# Buckling Analysis of Laminated Stiffened Plates with Material Anisotropy Using the Rayleigh–Ritz Approach

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## Abstract

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## 1. Introduction

## 2. Extension of Rayleigh–Ritz Buckling Solutions for Partially Anisotropic Stiffened Plates

#### 2.1. Rayleigh–Ritz Formulation of a Stiffened Anisotropic Plate Buckling Problem

_{skin}). The plate is reinforced by several orthotropic stiffeners (N

_{stiffener}), as shown in Figure 1. The plate is compressively loaded with a uniform or linearly varying load N

_{x}, in the x-direction. The stiffeners are also loaded with a compressive force F in the same direction.

_{skin}and the stiffeners’ potential energy Π

_{stiffener}, as shown in Equation (1). The extension of the generalized Rayleigh–Ritz formulation, currently presented, is based on the consideration of the potential energy of the stiffeners.

_{x}, ε

_{y}, and γ

_{xy}by virtue of the Kirchhoff–Love hypothesis that ε

_{z}, γ

_{yz}, and γ

_{xz}are assumed to be zero. Considering small strains and linear elasticity, the stress resultants versus the strains are given by the following equations:

_{ij}, B

_{ij}, and D

_{ij}(i, j = 1, 2, and 6) represent the extensional, coupling, and bending stiffness, respectively. The plate middle surface strains and curvatures are denoted by ε

_{x}, ε

_{y}, γ

_{xy}and k

_{x}, k

_{y}, k

_{xy}. The subscript k denotes the kth layer while N

_{ply}represents the total number of plies in the laminate. The stiffness of each layer, denoted by Q

_{ij}, is transformed in the global coordinate system and depends on the layer’s mechanical properties in the material principal axes and orientation angle. Finally, the distance Z

_{k}is illustrated in Figure 2.

_{skin}of an anisotropic flat skin can be written as below. It is worth mentioning that this is the generalized expression of the potential skin energy as documented in [29], which considers the extension, coupling, and all bending stiffness matrix terms, as well as the in-plane displacements.

_{stiffener}of the longitudinal stiffener in the deformed state is written as:

_{y}is the stiffener’s second moment of inertia about the skin’s mid-plane, and N

_{stiffener}is the number of stiffeners considered.

_{skin}) carried out upon the application of compressive loading in the x-direction (N

_{x}) on the skin middle plane is:

_{stiffener}) carried out during the application of a compressive force (F) acting on the longitudinal stiffeners is:

_{x}acting on the plate, as shown in Equation (8):

_{stiffener}and t

_{skin}are the stiffener’s cross section area and the plate’s thickness, respectively. For the global stiffness matrix of the skin and stiffener refer to Appendix A.

_{mn}, B

_{mn}, and C

_{mn}are the un-determined coefficients of the functions and m and n are the number of terms considered in the finite deflection series. The assumed deflections U

_{mn}, V

_{mn}and W

_{mn}are functions of the spatial coordinates x and y in a variable separable form, i.e., X

_{m}(x)Y

_{n}(y). Additionally, since the buckling shape, W

_{mn}, of a laminated plate under compression is typically described by a single sin-shape in the x-direction, it is assumed that the buckling mode of the stiffened plate is characterized by m buckling half waves in the longitudinal direction. It is also assumed that, in the transverse y-direction, the plate is assumed to buckle in n buckling half waves, as described in [15]. The specific mode shapes in this study are available in Section 2.2.

_{x}is given by Equation (11):

#### 2.2. Boundary Conditions

_{x}vanish. The mathematical expressions of the conditions are shown in Equations (12) and (15).

_{y}must attain zero values for the unloaded longitudinal edges of the skin at y = 0 and y = b of Figure 1, as shown in Equations (14) and (15).

## 3. Buckling Solution of Stiffened and Unstiffened Plates with Varying Degree of Anisotropy Using the Rayleigh–Ritz Approach

#### 3.1. Unstiffened Plates

#### 3.1.1. Uniform Loaded Plates

_{ply}= 0.134 mm. The material properties of the plate are the following: modulus of elasticity E

_{11}= 130 GPa, E

_{22}= 10.5 GPa; shear modulus G

_{12}= 6 GPa; and Poisson’s ratio ν

_{12}= 0.28. The applied load comprises a uniform compressive load (N

_{x}).

_{2}/45

_{2}/0

_{2}/−45

_{2}/0

_{2}] and [0

_{6}/60

_{6}] are presented in Figure 3 and Figure 4.

_{6}/60

_{6}], where a high degree of mechanical coupling exists, the boundary conditions are partially satisfied; according to the selected trigonometric functions of Equations (16) and (17), only the out-of-plane displacements obtain zero values in the boundaries. Therefore, higher differences are observed between experimental, numerical and the Rayleigh–Ritz method’s results for fully anisotropic plates.

_{6}/60

_{6}]. This can be justified by considering the extensional–twisting and bending–twisting mechanical coupling that this specific lamination demonstrates.

#### 3.1.2. Linear Varying Loaded Plates

_{skin}= 100, i.e., it is a thin laminate. The laminate stacking sequence is [0/90/90/0/0] with lamina material properties of E

_{11}/E

_{22}= 10, G

_{12}/G

_{22}= 0.5, and Poisson’s ratio ν

_{12}= 0.25. The buckling load obtained for this case is compared to the respective results from [13]. Once again, the number of terms used for the double sine series in the Rayleigh–Ritz approach is M × N = 30 × 30.

#### 3.2. Stiffened Plates

#### 3.2.1. Stiffened Plates with One Stiffener

_{stiffener}= 9t

_{skin}. Two different loading conditions are assumed for the plate: (a) uniform load and (b) linear varying load. The stiffener is unloaded and loaded for the aforementioned loading case.

#### 3.2.2. Plates Stiffened with Multiple Stiffeners

_{s}lamination with a ply thickness of 0.134 mm.

_{6}/60

_{6}], where a high degree of mechanical coupling exists, the boundary conditions are not fully satisfied by the selected trigonometric functions of Equations (13) or (15). Therefore, an error is introduced to the examined cases of this lamination, leading to unconservative results for all the stiffener-to-skin stiffness ratios.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Buckling modes of the [0

_{2}/45

_{2}/0

_{2}/−45

_{2}/0

_{2}] plate. (

**a**) FE model (ANSYS), (

**b**) R–R method.

**Figure 4.**Buckling mode of the [0

_{6}/60

_{6}] plate calculated. (

**a**) FE model (ANSYS), (

**b**) R–R method.

**Figure 5.**(

**a**) Mode shape comparison for the case of a [0/90/90/0/0] laminated stiffened plate with unloaded stiffener. (

**a**) Uniform loaded skin, (

**b**) linearly varying loaded skin.

**Figure 6.**(

**a**) Mode shape of a [0/90/90/0/0] laminated stiffened plate with a loaded stiffener. (

**a**) Skin uniformly loaded, (

**b**) skin linearly varying loaded.

**Figure 7.**Critical buckling loads versus percentage stiffener-to-skin stiffnesses for different skin laminations.

**Figure 8.**Global buckling of stiffened plates with skin lamination. (

**a**) [0

_{2}/45

_{2}/0

_{2}/45

_{2}/0

_{2}], (

**b**) [0

_{6}/60

_{6}].

Mechanical Coupling | |||||
---|---|---|---|---|---|

Laminate | Extension-Shear | Extension-Bending | Extension-Twisting and Shear-Bending | Shear-Twisting | Bending-Twisting |

[0_{3}/90_{3}]_{s} | - | - | - | - | - |

[0_{3}/90_{3}/0_{3}/90_{3}] | - | Yes | - | - | - |

[0_{2}/45_{2}/0_{2}/45_{2}/0_{2}] | Yes | - | - | - | Yes |

[0_{2}/45_{2}/0_{2}/-45_{2}/0_{2}] | - | - | Yes | - | - |

[0/90/90/0/0] | - | Yes | - | - | - |

[0_{6}/60_{6}] | Yes | Yes | Yes | Yes | Yes |

Buckling Load, N_{x} in (N/mm) | ||||
---|---|---|---|---|

Lamination | R–R Method | FE Model (ANSYS) | Lagace, [9] R–R Results | Experimental Results, Lagace, [9] |

[0_{3}/90_{3}]_{s} | 26.475 | 26.367 | 27.150 | 19.650 |

[0_{3}/90_{3}/0_{3}/90_{3}] | 17.514 | 15.305 | 20.439 | 14.970 |

[0_{2}/45_{2}/0_{2}/45_{2}/0_{2}] | 22.789 | 21.675 | 18.389 | 23.440 |

[0_{2}/45_{2}/0_{2}/-45_{2}/0_{2}] | 20.426 | 20.002 | 17.770 | 21.480 |

[0_{6}/60_{6}] | 17.788 | 10.647 | 18.00 | 11.00 |

**Table 3.**Buckling load comparison between the R–R solution, FE model, and reference results [13] for the case of a linearly varying loaded plate.

[0/90/90/0/0] | Buckling Load, N_{x} in (N/mm) | ||
---|---|---|---|

Modes (m, n) | R–R Solution | FE Model (ANSYS) | Papazoglou et al. [13] |

(1, 1) | 67.749 | 59.476 | 65.00 |

(2, 1) | 69.298 | 65.803 | - |

(3, 1) | 113.31 | 108.266 | - |

**Table 4.**Buckling load comparison between R–R solution, FE model, and the reference results of [33] work for the case of a [0/90/90/0/0] laminated stiffened plate with unloaded stiffener.

Unloaded Stiffener | Buckling Load, N_{x} in (N/mm) | ||
---|---|---|---|

R–R Solution | FE Model (ANSYS) | Kumar and Mukhopadhyay [33] | |

Uniform loaded skin | 119.6 | 116.265 | - |

Linearly varying loaded skin | 172.92 | 171.755 | 260.00 |

**Table 5.**Buckling load comparison between the R–R solution and FE model for the case of a [0/90/90/0/0] laminated stiffened plate with a loaded stiffener.

Buckling Load, N_{x} in (N/mm) | ||
---|---|---|

R–R Solution | FE Model (ANSYS) | |

Uniform loaded skin | 118.45 | 112.542 |

Linearly varying loaded skin | 184.05 | 168.747 |

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**MDPI and ACS Style**

Stamatelos, D.G.; Labeas, G.N.
Buckling Analysis of Laminated Stiffened Plates with Material Anisotropy Using the Rayleigh–Ritz Approach. *Computation* **2023**, *11*, 110.
https://doi.org/10.3390/computation11060110

**AMA Style**

Stamatelos DG, Labeas GN.
Buckling Analysis of Laminated Stiffened Plates with Material Anisotropy Using the Rayleigh–Ritz Approach. *Computation*. 2023; 11(6):110.
https://doi.org/10.3390/computation11060110

**Chicago/Turabian Style**

Stamatelos, Dimitrios G., and George N. Labeas.
2023. "Buckling Analysis of Laminated Stiffened Plates with Material Anisotropy Using the Rayleigh–Ritz Approach" *Computation* 11, no. 6: 110.
https://doi.org/10.3390/computation11060110