2.1.1. Time Synchronization

The TLS must be able to provide a timestamp, in order to have the acquisition time of each point and assign a time to each surveyed line. The use of the GNSS receiver provided with the TLS, can allow us to express the timestamp as GPS time; in this way the acquisitions obtained with di fferent instruments (videos, point sensors, etc.) can be synchronized and, above all, by positioning a receiver on a mobile load, its positions can be correlated to the acquired deformations.

The TLS are able to scan many lines per second (the instrument used, for example, can scan up to 120 lines per second), so it is possible to assign to a line the time derived from the timestamp of a point of the central zone with the precision of about 0.01 s.

As for the TLS/Video synchronization, we need to compose the approximation of the time assigned to the lines with the frame rate which, typically, is 30 fps. Therefore, we can consider conservatively a mean synchronization error equal to (0.0167<sup>2</sup> + 0.012) 1/2 = 0.0194 sec., and a 95% confidence interval of 0.039 sec. With regard to the TLS/GNSS synchronization, given that the time given by the GNSS receiver can be easily obtained with a precision of 0.001 sec, we can assume a mean error equal to (0.01<sup>2</sup> + 0.0012) 1/2 = 0.010 sec., and a 95% confidence interval of 0.020 sec. The synchronization error involves a positioning error proportional to the speed of the mobile load. For the TLS/GNSS combination, by taking into account a speed of 20 m/sec, an error of ±0.20 m is obtained, with a maximum of ±0.40 m. For the TLS/video pair, an error of ±0.39 m is obtained, with a maximum of ±0.78 m. This error, that can be dramatically reduced by using higher frame rates (60 to 120 fps), should be composed with the error due to the approximation in the positioning of the mobile load in the image, depending on the Ground Sample Distance (GSD). This last error is negligible (a few cm) even for frames taken from a distance of 100 m, given the high resolution of the recent cameras.

Ultimately we can say that the error due to imperfect synchronization of acquisitions produces an error in the positioning of the load on average equal to a few decimetres. This implies a variation of the displacement from 1 to 3 percent for medium-span bridges (from 20 to 50 m) and lower for larger spans.

#### 2.1.2. Use of Theoretical Models to Extract the Elastic Curves

 axis)

From a theoretical point of view, if we have the detailed geometric and physical data, we could obtain analytical models that represent the physical bending of a structure and, in particular, of a beam. For a beam, the governing di fferential equation for the elastic curve, is given by:

$$\frac{\partial^2 z}{\partial \mathbf{x}^2} = \frac{M(\mathbf{x})}{EI} \tag{1}$$

where:

*z* = deflection *x* = abscissa (along the longitudinal

*M* = bending moment;

*E* = Modulus of Elasticity;

*I* = Area moment of Inertia cross-section.

The displacement values are a function of the structural scheme and of the position in which the loads are applied [29]. In many cases, the displacement model can be assimilated to polynomials. With reference to Figure 1, e.g., taking into account a simply supported beam with uniform section, a structural scheme currently used in many bridges, a punctual load produces a displacement given by:

$$\delta\_{\mathbf{x}} = \frac{\text{Fax} \{l^2 - a^2 - \mathbf{x}^2\}}{6lEI} \tag{2}$$

for 0 < *x* < *b*, and by:

$$\delta\_{\mathbf{x}} = \frac{Fb(l-\mathbf{x})\left[l^2 - b^2 - \left(l-\mathbf{x}\right)^2\right]}{6lEI} \tag{3}$$

for *b* < *x* < *l* where:

*F* = Force acting on the beam;

δ*x* = displacement at a distance x from the support 1;

*a* = distance from the load to the support 2;

*b* = distance from the load to the support 1;

*l* = distance between supports.

**Figure 1.** The elastic line of a simply supported uniform cross-section beam subject to a point load.

The generalized form of the cubic polynomial is given in Equations (4) and (5) [30].

$$
\delta \delta\_{\mathbf{x}} = a\_{\overline{3}0} \mathbf{x}^{\overline{3}} + a\_{10} \mathbf{x} + a\_{00} + a\_{01} \mathbf{y} \tag{4}
$$

for 0 < *x* < *b*, and by:

$$\delta\_{\mathbf{x}} = \,^b \mathbf{x}^3 + b \mathbf{\_{20}x}^2 + b \mathbf{\_{10}x} + b \mathbf{\_{00}} + a\_{01} \mathbf{y} \tag{5}$$

for *b* < *x* < *l*, where:

> *y* = horizontal distance from the longitudinal axis.

The best fitting line of the 2D point cloud provided by TLS in line scanner mode will be obtained by finding the coefficients *ai*0 and *bi*0 of the previous formulas with a least-squares procedure. The coefficient *a*01 has been introduced in [30] to take into account rotation about x axis.

## 2.1.3. Structures with Complex Shape

The use of a polynomial interpolation line is not possible for those structures characterized by complex structural patterns, or by non-uniformity in the geometric and physical characteristics of the structural elements, or that have suffered yielding or phenomena such as relaxation and creep.

In this cases structural calculations cannot be based on simple analytical expressions, but are performed through Finite Element Method (FEM); for this aim a 3D survey performed by TLS can be very useful when the project and the *as built* of the structures to be monitored are not available. The discrete values of the displacements obtained through FEM analysis will be interpolated using splines for longitudinal or transverse sections. Splines will be used also to obtain the interpolating lines of the 2D point cloud provided by the TLS in line scanner mode. For new structures, which do not have local irregularities due, for example, to material detachments, it may be considered to eliminate the points with a distance from the spline greater than the precision indicated by the instrument manufacturer (2 sigma). For dated structures, this procedure could worsen the results.
