# Effective Stiffness of Thin-Walled Beams with Local Imperfections

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. General Information (Workflow)

**ABDR**matrix [18] (according to a lamination theory) was computed—this result is considered in the study as the reference result. Next, the buckling analyses (see Section 2.3) were performed for different deformation modes, received from applying typical loads: compression and bending/shearing forces in two directions. The buckling modes received with different scale ratios were then used to compute the weakened mechanical properties of the beams by applying the homogenization method (see Section 2.4). Later, those results were compared to the reference one in order to select one deformation mode that could be representative for all cases.

#### 2.2. Shell-to-Beam Numerical Homogenization

#### 2.3. Buckling Analysis

#### 2.4. Reference Model and Models with Geometric Imperfections

## 3. Results

#### 3.1. Buckling due to Compression

#### 3.2. Buckling due to Bending about the Horizontal Axis

#### 3.3. Buckling due to Bending about the Vertical Axis

#### 3.4. Buckling due to Shear

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the study workflow: shell-to-beam homogenization (Garbowski et al. 2021 [18]) for the reference beam and its counterparts for beams with imperfections due to different modes.

**Figure 2.**Z profile considered: (

**a**) cross-section (units in mm); (

**b**) finite element mesh with condensed nodes selected for a 100 mm case.

**Figure 4.**Buckling in compression for 100 mm depth: (

**a**) mode 1; (

**b**) mode 2; (

**c**) plot of the stiffness reduction of $\mathrm{E}\mathrm{A}$, depending on the size of imperfections.

**Figure 5.**Buckling in compression for 150 mm depth: (

**a**) mode 1; (

**b**) mode 2; (

**c**) plot of the stiffness reduction of $\mathrm{E}\mathrm{A}$, depending on the size of imperfections.

**Figure 6.**Buckling in compression for 200 mm depth: (

**a**) mode 1; (

**b**) mode 2; (

**c**) plot of the stiffness reduction of $\mathrm{E}\mathrm{A}$, depending on the size of imperfections.

**Figure 7.**Buckling due to bending about the horizontal axis, top flange in tension ((i) case) for a depth of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of $E{I}_{x}$, depending on the size of imperfections.

**Figure 8.**Buckling due to bending about the vertical axis for case (ii) (tension of the lower part of the cross-section), for a depth of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of $E{I}_{x}$, depending on the size of imperfections.

**Figure 9.**Buckling due to bending about the horizontal axis for (i) case, for a depth of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of $E{I}_{y}$, depending on the size of imperfections.

**Figure 10.**Buckling due to bending about the vertical axis for case (ii), for a depth of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of $E{I}_{y}$, depending on the size of imperfections.

**Figure 11.**Buckling due to shearing for case (i) for the Z profile with an elongation of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of ${G}_{zx}A$, depending on the size of imperfections.

**Figure 12.**Buckling due to shearing for case (ii), for a depth of 100 mm: (

**a**) mode 1; (

**b**) plot of the stiffness reduction of ${G}_{zy}A$, depending on the size of imperfections.

**Table 1.**Effective stiffness of the Z profile with a 5 mm mesh depending on the elongation (beam axis).

Depth $\left(\mathbf{m}\mathbf{m}\right)$ | $\mathbf{E}\mathbf{A}$$\left({10}^{7}\mathbf{P}\mathbf{a}{\mathbf{m}}^{2}\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left({10}^{4}\mathbf{P}\mathbf{a}{\mathbf{m}}^{4}\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left({10}^{5}\mathbf{P}\mathbf{a}{\mathbf{m}}^{4}\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left({10}^{6}\mathbf{P}\mathbf{a}{\mathbf{m}}^{2}\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left({10}^{7}\mathbf{P}\mathbf{a}{\mathbf{m}}^{2}\right)$ |
---|---|---|---|---|---|

200 | 9.135 | 6.571 | 1.271 | 6.037 | 9.350 |

150 | 9.160 | 6.542 | 1.269 | 7.604 | 1.098 |

100 | 9.211 | 6.521 | 1.270 | 9.801 | 1.308 |

**Table 2.**Stiffness reduction of the Z profile depending on the elongation depth (beam axis) and buckling mode in compression.

Depth $\left(\mathbf{m}\mathbf{m}\right)$ | Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction (Mode 1/Mode 2) | ||||
---|---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | ||

100 | 1.0 | −2.22/−5.98 | 0.04/−0.36 | −0.19/−0.71 | −0.05/−0.14 | −0.32/−2.44 |

2.5 | −9.00/−17.51 | −0.27/−1.32 | −1.14/−3.30 | −0.31/−0.82 | −1.81/−12.13 | |

5.0 | −17.34/−26.73 | −1.15/−2.81 | −3.86/−7.97 | −1.09/−2.78 | −6.38/−30.39 | |

150 | 1.0 | −4.31/−7.23 | −0.27/−0.31 | −0.40/−0.94 | −0.23/−0.39 | −1.49/−2.72 |

2.5 | −14.40/−19.14 | −1.17/−1.35 | −2.18/−3.90 | −1.33/−1.96 | −8.25/−13.46 | |

5.0 | −23.79/−28.22 | −2.92/−3.13 | −6.38/−9.10 | −4.46/−5.89 | −24.15/−33.63 | |

200 | 1.0 | −5.59/−2.71 | −0.25/−0.17 | −0.64/−0.22 | −0.50/−0.25 | −1.90/−0.90 |

2.5 | −16.83/−10.59 | −1.26/−0.82 | −2.95/−1.28 | −2.47/−1.46 | −10.13/−5.30 | |

5.0 | −26.28/−19.66 | −3.24/−2.25 | −7.84/−4.28 | −7.44/−5.04 | −28.05/−17.52 |

**Table 3.**Stiffness reduction of a Z profile with an elongation of 100 mm due to bending about the horizontal axis for (i) case, depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0. | −2.47 | −1.79 | −1.66 | −0.21 | −0.79 |

2.5 | −10.85 | −7.91 | −7.99 | −1.19 | −4.57 |

5.0 | −21.95 | −16.40 | −18.90 | −3.67 | −14.59 |

**Table 4.**Stiffness reduction of Z profile with an elongation of 100 mm due to bending about the horizontal axis for case (ii), i.e., the tension of the lower part of the cross-section, depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0 | −3.51 | −3.64 | −4.08 | −1.22 | −0.30 |

2.5 | −11.76 | −13.89 | −14.06 | −6.57 | −1.54 |

5.0 | −20.67 | −26.44 | −24.34 | −18.37 | −4.73 |

**Table 5.**Stiffness reduction of Z profile with an elongation of 100 mm due to bending about the vertical axis for (i) case, depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0 | −1.04 | −2.53 | −0.95 | −0.28 | −0.08 |

2.5 | −4.70 | −11.11 | −4.64 | −1.63 | −0.45 |

5.0 | −10.22 | −22.83 | −11.48 | −5.43 | −1.36 |

**Table 6.**Stiffness reduction of the Z profile with elongation of 100 mm due to bending about the vertical axis for (ii) case, depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0 | −0.54 | −1.30 | −0.57 | −0.10 | −0.06 |

2.5 | −2.32 | −5.53 | −2.56 | −0.56 | −0.34 |

5.0 | −4.83 | −10.86 | −5.75 | −1.71 | −1.01 |

**Table 7.**Stiffness reduction of the Z profile with elongation of 100 mm due to shearing for case (i), depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction (Method I/Method II) | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0 | −2.30/−2.39 | −0.02/−0.04 | −0.25/−0.26 | −0.09/−0.09 | −1.60/−1.68 |

2.5 | −8.34/−8.53 | −0.25/−0.28 | −1.28/−1.34 | −0.43/−0.43 | −6.65/−7.12 |

5.0 | −14.87/−15.03 | −0.56/−0.60 | −3.44/−3.47 | −1.26/−1.24 | −15.18/−16.68 |

**Table 8.**Stiffness reduction of the Z profile with an elongation of 100 mm due to shearing for case (ii), depending on the size of imperfections.

Size of Imperfections $\left(\mathbf{m}\mathbf{m}\right)$ | Stiffness Reduction (Method I/Method II) | ||||
---|---|---|---|---|---|

$\mathbf{E}\mathbf{A}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{y}}$$\left(\%\right)$ | $\mathit{E}{\mathit{I}}_{\mathit{x}}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{y}}\mathit{A}$$\left(\%\right)$ | ${\mathit{G}}_{\mathit{z}\mathit{x}}\mathit{A}$$\left(\%\right)$ | |

1.0 | −1.66/−1.71 | −2.05/−1.91 | −1.74/−1.70 | −1.18/−1.28 | −0.11/−0.14 |

2.5 | −7.11/−6.89 | −8.55/−7.49 | −7.46/−7.04 | −5.25/−5.53 | −0.82/−0.91 |

5.0 | −15.00/−14.47 | −18.31/−16.21 | −15.16/−14.03 | −12.48/−12.99 | −2.77/−3.00 |

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**MDPI and ACS Style**

Staszak, N.; Gajewski, T.; Garbowski, T.
Effective Stiffness of Thin-Walled Beams with Local Imperfections. *Materials* **2022**, *15*, 7665.
https://doi.org/10.3390/ma15217665

**AMA Style**

Staszak N, Gajewski T, Garbowski T.
Effective Stiffness of Thin-Walled Beams with Local Imperfections. *Materials*. 2022; 15(21):7665.
https://doi.org/10.3390/ma15217665

**Chicago/Turabian Style**

Staszak, Natalia, Tomasz Gajewski, and Tomasz Garbowski.
2022. "Effective Stiffness of Thin-Walled Beams with Local Imperfections" *Materials* 15, no. 21: 7665.
https://doi.org/10.3390/ma15217665