PSI - Issue 84

Marco Pirrò et al. / Procedia Structural Integrity 84 (2026) 552–559

553

1. Introduction In bridge assessment, the deflection is usually considered as a basic parameter that must be measured because it is closely related to the bridge bearing capacity and changes in the deflection associated to target loads indicate the occurrence of structural changes and damages. In many cases, bridge deflections are measured directly by displacement meters, which can provide with stable and accurate results: Linear Variable Displacement Transducers (LVDT) are widely adopted for this purpose (Cavadas et al., 2013), but direct measurements cannot apply to bridges crossing rivers or deep valleys. Other deflection measurement methods (see, e.g., Huang et al., 2022) mainly rely on dial gauges, precision levels, digital levels, total stations, global positioning system (GPS) and microwave radars. Among those techniques, the radar interferometry (Gentile and Bernardini, 2010) can be applied to remote sensing of bridges’ displacements in both static and dynamic conditions. In recent years, indirect measurements of displacement by means of inclinometers have been employed, as these devices exhibit low cost, can be always used and do not require any fixed observation positions (Yang and Ma, 2002); in addition, the measurements are not affected by climatic conditions. In (Sanli et al., 2000), the authors use inclinometers with capability for operating under adverse thermal and high humidity environments in live load tests of the Drweca River steel girder bridge: the deflection evaluation involves the use of cubic spline and the corresponding results compare well with the results from LVDT measurements. In (Helmi et al., 2015), 3 bridges were instrumented with inclinometers and strain gauges (i.e., two spans of a pre-stressed concrete box girder bridge in California and the concrete slab on steel girder bridge in Illinois): again, the indirect deflection measurements were in good agreement with the reference provided by conventional approaches. Furthermore, the deflection time histories of the Taolaizhao bridge at the controlling position of a photoelectric gauge (Hou et al., 2005) are computed and verified with experimental data. The paper presents two methods for estimating bridge deflection using biaxial inclinometers. The first approach involves interpolating and integrating the longitudinal rotations measured on each side of the bridge to estimate the vertical deflections (after properly applying the boundary conditions, such as zero displacement at supports); subsequently, the vertical displacements of the two sides are averaged to get the mid-line curve. On the contrary, the second approach firstly averages the longitudinal rotations on the opposite sides of each instrumented cross-section to establish the mid-line deflection; afterwards, the measured transverse rotations are used to estimate the torsion component of the vertical displacements at the opposite sides of the deck. The accuracy of the proposed procedure is verified in the paper by checking the deflections measured using conventional LVDTs at some locations during load tests on a 3-span overpass. The load tests have involved centered and eccentric positions of the loading trucks along the bridge, to attain about 75% of the bending/torsion capacity of the bridge. Furthermore, since inclinometers are suitable for the application in long-term monitoring of bridges, an automated strategy for real time monitoring of bridge is defined: when local rotations exceed pre-selected threshold values (that can be estimated during preliminary experimental verification), the corresponding deformed shape is evaluated in real time from measured rotations so that comparison with numerical (FE) simulations might provide with useful indications on the structural condition. 2. General procedure for estimating deflections Let us assume that the rotation  ( x ) measured at point x of the beam represented in Fig. 1 is represented by the following polynomial function of order n :

(1)

 ( x ) = a 1 x

n + a

n–1 +…+ a

2 x

n x + a n+1

where a i are unknown coefficients. Equation 1 allows to define similar expressions also for the curvature  ( x ) and the deflection  ( x ):

(2)

n–1 + (n–1)a

n–2 +…+ a

 (x) =  (x) = na 1 x

2 x

1 x + a n