# Concurrent Lamination and Tapering Optimization of Cantilever Composite Plates under Shear

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Cantilever Laminated Plate

#### 2.2. Stiffness Formulation

#### 2.3. Finite Element Analysis

#### 2.4. Optimization

## 3. Results and Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QI-R | Quasi-isotropic and rectangular |

QI-T | Quasi-isotropic and tapered |

ST-R | Stiffness-tailored and rectangular |

ST-T | Stiffness-tailored and tapered |

## Nomenclature

A | Plate area |

${C}_{p}$ | Constitutive matrix for in-plane deformation |

${C}_{t}$ | Constitutive matrix for transverse shear deformation |

${E}_{11}$ | Longitudinal modulus |

${E}_{22}$ | Transverse modulus |

${G}_{12}$ | In-plane shear modulus |

${G}_{13}$, ${G}_{23}$ | Transverse shear moduli |

F | Tip shear force |

f | Nodal force vector |

K | Nodal stiffness matrix |

l | Plate length |

N | Number of layers |

t | Plate thickness |

${t}_{i}$ | Layer thicknesses |

${U}_{i}$ | Material invariants |

u | Nodal displacement vector |

${V}_{1}$, ${V}_{3}$ | Lamination parameters |

w | Plate width |

$\alpha $ | Taper angle |

${\nu}_{12}$ | Major Poisson’s ratio |

${\theta}_{i}$ | Layer angles |

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**Figure 1.**Schematic diagrams of (

**a**) non-tapered and (

**b**) width-tapered cantilever laminated composite plates with identical thickness, length, and surface areas.

**Figure 2.**Feasible region of lamination parameters and sample design points with corresponding layer angles.

**Figure 3.**Exemplary finite element meshes for (

**a**) non-tapered and (

**b**) ${25}^{\circ}$ width-tapered plates with $l/w=1.0$.

**Figure 4.**Optimal lamination parameters with corresponding layer angles and taper angles for different design approaches and aspect ratios.

**Figure 6.**The influences of aspect ratio and design strategy on (

**a**) the normalized tip displacements of cantilever plates under shear load and (

**b**) displacement reduction percentage compared to the quasi-isotropic rectangular plates.

${E}_{11}$ | $25{E}_{22}$ |

${G}_{12}={G}_{13}$ | $0.5{E}_{22}$ |

${G}_{23}$ | $0.2{E}_{22}$ |

${\nu}_{12}$ | $0.25$ |

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**MDPI and ACS Style**

Serhat, G. Concurrent Lamination and Tapering Optimization of Cantilever Composite Plates under Shear. *Materials* **2021**, *14*, 2285.
https://doi.org/10.3390/ma14092285

**AMA Style**

Serhat G. Concurrent Lamination and Tapering Optimization of Cantilever Composite Plates under Shear. *Materials*. 2021; 14(9):2285.
https://doi.org/10.3390/ma14092285

**Chicago/Turabian Style**

Serhat, Gokhan. 2021. "Concurrent Lamination and Tapering Optimization of Cantilever Composite Plates under Shear" *Materials* 14, no. 9: 2285.
https://doi.org/10.3390/ma14092285