Optimal Timing of Maintenance and Rehabilitation in 60-Year Concrete Pavement Design Using the Weibull Distribution

Transcription

1 Optimal Timing of Maintenance and Rehabilitation in 60-Year Concrete Pavement Design Using the Weibull Distribution By Bernard Igbafen Izevbekhai, P.E., Ph.D. Research Operations Engineer Minnesota Department of Transportation 25th Annual CTS Transportation Research Conference, May

2 MnDOT 60-Year Concrete Design Philosophy of Get in Get out & Stay out Demo project in 2000: Rangaraju Anticipated Traffic Levels Low Water cement Ratio < 0.4 High Pozzolan Slag /FA > 15% Air content 6.5 ± ft Non-Skewed Panels Steel Dowel Preformed Elastomeric Sealers. Demo on TH 35 W In Richfield MN.

3 Incremental Cost Philosophy Design Improvement and Incremental Benefit if n t s F 0 ttt t sss F 0 t n i=1 i=1 C60 t=0 Cj t=0 > 1 where ts is the time to reach terminal serviceability level trigger level acceptable performance level Ft60 is the performance function of pavements with respect to time based on 60-year design Ft is the performance function of regular pavements with respect to time CC t=0 is the base year referenced summation of the cost for defender C60 t=0 is the base year referenced summation of the cost for defender Chosen with LCCA IBCA, BCA.

4 MnDOT Institutional Repair Schedule For Life Cycle Cost Analysis Concrete Pavement (All Traffic Levels) Service Year Initial Construction 0 Joint Reseal and Minor CPR (partial depth repairs) 17 Minor CPR (PDR) and some full depth repairs (FDR) 27 Major CPR (Full Depth Repair & Diamond Grinding) 40 End of Analysis Period (5 years of remaining life) 50

5 Life Cycle Cost Scenario TH 100 Segment Time (Yrs)

6 LCCA & Incremental B/C A Option EAEC/* $32.64 $28.41 $27.38 $37.18 EAEB $9000 $8000 $10,000 $7000 Annualized B/C Annual Alternative Annual Cost Comp of Incremental Benefit Cost Incremental B/C Ratio (ΔB/ΔC) 3 $27.38 $10, $ and 2 $1.03 -$1, $ $28.41 $9, $ and 1 $5.26 -$2, $ $32.64 $8, $ and 4 $9.80 -$3, $ $37.18 $7, $188.27

7 Incremental Benefit/Cost Guaranteed? If Bi (B1 B2, B3, B4) represent the benefits corresponding to options with NPVi (NPV1; NPV2; NPV3; and NPV4) If B1 =6; B2 =10; C1=4; and C2=7; B1/C1 = 1.5 B2/C2 = Thus B1/C1 > B2/C2; However, incremental benefit B is 4 and incremental cost C is 3. When considering B2 against B1, B/ C= 4/3 which is >1. Lower LCCA does not imply Lower Incremental cost: TEST BOTH!!!

8 Classic 60 Year Design Features P00 P11 Performance Index P10 P22 P21 P33 P44 P32 P43 MTBMA1 MTBMA2 MTBMA 3 MTBMA4 Time (Yrs)

9 Problem Statement MMMMMM ttt CCCCCCC CMMMMMM Benefit f i,m R 0 t + R 1 (t) CCCCCCC + R CMMMMMM 2 (t) EEEEEEE + R CCCCCCC 3 (t) C = Cumulative and tt = terminal perf. time; i = discount rate; m= Market OPTIMIZE: MTBMA 1; MTBMA 2; MTBMA 3; MTBMA 4 Fixes that maximize (Pjj - P j (j-1) Corresponding annualized equivalent cost or LCC Maximize Benefit / Cost : Given the constraints Pjj< P(j-1) (j-1); 4 fixes are assumed. CMTBMA4 < 60 Years but Ctt5= 60 years CCCC + R CEEEEEEE 4 (t)

10 Consider a Somewhat Deterministic Approach Dr. E. H. Waloddi Weibull ( ) Swedish Scientist & Mathematician If I have seen further it is by Standing on the Shoulders of Giants Isaac Newton (1676)

11 Weibull Distribution To The Rescue

12 Typical Weibull Features Probability Density Function (Pdf) ƒ(t) = (β (t- t 0 ) / μ ) exp { - (t- t 0 / μ)} Survival or Reliability Function P(t) = exp { - (t- t 0 / μ) } Hazard Function Z(t) = (β/ μ ) (t- t 0 )

13 Important Weibull Characteristics t 0 Threshold time-to-failure, or guaranteed life. In many cases of wear-out the first failure does not appear until some significant running time t 0 has elapsed. μ= Characteristic life. When t t o = μ, P(t) = exp (-1) = 0.37, μ is the interval between t o and the time at which it can be expected that 63% of the items will have failed(it is pivotal).. β= Shape Factor

14 Characteristics of the Shape Factor Implications in Infrastructure A value of β< 1 indicates Failure Rate Decreases Over Time. Infant Mortality Defective items Failing Early Failure Rate Decreasing over Time A value of β = 1 Constant Failure Rate. Random Events A value of β > 1 Failure Rate Increases with Time. Aging Process, ( Wear out ) Increasing Failure Rate wrt time.

15 PDFS of Various Weibull Shapes Probability Probability PDF with β=0.5 (<1) PDF with β = 1.0 PDF with β =2 (>1) PDF with β =5 (>>1)

16 HAZARD FUNCTION OF A WEIBULL DISTRIBUTION β <1 β =1 β >1 β >>1

17 Finding the Parameters Analytically A single expression dependent on the shape factor β. Shape factor characterize the failure. TTT ppppppppppp ddddddd ffffffff f t = β(t t o) β 1 μ β e t to μ TTT CCCCCCCCCC ppppppppppp ffffffff F t = 1 e t t 0 μ β β

18 Finding the Parameters Analytically From Linear Plot obtain Shape factor LL (LL 1 F(t) 1 = βββ t t 0 + β ln μ β= slope of best line= Shape factor μ = e IIIIIIIII β

19 Data Processing

20 Ranking and Characteristic Life

21 Threshold Time To Failure To determine the actual t o, the curve that approximates best to a straight line has the correct t o. The t o for each MTBMA was then determined

22 Threshold Time To Failure If LL LL 1 F(t) 1 is plotted against ll t t 0 a family of curves is generated. t 0 = t m t m t 1 t 2 t m t 2 t m t m t 1 t m t 2

23 Characteristics of the Shape Factor Implications in Infrastructure A value of β< 1 indicates Failure Rate Decreases Over Time. Infant Mortality Defective items Failing Early Failure Rate Decreasing over Time A value of β = 1 Constant Failure Rate. Random Events A value of β > 1 Failure Rate Increases with Time. Aging Process, ( Wear out ) Increasing Failure Rate wrt time.

24 DATA

25 Closing Remarks & Summary of Results MTBMA1 MTBMA2 MTBMA3 MTBMA4 MnDOT Characteristic Life (µ) Threshold TTF (to) Shape Factor (β) Performance Type Random to Wear Out Log Normal Crack Development Random Random to Wear Out Log Normal Crack Development Wear Out Spalling Punchout Faulting Rocking Panels

26 Shape Factor Implication

27 Closing Remarks A tenable and procedurally deterministic process Meaningful parameters of failure pattern, such as probable minimum time to failure. Characteristic Life and Failure Mode Simple graphical techniques available. Single p.d.f which can be made to represent any of the three types of failure distribution (running-in, purely random, or wearout modes of failure). Meaningful parameters of failure pattern, such as probable minimum time to failure. Simple graphical techniques available.

28 Disclaimer (Weibull Distribution QUACKERY) This Weibull Solves all Your Pavement Performance Problems

29 Multivalent Intellectualism Sir Isaac Newton Natural Philosopher Father of Science Great Thinker & Problem Solver Frederick Winslow Taylor Engineer Father of Scientific Management Great Thinker & Problem Solver

30 QUESTIONS