# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 3 Fatigue Load Models
#### Fatigue Load Models LM2 and LM4 of Eurocode EN1991-2  
The Eurocode EN1991-2 provides five load models for fatigue verifications of road bridges. Fatigue load model no. 1 (LM1), derived from the main model used for static verifications, is a very rough and simplified fatigue model intended to assess unlimited fatigue life. This model is out of the scope of the present study, therefore it will not be discussed further. Fatigue load model no. 3 (LM3) is a simplified model for fatigue damage calculations, which is described in the following Section 5. Fatigue load model no. 5 (LM5), which is the most general one, consists of a representative real traffic sample directly derived from traffic registrations. Evidently, in the present case LM5 represents, in turn, one of the three traffic measurements. Fatigue load model no. 2 (LM2) is intended to assess whether fatigue life is limited or not, therefore it can be used only when a constant amplitude fatigue limit \(\Delta\sigma_{D}\) exists, like for steel details. Fatigue load model no. 4 (LM4) is specifically devoted to damage calculations. It must be remarked that LM2 and LM4 are notional models, which have been calibrated with different purposes, as explained later. For that reason, to avoid inconsistencies they should be applied in sequence. In any case, it should be borne in mind that compliance with fatigue verification for unlimited fatigue life, if relevant, automatically implies that fatigue damage cannot occur, as explained better in the following. In fact, if \(\Delta\sigma_{D}\) exists, LM2 should be used first. Said \(\gamma_{F,f}\) the partial factor on fatigue actions; \(\gamma_{M,f}\) the partial factor on fatigue resistance and \(\Delta\sigma_{\text{max},f}\) the maximum stress range caused by LM2, if it results as:  
\[\gamma_{F,f}\Delta\sigma_{\text{max},f}\leq\frac{\Delta\sigma_{D}}{\gamma_{M, f}}, \tag{1}\]  
the fatigue verification is satisfied and no further verification is needed. If Inequality (1) is not verified, the fatigue damage \(D\) should be assessed using LM4 and the Palmgren-Miner formula [35; 36]:  
\[D=\sum_{\Delta\sigma_{D}\geq\Delta\sigma_{L}}\frac{n_{i}}{N_{i}}\leq 1, \tag{2}\]where \(n_{i}\) is the number of cycles in the stress spectrum at \(\Delta\sigma_{i}\) stress range, \(N_{i}\) the number of cycles to failure at \(\Delta\sigma_{i}\); and the summation is extended to all cycles for which \(\Delta\sigma_{i}>\Delta\sigma_{L}\), being \(\Delta\sigma_{L}\) the cut-off limit. Evidently, as already remarked, when verifications are carried out using both LM2 and LM4, while Inequality (2) implies Inequality (1), the opposite seems not always be true; but this contradiction is not real, since \(\Delta\sigma_{D}\) is a threshold for fatigue damage: if the maximum stress range is below that threshold, it cannot cause fatigue damage. The general statement is that the fatigue assessment is governed by the most favorable, or the less onerous, between Inequalities (1) and (2). It must be stressed that, according to the basic hypotheses of the Eurocodes, characteristic fatigue strength \(S\)-\(N\) curves are considered in the present study. The \(S\)-\(N\) characteristic curve corresponds to the lower bound of the prediction interval 5%-95%, i.e., 5% probability of failure. It is obtained by a downward vertical translation of two standard deviation of the mean \(S-N\) curve. Fatigue load models LM2 (Figure 3) and LM4 (Figure 4) are sets of standardized lorries, representing the most common axle arrangements of European lorries, characterized by \"_frequent_\" (LM2) or \"_damage equivalent_\" (LM4) values of axle loads. The axles of the standardized lorries are distinct in three typologies: \(A\); \(B\); \(C\); depending on the dimensions of the contact surface of the wheels, as illustrated in Figure 5. In Figures 3 and 4, the axle types are indicated in the last column.