# Mode Pressure Coefficient Maps as an Alternative to Mean Pressure Coefficient Maps for Non-Gaussian Processes: Hyperbolic Paraboloid Roofs as Cases of Study

## Abstract

**:**

## 1. Introduction

## 2. Geometrical Sample and Experimental Setup

_{1}, f

_{1}, l

_{2}, and f

_{2}are the upward and downward parabolas’ sags and spans, respectively. For the square plan, l

_{1}is equal to l

_{2}; H is the sum of f

_{1}+ f

_{2}and in this research, it was equal to 1/10 l

_{1}(for model 1 and 2) and 1/6 l

_{1}(for model 3 and 4 of l

_{1}[27]); and H

_{B}is the distance between the ground and the lower point of the roof.

_{0}= 0.247 cm, model scale. This value was calculated by fitting the experimental points in the range of heights of interest, that is, from the ground level to the highest point of the roof. In a geometric scale of 1:100, this corresponds to a roughness length of 0.247 m, which is slightly lower than that of exposure category III of EN-1991 (z

_{0}= 0.3 m), described as “area with regular cover of vegetation or buildings or with isolated obstacles with separations of maximum 20 obstacle heights (such as villages, suburban terrain, permanent forest)”.

^{5}, and 1.7 × 10

^{8}at real scale. Turbulence intensity is about 10%. It is known that the Reynolds scale effect can affect results. However, it is expected that no Reynolds effects take place, because of the sharpness of the roof edges [33]. The longitudinal integral length scale at the roof height is about 30 cm at model scale, which would bring a full scale value lower than actual ones. This mismatch, common in most wind tunnel tests, is assumed to have minimal consequences on the results.

^{2}m is the dynamic pressure measured by pressure taps.

## 3. Experimental Data

_{3}and the excess kurtosis γ

_{4}(i.e., kurtosis-3) should be less than or equal to 0.5.

^{2}of the roof (about 8.5 m × 8.5 m). Considering a pressure in suction equal to 0.45 kN/m

^{2}(0.5 × 1.25 × 27

^{2}), and considering pressure coefficients equal to 1.5, cables are loaded by wind with forces equal to about 49 kN. Finally, considering that the dead and permanent loads are about 0.2 kN/m

^{2}, in the pressure tap area, there is a gravitational force equal to at least 14 kN (i.e., 0.9 kN for each node). The gravitational force is about 1/3 of the wind action. In this case, it corresponds to a uniformly distributed load on cables with a spacing equal to 2 m equal to (49 kN − 14 kN)/2 m = 17.5 kN/m or (49 kN × 1.05 − 14 kN)/2 m = 18.7 kN/m [27]. The difference is bigger than 7%. The cables traction load supports structures that are generally made of steel. It is important to consider that for steel structures, safety coefficients for material (i.e., material factors) given by Eurocode for the ultimate limit state are between 1 and 1.33. In particular, the safety coefficient for yielding of a metal face, shear failure of a profiled face, and support reaction capacity of a profiled face is equal to 1.1 (i.e., 10%). This value is dangerously close to 7%.

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Gaussian process, model 1, #19 = 0° pdf (probability density function) (

**a**); non-Gaussian process #2 = 0° (

**b**).

**Figure 5.**Gaussian process distribution on the roofs, model 1 α = 0° (

**a**), α = 45° (

**b**), and α = 90° (

**c**); model 2 α = 0° (

**d**), α = 45° (

**e**), and α = 90° (

**f**); model 3 α = 0° (

**g**), α = 45° (

**h**), and α = 90° (

**i**); model 4 α = 0° (

**j**), α = 45° (

**k**), and α = 90° (

**l**).

**Figure 6.**Ratio $\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}$ distribution on the roofs, model 1 α = 0° (

**a**), α = 45° (

**b**), and α = 90° (

**c**); model 2 α = 0° (

**d**), α = 45° (

**e**), and α = 90° (

**f**); model 3 α = 0° (

**g**), α = 45° (

**h**), and α = 90° (

**i**); model 4 α = 0° (

**j**), α = 45° (

**k**), and α = 90° (

**l**).

**Figure 7.**Model 1, mean pressure coefficient maps for α = 0° (

**a**), α = 45° (

**c**), and α = 90° (

**e**); maps using mode of pressure coefficients for α = 0° (

**b**), α = 45° (

**d**), and α = 90° (

**f**).

**Figure 8.**Model 2, mean pressure coefficient maps for α = 0° (

**a**), α = 45° (

**c**), and α = 90° (

**e**); maps using mode of pressure coefficients for α = 0° (

**b**), α = 45° (

**d**), and α = 90° (

**f**).

**Figure 9.**Model 3, mean pressure coefficient maps for α = 0° (

**a**), α = 45° (

**c**), and α = 90° (

**e**); maps using mode of pressure coefficients for α = 0° (

**b**), α = 45° (

**d**), and α = 90° (

**f**).

**Figure 10.**Model 4, mean pressure coefficient maps for α = 0° (

**a**), α = 45° (

**c**), and α = 90° (

**e**); maps using mode of pressure coefficients for α = 0° (

**b**), α = 45° (

**d**), and α = 90° (

**f**).

Model | l_{1} [cm] | l_{2} [cm] | f_{1} [cm] | f_{2} [cm] | H [cm] | H_{B} [cm] |
---|---|---|---|---|---|---|

1 | 80.00 | 80.00 | 2.67 | 5.33 | 8.00 | 13.33 |

2 | 80.00 | 80.00 | 2.67 | 5.33 | 8.00 | 26.66 |

3 | 80.00 | 80.00 | 4.44 | 8.89 | 13.33 | 13.33 |

4 | 80.00 | 80.00 | 4.44 | 8.89 | 13.33 | 26.66 |

Model | α = 0° | α = 45° | α = 90° |
---|---|---|---|

1 | 44.9 | 27.0 | 32.6 |

2 | 29.2 | 40.4 | 28.1 |

3 | 41.6 | 41.6 | 21.3 |

4 | 34.8 | 46.1 | 16.9 |

**Table 3.**Mean value of $\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}$ and percentage of cases where $\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}<1$ and $\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}>1$.

Model | α = 0° | α = 45° | α = 90° |

Mean of $\left(\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}\right)$ ratio | |||

1 | 1.03 | 1.01 | 1.05 |

2 | 1.03 | 1.01 | 1.05 |

3 | 1.02 | 1.02 | 1.01 |

4 | 1.00 | 1.01 | 1.02 |

Percentage of cases where | |||

$\left(\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}\right)<1$ (%) | |||

Value in parenthesis is the mean value of the ratio restricted to cases where ratio <1 | |||

1 | 30.3 | 41.6 | 32.6 |

(0.98) | (0.97) | (0.94) | |

2 | 36.0 | 46.1 | 28.1 |

(0.99) | (0.98) | (0.95) | |

3 | 34.8 | 28.1 | 38.2 |

(0.98) | (0.97) | (0.87) | |

4 | 56.2 | 41.6 | 38.2 |

(0.97) | (0.98) | (0.92) | |

Percentage of cases where | |||

$\left(\frac{{\mathit{\mu}}_{{\mathit{c}}_{\mathit{p}}}}{{\mathit{v}}_{{\mathit{c}}_{\mathit{p}}}}\right)>1$ (%) | |||

Value in parenthesis is the mean value of the ratio restricted to cases where ratio >1 | |||

1 | 69.7 | 58.4 | 67.4 |

(1.05) | (1.04) | (1.10) | |

2 | 64.0 | 53.9 | 71.9 |

(1.05) | (1.03) | (1.09) | |

3 | 65.2 | 71.9 | 61.8 |

(1.04) | (1.04) | (1.10) | |

4 | 43.8 | 58.4 | 61.8 |

(1.03) | (1.03) | (1.08) |

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**MDPI and ACS Style**

Viskovic, A.
Mode Pressure Coefficient Maps as an Alternative to Mean Pressure Coefficient Maps for Non-Gaussian Processes: Hyperbolic Paraboloid Roofs as Cases of Study. *Computation* **2018**, *6*, 64.
https://doi.org/10.3390/computation6040064

**AMA Style**

Viskovic A.
Mode Pressure Coefficient Maps as an Alternative to Mean Pressure Coefficient Maps for Non-Gaussian Processes: Hyperbolic Paraboloid Roofs as Cases of Study. *Computation*. 2018; 6(4):64.
https://doi.org/10.3390/computation6040064

**Chicago/Turabian Style**

Viskovic, Alberto.
2018. "Mode Pressure Coefficient Maps as an Alternative to Mean Pressure Coefficient Maps for Non-Gaussian Processes: Hyperbolic Paraboloid Roofs as Cases of Study" *Computation* 6, no. 4: 64.
https://doi.org/10.3390/computation6040064