Risk Analysis for Engineering 5 docx

• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering CHAPTER 4a CHAPMAN HALL/CRC Risk Analysis in Engineering and Economics Risk Analysis for Engineering Department of Civil and Environmental Engineering University of Maryland, College Park RELIABILITY ASSESSMENT CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 1 Introduction “The reliability of an engineering system can be defined as its ability to fulfill its design purpose defined as performance requirements for some time period and environmental conditions. The theory of probability provides the fundamental bases to measure this ability.” CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 2 Introduction ̈ The reliability assessment methods can be based on 1. Analytical strength-and-load performance functions, or 2. Empirical life data. ̈ They can also be used to compute the reliability for a given set of conditions that are time invariant or for a time-dependent reliability. CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 3 Introduction ̈ The reliability of a component or system can be assessed in the form of a probability of meeting satisfactory performance requirements according to some performance functions under specific service and extreme conditions within a stated time period. ̈ Random variables with mean values, variances, and probability distribution functions are used to compute probabilities. CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 4 Analytical Performance-Based Reliability Assessment ̈ First-Order Second Moment (FOSM) Method. ̈ Advanced Second Moment Method ̈ Computer-Based Monte Carlo Simulation CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 5 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method Demand -Supply ),,,( 21 = = n XXXZZ K Z Z X X X n = = (, ,, ) 12 K Structural strength - Load effec t R-LXXXZZ n = = ),,,( 21 K (1a) (1b) (1c) Z = performance function of interest R = the resistance or strength or supply L = the load or demand as illustrated in Figure 1 CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 6 Analytical Performance-Based Reliability Assessment Load Effect (L) Strength (R) Density Function Origin 0 Random Value Failure Probability (Area for g < 0) Performance Function (Z) Figure 1. Performance Function for Reliability Assessment CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 7 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method – The failure surface (or the limit state) of interest can be defined as Z = 0. – When Z < 0, the element is in the failure state, and when Z > 0 it is in the survival state. – If the joint probability density function for the basic random variables ’s is , then the failure probability P f of the element can be given by the integral ∫∫ = nnXXXf dxdxdxxxxfP n KKL K 2121,,, ),,,( 21 (2) i X f x x x XX X n n12 12,,, (, , , ) K K CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 8 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method – Where the integration is performed over the region in which Z < 0. – In general, the joint probability density function is unknown, and the integral is a formidable task. – For practical purposes, alternate methods of evaluating Pf are necessary. Reliability is assessed as one minus the failure probability. CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 9 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method – Reliability Index • Instead of using direct integration (Eq. 2), performance function Z in Eq. 1 can be expanded using Taylor series about the mean value of Xs and then truncated at the linear terms. Therefore, the first-order approximation for the mean and variance are as follows: ),,,( 21 n XXXZ Z µ µ µ µ K ≈ ),())(( 11 2 ji j n i n j i Z XXCov X Z X Z ∂ ∂ ∂ ∂ σ ∑∑ == ≈ (3) (4a) CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 10 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method – Reliability Index (cont’d) Where variablerandom ofmean at the evaluated derivative partial and of covariance ),( of variance ofmean variablerandom ofmean 21 2 = ∂ ∂ = = = = i ji Z Z X Z XXXXCov Z Z σ µ µ CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 11 Analytical Performance-Based Reliability Assessment ̈ Advanced Second-Moment Method – Reliability Index (cont’d) • For uncorrelated random variables, the variance cab be expressed as • The reliability index β can be computed from: 2 1 22 )( i n i XZ X Z i ∂ ∂ σσ ∑ = ≈ (4b) 22 LR LR Z Z µµ µ µ σ µ β + − == )(1 β Φ − = f P (5) (6) If z is assumed normally distributed. CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 12 Analytical Performance-Based Reliability Assessment Load (L) Strength or Resistance (R) Failure Region L > R Survival Region L < R Limit State L = R Figure 2. Performance Function for a Linear, Two-Random Variable Case CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 13 ̈ Advanced Second-Moment Method – Nonlinear Performance Functions • For nonlinear performance functions, the Taylor series expansion of Z in linearized at some point on the failure surface referred to as the design point or checking point or the most likely failure point rather than at the mean. • Assuming X i variables are uncorrelated, the following transformation to reduced or normalized coordinates can be used: Analytical Performance-Based Reliability Assessment i i X Xi i X Y σ µ − = (8a) CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 14 ̈ Advanced Second-Moment Method – Nonlinear Performance Functions (cont’d) • It can be shown that the reliability index β is the shortest distance to the failure surface from the origin in the reduced Y-coordinate system. • The shortest distance is shown in Figure 3, and the reduced coordinates are Analytical Performance-Based Reliability Assessment L LR R L R L YY σ µ µ σ σ − += (8b) CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 15 Analytical Performance-Based Reliability Assessment Figure 3. Performance Function for a Linear, Two-Random Variable Case in Normalized Coordinates Failure Region L > R Survival Region L < R Limit State: L L L L Y σ µ − = R R R R Y σ µ − = L LR R L R L YY σ µ µ σ σ − += Intercept = L LR σ µ µ − β Design or Failure Point CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 16 ̈ Advanced Second-Moment Method – Nonlinear Performance Functions (cont’d) • The concept of the shortest distance applies for a nonlinear performance function, as shown in Figure 4. • The reliability index β and the design point, can be determined by solving the following system of nonlinear equations iteratively for β: Analytical Performance-Based Reliability Assessment ),,,( ** 2 * 1 n XXX K CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 17 β Limit State in Reduced Coordinates R = Resistance or Strength L = Load Effect L L L L Y σ µ − = R R R R Y σ µ − = Design or Failure Point Analytical Performance-Based Reliability Assessment Figure 4. Performance Function for a Nonlinear, Two-Random Variable Case in Normalized Coordinates CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 18 2/1 1 22 )( )(       = ∑ = n i X i X i i i i X Z X Z σ ∂ ∂ σ ∂ ∂ α ̈ Advanced Second-Moment Method – Nonlinear Performance Functions (cont’d) Analytical Performance-Based Reliability Assessment X iXiX ii * =− µαβσ ZX X X n (,,,) ** * 12 0K = (9) (10) (11) CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 19 ̈ Advanced Second-Moment Method – Nonlinear Performance Functions (cont’d) • Where α i is the directional cosine, and the partial derivatives are evaluated at the design point. • Eq. 6 can be used to compute P f . • However, the above formulation is limited to normally distributed random variables. • The directional cosines are considered as measure of the importance of the corresponding random variables in determining the reliability index β. Analytical Performance-Based Reliability Assessment [...]... to 5: (1) (5) 4 5 2 3 Z (5) 2 (0. 25) 2 (1) 2 (0. 25) 2 ( 0 .5 / 4 ) 2 (0.8) 2 Z 1 .56 25 0.06 25 0.04 1.2903 Z Z 3 1.2903 2.3 25 CHAPTER 4a RELIABILITY ASSESSMENT Slide No 35 Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • These values are applicable to both cases (a) and (b) Using advanced second-moment reliability analysis, ... Point X Xi Variable X1 4.242E-01 1.221E+00 X2 4.885E+00 1.061E-01 X3 4.295E+00 -1.930E-01 i Directional Cosines (a) 9.841E-01 8 .54 7E-02 -1 .55 5E-01 Slide No 38 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 2.3628 for this iteration Case (a): Iteration 3 Z Random... ( ) N Xi X1 4.202E-01 1.035E-01 7.718E-01 5. 050 E-01 9.118E-01 X2 4.881E+00 2.439E-01 4.992E+00 1.025E-01 1. 850 E-01 X3 4.206E+00 8.330E-01 3.912E+00 -2.031E-01 -3.667E-01 Slide No 45 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 3.3224 for this iteration Case (b):... Directional Cosines ( ) X1 4 .58 4E-01 1.129E-01 8.020E-01 5. 465E-01 9.118E-01 X2 4.843E+00 2.420E-01 4.991E+00 1.109E-01 1. 850 E-01 X3 4.927E+00 9. 758 E-01 3.803E+00 -2.198E-01 -3.667E-01 Slide No 46 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 3.3126 for this iteration Case... 8.041E-01 5. 500E-01 9.118E-01 X2 4.843E+00 2.420E-01 4.991E+00 1.116E-01 1. 850 E-01 X3 4.989E+00 9.880E-01 3.789E+00 -2.212E-01 -3.667E-01 Slide No 48 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 3.31 25 for this iteration which means that has converged to 3.31 25 • The... 1.136E-01 8.041E-01 5. 499E-01 9.118E-01 X2 4.843E+00 2.420E-01 4.991E+00 1.116E-01 1. 850 E-01 X3 4.989E+00 9.880E-01 3.789E+00 -2.212E-01 -3.667E-01 Slide No 47 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 3.31 25 for this iteration Case (b): Iteration 5 Equivalent Normal... constructed for cases (a) and (b): Case (a): Iteration 1 Directional Z X Cosines ( ) Xi i 1.000E+00 1. 250 E+00 9.687E-01 5. 000E+00 2 .50 0E-01 1.937E-01 4.000E+00 -2.000E-01 -1 .55 0E-01 Random Failure Point Variable X1 X2 X3 Slide No 36 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment Case (a): Iteration 1 ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance... function Slide No 55 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Monte Carlo Simulation Methods – Conditional Expectation (cont’d) • For the following performance function: Z R L (22) and for a randomly generated value of L or R, the failure probability for each cycle is given, respectively, as Pf i Pf i FR (l i ) (23) 1 FL ( ri ) (24) Slide No 56 CHAPTER 4a... can be determined by solving for the root according to Eq 11 for the limit state of this example using the following equation: Z N X1 1 N X1 N X2 2 N X2 N X3 3 N X3 0 Slide No 44 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 2.3 053 0 for this iteration Case (b):... index can be determined by solving for the root according to Eq 11 for the limit state of this example using the following equation: Z X1 1 X1 X2 2 X2 X3 3 X3 0 Slide No 37 CHAPTER 4a RELIABILITY ASSESSMENT Analytical Performance-Based Reliability Assessment ̈ Example 1 (cont’d): Reliability Assessment Using a Nonlinear Performance Function • Therefore, = 2.377 35 for this iteration Case (a): Iteration . Assessment 3 254 )5) (1( =−=−≅ Z µ 2903.104.006 25. 056 25. 1 )8.0()4 /5. 0() 25. 0()1() 25. 0( )5( 222222 =++= −++≅ Z σ 3 25. 2 2903.1 3 ==≅ Z Z σ µ β CHAPTER 4a. RELIABILITY ASSESSMENT Slide No. 35 ̈ Example. Performance Function • Therefore, β = 2.377 35 for this iteration. Analytical Performance-Based Reliability Assessment i X i X Z σ ∂ ∂ -1 .55 5E-01-1.930E-014.295E+00X 3 8 .54 7E-021.061E-014.885E+00X 2 9.841E-011.221E+004.242E-01X 1 Directional. J. Clark School of Engineering •Department of Civil and Environmental Engineering CHAPTER 4a CHAPMAN HALL/CRC Risk Analysis in Engineering and Economics Risk Analysis for Engineering Department