J. Cent. South Univ. (2013) 20: 2863−2869 DOI: 10.1007/s11771­013­1807­1

Failure mode classification of reinforced concrete column using Fisher method

QI Yong­le(戚永乐) 1. 2 , HAN Xiao­lei(韩小雷) 2 , JI Jing(季静) 2

1. Guangdong Electric Power Design Institute, China Energy Engineering Group Co. Ltd., Guangzhou 510663, China; 2. School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China

© Central South University Press and Springer­Verlag Berlin Heidelberg 2013

Abstract: In order to apply the performance­based seismic design, an engineer must first find out whether the column is expected to fail in shear before or after flexural yielding. According to column failure characteristics and failure mode of reinforced concrete column, the UW­PEER structure performance database was discussed and analyzed. In order to investigate the relevance of failure mode and factors such as longitudinal reinforcement ratio, transverse reinforcement ratio, hoop spacing to depth ratio, aspect ratio, shearing resistance demand to shear capacity ratio and axial load ratio, Fisher’s discriminant analysis (FDA) of the above factors was carried out. A discriminant function was developed to identify column failure mode. Results show that three factors, i.e., Vp/Vn, hoop spacing to depth ratio and aspect ratio have important influence on the failure mode. The failure mode has less to do with longitudinal reinforcement ratio, transverse reinforcement ratio and axial load ratio. Through using these three factors and the model proposed, over 85.6% of the original grouped cases were correctly classified. The value of coefficient of Vp/Vn is the largest, which means that discriminant equation is most sensitive to the shearing resistance demand to shear capacity ratio.

Key words: Fisher’s discriminant analysis (FDA); concrete column; failure mode identification; performance­based seismic design

1 Introduction

Performance­based earthquake engineering (PBEE) aims to improve structure engineering by providing engineers with the capability of designing structures to achieve a variety of performance levels [1−2]. In severe earthquakes, some columns failed in shear after flexure yielding while others failed in shear without flexure yielding, and their seismic performance creates sharp distinction [3−5]. To implement PBEE, it is necessary to classify the failure modes firstly, and then give different performance acceptance criteria under different failure modes.

Columns are the primary members of frame structures that dominate the frame response during earthquakes. Most building collapses in the Wenchuan earthquake for poor column performances [6]. Column failures in buildings are either due to insufficient shear resistance (shear failure) or due to insufficient deformation capacity (flexure­shear and flexure failure) [7]. New generation of performance­based seismic design [8−9] expresses the performance of column in terms of plastic rotation capacities of the critical end regions. In order to apply the performance­based seismic design, an engineer must first determine which kind of

failure mode the column is expected to fail in. Because the modeling parameters and numerical acceptance criteria for nonlinear procedure of reinforced concrete columns are different in different failure modes, ASCE/SEI 41—06 and FEMA 356 classified modeling parameters for reinforced concrete columns according to whether they are flexure failure, shear failure, or flexure­shear failure [10]. Columns controlled by shear had zero permissible plastic deformation and were evaluated using lower­bound material strengths. A column is to be classified into one of three failure modes based on the nominal shear strength, the plastic shear demand on the column, and the transverse reinforcement detailing. But due to failure mode depended on many other variables, it should not be expected that classification in FEMA 356 and ASCE/SEI 41—06 will correctly predict the failure mode of a column in every case. Therefore, the primary objective of this work is to propose a discriminant model more precise to investigate and predict the failure type of reinforced concrete (RC) column before damage.

2 Column failure modes

RC columns under both lateral and axial loads are damaged in one of the following forms: 1) flexure

Foundation item: Project(2011ZA05) supported by the State Key Laboratory’s Autonomous Project of Subtropical Building Science in South China University of Technology

Received date: 2012−05−23; Accepted date: 2012−11−10 Corresponding author: QI Yong­le, PhD; Tel: +86−20−87113349; E­mail: q.yongle@mail.scut.edu.cn

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failure, where degradation in the lateral load capacity occurs after yielding of the longitudinal reinforcement due to damage related to flexural deformations (i.e., spalling of concrete, buckling of longitudinal bars, concrete crushing); 2) shear failure, where degradation in the lateral load capacity occurs before yielding of the longitudinal reinforcement due to shear distress (i.e., diagonal cracking) in the column; 3) a combined flexure­shear failure, where degradation in the lateral load capacity occurs after yielding of the longitudinal reinforcement but results from shear distress in the column [10].

The PEER structure performance database has been assembled to provide researchers with the data needed to evaluate and develop seismic performance models for reinforced concrete columns. The database is available on the World Wide Web from the University of Washington and from PEER (UW­PEER).

In this database [11], the nominal column failure mode was classified as flexure­critical, flexure­shear critical, or shear­critical. If no shear damage was reported by the experiment, the column was classified as flexure­critical. If shear damage was reported, the absolute maximum effective force (Feff), was compared with the calculated force corresponding to a maximum strain of 0.004 (F0.004). The failure displacement ductility at the 80% effective force, μfail, was also considered. If the maximum effective force was less than 95% of the ideal force (Feff<0.95F0.004) or if the failure displacement ductility was less than or equal to 2 (μfail≤2), the column was classified as shear failure. Otherwise, the column was classified as flexure shear failure [11].

Above categorization method is given according to the experimental phenomena after column failure. Nevertheless, in order to apply the performance­based seismic design, an engineer must first find out which mode the column is expected to fail before the experiment. Hence, the effort is to provide a relation which determines the column failure mode before the failure.

3 Data distribution

In total, 111 columns obtained from PEER structure performance database are considered to develop the column classification method. The 111 columns considered in this work have properties within the following ranges. Figure 1 gives their distribution in transverse reinforcement ratio, longitudinal reinforcement ratio, hoop spacing to depth ratio, aspect ratio, axial load ratio and failure modes. There are many factors affecting the columns failure modes. To avoid the error caused by choosing different factors, all the gathered factors were assumed to influence the failure

modes. Through the stepwise discriminant analysis method, the items were ranked according to their importance, from most to least significance, and then the discriminant equations were established.

4 Column failure mode classification

4.1 Based on aspect ratio In a much generalized statement, the type of failure

can be said to be mainly dependent on the aspect ratio of the column [12]. If a/d≤ 2, the column fails in shear, whereas for a/d ≥4, it exhibits “flexure failure”. For the range of 2<a/d<4, shear strength and shear demand tend to be close and the type of failure is generally uncertain. This range is a grey area of so called “flexure­shear” which has to be explored.

Figure 2 compares the observed column failure mode and the value of aspect ratio of the selected database. The plot shows relatively high dispersion for all three failure modes. Only 63% of columns with aspect ratio a/d≤2 experienced pure shear failures. Only 51% of columns with aspect ratio a/d≥4 experienced pure flexure failures. In contrast, 92% of column with aspect ratio 2<a/d<4 experienced flexure­shear failure. It is apparent that the boundaries of a/d=2 and a/d=4 are not sufficient to distinguish the three failure modes. Hence, the classification of column failure modes based on the aspect ratio is not adequate.

4.2 Based on shear strength Here, only the transverse reinforcement details of

ACI conforming details with 135° hooks are researched. It is well recognized that the relation between plastic shear demand and shear strength provides useful information in the determination of column failure modes [13]. Here, the column shear demand is determined by its maximum moment capacity divided by the shear span, Vp=Mmax/a. The maximum moment capacity, Mmax, is computed through a moment­curvature analysis for the column’s cross section. The column shear strength, Vn, is calculated according to a shear strength model proposed by SEZEN and MOEHLE [14]. For columns, the shear strength, Vn, is calculated according to Eq. (1):

g g c

u c y v n 8 . 0 )

6 1

/ 6

( A A f

N Vd M f

k s d f A

k V ′

+ ′

+ = λ (1)

≥

< < −

≤

= 0 . 6 , 7 . 0

0 . 6 2.0 , 075 . 0 15 . 1 0 . 2 , 0 . 1

δ

δ δ

δ

µ µ µ

µ k (2)

where c f ′ is in MPa and the coefficient k defines the shear strength degradation with increasing displacement

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Fig. 1 Parameter histograms of specimens

Fig. 2 Relation between observed column failure mode and aspect ratio

ductility, δ µ ; Nu is the axial compression force (Nu=0 for tension force); M/Vd is the largest ratio of moment to shear times effective depth under design loadings for the column but shall not be taken greater than 4 or less than 2; d is the effective depth; Ag is the gross cross­sectional area of the column. It shall be permitted to assume d=0.8h, where h is the dimension of the column in the direction of shear.

To reduce the likelihood of non­conservatively misclassifying a column as flexure­critical when it might actually sustain a flexure­shear failure, the upper bound on Vp/(Vn/k) for flexure failure mode was set at 0.6, rather than 0.7, as might be inferred from Eq. (1) [13].

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The above method was used in ASCE/SEI 41—06, and the detail is given in Table 1.

Table 1 Classification of columns for determination of modeling parameters

Condition Conforming details with 135° hooks

Vp/(Vn/k)≤0.6 Flexure failure

(flexural yielding without shear failure)

0.6＜Vp/(Vn/k)≤1.0 Flexure­shear failure (shear failure

following flexural yielding)

1.0＜Vp/(Vn/k) Shear failure (shear failure before

flexural yielding)

Figure 3 compares the observed column failure mode and the value of Vp/(Vn/k) for the selected database. The plot also shows relatively high dispersion for all three failure modes. It is also apparent that the boundaries of Vp/(Vn/k)=1.0 and Vp/(Vn/k)=0.6 are not sufficient to distinguish the three failure modes. Hence, the classification of column failure modes based only on the shear strength model is also not adequate; other column parameters, which may also influence the observed failure mode, should be considered.

Fig. 3 Relation between observed column failure mode and Vp/(Vn/k)

5 Fisher discriminant analysis

5.1 Basic principle of Fisher’s linear discriminant analysis Fisher’s linear discriminant analysis (FDA) is a

method to find a linear combination of features which characterize or separate two or more classes of objects or events [15]. Fisher’s discriminant analysis is a linear dimensionality reduction technique, optimal in terms of maximizing the separation among different classes. Through a series of linear transformation, FDA technique can maximize the scatter between the classes and minimize the scatter within the classes. Consequently,

various classes can be re­arrayed and separated in the transformed data space [16]. This property of FDA can be used to isolate the fault source.

All the operation data from the measurement system, including normal operation and faulty operation, can be classified into different data classes Gi(i=1, …, k), where G1 refers to the normal operation data class, and G2, …, Gk refer to the various faulty data classes. With the ni rows of samples from class Gi, supposing i x is a mean m­dimensional sample from class i that can be denoted

as 1

i

i i G n ∈

= ∑ x

x x . The mean of all the samples is

1

1

i

k

i G n = ∈

= ∑ ∑ x

x x , where n is the total number of all the

samples, then the within­class scatter matrix can be given

by T w

1 ( )( ) .

i

k

i i i G = ∈

= − − ∑ ∑ x

S x x x x And the

between­class scatter matrix can be given

by T b

1 = ( )( ) .

k

i i i i n

= − − ∑ S x x x x Therefore, the optimal

discriminant direction is obtained by maximizing the

Fisher criterion: T

b

w

( ) T

J ϕ = S S

ϕ ϕ

ϕ ϕ , where φ is the Fisher

optimal discriminant direction which maximizes the between­class scatter but minimizes the within­class scatter. If Sw is nonsingular, the optimal problem can be transferred into a conventional eigenvalue problem by writing 1

w b − = S S ϕ λϕ , where λ=(λ1, λ2, …, λr) are the

eigenvalues of 1 w b , − S S and the corresponding

eigenvectors can be denoted as l=(l1, l2, …, lr). So, the discriminant function can be given by y=l T x . With the Fisher transformation shown in Fig. 4, the various classes from the measurement space can be arrayed again and separated in another space.

5.2 Choice of discriminant fact The above analysis indicates that it is not adequate

to classify column failure modes based only on one factor such as aspect ratio or shear strength. Therefore, the classification of column failure modes should be based on various factors. Because traditional methods cannot avoid human interference in selection of factors, they could lead to an imprecise result. To avoid the subjective error, longitudinal reinforcement ratio, transverse reinforcement ratio, hoop spacing to depth ratio, aspect ratio, shearing resistance demand to shear capacity ratio and axial load ratio are all selected for classifying the column failure modes. And the stepwise discriminant analysis is designed to reduce the variables of the discriminant function for the classification of column failure modes.

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Fig. 4 Classes separation based on Fisher transformation

6 Evaluation of Fisher discriminant analysis

6.1 Divided into three failure modes Three failure modes are considered (flexure failure,

flexure­shear failure and shear failure) here. Through stepwise statistics analysis, structure matrix is obtained in Table 2. From the coefficient of the discriminant function, transverse reinforcement ratio, axial load ratio and longitudinal reinforcement ratio cannot cause significant influence on the failure modes. And this variable will not be used in the analysis.

Table 2 Structure matrix Influencing factor Function 1 Function 2

Vp/Vn 0.805 * −0.221

Aspect ratio −0.611 * 0.476 Transverse

reinforcement ratio a −0.469 * −0.061

Axial load ratio a −0.390 * 0.185 Longitudinal

reinforcement ratio a 0.042 * 0.018

Hoop spacing to depth ratio 0.328 0.927 *

1) Pooled within­groups correlation is between discriminating variables and standardized canonical discriminant functions; 2) Variables ordered by absolute size of correlation within function. *: Largest absolute correlation between each variable and any discriminant function; a: This variable not used in analysis.

Therefore, the following three discriminant functions obtained through the discriminant analysis respectively were used to determine the failure mode. Substituting the observed value into each discriminant function, the failure mode is determined by the largest function value. Therefore, the following three discriminant functions can be drawn respectively through discriminant analysis. An observed value can be

classified into three types (flexure failure, flexure­shear failure and shear failure) as follows: substitute the value of Hoop spacing to depth ratio, aspect ratio and Vp/Vn into these three functions. Then the failure mode belongs to the very type that corresponds to the function which gets the largest results. Table 3 gives the coefficients and constants in three discriminant functions.

Table 3 Classification function coefficients Failure mode

Influencing factor Flexure Flexure­shear Shear

Hoop spacing to depth ratio −7.664 −0.168 −2.935

Aspect ratio 6.448 4.942 4.038

Vp/Vn 16.860 21.232 25.726

(Constant) −19.455 −20.815 −23.23 Fisher’s linear discriminant functions.

In Fisher discriminant analysis, the explained quality of discriminant function can be determined by the cumulative variance. In Table 4, the variance of Function 1 accounts for 93.9%, which indicates that Function 1 can explain the information of sample in 93.9% degree. Only by this function, most samples can be explained very well. In the discriminant Function 1, the coefficient of Vp/Vn is the largest, which indicates that the function is mostly sensitive to Vp/Vn. Vp/Vn can better reflect the failure modes of RC columns. With the increase of a/d, the columns are most likely to experience flexure failure, and with the increase of s/d, the columns are most likely

Table 4 Eigenvalues

Function Eigenvalue Variance Cumulative Canonical correlation

1 1.800 a 93.9% 93.9% 0.802

2 0.117 a 6.1% 100.0% 0.324 a: First 2 canonical discriminant functions are used in analysis.

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to experience shear failure. In Table 5, it can be concluded that the center value

of zone flexure failure is −1.093, the center value of zone flexure­shear failure is 1.187, and the center value of zone shear failure is 2.472. A new sample can be adopted into one of these types by comparing its distance with the center points of these three groups of value, thus, the classification of failure modes of RC columns can be realized. Figure 5 shows the canonical discriminant functions.

Table 5 Functions at group centroids Failure mode Function 1 Function 2

Flexure failure −1.093 −0.077

Flexure­shear failure 1.187 0.381

Shear failure 2.472 −0.798 Unstandardized canonical discriminant functions are evaluated at group means.

Fig. 5 Grouping using first and second discriminant functions

Table 6 shows that the accuracy of predicted result of the functions reaches 85.6%. As can be seen, by using the Fisher discriminant analysis method in statistics, the classification and failure problem of RC columns can be solved to a great extent. The result also shows that the accuracy is above 90% in flexure failure and shear failure

Table 6 Classification results Predicted group membership Failure mode

(Original) Flexure Flexure­shear Shear Total

Flexure 59 5 0 64

Flexure­shear 5 26 5 36

Shear 0 1 10 11

Flexure 92.2% 7.8% 0.0% 100.0%

Flexure­shear 13.9% 72.2% 13.9% 100.0%

Shear 0.0% 9.1% 90.9% 100.0% 85.6% of original grouped cases are correctly classified.

mode. Some discriminations of shear­flexure failure modes go wrong. One of the reasons is that there is no obvious boundary between them when the test results are recorded.

6.2 Divided into two failure modes If adopting only two failure modes to distinguish

the failures classifications of RC columns, i.e. merging the shear­flexure failure mode of RC columns into shear failure mode, and working out the performance indices of this mode according to shear failure mode, the values are more conservative and safer. The performance indices of classified columns in FEMA356 are worked out in this way. As given in Table 7, only one function can explain all the variables.

Table 7 Eigenvalues

Function Eigenvalue Variance Cumlative Canonical correlation

1 1.528 a 100.0% 100.0% 0.777

a: First canonical discriminant functions are used in analysis.

Table 8 gives the coefficients and constants in the first and second discriminant functions. The variance of Function 1 accounts for 100%, which indicates that Function 1 can explain the information of sample in 100% degree, and only by this function, the failure mode can be determined. Figure 6 shows the histograms from the discriminant functions. It can be concluded that the center value of zone flexure is −1.050, and the centroids value of zone shear is 1.43. A new sample can be adopted into this type by comparing its distance with the centroids points of these two groups of value, thus the classification of failure modes of RC columns can be realized.

Table 8 Classification function coefficientsFailure mode

Influencing factor Flexure failure Shear failure

Hoop spacing to depth ratio −7.998 −0.541

Aspect ratio 6.422 4.866

Vp/Vn 17.444 22.032

(Constant) −19.149 −20.805 Fisher’s linear discriminant functions.

Table 9 shows that the accuracy of predict result of the functions reaches 90.1%. As can be seen, by using the Fisher discriminant analysis method in statistics, the classification and failure problem of RC columns can be solved to a great extent. Some discriminations of shear­flexure failure modes go wrong.

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Fig. 6 Parameter histograms of zone flexure and zone shear

Table 9 Classification results Predicted group membership Failure mode

(Original) Flexure Flexure­shear Total

Flexure failure 59 5 64

Flexure­shear 6 41 47

Flexure 92.2% 7.8% 100.0%

Flexure­shear 12.8% 87.2% 100.0% 90.1% of original grouped cases are correctly classified.

7 Conclusions

1) The failure modes of RC columns can be identified by using the discriminant function. In the discriminant function, the coefficient of Vp/Vn is the largest, which indicates that the function is mostly sensitive to Vp/Vn. Vp/Vn can better reflect the failure modes of RC columns. In the condition that the minimum hooped reinforcement ratio is met, the failure mode of RC columns has little relationship with hooped reinforcement ratio, axial compression ratio, and axial reinforcement ratio.

2) In order to avoid shear failure mode, we can increase the value of Vp/Vn, aspect ratio (to avoid

short­column failure), or decrease the space of hooped reinforcement.

3) The structure system contains many other components such as shear wall and beam. In the future work, the FDA method will also be used in these components to propose a discriminant model.

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(Edited by HE Yun­bin)