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Reliability analysis of structural concrete elements Jonsson, Gisli 1992

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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.  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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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