TKT4196 Aspects of Structural Safety Compendium 2023 PDF

ASPECTS OF STRUCTURAL SAFETY THE COMPENDIUM FOR TKT4196 - Prosjektering Sikkerhetsforhold Jochen Köhler Høstsemester 2023 Norges teknisk-naturvitenskapelige universitet Institutt for konstruksjonsteknikk The Compendium TKT4196 - Prosjektering/Sikkerhetsforhold Aspects of Structural Safety Prof. Dr.-Ing. Jochen Köhler Synopsis: This compendium is intended for the students of the Norwegian University of Science and Technology (NTNU) that take part in the course TKT4196 - Prosjektering Sikkerhetsforhold. It is still under continuous improvement and development. Any critical remarks and suggestions for improvement is welcome and highly appreciated. Acknowledgement: The present version of the compendium is produced in the period 2015-2022. The help and contribution of Jorge Mendoza, Michele Baravalle and Klodian Gradeci is highly acknowledged. Table of Contents Page Chapter 0. Introduction to TKT4196.......................... 1 1. Organization basics................................................. 1 2. Learning strategy................................................... 2 3. Literature & Standards............................................... 3 4. Rationale of the course............................................... 4 5. Course content..................................................... 5 6. Safety - a requirement for structures.................................... 6 7. Summary......................................................... 10 8. Motivating Example (Exercise 1):..................................... 10 8.1 Problem statement............................ 10 8.2 Tasks................................... 11 8.3 Solution.................................. 12 Chapter 1. Structural optimization............................ 17 1. Theoretical aspects.................................................. 17 1.1 Introduction................................ 17 1.2 Optimization............................... 18 1.3 Interest.................................. 22 1.4 Continuous renewal assumption..................... 23 1.5 Risk acceptance.............................. 25 1.6 Conclusion of Chapter 1......................... 27 5 Section: TABLE OF CONTENTS TKT4196 - Aspects of Structural Safety 2. Application examples................................................ 28 3. Practical exercises.................................................. 34 Chapter 2. Structural reliability - Problem Statement and Monte Carlo Method.. 37 1. Theoretical aspects.................................................. 37 1.1 Prelude.................................. 37 1.2 Introduction: The limit state principle.................. 37 1.3 A simple case - the Fundamental Reliability Problem.......... 39 1.4 Definitions................................ 41 1.5 Monte Carlo Method........................... 42 1.6 Sampling of correlated random variables................. 44 1.7 Generalization for multivariate distributions............... 47 1.8 Variance reducing methodologies.................... 48 2. Application examples................................................ 52 3. Practical exercises.................................................. 57 Chapter 3. First Order Reliability Method....................... 61 1. Theoretical aspects.................................................. 61 1.1 Linear Limit State Functions and Normal Distributed Variables..... 61 1.2 Non-linear limit state function...................... 62 1.3 Correlated and Dependent Random Variables.............. 65 1.4 Non-Normal and Dependent Random Variables............. 68 2. Application examples................................................ 71 3. Practical exercises.................................................. 77 Chapter 4. System Reliability............................... 79 1. Theoretical aspects.................................................. 79 1.1 Introduction................................ 79 1.2 Simple bounds for system reliability of failure.............. 82 1.3 System Reliability using FORM..................... 83 2. Application examples................................................ 87 3. Practical exercises.................................................. 91 Chapter 5. Introduction to Probabilistic Load and Material Modelling........ 95 Page 6 Chapter 0. TABLE OF CONTENTS 1. Theoretical aspects.................................................. 95 1.1 Recapturing basic principles of probability theory............ 95 1.2 Interpretation of Probability....................... 96 1.3 Random Variables............................ 98 2. Uncertainties in Engineering Problems.................................. 106 2.1 Load bearing behaviour of materials................... 108 2.2 Scales of (variability) modelling..................... 108 2.3 Brittle material.............................. 109 2.4 Probabilistic load modelling....................... 110 2.5 Time variable load............................ 112 2.6 Load combination............................. 118 3. Practical exercises.................................................. 121 Chapter 6. Structural Design Standards......................... 125 1. Concepts.......................................................... 125 1.1 Levels of design.............................. 125 1.2 Risk informed decision making...................... 126 1.3...................................... 126 1.4 Semi-probabilistic design......................... 127 1.5 Relevance and correspondence...................... 127 2. EUROCODE semi-probabilistic design.................................. 128 2.1 General framework and design values.................. 128 2.2 Partial factors............................... 129 2.3 Reliability requirements in the Eurocodes................ 129 2.4 Direct correspondence between reliability and design values - the design value approach.............................. 130 2.5 Derivation of generalized α-values.................... 134 2.6 The generalised α-values in the Eurocodes................ 135 3. Application examples................................................ 136 Chapter 7. Engineering Decision Making........................ 145 1. Concept.......................................................... 145 2. Bayes’ Rule....................................................... 147 2.1 Conditional Probability and Bayes’ Rule................. 147 2.2 Example 2 - Using Bayes’ Rule for Bridge Upgrading.......... 149 2.3 Example 3 - Covid Test.......................... 150 Page 7 3. Engineering application.............................................. 151 3.1 A priori analysis............................. 151 3.2 A posteriori analysis........................... 152 3.3 A pre-posteriori analysis......................... 153 3.4 Summary................................. 156 Chapter 8. Engineering uncertainty quantification................... 159 1. Engineering uncertainty quantification.................................. 159 1.1 Cumulative Distribution and Probability Density Functions....... 159 1.2 Sampling and representation of data................... 167 1.3 Model building and parameter estimation................ 171 2. Application examples................................................ 178 Bibliography........................................ 193 Chapter 0 Introduction to TKT4196 In this first chapter of TKT4196 - Aspects of Structural Safety, a general overview of the course is given. After going through the practical and organisational aspects related to the course imple- mentation, we will have a brief discussion on the role of safety aspects in structural engineering and in civil engineering in general. This chapter is meant to be a soft and “not too technical” start to the course and to the topic. 1. Organization basics Contacts Lecturer: Jochen Köhler [email protected] Tutor: Lombe Mutale [email protected] Student Assistant: Petter Sunde [email protected] Reference group: TBA TBA TBA Lectures and Exercises The main language of the course is English. The lectures will be given in English and the written material and suggested literature is in English. The lecturers masters Norwegian to some degree and the students are very welcome to ask questions and start discussion in Norwegian. Timing and location: Mondays 08.15 - 10.00 R4 Thursdays 08.15 - 10.00 B1 Fridays 10.15 - 12.00 B1 1 Section: Learning strategy TKT4196 - Aspects of Structural Safety The lectures run from week 34 - 47. The first lecture is taking place on Thursday, August 26, at 08:15 in room B1. It is recommended that the students are present, both, physically and mentally in all the announced classes. Compulsory problem tasks Three exercises must be delivered and accepted in order to be allowed to take the exam. The ex- ercises will not be graded but evaluated with pass/fail. Solutions will be published and presented during exercise classes. The problem tasks will be solved and documented in groups by 3-5 students. Exercises hand in: directly in the lecture or electronically following the lecture program and announcements. Exam The online exam will take place 21.12.2022 at 1500h. It will be written exam with a duration of 4 hours. The exam is normally held in Norwegian but English is also possible, of course. 2. Learning strategy One of the important premises of this course (and in general) is that learning will be achieved solely by the students themselves and this requires corresponding commitment. The successful participation of this course gives 7.5 credit points and it is expected that an average student has to work 10 hours per week for this course in order to reach an average grade. (Note that this rule holds generally for all courses with a corresponding number of credits). Learning is facilitated by proper supervision and teaching. For this course the concept in- cludes five major components. As a rule of thumb, 2 hours/week should be invested for each of the components. 1. Compendium: An accompanying compendium, which this chapter is a part of, is avail- able to the students. It is divided into chapters for the different learning topics of the course. Each chapter is divided into three sections: 1. Theoretical aspects, 2. Applica- tion examples and 3. Practical exercises. 1. Theoretical aspects covers the content of the lectures. The material of some chapters is introduced during several lectures. 2. Ap- plication examples covers some applications of the theory, including the solution. These will be discussed during the exercise classes. 3. Practical exercises include some addi- tional exercises, without the solutions, that the students should solve during the practical classes. (The exercises that are to be handed in are following more or less directly from these practical classes.) 2. Lectures: All content of this course will be transmitted in the lectures. The theoretical concepts will be illustrated by examples. The lectures are oral and will be supported by blackboard demonstration and overhead slides. The Students are asked to follow actively the lectures and rise questions if something is unclear. An open dialog in the lecture will clearly increase the quality of the lecture. Keep in mind that no questions are “stupid” Page 2 Chapter 0. Introduction to TKT4196 and that many other students may benefit from answering the same question. It is strongly recommended to make comprehensive notes during the lectures. 3. Exercise classes: In the exercise classes, the application examples will be demonstrated and discussed. Vital participation of the students (in terms of questions and comments) is absolutely required. 4. Practical classes: It is very important that the new obtained knowledge becomes an ex- ecutable part of the students’ skills. We will therefore spend some time on “hands-on” practical classes where students try to implement gained knowledge and understanding in practical problems. These exercises are largely based on the application examples. The main difference is that the solutions will not be provided beforehand. This brings an op- portunity to the students to test the understanding of the course material avoiding any “narrative falacy”: exercises seem easier and more logical after the solution is given, but this cancels the student’s creativity to think for herself on different ways how to answer a question. We will use some IT-tools for that (e.g. Python or Excel) and students should work in groups on their own computers. The classes will be tutored. 5. Self-study: Students are expected to constantly revise the current course material and read additional literature as it will be given in the lectures. 3. Literature & Standards The following literature is recommended for further reading. Special chapters of some of the books will be defined as pensum during the lecture. Reliability Basics Schneider J., 2006. Introduction to safety and reliability of structures. [Zürich] International Association for Bridge and Structural Engineering. [Available on Blackboard]. Reliability Melchers R. E., Beck A.T., 2017. Structural Reliability Analysis and Prediction. Further Reading Wiley. Structural Codes Gulvanessian, H; Formichi, P; Calgaro, J. A 2009. Designers’ guide to Eurocode 1: EN 1991-1-1 and -1-3 to -1-7, Actions on buildings / H. Gulvanessian, P. Formichi and J.-A. Calgaro [ebook at NTNU Library]. Standard – EN 1990, Eurocode: Basis of structural design [via Standard Norge at NTNU Library]. Standard – EN 1991, Eurocode 1: Actions on structures [via Standard Norge at NTNU Library]. Uncertainty Benjamin, J. R. and C. A. Cornell (1970). Probability, Statistics, and Modelling Decision for Civil Engineers. Page 3 Section: Rationale of the course TKT4196 - Aspects of Structural Safety 4. Rationale of the course Civil Engineers play and will play an important role in our society. Many current and future challenges, as related to the efficient management of (limited) financial and natural resources or the appropriate mitigation to the possible effects of climate change, are closely related to the build environment and require good civil engineering. As the challenges will become bigger, the demand for new and innovative solutions, i.e. well beyond traditional engineering practice, will increase. The role of future civil engineers is very nicely defined by the American Society of Civil Engineers (ASCE)1 as a professional who is2 : “Entrusted by society to create a sustainable world and enhance the global quality of life, civil engineers serve competently, collaboratively, and ethically as master: planners, designers, constructors, and operators of society’s economic and social engine, the built environment; stewards of the natural environment and its resources; innovators and integrators of ideas and technology across the public, private, and aca- demic sectors; managers of risk and uncertainty caused by natural events, accidents, and other threats; and leaders in discussions and decisions shaping public environmental and infrastructure pol- icy. As used in the vision, “master” means one who possesses widely recognized and valued knowledge, skills, and attitudes acquired as a result of education, experience, and achievement. Individuals within a profession who have these characteristics are willing and able to serve society by orchestrating solutions to society’s most pressing current needs while helping to create a more viable future.” Many will agree that proper university education in the field of civil engineering develops a good understanding of the basic principles in science (being Mathematics, Physics, Applied Me- chanics, and Chemistry) combined with good understanding of the practical engineering prob- lems at hand. This combination of fundamentals and practical implications is very well reflected in the current study curriculum for “Bygg- og miljøteknikk” at NTNU, i.e. with fundamental classes and more practical classes for the individual study directions (that take up fundamentals and make them executable for practical engineering applications). The understanding of uncertainty, risk and safety is an essential skill for civil and structural engineers. A basic and central reality of practical civil engineering is the fact that our world (especially the future side of it) is not exactly known but can be estimated or predicted by taking into account the uncertainties that are associated to this incomplete knowledge. Safety related 1 ASCE is the permanent organization representing the civil engineering profession in the United States. http://www.asce.org/ 2 Part of “ASCE - Civil Engineering Body of Knowledge for the 21st Century. Preparing the Civil Engineer for the Future” LINK Page 4 Chapter 0. Introduction to TKT4196 engineering decisions must take into account uncertainties and risks. Every responsible engi- neer should have at least basic knowledge on how to treat uncertainties in practical engineering problems and understand the background of the generally used regulations and design standards. This course aims at the procurement of basic knowledge in the topics uncertainty represen- tation, risk and safety and makes students develop a new mind-set expendable for interpreting all kinds of engineering problems. (Expert knowledge might be attained by specializing in master or PhD projects). 5. Course content The course will develop basic knowledge, understanding and practical skills. The course content is organized in different thematic main blocks: Optimization and Acceptance. Evaluation of funda- mental risk based performance criteria. An objective function for engineering structures is developed. It is A. discussed how human safety can be incorporated. It is seen that the structural failure probability is of par- ticular interest. Methods for structural reliability assessment. Dif- ferent methods for the estimation of the structural fail- ure probability are introduced. The methods facili- B. tate a realistic representation of structural engineering problems, however, we will learn that a proper engi- neering uncertainty representation is necessary. Basic engineering uncertainty modelling. It is demonstrated how uncertainties can be represented C. for different relevant phenomena. Practical methods are introduced to represent material resistance and time varying loads. Standards and code calibration. The methodical D. basis of structural codes is introduced and the general background of the overall safety concept is examined. Page 5 Section: Safety - a requirement for structures TKT4196 - Aspects of Structural Safety 6. Safety - a requirement for structures The built environment, i.e. infrastructural elements, industrial buildings and facilities as well as residential buildings, constitutes the basis for our economy and the continuous development of our society. In this respect structures play an important role, since the primary purpose of structures is to provide the functionality of the built environment. That structures are safe is an important premise for the success of our society, and structural safety is clearly a basic require- ment of the society. Correspondingly, structural safety is a requirement that is defined in legislation on several hierarchical levels – structures have to be safe by law. In Norway a general safety requirement can be found in the ”Plan og Byggningsloven” (§29.5); more detailed prescriptions for safety are found in ”Byggteknisk forskrift (TEK17)”, Chapter 5 on ”Konstruksjonssikkerhet”. Here one can read: ”Byggverket skal plasseres, prosjekteres og utføres slik at det oppnås tilfredsstillende sikker- het for personer og husdyr, og slik at det ikke oppstår sammenbrudd eller ulykke som fører til uakseptabelt store materielle eller samfunnsmessige skader.” Several formal definitions of safety exist and accordingly, safety is used as absolute or relative characteristic in our regular language use. (“It is safe to use air traffic” is an absolute statement; “Air traffic is safer as car traffic” is a relative statement). In the context of structural engineering the following definition can be found in ISO 2394:20153 : Structural safety: “ability of a structure or structural member to avoid exceedance of ultimate limit states4 , (... ), with a specified level of reliability, during a specified period of time.” Remembering that structural reliability per time unit is defined as the complement of failure probability of structural failure per time unit; structural safety is an absolute description for situations where the probability of failure per time unit is sufficiently low. QUESTIONS: What further structural requirements does ”safety” include? What require- ments are independent from safety? (space for your notes) Attaining structural safety Structural engineers are responsible that structures are safe. Structural engineers therefore make proper decisions during the entire life-cycle of the structure. The typical phases of the life-cycle of civil engineering infrastructure are illustrated in Figure 1. For decision making, structural engineers take into account information e.g. about: 3 ISO 2394:2015: General principles on reliability for structures. 4 The exceedance of an ultimate limit states corresponds e.g. to structural failure. Page 6 Chapter 0. Introduction to TKT4196 Figure 1: The life-cycle of engineering structures. Traffic volume Loads Resistances (material, soil,...) Degradation processes Service life Manufacturing costs Execution costs Decommissioning costs As a matter of fact, the information that has to be integrated is associated with uncertainties. Thus, the uncertainties have to be taken into account in the decisions that are made in the life- cycle of the structure. The activities and the decision making of structural engineers are very well illustrated in the Swiss structural standard SIA 260 with the graph presented in figure 2. Here, a structure is defined by its service criteria and the environment it is embedded into. The structure then has the following phases during its life cycle: design Page 7 Section: Safety - a requirement for structures TKT4196 - Aspects of Structural Safety execution use conservation decommissioning QUESTION: What are examples for typical decisions that are made during the life-cycle of a structure? What is uncertain for these decisions? (space for your notes) Structural Design Phase In this course we will focus on the design phase of the structure. In figure 2 the design phase is further subdivided into “conceptual design”, “structural analysis” and “dimensioning”. In conceptual design a structural concept is developed that is best to meet the requirements to the structure, i.e. in regard to the intended societal activity that should be supported by the structure these are requirements in regard to space and shelter created by the structure, requirements on limitation of vibrations and deflections (serviceability), requirement on the durability of the structure and the intended service-life, safety, i.e. preventing damage, loss of functionality, damage to the environment and to personnel, cost efficiency. In the conceptional design phase, major decision are made in regard to principle structural layouts and material use. Build-ability is always a very important aspect here. Structural analysis is intended to create a link between the external loads on the structure and the corresponding structural response in terms of stresses, deformations and deterioration. Page 8 Chapter 0. Introduction to TKT4196 Figure 2: Schematic representation of structural engineering activities. Mechanical models (analytical and numerical) are utilized and structural analysis is generally performed at a rather high level of modelling detail. For structural dimensioning the structural dimensions are chosen such that the structure complies with some specified limit states regarding safety and serviceability. The compliance is demonstrated for several load scenarios (“design situations”). In general it can be observed that the effort and level of detail is not consistently distributed Page 9 Section: Summary TKT4196 - Aspects of Structural Safety among the different steps of structural design: Whereas a lot of attention is attributed to structural analysis, rather rudimentary criteria are used in dimensioning. 7. Summary In this first lecture notes some organization information is shared. The contents and ob- jectives of the course are defined and an initial approach to the term ”safety” is given. It is demonstrated that the information that is necessary for structural engineering decisions is un- certain, and that this uncertainty has to be taken into account by the structural engineer when evaluating safety, i.e. the probability of failure has to be quantified. But how safe is safe enough? This question will be followed up in the next lectures. 8. Motivating Example (Exercise 1): Design of a timber beam based on the Eurocodes. 8.1. Problem statement Your task is to design a timber roof beam (Fig.3) according to the semi-probabilistic safety format (partial factor format) in the Eurocodes. The distance D between the simply supported beams spanning over length L is D=L/8. In addition to their self-weight (SW ), the beams supports a uniformly distributed snow load (S) and a concentrated load at mid-span due to a permanent installation (P ). Their rectangular cross-section is characterised by a depth to width ratio of h/b = 3. The design should comply with the ultimate limit state corresponding to bending failure at the beam’s mid-span cross-section (ULS). Serviceability limit states (SLS) and durability re- quirements are not considered in this example. Figure 3: Simply supported beam Page 10 Chapter 0. Introduction to TKT4196 8.2. Tasks 8.2.1. Task 1.1 Determine the strictly required cross-section depth h of the beam by means of an Excel spread- sheet, considering a span length L = 10 m. The required input data for solving the exercise is given below: Characteristic value of the snow load: Sk = 3.5 kN /m2 Characteristic value of the permanent load: Pk = 30 kN Characteristic value of the material density: ρk = 380 kg/m3 Characteristic value of the flexural strength: fm,k = 24 N /mm2 Partial factor for permanent loads: γG = 1.35 Partial factor for variable loads: γQ = 1.5 Combination factor for variable loads: ψ0 = 0.7 Reduction factor for unfavourable permanent loads: ξ = 0.89 Partial factor for material strength: γM = 1.3 Action effects are to be determined for load combination 6.10a/b of EN1990. Hint: Consider that due to the influence of the self-weight, the design procedure is iterative. The procedure can be stopped when the difference between the cross-section depths obtained in two successive iteration steps is equal or less than ∆h = 1 mm. 8.2.2. Task 1.2 Instead of using a spread-sheet, now write a small program (in Python) to determine the required cross-section dimensions considering beam span lengths L varying between 5 and 25 m (∆L = 5 m). For the design procedure, use the same input data and convergence criterion (∆h = 1 mm) as in the previous task. Provide two graphs which show, respectively, the relationship between the required cross- section modulus (W ) and beam span (L) and the relationship between the required cross-section depth (h) and L. It is recommended to structure your code with the help of functions. 8.2.3. Task 1.3 Write a Python script which determines the failure probability Pf and the reliability index β of the beams designed under Task 1.2 (considering beam span lengths L varying between 5 and 25 m). Consider the snow load S, permanent load P , material density ρ and the flexural strength fm as normally distributed random variables. All other variables can be assumed as deterministic. Page 11 Section: Motivating Example (Exercise 1): TKT4196 - Aspects of Structural Safety Provide two graphs which show, respectively, the relationship between the failure probability (Pf ) and beam span (L) and the relationship between the reliability index β and L. Hint: First compute the mean values µ for each of the four random variables based on their characteristic values (see task 1.1), the corresponding fractile value (p) and the coefficient of variation (cov), given below: pS = 0.95; covS = 0.3 pP = 0.5; covP = 0.1 pρ = 0.5; covρ = 0.1 pfm = 0.05; covfm = 0.15 It is recommended to make use of the function you developed in task 1.2. 8.2.4. Task 1.4 Write a code to perform an economic optimization of the beam (considering beam span lengths L varying between 5 and 25 m). The decision parameter is the cross-section depth h. Con- sider fixed construction costs (C0 ), material costs depending on the decision parameter (C1 ) and compensation costs for fatalities due to a potential collapse of the beam H. The corresponding assumptions are given below: cm = 4E-6 N OK/mm3 C0 /L = 10000 N OK/m H/C0 = 20 Provide four graphs which show, respectively, the optimal cross-section depth (hopt ), the cor- responding failure probability (Pf,opt ), the reliability index βopt and the minimum expected total costs (ECtot,min ) as a function of the beam span L. In the same graphs, plot the corresponding solution based on the Eurocode design of the beam. 8.3. Solution 8.3.1. Task 1.1 Figure (4) shows the results of the Eurocode design of the beams. As was to be expected, both the cross-section depth h and the section modulus W increase with the beam span L. 8.3.2. Task 1.2 Figure (5) shows the reliability index (left) and failure probability (right) of the beam. Note that in spite of the fact that the beams have been strictly designed according to the decision rules in the Eurocode (Task 1.1), the reliability level varies significantly. Page 12 Chapter 0. Introduction to TKT4196 Figure 4: Cross-section depth h (left) and section modulus W (right) vs. the span length L Figure 5: Reliability index β (left) and failure probability Pf (right) vs. span length L 8.3.3. Task 1.3 As an example, Fig. (6) shows the expected total costs of the L=10 m beam versus the decision parameter, i.e. its cross-section depth h. An optimum depth of approximately hopt = 0.54 m is found. Observe the steep slope of the cost function on the left of the optimum, what indicates the importance of the expected failure costs in relation to the relatively low safety costs. Figure (7) compares the cross-section depth h and the expected total (minimum) costs EC associated with the Eurocode design and the economic optimisation. For larger spans, the Eu- rocode design leads to notably higher h whereas the EC differs only marginally. Figure (8) shows the corresponding comparisons of the reliability index and the failure prob- ability. In line with the observed behaviour for the cross-section depth h, the reliability level according to Eurocode design increases with larger span lengths, while the opposite is the case for the optimisation study. Finally, Figure (9) illustrates the saving potential of the optimisation study in comparison to the Eurocode design. While the reduction in expected costs is rather small, the optimisation entails significant material savings of up to around 20%, depending on the beam span. Taking Page 13 Section: Motivating Example (Exercise 1): TKT4196 - Aspects of Structural Safety Figure 6: Expected total costs EC versus the cross-section depth h of the beam with L=10m span Figure 7: Cross-section depth h (left) and expected total costs EC (right) vs. span length L. Red: Eurocode design; Black: Economic optimization into account the large number of structures designed all over the globe, this example nicely illustrates the large potential of explicit optimisation in regard to sustainable development, as opposed to the generalised and sub-optimal design rules in the structural codes and standards. petter Page 14 Chapter 0. Introduction to TKT4196 Figure 8: Reliability index β (left) and failure probability Pf (right) vs. span length L. Red: Eurocode design; Black: Economic optimization Figure 9: Difference in expected total costs (left) and material volume (right) vs. span length L. Page 15 Section: Motivating Example (Exercise 1): TKT4196 - Aspects of Structural Safety Page 16 Chapter 1 Structural optimization It is demonstrated how a simple structure can be designed in order to minimize the risk. The op- timization could be the foundation for identifying sufficiently low failure probabilities. However, it should be critically assessed whether the optimality criterion is adequate for that. The aspects of net present value are introduced. The simple optimization example is general- ized and the infinite renewal assumption is introduced and discussed. Aspects that go beyond monetary optimization are argued and an acceptance criteria for life safety is introduced and discussed. 1. Theoretical aspects 1.1. Introduction In the Chapter 0, a safe structure was defined as a structure with sufficiently low failure prob- ability per time unit. We will now elaborate on the question “how safe is safe enough?”. How safe is safe enough? The question “how safe is safe enough?” is clearly related to the term sufficiently low failure probability in the definition above, i.e. to the meaning and quantification of sufficiently low. Let us first look at the requirements for structures again1. Safety is an overall requirement, but which further requirements does this include? Structures are used by people in many ways (residents in buildings, car drivers on bridges, workers on offshore platforms). The risk to life and limp induced by these structures should be sufficiently low. Furthermore, structures should not endanger the qualities of the natural environment excessively. And, it has to be considered that a large proportion of the societal wealth is invested into the development and maintenance of structures as a key part of the build environment. In summary, structures have to fulfil the following requirements2 : 1 Note that the functionality of the structure is THE basic requirement, since providing functionality for societal activities is the reason for creating structures. Functionality might include a multitude of further requirements like aesthetics, etc. However, in the present context we are primary interested in the requirements related to the load bearing behaviour of structures. 2 In regard to the load bearing behaviour. 17 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety 1. Safety of personnel 2. Safety of the environment 3. Cost effectiveness Let’s look at the third requirement first. Cost effectiveness might be assessed with a cost- benefit analysis, i.e. where the expected benefit of the structure is maximized, or, the expected cost of the structure is minimized. 1.2. Optimization We will approach this topic by introducing a practical example. Consider the following task that could have given to you in connection to a consulting project: You have to design a beam that has to span l = 10 m and has to carry a load Q. The material that is available is glued laminated timber (Glulam) and the cross-section is specified to be rectangular with a width of b = 300 mm and height h. For a project like this, you would work through the following steps: 1. Conceptual Design: this step includes the definition of the requirement for the structure and the setup of the structural concept. Requirements: after a careful discussion with the client it is specified that neither people nor environment are endangered by a possible failure of this structure. Thus the requirement is defined to be cost effectiveness only. The functionality does not require any permissions on deflection. Structural failure is the scenario that leads to conse- quences. The service period of the structure is specified to be 50 years. Structural concept: You might choose simple support instead of fixed support due to eco- nomic reasons. (a moment resisting support is more expensive). 2. Structural analysis: in this phase, the loads are specified and the load effects and the load bearing capacity are estimated. The load is given in this project to be a uniform distributed load that is represented by its 50 years maximum value Q. The material property of interest in this case is the bending strength of the Glulam Fm. The situation is illustrated in Figure 1.1. Figure 1.1: Beam geometry. The maximum effect of the load S is the bending moment at mid-span that is given with Ql /8. The elastic bending load bearing capacity of the rectangular cross-section R is Fm bh2 /6. 2 Page 18 Chapter 1. Structural optimization 3. Dimensioning: in this phase, limit states are formulated. Limit states are equations that describe events like, ”failure” or ”excessive deflection” that are relevant in regard for the de- scription of requirements to the structure. Structural failure, for example, is defined as the event when the load on a structure is larger than its load bearing capacity, or the difference between the load bearing capacity and the load becomes negative. The corresponding limit state is general referred to as Ultimate Limit State (ULS) and for this example expressed as: 2 2 bh l g (R, S) = R − S = Fm − Q≤0 (1.1) 6 8 The limit state equation contains different variables, some of them are uncertain or random, as Q, Fm , some of them are given or can be controlled, as l, b, h. We can now formulate the failure probability or the reliability index β as a function of the design choice h (we have chosen h as being the most effective design variable). If we assume that Q and Fm are represented as independent and both normal distributed random variables, the reliability index corresponding to a 50 years period3 can be assessed as a function of h using Eq. (1.2): 2 l2 µR (h) − µS µ Fm · bh /6 − µ Q · β (h) = p 2 = s 8 (1.2) 2 σR (h) + σS l 2 2 2 (σFm · bh2 /6) + σQ 8 Mean value and standard deviation for the two random variables are given as in Table 1.1. Based on this and remembering that the failure probability is related to the reliability index as Pf = Φ(−β), we can graphically represent the relation of h to the failure probability and to the reliability index correspondingly, cf. Figure 1.2. As it can be seen, a functional relationship be- tween the design variable h and the failure probability can be established. However, the question remains how it can be decided what failure probability is sufficiently low. Table 1.1: Representation of load and resistance. The coefficient of variation VX is defined as VX = σX /µX. Variable Distribution Ch. value µX VX Fm Normal 20 MPa 26.6 MPa 0.15 Q Normal 39.4 N/mm 24.1 N/mm 0.30 2 R =Wel fm Normal W el2µfm = 1330h Nmm 0.15 l2 l S= Q Normal µQ = 301.25 · 106 Nmm 0.30 8 8 1.2.1. Identification of the optimal decision by minimizing expected costs If cost effectiveness is taken as a criterion, as in this example, the optimal choice of h can be identified by minimizing the expected costs of the structure. The following costs are considered. The parameter values, units and symbols are summarized in Table 1.2. 3 Note that in the entire example the reference period for the reliability index and the failure probability is 50 years. Page 19 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety 6 100 4 -5 10 2 0 10-10 500 1000 1500 500 1000 1500 Figure 1.2: Reliability index β (left) and probability of failure Pf (right) as a function of h. Table 1.2: Assumed values and relations. Variable Symbol Value, relations and units Cost of timber cglulam 4,000 NOK/m3 Beam length l 10,000 mm Fixed construction costs C0 50,000 NOK Variable construction costs coefficient CI b · l · cglulam Indirect costs of failure H rHC0 · C0 = 20 · C0 Construction costs: These are the costs for implementing the structure. It covers transport and installation cost but also the material cost of the structural element. Correspondingly, the construction costs consist of a part depending on the chosen beam height and a fixed part C0 (that is independent from the chosen height h). The construction cost is a function of h, i.e. the possible decision alternatives for risk reduction. See Eq. (1.3). Cconstr (h) = C0 + l · cglulam As = C0 + l · cglulam (b · h) = C0 + CI h (1.3) Expected failure costs: The failure costs include the costs for re-installing the structure after failure, i.e. the construction costs. Additionally a cost term H is introduced. H contains all failure costs beyond the pure re-installation costs, e.g. damage to equipment, cleaning costs, downtime of services that are supported by the structure, etc. The failure cost is multiplied with the probability of failure in order to express the expected failure cost. Note that another term for the expected failure cost would be Risk. E [Cf,U LS ] (h) = Cf,U LS (h) · Pf (h) = [Cconstr (h) + H] · Pf (h) (1.4) Objective function: The objective function to be minimized represents the total expected costs, cf. Eq. (1.5). The minimum is identified subject to h (the decision variable) and represents the optimal balance between investments into the structure and the expected adverse consequences. Page 20 Chapter 1. Structural optimization E [Ctot ] (h) = Cconstr (h) + E [Cf,U LS ] (h) = [C0 + CI h] + [Cconstr (h) + H] Pf (h) (1.5) = [C0 + CI h] + [CI h(h) + C0 (rHC0 + 1)] Pf (h) hopt = arg min E [Ctot ] (h) (1.6) h The numerical minimization of the total expected costs can be performed with Matlab® or Python. The optimal height of the beam is found to be 781 mm. 104 12 10 8 6 4 2 0 500 700 900 1100 1300 1500 Figure 1.3: Expected monetary costs with economic optimum and input values. The sensitivity of the optimal cross-section height is plotted in Figure 1.4 for different values of the input parameters. Figure 1.4: Sensitivity analysis for hopt. By formulating the objective function as in Eq. (1.5), it is assumed that the structure is re- placed if it fails within the 50 years life cycle and therefore the construction costs have to be paid Page 21 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety (assuming we would then implement the same design variable h and the parameters of the cost function, C0 and CI are still valid) in addition to the damages to property beyond the construction itself, H. 1.3. Interest What we did not take into account so far in this simple example is the fact that the expenses that are contained in the objective function take place at different times. I.e. when the structure is constructed and when the structure fails. In order to be comparable, the expenses have to be transformed to the value at the same point in time. The logical choice is the time when the decision takes place, i.e. when the structure is constructed that is now, presently. The failure costs take place in the future and have to be transferred back in time, given an assumed interest rate. 100 i = 0.01 90 i = 0.03 i = 0.05 Present value of 100 MU at time 80 70 60 50 40 30 20 10 0 0 50 100 150 200 in [years] Figure 1.5: Illustration of the net present value: The graph shows the monetary units (MU) at time τ that correspond to 100 MU at present. The three lines correspond to the three different interest rates i = 0.01, i = 0.03, i = 0.05. The net present value (NPV) of the cost Ct=τ that is generated in τ years can be generally transferred back as: 1 Ct=0 = Ct=τ (1.7) (1 + i)τ where i is the expected average interest rate. See the illustration in Figure (1.5). The objective function in Eq. (1.5) can be modified correspondingly, i.e. all cost terms that are paid in the future are scaled down by (1 + i)−τ : 1 E [Ctot ] (h) = Cconstr (h) + E [Cf,U LS ] (h) (1 + i)τ 1 (1.8) = [C0 + CI h] + [Cconstr (h) + H] Pf (h) (1 + i)τ Page 22 Chapter 1. Structural optimization Accounting for the interest has a significant effect on the result. Let us consider the beam example above. Assuming an interest rate of 3%, i.e. i = 0.03 and an average failure time after 25 years, the result presented in Figure 1.6 is obtained. Compared to the result without interest, both, the optimal height and the reliability are smaller. (hopt = 757 mm instead of 781 mm and βopt = 3.16 instead of 3.36). 104 10 8 6 4 2 0 500 700 900 1100 1300 1500 Figure 1.6: Expected monetary costs for net present value. QUESTION: Why is hopt and the optimal reliability index βopt getting smaller when the interest is considered? (space for your notes) 1.4. Continuous renewal assumption In the example above, a simple structure was optimized for single use in a finite service period of 50 years. This perspective is valid if we want to optimize for the owner of the structure and for the case where the specific structure is demolished after the planned service period. A more general perspective is the perspective from legislation and code writing. We are more generally interested in the reliability level that structures should have, from a societal perspective, in order to specify the target reliability level for structural codes and standards. If we do so, the structures are no longer optimized individually for their finite service periods. What is optimized is the decision on structures that support a societal activity. It is assumed that the societal activities are continuously ongoing, whereas the structures that support these activities are continuously renewed. Hereby, the act of renewal can become necessary for two reasons: Page 23 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety a) the structure becomes obsolete and is replaced by a newer, more modern one; b) the structure failed. The optimization is then performed on a yearly basis (with yearly probability of failure and yearly renewal costs). Regarding the simple previous beam example and representing the height of the beam h with a design parameter h = p, the interest rate with γ = i the objective function would then be written as: 1 1 E [Ctot (p)] = Cconstr (p) + E [Cf (p)] + E [Cobs (p)] γ γ (1.9) (1a) λPf (p) ω = [C0 + CI p] + [C0 + CI p + H] γ + [C0 + CI p + D] γ Equation (1.9) consist of the following: Construction costs: Cconst (p) = C0 + CI p are the construction cost with a part CI p that is proportional to p, and a part C0 that is independent of p. (1a) Expected annual failure cost: E [Cf (p)] = [Cconst (p) + H] λPf (p), here H is intro- duced as a nonstructural failure cost, the construction costs are part of the failure cost as a similar decision (similar investment into p) is expected after failure. The failure cost are considered as annual expectations, so they are multiplied by the annual failure probability (1a) Pf (p). λ is introduced as the rate of a Poisson process with λ = 1 in order to interpret the annual failure probability as a rate of occurrence. γ is the annual interest rate that is selected as the societal interest rate γ = γS if the preferences of the society are represented, or as the private interest rate γ = γE , if the optimisation is done relative to entrepreneurial preferences. Expected obsolescence cost: E [Cobs ] = [Cconst (p) + D] ω, here D represents the demo- lition cost and ω is representing the expected rate at which the decision on p becomes obsolete. The design parameter p = p∗ that is minimising the expected total cost is found by (1a) ( ) d λPf (p) ω C0 + CI p + [C0 + CI p + H] + [C0 + CI p + D] ≡0 dp γ γ p=p∗ (1a) C 0 + C I p∗ + H 1 + Pf (p∗ ) γ1 + ω γ (1.10) ⇒ = CI dP (1a) (p) 1 − fdp γ p=p∗ A simplified approximation can be formulated by considering that for typical structural en- gineering decision situations it is Pf (p∗ ) ≪ ω + γ: (1a) CI · (γ + ω) dPf (p) ≈ − (1.11) C 0 + C I p∗ + H dp p=p∗ Page 24 Chapter 1. Structural optimization Note that the expression for the ratio between the marginal safety costs (CI ) and the total failure costs (C0 +CI p∗ +H) can be further simplified. Usually, CI p∗ ≪ C0 since the construction costs are dominated by the fixed costs. In these cases, the ratio on the left side of 1.11 is approximately equal to CI /(C0 + H). (1a) CI · (γ + ω) dPf (p) ≈− (1.12) C0 + H dp ∗ p=p The optimal design p∗ can be estimated from Eq. (1.11) and (1.12) when the functional relation- (1a) ship Pf (p) is known analytically in the vicinity of p = p∗. In other cases, the equation must be solved numerically. It is interesting to note that the optimal choice of p does only depend on (1a) (1a) dPf (p) /dp and not on the absolute values of Pf (p), i.e. in the optimisation subject to p it (1a) is only necessary to represent the gradual change of Pf by changing p. This makes the results of the optimisation insensitive against the non-inclusion of failure scenarios those probability of occurrence cannot be influenced by gradually changing p. An example is the potential presence of gross human error. This definitely has a strong effect on the estimated absolute probability of failure, however, if it is assumed that the probability of failure conditional on gross human error is not (or very weakly) influenced by the particular choice of p, the optimum choice of p = p∗ derived by neglecting gross human error is still valid. A reliability requirement can be determined from the above optimisation, i.e. if the choice of (1a) (1a) p = p∗ is minimising the expected total costs, then Pf (p∗ ) and β(p∗ ) = −Φ−1 (Pf (p∗ )) are the corresponding failure probability and reliability index of the optimal choice. If for a decision problem at hand, it can be assumed that the attributes of the decision problem are sufficiently similar to the assumed attributes contained in Eq. (1.11), this probability / reliability should be chosen as a target. However, since such a target is expressed in terms of an absolute value it is of high importance that the reliability target is only applied to sufficiently similar scenarios. The example will be worked further in the exercise class. 1.5. Risk acceptance The economic optimisation that is obtained from Eq. (1.11) ensures that resources are ex- pended optimally from a financial point of view. But it does not guarantee that the optimum decision p = p∗ is consistent with the societal preferences in regard to life safety. I.e. if human lives are at risks it has to be ensured that societal resources are allocated efficiently for preventing possible fatalities associated with structural failure. The marginal life-saving costs principle can be applied to derive risk acceptance limits (ISO 2015; Blomquist 1979). The principle ensures that the societal resources are allocated to efficient risk-reducing measures, i.e. measures that can save one additional life at a cost that the society is willing (or able) to pay. This is here re- ferred to as the Societal Willingness To Pay (SWTP). The SWTP is multiplied with the expected number of fatalities given structural failure and inserted to the objective function in Eq. (1.8) which leads to the specification of the acceptable domain in Eq. (1.13), ISO 2015; Köhler et al. 2019. (1a) dPf (p) CI (γS + ω) − ≤ = K1 (1.13) dp SW T P · NF Page 25 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety The constant K1 is introduced in (ISO 2015) as an indicator. Here, also typical values of K1 are given. The minimum acceptable design (pacc ) just satisfies the inequality in Eq. (1.13) and depends on:. (1a) 1. the uncertainty involved in the problem through the term dPf (p) dp, 2. the marginal safety costs CI measured in monetary units, 3. the societal ability to pay for saving one statistical life SW T P , measured in monetary units, 4. the number of expected fatalities given structural failure NF , and 5. the obsolescence rate ω and the societal interest rate γS. The condition for which the optimal design is within the acceptable domain (i.e. p∗ ≥ pacc ) is obtained inserting Eq. (1.13) into Eq. (1.12) : CI · (γS + ω) ≤ K1 (1.14) C0 + H How the SWTP can be quantified is discussed in the following: 1.5.1. The societal willingness to pay for saving one additional life (SWTP) The SW T P corresponds to the amount of money that should be invested into saving one addi- tional life for a given life risk reduction activity. The SW T P is expressed in monetary terms and can be derived e.g. by means of the Life Quality Index (LQI) Nathwani et al. 1997. The LQI index informs about the revealed preferences of the society for investments into life safety by estimating the ability of a national economy to allocate monetary resources in risk reducing activities. Values of the SW T P that are derived from the LQI index are shown in Table 1.3 for some selected countries, a more complete Table is given in ISO 2015. The values in the Table are provided for different societal interest rates γS for considering the discounting of the future costs and benefits of the risk-reducing measures. Table 1.3: SWTP for five selected countries from ISO 2015 for γS equal to 2 %, 3 % and 4 % (all numbers in thousands purchasing power parity US dollars). Country 2008 GDP SWTP γS =2% γS =3% γS =4% Australia 35 624 4 840 4 298 3 843 Brazil 9 517 804 712 634 Norway 49 416 3 937 3 500 3 129 Mali 1 043 54 48 43 US 42 809 3 187 2 833 2 543 The use of monetary units in the context of risk acceptance should not be misunderstood. Indicators such as the cost of a statistical life or the money the society is willing (or better “able”) Page 26 Chapter 1. Structural optimization to pay for saving one additional life (SW T P ) are not to be considered the monetary value of a individual person’s life, which has of course a value that cannot be expressed in monetary terms. The mentioned indicators are derived from small marginal changes in the probability of having a fatality caused by failure. 1.5.2. Expected values The cost terms H, C0 , CI and NF but also ω and γS are in general uncertain. A similar failure event might lead to different consequences depending, for example, on the occupancy of the building. All these terms need to be evaluated up to their expected value only since they affect the expected total cost linearly (compare Equation (1.8)). If in addition it can be assumed that they are independent of each other their probability density functions are irrelevant for the opti- misation, see Raiffa et al. 1961. It follows that the values for the optimisation are not the highest or extreme values associated with the corresponding events, but the expected ones. However, the expectation operator is omitted in the text in order to ease the notation. When evaluating the expected values of H and NF , attention should be paid in cases where the assets and the lives lost in the event of failure might be themselves a significant cause of failure. A representative example might be the failure of a grandstand in a stadium due to static or dynamic load induced by the occupants. In this case, the likelihood of failure increases with the occupancy. This effect should be considered both in the formulation of the objective function and in the representation of the variables. 1.6. Conclusion of Chapter 1 The principles of structural optimisation and risk acceptance criteria for direct structural de- sign and for code-making have been briefly discussed with the aim of providing a transparent background risk and reliability based design. The design requirements are derived by monetary optimisation minimising the expected costs over time. The analysis of the optimisation prob- lem showed the four most important aspects that influence the optimal reliability levels. These aspects are the ratio between total failure costs and marginal safety costs; the uncertainty involved in the problem; the obsolescence rate; and the discounting rate. The acceptability of the monetary optimum is not always satisfied. The acceptance criterion can be derived by the Marginal Life-Saving Cost Principle that was here implemented using the Life Quality Index (LQI). It was demonstrated that the acceptable threshold depends on: the marginal safety costs; the societal willingness to pay for saving one additional life; the number of expected fatalities given failure; and Page 27 Section: Application examples TKT4196 - Aspects of Structural Safety the obsolescence and the societal interest rates. All considerations in this chapter depend on reasonable assumptions for the consequence and for the probability of failure as a function of the design decision variables. Whereas it is sufficient to represent the consequences as expected values, the accurate representation of the failure probability is of utmost importance. Accordingly, more advanced reliability methods and probabilistic load and resistance modelling will be treated further in the upcoming chapters of this lecture. 2. Application examples E1.1. Consider the structure in Figure 1.1. The rectangular cross-section dimensions are paramet- ric on the cross-section height h. The width b and span l are known without uncertainty, being b = 300 mm and l = 10 m. The resistance fm and load Q are random variables, whose distri- butions were given in Table 1.2. In order to get started with a Python script you may import the necessary Python packages and do some presettings for the output: 1 import numpy as np 2 import scipy as sp 3 import matplotlib. pyplot as plt 4 import scipy. stats 5 fontsizes = 18 6 plt. rcParams. update ( { ’ font. size ’: fontsizes } ) 7 plt. rcParams. update ( { " font. family " : " serif " } ) 8 plt. rcParams. update ( { " mathtext. fontset " : " cm " } ) 9 plt. rcParams. update ( { ’ font. serif ’: ’ Times New Roman ’} ) 10 plt. close ( ’ all ’) Then you specify some input parameters to the problem and compute the failure probability and the reliability index as functions of the height h: 1 # Geometry 2 l = 10000 # [ mm ] span 3 b = 300 # [ mm ] width 4 # ===================================== 5 # Material properties 6 mu_fm = 26. 6 # [ MPa ] mean material resistance 7 cov_fm = 0. 15 # coeff. of variation 8 std_fm = mu_fm * cov_fm # [ MPa ] standard deviation 9 10 mu_R = mu_fm * b / 6 11 std_R = std_fm * b / 6 12 # ====================================== 13 # Load 14 mu_q = 24. 1 # [ N / mm ] mean load Page 28 Chapter 1. Structural optimization 15 cov_q = 0. 3 # coeff. of variation 16 std_q = mu_q * cov_q # [ MPa ] standard deviation 17 18 mu_S = ( l ** 2 / 8 ) * mu_q 19 std_S = ( l ** 2 / 8 ) * std_q 20 # ======================================= 21 # Cost model 22 c_gl = 4000 * 1E - 9 # [ NOK / mm ^ 3 ] Cost Glulam 23 C0 = 50000 # [ NOK ] Fixed cost of construction 24 H = C0 * 20 # Direct cost of failure 25 C1 = l * b * c_gl # [ NOK / mm ] Variable cost 26 i = 0. 03 # interest rate 27 T = 25 # Service life 28 C_tau = 1 / (( 1 + i ) ** T ) # discount factor 29 30 # ===================================== 31 # Reliability index as a function of the decision variable 32 BETA = lambda h , mu_S , std_S : ( mu_R * h ** 2 - mu_S ) / ((( std_R * h ** 2 ) ** 2 + ( std_S ) ** 2 ) ** 0. 5 ) 33 h1 = np. linspace ( 500 , 1500 , num = 10000 ) 34 beta = BETA ( h1 , mu_S , std_S ) 35 PF = sp. stats. norm. cdf ( - beta ) The results can now be plotted: 1 # Plot 2 plt. figure () 3 4 plt. subplot ( 121 ) 5 plt. plot ( h1 , beta , color = ’ black ’ , lw = 2 ) 6 plt. xlabel ( ’ $h$ [ m ] ’ , fontsize = fontsizes ) 7 plt. ylabel ( r ’$ \ beta$ ’ , fontsize = fontsizes ) 8 plt. xlim ( 500 , 1500 ) 9 plt. ylim (0 , 6 ) 10 plt. subplot ( 122 ) 11 plt. plot ( h1 , PF , color = ’ black ’ , lw = 2 ) 12 plt. yscale ( ’ log ’) 13 plt. xlabel ( ’ $h$ [ m ] ’ , fontsize = fontsizes ) 14 plt. ylabel ( ’ $P_f$ ’ , fontsize = fontsizes ) 15 plt. xlim ( 500 , 1500 ) 16 plt. ylim ( 1e - 10 , 1e0 ) 17 plt. tight_layout () 18 plt. show () Output: Page 29 Section: Application examples TKT4196 - Aspects of Structural Safety E1.1.1. Structure with finite life (owner perspective) – neglecting interest rate Make your own Python script to reproduce the results in Figure 1.3. The minimum expected costs are obtained by minimization: 1 # Not discounted costs 2 C_c = C0 + C1 * h1 # Construction cost 3 EC_f = ( C_c + H ) * PF # Expected failure cost 4 ECtot = C_c + EC_f # Objective function 5 # Compute optimum 6 idx_opt = ECtot. tolist (). index ( min ( ECtot ) ) 7 h_opt = h1 [ idx_opt ] 8 beta_opt = beta [ idx_opt ] 9 Pf_opt = PF [ idx_opt ] 10 ECtot_min = ECtot [ idx_opt ] 11 # print (" Optimum cross - section height : %. 2f m " % ( h_opt ) ) 12 s1 = " Optimum cross - section height : { h :. 0f } m \ n ". format ( h = h_opt ) 13 s2 = " Optimum probability of failure : { pf :. 2e } \ nOptimum beta : { b :. 2f } \ n ". format ( pf = Pf_opt , b = beta_opt ) 14 s3 = " Minimum expected total cost = { c :. 0f } NOK \ n ". format ( c = ECtot_min ) 15 print ( s1 + s2 + s3 ) Optimum cross-section height: 781 m Optimum probability of failure: 3.84e-04 Optimum beta: 3.36 Minimum expected total cost = 59778 NOK And plot the objective function. 1 # Plot function 2 def pltECost ( ECT_list , EC_f_list , EC_list ) : 3 # plt. figure ( idx_plt ) 4 plt. plot ( h1 , ECT_list [ 0 ] , ECT_list [ 1 ] , label = ECT_list [ 2 ] ) 5 plt. plot ( h1 , EC_f_list [ 0 ] , EC_f_list [ 1 ] , label = EC_f_list [ 2 ] ) 6 plt. plot ( h1 , EC_list [ 0 ] , EC_list [ 1 ] , label = EC_list [ 2 ] ) 7 plt. ylim (0 , 200000 ) 8 plt. xlim ( 500 , 1500 ) 9 plt. xlabel ( ’ $h$ [ m ] ’ , fontsize = fontsizes ) 10 plt. ylabel ( ’$ \ mathrm { E } [ C_ { tot } ] $ ’ , fontsize = fontsizes ) 11 plt. tight_layout () Page 30 Chapter 1. Structural optimization 12 plt. ticklabel_format ( style = ’ sci ’ , useMathText = True ) 13 return 14 15 # Plot undiscounted costs 16 plt. figure () 17 pltECost ( [ ECtot , ’r ’ , ’ Undiscounted $ \ mathrm { E } [ C_T ] $ ’] ,[ EC_f , ’b ’ , ’ Undiscounted $ \ mathrm { E } [ C_F ] $ ’] ,[ C_c , ’k ’ , ’ $C_c$ ’] ) 18 # plt. plot ( np. array ([ h_opt , h_opt , 0 ]) , np. array ([ 0 , ECtot_min , ECtot_min ]) , ’: b ’) 19 plt. plot ( h_opt , ECtot_min , ’ or ’) 20 plt. show () Output: E1.1.2. Structure with finite life (owner perspective) – present value of costs Optimize the cross-section of the beam from the perspective of the owner, accounting for an interest rate of 3% and an average failure time τ of 25 years, i.e. reproduce the results in Fig- ure 1.6. E1.1.3. Structures with infinite renewal (societal perspective) – present value of costs The optimization is now performed from the societal point of view. Under the infinite renewal assumption, structures are considered to be re-constructed after failure and demolished and re- constructed once they become obsolete. Estimate the minimum expected cost and the optimal cross-section, probability of failure Pf,50yr and reliability β50yr (related to a 50 year period) by following this perspective. Some additional values to be used in the calculation are collected in Table 1.4. The 1 year max- ima distributed load Q is assumed to be Normal with parameters µQ,1yr = 15.20 N/mm and σQ,1yr = 6.11 N/mm. Solutions The expected risk is plotted in Figure 1.7. The optimum cross-section height is hopt = 753 mm. The reliability and failure probability associated with a 50 year period are computed assuming that failure between different years is independent. They result in β50yr = 3.12 and Pf,50yr = 9.09 · 10−4 , respectively. Page 31 Section: Application examples TKT4196 - Aspects of Structural Safety Table 1.4: Assumed values and relations. Variable Symbol Value, relations and units Demolition costs D rDC0 · C0 = 0.2C0 Design life T 50 years Interest rate i 3% Rate of obsolescence ω 1/T Societal interest rate γs 2% Gross domestic product g (ISO 2398:) Societal Willingness To Pay to save an 32,100,000 NOK (ISO 2394:2015 Table SW T P additional life G.2, delta-regime) 5 10 2 1.5 1 0.5 0 500 700 900 1100 1300 1500 Figure 1.7: Expected cost assuming infinite renewal (E1.1.3). E1.1.4. Risk acceptance, part 1 The principle of the marginal life-saving costs is applied to this example in order to evaluate the maximum accepted risk. The marginal life-saving cost principle is implemented through the use of the life quality index (LQI). The definition of acceptability is based on the requirement that all efficient life-saving measures have to be performed for societal risk acceptance. The accept- able region (i.e. acceptable values of h in our example) is equivalent to sufficient expenditure for life saving investments. The acceptable region is defined by the values of h satisfying the following inequality in Eq. (1.13). The criterion depends on the expected number of fatalities given structural failure NF. Define the boundaries of the acceptable region for NF = 0.001 and recalculate, hopt , if necessary. Solutions The solution is presented in Figure 1.8. Assuming NF = 0.001, the acceptable region is h > 656 mm. This means that the optimal design found minimizing the expected costs is acceptable. Therefore, h∗ = hopt = 753 mm simultaneously minimizes monetary costs and ensures an acceptable level of risk. Page 32 Chapter 1. Structural optimization Figure 1.8: Monetary optimization with acceptance criteria NF = 0.001 (E1.1.4). E1.1.5. Risk acceptance, part 2 Re-do the calculations in the previous example for NF = 10. Solutions The solution is presented in Figure 1.9. Assuming NF = 10 the acceptable region is h > hmin = 971 mm, meaning that the optimal design found minimizing the expected costs, i.e. hopt = 753 mm is not acceptable. It follows that the design that simultaneously minimizes monetary costs and ensures an acceptable level of risk corresponds to hopt = hmin = 971 mm. Figure 1.9: Monetary optimization with acceptance criteria NF = 10 (E1.1.5). Page 33 Section: Practical exercises TKT4196 - Aspects of Structural Safety 3. Practical exercises P1.1. Fixed-end beam optimization Consider the beam in Figure 1.10. In this exercise, the beam is fixed at both ends. The span and the square cross-section are l = 10 m and b = 300 mm, as before. The basic random variables are also Normal distributed with mean and coefficient of variation given in Table 1.1. All other inputs are given in the example presented in the lecture. Figure 1.10: Boundary conditions of the beam in P1.1. P1.1.1. Fixed-end beam optimization Find the new optimum beam height h assuming finite life and using discounted costs. Solutions The new optimum cross-section height is lower than for the simply supported beam: hopt = 623 mm. However, it yields a higher optimum reliability index: βopt = 3.22. This is due to the fact that the load effects of this boundary conditions are ca. 66% lower than for the simply supported beam. P1.1.2. Additional efficient investment Fixing the extremes of a beam is a more costly activity when compared to simply supporting them. The costs of producing a fix joint for beams of relatively similar cross-section dimensions can be assumed to be equal. Therefore, this cost can be considered part of the fixed construction costs C0. Calculate the maximum additional investment that is sensible to make in fixing the extremes of the beam before it becomes a less optimum alternative. Solutions The maximum additional investment in the fixed cost is found to be 2003 NOK. Note that this solution provides a higher reliability (β = 3.22) for the same expected optimum cost. P1.2. Uncertainty in the modelling of the supports A visual inspection of the beam that was designed in the previous practical exercise (see Figure 1.10) is conducted after construction due found defects in the joints. The uncertainties associated with the outcomes of the evaluation are given by the probability of both joints not Page 34 Chapter 1. Structural optimization Figure 1.11: Lost of stiffness. being stiff enough being Pjoint ≈ 10−1. This is equivalent to having a simply supported static scheme instead of having the ends fixed, see Figure 1.11. According to the the law of total probability, the probability of failure is now: Pf∗ = Pr (stiff joints ∩ (MR (hopt ) ≤ MQ |fix-fix)) + (1.15) Pr (non-stiff joints ∩ (MR (hopt ) ≤ MQ |hinge-hinge)) For independent events, it can be rewritten as: Pf∗ = Pr (stiff joints) · Pr (MR (hopt ) ≤ MQ |fix-fix) + (1.16) Pr (non-stiff joints) · Pr (MR (hopt ) ≤ MQ |hinge-hinge) 1. Evaluate the probability of failure and the reliability index, and comment the obtained values. 2. Evaluate, the total expected costs and compare them to the ones obtained in P1.1.1. Com- ment the results. Evaluate and compare the expected failure costs only. 3. Imagine the reparation of the joints will cost ν NOK, what can be a possible decision criterion for deciding if repairing the joints or not? Page 35 Section: Practical exercises TKT4196 - Aspects of Structural Safety Question:: How safe is safe enough? Describe how a reliability criterion may be established based on optimization. Give an example for a possible objective function Represent the objective function graphically and explain the shape of the graph. What is the difference between the optimization of a single structure and the optimiza- tion of a large portfolio of (similar) structures? How are the objective functions differ- ent? How can the concept be adapted in order to represent societal preferences to prevent human fatalities? Page 36 Chapter 2 Structural reliability - Problem Statement and Monte Carlo Method The fundamental structural reliability problem is revisited and extended to linear limit states of n normal distributed random variables. The Monte Carlo simulation method is introduced as the simplest way to approach more realistic reliability problems. It is demonstrated how samples of correlated random variables can be generated. Latin Hypercube Sampling is introduced as a variance reduction method for Monte Carlo sim- ulations. 1. Theoretical aspects 1.1. Prelude Probability of failure is, as seen in the previous chapter, a crucial factor within the assessment of cost optimality and risk acceptability of civil engineering structures. In the introductory ex- amples, we estimated the failure probability under strong assumptions. Namely, we represented failure by a linear limit state function (R − S) of Normal distributed variables and independent resistance and load. However, the physical phenomena that generate structural resistance and structural loads are not very well represented by the Normal distribution and other distribution types should be used. Many relevant failure scenarios are better represented by non-linear limit state functions and the different variables of a structural problem might be correlated. In short, practical engineering situations are more complex and cannot be represented by the strong simplifications that we applied in the previous chapter. In this chapter, methods that allow for a more realistic represen- tation of practical engineering situations are introduced. 1.2. Introduction: The limit state principle The safety of an engineering structure depends on the type and magnitude of the applied load and the structural strength and stiffness. Whether the performance is considered satisfactory depends on the requirements which must be satisfied. Among others, these include reliability of 37 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety the structure against collapse, limitation of damages or of deflections, or other criteria. In general, any state that may be associated with consequences in terms of costs, loss of lives and impact to the environment are of interest. In the following, it is not differentiated between these different types of states. For simplicity, it is referred to all of them as being failure events. It is convenient to describe failure events in terms of functional relations, which if fulfilled, define that the failure event F will occur: F = {g (x) ≤ 0} (2.1) where g (x) is denominated as the limit state function. The components of the vector x are the realizations of the so-called basic random variables X, representing all relevant uncertainties influencing the problem at hand1. The failure event F is defined as the set of realizations of the limit state function g (x), which are zero or negative. Some typical limit states are given in Table 2.1. Table 2.1: Typical limit states for structures. Limit State Type Description Examples Ultimate Collapse of the structure Rupture, plastic mechanism, instability, or part of it progressive collapse, fatigue, deteriora- tion, fire. Damage (often included in above) Excessive permanent cracking, perma- nent irreversible deformation. Serviceability Disruption of normal use Excessive deflection, vibration, local damage. Due to the associated consequences, failure events linked with the most serious limit states, such as collapse and major damage, should be relatively rare events. The study of structural reliability is concerned with the assessment of the probability of failure of engineered structures related events at any stage during its service life. The probability of occurrence of a failure event is a measure of the chance of its occurrence. This chance may be quantified by observing the long term frequency of the event for gener- ally similar structures or components, or may be simply a subjective estimate of its numerical value. Engineering structures are mostly exclusive in regard to the structural assembly and their exposure to loads and environment, which in general precludes the assignment of relative fre- quencies of events within many similar structures. However, a more generic description can be given for structural components and materials. Thus, a combination of subjective estimates and frequency observations about structural components and structural assemblies is utilized in practice to assess the probability of limit state violation of a structure. The probability of failure pf may be generally determined by the following integral: Z pf = Pr (g (X) ≤ 0) = fX (x) dx (2.2) g(x)≤0 1 Note that in general a capital letter is used for the Random Variables X, and a small letter is used for the realisations x. Page 38 Chapter 2. Structural reliability - Problem Statement and Monte Carlo Method where g (X) is the limit state function, X is a vector of basic random variables and fX (.) is the joint probability function of the variables X. Figure 2.1: Here, the joint probability distribution of two random variables, X1 and X2 , is illustrated. The reliability integral is the volume that is “cut away” by the limit state function g (x1 , x2 ) = 0. 1.3. A simple case - the Fundamental Reliability Problem Several methods can be found in the literature to calculate the probability of failure by solving the integral in Eq. (2.2). An overview and a discussion of different approaches is presented in e.g. J.Schneider 2006. The most straightforward method is that of Monte Carlo simulation, while probably the more efficient are the so-called approximate methods based on the calculation of the reliability index β, e.g. the first order reliability method (FORM) and the second order reliability method (SORM). There are also methods to increase the efficiency of the Monte Carlo simulation like Importance Sampling or Adaptive Sampling. In this chapter and the next one, only the main ideas of Monte Carlo Simulation and FORM are briefly outlined. Linear Limit State Functions and Normal Distributed Variables First, we consider the two dimensional case (i.e. two basic random variables), which is aligned to the simplification we applied in the previous chapter. A limit state function can be g (r, s) = r − s, and the corresponding safety margin is M = R − S. For R and S uncorrelated, the reliability index is: Page 39 Section: Theoretical aspects TKT4196 - Aspects of Structural Safety E [M ] E [R] − E [S] µR − µS βC = =p =p 2 (2.3) D [M ] V ar [R] + V ar [S] σR + σS2 where E [M ] is the expected value of M and D [M ] is its standard deviation. In the multidi- mensional space (i.e. a linear combination of more than two basic random variables), the limit state function is defined as: n X g (x) = a0 + ai x i = a0 + a T x (2.4) i=1 where a vector notation has been introduced; aT is a row vector with elements ai , and x is a column vector with elements xi. The so-called safety margin M is defined as: n X M = a0 + ai X i = a0 + a T X (2.5) i=1 M is Normal distributed with expected value E [M ] = a0 + ni=1 ai E [Xi ] = a0 + aT E [X] P √ and standard deviation D [M ] = aT CX a. Here E [X] is the vector of expected values (with 1 × n elements): E [X] = [µX1 , µX2 ,..., µXn ] (2.6) and CX is the matrix of covariance of X (symmetric, with n × n elements). For uncorrelated basic random variables it is: 2   σX 1 0 ··· 0 2  0 σX 2 ··· 0  CX =   .........  (2.7) ...  2 0 0 · · · σXn The reliability index can be written in a matrix form as: E [M ] a0 + aT E [X] βC = = √ T (2.8) D [M ] a CX a Defining the failure event as in Eq. (2.1), the probability of failure can be defined as: pf = P (g (X) ≤ 0) = P (M ≤ 0) = Φ (−βC ) (2.9) If the two-dimensional case of two independent Normal distributed random variables is con- sidered, the reliability index β has a geometrical interpretation as illustrated in Figure 2.2. Page 40 Chapter 2. Structural reliability - Problem Statement and Monte Carlo Method Figure 2.2: The safety margin M = R − S for the case of independent and Normal distributed R and S. The reliability index β defines the mean value of M as a function of its standard deviation: µM = βσM. 1.4. Definitions Second-moment reliability theory: reliability theory based on the representation of uncer- tainties solely in terms of expected values (first moments) and covariance (second moments). Basic variables: fundamental variables which define and characterize the behaviour and safety of a structure. They are denoted b

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