# Impact of Road Traffic Tendency in Europe on Fatigue Assessment of Bridges
## 5 Damage-Equivalence Factors  
In the damage-equivalent factors method, also referred to as the \(\lambda\) method, the fatigue verification is carried out by means of an \"equivalent\" static verification. The rationale of the method is to define an equivalent constant amplitude stress history reproducing the actual fatigue damage \(D\). In principle, there are infinite stress histories able to assure that equivalence; in fact, given \(D\), it is possible to derive the number of cycles to failure from the stress range of each cycle and vice versa but, fixing conventionally a reference value for one of the two parameters, the other parameter results in being univocally determined. In the Eurocodes, the equivalence has been established considering as reference value the number of cycles to failure identifying the fatigue category. Therefore, for structural steel details the reference value is \(2\times 10^{6}\) cycles and for reinforcing steel and pre-stressing steel details it is \(N^{*}\) (see Section 4.2). The verification consists thus in comparing a suitably calculated \"equivalent\" stress range, \(\Delta\sigma_{eq}\), with the fatigue category of the detail:  
\[\gamma_{Ff}\Delta\sigma_{eq}\leq\frac{\Delta\sigma_{C}}{\gamma_{Mf}}. \tag{13}\]  
The most general expression of the \"_equivalent_\" stress range, \(\Delta\sigma_{eq}\), is:  
\[\Delta\sigma_{eq}=\Delta\varphi_{fat}\Delta\overline{\sigma}\prod_{\lambda_{i }=1}^{n}\lambda_{i}=\Delta\varphi_{fat}\lambda\Delta\overline{\sigma}, \tag{14}\]  
where \(\Delta\varphi_{fat}\geq 1.0\) is an additional amplification factor for fatigue verifications, which takes into account, if relevant, dynamic effects, \(\Delta\overline{\sigma}\) is the basic stress range to be considered, and \(\lambda\) is the damage equivalent factor. Obviously, in Equation (14) it is:  
\[\lambda=\prod_{\lambda_{i}=1}^{n}\lambda_{i}, \tag{15}\]  
being \(\lambda_{i}\) the damage equivalent factors, accounting for the most relevant effects influencing the fatigue verification. According what was discussed in Section 3 about fatigue load models LM2 and LM4, for details characterized by constant amplitude fatigue limit, an upper bound \(\lambda_{\text{max}}\) should be introduced for \(\lambda\), to avoid that, unrealistically, the most unfavorable outcome between Equations (1) and (2) prevails. In the bridge parts of Eurocodes [24; 26], Equation (14) for the \"equivalent\" stress range is expressed by:  
\[\Delta\sigma_{eq}=\Delta\varphi_{fat}\Delta\sigma_{EC}\prod_{\lambda_{i}=1}^{ 4}\lambda_{i}=\Delta\varphi_{fat}\lambda\Delta\sigma_{EC}, \tag{16}\]  
being \(\Delta\sigma_{EC}\) the maximum stress range,  
\[\Delta\sigma_{EC}=\sigma_{\text{max}}-\sigma_{\text{min}}, \tag{17}\]  
caused by fatigue load model LM3 of EN1991-2 in the considered detail. Obviously, in the Eurocodes it is assumed \(\Delta\overline{\sigma}=\Delta\sigma_{EC}\). The fatigue load model LM3 is a single \"fatigue equivalent lorry\" used to calculate \(\Delta\sigma_{EC}\). It consists in a simplified and conventional model, constituted by a four axle symmetrical lorry. The four axles, weighing 120 kN each, are spaced 1.2 m, 6.0 m and 1.2 m, respectively (Figure 12). The total weight of LM3, 480 kN, corresponds to the equivalent lorry weight of the Auxerre traffic. In the Eurocodes, the rationale of the individual damage-equivalence coefficients in Equation (16) is as follows:  
* \(\lambda_{1}\) considers the effects of the shape and of the basic length of the influence line on the stress history. In fact, since Equation (17) disregards secondary cycles, \(\Delta\sigma_{EC}\) must be modified to duly include the increased fatigue damage caused by secondary cycles, if any. Clearly, \(\lambda_{1}\geq 1.0\) depends on the static scheme, on the reference length, on the material and on the detail and it requires specific calibration. The scrutiny of this factor is out of the scope of the present work, so it will not be discussed further.