1 Effects of Error, Variability, Testing and Safety Factors on Aircraft Safety Erdem Acar, Amit Kale and Raphael T. Haftka Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida

2 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida  The FAA makes a distinction between error and variability through use of A-basis and B-basis properties.  A-Basis property is the value exceeded by 99% of population with 95% confidence.  Problems in acceptance of probabilistic design. We are interested to see whether the differentiating errors and variability may help.  We are interested to see how epistemic and alleatory uncertainty interact in determining the safety factor of aircraft. Motivation

3 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Definition of uncertainties and safety measures considered The error model Simulation process for certification testing Certification test effectiveness in terms of error, variability and average safety factor Uncertainty in probability of failure Concluding remarks Outline

4 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Error and Variability

5 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Safety measures FAA requirements Safety factor (S F ) = 1.5 Certification tests: Testing the structural design for failure to compensate for ERROR (our interpretation!) A-basis and B-basis material properties to account for VARIABILITY

6 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Approach to the problem Structural failure due to stress failure without damage propagation  =P/A  f (  is the point stress in any structural component) A single test, which is a pass-fail certification test Simulation of variability and error requires simulating the design of multiple aircraft and multiple models. Monte Carlo simulation and analytical approximation used.

7 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Error model Error in load calculation Error in point stress analysis (1) (2) The designer uses Eq. (2) to calculate design thickness  The deviation of actual load and stress values (fleet-average value) from the values calculated by the designer

8 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Error in implementation Error in geometric parameters  Deviation of average actual geometry and material properties from design specification Error in material properties

9 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Fleet-Average safety factor where Fleet average of stress in a panel under correct design loads Fleet-average safety factor cumulative error in safety factor for the average airplane (fleet-average) built by a company is safety margin for variability (brings 1.27 additional safety factor)

10 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Error factor distributions Error Factors Bounds (Nominal Values) eσeσ 5% ePeP 10% ewew 1% etet 2% emem 20% Uniform distribution with zero mean

11 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Variability VariablesDistributionMeanScatter Actual Service Load, P act LognormalP d = 10010% c.o.v. Actual Panel width, w act Uniformw built = 11% bounds Actual Panel thickness, t act Uniformt built 3% bounds Actual Failure stress, σ f act Lognormalσ f built = 15010% c.o.v.  Variation from one aircraft to another in the fleet. For example,

12 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Monte Carlo simulation N different aircraft models (Boeing 777, Airbus 320A)

13 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Effect of certification on S F fleet The model is certified if Mean initial : 1.907=1.5 x 1.27 Mean updated : SAFETY IS IMPROVED ! Since some unsafe designs fail in certification test. The use of A-basis properties gives and additional safety factor of 1.27.

14 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Comparison of Monte Carlo and analytical approximation Bayes Theorem is used to compute analytical approximation -Variability in geometric variables are approximated as lognormal -Certification testing does not affect error term e p

15 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida For low variability errors lead to safer design e t design before cert. Mean and c.o.v. t design after cert. Mean and c.o.v Probability of failure (0.06)1.05 (0.03)7.95 x (0.29)1.25 (0.08)2.63 x10 -6 SAFETY IS IMPROVED ! When the variability is very small!

16 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Effect of certification on P f  With variability, increase of k leads to increase of probability of failure  As error grows, P f ratio becomes smaller indicating that the certification tests become more effective Introduce a new parameter k,

17 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Effect of variability Increase of variability leads to  Increase in probability of failure (A-basis not sufficient?!)  Increase in P f ratio indicating that certification testing loses its efficiency

18 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Effect of different safety measures (simpler error model) The usefulness of certification tests increases with - low safety factor - low variability - high error

19 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Coefficient of Variation of P f  Coefficient of variation of probability of failure is huge.  It may be difficult for an individual company to use the computed probability of failure.  However, for FAA it is O.K. since they are judged based on national average.

20 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Small changes in S F may be sufficient for reliability based design  Deterministic and probabilistic design optimization of a simplified wing model.  For deterministic optimization, S F =1.5 and A-basis properties used. The use of both safety measure translates into an effective safety factor of  The probabilistic optimization for fixed weight corresponding to deterministic optimum. Section Safety factor in Deterministic Design Safety factor in Probabilistic Design 5 (root) 1.5 x x x x x x x x (tip) 1.5 x x 1.57 Pf 8.0 x x10 -5  Aircraft companies may be given freedom to select conservative material properties to account for variability.

21 Structural and Multidisciplinary Optimization Group Dept. of Mechanical and Aerospace Engineering University of Florida Concluding Remarks  Safety is determined by error and variability. for error: S F and Cert. test --- for variability: A-basis Hence, certification tests are most effective for low safety factors high errors low variability  Large coefficient of variation in probability of failure is found.  Safety factor may be useful for FAA to manage error. Aircraft companies may be given freedom to select conservative material properties to account for variability to improve safety.