*5.2. Prefabricated Beam*

A scheme of a beam is shown in Figure 23. Its dimensions were 20 cm × 40 cm × 360 cm and it was made of concrete with a mean compression strength after 28 days of *f*cm,cube = 38 MPa. The maximal aggregate diameter was *d*a max = 16 mm. The reinforcement was made of steel with characteristic yield point declared by the manufacturer of *f*yk = 500 MPa. The longitudinal lower reinforcement consisted of four bars with a diameter of 12 mm, top reinforcement of two bars with a diameter of 10 mm and the transverse reinforcement was made of bi-armed stirrups with a diameter of 8 mm with a spacing of 125 mm. As mentioned at the beginning of this point, the beam was damaged during transport and there were three cracks crosswise to its axis, two of which were in the central area of the beam selected for tomographic imaging. The shape and width of this defects are shown in Figure 24. The studies consisted in the measurement of the time of propagation of the longitudinal ultrasound wave using a Pundit-Lab tester and transreceiver heads with a frequency of 54 kHz. The coupling of the heads and beam was provided by special gel for ultrasonic testing. Taking into account the measured longitudinal wave propagation times and basic frequency of the ultrasonic pulses, the average wavelength was ∼ 7 cm and it allowed satisfying the basic requirements described in ASTM D2845-08 regarding the selection of frequency from the point of view of average grain size (as in point 5.1). The adopted system of transmitting/receiving points distant from each other by Δ<sup>p</sup> = 10 cm is shown in Figure 23. Rays were assumed between the opposite points and those lying diagonally at an angle of ∼ 26.6◦ and ∼ 116.6◦ in relation to the beam axis (Figure 23). These angles have been changed from those used in computational examples and experiments on the laboratory beams to shorten to a reasonable minimum the length of diagonal paths taking account of the attenuation of ultrasound signals and to ensure the most correct reading of the longitudinal wave propagation times. In turn, the section for tomographic examinations was the vertical longitudinal plane of symmetry of the system running simultaneously between the longitudinal reinforcement bars. Measured and interpolated longitudinal wave propagation times *t*path int,*<sup>i</sup>* are shown in Figure 25, where interpolated times are determined using a cubic Hermite spline. The diagrams also show *t*ray approx,*<sup>i</sup>* determined according to relation (28) with β = βopt according to formulas (24) and (25). On the other hand, Figure 26 shows an example of a recorded signal by the receiving head together with a reading of the time of longitudinal wave propagation. As in the case of the tests presented in point 5.1, Figure 26 shows first the signal caused by the propagation of the longitudinal wave and then the signal induced by the propagation of transverse and Rayleigh waves of much higher amplitude.

**Figure 23.** Scheme of the beam and the assumed system of transmitting/receiving points and rays.

**Figure 24.** (**a**) Picture of the beam (the places of visible cracks are marked with the slats put on the upper surface of the beam). (**b**) Shape and width of the visible cracks.

**Figure 25.** Propagation times *t*path int,*<sup>i</sup>* and *t*ray approx,*<sup>i</sup>* between the transmitting/receiving points in the beam. The results are presented as a function of the position of the transmitting points: *t*path int,*<sup>i</sup>* grey lines with nodes marked with circles, and *t*ray approx,*<sup>i</sup>* a black dashed line. The top diagram refer to the paths connecting the opposite points and the bottom one to the points lying diagonally at an angle of 63.4◦ and 116.6◦ to each other in relation to the axis *x*.

**Figure 26.** Example of a signal recorded by the receiving head (transmitting point No. 3 and receiving point No. 1).

Figure 27 shows a tomographic reconstruction of the longitudinal wave velocity map in the longitudinal section of the beam which was determined by a randomized Kaczmarz method according to the information presented in point 2. The maximum measured average velocity of a longitudinal wave along all rays, i.e., 3913 m/s, was assumed to be *c*L ref. The resolution of δ<sup>1</sup> × δ<sup>2</sup> = 3.33 cm × 3.33 cm was applied and the arrangement of rays as in Figure 23 with addition of rays between real ones connecting fictitious transmitting/receiving points at a distance of every 6.25 mm. The results are shown only for the central section of the beam separated by a red dashed line through which all types of rays passed due to their inclination. The maps presented here are calculated on the basis of Equation (3) with propagation times *t*ray,*<sup>i</sup>* = *t*ray approx,*<sup>i</sup>* in accordance with (28) for the paths connecting opposite points and *t*ray,*<sup>i</sup>* = *t*path int,*<sup>i</sup>* for the diagonal paths. For this purpose, the values of *t*path int,*<sup>i</sup>* are taken, as shown in Figure 25. The optimal coefficient β necessary for the determination of *t*ray approx,*<sup>i</sup>* was calculated in accordance with formulas (24)–(25) and amounted to 0.25. At the same time *d*cL ray max according to Equation (29) amounted to 0.007, which, in the considered case, proves the lack of significant influence of the generated cracks on the deflecting the fastest ultrasound wave propagation paths. This may also demonstrate the high degree of homogeneity of the concrete in the prefabrication plant.

**Figure 27.** Tomographic reconstructions of the distribution of wave velocity according to Equation (3) in the longitudinal section of the beam (red dotted line—explanation in the text).

In Figure 27, on the left side, clearly formed 2 elastically degraded zones can be seen which were created around the visible cracks (at *x* ≈ −0.6 m and *x* ≈ −0.15 m). In addition, there are also two other zones of this type which can be tomographically observed and were not signaled by visible defects (at *x* ≈ 0.2 m and *x* ≈ 0.5 m) and a few smaller ones, reaching up to about 6 ÷ 9 cm deep into the beam from its lower and upper surfaces. The latter may have been created before the beam was damaged as a result of shrinkage stresses occurring while the element was drying out after dismantling the beam formwork. Based on Equation (20), the maximal change of Young's tangent modulus defined by the min(*E*D/*E*0) ratio at the 0.77 level can be estimated, if one assumes in this case that *c*L0 = *c*L ref (in a zone that goes across the beam at *x* ≈ −0.15 m). The width of the defect at this point is approximately 17 cm. It is very close to the width of the elastically degraded zone, which was calculated theoretically in point 3. For the min(*E*D/*E*0) = 0.77 ratio, this corresponds to a damaged area width of approximately 18 cm, which can be read from Figure 5b or Figure 6b. This result also indirectly pre-confirms the validity of the identification method for the internal length *l*c of concrete proposed in point 3.
