# Load-Carrying Capacity of Bailey Bridge in Civil Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Outline of the Study

- Double-truss, single-story (DS) for a span with length of 12.196 m;
- Triple-truss, single-story (TS) for spans: 12.192 m, 15.240 m, 18.288 m and 21.336 m;
- Double-truss, double-story (DD) for spans: 21.336 m, 24.384 m and 27.432 m;
- Triple-truss, double-story (TD) for spans: 27.432 m, 30.480 m, 33.528 m, and 36.576 m.

- Stringers (Figure 4)
- -
- Old stringers made of the original cross-section.
- -
- New stringers made of the cross-section IPN100.
- -
- Alternative new stringers made of the cross-section IPN120.

- Cross beams—‘transoms’.
- Panel of truss girders.
- -
- Upper chord.
- -
- Bottom chord.
- -
- Diagonals with U-shaped cross-section.
- -
- Alternative diagonals with I-shaped cross-section.
- -
- Verticals with U-shaped cross-section.
- -
- Alternative verticals with I-shaped cross-section.

- End verticals—‘end posts’.
- Inclined struts—‘rakers’.
- Panel pin (hinged connection between panels).
- Bottom bracings—‘sway brace’.
- Floor bolts (in double-storeys arrangement)—‘chord bolts’.

- a)
- Three cross beams within each panel of the bridge—the spans of the stringers are then 1440 + 1290 + 318 mm.
- b)
- Two cross beams within each panel of the bridge—the spans of the stringers are then 1608 + 1440 mm (this alternative was applicable only for single-story systems).

- a)
- Standard solution of two layers of 2 × 50 mm thick timber boards, where the bottom boards are placed perpendicular to the stringers, while the top layer is oriented in 45° degrees to the bridge axis;
- b)
- Stiffer roadway, where bottom supporting layer of the bridge deck is made of timber beams of 100 mm section height placed side by side perpendicular to the stringers, while the top layer stays the same as in the alternative a).

## 3. Global Analyses

#### 3.1. Numerical Models

_{y}= 360 MPa was considered for the needs of the presented study. Modulus of elasticity E = 210,000 MPa was utilized in the analysis as well.

_{deck}= 10 MPa]. This almost completely prevented the interaction of the deck with the steel elements of the bridge deck, but it made it possible to place a traffic load anywhere on the surface of the bridge deck. The second advantage of this modelling approach is that it allows the redistribution of a modelled traffic load to the corresponding stringers. Figure 8 and Figure 9 present the finished models for the shortest and the longest span analysed in the study.

#### 3.2. Loads in Global Analysis

_{w}

^{*}according to Eurocode 1 [17] and were properly combined with the traffic loads. Accordingly, the wind pressure value of q

_{p,z}= 0.50 kPa was considered acting uniformly for all analysed BB models. The effect of wind pressure acting on bridge deck combined with pressure on traffic load was transformed into the edge horizontal load, surface horizontal load and vertical surface load producing torsion of the deck around its longitudinal axis, as shown in Figure 10a. The values presented in Figure 10a were derived by utilizing the reference height of traffic load 2.0 m consistent with [17]. Accordingly, the wind load on structural members of BB panels was calculated as uniformly distributed load using the force coefficient c

_{f}given in [17] with dependence on the shape of each cross-section, as shown in Figure 10b.

_{Qi}and α

_{qi}[19]. Moreover, in the case of short spans, LM1 causes disproportionately high effects because the dominant ‘uniformly distributed load’ (UDL) as well as the dominant ‘tandem system’ (TS) are concentrated together in the same lane on the relatively small length section. The abovementioned phenomenon is even more visible in narrow bridges. Findings that the load schemes given in the Eurocode [19] are not very suitable for load-carrying capacity calculations of existing bridges are also supported by the conclusions in [22], which provides an extensive state of the art presentation of the traffic load models.

_{n}. The ‘normal load-carrying capacity’ represents the maximum permissible weight of any vehicle on the bridge without any restrictions or addition regulations in traffic within the road section on the bridge. Geometric parameters of the load scheme are presented in Figure 11.

_{e}represents maximum permissible weight of a single lorry on the bridge, while no other vehicles on the bridge are allowed at the same time, e.g., by other traffic signs. In the case of analysed Bailey bridges, which are the narrow one-lane bridges, this can be easily specified by the traffic sign of minimum following distance between vehicles. Alternatively, in some countries, the additional text table to normal load carrying capacities sign can be installed where the text concerning the maximum weight of the only vehicle on the bridge is specified.

_{d}represents the effective length in meters for each type of member. In the case of main girders and their components, the theoretical span can be taken into account. Thus, the dynamic factor applied to main girders internal forces varied from the value ϕ = 1.17 in the case of longest analysed span with the length of 36.576 m, up to ϕ = 1.30 in the case of span 12.196 m long. Similarly, the dynamic factors ϕ = 1.48 for inclined struts (rakers) and ϕ = 1.50 for all other members of BB superstructure were considered.

#### 3.3. Stability Analysis

_{cr}applied in order to reach the elastic global instability.

_{cr}represents an amplifier by which the design load would have to be increased to cause elastic instability [16]. Table 2 summarizes critical buckling load factors α

_{cr,z}for all alternatives of Bailey bridges analysed in the study as obtained for design load combinations. Based on the Euler’s critical force, the buckling length L

_{cr,z}for out-of-plane buckling can be calculated according to the following equation:

_{z}represents quadratic moment of the cross-sectional area of top chord (about the vertical axis), α

_{cr,z}is the critical buckling load factor to reach the out-of-plane elastic global instability of top chords, and N

_{Ed}is the design value of compressive force in a top chord member.

_{cr}is between 3.0 and 10.0 in all cases, second order effects can be included into verification process indirectly, for example by the ‘equivalent column method’, [16]. Usually, if the value of α

_{cr}is smaller than 3.0, a geometrical nonlinear analysis with imperfections included is required. Such an analysis is very time consuming in the case of a numerical model with large number of elements [13].

_{cr,z}, it is clear that the buckling lengths seem to be smaller than panel length, as they reach values from 2.05 to 2.50 m in the presented numerical analyses. Therefore, provided that all bracings (both bracing frames and sway bracings), rakers and all mutual connections are feasible in accordance with BB documentation, there will be probably no problems with stability of top chords, and a well-known equivalent column method can be utilized.

_{cr}are also valid for the traffic loads smaller than load-carrying capacities of chords under compression, presented in the following sections, since data in Table 2 were obtained iteratively for ultimate value of compressive buckling resistance in the critical element of top chord, individually for each variant of BB structure.

## 4. Load-Carrying Capacity

_{n}or V

_{e}(V

_{n(e)}) can be calculated according to the Equation (3), which is equivalent to equations according to regulations used in other countries [22,29]:

_{d}is the design value of the resistance of the cross-section or bridge member; E

_{Vn(e),d,REP}is the value of static quantity due to effects of variable load model defined in Figure 11 or Figure 12, respectively, but only those parts of them, which depend on value of V

_{n,REP}or V

_{e,REP}(i.e., without uniformly distributed load 2.5 kN/m

^{2}in the case on normal load-carrying capacity); ${\sum}_{i=1}^{n-1}{E}_{rs,d,i}$ are the design, combination or group values of the effects of other (the rest) loads acting, including those parts of the variable road load model independent of value of V

_{n,REP}or V

_{e,REP}, respectively; finally designation V

_{n(e),REP}express representative values of weight of vehicles implemented into global analysis by which static entities E

_{Vn(e),d,REP}were produced.

_{n,REP}= 32 tons and implemented into all particular load cases, while for the representative weight of ‘exclusive’ vehicle, the value of V

_{e,REP}= 40 tons was used. Thus, those values correspond to relative load-carrying capacity Z = 1.0. The basic formulae for obtaining relative load-carrying capacity Z can by then written as:

_{n,REP}or V

_{e,REP}, respectively, the load carrying capacity can be then calculated according to Equations (5) and (6), as follows:

_{n}and Z

_{e}, or final load-carrying capacities of verified elements V

_{n}and V

_{e}, respectively. As calculations of load-carrying capacity have to be done for many of verification procedures according to Eurocode 1 [16,19], the whole procedure was programmed. If stability was the issue, the verification of buckling resistances of members under compression and/or lateral-torsional buckling of members under bending moment were executed repeatedly as well.

## 5. Results and Discussion

#### 5.1. Main Girders

_{n}and 40 tons in the case of exclusive load-carrying capacity V

_{e}, respectively. In addition, the values of V

_{n}smaller than 22 tons and of V

_{e}smaller than 32 tons are underlined to point out that they represent values very far from the required level defined at the beginning of the study. Thus, the applicability of main girders is clear from both tables.

_{n}= 28 tons). Despite of that, these results do not lead to the conclusion that the BB system cannot be utilized for such variants and spans at all. These BB system configurations can still be utilized; however, the weight of the vehicles allowed to cross the bridge has to be limited to given levels. The similar conclusions can be derived for exclusive load-carrying capacity according to Table 4.

#### 5.2. Deck Members

_{n}< 28 tons and V

_{e}< 40 tons, respectively. As in Section 5.1, the underlined values are those less than 22 tons for the normal load-carrying capacity and less than 32 tons in the case of the exclusive load-carrying capacity. Accordingly, the limit values for highlighting or underlining the values applicable to the maximum permissible weight per axle were considered to be 12 tons, and 10 tons respectively.

#### 5.2.1. Stringers

#### 5.2.2. Cross Beams

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**‘Long-term’ temporary bridge Bailey bridge (DS-15 + 21 + 27 + 21) over Lužnice river (Czech Republic).

**Figure 5.**Two alternatives of construction arrangement of cross beams: (

**a**) 3 cross beams within each panel; and (

**b**) 2 cross beams within each panel.

**Figure 6.**Two alternatives of timber roadway deck: (

**a**) standard solution of timber deck; and (

**b**) more stiff timber deck.

**Figure 8.**Geometry of complete spatial FEM model in the case of the shortest analysed span TS-12 (40 ft): (

**a**) model from beam and shell elements; and (

**b**) front view visualization.

**Figure 9.**Geometry of compete spatial FEM model in the case of the longest analysed span TD-36 (120 ft): (

**a**) model from beam and shell elements; and (

**b**) front view visualization.

**Figure 10.**Example view from approximation of wind load: (

**a**) left wind action on deck with traffic load on it; and (

**b**) wind load assumed on main girders BB panels.

**Figure 13.**Positions of heavy vehicles within cross-section: V

_{n}on the left, and V

_{e}on the right.

**Figure 14.**In situ verification of pin joint dimensions and its deformation capacity in horizontal direction: (

**a**) measurement of horizontal deformation capacity of the joint; and (

**b**) stored Bailey bridge panels.

**Figure 15.**Examples of buckling modes of top chords in the case of BB composed as TS-12 (top right) and DD-24 (bottom left).

Span | Truss | Story | Abbreviation |
---|---|---|---|

40 ft/12.192 m | triple | single | TS-12 |

60 ft/18.288 m | triple | single | TS-18 |

70 ft/21.336 m | triple | single | TS-21 |

80 ft/24.384 m | double | double | DD-24 |

90 ft/27.432 m | double | double | DD-27 |

110 ft/33.528 m | triple | double | TD-33 |

120 ft/36.576 m | triple | double | TD-36 |

Traffic Load in Combinations | V_{n} | V_{e} | ||||
---|---|---|---|---|---|---|

BB Arrangement | α_{cr,z}[−] | N_{cr,z}[kN] | L_{cr,z}[m] | α_{cr,z}[−] | N_{cr,z}[kN] | L_{cr,z}[m] |

TS-12 | 7.53 | 4021 | 2.05 | 6.62 | 3743 | 2.12 |

TS-18 | 5.73 | 3182 | 2.30 | 5.65 | 3128 | 2.32 |

TS-21 | 4.79 | 2741 | 2.48 | 5.27 | 2967 | 2.38 |

DD-24 | 4.93 | 2925 | 2.40 | 5.22 | 3149 | 2.32 |

DD-27 | 4.83 | 2936 | 2.40 | 5.13 | 3088 | 2.34 |

TD-33 | 4.47 | 2695 | 2.50 | 4.90 | 2952 | 2.39 |

TD-36 | 4.38 | 2661 | 2.52 | 4.84 | 2928 | 2.40 |

Normal Load-Carrying Capacity [tons] | Configuration | TS-12 | TS-18 | TS-21 | DD-24 | DD-27 | TD-33 | TD-36 |
---|---|---|---|---|---|---|---|---|

Span Length [m] | 12.192 | 18.288 | 21.336 | 24.384 | 27.432 | 33.528 | 36.576 | |

Predominant Stress | V_{n} | V_{n} | V_{n} | V_{n} | V_{n} | V_{n} | V_{n} | |

Chords | T + B (bottom) | 21.1 | 22.9 | 18.4 | 22.1 | 16.3 | 17.7 | 15.6 |

C (top) | 54.8 | 38.0 | 30.0 | 41.2 | 30.9 | 27.5 | 20.4 | |

Diagonals U | C | 22.6 | 19.8 | 18.6 | 23.0 | 21.5 | 22.3 | 21.3 |

T | 39.1 | 42.1 | 40.1 | 44.0 | 42.4 | 43.5 | 42.7 | |

Diagonals I | C | 18.9 | 16.5 | 15.4 | 19.0 | 17.7 | 18.4 | 17.7 |

T | 45.8 | 45.0 | 43.0 | 47.1 | 45.2 | 46.7 | 45.6 | |

Verticals U | C | 30.1 | 26.8 | 25.9 | 27.7 | 25.0 | 24.8 | 25.0 |

T | 40.0 | 40.3 | 38.6 | 40.0 | 40.4 | 41.0 | 41.2 | |

Verticals I | C | 25.4 | 22.4 | 21.7 | 22.7 | 20.1 | 20.6 | 19.9 |

T | 42.3 | 42.5 | 42.2 | 43.6 | 44.1 | 44.6 | 44.9 | |

Rakers | C | 31.3 | 34.3 | 31.3 | 37.2 | 37.2 | 40.7 | 43.5 |

T | 92.6 | 91.4 | 67.3 | 95.2 | 79.4 | 83.6 | 80.0 | |

End posts | C | 65.3 | 62.4 | 62.4 | 46.5 | 44.5 | 53.5 | 52.2 |

Sway braces | T | 66.1 | 58.1 | 54.0 | 61.0 | 55.5 | 51.7 | 47.0 |

Pin joint | S + B | 92.4 | 64.1 | 51.4 | 58.7 | 47.6 | 44.5 | 35.2 |

Chord bolts | S + T | 66.7 | 61.8 | 63.7 | 60.9 |

Exclusive Load-Carrying Capacity [tons] | Configuration | TS-12 | TS-18 | TS-21 | DD-24 | DD-27 | TD-33 | TD-36 |
---|---|---|---|---|---|---|---|---|

Span Length [m] | 12.192 | 18.288 | 21.336 | 24.384 | 27.432 | 33.528 | 36.576 | |

Predominant Stress | V_{e} | V_{e} | V_{e} | V_{e} | V_{e} | V_{e} | V_{e} | |

Chords | T + B (bottom) | 53.1 | 47.7 | 37.0 | 45.7 | 40.0 | 46.9 | 40.0 |

C (top) | 76.2 | 50.6 | 42.0 | 59.8 | 51.4 | 55.2 | 47.8 | |

Diagonals U | C | 33.2 | 34.1 | 33.4 | 46.5 | 45.9 | 47.4 | 47.0 |

T | 56.9 | 64.0 | 60.8 | 78.5 | 78.2 | 81.2 | 80.9 | |

Diagonals I | C | 28.9 | 29.9 | 29.3 | 40.5 | 39.8 | 41.1 | 40.6 |

T | 64.6 | 66.7 | 65.3 | 82.4 | 82.2 | 85.2 | 85.0 | |

Verticals U | C | 46.2 | 44.7 | 45.1 | 49.2 | 48.8 | 51.8 | 52.1 |

T | 62.5 | 64.2 | 61.0 | 91.0 | 92.0 | 93.2 | 93.8 | |

Verticals I | C | 40.6 | 39.0 | 38.9 | 41.9 | 41.5 | 45.2 | 44.6 |

T | 73.2 | 75.3 | 71.3 | 98.1 | 99.1 | 100.4 | 101.1 | |

Rakers | C | 42.1 | 30.7 | 30.7 | 51.2 | 51.3 | 51.8 | 56.2 |

T | 106.2 | 99.0 | 87.7 | 129.5 | 125.6 | 152.8 | 148.0 | |

End posts | C | 75.8 | 76.3 | 74.6 | 68.9 | 70.1 | 77.0 | 76.5 |

Sway braces | T | 80.3 | 71.2 | 67.2 | 87.6 | 80.9 | 79.6 | 76.7 |

Pin joint | S + B | 68.8 | 72.3 | 73.4 | 74.4 | 75.1 | 76.2 | 76.7 |

Chord bolts | S + T | 113.2 | 109.8 | 115.4 | 114.1 |

Stringers with Spans of 1608 + 1440 mm (2 Cross Beams within Each Panel) | |||||||
---|---|---|---|---|---|---|---|

Load-Carrying Capacity [tons] | Normal LCC V_{n} | Exclusive LCC V_{e} | Axle Weight | ||||

Standard Deck | Stiffer Deck | Standard Deck | Stiffer Deck | Standard Deck | Stiffer Deck | ||

original | mid-span | 10.0 | 11.9 | 31.7 | 38.1 | 7.5 | 8.9 |

mid-support | 12.8 | 15.0 | 38.3 | 42.7 | 9.6 | 10.7 | |

IPN100 | mid-span | 13.1 | 15.5 | 41.2 | 49.1 | 9.8 | 11.6 |

mid-support | 16.9 | 19.9 | 49.7 | 55.9 | 12.4 | 14.0 | |

IPN120 | mid-span | 20.8 | 24.4 | 64.3 | 76.2 | 15.6 | 18.3 |

mid-support | 26.2 | 31.2 | 76.3 | 86.2 | 19.1 | 21.5 |

Stringers with Spans of 1440 + 1290 + 318 mm (3 Cross Beams within Each Panel) | |||||||
---|---|---|---|---|---|---|---|

Load-Carrying Capacity [tons] | Normal LCC V_{n} | Exclusive LCC V_{e} | Axle Weight | ||||

Standard Deck | Stiffer Deck | Standard Deck | Stiffer Deck | Standard Deck | Stiffer Deck | ||

original | mid-span | 12.6 | 12.4 | 37.2 | 38.4 | 9.3 | 9.3 |

mid-support | 18.1 | 17.6 | 51.0 | 53.8 | 12.8 | 13.2 | |

IPN100 | mid-span | 16.6 | 16.3 | 48.2 | 49.6 | 12.1 | 12.3 |

mid-support | 24.6 | 24.2 | 66.7 | 63.0 | 16.7 | 15.8 | |

IPN120 | mid-span | 26.5 | 26.2 | 75.3 | 77.6 | 18.8 | 19.4 |

mid-support | 39.7 | 39.7 | 99.4 | 90.5 | 24.9 | 22.6 |

Three Cross Beams within Each Panel (Stringers with Spans of 1440 + 1290 + 318 mm) | ||||
---|---|---|---|---|

Load-Carrying Capacity [tons] | Normal LCC V _{n} | Exclusive LCC V _{e} | Axle Weight | |

Cross beam | DS | 28.3 | 57.1 | 14.3 |

TS | 29.8 | 56.1 | 14.0 | |

DD | 28.2 | 56.7 | 14.2 | |

TD | 28.9 | 55.6 | 13.9 |

Two Cross Beams within Each Panel (Stringers with Spans of 1608 and 1440 mm) | ||||
---|---|---|---|---|

Load-Carrying Capacity [tons] | Normal LCC V _{n} | Exclusive LCC V _{e} | Axle Weight | |

Cross beam | DS | 21.5 | 40.0 | 10.0 |

TS | 24.5 | 47.3 | 11.8 |

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Prokop, J.; Odrobiňák, J.; Farbák, M.; Novotný, V.
Load-Carrying Capacity of Bailey Bridge in Civil Applications. *Appl. Sci.* **2022**, *12*, 3788.
https://doi.org/10.3390/app12083788

**AMA Style**

Prokop J, Odrobiňák J, Farbák M, Novotný V.
Load-Carrying Capacity of Bailey Bridge in Civil Applications. *Applied Sciences*. 2022; 12(8):3788.
https://doi.org/10.3390/app12083788

**Chicago/Turabian Style**

Prokop, Jozef, Jaroslav Odrobiňák, Matúš Farbák, and Vladimír Novotný.
2022. "Load-Carrying Capacity of Bailey Bridge in Civil Applications" *Applied Sciences* 12, no. 8: 3788.
https://doi.org/10.3390/app12083788