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IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 2
Ultrasonic tomography in concrete
where:
Pj: wave slowness in element j.
Equation 3 could also be regarded as a summation, as shown in
Equation 4, where n is the total elements.
(4)
T=
Σ
p
j
*dL
j
n
j=1
Each ultrasound reading can be expressed by Equation 4. The
total number of ultrasound readings (m) is represented in Equation
5, which is graphically represented in Figure 3-c.
(5)
Σ
T
i
= p
j
*dL
i,j
n
j=1
(i=1,…,m)
Equation 5 could be expressed in matricial form according to
Equation 6.
(6)
T
m
=D
m,n
*P
n
where:
m: total number of ultrasound readings;
n: total number of elements;
D: a rectangular matrix with m lines and n columns that records the
travel distances of the ultrasound waves in element j, when read-
ing i is performed;
P: a column vector with n lines that records the wave slowness in
each different element j;
of the ultrasound reading is the total travel-time T of the ultrasound
wave, thus the ultrasound velocity is automatically known (MAL-
HOTRA
et al.
[18]).
(1)
V= L T ∴ T= L V
One could divide the section to be analyzed in a grid of n elements
(Figure 3-b) in that case the ultrasonic pulse travels through several
elements at known distances. The total travel path L is the sum of all
distances travelled in each element. Similarly, the sum of the individu-
al time to travel each element is the total travel-time T. Therefore, the
total travel-time T could be determined as presented in Equation 2.
(2)
T=
⎰
1 V
j
R
E
*dL
j
where:
T: total travel-time of the ultrasound wave from the transmitter to
the receiver;
Vj: ultrasound velocity in element j;
dLj: travel distance in element j.
Equation 2 could be rewritten considering the term wave slowness
(p) as the inverse of wave velocity, as expressed in Equation 3
(JACKSON
et al.
[19]).
(3)
⎰
T= p
j R
E
*dL
j
Figura 3 – Representação das leituras ultrassônicas – (a) leitura – (b) leitura com
elementos discretizados – (c) várias leituras com elementos discretizados
A
B
C
