Structural and Multidisciplinary Optimization

, Volume 57, Issue 6, pp 2109–2125 | Cite as

An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake pedal case study

  • José Romero
  • Nestor V. Queipo


Accounting for uncertainty reduction measures (URMs) is critical to maximize the potential benefits of probabilistic design methods such as reliability-based design optimization (RBDO) and tackle the challenges in the design and construction of lightweight, high quality and reliable products. This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress-Smax and buckling-fbuck) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle while establishing the URMs global relative contributions to weight savings (expected value and variability) and computational expense. Given a set of URMs such as number of coupon tests, mesh refinement and manufacturing control, the solution approach includes: i) modeling structural analysis errors, ii) construction of surrogate models for the functions of interest, e.g., mass-M, Smax, fbuck and the corresponding error functions, iii) modeling pre-design and post-design URMs, such as material property density functions from coupon tests, and manufacturing tolerances (quality control), iv) solving the RBDO problems associated with each of the entries in a DOE with replication, and v) using ANOVA to compute main effects of most significant URMs on selected performance measures, i.e., mean and standard deviation of brake pedal mass, and computational expense. Results show that in the context of the brake pedal case study: the adoption of URMs led to reductions of up to 15 and 85% of mass mean and standard deviation, respectively, design and post-design URMs were responsible for 77 and 19% of the maximum mass reduction, respectively, and it was possible to set preliminary guidelines for URMs allocation and meet a particular performance objective under alternative URMs.


Brake pedal FSAE Design under uncertainty Reliability-based design optimization Uncertainty reduction measures Surrogate models ANOVA 



Design parameter i


Covariance function


Computational time


Vector of random design variables


Random design variable i


Young’s modulus


Regression function in Kriging model


Buckling load factor function


Limit state function for failure mode j


Limit state function of stress condition


Limit state function of buckling condition


Indicator function


Vector of random design parameters


Brake pedal mass function

\( \hat{\mathrm{M}}(.) \)

Random brake pedal mass function


Number of sample points


Number of random variables


Probability of the statement within the braces to be true


Probability of system failure


Target probability of system failure


Correlation function


Material yield strength


Maximum von Mises stress function


Manufacturing tolerance probability distribution of di


Vector of random variables, x = [d, p] or input variables vector


Random variables or vector of the input variable at the ith sample point


Lower [−1] or higher [1] level correponding to number of coupon tests


Lower [−1] or higher [1] level correponding to the mesh refinement


Lower [−1] or higher [1] level correponding to the DOE size for the construction of surrogate model


Lower [−1] or higher [1] level correponding to the degree of accuracy of the selected surrogate model


Lower [−1] or higher [1] level correponding to the manufacturing quality control


Set of uncertainty reduction measures level combination x URM  = {xA, xB, xC, xD, xE}


Response surface model of computational expense – total run time of RBDO


Response surface model of the mean of the brake pedal mass


Response surface model of the standard deviation of the brake pedal mass

\( \hat{\mathrm{y}}(.) \)

Surrogate prediction function


Random function with mean zero, and nonzero covariance

β0, βi, βiiij

Polynomial regression coefficients


Buckling load factor surrogate model error


Mass surrogate model error


Maximum von Mises stress surrogate model error


Buckling prediction error due to the mesh refinement level


Maximum von Mises stress prediction error due to the mesh refinement level


Vector of mean values of random design variables


Mass mean value


Vector of mean values of random design parameters


Poisson’s ratio


Process variance in Kriging model


Mass standard deviation value



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of ZuliaMaracaiboVenezuela
  2. 2.Applied Computing InstituteUniversity of ZuliaMaracaiboVenezuela

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