# Dynamic Characterisation and Finite Element Updating of a RC Stadium Grandstand

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and State-of-the-Art

## 2. Research Objectives and Methods

#### Description of the Grandstand

^{2}) and ring beams (0.70 × 0.60 mm

^{2}). The ring beams have spans ranging between 10.5 m and 12.5 m, while pre-stressed hollow-core slabs with RC topping are used in the floors.

^{2}its size), and a horizontal slab (0.82 m the width, with an average thickness of 0.13 m). The beam comprises then a 0.16 m stem, which is interrupted at the supports. Multiple riser units are then connected between themselves, at mid-span, through a steel dowel having a nominal diameter of 20 mm (Figure 3b).

## 3. Preliminary Finite Element Model of the Grandstand

^{3}) were finally considered for all the RC and pre-stressed structural components.

^{7}kN/m. As the free length of the dowels (i.e., the gap between the riser and the raker beam) amounted to 20 mm only and the cross-section of the dowels was relevant, this results in an extremely high flexural stiffness for the connection. It was shown in [26] that dynamic estimates for structural systems may be highly affected by input assumptions in the supports and connection details, especially when elastomeric components are used [27], hence requiring careful attention, with respect to the nominal mechanical features (see also Section 5).

## 4. Experimental Dynamic Identification of the Grandstand

#### 4.1. Test Procedure and Vibration Measurement

- (I)
- natural excitation (wind), and
- (II)
- combined, natural (wind) and artificial (random movement of a group of volunteers) excitation, in order to identify the corresponding dynamic performance, with careful consideration for the the seating deck modes.

#### 4.2. Pre-Processing and Data Relevance Analysis of the Measurements Records

_{i}> x

_{j}) for all pairs (i, j)

_{(i = 1, N)}with i < j represents the reverse arrangements [33,34]. The observed number of reverse arrangements A is used to identify the non-stationarity trends. When such a number is in the interval from A

_{N}

_{;1α/2}to A

_{N}

_{;α/2}(where α is the confidence level), the time record can be considered as stationary to the degree of significance α. From a theoretical point of view, the Root Mean Square (RMS) values from adjacent segments of a stationary signal are independent observations from a random variable. The presence of time trends in the RMS values may indicate a non-stationarity signal [33].

#### 4.3. Data Processing for Modal Identification

## 5. Sensitivity Analysis and Finite Element Model Updating

- (a)
- the riser and raker beams,
- (b)
- the beams and columns belonging to the main frame,
- (c)
- the presence of non-structural elements, and
- (d)
- the mechanical features of the connections between multiple RC components.

_{1}, f

_{2}and f

_{3}) hence allowing to emphasise the major parameters affecting the structural dynamics of the grandstand, hence required to be considered in the updating of the initial FE assembly. The final values of the so selected parameters, through the sensitivity study that is only briefly described in this paper, were in fact chosen using good engineering reasoning, in order to maximise the correlation between the numerical estimates and the experimentally derived dynamic properties of the structure.

#### 5.1. Riser Beams

#### 5.2. Racker Beams

#### 5.3. Main Structural Frame

#### 5.4. Connections

^{7}kN/m) proved to be numerically unreliable, however, a lower value (10

^{5}kN/m) was taken into account in the FE updated model.

#### 5.5. Non-Structural Components

#### 5.6. Optimal FE Parameters

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Structural features of the South grandstand (M7): (

**a**) location (plan view) and (

**b**) detailing of the radial frame (nominal dimensions given in m).

**Figure 3.**Geometrical features of the risers: (

**a**) supports on the raker beams and (

**b**) mid-span cross-section (nominal dimensions given in m).

**Figure 4.**Riser beam typologies: schematic view of (

**a**) single, (

**b**) two and (

**c**) three-element riser beams.

**Figure 6.**Numerically estimated vibration modes of the lower tier (reference FE model, SAP2000). (

**a**) First, (

**b**) second and (

**c**) third modal shape and corresponding frequency.

**Figure 7.**AVT of the empty structure: (

**a**) detail of the monitored zone for the M7 grandstand (lower tier) and (

**b**) corresponding test grid (with grid point #24 representing the reference geophone).

**Figure 8.**Data relevance for a representative time record for grid point #24: (

**a**) velocity time history for each signal segment; (

**b**) auto-power spectral density estimate for each signal segment; (

**c**) cross-spectral density, phase angle and coherence for the two signal segments.

**Figure 9.**(

**a**) Typical time history and (

**b**) auto-power spectral density for selected representative grid points (vertical direction).

**Figure 10.**Coherence of the cross spectral density function for representative grid point pairs: (

**a**) grid points #33 and #24; (

**b**) #15 and #24; (

**c**) #22 and #24.

**Figure 12.**FE sensitivity study of the grandstand. In evidence, the variation of the first three vibration frequencies of the structure. (

**a**–

**i**) plots according to Table 4.

**Figure 13.**(

**a**) Numerical (updated FE model) and (

**b**) experimental fundamental modal shapes, with evidence of the corresponding vibration frequencies.

**Table 1.**Experimentally identified fundamental frequencies of the lower tier of the grandstand. n.a. = not available.

f [Hz] | ||
---|---|---|

Vibration Mode | EFDD | SSI |

1st mode | n.a. | 9.40 |

2nd mode | 9.66 | 9.67 |

3rd mode | 10.96 | 10.73 |

Reference FE Model | 1st mode | 2nd mode | 3rd mode |

1st mode | 1.00 | 0.03 | 0.12 |

2nd mode | 0.03 | 1.00 | 0.24 |

3rd mode | 0.12 | 0.24 | 1.00 |

Experiment | 1st mode | 2nd mode | 3rd mode |

1st mode | 1.00 | 0.28 | 0.00 |

2nd mode | 0.28 | 1.00 | 0.47 |

3rd mode | 0.00 | 0.47 | 1.00 |

Reference FE Model/Experiment | 1st Mode | 2nd Mode | 3rd Mode |
---|---|---|---|

1st mode | 0.98 | 0.21 | 0.01 |

2nd mode | 0.00 | 0.64 | 0.24 |

3rd mode | 0.00 | 0.52 | 0.87 |

**Table 4.**Key parameters for the sensitivity analyses and FE model updating. In evidence, the reference input and the updated (optimal) values. ∆ = 100 × (v

_{up}− v

_{ref})/v

_{ref}.

FE Parameter | Reference Value | Updated Value | ∆ [%] | Reference Figure | ||
---|---|---|---|---|---|---|

Riser beams | Elastic modulus E | GPa | 35 | 38.5 | +10 | 12a |

Specific weight γ | kN/m^{3} | 25 | 24.5 | −2 | 12b | |

Raker beams | Elastic modulus E | GPa | 34 | 35 | +3 | 12c |

Specific weight γ | kN/m^{3} | 25 | 25 | 0 | 12d | |

Main frame beams | Elastic modulus E | GPa | 33 | 29.7 | −10 | 12e |

Main frame columns | Elastic modulus E | GPa | 34 | 30.6 | −10 | 12f |

Dowels | Diameter φ | mm | 20 | 18 | −10 | 12g |

Connections | Stiffness k (riser-to-raker beams) | kN/m | 10^{7} | 10^{5} | −100 | 12h |

Stiffness k (walls-to-riser beam) | kN/m | 10^{7} | 10^{5} | −100 | 12i |

**Table 5.**Experimental and numerical frequencies, as obtained from the reference and updated FE models. ∆ = 100 × (v

_{FE}− v

_{exp})/v

_{exp}.

Experiment | Reference FE Model | Updated FE Model | |||||
---|---|---|---|---|---|---|---|

Vibration Mode | f [Hz] | f [Hz] | ∆ [%] | MAC | f [Hz] | ∆ [%] | MAC |

1st | 9.40 | 9.50 | 1.06 | 0.98 | 9.41 | 0.11 | 0.98 |

2nd | 9.66 | 10.22 | 5.69 | 0.64 | 9.99 | 3.31 | 0.75 |

3rd | 10.73 | 11.76 | 9.60 | 0.87 | 10.65 | 0.75 | 0.88 |

Updated FE Model/Experiment | 1st Mode | 2nd Mode | 3rd Mode |
---|---|---|---|

1st mode | 0.98 | 0.21 | 0.01 |

2nd mode | 0.07 | 0.75 | 0.20 |

3rd mode | 0.04 | 0.42 | 0.88 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Santos, F.; Cismaşiu, C.; Cismaşiu, I.; Bedon, C.
Dynamic Characterisation and Finite Element Updating of a RC Stadium Grandstand. *Buildings* **2018**, *8*, 141.
https://doi.org/10.3390/buildings8100141

**AMA Style**

Santos F, Cismaşiu C, Cismaşiu I, Bedon C.
Dynamic Characterisation and Finite Element Updating of a RC Stadium Grandstand. *Buildings*. 2018; 8(10):141.
https://doi.org/10.3390/buildings8100141

**Chicago/Turabian Style**

Santos, Filipe, Corneliu Cismaşiu, Ildi Cismaşiu, and Chiara Bedon.
2018. "Dynamic Characterisation and Finite Element Updating of a RC Stadium Grandstand" *Buildings* 8, no. 10: 141.
https://doi.org/10.3390/buildings8100141