# Load-Carrying Capacity of Bailey Bridge in Civil Applications
## 4 Load-Carrying Capacity  
As the probabilistic approach would be quite inefficient, [26], the semi-probabilistic engineering approach covered by Eurocode 3 was applied in this research [16; 27]. All relevant verifications of cross-sections and members were performed. The possibility of local yielding was allowed only in cross-sections of class 1 and 2. In accordance to tests performed in [28], the cross-sections of BB members are able to resist such local yielding stresses in the case of ultimate loading. The normal or exclusive load-carrying capacity \(V_{n}\) 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]:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Traffic Load in** & & & & & & \(V_{e}\) & \\
**Combinations** & & & & & & \\ \hline \multirow{2}{*}{**BB Arrangement**} & \(\mathbf{a}_{cr,z}\) & \(N_{cr,z}\) & \(L_{cr,z}\) & \(\mathbf{a}_{cr,z}\) & \(N_{cr,z}\) & \(L_{cr,z}\) \\  & [–1] & [**kN**] & [**m**] & [–1] & [**kN**] & [**m**] \\ \hline TS-12 & 7.53 & 4021 & 2.05 & 6.62 & 3743 & 2.12 \\ TS-18 &  
\[V_{n(e)}=\left[\left(R_{d}-\sum_{i=1}^{n-1}E_{rs,d,i}\ \right)/E_{Vn(e),d,REP} \right]\cdot V_{n(e),REP} \tag{3}\]  
where: \(R_{d}\) 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. In order to apply the principle of proportionality and to simplify the calculation process, the relative values for load-carrying capacities were considered in a linear global analysis. Thus, load schemes were modelled in FEM analysis with its specified so-called'representative' values and relative load-carrying capacity \(Z\) were calculated according to Equation (4). Namely, the representative weight of the modelled 'normal' vehicle was set to \(V_{n,REP}\) = 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:  
\[Z_{n(e)}=\left(R_{d}-\sum_{i=1}^{n-1}E_{rs,d,i}\ \right)/E_{Vn(e),d,REP} \tag{4}\]  
where all symbols were defined under Equation (3). The use of relative values of load-carrying capacity \(Z\) is common for railway bridges, where it represents a relative ratio to the effects produced by railway traffic load models [30]. Anyway, for the public roads, the values of maximum permissible weight of a vehicle passing the bridge in tons are commonly utilised, as they can be controlled and defined by traffic signs. As relative value \(Z\) is a function of specified value of the modelled representative vehicles \(V_{n,REP}\) or \(V_{e,REP}\), respectively, the load carrying capacity can be then calculated according to Equations (5) and (6), as follows:  
\[V_{n}=Z_{n}\cdot V_{n,REP}=Z_{n}\cdot 32\ \text{t} \tag{5}\]  
\[V_{e}=Z_{e}\cdot V_{e,REP}=Z_{e}\cdot 40\ \text{t} \tag{6}\]  
Combinations of internal forces in a critical cross-section have been taken into account.