PSI - Issue 10

P.A. Kakavas-Papaniaros et al. / Procedia Structural Integrity 10 (2018) 311–318 P.A. Kakavas-Papaniaros et al. / Structural Integrity Procedia 00 (2018) 000 – 000

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conventional static compressive stress tests. With f c known, the expected value of the longitudinal ultrasonic wave propagation velocity, u m,exp , was obtained using Eqs.2(b-d). The results of this process are summarized in Table 1.

Table 1. Calculation of the longitudinal ultrasonic wave propagation velocity, u m,exp , using compressive strength test results. Density, ρ (kg/m 3 ) 2312.00 Strength, f c (MPa) 23.02 Static modulus of elasticity, E st (Eq. (2c), GPa) 22.69 Dynamic modulus of elasticity, E d (Eq. (2b), GPa) 27.34 Expected ultrasound velocity, u m,exp (Eq. (2d), km/sec) 3.62

The empirical coefficient, n , can be calculated using Eq. (7), modified as follows:

  

   

u

log

m

,



n

2 1   

(8)

u

m

,exp

The value of u θ ,m required in Eq. (8) was calculated from the proposed method as described in the previous section. The method was applied using wooden inserts as shown in Fig.3. The ultrasonic wave propagation velocities along the principal directions were obtained from ultrasound measurements on cubic specimens of the same wood, which led to u 0 = 4762 m/sec and u 90 = 2046 m/sec. These values are in agreement with those proposed in Xu et al. (2014). Assuming that θ = 45 ο , Eq.(3) led to u θ ,w = 2862 m/sec. Given the dimensions of the wooden prisms (d 1 = 6cm, Fig. (2)), the ultrasonic wave “travel - distance” in each insert was calculated as d w = 3 cm. The respective “travel -time s” were then obtained using Eq.(4), t w = 10.48 μ sec. The total ultrasound “travel - time” was measured as t tot = 53.20 μ sec; hence for the concrete element we obtain t m = 32.24 μ sec. The results of the last steps of the procedure, leading to u θ ,m (from Eq. (6)) and to an estimation for the empirical coefficient, n , are presented in Table 2. With the value of n for masonry defined, it is possible to apply the proposed setup and the calculation procedure directly in order to obtain the strength of thick structural elements or masonries. However, it should be stressed that the value of n in Table 2 should be used with caution, as it is valid only for masonries with material properties similar to those used in the conducted experiments. Widely applicable values of n can be obtained by further experimental and theoretical investigation of ultrasonic wave propagation in an extended typology of concrete elements and masonries.

Table 2. Results of the experimental procedure for the calibration of the empirical coefficient, n . Prism location, d 2 (mm) 75.00 Ultrasonic wave “travel distance”, d c (mm) 106.00 Respective “travel - time”, t m (Eq.(5), μ sec) 32.24 Ultrasound velocity, u θ , μ (Eq.(6), km/sec) 3.29 Empirical coefficient, n (Eq.(8)) 3.85

3. Consideration of wave scattering using slowness curves

The previously discussed procedure was based on several arbitrary assumptions. E.g. in order to simplify the re quired calculation steps, it was assumed that θ (m) = θ (w) . Snell’s law relates the direction of propagation of the ultra sonic wave in the masonry with that in the prisms, θ (w) , and the respective velocities u θ ,m and u θ ,w as follows (Nayfeh (1995)):

( ) w

( ) m

sin

sin





(9)



u

u

w

m

,

,



