The Eurocode fatigue verification for steel road bridges rests on damage-equivalence factors (λ) and partial factors that Leander (2018) showed, in a fully probabilistic analysis with Swedish bridge weigh-in-motion (BWIM) traffic, produce a large scatter of achieved reliability (β from 0.86 to 4.9) — and that recalibrating λ1 and λ2 to a target β = 3.1 largely removes it. That calibration, however, assumed one vehicle on the bridge at a time on a two-dimensional beam model, kept a deterministic dynamic impact factor in the design format, and used only Swedish traffic statistics. Multi-vehicle presence effects on fatigue damage have been quantified at the deterministic damage-equivalence level (Sjaarda, Nussbaumer and Walbridge 2024, who define multiple-presence factors); recorded same-lane vehicle sequences and traffic jams are now included in WIM-based load spectra (Maljaars 2020, 2022); measured dynamic-amplification distributions exist; and Eurocode fatigue safety factors have been shown to need raising (Maljaars et al. 2022). No published work integrates multi-vehicle interaction — overtaking, side-by-side presence in adjacent lanes, closely spaced vehicles in the same lane — and a probabilistic treatment of vehicle-induced dynamics into the reliability-based calibration of the Eurocode verification format itself, with three-dimensional structural modeling where transverse effects matter, multi-vehicle event statistics calibrated to measured traffic, in-situ validation, and regional transfer. A 2025 review (Maljaars, Leander, Nussbaumer, Sørensen and Straub) still lists dynamic-interaction modeling and probabilistic fatigue load models as open. That integration, with the resulting recalibrated factors, is the open problem.

Background

Fatigue — the initiation and slow growth of cracks under repeated stress cycles — governs the design of welded steel bridges. The European design standard for steel structures, Eurocode 3 (EN 1993-1-9), verifies fatigue with nominal stress ranges: the damage-equivalent stress range computed at a detail must not exceed the detail's fatigue strength, defined by an S–N curve (the curve relating constant-amplitude stress range to the number of cycles to failure), divided by a partial safety factor. Instead of explicit cycle counting of real traffic, EN 1991-2 provides simplified fatigue load models: FLM3, a single notional 480 kN lorry, and FLM4, a set of five standard lorries, together with damage-equivalence factors λ (λ1 depending on the critical length of the influence line — the length of the loaded region that maximizes the load effect at the section considered — and λ2 on the annual number of heavy vehicles). These factors were calibrated so that the notional single-lorry crossing reproduces the damage caused by real traffic, using weigh-in-motion (WIM) measurements — roadside systems recording axle loads of vehicles at highway speed; when the sensors are mounted on a bridge and the bridge itself serves as the scale, the technique is called bridge WIM (BWIM) — performed in 1986 at Auxerre, France.

Because this deterministic format ignores the statistical scatter of vehicle weights, traffic composition, fatigue resistance, and analysis models, Leander (2018) built a fully probabilistic model for Swedish conditions: linear damage accumulation by the Palmgren–Miner rule (each stress range consumes a fraction n⋅Δσm/Kn\cdot\Delta\sigma^m/K of the fatigue life) with a bilinear S–N curve, stochastic fatigue-resistance variables, and stress-range spectra from single passages of 30 vehicle types whose gross-weight distributions were fitted to 872,090 heavy vehicles recorded at 152 Swedish BWIM sites in 2005–2009. Stress histories were rainflow-counted (the standard cycle-counting procedure for irregular loading histories); vehicle-induced dynamics entered only through a generic stochastic correction factor while the design format kept the Eurocode's deterministic damage-equivalent impact factor; reliability was computed by Monte Carlo simulation and expressed through the reliability index β (larger β means smaller failure probability). Evaluating the existing Eurocode format for nine bridges and four traffic categories, the study found a large scatter of achieved reliability — β from 0.86 to 4.9 — governed mainly by critical influence-line length and traffic volume, and recalibrated the λ1 and λ2 functions to a target β = 3.1 over a 100-year reference period (the partial-factor product of the format having been calibrated earlier by Leander, Norlin and Karoumi (2015)), reducing the scatter to roughly 2.4–3.4.

The 2018 calibration rested, however, on simplified two-dimensional beam models loaded by one vehicle at a time: vehicles never overtook, never travelled side by side in adjacent lanes, and never followed each other closely on the influence line; the traffic statistics were exclusively Swedish; and vehicle-induced dynamics had no physics-based probabilistic treatment. Fatigue has become more decisive in new Swedish bridge designs, so whether these simplifications make the calibrated verification format unsafe or overly conservative is a live question for the ongoing European re-calibration of fatigue safety factors (the CEN technical committee responsible for EN 1993-1-9 has explicitly formulated the task of re-calibrating them).

Problem Statement

The open problem is the reliability-based recalibration of the Eurocode fatigue verification format for steel road bridges under realistic multi-vehicle loading. Concretely, a recalibration answering that question must address what the single-vehicle, two-dimensional, Swedish-traffic and deterministic-dynamics basis of the 2018 calibration leaves open: (1) a statistical model for the occurrence and composition of multi-vehicle events — overtaking and simultaneous presence in adjacent lanes, closely spaced vehicles in the same lane, and platoons — calibrated to measured WIM/BWIM traffic, combined with structural models that represent what two-dimensional beam models cannot: transverse load distribution and simultaneous multi-lane loading; (2) a probabilistic treatment of the dynamic amplification of stress ranges informed by measurements under real traffic rather than the deterministic code value retained in the verification format; (3) a recomputation of the reliability indices achieved by the verification format with this enriched load model over the relevant ranges of critical influence-line lengths — including a refinement of the critical-length definition for support sections, where the existing calibration shows systematic scatter — and of traffic volumes and compositions, covering regional traffic other than the Swedish 2005–2009 BWIM population; and (4) a recalibration of the damage-equivalence factors λ1 and λ2 and/or the partial factors to the target reliability. The completion boundary is a defensible quantitative statement of how, and by how much, multi-vehicle interaction and probabilistic dynamics change the achieved reliability level, and the recalibration of the damage-equivalence factors λ1 and λ2 and/or the partial factors of the Eurocode fatigue verification for steel road bridges that this entails. These components are jointly in service of that single objective: the traffic statistics, the structural models that capture transverse and multi-lane load effects, the probabilistic dynamics, and the measurement-based validation are the means by which the recalibrated factors become trustworthy.

References

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