Author Contributions
Conceptualization, F.T., Z.H., Q.W. and W.M.; Software, F.T., Z.H., Q.W. and W.M.; Validation, F.T., Z.H. and W.M.; Formal analysis, Z.H., Q.W. and W.M.; Investigation, F.T., Z.H., Q.W. and W.M.; Resources, F.T., Z.H.; Data curation, W.M.; Writing—original draft, Z.H., Q.W. and W.M.; Writing—review & editing, F.T.; Supervision, F.T.; Project administration, F.T. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Equivalent forces of partial tendon profile.
Figure 1.
Equivalent forces of partial tendon profile.
Figure 2.
The ideal type of parabolic shape of the prestressing tendons for reference model.
Figure 2.
The ideal type of parabolic shape of the prestressing tendons for reference model.
Figure 3.
The specific prestressing tendons layout for reference model.
Figure 3.
The specific prestressing tendons layout for reference model.
Figure 4.
The dimetric view of the reference model in Strand7.
Figure 4.
The dimetric view of the reference model in Strand7.
Figure 5.
The Effect of the Long-Span Length on Natural Frequency.
Figure 5.
The Effect of the Long-Span Length on Natural Frequency.
Figure 6.
The Effect of the Short-Span Length on Natural Frequency.
Figure 6.
The Effect of the Short-Span Length on Natural Frequency.
Figure 7.
The Effect of Slab Thickness on Natural Frequency.
Figure 7.
The Effect of Slab Thickness on Natural Frequency.
Figure 8.
The Effect of Slab Damping Ratio on Natural Frequency.
Figure 8.
The Effect of Slab Damping Ratio on Natural Frequency.
Figure 9.
The Effect of the Long-Span Length on Peak Acceleration.
Figure 9.
The Effect of the Long-Span Length on Peak Acceleration.
Figure 10.
The Effect of the Short-Span Length on Peak Acceleration.
Figure 10.
The Effect of the Short-Span Length on Peak Acceleration.
Figure 11.
The Effect of Slab Thickness on Peak Acceleration.
Figure 11.
The Effect of Slab Thickness on Peak Acceleration.
Figure 12.
The Effect of Slab Damping Ratio on Peak Acceleration.
Figure 12.
The Effect of Slab Damping Ratio on Peak Acceleration.
Figure 13.
The Effect of the Long-Span Length on Response Factor R.
Figure 13.
The Effect of the Long-Span Length on Response Factor R.
Figure 14.
The Effect of the Short-Span Length on Response Factor R.
Figure 14.
The Effect of the Short-Span Length on Response Factor R.
Figure 15.
The Effect of the slab thickness on Response Factor R.
Figure 15.
The Effect of the slab thickness on Response Factor R.
Figure 16.
The Effect of the damping ratio on Response Factor R.
Figure 16.
The Effect of the damping ratio on Response Factor R.
Table 1.
The international standards and limits for vibration comfort.
Table 1.
The international standards and limits for vibration comfort.
Standards Name | Assessment Criteria | Category | Limitations |
---|
GB50010-2010 [19] (Chinese code) | Natural frequency | Residential and Apartment | f ≥ 5 Hz |
Office and Hotel | f ≥ 4 Hz |
Public buildings | f ≥ 3 Hz |
IS02631-2 [20] (International) | Peak acceleration limit | Office and residences | 10× baseline |
Indoor footbridges, shopping mall | 28× baseline |
Rhythmic activities | 100× baseline |
SCI [21] (European code) | Response factor R | Office | 8–10 |
Residential | 4–8 |
Operating room | 1–4 |
Table 2.
The span-to-depth ratios limitation.
Table 2.
The span-to-depth ratios limitation.
Floor System Type | Span-to-Depth Ratio (L/D) |
---|
Flat plate | 45 |
One-way slab | 48 |
Flat slab with drop panels | 50 |
Edge-supported slab | 55 |
Waffle slab | 35 |
Table 3.
The strength and serviceability limit states requirements.
Table 3.
The strength and serviceability limit states requirements.
Check Limit States | Limit Value | Code |
---|
Average prestress P/A | 1.2–2.6 MPa | Gilbert et al. [31]. |
Ultimate strength in bending | | Clause 8.1.6.1 |
Maximum Compressive Stress | | Clause 18.1 |
Shear stress | | Clause 9.3.4 |
Deflection | | AS3600-2018 Table 2.3.2 |
Table 4.
The concrete material properties used in models.
Table 4.
The concrete material properties used in models.
Concrete Grade | Modules (MPa) | Density (kg/m3)
| Poisson’ Ratio |
---|
40 | 32,800 | 2400.0 | 0.2 |
Table 5.
The properties and layout limitations of the FMA505 tendons [
34].
Table 5.
The properties and layout limitations of the FMA505 tendons [
34].
FMA505 | No. Strands | 5 No. 12.7 mm Strands |
Modules (MPa) | 200,000 |
Nominal tensile strength (MPa) | 1870 |
Breaking Load (kN) | 920 |
Proof Force (kN) | 780 |
Minimum Concrete Thickness (mm) | 140 |
Minimum Anchor Spacing (mm) | 300 |
Table 6.
The geometric and performance parameters of each model.
Table 6.
The geometric and performance parameters of each model.
| Length (m) | Width (m) | Slab Thickness (mm) | Damping Ratio | Concrete Characteristic Strength (MPa) |
---|
Reference Model | 10 | 9 | 270 | 0.02 | 40 |
Model 2.1 | 12 | 9 | 270 | 0.02 | 40 |
Model 2.2 | 8 | 9 | 270 | 0.02 | 40 |
Model 3.1 | 10 | 11 | 270 | 0.02 | 40 |
Model 3.2 | 10 | 8 | 270 | 0.02 | 40 |
Model 4.1 | 10 | 9 | 300 | 0.02 | 40 |
Model 4.2 | 10 | 9 | 330 | 0.02 | 40 |
Model 5.1 | 10 | 9 | 270 | 0.03 | 40 |
Model 5.2 | 10 | 9 | 270 | 0.04 | 40 |
Table 7.
The relationship between dynamic load factor and frequency.
Table 7.
The relationship between dynamic load factor and frequency.
N.O. Fourier Harmonic Function | Frequency (Hz) | Design Dynamic Load Factor |
---|
1st harmonic Design dynamic load factor | 1.0 | 0.0205 |
2.316 | 0.56 |
2.8 | 0.56 |
2nd harmonic Design dynamic load factor | 2.0 | 0.0802 |
5.6 | 0.10036 |
3rd harmonic Design dynamic load factor | 3.0 | 0.0522 |
8.4 | 0.08676 |
4th harmonic Design dynamic load factor | 4.0 | 0.039 |
11.2 | 0.0858 |
Table 8.
The average values of the Fourier coefficients for the walking activity.
Table 8.
The average values of the Fourier coefficients for the walking activity.
N.O. | First
| Second
| Third
| Fourth
|
---|
Average | 0.33 | 0.14 | 0.05 | 0.05 |
Table 9.
The results of natural frequency calculated by change the long-span lengths.
Table 9.
The results of natural frequency calculated by change the long-span lengths.
| Long-Span Length (m) | CSTR43 Solution (Hz) | Strand7 FEA Solution (Hz) |
---|
Reference Model | 10 | 5.5078 | 5.9886 |
Model 2.1 | 12 | 4.9947 | 4.3577 |
Model 2.2 | 8 | 6.5213 | 7.2540 |
Table 10.
The results of natural frequency calculated by change the short-span lengths.
Table 10.
The results of natural frequency calculated by change the short-span lengths.
| Short-Span Length (m) | CSTR43 Solution (Hz) | Strand7 FEA Solution (Hz) |
---|
Reference Model | 9 | 5.5078 | 5.9886 |
Model 3.1 | 11 | 4.7329 | 5.2034 |
Model 3.2 | 8 | 6.1459 | 6.2869 |
Table 11.
The results of natural frequency calculated by change the slab thickness.
Table 11.
The results of natural frequency calculated by change the slab thickness.
| Slab Thickness (m) | CSTR43 Solution (Hz) | Strand7 FEA Solution (Hz) |
---|
Reference Model | 0.270 | 5.5078 | 5.9886 |
Model 4.1 | 0.300 | 6.1198 | 6.6534 |
Model 4.2 | 0.330 | 6.7318 | 7.3182 |
Table 12.
The results of natural frequency calculated by change damping ratio.
Table 12.
The results of natural frequency calculated by change damping ratio.
| Damping Ratio | CSTR43 Calculation (Hz) | Strand7 FEA Solution (Hz) |
---|
Reference Model | 0.02 | 5.5078 | 5.9886 |
Model 5.1 | 0.03 | 5.5078 | 5.9886 |
Model 5.2 | 0.04 | 5.5078 | 5.9886 |
Table 13.
The results of peak acceleration calculated by change long-span length.
Table 13.
The results of peak acceleration calculated by change long-span length.
| Long-Span Length (m) | The SCI Approach(%g) | Strand7 FEA Solution (%g) |
---|
Reference Model | 10 | 0.420 | 0.399 |
Model 2.1 | 12 | 0.345 | 0.280 |
Model 2.2 | 8 | 0.536 | 0.508 |
Table 14.
The results of peak acceleration calculated by change short-span length.
Table 14.
The results of peak acceleration calculated by change short-span length.
| Short-Span Length (m) | The SCI Approach (%g) | Strand7 FEA Solution (%g) |
---|
Reference Model | 9 | 0.420 | 0.399 |
Model 3.1 | 11 | 0.337 | 0.341 |
Model 3.2 | 8 | 0.478 | 0.428 |
Table 15.
The results of peak acceleration calculated by change slab thickness.
Table 15.
The results of peak acceleration calculated by change slab thickness.
| Slab Thickness (m) | The SCI Approach (%g) | Strand7 FEA Solution (%g) |
---|
Reference Model | 0.27 | 0.420 | 0.399 |
Model 4.1 | 0.3 | 0.378 | 0.380 |
Model 4.2 | 0.33 | 0.343 | 0.363 |
Table 16.
The results of peak acceleration calculated by change damping ratio.
Table 16.
The results of peak acceleration calculated by change damping ratio.
| Damping Ratio | The SCI Approach (%g) | Strand7 FEA Solution (%g) |
---|
Reference Model | 0.02 | 0.420 | 0.399 |
Model 5.1 | 0.03 | 0.280 | 0.325 |
Model 5.2 | 0.04 | 0.210 | 0.278 |
Table 17.
The results of response factor R calculated by long-span length.
Table 17.
The results of response factor R calculated by long-span length.
| Long-Span Length (m) | The SCI Approach | Strand7 FEA Solution |
---|
Reference Model | 10 | 4.316 | 4.47 |
Model 2.1 | 12 | 3.545 | 2.65 |
Model 2.2 | 8 | 5.515 | 5.78 |
Table 18.
The results of response factor R calculated by short-span length.
Table 18.
The results of response factor R calculated by short-span length.
| Short-Span Length (m) | The SCI Approach | Strand7 FEA Solution |
---|
Reference Model | 9 | 4.316 | 4.47 |
Model 3.1 | 11 | 3.470 | 3.59 |
Model 3.2 | 8 | 4.916 | 4.56 |
Table 19.
The results of response factor R calculated by slab thickness.
Table 19.
The results of response factor R calculated by slab thickness.
| Slab Thickness (m) | The SCI Approach | Strand7 FEA Solution |
---|
Reference Model | 0.27 | 4.316 | 4.47 |
Model 4.1 | 0.3 | 3.885 | 4.26 |
Model 4.2 | 0.33 | 3.620 | 3.92 |
Table 20.
The results of response factor R calculated by damping ratio.
Table 20.
The results of response factor R calculated by damping ratio.
| Damping Ratio | The SCI Approach | Strand7 FEA Solution |
---|
Reference Model | 0.02 | 4.316 | 4.47 |
Model 5.1 | 0.03 | 2.949 | 3.44 |
Model 5.2 | 0.04 | 2.212 | 2.79 |
Table 21.
Possible error of the natural frequency in Strand7 compared to CSTR43 results.
Table 21.
Possible error of the natural frequency in Strand7 compared to CSTR43 results.
| Long-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 10 | 8.73% | 10.91% |
Model 2.1 | 12 | 12.75% |
Model 2.2 | 8 | 11.24% |
| Short-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 8 | 2.29% | 6.99% |
Model 3.1 | 9 | 8.73% |
Model 3.2 | 11 | 9.94% |
| Slab thickness (m) | Strand7 error ratio | Average error ratio |
Reference Model | 0.27 | 8.73% | 8.72% |
Model 4.1 | 0.30 | 8.72% |
Model 4.2 | 0.33 | 8.71% |
| Damping ratio | Strand7 error ratio | Average error ratio |
Reference Model | 0.02 | 8.73% | 8.73% |
Model 5.1 | 0.03 | 8.73% |
Model 5.2 | 0.04 | 8.73% |
Table 22.
Possible error of the peak acceleration in Strand7 compared to the SCI results.
Table 22.
Possible error of the peak acceleration in Strand7 compared to the SCI results.
| Long-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 10 | 4.92% | 9.65% |
Model 2.1 | 12 | 18.77% |
Model 2.2 | 8 | 5.26% |
| Short-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 9 | 4.92% | 5.48% |
Model 3.1 | 11 | 1.08% |
Model 3.2 | 8 | 10.45% |
| Slab thickness (m) | Strand7 error ratio | Average error ratio |
Reference Model | 0.27 | 4.92% | 3.75% |
Model 4.1 | 0.30 | 0.61% |
Model 4.2 | 0.33 | 5.72% |
| Damping ratio | Strand7 error ratio | Average error ratio |
Reference Model | 0.02 | 4.92% | 17.86% |
Model 5.1 | 0.03 | 16.17% |
Model 5.2 | 0.04 | 32.49% |
Table 23.
Possible error of the response factor R in Strand7 compared to the SCI results.
Table 23.
Possible error of the response factor R in Strand7 compared to the SCI results.
| Long-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 10 | 3.56% | 11.21% |
Model 2.1 | 12 | 25.26% |
Model 2.2 | 8 | 4.80% |
| Short-span length (m) | Strand7 error ratio | Average error ratio |
Reference Model | 9 | 3.56% | 4.75% |
Model 3.1 | 11 | 3.46% |
Model 3.2 | 8 | 7.24% |
| Slab thickness (m) | Strand7 error ratio | Average error ratio |
Reference Model | 0.27 | 3.56% | 7.17% |
Model 4.1 | 0.30 | 9.66% |
Model 4.2 | 0.33 | 8.29% |
| Damping ratio | Strand7 error ratio | Average error ratio |
Reference Model | 0.02 | 3.56% | 15.44% |
Model 5.1 | 0.03 | 16.63% |
Model 5.2 | 0.04 | 26.12% |