Gestión de Configuración del Software en los Modelos de Calidad

The calibration method, the underlying philosophy, the methodological approach and the main features of the fatigue models of Eurocode 1 are summarised in the following.

2.2.1 Calibration method

Fatigue load models have been calibrated referring to reference influence surfaces relative to simply supported and continuous bridges, spanning in the range 3 m to 200 m.

In agreement with the general fatigue assessment procedures, calibration has been set-up according to the following scheme,

- choice of the most significant European traffic data;

- selection of appropriate S-N curves;

- evaluation of the stress histories in reference bridges;

- cycle counting and stress spectra computation;

- first identification of fatigue models;

- definition of standardised lorry silhouettes;

- calibration of frequent load models, best fitting the maximum stress range ∆σmax

induced by real traffic;

- calibration of equivalent load models, best fitting the fatigue damage D induced by the traffic.

2.2.2 Reference S-N curves

Reference S-N curves pertain to steel details, characterised by endurance limit.

As known, in the logarithmic S-N chart these curves are represented by a bilinear curve, characterised by a sloping branch of constant slope, m=3, (figure 1), or by a trilinear curve, characterised by two sloping branches, m=3 and m=5, (figure 2), according as boundless fatigue life or fatigue damage is to be assessed.

Since steel details exhibit fatigue limit ∆σD, two cases can be envisaged, according to the maximum stress range ∆σmax of the real stress history is higher than ∆σD or not.

To be significant for fatigue, ∆σmax must be exceeded several times during the design working life of the bridge and its definition is not trivial. Two different approaches have been proposed, leading to similar results: in the former, ∆σmax is defined as the stress range such that the 99% of the total fatigue damage results from all stress ranges below ∆σmax; in the latter ∆σmax is the stress range exceeded approximately 5⋅10⁴ times during the bridge life.

O N

S

m=3

?σ 5⋅¹⁰⁶

∆σD

Figure 1. Bilinear S-N curve

O N

S

m=3

?σ 5⋅¹⁰⁶

D m=5

L

10⁸

∆σ

∆σ

Figure 2. Trilinear S-N curve

This last definition implies that the return period of ∆σmax is about half a day, giving so direct explanation of frequent load spectrum denomination.

In EN 1991-2 studies, to derive equivalent load spectra independently from the fatigue classification of details, cumulative damage has been computed referring generally to simplified S-N curves with unique slope, in turn m=3 (figure 3) or m=5 (figure 4). S-N curves with double slope (figure 5) and without endurance limit have been used for some additional calculations.

Some comparisons show that load spectra obtained using the simplified curve m=5 are free from significant errors and reproduce generally well the actual fatigue damage.

O N

S

m=3

Figure 3. Single slope S-N curve (m=3)

O N

S

m=5

Figure 4. Single slope S-N curve (m=5)

O N

S

m=3

?σ 5⋅¹⁰⁶

D m=5

∆σ

Figure 5. Double slope m=3- m=5 S-N curve without endurance limit

2.2.3 Fatigue load models

From the above-mentioned considerations, it derives that at least two conventional fatigue load models must be considered: the one for boundless fatigue life assessments, the other for fatigue damage calculations. Besides, since an adequate fitting of the effects induced by the real traffic requires very sophisticated load models, whose application is often difficult, the introduction of simplified and safe-sided models, to be used when sophisticated checks are unnecessary, seems very opportune.

For this reason in EN 1991-2 two fatigue load models are foreseen for each kind of fatigue verification: the former is essential, safe-sided and easy to use, the latter is more refined and accurate, but also more complicated. In conclusion, four conventional models are given:

- models 1 and 2 for boundless fatigue checks;

- models 3 and 4 for damage calculations.

Fatigue load model 1 is extremely simple and generally very safe-sided. It directly derives from the main load model used for assessing static resistance, where the load values are simply reduced to the frequent ones (figure 6.a), multiplying the tandem axle loads Q_ik by 0.7 and the weight density of the uniformly distributed loads qik by 0.3.

Obviously, for local verifications, the fatigue load model n. 1 is constituted by the isolated concentrated axle weighing Q=280 kN (frequent value - figure. 6.b).

Lane n. 1 Q =210 kN

q =2.7 kN/m Q_ik

Q_ik

q_ik

Lane n. 2

0.5 2.0 0.5

Q =140 kN_2k q =0.75 kN/m_2k ²

Lane n. 3 ^{Q =70 kN}q =0.75 kN/m_3k^{3k 2}

Remaining area q =0.75 kN/m_rk ²

1k 1k 0.5

2.0 0.5

0.5 2.0 0.5 w

2

Figure 6.a. Fatigue load model n. 1

Figure 6.b. Fatigue load model n. 1 for local

verifications The verification consists of checking that the maximum stress range ∆σmax induced by the model is smaller of the fatigue limit ∆σD. The application rules for the load model n. 1 agree exactly with those given for the main load model, so that the absolute minimum and maximum stresses correspond as rule to different load configurations. The model allows making “coarse” verifications also in multi-lane configurations, generally resulting extremely safe-sided.

The simplified fatigue model n. 3, conceived for damage computation, is constituted by a symmetrical conventional four axle vehicle, also said fatigue vehicle (figure 7). The equivalent load of each axle is 120 kN. This model is accurate enough for spans bigger than 10 m, while for smaller spans it results safe-sided.

Figure7. Fatigue load model n. 3

Fatigue load models n. 2 and n. 4 are the most refined one and they are load spectra constituted by five standardised vehicles, representative of the most common European lorries.

Fatigue load model n. 2, which is a set of lorries with frequent values of axle loads, and fatigue model n. 4, which is a set of lorries with equivalent values of the axle loads, are illustrated in tables 1 and 2, respectively. They allow to perform very precise and sophisticated verifications, provided that the interactions amongst vehicles simultaneously crossing the bridge are negligible or opportunely considered.

Table 1. Fatigue load model n. 2 – frequent set of lorries LORRY

Table 2. Fatigue load model n. 4 – equivalent set of lorries

LORRY SILHOUETTE TRAFFIC TYPE

Long

The types of wheels pertaining to each standardised lorries of fatigue load models n. 2 and n. 4 are indicated in table 1, referring to table 3.

The number of lorries to be taken into account for damage assessments depends on the traffic category: indicative values of Nobs, representing the number of lorries of year per slow lane, are given in table 4. The additional traffic on the fast lane can be assumed to be 10% of the slow lane traffic.

In fact, in EN 1991-2 a further general purpose fatigue model is anticipated too, denominated fatigue model n. 5. This model is constituted by a sequence of consecutive axle loads, directly derived from recorded traffic, duly supplemented to take into account vehicle interactions, where relevant.

Fatigue model n. 5 is aimed to allow accurate fatigue verifications in particular situations, like suspended or cable-stayed bridges, important existing bridges or bridges carrying unusual traffics, whose relevance justifies ad hoc investigations [2].

2.2.4 Accuracy of fatigue load models

In the following, some significant results obtained using the fatigue load models are compared with those pertaining to the reference traffic, allowing to highlight the accuracy and the field of application of the each conventional model.

Table 3. Definition of wheels and axles of standard lorries Wheel axle

type Geometrical definition

A

Longitudinal axis of the bridge

2 1.78

0.32 0.22

0.22 0.32

B

Longitudinal axis of the bridge

2 0.22

0.32

0.22

0.32

0.22 0.22

0.54 0.54

C

Longitudinal axis of the bridge

2 1.73

0.32 0.27

0.27 0.32

Table 4. Indicative number of lorries expected per year and for a slow lane

Traffic categories Nobs per year and per slow lane 1 Roads and motorways with 2 or more lanes per direction

with high flow rates of lorries

2.0⋅10⁶ 2 Roads and motorways with medium flow rates of lorries 0.5⋅10⁶ 3 Main roads with low flow rates of lorries 0.125⋅10⁶ 4 Local roads with low flow rates of lorries 0.05⋅10⁶

Essentially, the comparison concerns the four influence surfaces shown in figure 8, for bridges span L varying between 3 m and 100 m. The influence surfaces pertain to bending moment M₀ at midspan of simply supported beams, bending moments M₁ and M₂ at midspan and on the support, respectively, of two span continuous beams and bending moment M3 at midspan of three span continuous beams.

Figure 8. Reference influence lines

In figures 9 to 14 the outcomes of different fatigue load models for the aforesaid influence lines are compared with the real traffic (Auxerre traffic) effects, in function of the span length L, considering only one notional lane.

For unlimited fatigue life assessments, the ratios between the maximum bending moment ranges due to fatigue load model 1, ∆Mmax,LM1, and fatigue load model 2, ∆Mmax,LM2, and the maximum bending moment ranges due to real traffic, ∆Mmax,real, are diagrammatically reported in figures 9 and 10, respectively.

For fatigue damage assessments reference can be made to the equivalent bending moment range corresponding to 2⋅10⁶ cycles.

The ratios between the equivalent bending moment ranges due to fatigue load model 3, ∆M_eq,LM3, and the equivalent bending moment ranges due to real traffic, ∆M_eq,real, are shown in figures 11 and 12, considering one slope S-N curves characterised by m=3 and m=5, respectively. Analogous diagrams for equivalent bending moment ranges due to fatigue load model 4, ∆Meq,LM4, are illustrated in figure 13, for m=3 S-N curve.

Figure 9. Accuracy of fatigue load model n. 1

Figure 10. Accuracy of fatigue load model n. 2

Figure 11. Accuracy of fatigue load model n. 3 – m=3

Figure 12. Accuracy of fatigue load model n. 3 – m=5

Figure 13. Accuracy of fatigue load model n. 4

Analysis of the diagrams shows that:

- as just said, fatigue load model n. 1 appears very safe-sided, especially for short spans;

- load model n. 2 results much more reliable; values little below the actual ones are obtained for M2 in the span range 20 to 50 m, but this depends on the particular shape of the influence line;

- as expected, model n. 4 fits very good actual results for short influence lines;

- fatigue model n. 3 looks unsafe for M2 influence lines when spans are above 30 m, in particular for higher m values. To solve the problem it has been proposed to modify the model n. 3 taking into account an additional fatigue vehicle each time that the influence surface exhibits two contiguous areas of the same sign. This additional fatigue vehicle, having equivalent axle loads set to 40 kN, should run on the same lane of the basic fatigue vehicle, 40 m away from it. The adoption of such an additional vehicle should mitigate the error in computation of ∆M2,eq, as it appears evident in figure 14, where damage calculations for M2 influence line and for m=3 and m=9 S-N curves, considering additional fatigue vehicle.

Figure 14. Accuracy of improved fatigue load model n. 3 – 2 vehicles