# Numerical Determination of the Load-Bearing Capacity of a Perforated Thin-Walled Beam in a Structural System with a Steel Grating

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- Laboratory tests of a 3 m-span structural system with a load transferred to the system through a single steel grating.
- Laboratory tests of a 6 m-span structural system with a load transferred to the system through a single steel grating.
- Laboratory tests of the strength properties of the steel from which the beams and connectors were made.

- Development and validation of a 3 m-span structural system model.
- Development and validation of a 6 m-span structural system model.
- Development of structural system models for spans of 3.0 m, 3.5 m, 4.0 m, 4.5 m, 5.0 m, 5.5 m, and 6.0 m with a load transferred by steel gratings placed over the entire span.
- Conducting simulations to determine load-bearing curves.

#### 2.1. Numerical Model

- Connectors were fixed at the location of mounting holes on a surface with a diameter of 40 mm.
- Connectors were connected to the beams using a “Tie”-type contact [19], which corresponds to a rigid connection of the connector to the beam.
- Symmetric boundary conditions were applied to the axis of symmetry of the system on the steel grating and beams.

- Normal Behavior—“Hard” Contact.
- Tangential Behavior—Penalty, Friction coefficient = 0.3.

- Young’s modulus E = 205.5 GPa, determined based on laboratory tests.
- Poisson’s ratio ν = 0.3.
- Yield strength R
_{H}= 490 MPa, determined based on laboratory tests.

- Young’s modulus E = 200 GPa, determined based on laboratory tests.
- Poisson’s ratio ν = 0.3.
- Yield strength R
_{H}= 362 MPa, determined based on laboratory tests.

#### 2.2. Numerical Model Used to Develop Load-Bearing Curves for the Central Beam

^{2}] acting on a band 160 mm wide, and x is a scaling factor for the load q in relation to the actual percentage value of the load taken over by the central beam, e.g., x = 0.6 for 60% according to Table 1. The percentage value of the load borne by the central beam was determined based on a comparison of deflections. This was achieved when LE Max. Principal (Abs) = 5% [19] was reached in the model, where LE Max. Principal (Abs) is the logarithmic measure of strain. The use of logarithmic strains is particularly useful in analyses where traditional, small linear strains are not accurate enough.

#### 2.3. Laboratory Tests

- Loading to achieve a displacement value of the actuator of the strength testing machine equal to 7.0 mm over 70.0 s for the 3 m beam and 15.0 mm over 150.0 s for the 6 m beams (speed: 0.1 mm/s);
- Unloading to a displacement value of the actuator of the machine equal to 0.0 mm over 70.0 s for the 3 m beam and 150.0 s for the 6 m beams (speed: 0.1 mm/s);
- Elastic phase, in which the displacement value of the actuator was increased from 0.0 mm to 7.0 mm over 70.0 s for the 3 m beam and 15.0 mm over 150.0 s for the 6 m beams (speed: 0.1 mm/s); during this phase, measurements of beam and connector displacements and force were made, and a deflection–force diagram was plotted;
- Destruction phase; after conducting the test of beam operation in the elastic range, the sample was unloaded to a displacement value of the actuator of the strength testing machine equal to 0.0 mm and then loaded until destruction at a speed of 0.1 mm/s. During the test, a deflection–force diagram was plotted.

- Loading to achieve a displacement value of the actuator of the strength testing machine equal to 0.3 mm over 60.0 s (speed: 0.3 mm/min);
- Unloading to a displacement value of the actuator of the machine equal to 0.0 mm over 60.0 s (speed: 0.3 mm/min);
- Main phase, in which the displacement value of the actuator was increased from 0.0 mm to 0.3 mm over 60.0 s (speed: 0.3 mm/min); during this phase, measurements of the elongation of the sample and the increase in force were made, and a deflection–force diagram was plotted.

## 3. Results and Discussion

#### 3.1. Results of Laboratory Tests

#### 3.2. Results of Numerical Simulations and Model Validation

#### 3.3. Results of Targeted Numerical Simulations

- Young’s modulus E = 200 GPa.
- Poisson’s ratio ν = 0.3.
- Yield strength R
_{H}= 350 MPa.

- Young’s modulus E = 200 GPa.
- Poisson’s ratio ν = 0.3.
- Yield strength R
_{H}= 235 MPa.

**a**and

**b**on the load-bearing curves can be termed as turning points, defining the actual progression of the load-bearing curves. For beams with lengths from 3 m to 3.5 m, we can estimate the limit load for the criteria L/250 and L/350 from Figure 24. For beams longer than 3.5 m, the L/250 criterion becomes unattainable because the 5% LE Max. Principal (Abs) in the connector will be reached before such deflection at the middle of the central beam’s span is achieved. From point

**a**to point

**b**, the load-bearing capacity is limited by the L/350 condition and the limit deformation value in the connector. Meanwhile, from point

**b**, i.e., for beam spans around 4.1 m, the load-bearing capacity is determined solely by the limit deformation of the connector.

## 4. Conclusions

- The study of 6 m beams showed a pattern of destruction through loss of stability, while 3 m beams were destroyed through yielding.
- Numerical simulations and laboratory tests demonstrated analogous destruction patterns.
- Measurements of horizontal displacements in all tests showed greater displacements in the lower flange of the beam. This is consistent with expectations, as the upper part of the beam is somewhat “braced” by the bridge deck.
- This observation was also taken into account in the numerical model by applying normal and tangential contact with friction between the bridge deck and beams.
- Based on the conducted tests, it is recommended to place additional bracings between the lower flanges of the beams.
- The connector and the support zone of the beam are the weakest elements of the tested system. Each time, the beam’s destruction process started from the yielding of the connector, which was observed in experimental studies and confirmed in the numerical model for all tested lengths.
- The presented study contributes to a better understanding of the operation of such structural systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Static scheme of beams during testing in the elastic range: (

**a**) 3 m beam, (

**b**) 6 m beams, and (

**c**) view of the test station for the 3 m beam.

**Figure 5.**Displacement measurement system for horizontal movements during operation in the elastic range.

**Figure 19.**Distribution of reduced Mises stresses and deformation of the system with a span of 3 m—cross-section.

**Figure 20.**Distribution of reduced Mises stresses and deformation of the system with a span of 3 m—view without the bridge deck.

**Figure 22.**Distribution of reduced Mises stresses and deformation of the system with a span of 6 m—cross-section.

**Figure 23.**Distribution of reduced Mises stresses and deformation of the system with a span of 6 m—view without the bridge deck.

**Figure 24.**Load-bearing curves developed based on simulations of structural systems with spans ranging from 3 m to 6 m.

**Table 1.**Percentage amount of load taken over by individual beams calculated based on the comparison of deflections.

Load Taken over by the Right Beam [%] | Load Taken over by the Central Beam [%] | Load Taken over by the Left Beam [%] | Length of the Beam [m] |
---|---|---|---|

1.284 | 89.514 | 9.203 | 3.0 |

1.903 | 89.582 | 8.515 | 3.5 |

3.481 | 86.969 | 9.55 | 4.0 |

4.368 | 85.925 | 9.706 | 4.5 |

3.682 | 87.064 | 9.254 | 5.0 |

3.593 | 87.799 | 8.608 | 5.5 |

3.501 | 87.825 | 8.674 | 6.0 |

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**MDPI and ACS Style**

Denisiewicz, A.; Socha, T.; Kula, K.; Macek, W.; Błażejewski, W.; Lesiuk, G.
Numerical Determination of the Load-Bearing Capacity of a Perforated Thin-Walled Beam in a Structural System with a Steel Grating. *Appl. Sci.* **2024**, *14*, 1505.
https://doi.org/10.3390/app14041505

**AMA Style**

Denisiewicz A, Socha T, Kula K, Macek W, Błażejewski W, Lesiuk G.
Numerical Determination of the Load-Bearing Capacity of a Perforated Thin-Walled Beam in a Structural System with a Steel Grating. *Applied Sciences*. 2024; 14(4):1505.
https://doi.org/10.3390/app14041505

**Chicago/Turabian Style**

Denisiewicz, Arkadiusz, Tomasz Socha, Krzysztof Kula, Wojciech Macek, Wojciech Błażejewski, and Grzegorz Lesiuk.
2024. "Numerical Determination of the Load-Bearing Capacity of a Perforated Thin-Walled Beam in a Structural System with a Steel Grating" *Applied Sciences* 14, no. 4: 1505.
https://doi.org/10.3390/app14041505