# Determination of Forces and Moments Per Unit Length in Symmetric Exponential FG Plates with a Quasi-Triangular Hole

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Formulations

^{th}layer is:

^{th}layer as function of the mid surface curvatures (k

_{x}, k

_{y}and k

_{xy}) and mid surface strain (${\epsilon}_{x}^{0}$,${\epsilon}_{y}^{0}$ and ${\gamma}_{xy}^{0}$) are in accordance with Equation (4) [19]:

_{x}, k

_{y}, k

_{xy}) and ${\epsilon}_{x}^{0}$, ${\epsilon}_{y}^{0}$, ${\epsilon}_{xy}^{0}$ are the k

^{th}layer mid surface curvatures and strains.

^{th}layer to to the middle surface (Figure 2). Every layer has a fixed thickness ${t}_{k}$. For a laminate that possesses $N$ layers, the overall thickness $H$ is:

_{x}, f

_{y}, p(Γ), m(Γ) and respectively Γ are internal forces, transverse force, bending moment, and arc hole length to the current point, respectively.

## 3. Numerical Results

_{y}acquired from the provided analytical solution including one provided by the numerical solution. Figure 5 presents a resembling comparison of the stress resultant N

_{xy.}The angle α denotes the points’ position on the hole boundary in regard to the horizontal axis. The mesh and loading patterns are depicted in Figure 6 and Figure 7, correspondingly. The maximum resultant forces’ ratio at the hole corner to the implemented load is considered as the normalized stress resultant. Evidently, when the implemented load is 1, i.e., (${N}_{x}=1N/mm$), hence the normalized stress resultant is the resultant forces. Furthermore, the normalized moment resultants (${m}_{x},{m}_{y}$ and ${m}_{xy}$) are defined as ($\frac{{M}_{x}*1000}{\sqrt{{Q}_{11}{N}_{x}H}}$,$\frac{{M}_{y}*1000}{\sqrt{{Q}_{11}{N}_{x}H}}$,$\frac{{M}_{xy}*1000}{\sqrt{{Q}_{11}{N}_{x}H}}$) correspondingly [25], such that H is the overall laminate thickness.

## 4. Results

_{x,}n

_{y,}n

_{xy}) occur at approximately rotation angle of 45° and the rotation angle of 20° leads to the maximum normalized resultant forces. For example, the normalized resultant forces (n

_{xy}) presented in Figure 9.

_{y,}m

_{y}) are selected excellently.

_{x}and n

_{y}occur at rotation angle of 30° however, for n

_{y}this value occurs at rotation angle of zero degrees. The rotation angle of 30° causes the minimum values of ${m}_{y}$ and ${m}_{xy}$, however, the minimum value of ${m}_{x}$ occurs at rotation angle of zero degrees.

## 5. Conclusions

- (1)
- By comparing the provided analytical solution in addition to the numerical solution acquired via finite element modeling, favorable compatibility was exhibited between these outcomes.
- (2)
- When a perforated plate subjected to uniaxial loading along the x-direction, the normalized forces and moments per unit length at the proximity of the holes of various aspect rations will be greater at the loading direction compared to other directions.
- (3)
- The hole corner’s curvature radius does not only determine the values of the forces and moments per unit length, but also hole orientation and aspect ratio have substantial roles in this regard. Therefore, cautious modifications of such parameters may substantially decrease the forces and moments concerning every hole at all provided corner curvatures.
- (4)
- Most forces and moments per unit length change linearly with aspect ratio of triangular hole (c).
- (5)
- The lowest values of the normalized forces and moments occur in ω = 0 which is equivalent to a circular hole this means that for a triangular hole, the values of the normalized forces and moments are not less than the circular hole in any value of the parameters discussed.
- (6)
- As bluntness parameter (ω) increases, the normalized forces and moments increase at different rotation angles.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$Z{p}_{j}$ | Mapping function |

$\zeta $ | Complex variable |

${a}_{j}$,${b}_{j}$, ${c}_{j}$, ${d}_{j}$ | Complex constants related to the roots of the characteristic equation |

$\omega $ | Bluntness |

$n$ | Hole geometry |

$i$ | Imaginary constant |

$j$ | Counter |

${\gamma}_{xy}$ | Shear strain |

$k$ | Layers counter |

${\overline{Q}}_{ij}$ | Reduced stiffness matrix |

${\gamma}_{xy}^{0}$ | Mid-plate shear strain |

$z$ | Thickness |

$H$ | Total thickness |

${z}_{k}$ | Distance to mid plane |

${t}_{k}$ | Thickness of each layer |

$MP(z)$ | Exponential function |

${P}_{t}$ | Material properties in the upper layers |

${P}_{b}$ | Material properties in the lower layers |

$e$ | Napier number (2.718) |

$\eta $ | Exponential function constant |

$Re$ | Real part |

${\phi}_{j}$ | Potential function |

$c$ | Hole aspect ratio |

${s}_{j}$ | The roots of the characteristic equation |

$\beta $ | Hole rotation angle |

$\lambda $ | The hole size |

${\sigma}_{x}$,${\sigma}_{y}$ | Normal stress |

${\tau}_{xy}$ | Shear stress |

${\epsilon}_{x}$,${\epsilon}_{y}$ | Longitudinal strain |

${\epsilon}_{x}^{0}$,${\epsilon}_{y}^{0}$ | Midline longitudinal strains |

${k}_{x}$,${k}_{y}$, ${k}_{xy}$ | Intra-curvature curvature |

${N}_{x}$,${N}_{y}$, ${N}_{xy}$ | Resultant forces |

${M}_{x}$, ${M}_{y}$, ${M}_{xy}$ | Moments resultants |

$N$ | Number of layers |

${A}_{ij}$ | Tensile stiffness matrix |

${B}_{ij}$ | Coupling matrix |

${D}_{ij}$ | Bending stiffness matrix |

${f}_{1j}$, ${f}_{2j}$, ${f}_{3j}$ | Complex constant in calculating force and moments |

${\mu}_{j}$ | Anisotropy factor |

${f}_{x},{f}_{y}$ | Components of internal force |

$f$ | Force at the boundary of hole |

$m$ | Normal bending at the boundary of hole |

$p$ | Transverse force at the boundary of hole |

$\mathsf{\Gamma}$ | length of the hole arc |

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**Figure 3.**FGP with a hole [15].

**Figure 4.**Comparing the moments m

_{x}provided by the analytical approach and the numerical solution (ω = 0.125).

**Figure 5.**Comparing the resultant force n

_{x}provided by the analytical approach and the numerical solution (ω = 0.125).

**Figure 9.**Discrepancies of the normalized forces and moments per unit length with the rotation angle of triangular hole (β).

**Figure 10.**The effect of c on the normalized forces and moments per unit length in different values of ω.

Material | E_{1}(GPa) | E_{2}(GPa) | G_{12}(GPa) | ν_{12} |
---|---|---|---|---|

Graphite BMI | 124 | 8.46 | 4.59 | 0.28 |

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**MDPI and ACS Style**

Jafari, M.; Chaleshtari, M.H.B.; Abdolalian, H.; Craciun, E.-M.; Feo, L.
Determination of Forces and Moments Per Unit Length in Symmetric Exponential FG Plates with a Quasi-Triangular Hole. *Symmetry* **2020**, *12*, 834.
https://doi.org/10.3390/sym12050834

**AMA Style**

Jafari M, Chaleshtari MHB, Abdolalian H, Craciun E-M, Feo L.
Determination of Forces and Moments Per Unit Length in Symmetric Exponential FG Plates with a Quasi-Triangular Hole. *Symmetry*. 2020; 12(5):834.
https://doi.org/10.3390/sym12050834

**Chicago/Turabian Style**

Jafari, Mohammad, Mohammad Hossein Bayati Chaleshtari, Hamid Abdolalian, Eduard-Marius Craciun, and Luciano Feo.
2020. "Determination of Forces and Moments Per Unit Length in Symmetric Exponential FG Plates with a Quasi-Triangular Hole" *Symmetry* 12, no. 5: 834.
https://doi.org/10.3390/sym12050834