# Vibration Analysis of a Composite Concrete/GFRP Slab Induced by Human Activities

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Human Sensitivity to Vibrations—Brief Literature Review

^{2}, and a smallest natural frequency of 5 Hz for comfort criteria. ISO 10137 [10] guidelines for serviceability in buildings suggests using the curves presented in Figure 3. This guideline permits higher RMS accelerations when a general assessment of footbridge vibrations is performed than to the scenario of a person standing at midspan while another crosses the footbridge.

#### Dynamic Loads Generated by Human Activities

#### Mathematical Modeling for Walking and Jumping

^{th}harmonic; $n$ is the number of harmonics to be considered; ${\alpha}_{i}$ represents the dynamic coefficient for the i

^{th}harmonic; ${f}_{P}$ is the step frequency and ${\varphi}_{i}$ is the phase angle, between the i

^{th}harmonic and the first one.

_{m}is the maximum value of the Fourier series, which is given by Equation (3); f

_{mi}is the amplification factor due to heel impact; T

_{p}is the step period; and C

_{1}and C

_{2}are coefficients defined by Equations (4) and (5), respectively.

_{i}, Varela [12] presented the best-fit polynomial functions (Equations (6) to (9)) obtained from the data presented by Rainer, Pernica and Allen [15]:

_{1}= 0; ϕ

_{2}= π/2; ϕ

_{3}= π; ϕ

_{4}= 3π/2. By using this model, the loads induced by walking observed by Ohlsson [16] could be appropriately described, as can be seen in Figure 4. In this work, Equations (2) to (9) will be used to model the walking load in the numerical analysis (subsection 4.2).

_{1}= 0; ϕ

_{2}= ϕ

_{3}= π [1 − (f

_{p}t

_{c})] = 1, with the contact time with the structure, t

_{c}as 0.2 s. Figure 5 presents the resultant load function proposed, F

_{i}(t). The jumping transient load can then be obtained by multiplying F

_{i}(t) by the person’s weight, G. This resulting load will be used in the numerical analysis (subsection 4.3) to model the jumping cases.

## 3. Composite Slab Concrete/GFRP Profiles

#### 3.1. Material Properties

_{c}) was obtained according to ACI 318 recommendation [24] yielding a value of 26.07 GPa. The usual value of 0.2 was adopted for Poisson´s coefficient, resulting in a shear modulus (G

_{c}) of 10.86 GPa.

#### 3.2. Design and Static Analysis

^{2}, were considered acting along the span of the simply supported slab strip. One-way action was assumed and the deflections in the slab were calculated using the Timoshenko Beam Theory (TBT), combined with the Transformed Area Approach. The maximum deflection in the slab, at midspan, considering shear deformation, can be found by Equation (10):

_{s}is the shear factor.

_{c}, G

_{c}, defined in Section 3.1, the resulting properties of the transformed section were calculated in Table 2.

_{u1}was obtained using an estimated theoretical value for the ultimate shear stress in the GFRP laminae, obtained from Halphin-Tsai equations [21], and V

_{u2}was found from the bond strength at the interface concrete/GFRP, which has been measured experimentally [1].

_{s1}= 0.51). Although the required safety factors have not yet been established in the Design Codes for the case of concrete/GFRP composite slabs, a safety factor of at most 0.50 should be adopted, since a brittle kind of failure is expected in such structures. Hence, in order to attend both serviceability and ultimate limit states, the span of the composite slab was reduced to 4.0 m (Φ

_{s1}= 0.44).

_{s}is the shear stiffness of the transformed section (see Table 2).

_{I}= 38.5 kN and V

_{II}= 39.0 kN), close to the estimated one (V

_{u}= 40.1 kN).

## 4. Dynamic Behavior of the Composite Slab

#### 4.1. Theoretical Fundamental Frequency of the Composite Slab

_{1}) can be obtained from Equation (12), using the equation based on Fourier series [25].

_{1}), which corresponds to a frequency f

_{1}of 11.88 Hz.

#### 4.2. Experimental Analysis of the Composite Slab

#### 4.2.1. Heel-Drop Test

#### 4.2.2. Walking Test

^{2}(≈ 0.4 to 0.7 g) were attained in both prototype responses, considering all the volunteers, which for a frequency around 12 Hz lie in the unbearable range of vibration perception in Goldman’s scale (see Figure 2). In terms of RMS accelerations, the results varied from 0.5 to 1.4 m/s

^{2}, with an average of 1.05 m/s

^{2}for both prototypes. These results are above the RMS acceleration limit (using general assessment) for human comfort by ISO 10137 [10], according to Figure 3.

#### 4.2.3. Jumping Test

^{2}for both prototypes. Again, all the values are above the limit recommended for human comfort by ISO 10137 [10], as shown in Figure 3.

#### 4.3. Numerical Analysis

^{2}(around 0.4 g), and corresponding RMS acceleration of 0.97 m/s

^{2}were obtained. These values lie in the unbearable range of vibration perception in Goldman’s scale (Figure 2) for a structure with fundamental frequency of 12 HZ. The RMS value is much higher than the limit given by ISO 10137 [10].

_{i}(t), shown in Figure 5, by the person’s weight, G) was applied distributed over two areas, corresponding to the shoe/slab contact area, at the center of the numerical model, in the same position as applied in the experiments. The results for jumping are shown in Figure 16, for the test performed by Volunteer 4, in terms of graphs of acceleration along time and its respective spectrum. From the acceleration spectrum it can be observed, again, that the slab responds primarily at its fundamental mode. The peak accelerations obtained numerically reached at most 9.47 m/s

^{2}with RMS of 3.29 m/s

^{2}. These values are much higher than the limit values given by Figure 2 and Figure 3.

## 5. Analysis and Discussion of Results

^{2}) are around 35% greater than the results from the numerical model (3.6 m/s

^{2}). This difference is also observed for the RMS accelerations, when comparing 1.30 to 0.97 m/s

^{2}, found from the experimental and numerical analysis, respectively.

^{2}as compared to 14.38 and 17.74 m/s

^{2}), the RMS accelerations are similar (3.29 m/s

^{2}for the numerical model and 3.26 and 4.94 m/s

^{2}for the experimental test). This similarity in RMS values may be due to the difficulty of imposing a constant load rhythm on the slab by the volunteers, which does not occur when the slab was analyzed numerically.

## 6. Conclusions

- The slab responds primarily at its first vibration mode (flexural). The other vibration modes did not show significant amplitudes. This indicates that in a simplified approach, even considering just the first mode in the analysis, should lead to good results
- Results from experimental tests in terms of accelerations were always higher than the ones obtained from the numerical model. In addition, in the experimental tests the peak accelerations did not follow a uniform pattern as the ones observed in the numerical analysis. This may be due to the fact that the volunteers were not able to maintain a constant load rhythm.
- The composite slab was designed under static loads to attend both ULS and SLS requirements, and it was verified experimentally under 4-point bending tests. However, in spite of its apparent stiffness, with a fundamental frequency above 5 Hz, the concrete/GFRP composite slab under study was shown to be sensitive to human activities. Therefore, in addition to the usual verification of ultimate and serviceability limit states, the dynamic behavior of the slab must also be considered in the design.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Composite slab concrete/glass fiber-reinforced polymer (GFRP) profiles. Reproduced with permission from Santos Neto, A.B.S., published by Federal University of Santa Catarina, 2006.

**Figure 2.**Scale used to measure human sensitivity to vibrations suggested by [8].

**Figure 3.**Curves for level of vibrations in the vertical direction for walkways of [10].

**Figure 5.**Load function induced by jumping used by [17].

**Figure 6.**(

**a**) Dimension of I-section GFRP profile; (

**b**) Cross-section of the representative strip utilized to study the composite slab behavior.

**Figure 8.**Response of Prototype II to heel-drop performed by Volunteer 4 in terms of graphs: (

**a**) acceleration along time; (

**b**) respective spectrum.

**Figure 10.**Response of Prototype I to walking performed by Volunteer 4 in terms of graphs: (

**a**) acceleration along time; (

**b**) respective spectrum.

**Figure 12.**Response of Prototype II to jumping performed by Volunteer 4 in terms of graphs: (

**a**) acceleration along time; (

**b**) respective spectrum.

**Figure 14.**Natural frequencies and respective vibration modes of the composite slab: (

**a**) first mode (bending about y axis); (

**b**) second mode (torsion about x axis) and; (

**c**) third mode (bending about y axis).

**Figure 15.**Numerical response of the composite slab to walking loading in terms of graphs: (

**a**) acceleration along time; (

**b**) respective spectrum.

**Figure 16.**Numerical response of the composite slab to jumping loading in terms of graphs: (

**a**) acceleration along time; (

**b**) respective spectrum.

**Figure 17.**Comparison between acceleration graphs obtained numerically and experimentally from the test on Prototype I performed by Volunteer 4 walking.

**Figure 18.**Comparison between acceleration graphs obtained numerically and experimentally from test on Prototype II performed by Volunteer 4 jumping.

**Table 1.**Elastic properties of the laminates that form the GFRP profile web and flanges (considered as an equivalent orthotropic material)

^{a}.

Elastic Properties | E_{1} (GPa) | E_{2} (GPa) | G_{12} (GPa) | ν_{12} (GPa) |
---|---|---|---|---|

Laminates of GFRP profiles (I section) | 26.73 | 7.19 | 2.44 | 0.341 |

^{a}where 1 is the longitudinal direction, parallel to the longitudinal profile axis, and 2 is the transversal direction of the profile web and flanges.

x (mm) | EI (kNm^{2}) | GĀ (kN) | n |
---|---|---|---|

30.04 | 1.263 × 10^{3} | 1.376 × 10^{4} | 0.995 |

^{b}where x is the neutral axis depth; EI is the flexural stiffness; GĀ is the shear stiffness, in which Ā = A/f

_{s}and fs is the shear factor; n = E

_{c}/E

_{GFRP}is the moduli ratio.

**Table 3.**Ultimate Limit State (ULS): Design and ultimate efforts for the composite slab under uniform load (L = 4.65 m).

Effort | (a) Bending | (b) Shear Web/Flange | (c) Bond Shear (Interface) |
---|---|---|---|

Design Effort | M_{d} = 23.93 kN·m | V_{d1} = 20.60 kN | V_{d2} = 20.60 kN |

Ultimate Effort | M_{u} = 92.26 kN·m | V_{u1} = 40.10 kN | V_{u2} = 58.02 kN |

Safety Factor | Φ_{b} = 0.26 | Φ_{s1} = 0.51 | Φ_{s2} = 0.36 |

**Table 4.**Serviceability Limit State (SLS): Comparison of maximum displacement obtained theoretically and experimentally for the composite slab under 4-point bending (L = 4.00 m; P = 4.63 kN).

Theoretical (TBT) | Experimental | Limit (L/250) |
---|---|---|

9.69 mm | Prototype I = 10.00 mm | 16.00 mm |

Prototype II = 7.92 mm | ||

- | Average value = 8.96 mm | - |

**Table 5.**ULS: Comparison between ultimate efforts obtained theoretically and experimentally for the composite slab under 4-point bending (L = 4.00 m).

Effort | (a) Bending | (b) Shear Web/Flange | (c) Bond Shear (Interface) |
---|---|---|---|

Theoretical Ultimate effort | M_{u} = 92.26 kN.m | V_{u1} = 40.10 kN | V_{u2} = 58.02 kN |

Experimental Ultimate Effort^{a} | - | - | V_{I} = 38.50 kN |

- | V_{II} = 39.00 kN^{c} | - |

^{c}I and II refer to Prototype I and II, respectively.

Volunteer | Mass (kg) | Height (m) | Gender |
---|---|---|---|

1 | 70 | 1.83 | M |

2 | 67 | 1.70 | M |

3 | 83 | 1.92 | M |

4 | 53 | 1.63 | F |

5 | 66 | 1.72 | M |

Volunteer | Natural Frequency (Hz) | Damping Factor (%) | ||
---|---|---|---|---|

Prototype I | Prototype II | Prototype I | Prototype II | |

1 | 11.61 | 12.86 | 4.50 | 2.88 |

2 | 13.03 | 12.79 | 4.81 | 6.11 |

3 | 12.86 | 13.02 | 6.31 | 4.85 |

4 | 12.30 | 12.93 | 4.48 | 5.44 |

5 | 12.63 | 12.36 | 3.77 | 6.42 |

Mean value | 12.64 Hz | 4.96% |

Volunteer | High Peak Accelerations | RMS Accelerations | ||
---|---|---|---|---|

Prototype I | Prototype II | Prototype I | Prototype II | |

1 | 5.27 | 4.86 | 0.86 | 0.83 |

2 | 4.37 | 3.49 | 0.95 | 0.52 |

3 | 6.43 | 6.10 | 1.25 | 1.09 |

4 | 5.03 | 4.85 | 1.28 | 1.30 |

5 | 5.56 | 3.51 | 1.41 | 0.94 |

Mean | 5.33 | 4.56 | 1.15 | 0.94 |

Volunteer | High Peak Accelerations | RMS Accelerations | ||
---|---|---|---|---|

Prototype I | Prototype II | Prototype I | Prototype II | |

1 | 9.82 | 14.71 | 2.09 | 3.04 |

2 | 14.26 | 12.46 | 3.36 | 2.69 |

3 | 7.70 | 11.42 | 1.41 | 2.03 |

4 | 14.38 | 17.74 | 3.26 | 4.94 |

5 | 5.09 | 10.83 | 1.19 | 2.06 |

Mean | 10.25 | 13.43 | 2.26 | 2.95 |

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**MDPI and ACS Style**

Junges, P.; Rovere, H.L.L.; Pinto, R.C.d.A.
Vibration Analysis of a Composite Concrete/GFRP Slab Induced by Human Activities. *J. Compos. Sci.* **2017**, *1*, 11.
https://doi.org/10.3390/jcs1020011

**AMA Style**

Junges P, Rovere HLL, Pinto RCdA.
Vibration Analysis of a Composite Concrete/GFRP Slab Induced by Human Activities. *Journal of Composites Science*. 2017; 1(2):11.
https://doi.org/10.3390/jcs1020011

**Chicago/Turabian Style**

Junges, Paulo, Henriette Lebre La Rovere, and Roberto Caldas de Andrade Pinto.
2017. "Vibration Analysis of a Composite Concrete/GFRP Slab Induced by Human Activities" *Journal of Composites Science* 1, no. 2: 11.
https://doi.org/10.3390/jcs1020011