Component Reliability Analysis Chapter 3 Component Reliability Analysis of Structures

Chapter 3: Element Reliability Analysis of Structures Contents 3.1 MVFOSM — Mean Value First Order Second Moment Method 3.2 AFOSM — Advanced First Order Second Moment Method 3.3 JC Method — Recommended by the JCSS Committee 3.4 MCS — Monte Carlo Simulation Method

Chapter 3 Component Reliability Analysis of Structures 3.1 MVFOSM — Mean Value First Order Second Moment Method

3.1 MVFOSM — Mean Value First Order Second Moment Method …1 Mean Value or Center Point: The Taylor series expansion is on the means values. First Order: The first-order terms in the Taylor series expansion is used. Second Moment: Only means and variances of the basic variables are needed. This method is also named Mean Value Method or Center Point Method.

3.1 MVFOSM — Mean Value First Order Second Moment Method …2 3.1.1 Linear Limit State Functions 1. Assumptions Consider a linear limit state function of the form where, the terms are constants; The terms are uncorrelated random variables. 2. Formula According to the linear functions of uncorrelated random variables introduced in Chapter 1, the mean and standard deviation of Z are:

3.1 MVFOSM — Mean Value First Order Second Moment Method …3 According to the central limit theorem, as n increases, the random variable Z will approach a normal probability distribution. Formula of Reliability Index If the random variables are all normally distributed and uncorrelated, then the above formula is exact. Otherwise, it provides only an approximate estimate on the failure probability.

3.1 MVFOSM — Mean Value First Order Second Moment Method …4 Example 3.1 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. Turn to Page 102, look at the example 5.1 carefully!

3.1 MVFOSM — Mean Value First Order Second Moment Method …5 3.1.2 Nonlinear Limit State Functions 1. Assumptions Consider a nonlinear limit state function of the form where, the terms are uncorrelated random variables, and its mean and standard deviation are , respectively . 2. Formula We can obtain an approximate solution by linearizing the nonlinear function using a Taylor series expansion. The result is

3.1 MVFOSM — Mean Value First Order Second Moment Method …6 where, is the point about which the expansion is performed. From now on ,this point is represented by . Therefore, the above formula can be rewritten briefly as follows: One choice for this linearization point is the point corresponding to the mean values of the random variables. The point is also called mean value point or central point.

3.1 MVFOSM — Mean Value First Order Second Moment Method …7 Moments of the performance function Z where, Formula of Reliability Index

3.1 MVFOSM — Mean Value First Order Second Moment Method …8 Example 3.2 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. Turn to Page 104, look at the example 5.2 carefully!

3.1 MVFOSM — Mean Value First Order Second Moment Method …9 3.1.3 Comments on MVFOSM 1. Advantages It is very easy to use. It does not require knowledge of the distributions of the random variables. 2. Disadvantages Results are inaccurate if the tails of the distribution functions cannot be approximated by a normal distribution. There is an invariance problem: the value of the reliability index depends on the specific form of the limit state function. That is to say, for different forms of the limit state equation which have the same mechanical meanings, the values of reliability index calculated by MVFOSM may be different !

3.1 MVFOSM — Mean Value First Order Second Moment Method …10 The invariance problem is best clarified by Example 3.3 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. Turn to Page 107, look at the example 5.3 carefully!

Chapter 3 Component Reliability Analysis of Structures 3.2 AFOSM — Advanced First Order Second Moment Method

3.2 AFOSM — Advanced First Order Second Moment Method …1 To overcome the invariant problem, Hasofer and Lind propose an advanced FOSM method in 1974 , which is called AFOSM . The “correction” is to evaluate the limit state function at a point known as the “design point” instead of the mean values. Therefore, this method is also called “design point method” or “checking point method”. The “design point” is a point on the failure surface . Since the design point is generally not known a priori, an iteration technique is generally used to solve for the reliability index.

3.2 AFOSM — Advanced First Order Second Moment Method …2 3.2.1 Principles of AFOSM 1. Assumptions Consider a nonlinear limit state function of the form where, the terms are uncorrelated random variables, and its mean value and standard deviation are known. 2. Transformation from X space into U space The general random variable is transformed into its standard form as follows:

3.2 AFOSM — Advanced First Order Second Moment Method …3 The X space is then transformed into U space: The limit equation in X space is transformed to U space as follows. The design point in X space is then transformed to in U space.

3.2 AFOSM — Advanced First Order Second Moment Method …4 3. Reliability Index in U Space In U space, the tangent plane equation through the design point on failure surface is Since the design point is a point on the failure surface , then we have The hyper-plane equation can therefore be simplified as follows:

3.2 AFOSM — Advanced First Order Second Moment Method …5 The distance from the origin of U space to the tangent plane is actually the reliability index Design point Tangent Failure surface

3.2 AFOSM — Advanced First Order Second Moment Method …6 From the geometric meaning of the reliability index, we know Let is actually the direction cosine of the distance

3.2 AFOSM — Advanced First Order Second Moment Method …7 4. Reliability Index in X Space The design point in X space Since we have The direction cosine in X space

3.2 AFOSM — Advanced First Order Second Moment Method …8 The reliability index in X space

3.2 AFOSM — Advanced First Order Second Moment Method …9 Comparison of Formulas in X Space MVFOSM: linear case MVFOSM: nonlinear case center point AFOSM: nonlinear case design point

3.2 AFOSM — Advanced First Order Second Moment Method …10 3.2.2 Computation Formulas of AFOSM … … … … …(1) … … … … … …(2) … … … … … … … … … …(3) … … … … … … … … … …(4)

3.2 AFOSM — Advanced First Order Second Moment Method …11 3.2.3 Iteration Algorithm of AFOSM Formulate the limit state equation Give the distribution types and appropriate parameters of all random variables. Assume the initial values of design point and reliability index In general, the initial value of design point is taken as mean value . Then the initial value of is 0. Using Eq.(1) to calculate the n values of direction cosine . Using Eq.(2) to calculate the n values of design point . Using Eq.(3) to calculate the reliability index . Using Eq.(2) to calculate the new design point .

3.2 AFOSM — Advanced First Order Second Moment Method …12 Go back to Step 3 and repeat. Iterate until the values converge. Flowchart Begin Assume Calculate Calculate Calculate from Output and No Yes

3.2 AFOSM — Advanced First Order Second Moment Method …13 Example 3.4 Assume that a steel beam carry a deterministic bending moment , The plastic section modulus and the yield strength of the beam are statistically independent, normal random variables. It is known that The limit state equation is Calculate the reliability index of the beam as well as the checking points of and by AFOSM method.

3.2 AFOSM — Advanced First Order Second Moment Method …14 Solution: (a)

3.2 AFOSM — Advanced First Order Second Moment Method …15 (b) (c) (d) Iteration cycle 1 (1) Let (2) Solve and from formula (a) Checking (3) Solve from formula (d)

3.2 AFOSM — Advanced First Order Second Moment Method …16 Iteration cycle 2 (1) Solve and from formula (b) (2) Solve and from formula (a) Checking (3) Solve from formula (d)

3.2 AFOSM — Advanced First Order Second Moment Method …17 Iteration cycle 3 (1) Solve and from formula (b) (2) Solve and from formula (a) (3) Solve from formula (d) The final results:

Chapter 3 Component Reliability Analysis of Structures 3.3 JC Method — Recommended by the JCSS Committee

3.3 JC Method — Recommended by the JCSS Committee …1 The AFOSM method can only treat with the limit state equation with normal random variables. To overcome this problem, Rackwitz and Fiessler propose a procedure which can deal with the general random variables in 1978. This method is then recommended by the Joint Committee of Structural Safety, Therefore it is also named JC Method. The reliability index calculated by JC method is also called Rackwitz—Fiessler reliability index. The basic idea of JC method is to convert each non-normal random variable into an equivalent normal random variable by using the Principle of Equivalent Normalization.

3.3 JC Method — Recommended by the JCSS Committee …2 3.3.1 Basic Idea of JC Method Convert each non-normal random variable into an equivalent normal random variable by using the Principle of Equivalent Normalization. After this transformation, the problem can then be solved by AFOSM method. 3.3.2 Principle of Equivalent Normalization 1. Transformation Conditions of Equivalent Normalization (1) At the design checking point , the CDF value of the equivalent normal random variable is equal to that of the original non-normal random variable. (2) At the design checking point , the PDF value of the equivalent normal random variable is equal to that of the original non-normal random variable.

3.3 JC Method — Recommended by the JCSS Committee …3 PDF of non-normal RV PDF of equivalent normal RV

3.3 JC Method — Recommended by the JCSS Committee …4 2. Formulas of Equivalent Normalization … … … … …(1) … … … … …(2)

3.3 JC Method — Recommended by the JCSS Committee …5 3. Formulas of Equivalent Normalization for lognormal RV … … … … …(3) … … … … …(4) Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. Turn to Page 122, look at the example 5.8 carefully!

3.3 JC Method — Recommended by the JCSS Committee …6 3.3.3 Procedure of JC Method Formulate the limit state equation Determine the distribution types and appropriate parameters of all random variables. Assume the initial values of design point and reliability index In general, the initial value of design point is taken as mean value . Then the initial value of is 0. For non-normal RV , the mean and standard deviation should be calculated, and then, they replace the mean and standard deviation of the non-normal RV.

3.3 JC Method — Recommended by the JCSS Committee …7 Calculate the direction cosine using Calculate the design point using Calculate the reliability index using Calculate the new design point using Repeat Steps 3-7 until and the design points converge.

3.3 JC Method — Recommended by the JCSS Committee …8 Example 3.5 Assume that a reinforced concrete short column that carry a dead load and a live load. The limit state equation is The random variables are dead load effect G, live loaf effect Q, and section resistance . The parameters of these RV are listed in the following table: Random Variables Types of Distribution Mean (kN) Standard deviation (kN) C.o.V Normal 50 2.5 0.05 Extreme Ⅰ 85 17 0.2 Lognormal 250 25 0.1 Calculate the reliability index of the column by JC method .

Chapter 3 Component Reliability Analysis of Structures 3.4 MCS — Monte Carlo Simulation

3.4 MCS — Monte Carlo Simulation …1 3.4.1 Procedure of MCS 1. Formulate the limit state equation: 2. Determine the necessary distribution information. 3. Determine the number N of simulated values of the limit state equation to be generated according to the following formula: 4. Generate the random number values of the basic variables in the limit state equation. 5. Calculate a simulated value z of Z of the limit state function for each set of random number values of the basic variables. 6. Calculate the times of the simulated are less than zero. Assume that it is denoted as . 7. Calculate the estimated probability of failure according to the following formula:

3.4 MCS — Monte Carlo Simulation …2 3.4.2 Application Area of MCS 1. It is used to solve complex problems for which closed-form solutions are either not possible or extremely difficult. 2. It is used to solve complex problems that can be solved in closed form if many simplifying assumptions are made. 3. It is used to check the results of other solution techniques. 3.4.3 Accuracy of Probability Estimate of MCS Let be the theoretical correct probability that we are trying to estimate by calculating . The probability estimate accuracy is:

3.4 MCS — Monte Carlo Simulation …3 Example 3.6 Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak. Turn to Page 138, look at the example 5.16 carefully! We will demonstrate this example in MATLAB immediately……

3.4 MCS — Monte Carlo Simulation …4 Solution: Lognormal Normal Extreme Ⅰ

3.4 MCS — Monte Carlo Simulation …5 Simulated values of RVs in MATLAB Lognormal Normal Extreme Ⅰ

Chapter3: Homework 3 Homework 3.1 3.1 Programming the AFOSM in MATLAB environment according to the flow chart proposed by this course. (1) By using your own handwork, re-calculate the example 5.4 in text book on P.112 (2) By using your own subroutine, calculate the problem 5.3 in text book on P.142

Chapter3: Homework 3 Homework 3.2 3.2 Programming the JC Method in MATLAB environment according to the procedure proposed by this course. (1) By using your own handwork, re-calculate the example 3.5 by assuming the initial iteration value at the means. (2) By using the procedure proposed by this course, re-calculate the example 5.9 on Page 123 and the example 5.10 in the textbook on Page 125. (3) By using your own subroutine, calculate the example 5.11 on P.127 and the problem 5.4 in text book on P.142

Chapter3: Homework 3 Homework 3.3 3.3 Programming the MCS Method in MATLAB environment according to the procedure proposed by this course. By using your own subroutine, re-calculate the example 5.11 in P.127 and the problem 5.4 in text book on P.142 by Monte Carlo Simulation.

End of Chapter 3