Prediction of Lateral Confinement Coefficient in Reinforced Concrete ...

KSCE Journal of Civil Engineering (2013) 17(7):1714-1719 Copyright ⓒ2013 Korean Society of Civil Engineers DOI 10.1007/s12205-013-0214-3

Structural Engineering

pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205

Prediction of Lateral Confinement Coefficient in Reinforced Concrete Columns using M5’ Machine Learning Method Mojtaba Naeej*, Meysam Bali**, Mohamad Reza Naeej***, and Javad Vaseghi Amiri**** Received April 28, 2012/Accepted February 1, 2013

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Abstract Predicting the lateral confinement coefficient in reinforced concrete columns is a very important issue in structural engineering. Therefore, several experimental formulas have developed to predict it. Recently, soft computing tools such as artificial neural networks have been used to predict the confinement coefficient. However, these tools are not as transparent as empirical formulas. In this study, another soft computing approach, i.e. model trees have been used for predicting the confinement coefficient. The main advantage of model trees is that, unlike the other data learning tools, they are easier to use and more importantly they represent understandable mathematical rules. In this paper, a new formula that includes some structural parameter is derived using dimensionless parameter for estimating the confinement coefficient. A comparison is made between the estimated confinement coefficient by this new formula and formula given by others researches shows the accuracy of prediction. Keywords: confinement coefficient, reinforced concrete columns, M5’ machine learning method, model tree, new formula ··································································································································································································································

1. Introduction As a constant stress to some material applied, a strain is produced which remains exactly the same for the period of time that the stress is maintained. When the load is released, the strain is totally recovered. That is, the material is the same as it was before the stress was applied, and this recovery happens immediately. Inelastic deformation occurs when this recovery does not happen immediately, but takes a certain amount of time. Inelastic deformability of reinforced concrete columns is essential for overall stability of structures in order to sustain strong earthquakes. Proper confinement of reinforced concrete leads to deformability of columns. The magnitude of the increase in strength of the R/C columns can be defined by a confinement coefficient, Ks. For this purpose, different behavior models are proposed to describe the confinement degree of confined concrete according to various parameters by many researchers. Kent and Park (1971) proposed a stress–strain curve for concrete confined by rectangular steel hoops. This curve is depicted by a second degree parabola up to maximum stress; a linear falling branch and a horizontal linear portion with stress constant at 0.2 of the maximum stress. The slope of the linear falling branch was found to depend on the concrete cylinder strength, the ratio of width of confined concrete to spacing of hoops and the ratio of volume of hoop steel to volume of concrete core. Park et al.

(1982), modified the stress–strain relation proposed by Kent and Park (1971). Sheikh and Uzumeri (1980) tested 24 short tied column sunder monotonic axial compression experimentally to evaluate the impact of various parameters on the behavior of tied columns. The main variables included the distribution of longitudinal steel, the column perimeter, the tie configuration, the amount of lateral confinement and longitudinal column steel. Also, Sheik and Uzumeri (1982) carried out a complete stress– strain curve for confined concrete based on the experimental results. In this model, they showed that the distribution of longitudinal steel and smaller tie spacing lead to higher strength and ductility of concrete. Ductility is the material's ability to deform under tensile stress. Actually the ductile materials are easily stretched without breaking or lowering in strength. Saatcioglu and Razvi (1992, 1999) proposed a stress–strain relationship for confined concrete. The relationship consists of a parabola for the ascending branch, and a linear portion for the descending branch. The descending branch is defined as the strain corresponding to 85% of the peak stress. A constant residual strength has been assumed at 20% strength level beyond the descending branch. The parameters of the analytical model have been developed from a large volume of experimental data, including poorly confined and well-confined concrete. The model has been compared with a large number of columns tested experimentally. Chung et al. (2002) proposed a stress–strain

*Master’s Student, Dept. of Civil Engineering, Babol University of Technology, Babol, Iran (Corresponding Author, E-mail: [email protected]) **Ph.D. Student, Dept. of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran (E-mail: [email protected]) ***Master’s Student, Dept. of Civil Engineering, Shahrood University of Technology, Shahrood, Iran (E-mail: [email protected]) ****Associate Professor, Dept. of Civil Engineering, Babol University of Technology, Babol, Iran (E-mail: [email protected]) − 1714 −

Prediction of Lateral Confinement Coefficient in Reinforced Concrete Columns using M5’ Machine Learning Method

relation of confined concrete from an empirical study of 65 columns. The experimental parameters were the volumetric ratio, the strength of concrete, the distribution of longitudinal bars and the confinement type of rectilinear ties. Sema Noyan Alacali et al. (2010) predicted the lateral confinement coefficient in reinforced concrete columns using neural network simulation. The main parameters were yield strength of transverse reinforcement, ultimate compressive strength, number of tie legs, volumetric ratio of lateral ties, center-to-center spacing between lateral ties and transverse steel area. In this study, M5’ algorithm (1997) is employed to predict the lateral confinement coefficient. M5’ model tree is a new soft computing method, first introduced by Quinlan (1992). The main advantage of the model trees is that, they provide some rules that are comprehensible and transparent. The trees obtained from M5, called model trees, are binary decision trees that can have linear regression equations at the leaves. The model trees would be used successfully in the prediction of lateral confinement coefficient in reinforced concrete columns. However, to the authors’ knowledge, this method has not been applied in the prediction of lateral confinement coefficient in reinforced concrete columns. In this research, the models have been developed by some reported Chung et al. (2002) data set (57 data). It has been trained by 2/3 data (37 data) and it has been tested by rest of data (20 data). This paper is outlined as follows: Previous approaches for lateral confinement coefficient given in Section 2. The used data set is described in Section 3. Evaluation of formulas is outlined in Section 4. M5 model trees are presented in section5. Development of new formula is described in section 6. Then, the conclusions are given in Section 8. Finally, the acknowledgments are presented in Section 9.

2.2 Sheikh and Uzumeri’s Approach Sheikh and Uzumeri (1982) carried out an experimental study on the stress-strain relation and confinement. In their study parameter Ks was affected by the gain in the concrete strength due to confinement provided by rectilinear reinforcement. Besides, they considered the effect of rectilinear reinforcement as a product of the square root of volumetric ratio of steel. From a regression analysis, Ks can be written as: 2

2 mc1 ⎞ ⎛ 22.9bc ⎛ s 2 -2 1 – -------⎞ Ks = 1 + ---------------- 1 – ------------Pocc ⎝ 5.5bc ⎠ ⎝ 2bc⎠

Pocc = 0.85f c′ ( Ack – Ast )

ρ s fyh

(3) (4)

where Ack is the area of confined core concrete, Ast is the area of total longitudinal reinforcement, m is the number of the longitudinal steels, and c1 is the net distance between longitudinal steels bounded by perimeter. 2.3 Fafitisand Shah’s Approach Fafitis and Shah (1985), proposed a stress-strain relationship for unconfined and confined concrete. A parametric study was conducted to assess the influence of the concrete strength, the level of axis load, the degree of confinement, and the shape of the section on the capacity of columns subjected to large deformations. According to their study a relatively good agreement between theoretical and experimental results was perceived. But it was not accurate enough. The empirical equation can be written as: 214.27 f Ks = 1 + ⎛ 1.15 + ----------------⎞ ----r⎝ f′c ⎠ f ′c

(5)

where fr is expressed by:

2. Previous Approaches for Lateral Confinement Coefficient 2.1 Kent and Park’s Approach Kent and Park (1971) proposed the modified stress–strain relation in confined concrete and observed a new relation for passive confinement controlled by KS. However, the location of longitudinal reinforcement and lateral reinforcement arrangements were not considered. An empirical equation can be written as: ρ s fyh K s = 1 + ---------f ′c

(1)

In this equation, ρs(Volumetric ratio of lateral ties) can be expressed by: Ashls ρ s = ----------sbc dc

(2)

where bc and dc are the width and depth of the confined core measured to outside of ties respectively, and S is the center to center spacing of ties. ls is the total length of the lateral ties. Ash is the area of lateral ties. fyh is the yield strength of lateral tie and fc is the compressive strength of concrete cylinder. Vol. 17, No. 7 / November 2013

Ashfyh fr = -----------sD

(6)

where D is the width of column section. 2.4 Soft Computing Approaches Artificial Neural Network (ANN) consists of very simple mathematical structures solving highly non-linear problems readily. ANN modeling has been used as an alternative approach for establishing nonlinear empirical equations in engineering problems comprehensively. ANNs are able to compensate for changing circumstances and to adapt solutions over time. They can assess empirical, theoretical or experimental data based on good and reliable past experience or a combination of these. Sema Noyan Alacali et al. (2010) predicted the confinement degree for confined concrete by using an ANN analysis. The accuracy of an ANN model is significantly affected by the input variables. They selected the six design parameters for the input layer to evaluate the Ks (output data) in R/C columns. The parameters concluded the compressive strength of unconfined concrete, the number of longitudinal bars, the yield strength of lateral tie, the diameter of lateral ties, the volumetric ratio of lateral ties and the spacing

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Mojtaba Naeej, Meysam Bali, Mohamad Reza Naeej, and Javad Vaseghi Amiri

between lateral ties. A nonlinear regression analysis was also carried out and the following nonlinear equation with a coefficient correlation of 0.60 was obtained by using statistical software (2010). ANN methods, however, are not as transparent and understandable as empirical formulas.

3. The Used Data Set In this study, Chung et al.’s (2002) laboratory data set was used for the evaluation of mentioned formula. A sample of their reinforced concrete columns with reinforcement details is shown in Fig. 1. Chung et al. (2002) proposed a stress-strain relation of confined concrete from an empirical study of 65 columns and investigated the confinement effects of concrete columns by lateral ties. Test columns have reinforced diameter. Table 1 shows the ranges of different parameters of the data set.

4. Evaluation of Formulas 57 data have been used to evaluate existing formulas. Fig. 2 shows the scatter between the measured and predicted Ks for the existing formulas. Fig. 2(a), shows that Sheikh and Uzumeri’s (1982) formula overestimates Chung’s data. Moreover, a noticeable deviation between measured and predicted Ks have been shown in Fafitis and Shah’s (1985) formula (Fig. 2(b)). Although, Kent and Park’s (1971) formula is more accurate than others, less deviation between measured and predicted Ks has been seen in this formula (Fig. 2(c)).

Fig. 1. A Sample of Chung et al.’s Column Section Details Table 1. Range of Dimensional Parameters parameters fc ′ (MPa) m fyh(MPa) fs(mm) s(mm) ρs Ks

range 19.6-56.4 0-12 550-1300 5-8 30-100 0.007-0.051 0.81-3.6

Fig. 2. Comparison Between Measured and Predicted Ks for All Data by: (a) Sheikh and Uzumeri, (b) Fafitis and Shah, (c) Kent and Park, (d) Ann Model − 1716 −

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5. M5 Model Trees Decision trees, widely used in classification problems, can be generalized to regression trees and modal trees that can deal with continuous attributes. Trees-structured regression is built on the assumption that the functional dependency is not constant in the whole domain but can be approximate as such on smaller sub domains. Depending on the nature of such model, there are several types of trees used for numerical prediction, like regression tree, model tree etc. M5 Model tree were first introduced by Quinlan (1992) and his idea was improved in a method called M5’ by Wang and Witten (1997). Model trees, like regression trees, are efficient for large data sets. At first, M5 model trees algorithm constructs a regression tree by recursively splitting the instance space. The splitting condition is used to minimize the intra-subset variability in the values down from the root through the branch to the node. The variability is measured by the standard deviation of the values that reach that node from the root through the branch with calculating the expected reduction in error as a result of testing each attribute at that node. The attribute which maximizes the expected error reduction is chosen. The splitting process is terminated if the output values of all the instances that reach the node vary only slightly or only a few instances remain. The standard deviation reduction (SDR) is calculated as: T SDR = sd ( T ) – ∑ -----i- × sd( T i ) i T

(7)

where T is the set of data point before splitting, Ti is data point which is the result of splitting the space and fall into one subspace according to the chosen splitting parameter and sd is the standard deviation. Standard deviation is used as an error measure for the data points of a subspace. M5’ model tree tests different splitting points by calculating sd for sub-spaces before dividing the space. As SDR is maximized in a point, the point is selected as the splitting point (node). The splitting stops when SDR change is less than a certain value or a few data points remain in sub-space. The accuracy of the model for training set increases uniformly as tree grows. However, over-fitting may be inevitable while a model tree is being built. Hence, in the second step, pruning is used to avoid over-fitting. Prediction of the 11 expected error at each node for the test data is used for the pruning procedure. For each training data, the predicted value is calculated. To prevent underestimation of the expected error , the predicted value is multiplied by the factor of (n+v)/(n−v) where n is the number of training data that reach the node and is the number of parameter in the model that represent the value at the node (Wang and Witten, 1997). The sub-space can be pruned if predicted error is lower than the expected one (Witten and Frank, 1997). The final step, smoothing, is the regularization process to compensate any probable discontinuities among adjacent linear models. The smoothing procedure uses the models built in each sub-space (leaf) to compute the predicted value. The obtained value is then modified along the route back to the root of the tree by smoothing it at each node. The value at each node is Vol. 17, No. 7 / November 2013

Fig. 3. Splitting of Input Space and Prediction by the Model Trees for a New Dataset: (a) Splitting of the Input Space such as X1×X2 by M5 Model Tree Algorithm, each Model is a Linear Regression Model Y = a0 + a1X1 + a2X2, (b) Prediction for New Instance by Model Tree

combined with the output value by the linear model for that node as P' = (np + kq)/(n + k), where is prediction passed up to the next higher node, is prediction passed node from the below, is the value predicted by the model at this node, n is the number of training data points that reach the node below and k is a constant (Wang and Witten, 1997). The predicted value by the leaf is combined with that of linear model for each node on the top of the leaf to the root. This smoothing process usually improves the predictions. Fig. 3 illustrates splitting the space for building a tree and eliciting knowledge from the structure. The record finds its way down by passing through the nodes. Nodes in the tree represent the testing of a particular parameter. This testing process involves comparing the given parameter with a constant value. These nodes are arranged on the basis of dividing condition of the first process (i.e., the process of building the tree). The related prediction of the introduced record is obtained when a leaf is reached, and it is recognized as an output. That record is indeed classified on the basis of the class appointed to that leaf. More explanations are available in Jung et al.’s (2009) research.

6. Development of New Formula Chung’s data has been used to develop new formula. From this data, 37 data points were selected for model training. The rest data has been used to test model. Dimensionless parameters have been used to develop new formula. The ranges of different parameters used for test and train are presented in Table 2. As

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Mojtaba Naeej, Meysam Bali, Mohamad Reza Naeej, and Javad Vaseghi Amiri

Table 2. Range of Non-Dimensional Parameters. Parameters m ρsfyh/ f ′c db/c1 1−S/2D

Range (train) 8-12 0.07-2.76 3.86-14.46 0.75-0.92

Range (test) 4-12 0.1-1.88 3.86-14.46 0.75-0.92

seen, the selected parameters are the same as those used by previous investigators and have wide ranges. The M5 model tree produced only linear relationships between the input and output parameters, i.e. Ks = a(m)b(ρsfyh/ f ′c )c(db/c1)d(1− (S/2D))e. To overcome this limitation, the parameters have been used in their logarithmic forms. The developed formulas (MT) were: K s = 6.17 ( m )

–0.15

s f yh⎞ ⎛ρ --------⎝ f ′c ⎠

0.22

⎛d ----b⎞ ⎝ c1 ⎠

0.33

S ⎞ 0.77 ⎛ 1 – -----⎝ 2D⎠

(8) N

where db is the diameter of longitudinal bars and other parameters were defined before. As seen in Eq. (8), the lateral confinement coefficient depends on four dimensionless terms. Width of column section and center-to-center spacing between lateral ties were found to be the most effective parameters on lateral confinement coefficient. These results were achieved by some researches like Sheikh and Uzumeri (1982). However, this simple and compact formula is in good agreement with engineering sense and previous knowledge about role of different parameters in the lateral confinement coefficient. For example, increasing the volumetric ratio of lateral ties increases Ks. This issue can be achieved by Kent and Park (1971). Scatter diagrams of measured and predicted Ks by MT are shown in Fig. 4 (all data) and 5 (testing data). As seen in these figures, the data points are more concentrated on the optimal line compared to the corresponding diagram of others formula (Fig. 3). The performance of different approaches were also judged quantitatively using error measures such as BIAS, Scatter Index (SI), Correlation Coefficient (CC) and agreement index (Ia) defined as: N

BIAS =

1

- ( Yi – Xi ) ∑ --N

Fig. 5. Comparison Between Measured and Predicted Ks for Test Data by MT

(9)

i=1

2 1 ---- ∑ ( Y i – Xi ) Ni = 1 SI = ----------------------------------Xi

(10)

N

∑ ( Xi – X )( Yi – Y )

i=1 CC = ------------------------------------------------------N

2

N

∑ ( Xi – X ) ∑ ( Y i – Y )

(11)

2

i=1

i=1

N

∑ ( Yi – Xi )

2

i=1

Ia = 1 – --------------------------------------------N 2 ( Y – X ) ( X – X ) i ∑ i

(12)

i=1

where Xi and Yi denote the measured and predicted values respectively and N is the number of observations. Xand Y are the mean values of the measured and predicted parameters, respectively. Table 3 shows the error measures of MT and previous equations for the used data set. It can be seen the BIAS and SI value of MT are less than those of other formulas. The BIAS and SI have largely reduced using MT. In addition, the CC and Ia of MT are larger than those of others. Generally, it can be concluded that MT is more accurate than the previous equations. Although ANN method has same results with the new formula but ANN methods could not offer insight into the developed model.

Table 3. Error Measures of Different Formulas for All Irregular Wave Data

Fig. 4. Comparison Between Measured and Predicted Ks for All Data by MT

Formulas Kent and Park Sheikh and Uzumeri Fafitis and Shah ANN MT(test) MT(all data)

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BIAS -0.01 1.24 0.30 -0.13 -0.08 -0.01

SI 0.200 1.291 0.490 0.220 0.160 0.120

CC 0.63 0.72 0.33 0.60 0.78 0.84

Ia 0.89 0.47 0.66 0.85 0.96 0.95

KSCE Journal of Civil Engineering


7. Conclusions

References

In this study, M5 model tree was used to predict the confinement coefficient in R/C rectangular columns and its skill was compared with the previous formula and soft computing methods. To compare the performance of the model trees for confinement coefficient prediction, a MT models were developed. In this study, to develop new formula, Chung et al.’s (2002) laboratory data set (57 data point) was used The data of 37 data point were used to train of M5. On the other hand, the testing of the M5 was carried out by the data of 20 data point. Based on the results presented, the following conclusions are drawn: 1. MT showed high performance for all cases leading to the reduction of systematic errors. In addition, the uncertainties of the given formulas were given for the prediction of confinement coefficient when a certain level of risk is desired. 2. It is found that the performances of the proposed models were better than those of previous empirical and the soft computing methods. In addition to the higher accuracy (CC of 0.84), the other advantage of the model trees (compared to previous soft computing approaches) was the ability to generate simple and meaningful formulas. 3. Width of column section and center-to-center spacing between lateral ties were found to be the most effective parameters on lateral confinement coefficient. These results can be achieved by Eq. (8) and evaluating the parameters. 4. The approach employed in this study was shown to be capable of providing the best accurate estimates of Ks by using the six design parameters. The results are considered to be encouraging for further research of expanded data sets. By setting up some random variations in the design parameters, it is possible to estimate the Ks for new or existing R/C rectangular columns. To study modeling and prediction methods, and construct an accurate prediction model of Ks, there is a need for an experimental research in which full-scale data of R/C rectangular columns with various configurations are considered. This type of a broad range of experimental data would also be helpful for evaluating the sensitivity analyses results.

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Acknowledgements The authors wish to express their sincere thanks to Chung et al. for their comprehensive and freely database of the lateral confinement coefficient reinforced concrete columns.

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