RESPONSE-BASED STATISTICS Part 2 Steve Winterstein Probability-Based Engineering, Menlo Park, CA September 18, 2014
ENVIRONMENTAL CONTOURS How to estimate N-Year Wind/Wave Design Loads?
Deep Water LRFD Design Load = L (Hs, Tp ) Seek 100-yr contour [Hs, Tp]100 so that L100 = worst load along [ Hs, Tp]100 How to construct [Hs, Tp]100? Wind: Interest in L50… Similar contours of V=mean wind speed, I=turbulence intensity
The Problem and FORM solution • Load = Y (Hs, Tp ) Y (U1, U2) where U1, U2 are standard normal
The Problem and FORM solution • Load = Y (Hs, Tp ) Y (U1, U2) where U1, U2 are standard normal • Failure if Y > yCAP M = yCAP -Y (U1, U2) <0
The Problem and FORM solution • Load = Y (Hs, Tp ) Y (U1, U2) where U1, U2 are standard normal • Failure if Y > yCAP M = yCAP -Y (U1, U2) <0 • Given yCAP pF = P[ yCAP -Y (U1, U2)<0 ] estimated with FORM
The Problem and FORM solution • Load = Y (Hs, Tp ) Y (U1, U2) where U1, U2 are standard normal • Failure if Y > yCAP M = yCAP -Y (U1, U2) <0 • Given yCAP pF = P[ yCAP -Y (U1, U2)<0 ] estimated with FORM • Given pF = pF,TARGET (e.g., 10-2 / year) want consistent yCAP (e.g., y100)
Brute Force FORM solution
Brute Force FORM solution
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Assume yCAP = y1
Brute Force FORM solution
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Assume yCAP = y1
Brute Force FORM solution
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Assume yCAP = y1
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FORM: find β1, pF1 = Φ(-β1)
Brute Force FORM solution
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Assume yCAP = y1
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FORM: find β1, pF1 = Φ(-β1)
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Suppose pF1 < pF,TARGET
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Reduce yCAP from y1 to y2
Brute Force FORM solution
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Assume yCAP = y1
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FORM: find β1, pF1 = Φ(-β1)
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Suppose pF1 < pF,TARGET
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Reduce yCAP from y1 to y2
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FORM: find β2, pF2 = Φ(-β2)
Brute Force FORM solution
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Assume yCAP = y1
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FORM: find β1, pF1 = Φ(-β1)
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Suppose pF1 < pF,TARGET
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Reduce yCAP from y1 to y2
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FORM: find β2, pF2 = Φ(-β2)
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Suppose pF2 > pF,TARGET
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Increase yCAP from y2 to y3
Brute Force FORM solution
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Iterate until we find yCAP for which β = target value (here, β = 3.7)
Brute Force FORM solution
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Iterate until we find yCAP for which β = target value (here, β = 3.7)
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Observation: This is silly
Brute Force FORM solution
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Iterate until we find yCAP for which β = target value (here, β = 3.7)
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Observation: This is silly
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Better: just search circular contour, with |u| = β , to find maximum value of load/response Y(U1, U2)
Brute Force FORM solution
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Iterate until we find yCAP for which β = target value (here, β = 3.7)
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Observation: This is silly
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Better: just search circular contour, with |u| = β , to find maximum value of load/response Y(U1, U2)
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Result: Environmental Contours
Contours 1: What are They? Four sites considered (Winterstein et al 1999):
H100=11.7
• GOM1: Generic GOM site (API TLP code calibration JIP) • GOM2: Sudden GOM storms (smaller h100, can’t evacuate) • Southern NS: Ekofisk • Northern NS: Statfjord Our base case shown here: GOM1 H100=11.7m
Contours 1: What are They? Gulf of Mexico site: H100=11.7m
H100=11.7
100-yr PDF contour: P[ fall outside ] = .01 / yr
Contours 1: What are They? Gulf of Mexico site: H100=11.7m
H100=11.7
100-yr PDF contour: P[ fall outside ] = .01 / yr 100-yr FORM contour: P[fall outside any tangent line] = .01 /yr So: 100-yr FORM contour includes H100 = 11.7m
Contours 2: How to Draw Them?
• Start with Standard Normal Variables (U1, U2) • From target pf, get corresponding normal fractile β=Φ-1 (1-pF) • Contour = circle with radius β (sphere if n=3 vars)
Contours 2: How to Draw Them?
Contours 3: From U to X space Plus: Rosenblatt Transformation
Yields:
Contours 4: Including Response Variability PROBLEM X100 = 100-Year Load/Response is NOT SAME as Expected Load in Worst 100-Year Environment (e.g., worst median X50 along [Hs,Tp]100) SOLUTIONS 1. Exact: Construct, search 3-D contour [Hs,Tp,X]100 2. Approximate: • Search [Hs,Tp]100 for worst higher fractile load Xp • Calibration to exact results suggests p=.85-.90
Predicting 100-Year Wave Crests: the 85% solution works…
EXAMPLE: SURGE MOTION OF A SPAR • SPAR GEOMETRY: “Consensus Spar” established by OTRC • FORCES: Linear and quadratic transfer functions (LTFs and QTFs) found for 6DOF from diffraction analysis (OTRC) • MOTIONS: LTFs and QTFs found for surge motion • STATISTICS: First 4 moments of surge found Hermite model for extremes
EXAMPLE: SURGE MOTION OF A SPAR • RESULT: Isolines of constant surge [m] as a function of HS and TP
EXAMPLE: SURGE MOTION OF A SPAR • RESULT: Isolines of constant surge [m] as a function of HS and TP • NEED: worst value along 100-yr contour y100 about 16m
EXAMPLE: SURGE MOTION OF A SPAR • RESULT: Isolines of constant surge [m] as a function of HS and TP • NEED: worst value along 100-yr contour y100 about 16m • NOTE: don’t need isolines; just search contour (any response)
EXAMPLE: SURGE MOTION OF A SPAR • RESULT: Isolines of constant surge [m] as a function of HS and TP • NEED: worst value along 100-yr contour y100 about 16m • IF: We applied only H100 = 11.7m, we’d get y of about 14m
SURGE MOTION: SPAR VS TLP • ENVIRONMENT: North Sea site; storms above HS=8m
SURGE MOTION: SPAR VS TLP • ENVIRONMENT: North Sea site; storms above HS=8m • ISOLINES: TLP especially period-sensitive (TP around 10s)
SURGE MOTION: SPAR VS TLP • ENVIRONMENT: North Sea site; storms above HS=8m • ISOLINES: TLP especially period-sensitive (TP around 10s) • CONTOURS: Depend only on environment (same in both cases)
HS-TP CONTOURS AT OTHER SITES
HS-TP CONTOURS AT OTHER SITES
HS-TP CONTOURS AT OTHER SITES
HS-TP CONTOURS AT OTHER SITES
LRFD Study: Gulf of Mexico vs North Sea
GOM2 GOM1 SNS NNS
• GOM1: Generic GOM site (API TLP code calibration JIP) • GOM2: Sudden GOM storms (smaller h100, can’t evacuate) • Southern NS: Ekofisk • Northern NS: Statfjord
LRFD Study: Gulf of Mexico vs North Sea • GOM environment relatively harsher above design (100-yr) conditions • Load factor 1.3 * L100 gives pf about 10-3 (GOM) ………………..10-4 (NS)
Higher-Dimension Contours • Include (scalar) current or response variability n=3 • Include directional effects (n =3 or more) • No longer easy to visualize as in 2D • Still easy to construct/search for worst (N-year) load/ response • Optimization problem easier here (inverse FORM) than in standard FORM: need only to find optimum given “box-like” constraints in n-1 direction cosines (n-1 angles between 0 and 2*pi). Both optimizations (FORM and inverse FORM) available in MATLAB.
Environmental Contours: Summary • Easy to construct • Not impossible to understand (FORM knowledge helps) • Useful for structures sensitive to multiple environmental parameters (floating structures/ships, wind turbines) • Convenient basis for simulation or experimental studies • Wave Examples: Hs-Tp, Hs-Current, Hs-Wind Speed • Wind Examples: Mean Wind Speed, Turbulence Intensity • Response Variability: Choose higher fractile Xp (p=.85-.90)
Contour References Haver, Sverre and Winterstein, Steven R., "Environmental Contour Lines: A Method for Estimating Long Term Extremes by a Short Term Analysis," Transactions, Society of Naval Architects and Marine Engineers, Vol. 116, 2009. Saranyasoontorn, Korn and Manuel, Lance, “Design Loads for Wind Turbines using the Environmental Contour Method,” Journal of Solar Engineering including Wind Energy and Building Energy Conservation, Transactions of the ASME, Vol. 128, No. 4, November 2006. Winterstein, Steven R., Jha, Alok K., and Kumar, Satyendra, ``Reliability of Floating Structures: Extreme Response and Load Factor Design," Journal of Waterway, Port and Coastal Engineering, ASCE, Vol. 125, No. 4, July/August, 1999. Winterstein, Steven R., Ude, T.C., Cornell, C.A., Bjerager, P., and Haver, S, "Environmental Parameters for Extreme Response: Inverse FORM with Omission Factors," Proceedings, ICOSSAR-93, Innsbruck, Austria, 1993.
