# Effective Stiffness of Thin-Walled Beams with Local Imperfections

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- Boswell, L.F.; Zhang, S.H. A box beam finite element for the elastic analysis of thin-walled structures. Thin-Walled Struct.
**1983**, 1, 353–383. [Google Scholar] [CrossRef] - Murray, N.W.; Lau, Y.C. The behaviour of a channel cantilever under combined bending and torsional loads. Thin-Walled Struct.
**1983**, 1, 55–74. [Google Scholar] [CrossRef] - Jørgensen, F. Buckling behaviour of a liquid storage tank. Thin-Walled Struct.
**1983**, 1, 309–323. [Google Scholar] [CrossRef] - Rosmanit, M.; Parenica, P.; Sucharda, O.; Lehner, P. Physical Tests of Alternative Connections of Different High Roof Purlins Regarding Upward Loading. Buildings
**2021**, 11, 512. [Google Scholar] [CrossRef] - Taranu, G.; Toma, I.-O. Experimental Investigation and Numerical Simulation of C-Shape Thin-Walled Steel Profile Joints. Buildings
**2021**, 11, 636. [Google Scholar] [CrossRef] - Szewczak, I.; Rozylo, P.; Rzeszut, K. Influence of Mechanical Properties of Steel and CFRP Tapes on the Effectiveness of Strengthening Thin-Walled Beams. Materials
**2021**, 14, 2388. [Google Scholar] [CrossRef] - Szewczak, I.; Rozylo, P.; Snela, M.; Rzeszut, K. Impact of Adhesive Layer Thickness on the Behavior of Reinforcing Thin-Walled Sigma-Type Steel Beams with CFRP Tapes. Materials
**2022**, 15, 1250. [Google Scholar] [CrossRef] - Eurostat Report. Renewable Energy Statistics. Available online: https://ec.europa.eu/eurostat/statistics-explained/index.php?title=Renewable_energy_statistics#Share_of_renewable_energy_more_than_doubled_between_2004_and_2020. (accessed on 21 April 2022).
- Gajewski, T.; Staszak, N.; Garbowski, T. Parametric Optimization of Thin-Walled 3D Beams with Perforation Based on Homogenization and Soft Computing. Materials
**2022**, 15, 2520. [Google Scholar] [CrossRef] - Ciesielczyk, K.; Studziński, R. Experimental and numerical investigation of stabilization of thin-walled Z-beams by sandwich panels. J. Constr. Steel Res.
**2017**, 133, 77–83. [Google Scholar] [CrossRef] - Ciesielczyk, K.; Studziński, R. Experimental Investigation of the Failure Scenario of Various Connection Types between Thin-Walled Beam and Sandwich Panel. Materials
**2022**, 15, 6277. [Google Scholar] [CrossRef] - Gosowski, B. Spatial stability of braced thin-walled members of steel structures. J. Constr. Steel Res.
**2003**, 59, 839–865. [Google Scholar] [CrossRef] - Ferreira, F.P.V.; Tsavdaridis, K.D.; Martins, C.H.; De Nardin, S. Steel-Concrete Composite Beams with Precast Hollow-Core Slabs: A Sustainable Solution. Sustainability
**2021**, 13, 4230. [Google Scholar] [CrossRef] - Mrówczyński, D.; Gajewski, T.; Garbowski, T. Application of the generalized nonlinear constitutive law in 2D shear flexible beam structures. Arch. Civ. Eng.
**2021**, 67, 157–176. [Google Scholar] - Wang, S.; Wang, Z.; Ping, C.; Wang, X.; Wu, H.; Feng, J.; Cai, J. Structural Performance of Thin-Walled Twisted Box-Section Structure. Buildings
**2022**, 12, 12. [Google Scholar] [CrossRef] - Marek, A.; Garbowski, T. Homogenization of sandwich panels. Comput. Assist. Methods Eng. Sci.
**2015**, 22, 39–50. [Google Scholar] - Biancolini, M.E. Evaluation of equivalent stiffness properties of corrugated board. Compos. Struct.
**2005**, 69, 322–328. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T. Determination of transverse shear stiffness of sandwich panels with a corrugated core by numerical homogenization. Materials
**2021**, 14, 1976. [Google Scholar] [CrossRef] - Staszak, N.; Garbowski, T.; Szymczak-Graczyk, A. Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs. Materials
**2021**, 14, 4120. [Google Scholar] [CrossRef] - Garbowski, T.; Knitter-Piątkowska, A.; Mrówczyński, D. Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation. Materials
**2021**, 14, 3786. [Google Scholar] [CrossRef] - Staszak, N.; Gajewski, T.; Garbowski, T. Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams. Materials
**2022**, 15, 1827. [Google Scholar] [CrossRef] - Tserpes, K.I.; Chanteli, A. Parametric numerical evaluation of the effective elastic properties of carbon nanotube-reinforced polymers. Compos. Struct.
**2013**, 99, 366–374. [Google Scholar] [CrossRef] - Grimal, Q.; Rus, G.; Parnell, W.J.; Laugier, P. A two-parameter model of the effective elastic tensor for cortical bone. J. Biomech.
**2011**, 44, 1621–1625. [Google Scholar] [CrossRef] [PubMed] - Allaire, G.; Geoffroy-Donders, P.; Pantz, O. Topology optimization of modulated and oriented periodic microstructures by the homogenization method. Comput. Math. Appl.
**2019**, 78, 2197–2229. [Google Scholar] [CrossRef][Green Version] - Geoffroy-Donders, P.; Allaire, G.; Pantz, O. 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method. J. Comput. Phys.
**2020**, 401, 108994. [Google Scholar] [CrossRef] - Abaqus Unified FEA Software. Available online: https://www.3ds.com/products-services/simulia/products/abaqus (accessed on 21 August 2022).

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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