# Using Tolerance Bounds for Estimation of Characteristic Fatigue Curves for Composites with Confidence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}is a constant, $\beta $ describes the slope of the curve, and S is the applied stress amplitude. In some cases S can also be the stress range or the maximum stress. The definition of S does not influence the statistical procedures described here.

## 2. Theoretical Methods

#### 2.1. Theory of Tolerance Bounds for Random Variables

_{k}is established through the probability equation

_{1−α}. Provided n observations of Y are available, the probability equation defining the random estimation interval can be rewritten as follows:

_{1−γ}is the 1 − γ quantile in the standard normal distribution function, and (Y − μ)/σ is standard normally distributed.

#### 2.1.1. Independent Variables

_{1−α}can be interpreted as the 1 − α quantile in the distribution of a variable

^{2}/σ

^{2}is chi-square distributed with n − 1 degrees of freedom. It is noted that $C\sqrt{n}$ is a non-central t-distributed variable.

_{1−α}can be read off from tables; see Resnikoff and Lieberman (1957) [34], Owen (1958) [23] and Pearson and Hartley (1976) [35]. Alternatively, it can be interpreted as the 1 − α quantile in the simulated distribution of the variable C as obtained from Monte Carlo simulation of its two parent random variables U and W. Yet another alternative is to calculate the quantile c

_{1−α}by means of Hald’s approximation, see Hald (1952) [36] and Madsen et al. (1986) [4],

^{−1}() denotes the inverse standard normal distribution function.

_{1−α}leads to the construction of lower tolerance bounds that are identical to those defined according to the method of DIN55303 [28]. The so-called ‘A’ basis and ‘B’ basis design allowables, according to MIL-HDBK-5B [5], are also based on this method.

#### 2.1.2. Dependent Variables

_{i}values. Further, μ(${x}_{0}$) is the mean value of Y for X = ${x}_{0}$, σ is the standard deviation of the residuals of Y about the fitted linear relationship between μ and X, and $\overline{Y}\left({x}_{0}\right)$ and S are the corresponding uncertain estimators. Note that Y is assumed to be homoscedastic, i.e., the standard deviation σ is assumed to be a constant, independent of the value of X.

_{1−α}can be interpreted as the 1 − α quantile in the distribution of a variable

^{2}/σ

^{2}is chi-square distributed with n − 2 degrees of freedom.

_{1−α}can be interpreted as the 1 − α quantile in the simulated distribution of the variable C as obtained from Monte Carlo simulation of its two parent random variables U and W.

_{k}for X = ${x}_{0}$ takes on the numerical realization

_{1−α}, which is obtained as the solution of the probability equation, is a function, not only of the proportion γ, the confidence level 1 − α, and the number of observations n of Y, but also of the particular values of X for which Y is observed, and of the value ${x}_{0}$ of X for which the lower tolerance bound for Y is sought.

#### 2.2. Tolerance Bounds for Dependent Variables

#### 2.2.1. Theory

_{10}S values for the various tests are fairly evenly distributed over the interval for logS which is covered by the tests.

_{10}S in the following. With the assumption of uniformly distributed x

_{i}values, i = 1, … n, over an interval of length L

_{x}along the x axis, the factor h

_{n}used in the expression for the variable C in Equation (13)

_{0}− $\overline{x}$, where x

_{0}is the logS value of interest for prediction of the 1 − γ quantile of logN with confidence, and where the mean value $\overline{x}$ of the n x

_{i}values, under the assumption of uniformly distributed x

_{i}values, is also the midpoint of the interval of length L

_{x}that covers the n x

_{i}values. For further explanation, see Figure 1.

_{x}of the logS interval can be calculated as

_{max}is the maximum and logS

_{min}is the minimum of logS in the test set, and n is the number of tests, each leading to a data pair (S, N).

#### 2.2.2. Graphical Representation of the Quantile ${c}_{1-\alpha}$

_{x}from the midpoint of the considered X = logS interval over which available test data are uniformly distributed, simulations of the variable C in Equation (15) are carried out by simulating the parent variables U and W. A Monte Carlo simulation procedure, as described in Tvedt (2006) [43], is used for this purpose. For each combination of tolerance γ and number of fatigue tests n, 200,000 simulations of C are carried out. The probability distribution of C results from these simulations and the quantile c

_{1−α}can be extracted from this distribution for given cumulative probability 1 − α, i.e., for specified confidence level 1 − α.

_{x}from the midpoint. An example of applying ${c}_{1-\alpha}$ for a specific Δx/L

_{x}is graphically indicated in Figure 1. Figure 2 shows how ${c}_{1-\alpha}$ increases with Δx/L

_{x}. The tolerance bound curve in Figure 1 is given by the regression curve of the data shifted horizontally by ${c}_{1-\alpha}$ for the relevant Δx/L

_{x}value. Therefore, the shape of the tolerance bound curve in Figure 1 is directly related to the shape of the ${c}_{1-\alpha}$ vs. Δx/L

_{x}from Figure 2.

#### 2.2.3. Mathematical Representation of the Quantile c_{1−α}

_{10}N are uniformly distributed over the interval of length L

_{x}, it can be deduced that the quantile c

_{1−α}can be represented well by a hyperbolic expression in Δx/L

_{x},

_{x}, it can be deduced that

_{n}

_{− 2}(1 − α) is the 1 − α quantile in the Student’s t distribution with n − 2 degrees of freedom. This quantile can be read off from statistical tables, see for example Snedecor and Cochran (1989) [44] and DNV-RP-C207 [45]. The most commonly needed quantiles are summarized in Table 2.

_{1}and c

_{2}can be derived

^{−1}(γ) as a function of γ. Table 4 tabulates ${c}_{1-a,k}\left(n-1\right)$ as a function of n, 1 − α and k. Interpolation may be necessary, if ${c}_{1-a,k}\left(n-1\right)$ is needed for other values of n, 1 − α and k than those tabulated.

_{1}and c

_{2}. However, based on Hald’s approximation in Equation (8), generalized to dependent variables, the following expression can be derived and used as an approximation to the coefficient c

_{1}

^{−1}denotes the inverse standard Gaussian distribution function. When used together with the expressions for c

_{1}+ c

_{2}and c

_{3}, this approximation will lead to rather accurate results for the sought-after quantile c

_{1−α}by Equation (18). The inaccuracy in the prediction of c

_{1−α}when using this approximation for c

_{1}will decrease for increasing sample size n and will eventually vanish as n approaches infinity.

## 3. Results

#### 3.1. Verification

_{1−α}in Equation (18) with the expressions for the coefficient sum c

_{1}+ c

_{2}and the coefficient c

_{3}in Equations (19) to (21) has been carried out for an example case based on n = 8 observations of logN. The verification was conducted by comparing with the true c

_{1−α}values obtained from the distribution of the variable C, established according to Equations (15) and (16) by simulating the parent variables U and W. The verification was carried out for a survival probability γ = 0.95 and a confidence level 1 − α = 0.95. The results are presented in Figure 3. Excellent agreement with simulation results (red full curve) was found when the coefficient c

_{1}was back-calculated from the asymptotic value of c

_{1−α}for a large value of Δx/L

_{x}and used in conjunction with the expressions for c

_{1}+ c

_{2}and c

_{3}(blue dashed curve), assuming a straight line for the asymptote to the intercept with the c

_{1−α}axis. The asymptotic value of c

_{1−α}for a large value Δx/L

_{x}was taken as the simulated value of c

_{1−α}, assuming that the true c

_{1−α}curve for all practical purposes is “identical” to the asymptote for this large value of Δx/L

_{x}. Here, Δx/L

_{x}= 4 was used. Just as excellent an agreement was found when the coefficient c

_{1}was approximated by the expression in Equation (22) (green dotted curve). Similar verifications have been carried out for other combinations of the sample size n, survival probability γ and confidence level 1 − α and show the same level of accuracy.

#### 3.2. Numerical Example

^{3}to 2·10

^{6}cycles in the air. During cyclic fatigue testing the specimens were loaded in both tension and compression, at a load ratio of R = −1, i.e., the maximum stress in compression is equal to the (negative) maximum stress in tension, and the mean stress is zero. For further details about the material and test set-up, see Echtermeyer et al. (2004) [46].

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- DNV Standard. Submarine Pipeline Systems; DNV-OS-F101; DNV Offshore Standard: Høvik, Norway, 2015. [Google Scholar]
- DNV Standard. Composite Components; DNVGL-OS-C501; DNV Offshore Standard: Høvik, Norway, 2013. [Google Scholar]
- DNV Standard. Thermoplastic Composite Pipes; DNVGL-ST-F119; DNV Standard: Høvik, Norway, 2018. [Google Scholar]
- Madsen, H.O.; Krenk, S.; Lind, N.C. Methods of Structural Safety; Prentice-Hall Inc.: Englewood Cliffs, NJ, USA, 1986. [Google Scholar]
- Metallic Materials and Elements for Aerospace Vehicle Structures. In Military Standardization Handbook; MIL-HDBK-5B; Department of Defense: Washington, DC, USA, 1971.
- Jones, D.R.; Ashby, M.F. Engineering Materials 1: An Introduction to Properties, Applications and Design; Elsevier: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Callister, W.D., Jr.; Rethwisch, D.G. Fundamentals of Materials Science and Engineering: An Integrated Approach; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Tsai, S.W. Theory of Composites Design; Think Composites: Dayton, OH, USA, 1992. [Google Scholar]
- Carlsson, L.A.; Gillespie, J.W.; Zweben, C.H. Delaware Composites Design Encyclopedia: Mechanical Behavior and Properties of Composite Materials; CRC: Boca Raton, FL, USA, 1989. [Google Scholar]
- Mayer, R.M. (Ed.) Design of Composite Structures against Fatigue: Applications to Wind Turbine Blades; Mechanical Engineering Publications: Bury St Edmunds, UK, 1996. [Google Scholar]
- Nijssen, R.P.L. Fatigue Life Prediction and Strength Degradation of Wind Turbine Rotor Blade Composites; Sandia Report, SAND2006-7810P; Sandia National Laboratories: Albuquerque, NM, USA, 2007.
- Bond, I.P. Fatigue life prediction of GRP subject to variable amplitude loading. Compos. Part A
**1999**, 30, 961–970. [Google Scholar] [CrossRef] - Westphal, T.; Nijssen, R.P.L. Fatigue life prediction of rotor blade composites: Validation of constant amplitude formulations with variable amplitude experiments. J. Phys. Conf. Ser.
**2014**, 555, 012107. [Google Scholar] [CrossRef][Green Version] - Sanchez, H.; Sankararaman, S.; Escobet, T.; Puig, V.; Frost, S.; Goebel, K. Analysis of two modeling approaches for fatigue estimation and remaining useful life predictions of wind turbine blades. In Proceedings of the Third European PHM, Bilbao, Spain, 5–8 July 2016. [Google Scholar]
- Passipoularidis, V.A.; Brønsted, P. Fatigue Evaluation Algorithms: Review; Risø-R-Report-1740 (EN); NLSE: Risø, Denmark, 2009. [Google Scholar]
- API; Recommended Practice. Design of Risers for Floating Production Systems (FPS) and Tension-Leg Platforms (TLPs); American Petroleum Institute: Washington, DC, USA, 1998. [Google Scholar]
- Lotsberg, I. Fatigue Design of Marine Structures; Cambridge University Press: New York, NY, USA, 2016. [Google Scholar]
- Post, N.L. Reliability Based Design Methodology Incorporating Residual Strength Prediction of Structural Fiber Reinforced Polymer Composites under Stochastic Variable Amplitude Fatigue Loading. Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2008. [Google Scholar]
- Castro, O.; Branner, K.; Dimitrov, N. Assessment and propagation of mechanical property uncertainties in fatigue life predictions of composite laminates. J. Compos. Mater.
**2018**, 52, 3381–3398. [Google Scholar] [CrossRef] - Nishijima, S. Statistical Fatigue Properties of Some Heat-Treated Steeis for Macliine Structural Use. In Statistical Analysis of Fatigue Data; ASTM STP 744; Little, R.E., Ekvall, J.C., Eds.; American Society for Testing and Materials: Philadelphia, PA, USA, 1981; pp. 75–88. [Google Scholar]
- Sudret, B.; Hornet, P.; Stephan, J.M.; Guede, Z.; Lemaire, M. Probabilistic assessment of fatigue life including statistical uncertainties in the S-N curve. In Proceedings of the 17th International Conference on Paper # M03-4 Structural Mechanics in Reactor Technology (SMiRT 17), Prague, Czech Republic, 17–22 August 2003. [Google Scholar]
- Schneider, C.R.A.; Maddox, S.J. Best Practice Guide on Statistical Analysis of Fatigue Data; TWI: Cambridge, UK, February 2003. [Google Scholar]
- Owen, D.B. Tables of Factors of One-Sided Tolerance Limits for a Normal Distribution; SCR-13; Sandia Corporation Monograph: Washington, DC, USA, 1958. [Google Scholar]
- Wirsching, P.H. Statistical Summaries of Fatigue Data for Design Purposes; NASA Contractor Report 3697; NASA: Washington, DC, USA, 1983.
- Wei, Z.; Dogan, B.; Luo, L.; Lin, B.; Konson, D. Design Curve Construction Based on Tolerance Limit Concept. In Proceedings of the ASME 2012 Pressure Vessels & Piping Conference (PVP 2012), Toronto, ON, Canada, 15–19 July 2012. [Google Scholar]
- Ronold, K.O.; Echtermeyer, A.T. Estimation of fatigue curves for design of composite laminates. J. Compos. Part A
**1996**, 27A, 485–491. [Google Scholar] [CrossRef] - Echtermeyer, A.T. Integrating Durability in Marine Composite Certification. In Durability of Composites in a Marine Environment; Solid Mechanics and Its Applications 208; Springer Science + Business Media: Dordrecht, The Netherlands, 2014; pp. 179–194. ISBN 978-94-007-7416-2. [Google Scholar]
- DIN. Statistische Auswertung von Daten, Bestimmung Eines Statistischen Anteilsbereichs (Statistical Interpretation of Data; Determination of a Statistical Tolerance Interval); DIN55303, Part 5; Deutsches Institut für Normung E.V. (DIN): Berlin, Germany, 1987. (In Germany) [Google Scholar]
- ASTM. Standard Practice for Obtaining Hydrostatic or Pressure Design Basis for “Fiberglass (Glass-Fiber-Reinforced Thermosetting-Resin) Pipe and Fittings; ASTM Standard D2992-12; American Society for Testing Materials: Philadelphia, PA, USA, 2012. [Google Scholar] [CrossRef]
- Mandell, J.F.; Samborsky, D.D.; Wahl, N.K.; Sutherland, H.J. Testing and Analysis of Low Cost Composite Materials Under Spectrum Loading and High Cycle Fatigue Condition. Conference Paper, ICCM14, Paper # 1811; SME/ASC. 2003. Available online: https://www.semanticscholar.org/paper/Testing-and-Analysis-of-Low-Cost-Composite-Under-Mandell-Samborsky/8cef98d9ccec02d1e54489c597c0eb536de26bdb (accessed on 12 November 2021).
- Echtermeyer, A.T.; Engh, B.; Buene, L. Influence of matrix and fabric. In Design of Composite Structures Against Fatigue: Applications to Wind Turbine Blades; Mayer, R.M., Ed.; Mechanical Engineering Publications Ltd.: Suffolk, UK, 1996; pp. 33–49. [Google Scholar]
- Echtermeyer, A.T.; Kensche, C.; Bach, P.; Poppen, M.; Lilholt, L.; Andersen, S.I.; Brønsted, P. Method to Predict Fatigue Lifetimes of GRP Wind Turbine Blades and Comparison with Experiments. In Proceedings of the 1996 European Union Wind Energy Conference and Exhibition, Gothenburg, Sweden, 20–24 May 1996; H.P. Stephens & Associates: Bedford, UK, 1996; pp. 907–913. [Google Scholar]
- Ronold, K.O.; Lotsberg, I. On the estimation of characteristic S–N curves with confidence. J. Mar. Struct.
**2012**, 27, 29–44. [Google Scholar] [CrossRef] - Resnikoff, G.J.; Lieberman, G.J. Tables of the Noncentral t-Distribution; Stanford University Press: Stanford, CA, USA, 1957. [Google Scholar]
- Pearson, E.S.; Hartley, H.O. Biometrika Tables for Statisticians, 3rd ed.; Biometrika Trust: London, UK, 1976. [Google Scholar]
- Hald, A. Statistical Theory with Engineering Applications; John Wiley: Hoboken, NJ, USA, 1952. [Google Scholar]
- Guttman, I. Statistical Tolerance Regions: Classical and Bayesian; Griffin: London, UK, 1970. [Google Scholar]
- Zacks, S. The Theory of Statistical Inference; John Wiley and Sons, Inc.: New York, NY, USA, 1970. [Google Scholar]
- Little, R.E. Review of statistical analysis of fatigue life data using one-sided lower statistical tolerance limits. In Statistical Analysis of Fatigue Data; ASTM STP 744; Little, R.E., Ekvall, J.C., Eds.; American Society for Testing and Materials: Philadelphia, PA, USA, 1981; pp. 3–23. [Google Scholar]
- Beaumont, P.W.R.; Schultz, J.M. Statistical Aspects of Fracture. In Failure Analysis of Composite Materials; Chapter 4.6; Beaumont, P.W.R., Schultz, J.M., Friedrich, K., Eds.; Technomic Publishing Co., Inc.: Lancaster, PA, USA, 1990; Volume 4. [Google Scholar]
- Ang, A.H.-S.; Tang, W.H. Probability Concepts in Engineering Planning and Design, Vol. I, Basic Principles; John Wiley & Sons: New York, NY, USA, 1975. [Google Scholar]
- Pascual, F.G. Planning Fatigue Experiments and Analyzing Fatigue Data with the Random Fatigue-Limit Model and Modified Sudden Death Tests. Ph.D. Thesis, Iowa State University, Ames, IA, USA, 1997. [Google Scholar]
- Tvedt, L. Proban—Probabilistic Analysis. Struct. Saf.
**2006**, 28, 150–163. [Google Scholar] [CrossRef] - Snedecor, G.W.; Cochran, W.G. Statistical Methods; Iowa State University Press: Ames, IA, USA, 1989. [Google Scholar]
- Veritas, D.N. Statistical Representation of Soil Data; DNV-RP-C207; Recommended Practice: Høvik, Norway, 2012. [Google Scholar]
- Echtermeyer, A.T.; Ekeberg, T.S.; Sund, O.E. Long-Term Testing of Composite Through-Thickness Properties; Research Report 131; Health & Safety Executive, HSE: Norwich, UK, 2004. [Google Scholar]
- American Society for Testing and Materials. Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data; ASTM E739–91; American Society for Testing and Materials: West Conshohocken, PA, USA, 2004. [Google Scholar]
- Little, R.E.; Jebe, E.H. Statistical Design of Fatigue Experiments; Applied Science Publishers Ltd.: London, UK, 1975. [Google Scholar]
- Bowden, D.C. Query 26: Tolerance Interval in Regression. Technometrics
**1968**, 10, 207–209. [Google Scholar] [CrossRef]

**Figure 1.**Explanation of measures and symbols. modified from [26].

**Figure 2.**Factor c

_{1−α}for nine different combinations of tolerance γ and confidence level 1 − α: (

**a**) $\gamma =0.9$ and $1-\alpha =0.75$; (

**b**) $\gamma =0.9$ and $1-\alpha =0.90$; (

**c**) $\gamma =0.9$ and $1-\alpha =0.95$; (

**d**) $\gamma =0.95$ and $1-\alpha =0.75$; (

**e**) $\gamma =0.95$ and $1-\alpha =0.90$; (

**f**) $\gamma =0.95$ and $1-\alpha =0.95$; (

**g**) $\gamma =0.97725$ and $1-\alpha =0.75$; (

**h**) $\gamma =0.97725$ and $1-\alpha =0.90$; (

**i**) $\gamma =0.97725$ and $1-\alpha =0.95$. The meaning of the axis is explained in Figure 1.

**Figure 3.**Comparison between the true (simulated) c

_{1−α}values and the c

_{1−α}values obtained by the hyperbolic model for n = 8, γ = 0.95 and 1 − α = 0.95.

**Figure 4.**Estimated mean S–N curve on log-log scale and estimated characteristic S–N curve (1) by theoretically correct approach and (2) by Figure 2.

Symbol | |
---|---|

α | probability complement of confidence level |

1 − α | statistical confidence level |

Γ | gamma function |

ε | strain |

Φ | standard Gaussian cumulative distribution function |

σ | standard deviation |

Δx | increment of x |

_{Lx} | length of finite interval of x = logS |

h_{n} | function of n |

m | material property of fiber strength |

n | number of fatigue tests |

W | chi-square distributed variable with specific number of degrees of freedom |

C | non-central t-distributed variable |

S | stress range |

s | estimate of standard deviation σ |

S | stochastic standard deviation |

N | number of stress cycles to failure at stress range S |

$\overline{\sigma}$ | mean fibre strength |

σ_{f} | fibre bundle strength |

# Test Data | t_{n}_{−2}(1 − α) | ||
---|---|---|---|

n | 1 − α = 0.75 | 1 − α = 0.90 | 1 − α = 0.95 |

4 | 0.816 | 1.886 | 2.920 |

8 | 0.718 | 1.440 | 1.943 |

10 | 0.706 | 1.397 | 1.860 |

12 | 0.700 | 1.372 | 1.812 |

15 | 0.694 | 1.350 | 1.771 |

20 | 0.688 | 1.333 | 1.734 |

30 | 0.683 | 1.313 | 1.701 |

50 | 0.679 | 1.300 | 1.676 |

100 | 0.677 | 1.290 | 1.660 |

∞ | 0.674 | 1.282 | 1.645 |

Survival Probability (Tolerance) γ | γ Quantile of Standard Normal Variate Φ ^{−1} (γ) |
---|---|

0.50 | 0.000 |

0.75 | 0.674 |

0.90 | 1.282 |

0.95 | 1.645 |

0.97725 | 2.000 |

0.99 | 2.326 |

**Table 4.**Quantiles ${c}_{1-\alpha ,k}\left(n-1\right)$ for various auxiliary proportions $k$ and confidence levels $1-\alpha $.

No. of Observations, n | Auxiliary Proportion k = 0.90 | Auxiliary Proportion k = 0.95 | Auxiliary Proportion k = 0.99 | ||||||
---|---|---|---|---|---|---|---|---|---|

1 − α = 0.75 | 1 − α = 0.90 | 1 − α = 0.95 | 1 − α = 0.75 | 1 − α = 0.90 | 1 − α = 0.95 | 1 − α = 0.75 | 1 − α = 0.90 | 1 − α = 0.95 | |

4 | 2.501 | 4.258 | 6.158 | 3.152 | 5.312 | 7.657 | 4.396 | 7.340 | 10.552 |

8 | 1.791 | 2.333 | 2.755 | 2.251 | 2.904 | 3.404 | 3.126 | 3.972 | 4.641 |

10 | 1.702 | 2.133 | 2.454 | 2.147 | 2.660 | 3.038 | 2.977 | 3.641 | 4.143 |

12 | 1.646 | 2.012 | 2.275 | 2.078 | 2.511 | 2.825 | 2.885 | 3.444 | 3.852 |

15 | 1.591 | 1.895 | 2.108 | 2.012 | 2.366 | 2.621 | 2.796 | 3.257 | 3.585 |

20 | 1.536 | 1.781 | 1.949 | 1.947 | 2.237 | 2.429 | 2.710 | 3.078 | 3.331 |

30 | 1.479 | 1.664 | 1.788 | 1.877 | 2.094 | 2.233 | 2.619 | 2.895 | 3.079 |

50 | 1.428 | 1.563 | 1.651 | 1.817 | 1.976 | 2.075 | 2.540 | 2.740 | 2.870 |

100 | 1.380 | 1.471 | 1.528 | 1.758 | 1.862 | 1.929 | 2.470 | 2.601 | 2.683 |

∞ | 1.282 | 1.282 | 1.282 | 1.645 | 1.645 | 1.645 | 2.326 | 2.326 | 2.326 |

Stress Amplitude S (MPa) | Number of Cycles to Failure N | logS | logN |
---|---|---|---|

2.60 | 1,591,872 | 0.415 | 6.202 |

3.20 | 1,140,319 | 0.505 | 6.057 |

3.20 | 2,680,000 | 0.505 | 6.428 |

3.85 | 19,550 | 0.585 | 4.291 |

3.85 | 802,398 | 0.585 | 5.904 |

3.85 | 204,100 | 0.585 | 5.310 |

5.80 | 15,639 | 0.763 | 4.194 |

6.45 | 4595 | 0.810 | 3.662 |

6.45 | 2137 | 0.810 | 3.330 |

6.45 | 2330 | 0.810 | 3.367 |

7.10 | 2034 | 0.851 | 3.308 |

Stress Amplitude S (MPa) | logS Covered by Tests | By Theory (Simulation by Equation (13)) ${c}_{1-\alpha}$ | By Figure 2 | |
---|---|---|---|---|

$\frac{\Delta x}{{L}_{x}}$ | ${c}_{1-\alpha}$ | |||

2.60 | 0.415 | 3.79 | 0.455 | 3.75 |

3.20 | 0.505 | 3.59 | 0.267 | 3.57 |

3.85 | 0.585 | 3.48 | 0.099 | 3.46 |

5.80 | 0.763 | 3.52 | 0.272 | 3.57 |

6.45 | 0.810 | 3.59 | 0.368 | 3.66 |

7.10 | 0.851 | 3.67 | 0.455 | 3.77 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ronold, K.O.; Echtermeyer, A.T.
Using Tolerance Bounds for Estimation of Characteristic Fatigue Curves for Composites with Confidence. *Safety* **2022**, *8*, 1.
https://doi.org/10.3390/safety8010001

**AMA Style**

Ronold KO, Echtermeyer AT.
Using Tolerance Bounds for Estimation of Characteristic Fatigue Curves for Composites with Confidence. *Safety*. 2022; 8(1):1.
https://doi.org/10.3390/safety8010001

**Chicago/Turabian Style**

Ronold, Knut O., and Andreas T. Echtermeyer.
2022. "Using Tolerance Bounds for Estimation of Characteristic Fatigue Curves for Composites with Confidence" *Safety* 8, no. 1: 1.
https://doi.org/10.3390/safety8010001