# Load-Carrying Capacity of Bailey Bridge in Civil Applications
## 3 Global Analyses
### Stability Analysis  
For the estimation of the resistance of top chord of the main girders, the out-of-plane buckling phenomena had to be carefully estimated. Except for the influences connected to traffic and corrosion damages [23], cases of collapsed Bailey bridges confirmed buckling instability as one of the most important failure modes of the structure [24]. Thus, attention was paid to the correct stability estimation of compression chords of the truss panels. The elastic analysis is suitable for the main girder behaviour, as failure mode of elastic buckling is occurring [25]. Figure 12: Scheme of vehicle used for exclusive load-carrying capacity. Figure 13: Positions of heavy vehicles within cross-section: \(V_{n}\) on the left, and \(V_{e}\) on the right. In order to identify the deformation ability of the pin joint in an out of plane direction, an in situ verification was executed (Figure 14). It was found that the tolerances in the pin joint permit horizontal displacement of the joint from the straight direction up to 55-60 mm in the case of two adjacent panels. Of course, such values are unrealistic on the construction site, but they indicate that the pin joint of the panels can be considered hinged even for horizontal moments. Therefore, the pin joints of panels were modelled as hinged joints in both horizontal and vertical directions. The results of the elastic buckling analysis are usually expressed by the shapes of buckling modes (Figure 15), which are quantified by the critical buckling load factor \(\alpha_{cr}\) applied in order to reach the elastic global instability. Thus, the critical buckling load factor \(\alpha_{cr}\) 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 \(\alpha_{cr,z}\) for all alternatives of Bailey bridges analysed in the study as  
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). 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:  
\[L_{cr,z}=\pi\sqrt{\frac{EI_{z}}{\alpha_{cr,z}\ N_{Ed}}} \tag{2}\]  
where \(E\) is modulus of elasticity of steel, \(I_{z}\) represents quadratic moment of the cross-sectional area of top chord (about the vertical axis), \(\alpha_{cr,z}\) 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. Based on presented results it can be concluded that the main girders are not prone to out-of-plane instability. As the value of critical buckling load factor \(\alpha_{cr}\) 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 \(\alpha_{cr}\) 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]. Moreover, from the values of derived critical buckling lengths \(L_{cr,z}\), 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.