1. Introduction
Devastating casualties could result from a concrete slab’s failure under punching shear, which is due to it occurring suddenly with no warning. Many recent punching shear failures have been reported worldwide [
1,
2,
3,
4,
5]. Several design models were developed for punching shear strength; however, these models vary significantly in the considered parameters and mechanism in developing the model [
6,
7,
8]. For example, the European concrete design code (EC2) [
6] model is semi-empirical. In contrast, the FIB model design code (MC) [
7] is physically based. Thus, punching shear strength continues to be a dilemma that involves many parameters and mechanisms affecting the strength [
9,
10,
11,
12,
13,
14]. The load transfer mechanisms for punching shear strength consists of the following components: (1) the direct shear resistance of the un-cracked compression zone; (2) the friction shear across the diagonal crack from the aggregate interlock and interface shear; (3) the bending and direct shear resistance of the flexure reinforcement crossing the cracks (i.e., dowel action). The effective parameters are as follows: (1) yield strength of steel; (2) concrete properties; (3) flexure reinforcements; (4) slab dimensions; and (5) column dimensions.
The cost of slabs is relatively high; thus, the engineering community tends to look at reducing the cost via implementing new materials, such as concrete with fiber-reinforced polymer, lightweight concrete (LWC), fiber-reinforced concrete, or high-strength concrete [
15,
16,
17,
18,
19,
20]. However, LWC is an overwhelming choice, different from conventional normal-weight concrete (NWC) in the aggregate type. In NWC, diagonal shear cracks propagate around the strong aggregates; thus, sharp aggregate sticks out across the crack edges. On the other hand, diagonal shear cracks in LWC will go through the weak aggregate [
21,
22,
23]. Therefore, the LWC punching shear strength could be different from that of NWC.
For the case of LWC slabs under punching shear, design codes are based on research studies focused on investigating either LWC beams under one-way shear [
23,
24] or NWC slabs under punching shear [
25,
26,
27,
28,
29]. In addition, there are very limited studies in the available literature verifying the most recent revision of the American concrete design code (ACI) [
8] design code for the punching shear of LWC slabs. Globally, researchers have tackled the punching shear design of NWC and LWC slabs for more than six decades. Most of the research studies were experimental ones [
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45]. Based on this pioneering work of Moe [
30], the ACI [
8] punching shear design provisions for NWC slabs were developed. Later, based on the pioneering work of Hognestad [
31] and Ivy [
32], the ACI was modified to fit LWC slabs. Those modifications used either the actual tensile splitting strength or a reduction factor (λ) for calculating the shear strength of LWC slabs. A more recent study by Caratelli [
39] concluded that the MC is conservative for the case of LWC under punching shear; thus, increasing the aggregate factor by 40–60% of the nominal maximum aggregate size was proposed. On the other hand, the ACI [
8] proposed a modified reduction factor using the concrete density, which was developed to investigate LWC beams under one-way shear [
22,
23].
This current study investigates the punching shear strength of LWC slabs. An extensive experimental database for LWC slabs tested under punching shear loading is collected. Effective parameters were selected using statistical analysis: concrete density, columns dimensions, slab effective depth, concrete strength, flexure reinforcement ratio, and steel yield stress. Three artificial intelligence (AI) techniques were selected and used to develop punching shear strength models for LWC slabs. Strength predictions were compared with each other and with selected design codes concerning the experimentally measured strength. Recommendations and concluding remarks were outlined and discussed.
2. Experimental Database
An extensive experimental database for the experimental punching capacity of LWC slabs, which consists of 116 records, was collected [
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45]. Each record contains the following data: (1) concrete density (γ) in kN/m
3; (2) column width (short side) (
) in m; (3) column length (long side) (
) in m, (for circular columns
); (4) slab depth (
) in m; (5) 28-day cylinder compressive strength of concrete (
) in MPa; (5) longitudinal reinforcement ratio x yield strength of steel (
) in MPa; (6) ultimate punching capacity (
) in kN. These collected records were divided into a training set (90 records) and a validation set (26 records).
Table 1 summarizes their statistical characteristics. In addition,
Figure 1 shows the histograms for both inputs and outputs.
5. AI Model Development
Artificial intelligence (AI) techniques are searching algorithms that aim to find the best solution for certain problems according to certain criteria within the available time and resources. (AI) techniques may be classified into statistically based techniques (such as fuzzy logic), decision-tree-based techniques (such as expert systems), human brain simulation techniques (such as neural networks), evolutionary-based techniques (such as genetic algorithm and genetic programming), and mimicking natural creature behavior techniques (such as ant colony and particle swarm) [
46].
ANN is a well-known technique that simulates the human brain activity. The model consists of a number of cells (nodes) arranged in groups (layers), where the cells of each group are connected to cells of other groups with links. Each link has its importance (weight) and each cell has its triggering mechanism (activation function). During the training stage, information propagates from the first group (input layer) to the last group (output layer) through the intermediate groups (hidden layers). The errors in the predicted values are backpropagated to the first group (input layer). During this process, the weights of links are updated to enhance the performance. ANN is a very powerful and flexible tool that can simulate any non-linear behavior, but its output is a weights matrix that cannot be utilized manually.
Figure 2 presents a general schematic for the artificial neural network.
Genetic algorithms (GA) are AI-optimizing techniques that mimic the evolution procedure of living creatures. It starts with generating a random set of solutions, and then the fitness of each solution is evaluated, the most fitting solutions are selected, and the rest are deleted. Those selected solutions are used to produce the next generation of solutions, and the cycle continues until the desired accuracy is achieved. This technique was used as a base to develop more AI techniques, such as genetic programming (GP) and EPR.
GP is a technique that uses GA to optimize the fitness of the mathematical formula to a certain dataset. The output of this technique is a closed-form equation that could be utilized manually. EPR is another technique based on GA, where GA is used to optimize the traditional polynomial regression by selecting the most affecting terms and deleting the rest. The output of this technique is also a closed-form equation (polynomial) that could be utilized manually [
47].
Figure 3 shows the mathematical formula in a tree and genetic form, whereas
Figure 4 illustrates the flow chart of the EPR technique.
Three different artificial intelligent (AI) techniques were used to predict the ultimate punching capacity of LWC slabs using the collected database. These techniques are genetic programming (GP), artificial neural network (ANN), and polynomial regression optimized using a genetic algorithm, which is known as evolutionary polynomial regression (EPR). All of the three developed models were used to predict the ultimate punching capacity (Vu in kN) using the concrete density (γ in kN/m3), column width (a in m), column length (b in m), slab depth (d in m), 28-day concrete cylinder strength ( in MPa), and reinforcement ratio by steel yield stress (in MPa).
Each model on the three developed models was based on a different approach (evolutionary approach for GP, mimicking biological neurons for ANN, and optimized mathematical regression technique for EPR). However, for all developed models, the prediction accuracy was evaluated in terms of the sum of squared errors (SSE), which is calculated such that:
The following section discusses the results of each model. The accuracies of the developed models were evaluated by comparing the (SSE) between predicted and calculated shear strength parameters values. The results of all developed models are summarized in
Table 4.
5.1. GP Model
The developed GP model has four levels of complexity. The population size, survivor size, and number of generations were 100,000, 25,000, and 100, respectively. Equation (4) presents the output formulas for (
Vu), whereas
Figure 5a shows its fitness. The average error % of total dataset is (32.4%), while the (R
2) value is (0.823).
5.2. ANN Model
A backpropagation ANN with one hidden layer and nonlinear activation function (Hyper Tan) was used to predict (Vu) values. The used networks layout and their weights are illustrated in
Figure 6 and
Table 5. The average error % of total dataset was (26.1%) and the (R
2) value was (0.890). The relative importance values for each input parameter are illustrated in
Figure 7, which indicates that the slab thickness is the most important factor, whereas the concrete strength and flexure reinforcement ratio is second after the depth. The relationship between the calculated and predicted values are shown in
Figure 5b.
5.3. EPR Model
Finally, the developed EPR model was limited to the quadrilateral level. For six inputs, there are 210 possible terms (126 + 56 + 21 + 6+1 = 210) as follows:
The GA technique was applied on these 210 terms to select the most effective ten terms to predict the values of (Vu). The outputs are illustrated in Equation (5), and its fitness is shown in
Figure 7c. The average error % and (R
2) values were (26.4%—0.888) for the total datasets.
8. Conclusions
This research presents three models using three (AI) techniques (GP, ANN, and EPR) to predict the punching capacity of LWC slabs (Vu) using the concrete density (γ), columns dimensions (a & b), slab depth (d), concrete strength (), and reinforcement ratio by steel yield stress (). Although concluding remarks are limited to the range of parameter values in the database, which can improve with more testing of slabs, concluding remarks are as follows:
Both (ANN) and (EPR) have the greatest prediction accuracy (73.9% and 73.6%, respectively), whereas the (GP) model has the lowest prediction accuracy (67.6%);
(GP) and (EPR) have almost the same level of accuracy (65.3% and 68.1%, respectively);
Although the error% of the (ANN) and (EPR) models are so close, the output of (EPR) is closed-form equations, which could be used manually or as software, unlike the (ANN) output, which cannot be used manually;
The summation of the absolute weights of each neuron in the input layer of the developed (ANN) model indicates that the slab depth (d) has a major influence on the punching capacity; other parameters have a minor effect, especially the compressive strength of the concrete;
The formula developed using (EPR) did not include the parameter (), which indicates its minor effect on the punching capacity.
The GA technique successfully reduced the 210 terms of the conventional polynomial regression quadrilateral formula to only ten terms without significant impact on its accuracy.
AI models captured the true behavior and overcame the variability of the traditional design codes concerning the effective parameters.