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Journal of Mechanical Design | Volume 129 | Issue 3 | Research Paper
RESEARCH PAPERS

Tradeoff of Uncertainty Reduction Mechanisms for Reducing Weight of Composite Laminates

[+] Author and Article Information
Erdem Acar1

Mechnical and Aerospace Engineering Department, University of Florida, Gainesville, FL 32611eacar@ufl.edu

Raphael T. Haftka

Mechnical and Aerospace Engineering Department, University of Florida, Gainesville, FL 32611haftka@ufl.edu

Theodore F. Johnson

 NASA Langley Research Center, Hampton, VA 23681theodore.f.johnson@nasa.gov

The term design response surface (DRS) follows Qu et al. (19) and indicates approximations to the probability of failure or other measures of safety as a function of design variables.

1

Corresponding author. Currently Post-doctoral research associate at the Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39579.

J. Mech. Des 129(3), 266-274 (Feb 23, 2006) (9 pages) doi:10.1115/1.2406097 History: Received September 19, 2005; Revised February 23, 2006
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Inspired by work on allocating risk between the different components of a system for a minimal cost, we explore the optimal allocation of uncertainty in a single component. The tradeoffs of uncertainty reduction measures on the weight of structures designed for reliability are explored. The uncertainties in the problem are broadly classified as error and variability. Probabilistic design is carried out to analyze the effect of reducing error and variability on the weight. As a demonstration problem, the design of composite laminates at cryogenic temperatures is chosen because the design is sensitive to uncertainties. For illustration, variability reduction takes the form of quality control, while error is reduced by including the effect of chemical shrinkage in the analysis. Tradeoff plots of uncertainty reduction measures, probability of failure and weight are generated that could allow choice of optimal uncertainty control measure combination to reach a target probability of failure with minimum cost. In addition, the paper also compares response surface approximations to direct approximation of a probability distribution for efficient estimation of reliability.
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Copyright © 2007 by American Society of Mechanical Engineers
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References

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Figures

Grahic Jump Location
+Material properties E1, E2, G12, and ν12 as a function of temperature
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Material properties E1, E2, G12, and ν12 as a function of temperature
Figure 7

Material properties E1, E2, G12, and ν12 as a function of temperature

Grahic Jump Location
+Material properties α1 and α2 as a function of temperature
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Material properties α1 and α2 as a function of temperature
Figure 8

Material properties α1 and α2 as a function of temperature

Grahic Jump Location
+Tradeoff plot for the probability of failure, design thickness, and uncertainty reduction measures: (ER) error reduction (reducing from 20% to 10%), (QC) quality control to −2σ
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Tradeoff plot for the probability of failure, design thickness, and uncertainty reduction measures: (ER) error reduction (reducing from 20% to 10%), (QC) quality control to −2σ
Figure 5

Tradeoff plot for the probability of failure, design thickness, and uncertainty reduction measures: (ER) error reduction (reducing from 20% to 10%), (QC) quality control to −2σ

Grahic Jump Location
+Tradeoff of probability of failure and uncertainty reduction. Probabilities of failure are calculated via MCS (sample size of 1,000,000). The crosses in the figure indicate the optimal uncertainty control combination that minimizes the cost of uncertainty control for a specified probability of failure.
View Large  |  Save Figure  |  Download Slide (.ppt)
Tradeoff of probability of failure and uncertainty reduction. Probabilities of failure are calculated via MCS (sample size of 1,000,000). The crosses in the figure indicate the optimal uncertainty control combination that minimizes the cost of uncertainty control for a specified probability of failure.
Figure 6

Tradeoff of probability of failure and uncertainty reduction. Probabilities of failure are calculated via MCS (sample size of 1,000,000). The crosses in the figure indicate the optimal uncertainty control combination that minimizes the cost of uncertainty control for a specified probability of failure.

Grahic Jump Location
+Geometry and loading of the laminate with two ply angles [±θ1∕±θ2]s (x-is the hoop direction and y is the axial direction)
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Geometry and loading of the laminate with two ply angles [±θ1∕±θ2]s (x-is the hoop direction and y is the axial direction)
Figure 1

Geometry and loading of the laminate with two ply angles [±θ1∕±θ2]s (x-is the hoop direction and y is the axial direction)

Grahic Jump Location
+Comparison of CDF obtained via 1000 MCS, the approximate normal distribution and conservative approximate normal distributions for ε2 on θ1 corresponding to the deterministic optimum: (a) CDF versus strain, (b) actual (empirical) CDF versus fitted CDF
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Comparison of CDF obtained via 1000 MCS, the approximate normal distribution and conservative approximate normal distributions for ε2 on θ1 corresponding to the deterministic optimum: (a) CDF versus strain, (b) actual (empirical) CDF versus fitted CDF
Figure 2

Comparison of CDF obtained via 1000 MCS, the approximate normal distribution and conservative approximate normal distributions for ε2 on θ1 corresponding to the deterministic optimum: (a) CDF versus strain, (b) actual (empirical) CDF versus fitted CDF

Grahic Jump Location
+Reducing laminate thickness (hence weight) by error reduction (no variability reduction)
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Reducing laminate thickness (hence weight) by error reduction (no variability reduction)
Figure 3

Reducing laminate thickness (hence weight) by error reduction (no variability reduction)

Grahic Jump Location
+Reducing laminate thickness by error reduction (ER) and quality control (QC)
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Reducing laminate thickness by error reduction (ER) and quality control (QC)
Figure 4

Reducing laminate thickness by error reduction (ER) and quality control (QC)

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