The free vibration analysis of simply supported box-girder bridges is carried out using the finite element method. The fundamental frequency is determined in straight, skew, curved and skew-curved box-girder bridges. It is important to analyse the combined effect of skewness and curvature because skew-curved box-girder bridge behaviour cannot be predicted by simply adding the individual effects of skewness and curvature. At first, an existing model is considered to validate the present approach. A convergence study is carried out to decide the mesh size in the finite element method. An exhaustive parametric study is conducted to determine the fundamental frequency of box-girder bridges with varying skew angle, curve angle, span, span-depth ratio and cell number. The skew angle is varied from 0° to 60°, curve angle is varied from 0° to 60°, span is changed from 25 to 50 m, span-depth ratio is varied from 10 to 16, and single cell & double cell are used in the present study. A total of 420 bridge models are used for parametric study in the investigation. Mode shapes of the skew-curved bridge are also presented. The fundamental frequency of the skew-curved box-girder bridge is found to be more than the straight bridge, so, the skew-curved box-girder bridge is preferable. The present study may be useful in the design of box-girder bridges.

In general, the box-girder bridge is used when the problems of a larger span and wider deck occur. They have high strength and more torsional and flexural stiffnesses. The deck of a straight bridge is supported orthogonally to the traffic. But due to some reasons, i.e., the site conditions and land acquisition problems, the bridge axis may not be perpendicular to the supports in the plan and such bridges are termed as skewed, as shown in

Several commercial softwares are available based on the finite element method (FEM). For a bridge being a large structure or similar structures, FEM based software can be used to assess the natural frequencies in spite of the size and complexities of such structures. Mostly the shock absorber bearing is used in the bridge. Bearings are used to transfer forces from the superstructure to the substructure while tolerating or constraining relative movement. They provide vibration isolation. Hence, the bridge deck will only influence the natural frequencies of the bridge. Therefore, only the mass of the deck is considered in the analysis while taking the effect of the bridge. However, the dynamic response will be affected under the influence of external load like vehicle’s frequency, etc. It can be determined when forced vibration analysis on such structures is performed. Several studies have been carried out to investigate the vibration behaviour of box-girder bridges, and some of the important contributions are mentioned below.

The significant literature identified with this area is introduced as follows: A new formula is proposed for determining the natural frequency of a curved girder bridge with an asymmetrical cross-section and verified the proposed equations with the experimental studies [

The literature mentioned above mainly focuses on the free vibration analysis of straight box-girder bridges. It appears that the free vibration analysis of skew, curved and skew-curved RC box-girder bridges is not studied so far with the importance they deserve. Further, no such specifications and limitations for vibration are available in the literature on the skew, curved and skew-curved bridges. So, it is necessary to investigate the vibrational behaviour of such bridges. Thus, the present study aims to determine the frequencies and mode shapes of straight, skew, curved and skew-curved RC box-girder bridges using the finite element method. The comparison is also made between single-cell and double-cell box-girder bridges. The cross-section of the double-cell box-girder bridge is kept the same as that of the single-cell box-girder bridge. The bridges with different degrees of complexity can be modelled efficiently using the finite element method. Also, the results can be obtained quickly and precisely, so it is becoming quite popular nowadays compared to the experimental methods, which are rather costly and time-consuming. Thus, in the present study, a finite element based software CSiBridge [

In general, the modelling of any structure is done either in two-dimension or in three-dimension. But it should be done in three-dimension to evaluate the responses of the bridge deck. The modelling and analysis of the bridges are carried out using a finite element-based software CSiBridge v.20.0.0, as shown in

1. The span of the box-girder bridge should not be less than 25 m,

2. The span-depth ratio should be around 17–18 for RC bridge,

3. Cantilever arm length is equal to 0.45 times the distance between webs,

4. The minimum soffit slab thickness should be 1/20 times the distance between the girders.

The skew angle, curve angle, span, span-depth ratio, and the cell number are geometrical parameters that directly affect the results, thereby the design. Bridges are designed to satisfy the functional requirements as well as requirements arising due to site conditions. So, these parameters are required to be considered in the estimation of frequencies of skew-curved box-girder bridges. In the present study, the cross-sectional properties of box-girder bridge deck are as follows: Total width = 11.5 m, consisting of roadway = 7.5 m, Kerb = 0.5 m on both sides and Footpath = 1.5 m on both sides. The thickness of the web, top and bottom flanges, are different for different spans and span-depth ratios. The sections are chosen based on a trial procedure. After that, the double-cell box-girder bridge is modelled, keeping the same cross-section as that of a single-cell box-girder bridge. The straight bridge models are found to be safe under limit states of collapse and serviceability, i.e., stress, deflection, and vibration, according to the specifications [

Span-depth ratio (L/d) | Number of cell | Span | Cross-sectional properties | |||
---|---|---|---|---|---|---|

Thickness of top flange, t_{tf} (m) |
Thickness of bottom flange, t_{bf} (m) |
Thickness of web, t_{w} (m) |
Cross-sectional area, A (m^{2}) |
|||

10 | Single | 25 | 0.30 | 0.30 | 0.30 | 6.42 |

30 | 0.30 | 0.30 | 0.32 | 6.82 | ||

35 | 0.30 | 0.32 | 0.34 | 7.37 | ||

40 | 0.30 | 0.34 | 0.36 | 7.95 | ||

45 | 0.30 | 0.36 | 0.38 | 8.57 | ||

50 | 0.30 | 0.38 | 0.40 | 9.24 | ||

12 | Single | 35 | 0.30 | 0.33 | 0.35 | 7.07 |

14 | 0.36 | 0.45 | 0.55 | 8.77 | ||

16 | 0.32 | 0.48 | 0.45 | 7.87 | ||

10 | Double | 25 | 0.243 | 0.30 | 0.30 | 6.42 |

The material properties of concrete used in the present bridge models are: Characteristic strength = 40 MPa; Poisson’s ratio = 0.2; Density = 25 kN/m^{3}; Modulus of elasticity = 3.16 × 10^{7} kN/m^{2}. The material properties of reinforcing steel are as follows: Density = 77 kN/m^{3}; Yield strength = 500 MPa; Ultimate tensile strength = 545 MPa; Modulus of elasticity = 2 × 10^{8} kN/m^{2}.

Simply supported boundary condition is used to constraint the bridge, and for that two roller supported bearings on one end and two pin supported bearings on the other end are considered. For the straight bridge deck in the present study, the vibration analysis specifications are presented as per Lenzen’s criteria [

i) Estimation of the maximum deflection under dead load

Span = 25 m, L/d = 10

M-40 Grade concrete, so

Maximum deflection,

Natural frequency of vibration

ii)

iv) Ensuring that the product,

v) Vibration characteristic is found from the Lenzen’s criteria using the values of

The Vibration characteristics of other spans and span-depth ratios are computed similarly and presented in

Cell | L/d | L (m) | A_{provided}^{2}) |
I_{provided}^{4}) |
δ (mm) | N_{f} (Hz) |
Δ (mm) | a (mm/sec^{2}) |
Safe/ |
Zone |
---|---|---|---|---|---|---|---|---|---|---|

Single | 10 | 25 | 6.42 | 6.80 | 0.27 | 12.39 | 0.11 | 651.96 | Safe | Slightly perceptible |

30 | 6.82 | 10.35 | 0.30 | 10.30 | 0.12 | 511.44 | Safe | Slightly perceptible | ||

35 | 7.37 | 15.16 | 0.33 | 8.81 | 0.13 | 405.66 | Safe | Slightly perceptible | ||

40 | 7.95 | 21.16 | 0.35 | 7.67 | 0.14 | 329.06 | Safe | Slightly perceptible | ||

45 | 8.57 | 28.48 | 0.37 | 6.77 | 0.15 | 271.33 | Safe | Slightly perceptible | ||

50 | 9.24 | 37.26 | 0.39 | 6.04 | 0.16 | 226.49 | Safe | Slightly perceptible | ||

12 | 35 | 7.07 | 10.18 | 0.49 | 7.37 | 0.19 | 422.87 | Safe | Slightly perceptible | |

14 | 35 | 8.77 | 8.53 | 0.58 | 6.06 | 0.23 | 340.90 | Safe | Slightly perceptible | |

16 | 35 | 7.83 | 6.07 | 0.82 | 5.41 | 0.33 | 381.83 | Safe | Slightly perceptible | |

Double | 10 | 25 | 6.42 | 6.67 | 0.27 | 12.39 | 0.11 | 651.96 | Safe | Slightly perceptible |

A convergence study is performed to select the refinement in mesh needed to provide the nearest solution. It gives the size of the element required to achieve the minimum error in the evaluated fundamental frequency of bridges. The different models of a single-cell box-girder bridge, i.e., straight, curved, skew and skew-curved, are considered for the study. As presented in

Mesh size (mm) | Fundamental frequency (Hz) of bridges | |||
---|---|---|---|---|

Straight | Skew | Curved | Skew-curved | |

500 | 1.762 | 1.975 | 1.753 | 2.235 |

450 | 1.598 | 1.895 | 1.592 | 2.052 |

400 | 1.456 | 1.782 | 1.478 | 1.925 |

350 | 1.325 | 1.585 | 1.326 | 1.725 |

300 | 1.235 | 1.468 | 1.246 | 1.625 |

250 | 1.182 | 1.392 | 1.198 | 1.503 |

200 | 1.170 | 1.291 | 1.185 | 1.405 |

150 | 1.162 | 1.242 | 1.179 | 1.332 |

100 | 1.159 | 1.201 | 1.172 | 1.323 |

90 | 1.158 | 1.198 | 1.169 | 1.321 |

80 | 1.158 | 1.195 | 1.168 | 1.319 |

In the analysis, a three-dimensional linear elastic finite element model is used for all the bridge models.

The results on the frequencies of skew-curved RC box-girder bridges are not available in the literature. However, one may see that the present results are converged; hence, the developed modelling process can be accepted and is extended for further studies on the box-girder bridges.

A detailed parametric study is performed to investigate the natural frequencies of box-girder bridges. Skew angle, curve angle, span, span-depth ratio, and the cell number vary in the study because these are the primary factors that directly affect the bridges’ vibrational characteristics. Here, only the fundamental frequency of the bridge is presented, and the results of other modes of frequencies are circumvented for the sake of conciseness of the paper. The parameters stipulated are as follows: Skew angle (θ) = 0 to 60° at an interval of 10°; Curve angle (α) = 0 to 60° at an interval of 12°; Span (L) = 25 to 50 m at an interval of 5 m for span-depth ratio (L/d) 10; span-depth ratio = 10 to 16 at an interval of 2 for 35 m spans; and the number of cell = 1 and 2. The obtained results are presented in tabular and graphical forms; however, it is hard to interpret the results as the values are relatively closer. The effect of these parameters is shown separately in the following sections.

The skew angle effect on the fundamental frequency of bridges is investigated for different spans, span-depth ratios, and cells. A single-cell box-girder of the span-depth ratio of 10 is considered for the study.

Span (L) | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | |

25 m | 3.12 | 3.10 | 3.11 | 3.13 | 3.15 | 3.18 | 3.19 |

30 m | 2.30 | 2.29 | 2.29 | 2.31 | 2.33 | 2.35 | 2.38 |

35 m | 1.77 | 1.76 | 1.77 | 1.78 | 1.79 | 1.81 | 1.84 |

40 m | 1.42 | 1.41 | 1.41 | 1.42 | 1.43 | 1.45 | 1.47 |

45 m | 1.16 | 1.15 | 1.16 | 1.16 | 1.17 | 1.18 | 1.20 |

50 m | 0.97 | 0.96 | 0.96 | 0.97 | 0.97 | 0.98 | 1.00 |

The bridges’ fundamental frequency is evaluated for different skewness and span-depth ratio (10, 12, 14 and 16). A single-cell box-girder bridge of a 35 m span with varying depth is considered for the investigation. The spans less than 35 m are not considered because they do not satisfy the stress and deflection limitations (specified in IRC:21-2000) for all span-depth ratios. However, the effect of the span-depth ratios on the fundamental frequency will be similar if the spans more than 35 m are considered.

Span-depth ratio (L/d) | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | |

10 | 1.77 | 1.76 | 1.77 | 1.78 | 1.79 | 1.81 | 1.84 |

12 | 1.84 | 1.83 | 1.83 | 1.84 | 1.86 | 1.87 | 1.90 |

14 | 1.98 | 1.97 | 1.97 | 1.99 | 2.01 | 2.02 | 2.06 |

16 | 2.03 | 2.02 | 2.03 | 2.04 | 2.05 | 2.02 | 2.86 |

The frequency of single-cell and double-cell box-girder bridges is compared. For the investigation, a span of 25 m and a span-depth ratio of 10 are considered. A span of 25 m is considered in the study as it is specified in IRC:21-2000 to use a minimum span of 25 m for box-girder bridges. Span-depth ratios of more than 10 are not considered as they do not satisfy the serviceability criteria specified in IRC:21-2000.

Number of cells | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | |

1 | 3.12 | 3.10 | 3.11 | 3.13 | 3.15 | 3.18 | 3.19 |

2 | 3.21 | 3.18 | 3.23 | 3.29 | 3.42 | 3.60 | 3.90 |

The curve angle effect on the fundamental frequency of bridges is investigated for different spans, span-depth ratios, and cells.

Span (L) | Fundamental frequency (Hz) with curvature (α) | |||||
---|---|---|---|---|---|---|

0° | 12° | 24° | 36° | 48° | 60° | |

25 m | 3.12 | 3.12 | 3.12 | 3.13 | 3.13 | 3.15 |

30 m | 2.30 | 2.30 | 2.30 | 2.32 | 2.32 | 2.32 |

35 m | 1.77 | 1.77 | 1.78 | 1.78 | 1.79 | 1.80 |

40 m | 1.42 | 1.42 | 1.42 | 1.43 | 1.44 | 1.44 |

45 m | 1.16 | 1.16 | 1.16 | 1.16 | 1.17 | 1.18 |

50 m | 0.97 | 0.99 | 0.99 | 0.97 | 0.98 | 0.99 |

The influence of curve angle and span-depth ratio on the fundamental frequency is investigated and presented in

Span-depth ratio (L/d) | Fundamental frequency (Hz) with curvature (α) | |||||
---|---|---|---|---|---|---|

0° | 12° | 24° | 36° | 48° | 60° | |

10 | 1.77 | 1.77 | 1.78 | 1.78 | 1.79 | 1.80 |

12 | 1.84 | 1.84 | 1.84 | 1.85 | 1.86 | 1.87 |

14 | 1.98 | 1.98 | 1.98 | 1.99 | 2.00 | 2.01 |

16 | 2.03 | 2.04 | 2.04 | 2.04 | 2.06 | 2.06 |

Number of cells | Fundamental frequency (Hz) with curvature (α) | |||||
---|---|---|---|---|---|---|

0° | 12° | 24° | 36° | 48° | 60° | |

1 | 3.12 | 3.12 | 3.13 | 3.13 | 3.13 | 3.15 |

2 | 3.20 | 3.20 | 3.21 | 3.22 | 3.22 | 3.24 |

The fundamental frequency of skew-curved bridges is investigated.

Span (L) | Curve angle (α) | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | ||

25 m | 0° | 3.12 | 3.10 | 3.11 | 3.13 | 3.15 | 3.18 | 3.19 |

12° | 3.12 | 3.11 | 3.12 | 3.15 | 3.19 | 3.25 | 3.31 | |

24° | 3.12 | 3.12 | 3.08 | 3.18 | 3.23 | 3.31 | 3.52 | |

36° | 3.13 | 3.13 | 3.16 | 3.21 | 3.27 | 3.38 | – | |

48° | 3.13 | 3.15 | 3.19 | 3.24 | 3.31 | 3.48 | – | |

60° | 3.15 | 3.17 | 3.20 | 3.26 | 3.35 | – | – | |

30 m | 0° | 2.30 | 2.29 | 2.29 | 2.31 | 2.33 | 2.35 | 2.38 |

12° | 2.30 | 2.29 | 2.30 | 2.32 | 2.35 | 2.39 | 2.45 | |

24° | 2.30 | 2.30 | 2.31 | 2.34 | 2.37 | 2.43 | 2.58 | |

36° | 2.32 | 2.31 | 2.33 | 2.36 | 2.40 | 2.47 | 2.66 | |

48° | 2.32 | 2.32 | 2.34 | 2.37 | 2.42 | 2.52 | – | |

60° | 2.32 | 2.34 | 2.36 | 2.39 | 2.44 | 2.56 | – | |

35 m | 0° | 1.77 | 1.76 | 1.77 | 1.78 | 1.79 | 1.81 | 1.84 |

12° | 1.77 | 1.77 | 1.77 | 1.79 | 1.81 | 1.84 | 1.89 | |

24° | 1.78 | 1.77 | 1.78 | 1.80 | 1.82 | 1.86 | 1.98 | |

36° | 1.78 | 1.78 | 1.79 | 1.81 | 1.84 | 1.89 | 2.02 | |

48° | 1.79 | 1.79 | 1.81 | 1.83 | 1.86 | 1.92 | – | |

60° | 1.80 | 1.80 | 1.82 | 1.84 | 1.87 | 1.95 | – | |

40 m | 0° | 1.42 | 1.41 | 1.41 | 1.42 | 1.43 | 1.45 | 1.47 |

12° | 1.42 | 1.41 | 1.42 | 1.43 | 1.44 | 1.45 | 1.52 | |

24° | 1.42 | 1.42 | 1.42 | 1.44 | 1.45 | 1.48 | 1.57 | |

36° | 1.43 | 1.42 | 1.43 | 1.45 | 1.46 | 1.51 | 1.60 | |

48° | 1.44 | 1.43 | 1.45 | 1.46 | 1.48 | 1.53 | 1.63 | |

60° | 1.44 | 1.44 | 1.47 | 1.47 | 1.49 | 1.54 | – | |

45 m | 0° | 1.16 | 1.15 | 1.16 | 1.16 | 1.17 | 1.18 | 1.20 |

12° | 1.16 | 1.16 | 1.16 | 1.17 | 1.18 | 1.20 | 1.25 | |

24° | 1.16 | 1.16 | 1.17 | 1.17 | 1.19 | 1.21 | 1.28 | |

36° | 1.16 | 1.17 | 1.17 | 1.18 | 1.20 | 1.23 | 1.30 | |

48° | 1.17 | 1.17 | 1.18 | 1.19 | 1.21 | 1.24 | 1.32 | |

60° | 1.18 | 1.18 | 1.19 | 1.20 | 1.21 | 1.26 | 0.99 | |

50 m | 0° | 0.97 | 0.96 | 0.96 | 0.97 | 0.97 | 0.98 | 1.00 |

12° | 0.99 | 0.96 | 0.97 | 0.97 | 0.98 | 0.99 | 1.01 | |

24° | 0.99 | 0.97 | 0.97 | 0.98 | 0.99 | 1.01 | 1.03 | |

36° | 0.97 | 0.97 | 0.98 | 0.98 | 1.00 | 1.03 | 1.04 | |

48° | 0.98 | 0.98 | 0.98 | 0.99 | 1.00 | 1.04 | 1.05 | |

60° | 0.99 | 0.98 | 0.99 | 1.00 | 1.01 | 1.05 | 1.07 |

The influence of skew and curve angles on the fundamental frequency for different span-depth ratios is shown in

Span-depth ratios (L/d) | Curve angle (α) | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | ||

10 | 0° | 1.77 | 1.76 | 1.77 | 1.78 | 1.79 | 1.81 | 1.84 |

12° | 1.77 | 1.77 | 1.77 | 1.79 | 1.81 | 1.84 | 1.89 | |

24° | 1.78 | 1.77 | 1.78 | 1.80 | 1.82 | 1.86 | 1.98 | |

36° | 1.78 | 1.78 | 1.79 | 1.81 | 1.84 | 1.89 | 2.02 | |

48° | 1.79 | 1.79 | 1.81 | 1.83 | 1.86 | 1.92 | – | |

60° | 1.80 | 1.80 | 1.82 | 1.84 | 1.87 | 1.95 | – | |

12 | 0° | 1.84 | 1.83 | 1.83 | 1.84 | 1.86 | 1.87 | 1.90 |

12° | 1.84 | 1.83 | 1.84 | 1.86 | 1.88 | 1.91 | 1.98 | |

24° | 1.84 | 1.84 | 1.85 | 1.87 | 1.90 | 1.94 | 2.07 | |

36° | 1.85 | 1.85 | 1.87 | 1.89 | 1.92 | 1.98 | 2.12 | |

48° | 1.86 | 1.86 | 1.88 | 1.91 | 1.94 | 2.02 | – | |

60° | 1.87 | 1.88 | 1.90 | 1.92 | 1.96 | 2.05 | – | |

14 | 0° | 1.98 | 1.97 | 1.97 | 1.99 | 2.01 | 2.02 | 2.06 |

12° | 1.98 | 1.97 | 1.98 | 2.00 | 2.03 | 2.06 | 2.12 | |

24° | 1.98 | 1.98 | 2.00 | 2.02 | 2.06 | 2.10 | 2.24 | |

36° | 1.99 | 1.99 | 2.01 | 2.04 | 2.09 | 2.13 | 2.32 | |

48° | 2.00 | 2.01 | 2.03 | 2.07 | 2.12 | 2.17 | – | |

60° | 2.01 | 2.03 | 2.06 | 2.10 | 2.14 | 2.21 | – | |

16 | 0° | 2.03 | 2.02 | 2.03 | 2.04 | 2.05 | 2.02 | 2.86 |

12° | 2.04 | 2.03 | 2.04 | 2.05 | 2.08 | 2.11 | 2.16 | |

24° | 2.04 | 2.04 | 2.05 | 2.07 | 2.10 | 2.16 | 2.23 | |

36° | 2.04 | 2.05 | 2.07 | 2.09 | 2.13 | 2.20 | 2.31 | |

48° | 2.06 | 2.06 | 2.09 | 2.12 | 2.17 | 2.24 | – | |

60° | 2.06 | 2.08 | 2.11 | 2.15 | 2.20 | 2.27 | – |

Number of cells | Curve angle (α) | Fundamental frequency (Hz) with skewness (θ) | ||||||
---|---|---|---|---|---|---|---|---|

0° | 10° | 20° | 30° | 40° | 50° | 60° | ||

1 | 0° | 3.12 | 3.10 | 3.11 | 3.13 | 3.15 | 3.18 | 3.19 |

12° | 3.12 | 3.11 | 3.12 | 3.15 | 3.19 | 3.25 | 3.31 | |

24° | 3.12 | 3.12 | 3.08 | 3.18 | 3.23 | 3.31 | 3.52 | |

36° | 3.13 | 3.13 | 3.16 | 3.21 | 3.27 | 3.38 | – | |

48° | 3.13 | 3.15 | 3.19 | 3.24 | 3.31 | 3.48 | – | |

60° | 3.15 | 3.17 | 3.20 | 3.26 | 3.35 | – | – | |

2 | 0° | 3.21 | 3.18 | 3.23 | 3.29 | 3.42 | 3.60 | 3.90 |

12° | 3.20 | 3.20 | 3.25 | 3.35 | 3.51 | 3.84 | 4.23 | |

24° | 3.21 | 3.21 | 3.27 | 3.38 | 3.56 | 3.89 | 4.40 | |

36° | 3.22 | 3.23 | 3.30 | 3.41 | 3.61 | 3.93 | – | |

48° | 3.22 | 3.25 | 3.33 | 3.45 | 3.66 | 3.77 | – | |

60° | 3.24 | 3.28 | 3.36 | 3.49 | 3.70 | – | – |

A bridge with 45 m span and a span-depth ratio of 10 is selected herein to investigate the effect of skew and curve angles. The combined effect of skew and curve angles on the fundamental frequency is illustrated in

The bridge has an endless number of possible modes because it has infinite degrees of freedom with continuous mass. According to IS 1893:2016, part 1 [

Mode number | Natural frequency (Hz) | Modal participating mass ratio | Cumulative modal participating mass ratio (%) |
---|---|---|---|

1 | 1.32 | 0.14411 | 14.441 |

2 | 2.99 | 0.00485 | 14.896 |

3 | 5.24 | 0.29992 | 44.888 |

4 | 7.09 | 0.00063 | 44.951 |

5 | 7.18 | 0.01132 | 46.083 |

6 | 8.97 | 0.26502 | 72.584 |

7 | 11.03 | 0.00793 | 73.377 |

8 | 12.53 | 0.01309 | 74.686 |

9 | 13.10 | 0.10506 | 85.193 |

10 | 13.93 | 0.00883 | 86.076 |

11 | 14.87 | 0.05196 | 91.272 |

The fundamental frequency of different box-girder bridges was evaluated, and it is mostly in the range of 1 to 4.5 Hz. Here, the fundamental frequency of the bridge and the external agencies’ vibration are correlated. The bridge vibrates due to the moving loads. The exciting frequency generated due to walking over the bridge is generally in the range of 1.6 to 2.4 Hz [

Also, the maximum dynamic response occurs when the vehicle’s speed is such that the time it takes to cross a span is approximately equal to the fundamental period of vibration of the structure. However, the vehicle speed is not a significant factor in determining the response of the bridges. A larger response may be produced by heavy vehicles due to accelerating or braking. That is why the vehicle speed is restricted on large span bridges as it produces maximum dynamic loading, except for a very short span (less than 15 m).

Because of the above, the box-girder bridges should be designed considering the critical range occupied by most of the vehicles, i.e., 2 to 5 Hz. Therefore, the present study is important as it provides the results of the fundamental frequency of different box-girder bridges.

The fundamental frequency of different box-girder bridges is investigated via a parametric study using the finite element method. The parameters chosen are skew angle, curve angle, span, span-depth ratio, and cell number, which influence the fundamental frequency of these bridges (skew, curved and skew-curved). The following conclusions are drawn from the present study:

The effect of the curve angle on the fundamental frequency is insignificant. But the fundamental frequency increases with the skew angle. So, the skew bridge is more effective than the straight and curved bridges where vibrations are dominant. Hence, in those situations, skew bridges are preferable compared to straight and curved bridges.

The fundamental frequency decreases with the increase in the span. However, it increases with the span-depth ratio.

The fundamental frequency of the double-cell bridge is more compared to that of a single-cell bridge. So, a double-cell bridge is preferable compared to a single-cell bridge. The double-cell bridge’s fundamental frequency increases with the skew angle, but the effect of the curve angle on the double-cell bridge is insignificant.

The fundamental frequency is more in the case of skew-curved bridges than that in the straight bridges for different spans, span-depth ratios and the number of cells. But the fundamental frequency of the bridge with more skewness (60°) and curvature (60°) is smaller in comparison to the straight bridge.

The increment in the fundamental frequency of a skew-curved bridge (α = 48°, θ = 60°) is about 13% and 15% for single-cell and double-cell bridges, respectively, compared to the straight bridge. So, the double-cell skew-curved box-girder is preferable.

The bridges need to be strengthened against vibration if the range of fundamental frequency is small (1.0–4.5 Hz).

Skew angle (°)

Curve angle (°)

Radius of curve bridge (m)

Span of central bridge girder (m)

_{o}

Span of outer girder bridge deck (m)

_{i}

Span of inner girder bridge deck (m)

_{tf}

Thickness of top flange (m)

_{bf}

Thickness of bottom flange (m)

_{w}

Thickness of web (m)

Cross-sectional area

Span-depth ratio

Modulus of elasticity of concrete (kN/m^{2})

Moment of inertia (m^{4})

Maximum deflection (m)

_{f}

Natural frequency (Hz)

Acceleration due to gravity (m/sec^{2})

_{d}

Dead weight of bridge (kN/m)

Maximum acceleration (m/sec^{2})

Maximum amplitude of vibration (m)

Fundamental frequency (Hz)