# Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

- -
- Model constructed with volume-type elements: The most complex and accurate modeling element type. Although unattractive for practical use in the industry, it was used as a comparison basis.
- -
- Ordinary beam element type model: used to evaluate the improvement of the alternative beam models.
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- Alternative beam element type: As will be shown in the next section, the assignment of the joint type (T1 or T2) might not be obvious in some specific cases. To analyze the influence of choosing between different criteria, three models are presented and evaluated in the paper.

## 3. Experimental Validation

#### 3.1. Validation Structure Description

#### 3.2. Experimental Setup

- To attach the dial gauges, the back clamping system was used in order to avoid any differences in measurement due to friction between the sensor and the standardized clamping system. By way of example, Figure 8 presents the mounting of the dial gauge for the displacement measurement of the a2 section.
- A semi-automatic system was used to read and acquire the data from the dial gauges by using a high-resolution photographic camera with an external trigger in order to avoid estimation errors between measurements.
- The defined sections were cleaned and smoothed using solvents and scouring pads with rough polymer fibers so the surfaces would not show any defects. Adhesive strips were also attached to each of these sections. As an example, Figure 9 shows a detail of the region adjacent to the a13 measurement section.

#### 3.3. FEM Model of the Validation Structure Modeled with Beam-Type Elements

#### 3.4. Characteristics of the Validation Structure Modeled with Alternative Beam T-Junctions

^{5}–1 × 10

^{6}(N/m). These stiffnesses in some manner quantify the contribution of the joint to the deformation of the T-junctions in the linear domain. Ideally, it would be desired to improve each junction with sets of elastic elements so that the complex beam structure provides the most accurate possible results. Despite that the assimilation to a joint type in 5 out of the 38 joints of the validation structure is not obvious, it was demonstrated in [8] that although between the T1 and T2 junctions there are significant differences, these differences were comparatively lower than the ones in comparison to the regular beam junction. In other words, assigning a junction as T1 when it is a T2 would induce fewer deviations than just having it as a regular beam.

#### 3.5. Characteristics of the Validation Structure Modeled with Volume-Type Elements

## 4. Results and Discussion

- The best approximations obtained correspond to the detailed volume-type element model.
- The beam and the alternative beam models show significant deviations for sections a3, a4, a8, and a9 (Figure 7). These sections are located at short distances from the clamps, for which local effects are influencing the results, which cannot be characterized using beam-type elements. It can also be seen that these sections do not undergo any significant changes in the alternative beam models.
- Focusing on alternative beam models, variant (c) shows the best approximations to the experimental validation structure and to the detailed volume element model. The calculations are even better displacement estimations than those of the latter at a5 and a10 sections.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Adams, V.; Askennazi, A. Building Better Products with Finite Element Analysis; OnWord Press: Jefferson, NY, USA, 1999. [Google Scholar]
- Lee, K.; Nikolaidis, E. Identification of Flexible Joints in Vehicle Structures. AIAA J.
**1992**, 30, 482–489. [Google Scholar] [CrossRef] - Eriksson, P. Optimisation of a Bus Body Structure. Heavy Veh. Syst.
**2001**, 8, 1–16. [Google Scholar] [CrossRef] - Kim, H.S.; Hwang, Y.S.; Yoon, H.S. Dynamic Stress Analysis of a Bus Systems. In Proceedings of the 2nd MSC Worldwide Automotive Conference, Dearborn, MI, USA, 9–11 October 2000. [Google Scholar]
- Gauchia, A.; Diaz, V.; Boada, M.J.L.; Boada, B.L. Torsional Stiffness and Weight Optimization of a Real Bus Structure. Int. J. Automot. Technol.
**2010**, 11, 41–47. [Google Scholar] [CrossRef] - Lan, F.; Chen, J.; Lin, J. Comparative Analysis for Bus Side Structures and Lightweight Optimization. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2004**, 218, 1067–1075. [Google Scholar] [CrossRef] - Gombor, B. Dynamic Analysis of a Bus Body Frame: Determination of the Loads and Stresses. Veh. Syst. Dyn.
**2005**, 43, 807–822. [Google Scholar] [CrossRef] - Alcalá, E.; Badea, F.; Martin, Á.; Aparicio, F. Methodology for the Accuracy Improvement of FEM Beam Type T-Junctions of Buses and Coaches Structures. Int. J. Automot. Technol.
**2013**, 14, 817–827. [Google Scholar] [CrossRef] - Garcia, A.; Vicente, T. Characterization and Influence of Semi-Rigid Joints in the Buses and Coaches Structural Behavior. In Proceedings of the Bus & Coach Experts Meeting, 33rd International Conference on Vehicle Safety, Keszthely, Hungary, 2–4 September 2002. [Google Scholar]
- Arribas, D.; Badea, F.; Perez, J.Á. Análisis y Optimización Estructural de Autobuses Mediante Modelos Matemáticos. X Congr. Ing. del Transp.
**2012**. [Google Scholar] - GMBH, R.B. Bosch Automotive Handbook; Robert Bosch GmbH: Gerlingen, Germany, 2011. [Google Scholar]
- Badea, F.M.; Alcala, E.; Grimaldi, R.; Ogando, A.; Aparicio, F. Optimización de Uniones y Estructuras de Autobuses. In Proceedings of the XVII Congreso Nacional de Ingeniería Mecánica, Gijón, España, 24 November 2010; p. 8. [Google Scholar]
- Horton, B.; Gurgenci, H.; Veidt, M.; Friswell, M.I. Finite Element Model Updating of a Welded Space Frame. Proc. Int. Modal Anal. Conf.-IMAC
**2000**, 1, 529–535. [Google Scholar] - Skrinar, M. Improved Beam Finite Element for the Stability Analysis of Slender Transversely Cracked Beam-Columns. Comput. Mater. Sci.
**2009**, 45, 663–668. [Google Scholar] [CrossRef] - Donders, S.; Takahashi, Y.; Hadjit, R.; Van Langenhove, T.; Brughmans, M.; Van Genechten, B.; Desmet, W. A Reduced Beam and Joint Concept Modeling Approach to Optimize Global Vehicle Body Dynamics. Finite Elem. Anal. Des.
**2009**, 45, 439–455. [Google Scholar] [CrossRef] - Moroncini, A.; Cremers, L.; Baldanzini, N. Car Body Concept Modeling for NVH Optimization in the Early Design Phase at BMW: A Critical Review and New Advanced Solutions. In Proceedings of the International Conference on Noise and Vibration Engineering 2012, ISMA 2012, including USD 2012: International Conference on Uncertainty in Structure Dynamics, Leuven, Belgium, 17–19 September 2012; Volume 5, pp. 3809–3823. [Google Scholar]
- Liu, F.; Wang, L.; Jin, D.; Wen, H. Equivalent Continuum Modeling of Beam-like Truss Structures with Flexible Joints. Acta Mech. Sin. Xuebao
**2019**, 35, 1067–1078. [Google Scholar] [CrossRef]

**Figure 1.**T-junctions equivalent beam-type element model. T1 junction with two elements (green and purple) and T2 with three elements (green, purple and dark blue).

**Figure 2.**Diagrams comparing the behavior of a standard beam T-junction and the alternative beam T-junction model for the same load state. (

**a**) Rigid junction (

**b**) flexible junction.

**Figure 6.**Three-dimensional CAD model with the characteristics of the experimental assembly corresponding to the testing of the validation structure.

**Figure 10.**Validation structure modeled with beam-type elements. Beam colors refer to the following sections employed: Light blue: 80 mm × 80 mm × 3 mm, red: 80 mm × 60 mm × 3 mm, purple: 80 mm × 60 mm × 2 mm, green: 80 mm × 40 mm × 3 mm, lilac: 80 mm × 40 mm × 2 mm, dark blue: 40 mm × 40 mm × 3 mm.

**Figure 11.**Characteristics of the structure and calculation of the moment of inertia equivalent to the cross-section in the regions of the clamping devices.

**Figure 16.**Von Mises stress maps for the validation structure modeled with volume, beam, and alternative beam type elements [MPa].

**Table 1.**Deflection relative differences of the FEM models with respect to the experimental results.

Exp Volume Deviation | Exp Beam Deviation | Exp Alt Beam Var (a) Deviation | Exp Alt Beam Var (b) Deviation | Exp Alt Beam Var (c) Deviation | |
---|---|---|---|---|---|

Characterized sections | [%] | [%] | [%] | [%] | [%] |

a1 | 6.03 | 43.96 | 15.80 | 33.21 | 13.89 |

a2 | 5.93 | 43.39 | 15.91 | 33.87 | 15.57 |

a3 | 15.38 | 67.35 | 66.88 | 65.63 | 60.54 |

a4 | 8.43 | 97.35 | 97.34 | 97.21 | 97.29 |

a5 | 13.46 | 45.58 | 13.14 | 23.15 | 11.20 |

a6 | 6.01 | 88.95 | 69.83 | 64.87 | 23.54 |

a7 | 6.59 | 35.49 | 15.43 | 12.98 | 9.31 |

a8 | 44.64 | 71.09 | 81.14 | 82.46 | 82.24 |

a9 | 21.55 | 94.57 | 89.25 | 82.22 | 82.25 |

a10 | 6.22 | 40.25 | 1.16 | 3.30 | 2.87 |

a11 | 6.68 | 32.40 | 15.58 | 10.73 | 8.54 |

a12 | 12.37 | 49.23 | 14.25 | 13.16 | 14.21 |

a13 | 17.41 | 56.96 | 26.81 | 22.89 | 23.50 |

a14 | 14.46 | 53.30 | 48.95 | 27.48 | 21.28 |

Absolute average deviation | 13.23 | 58.56 | 40.81 | 40.94 | 33.30 |

Absolute average deviationwithout clamps (discarding the a3, a4, a8, a9 values) | 9.52 | 48.95 | 23.69 | 24.56 | 14.39 |

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**MDPI and ACS Style**

Badea, F.; Olazagoitia, J.; Perez, J. Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures. *Materials* **2022**, *15*, 6468.
https://doi.org/10.3390/ma15186468

**AMA Style**

Badea F, Olazagoitia J, Perez J. Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures. *Materials*. 2022; 15(18):6468.
https://doi.org/10.3390/ma15186468

**Chicago/Turabian Style**

Badea, Francisco, JoseLuis Olazagoitia, and JesusAngel Perez. 2022. "Validation of Alternative Beam T-Junction Fem Models for Complex Tubular Structures" *Materials* 15, no. 18: 6468.
https://doi.org/10.3390/ma15186468