# Assessing the accuracy of RC design code predictions through the use of artificial neural networks

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## Abstract

In light of recently published work highlighting the incompatibility between the concepts underlying current code specifications and fundamental concrete properties, the work presented herein focuses on assessing the ability of the methods adopted by some of the most widely used codes of practice for the design of reinforced concrete structures to provide predictions concerning load-carrying capacity in agreement with their experimentally established counterparts. A comparative study is carried out between the available experimental data and the predictions obtained from (1) the design codes considered, (2) a published alternative method (the compressive force path method), the development of which is based on assumptions different (if not contradictory) to those adopted by the available design codes, as well as (3) artificial neural networks that have been calibrated based on the available test data (the later data are presented herein in the form of a database). The comparative study reveals that the predictions of the artificial neural networks provide a close fit to the available experimental data. In addition, the predictions of the alternative assessment method are often closer to the available test data compared to their counterparts provided by the design codes considered. This highlights the urgent need to re-assess the assumptions upon which the development of the design codes is based and identify the reasons that trigger the observed divergence between their predictions and the experimentally established values. Finally, it is demonstrated that reducing the incompatibility between the concepts underlying the development of the design methods and the fundamental material properties of concrete improves the effectiveness of these methods to a degree that calibration may eventually become unnecessary.

## Keywords

Artificial neural network Design codes Ultimate limit state RC beams Compressive force path method Physical models## Abbreviations

- CFP
Compressive force path

- ANN
Artificial neural network

- ULS
Ultimate limit state

## List of symbols

- \(a_{\text{v}}\)
Shear span

- \(b\)
Beam width

- \(d\)
Effective depth

- \(x\)
Depth of the compressive zone

- \(A_{\text{s}}\)
Area of tensile reinforcement

- \(A_{{{\text{s}}^{\prime } }}\)
Area of compressive reinforcement

- \(A_{\text{sw}}\)
Area of transverse reinforcement

- \(\alpha_{\text{v}} /d\)
Shear span-to-depth ratio

- \(f_{\text{c}}\)
Uniaxial compressive strength of concrete

- \(f_{\text{yl}}\)
Longitudinal reinforcement yield stress

- \(f_{\text{yw}}\)
Transverse reinforcement yield stress

- \(s\)
Spacing between shear links

- \(\rho_{\text{l}}\)
Ratio of tensile reinforcement (\(\rho_{\text{l}} = A_{\text{s}} /b \times d\))

- \(\rho_{\text{t}}\)
Ratio of compressive reinforcement (\(\rho_{\text{t}} = A_{{{\text{s}}^{\prime } }} /b \times d\))

- \(\rho_{\text{w}}\)
Ratio of transverse reinforcement (\(\rho_{\text{w}} = A_{\text{sw}} /b \times s\))

- \(V_{\text{c}}\)
Shear resistance of the RC beam without the contribution of the shear links

- \(V_{\text{s}}\)
Shear resistance offered by of shear links

- \(V_{\text{u}}\)
Shear developing along the span of the RC beam at failure

- \(M_{\text{u}}\)
Bending developing along the span of the RC beam at failure

- \(M_{\text{f}}\)
Flexural moment capacity of the cross section of the RC beam

## Introduction

It has recently been shown that the concepts underlying code specifications for the design of reinforced concrete structures (RC) is incompatible with fundamental concrete properties such as post-peak stress–strain characteristics, mechanism of crack extension, and sensitivity of strength and deformation characteristics to the presence of small confinement stresses (acting orthogonal to the direction considered) (Kotsovos 2015). Consequently, the need of calibrating these specifications with experimental data describing the behaviour of RC beam/column elements leads to an ever increasing complexity of the code-adopted design formulae. Moreover, it has been shown that compliance with the fundamental concrete properties can lead to the development of alternative design methods, such as the compressive force path (CFP) method, which may be, not only simpler and more efficient, but also rely on failure criteria derived from first principles (of mechanics), without the need of calibration through the use of experimental data (Kotsovos 2014).

To this end, the present work has been aimed at evaluating the shear design specifications of some of the most widely adopted codes of practice (ACI 445R-99 1999, ACI 318 2014, Eurocode 2 2004, JSCE 2007, CSA 2007, NZ 2006) and the CFP method (Kotsovos 2014) as regards, not only the closeness of the correlation of the calculated values of load-carrying capacity with their experimentally established counterparts, but also the validity of the concepts underlying the development of the method of calculation. The present study focuses on comparing the values of load-carrying capacity predicted by the design codes with those obtained through the use of a purely empirical (soft computing) method that employs artificial neural networks (ANNs) (Cladera and Mari 2004a, b; Mansour et al. 2004; Abdalla et al. 2007; Yang et al. 2007; Jung and Kim 2008) capable of providing predictions concerning load-carrying capacity that correlate closely to the available test data. In what follows the description of the ANN approach developed herein and the presentation and discussion of the results of the comparative study are preceded by a concise description of the methods of calculation employed and the presentation of the experimental information used for the training and verification of the ANNs.

## Experimental information on RC structural response at ULS

To date, a large number of experiments have been conducted on simply supported RC beam specimens to study the mechanics underlying RC structural response throughout the loading process and establish the effect of a wide range of parameters [e.g., width (*b*), effective depth (*d*), shear span-to-depth ratio (*α*_{v}*/d*), compressive strength of concrete (*f*_{c}), longitudinal (*ρ*_{l}) and transverse (*ρ*_{w}) steel reinforcement ratios, yield strength of longitudinal (*f*_{yl}) and transverse (*f*_{yw}) reinforcement bars, etc.] on the exhibited behaviour (Κani 1964; Panagiotakos and Fardis 2001; Reineck 2013; Reineck et al. 2014; Kotsovos 2014). During testing, attention is focused on measuring certain aspects of the exhibited behaviour such as the applied load and reaction forces, the displacement of certain points along the element span as well as the strain at specific critical locations (i.e., the compressive region or on the steel longitudinal and transverse reinforcement bars). In addition, the deformation profiles and crack patterns forming as well as the exhibited mode of failure are also recorded, as they provide important information concerning the internal stress state developing within the members considered. The test data published and the different parameters considered have been used to form an extensive database describing the effect of the above parameters on the exhibited behaviour mainly at ULS (see Tables A1 and A2 presented as supplementary material).

## Design methods

Current design methods are intended to size the cross section of structural members and specify an amount and arrangement of reinforcement that will safeguard desired performance characteristics such as load-carrying capacity and ductility. This objective can only be achieved by ensuring that the load corresponding to shear types of failure is always higher than that corresponding to flexural capacity. To this end, the sizing of the cross section and the assessment of the longitudinal reinforcement are linked to flexural capacity, whereas the associated shear-force diagram underlies the method adopted for specifying an amount and arrangement of transverse reinforcement that will safeguard against shear failure occurring before flexural capacity is exhausted.

### Code-adopted methods

Formulae adopted by the RC design codes for predicting shear capacity

Physical models | Contribution of the specimen cross section without stirrups | Contribution of the stirrups | Limitations | Factors used for multiplying shear capacity for the case of small shear spans |
---|---|---|---|---|

\(V_{\text{c}}\) | \(V_{\text{s}}\) | |||

ACI-318-11 (SI units) | \(V_{\text{c}} = \left( {0.16f_{\text{c}}^{0.5} + 17\frac{{A_{\text{s}} \times V_{\text{u}} }}{{M_{\text{u}} }}} \right)bd\) | \(V_{{{\text{s}},45^\circ }} = \frac{{A_{\text{sw}} f_{\text{yw}} d}}{s}\) | \(V_{\text{c}} \le 0.29\sqrt {f_{\text{c}} } bd\) \(d \le 500\,{\text{m}}\) \(f_{\text{y}}\) ≤ 420 MPa \(f_{\text{c}}\) ≤ 70 MPa | If [3.5–2.5 ( If [3.5–2.5 ( If [3.5–2.5 ( |

EC2 | \(V_{\text{c}} = 0.18\,k\,(100\,\rho_{\text{l}} f_{\text{c}} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} bd\) | \(V_{\text{s}} = \frac{{A_{\text{sw}} f_{\text{yw}} 0.9d\cot \theta }}{s},\) where 1 ≤ \(\cot \theta\) ≤ 2.5 | \(V_{\text{c}} \le v_{\hbox{min} } b \times d\) \(v_{\hbox{min} } = 0.035f_{\text{c}}^{1/2} k^{3/2}\) \(\rho_{\text{l}} = \frac{{A_{\text{s}} }}{bd} \le 0.02\) \(k = (1 + \sqrt {200/d} ) \le 2\) \(f_{\text{ck}}\) < 90 MPa | If ( |

JSCE (SI units) | \(V_{\text{c}} = \beta_{\text{d}} \beta_{\text{p}} \beta_{\text{n}} f_{\text{vcd}} bd/\gamma_{\text{b}}\) | \(V_{\text{s}} = \frac{{A_{\text{sw}} f_{\text{yw}} (\sin \alpha + \cos \alpha )z}}{{s\gamma_{\text{b}} }}\)
| \(\beta_{\text{p}} = (100\rho_{\text{l}} )^{1/3} \le 1.5\) 0≤ \(\beta_{\text{n}}\) ≤ 2.0 \(\beta_{\text{d}} = (1000/d)^{1/4} \le 1.5\) \(f_{\text{vcd}} = 0.20 \times f_{\text{c}}^{1/3}\) ≤ 0.72 \(f_{\text{y}}\) ≤ 400 Mpa \(f_{\text{c}}\) ≤ 60 MPa | No change required (see Section 9.2.2.2 of the revevant code) |

KBCS-(SI units) | \(V_{\text{c}} = \alpha_{\text{d}} \left( {0.16f_{\text{c}}^{0.5} + 17.6\rho_{\text{l}} \frac{{V_{\text{u}} d}}{{M_{\text{u}} }}} \right)bd\) | \(V_{\text{s}} = f_{\text{yw}} \rho_{\text{w}} bd\) | \(V_{\text{c}} \le 0.29\sqrt {f_{\text{c}} } bd\) | \(\alpha_{\text{d}} = [3.5 - 2.5(a_{\text{v}} /d)]\) ≤ 2.5 |

CSA (SI units) | \(V_{\text{c}} = \varPhi_{\text{c}} \lambda \beta f_{\text{c}}^{0.5} bd\) | \(V_{\text{s}} = \frac{{\varPhi_{\text{s}} A_{\text{sw}} f_{\text{yw}} d\cot \theta }}{s}\) |
\(f_{\text{y}}\) ≤ 400 Mpa \(f_{\text{c}}\) ≤ 60 MPa | No change required (see Section 11.3.2 of the revevant code) |

NZ (SI units) | \(V_{\text{c}} = v_{\text{c}} A_{\text{cv}}\), \(A_{\text{cv}} = bd\) \(v_{\text{c}} = k_{\text{d}} k_{\text{a}} v_{\text{b}}\), \(k_{\text{a}} = 1.0\) \(v_{\text{b}} = \hbox{min} [(0.07 + \rho_{\text{w}} )\sqrt {f_{\text{c}} }\), 0.2 \(\sqrt {f_{\text{c}} } ]\) | \(V_{{{\text{s}}^{\prime } }} = \frac{{A_{\text{sw}} f_{\text{yw}} d}}{s}\) | For \(d \le 400\,{\text{m}}\,k_{\text{d}} = 1.0\) for \(f_{\text{y}}\) ≤ 500 Mpa \(f_{\text{c}}\) ≤ 50 Mpa | No change required (see Section 9.3.9.3.1 of the revevant code) |

### The CFP method

The CFP theory attributes what is commonly referred to as ‘shear’ failure to the development of tensile stresses at particular locations [dependent on the shear span-to-depth ratio (*a*_{v}/*d*)] along the compressive zone. For an RC beam/column element with flexural reinforcement only, the values of the shear force corresponding to such types of failure are obtained from.

*a*

_{v}/

*d*> 2.5, failure occurs when either the shear force at a distance equal to 2.5

*d*from a simple support or a point of contra-flexure attains a critical value:

*f*

_{t}is the tensile strength of concrete and

*b*,

*d*are the width and depth of the cross section, or after yielding of the flexural reinforcement in regions, where the value of the shear force attains a critical value:

*F*

_{c}is the value of the compressive force developing on account of bending when

*M*

_{f}is attained and

*f*

_{c}and

*f*

_{t}are the values of the compressive and tensile, respectively, strengths of concrete.

*a*

_{v}/

*d*≤ 2.5, failure is considered to occur when the bending moment at the cross section located at a distance equal to

*a*

_{v}from the closest simple support or point of contra-flexure exceeds a critical value:

*M*

_{II}=

*V*

_{II}(2.5

*d*) and

*M*

_{IV}=

*M*

_{f}(flexural capacity) at cross sections at distances of 2.5

*d*and

*d*, respectively, from the closest simple support or point of contra-flexure.

*a*

_{v}/

*d*< 1, failure occurs when the shear force exceeds a critical value provided by Eq. (4):

*a*

_{v}/

*d*> 2.5, the transverse tension developing at a distance of 2.5

*d*from the location of the simply support or the point of contra-flexure has been shown to be numerically equal to the shear force

*V*

_{f}developing at this location and, then, the amount of transverse reinforcement required to prevent failure is obtained from:

Such reinforcement is spread over a length of 2*d* extending symmetrically on either side of the location, where transverse tension is considered to develop (i.e., the location at a distance from a simple support or a point of contra-flexure).

This method is also used to calculate the amount of transverse reinforcement required to prevent failure in the region of points of contra-flexure.

*V*

_{f}>

*V*

_{II,2}(where

*V*

_{f}is the shear force corresponding to

*M*

_{f}), the transverse tensile stresses

*σ*

_{t}exceed the tensile strength of concrete

*f*

_{t}in the compressive zone of the regions of beam/column elements, where yielding of the flexural reinforcement occurs. Then, these tensile stresses can be calculated from Eq. (2) by replacing

*V*

_{II,2}with

*V*

_{f}and

*f*

_{t}with

*σ*

_{t}, and solving for

*σ*

_{t}, that is

The second term of Eq. (7) is divided by 2 to allow for the non-uniform distribution of the transverse stresses within the compressive zone.

*T*

_{II,2}is equal to:

*a*

_{v}/

*d*≤ 2.5, this is obtained from:

It should be noted that this reinforcement is uniformly distributed within the shear span and, in accordance with the CFP theory, is intended to increase flexural capacity from *M*_{III} to *M*_{f} rather than prevent shear failure.

It is also important to note that in accordance with the CFP theory, the specified transverse reinforce is effective when its spacing is smaller than *d*/2.

The above expression has been derived by considering the triaxial stress conditions invariable developing in the compressive zone when ULS is approached.

## Artificial neural networks

Artificial neural networks mimic their biological counterparts in the nervous system and the brains of animals and humans. They are used to estimate or approximate functions that depend on a large number of parameters, the effect of which is not clearly established or quantified. Due to their adaptive nature and ability to remember information introduced to them during training (calibration), ANNs can learn, generalize, categorize, and empirically predict values.

*x*

_{j}(describing selected parameters associated with the design details and strength characteristics of the structural elements investigated), each of them being attributed an (initially assumed) weight

*w*

_{ji}and a bias

*Θ*

_{i}. This input information is used to form the sum

*n*

_{i}of the products

*x*

_{j}×

*w*

_{ji}and

*Θ*

_{i}, that is

*n*

_{i}is inserted as argument into an adopted

*activation*function

*g*, that is

*g*,

*y*

_{i}=

*g*(

*n*

_{i}) is the output

*O*

_{i}of the AN.

*w*

_{kj}) and bias (

*Θ*

_{j}) are initially randomly assigned, but obtain their final values through an iterative training (calibration) process (described later) which aims at reducing to an acceptable value the difference between the output values

*O*

_{i}(obtained from the ANN) and the target values

*T*

_{i}(provided in the database) (Svozil et al. 1997; Basheer and Hajmeer 2000). A measure of the deviation of

*O*

_{i}from

*T*

_{i}is obtained from:

Activation functions used in the ANN models

Activation functions | Layers | Mathematical expression | Range of values of | Graphical expression |
---|---|---|---|---|

Sigmoid | B/w input layer and first hidden layer B/w first hidden layer and second hidden layer | \(f(x) = \frac{1}{{1 - e^{ - x} }}\) | (0, + 1) | |

Hyperbolic (tanghn) | B/w second hidden layer and out put layer | \(f(x) = \frac{{e^{x} - e^{ - x} }}{{e^{x} + e^{ - x} }}\) | (− 1, + 1) |

Convergence is considered to be achieved when *E* becomes smaller than an acceptable value.

### Experimental database

Statistical information concerning the test data included in DB-I AND DB-II

Min | Max | Diff | Avg. | St. dev | COV | ||
---|---|---|---|---|---|---|---|

For beams without stirrups (BWOS) DB-I | |||||||

| (mm) | 50 | 500 | 450 | 171.51 | 68.71 | 0.4 |

| (mm) | 65 | 483 | 418 | 255.15 | 73.6 | 0.29 |

| 0 | 0.98 | 9.73 | 8.75 | 3.53 | 1.4 | 0.4 |

| (%) | 0.250261 | 7.462626 | 7.212365 | 2.14 | 1.09 | 0.51 |

| (MPa) | 283 | 1779 | 1496 | 444.83 | 189.15 | 0.43 |

| (MPa) | 12.2 | 110.9 | 98.7 | 39.07 | 21.43 | 0.55 |

| (kN mm) | 2000 | 844,000 | 842,000 | 101,307 | 109,891.2 | 1.08 |

| (kN) | 7 | 360 | 353 | 69.09 | 47.07 | 0.68 |

For beams with stirrups (BWS) DB-II | |||||||

| (mm) | 100 | 510 | 410 | 207.27 | 67.64 | 0.33 |

| (mm) | 113 | 975 | 862 | 342.87 | 157.03 | 0.46 |

| 1 | 7.002188 | 6.002188 | 3.43 | 1.35 | 0.39 | |

| (%) | 0.18 | 5.57 | 5.39 | 2.35 | 1.05 | 0.45 |

| (MPa) | 250 | 888.48 | 638.48 | 411.33 | 77.88 | 0.19 |

| (MPa) | 13.8 | 126.2 | 112.4 | 44.51 | 22.41 | 0.5 |

| (%) | 0.08 | 2.24 | 2.16 | 0.57 | 0.5 | 0.88 |

| (MPa) | 224.08 | 875.5 | 651.42 | 399.4 | 112.62 | 0.28 |

| (kN mm) | 2648 | 1,738,423 | 1,735,775 | 283,484.8 | 378,729.4 | 1.34 |

| (kN) | 4.0967 | 760.1942 | 756.0975 | 185.54 | 134.25 | 0.72 |

### Selection of input parameters

*r*) described by Eq. (14) (Mojtaba et al. (2013)):

*n*represents the total number of samples and

*x*,

*y*are the two variables considered.

*r*) can obtain values between 1 and − 1 (1 >

*r*> − 1). Negative values of

*r*describe an inverse relation between the two associated parameters (i.e., an increase in the value of one parameter results in the decrease in the value of the other). Positive values of

*r*have a similar effect on the values of the associated parameters (i.e., an increase in the value of one parameter results in an increase of the other). Higher absolute values of |

*r|*indicate a more pronounced relation between the two parameters considered. Depending on

*r*and on the basis of current knowledge on RC structural response at ULS, different combinations of input parameters are selected for the case of the beams without and with stirrups, as shown in Tables 6 and 7.

Values of *r* expressing the effect between parameters considered in the database for the case of beams without stirrups

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

| 1.00 | |||||||

| 0.39 | 1.00 | ||||||

| − 0.02 | 0.04 | 1.00 | |||||

| − 0.14 | − 0.16 | 0.23 | 1.00 | ||||

| 0.09 | 0.03 | 0.02 | − 0.20 | 1.00 | |||

| 0.20 | − 0.08 | − 0.09 | 0.25 | 0.09 | 1.00 | ||

| 0.55 | 0.68 | 0.11 | 0.23 | 0.09 | 0.22 | 1.00 | |

| 0.56 | 0.48 | − 0.30 | 0.17 | − 0.04 | 0.30 | 0.67 | 1.00 |

Values of *r* expressing the effect between parameters considered in the database for the case of beams with stirrups

| | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

| 1 | |||||||||

| 0.71 | 1 | ||||||||

| 0.12 | − 0.10 | 1 | |||||||

| − 0.17 | 0.08 | − 0.11 | 1 | ||||||

| 0.07 | − 0.24 | 0.57 | − 0.45 | 1 | |||||

| − 0.07 | − 0.33 | 0.36 | 0.31 | 0.28 | 1 | ||||

| − 0.23 | − 0.36 | − 0.37 | 0.10 | − 0.07 | 0.17 | 1 | |||

| 0.09 | − 0.36 | 0.21 | − 0.25 | 0.36 | 0.29 | 0.20 | 1 | ||

| 0.71 | 0.74 | 0.15 | 0.33 | − 0.03 | 0.05 | − 0.14 | − 0.10 | 1 | |

| 0.26 | 0.46 | − 0.62 | 0.29 | − 0.60 | − 0.23 | 0.25 | − 0.26 | 0.33 | 1 |

Architecture of ANN models

Sr. no | Model | Input | No of input neurons | Act. function b/w first two layers | Act. function in last layers | No of neurons in each hidden layer |
---|---|---|---|---|---|---|

For beams without stirrups (BWOS) DB-I | ||||||

1 | BWOS-1 |
| 6 | Sigmoid | tanghn | 12–12 |

2 | BWOS-2 |
| 5 | Sigmoid | tanghn | 10–10 |

3 | BWOS-3 |
| 3 | Sigmoid | tanghn | 6–6 |

4 | BWOS-4 |
| 4 | Sigmoid | tanghn | 8–8 |

5 | BWOS-5 |
| 4 | Sigmoid | tanghn | 8–8 |

6 | BWOS-6 |
| 5 | Sigmoid | tanghn | 10–10 |

For beams with stirrups (BWS) DB-II | ||||||

1 | BWS-1 |
| 8 | Sigmoid | tanghn | 16–16 |

2 | BWS-2 |
| 7 | Sigmoid | tanghn | 14–14 |

3 | BWS-3 |
| 5 | Sigmoid | tanghn | 10–10 |

4 | BWS-4 |
| 4 | Sigmoid | tanghn | 8–8 |

5 | BWS-5 |
| 5 | Sigmoid | tanghn | 10–10 |

6 | BWS-6 |
| 6 | Sigmoid | tanghn | 12–12 |

7 | BWS-7 |
| 7 | Sigmoid | tanghn | 14–14 |

8 | BWS-8 |
| 6 | Sigmoid | tanghn | 16–16 |

Results of different ANN models

| Min | Max | Diff | Avg. | St. dev | COV | |
---|---|---|---|---|---|---|---|

For beams without stirrups (BWOS) DB-I | |||||||

Exp | (kN) | 7 | 360 | 353 | 69.09 | 47.07 | 0.68 |

BWOS-1 | (kN) | 5 | 307 | 302 | 69.53 | 46.61 | 0.67 |

BWOS-2 | (kN) | 0 | 300 | 300 | 69.57 | 45.75 | 0.66 |

BWOS-3 | (kN) | 6 | 394 | 388 | 78.95 | 63.04 | 0.8 |

BWOS-4 | (kN) | 7 | 371 | 364 | 69.29 | 46.14 | 0.67 |

BWOS-5 | (kN) | 7 | 319 | 312 | 70.72 | 50.48 | 0.71 |

BWOS-6 | (kN) | 7 | 306 | 299 | 68.38 | 44.46 | 0.65 |

For beams with stirrups (BWS) DB-II | |||||||

Exp | (kN) | 4 | 761 | 757 | 185.54 | 134.25 | 0.72 |

BWS-1 | (kN) | 3 | 757 | 754 | 186.41 | 134.36 | 0.72 |

BWS-2 | (kN) | 13 | 780 | 767 | 184.99 | 130.65 | 0.71 |

BWS-3 | (kN) | 0 | 767 | 767 | 185.34 | 137.65 | 0.74 |

BWS-4 | (kN) | 4 | 656 | 652 | 184.5 | 133.08 | 0.72 |

BWS-5 | (kN) | 2 | 763 | 761 | 186.74 | 137.71 | 0.74 |

BWS-6 | (kN) | 7 | 846 | 839 | 187.31 | 137.19 | 0.73 |

BWS-7 | (kN) | 4 | 863 | 859 | 186.9 | 137.83 | 0.74 |

BWS-8 | (kN) | 3 | 834 | 831 | 187.31 | 141.15 | 0.75 |

### Normalization of database

*x*

_{max}and

*x*

_{min}represent the minimum and maximum values of a specific parameter included in the database considered.

### Training process

*E*as defined by Eq. (13)] is larger than a predefined acceptable value), then the values of the weights and bias are re-adjusted through the “gradient descent” method in which the change in weights (Δ

*w*

_{kj}) is expressed in the form:

*a*is the learning rate.

*α*) result in the values of the weights (

*w*

_{jk}) changing more drastically (large values of Δ

*w*

_{jk}) during each iteration. This can result in the training process not converging to the most optimum combination of values of weights for which the error attains its minimum value for the whole database considered (global minima). Small values of

*α*result in small changes to the weights (

*w*

_{jk}) (i.e., small values of Δ

*w*

_{jk}) for each iteration. In the latter case, although more training time is required (due to the larger number of iterations that have to be carried out to satisfy the convergence criteria), the iterative process is able to identify more effectively the optimum combination of values

*w*

_{jk}for which the error function obtains its minimum value for the whole database considered allowing the ANN to provide more accurate predictions. To facilitate this process, the values of Δ

*w*

_{kj}calculated for each iteration (

*n*> 1) are provided as a function of the corresponding adjustment carried out in the previous iteration (

*n*− 1) and the momentum factor (0 <

*η*<1) as follows:

*R*), the mean squared error (MSE), and the mean absolute error (MAE) which are analytically expressed by Krogh and Vedelsby (1995), Utans et al. (1995) and Castellano and Fanelli (2000):

*T*

_{i}and

*O*

_{i}values, respectively, and

*n*is the total number of samples in databases, i.e., DB-I and DB-II (see Tables A1 and A2 which provided in the form of supplementary material).

- 1.
For 100 iterations (epochs) including the full MLFFBP process.

- 2.
Maximum of 6 validation failures are exhibited. Validation failure occurs when the performance of the ANN (assessed during each iteration) fails to improve or remains constant.

- 3.
The values of performance goal indicators, expressing the difference between the target and output values (see Eqs. 18a–18c) attain small values (e.g., 10

^{−6}), indicating that convergence has been achieved. - 4.
Minimum performance gradient (related to the rate at which the weights are adjusted through the MLFFBP process) becomes 10

^{−10}.

A number of ANN models have been presently considered employing different parameters, activation functions, and number of hidden neurons and layers (see Table 6) to study their effect on the ability of the subject models to provide accurate predictions. The values of the input and output parameters are normalized between 0.1 and 0.9 at the start of the training process, whereas the values of the weight and biases vary between − 1.0 and 1.0, at the end of the training process. The previous work (Ahmad et al. 2016) has revealed that in such cases, the use of (1) the sigmoid activation function between the first three layers of the ANN (e.g., the input and the two hidden layers) and (2) the hyperbolic (tan*h*) activation function between the second hidden layer and the output layer allows the ANN to provide accurate predictions (see Tables 2 and 6).

*R*(correlation function) in combination with the lowest values of MSE and MAE (Basheer and Hajmeer 2000). Figure 7b provides the values of

*R*, MSE, and MAE calculated for the different ANN models developed for database DB-I (including beams without stirrups). From Table 7 and Fig. 8, it is observed the BWOS-4 exhibited the highest value of

*R*(99.2%) in combination with the smallest values of MSE (0.05%) and MAE (1.5%), while, at the same time, employing the smallest number of parameters (i.e.,

*d*,

*a*

_{v}

*/d*,

*M*

_{f}

*/f*

_{c}

*bd*

^{2}, and

*f*

_{c}). Similarly, based on the information provided in Table 7 and Fig. 8 for the case of database DB-II (including beams with stirrups), BWS-8 is the most efficient ANN model (with

*R*= 99.2%, MSE = 0.08%, and MAE = 1.9%) and employs only seven parameters (i.e.,

*d*,

*b/d*,

*a*

_{v}

*/d*,

*M*

_{f}

*/f*

_{c}

*bd*

^{2},

*f*

_{c}, and

*ρ*

_{w}

*× f*

_{yw}).

In view of the above, ANN models BWOS-4 and DWS-8 have been selected for use in what follows.

## Comparative study

Results of *V*_{Exp}/*V*_{Pred} for RC beams

Min | Max | Mean | St. dev | CV% | |
---|---|---|---|---|---|

For beams without stirrups (BWOS) DB-I | |||||

| 0.48 | 2.00 | 1.00 | 0.16 | 0.16 |

| 0.57 | 2.26 | 1.12 | 0.26 | 0.23 |

| 0.67 | 6.47 | 1.51 | 0.67 | 0.44 |

| 0.72 | 6.62 | 1.27 | 0.65 | 0.51 |

| 0.72 | 6.62 | 1.27 | 0.65 | 0.51 |

| 0.63 | 8.73 | 1.58 | 0.86 | 0.54 |

| 0.67 | 6.36 | 1.49 | 0.65 | 0.44 |

| 0.66 | 12.93 | 1.57 | 1.24 | 0.79 |

| 0.56 | 11.53 | 1.60 | 1.08 | 0.67 |

For beams with stirrups (BWS) DB-II | |||||

| 0.26 | 1.79 | 1.01 | 0.14 | 0.14 |

| 0.59 | 5.15 | 1.12 | 0.41 | 0.36 |

| 0.66 | 4.97 | 1.24 | 0.39 | 0.31 |

| 0.75 | 4.69 | 1.26 | 0.39 | 0.31 |

| 0.55 | 4.69 | 1.14 | 0.33 | 0.29 |

| 0.82 | 5.79 | 1.53 | 0.61 | 0.40 |

| 0.65 | 4.97 | 1.24 | 0.39 | 0.31 |

| 0.74 | 4.97 | 1.24 | 0.41 | 0.33 |

| 0.04 | 4.15 | 0.92 | 0.51 | 0.56 |

It should be noted at this stage that, unlike the other methods investigated, the CFP method is, by nature, capable of providing close predictions of not only the values of load-carrying capacities, but also of the locations and the causes of failure. In fact, the predictions of the latter have been successful in all cases investigated in the present work. Naturally, similar predictions cannot be expected to be obtained from a purely empirical method such as the ANN technique employed in the present work, which, however, has been found capable of producing the best possible numerical predictions of load-carrying capacity.

It is also important to note that, as already discussed, the code methods are also heavily dependent on the use of experimental data for the calibration of the formulae underlying the assumed mechanisms of load transfer. The use of the code-adopted formulae, when compared with the use of the ANN technique employed in the present work, appears to reduce the effectiveness of the calibration process and this may be considered to reflect shortcomings of the underlying theory in describing the actual mechanism of load transfer. On the other hand, the ability of the CFP method to produce numerical predictions exhibiting a relatively small deviation from those resulting from the ANN procedure, without the need of prior calibration, may be considered as an indication of the validity of the concepts underlying the method.

## Conclusions

The accuracy of the predictions of the RC design methods considered is assessed primarily against the available test data. The calibrated ANNs developed were successful in providing predictions very close to their experimentally established counterparts. The good correlation between the available test data and the predictions of the ANNs demonstrate that the latter soft computing method provides an effective tool capable of objectively analyzing the available test data and quantifying the effect of specific parameters on certain important characteristics of RC structural behaviour (e.g., the load-carrying capacity and mode of failure)

The ANN procedure, which is purely empirical in nature and follows a detailed mathematical approach within a heuristic scheme allowing for all parameters widely assumed to affect load-carrying capacity, is found to produce a nearly perfect fit to the test data available in the literature on shear capacity of simply supported beam/column elements.

Current code methods for calculating shear capacity are semi-empirical in that, unlike the ANN procedure, the test data are used for calibrating the formulation of the theoretical basis of the methods. The resulting formulae are found conservative: they underestimate shear capacity by an amount ranging between approximately 15% and 60%.

The inability of the code methods to provide a fit to the test data as close as that of the ANN procedure is considered to reflect the lack of effectiveness of the code formulae to describe the trends of behaviour exhibited by the test data and this may be viewed as an indication of shortcomings of the theoretical basis of the code-design methods.

In contrast with code methods, the formulation of the theoretical basis of the CFP method does not require calibration through the use of test data. The failure criteria have been derived from first principles and are functions of the strength of concrete of uniaxial compression and tension. Moreover, unlike the code methods, the CFP is capable of identifying both the location and the causes of failure.

An indication of the validity of the method is provided not only by the fit to the test data which is nearly as close as that of the ANN procedure for the case of simply supported beams, but also by the realistic predictions of the load-carrying capacity, location, and causes of failure of the structural elements exhibiting points of contra-flexure. However, since test data on the behaviour of the latter structural elements are rather sparse, additional work is required for drawing definitive conclusions on the method.

## Notes

### Acknowledgements

This research was supported by the HORIZON 2020 Marie Skłodowska-Curie Research Fellowship Programme H2020-660545, titled: Analysis of RC Structures Employing Neural Networks (ARCSENN).

## Supplementary material

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