Assessment of RC exterior beam-column Joints based on artificial neural networks and other methods

Abstract A database on the behaviour of reinforced concrete external beam-column joint sub-assemblages established from the results of over 150 tests is developed and used for the development, training and validation of an artificial neural network (ANN) based model. The ANN model predictions on the mode of failure and load-carrying capacity of the joints, together with the predictions of widely used code methods and those of a recently proposed method, which does not require calibration through the use of test data, are compared with their counterparts stored in the database developed herein. The comparison confirms the already reported shortcomings of current code methods and demonstrates that both ANN model and the recently proposed method can provide reliable alternatives to the code methods.


Introduction
The structural assessment of reinforced concrete (RC) beamcolumn joints is normally based on the design methods incorporated in current codes of practice such as, for example, ACI 318 [1] and EC2 [2]-EC8 [3]. However, in spite of the considerable research carried out to date on the subject, the code specified amount of stirrup reinforcement exhibits significant discrepancies. An example of such discrepancies has been reported in Ref. [4] where it was shown that the amount of stirrup reinforcement specified by EC2 and EC8 can be more than four times the amount specified by ACI 318. And yet, even for such a case, designing a beam-column joint in accordance with the EC2 and EC8 provisions for the design of earthquake-resistant structures cannot safeguard structural performance which satisfies the code requirements; in fact, not only does the joint suffer considerable cracking before the formation of a plastic hinge in the adjacent beam, but also such cracking occurs at early load stages, and thus violates the assumption of 'rigid joint' which underlies the methods adopted in practice for structural analysis [5]. Similar results have been obtained from a number of investigations [6][7][8][9]. It appears from the above, therefore, that the need for a reliable structural assessment method is as urgent as ever.
To this end, the present work has been aimed at developing a procedure suitable for the structural assessment of RC external beam-column joints in the form of an analytical algorithm derived through the use of artificial neural networks (ANNs). In fact, the use of ANNs has already led to the development of analytical algorithms found to produce very accurate predictions of the behaviour of simply-supported RC beams [10]. In contrast with most methods proposed to date, those incorporated in current code methods included, the development of an ANNs-based procedure does not rely on preconceived theories describing the mechanism of load transfer within a structural element or structure which are subsequently calibrated through the use of experimental data, but on the ability of the ANNs to produce the closest possible fit to such data.
In what follows, the description of the development of the proposed procedure is preceded by the presentation of the database used in the subsequent stages of the work and followed by a discussion of the results obtained from the application of the proposed procedure for predicting the mode of failure and loadcarrying capacity of the external RC beam-column joints comprising the database. Finally, the proposed procedure's predictions are compared with their ACI 318 and EC2-EC8 counterparts in an attempt to obtain an indication of the effectiveness of the current code methods. The procedure's predictions are also used to establish the effectiveness of one of the methods proposed as an alternative to those adopted by current codes, this method being selected not only because it has been claimed to be effective (on the basis of a comparative study of its predictions with experimental information, not however as extensive as that included in the database developed in the present work) [5], but also, unlike any other method proposed to date, has been derived without the need of calibration with data obtained from tests on RC beam-column joints [10].

Experimental database
Τhe database developed for the needs of the present work comprises experimental information on 153 exterior reinforced concrete beam-column joint sub-assemblages, such as those schematically shown in Fig. 1, obtained from the literature. This experimental information includes design details, material properties and structural response to the imposed loading history such as load-deflection curves, failure mode and load-carrying capacity. The linear elements (beam and columns) of these subassemblages are considered to represent the portions of frametype structures extending between the joint-element interface and the nearest point of contra-flexure (point of zero bending moment). In most cases, the beam-column sub-assemblages were subjected to transverse loading (cyclic or monotonic) usually applied near the beam end, and only in a small number of cases near the end of the upper column. In most of the subassemblages the transverse load was combined with a constant axial compressive force exerted concentrically at the end of the upper column.
The parameters widely considered to affect the behaviour of the joints are: the dimensions of the beam/column elements and the joint; the compressive strength of concrete; the amount, arrangement and yield stress of the longitudinal (tensile and compressive) reinforcement of beams and columns; the amount, arrangement and yield stress of the transverse and diagonal reinforcement of the joints; the amount of the imposed axial load on the upper column element; the height of the column; and the distance between the beam/column-joint interface and the point of application of the transverse load.
Full design details and description of the experimentallyestablished behaviour of the external RC beam-column joint subassemblages forming the database developed can be found in the relevant publications [4,7,9, which describe most of the work carried out to date on the subject.

Background information
Artificial neural networks is a type of artificial intelligence technique that mimics the manner in which the human brain functions. ANNs are used to estimate or approximate functions depending on a large number of input parameters the effect of which is not clearly established or quantified. They have the ability to learn, generalize, categorize and predict values due to their adaptive nature and their ability to remember information introduced to them during their training. Over the last years they have been applied in various aspects of science and engineering; in the present work ANNs are used to predict the mode of failure and load-carrying capacity of external RC beam-column joint sub-assemblages, with the load-carrying capacity being also expressed in the form of the horizontal shear force developing at the joint mid height.
Free-forward multilayered networks (MLP), as shown in Fig. 2, are used in the present work. These ANNs consist of a number of layers (input, several hidden and output layers) organized strictly in one direction, each of them being a system of interconnected ''neurons" (or otherwise referred to as nodes). Fig. 2 depicts a typical MLP ANN with one of its constituent neurons being shown in Fig. 3. From Fig. 2, it can be seen that the ANN consists of three layers -input, hidden and output layers -with the input layer comprising k neurons, the hidden layer three neurons and the output layer two neurons.
From Figs. 2 and 3, it can also be seen that the link forming between consecutive neurons is assigned a specific weight, which is multiplied with the input values generated by the neurons. The values obtained from all neurons of a specific layer are then transferred through the links and summed up with the bias (see Fig. 2). This latter sum is introduced into a predefined activation function representing the relationship between layers and is described as follows: where y i is the output from the ANN; x j are the input values; w ji are the weight coefficients; H i is the bias value; and g is the activation function which is usually a sigmoid, a tan-sigmoid or a linear function. In the present work, sigmoid activation functions are used for the input and hidden layers and tan-sigmoid for the output layer.

Input and output data
The input and output/target data have been obtained from the database and organized in a tabular form. The selection of the parameters used as input data has been based on the commonly shared view that these parameters have a dominant effect on beam-column joint behaviour [1][2][3]. They are the following: the uniaxial cylinder compressive strength (f c ) of concrete; the crosssectional beam width (b b ) and height (h b ); the cross-sectional column width (b c ) and height (h c ); the beam (M Rb ) and column (M Rc ) flexural capacities; column's longitudinal reinforcement ratio (q c ), the beam's tensile (q bt ) and compressive (q bc ) longitudinal reinforcement ratio; the product (q s f ys ) of the yield stress (f ys ) and the reinforcement ratio (q s ) of the joint stirrups; the product (q d f yd ) of the yield stress (f yd ) and the reinforcement ratio (q d ) of the horizontal component of the diagonal reinforcement of the joints; the vertical reinforcement ratio (q sv ) of the joints; the column height (H c ) (see Fig. 1); and the distance of the beam-joint interface from the point of application of the transverse load (L b ).
Since the code structural performance requirements for the beam-column joints are linked with their mode of failure and load-carrying capacity, these two parameters have been used as target data. The modes of failure (FM exp ) are classified as joint failure (JF) or beam failure (BF); the load-carrying capacity is represented by the transverse load (P max ) applied on the beam-column sub-assemblage and by the shear capacity (V jh ) of the horizontal mid-height cross section of the joint (see Appendix). The maximum and minimum values of the input and target values are shown in Table 1, whereas the values of all input parameters are shown in Table A1 in Appendix A. It should also be noted that, as discussed later, in order to minimize the likelihood of numerical instabilities and/or low convergence rates, the values of the input and target parameters have been normalized in the range [0,1] or [0.1,0.9].

ANN procedure architecture
The number of neurons of the input layer is determined from the number of the parameters considered to have a significant effect on structural behaviour. In the present work, two sets of parameters are investigated: set 1 consisting of fifteen parameters, those in lines 1 to 15 in Table 1, and set 2 consisting of twelve parameters, those of set 1 except for parameters q c , q tb and q cb in lines 4, 9 and 10 of Table 1, since these parameters are allowed for in the calculated values of the beam and column flexural capacities. Therefore, as regards the number k of the neurons in the input layer (see Fig. 1), two cases are investigated: k = 15 and k = 12.
The selection of the number of neurons comprising the hidden layers is usually based on the outcome of a parametric study. Such a parametric study carried out in previous work concerned with the use of ANNs for assessing the load-carrying capacity and mode of failure of RC beams has shown that the use of nine neurons produces very close predictions [11]. Therefore, the same number of neurons has been adopted in the present work.
As regards the output layer, this consists of one neuron, which is either the load-carrying capacity or the mode of failure. The loadcarrying capacity is calculated in the form of the transverse load (P max ) applied on the beam-column sub-assemblage and of the shear capacity of the horizontal cross section at mid-height of the joint, whereas the output values for BF and JF are 1 and 2, respectively.
In view of the above, the ANNs models adopted for the present work are models 1 and 2 with the same number of neurons (9 and   1) in the hidden and output layers, but differing in the number of neurons (15 against 12) of the input layer. In order to investigate the effect of the use of the normalization equations on the ANNs predictions, each of the models 1 and 2 are subdivided into models 1(a) (ANN1) and 1(b) (ANN2) and models 2(a) (ANN3) and 2(b) (ANN4) in which the input data are normalized in the range [0,1] or [0.1,0.9], respectively.

Training, validation and testing of the ANNs
The process of training a neural network involves tuning the values of the weights and biases of the network, which are initially randomly assigned, to optimize network performance according to an iterative process. The default performance function for feed forward networks is the mean square error msethe average squared error between the network outputs. It is defined as follows: where T and O are the target and output values respectively. The error of each output node is computed from the difference between the computed output and the desired output (target). To minimize the error obtained at each iteration, such as the output values of the ANN essentially agree with the target values, the back-error propagation algorithm (or Delta rule) is employed. This process is carried out from right to left of the ANN and makes use of the information provided from the database. According to this process, after the error is computed, the weights and biases are readjusted in order to obtain more accurate outputs and thus smaller errors. To enhance the performance and accurate predictions of the ANN, the normalized data (inputs and targets) is initially and randomly divided into three sub-sets for training, validation and testing purposes. In the present work 60% of the sets of inputs and target outputs is used for training, 20% for validation and 20% for testing.
Learning algorithms are classified as local or global algorithms; furthermore the methods used for dealing with the problem of weights and biases calibration may also be classified as deterministic or probabilistic. Global algorithms make use of knowledge regarding the state of the entire network, such as the direction of the overall weight update vector. In the widely used backpropagation global learning algorithm the gradient descent algorithm is used. As regards local adaptation strategies, these are based on specific information of the weight values such as the temporal behaviour of the partial derivative of the weights. The local approach is better related to the ANNs concept of distributed processing, where the computations are performed independently. Moreover, it appears that for many applications local strategies achieve faster and more reliable predictions than global techniques [40], this is why such a method was adopted in the current study of deterministic nature.
This iterative process, which can be schematically be seen in Fig. 4 is repeated until one of the following conditions is met: I. The maximum number of 100 iterations (epochs) of training after which the algorithm terminates the training process II. A maximum of 6 validation failures are exhibited. Validation failure occurs when the performance of the ANN during each iteration fails to improve or remains constant. III. The value of performance goal becomes 10 À6 expressing the difference between the target and output values and IV. The minimum performance gradient becomes 10 À10 .

Comparative studies
The four ANN models investigated have been used to predict the transverse load-carrying capacities (P ANN1, P ANN2 , P ANN3 , P ANN4 ) of the beam-column sub-assemblages of the database developed in the present work. The mean values and standard deviations of the results obtained are summarized in a normalized form (resulting by dividing the calculated values of load-carrying capacity with their experimental counterparts P EXP ) in Table 2; the full results are presented in Table A2 in Appendix A.
From Table 2, it can be seen that both the number of neurons of the input layer and the expressions used for normalizing the input data investigated have a negligible for practical purposes effect on the models' predictions. However, in contrast with widely held views, normalization within the range [0,1], rather than within the range [0.1,0.9], resulted the smallest values of standard deviation. Therefore, in what follows, the ANNs predictions have been based on the use of Model 4.
The number of successful ANN predictions of the types of failure suffered by the structural forms investigated are shown in Table 3; detailed predictions are provided in Table A3. Both tables also include the predictions obtained from the methods incorporated in two widely used codes of practice, ACI 318 and EC2-EC8, together with those of the method proposed by Kotsovou & Mouzakis [41]. A concise description of the latter methods is provided in Appendix B. It should be noted that the code methods are calibrated through the use of values which form a safe envelope to the available experimental data and not mean values such   Table 3, it can be seen that, in spite of the conservative nature (excluding the use of safety factors assumed to be equal to 1 for the purposes on the present work) of the code methods, the successful predictions of the type of failure suffered by the structural elements investigated are significantly fewer than those of the Kotsovou& Mouzakis method which exhibited a rather small deviation (of the order of 5%) from those of the ANNs, the latter providing the best predictions. The methods used to predict the type of failure suffered by the structural elements contained in the database have also been used to predict the 'shear' capacity of the elements characterized by joint failure. From Table A3, it can be seen that 101 of the 153 elements of the database exhibited joint failure. The mean values and the standard deviations of the models' predictions are shown in Table 4which also includes the number of the predictions used to obtain the mean values and corresponding standard deviations. The detailed results are shown in Table A4 in Appendix A.
It should be noted that the values obtained through the use of the ACI 318 and EC2-EC8 methods are not predictions of 'shear' capacity, but maximum permissible values of the shear force developing at the joint mid height under the load applied to the beam-column joint sub-assemblages investigated. It is the specified stirrup reinforcement of the joint that is expected to delay joint failure until plastic hinge formation at the beam joint interface, but, unlike the Kotsovou & Mouzakis method, the codes do not provide any formula linking the amount of such reinforcement with the 'shear' capacity of the joint. Therefore, as regards the ACI 318 and EC2-EC8 methods, the values included in Tables 4 and A4 are the upper limits of 'shear' capacity calculated only for the structural elements with stirrup reinforcement in accordance with the code specifications. From the tables, it can be seen that the number of such elements is rather small.
Unlike the code specifications, the Kotsovou & Mouzakis method links the amount of stirrup reinforcement with the value of 'shear' force that should be sustained by the joint so as to prevent failure before the formation of plastic hinge formation at the beam-joint interface. From both the detailed results (Table A4) and their summary (Table 4), it can be seen that the method produces safe predictions of 'shear' capacity in all cases investigated.
Finally, ANNs produce the closest estimate of joint 'shear' capacity. However, it should be reminded that the method is empirical by nature and, therefore, it is only by implication that the predicted values refer to 'shear' capacity, since the method has been trained using values of 'shear' capacity. Moreover, the predicted values obtained are as valid as the experimental information used to train and validate the method.

Conclusions
The development of ANNs, which is based on a detailed mathematical approach allowing for all parameters found experimentally to affect joint behaviour but not considering any mechanism of load transfer such as those underlying most of the seismic design beam-column joints methods introduced to date, is found to produce predictions of the joint strength which correlate very closely with the experimentally-established values. It is also found capable of predicting the mode of failure of beam-column subassemblages in over 95% of the 153 cases investigated.
On the other hand, not only it is not possible to assess joint strength through the use of current code methods, but also these methods are found unable to predict the mode of failure of beam-column joint sub-assemblages, which is considered an indication of their inability to safeguard the code requirement for failure of the sub-assemblages due to the formation of a plastic hinge at the beam-column interface, before the joint suffering significant cracking. These findings confirm already reported similar findings.
In contrast with current code methods, the method proposed by Kotsovou & Mouzakis method, which unlike any other method proposed to date does not require calibration through the use of experimental data on joint behaviour, is found capable of predicting the mode of failure of external beam-column joint sub- where A sh is the total area of horizontal hoops, V jhd is the design joint shear force, h jw is the distance between the top and bottom beam reinforcement, and m d is the normalized axial force of the column above. where h c is the column cross section height, x c the depth of the column compressive zone depth, z c ¼ h c À x c , when x c > z c =3, then z c =3 is replaced with x c , w j the joint width, and a is the inclination of the joint diagonal The total amount of stirrup reinforcement is obtained from: where T j ¼ F j tanb ¼ The shear force F js,h that can be sustained at the joint mid height when the transverse reinforcement is at yield is obtained from