1、Chapter 2,Basic Concepts of Structural Reliability Theory,Chapter 2: Basic Concepts of Structural Reliability Theory,2.2 Failure Probability of Structures,2.3 Reliability Index of Structures,2.1 Definitions of Structural Reliability,2.5 Relationship between Reliability Index and Safety Factor,Conten
2、ts,2.4 Geometric Meaning of Reliability Index,2.1 Definitions of Structural Reliability,Chapter 2 Basic Concepts of Structural Reliability Theory,2.1 Definitions of Structural Reliability 1,Structural Reliability is the ability of a structure or structural element to fulfill the specified performanc
3、e requirements under the prescribed conditions during the prescribed time.,2.1.1 Reliability of Structures,1. The Prescribed Time,the prescribed time = the design working lifethe design working life the design reference period,Design working life is assumed period for which a structure or a structur
4、al element is to be be used for its intended purpose without major repair being necessary.,Design reference period is a chosen period of time which is used as a basis for assessing values of variable actions, time-dependent material properties, etc.,2.1 Definitions of Structural Reliability 2,Table2
5、.1 Design Working Life,2.1 Definitions of Structural Reliability 3,2. The Prescribed Conditions,3. The Specified Performance Requirements,Ultimate limit state requirement,They shall withstand extreme and/or frequently repeated actions occurring during their construction and anticipated use.,Servicea
6、bility limit state requirement,They shall perform adequately under all expected actions.,Structural robustness requirement,They shall not be damaged by events like flood, land slip, fire, explosion, impact, earthquake, tornado, etc.,Structural durability requirement,They shall remain fit for use dur
7、ing their design working lives in their environment, given appropriate maintenance.,2.1 Definitions of Structural Reliability 4,A limit state is a state beyond which a structure or part of it no longer satisfies the design performance requirements.,2.1.2 Limit States of Structures,1. Definition of L
8、imit State,The concept of a limit state is used to help define failure in the context of structural reliability theory.,We could say that a structure fails if it cannot perform its intended function, or violation of a limit state.,A limit state is a boundary between desired and undesired performance
9、 of a structure.,The boundary between desired and undesired performance of a structure is often represented mathematically by a limit state function or performance function.,2.1 Definitions of Structural Reliability 5,2. Types of Limit State Classification 1,Ultimate limit states are mostly related
10、to the loss of load-carrying capacity. Examples of modes of failure in this category include:,(1) Ultimate Limit State (ULS),Exceeding the moment carrying capacityFormation of a plastic hingeCrushing of concrete in compressionShear failure of the web in a steel beamLoss of the overall stabilityBuckl
11、ing of flangeBuckling of webWeld ruptureLoss of foundation load-carrying,2.1 Definitions of Structural Reliability 6,Serviceability limit states are mostly related to gradual deterioration, users comfort, or maintenance. They may or may not be directly related to structural integrity. Examples of mo
12、des of failure in this category include:,(2) Serviceability Limit State (SLS),Excess deflectionExcess vibrationPermanent deformationsCracking,2.1 Definitions of Structural Reliability 7,(3) Collapse Limit State (CLS),ULS is for the largest load-carrying capacity of intact structures, while CLS is fo
13、r that of local damaged structures.,Failure-Safe or Conditional Limit State,Transformation of the local damaged structure into a mechanism.The key parts of a local damaged structure fail when the strength of material is exceeded.Instability of the elements of a local damaged structure (e.g. buckling
14、).Loss of equilibrium of the local damaged structure or of a part of it, considered as a rigid body.,2.1 Definitions of Structural Reliability 8,(3) Collapse Limit State (CLS),2.1 Definitions of Structural Reliability 9,(3) Collapse Limit State (CLS),2.1 Definitions of Structural Reliability 10,(3)
15、Collapse Limit State (CLS),2.1 Definitions of Structural Reliability 11,(3) Collapse Limit State (CLS),(March, 2007),(Jan., 2002),2.1 Definitions of Structural Reliability 12,2. Types of Limit State Classification 2,Irreversible limit state is a limit state which will remain permanently exceeded whe
16、n the actions which caused the excess are removed.,(1) Irreversible Limit State,The effect of exceeding an ultimate limit state is almost always irreversible and the first time that this occurs it causes failure.In the cases of permanent local damage or permanent unacceptable deformations, exceeding
17、 a serviceability limit state is irreversible and the first time that this occurs it causes failure.,Reversible limit state is a limit state which will not be exceeded when the actions which caused the excess are removed.,(2) Reversible Limit State,In the cases of temporary local damage, temporary l
18、arge deformations and vibrations, exceeding a serviceability limit state is reversible.,2.1 Definitions of Structural Reliability 13,2.1.3 Limit State Functions (Performance Functions),1. Form of Comprehensive Variables,In general, the factors influencing structural reliability can be put into two k
19、inds of comprehensive variables, that is, load effect S (demand) and structural resistance R (supply).,A performance function, or limit state function, can be defined as,The limit state equation is defined as,2.1 Definitions of Structural Reliability 14,Each limit state function is associated with a
20、 particular limit state.,Even for a particular limit state, its limit state functions may be different.,Different limit states may have different limit state functions.,Z is also named safety margin or margin of safety.,Properties of Limit State Functions,2.1 Definitions of Structural Reliability 15
21、,2. Form of Basic Variables,In general, the performance function can be a function of many variables: load components, environment influence, resistance parameters, material properties, dimensions, analysis factors, and so on.,The above random variables are called basic random variables, or basic va
22、riables.,The general form of a performance function:,The basic variable space is a specified set of basic variables. It is also called state space:,The general form of a limit state equation:,2.1 Definitions of Structural Reliability 16,2.1.4 Measures of Structural Reliability,1. Safety (Survival) P
23、robability of Structures,The probability of the event that the structure is to be safe is defined as safety probability, or survival probability. Mathematically,The probability of the event that the structure is to be unsafe is defined as failure probability. Mathematically,2. Failure Probability of
24、 Structures,2.1 Definitions of Structural Reliability 17,The reliability index of structures is defined as,3. Reliability Index of Structures,Relationship between and,where, is the inverse standardized normal distribution function.,2.2 Failure Probability of Structures,Chapter 2 Basic Concepts of St
25、ructural Reliability Theory,2.2 Failure Probability of Structures 1,2.2.1 General Basic Variables,1. Assumptions,Consider the performance function,where,is a n-dimension random vector.,It is assumed that the joint PDF of is,2. Formula,The failure probability of structures with general basic variable
26、s:,2.2 Failure Probability of Structures 2,In the above formula, is called as failure domain of structures.,3. Calculation Method,Analytical Method,Simulation Method,Integral Method,Precise Analytical Method:,FOSM,SORM,Probability Interference Method (PIM),Approximate Analytical Method,Monte Carlo M
27、ethod (MCS),Latin Hypercube Sampling,Important Sampling,FORM,2.2 Failure Probability of Structures 3,2.2.2 Two Comprehensive Variables,1. Assumptions,(1) S is the total load effect, known:,2. Probability Interference Method (PIM),The limit state function:,(2) R is the member resistance, known:,(3) R
28、 and S are statistical independent, that is,The failure domain:,2.2 Failure Probability of Structures 4,2. Probability Interference Method (PIM) ,The failure probability:,2.2 Failure Probability of Structures 5,Formula 2 of PIM,First integration for s, then for r,Formula 1 of PIM,First integration f
29、or r, then for s,2.2 Failure Probability of Structures 6,3. Physical Meaning of Probability Interference,The probability of load effect S being in space :,The probability of resistance R being less than a specific load effect s:,Since R & S are statistical independent, the probability of R & S simul
30、taneously occurring in space is:,2.2 Failure Probability of Structures 7,Actually, it has been verified that there exists a relationship as follows:,2.2 Failure Probability of Structures 8,Example 2.1,Consider the performance function of the square section strength of a structural element,where R an
31、d S are random variables.,The distribution parameters of R & S are shown below:,R is a normal RV,S is a exponent RV,The PDF of S is,Calculate the failure probability of the element.,2.2 Failure Probability of Structures 9,Solution:,Let, then,2.2 Failure Probability of Structures 10,If we place the p
32、ractical values of , , , into the above formula, then we will obtain the failure probability:,End of Example 2.1,2.3 Reliability Index of Structures,Chapter 2 Basic Concepts of Structural Reliability Theory,2.3 Reliability Index of Structures 1,2.3.1 R & S are Independent Normal Variables,1. Assumpt
33、ions,Consider the performance function,where,are normal random variables.,2. Formula,Since R & S are all normal random variables, then we know that Z is also a normal RV. Therefore, we have,2.3 Reliability Index of Structures 2,The failure probability is:,Let , then,2.3 Reliability Index of Structur
34、es 3,Let , then,Formula of Reliability Index,. is called Reliability Index., format short e beta = 1.0:0.5:5.0; Pf = normcdf( -beta,0,1),MATLAB Programs, n=1:9 Pf = 10.(-n); beta = -norminv(Pf),2.3 Reliability Index of Structures 4,Table2.2 Relationship between and,2.3 Reliability Index of Structure
35、s 5,2.3.2 R & S are Independent Lognormal Variables,1. Assumptions,Consider the performance function,where,are lognormal random variables.,2. Formula,2.3 Reliability Index of Structures 6,Example 2.2,Consider the performance function of a structural element ,Calculate the reliability index of the el
36、ement.,2.3 Reliability Index of Structures 7,where R and S are normal random variables.,Solution:,Example 2.3,Consider the performance function of a structural element ,Calculate the reliability index of the element.,2.3 Reliability Index of Structures 8,where R and S are lognormal random variables.
37、,Solution:,2.3 Reliability Index of Structures 9,If we use the approximate formula, then we will have,The relative error of these two methods is:,2.4 Geometric Meaning of Reliability Index,Chapter 2 Basic Concepts of Structural Reliability Theory,2.4 Geometric Meaning of Reliability Index 1,2.4.1 Re
38、duced Variables,The standard forms of the basic variables R & S can be expressed as:,The variables and are called reduced variables .,2.4 Geometric Meaning of Reliability Index 2,We can find a straight line equation in the space of reduced variables:,The above equation can be transformed into a norm
39、al equation :,2.4 Geometric Meaning of Reliability Index 3,2.4.2 Geometric Meaning of Reliability Index,The Reliability Index is the shortest distance from the origin of reduced variables to the limit state equation.,The definition can be generalized for n variable space.,is called the design point.
40、,the coordinate values of,2.5 Relationship between Reliability Index and Safety Factor,Chapter 2 Basic Concepts of Structural Reliability Theory,2.5 Relationship between Reliability Index and Safety Factor1,2.5.1 Problems of Safety Factor,For traditional design, structural safety is expressed as saf
41、ety factor:,The design formula of traditional design is,Problems of traditional design,The values of K is determined by experience and engineering judgmentsK is only related to the mean values of R & S, therefore it cannot reflect the practical failure events of structures.,2.5 Relationship between
42、Reliability Index and Safety Factor2,However,is not only related to the center position of PDF of R & S,Safety factor K cannot reflect this fact !,Reliability index can reflect this fact !,it is also related to the degree of disperse relative to the means.,2.5 Relationship between Reliability Index
43、and Safety Factor3,For two independent normal RVs,2.5.2 Relationships between K and,Chapter2: Homework 2,Homework 2,2.1 Deduce the formula of safety probability of probability interference method.,Required: (1) Give two types of just like .,2.2 By using the formula that you deduce in homework 2.1, calculate the safety probability of the performance function shown in Example 2.1.,Known: All statistical information is identical to that in Example 2.1.,(2) The key point of your deduction should be shown by figure.,End ofChapter 2,
