Document Type : Regular Article
Authors
^{1} Ph.D. Candidate, Faculty of Civil Engineering, Semnan University, Semnan, Iran
^{2} Ph.D., Florida International University, Miami, FL, USA
Abstract
Highlights
Keywords
Main Subjects
In structural design, usually the effect of the torsional moment is neglected, and members are designed only for stresses due to bending, shear and axial loads. However, the amount of torsion in lateral beams or the beams connected to a slab or another beam in one side is significant, and even a small amount of torsion can cause high stresses. Since failure due to torsion is a brittle and nonpredictable failure, therefore designing against torsion becomes important.
There are not several studies conducted on the reinforced concrete beams behavior under torsion up to now. H.J. Chiu et al. have studied 13 reinforced concrete beams specimens in two types of high strength concrete and normal concrete. The purpose of this study was to investigate the cracks pattern, torsion strength and deformation of the beams under pure torsion loading. The results showed that hollow beams have less torsion strength against cracking with respect to filled rectangular beams and increasing the ratio in the beams section causes lesser torsion strength, cracking and increasing of the cracks widths [1].
Victor and Muthukrishnan have investigated the effect of varying number of stirrups on the torsion capacity of the reinforced concrete beams and presented an experimental relation for the share of stirrups in the torsion capacity [2].
Rasmussen and Baker have studied the behavior of high strength concrete, and normal concrete beams under pure torsion [3, 4]. The results showed that high strength concrete increases the torsion capacity and stiffness.
McMullen and Rangan have investigated rectangular reinforced concrete beams by varying dimensional ratios and the number of stirrups [5]. Authors concluded that longitudinal stirrups are more effective in controlling traverse cracks than traverse stirrups [6, 7].
Although many relations have been presented for reinforced concrete beams under pure torsion, in this study, the behavior of rectangular reinforced concrete beams under pure torsion load has investigated using the neural network, and finally, the weight effect of each of the input parameters on the target function (ultimate torsion capacity) has studied.
Neural networks can be assumed as a very simplified electronic model of the human brain neural structure. The learning and training mechanism of the brain is experimental. The electronic neural network models are based on the same pattern, and the analysis method of these models is different from the usual calculation method of computer systems. Artificial neural networks (ANN) are suitable tools for predicting of the nonexisting situations based on the existing conditions. In other words, artificial neural networks are able to interpret the relations between the parameters of one phenomenon and the phenomenon using the training based on experience. In recent years the neural networks have been used by many researchers for many of the civil engineering systems including predicting of surrounded and nonsurrounded concrete compressive strength [8, 9], predicting of ultimate shear strength of concrete beams reinforced with FRP [10, 11] and their free vibration analysis [12] and predicting of shear capacity of concrete beams reinforced with steel plates [13].
In general, artificial neural networks are consisting of three layers including input layer, middle or hidden layer and an output layer. The presented data to the network including the variables and target function resulted from them are placed in the input and output layers respectively. Then by regulating the weights in the middle layer, a pattern is obtained to achieve the target values from the input data. This trend is called training the network for predicting the target values. The number of data and less number of input variables result in better training of the network and more reliable obtained weights and more accurate network prediction.
Therefore the most important step in modeling a reliable and proper artificial neural network is collecting an appropriate number of experimental accurate and homogenous data (the more number of used data in modeling will result in better specifying relations among variables by the network). In this study, 112 homogenous experimental data have been gathered [35, 7, 1419]. The homogenous data is defined as the data in which the behaviors of the specimens are similar, and the failure is caused by shear under torsion moment. For example, beams that are failed under combination of torsion and bending moment cannot be considered for torsion strength study. After gathering suitable data for using in the network, selection of effective parameters in the target values shall be concerned. Studying the existing researches and the relations in the certified regulations [20, 21], the following parameters have specified to be effective in torsion capacity of the reinforced concrete beam:
To reduce the number of input variables, some of the related parameters are combined with each other, and finally, the following five parameters are introduced to the network as the input variables:
The ultimate bending moment for the beams (T_{u}) which is obtained from the experiments is considered as the target value for each input.
The number of the neuron in the hidden layer is considered as 8. The MSE graph of this network is shown in figure 1. As it is seen in the figure, MSE starts from high values and decreases to lower values. This shows the learning process of network is successful.
In the beginning of learning, the network has a rather high error and with continuation of learning and changing the used weights in the 31^{st} step the amounts of errors reaches to 0.005, 0.09 and 0.1 for Training, Validation and Test respectively. This graph has 3 curves, each of them represents a group of Training, Validation and Test data.
The graphs of procedure of learning and data regression values are presented in Figures 2 and 3 respectively. The reducing of the gradient in figure 2 represents the network procedure of learning. The reduction of gradient continues until MSE value reaches its minimum. From this point on the network, learning stops and the gradient value becomes constant. The amount of regression shown in figure 3 indicates a good network learning and close relation between the target vector and the network output.
Fig. 1. MSE graph during training of the trained artificial neural network
Fig. 2. The learning graph of the trained artificial neural network
It can be concluded that the modeled network is trained well with respect to the input and output data. Therefore, this network is chosen and its results are compared with the existing relations for ultimate torsion moment.
To validate the artificial neural network results, a comparison is made between the experimental results and the network results. Figure 4 shows the experimental results for the ultimate torsion moment, which are the target vector in network training, versus the output values, resulted from neural network simulation. In the presented curves the points corresponding to the 45degree line, indicate the proper prediction of the model and no difference between the experimental results and the estimated values by the network and distance from this line indicates the error percentage of the model. As it can be seen, most of the points of the network prediction fall in the neighborhood of 45degree line which indicates the accuracy of the network. %83 of the data have less than 10% error with respect to the experimental results. Also, % 97.3of the data have less than 20% error with respect to the experimental results. Summary of the network function is presented in Table 1. Considering figure 4 and Table 1, it can be concluded that the maximum error between the experimental results and the network output results is 30%.
Fig. 3. Fitting of the neural network data related to Training, Validation, and Test
Fig. 4. comparison between artificial neural network data and the experimental results
Table 1. comparison between artificial neural network error and the existing relations
Range of error (%) 
Number of data in the range of error 
Percentage of data in the range of error (%) 
±5 
60 
53.5 
±10 
93 
83 
±15 
102 
91 
±20 
109 
97.3 
±25 
111 
99.1 
±30 
112 
100 
In the artificial neural network system, each neuron has an internal weight, which affects the input values of the neuron and directs the weight vectors to the excitation functions. It may be required for a vector to displace in the vector space in addition to changing its weight; this can be done by adding a bias to the weight matrix. Then the weight values are transferred to the excitation functions, and the output function achieve its main value and considering obtaining a proper response, these values are compared to the target vector, and if there is the difference, the values are returned to select better weights for the vectors.
By increasing the number of input parameters, layers and neurons, calculation of each parameter effect on the network becomes more complex. Since 1980, many methods are presented for interpreting the effect of each parameter on the network. In this study analysis with respect to the weights size is used.
Analysis with respect to the weights size is based on the stored values in the weight matrix to determine the effect of input parameters on outputs. In this study, the presented relation by Garson [22] is used to evaluate the effect of input parameters on outputs (equation 1).
(1)
Where:
N: the input neurons,
L: the hidden neurons,
: The sum of N input neurons and j hidden neurons weights
Q_{ik}: the effect percent of input parameter x_{i }on the output y_{k}
For calculation of the effect of input parameters on the outputs, the weight of hidden layer shall be used. Input and output weights of ANN are presented in Table 2 and 3:
Table 2. Weight of input layer
Input nodes 
X (mm) 
Y (mm) 
f’c(MPa) 
Al Fy(KN) 
At Fyt /S (N/mm) 
Input Weights 
0.90496 
0.00676 
1.0286 
0.84106 
0.16771 
1.1542 
0.69404 
1.779 
0.82361 
1.3617 

0.15887 
0.81979 
0.80089 
0.48162 
2.0927 

2.0442 
0.5652 
0.88515 
5.1367 
0.55387 

2.0988 
0.76904 
0.44781 
1.8936 
1.7398 

0.93063 
1.1389 
1.0987 
1.7881 
0.68459 

0.39337 
0.49757 
0.35269 
0.58626 
1.0928 

0.06803 
0.59484 
1.0416 
0.17973 
1.6589 
Table 3. weights of output layer
Input nodes 
X (mm) 
Y (mm) 
f’c(MPa) 
Al Fy(KN) 
At Fyt /S (N/mm) 
Target Weights 
1.8417 
1.2028 
0.66981 
0.86408 
0.88664 
Substituting the weight values of the hidden layer in equation (1), the effect of input parameters on network outputs can be determined. The values of relative effect percentage of each of the 5 input parameters are presented in figure 5 and Table 4.
Fig. 5. effect percentage of input parameters on network outputs
Table 4. Effect percentage of input parameters on network outputs
Parameter 
X (mm) 
Y (mm) 
f’c(MPa) 
Al Fy(kN) 
At Fyt /S (N/mm) 
Effect (%) 
15.51 
13.07 
20.25 
24.93 
26.24 
According to the above table, it can be concluded that A_{t}F_{yt}/s parameter with relative effect percentage of 26.24% has the most effect on network output. It means that the target vector (ultimate torsion moment) has high sensitivity to A_{t}F_{yt}/s variations. On the other hand, Y (the larger dimension of the beam) parameter with relative effect percentage of 13.07% has the least effect on the ultimate torsion moment.
In this study, a model presented for calculation of reinforced concrete beams ultimate torsion moment using artificial neural network algorithm. Using the existing technical data, the results of experiments on 112 reinforced concrete beams under pure torsion have been gathered. After network training for validation of the network outputs, simulation of the existing data was performed and resulted in high accurate outputs and proved the ability of the trained neural network for estimation of torsion capacity of the reinforced concrete beams. The maximum error between the experimental results and the network output results was 30%.
Also by analyzing the network weights, it was concluded that in reinforced concrete beams, a variation of traverse reinforcements has the most effect on the torsion moment.