# Assessing Static and Dynamic Response Variability due to Parametric Uncertainty on Fibre-Reinforced Composites

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fibre-Reinforced Composites

#### 2.2. Constitutive Relations and Equilibrium Equations

#### 2.3. Simulation of Modelling Parameters Uncertainty

#### 2.4. Forward Propagation of the Uncertainty

#### 2.5. Multivariable Linear Regression Model

## 3. Results and Discussion

#### 3.1. Uncertainty in the Material Properties

#### 3.2. Uncertainty in the Layer Orientation

_{1}–θ

_{4}), as shown in Figure 6. Note that the sample with the maximum transverse displacement given in Figure 5 is the one in Figure 6 with the combination of all stacking angles being uncertain (All).

_{4}and must be further evaluated. To evaluate the results for other stacking sequences, the case studies presented in Table 2 are considered.

_{S}laminate, although its significance is now shared with the first layer. Note that both are external layers.

_{4}(Table 4) to almost 0.30 for [0/90]

_{S}(Table 5), and to an inverse correlation in the [0/90]

_{2}laminate (Table 6).

_{S}presents higher values for all stacking angles, with the value for ${\mathsf{\theta}}_{4}$ remaining the highest.

#### 3.3. Uncertainty in the Layer Thickness

#### 3.4. Regression Models

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Matrix plot of the modelling parameters and the resulting maximum deflection $({\mathrm{w}}_{\mathrm{max}})$ and fundamental frequency $({\mathrm{f}}_{1})$ (unidirectional plate, a/h = 20, all input parameters uncertain).

**Figure 3.**Matrix plot of the maximum deflection $({\mathrm{w}}_{\mathrm{max}}(\mathrm{m}))$ for different sets of uncertain parameters (unidirectional plate, a/h = 20).

**Figure 4.**Matrix plot of the modelling parameters and the resulting maximum deflection $({\mathrm{w}}_{\mathrm{max}})$ and fundamental frequency $({\mathrm{f}}_{1})$ (unidirectional plate, a/h = 20, all modelling parameters uncertain except the ply thickness).

**Figure 5.**Matrix plot of the stacking angles (θ

_{1}–θ

_{4}) and the resulting maximum deflection $({\mathrm{w}}_{\mathrm{max}})$ and fundamental frequency $({\mathrm{f}}_{1})$ for Case 1.a (a/h = 20, [0]

_{4}).

**Figure 6.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain stacking angles for Case 1.a (a/h = 20, [0]

_{4}).

**Figure 7.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain stacking angles for Case 2.a (a/h = 100, [0]

_{4}).

**Figure 8.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain stacking angles for Case 2.b (a/h = 100, [0/90]

_{S}).

**Figure 9.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain stacking angles for Case 1.c (a/h = 20, [0/90]

_{2}).

**Figure 10.**Matrix plot of the ply thicknesses (h

_{1}–h

_{4}) and the resulting maximum deflection $({\mathrm{w}}_{\mathrm{max}})$ and fundamental frequency (f

_{1}) for Case 3.a (a/h = 20, [0]

_{4}).

**Figure 11.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain ply thicknesses for Case 3.c (a/h = 20, [0/90]

_{2}).

**Figure 12.**Matrix plot of the maximum transverse displacement $({\mathrm{w}}_{\mathrm{max}})$ considering different sets of uncertain ply thicknesses for Case 4.b (a/h = 100, [0/90]

_{S}).

${\mathbf{E}}_{\mathbf{11}}$ (GPa) | ${\mathbf{E}}_{\mathbf{22}}\mathbf{,}{\mathbf{E}}_{\mathbf{33}}$ (GPa) | ${\mathbf{G}}_{\mathbf{12}}\mathbf{,}{\mathbf{G}}_{\mathbf{13}}$ (GPa) | ${\mathbf{G}}_{\mathbf{23}}$ (GPa) | ${\mathsf{\nu}}_{\mathbf{12}}\mathbf{,}{\mathsf{\nu}}_{\mathbf{13}}$ | ${\mathsf{\nu}}_{\mathbf{23}}$ | $\mathsf{\rho}$ (kg/m^{3}) |
---|---|---|---|---|---|---|

161 | 11.38 | 5.17 | 3.98 | 0.32 | 0.44 | 1500 |

Case | $\mathbf{a}\mathbf{/}\mathbf{h}$ | Stacking Sequence | ${\mathsf{\mu}}_{{\mathsf{\theta}}_{\mathbf{p}\mathbf{l}\mathbf{y}}}$ | ${\mathsf{\sigma}}_{{\mathsf{\theta}}_{\mathbf{p}\mathbf{l}\mathbf{y}}}$ |
---|---|---|---|---|

1.a | 20 | [0]_{4} | nominal values | 2° |

1.b | [0/90]_{s} | |||

1.c | [0/90]_{2} | |||

2.a | 100 | [0]_{4} | nominal values | 2° |

2.b | [0/90]_{s} | |||

2.c | [0/90]_{2} |

Case | $\mathbf{a}\mathbf{/}\mathbf{h}$ | Stacking Sequence | ${\mathsf{\mu}}_{{\mathbf{h}}_{\mathbf{p}\mathbf{l}\mathbf{y}}}$ | $\mathbf{C}\mathbf{o}{\mathbf{V}}_{{\mathbf{h}}_{\mathbf{p}\mathbf{l}\mathbf{y}}}$ |
---|---|---|---|---|

3.a | 20 | [0]_{4} | 0.131 mm | 7.5% |

3.b | [0/90]_{s} | |||

3.c | [0/90]_{2} | |||

4.a | 100 | [0]_{4} | 0.131 mm | 7.5% |

4.b | [0/90]_{s} | |||

4.c | [0/90]_{2} |

**Table 4.**Correlation coefficients obtained with uncertain stacking angles for Case 1.a (left) and Case 2.a (right).

θ_{all} | 0.12 | −0.23 | 0.01 | 0.33 | θ_{all} | 0.16 | 0.18 | −0.07 | 0.35 |

θ_{1} | −0.13 | −0.01 | −0.12 | θ_{1} | 0.26 | −0.09 | −0.12 | ||

θ_{2} | −0.22 | −0.04 | θ_{2} | −0.18 | 0.04 | ||||

[0]_{4} | θ_{3} | −0.02 | [0]_{4} | θ_{3} | −0.05 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | θ_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | θ_{4} |

**Table 5.**Correlation coefficients obtained with uncertain stacking angles for Case 1.b (left) and Case 2.b (right).

θ_{all} | 0.31 | 0.17 | 0.33 | 0.46 | θ_{all} | 0.32 | 0.14 | 0.33 | 0.49 |

θ_{1} | 0.17 | 0.29 | −0.12 | θ_{1} | 0.16 | 0.29 | −0.12 | ||

θ_{2} | −0.15 | −0.28 | θ_{2} | −0.15 | −0.27 | ||||

[0/90]s | θ_{3} | 0.27 | [0/90]s | θ_{3} | 0.27 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | θ_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | θ_{4} |

**Table 6.**Correlation coefficients obtained with uncertain stacking angles for Case 1.c (left) and Case 2.c (right).

θ_{all} | 0.35 | 0.00 | −0.10 | 0.33 | θ_{all} | 0.36 | −0.01 | −0.09 | 0.33 |

θ_{1} | 0.00 | 0.26 | −0.13 | θ_{1} | 0.00 | 0.26 | −0.13 | ||

θ_{2} | −0.19 | 0.19 | θ_{2} | −0.20 | 0.19 | ||||

[0/90]_{2} | θ_{3} | −0.19 | [0/90]_{2} | θ_{3} | −0.20 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | θ_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | θ_{4} |

**Table 7.**Correlation coefficients obtained with uncertain ply thicknesses for Case 3.a (left) and Case 4.a (right).

h_{all} | 0.10 | 0.17 | 0.24 | 0.97 | h_{all} | 0.10 | 0.17 | 0.24 | 0.97 |

h_{1} | −0.01 | −0.01 | 0.02 | h_{1} | −0.01 | −0.01 | 0.02 | ||

h_{2} | 0.04 | 0.09 | h_{2} | 0.04 | 0.10 | ||||

[0]_{4} | h_{3} | 0.02 | [0]_{4} | h_{3} | 0.02 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | h_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | h_{4} |

**Table 8.**Correlation coefficients obtained with uncertain ply thicknesses for Case 3.b (left) and Case 4.b (right).

h_{all} | 0.06 | 0.25 | 0.45 | 0.88 | h_{all} | 0.06 | 0.26 | 0.45 | 0.88 |

h_{1} | −0.02 | 0.00 | 0.02 | h_{1} | −0.02 | 0.00 | 0.02 | ||

h_{2} | 0.05 | 0.10 | h_{2} | 0.05 | 0.10 | ||||

[0/90]_{S} | h_{3} | 0.01 | [0/90]_{S} | h_{3} | 0.01 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | h_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | h_{4} |

**Table 9.**Correlation coefficients obtained with uncertain ply thicknesses for Case 3.c (left) and Case 4.c (right).

h_{all} | 0.10 | 0.18 | 0.24 | 0.97 | h_{all} | 0.09 | 0.18 | 0.22 | 0.97 |

h_{1} | −0.01 | 0.00 | 0.02 | h_{1} | −0.01 | 0.00 | 0.02 | ||

h_{2} | 0.05 | 0.10 | h_{2} | 0.05 | 0.10 | ||||

[0/90]_{2} | h_{3} | 0.02 | [0/90]_{2} | h_{3} | 0.02 | ||||

$\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{20}$ | h_{4} | $\mathbf{a}\mathbf{/}\mathbf{h}\mathbf{=}\mathbf{100}$ | h_{4} |

${\mathbf{w}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ | ${\mathbf{f}}_{\mathbf{1}}$ | |||
---|---|---|---|---|

Adj. ${\mathbf{R}}^{\mathbf{2}}$ | 97.44% | 99.77% | ||

Model | F-test | p-value | F-test | p-value |

158.4 | <2.2 × 10^{−16} | 1543 | <2.2 × 10^{−16} | |

Estimate | p-value | Estimate | p-value | |

$\mathsf{\beta}0$ | −5.856 × 10^{−4} | 3.67 × 10^{−15} *** | −6.162 × 10^{−1} | 0.07120 . |

$\mathsf{\beta}1$ | 5.492 × 10^{−16} | 2.96 × 10^{−8} *** | 2.626 × 10^{−11} | <2 × 10^{−16} *** |

$\mathsf{\beta}2$ | 9.294 × 10^{−16} | 0.3557 | 5.999 × 10^{−11} | 4.47 × 10^{−6} *** |

$\mathsf{\beta}3$ | 9.081 × 10^{−5} | 0.0165 * | 6.774 × 10^{−1} | 0.06491 . |

$\mathsf{\beta}4$ | −1.723 × 10^{−15} | 0.4396 | 1.974 × 10^{−11} | 0.37622 |

$\mathsf{\beta}5$ | −5.241 × 10^{−16} | 0.8048 | 7.866 × 10^{−11} | 0.00111 ** |

$\mathsf{\beta}6$ | −4.662 × 10^{−1} | 0.0822 . | 3.407 × 10^{−11} | 0.19487 |

$\mathsf{\beta}7$ | 1.829 × 10^{−1} | <2 × 10^{−16} *** | 5.360 × 10^{3} | <2 × 10^{−16} *** |

$\mathsf{\beta}8$ | - | - | −3.563 × 10^{−3} | <2 × 10^{−16} *** |

Residuals | Independence/normality rejected | OK |

${\mathbf{w}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ | ${\mathbf{f}}_{\mathbf{1}}$ | |||
---|---|---|---|---|

Adj. ${\mathbf{R}}^{\mathbf{2}}$ | 90.46% | 99.317% | ||

Model | F-test | p-value | F-test | p-value |

276.1 | 4.917 × 10^{−16} | 1401 | <2.2 × 10^{−16} | |

Estimate | p-value | Estimate | p-value | |

$\mathsf{\beta}0$ | −4.829 × 10^{−4} | <2 × 10^{−16} *** | 1.028 | 0.00254 ** |

$\mathsf{\beta}1$ | - | - | 2.606 × 10^{−11} | <2 × 10^{−16} *** |

$\mathsf{\beta}7$ | 1.806 × 10^{−1} | 4.92 × 10^{−16} *** | 5.343 × 10^{3} | <2 × 10^{−16} *** |

$\mathsf{\beta}8$ | - | - | −3.586 × 10^{−3} | <2 × 10^{−16} *** |

Residuals | OK | OK |

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**MDPI and ACS Style**

Carvalho, A.; Silva, T.A.N.; Loja, M.A.R.
Assessing Static and Dynamic Response Variability due to Parametric Uncertainty on Fibre-Reinforced Composites. *J. Compos. Sci.* **2018**, *2*, 6.
https://doi.org/10.3390/jcs2010006

**AMA Style**

Carvalho A, Silva TAN, Loja MAR.
Assessing Static and Dynamic Response Variability due to Parametric Uncertainty on Fibre-Reinforced Composites. *Journal of Composites Science*. 2018; 2(1):6.
https://doi.org/10.3390/jcs2010006

**Chicago/Turabian Style**

Carvalho, Alda, Tiago A.N. Silva, and Maria A.R. Loja.
2018. "Assessing Static and Dynamic Response Variability due to Parametric Uncertainty on Fibre-Reinforced Composites" *Journal of Composites Science* 2, no. 1: 6.
https://doi.org/10.3390/jcs2010006