CC BY
41Magazine of Civil Engineering
journal homepage: http://engstroy.spbstu.ru/
ISSN 2712-8172
DOI: 10.34910/MCE.110.7
Application of artificial neural networks for prediction of concrete properties
N.A. Abdulla
University of Salahaddin, Erbil, Iraq E-mail: anwzad@yahoo.com
Keywords: slump, aggregate/cement ratio, artificial neural network, concrete stress, strain at peak stress, regression analysis
Abstract. The effect of different mix ratios on the mechanical properties of concrete was investigated. The strength and deformation in terms of the strain of normal strength concrete were evaluated under concentric loading. The artificial neural network (ANN) technique was used for predicting the compressive stress and strain at peak stress of concrete. The input parameters for ANN architectures included water/cement ratio, aggregate/cement ratio, and slump values. An equation for predicting the strain of concrete at peak stress was proposed based on ANN output values for compressive stress and strain at peak stress. The capability and performance of the proposed equation are compared with actual experimental results and predictions from existing fifty-three empirical equations, including several design codes and various strain models for normal and high strength concretes, using several statistical indexes. The results showed that ANN technique have good potential for predicting the compressive strength and strain at peak stress of concrete, yielding close predictions and good agreement with the original ones.
In the last nine decades, considerable effort has been spent to understand the inelastic behavior of concrete and the resulting shape of the stress-strain curve. The compressive strength and the strain at peak stress of concrete were found to have a significant interrelationship. Several studies have reported the axial strain capacity to increase significantly in concretes with improved compressive strength. Modern technology and new additives assisted in the improvement of concrete properties. Several parameters influence the strain at peak stress, including water/binder ratio and workability. One measure of quality control and uniformity of concrete from batch to batch is the slump test. Workability extends the fresh use of concrete to achieve full compaction, increase the resistance to bleeding, harshness, and segregation. A significant component that makes up concrete and contributes to its physical improvements is aggregate. A high aggregate/cement ratio indicates lower compressive strength and vice versa.
Some design codes have adopted a constant value for the strain at peak stress of concrete tested under axial compression load. However, such approaches may not yield acceptable results with more minor errors since the compressive strength of concrete and its strain at peak stress are influenced by several parameters such as mixed ingredients, specimen geometry, testing conditions, and environmental conditions.
Several studies have highlighted the potential use of artificial neural networks (ANN) to predict the compressive strength of concrete. In a research, ANN procedure was employed for evaluating the compressive strength of concretes containing metakaolin and silica fume [1] using the experimental test results from 195 specimens produced with 33 different mixture proportions. The obtained test data was used in the multi-layer feed-forward neural network models and were arranged in a format of eight input parameters. In another study, ANNs and genetic programming (GP) were used for predicting the strength of concrete [2]. The ANN model with the training function, Levenberg-Marquardt (LM), was found to be a
Abdulla, N.A. Application of artificial neural networks for prediction of concrete properties. Magazine of Civil Engineering. 2022. 110(2). Article No. 11007. DOI: 10.34910/MCE.110.7
© Abdulla, N., 2022. Published by Peter the Great St. Petersburg Polytechnic University. This work is licensed under a CC BY-NC 4.0
1. Introduction
useful tool for compressive strength predictions of concrete. Lai and Serra [3] used the ANN model for assessing the compressive strength of cement conglomerates by constructing models in which a variety of different mix-design parameters associated with cement conglomerates were considered.
Unlike normal concrete, high-strength concrete (HSC) has been reported to be a highly sophisticated material that makes its modeling difficult. ANN models [4] have shown to be a powerful tool for calculating the compressive strength and slump of HSC. The ANN models for compressive strength and slump were constructed, trained, and tested using a database of 187 test results arranged in a format of seven input parameters. The mean absolute percentage error found to be less than 1 % for compressive strength and 5 % for slump values, and the corresponding R2 values were 99.93 % and 99.34 %, respectively. Naderpour and Mirrashid [5] utilized the ANN technique to assess the effect of micro-silica and calcium in silicate minerals on the compressive strength of mortars. The ANN modeling showed high accuracy, functional ability, and acceptable performance in predicting the compressive strength of the tested mortars. Asteris et al. [6] reported ANN to be a proper simulation technique for predicting concrete properties. Duan et al. [7] suggested an ANN model-based explicit formulation for predicting the loss in compressive strength of recycled aggregate concrete. The authors used one hundred forty-six available sets of data from sixteen different published literature sources to construct ANN models with fourteen input parameters. They concluded that with varying the types and sources of recycled aggregates, ANN showed excellent potential as a technique for evaluating the compressive strength of recycled aggregate concretes.
An experimental program [8] consisting of a direct and indirect evaluation of unconfined compressive strength (UCS) of sixty-six granite and limestone sample sets of rocks was carried out. Point load index test, Schmidt hammer rebound number, p-wave velocity test, and dry density test were used as inputs of the network while the output was the UCS values. A PSO-based ANN techniques hybrid model was proposed. Several studies have reported the predictions of the alternative evaluation methods to be often closer to the experimental test data than the predictions from design codes [9]. ANN method performed better and yielded more accurate predictions than the Multiple Linear Regression (MLR) technique for both slump and compressive strength of concrete [10]. From the above review, most of the previous studies were on the compressive strength of concrete, and the application of the ANN technique to predict the concrete deformation in terms of strain at peak stress still needs to be covered.
The objectives of the present study are to evaluate the previously published models relating the strain at peak stress to the compressive strength; to perform experimental tests to assess the stress-strain relationship parameters, compressive strength and its corresponding compressive strain at maximum stress, of normal strength concrete; to use ANN technique to predict the above two parameters based on the experimental results. Finally, the paper sets to develop a model to predict the strain at peak stress based on regression analysis of ANN predicted values.
2. Materials and Methods
2.1. Materials
The following ingredients were used for the twelve concrete mixtures: water, coarse river aggregate, fine river aggregate, and cement.
Ordinary Portland cement with a specific gravity of 3.15 and conforming to ASTM C150 Type I was used for making concrete.
The coarse aggregate for all the mixes was rounded and well-graded river gravel with a maximum aggregate size of 20 mm and specific gravity of 2.72.
The fine aggregate was river sand from the Eski-kalak region north of Iraq with a maximum size of 4.75mm and specific gravity of 2.7.
The quantity of coarse aggregate was adjusted, yielding twelve mixes with aggregate/cement (a/c) ratios ranging from 3 to 8 in increments of 0.5. Tap water was used to hydrate the ordinary Portland cement in a drum-type rotary mixer. All the mix details are summarized in Table 1.
2.2. Methods of Testing
2.2.1. Fresh concrete
The slump test was used to evaluate the consistency of fresh concrete and the effect of change in the mix on fresh concrete properties. The truncated steel cone was filled with the fresh concrete in three equal layers, and each layer was tamped twenty-five times with a steel rod to ensure compaction. The workability of new concrete decreased as the a/c ratio increased, resulting in harsh combinations with increased compaction difficulties. The details of slump results are presented in Table 1. The unit weight of
fresh concrete was measured and computed in Table 1. In general, as the amount of fine material or sand decreased, the unit weight decreased, too.
2.2.2. Hardened concrete
After 28 days of curing, the rough surface of concrete cylinders was capped with a filling material. The specimens were tested in compression under a displacement control mode at a rate of approximately 0.5 mm/min. The relative displacement between the upper and lower loading platens was measured using two linear variable differential transformer (LVDT) and one dial gauge (Fig. 1). Two electrical strain gauges on the surface of cylinders used for longitudinal and lateral strain measurements. There was an inverse relationship between compressive strength and the three test parameters for hardened concrete depending on variations in constituent materials, Table 1.
2.3. Intelligent systems and methods 2.3.1. Artificial neural networks
The neural network modeling process involves five main aspects: data acquisition, analysis, and problem representation; architecture determination; learning process determination; training of the networks; and testing of the trained system for generalization evaluation [1, 11]. ANN procedure handles severe problems via the interaction between nodes (artificial neurons). The ANN model has elements arranged in layers and trained with the data from the experimental results. The first layers of nodes (neurons) get data from the outside environment and transfer it to the nodes of the hidden layer without performing any computation, where the data is processed to draw out useful features. Weights interconnect other nodes in other layers. The output layer neurons produce network predictions. In a three-layer ANN, training data relates between input and output nodes. Due to the ability of neurons to pass and remember the data from experimental results during the training process, the network can learn, categorize, and predict values.
Table 1. Mix details and properties of fresh and hardened concrete.
Specimen Mix ratio w/c ratio Aggregate/ cement ratio Slump (mm) Y (kg/m3) fc (MPa) Zc
1-1 1:2:1 0.435 3.0 12 2400 38.7 0.00197
1-2 1:2:1 0.435 3.0 12 38.9 0.0019
1-3 1:2:1 0.435 3.0 12 38.5 0.00192
2-1 1:2:1.5 0.450 3.5 10 2400 36.5 0.00187
2-2 1:2:1.5 0.450 3.5 10 38.1 0.00188
2-3 1:2:1.5 0.450 3.5 10 37.5 0.0018
3-1 1:2:2 0.485 4.0 9.5 2380 34.6 0.00179
3-2 1:2:2 0.485 4.0 9.5 34.3 0.00188
3-3 1:2:2 0.485 4.0 9.5 33.7 0.00182
4-1 1:2:2.5 0.505 4.5 9.5 2375 34 0.00177
4-2 1:2:2.5 0.505 4.5 9.5 32 0.00173
4-3 1:2:2.5 0.505 4.5 9.5 32.7 0.00179
5-1 1:2:3 0.51 5.0 8 2370 31.1 0.00168
5-2 1:2:3 0.51 5.0 8 33.3 0.00176
5-3 1:2:3 0.51 5.0 8 32.2 0.00169
6-1 1:2:3.5 0.51 5.5 7 2355 30.8 0.00172
6-2 1:2:3.5 0.51 5.5 7 32 0.00174
6-3 1:2:3.5 0.51 5.5 7 30.8 0.00171
7-1 1:2:4 0.52 6.0 5 2340 28.5 0.00169
7-2 1:2:4 0.52 6.0 5 29.7 0.00176
7-3 1:2:4 0.52 6.0 5 27.6 0.00169
8-1 1:2:4.5 0.535 6.5 5 2325 25.2 0.00157
8-2 1:2:4.5 0.535 6.5 5 24.8 0.00172
8-3 1:2:4.5 0.535 6.5 5 26.5 0.00182
9-1 1:2:5 0.56 7.0 4 2300 20.6 0.00164
Specimen Mix ratio w/c ratio Aggregate/ cement ratio Slump (mm) Y (kg/m3) fc (MPa) ^c
9-2 1:2:5 0.56 7.0 4 19.6 0.00165
9-3 1:2:5 0.56 7.0 4 21.6 0.00165
10-1 1:2:5.5 0.585 7.5 2.5 2245 19.9 0.00169
10-2 1:2:5.5 0.585 7.5 2.5 18.2 0.00151
10-3 1:2:5.5 0.585 7.5 2.5 19.5 0.00158
11-1 1:2:6 0.6 8.0 0.0 2270 16.3 0.00153
11-2 1:2:6 0.6 8.0 0.0 16.1 0.00154
11-3 1:2:6 0.6 8.0 0.0 17.7 0.00169
12-1 1:2:6.5 0.605 8.5 0.0 2225 15.6 0.00153
12-2 1:2:6.5 0.605 8.5 0.0 16.85 0.00164
12-3 1:2:6.5 0.605 8.5 0.0 16.4 0.00167
A typical node (AN) designed to carry out specific tasks, as shown schematically in Fig. 2. The network supplied with the values of parameters Xj (selected parameters related to the strength
characteristics of the concrete) [9]. Each parameter assigned weight wH and bias 0. and this yields the
jl l
sum nt of the multiplication:
nt =Z(Xj )(wJl ) + (1
The n is integrated into an established activation function (g):
Figure 1. Testing of concrete specimens.
y. = gi = g (Kxi )(wji) + %)
(2)
yt =Si +
X
1
2
3
Figure 2. A simple neuron model.
And the result is the output Oi of the AN. Several layers assembled to establish the artificial neural
networks used in the present study. The randomly assigned values of the weights and bias undergo iterative training to yield the final results. The architecture of the ANN established through a trial-and-error process related to the type of activation function and the number of neurons in each hidden layer. The goal of this process is to reduce the gap between the ANN output Oi values and the target values Tt [12, 13, 1]. A
measure of the deviation of Oi from Tt is given by:
i 2
E = --Ot )2 (3)
2.3.2. Adopted ANN models
The three-layer feed-forward type of ANNs was adopted in the present study, Figs. 3 to evaluate the peak stress and Fig.4 to assess the strain at peak stress. The transferred information is processed to extract useful features to reconstruct the architecture from the input space to the output space [15]. If convergence is not achieved, the calculation repeated. Weights fully interconnect the neighboring layers. The network architecture is dictated by the interconnected input, hidden, and output neurons. The weights and processing function of each neuron influence the output layer.
Input layer
mJukm
_Y_
Hidden layer
Output layer
Figure 3. Architecture of ANN model to evaluate peak stress.
Most studies adopted a neural network with one hidden layer that was found sufficient to solve most problems in civil engineering [16-18]. When considering computational efforts required for more hidden layers and an additional number of neurons, one hidden layer reported to be sufficient to produce an acceptable ANN model [19] and to predict the elastic modulus of concrete. When increasing the number of hidden layers, a marginal change in results was obtained [20]. Two ANN models, ANN-1 and ANN-2, with one hidden layer were constructed, trained, and tested using the current experimental test results. The experimental test data was based on two different mix parameters (i.e., w/c and a/c ratios) in addition to slump results. The experimental data was divided into three subcategories of training, validation, and test. Subsequently, the network was trained to minimize the error between the experimental and ANN predicted values.
Input layer
e up a
Hidden layer
do eTooo
, m t
Output layer
Figure 4. Architecture of ANN model to evaluate strain at peak stress.
The two concrete mix parameters (w/c ratio and a/c ratio) and slump values were used as the four input variables and the bias. The compressive strength and strain at peak stress were the output parameters for the two models, as shown in Figs. 3 and 4, respectively. The hidden layer of the ANN-1 Model to evaluate peak stress (Fig. 3) consisted of seven neurons. In contrast, the same number for the ANN-2 Model to evaluate strain consisted of six neurons (Fig. 4).
3. Results and Discussions
3.1. Strain models
The size and shape of the specimen and concrete physical characteristics influence the strain at peak stress of concrete [21]. Strain is needed to determine the response of a structure to crack control and assess cracking risk [22]. Generally concrete strain is expressed as a function of compressive strength in standard codes and empirical expressions. Different function forms of sc - fc relationship is found in the
literature which can be classified into three groups. The experimental test results were checked against the three groups containing fifty-three existing expressions for strain at peak stress [21-68]. It included the following:
3.3.1. Group one (G-1) with polynomial function
In models with polynomial functions [21, 22, 27], the compressive strength is raised to a non-negative integer power. The leading term or the highest power of the variable in the polynomial function was 3. Among the three models in group one is the model by Tasdemir et al. [22], which was based on tests on concrete with a wide range of strengths from 6 to 105 N/mm2. Using a total of 228 test results for specimens tested under uniaxial compressive loading conditions, the authors reported a polynomial function to best fit the experimental data with a reasonably good correlation coefficient of 0.75, Table 2.
3.3.2. Group two (G-2) with linear function
Twenty-one models [24-26, 28, 30, 32, 33, 35, 36, 38-40, 47, 49, 54, 56, 59, 60, 62, 63, 65] among the fifty-three models have a linear function. Group two linear models with large database include two models; Chen et al. [60] and Chen et al. [63] with 380 test data from 15 studies.
3.3.3. Group three (G-3) with power functions
The largest group with twenty-nine models [23, 29, 32, 34, 37, 39, 41-46, 48, 50-53, 55, 57, 58, 61, 64, 66a, 67, 68 ]. Several of these models based on a large database, including De Nicolo et al. [39] using 17 studies, Arioglu [42] with 41 sets of test data from 8 studies, Wee et al.[44] with 163 test data, Mansur et al. [48] with 54 test data, Lu and Zhao [55] with 75 test data, Ding et al. [57] with 165 test data, Kumar et al. [58] with 162 test results, Lim and Ozbakkaloglu [61] with 147 test data, Hoang and Fehling [67] with 132 test data from 6 studies. All the equations with a power function have a variable base as a compressive strength raised to power with a small, even positive number. The power ranges from 0.19 [66] (a) to 0.53 [52] and [66]c for most of the studies. However, the model [67] uses power as high as 0.96. On the other hand, the model [68] incorporates negative power (-2.256).
Other studies, not included in Table 2, employed constant values of strain such as 0.002 [44, 69], 0.0022 [70-73], 0.0024 [74], and 0.003 [75, 76]. The sc =0.003 was suggested to be used for design purposes, for concrete compressive strengths up to 124MPa [75, 76]. Table 2. Models used to predict the strain at peak stresses.
Year
Source
Model
Strength range MPa
No. of test data
Comments
1993 Collins et al. [21]
1998
Tasdemir et al[22]
£c =
fc (0.0588/c + 0.8)
(3220fc0 5 + 6900)(0.0588fc - 0.2)
Sc is strain at peak stress
^c =
fc is compressive strength
if2 f ^ -0.067+ 29.9^*- +1053
fc fc
f*= 1 MPa
10"
21 to 83MPa
6 to 105MPa
14
228 from 12 studies
R2= 0.75
Year Source
Model
Strength No. of test Comments range MPa data
1936 Emperger [23]
1994 Hsu and Hsu [38]
1994 De Nicolo et al. [39]
Brandtzaeg
[39]
1995 Almusallam and Alsayed
[40]
1995 CEB-FIB
[41]
1995 Arioglu [42]
Sc =
10
-2
1950 Ros [24] Sc =
1955 Hognestad et al. [25]
1962 Liebenberg [26] Sc
1964 Saenz [27] Sc ='
1967 Soliman et al.[28]
1970 Popovics [29]
1970 Tadros [30]
1973 Popovics [31]
1976 Bashur and Darwin[32]
1982 Ahmad-Shah [33] Sc
1984 1985 Tomaszewic z [34] Shah-Fafitis [35] Sc :
1986 Carreira-Chu [36] Sc
1990 Ali et al. [37]
(0.0445(fc )0 5)l sc =(0.0546 + 0.003713fc )l0-2
sc =1 0.004 + 10-6 c 1 6.5c
f
0.004 +
fc
( 0.73)(8.3)
10"
10"
sc = 2 (fc ) / 25097* *MPa
sc = 0.000735 (fc )'
0.25
Sc =(1.6 + 0.01fc )10
sc = 0.000937(fc )'
fc
-3
0.25
363000 + 400fc
( fc in Psi)
0.31, _-6
= 700 (fc ) 10
^(10-5)(fc )
+ 0.00195
sc = 0.000875 (fc )'
0.25
Sc = 1.29 (10-5 )(fc ) + 2.114 (10-3 )
5 to 52.4
7 to 69
sc = 0.00076-
0.626 ^^ - 4.33 110-7 f *
0.5
( fc in Psi) 20 to 75 fc< 85MPa
8.96 to
52.4
16.7 to
43.5
65.8 to 91.4
10 MPa<
fc^ 100
MPa
fc
-10
-2
46.886 + 2.6fc Sc =(0.398fc +18.147 )10-4
Sc =(0.1 fc )0 31 (10-3 )
1 -7CO/V \0.2775^ \-0.09314 Sc = 1 753 (fc ) (V )
20 to 110MPa
fc<
100MPa V=volume
26
16
9 12 14
Not given 17 studies
41 from 8 studies
For concrete with a low level of sand
Curve fitting using data from another
reference Curve fitting using data from another
High-strength
Data from two studies
R2= 0.874
Year
Source
Model
Strength No. of test Comments range MPa data
1996 Attard and Setunge [43]
1996 Wee et al.[44]
1997 Guo [45]
1998 Xu [46]
1999 CEB-FIB [47]
1999 Mansur et al. [48] a
1999 Mansur et al. [48] b
2002 Lee [49] c
2003 NS 3473[50]
2003 Yu and Ding
[51]
2004 EC2 [52]
2004 Tasnimi [53]
2006 Mertol [54]
2008 Lu and Zhao [55]
2010 Arslan and Cihanli[56]
2011 Ding et al. [57] a
2011 Ding et al. [57] b
2011 Kumar et al.
[58]
2013 Hussin et al.
[59]
2013 Chen et al.[60]
Sc =
3.78fc
\0.25
Ec tffc
Sc = 0.00078(fc )0 5c =(700 +172 ( f ))10-3
S =(966 +155.64(2fc -13.77))
10
-3
1.7 +
fc
fc
-3
V Jcmoy
10
\0.35
sc = 0.0005 (fc ) Sc = 0.00048 (fc )0 35
■ = fc
c (46.886 + 2.6fc )102
0.7 /, x0.31 S =--(fc )
1000
Sc =1
Ï-6
Sc =
(383fcM % ) 10 Sc = 2 + 0.085(fc - 50)' (65.57f sc = 0.0033 -13.793 (10-5 f sc = 430(fc )0 38 (10-6) 0.001 (fc - 20)
0.53
0 44 - 6.748)10-5
sc = 0.002 +
V
70
y
Sc = 520 (fc )% (10-3 ) Sc = 383 (fcU (10-6 )
Sc = 0.0006(fc )X (10-3)
sc = 2 (fc )(10-5 ) + 0.0008
Sc = 174(fc )(10-6) + (2.41)(10-3)
> 40MPa
Up to 120MPa
fcmo=~70,
fc<100MPa
fc<100MPa
70 to 120MPa
70 to 120MPa
75 to 78
fc=peak stress
50MPa<fc< 100MPa
69 to 124
42.7 to 125.6
50MPa<fc< 100MPa
20 to 150MPa from five studies 30 to 150MPa from four
35 to 70MPa
9.25 to 38
163
54 54 20 HPC
21
75
highperformance concrete
For cylinders For prisms
fc = 0.4fc/
0.002 for fc<50MPa
165groups
58 groups
162 cylinders
26
fc = 0.4fcu/6
self-compacting concrete
HSC 380 from C0V=0.176 considerin 15 studies g size effect
Year
Source
Model
Strength range MPa
No. of test data
Comments
2014 Lim and Ozbakkalogl
u[61]
2015 Ahmed et
al.[62]
2015 Chen et
al.[63]
2016 Wang et al.
[64]
2016 Shanaka
[65]
2016 Nematzadeh et al.[66] a
2016 Nematzadeh et al.[66] b
2016 Nematzadeh et al.[66] c
2017 Hoang and Fehling[67]
2017 Aslam et al.[68] Proposed
s =
y co
(fco"22>t' )(10-3)t.t.
ks = 1, ka = 1
sc = 0.00003(fc ) + 0.001
sc = 4.76 (fc )10-6 + (2.13) 10-3 , = 0.5 (1.95 + 0.0149fc + 0.763f )
Sc = 1.1
¿L
v^ c
Sc =
(1074fc 019 )
10
-6
Sc = 402 (10-6 )(fc041 )
Sc = 225 (10-6 )(fc 053 ) Sc = 0.0257 (10-2 )(fc 096 )
S = 18.938 (fc )
Sc = 0.001(fc )
-2.256
0.17
10MPa<fc< 150MPa
5.9 to 26.5 20 to 105
fc<200MPa
95 to 147MPa
17.9-52.6MPa RC
43-83MPa LPCC
44-91MPa SPCC
100<fc
<1 50MPa
42.5 to 52
16.1 to 38.9MPa
147
78
380 from 15 studies
18
30
30
30
132 from 6 studies
9 LWA 36
R2=0.626 NWC
R2=0.49 C0V=0.14
R2=0.62, reference concrete
(RC) R2=0.55, long-term pressure-compressed
concrete R2=0.74,Sho rt-term pressure-compressed concrete R2=0.86, UHPC; Ultrahigh performance concrete R2=0.96
R2=0.82
3.2. ANN models
A total of 36 data sets with minimum and maximum concrete strengths of 15.6 MPa and 38.9MPa were unequally divided and used for training (approximately 80% of modeling) and the rest for testing (about 20%) of the model. The ANN approach considers several parameters affecting the strength of
■a «
o Cl
40
35
30
25
20
15
---Experiment
©-- ANN-1
10 20 Data order
30
40
Figure 5. Comparison of predicted peak stress (ANN-1) with experimental peak stress.
0
concrete at the same time [19]. The experimental compressive strength and predicted strength values using the ANN-1 model were plotted in Fig.5. A similar procedure followed for strain at peak stress, using ANN-2 model, as shown in Fig. 6. The results showed that, on average, the peak stress decreased as aggregate/cement ratio and w/c ratio increased.
In Figure 6, the trend for the experimental strain was more to increase with a decrease in the aggregate/cement ratio, and this follows the same trend for predicted strain at peak stress using the ANN-2 model. ANN models have captured the inter-relationships between input and output data pairs. The curves plotted in Fig. 5 for peak stress and Fig. 6 for corresponding strain demonstrate that the neural network was effective in learning the relationship between the different input parameters and the outputs, compressive strength, and strain at peak stress. The error between the predicted and experimental values then was computed. The output error can be minimized by modifications on the weights and bias at each neuron. The connections which could be positive or negative were not shown in Figs. 3 and 4. Instead, all the connection weights and biases used to predict the peak stress of concrete were computed in Table 3, and the connection weights and biases used to predict the strain at peak stress of concrete were summarized in Table 4.
Table 3. Connection weights and biases used to predict the strength of concrete.
Neuron_w/C_a/c_Slump_(Bias)
1 -0.423 0.242 0.205 -0.152
2 .171 .039 -.208 .338
3 -.284 -.205 .190 .380
4 -.868 -.319 -.083 .046
5 .347 .293 -.109 .230
6 .437 .404 .066 .075
Neuron 1 2 3 4 5 6 (Bias)
fco 0.144 -0.177 0.326 0.673 0.027 -0.481 -0.010
Table 4. Connection weights and biases used to predict the peak strain of concrete.
Neuron w/c a/c slump (Bias)
1 -.346 -.246 .509 -.732
2 -.729 .447 .136 -.078
3 .703 -.697 .174 -.034
4 .568 -.286 .072 .132
5 .053 -.340 -.915 .951
Neuron 1 2 3 4 5 (Bias)
Strain 0.538 0.709 -0.820 -0.441 -1.176 1.013
In normal strength concretes the elastic mismatch of aggregate and the matrix is significant, leading to large tangential, radial, and shear stresses at the paste-aggregate interface [22]. In the current study, the predicted peak strain values, using ANN-2 model, were plotted versus the corresponding predicted maximum stress, using the ANN-1 model and the graph was fitted with a nonlinear curve to find the relationship between the two variables. Consequently, an equation obtained and proposed for peak strain predictions, as summarized in Table 2.
3.3. Statistical indices
Several standard statistics used to assess the performances of the existing fifty-three equations for the strain at peak stress which include:
3.3.1. Normalized root-mean-square error (NRMSE)
Figure. 6. Comparison of predicted strain at peak stress (ANN-2) with the corresponding
experimental values.
NRMSE = ■
Vzf=1 (exP., - modeli )2
N_
(4)
srN
Where the model prediction is expressed by (model.), the experimental value is represented by (exp.i), and N is the total number of data.
3.3.2. Average absolute error (AAE)
VN
Xi =1
AAE =
modelt - exp.;-
exp.i
(5)
N
Lower AAE values indicate excellent model performance. 3.3.3. Nash-Sutcliffe efficiency (E)
E = 1 -
Z,-=1 (exP i - mod eli )
Zf=1 (exP i - aver exP i )2
(6)
An E value of over 0.80 is considered good. 3.3.4. Modified Nash-Sutcliffe efficiency (El)
E1 = 1 X |exp.j - modeli|
X |exp.i - aver .exp.i
(7)
The Modified Nash-Sutcliffe efficiency is based on absolute deviations instead of squares of the deviations.
3.3.5. Coefficient of correlation (R2)
f ^N ^
R 2 =
XN=1 ( exp.i - aver .exp.i )(mod eli - mod eliaver )
Xf=1(exp.i -aver.exp.i ) Xf= :1(modeli - modeliaver )
2 v^ N
(8)
R2 is the rate of association between the two variables, and many outliers result in weak R2. The statistical measures also used to determine the performance of the two trained ANN models.
3.4. Strain at peak stress
The fifty-three existing strain models found in the literature [21-68] and the proposed ANN-based
model were used to predict the strains (sc) at the peak stress and were compared with the experimental
results. All the models for the strain at peak stress prediction summarized in Table 2 including the proposed ANN-based model. Some models have limitations that make these models not applicable to the strength range of the present study, but were included for comparison purposes. The predicted strains and the best fitting line or curve for each model shown in Fig. 7 for G-1 models with polynomial functions. Also shown in Fig.7 is the predictions by the proposed ANN-based model. A significant trend is observed as the compressive stress of concrete increases, the strain at peak stress increases, too [21, 22, 27]. The same thing is correct for the proposed equation. Model [22] exhibit good predictions for concrete strengths up to 25MPa. However, at strengths beyond 25Mpa the error increases. The model [27] show a reasonable performance for all the strength range tested in the present study. The third model in G-1 over predicts the strain values, Fig.7. The proposed strain equation out performed all the three models in G-1 with very good predictions, Fig.7.
Fig. 8 shows predictions and best-fitting lines for the models with linear function in addition to the proposed model. The predicted strains spread over a wide strain range, 0.0001 to 0.004. The best-fitting lines or curves for some of the models [28, 40,49, 54] become steeper away from the origin. All models prediction show an increase in compressive strength, except the EC2 model, which shows a reverse trend because it is proposed for strengths higher than 50 MPa, which is beyond the experimental strength range covered in the present study. As can be seen, most of the models overestimated the strain values, and some models largely over-predicted the experimental strain values [25, 26, 40]. Using model [24], a better prediction of strain at lower values of fc is observed. The opposite can be said for other models [28].
The predictions of models using power functions are displayed in Fig. 9. The predictions with best-fitting curves in Fig. 9 compared with Fig. 8 are more compact with fewer models overestimating the predicted strain values, except for model 67 which was proposed for ultra-high-strength concrete. In contrast to models with a linear function, only a couple of models with power functions underestimated the values for the strain at peak stress, [57]a and [66]c. Some models showed a better prediction of strain at lower values of fc , example model [23]. This trend is reversed in the predictions of other models, such as
model [55]. Over all, the models with linear functions, Fig. 8, displayed more change in the best-fitting line slope than models in Fig.9. Most existing model predictions are higher than the test data and the proposed model, this is ascribed to the fact that these models were developed for concretes with higher strength.
3.5. sc Pre/sc Exp ratio
To further assess the performance of the fifty-three existing models and the proposed model, the
ratio of the predicted strain (scPre) to the experimental strain (scExp) at peak stress was computed and
shown in Table 5. Five statistical indexes used to evaluate the percentage of error or the degree of association between the experimental test results and their corresponding predicted values. The difference between the experimental and predicted values of the strain at peak stress was summarized in Table 6. Other statistical indexes including minimum predictions (Min.), maximum predictions (Max.), mean, standard deviation (STD), and coefficient of variance (COV) are also shown in Table 6. Lower values of NRMSE and AAE show good performance of the models. In contrast, higher values of E, El, and R2 show less error measured between the predicted and experimental values.
Statistics on the performance of fifty-three existing models and the proposed model illustrate that the values of NRMSE ranged from 0.0011(proposed model) to 0.27 [68] and for AAE ranged from 3.46% (proposed model) to 785.81% [68]. A similar observation made for the other indexes. The six models with least values of NRMSE and AAE included the proposed model, [44], [51], [58], [33], and [27] with values of 0.0011, 0.0014, 0.0019, 0.0018, 0.0021, 0.0025 and 3.46%, 4.445% , 5.34%, 5.712%, 6.072%, 6.761% respectively. Among the six models with the least values of NRMSE and AAE, four models have power function, one model with a linear function, and one model with a polynomial function. This indicates that models with power functions showed better performance compared with the other two types of models. Ahmed-Shah's [33] model with a linear function yielded nearly a horizontal best-fitting line and shows reasonable predictions with NRMSE and AAE values of 0.0025 and 6.761%, respectively. For the range of strength considered in the present study, the constant value of the model [33] controls the outcome of the strain values with the variable parameter (fc) having a minor role. The model [44] with AAE values of
4.445% displayed reasonable predictions of strain. The proposed model with AAE values of 3.46% yielded the lowest AAE values with predictions of strain at peak stress very close to the experimental strain values. The proposed model showed good performance with the highest values of E and E1. Furthermore, the
proposed model yielded the lowest values of NRMSE among all the models, as shown in Fig. 10. The R2 value for the proposed model (0.708) was slightly lower than for models [35, 36] with similar values of 0.736, Table 6. Few equations found to reasonably predict the strains at peak stress for the different concrete mixtures. As it was explained before, several existing strain models have been calibrated for high-strength concrete and show poor performance when used to predict the strain of normal strength concrete. Furthermore, most of these models derived from simple regression analysis based on test results carried out by the generators of the models and applicable for a particular type of concrete. Several of these expressions are valid within limited ranges of strength and not for other varieties. The complex relations developed between the mix proportions of concrete and the compressive strength can be captured better by the ANN-based model, which trained to yield low mean squared error and absolute average error between the experimental results and the network predicted values.
Table 5. The ratio of £c Pre/^c Exp using existing fifty-three models and the proposed model.
£c Pre/^c Exp
Specimen [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
1-1 1.081 1.071 1.405 1.007 2.033 2.034 0.988 1.565 0.931 1.009 1.186 1.092 1.161 1.104 1.283
1-2 1.122 1.113 1.461 1.048 2.108 2.108 1.026 1.632 0.966 1.047 1.232 1.133 1.205 1.146 1.332
1-3 1.107 1.096 1.438 1.029 2.086 2.086 1.013 1.598 0.954 1.034 1.216 1.12 1.189 1.131 1.315
2-1 1.121 1.099 1.438 1.017 2.142 2.142 1.032 1.555 0.966 1.051 1.232 1.141 1.203 1.142 1.334
2-2 1.128 1.114 1.461 1.043 2.131 2.131 1.033 1.615 0.971 1.054 1.238 1.142 1.211 1.151 1.339
2-3 1.173 1.156 1.514 1.077 2.225 2.225 1.077 1.66 1.01 1.097 1.288 1.19 1.259 1.196 1.394
3-1 1.156 1.121 1.462 1.023 2.238 2.238 1.069 1.54 0.996 1.087 1.27 1.183 1.24 1.173 1.378
3-2 1.098 1.064 1.386 0.968 2.13 2.13 1.016 1.454 0.946 1.034 1.206 1.125 1.178 1.114 1.309
3-3 1.13 1.09 1.419 0.988 2.201 2.201 1.047 1.476 0.973 1.064 1.24 1.158 1.211 1.144 1.348
4-1 1.164 1.126 1.466 1.022 2.263 2.263 1.078 1.531 1.003 1.096 1.278 1.193 1.248 1.18 1.388
4-2 1.174 1.122 1.455 1.002 2.315 2.315 1.092 1.474 1.01 1.11 1.288 1.209 1.258 1.185 1.403
4-3 1.141 1.094 1.422 0.983 2.237 2.237 1.059 1.456 0.982 1.077 1.252 1.172 1.222 1.153 1.362
5-1 1.202 1.142 1.477 1.012 2.384 2.384 1.12 1.475 1.033 1.138 1.317 1.239 1.286 1.209 1.437
5-2 1.165 1.122 1.459 1.013 2.276 2.276 1.081 1.508 1.003 1.098 1.279 1.196 1.249 1.179 1.39
5-3 1.204 1.152 1.494 1.031 2.37 2.37 1.119 1.518 1.036 1.137 1.321 1.239 1.29 1.215 1.438
6-1 1.172 1.111 1.436 0.982 2.328 2.328 1.092 1.427 1.007 1.109 1.283 1.208 1.254 1.178 1.401
6-2 1.168 1.116 1.447 0.997 2.302 2.302 1.086 1.466 1.005 1.103 1.281 1.202 1.251 1.178 1.395
6-3 1.179 1.117 1.444 0.988 2.342 2.342 1.098 1.435 1.013 1.116 1.291 1.215 1.261 1.185 1.409
7-1 1.174 1.095 1.406 0.949 2.369 2.369 1.097 1.344 1.005 1.115 1.281 1.213 1.253 1.17 1.405
7-2 1.137 1.069 1.378 0.937 2.275 2.275 1.061 1.345 0.975 1.078 1.243 1.173 1.215 1.138 1.36
7-3 1.168 1.081 1.383 0.929 2.369 2.369 1.092 1.301 0.997 1.11 1.271 1.206 1.245 1.158 1.397
8-1 1.239 1.124 1.423 0.944 2.55 2.55 1.158 1.279 1.049 1.18 1.337 1.276 1.315 1.212 1.481
8-2 1.128 1.019 1.288 0.853 2.328 2.328 1.054 1.149 0.954 1.074 1.216 1.161 1.196 1.101 1.349
8-3 1.077 0.988 1.259 0.841 2.2 2.2 1.007 1.16 0.916 1.025 1.168 1.111 1.146 1.062 1.289
9-1 1.161 1 1.232 0.799 2.441 2.441 1.071 1.001 0.955 1.101 1.217 1.169 1.212 1.09 1.376
9-2 1.151 0.978 1.194 0.772 2.426 2.426 1.055 0.947 0.937 1.088 1.195 1.148 1.195 1.067 1.359
9-3 1.158 1.011 1.253 0.817 2.426 2.426 1.073 1.043 0.96 1.101 1.224 1.175 1.215 1.1 1.377
10-1 1.125 0.959 1.175 0.76 2.369 2.369 1.033 0.938 0.919 1.064 1.171 1.125 1.169 1.047 1.329
10-2 1.255 1.043 1.257 0.809 2.651 2.651 1.138 0.961 1.005 1.18 1.282 1.232 1.29 1.14 1.471
10-3 1.202 1.019 1.244 0.804 2.534 2.534 1.101 0.984 0.978 1.136 1.246 1.198 1.247 1.113 1.418
11-1 1.241 0.995 1.174 0.752 2.616 2.616 1.102 0.849 0.965 1.152 1.231 1.181 1.253 1.087 1.433
£c Pre/^c Exp
Specimen [211 [221 [231 [241 [251 [261 [271 [281 [291 [301 [311 [321 [331 [341 [351
11-2 1.234 0.985 1.159 0.743 2.599 2.599 1.092 0.833 0.956 1.144 1.219 1.169 1.243 1.076 1.422
11-3 1.121 0.924 1.108 0.712 2.368 2.368 1.012 0.835 0.892 1.051 1.137 1.093 1.148 1.009 1.31
12-1 1.244 0.982 1.149 0.735 2.616 2.616 1.093 0.813 0.955 1.148 1.217 1.166 1.245 1.072 1.427
12-2 1.157 0.938 1.114 0.714 2.441 2.441 1.034 0.819 0.908 1.078 1.158 1.112 1.174 1.024 1.342
12-3 1.137 0.913 1.079 0.692 2.397 2.397 1.01 0.783 0.886 1.056 1.129 1.084 1.149 0.998 1.314
Table 5 continued.
£c Pre/£ c Exp
Specimen [361 [371 [381 [391 [391b [401 [411 [421 [431 [441 [451 [461 [471 [481a [481b
1-1 0.992 1.108 1.327 1.102 1.332 1.703 1.104 1.237 1.218 0.988 0.898 0.885 1.144 0.912 0.876
1-2 1.03 1.15 1.377 1.145 1.383 1.77 1.146 1.284 1.265 1.025 0.933 0.919 1.187 0.948 0.91
1-3 1.017 1.135 1.36 1.128 1.364 1.743 1.131 1.267 1.247 1.012 0.92 0.906 1.172 0.935 0.897
2-1 1.037 1.15 1.382 1.134 1.377 1.747 1.142 1.282 1.255 1.025 0.93 0.913 1.188 0.942 0.904
2-2 1.038 1.156 1.386 1.147 1.389 1.772 1.151 1.29 1.269 1.031 0.937 0.922 1.194 0.951 0.913
2-3 1.081 1.203 1.443 1.191 1.443 1.837 1.196 1.342 1.317 1.072 0.974 0.958 1.242 0.988 0.948
3-1 1.076 1.186 1.43 1.16 1.413 1.783 1.173 1.32 1.285 1.057 0.956 0.937 1.226 0.966 0.927
3-2 1.023 1.126 1.36 1.101 1.341 1.691 1.114 1.253 1.219 1.004 0.908 0.889 1.165 0.917 0.88
3-3 1.055 1.158 1.4 1.129 1.377 1.734 1.144 1.288 1.251 1.033 0.933 0.913 1.199 0.941 0.903
4-1 1.086 1.194 1.442 1.165 1.42 1.79 1.18 1.328 1.291 1.064 0.962 0.941 1.235 0.971 0.932
4-2 1.102 1.203 1.461 1.164 1.422 1.785 1.185 1.336 1.29 1.072 0.967 0.943 1.247 0.972 0.933
4-3 1.068 1.169 1.417 1.134 1.385 1.741 1.153 1.299 1.257 1.042 0.941 0.918 1.211 0.947 0.909
5-1 1.131 1.23 1.497 1.185 1.449 1.817 1.209 1.365 1.314 1.096 0.988 0.961 1.276 0.991 0.951
5-2 1.089 1.194 1.445 1.162 1.418 1.784 1.179 1.328 1.288 1.065 0.962 0.94 1.236 0.969 0.93
5-3 1.129 1.233 1.497 1.194 1.459 1.832 1.215 1.37 1.324 1.099 0.992 0.967 1.278 0.997 0.957
6-1 1.104 1.198 1.46 1.153 1.41 1.768 1.178 1.33 1.279 1.068 0.962 0.935 1.244 0.965 0.926
6-2 1.096 1.196 1.452 1.157 1.414 1.775 1.178 1.328 1.283 1.066 0.961 0.937 1.24 0.967 0.928
6-3 1.11 1.205 1.469 1.159 1.419 1.778 1.185 1.337 1.286 1.075 0.968 0.941 1.251 0.97 0.932
7-1 1.114 1.196 1.468 1.137 1.394 1.745 1.17 1.324 1.263 1.066 0.958 0.925 1.247 0.956 0.917
7-2 1.074 1.161 1.419 1.11 1.36 1.703 1.138 1.286 1.232 1.035 0.93 0.902 1.207 0.931 0.894
7-3 1.11 1.187 1.462 1.123 1.376 1.724 1.158 1.313 1.247 1.058 0.949 0.914 1.239 0.945 0.907
8-1 1.184 1.249 1.554 1.165 1.428 1.795 1.212 1.378 1.295 1.113 0.996 0.95 1.312 0.985 0.946
8-2 1.079 1.135 1.415 1.057 1.295 1.629 1.101 1.252 1.175 1.012 0.905 0.862 1.194 0.894 0.859
8-3 1.026 1.091 1.349 1.026 1.258 1.577 1.062 1.205 1.14 0.972 0.871 0.836 1.142 0.865 0.83
9-1 1.114 1.137 1.451 1.028 1.251 1.606 1.09 1.247 1.143 1.013 0.903 0.837 1.216 0.879 0.844
9-2 1.103 1.116 1.434 1.001 1.214 1.573 1.067 1.223 1.114 0.995 0.886 0.813 1.2 0.859 0.824
9-3 1.111 1.143 1.45 1.042 1.27 1.621 1.1 1.256 1.158 1.019 0.909 0.849 1.217 0.888 0.853
10-1 1.078 1.094 1.403 0.983 1.194 1.542 1.047 1.199 1.094 0.975 0.868 0.8 1.174 0.843 0.809
10-2 1.198 1.197 1.555 1.06 1.279 1.681 1.14 1.309 1.18 1.067 0.95 0.857 1.298 0.914 0.878
10-3 1.151 1.164 1.497 1.043 1.265 1.64 1.113 1.275 1.16 1.037 0.924 0.847 1.252 0.895 0.859
11-1 1.174 1.149 1.519 0.998 1.193 1.61 1.087 1.253 1.112 1.024 0.911 0.793 1.263 0.868 0.833
11-2 1.165 1.138 1.508 0.986 1.178 1.594 1.076 1.24 1.1 1.015 0.903 0.782 1.253 0.859 0.824
11-3 1.068 1.062 1.386 0.936 1.127 1.491 1.009 1.16 1.042 0.947 0.842 0.754 1.156 0.809 0.777
12-1 1.17 1.137 1.513 0.979 1.166 1.592 1.072 1.238 1.092 1.013 0.902 0.769 1.257 0.855 0.821
12-2 1.097 1.081 1.422 0.944 1.133 1.515 1.024 1.179 1.052 0.964 0.857 0.756 1.183 0.819 0.787
12-3 1.076 1.054 1.393 0.916 1.097 1.477 0.998 1.15 1.022 0.94 0.836 0.73 1.158 0.797 0.765
Table 5 continued.
£c Pre/^c Exp
Specimen [49] [50] [51] [52] [53] [54] [55] [56] [57] a [57] b [58] [59]
1-1 1.332 1.104 0.939 1.171 1.629 1.034 0.876 1.151 0.892 0.806 1.018 0.799
1-2 1.383 1.146 0.975 1.213 1.692 1.087 0.91 1.195 0.926 0.837 1.057 0.831
1-3 1.364 1.131 0.962 1.203 1.667 1.047 0.897 1.179 0.913 0.825 1.042 0.818
2-1 1.377 1.142 0.97 1.25 1.671 0.928 0.902 1.196 0.921 0.83 1.052 0.818
2-2 1.389 1.151 0.979 1.232 1.695 1.04 0.912 1.201 0.93 0.839 1.061 0.831
2-3 1.443 1.196 1.017 1.291 1.757 1.04 0.947 1.25 0.966 0.871 1.102 0.861
3-1 1.413 1.173 0.995 1.32 1.704 0.823 0.924 1.234 0.946 0.849 1.079 0.834
3-2 1.341 1.114 0.945 1.258 1.616 0.761 0.876 1.172 0.898 0.806 1.025 0.79
3-3 1.377 1.144 0.971 1.304 1.656 0.741 0.899 1.206 0.922 0.826 1.052 0.81
4-1 1.42 1.18 1.001 1.339 1.71 0.785 0.928 1.243 0.951 0.853 1.085 0.836
4-2 1.422 1.185 1.004 1.383 1.703 0.644 0.928 1.255 0.953 0.852 1.088 0.832
4-3 1.385 1.153 0.977 1.332 1.662 0.676 0.904 1.219 0.928 0.83 1.059 0.812
5-1 1.449 1.209 1.024 1.431 1.731 0.589 0.945 1.285 0.972 0.868 1.11 0.846
5-2 1.418 1.179 1 1.351 1.704 0.735 0.926 1.244 0.949 0.851 1.084 0.833
5-3 1.459 1.215 1.029 1.415 1.748 0.675 0.952 1.287 0.978 0.874 1.116 0.854
6-1 1.41 1.178 0.997 1.399 1.683 0.551 0.92 1.252 0.947 0.844 1.081 0.823
6-2 1.414 1.178 0.998 1.375 1.693 0.64 0.922 1.248 0.948 0.847 1.082 0.828
£c Pre/^c Exp
Specimen [491 [501 [511 [521 [531 [541 [551 [561 [571 a [571 b [581 [591
6-3 1.419 1.185 1.002 1.408 1.693 0.555 0.925 1.26 0.952 0.849 1.087 0.828
7-1 1.394 1.17 0.988 1.439 1.654 0.373 0.909 1.255 0.939 0.834 1.072 0.811
7-2 1.36 1.138 0.962 1.375 1.618 0.453 0.886 1.215 0.914 0.814 1.044 0.792
7-3 1.376 1.158 0.978 1.445 1.63 0.3 0.898 1.248 0.929 0.824 1.061 0.8
8-1 1.428 1.212 1.021 1.571 1.685 0.112 0.933 1.321 0.97 0.856 1.108 0.831
8-2 1.295 1.101 0.927 1.436 1.527 0.07 0.847 1.203 0.881 0.776 1.006 0.753
8-3 1.258 1.062 0.896 1.348 1.486 0.195 0.821 1.15 0.851 0.753 0.972 0.731
9-1 1.251 1.09 0.914 1.531 1.472 0.28 0.828 1.225 0.868 0.757 0.993 0.739
9-2 1.214 1.067 0.894 1.527 1.431 0.362 0.807 1.209 0.849 0.738 0.971 0.722
9-3 1.27 1.1 0.923 1.516 1.495 0.194 0.838 1.226 0.877 0.767 1.002 0.747
10-1 1.194 1.047 0.877 1.489 1.407 0.329 0.793 1.183 0.833 0.725 0.953 0.709
10-2 1.279 1.14 0.953 1.677 1.512 0.523 0.858 1.307 0.905 0.784 1.035 0.771
10-3 1.265 1.113 0.932 1.595 1.491 0.386 0.841 1.261 0.885 0.77 1.012 0.753
11-1 1.193 1.087 0.906 1.666 1.419 0.687 0.812 1.273 0.861 0.741 0.985 0.736
11-2 1.178 1.076 0.897 1.656 1.402 0.701 0.803 1.263 0.852 0.733 0.975 0.729
11-3 1.127 1.009 0.843 1.501 1.334 0.508 0.758 1.164 0.801 0.693 0.916 0.683
12-1 1.166 1.072 0.893 1.67 1.391 0.751 0.798 1.266 0.848 0.729 0.971 0.727
12-2 1.133 1.024 0.855 1.551 1.344 0.595 0.767 1.192 0.812 0.7 0.929 0.693
12-3 1.097 0.998 0.832 1.525 1.304 0.622 0.745 1.167 0.79 0.681 0.904 0.675
Table 5 continued.
£c Pre/^c Exp
Specimen [601 [611 [621 [631 [641 [651 [661 a [661 b [661 c [671 [681 Proposed
1-1 1.258 1.155 1.097 1.175 1.124 0.784 1.092 0.914 0.793 4.362 2.518 0.945
1-2 1.304 1.199 1.141 1.219 1.167 0.816 1.133 1.133 0.824 4.545 2.58 0.981
1-3 1.29 1.184 1.122 1.205 1.152 0.802 1.119 1.119 0.811 4.453 2.614 0.969
2-1 1.323 1.201 1.12 1.232 1.168 0.796 1.138 1.138 0.81 4.344 3.027 0.986
2-2 1.317 1.207 1.14 1.229 1.174 0.814 1.141 1.141 0.824 4.503 2.733 0.988
2-3 1.375 1.256 1.181 1.283 1.221 0.842 1.188 1.188 0.853 4.632 2.958 1.029
3-1 1.38 1.24 1.139 1.282 1.206 0.805 1.176 1.176 0.822 4.311 3.567 1.02
3-2 1.314 1.178 1.079 1.22 1.146 0.762 1.118 1.118 0.779 4.071 3.464 0.97
3-3 1.356 1.212 1.105 1.258 1.179 0.778 1.151 1.151 0.798 4.134 3.723 0.999
Sc Pre/Sc Exp
Specimen [60] [61] [62] [63] [64] [65] [66] a [66] b [66] c [67] [68] Proposed
4-1 1.395 1.249 1.141 1.295 1.215 0.805 1.186 1.186 0.824 4.287 3.753 1.029
4-2 1.425 1.261 1.133 1.319 1.226 0.792 1.199 1.199 0.816 4.138 4.402 1.042
4-3 1.378 1.224 1.107 1.277 1.191 0.776 1.164 1.164 0.798 4.084 4.052 1.011
5-1 1.467 1.29 1.151 1.356 1.255 0.801 1.228 1.228 0.828 4.146 4.835 1.068
5-2 1.402 1.25 1.136 1.3 1.216 0.799 1.188 1.188 0.82 4.226 3.955 1.031
5-3 1.459 1.292 1.163 1.351 1.257 0.814 1.229 1.229 0.838 4.262 4.443 1.068
6-1 1.432 1.257 1.119 1.324 1.223 0.778 1.198 1.198 0.805 4.012 4.827 1.041
6-2 1.417 1.253 1.126 1.312 1.219 0.788 1.192 1.192 0.812 4.115 4.377 1.036
6-3 1.441 1.265 1.125 1.331 1.23 0.782 1.205 1.205 0.809 4.036 4.855 1.047
7-1 1.455 1.257 1.098 1.341 1.224 0.753 1.201 1.201 0.786 3.791 5.852 1.046
7-2 1.399 1.219 1.074 1.291 1.186 0.743 1.162 1.162 0.771 3.787 5.12 1.011
7-3 1.454 1.248 1.082 1.338 1.216 0.738 1.194 1.194 0.773 3.676 6.292 1.04
8-1 1.563 1.316 1.118 1.433 1.285 0.749 1.263 1.263 0.793 3.626 8.315 1.102
8-2 1.426 1.197 1.014 1.307 1.169 0.677 1.149 1.149 0.717 3.259 7.869 1.004
8-3 1.35 1.149 0.986 1.24 1.12 0.668 1.1 1.1 0.702 3.282 6.404 0.959
9-1 1.491 1.204 0.987 1.359 1.184 0.629 1.164 1.164 0.682 2.86 12.54 1.02
9-2 1.481 1.184 0.962 1.347 1.166 0.605 1.146 1.146 0.66 2.71 13.95 1.005
9-3 1.483 1.21 0.999 1.353 1.187 0.645 1.167 1.167 0.695 2.975 11.2 1.022
10-1 1.447 1.16 0.945 1.316 1.141 0.597 1.122 1.122 0.65 2.685 13.16 0.984
10-2 1.617 1.272 1.024 1.468 1.257 0.629 1.234 1.234 0.693 2.758 18.02 1.085
10-3 1.547 1.235 1.003 1.407 1.216 0.63 1.195 1.195 0.687 2.816 14.74 1.049
11-1 1.594 1.225 0.973 1.443 1.218 0.577 1.193 1.193 0.646 2.449 22.8 1.05
11-2 1.583 1.213 0.963 1.433 1.207 0.569 1.182 1.182 0.637 2.404 23.29 1.041
11-3 1.444 1.13 0.906 1.31 1.118 0.552 1.097 1.097 0.611 2.399 17.14 0.964
12-1 1.593 1.213 0.959 1.441 1.209 0.56 1.183 1.183 0.631 2.348 25.17 1.043
12-2 1.487 1.151 0.918 1.348 1.142 0.551 1.12 1.12 0.613 2.358 19.74 0.986
12-3 1.46 1.124 0.893 1.322 1.117 0.531 1.094 1.094 0.593 2.257 20.6 0.963
When R2 equals the value of one that does not mean a perfect prediction because other indices influence the selection. In Table 6, the predictions of two strain models [36, 62] yielded equal values of R2 (0.736) but with higher values of AAE (9.4% and 9.02%) and lower values of E1 (0.99 and 0.988) compared with 3.46% and 0.998 for the proposed model. Therefore, for this kind of evaluation, R2 is not the most critical index.
Figure. 7. Predicted strain at peak stress versus the fc for G-1 strain models with polynomial functions.
3.6. Applicability of strain models
The five best performing models [27, 33, 44, 51, 58] and the proposed ANN-based model were checked against the test results reported by Woldemariam et al. [77] for concrete strain at peak stress, specimens C1 (sc
=0.0021), C2 (sc =0.0026), C3 (sc =0.0029), C4 (sc =0.0031), C5 (sc =0.0033), respectively. The values of AAE
in percentage were 38.7 [27], 39.6 [33], 43.21 [44], 49.7 [58], and 41.5 for the proposed model, respectively. The corresponding values of NRMSE were close, 0.021 [27], 0.022 [33], 0.021 [44], 0.026 [58], and 0.022 for the proposed model, respectively. However, it was slightly higher for model [51] with a value of 0.026, (Fig. 11). For the coefficient of correlation, the corresponding values were (0.985, 0.945, 0.98, 0.992, 0.992) for the five models [27, 33, 44, 51, 58] and 0.988 for the proposed model. However, other statistical measures show more measured errors (standard deviations with values of 0.07, 0.102, 0.052, 0.034, and 0.037 for the five models and 0.068 for the proposed model), Fig. 11.
Figure. 8. Predicted strain at peak stress versus the fc for G-2 strain models with linear functions.
The corresponding values of covariance are 0.005, 0.01, 0.003, 0.001, and 0.0014 for the five models [27, 33, 44, 51, 58] and 0.004 for the proposed model, respectively. Therefore the efficiency of the models in predicting the strain at peak stress can be better represented by the four indexes AAE, NRMSE, E, and E1 compared with the R2 index.
Figure. 9. Predicted strain at peak stress versus the fc for G-3 strain models with power functions.
Figure 10. NRMSE values in model predictions of strain at peak stress [21-68] and the proposed model. Table 6. Performance of the strain models using statistical indexes.
Model Min. Max. Mean COV STD NRMSE AAE(%) E E1 R2
Collins et al. [211 1.077 1.255 1.161 0.002 0.045 0.005 16.09 0.973 0.837 0.744
Tasdemir et al.[221 0.913 1.156 1.060 0.005 0.069 0.003 7.85 0.991 0.918 0.732
Emperger [231 1.079 1.514 1.343 0.017 0.131 0.010 34.33 0.854 0.642 0.720
Ros [241 0.929 1.446 1.118 0.025 0.158 0.004 10.70 0.980 0.894 0.736
Hognestad et al. [251 2.033 2.651 2.336 0.024 0.157 0.037 133.6 -0.818 -0.36 0.706
Liebenberg [261 2.034 2.651 2.336 0.024 0.157 0.037 133.6 -0.818 -0.36 0.706
Saenz [271 0.988 1.158 1.067 0.002 0.040 0.0025 6.761 0.994 0.932 0.701
Model
Min.
Max.
Mean
COV
STD NRMSE AAE(%)
E
E1
R2
Soliman et al.[28] 0.783 1.660 1.271 0.083 0.290 0.0110 34.02 0.823 0.64 0.736
Popovics [29] 1.054 1.249 1.158 0.002 0.047 0.005 15.79 0.972 0.838 0.711
Tadros [30] 1.009 1.18 1.093 0.002 0.042 0.003 9.29 0.99 0.907 0.736
Popovics [31] 1.129 1.337 1.24 0.002 0.050 0.007 24.00 0.938 0.735 0.711
Bashur and Darwin [32] 1.084 1.276 1.171 0.002 0.045 0.0048 17.10 0.968 0.826 0.675
Ahmad-Shah [33] 0.839 1.093 0.963 0.004 0.064 0.0021 6.08 0.994 0.936 0.736
Tomaszewicz [34] 0.998 1.215 1.129 0.003 0.058 0.004 12.87 0.979 0.867 0.713
Shah-Fafitis [35] 1.283 1.481 1.375 0.002 0.049 0.01 37.53 0.855 0.616 0.736
Carreira-Chu [36] 0.992 1.198 1.093 0.002 0.049 0.003 9.37 0.99 0.907 0.736
Ali et al. [37] 1.054 1.249 1.158 0.002 0.047 0.0046 15.8 0.972 0.834 0.711
Hsu and Hsu [38] 1.327 1.555 1.439 0.003 0.056 0.0121 43.9 0.803 0.522 0.736
De Nicolo et al. [39] 0.916 1.194 1.09 0.006 0.08 0.003 10.44 0.984 0.891 0.714
Brandtzaeg [39] B 1.097 1.459 1.325 0.011 0.104 0.01 32.47 0.876 0.663 0.693
Almusallam & Alsayed [40] 1.477 1.837 1.694 0.01 0.101 0.02 69.35 0.484 0.284 0.736
CEB-FIB [41] 0.998 1.215 1.129 0.003 0.058 0.004 12.87 0.979 0.867 0.713
Arioglu [42] 1.15 1.378 1.28 0.003 0.057 0.008 27.97 0.915 0.712 0.712.
Attard and Setunge [43] 1.022 1.324 1.21 0.007 0.086 0.006 20.99 0.945 0.781 0.711
Wee et al. [44] 0.94 1.113 1.032 0.002 0.042 0.001 4.45 0.997 0.955 0.711
Guo [45] 0.836 0.996 0.927 0.002 0.041 0.0023 7.24 0.993 0.925 0.72
Xu [46] 0.73 0.967 0.88 .0045 0.068 0.0037 11.95 0.982 0.879 0.697
CEB-FIB [47] 1.142 1.312 1.22 0.002 0.043 0.006 21.96 0.95 0.776 0.736
Mansur et al. [48] a 0.797 0.997 0.92 0.003 0.055 0.0026 8.04 0.991 0.919 0.715
Mansur et al. [48] b 0.765 0.957 0.883 0.003 0.053 0.0035 11.71 0.984 0.881 0.715
Lee [49] 1.097 1.459 1.325 0.011 0.104 0.01 32.47 0.876 0.663 0.693
NS 3473 [50] 0.998 1.215 1.129 0.003 0.058 0.004 12.87 0.979 0.867 0.713
Yu and Ding [51] 0.832 1.029 0.952 0.0028 0.053 0.0019 5.34 0.995 0.946 0.714
EC2 [52] 1.171 1.677 1.422 0.019 0.139 0.012 42.20 0.812 0.575 0.755
Tasnimi [53] 1.304 1.757 1.584 0.018 0.134 0.017 58.38 0.618 0.395 0.718
Mertol [54] 0.072 1.087 0.605 0.073 0.727 0.013 40.85 0.775 0.589 0.436
Lu and Zhao [55] 0.745 0.952 0.873 0.0033 0.058 0.0038 12.69 0.981 0.871 0.716
Arslan G, Cihanli E [56] 1.15 1.321 1.228 0.002 0.043 0.006 22.79 0.946 0.767 0.736
Ding et al. [57] a 0.79 0.978 0.904 0.0025 0.051 0.0029 9.57 0.988 0.903 0.714
Ding et al. [57] b 0.68 0.874 0.801 0.003 0.055 0.0057 19.92 0.957 0.797 0.716
Kumar et al. [58] 0.904 1.117 1.033 0.0032 0.057 0.0018 5.71 0.995 0.941 0.714
Hussin e t al. [59] 0.675 0.861 0.786 0.003 0.053 0.006 21.43 0.951 0.781 0.736
Chen et al. [60] 1.175 1.468 1.318 0.005 0.071 0.009 31.84 0.896 0.677 0.736
Lim and Ozbakkaloglu [61] 1.123 1.316 1.219 0.0021 0.046 0.0062 21.89 0.949 0.776 0.71
Ahmed et al.[62] 0.893 1.181 1.062 0.007 0.083 0.003 9.02 0.988 0.906 0.736
Chen et al. [63] 1.175 1.468 1.318 0.005 0.071 0.009 31.84 0.896 0.677 0.736
Wang et al. [64] 1.117 1.285 1.193 0.002 0.042 0.005 19.25 0.961 0.803 0.724
Shanaka [65] 0.531 0.842 0.715 0.0094 0.097 0.0081 28.51 0.912 0.711 0.722
Nematzadeh [66] a 1.091 1.2629 1.167 0.0017 0.042 0.005 16.7 0.97 0.83 0.709
Nematzadeh [66] b 0.757 0.9875 0.901 0.0044 0.067 0.003 9.88 0.986 0.901 0.717
Nematzadeh [66] c 0.593 0.8533 0.750 0.0061 0.079 0.007 24.99 0.932 0.746 0.721
Hoang and Fehling [67] 2.257 4.632 3.586 0.602 0.782 0.077 258.6 -6.96 -1.7 0.735
Aslam et al. [68] 2.518 25.17 8.858 47.28 6.924 0.27 785.8 -96.51 -6.71 0.598
Proposed 0.945 1.102 1.018 0.001 0.037 0.0011 3.46 0.998 0.965 0.708
♦ Woldemariam et al. [77]
Wee et al. [44] A Proposed
O Saenz [27] X Yu and Ding [51]
Ahmad-Shah [33] ■ Kumar et al. [58]
0,0028 (0,002
0,0012
7
11
15 fco (MP)
23
27
[27]
NRMSE=0.021
AAE=38.7%
E=0.828
E1=0.59
R2 =0.985
STD=0.07
COV=0.005
[33] NRMSE=0.022 AAE=39.6% E=0.78 E1=0.58 R2 =0.945 STD=0.102 COV=0.01
[44]
NRMSE=0.021 AAE=43.21% E=0.795 E1=0.55 R2 =0.98 STD=0.052 COV=0.003
[51]
NRMSE=0.026
AAE=49.7%
E=0.737
E1=0.54
R2 =0.992
STD=0.034
COV=0.001
[58]
NRMSE=0.023
AAE=44.9%
E=0.784
E1=0.534
R2 =0.992
STD=0.037
COV=0.0014
Proposed
NRMSE=0.022
AAE=41.5%
E=0.804
E1=0.56
R2 =0.988
STD=0.068
COV=0.004
Figure 11. Strain predictions of test results of [77] using models [27, 33, 44, 51, 58] and the proposed
model.
4. Conclusions
1. Several parameters such as aggregate/cement ratio, w/c ratio, and slump values that influence the compressive strength and strain at peak stress of concrete were considered as input for the two ANN models.
2. The developed ANN models successfully yielded good predictions of the test results presented in the current study.
3. The AAE value was found to be less than 3.46% for the proposed model, and the NRMSE value was the lowest, 0.0011.
4. The predicted strains obtained from the regression of ANN output data for stress and strain at peak stress were more accurate than those obtained from the fifty-three existing expressions for predicting the strain at peak stress.
5. Both the NRMSE and AAE indexes allow the assessing of the performance of the strain models for the present study more properly than the R2 index. ANN procedure is a valuable modeling technique for practicing engineers interested in concrete technology.
6. The models with power function show better performance in predicting the strain at peak stress than models with linear or polynomial functions.
New research should be carried out along these lines to include input parameters outside the range considered in the present study and to improve the prediction capability of the proposed model and its application to test data with higher strengths.
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Contacts:
Nwzad Abduljabar Abdulla, anwzad@yahoo.com
Received 27.05.2020. Approved after reviewing 11.05.2021. Accepted 12.05.2021.
