from the reliability analysis for the example runs are listed in Table 6.4 and Table 6.5. The difference between the concentric and the eccentric load case is rather large, which was partly expected because of the size difference between the resultant load vectors. Table 6.4 RELAN Results: Wall Element Type of Ice-Impact 13 Pf 555 MN Concentric/uncorrelated 5.378 0.376E-7 555 MN Concentric/correlated 4.441 0.448E-5 392.4 MN Eccentric/uncorrelated 1.395 0.815E-1 392.4 MN Eccentric/correlated 0.467 0.320 It should also be noted as mentioned earlier, because the change in direction of the ultimate resultant components is totally ignored and an assumption was made for the ice impact, the reliability results should not be taken out of context. 85 CHAPTER 6 Reliability of Concrete Offshore Structures Table 6.5 Sensitivity Factors. Wall Element Definition of Variables I Units The Main Sensitivity Factors from Each Run Con./unc. Con./cor. Ecc.Iunc. Ecc./cor. 0.346E-1 0.272E-1 0.382E-1 0. 157E-1 0.0 0.0 0.758E4 0.101 51 of Steel in X-dir. A -Area 0.183 0.173 0. 143E3 0. 190E3 -Area of Steel in X-dir. 2 A 0.543E3 0.555E4 0.295 0.191 -Spacing of Bars in X-dir. 1 S 0.137 0.135 0.102E3 0.165E3 -Spacing of Bars in X-dir. 2 S 0.275&3 0. 158E3 0.239 0.157 -Location of X-Bars in Z-dir. 1 Z 0.184E1 0.172E4 0.548E3 0.778&3 -Location of X-Bars in Z-dir. 2 Z 0.288&2 0.245E2 0. 134E4 0. 178E4 1 -Steel Yield Strength f 0.209 0.205 0.0 0.0 f 2 Steel Yield Strength Ice -Ice Impact 0.0 0.0 0.419 0.273 0.950 0.953 0.821 0.924 f. -Compression Strength f-Yield Strength of Stirrups 86 CHAPTER 6 Reliability of Concrete Offshore Structures 6.4. Development of PDF-Functions for Load and Resistance In order to represent the probability of failure with geometric representation, the PDF-Functions can be derived from the Gumbel extreme distribution described in Equation 4.21. The corresponding probability density function, which is basically the derivative of Equation 4.21, can be written as following: f(x) = Aexp{—A(x— B)}exp{—expf—A(x —B)}} (6.5) L PDF-CURVES FOR ANNUAL CONCENTRIC ICE-IMPACT 0.006 0.005 0.004 Freq. 0.003 0.002 0.001 0 -500 0 500 LOAD (MN) Figure 6.6: PDF-Curves for Annual Concentric ice-Impact 87 1000 CHAPTER 6 Reliability of Concrete Offshore Structures To plot the PDF-Curves (see Figure 6.6 and 6.7) for annual concentric and eccentric ice impact, Equation 6.5 and the corresponding A and B listed in Table 6.2, respectively, the location and the scale parameters, can be used. PDF-CURVES FOR ANNUAL ECCENTRIC ICE-IMPACT 0.006 0.005 0.004 Freq. 0.003 0.002 0.001 0 -500 1000 500 LOAD (MN) Figure 6.7: PDF-Curves for Annual Eccentric Ice-Impact If the annual PDF-Function is now expanded for N-years we get the following: f(x) = NAexp{—A(x — B)}exp{—Nexp{—A(x 88 — B)}} (6.6) CHAPTER 6 Reliability of Concrete Offshore Structures Now the PDF-Curves for 100 year concentric and eccentric ice impact can be plotted as shown in Figure 6.8 and 6.9. PDF-CURVES FOR 100 YEAR CONCENTRIC ICE-IMPACT 0.006 0.005 0.004 Freq. 0.003 0.002 0.00 1 0 0 200 400 600 800 1000 LOAD (MN) Figure 6.8: PDF-Curves for 100 Year Concentric Ice-Impact 89 1200 1400 CHAPTER 6 Reliability of Concrete Offshore Structures L PDF-CURVES FOR 100 YEAR ECCENTRIC ICE-IMPACT 0.006 0.005 0.004 Freq. 0.003 0.002 0.00 1 0 0 400 200 600 1000 800 1200 1400 LOAD (MN) Figure 6.9: PDF-Curves for 100 Year Eccentric Ice-Impact In order to derive the CDF-Function for element resistance, the performance function for the modified SHELL474 can be written in the same manner as Equation 5.24 or in terms of load ratio versus certain load constant: = G — 0 L (6.7) If the probability of failure is calculated for number of load constants, the CDF-Curves can be plotted. 90 CHAPTER 6 Reliability of Concrete Offshore Structures Using the Least Square Method to fit to the data results, the location and scale parameters A and B can be derived. If now the PDF-Curves for element resistance and 100 year uncorrelated/correlated eccentric ice impact are plotted (see Figure 6.10 and 6.11), the corresponding reliability calculation results can be found in Table 6.4. PDF-CURVES FOR RESISTANCE OF WALL ELEMENT AND 100 YEAR ECCENTRIC ICE-IMPACT 4 3 Freq. 2 00 0.5 1 1.5 2 2.5 3 LOAD RATIO Figure 6.10: Element Resistance and 100 Year Uncorrelated Eccentric Ice-Impact 91 CHAPTER 6 Reliability of Concrete Offshore Structures PDF-CURVES FOR RESISTANCE OF WALL ELEMENT AND 100 YEAR ECCENTRIC ICE-IMPACT 4 3 Freq. 2 00 0.5 1 1.5 2 2.5 3 LOAD RATIO Figure 6.11: Element Resistance and 100 Year Correlated Eccentric Ice-Impact 6.5. System Performance Using the Joint PDF-Functions If the joint probability functions are known for the element resistance and the applied load, or can be derived as in Section 6.4, the performance function can be written in terms of two random variables instead of 28 as for the example runs: G=R—D 92 (6.8) CHAPTER 6 Reliability of Concrete Offshore Structures In this case, R represents the derived joint probability distribution for the resistance of the element with its mean and standard deviation, and D represents the derived joint probability distribution for the applied load with its mean and standard deviation. With joint probability distributions known, the evaluation of the reliability for a certain element with a certain load can be simplified somewhat, but since the joint probability distributions are rarely known programs like TIN and the modified SHELL474 are still needed. 93 CHAPTER 7 Concluding Remarks and Further Study Probabilistic methods provide a logical framework for uncertainty analysis and safety evaluation in structural engineering. Through reliability calculations, designers are provided with a powerful tool, which can help them understand the most complex behavior of structural systems. One very important by-product of reliability They help designers to understand the calculations, are the sensitivity factors. importance of different design variables and various modes of failure, which the system might undergo when considering different strength and geometric properties. Linking concrete design programs such as TIN and the state-of-the-art program SHELL474 to reliability evaluation programs like RELAN, makes it possible to deal with the influence of the variability of all intervening variables on the theoretical strength. In order to conduct probability-based design calculations, basic information on each random variable, such as the probability distribution and estimates of the mean and standard deviation are needed. While frequently there are sufficient data to obtain reasonable estimates of the probability distributions, convenience must be used in many other cases. 94 physical argument and CHAPTER 7 Concluding Remarks and Further Study Further study seems to be needed in case of the concrete compression strength, because in the work by Mirza, Hatzinikolas and MacGregor [38] it is suggested that the mean value should be adjusted for rate of loading, when at the same time the increase due to additional hydration is ignored, and the actual concrete compression strength is usually 10-15% higher than the specified value. In the study how different spans and steel ratios effect the reliability of a beam, the suggestions by Mirza, Hatzinikolas and MacGregor were used. The results indicate that the code limits for maximum reinforcement ratio to prevent premature concrete compression failure is too high when the variability of intervening variables is taken into account. While the statistical data for applied load, i.e. gravity, snow and service load, seems to be well established, and easily applied through the code equation for the beam case, following needs to be considered to establish the probabilistic ice impact in order to use it as a load on wall elements: i. The ice impact is influenced by the variability of its thickness, diameter, velocity and compression strength, all affecting the contact area. ii. It can be assumed that there is a correlation between the ice parameters even though there is no data which confirms it. iii. Based on test results, the ice compression strength depends on contact area and decreases with increase in contact area [47]. 95 CHAPTER 7 Concluding Remarks and Further Study Based on this, it can be seen that increasing penetration means increasing contact area and therefore changes in the eight sectional forces, i.e. the two normal forces N and N, a membrane shear force N, two flexural bending moments M and M, a torsional bending moment M, as well as two transverse shear forces V and l. The runs made in this study did not account for the change in element sectional forces. The ratios of the eight sectional forces (stress resultants), which were calculated using finite element program COSMOS, were assumed to be constant as the ice impact and the penetration progressed. As a recommendation for further study, a finite element program such as COSMOS needs to be linked to both the modified SHELL474 and P.ELAN in order to take the variability of ice impact and other loads on reliability calculation of wall elements into account. 96 Bibliography 1. Adebar, P., and Collins, M. P., (July 1992) “Shear Design of Concrete Qffshore Structures”, ACI Structural Journal. 2. Allen, D. E., (Sept. 1968) “Choice of Failure Probabilities”, ASCE, J. Structural Div., V94, ST9, pp. 2169-2173. 3. Allen, D. E., (Dec. 1970) “Probabilistic Study of Reinforced Concrete in Bending”, ACI Journal, V67, pp. 989-993. 4. Allen, D. E., (1975) “Limit States Design-A Probabilistic Study”, CJCE, V2, No.1, pp. 36-49. 5. Allyn, N., Yee, S., and Adebar, P., (June 1992) “A Verification Study of the New Canadian Standard for Concrete Qffshore Structures”, Proceedings of the 11 .th International Conference on Offshore Mechanics and Arctic Engineering. 6. Ang, A. 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Collins, M. P., and Mitchell, D., (1987) “Prestressed Concrete Basics”, l.st. edition, Published by CPCI. 14. Cornell, C. A., (Dec. 1969) “A Probability-Based Structural Code”, ACI Journal, V66, No.12, PP. 974-985. 15. Costello, J. F., and Chu, K., (Oct. 1969) “Failure Probabilities of Reinforced Concrete Beams”, ASCE, J. Structural Div., V95, No. ST1O, pp. 2281-2304. 16. Ellingwood, B., (April 1979) “Reliability of Current Reinforced Concrete Designs”, ASCE, J. Structural Div., V105, No. ST4, pp. 669-713. 17. Ellingwood, B., (April 1979) “Reliability Based Criteria for Reinforced Concrete Design”, ASCE, J. Structural Div., V105, No. ST4, pp. 713-727. 18. Fardis, M. N., Cornell, C. A., and Meyer, J. E., (Jan. 1979) “Accident and Seismic Containment Reliabili”, ASCE, J. Structural Div., V105, No. ST1, pp. 67-83. 19. Foschi, R. 0., (1988) “User Manual: RELAN ‘RELiabilitv ANalysis)”, Civil Engineering Department, University of British Columbia, Vancouver, Canada. 20. Foschi, R. 0., Folz, B. R., and Yao, F. Z., (1989) “Reliability-Based Design of Wood Structures”, Structural research series, Report No.34, Department of Civil Engineering University of British Columbia Vancouver Canada. 21. Freudentahi, A. C., (1956) “Safety and the Probability of Structural Failure”, ASCE, Transactions, V121, pp. 1337-1375. 22. Galambos, T., Ellingwood, B., MacGregor, J. G., and Cornell, C. A., (May 1982) “Probability Based Load Criteria: Assessment of Current Design Practice”, ASCE, J. Structural Div., V108, No. ST5, pp. 959-977. 23. Galambos, T., Ellingwood, B., MacGregor, J. G., and Cornell, C. A., (May 1982) “Probability Based Load Criteria: Load Factors and Load Combinations”, ASCE, J. Structural Div., V108, No. ST5, pp. 978-997. 24. Grant, L. H., Mirza, S. A., and MacGregor, J. G., (August 1978) “Monte Carlo Study of Strength of Concrete Columns”, ACT Journal, V75, No.8, pp. 348-358. 98 Bibliography 25. Harris, H. G., Sabnis, G. M., and White, R. N., (Sept. 1966) “Small Scale Direct Models of Reinforced and Prestressed Concrete Structures”, Report No. 326, School of Civil Engineering, Cornell University, Ithaca, N.Y. 26. Hasofer, A. M., and Lind, N., (Feb. 1974) “An Exact and Invariant First Order Reliability Format”, ASCE, 3. Engineering Mechanics, V100, No. EM1, pp. 111121. 27. Jordaan, I. J., and Associates Inc. St. John’s Newfoundland (March 1988) “Tii Application of AES Environmental Data to Ice Interaction Modeling”, Submitted to Canadian Climate center Atmospheric Environment Service Environment Canada. 28. Jordaan, I. J., and Maes, M. A., (Oct. 1991) “Rationale for Load Specifications and Load Factors in the new CSA Code for Fixed Qffshore Structures”, CJCE, V18, pp. 454-464. 29. Kent, D. C., and Park, R., (July 1971) “Flexural Members with Confined Concrete”, ASCE, I. Structural Div., V97, No. ST7, pp. 1969-1990. 30. Lind, N. C., (June 1971) “Consistent Partial Safety Factors”, ASCE, J. Structural Div., V97, No. ST6, pp. 1651-1669. 31. MacGregor, J. G., (Dec. 1976) “Safety and Limit States Design for Reinforced Concrete”, CJCE, V3, No.4, pp. 484-513. 32. MacGregor, 3. 0., (July-August 1983) “Load and Resistance Factors for Concrete Design”, ACI Journal, V80, No.4, pp. 279-287. 33. Madsen, H. 0., Krenk, S., and Lind, N. C. (1986) “Methods of Structural Safety”, Published by Prentice-Hall, Inc., Englewood Ciffs, New Jersey. 34. Maes, M. A., (March 1986) “Study of a Calibration of the new CSA Code for Fixed Qfi’shore Structures”, Technical Report No.7 For Environmental Protection Branch Canada Oil and Gas Lands Administration. 35. Mirza, S. A., and MacGregor, J. G., (April 1979) “Variations in Dimensions of Reinforced Concrete Members”, ASCE, J. Structural Div., V 105, No. ST4, pp. 751-765. 99 Bibliography 36. Mirza, S. A., and MacGregor, J. G., (May 1979) “Variability of Mechanical Properties of Reinforcing Bars”, ASCE, J. Structural Div., V105, No. ST5, pp. 921-937. 37. Mirza, S. A., and Skrabek, B. W., (August 1991) “Reliability of Short Composite Beam-Column Strength Interaction”, ASCE, J. Structural Div., Vi 17, No.8, pp. 2320-2339. 38. Mirza, S. A., Hatzinikolas, M., and MacGregor, J. G., (June 1979) “Statistical Descriptions of Strength of Concrete”, ASCE, J. Structural Div., Vi05, No. ST6, pp. 1021-1037. 39. Mirza, S. A., Kikuchi, D. K., and MacGregor, 3. G., (July-August 1980) “Flexural Strength Reduction Factor for Bonded Prestressed Concrete Beams”, ACI Journal, V77, No.4, pp. 237-246. 40. Mirza, S. M., (1967) “An Investigation of Combined Stresses in Reinforced Concrete Beams”, thesis presented to McGill University, at Montreal, Canada, in partial fulfillment of the requirements for the degree of Ph.D. 41. Pahl, P. J., and Soosaar, K., (Feb. 1964) “Structural Models for Architectural and Engineering”, Massachusetts Institute of Technology, Cambridge, Mass. 42. Park, R., and Paulay, T., (1975) “Reinforced Concrete Structures”, Wiley Interscience Publication, John Wiley and Sons, New York. 43. Pillai, S. U., and Kirk, D. W., (1988) “Reinforced Concrete Design”, 2.nd. edition, Published by McGraw-Hill Ryerson Limited. 44. Preliminary Standard S471-M1989 (May 1989) “General Requirements Design Criteria, the Environment, and Loads”, Published by Canadian Standards Association. 45. Proceedings from 2.nd. International Conference “Application of Statistics and Probability in Soil and Structural Engineering”, 15.th-18.th Sept. 1975 in Aachen F.R. Germany, V1-V3, Published by Deutsche Gesellschaft fur Erd-und Grundbau e.V. 46. Sabnis, G. M., and Mirza, S. M., (June 1979) “Size Effects in Model Concretes?”, ASCE, 3. Structural Div., Vi05, No. ST6, pp. 1007-1021. 100 Bibliography 47. Sanderson, T. J. 0., (1988) “Ice Mechanics Risks to Offshore Structures”, Published by Graham and Trotman Inc. 48. Schueller, G. I., and Shinozuka, M., (1987) “Stochastic Methods in Structural Dynamics”, Published by Martinus Nijhoft Publishers, Boston. 49. Vecchio, F. J., and Collins, M. P., (March-April 1986) “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear”, ACI Journal, V83, No.2, pp. 219-231. 50. Warner, R. F., and Kabaila, A. P., (Dec. 1968) “Monte Carlo Study of Structural Safety”, ASCE, J. Structural Div., V94, No. ST12, pp. 2847-2859. 51. Zech, B., and Wittmann, F. H., (Sept.-Oct. 1980) “Variability and Mean Value of Strength of Concrete as Function ofLoad”, ACI Journal, V77, No.5, pp. 358363. 101 Appendix Al Beam Program Subroutines Appendix Al C $debug SUBROUTINE MAIN C C MAIN.FOR C C FY-YIELD STRENGTH OF STEEL (N/mm ) 2 C FP-YIELD STRENGTH OF LOW RELAXATION STEEL (N/mm ) 2 C FCPP-PEAK STRESS OBTAINED FROM A CYLINDER TEST (N/mm ) 2 C ECT-INITIAL TANGENT MODULUS (N/mm ) 2 C FCR-LONGITUDINAL STRESS (N/mm ) 2 C DEP-SRAIN DIFFERENCE C ES-MODULUS OF ELASTICITY FOR STEEL (N/mm ) 2 C EP-MODULUS OF ELASTICITY FOR LOW RELAXATION STEEL (N/mm ) 2 C NLAY-NUMBER OF LAYERS C TOL-SPECIFIED TOLERANCE ON AXIAL LOAD (N) C A(I)-CROSS SECTIONAL AREA OF CONCRETE (mm ) 2 C A(I)-CROSS SECTIONAL AREA OF STEEL (mm ) 2 C A(I)-CROSS SECTIONAL AREA OF TENDON (mm ) 2 C Z(I)-LOCATION OF THE FORCE (mm) C MTYP(I)-TYPE OF MATERIAL C ICURV-INITIAL CURVATURE (rad/mm) C CSTEP-CURVATURE STEP C CMAX-MAXIMUM ALLOWABLE CURVATURE C AXL-AXIAL FORCE (N) C MCAP-MOMENT CAPACITY OF THE SECTION (Nmm) IMPLICIT REAL*8 (A-H,O-Z) REAL*8 MCAP, ICURV, ISTRN,MISM INTEGER , COUNT2 COMMON/LAYER/A(50) ,Z(50) ,MTYP(50) COMMON/B /NLAY COMMON/B2/PHIC, PHIS, PHIP COMMON/B3 /MCAP COMMON/ B4 /AXL COMMON/B7 /COUNT, COUNT2 COMMON/BlO/BETA1, FPCN, FYN, BN, ASN, DEFFN C TOLERANCE FOR MAXIMUM MOMENT CAPACITY MISM = 0.001 C INITIALIZE COUNTER AND BENDING MOMENTS 1=0 BEND1 = 0.0D0 MCAP = 0.ODO C INITIALIZE STRAIN ISTRN = 0.ODO STRN = ISTRN C INITIALIZE CURVATURE ICURV = 0.000 CURV = ICURV CURVATURE STEP INITIALIZED CSTEP = 1.0 C C C C C MAXIMUM CURVATURE FOR THE SECTION EC = 0.003,ES = 0.02 THEN CMAX IS MULTIPLIED WITH 10 TO MAKE SURE THE PROGRAM IS GOING TO RUN FOR HIGH ENOUGH CURVATURE 103 Appendix Al CMAX = (0.023/(DEFFN))*1E7 IF (AXL NE 0. 0) THEN TOL = 0.01*AXL ELSE TOL = 0.001*FPCN*BN*DEFFN END IF 2 CURV = CURV + CSTEP CALL ITER(CURV,STRN,AXL,BEND,TOL,NN) 5 10 IF(BEND.GT.0.000)GOTO 5 CSTEP = CSTEP/2 CURV = 0.ODO GOTO 2 CURV = 0.000 IF(CURV.LE.CMAX)THEN CURV = CURV + CSTEP CALL ITER(CURV,STRN,AXL,BEND,TOL,NN) BEND2 = BEND 1=1+1 IF(I.GE.2)GOTO 15 SLOPE1 = (BEND2-BEND1)/CSTEP BEND 1 = BEND2 GOTO 10 15 SLOPE2 = (BEND2-BEND1)/CSTEP P = SLOPE1*SLOPE2 IF(P.LE.0.ODO)GOTO 50 SLOPE 1 = SLOPE2 BEND1 = BEND2 GOTO 10 50 IF(DABS(BEND1-MCAP).LE. (MISM*MCAP))GOTO 100 MCAP = BEND1 2.ODO*CSTEP CURV = CURV CSTEP = CSTEP/2.ODO - CALL ITER(CURV,STRN,AXL,BEND,TOL,NN) BEND1 = BEND 1=0 GOTO 10 100 CURV 110 IF(COUNT2 .LE.43)THEN OPEN(UNIT=4,FILE=’gis.out’ ,ACCESS=’APPEND’ WRITE(4, 110)CURV,MCAP FORMAT(’ ,2X,F15.5,SX,F25.5) COUNT2 = COUNT2 + 1 CLOSE (4) ENDIF = CURV - CSTEP 104 , STATUS=UNKNOWN) Appendix Al 120 ELSE OPEN(UNIT=4,FILE=’gis.out ,ACCESS=APPEND ,STATUS=’UNKNOWN) WRITE(4,120)’CURVATURE EXCEEDED CMAX’ FORMAT(’ ‘,A) CLOSE (4) ENDIF RETURN END C C C C C STRESS.FOR SUBROUTINE WHICH CALCULATES THE STRESS FOR A GIVEN STRAIN SUBROUTINE STRES (STRN, MTYP, STRS) IMPLICIT REAL*8(A_H,O_Z) COMMON/B1/FCPP, FY, FP,ECT, FCR, ES, EP COMMON/B6/DEP C FCP IS NEGATIVE FCP = (_1.0)*(DABS(FCPP)) C CONCRETE, C CONCRETE WITH TENSION STIFFENING IF(MTYP.EQ. 1)THEN ECR = FCR/ECT ECP = 2.0*FCP/ECT ECF = 2.0*ECP IF(STRN.LE.ECF)THEN STRS = 0.0 ELSE IF(STRN.GT.ECF.AND.STRN.LE. 0.0)THEN STRS = FCP*((2.0*STRN/ECP)_(STRN/ECP)**2) ELSE IF(STRN.GT.0.0.AND.STRN.LE.ECR)THEN STRS = STRN*ECT ELSE IF(STRN.GT.ECR)THEN STRS = FCR/(1+SQRT(500.0*STRN)) END IF C CONCRETE WITHOUT TENSION STIFFENING ELSEIF(MTYP .EQ. 2 )THEN ECR = FCR/ECT ECP = 2.0*FCP/ECT ECF = 2..0*ECP IF(STRN.LE.ECF)THEN STRS = 0.0 ELSE IF(STRN.GT.ECF.AND. STRN.LE.0.0)THEN STRS = FCP*((2.0*STRN/ECP)_(STRN/ECP)**2) ELSE IF(STRN.GT. 0.0.AND. STRN.LE.ECR)THEN STRS = STRN*ECT ELSE IF(STRN.GT.ECR)THEN ELSE IF(STRN.GT.0.0)THEN STRS = 0.0 ENDIF C C C STEEL, PRESTRESSED STEEL 105 Appendix Al C REINFORCEING STEEL ELSEIF(MTYP. EQ. 3)THEN IF(STRN.LE.-0.002)THEN STRS = -FY ELSE IF(STRN.GT.-0.002.AND.STRN.LE.0.002)THEN STRS = ES*STRN ELSE IF(STRN.GT.0.002)THEN STRS = FY ENDIF C PRESTRESSED STEEL ELSE IF(MTYP.EQ.4)THEN STRNP = STRN+DEP IF(STRNP.LE.-0.008)THEN STRS = -FP ELSE IF(STRNP.GT.-0.008.AND.STRNP.LE.0.0)THEN STRS = EP*STRNP ELSE IF(STRNP.GT.0.0)THEN STRS = STRNP*EP*(0.025+0.975/(1+(118*STRNP)**10)**0.10) ENDIF IF(STRS.GT.FP)THEN STRS = 0.0 ENDIF ENDIF RETURN END C C C ITER.FOR * * * * * * * * * * * * * * * * * * * Subroutine * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ITER * * * * * * * C* C* C* Version 1.10 Written by February 10, P.E. Adebar 1990 * * ITERates for the strain at the centroid which results in the required axial load at a specified curvature. Calls subroutine FORMOM to calculate the axial load and bending moment associated with a given curvature and strain at the centroid. Input Variables: specified CURVature (rad/mm) CURV STRaiN at the section centroid STRN found for the previous curvature AXL required AXiaL load (N) specified TOLerance on the axial TOL load (kN) Output Variables: STRN STRaiN at the centroid BEND BENDing moment associated with the specified curvature and axial load Number of iterations NN Description: C* C* c* C* C’ C* C* C C* C* * * * * * * * * * * * * * * * * * * C* * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE ITER(CURV,STRN,AXL,BEND,TOL,NN) IMPLICIT REAL*8(A_H,O_Z) 106 * * * * * * * * * * * Appendix Al C C C Set strain increment DSTRN = 0.0005 Set limit on maximum iterations MAXNN = 250 Check that tolerance is not zero TOL = DABS(TOL) IF(TOL .LT. l.OE-l0)THEN WRITE(*,*)’**** Specified TOLerance Too Small STOP ENDIF NN = 0 C First try to bound solution 10 NN = NN+1 IF(NN .GT. MAXNN)THEN WRITE(*,*)’**** No Solution Found ****1’ C BEND = 0.0 STRN = 0.0 GOTO 1000 END IF CALL FORMOM ( CURV, STRN, FOR, BEND) C PDIF = DIF DIF = AXL-FOR Check tolerance just in case IF(DABS(DIF) .LE. TOL)GOTO 1000 C Decide on direction to increment strain IF(FOR .LT. AXL)THEN KK = +1 ELSE KK = -1 ENDIF C If first iteration step IF(NN .EQ. 1)THEN PSTRN = STRN STRN = STRN + KK*DSTRN GOTO 10 END IF Check if solution bounded IF(PDIF*DIF)20, 21,22 WRITE(*,*) ‘‘ No Solution Found ****2’ C 21 BEND = 0.0 STRN = 0.0 GOTO 1000 PSTRN = STRN 22 STRN = STRN + KK*DSTRN GOTO 10 C C Solution is now bounded 20 Xl = PSTRN X2 = STRN Fl = PDIF F2 = DIF 30 NN = NN+1 107 ****‘ Appendix Al WRITE(*,*)’**** No Solution Found ****3 IF(NN .GT. MAXNN)THEN BEND = 0.0 STRN = 0.0 GOTO 1000 ENDIF C C Linearly interpolate for new guess X3 = X2_((X2_Xl)*F2/(F2_F1)) STRN = X3 CALL FORMOM ( CURV, STRN,FOR, BEND) F3 C = AXL-FOR Check tolerance IF(DABS(F3) .LT. TOL)GOTO 1000 F13 = F1*F3 IF(F13 .LT. 0)THEN X2 = X3 F2 = F3 ELSE Xl = X3 Fl = F3 END IF GOTO 30 1000 C C C C RETURN END FORMOM.FOR SUBROUTINE WHICH CALCULATES THE FORCE AND MOMENT SUBROUTINE FORMOM(CURV, STRAIN,AXL,BEND) IMPLICIT REAL*8(A_H,O_Z) COMMON/LAYER/A( 50 ) ,Z C 50) , MTYP ( 50) COMMON/B/NLAY STRN = 0.0 AXL = 0.0 BEND = 0.0 DO 10 I = 1,NLAY IF(CURV.EQ.0.0)THEN STRN = STRAIN ELSE STRN = STRAIN_CURV*Z(I)/1E6 END IF CALL STRES(STRN,MTYP(I) ,STRS) AXL = A(I)*STRS+AXL BEND= (_1.0)*Z(I)*A(I)*STRS+BEND 10 CONTINUE RETURN END 108 Appendix Al CSUBROUTINE DETERM (IMODE) C C AXL-AXIAL LOAD C GAMMA-RATIO BETWEEN NOMINAL DEAD AND LIVE LOAD PHIC-MATERIAL FACTOR FOR CONCRETE C C PHIS-MATERIAL FACTOR FOR STEEL C BETA1-STRESS BLOCK FACTOR C DEP-STRAIN DIFFERENCE BETWEEN CONCRETE AND TENDONS C FPCN-NOMINAL COMPRESSION STRENGTH OF CONCRETE C FYN-NOMINAL YIELD STRENGTH OF STEEL C BN-NOMINAL SECTION WIDTH C ASN-NOMINAL TOTAL STEEL AREA DEFFN-NOMINAL EFFECTIVE DEPTH C IMPLICIT REAL*8 (A H, 0 Z) REALL*8 LN,N1,N2,N3,N4 INTEGER COUNT, COUNT 2 CHARACTER*5 PASS COMMON! B/NLAY COMMON/B2 /PHIC, PHIS, PHIP COMMON/B 4 /AXL COMMON/B5/DN,LN, SN,N1,N2,N3,N4 COMMON/B6/DEP COMMON/B 7/COUNT, COUNT2 COMMON/BlO/BETA1 , FPCN, FYN, BN, ASN, DEFFN COUNT = 1 COUNT2 = 1 DO 20 TALA = 1,3 CALL CLS CALL MOVCUR(3,0) - WRITE (* * (* * ) ****** **** ********************* ****** *** ************ ) ******* ********************************************* ) ***************************************************** WRITE(*,*)’ WRITE 20 30 - PLEASE ENTER YOUR PASSWORD READ(*, ‘(A5) )PASS IF( (PASS.EQ. SIGGA’ ) .OR. (PASS.EQ. ‘sigga ) )GOTO 30 WRITE(*,*)CHAR(7) WRITE ( *, * ) CHAR( 7) WRITE(*,*)CHAR(7) CONTINUE WRITE(*,*)’ ### ACCESS DENIED ### STOP CALL CLS CALL MOVCUR (3,0) OPEN(UNIT=3,FILE=’gisli.in ,STATUS=OLD’) OPEN(UNIT=4,FILE=gis.out ,STATUS=’UNKNOWN’) WRITE (4, * WRITE(4,*)’ CURV MCAP WRITE (4, *) ***************************************************** CLOSE (4) OPEN(UNIT=7,FILE=gis.con ,STATUS=’UNKNOWN’) WRITE ( 7 , * ) ‘ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * WRITE(7,*)’ MCAP MCODE GXP WRITE ( 7 * ) ‘ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * CLOSE (7) READ(3, *)DN,LN,SN,N1,N2,N3,N4 READ(3, *)PHIC, PHIS,PHIP, BETA1 READ(3, *)NLAY,DEP,AXL READ(3, *)FPCN,FYN,BN,ASN,DEFFN 109 Appendix Al 100 WRITE ( * * ) • * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * WRITE(*, 100) RELIABILITY ANALYSIS OF CONCRETE BEAMS FORMAT( ) WRITE(*,*) RETURN END C SUBROUTINE GFUN C C C C C C C C C C C C C (X, N, IMODE, GXP) B-BEAM WIDTH H-BEAM HEIGHT C-CONCRETE COVER PHIl-DIAMETER OF STIRRUP PHI2-DIAMETER OF STEEL IN THE 1.ST LAYER PHI3—DIAMETER OF STEEL IN THE 2.ND LAYER El-DISTANCE BETWEEN 1.ST AND 2.ND LAYER DR-RATIO BETWEEN ACTUAL DEAD LOAD AND NOMINAL LR-RATIO BETWEEN ACTUAL LIVE LOAD AND NOMINAL ATOT-TOTAL AREA WY-SECTION MODULUS FOR THE SECTION YO-CENTER OF GRAVITY FOR THE SECTION IMPLICIT REAL*8 (A-H,O-Z) REAL*8 LR,LN,MCAP,MCODE,MULT,N1,N2,N3,N4 , COUNT2 INTEGER DIMENSION X(N) COMMON/LAYER/A(50) , Z(50) ,MTYP(50) COMMON/B INLAY COMMON/B1/FCPP, FY, FP, ECT, FCR,ES , EP COMMON/B2/PHIC, PHIS, PHIP COMMON/B 3 /MCAP COMMON/B4/AXL COMMON/B5 /DN , LN , SN, Ni, N2 , N3 , N4 COMMON/B7 /COUNT, COUNT2 COMMON/B10/BETA1, FPCN, FYN, BN, ASN, DEFFN FCPP = X(l) FY = X(2) FP = X(3) ECT = X(4) FCR = X(5) ES = X(6) EP = X(7) B = X(8) H = X(9) C = X(l0) PHIl = X(ll) PHI2 = X(12) PHI3 = X(13) PHI4 = X(14) PHI5 = X(15) El = X(l6) E2 = X(17) E3 = X(l8) DR = X(19) SR = X(20) LR = X(21) 110 Appendix Al Al = Nl*(22/7)*X(12)**2/4 A2 = N2*(22/7)*X(13)**2/4 A3 = N3*(22/7)*X(14)**2/4 A4 = N4*(22/7)*X(15)**2/4 ATOT = X(8)*X(9)+Al+A2+A3+A4 WY =Al*(X(lO)+X(ll)+X(12)/2) WY = WY+A2*(X(l0)+X(ll)+X(12)+X(16)+X(13)/2) WY = WY+A3*(X(l0)+X(ll)+X(12)+X(13)+X(16)+X(17)+X(14)/2) WY = WY+A4*(X(l0)+X(ll)+X(12)+X(13)+X(14)+X(16)+X(17)+X(18)+ + X(15)/2)+X(8)*X(9)**2/2 YO = WY/ATOT Z(l)=(X(lO)+X(ll)+X(12)/2)—YO A(l) = Al MTYP(l) = 3 Z(2)=(X(lO)+X(ll)+X(12)+X(l6)+X(13)/2)—Y0 A(2) = A2 MTYP(2) = 3 Z(3)=(X(l0)+X(ll)+X(12)+X(13)+X(16)+X(17)+X(14)/2)—Y0 A(3) = A3 MTYP(3) = 3 Z(4)=(X(l0)+X(ll)+X(12)+X(13)+X(14)+X(16)+X(17)+X(18)+X(15)/2)—Y0 A(4) = A4 MTYP(4) = 3 Z(5)=—YO/4 A(5) = X(8)*Y0/2 MTYP(5) = 2 Z(6)=_3*Y0/4 A(6) = X(8)*Y0/2 MTYP(6) = 2 Z(7)=(X(9)—YO)/4 A(7) = X(8)*(X(9)_Y0)/2 MTYP(7) = 2 Z(8)=3*(X(9)_Y0)/4 A(8) = X(8)*(X(9)_Y0)/2 MTYP(8) = 2 Md 1 2 ((BETA1*ASN*PHIS*FYN)/(0.85*BETA1*PHIC*FPCN*BN)) = CODE = (0. 85*PHIC*FPCN*BN*Mdl* (DEFFN—MCl/2 .0) * ((X(19)*(DN/(SN+LN))*((SN+LN)/SN)+X(20)+(LN/SN)*X(21) (l.25*(DN/(SN+LN))*((SN+LN)/SN)+1.5*(l+LN/SN)))) MCODE = )/ CODE/1E6 CALL MAIN MULT GXP 210 = = MCAP/1E6 MULT-MCODE IF(COUNT.LE.43)THEN OPEN(UNIT=7,FILE=gis.con’ ,ACCESS=APPEND ,STATUS=UNKNOWN’) WRITE(7, 210)MULT,MCODE,GXP FORMAT(’ ‘,F15.4,3X,F15.4,3X,Fl5.4) COUNT = COUNT + 1 CLOSE (7) END IF RETURN END 111 ‘IppendLr Al _ C E DFUN (X, N, IMODE, DELTA) SUBROUTIN C H, 0 Z) IMPLICIT REAL*8 (A DIMENSION X(N), DELTA(N) RETURN END - - C SUBROUTINE D2FUN (X, N, IMODE, D2, C IMPLICIT REAL*S (A H, 0 DIMENSION X(N), D2(N2,N2) RETURN END - - 2) C 112 N2) Appendix A2 Modified Main Subroutine in SHELL474 Appendix A2 CModified SHELL474 Version C 3.00 Jan. 1992 C SUBROUTINE SKEL(FC,FCRK,EC,ECP,AGG,THICK,DV, 1 DIAX,ASX,SPX,ZSX,FYX,DEPSX, 2 DIAY,ASY,SPY,ZSY,FYY,DEPSY, 3 RHOSZ,ROWSZ,DIAZ,NASX,NASY,SPZX,SPZY,FYZ, 4 RNXCON, RNYCON, RNXYCON, RMXCON, RMYCON, RMXYCON, RVXCON, RVYCON, 5 RNX,RNY,RNXY,RMX,RMY,RMXY,RVX,RVY, 6 PHIC,PHIS,PHIP, 7 IPRINT, B RESULT) C REAL*4 FC,FCRK,EC,ECP,AGG,THICK,DV, • DIAX(10),ASX(10),SPX(10),ZSX(10),FYX(10),DEPSX(10), DIAY(10),ASY(10),SPY(10),ZSY(10),FYY(10),DEPSY(10), RHOSZ ,ROWSZ ,DIAZ, SPZX, SPZY, FYZ, RNXCON, RNYCON, RNXYCON, RMXCON, RMYCON, RMXYCON, RVXCON, RVYCON, RNX , RNY I RNXY , RMX , RMY , RMXY , RVX , RVY PHIC, PHIS, PHIP, • RESULT(8) INTEGER NASX, NASY, IPRINT CHARACTER* 35 SPNAME C C Local variable declaration Common blocks REAL t, epsc0, fcp, epscr, fcr, Eec COMMON /sepcon/ t, epsco, fcp, epscr, INTEGER Nnlay, nip REAL zc(21), dz(20) COMMON /seplay/ Nnlay, nip, INTEGER nas(3) REAL deps(3,10), as(3,10), COMMON /sepste/ nas, deps, zc, fcr, Eec dz es(3,10), fy(3,10), as, es, fy, zs zs(3,10) REAL rhos(2), rhoc(2), sm(2), smodc, fcrack, magg, ffc COMMON /fcl/ rhos, rhoc, Sm, smodc, fcrack, magg, ffc REAL RROWCX, RROWCY,RROWCZ, SSMX, SSMY, SSMZ, RROWSX, RROWSY, RROWSZ, FFYZ, FFFC, SSMODC, EECP, FFCR, AAGG COMMON /SPC/ RROWCX, RROWCY, RROWCZ, SSMX, SSMY, SSMZ, RROWSX, RROWSY, RROWSZ , FFYZ, FFFC, SSMODC, EECP, FFCR, AAGG REAL DDV COMMON /CHK/ DDV integer*2 ikey common /key/ ikey REAL*4 GAMC,ISS(9) ,RNN(8) ,BSS(9) ,BNN(8) ,ACC, SPNN(8),RNNMAX(8),DNN(8),ISSMAX(9), STORN ( 8, 1000) , STORS (9, 1000) , DATA ( 28) SPNCON(8) ,RNNP(7) 114 Appendix A2 LOGICAL*2 GOTS8 INTEGER IN(5),MAXIT,NDB CALL REZERO() C Program version number vernum=4. 00 DDV=DV IF(IPRINT.EQ. 1)THEN OPEN(l1,FILE=SHELL.OUT’ ,ACCESS=APPEND ENDIF 20 NTYPE=1 C Apply material resistance factors FC=FC*PHIC DO 42 I=l,NASX IF(DEPSX(I) .EQ. 0.)THEN FYX(I)=FYX(I)*PHIS ELSE FYX(I)=FYX(I)*PHIP ENDIF 42 CONTINUE DO 43 I=l,NASY IF(DEPSY(I) .EQ. 0.)THEN FYY(I)=FYY(I)*PHIS ELSE FYY(I)=FYY(I)*PHIP ENDIF 43 CONTINUE FYZ=FYZ*PHIS CALL PARA(NASX,DIAX,ASX,SPX,ZSX, NASY,DIAY,ASY,SPY, ZSY, ROWSZ, DIAZ, SPZX, SPZY, THICK, ROWSX, ROWSY , ROWCX, ROWCY, ROWCZ, SMX, SMY, SMZ, SMXTOP, SMXBOT, SMYTOP, SMYBOT) Output data DATA(l)=FC DATA(2)=FCRK DATA(3)=EC DATA(4)=ECP DATA (5) =AGG DATA (6) =THICK DATA( 7) =ROWSZ DATA(8)=DIAZ DATA(9)=SPZX DATA(l0)=SPZY DATA(ll)=FYZ DATA(12 )=DV DATA (13) =ROWCX DATA ( 14) =ROWCY DATA( 15) =ROWCZ DATA(16)=SMX DATA(17)=SMY 115 ,STATUS=UNKNOWN’) Appendix A2 DATA(18)=SMZ DATA( 19) =SMXTOP DATA(20)=SMXBOT DATA (21 ) =SMYTOP DATA( 22) =SMYBOT DATA(23)=DELFYX DATA(24)=DELFYY DATA(25)=DELFYZ DATA(26)=ROWSX DATA(27)=ROWSY SPNAME=’RELIABILITY OF CONCRETE ELEMENTS’ IF(IPRINT.EQ. 1)THEN CALL OUTPUT ( SPNAME , NASX, NASY, DATA, DIAX,ASX, SPX, ZSX,FYX,DEPSX, DIAY,ASY,SPY, ZSY,FYY,DEPSY) ENDIF C Set switches NDB=0 ACC=. 01 MAXIT=10 C Common /sepcon/ T=THICK FCP—FC FCR=FCRK C Common /nlay/ IF(RMX .NE. 0. • RMY •NE. 0. • RMXY .NE. 0. • RMXCON .NE. • RMYCON .NE. • RMXYCON .NE. NNLAY=9 ELSE NNLAY= 1 END IF .OR. .OR. .OR. 0. .OR. 0. .OR. 0.)THEN Common /sepste/ NAS(1)=NASX NAS(2)=NASY NAS(3)=0. DO 10 I=1,NASX AS(1, I)=ASX(I)/SPX(I) FY(1,I)=FYX(I) ZS(1,I)ZSX(I) DEPS(1, I)=DEPSX(I) ES(1, I)=200000. CONTINUE 10 DO 11 I=1,NASY AS(2, I)=ASY(I)/SPY(I) FY (2 , I) =FYY (I) ZS(2,I)=ZSY(I) DEPS(2,I)=DEPSY(I) ES(2, I)=200000. CONTINUE 11 C 116 Appendix A2 C Common /fcl/ RHOS(1)=ROWSX RHOS(2)=ROWSY RHOC(1)=ROWCX RHOC (2 ) =ROWCY SM(1)=SMX SM(2)=SMY SMODC=EC FCRACK=FCRK MAGG=AGG FFC=FC C Common /SPC/ RROWCX=ROWCX RROWCY=ROWCY RROWCZROWCZ SSMX=SMX SSMY=SMY SSMZ=SMZ RROWSX=ROWSX RROWSY=ROWSY RROWS Z=R0WS Z FFYZ=FYZ FFFC=FC SSMODC=EC EECP=ECP FFCR=FCRK AAGG=AGG CALL SEPINI( C Common /sepcon/ EPSCO=ECP EEC=EC C Convert to notation and units of loading used by subprogram SEP SPNN(l)=RNX SPNN(2)RNY SPNN(3)=RNXY SPNN(4)=_RNX*l000. SPNN(5)=_RMY*1000. SPNN(6)=_RNXY*l000. SPNN(7)=RVX SPNN(8)=RVY SPNCON (1) =RNXCON SPNCON (2 ) =RNYCON SPNCON (3) =RNXYCON SPNCON (4) =_RMXCON* 1000. SPNCON( 5 )=_RMYCON* 1000. SPNCON( 6 )=_RMXYCON* 1000. SPNCON (7) =RVXCON SPNCON (8) =RVYCON 1000 RLFO. NCON=0 NUMIT=0 DO 50 1=1,8 RNNMAX ( I ) =SPNCON ( I) ISSMAX(I)=0. 117 Appendix A2 50 51 200 52 CONTINUE ISSMAX(9)0. RK=0.5 DO 100 JJ=1,11 pJ(=pK*2. DLF1. /R1c PLF=RLF DO 51 1=1,8 DNN(I)=SPNN(I)/RK RNN ( I ) =RNNMAX ( I) ISS(I)=ISSMAX(I) CONTINUE ISS(9)=ISSMAX(9) NUMIT=NUMIT+1 PLF=PLF+DLF DO 52 1=1,8 RNN(I)RNN(I)+DNN(I) CONTINUE CALL SHELL(ISS,RNN,ACC,MAXIT,NDB,DV, BSS,BNN,GOTS8) IF (GOTS8 )THEN RLF=PLF Store load deformation data NCON=NCON + 1 DO 57 1=1,3 STORN(I,NCON)=BNN(I) STORS(I,NCON)=BSS(I) CONTINUE 57 DO 58 1=4,6 STORN(I,NCON)=_0.001*BNN(I) STORS(I,NCON)=—1000. *BSS(I) CONTINUE 58 DO 59 1=7,8 STORN(I,NCON)=BNN(I) STORS(I,NCON)=BSS(I) CONTINUE 59 STORS(9,NCON)=BSS(9) C Store data to iterate to the peak DO 53 1=1,8 RflNMAX(I)=RNN(I) ISSMAX(I)=BSS(I) CONTINUE 53 ISSMAX(9)=BSS(9) C 100 GOTO 200 END IF CONTINUE Create result file by reading data from STORN(I) DO 70 1=1,8 RESULT(I)=STORN(I,NCON) 70 CONTINUE C 118 Appendix A2 C Print load deformation data for ULS analysis IF(IPRINT.EQ. 1)THEN *******‘ WRITE(ll,*)’ ******* RESULTS FROM ULS ANALYSIS WRITE(ll,522)’ ‘, • ‘Membrane Forces (kN/m)’, ‘Membrane Strains (mm/mm)’ 61 WRITE(1l,524)’ ‘, • ‘LS’ , ‘Nx’ , ‘Ny’, ‘Vxy’ , ‘Ex,o’ , ‘Ey,o’ , ‘Gxy,o’ DO 61 I=l,NCON WRITE(ll,530)I,STORN(l,I),STORN(2,I),STORN(3,I), • STORS(l,I),STORS(2,I),STORS(3,I) CONTINUE 62 WRITE(ll,523)’ ‘, • ‘Bending Moments (kNm/m)’, ‘Curvatures (rad/m)’ WRITE(1l,524)’ ‘, • ‘LS’,’Mx’, ‘My’, ‘Txy’, ‘phix’, ‘phiy’, ‘phixy’ DO 62 I=1,NCON WRITE(11,530)I,STORN(4,I),STORN(5,I),STORN(6,I), • STORS(4,I),STORS(5,I),STORS(6,I) CONTINUE 63 534 WRITE(11,525)’ ‘, •‘Transverse Shears (kN/m)’,’Transverse Strains WRITE(l1,526)’ ‘, • ‘LS’, ‘Vxz’, ‘Vyz’, ‘Gxz’, ‘Gyz’, ‘Ez’ DO 63 I=1,NCON WRITE(11, 532)I,STORN(7, I) ,STORN(8, I), • STORS(7,I) ,STORS(8,I),STORS(9,I) CONTINUE write(11,*)’ write(11,*)’ write(11,534)’ Maximum Load Factor format(A,F8.3) if(ikey •eq. write(ll,*)’ write(11,*)’ write(11,*)’ endif CLOSE (11) 522 523 524 525 526 530 532 536 538 = (mm/mm)’ ‘,RLF 27)then >> Note Analysis Terminated by User FORMAT(//,A,9X,A,12X,A,/) FORMAT(//,A,9X,A,l5X,A,/) FORMAT(A,A,7X,A,1OX,A,9X,A,8X,A,9X,A,8X,A,/) FORMAT(//,A,7X,A,13X,A,/) FORMAT(A,A,8X,A, 11X,A, 15X,A, 9X,A, 11X,A, I) FORMAT(’ ‘,13,2X,F7.0,2(5X,F7.0),3(5X,F8.6)) FORMAT(’ ‘,I3,5X,F7.O,7X,F7.O,6X,3(5X,F8.6)) FORMAT(4X,A,7X,A,6X,A,7X,A,7X,A,6X,A,6X,A,6X,A,/) FORMAT(F7.0,7(2X,F7.O)) ENDIF RETURN END C 119 <<‘ Appendix A3 Subroutines to Link RELAN and SHELL474 t2-U Appendix A3 C $debug SUBROUTINE DETERM (IMODE) C OPTION FOR OUTPUT DATA C IPRINT PHIC MATERIAL FACTOR FOR CONCRETE C C PHIS MATERIAL FACTOR FOR STEEL PHIP MATERIAL FACTOR FOR TENDONS C AGG MAXIMUM SIZE OF AGGREGATE C NASX AND NASY C NUMBER OF LAYERS IN X AND Y-DIRECTION DEPSX AND DEPSY STRAIN DIFFERENCE IN X AND Y-DIRECTION C RHOSZ (ROWSZ) TRANSVERSE REINFORCEMENT RATIO IN Z-DIRECTION C RNX AND RNY AXIAL FORCES (kN/m) C RNXY SHEAR FORCE (kN/m) C RMX AND RMY BENDING MOMENTS (kNm/m) C RMXY TORSION (kNm/m) C RVX AND RVY TRANSVERSE SHEAR (kN/m) C VECTOR WHICH MEASURES THE DEMAND FOR THE STRUCTURE (kN/m) C DEMAND REAL*4 AGG,RHOSZ,ROWSZ,DEPSX(10) ,DEPSY(10) ,DV,SPZX,SPZY, • RNXCON, RNYCON, RNXYCON, RMXCON, RMYCON, RMXYCON, RVXCON, RVYCON, • PHIC,PHIS,PHIP, • RNX,RNY,RNXY,RMX,RMY,RMXY,RVX,RVY, • SCALE,DEMAND INTEGER IMODE, IPRINT, IPRINT2, ICOUNT,NASX, NASY, I COMMON/B/IPRINT, IPRINT2 ,PHIC, PHIS, PHIP COMMON/B1/AGG,NASX,NASY,DV,SPZX,SPZY,RHOSZ,ROWSZ COMMON/B2 /DEPSX, DEPSY COMMON/B3 /RNXCON, RNYCON, RNXYCON, RMXCON, RMYCON, RMXYCON 1 RVXCON,RVYCON COMMON/B4/RNX,RNY,RNXY,RMX,RMY,RMXY,RVX,RVY COMMON/B 5 /DEMAND COMMON/BlO/ICOUNT, SCALE ICOUNT=0 IF(IMODE.GT.1)GOTO 250 READ IN SECTIONAL INFORMATION C OPEN(4,FILE=’ALAG.DAT’ ,STATUS=OLD’) OPEN(7,FILE=INN.DAT’ ,STATUS=’OLD’) READ(7,*)IPRINT,IPRINT2,PHIC,PHIS,PHIP READ(7, *)AGG,DV,SPZX,SPZY READ(7, *)RHOSZ,NASX,NASY,SCALE DO 38 I=1,NASX READ(7,*)DEPSX(I) CONTINUE 38 DO 39 I=1,NASY READ(7,*)DEPSY(I) CONTINUE 39 CLOSE (7) - - - - - - - - - - - - - - ROWSZ=RHOSZ/1000000.0 C C C AXIAL FORCES: Nx,Ny (kN/m), SHEAR FORCE: Vxy BENDING MOMENTS: Mx,My (kNm/rn), TORSION: Txy TRANSVERSE SHEAR FORCE: Vxz,Vyz (kN/m) C C INPUT CONSTANT COMPONENT OF SECTIONAL FORCES Fig. 8.2 for sign convention See CSA S474 READ (4, * ) RNXCON, RNY CON, RNXY CON READ(4, *)RMXCON,RMYCON,RMXYCON READ(4, *)RVXCON,RVYCON 121 (kN/rn) (kNm/m) Appendix A3 C C INPUT SECTIONAL FORCES TO BE INCREASED PROPORTIONALLY Fig. 8.2 for sign convention See CSA S474 P.EAD(4, *)RNX,RNY,RNXY READ(4, *)p34x,R4Y,py.Xy READ (4, * ) RVX, RVY C INPUT CONSTANT LOAD VECTOR 250 READ(4,*)DEMAND IF(IMODE.EQ.20)CLOSE(4) 100 CALL CLS CALL MOVCUR(3,0) WRITE (*,lQ0) FORMAT (17X,*************************************************/ */ • 17X,’* */ • 17X,’* University of British Columbia 1991—1992 • l7X,’* Part of M.A.Sc Thesis by Gisli Jonsson */ • 17X,’* In The Department of Civil Engineering *‘/ 17X,’* • */ 17X,’* • Reliability Analysis of Concrete Elements */ • 17X,’* Iceberg Impact Based on Energy Principles */ 17X,* • Capacity calculation: SHELL 474 P.E.ADEBAR */ • 17X,’* Reliability calculation: RELAN R.O.FOSCHI *,/ • 17X,’* 17X, • RETURN END C SUBROUTINE GFUN C C C C C C C C C C C C C C C C C C C C C C C C C (X, N, IMODE, GXP) OPTION FOR OUTPUT DATA IPRINT MAXIMUM SIZE OF AGGREGATE AGG NUMBER OF LAYERS IN X AND Y-DIRECTION NASX AND NASY CYLINDER COMPRESSIVE STRENGTH (MPa) FC FCRK TENSILE STRENGTH FOR CONCRETE (MPa) SECANT MODULUS (MPa) EC STRAIN AT PEAK COMPRESSIVE STRESS ECP THICK THICKNESS OF SECTION (mm) DV SHEAR DEPTH OF SECTION (mm) DIAMETER OF BARS AND STIRRUPS (mm) DIAX,DIAY OR DIAZ AREA OF BARS IN X AND Y-DIRECTION (mm2) ASX OR ASY SPACING OF BARS IN X AND Y-DIRECTION (mm) SPX OR SPY SPACING BETWEEN STIRRUPS IN X AND Y-DIRECTION (mm) SPZX OR SPZY LOCATION OF BARS IN Z-DIRECTION FROM MIDPLANE (mm) ZSX OR ZSY YIELD STRENGTH OF BARS AND STIRRUPS (MPa) FYX,FYY OR FYZ STRAIN DIFFERENCE IN X AND Y-DIRECTION DEPSX AND DEPSY TRANSVERSE REINFORCEMENT RATIO IN Z-DIRECTION RHOSZ (ROWSZ) AXIAL FORCES (kN/m) RNX AND RNY RNXY SHEAR FORCE (kN/m) BENDING MOMENTS (kNm/m) RMX AND RHY TORSION (kNm/m) RMXY TRANSVERSE SHEAR (kN/m) RVX AND RVY VECTOR WHICH MEASURES THE STRUCTURAL RESISTANCE (kN/m) RESIST VECTOR WHICH MEASURES THE DEMAND FOR THE STRUCTURE (kN/m) DEMAND - - - — - - - - - — - - - - - - - - - - - - - - 122 Appendix A3 REAL*4 FC,FCRK,EC,ECP,THICK, ASX(10),SPX(10),ZSX(10),FYX(10),DIAX(10), ASY(10),SPY(10),ZSY(10),FYY(10),DIAY(10), DEPSX(10),DEPSY(10),DIAZ,FYZ, AGG,DV,SPZX,SPZY,RHOSZ,ROWSZ,PHIC,PHIS,PHIP, RNXCON, RNYCON, RNXYCON,RMXCON, RMYCON, RMXYCON, RVXCON,RVYCON, RNX,RNY, RNXY, RMX,RMY,RMXY, RVX,RVY, PESULT(8), RATIO1,RATIO2 ,RATIO3 ,RATIO4,RATIO5 ,RATIO6 ,RATIO7, RATIO8, SCALE,RESIST,LOAD,CAPACITY,DEMAND REAL*8 GXP,X(N) INTEGER IMODE, IPRINT, IPRINT2, ICOUNT, ISTART,NASX, NASY, I INTRINSIC SQRT COMMON/B/IPRINT, IPRINT2 ,PHIC,PHIS,PHIP COMMON/B1/AGG,NASX,NASY,DV,SPZX,SPZY,RHOSZ,ROWSZ COMMON/B2 /DEPSX, DEPSY COMMON/B 3 /RNXCON, RNY CON, RNXYCON, RMXCON, RMYCON, RMXY CON 1 RVXCON,RVYCON COMMON/B4/RNX, RNY,RNXY,RMX, RMY, RMXY, RVX, RVY COMMON/B5/DEMAND COMMON/BlO/ICOUNT, SCALE ICOUNT=ICOUNT+1 FC=SNGL(X(1)) THICK=SNGL(X(2)) FYZ=SNGL(X(3)) • • • • • • • • • • 10 12 14 16 18 20 22 24 ISTART = 3 DO 10 I=1,NASX ASX(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + NASX DO 12 I=1,NASX SPX(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (2 * NASX) DO 14 I=1,NASX ZSX(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (3 * NASX) DO 16 I=1,NASX FYX(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (4 * NASX) DO 18 I=1,NASY ASY(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (4 * NASX) + DO 20 I=1,NASY SPY(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (4 * NASX) + DO 22 I=1,NASY ZSY(I)=SNGL(X(ISTART+I)) CONTINUE ISTART = 3 + (4 * NASX) + DO 24 I=1,NASY FYY(I)=SNGL(X(ISTART+I)) CONTINUE NASY (2 * NASY) (3 * NASY) 123 Appendix A3 38 DO 38 I=1,NASX DIAX(I)=SQRT(1.2732*ASX(I)) CONTINUE 39 DO 39 I=1,NASY DIAY(I)=SQRT(1.2732*ASY(I)) CONTINUE EC5000. O*SQRT(FC) ECP—1. 7*FC/EC FCRK=O. 33*SQRT(FC) DIAZ=SQRT (1.2732 *ROWSZ*SPZX*SPZY) 1 2 3 4 5 6 7 8 C C CALL SKEL(FC,FCRK,EC,ECP,AGG,THICK,DV, DIAX,ASX,SPX,ZSX,FYX,DEPSX, DIAY,ASY,SPY,ZSY,FYY,DEPSY, RHOSZ,ROWSZ,DIAZ,NASX,NASY,SPZX,SPZY,FYZ, RNXCON, RNYCON, RNXYCON, RMXCON, RMYCON, RMXYCON, RVXCON, RVYCON, RNX,RNY,RNXY,RMX,RMY,RMXY,RVX,RVY, PHIC,PHIS,PHIP, IPRINT, RESULT) CAPACITY OF THE ELEMENT WHERE THE MOMENTS ARE SCALED DOWN BECAUSE OF DIFFERENT UNITS RESIST = SQRT( (RESULT(1) )**2+(RESULT(2) )**2+(pFSULT(3) )**2 1 +(RESULT(4)/SCALE)**2+(RESULT(5)/SCALE)**2+(RESULT(6)/SCALE)**2+ 2 +(RESULT(7) )**2+(RESULT(8) )**2) C APPLIED LOAD ON THE ELEMENT LOAD = SQRT((RNX)**2+(RNY)**2+(RNXY)**2+(RMX/SCALE)**2 1 +(RMY/SCALE)**2+(RMXY/SCALE)**2+(RVX)**2+(RVY)**2) C PERFORMANCE FUNCTION FOR THE ELEMENT WHERE GXP<O MEANS FAILURE CAPACITY GXP C = 533 48 RESIST/LOAD DBLE(CAPACITY) - DBLE(DEMAND) IF(IPRINT2 .EQ. 1)THEN WRITE OUT SECTIONAL INFORMATION TO DATA FILE OPEN(8,FILE=’UT.DAT ,ACCESS=APPEND’ ,STATUS=UNKNOWN) WRITE (8, * 532 = ) ***************************************************** WRITE(8,532) SECTIONAL INFORMATION’ FORMAT(10X,A) WRITE(8,533)’ITERATION #‘,ICOUNT FORMAT(/,1OX,A,14) WRITE(8,*) WRITE(8,*)IPRINT,IPRINT2,NASX,NASY,PHIC,PHIS,PHIP,AGG WRITE(8, *)FC, THICK,FCRK,EC,ECP WRITE(8,*)FYZ,SPZX,SPZY,DV,DIAZ,RHOSZ DO 48 I=1,NASX WRITE(8,*)DIAX(I),ASX(I),SPX(I),ZSX(I),FYX(I),DEPSX(I) CONTINUE 124 Appendix A3 49 550 551 536 538 542 543 DO 49 I=1,NASY WRITE(8, * )DIAY(I) ,ASY(I) ,SPY( I) , ZSY(I) ,FYY( I) ,DEPSY(I) CONTINUE WRITE(8,550) ‘RNX, ‘RNY, RNXY, ‘RMX’, RMY’, RMXY’, • ‘RVX,RVY’ FORMAT(/,4X,A,6X,A,5X,A,6X,A,6X,A,5X,A,5X,A,5X,A,/) WRITE(8,551)RNX,RNY,RNXY,RMX,RMY,RMXY,RVX,RVY FORMAT(F7.0,7(2X,F7.0)) WRITE(8,536) NX’, NY, ‘VXY’, MX, MY, MXY, • vxz,,,vYz, FORMAT(/,4X,A,7X,A,6X,A,7X,A,7X,A,6X,A,6X,A,6X,A,/) WRITE(8,538)RESULT(1),RESULT(2),RESULT(3),RESULT(4), • RESULT(5) ,RESULT(6) ,RESULT(7) ,RESULT(8) FORMAT(F7.0,7(2X,F7.0)) WRITE(8,542) ‘RESIST=’ ,RESIST, LOAD=’ ,LOAD WRITE(8,543) ‘CAPACITY=’ ,CAPACITY, ‘DEMAND=’ ,DEMAND, GXP=’ ,GXP FORMAT(/,4X,A,F10.4,3X,A,F10.4,3X,A,F10.4) FORMAT(/,4X,A,F10.4,3X,A,F10.4,3X,A,F10.4) CLOSE (8) ENDIF RETURN END C SUBROUTINE DFUN (X, N, IMODE, DELTA) C IMPLICIT REAL*8 (A H, 0 DIMENSION X(N), DELTA(N) RETURN END - - Z) C SUBROUTINE D2FUN (X, N, IMODE, D2, C IMPLICIT REAL*8 (A H, 0 DIMENSION X(N), D2(N2,N2) RETURN END - - Z) C 125 N2) Appendix A4 Subroutines for Probabilistic Evaluation of Ice-Impact Appendix A4 C $DEBUG SUBROUTINE DETERM ( IMODE) C N NUMBER OF GAUSS POINTS C C P1 THE ICEBERG MASS DENSITY (kg/m3) C Z THE WATER DEPTH (m) REFERENCE AREA (100 m2) C AO THE ICEBERG ANGLE AT THE TOP (DEGREES) C A B THE ICEBERG ANGLE AT THE BOTTOM (DEGREES) C C FO APPLIED LOAD (MN) C LOCATION OF POINT E(I) WEIGHT OF POINT C H(I) REpJ* Pi,Z,AO,A,B,FO,E(32),H(32) INTEGER IMODE,N COMMON/G1/N,Pi, Z COMMON/G2/AO,A, B COMMON/G3 /FO COMMON/G4/E, H IF(IMODE.GT.1)GOTO 10 C READ IN SECTIONAL INFORMATION OPEN(4,FILE=’PROB.DAT’ ,STATUS=’OLD’) OPEN(5,FILE=’NID.DAT’ ,STATUS=’UNKNOWN’) - - — - - - - - - WRITE(S,*) ‘****************************************************‘ WRITE(5,*)’ A2C 2CM FX WRITE(5,*) ‘****************************************************‘ READ(4, *)N READ(4, *)Pj Z,AO,A, B CALL GAUSS(N,E,H,IERR) 10 READ(4,*)FO IF(IMODE.EQ. 20)CLOSE(4) RETURN END C SUBROUTINE GFUN(X,N, IMODE,GXP) C C C C C C C C C C C C C C COMPRESSIVE STRENGTH OF THE ICEBERG (MPa) SM THE VELOCITY OF THE ICEBERG (mis) VI LI THE WIDTH OF THE ICEBERG (m) HI THE ICEBERG THICKNESS (m) REFERENCE AREA (100 m2) AO A THE ICEBERG ANGLE AT THE TOP (DEGREES) THE ICEBERG ANGLE AT THE BOTTOM (DEGREES) B FACTOR TO EXPRESS THE AVERAGE RATIO OF MEAN TO PEAK LOAD LM MAXIMUM PENETRATION (m) XM CONTACT AREA BASED ON 2CM (m2) AX MAXIMUM ICEBERG IMPACT LOAD (MN) FX APPLIED LOAD (MN) FO PERFORMANCE FUNCTION GXP REAL*8 SM,VI,LI,HI, AO,A,B,FO, • LM,XM,AX,FX,GXP, • X(N) • INTEGER IMODE - - - - - - - - - - - - - 127 Appendix A4 INTRINSIC DTAN COMMON/G2 /AO, A, B COMMON/G3 /FO SM=X(1) VI=X(2) LI=X(3) HIX(4) LM = 2.0/3.0 CALL DELTA(XM,SM,VI,LI,HI,LM) DSQRT(2.0)*(2.0*LI*XM+(DTAN(A*22.O/7.0*1.0/180.0) +DTAN(B*22.O/7.0*1.O/180.0))*(XM**2.0)) FX = (LM*SM*DSQRT(AO) *DSQRT(AX)) GXP = FX-FO WRITE(5,200)XM,AX,FX FORMAT( ,F15.4,3X,F15.4,3X,F15.4) AX 1 200 = RETURN END C SUBROUTINE DFUN (X, N, IMODE, DELTA) C IMPLICIT REAL*8 (A H, 0 DIMENSION X(N), DELTA(N) RETURN END - - Z) C SUBROUTINE D2FUN (X, N, IMODE, D2, N2) C H, 0 IMPLICIT REAL*8 (A DIMENSION X(N), D2(N2,N2) RETURN END - - Z) C SUBROUTINE DELTA(XM,SM,VI,LI,HI,LM) C C C C C C C C C C C C C C C VI THE ICEBERG VELOCITY (m/s) HI THE ICEBERG THICKNESS (m) LI THE WIDTH OF THE ICE (m) COMPRESSIVE STRENGTH OF THE ICE (MPa) SM Z WATER DEPTH (m) CM THE ADDED MASS FACTOR THE ICEBERG MASS DENSITY (kg/m3) Pi LM FACTOR TO EXPRESS THE AVERAGE RATIO OF MEAN TO PEAK LOAD MAXIMUM PENETRATION (m) XM AO REFERENCE AREA (100 m2) A THE ICEBERG ANGLE AT THE TOP (DEGREES) B THE ICEBERG ANGLE AT THE BOTTOM (DEGREES) LOCATION OF POINT E(I) H(I) WEIGHT OF POINT - - — - - - - - - - - - — - 128 Appendix A4 • • 10 REAL*8 XM,D,DD,E(32),H(32),SUM, A,B,AO,CM,Z,Pi,KE,Y,YP,F, LM,SM,LI,HI,VI INTEGER I,N,NINT INTRINSIC DTAN, DSQRT, DABS COMMON/GuN, Pi, Z COMMON/G2/AO,A, B COMMON/G4/E, H D = 0.0 DD = 0.01 NINT = 0 SUM=0.0 DO 10 I = 1,N SUM = SUM+DSQRT(DSQRT(2.0)*(HI*(1.0+E(I)) 1 +(DTAN(A*22.0/7.0*1.0/180.0)+DTAN(B*22.0/7.0*1.0/180.0)) 2 *(D/4.0)*((1.O+E(I))**2)))*H(I) CONTINUE CM = (0.9*HI)/((2.Q*Z)_(0.9*HI)) KE = (0.5*0.66*(LI**2.0)*HI*Pi*(1.0+CM)*(VI**2.0))/1.0E6 KE Y = (((D**(3.0/2.0))*LM*SM*DSQRT(AO))/2.0)*SUM IF(NINT.EQ.0)GOTO 5 F = y*yp IF(F.LE.0.0)GOTO 20 YP=Y D = D + DD NINT = NINT + 1 GOTO 1 — 5 20 D =(D_DD)+DD*DABS(YP)/(DABS(YP)+DABS(Y)) XM = D RETURN END C SUBROUTINE GAUSS(N,E,H, IERR) C REAL*8 E(32), H(32) M = (N_2)*(N_3)*(N_4)*(N_5)*(N_6)*(N_7)*(N_8) M = M*(N_9)*(N_10)*(N_11)*(N_12)*(N_15)*(N_16)*(N_32) IF (M.NE.0) GO TO 50 IERR = 0 IF (N.EQ.32) GO TO 40 IF (N.EQ.16) GO TO 30 IF (N.EQ.15) GO TO 29 IF (N.EQ.12) GO TO 28 IF (N.EQ.11) GO TO 27 IF (N.EQ.10) GO TO 26 IF (N.EQ.9) GO TO 25 IF (N.EQ.8) GO TO 23 IF (N.EQ.7) GO TO 20 IF (N.EQ.6) GO TO 18 IF (N.EQ.5) GO TO 15 IF (N.EQ.4) GO TO 13 129 Appendix A4 12 13 15 2 18 3 20 4 23 IF (N.EQ.3) GO TO 12 E(1) = 0.57735026918962600 E(2) = —E(1) H(1) = 1.000 H(2) = H(1) RETURN E(1) = 0.77459666924148300 E(2) = 0.000 E(3) = —E (1) H(1) = 0.55555555555555600 H(2) = 0.88888888888 888900 H(3) = H(l) RETURN E(1) = 0.86113631159405300 E(2) = 0.33998104358485 600 H(1) = 0. 347854845137454D0 H(2) = 0.65214515486254600 DO 1 I = 1,2 E(5—I) = —E(I) H(5—I) = H(I) RETURN E(1) = 0.90617984593866400 E(2) = 0.5384693 1010568300 E(3) = 0.000 H(1) = 0. 236926885056189D0 H(2) = 0.47862867049936600 H(3) = 0.56888888888888900 DO 2 I = 1,2 E(6—I) = —E(I) H(6—I) = H(I) RETURN E(1) = 0.93246951420315200 E(2) = 0.66120938646626500 E(3) = 0.23861918608319700 H(1) = 0.17132449237917000 H(2) = 0.36076157304813900 H(3) = 0.46791393457269100 DO 3 I = 1,3 E(7—I) = -E(I H(7—I) = H(I RETURN E(1) = 0. 949107912342759D0 E(2) = 0. 741531185599394D0 E(3) = 0.40584515137739700 E(4) = 0.000 H(1) = 0.12948496616887000 H(2) = 0.27970539148927700 H(3) = 0. 38183005050511900 H(4) = 0.41795918367346900 DO 4 I = 1,3 E(8—I) = —E(I) H(8—I) = H(I) RETURN E(1) = 0.96028985649753600 E(2) = 0. 796666477413627D0 E(3) = 0.52553240991632900 E(4) = 0.18343464249565000 14(1) = 0.10122853629037600 14(2) = 0. 22238103445337400 130 Appendix A4 5 25 6 26 7 27 77 28 H(3) = 0.313 706645877887D0 H(4) = 0. 362683783378362D0 DO 5 I = 1,4 E(9—I) = —E(I) H(9—I) = H(I) RETURN E(1) = 0. 968160239507626D0 E(2) = 0.83603110732 6636D0 E(3) = 0. 613371432700590D0 E(4) = 0. 324253423403809D0 E(5) = 0.ODO H(1) = 0.0812 74388361574D0 H(2) = 0. 180648160694857D0 H(3) = 0. 260610696402935D0 H(4) = 0. 312347077040003D0 H(5) = 0. 330239355001260D0 DO 6 I = 1, 4 E(1O—I) = —E(I) H(1O—I) = H(I) RETURN E(1) = 0. 973906528517172D0 E(2) = 0.865063366688985D0 E(3) = 0.679409568299024D0 E(4) = 0.433395394129247D0 E(5) = 0.14887433898163 1DO H(1) = 0.06667 1344308688D0 H(2) = 0. 149451349150581D0 H(3) = 0. 219086362515982D0 H(4) = 0. 269266719309996D0 H(5) = 0. 295524224714753D0 DO 7 I = 1, 5 E(11—I) = —E(I) H(11—I) = H(I) RETURN E(1) = 0. 978228658146057D0 E(2) = 0. 887062599768095D0 E(3) = 0. 730152005574049D0 E(4) = 0. 519096129206812D0 E(5) = 0. 269543155952345D0 E(6) = 0.ODO H(1) = 0. 055668567116174D0 H(2) = 0. 125580369464905D0 H(3) = 0. 186290210927734D0 H(4) = 0. 233193764591990D0 H(5) = 0. 262804544510247D0 H(6) = 0. 272925086777901D0 DO 77 I = 1, 5 E(12—I) = —E(I) H(12—I) = H(I) RETURN E(1) = 0. 981560634246719D0 E(2) = 0. 904117256370475D0 0. 769902674194305D0 E(3) E(4) = 0. 587317954286617D0 E(5) = 0. 367831498998180D0 E(6) = 0. 125233408511469D0 H(1) = 0. 047175336386512D0 H(2) = 0.10693932 5995318D0 H(3) = 0.160078328543346D0 131 Appendix A4 8 29 88 30 9 40 H(4) = 0.203 167426723066D0 H(5) = 0. 2334925365383 55D0 H(6) = 0. 249147045813403D0 DO 8 I = 1,6 E(13—I) = —E(I) H(13—I) = H(I) RETURN E(1) = 0.987992 518020485D0 E(2) = 0. 937273392400706D0 E(3) = 0. 848206583410427D0 E(4) = 0. 724417731360170D0 E(5) = 0. 570972].72608539D0 E(6) = 0.39415134707 7563D0 E(7) = 0.20119409399 7435D0 E(8) = 0.ODO H(1) = 0. 030753241996117D0 H(2) = 0. 070366047488108D0 H(3) = 0. 107159220467172D0 H(4) = 0.13957067792 6154D0 H(5) = 0. 166269205816994D0 H(6) = 0. 186161000015562D0 H(7) = 0.19843148532 7112D0 H(8) = 0.20257824192 5561D0 DO 88 I = 1, 7 E(16—I) = -E(I) H(16—I) = H(I) RETURN E(1) = 0.98940093499 1650D0 E(2) = 0.94457 5023073233D0 E(3) = 0.865631202387832D0 E(4) = 0. 755404408355003D0 E(5) = 0. 617876244402644D0 E(6) = 0. 458016777657227D0 E(7) = 0. 281603550779259D0 E(8) = 0. 095012509837637D0 H(1) = 0. 027152459411754D0 H(2) = 0.062253523938648D0 H(3) = 0.095158511682493D0 H(4) = 0. 124628971255534D0 H(5) = 0.149595988816577D0 H(6) = 0. 169156519395003D0 H(7) = 0. 182603415044924D0 H(8) = 0. 189450610455068D0 DO 9 I = 1,8 E (17—I) = —E(I) H(17—I) = H(I) RETURN E(1) = 0.997263861849482D0 E(2) = 0.985611511545268D0 E(3) = 0.964762255587506D0 E(4) = 0.934906075937740D0 E(5) = 0.896321155766052D0 E(6) = 0.849367613732570D0 E(7) = 0.794483795967942D0 E(8) = 0.732182118740290D0 E(9) = 0.663044266930215D0 E(10) = 0.587715757240762D0 E(11) = 0.506899908932229D0 E(12) = 0.421351276130635D0 132 Appendix A4 10 50 1000 E(13) = 0. 331868602282128D0 E(14) = 0. 239287362252137D0 E(15) = 0. 144471961582796D0 E(16) = 0. 048307665687738D0 0. 007018610009471D0 H(1) = 0. 016274394730906D0 H(2) = 0. 025392065309262D0 H(3) = H(4) = 0. 034273862913021D0 0.04283589802222 7D0 H(S) = 0. 050998059262376D0 H(6) = H(7) = 0. 058684093478536D0 H(8) = 0. 065822222776362D0 H(9) = 0. 072345794108849D0 H(10) = 0. 078193895787070D0 H(11) = 0.08331192422 6947D0 H(12) = 0. 087652093004404D0 0. 091173878695764D0 H(13) = H(14) = 0. 093844399080805D0 H(15) = 0. 095638720079275D0 H(16) = 0. 096540088514728D0 DO 10 I = 1, 16 E(33—I) = —E ( I) H(33—I) = H(I) RETURN WRITE(*,1000) FORMAT( WRONG CHOICE FOR NUMBER OF GAUSS INTEGRATION POINTS IERR = 1 RETURN END C 133 I) Appendix B Design of Concrete Beam According to the CSA Appendix B In design of the beam section we have to make sure the applied moment will not exceed the critical moment, and that the minimum and maximum reinforcement criteria is met. From Equations 4.11 and 4.27 we know that the maximum reinforcement can be written as: Pmax JafiicfcY 600 )6OO+f) (B 1) and also from Equation 4.10 and 4.11 the critical moment can be written as: 2 Mcr=Krbd (B.2) where Kr is: Kr =pØfI’i. ‘‘ q5j 1 2a (B.3) and the reinforcement ratio p: (B.4) 135 Appendix B In order to design a beam section, we need to assume certain number of steel layers. As an initial assumption, the beam will be assumed to need one layer of steel bars, with spacing according to the CSA code (see Figure B. 1). b Clear Cover Stirrup Holder Stirrup DIA. (Usually No.10 or 15) Centroid of Tension Steel Area MIN. 25 mm Clear Cover, MIN. 40 mm d S S S db Clear Spacing (db, 25 mm, 1 .33 MAX. AGGR. Size) = 50 mm + db/2 For No. 10 Stirrups d = 40 mm + DIA. of Stirrup + Inside RAD. of Stirrup Bend,For Larger than No.10 S = MIN. Spacing = Sc+db MAX. Number of Bars = (( b 2dc)/S) + 1 = - - Figure B.1: Code Requirements for Reinforcement Placing First of all we need to find the applied load, which the simply supported beam has to resist. By using the values from Table 4.1, Equation 4.13 gives us: Mm = ji(l.25x42,5+ l.5x(25.O+12.5))172 = 3951.2 (kNm) Then maximum reinforcement ratio according to Equation B. 1 is: max (O.85x0.816xO.6x35”( 600 “‘1=0.0257 0.85x400 J’,600+400) 136 Appendix B By introducing Pm as an initial value into Equation B.3 we get: K = 0.0257x0.85x400(1_ 0.0257x0.85x400) 1.7x0.6x35 = 6.5993 (MPa) Assuming that the beam will have only one layer, we can find the required effective depth from Equation B.2: I3951.2x106 d= 1 V 6.5993x500 = 1094 (mm) but assuming only one layer of No.35 steel bars and beam dimensions 500 x 1300 mm the effective depth becomes: d=1300_(40+11.3+35.4)=1231 (mm) Now the new Kr can be found from Equation B.2: Kr = 39512 X 106 2 500x1231 = 5.2149 (MPa) and by rearranging Equation B.3 we get the actual reinforcement ratio: 0.85x0.6x35( 1 0.85x400 L\ /_ V 2x5.2149 =0.0187 0.85x0.6x35) 137 Appendix B The required steel area for the beam can now be calculated from Equation B.4: 5 A = 0.0187x500x1231= 11510 (mm ) 2 which is approximately 12 No.35 bars. The minimum spacing criteria between bars in the CSA code, forces us to have two steel layers (see Figure B.1): d=50+35.7,4=67.85(mm) S=35.7+35.7=71.4(mm) , and therefore the maximum numbers of bars in each layer is: Max bars = 500— 2x67.85 +16 71.4 Because we have now two layers instead of one, the calculated effective depth changes according to that: d = 1300 (6x69.l5 + 6xl29.85 = 1200.5 (mm) 12 ) and in the same manner Kr: K = 39512 X l0_6 2 500x1200.5 = 138 5.4832 (MPa) Appendix B The actual reinforcement ratio is then: p= 2x5.4832 0.85x0.6x35( I ii— 11— 1=0.0199 <pmax 0.85x400 0.85x0.6x35) V and the required steel area for the beam consequently becomes: A = 0.0199x500x1200.5 = 11945 (mm ) 2 which is less than 12 No.35 steel bars so this section seems okay. 139 Appendix C Results from SHELL474 AppendLc C Program SHELL474 version 4.00 ************************************************************************ ************************ ECCENTRIC LOAD *********************** ************************************************************************ Concrete Properties Fac. Cylinder Comp. Strength Strain at peak stress x 1000 Cracking stress Maximum aggregate size Secant modulus of elasticity = = = = = 50.00 -2.40 2.33 20.00 35355. MPa MPa mm MPa Section Thickness = 1400.00 mm Shear Depth = 1200.00 mm In—Plane Reinforcement Dir. Bar dia. (mm) Area per bar (sq mm) X X 29.9 50.5 Y y Y Y 35.7 35.7 35.5 35.5 Fac. yld. stress (MPa) Spacing of bars (mm) Z Coord. of layer (mm) 702. 2000. 125. 125. 620. —615. 400. 400. .00 .00 1000. 1000. 990. 990. 235. 235. 470. 470. 585. —580. 560. —560. 400. 400. 1675. 1675. .00 .00 6.00 6.00 Transverse Reinforcement Amount (sq mm/ sq m) Bar Dia. (mm) 3414.0 11.3 Spacing of bars X (mm) Y 235.0 Reinforcement Ratios X direction Y direction Z direction - — — 1.544 % .909 % .341 % 141 125.0 Fac. yld. stress (MPa) 400.0 Prestrain (x 1000) Appendix C Effective Concrete Area Ratios X direction Y direction Z direction - - - 55. 58. 72. % % % Average Crack Spacings at Mid—Depth X direction Y direction Z direction ******* — — — 1420. mm 1410. mm 558. mm RESULTS FROM ULS ANALYSIS Membrane Forces LS 1 2 3 4 Nx 156. 224. 209. 213. Ny 1 2 3 4 Mx 4766. 4830. 4871. 4860. —2293. —2340. —2362. —2355. —433. —421. —442. —458. My 1 2 3 4 1689. 1712. 1724. 1735. .000606 . 000859 .000907 .000920 —13. —10. —12. —14. 2641. 2675. 2702. 2714. = = = .001445 .001711 .001770 .001788 (kN/m) —.000126 . 000131 —.000129 —.000127 —.000116 000188 —.000234 —.000254 — .003466 .006403 .007230 .007349 1.027 1.000 4 142 0 —. (rad/m) phiy .000179 .000188 .000196 .000200 Transverse Strains Gxz 6. 9. 6. 5. Gxy, Curvatures phix (mm/mm) Ey, o Ex, o (kNm/m) Vyz Vxz Maximum Load Factor Minimum Force Ratio Strain control index Membrane Strains Txy Transverse Shears LS (kN/m) Vxy Bending Moments LS ******* Gyz —.000250 —.000569 —.000762 —.000831 phixy —.000203 —.000324 —.000405 —.000440 (mm/mm) Ez .001778 .004677 .005657 .005771
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Reliability analysis of structural concrete elements Jonsson, Gisli 1992
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Title | Reliability analysis of structural concrete elements |
Creator |
Jonsson, Gisli |
Date Issued | 1992 |
Description | The reliability of reinforced concrete elements and design iceberg impact loads for offshore structures were studied. The reliability program RELAN was used to perform FORM reliability analysis for reinforced beams subjected to bending, and for a reinforced concrete wall from the Hibernia offshore structure subjected to complex loading. RELAN was also used to establish the probabilistic distribution of ice impact loads. To study the reliability of concrete beams accounting for the variability of the intervening variables, and in order to determine the theoretical flexural capacity of concrete beam, computer program TIN was developed. TIN uses a strain compatibility approach accounting for the non-linear stress-strain relationships of concrete and reinforcing steel. As a pilot study on the reliability of concrete elements, a beam designed according to the Canadian concrete code was analyzed with the objective of evaluating the effect of different spans and reinforcing steel ratios on the reliability of the beam. To study the reliability of more complex elements, an element from the icewall of the Hibernia offshore structure was used. The theoretical strength of the wall element was evaluated with program SHELL474. In order to link SHELL474 to RELAN for the reliability study, the main subroutine in SHELL474 was modified. Since one of the major factors in reliability studies of concrete offshore structures are the uncertainties associated with extreme environmental load conditions, the statistics for ice impact loads for the Hibemia structure were derived using RELAN and applications of energy conservation principles. For the purpose of deriving the ice load distributions to evaluate the reliability of the Hibemia icewall element, the program PROB, which is a product of the reliability program RELAN and the energy conservation principles, was developed. |
Extent | 3422798 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050470 |
URI | http://hdl.handle.net/2429/2971 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1992-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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