Prestressed concrete bridges rely on high-strength steel tendons that are tensioned during construction to keep the concrete in permanent compression. Over decades, that tension fades. Shrinkage and creep of the concrete, relaxation of the steel, friction along the ducts, corrosion, thermal swings and freeze–thaw cycles all conspire to bleed away the prestress force, and because that force directly governs a bridge’s load-carrying capacity and remaining service life, engineers have long wanted a way to measure it continuously under live traffic rather than during costly closures. A new study published in Advances in Bridge Engineering presents a method that does exactly this with just two strain sensors, while simultaneously identifying the moving vehicle forces crossing the span.
Existing dynamic identification techniques are largely insensitive to the prestress force itself, and their accuracy degrades badly under real operational conditions. Many approaches also demand dense sensor arrays, extensive computation, or in-situ tests — some destructive, such as strand-cutting, others intrusive, like drilled-hole or saw-cut methods — that disrupt traffic and cannot support continuous monitoring.
The core idea is deceptively simple: instead of using raw strain from a single sensor, the method works with the difference in strain between two measurement points along the bridge’s longitudinal direction. This differential strain cancels out much of the computational error and measurement uncertainty that plagues direct approaches. The prestress force is not measured directly; it is treated as an unknown that manifests as equivalent external pseudo-forces at element nodes, an effect arising from the compression-softening phenomenon in which axial compression lowers the effective flexural stiffness of a prestressed member. The team frames the whole problem as an inverse one: given measured differential strain, recover both the prestress force and the moving force time history.
Inverse problems of this kind are notoriously ill-conditioned — small amounts of noise in the measurements can produce wildly oscillating, physically meaningless solutions. The authors tackle this with a convolution matrix built from a Vandermonde matrix, essentially a Savitzky–Golay-style smoothing operator embedded directly into the least-squares identification. Rather than fixing the smoothing window arbitrarily, they introduce a diagnostic called the local pseudo-force ratio, computed from a pseudo-prestress force ratio response matrix, and compare it against a theoretical shape function derived from a Lagrangian polynomial-based Hermitian interpolation. The window size is chosen as the smallest value beyond which this ratio stabilises, balancing noise suppression against preservation of the true signal and keeping computational demand low. A quadratic polynomial was selected because higher-order fits offered negligible improvement while risking overfitting and oscillatory behaviour.
The method was validated in three stages of increasing realism. First, numerical studies on a two-span continuously supported prestressed concrete T-beam, each span 20 metres long and modelled in Abaqus with 40 mm solid elements, tested two prestress levels of 2000 kN and 3000 kN under vehicles crossing at various speeds over very good, average and poor road surfaces. Using a convolution window of 35, the relative percentage errors in prestress identification ranged from 1.95% to 7.32% depending on sensor pair, while moving force identification errors across six loading cases and five sensor pairs stayed between 0.75% and 6.99%. The method also handled a harder scenario: two vehicles travelling in opposite directions at 20 m/s, with overall errors of 2.46% and 1.88% for the two forces, though accuracy degraded as the vehicles closed on one another, with local errors peaking near 4.85%.
A striking pattern emerged from both the simulations and later experiments: sensor placement matters enormously. Accuracy was highest when the two gauges sat at or near mid-span, and errors grew steadily as the pairs moved toward the supports or as the element length between them increased. This gives bridge operators a practical, evidence-based rule — put your two strain sensors near mid-span, not near the bearings — and it held across prestress identification, moving force identification, and every excitation type tested.
Second came laboratory verification on a 6-metre simply supported prestressed box girder bridge model, excited at mid-span by a Moog control loading system delivering propulsive, periodic and retarding forces. Prestress was applied through two unbonded 15.2 mm tendons in three stages — 171.28 kN, 284.01 kN and 378.29 kN — and strain was sampled at 1000 Hz. With a window size of 25, prestress identification errors ranged from 2.20% to 7.42% across the three stages and three sensor pairs, and every identified excitation stayed under 5.02% error. Benchmark comparisons against two established inverse techniques were telling: the virtual distortion method with a load shape function produced highly fluctuating, unstable force estimates, while the modified preconditioned conjugate gradient method demanded far more computation and still oscillated more than the new approach, which achieved stable results from only two strain channels.
The third and most convincing stage used field data from a real three-span continuously supported prestressed concrete bridge with spans of 99, 180 and 99 metres — 378 metres in total. Electrical resistance strain gauges at two soffit locations and a mid-span accelerometer provided measurements sampled at 10 Hz, alongside dynamic weighing records for six vehicle categories captured overnight in September 2023. After finite element model updating, which matched the first three measured modal frequencies of 0.95, 2.12 and 2.78 Hz to within a maximum error of 4.25%, the team identified moving forces from the differential strain of passing trucks without any prior knowledge of the vehicles, comparing results against the equivalent gross vehicle weights from the weigh-in-motion system.
The authors are careful about scope. The method assumes uniform linear elastic behaviour with isotropic material properties and Rayleigh damping, and it currently resolves only overall axle forces, not finer wheel-level detail. Deep girders and wide box girders, where shear deformation and torsion matter, remain future work, as do experimental vehicle-crossing tests and long-term monitoring validated against standard prestress-loss formulas. The field prestress results, moreover, come from a calibrated finite element model rather than direct physical measurement of tendon tension on the real bridge, so they demonstrate feasibility rather than a fully independent field confirmation.
Even with those caveats, the practical implications are considerable. Two strain gauges, a calibrated model and a laptop-scale computation could give bridge managers a running estimate of both the traffic loads hitting their structure and the invisible prestress that keeps it safe — information that today requires closures, heavy instrumentation or destructive testing. The researchers suggest the dual-strain scheme could even serve as a simple bridge weigh-in-motion system. If long-term trials bear out the laboratory and simulated performance, the humble strain gauge pair near mid-span may become one of the most informative two-sensor investments a bridge owner can make.
Subject of Research: A differential strain-based inverse method for simultaneously identifying prestress force and moving vehicle loads in prestressed concrete bridges
Article Title: A differential strain-based approach for identifying prestress and moving forces in prestressed concrete bridges
Article References: Jeyamohan, K., Chan, T. H. T., Nguyen, K.-D., Thambiratnam, D. P., Zhu, X., Wu, D., & Zhang, E. (2026). A differential strain-based approach for identifying prestress and moving forces in prestressed concrete bridges. Advances in Bridge Engineering, 7(1), Article 26. https://doi.org/10.1186/s43251-026-00216-5
Image Credits: AI Generated
DOI: 10.1186/s43251-026-00216-5
Keywords: prestressed concrete bridges, prestress loss, moving force identification, strain gauges, structural health monitoring, inverse problems, Vandermonde matrix, Savitzky-Golay filtering, finite element model updating, bridge weigh-in-motion, sensor placement, time-domain method
News Source: Denise Maddox. (October 29, 2026). Two Strain Sensors Could Track Hidden Prestress Losses in Concrete Bridges. Scienmag.
