Synopses of Master Papers

Bulletin of IISEE, 47, 55-60, 2013

NUMERICAL MODELING FOR BACKBONE CURVEAND ULTIMATE POINT FOR FLEXURE-SHEAR
DOMINANT REINFORCED CONCRETE COLUMNS

Ahmed Abdullah NOOR∗
MEE11610

Supervisor:

Susumu KONO ∗∗ABSTRACTA displacement based evaluation approach, namely Axial Shear Flexure Interaction (ASFI), is used indetermining backbone curve and ultimate point for flexure-shear dominant reinforced concrete
columns, considering interaction among axial, shear and flexure mechanisms. Conventional section
analysis technique is employed to predict flexure behaviour and Modified Compression Field Theoryis adopted to determine the shear behaviour. Ultimate point and backbone curves were estimated using
this approach for four reinforced concrete columns which were critical in shear and flexure-shear, andthen compared with experiment results. Prediction of ultimate point and backbone curve are notsatisfactory using ASFI for the columns. Considering many aspects of material constitutive laws, theirdefinitions in the analytical model and different mechanisms acting inside the model, concrete
compression softening is thought to be a very influencing variable to predict the ultimate point andbackbone curve for shear and flexure-shear dominant reinforced concrete columns. A parametric study
is conducted considering different degrees of concrete compression softening with increasing principal
tensile strain to find out the sensitivity of strength reduction in cracked concrete for predicting thesestructural parameters. From this study, one of the options for simulating the continuous weakening
behaviour of cracked concrete found to be effective in determining ultimate point and backbone curvefor shear and flexure-shear dominant reinforced concrete columns.
Keywords: Concrete Compression Softening, Axial Strain at Inflection Section.

1. INTRODUCTION
On introduction of performance-based design concept, response estimation of structures has become
one of the main performance criteria in the design process considering ductility and deformability ofthe structure. So from the view point of performance and also economy, design of reinforced concrete
structures is dependent on the ductility of the structure. Therefore, developing suitable analytical tool
to estimate the ultimate deformation or ductility of reinforced concrete column with certain level ofprecision is a growing demand. To fix the level of required retrofit or assessment of the vulnerabilityof a structure for earthquake damage needs evaluation of strength and deformation capacity of thestructure. Recently a displacement-based analytical approach is developed by Mostafaei andKabeyasawa (2007) to estimate response of reinforced concrete elements considering interaction
among axial, shear and flexural mechanisms. The objective of this study is to predict the ultimate point
and backbone curve modifying the ASFI approach for reinforced concrete columns which are criticalin shear or shear after flexural yielding.
∗Public Works Department, Bangladesh.
Professor, Structural Engineering Research Center, Tokyo Institute of Technology, Japan.

∗∗

2. BASIC CONCEPT OF ASFI AND PREDICTION USING THIS APPROACH
The basic concept and methodology of the axial-shear-flexure interaction (ASFI) method is based on
the axial deformation and concrete compression softening interaction between the two models: aflexure model based on traditional uniaxial section analysis principles, and a shear model based on abiaxial shear element approach.
ASFI is applied to the four reinforced concrete column specimens, which were tested byKono et al. (2011) in the Kyoto University. The column specimens were critical in shear andflexure-shear. After application of ASFI in the specimens’ ultimate point and lateral load bearing
capacity were found too high than the experiment results shown in Figure 1. In the figure ASFI-Series
Calculation-Flex refers to backbone curve for flexural behaviour including shear effect in it.
ASFI-Series Calculation refers to the combination of shear and flexural behaviour. A large yieldplateau is obtained in all the specimens. No degradation of strength for flexure shear type specimen isachieved in backbone curve determined by ASFI method.
1500

1000
800

ASFI-Series Calculation

200

0-0.01 -0.005-200 0

0.005

0.01

0.015

ASFI-Series Calculation-Flex
ASFI-Ultimate Point-400

Lateral Load, KN

Lateral Load, KN

400

0

0.02

0.04

-500
-1000

ASFI-Ultimate Point

Drift Ratio

Column Kono et al. 2011, C03

Column Kono et al. 2011, C02
2000
Kono et al. 2011, C04

1000
500

ASFI-Series Calculation

00.01 0.02 0.03 0.04 0.05 0.06

ASFI-Series Calculation-Flex

Lateral Load, KN

1500

Lateral Load, KN

ASFI-Series Calculation-Flex

0
-0.02

Drift Ratio

-0.02

-1000
-1500

ASFI-Series Calculation

500

-600

-0.01 0-500

Kono et al. 2011, C03

1000

Kono et al. 2011, C02

600

ASFI-Ultimate Point

1500
1000

Column Kono et al. 2011, C04

ASFI-Series Calculation

500
0
-0.01
0
-500
-1000

Drift Ratio

Kono et al. 2011, C05

ASFI-Series Calculation-Flex
0.01

0.02

0.03

0.04

ASFI-Ultimate Point

Drift Ratio

Column Kono et al. 2011, C05

Figure 1. Backbone curve and ultimate point for different columns using ASFI method

3. PARAMETRIC STUDY WITH THE ASFI APPROACH
3.1. Probable options for parametric study
With a view to modify the ASFI approach developed by Mostafaei and Kabeyasawa (2007), threedifferent options for modifying concrete compression softening model are used for this parametricstudy, namely Opt-1, Opt-2 and Opt-3. All of these three options are thought considering different
level of concrete compressive strain. Shear reinforcement above 0.5% is assumed as high shear
reinforcement ratio. Basic idea of choosing these options is to take into account the effect of shear
cracks due to principal tensile strain in concrete starting from a very small range that is immediately
after cracking of concrete to ultimate state for a column specimen. It is assumed that the reduction ofcompressive strength in cracked concrete due to compression softening starts from this stage.
1.

Determine εci as shown in Figure 2.

2.

Using εci determine axial strain εxam at the inflection section of the column using
the following equations:
Opt-1
For Low Shear Reinforcement Ratio
εxam =

0.33fc′f′
�2 ε c + Es ρsx �

(1)

ci

For High Shear Reinforcement Ratio
εxam =

Figure 2. Determination of strain level fordeveloping different options for parametricstudy in the stress strain curve of concrete

0.33fc′f′
�1.5 c + Es ρsx �
εci

(2)

Opt-2
For Low Shear Reinforcement Ratio
εxam =

2X0.33fc′
f′
�2 c + Es ρsx �
εci

(3)

For High Shear Reinforcement Ratio
εxam =

3.
4.

2X0.33fc′
f′�1.5 ε c + Es ρsx �
ci(4)

Determine εpim by multiplying confinement factor K, εpim = Kεxam
The value εci obtained in step 1 is used in Opt-3 and values of εpim obtained instep 2 are used in Opt-1 and Opt-2 respectively, instead of peak strain ε′c in themodel developed by Vecchio and Collins (1986). Hognestad Parabola is used as abase curve and compressive strength reduction factor is determined modifying Eq.
(5) in the following form in Eq. (6) and Eq.(7):
Original Model developed by
Vecchio and Collins (1986)
For Opt-1 and Opt-2
For Opt-3

β=1

ε ≤ 1.00.80 − 0.34 c1ε′c1β=ε ≤ 1.00.80 − 0.34 c1
εpim
1
β=
ε ≤ 1.00.80 − 0.34 εc1
ci

(5)
(6)
(7)

3.2. Implementation of Parametric Options and Comparison of Results

1000

Kono et al. 2011, C02

800

ASFI-Series Calculation

600

ASFI-Series Calculation-Flex

400

PS-ASFI-Opt-1-Series Calculation200

PS-ASFI-Opt-1-Series Calculation-Flex

0

-0.01

-0.005
0-200

ASFI-Series Calculation

PS-ASFI-Opt-2-Series Calculation

0.005

0.01

0.015

PS-ASFI-Opt-2-Series Calculation-Flex

1000

ASFI-Series Calculation-Flex
PS-ASFI-Opt-1-Series Calculation

500

PS-ASFI-Opt-1-Series Calculation-Flex

0

-0.01

0

PS-ASFI-Opt-3- Series Calculation-600

PS-ASFI-Opt-3- Series Calculation-Flex

Lateral Load, KN

Lateral Load, KN

0.04

ASFI-Series Calculation-Flex
PS-ASFI-Opt-1-Series Calculation

0
0

0.01

0.02

0.03

0.04

0.05

PS-ASFI-Opt-1-Series Calculation-Flex
PS-ASFI-Opt-2-Series Calculation

-500

-0.02

PS-ASFI-Opt-2-Series Calculation-Flex-1000

PS-ASFI-Opt-2-Series Calculation-Flex

PS-ASFI-Opt-3- Series Calculation-Flex

Drift Ratio

2000

Kono et al. 2011, C05

1500

ASFI-Series Calculation
ASFI-Series Calculation-Flex

1000

PS-ASFI-Opt-1-Series Calculation

500

PS-ASFI-Opt-1-Series Calculation-Flex
PS-ASFI-Opt-2-Series Calculation

0
-0.01

0

0.01

0.02

0.03

-500

PS-ASFI-Opt-3- Series Calculation

Drift Ratio

PS-ASFI-Opt-2-Series Calculation

Column Kono et al. 2011, C03

ASFI-Series Calculation

500

-1500

0.03

Kono et al. 2011, C04

1000

-0.01

0.02

PS-ASFI-Opt-3- Series Calculation-1000

Column Kono et al. 2011, C02
1500

0.01

-500

-400

Drift Ratio

Kono et al. 2011, C03

1500

Lateral Load, KN

Lateral Load, KN

Using the parametric options described in section 3.1 concrete compression softening factor is
calculated for the column specimens selected for this study. After obtaining compressive strength
reduction factors using three options for each of the selected columns, these are applied to therespective specimens. Results are shown in Figure 3. Modifications are made to both the peak stressand strain at the peak stress by multiplying each of them with the reduction factor β. It is observed that
Opt-1 is giving better fit of the experimental load deformation data for all the column specimens.
Hence, this option is chosen for further investigation and analyses for all the column specimens.

PS-ASFI-Opt-2-Series Calculation-Flex
PS-ASFI-Opt-3- Series Calculation-1000

PS-ASFI-Opt-3- Series Calculation-Flex

Column Kono et al. 2011, C04

Drift Ratio

PS-ASFI-Opt-3- Series Calculation-Flex

Column Kono et al. 2011, C05

Figure 3. Backbone curve and ultimate point for selected columns using
ASFI and three different options of concrete compression softening.

Stress-strain relationship for cracked concrete
is derived by reducing both peak stress andpeak stress combined with strain at peakstress as shown in Figure 4 for column C02
as a representation of the procedure, in orderthat the ASFI method and Opt-1 can be used
to detect the sensitivity of reduction of strain
at peak stress. In Opt-1 a significant drop in
maximum stress and strain are noticed for all
the column specimens after occurring a large
tensile strain in the cracked concrete. This isbecause in the case of Opt-1, the degree ofsoftening is much higher than the softening

Principal Compressive Stress, MPa

5. RESULTS AND DISCUSSION

50
45
40
35
30
25
20
15
10
5
0
0.000

Deterioration of Compression Resistance in Compression Strut for
Cracked Concrete at principal tensile strain 0.003 for column C02
Without Considering
Comp. Softening
ASFI-Reduction of PeakStress Only
ASFI-Reduction of PeakStress-Strain
PS-ASFI-Opt-1--Reductionof Peak Stress Only
PS-ASFI-Opt-1-Reductionof Peak Stress-Strain

0.002
0.004
Principal Compressive Strain

0.006

Figure 4. Deterioration of compression resistance incompression strut for cracked concrete at an arbitrary
point of large principal tensile strain for column C02

mechanism adopted in ASFI. Figure 5 shows the final results obtained from ASFI and Opt-1 for thecolumns chosen for this parametric study. Ultimate point and backbone curve are shown for both theabove mentioned cases. From the result we can see that only in case of column C02 and C05,
prediction of ultimate points are almost same in terms of drift ratio both for ASFI and Opt-1, but in
estimating capacity, value obtained from ASFI is very high than experiment result and for Opt-1 thisestimation is close to experiment result. In case of column C02, C03 and column C04 ASFI could notpredict either capacity or the ultimate point at all. Whereas prediction of ultimate point and backbone
curve using Opt-1 is consistent with the experiment result in terms of capacity, but overestimation interms of drift ratio. In estimating ultimate point and backbone curve using both the ASFI method andOpt-1, there is an overestimation in terms of drift ratio. However this over estimation is sometime
very large in case of ASFI. Reduction of stress only (SO) and reduction of stress and strain at peakstress (SS) shown in Figure 4, that is softening of strength only and softening of both strength andstrain, were applied for ASFI and Opt-1. Reduction of peak stress and strain at peak stress could
predict a better ultimate point and backbone curve in terms of capacity among these four different
predictions for all the specimens. Therefore result obtained from softening of both the strength andstrain is considered as the final output from this study and these are shown in Figure 5.
ASFI & PS-ASFI-Reduction of Peak Stress-Strain(SS)
Kono et al. 2011, C02
1000

Lateral Load, KN

ASFI-Series Calculation

600

ASFI-Series Calculation-Flex

400

0.9 Qmax

200
0-0.01 -0.005-200 0

0.005

0.01

0.015

PS-ASFI-Series Calculation
PS-ASFI-Series Calculation-Flex

Lateral Load, KN

Qmax=559 KN

800

Drift Ratio-500

0.97 Qmax

0-0.01 0-500

0.01 0.02 0.03 0.04 0.05

-1500

ASFI-Series Calculation-Flex
PS-ASFI-Series Calculation

ASFI-Ultimate Point

Drift Ratio

PS-ASFI-Series Calculation Flex
ASFI-Ultimate Point
PS-ASFI-Ultimate Point

ASFI & PS-ASFI-Reduction of Peak Stress-Strain (SS)

Kono et al. 2011, C04

PS-ASFI-Series Calculation-Flex-1000

PS-ASFI-Series Calculation

0.04

Column Kono et al. 2011, C03

ASFI-Series Calculation

500

0.02

Drift Ratio

2000

Lateral Load, KN

Lateral Load, KN

1000

0

-1000

PS-ASFI-Ultimate Point

ASFI & PS-ASFI-Reduction of Peak Stress-Strain (SS)
Qmax=939 KN

ASFI-Series Calculation-Flex
1.12Qmax

0-0.02

Kono et al. 2011, C03
ASFI-Series Calculation

500

Column Kono et al. 2011, C02
1500

Qmax=718 KN

1000

ASFI-Ultimate Point

-400
-600

ASFI & PS-ASFI-Reduction of Peak Stress-Strain (SS)1500

Qmax=837 KN

1500

ASFI-Series Calculation

1000

ASFI-Series Calculation-Flex

500

0.89 Qmax

0
-0.02

-0.01
0
-500

PS-ASFI-Ultimate Point

Column Kono et al. 2011, C04-1000

Kono et al. 2011, C05

0.01

Drift Ratio

0.02

0.03

PS-ASFI-Series Calculation
PS-ASFI-Series Calculation-Flex
ASFI-Ultimate Point
PS-ASFI-Ultimate Point

Column Kono et al. 2011, C05

Figure 5. Backbone curve and ultimate point for different columns: ASFI and Opt-1 forcompression softening; Softening of both the strength and strain
5. CONCLUSIONS
Ultimate point and backbone curve are determined using the ASFI approach for four shear andflexure-shear dominant reinforced concrete columns, which were tested by Kono et al. (2011) in the
Kyoto University. Predictions of these parameters applying ASFI are found too high comparing withthe test result in terms of capacity and drift ratio. Therefore three different options have been examinedconsidering different degrees of concrete compression softening and Opt-1 is found to be the mostefficient among these options and ASFI approach. Using this option, prediction of backbone curve andultimate point in terms of capacity is satisfactory, but still overestimation in terms of drift ratio. At thesame time, only softening of strength for cracked concrete was found to be insufficient for predicting
the backbone curve properly, rather softening of both the strength and strain can predict better
backbone curve using Opt-1. On the other hand, prediction of ultimate point using Opt-1 is near
identical with each other both for softening of strength only and softening of both the strength andstrain for cracked concrete. After analyzing the calculated result, concrete compression softening isfound a key parameter to predict the ultimate point and the backbone curve for these types of columns.
Several parameters like concrete principal tensile strain, principal tensile stress, and ratio of principal
tensile to compressive strain are usually considered as the prime variables influencing the softening
effect. In addition to these, axial strain at inflection section is found to have significant influence indetermining the degree of softening for cracked concrete acting together with reinforcement. This inturn effects the prediction of ultimate point and backbone curve.
6. RECOMMENDATION
The degree of concrete compression softening with increasing principal tensile strain is very sensitivefor determining ultimate point and backbone curve for shear and flexure-shear dominant reinforced
concrete columns, hence further experimental studies are recommended in this regard. As the
behaviour of bond slip is not considered in the ASFI approach, it is also not considered in thisparametric study. So future attempt can be made to incorporate it in the ASFI approach.
ACKNOWLEDGEMENT
I am expressing heartfelt gratitude to my advisor Mr. Inukai for rendering continuous support andmany advices. I am very thankful to Mr. H Mostafaei for giving me some important suggestions anddocuments, which were very useful to conduct this study. Dr T. Saito has given me some valuableadvices and support throughout this individual study; I also like to acknowledge this with gratitude.
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