An Adaptive Neuro-Fuzzy Inference Model to Predict Punching Shear Strength of Flat Concrete Slabs

Abstract

An adaptive neuro-fuzzy inference system (ANFIS)-based model was developed to predict the punching shear strength of flat concrete slabs without shear reinforcement. The model was developed using a database collected from 207 experiments available in the existing literature. Five key input parameters were used to build the model, which were slab effective depth, concrete strength, reinforcement ratio, yield tensile strength of reinforcement, and width of square loaded area. The output parameter of the model was punching shear strength. The results from the adaptive neural fuzzy inference model were compared to those from the simplified punching shear equations of ACI, BS-8110, Model Code 2010, Euro-Code 2, and also experimental results. The root mean square error (RMSE) and the correlation coefficient (R) were used as evaluation criteria. Parametric studies were presented using ANFIS to assess the effect of each input parameter on the punching shear strength and to compare ANFIS results to those from the equations proposed in commonly used codes. The results showed that the ANFIS model is simple and provided the most accurate predictions of the punching shear strength of two-way flat concrete slabs without shear reinforcement.

1. Introduction

Generally, the contact surfaces between columns and slabs are very small in flab slab systems, and therefore high stresses are concentrated in the connections area. A punching shear failure may occur if the stresses exceed the limitations. This failure is brittle and may occur unexpectedly. To avoid this type of failure, various construction methods have been developed [1].

In the design and analysis of two-way flat slabs without shear reinforcement, the punching shear strength is an important parameter. Much research has been conducted throughout the current century, and the key variables affecting the punching shear strength of slabs have been identified [2,3,4,5]. Most of the research has been concerned with the generation of experimental data and the development of empirical equations in addition to the equations proposed by ACI 318-14 [6], BS-8110-97 [7], Model Code 2010 [8], and Euro-Code 2 [9]. However, the subject still needs further study to understand the complexity of punching shear behavior and to develop better prediction tools.

Fuzzy logic (FL) and neural network (NN) techniques have been widely used in civil engineering applications over the last two decades. In this study, an alternative model was developed within the framework of an adaptive neuro-fuzzy interface system (ANFIS) to predict the punching shear of two-way slabs without shear reinforcement. This model was developed using a large database (207 experimental results) compiled from 17 scientific studies. The predictions from this model were compared to those from the equations proposed in commonly used codes.

2. ANFIS: Literature Review

The solution of problems associated with engineering systems requires the use of several different disciplines implementing different methods of modeling and analysis. For a complex engineering system, often a physics-based mathematical model is used, which is extremely difficult to formulate. For such a system, several other approaches (neural networks, fuzzy inference systems, etc.) under the rubric of “soft computing” provide a useful alternative. Soft computing models are becoming popular and have been of increasing interest during the last three decades. This approach is based on human reasoning and learning and uses the human tolerance for uncertainty and imprecision and fuzziness in the decision-making processes [10]. Recently, artificial neural networks (ANNs) and ANFIS have been used extensively for various civil engineering applications in construction management, building materials, hydraulics, structural engineering, geotechnical and transportation engineering, etc. Here, a selected few recent works in the area related to our subject are presented. Kasperkiewicz et al. [11] developed an ANN to predict the compressive strength of high-performance concrete mixes. Takagi and Sugeno [12] developed a fuzzy inference system (FIS) model and applied it to modeling and controlling concepts. Topçu and Saridemir [13] applied ANN and FL to predict rubberized mortar properties. Bilgehan [14] used ANFIS and NN models to determine the critical buckling load. Tesfamariam and Najjaran [15] developed an ANFIS model to estimate the concrete strength of a given mix proportion based on existing datasets. Akbulut et al. [16] used ANFIS to predict the shear modulus and damping coefficient of sand and rubber mixtures. Inan et al. [17] used an adaptive neuro-fuzzy system to simulate nonlinear mapping in the sulphate expansion of Portland cement (PC) mortar. Experimental data that had previously been collected for various parameters were treated in the analysis. Fonseca et al. [18] developed a neuro-fuzzy model to classify and to predict the behavior of steel beams under concentrated loads. Wang and Elhag [19] applied ANFIS to assess bridge risk based on multiple bridge maintenance projects. Batenia and Jeng [20] used ANFIS to investigate the characteristics of a scour hole that develops around a group of piles in a well-defined field situation and to determine the parameters that control the scour hole. Mashrei [21] developed an ANFIS model to predict the shear strength of concrete beams reinforced with fiber-reinforced polymer (FRP) bars. Bilgehan and Kurtoglu [22] applied ANFIS to predict the moment capacities of reinforced concrete (RC) slabs exposed to fire. Mansouri et al. [23] investigated the ability of radial basis neural networks and ANFIS methods in the prediction of ultimate strength and strain of concrete cylinders confined with FRP sheets. Naderpour and Mirrashid [24] used ANFIS to determine the shear strength of RC beams with shear reinforcement. Basarir et al. [25] used an ANFIS model to predict the uniaxial compressive strength of cemented backfill.

3. Existing Equations Used for Two-Way Flat Slabs

For the design of a two-way flat slab–column connection, the shear stress is usually assumed to be a function of strength of concrete and the geometric parameters of the slab and column. The critical section for checking punching shear in slabs is usually situated between 0.5 and 2 times the effective depth from the edge of the load or reaction. Many empirical equations have been published to estimate the punching shear strength of two-way slabs, such as the equations proposed in ACI 318-14, BS-8110-97, Model-Code-2010, and Euro-Code 2 [6,7,8,9].

3.1. ACI 318-14 Building Code Equations

A set of simple equations were proposed in the ACI 318-14 code to calculate the shear strength provided by concrete. The control perimeter is half of the effective depth of the slab (0.5d) from the loaded area for punching shear stress. ACI 318-14 requires that the nominal shear resistance for slabs without shear reinforcement be approximated as the smallest value of $V_n$ calculated from the following expressions:

$$V_n= 0.083\left(2+\frac{4}{{\beta }_c}\right)\lambda \sqrt{f_c^{\prime }} b_o d ,$$

(1)

$$V_n=0.083\left({\alpha }_s\frac{d}{b_o}+2\right)\mathsf{\lambda } \sqrt{f_c^{\prime }} b_o d ,$$

(2)

$$V_n=0.33 \mathsf{\lambda } \sqrt{f_c^{\prime }} b_o d ,$$

(3)

where $V_n$ is the shear strength in N, b_o is the perimeter of the critical section in mm, $d$ is the effective depth of slab in mm, and λ = 1.0 for normal weight concrete and 0.75 for all lightweight concrete. Otherwise, λ is determined based on volumetric proportions of lightweight and normal weight aggregates, but does not exceed 0.85. Here, ${\alpha }_s$= 40 for interior columns, 30 for edge columns, and 20 for corner columns; ${\beta }_c$ is the ratio of the longer to the shorter dimension of the loaded area; and $f_c^{\prime }$ is the cylinder compressive strength of concrete in MPa.

3.2. Model Code 2010

The nominal punching shear strength is assumed to be proportional to ($f_{ck}$)^1/3 in Model Code 2010. The influences of the slab depth and steel reinforcement are also considered in this model. The punching strength according to Model Code 2010 is expressed by

$$V_n=0.18b_{o}d\times \xi \times \sqrt[3]{100\times \rho \times f_{ck}},$$

(4)

where $f_{ck}$ is the characteristic cylinder compressive strength in MPa, ξ = 1 + (200/d)^1/2 is a size effect coefficient, d is the slab effective depth in mm, ρ is the ratio of flexure reinforcement, and b_o is the length of the control perimeter at 2d from the column face in mm.

3.3. British Code: BS-8110-97

The British Code provisions proposed the following expression to estimate the shear strength of slabs:

$$V_n=0.79{\left(100\times \rho \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(400/d\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\times {\left(\frac{f_{cu}}{25}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{b_{o}d}{1.25},$$

(5)

where $f_{cu}$ is the cubic compressive strength in MPa. It should be noted that in the British Code, the critical section for shear is considered to be 1.5d from the face of the column. All terms were defined previously.

3.4. Euro-Code 2 (EC2)

The Euro-Code 2 (EC2) recommends that the punching shear resistance be expressed as proportional to ${\left(f_{ck}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$, where $f_{ck}$ is the compressive strength of concrete. In EC2, the influences of slab depth and steel reinforcement are also considered. The punching shear resistance according to EC2 may be calculated as

$$V_n=\frac{0.18}{{\gamma }_c}K{b}_{o}d{\left(1000\times \rho \times f_{ck}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{2d}{a_{crt}}\ge 0.035k^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{f}_{ck}{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{2d}{a_{crt}}{b}_{o}d,$$

(6)

where ${\gamma }_c$ is the material resistance factor for concrete = 1.5, d is the effective depth, $K=1+\sqrt{200/d}\le 2$ is the size factor of the effective depth, ρ is the flexural reinforcement ratio ≤ 2%, $f_{ck}$ is the cylinder compressive strength of concrete, and $a_{crt}$ is the distance from column face to the control perimeter.

It should be noted that some codes do not consider the size effect in estimates of the punching shear strength of slabs, such as ACI 318-14, while some common codes, such as Model-Code-2010 and Euro-Code 2, consider the size effect in the design of slabs for punching shear in the same form as presented in Equations (4) and (6). Deferent forms of the size effect have been presented by many researchers to consider the effect of this factor on punching shear strength: More details about the size effect can be found in References [26,27,28,29,30].

4. ANFIS: An Introduction

Recently, a fundamental change has occurred in the methodology of empirical analysis. Because of the nonlinearity and high degree of uncertainty associated with structural behavior, traditional mathematical models are difficult to develop. As an alternative, FIS- and ANN-based models (belonging to “soft computing”) are being used for many civil engineering problems. Nowadays, ANNs have been accepted as very useful tools for modeling nonlinear systems and are being widely used. FIS has emerged as a useful tool to represent and analyze complex systems [31,32,33]. Each method has its own advantages and disadvantages. Whereas in FIS there is no systematic procedure for designing a fuzzy controller, ANNs have the ability to map the input and output datasets through supervised learning and a self-organized structure. For this reason, it was proposed to combine an FIS and ANN together to get ANFIS, which enhances the efficiency of the systems and the modeling of problems using available data. ANFIS is thus an integration of an ANN and an FIS and uses basic FIS rules and the ANN network architecture to update system parameters using existing input and output pairs. ANFIS was first introduced by Jang [34]. In both an ANN and FIS, input parameters pass through the input layer using an input membership function, and the output parameters are seen in the output layer using output membership functions. In this method, the parameters are changed until an optimal solution is reached using a learning algorithm. A basic flow diagram of computations in ANFIS is illustrated in Figure 1. Several fuzzy inference systems have been developed by different researchers [12,35,36,37,38], who commonly use Mamdani-type and Takagi–Sugeno-type systems. In this study, a Takagi–Sugeno-type system was used.

5. ANFIS: This Study

In this study, an ANFIS model was developed using MATLAB R2013a [39] with five input parameters: The slab effective depth $\left(d\right)$, compressive strength of concrete $\left(f_c^{\prime }\right)$, reinforcement ratio $\left(\rho \right)$, yield strength of reinforcement $\left(f_y\right)$, and width of square loaded area $\left(c\right)$. The output variable is punching shear strength of a two-way slab $\left(V\right)$. A set of 207 experimental data points, collected from several sources [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56], was used to develop the model. The experimental data were randomly divided into two sets: The first one, with 164 data points, was used for training the model, and the second one, with 43 data points, was used for testing. A subtractive clustering technique produced by Chiu [57] was used to generate the ANFIS model with the (genfis2) function in MATLAB. Genfis2 is used to help in the creation of the initial set of membership functions for sets of input and output data. Genfis2 preforms this model by extracting a set of rules. The rule extraction method first uses the subclust function to determine the number of rules and antecedent membership functions. The type and the number of membership functions were evaluated when the training and testing datasets were giving good predictions according to the root mean square error (RMSE). After experimenting with different learning algorithms with a number of different epochs, the best correlations were found through a hybrid learning algorithm (a combination of least squares and back-propagation algorithms for membership function parameter estimations). The final errors of the model for training and testing were 0.45 and 0.52, respectively, and were achieved after 200 epochs. The structure of the ANFIS model is illustrated in Figure 2. In the model, 10 of the Gaussian membership functions (gaussmf) are selected for each input, and 10 rules define the relationship between inputs and outputs. A Gaussian membership function has two parameters: c, responsible for its center, and σ, responsible for its width, and the equation for this type is [39,58]

$$A{\left(x\right)}_{Gauss}=\exp \left[-{\left(\frac{x-c}{2\sigma }\right)}^2\right].$$

(7)

Readers are referred to Reference [58] for more details on this type of membership function. The numerical range of input parameters of the current study is listed in

Table 1. Range of input parameters in the database.

Parameters	Range
The slab effective depth $\left(d\right)$ (mm)	35–550
Concrete cylinder compressive strength ($f_c^{\prime }$) (MPa)	14.2–119
Reinforcement ratio $\left(\rho \right)$ (%)	0.25–5.01
Yield strength of reinforcement $\left(f_y\right)$ (MPa)	294–720
Width of square loaded area $\left(c\right)$ (mm)	80–500

. The data used to build the ANFIS model are summarized in

Table A1. Experimental data used to construct the ANFIS.

Test No.	$\mathit{d}$	$\mathit{f}\mathit{c}$	${\mathit{f}}_{\mathit{y}}$	$\mathit{\rho }$	$\mathit{c}$	${\mathit{V}}_{\mathit{n}}$	Reference
1	118	25.2	332	1.16	254	365	[43]
2	118	36.8	332	1.16	254	351
3	118	20.3	332	1.16	254	356
4	114	19.5	321	2.5	254	400
5	114	37.4	321	2.5	254	467
6	114	27.9	321	2.5	254	512
7	114	22.6	321	3.74	254	445
8	114	26.5	321	3.74	254	534
9	114	34.5	321	3.74	254	547
10	118	26.1	332	1.18	356	400
11	114	25	321	3.74	356	498
12	121	26.2	294	0.55	356	236
13	114	14.2	324	0.48	254	178
14	114	47.6	321	0.48	254	200
15	114	43.9	341	2	254	505
16	114	50.5	325	3.02	254	578
17	118	29	332	1.16	254	356
18	114	27.8	321	2.5	356	534
19	114	47.7	303	1.01	254	334
20	114	27.5	400	1.38	305	394	[44]
21	114	23.2	400	1.06	254	390
22	114	22	400	1.03	254	356
23	114	23.8	400	1.13	254	334
24	114	25.3	400	1.02	254	379
25	114	35.1	400	1.13	254	374
26	114	20.4	400	1.13	254	312
27	114	24.2	400	1.06	203	379
28	114	23	400	1.5	305	433
29	114	26.5	400	1.38	152	312
30	114	24.4	400	1.06	254	393
31	114	22.1	400	1.06	203	343
32	51	21.1	386	1.1	152	79	[41]
33	51	15.5	386	1.1	203	93
34	50	27.2	386	2.2	203	133
35	51	22.9	386	2.2	254	152
36	51	23	386	1.1	305	114
37	51	27.7	386	1.1	356	139
38	51	25	386	2.2	356	184
39	51	24.9	386	1.1	406	145
40	50	24.6	386	2.2	406	185
41	50	27	386	1.1	152	102
42	50	28.5	386	1.1	102	86
43	50	24.9	386	2.2	102	102
44	50	53.8	386	2.2	152	172
45	50	21.1	386	1.1	152	99
46	50	17	386	2.2	152	105
47	51	18	336	2.2	152	99
48	51	23.3	336	1.1	254	109
49	50	26.4	386	2.2	305	159
50	50	20	386	1.1	152	112
51	100	35.7	706	0.8	125	216	[42]
52	99	28.6	701	0.81	125	194
53	199	28.6	670	0.89	250	600
54	200	30.3	657	0.8	250	603
55	98	33.3	720	0.35	125	145
56	99	31.4	712	0.34	125	148
57	200	31.7	668	0.34	250	489
58	197	30.2	664	0.35	250	444
59	77	23.3	500	1.2	200	176	[45]
60	77	33.4	500	0.92	200	194
61	79	21.7	480	0.75	200	165
62	79	31.2	480	0.8	200	186
63	200	36.3	530	0.98	250	825
64	128	34.5	485	0.98	160	390
65	64	34.5	480	0.98	80	117
66	128	35.7	485	0.98	160	365
67	64	35.7	480	0.98	80	105
68	64	37.8	480	0.98	80	105
69	41	31.5	530	0.42	100	36	[46]
70	41	31.5	530	0.69	100	49
71	41	36.2	530	0.82	100	56
72	41	36.2	530	1.03	100	66
73	41	30.4	530	1.16	100	71
74	41	30.4	530	1.29	100	71
75	41	30.4	530	1.45	100	79
76	41	30.6	530	0.52	100	44
77	41	30.6	530	0.8	100	55
78	41	35.3	530	0.6	100	49
79	41	35.3	530	0.69	100	52
80	41	35.3	530	1.99	100	85
81	47	29.4	530	0.44	100	45
82	47	29.4	530	0.69	100	66
83	47	31.7	530	1.99	100	97
84	35	39.6	530	0.42	100	29
85	35	39.6	530	0.69	100	38
86	35	31.7	530	1.99	100	73
87	54	28.3	530	0.42	100	63
88	54	33.5	530	0.69	100	88
89	41	31.5	530	0.56	100	49
90	41	36.2	530	0.88	100	57
91	41	30.6	530	1.11	100	67
92	47	29.4	530	1.29	100	90
93	35	39.6	530	1.29	100	57
94	54	33.5	530	1.29	100	124
95	54	28.3	530	1.99	100	126
96	76	24.1	430	2.05	102	129	[47]
97	76	22.6	430	2.05	102	136
98	113	22.6	430	2.14	152	311
99	113	24.8	430	2.14	203	357
100	122	24.8	430	0.66	203	271
101	73	25	430	5.01	152	202
102	86	23.2	430	0.45	152	107
103	81	25.5	430	1.47	102	121
104	123	22.1	430	0.47	203	271
105	113	15.1	430	2.14	203	278
106	81	14.5	430	1.47	152	108
107	73	52.1	430	5.01	203	323
108	81	52.1	430	1.47	152	243
109	76	24.6	430	2.05	102	129
110	81	25	430	1.47	152	160
111	122	16.1	430	0.66	203	230
112	122	52.1	430	0.66	203	306
113	86	52.1	430	0.45	152	148
114	95	42	490	1.47	150	320	[40]
115	95	67	490	0.49	150	178
116	95	70	490	0.84	150	249
117	95	69	490	1.47	150	356
118	90	66	490	2.37	150	418
119	120	30	490	0.94	150	396
120	125	68	490	0.64	150	365
121	120	69	490	1.11	150	436
122	120	74	490	1.61	150	543
123	120	80	490	2.33	150	645
124	70	75	490	1.52	150	258
125	70	68	490	1.87	150	267
126	95	72	490	1.47	220	498
127	95	74	490	1.19	150	356
128	120	70	490	0.94	150	489
129	70	70	490	0.95	150	196
130	95	71	490	1.47	300	560
131	275	64	500	1.49	200	2050	[48]
132	275	112	500	1.49	200	2450
133	275	90	500	2.55	200	2400
134	200	88	500	1.75	150	1100
135	200	87	500	1.75	150	1300
136	200	119	500	1.75	150	1400
137	275	84	500	1.49	200	2250
138	200	70	500	1.75	150	1200
139	200	90	500	2.62	150	1450
140	200	98	500	2.62	150	1450
141	200	80	500	2.62	150	1250
142	200	108	500	2.62	150	1550
143	88	85	500	1.4	100	330
144	200	90	643	0.8	250	965	[50]
145	200	91	627	0.8	250	1021
146	200	92	596	1.19	250	1041
147	201	109	633	0.6	250	960
148	202	84	634	0.33	250	565
149	194	86	620	0.82	250	889
150	198	95	631	0.8	250	944
151	98	88.2	550	0.58	150	224	[49]
152	98	56.2	550	0.58	150	212
153	98	26.9	550	0.58	150	169
154	98	101.8	550	0.58	150	233
155	98	60.4	550	1.28	150	319
156	98	43.4	550	1.28	150	297
157	98	98.4	550	1.28	150	362
158	98	41.9	650	1.28	150	286
159	98	84.2	650	1.28	150	405
160	100	56.4	650	0.87	150	341
161	100	37.6	650	1.27	150	294
162	98	58.7	550	0.58	150	233
163	98	60.8	550	1.28	150	341
164	100	32.9	650	1.27	150	244
165	102	33.7	650	1.03	150	227
166	100	39.4	488	0.97	200	330	[51]
167	150	39.4	465	0.9	200	583
168	200	39.4	465	0.83	200	904
169	300	39.4	468	0.76	200	1381
170	400	39.4	433	0.76	300	2224
171	500	39.4	433	0.76	300	2681
172	210	27.6	400	1.5	260	1024	[52]
173	210	28.5	400	0.25	260	445
174	464	32.4	400	0.33	520	2153
175	210	32.2	400	0.25	260	408
176	210	29.3	400	0.33	260	550
177	96	34.7	400	1.5	130	236
178	100	34.7	400	0.75	130	243
179	102	34.7	400	0.25	130	118
180	210	40.5	400	0.25	260	439
181	102	34.7	400	0.33	130	141
182	210	28.5	400	0.33	260	540
183	100	24	718	0.8	250	270	[53]
184	100	24.4	718	0.8	250	250
185	125	27.2	718	0.64	150	265
186	124	33.1	488	1.54	250	483	[54]
187	190	33.5	531	1.3	300	825
188	260	31	524	1.1	350	1046
189	158	35	490	2.17	250	678	[55]
190	128	70	490	2.68	250	801
191	158	66.7	490	1.67	250	802
192	113	70	490	1.88	250	480
193	163	33	490	0.52	250	479
194	138	68.5	490	2.48	250	788
195	158	61.2	490	1.13	250	811
196	105	34	490	0.4	250	228
197	105	44.7	400	0.45	250	219	[56]
198	183	35	400	0.35	250	438
199	183	70	400	0.35	250	574
200	218	40	400	0.73	250	882
201	220	76	400	0.43	250	886
202	268	75	400	1.13	400	1721
203	263	65	400	1.44	400	2090
204	313	40	400	1.57	400	2234
205	313	60	400	1.57	400	2513
206	153	50.2	400	0.55	250	491
207	218	64.7	400	0.73	250	1023

in Appendix A. After the training procedure, the model was tested using the remaining data not used for the training. Figure 3 shows the performance for training and testing datasets. Figure 4 and Figure 5 show the matching of the experimental results with the results of the ANFIS model for both training and testing sets, respectively. Figure 6 shows a comparison between the experimental results of punching shear and the results predicted by the ANFIS model for all samples used in the model (training and testing sets). The adequacy of the developed ANFIS was evaluated by considering the coefficient of correlation (R), the average and standard deviation of the ratio of predicted to experimental punching shear strength, and the root mean square error (RMSE). The equations of the statistical parameter RMSE and the coefficient of correlation (R) that were used to compare the performance of each method are

$$\mathrm{RMSE}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(V_{ne}-V_{ni}\right)}^2}{N} ,}$$

(8)

$$R=1- \sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(V_{ne}-V_{ni}\right)}^2}{{{\displaystyle \sum }}_{i=1}^N{\left(V_{ne}\right)}^2} },$$

(9)

where $V_{ne}$ and $V_{ni}$ are the experimental and prediction nominal punching shear strength ($V_n)$ of two-way flat slabs, respectively, and $N$ is the total number of samples considered.

6. ANFIS: Results and Comparison

Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 show the comparison of the results obtained from the ANFIS model, ACI-14 code, Model Code 2010, British Code, and Euro-Code 2 for both training and testing datasets. A comparison of the results of the five models was also made with the experimental results. It was noted that the results of the ANFIS model were better than the results of four design codes: However, the results from BS-8110-97 were reasonable when compared to the experimental results.

Table 2. Comparison of punching shear between the experimental and predicted results for the training and testing sets. STDEV: Standard deviation.

Specimens	No.	Average of Vni/Vne					STDEV of Vni/Vne
		ANFIS	ACI-14 Code	Model-Code 2010	BS-8110 Code	Euro-Code 2	ANFIS	ACI-14 Code	Model-Code 2010	BS-8110 Code	Euro Code 2
Training set	164	1.0	0.88	1.10	1.01	1.45	0.11	0.30	0.16	0.14	0.20
Testing set	43	1.01	0.84	1.07	0.98	1.42	0.13	0.26	0.15	0.13	0.19

summarizes the average and standard deviation (STDEV) of the ratios of predicted punching shear strength ($V_{ni}$) to the experimental results ($V_{ne}$). The ANFIS model gave an average $V_{ni}$/$V_{ne}$ ratio for the training and test datasets of 1.0 and 1.01, respectively, and a standard deviation of 0.11 and 0.13, respectively. These results indicate that the ANFIS model could make more reliable predictions of the punching shear strength compared to those from the four design codes.

Table 3. Comparison summary of correlation (R) and root mean square error (RMSE %).

Type	Correlation (R)			RSME %
	Training	Testing	All Data	Training	Testing
ANFIS	0.996	0.995	0.995	0.45	0.52
ACI 318-14 Code	0.927	0.952	0.927	2.06	2.05
Model-Code-2010	0.986	0.992	0.986	0.93	0.72
BS-8110-97	0.986	0.992	0.987	0.83	0.93
Euro-Code 2	0.985	0.993	0.986	3.12	2.70

also confirms this conclusion when comparing the correlation coefficient for all models for training, testing, and the combined datasets. The values of 0.996, 0.995, and 0.995 for the ANFIS training, testing, and combined datasets, respectively, were very close to 1.0 and higher than those of the other four design codes. Finally, the same conclusion could be made from the root mean square error, as listed in Table 3: The minimum values of the RMSE were 0.45 and 0.52 for the training and testing sets, respectively.

7. Parametric Studies

After building and testing the ANFIS, and based on the comparison between the results obtained from the ANFIS model and the ACI 318-14 code, Model Code 2010, BS-8110, and Euro-Code 2, it could be concluded that the ANFIS was a suitable model in the prediction of the punching shear strength of two-way flat concrete slabs. The effect of each input parameter used to build the model was further investigated. The methodology of the parametric study was to vary one input parameter at a time, and the other input parameter were kept constant. Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23 show the predicted punching shear strength of a two-way flab slab as a function of each input variable. They show that the punching shear strength increased with an increase in the slab effective depth, concrete strength, and width of square loaded area. In general, the parametric tendencies of ANFIS agreed with the results from the ACI318-14 code, Model Code 2010, BS-8110, and Euro-Code 2, as shown in Figure 19, Figure 20 and Figure 21. The punching shear strength increased with an increase in the reinforcement ratio: This result agreed with the other models, except for the ACI code, as shown in Figure 22. Finally, the sensitivity of the punching shear strength to the yield strength of reinforcement is presented in Figure 23, where it can be seen that all models except ANFIS showed no effect on the punching shear strength. Interestingly, ANFIS predicted a slight increase in shear strength with increasing yield strength, which was in agreement with some of the experimental results used to build the ANFIS model.

8. Conclusions

An adaptive neuro-fuzzy inference system (ANFIS)-based model was developed to predict the punching shear strength of two-way flat concrete slabs without shear reinforcement. A database of 207 test results available in the literature was used to train and test the model. The database covered a rather wide range of two-way flat slab parameters, including slab thickness, concrete strength, reinforcement ratio, yield strength of reinforcement, and width of square loaded area. Five variables were selected as inputs into the ANFIS, with punching shear strength as the output variable. Within the framework of ANFIS, different models may be developed using different learning algorithms with different membership functions and epochs. After experimenting with several of these different models, a model was chosen that had the best potential to predict experimental results. An ANFIS model with a hybrid learning algorithm, 200 epochs, and 10 Gaussian membership functions was selected and then tested. The results from the ANFIS model were compared to the experimental results and to those from the equations recommended in ACI 318-14, BS-8110-97, Model Code 2010, and Euro-Code 2. For these comparisons, the correlation coefficient (R), the root mean square error (RMSE), and the average and standard deviations of the ratios of predicted ($V_{ni}$) to experimental ($V_{ne}$) punching shear strength were used as evaluation criteria. The values of R, RMSE, and average and standard deviations of $V{n}_i/V_{ne}$ for the training set were found to be 0.996, 0.45, 1.0, and 0.11, respectively, and for the testing set were 0.995, 0.52, 1.1, and 0.13, respectively, for the ANFIS model. This demonstrated that (i) the ANFIS model was capable of making highly reliable predictions of experimental results, (ii) the ANFIS model outperformed the equations recommended in four design codes currently used in practice, and (iii) the ANFIS model showed that it was a good tool for developing parametric studies to assess the influence of each parameter on the shear strength. In summary, the model developed in this study may serve as an economical, efficient, and reliable tool for the prediction of punching shear strength of flat concrete slabs.