# On the Identification of Sectional Deformation Modes of Thin-Walled Structures with Doubly Symmetric Cross-Sections Based on the Shell-Like Deformation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. One-Dimensional Formulation

#### 2.1. Displacement Fields

**D**= [U, V, W], is obtained as

**D**can be written in a one-dimensional way with a transformation matrix

**H**as

**x**, spanned by the set of amplitude functions χ, is separated into two column submatrices,

**x**

_{O}and

**x**

_{I}; the two submatrices correspond to the out-of-plane and in-plane deformation modes, respectively; and

**φ**,

**ψ**and

**ω**are the shape function vectors constituted by the set of φ, ψ and ω, respectively.

#### 2.2. Strain and Stress Fields

**C**and the constitutive matrix

**E**for the plane stress condition are respectively given by

#### 2.3. Beam Governing Equations

_{st}, the kinetic energy T

_{kn}and the potential energy U

_{pt}. By definition, the former two are respectively given by

**p**= [p, q, r]

^{T}. Here p, q and r represent the force densities in the axial, tangential and normal directions, respectively. The potential energy U

_{pt}can then be given by

_{1}and t

_{2}are the start time and the end time, respectively.

^{2}) and G = E/2(1 + ν). Equations (15) and (16) are the governing differential equations of thin-walled structures with a doubly symmetric cross-section.

## 3. Higher-Order Deformation Modes

#### 3.1. Shell-Like Deformation

_{1}= 0.3 m, flange width b

_{2}= 0.15 m, axial length L = 1.5 m, wall thickness τ = 0.01 m, Young’s modulus E = 2 × 10

^{11}Pa, Poisson’s ratio ν = 0.3 and material density ρ = 7850 kg/m

^{3}. It should be pointed out that the choice of these parameters is arbitrary; however, there should be noticeable cross-section deformation.

#### 3.2. Benchmark Points

#### 3.3. Identification of Deformation Modes

#### 3.3.1. Uncoupling Deformation Modes with a Novel Criterion

_{∞}means the infinite norm;

**Θ**

_{k}is the vector constructed by the corresponding nodal displacement components of a deformation shape to be identified; γ is a ratio to guarantee the displacement components of a corner benchmark point to be null after the process;

**Θ**

_{c}is the vector consisting of nodal displacements of an classical mode;

**Θ**

_{i}is the vector constituted with the nodal displacements of the newly identified deformation mode, and m is the node number of one corner benchmark point.

**Θ**

_{r}is the vector constructed by the nodal displacements of Vlasov distortion [30] and λ is the ratio to guarantee removing the corresponding deformation components.

#### 3.3.2. Higher-Order Deformation Modes

#### 3.3.3. Shape Functions of Sectional Deformation Modes

## 4. Applications and Illustrative Examples

#### 4.1. Convengence of the Finite Element

#### 4.2. Case Study 1: A Thin-Walled Structure Fixed at One End

_{i}in Equation (2)), also known as generalized displacements, fluctuate along the beam axis. Plainly, each modal shape consists of the components of several deformation modes. The most typical case is that a classical mode plays the predominant role with a secondary mode being auxiliary, such as the first, the third, the fourth, the seventh, the ninth and the tenth modes among the first 12 modes. The second case is that the participations of higher-order modes can almost be neglected compared with those of classical modes, such as the fifth, the sixth and the eighth modes. The third case is that the primary modes are dominant with the spare modes playing supplementary roles, such as the second and the eleventh modes. The results confirm that the identified deformation modes possess the hierarchic capability, being able to obtain a reduced model.

#### 4.3. Case Study 2: A Thin-Walled Structure Fixed at Two Ends

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The global (x, y, z) and local (s, n, z) coordinate systems of the thin-walled structure with a doubly symmetric cross-section.

**Figure 3.**Axial projections of the first 12 modal shapes of the thin-walled structure with a branched, doubly symmetric cross-section.

**Figure 6.**In-plane deformation mode family retrieved from the first 12 modal shapes of the thin-walled structure with a branched, doubly symmetric cross-section.

**Figure 7.**Out-of-plane deformation mode family retrieved from the first 12 modal shapes of the thin-walled structure with a branched, doubly symmetric cross-section.

**Figure 8.**Benchmark points on the doubly symmetric cross-section: (

**a**) 12 benchmark points; (

**b**) one in-plane mode and (

**c**) one out-of-plane mode indicated with benchmark points; (

**d**) equivalent form of the in-plane mode shown in (

**b**); (

**e**) equivalent form of the out-of-plane mode shown in (

**c**).

**Figure 9.**Benchmark points for different thin-walled cross-sections: (

**a**) eight benchmark points and (

**b**) four benchmark points for a rectangular cross-section; (

**c**) eight benchmark points for the I-section; and (

**d**) nine benchmark points for a dual-cell cross-section.

**Figure 10.**Classical modes indicated with benchmark points, numbered as modes I, II, III for the out-of-plane ones and modes i, ii, iii for the in-plane ones: (

**a**) rotation about z-axis; (

**b**) translation along y-axis; (

**c**) translation along x-axis; (

**d**) extension along z-axis; (

**e**) rotation about x-axis and (

**f**) rotation about y-axis.

**Figure 11.**Deriving the secondary deformation modes for the doubly symmetric thin-walled cross-section: (

**a**) in-plane mode iv from in-plane mode 1; (

**b**) out-of-plane mode V from out-of-plane mode 1.

**Figure 12.**Deriving the spare deformation modes for the doubly symmetric thin-walled cross-section: (

**a**) mode viii from mode 5; (

**b**) mode xiii from mode 10.

**Figure 13.**The flowchart providing a brief view of the process involved in uncoupling higher-order deformation modes of the doubly symmetric thin-walled cross-section.

**Figure 14.**Interpolation of the shape functions of the doubly symmetric thin-walled cross-section: (

**a**) one in-plane mode in symmetry and (

**b**) one in-plane mode in anti-symmetry described with cubic functions, respectively; (

**c**) one out-of-plane mode in symmetry described with quadratic functions; (

**d**) one out-of-plane mode in anti-symmetry described with linear functions.

**Figure 15.**The shape functions of the identified out-of-plane and in-plane higher-order deformation modes of the doubly symmetric thin-walled cross-section.

**Figure 16.**Convergence of the first 15 natural frequencies of the thin-walled structure varying along with the number of employed proposed elements: (

**a**) the 1st-5th modes; (

**b**) the 6th-10th modes and (

**c**) the 11th-15th modes.

**Figure 17.**Longitudinal variations of generalized displacements of the thin-walled structure in free vibration analyses.

**Figure 18.**Comparison of free vibration shapes of the cantilevered thin-walled structure between ANSYS shell model (

**right**) and proposed model (

**left**) concerning the first 12 modes.

**Figure 19.**Comparison of free vibration shapes of the cantilevered thin-walled structure between ANSYS shell model (

**right**) and proposed model (

**left**) concerning the 13th–15th modes.

**Figure 20.**Comparison of free vibration shapes of the fixed-fixed thin-walled structure between ANSYS shell model (

**right**) and proposed model (

**left**) concerning the first nine modes.

Mode | Present Model [Hz] | ANSYS Shell [Hz] | Relative Errors [%] |
---|---|---|---|

1st | 145.63 | 140.38 | 3.74 |

2nd | 178.26 | 171.89 | 3.71 |

3rd | 181.35 | 175.63 | 3.26 |

4th | 183.62 | 185.45 | −0.99 |

5th | 207.44 | 215.37 | −3.68 |

6th | 256.86 | 262.87 | −2.29 |

7th | 257.87 | 263.73 | −2.22 |

8th | 262.62 | 263.86 | −0.47 |

9th | 263.46 | 270.28 | −2.52 |

10th | 274.70 | 277.07 | −0.86 |

11th | 279.30 | 278.06 | 0.45 |

12th | 297.71 | 309.49 | −3.81 |

13th | 310.89 | 324.74 | −4.26 |

14th | 312.69 | 324.75 | −3.71 |

15th | 332.30 | 346.36 | −4.06 |

Mode | Present Model (Hz) | ANSYS Shell (Hz) | Relative Errors (%) |
---|---|---|---|

1st | 192.92 | 199.38 | −3.24 |

2nd | 236.09 | 247.04 | −4.43 |

3rd | 251.2 | 262.52 | −4.31 |

4th | 261.05 | 264.38 | −1.26 |

5th | 272.95 | 285.35 | −4.35 |

6th | 290.23 | 303.77 | −4.46 |

7th | 290.85 | 303.95 | −4.31 |

8th | 309.11 | 319.79 | −3.34 |

9th | 321.77 | 335.58 | −4.12 |

10th | 351.7 | 367.43 | −4.28 |

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**MDPI and ACS Style**

Zhang, L.; Ji, A.; Zhu, W.; Peng, L.
On the Identification of Sectional Deformation Modes of Thin-Walled Structures with Doubly Symmetric Cross-Sections Based on the Shell-Like Deformation. *Symmetry* **2018**, *10*, 759.
https://doi.org/10.3390/sym10120759

**AMA Style**

Zhang L, Ji A, Zhu W, Peng L.
On the Identification of Sectional Deformation Modes of Thin-Walled Structures with Doubly Symmetric Cross-Sections Based on the Shell-Like Deformation. *Symmetry*. 2018; 10(12):759.
https://doi.org/10.3390/sym10120759

**Chicago/Turabian Style**

Zhang, Lei, Aimin Ji, Weidong Zhu, and Liping Peng.
2018. "On the Identification of Sectional Deformation Modes of Thin-Walled Structures with Doubly Symmetric Cross-Sections Based on the Shell-Like Deformation" *Symmetry* 10, no. 12: 759.
https://doi.org/10.3390/sym10120759