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Neural Computing and Applications (2020) 32:11807–11826 https://doi.org/10.1007/s00521-019-04663-2 (0123456789().,-volV)(0123456789(). ,- volV) 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 123 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 123 11810 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 123 11812 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 11814 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. References 1. Bungey JH, Millard SG (1996) Testing of concrete in structures, 3rd edn. Blackie Academic & Professional, London 2. Trtnik G, Kavčič F, Turk G (2009) Prediction of concrete strength using ultrasonic pulse velocity and artificial neural networks. Ultrasonic 49(1):53–60 3. ASTM C 597-83 (1991) Test for pulse velocity through concrete. 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