Neural Computing and Applications (2020) 32:11807–11826
https://doi.org/10.1007/s00521-019-04663-2
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ORIGINAL ARTICLE
Concrete compressive strength using artificial neural networks
Panagiotis G. Asteris1
•
Vaseilios G. Mokos1
Received: 19 June 2019 / Accepted: 3 December 2019 / Published online: 10 December 2019
Ó Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract
The non-destructive testing of concrete structures with methods such as ultrasonic pulse velocity and Schmidt rebound
hammer test is of utmost technical importance. Non-destructive testing methods do not require sampling, and they are
simple, fast to perform, and efficient. However, these methods result in large dispersion of the values they estimate, with
significant deviation from the actual (experimental) values of compressive strength. In this paper, the application of
artificial neural networks (ANNs) for predicting the compressive strength of concrete in existing structures has been
investigated. ANNs have been systematically used for predicting the compressive strength of concrete, utilizing both the
ultrasonic pulse velocity and the Schmidt rebound hammer experimental results, which are available in the literature. The
comparison of the ANN-derived results with the experimental findings, which are in very good agreement, demonstrates
the ability of ANNs to estimate the compressive strength of concrete in a reliable and robust manner. Thus, the (quantitative) values of weights for the proposed neural network model are provided, so that the proposed model can be readily
implemented in a spreadsheet and accessible to everyone interested in the procedure of simulation.
Keywords Artificial neural networks Compressive strength Concrete Non-destructive testing methods Soft computing
List of symbols
B
Vector of bias values
fc
Compressive strength of concrete
IW Matrix of weights values for input layer
LW Matrix of weights values for hidden layer
R
Rebound hammer
Vp
Ultrasonic pulse velocity
Abbreviations
ANNs Artificial neural networks
BP
Back propagation
RH
Rebound hammer
UPV
Ultrasonic pulse velocity
& Panagiotis G. Asteris
asteris@aspete.gr; panagiotisasteris@gmail.com
1
Computational Mechanics Laboratory, School of Pedagogical
and Technological Education, 14121 Heraklion, Athens,
Greece
1 Introduction
Assessment of the bearing capacity of existing concrete
structures is an important issue, which is attracting the
interest of researchers, especially in recent years. Before
any design or intervention action is carried out, it is necessary to investigate and document the existing concrete
structure to a sufficient extent and depth, so as to obtain the
maximum amount of data with high reliability, on which to
base the assessment or redesign. This involves surveying
the structure and assessing its condition, recording any
wear or damage as well as conducting on site investigation
works and measurements. Notice that the tests available for
testing concrete range from (a) completely non-destructive,
(b) partially destructive tests, and (c) destructive tests, for
which the concrete surface has to be repaired after the test.
Measurements for concrete strength are usually performed
with non-destructive methods, as they do not require
destructive sampling, while their usage is simple and quick.
The investigation of concrete aims mainly at determining
the compressive strength for each area of the structure.
Other properties, such as modulus of elasticity, tensile
strength, etc., can be determined indirectly based on
compressive strength. The expected systematic differentiation of concrete strength must be taken into account,
123
11808
depending on its characteristic position in the structure, and
the conditions of concreting, compaction and maintenance.
It is possible that there are significant differences in
strength between slabs, beams, upper and lower parts of
columns, while in cases of poor workmanship in column
concreting, it cannot be ruled out that the lower part may
also develop lower strength due to segregation and
cavitations.
Non-destructive testing can be applied to both old and
new structures. For new structures, the principal applications focus on quality control and the resolution of issues
related to the quality of materials or construction. Testing
existing structures is focused on the assessment of structural integrity or adequacy. The ultrasonic pulse velocity
method and the rebound hammer test are the most commonly used non-destructive techniques for the estimation
of mechanical characteristics of concrete, both in the laboratory and in situ. In the international literature, a number
of relationships have been proposed via which the compressive strength of concrete is correlated with the speed of
ultrasound and the rate of bounce in case of rebound
hammer test. The main drawback of these methods is the
large dispersion of the values they predict, and the significant deviation from the actual (experimental) value of the
compressive strength of the concrete. The lack of appropriate and reliable empirical relationships to estimate the
compressive strength of concrete has attracted the interest
of researchers—over the past decade—toward the application of non-deterministic techniques.
Although research has been mainly concerned with the
determination of compressive strength values through nondestructive techniques, both on a theoretical and experimental levels, this important issue still remains unresolved.
The latter observation is manifested via various facts; most
available proposals result in the estimation of different
values, while predicted values are almost always found
either to be overestimated or underestimated in relation to
experimental values. This fact can be attributed to the
nonlinear behavior which governs the influence of the
parameters on a concrete material’s compressive strength.
Thus, the use non-deterministic techniques, such as soft
computing techniques, is of utmost importance in order to
achieve an optimum solution and reveal the complex
influence of each parameter (while at the same time taking
all parameters into account).
An artificial neural network (ANN) is a computational
model that is inspired by the biological neural networks in
the human brain, which process information. Neural networks are capable of ‘‘learning’’ and correlating large
datasets obtained from experiments or simulations. The
trained neural network serves as an analytical tool for
qualified prognoses of the actual results. There are efficient
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Neural Computing and Applications (2020) 32:11807–11826
methods for their training and validation, and they can
yield high accuracy scores in their predictions.
In this paper, the application of ANNs for predicting the
compressive strength of concrete structure has been
investigated. For the training of the ANN models, an
experimental database, based on ultrasonic pulse velocity
and Schmidt rebound hammer experimental results (available in the literature) has been utilized, in conjunction with
respective compressive strength tests, conducted on cores
of the same sample. The good comparison of the ANNderived results with the experimental findings and the
theoretical results demonstrates the ability of ANNs to
approximate the compressive strength of concrete, based
on non-destructive technique results, and in particular
ultrasonic pulse velocity measurements and Schmidt
hammer rebound tests, in a reliable and robust manner.
A MATLAB-Program has been developed, and representative examples have been studied in order to demonstrate
the efficiency, the accuracy and the range of applications of
the developed method. In addition to the architecture of the
proposed optimum neural system, a supplementary materials section is included which provides a simple design/
education tool which can assist both in teaching, as well as
the estimation of concrete compressive strength based on
non-destructive testing, By including all the necessary
information, anyone can test the proposed model. Additionally to the reliability it ensures, including this information also provides the means for other researchers,
students or field practitioners to further test the reliability
of the proposed model.
2 Literature review
The rebound hammer (RH) and the ultrasonic pulse
velocity (UPV) comprise the most utilized non-destructive
methods in determining the compressive strength of concrete. The results obtained by the RH and UPV tests can be
influenced by a sufficiently large set of factors, depending
on the investigated concrete element.
The rebound (Schmidt) hammer is one of the oldest and
best-known methods. It is used in comparing the concrete
in various parts of a structure and indirectly in assessing the
concrete strength. The rebound of an elastic mass depends
on the hardness of the surface against which its mass
strikes. The results of rebound hammer are significantly
influenced by factors such as: the smoothness of the test
surface; the size, the shape, and the rigidity of the specimens; the age of the specimen; the surface and internal
moisture conditions of the concrete; the type of coarse
aggregate; the type of cement; and the carbonation of
concrete surface [1]. It is worth noticing that the hammer
method can be used in the horizontal, vertically overhead
Neural Computing and Applications (2020) 32:11807–11826
or vertically downward positions as well as at any intermediate angle, provided that the hammer is perpendicular
to the surface under test. However, the position of the mass
relative to the vertical affects the rebound number due to
the action of gravity on the mass in the hammer. This
method has gained popularity due to its simple use and the
possibility of using it on a single concrete surface without
requiring access to the construction from both sides (which
is necessary for direct ultrasonic testing methods).
The ultrasonic pulse velocity (UPV) method is one of
the most popular non-destructive techniques used in the
assessment of concrete properties. The use of UPV to the
non-destructive assessment of concrete quality has been
extensively investigated for decades. Ultrasonic pulse
velocity testing of concrete is based on the pulse velocity
method to provide information on the uniformity of concrete, cracks and defects. The pulse velocity in a material
depends on its density and its elastic properties, which in
turn are related to the quality and the compressive strength
of the concrete. This test method is applicable to assess the
uniformity and relative quality of concrete, to indicate the
presence of voids and cracks, and to evaluate the effectiveness of crack repairs. It is also applicable to indicate
changes in the properties of concrete, and in the survey of
structures, to estimate the severity of deterioration or
cracking. However, it is difficult to accurately evaluate the
concrete compressive strength with this method since UPV
values are affected by a number of factors, which do not
necessarily influence the concrete compressive strength in
the same way or to the same extent [2]. The UPV test is
described in ASTM C 597-83 [3] and BS 1881-203 [4] in
detail.
It is worth noticing that a plethora of experimental data
and theoretical correlation relationships between compressive concrete strength and pulse velocity (as well as the
rebound hammer results) have been presented and proposed. Table 1, suggested by Whitehurst [5], shows how
the values of the obtained velocity can be utilized to
classify the quality of concrete. In addition, in Table 2 the
most accepted empirical relationships in the international
literature [2, 6–13], which allow the compressive strength
Table 1 Quality of concrete with respect to ultrasonic pulse velocity
[5]
Quality of concrete
Ultrasonic pulse velocity Vp in m/s
Above 4500
Excellent
3500–4500
Generally good
3000–3500
Questionable
2000–3000
Generally poor
Below 2000
Very poor
11809
of the concrete to be assessed by using (a) ultrasound rate
measurements Vp , (b) rebound (Schmidt) hammer results
R and (c) a combination of measurements by both methods
Vp ; R , are presented. Detailed and in-depth state-of-theart report can be found in the works of Erdal [13],
Mohammed and Rahman [14], Alwash et al. [15] as well as
in PhD thesis of Alwash [16].
Figures 1 and 2 present a comparison between some of
the aforementioned expressions (Table 2) for the evaluation of the concrete compressive strength. Especially,
Fig. 1 presents a comparison with respect to ultrasonic
pulse velocity value [single-variable equations (E1)–(E6)],
while Fig. 2 with respect to rebound hammer [singlevariable equations (E7)–(E9)]. Furthermore, Fig. 3 shows
the contours of the aforementioned expressions (Table 2)
for the evaluation of the concrete compressive strength
using the multi-variable equations (E10)–(E14). From
Figs. 1, 2 and 3, it is obvious that the concrete compressive
strength calculated based on these expressions shows
considerable variation, revealing the need for further
investigation and refinement of the proposals. The purpose
of this paper is to improve this situation through the use of
appropriate neural networks.
3 Materials and methods
3.1 Artificial neural networks
Artificial neural networks (ANNs) are models which process information and make predictions. They are designed
in order to learn from the available experimental or analytical/theoretical data. Such models have the ability to
classify data, predict values, and assist in decision-making
processes. A trained ANN achieves mapping of input
parameters onto a specific output; it presents similarities to
a response surface method. A trained ANN can achieve
more reliable results over conventional numerical analysis
procedures (e.g., regression analysis) and, furthermore, it is
produced with considerably less computational effort
[17–31].
ANNs work in a manner similar to that of the biological
neural network of the human brain [32–44]. The artificial
neuron is the basic building block of an ANN; in fact, it is a
mathematical model which attempts to mimic the behavior
of the biological neuron (Fig. 4).
Input data is passed into the artificial neuron and, after
being processed via a mathematical function, it can lead to
an output (same as the biological neuron’s fire-or-not situation). Before the input data enters the neuron, weights
are attributed to the input parameters in order to mimic the
random nature of the biological neuron. The ANN is
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Neural Computing and Applications (2020) 32:11807–11826
Table 2 Empirical relationships for estimating compressive strength of concrete ðfc Þ
Parameters
Vp
R
Equation
fc Vp ¼ 1:146e0:77Vp
fc Vp ¼ 1:119e0:715Vp
fc Vp ¼ 0:0854e1:288Vp
2
fc Vp ¼ 176:9 96:467Vp þ 13:906 Vp
1:7447
fc Vp ¼ 1:2 105 1000Vp
fc Vp ¼ 36:73Vp 129:077
Nr.
References
(E1)
Turgut [6]
(E2)
Nash’t et al. [7]
(E3)
Trtnik et al. [2]
(E4)
Logothetis [8]
(E5)
Kheder [9]
(E6)
Qasrawi [10]
fc ðRÞ ¼ 9:40 þ 0:52R þ 0:02R2
(E7)
Logothetis [8]
1:2083
VP ; R
fc ðRÞ ¼ 0:4030R
fc ðRÞ ¼ 1:353R 17:393
fc Vp ; R ¼ e1:78 lnðVp Þþ0:85 lnðRÞ0:02 0:0981
fc Vp ; R ¼ 18:6e0:515Vp þ0:019R 0:0981
2
fc Vp ; R ¼ ð0:10983 þ 0:00157R 0:79315 Vp =10 0:00002R2 þ 1:29261 Vp =10 Þ 103
fc Vp ; R ¼ 0:42R þ 13:166Vp 40:255
0:4254 1:1171
fc Vp ; R ¼ 0:0158 1000Vp
R
(E8)
Kheder [9]
(E9)
(E10)
Qasrawi [10]
Logothetis [8]
(E11)
Arioglu et al. [11]
(E12)
Amini et al. [12]
(E13)
Erdal 2009 [13]
(E14)
Kheder [9]
In the above relations, Vp in (km/s), R (rebound number) and fc in MPa
Fig. 1 Comparison of equations related to ultrasonic pulse velocity
for the evaluation of concrete compressive strength
consisted of these neurons in the same way as biological
neural networks work. Three main phases are necessary in
order to set up an ANN: (1) the ANN’s architecture must
be established; (2) the appropriate training algorithm,
which is necessary for the ANN learning phase, must be
defined; and (3) the mathematical functions describing the
mathematical model must be determined. The architecture
or topology of the ANN is related and describes the manner
in which the artificial neurons are organized in the group,
as well as the manner with which information flows within
123
Fig. 2 Comparison of equations related to rebound hammer for the
evaluation of concrete compressive strength
the network. When neurons are organized in more than one
layer, the network is called a multilayer ANN. The training
phase of the ANN is considered crucial; it is a function
minimization problem, where an error function is minimized, thus assisting in the determination of the optimum
value of weights. Different ANNs use different optimization algorithms for this purpose. The summation functions
and the activation functions are mathematical functions
that define the behavior of each neuron. In the present
study, a back propagation neural network (BPNN) is
implemented and described in the following section.
Neural Computing and Applications (2020) 32:11807–11826
30
30
fc(MPa)
28
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
26
25
24
23
22
29
fc(MPa)
29
28
70
28
27
100
55
26
50
45
25
40
35
24
30
23
90
26
80
70
25
60
24
50
40
23
20
30
15
22
20
6
20
22
10
10
21
20
20
5
110
27
60
21
4
120
25
21
3
fc(MPa)
65
R (Rebound Number)
27
30
R (Rebound Number)
29
R (Rebound Number)
11811
3
4
5
Vp(km/m)
Vp(km/m)
E10
E11
3
6
4
5
6
Vp(km/m)
E12
30
30
29
fc(MPa)
29
28
50
28
fc(MPa)
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
46
42
38
26
34
30
25
26
24
22
18
23
14
10
22
27
R (Rebound Number)
R (Rebound Number)
27
26
25
24
23
22
6
21
21
20
20
3
4
5
6
3
4
5
Vp(km/m)
Vp(km/m)
E13
E14
6
Fig. 3 Contours of the concrete compressive strength using multi-variable equations
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Neural Computing and Applications (2020) 32:11807–11826
Fig. 4 Schematic representation
of biological neuron structure
3.2 Architecture of BPNN
A BPNN is a feed-forward, multilayer [38] artificial neural
network. In BPNN, information flows from the input
toward the output with no feedback loops. Furthermore, the
neurons of the same layer are not connected to each other;
however, they are connected with all the neurons of the
previous and subsequent layer. A BPNN has a standard
structure. This structure can be described as
N H1 H2 HNHL M
ð1Þ
where N is the number of input neurons (input parameters);
Hi is the number of neurons in the i-th hidden layer for
i ¼ 1; . . .; NHL; NHL is the number of hidden layers; and
M is the number of output neurons (output parameters). For
example, the code 3-5-2-1 means that the model has an
architecture composed of an input layer consisting of 3
neurons, two hidden layers with 5 and 2 neurons, respectively, and an output layer with 1 neurons, i.e., a 3-5-2-1
BPNN.
Although multilayer NN models are most frequently
proposed by researches, it should be noted the ANN
models with only one hidden layer are capable to predict
any forecast problem in a reliable manner.
In Fig. 5, a notation for a single node (with the corresponding R-element input vector) of a hidden layer is
presented.
For each neuron i, the individual element inputs
p1 ; . . .; pR are multiplied by the corresponding weights
wi;1 ; . . .; wi;R . The weighted values are fed to the junction of
the summation function, in which the dot product (W p) of
the weight vector W ¼ wi;1 ; . . .; wi;R and the input vector
p ¼ ½p1 ; . . .; pR T is generated. The threshold b (bias) is
added to the dot-product forming the net input n, which is
the argument of the transfer function f:
n ¼ W p ¼ wi;1 p1 þ wi;2 p2 þ þ wi;R pR þ b
123
ð2Þ
Fig. 5 A neuron with a single R-element input vector
The transfer (or activation) function f which is chosen
may influence the complexity and performance of the ANN
to a great degree. Even though sigmoidal transfer functions
are the transfer functions most commonly come across,
different types of function may be used and may even be
more appropriate for certain problems. A large variety of
alternative transfer functions have been proposed in the
relevant literature [45, 46]. In this study, the Logistic
Sigmoid and the Hyperbolic Tangent transfer functions
were found to be the most appropriate accurately determining the issue investigated. The training data are fed into
the network during the training phase; during this phase,
the ANN achieves a mapping between the input values and
Neural Computing and Applications (2020) 32:11807–11826
11813
Table 3 Database
No
Sample
(1)
Batch
Nr.
(2)
1
1
Table 3 (continued)
Input
parameters
Specimen
Nr.
(3)
Vp
(km/s)
(4)
Output
parameter
R
fc (MPa)
(5)
(6)
Dataset
No
Sample
(7)
(1)
Batch
Nr.
(2)
Input
parameters
Output
parameter
Dataset
Specimen
Nr.
(3)
Vp
(km/s)
(4)
R
fc (MPa)
(5)
(6)
(7)
1
4.61
26.40
20.20
T
46
4
4.55
33.00
36.19
T
2
2
4.44
28.50
19.91
T
47
5
4.52
32.60
30.30
T
3
3
4.55
27.00
20.30
V
48
6
4.55
33.80
35.30
Test
4
4
4.58
26.60
22.16
T
49
1
4.60
28.00
22.16
T
5
5
4.62
27.00
20.89
T
50
2
4.52
28.00
22.26
T
6
4.51
26.80
21.67
Test
51
3
4.55
30.00
23.63
V
1
4.32
26.00
18.63
T
2
4.35
25.50
20.10
T
52
53
4
5
4.50
4.51
28.00
28.90
22.26
22.36
T
T
9
3
4.35
27.00
20.40
V
54
6
4.51
27.50
22.16
Test
10
11
4
5
4.37
4.37
26.00
26.10
19.22
21.57
T
T
55
1
4.62
31.00
27.85
T
56
2
4.67
31.20
27.26
T
12
6
4.32
27.30
19.22
Test
57
3
4.60
30.20
25.10
V
1
4.18
22.80
15.00
T
58
4
4.62
30.00
28.44
T
14
2
4.25
25.00
17.36
T
59
5
4.65
30.80
26.77
T
15
3
4.19
23.00
15.30
V
60
6
4.54
31.00
29.03
Test
16
4
4.20
24.50
15.89
T
61
1
4.74
28.60
23.63
T
17
5
4.13
23.80
14.22
T
62
2
4.70
30.70
25.69
T
6
4.26
24.00
16.67
Test
63
3
4.48
28.60
23.05
V
1
4.76
32.20
32.36
T
64
4
4.51
28.50
25.20
T
20
2
4.76
32.00
30.60
T
65
5
4.55
29.40
24.03
T
21
3
4.88
30.00
35.79
V
66
6
4.54
29.20
25.50
Test
22
4
4.78
31.80
31.58
T
23
5
4.73
32.50
30.89
T
67
68
1
2
4.69
4.73
30.00
30.00
28.44
25.30
T
T
6
4.77
32.00
32.36
Test
69
3
4.77
30.00
28.05
V
1
2
4.88
4.80
32.80
33.60
34.62
34.62
T
T
70
4
4.73
29.40
28.14
T
71
5
4.69
30.60
27.85
T
27
3
4.88
33.00
35.50
V
72
6
4.77
30.30
27.36
Test
28
4
4.85
33.80
35.50
T
73
1
4.65
25.10
17.16
T
29
5
4.80
33.50
34.32
T
74
2
4.48
26.00
19.61
T
6
4.85
34.00
37.46
Test
75
3
4.51
25.40
16.67
V
1
4.61
30.00
28.14
T
76
4
4.58
24.40
16.67
T
32
2
4.65
30.50
27.07
T
77
5
4.54
26.00
18.34
T
33
3
4.66
31.00
30.11
V
78
6
4.54
26.00
19.12
Test
34
4
4.66
30.80
28.14
T
79
1
4.48
28.60
22.16
T
35
5
4.64
29.00
28.44
T
80
2
4.52
28.00
23.14
T
6
4.64
31.00
28.44
Test
81
3
4.45
28.30
22.95
V
1
4.55
30.50
27.46
T
38
2
4.65
30.00
28.64
T
82
83
4
5
4.48
4.48
26.30
29.10
23.05
22.36
T
T
39
3
4.62
31.90
28.14
V
84
6
4.55
27.40
22.36
Test
40
4
4.62
30.80
28.93
T
85
1
4.45
26.60
19.61
T
41
42
5
6
4.65
4.62
31.30
30.80
29.62
29.52
T
Test
86
2
4.45
24.60
16.67
T
87
3
4.44
25.50
17.65
V
1
4.55
30.10
29.62
T
88
4
4.44
25.40
18.34
T
44
2
4.55
30.70
35.50
T
89
5
4.46
26.40
17.85
T
45
3
4.55
32.40
34.81
V
90
6
4.53
25.40
20.10
Test
6
7
2
8
13
3
18
19
4
24
25
26
5
30
31
6
36
37
43
7
8
9
10
11
12
13
14
15
123
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Neural Computing and Applications (2020) 32:11807–11826
Table 3 (continued)
No
Sample
(1)
Batch
Nr.
(2)
91
16
Specimen
Nr.
(3)
Table 3 (continued)
Input
parameters
Output
parameter
Vp
(km/s)
(4)
R
fc (MPa)
(5)
(6)
Dataset
No
Sample
(7)
(1)
Batch
Nr.
(2)
Input
parameters
Output
parameter
Dataset
Specimen
Nr.
(3)
Vp
(km/s)
(4)
R
fc (MPa)
(5)
(6)
(7)
1
4.63
29.70
24.32
T
136
5
4.10
26.20
14.42
T
92
2
4.58
29.30
24.52
T
137
6
3.85
22.90
14.81
T
93
3
4.77
30.80
25.20
V
138
1
5.05
38.20
43.25
Test
94
4
4.66
29.70
26.87
T
139
2
5.05
37.20
40.31
T
95
5
4.62
29.00
26.87
T
140
3
5.10
39.00
43.44
T
6
4.69
30.00
26.18
Test
141
4
5.10
38.00
39.32
V
1
2
4.17
4.15
23.00
23.50
14.51
16.18
T
T
142
143
5
6
5.05
5.05
36.80
37.10
42.27
41.58
T
T
99
3
4.19
23.50
14.42
V
144
1
4.80
36.00
47.56
Test
100
4
4.08
23.10
14.22
T
145
2
4.88
34.40
42.95
T
101
5
4.12
24.00
14.91
T
146
3
4.93
35.40
40.70
T
102
6
4.17
22.00
15.20
Test
147
4
4.81
34.60
41.97
V
1
4.03
21.60
12.85
T
148
5
4.92
35.20
44.33
T
104
2
3.94
20.50
12.16
T
149
6
4.92
35.50
43.64
T
105
3
3.90
21.80
12.45
V
150
1
5.22
41.50
50.21
Test
106
4
3.93
21.10
12.45
T
151
2
5.22
41.50
50.01
T
107
5
3.95
20.00
12.65
T
152
3
5.18
40.60
49.33
T
108
6
3.95
22.00
14.32
Test
153
4
5.22
42.00
50.01
V
2
4.35
24.90
15.79
T
154
5
5.22
40.00
45.40
T
110
3
4.36
25.80
16.18
T
155
6
5.22
41.40
50.01
T
111
4
4.35
24.30
15.89
V
156
1
5.00
41.00
50.41
Test
112
113
5
6
4.31
4.40
24.40
25.60
15.30
16.38
T
T
157
158
2
3
4.95
4.92
41.00
40.80
52.17
50.80
T
T
96
97
98
103
109
114
17
18
19
20
24
25
26
27
1
4.20
22.20
15.40
Test
159
4
5.00
41.00
50.01
V
115
2
4.17
22.00
14.42
T
160
5
4.98
40.00
49.13
T
116
3
4.22
21.90
14.32
T
161
6
4.99
41.20
50.80
T
117
4
4.20
22.10
15.10
V
162
1
4.96
38.90
42.95
Test
118
5
4.17
21.80
15.49
T
163
2
4.96
37.30
40.99
T
6
4.14
21.90
15.00
T
164
3
4.96
38.50
41.68
T
1
4.35
27.20
20.10
Test
165
4
4.96
38.30
40.99
V
121
2
4.35
28.40
19.81
T
166
5
4.96
37.30
41.19
T
122
3
4.45
27.90
20.89
T
167
6
5.00
37.00
41.78
T
123
4
4.48
28.60
22.85
V
168
1
4.92
36.20
39.91
Test
124
5
4.45
29.20
22.56
T
169
2
4.95
36.20
39.72
T
6
4.41
29.00
21.08
T
170
3
4.80
37.30
38.25
T
1
4.62
28.70
23.05
Test
171
4
4.80
38.10
40.40
V
127
128
2
3
4.55
4.58
28.00
29.80
24.22
23.54
T
T
172
173
5
6
4.78
4.85
37.20
35.20
39.72
39.72
T
T
129
4
4.62
28.40
24.52
V
174
1
4.77
36.10
38.83
Test
130
5
4.65
29.00
23.83
T
175
2
4.77
37.30
41.19
T
6
4.55
29.40
23.73
T
176
3
4.77
37.10
39.72
T
1
4.38
28.10
21.38
Test
177
4
4.73
36.80
38.05
V
133
2
4.45
28.20
20.99
T
178
5
4.77
37.10
39.42
T
134
3
4.40
28.60
21.67
T
179
6
4.88
36.10
40.21
T
135
4
4.45
28.00
24.42
V
180
1
4.84
33.00
31.87
Test
119
120
21
125
126
22
131
132
23
123
28
29
30
31
Neural Computing and Applications (2020) 32:11807–11826
11815
Table 3 (continued)
No
Sample
Input
parameters
Output
parameter
(1)
Batch
Nr.
(2)
Specimen
Nr.
(3)
Vp
(km/s)
(4)
R
fc (MPa)
(5)
(6)
(7)
181
2
4.75
33.90
33.05
T
182
3
4.72
33.80
33.93
T
183
4
4.75
34.60
33.93
V
184
5
4.72
33.30
34.81
T
185
6
4.75
33.80
34.81
T
186
33
Dataset
3.3 Experimental—database
1
4.52
30.00
26.38
Test
187
188
2
3
4.42
4.45
30.00
30.90
25.01
26.97
T
T
189
4
4.45
30.40
26.28
V
190
5
4.45
30.90
26.58
T
6
4.45
30.50
25.99
T
1
4.52
28.30
26.09
Test
193
2
4.48
30.00
26.58
T
194
3
4.45
30.00
27.46
T
195
4
4.45
29.90
27.46
V
196
5
4.45
30.90
27.46
T
197
6
4.45
31.00
27.46
T
1
4.26
28.00
22.85
Test
199
2
4.28
28.30
23.14
T
200
3
4.28
27.00
22.36
T
201
4
4.32
28.70
23.54
V
202
203
5
6
4.32
4.11
28.20
29.00
23.34
24.52
T
T
191
192
198
204
34
35
36
1
4.29
27.00
22.46
Test
205
2
4.26
28.50
22.16
T
206
3
4.20
28.20
21.38
T
207
4
4.29
29.60
22.85
V
208
5
4.29
28.00
23.54
T
209
6
4.29
28.00
23.73
T
the respective output values, by adjusting the weights in
order to minimize the following error function:
X
E¼
ðxi yi Þ2
ð3Þ
Table 4 The input and output
parameters used in the
development of BPNNs
where xi and yi are the measured and the prediction values
of the network, respectively, within an optimization
framework.
Variable
An extended and reliable database is a prerequisite for the
successful function of any artificial neuron network. The
database must be capable of training the system; it should
contain datasets which cover the whole range of possible
values of the parameters influencing the problem under
examination. Compiling such a database comes with a
number of difficulties. For one, it is impossible for one
researcher alone to obtain an amount of experimental data
large enough to be capable of adequately training the ANN.
Another issue is related to the reliability of available data,
as the optimum developed network is trained by the database; thus, if data is incorrect, the trained system will not
be able to predict correct values, thus confirming the
Garbage In, Garbage Out (GIGO) expression. Predictive
analytics demands good data. A higher amount of data can
be harmful, if the data is not correct. In order to predict one
requires, above all, accurate data. The optimization of the
ANNs proposed for the prediction of the compressive
strength of concrete is based on the experimental datasets
presented briefly in the following paragraphs.
An experimental database consists of experimental data
sets available in the literature has been prepared. Namely,
the database consists of 209 datasets based on experimental
results from the PhD thesis by Logothetis [8]. Leonidas
Logothetis under the supervision of Prof. Theodosios
Tassios, in order to support his PhD thesis entitled
‘‘Combination of three non-destructive methods for the
determination of the strength of concrete’’ at the National
Technical University of Athens, Athens, Greece, prepared
and tested 36 batches of cubic concrete specimens. Each
one from the 36 batches (except one) consisted of 6
specimens. Each specimen was initially measured through
non-destructive techniques; first ultrasonic pulse velocity
measurements were conducted and then Schmidt hammer
rebound tests. After the non-destructive measurements
were concluded, each specimen was subjected under uniaxial compressive test in order to measure its compressive
Unit
Parameter type
Data used in NN models
Min
Ultrasonic pulse velocity Vp
Rebound ðRÞ
Concrete compressive strength ðfc Þ
km/s
Input
–
MPa
Input
Output
Average
Max
STD
3.85
4.57
5.22
0.28
20.00
12.16
30.18
27.58
42.00
52.17
5.01
9.97
123
11816
Neural Computing and Applications (2020) 32:11807–11826
of rebound. The corresponding output training vectors are
of dimension 1 9 1 and consist of the value of the compressive strength of the concrete cubic specimens. Their
mean values together with the minimum, maximum values,
as well as standard deviation (STD) values are listed in
Table 4. Moreover, Fig. 6 demonstrates the frequency
histograms of the parameters.
The main advantages of this database are:
1. A sufficient number of experimental data is present
2. All cores have undergone the same environmental
conditions
3. All cores have been built and tested by the same
researchers
4. All cores have been built at the same time, complying
to the same standards
5. The tests were carried out by the same devices. This is
particularly important for the reliability of the
database, since it has been observed that for the same
concrete compositions significant variations/differences in the measured sizes are recorded when they
are performed by different laboratories, and
6. As shown in Table 4 and Fig. 6, all measured parameters cover almost all possible cases. Thus, values for
ultrasonic pulse velocity Vp range from 3.85 to 5.22
(km/s), for Rebound ðRÞ from 20.00 to 42.00 and for
the concrete compressive strength ðfc Þ from 12.15
(light concrete) to 52.17 MPa (usual concrete
material).
3.4 Training algorithms
Fig. 6 Histograms of the parameters
strength. This extended database is presented in detail, for
first time, in Table 3. Based on the above database, each
input training vector p is of dimension 1 9 2 and consists
of the value of the ultrasonic pulse velocity and the value
123
A large set of training functions, such as quasi-Newton,
Resilient, One-step secant, Gradient descent with momentum and adaptive learning rate and Levenberg–Marquardt
back propagation algorithms has been investigated for the
training phase of the BPNN models. Among these algorithms, the Levenberg–Marquardt implemented by levmar
seems capable to achieve the optimum prediction,
describing the nonlinear behavior of concrete compressive
strength. It should be noted that the difference of the
Levenberg–Marquardt implemented by levmar with the
other algorithms is vast [47]. This algorithm appears to be
the method which is the fastest for training moderate-sized
feed-forward neural networks (up to several hundred
weights), in addition to nonlinear problems. It also has an
efficient implementation in MATLABÒ software. In fact,
the solution of the matrix equation is a built-in function,
and therefore, its attributes are enhanced within a
MATLAB environment.
Neural Computing and Applications (2020) 32:11807–11826
11817
Table 5 Training parameters of BBNN models
Parameter
Value
Training algorithm
Levenberg–Marquardt algorithm
Normalization
Minmax in the range 0.10–0.90
Number of hidden layers
1; 2
Number of neurons per hidden layer
1 to 30 by step 1
Control random number generation
rand(seed, generator) where generator range from 1 to 10 by step 1
Training goal
0
Epochs
250
Cost function
MSE; SSE
Transfer functions
Tansig (T); Logsig (L); Purelin (P)
MSE mean square error, SSE sum square error
Tansig (T): Hyperbolic Tangent Sigmoid transfer function
Logsig (L): Log-sigmoid transfer function
Purelin (P): Linear transfer function
Table 6 Cases of NN
architectures based on the
number of input parameters that
were used
Case
Number of input parameters
Number of hidden layers
Input parameters
Vp
R
I
1
1
H
II
1
2
H
III
1
1
IV
1
2
V
2
1
H
H
VI
2
2
H
H
3.5 Normalization of data
The most crucial step for any type of problem in the field of
soft computing techniques, such as artificial neural networks techniques, is considered to be the normalization of
data. This is a pre-processing phase. In the present study,
the Min–Max [48] and the ZScore normalization methods
have been applied during the pre-processing stage. The two
input parameters (Table 4) and the single output parameter
have been normalized utilizing the Min–Max normalization method. Iruansi et al. [49] stated that in order to avoid
problems associated with low learning rates of the ANN,
the normalization of the data should be made within a
range defined by appropriate upper and lower limit values
of the corresponding parameter. The input and output
parameters, in the present study, have been normalized in
the range [0.10, 0.90].
3.6 Model validation
Three different statistical parameters were employed to
evaluate the performance of the derived NN models as well
as the available in the literature formulae for the
H
H
estimations of concrete compressive strength based on nondestructive measurements. Namely, root mean square error
(RMSE), the mean absolute percentage error (MAPE), and
the Pearson Correlation Coefficient R2 are used. The lower
RMSE and MAPE values represent more accurate prediction results. The higher R2 values represent a greater fit
between the analytical and predicted values. The aforementioned statistical parameters have been calculated by
the following expressions [50]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
1X
RMSE ¼
ð4Þ
ðxi y i Þ2
n i¼1
n 1X
xi yi MAPE ¼
ð5Þ
n i¼1 xi !
Pn
2
ð
x
y
Þ
i
i
R2 ¼ 1 Pi¼1
ð6Þ
n
Þ2
i¼1 ðxi x
where n denotes the total number of datasets, and xi and yi
represent the predicted and target values, respectively.
Furthermore, the following new engineering index, the
a20-inex, is proposed for the reliability assessment of the
developed ANN models:
123
11818
Neural Computing and Applications (2020) 32:11807–11826
Table 7 Statistical indexes of
the optimum BPNN for each
one from the five cases of NN
architectures based on input
parameters used (see also
Table 6)
Case
I
III
IV
V
VI
m20
M
1-28-1
1-21-29-1
1-26-1
1-21-14-1
2-25-1
2-20-14-1
ð7Þ
where M is the number of dataset sample and m20 is the
number of samples with the value of the ratio Experimental-value/Predicted-value falling between 0.80 and
1.20. Note that for a perfect predictive model, the values of
a20-index values are expected to be 1. The proposed a20index has the advantage that its value has a physical
engineering meaning: It represents the number of the
samples that predict values with a deviation of ± 20%
compared to experimental values.
4 Results and discussion
4.1 BPNN model development
In this work, a large number of different BPNN models
have been developed and implemented. Each one of these
ANN models was trained with over 140 data-points out of
the total of 209 data-points, (66.99% of the total number)
and the validation and testing of the trained ANN were
performed with the remaining 69 data-points. More
123
Dataset
Indexes
R
II
a20 - index ¼
Optimum BPNN model
RMSE
a20-index
Training
0.9576
2.8655
0.9429
Validation
0.8920
4.4661
0.8000
Test
0.9040
4.3668
0.7941
All
0.9379
3.4558
0.8947
Training
0.9616
2.7319
0.9500
Validation
0.9019
4.2697
0.8286
Test
0.9051
4.3270
0.7941
All
0.9423
3.3313
0.9043
Training
0.9878
1.5480
0.9857
Validation
0.9666
2.5857
0.9714
Test
All
0.9868
0.9838
1.6845
1.7850
1.0000
0.9856
Training
0.9911
1.3281
0.9929
Validation
0.9683
2.5026
0.9714
Test
0.9827
1.9120
0.9706
All
0.9857
1.6808
0.9856
Training
0.9929
1.1874
1.0000
Validation
0.9821
1.8948
1.0000
Test
0.9816
1.9344
1.0000
All
0.9891
1.4678
1.0000
Training
0.9938
1.1094
1.00
Validation
0.9849
1.7168
0.9714
Test
0.9803
2.0015
0.9412
All
0.9900
1.4035
0.9856
specifically, 35 data-points (16.75%) were used for the
validation of the trained ANN and 34 (16.27%) data-points
were used for the testing.
The architecture of the ANNs comprises a number of
hidden layers, usually ranging from 1 to 2, and with a
number of neurons ranging from 1 to 30 for each hidden
layer. Each one of the ANNs is developed and trained for a
number of different activation functions, such as the Logsigmoid transfer function (logsig), the Linear transfer
function (purelin) and the Hyperbolic tangent sigmoid
transfer function (tansig) [19, 20, 50–58].
The parameters used for the ANN training are summarized in Table 5. In order to achieve a fair comparison of
the predictions of the various ANNs used, the datasets are
split into training, validation and testing sets, using
appropriate indices to state whether the data belongs to the
training, validation or testing sets. In the general case, the
division of the data-points into the three groups is made
randomly.
Neural Computing and Applications (2020) 32:11807–11826
11819
Fig. 7 Architecture
of the optimum 2-25-1 BPNN model with one hidden layer and two input parameters the value of the ultrasonic pulse
velocity Vp and the value of the rebound ðRÞ
4.2 Optimum proposed NN model(s)
Based on the above, a total of 1,474,200 different BPNN
models have been developed and investigated in order to
find the optimum NN model for the prediction of the
compressive strength of concrete. Namely, six different
structures of ANNs (Table 6) based on the combinations of
the use of 1 or 2 hidden layers, as well the use of only the
ultrasonic velocity or only the rebound or both of them, as
input parameters, have been developed.
The developed ANN models were sorted in a decreasing
order based on the RMSE value and the optimum for each
of the six cases is presented in Table 7. Based on this
ranking, the optimum BPNN model based on RMSE index
for the prediction of the compressive strength is that of
2-25-1, which corresponds to a NN architecture with two
input parameters, the value of the ultrasonic pulse velocity
and the value of the rebound, and based on one hidden
layer (Fig. 7). As it is presented in Fig. 7, the transfer
functions are the Hyperbolic Tangent Sigmoid transfer
function for the hidden and output layer. In Table 7, the
values of statistical indexes R, RMSE and the value of the
proposed engineering index a20-inex are presented. At this
point, it is worth noting that all three optimal neural networks are useful (Figs. 7, 8, 9), and this is because one can
only measure ultrasound or only the rate of rebound or
123
Neural Computing and Applications (2020) 32:11807–11826
Hidden Layer
Input Layer
11820
Rebound (R)
b1
1
Hyperbolic tangent sigmoid
transfer function
1
2
3
...
4
23
24
25
26
b2
Output Layer
`
1
Hyperbolic tangent sigmoid
transfer function
Compressive Strength
Output Layer
Hidden Layer
Input Layer
Fig. 8 Architecture of the optimum 1-26-1 BPNN model with one hidden layer and one input parameter the value of the rebound ðRÞ
Ultrasonic Pulse Velocity (Vp)
b1
1
Hyperbolic tangent sigmoid
transfer function
1
2
3
4
...
25
26
27
28
b2
1
Linear transfer function
Compressive Strength
Fig.
9 Architecture of the optimum 1-28-1 BPNN model with one hidden layer and one input parameter value of the ultrasonic pulse velocity
Vp
both. Depending on the available data, one uses the corresponding neural network. Especially for the case where
measurements are only available for the rebound method,
123
the optimal neural network is 1-26-1 (Fig. 8), whereas if
only ultrasound measurements are available, it is suggested
to use 1-28-1 (Fig. 8).
Neural Computing and Applications (2020) 32:11807–11826
11821
Fig. 11 Experimental versus predicted values of the concrete compressive strength for the test datasets
Fig. 10 Experimental versus predicted values of compressive strength
for the training and test process
Figure 10 depicts the comparison of the exact experimental values with the predicted values of the optimum
BPNN model for the case of training and test datasets. It is
clearly shown that the proposed optimum 2-25-1 BPNN
reliably predicts the compressive strength of concrete
materials. It is worth noting that all samples used for the
testing process have a deviation less than ± 20% (points
between the two dotted lines in Fig. 10). The same conclusion is drawn from Figs. 11 and 12, where the experimental values are compared with the corresponding values
provided by the proposed optimum 2-25-1 NN model for
the case of test datasets.
Fig. 12 Experimental versus predicted values of the concrete compressive strength for the test datasets
4.3 Comparisons
In Table 8, the 14 bibliographic suggestions presented in
Table 2, as well as the results of the proposed neural network, are presented in descending order, based on the alpha
factor aligned with the RMSE index.
Additionally, in Fig. 13 the results of the ratio of the
predicted to the experimental value of the compressive
strength of the concrete for the 6 proposals that give the
best results compared to the experimental values are presented. Based on this classification, the proposed neural
123
11822
Table 8 Ranking of
Mathematical models for the
estimation of concrete
compressive strength based on
the value of the proposed
engineering index a-20
Neural Computing and Applications (2020) 32:11807–11826
Ranking
Mathematical model
Parameters
References
1
ANN 2-25-1
Vp ; R
Proposed herein
0.9891
1.4678
2
E.7
R
Logothetis [8]
0.9521
2.18
96.65
3
E10
Vp ; R
Logothetis [8]
0.9342
4.19
86.60
4
E4
Vp
Logothetis [8]
0.8198
4.23
80.86
RMSE
a20-index
100.00
5
E14
Vp ; R
Kheder [9]
0.9759
5.15
72.73
6
E8
R
Kheder [9]
0.9745
5.90
66.99
7
E9
R
Qasrawi [10]
0.9732
5.56
64.59
8
E3
Vp
Trtnik et al. [2]
0.8119
7.71
60.77
9
E2
Vp
Nash’t et al. [7]
0.9032
6.59
53.59
10
E13
Vp ; R
Erdal [13]
0.9421
7.09
51.20
11
E12
VP ; R
Amini et al. [12]
0.8961
8.27
49.28
12
E5
Vp
Kheder [9]
0.8942
7.46
45.45
13
E11
VP ; R
Arioglu et al. [11]
0.9535
8.13
43.06
14
E6
Vp
Qasrawi [10]
0.8893
12.24
19.14
15
E1
Vp
Turgut [6]
0.9035
12.84
13.88
network is preceded, followed by the three Logothetis
proposals (1978) and followed by Kheder’s two proposals
(1999).
Moreover in Fig. 14, based on the proposed NN model
1-26-1 that corresponds to the case with only one input
parameter, the value of the Rebound, the distribution of
compressive strength of the concrete with respect to
Rebound is presented, as it results from the presented
method, the Logothetis experimental results [8] and the
proposed Logothetis E7 [8] relationship. Also, in Fig. 15,
based on the proposed NN model 1-28-1 that corresponds
to the case with only one input parameter, the value of the
Ultrasonic Pulse Velocity, the distribution of compressive
strength of the concrete with respect to Ultrasonic Pulse
Velocity is presented, as it results from the presented
method, the Logothetis experimental results [8] and the
proposed Logothetis E7 [8] relationship. From Figs. 14 and
15, it can be seen that the proposed process approximates
the experimental results better and therefore the reliability
of the presented method results.
5 Final values of weights and bias of the NN
models
Even though it is common practice for authors to present
the architecture of an optimum NN model, without any
information related to the final values of NN weights, it
must be stressed that any architecture which does not
present these values is of limited assistance to others
researchers and practicing engineers. If, on the other hand,
a proposed NN architecture is accompanied by the
123
R
(quantitative) values of weights, it can be of great use,
making it possible for the NN model to be readily implemented in an MS-Excel file, thus available to anyone
interested in modeling issues.
With this in mind, in Table 9, the final weights for both
hidden layers and bias are stated. Based on Figs. 7, 8 and 9,
by employing the properties defined in Table 4 and
applying the weights and bias values between different
layers of ANN, it is possible to estimate the predicted value
of the concrete compressive strength.
6 Limitations
The (two) proposed ANN models can be applied only in
the case that the researcher, or the practitioner has the
experimental values of both ultrasonic velocity and Schmidt hammer rebound tests (or in the case with known
experimental value of the Schmidt hammer rebound test).
It should be stressed that the neural network models can be
reliably applied for parameter values ranging between the
lowest and highest values of each parameter (as presented
in Table 4); otherwise, the predicted value is unreliable.
7 Conclusions
A plethora of works based on non-destructive methods for
the estimation of concrete compressive strength has been
presented on the basis of conventional computational
techniques, such as multiple regression analysis. However,
the issue of the estimation of concrete compressive strength
Neural Computing and Applications (2020) 32:11807–11826
11823
Fig. 13 Comparison of the proposed NN model with available in literature analytical formulae for the estimation of concrete compressive
strength
123
11824
Neural Computing and Applications (2020) 32:11807–11826
Fig. 15 Concrete compressive strength versus ultrasonic pulse
velocity
Fig. 14 Concrete compressive strength versus rebound
Table 9 Final values of weights
and bias of the optimum NN
models
NN 1-26-1
NN 2-25-1
IW{1,1}
LW{2,1}S
B{1,1}
B{2,1}
IW{1,1}
LW{2,1}S
B{1,1}
B{2,1}
(2691)
(1926)
(2691)
(191)
(2592)
(1925)
(2591)
(191)
- 36.4000
- 0.3491
36.4000
0.5706
- 6.9186
- 1.0643
- 0.4535
- 7.0000
0.8432
36.4000
0.3117
- 33.4880
- 4.3197
5.5082
0.2887
6.4167
- 36.4000
0.4148
30.5760
1.7406
6.7801
- 0.2997
5.8333
- 36.4000
0.3779
27.6640
- 5.4113
- 4.4405
- 0.2244
5.2500
36.4000
0.0993
- 24.7520
4.0614
5.7013
0.3137
4.6667
36.4000
- 0.0521
- 21.8400
6.9743
- 0.5995
- 0.1784
- 4.0833
- 36.4000
0.2398
18.9280
5.7002
- 4.0630
- 0.1387
3.5000
- 36.4000
- 0.2703
16.0160
3.2325
6.2089
0.2014
2.9167
- 36.4000
0.0938
13.1040
- 3.2466
- 6.2016
0.2474
2.3333
36.4000
- 0.0265
- 10.1920
6.9728
0.6166
0.4585
1.7500
- 36.4000
- 0.0293
7.2800
- 4.1596
- 5.6301
- 0.4624
1.1667
0.1486
- 4.3680
2.2763
- 6.6195
0.2185
0.5833
36.4000
- 36.4000
- 0.3656
1.4560
- 1.5137
- 6.8344
0.3554
0.0000
- 36.4000
- 0.3397
- 1.4560
- 0.5814
- 6.9758
0.1435
0.5833
- 36.4000
36.4000
0.3776
- 0.2611
- 4.3680
7.2800
2.5492
- 6.9866
6.5193
0.4336
- 0.1393
0.1694
1.1667
- 1.7500
- 36.4000
- 0.2352
- 10.1920
- 4.8254
5.0711
0.0915
- 2.3333
- 36.4000
- 0.3490
- 13.1040
5.8421
- 3.8561
- 0.2345
2.9167
- 36.4000
- 0.2410
- 16.0160
- 5.8799
3.7982
0.3753
3.5000
36.4000
0.4441
18.9280
- 4.0396
- 5.7168
0.4255
4.0833
36.4000
- 0.1849
21.8400
- 6.8016
- 1.6549
- 0.1955
- 4.6667
36.4000
0.2360
24.7520
- 6.6740
2.1113
- 0.2942
- 5.2500
36.4000
- 0.0627
27.6640
5.4924
- 4.3397
- 0.1410
- 5.8333
- 36.4000
0.2329
- 30.5760
- 5.7825
- 3.9450
- 0.2063
- 6.4167
- 36.4000
0.3581
- 33.4880
5.1135
4.7804
0.2007
- 7.0000
- 0.2865
36.4000
36.4000
IW{1,1} = Matrix of weights values for between input layer and the first hidden Layer, LW{2,1} = Matrix
of weights values between the first hidden Layer and the 2nd hidden Layer
B{1,1} = Bias values for hidden layer, B{2,1} = Bias values for output layer
123
Neural Computing and Applications (2020) 32:11807–11826
is still open due to the fact that the available in the literature formulae depict a large dispersion of the values they
estimate, as well as a significant deviation from the actual
(experimental) value of the compressive strength of the
concrete.
In the present work, based on a large experimental
database, consisting of datasets from non-destructive tests
and compressive strength implemented on respective concrete cores, three different optimum ANN models are
proposed for the estimation of concrete compressive
strength. Namely, an optimum ANN model, for the case of
using only rebound test measurements, one for the case of
using only ultrasonic pulse velocity measurements and one
for the case where both ultrasonic and rebound methods
measurements are available, are developed and presented.
The comparison of the derived results with experimental
findings, as well as with available in literature analytical
formulae, demonstrates the promising potential of using
back propagation neural networks for the reliable and
robust approximation of the compressive strength of concrete based on non-destructive techniques measurements.
Also, the proposed NN models can continuously re-train
new data, so that it can conveniently adapt to new data in
order to expand the range of suitability of the ANN.
Using the architecture of the proposed optimum neural
network and the resulting values of final weights of the
parameters (see supplementary materials), a useful tool is
developed for researchers, engineers, and for supporting
the teaching and interpretation of the relationship between
non-destructive testing results and compressive strength
values.
Compliance with ethical standards
Conflict of interest The authors confirm that this article content has
no conflict of interest.
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