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fib Symposium Keep Concrete Attractive, Budapest 2005

MODELLING CONFINEMENT IN CONCRETE COLUMNS AND BRIDGE PIERS THROUGH 3D NONLINEAR FINITE ELEMENT ANALYSIS Vassilis K. Papanikolaou and Andreas J. Kappos Laboratory of Reinforced Concrete and Masonry Structures, Civil Engineering Department, Aristotle University of Thessaloniki, P.O. Box 482, 54124 Thessaloniki, Greece SUMMARY The goal of this analytical study is to evaluate the effectiveness of different confinement reinforcement arrangements in reinforced concrete columns and bridge piers under axial monotonic loading, in terms of structural strength and ductility. While in most previous studies the effect of confinement is handled by phenomenological models using semi-empirical confinement effectiveness factors, it is attempted here to shift the investigation to full three-dimensional (3D) finite element analysis. Constitutive model calibration and finite element modelling is followed by selected analysis results, compared with experimental results for validation purposes. A parametric study (part of which is a novel post-processing utility) is finally performed in order to evaluate the effectiveness of different confinement reinforcement configurations in rectangular solid and hollow sections. 1. INTRODUCTION The issue of confinement in reinforced concrete (R/C) columns and bridge piers has been studied both experimentally and analytically in the past decades. Experimental studies on confined concrete columns with solid section (e.g. Sheikh and Uzumeri, 1980; Scott et al., 1982) showed that the favourable effect of passive confinement depends on various parameters, the key ones being transverse reinforcement strength, amount, spacing and configuration. Calibrated on the basis of experimental results, numerous phenomenological confinement models have been proposed in the literature (e.g. Sheikh and Uzumeri, 1982; Park et al., 1982; Kappos, 1991, amongst several others), suggesting semi-empirical confinement effectiveness factors for scaling the uniaxial monotonic stress-strain relationship of unconfined concrete, in order to account for the triaxial stress state induced by the restrained (due to the transverse reinforcement) lateral expansion of the concrete core. However, the extension of the above models to hollow sections (usually adopted in R/C bridge piers) is not straightforward due to limited experimental support, which is mainly focused on lateral cyclic (e.g. Mander et al., 1983; Pinto, 1996) rather than axial monotonic loading. In order to overcome these limitations, recent studies suggest the application of nonlinear finite element analysis, aiming to explicitly capture the triaxial nature of passive confinement, through three-dimensional modelling of confined concrete (e.g. Abdel-Halim & Abu-Lebdeh, 1989; Liu & Foster, 2000; Kwon & Spacone, 2002; Montoya et al. 2001, 2004). Nevertheless, this area of research is still restricted to columns of solid section and furthermore, there are large discrepancies between employed constitutive theories and modelling techniques. In this paper, an R/C-focused finite element program is used for modelling R/C columns and bridge piers of solid and hollow section. Calibration of constitutive models is followed by validation of analysis results by comparisons with experimental data. Finally, the effectiveness of various confinement reinforcement configurations in solid and hollow columns under monotonic axial loading is evaluated.

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2. ANALYSIS SOFTWARE AND MODEL CALIBRATION 2.1 General Following a review of available software for nonlinear finite element modelling of reinforced concrete structures and trial analyses with some of them, the ATENA finite element software (Cervenka Consulting, 2004), was selected as the basis of this study. It features a variety of constitutive models for concrete, reinforcement, metals, rock and soil, two and three dimensional finite elements, and robust nonlinear solvers. 2.2 Concrete and reinforcement constitutive models The selected three-dimensional constitutive model for concrete combines fracture in tension with plasticity in compression (Cervenka and Cervenka, 1999). Fracture is modelled by an orthotropic smeared crack formulation and crack band model based on the Rankine tensile criterion with exponential softening (Fig.1a). The plasticity hardening/softening model for concrete in compression is based on the Mentrey-Willam three-parameter failure surface (Mentrey & Willam, 1995) and a non-associated flow rule of Drucker-Prager type (Fig.1b). Strains are separated into plastic and fracturing components and a recursive iterative algorithm combines the two aforementioned models by preserving stress equivalence.

a. Rankine failure surface F(,,) = t2cos 3f 0+ = or ii - ft = 0 Fracture model parameters Youngs modulus (MPa) ft Uniaxial tensile strength (MPa) GF Fracture energy (MN/m) FIXED 0 for rotating crack model, 1 for fixed, intermediate values for mixed behaviour

Tensile softening and characteristic length

Plastic potential function

G = 1I3

+ 22J = +

on-associated flow rule

pij =

ij

ij

G( )

b. Mentrey-Willam failure surface

F(,,) = 2

c c c

1.5 m r(, e) c 0f 6f 3f

+ + =

m = 2 2c t

c t

f ( f ) e3f f e 1 +

r(,e) = ( )

( ) ( )2 2 2

1/ 22 2 2 2

4 1 e cos (2e 1)

2 1 e cos (2e 1) 4 1 e cos 5e 4e

+ + +

Plasticity model parameters c Youngs modulus (MPa) Poissons ratio fc Uniaxial compressive strength (MPa) fco Onset of plastic hardening / elastic limit (MPa) ft Uniaxial tensile strength (MPa) cp Plastic strain at peak compressive stress fc wd Ultimate displacement for softening in compression (m) Tensile strength multiplier to satisfy intersection between M-W (plastic) and Rankine (fracture) surface e Failure surface eccentricity Plastic dilation parameter

Hardening function

In MW equation : fc = fc(eqp) eqp = min(ijp) Softening function

In MW equation : 2p

c eq

c

f ( )c

f =

Fig.1 Fracture (top) and plasticity models, used for tension and compression, respectively

The calibration of material parameters for concrete was mainly based on design code equations (CEB, 1993) and various parametric analyses. For simulating concrete dilatation, which is deemed to play an important role in capturing passive confinement, the hypothesis of

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zero volumetric strain at maximum compressive stress under uniaxial compression was assumed (Grassl et al., 2002, amongst others). Table 1 lists the suggested concrete model parameters for the present study. Finally, the reinforcement constitutive model is based on a uniaxial multilinear law, enabling to trace all stages of steel behaviour from elastic to fracture.

Tab.1 Concrete constitutive model parameters as functions of fc Parameter (see Fig. 1) Suggested value Calibration method and/or reference

c (MPa) 0.85215001/ 3

cf10

Ec that leads to 85% of Eci suggested by MC90 (CEB, 1993) 0.2 Adopted in MC90 (CEB, 1993)

fco (MPa) 0.07fc1.56 From parametric analysis based on MC90 equations (CEB, 1993)

ft (MPa) 1.42 / 3

cf 810 Adopted in MC90 (CEB, 1993)

cp = c - cel 0.31c c

c

0.7 f f1000 E Variable value suggested by Thorenfeldt et al. (1987)

wd (mm) -0.5 Suggested by Van Mier (1984) for normal concrete

2.0 Default value leading to intersection of plastic and fracture surfaces during hardening and softening (Cervenka Consulting, 2004) e 0.52 Leading to biaxial strength fbc = 1.14fc (Kupfer et al., 1969)

0.72 + 0.9 = c pc c

f (1 2)

From parametric analysis based on hypothesis of zero volumetric strain at maximum compressive stress under uniaxial compression

GF (MN/m)

0.7c

F0

f10G1000

Adopted in MC90 (CEB, 1993) GF0 is a function of the maximum aggregate size dmax For usual value of dmax = 16 mm, GF0 = 0.030 Nmm/mm2

FIXED 0 Rotating crack model adopted 2.3 Finite element modelling and analysis procedure Concrete was modelled using 8-node isoparametric solid elements and reinforcement by 2-node truss elements, embedded in concrete elements. Due to symmetry, one quarter of concrete sections was modelled, by applying appropriate boundary conditions, allowing horizontal expansion at top and bottom. Prescribed displacement was axially imposed in small steps, resulting in concentric compression. The Modified Newton-Raphson iterative scheme was applied with appropriate convergence criteria and maximum number of iterations. 2.4 Additional post-processing utilities Two additional software utilities were developed by the authors, in order to extend the capabilities of the existing software and considerably reduce the post-processing cost. The first one features a viewer of model data and analysis statistics, a user monitor data extractor, and a node data extractor from bulk binary results. The second one introduces a novel method namely optical integration, which performs stress averaging by optical recognition of coloured stress iso-areas, produced by the graphical postprocessor. With this method it is possible to extract the average stress value over a rectangular or hollow section cut and easily distinguish between confined and unconfined regions. It was verified that this method produced equivalent results to standard numerical integration (node stress times node tributary area) at a fraction of processing time.

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3. ANALYSIS OF RECTANGULAR COLUMNS WITH SOLID SECTION Four different rectangular columns in terms of material characteristics and confinement reinforcement arrangements were modelled, following the experimental setups of Sheikh and Uzumeri (1980) and the analytical results of Montoya et al. (2001), for comparison purposes. Based on the above models, a variety of different hoop configurations and bare concrete models were also analyzed, for evaluating confinement effectiveness. Table 2 lists all model properties in detail. Concrete constitutive model parameters were calculated according to Table 1. The model height was set to 6 times the hoop spacing and strains were measured at a test region of 102 mm at mid-height, as per the original experimental set-up.

Tab.2 Rectangular column model characteristics

Dimensions Conc. Longitudinal reinforcement (trilinear) Transverse reinf. (bilinear) Model b/2 = h/2

(m) Height

(m) fc

(MPa) Es

(MPa) fy

(MPa) sh Esh

(MPa) n d (mm)

Esw (MPa)

fyw (MPa)

dw (mm)

s (mm)

C1-x 0.1525 0.3426 31.88 200000 367 0.0077 9220 8 15.96 200000 540 4.76 57.1 C2-x 0.1525 0.6096 28.39 196400 392 0.0078 6200 12 19.01 199500 480 7.94 101.6 C3-x 0.1525 0.6096 27.97 196400 407 0.0091 8960 16 12.82 200000 480 7.94 101.6 C4-x 0.1525 0.2286 30.52 196400 392 0.0078 6200 12 19.01 199500 480 6.35 38.1

Model Config. x = 1 Config. x = 2 Config. x = 3 Config. x = 4 Config. x = 5

C1-x

2A1-1

C2-x

4B3-19

C3-x

2C5-17

C4-x

4D6-24

Original models tested by Sheikh and Uzumeri (1980) and analyzed by Montoya et al. (2001) Comparative plots of reaction forces versus strains between the current analysis, experimental results and Modified Compression Field Theory (MCFT) analysis (unsoftened) (Montoya et al., 2001) are presented in Fig. 2. Reasonable agreement between analysis and experimental results is observed and there is a general trend for analysis to slightly overestimate the experimentally measured axial capacity. These discrepancies may be attributed to the models inability to account for spalling of concrete cover, buckling of reinforcement bars, and reinforcement bond slip (Attarnejad and Amirebrahim, 2002). However, it should be noted that the different in principle MCFT approach, which was already calibrated on the basis of the selected experimental results, yields about the same amount of error.

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0

1000

2000

3000

4000

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Analysis

MCFT Analysis

Experimental

P (KN)

Specimen 21-1

0

1000

2000

3000

4000

5000

0 0.002 0.004 0.006 0.008 0.01 0.012

Analysis

MCFT Analysis

Experimental

P (KN)

Specimen 43-19

0

1000

2000

3000

4000

5000

0 0.005 0.01 0.015 0.02 0.025

Analysis

MCFT Analysis

Experimental

P (KN)

Specimen 2C5-17

0

1000

2000

3000

4000

5000

6000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Analysis

MCFT Analysis

Experimental

P (KN)

Specimen 4D6-24

Fig.2 Comparative plots for column models C1-4, C2-4, C3-5 and C4-4

To evaluate the effectiveness of different confinement patterns, a parametric study based on the four columns presented above was performed, by reducing the hoop configurations down to bare concrete models in a stepwise fashion (Tab. 2). Comparative plots of the reaction-strain response for the analyzed column models are presented in Figure 3.

0

1000

2000

3000

4000

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

P (KN)

C1-1

C1-4

C1-3C1-2Theoretical concrete

strength

0

1000

2000

3000

4000

5000

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

P (KN)

C2-1

C2-4

C2-3

C2-2Theoretical concrete strength

0

1000

2000

3000

4000

5000

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

P (KN)

C3-1

C3-4

C3-3

C3-2

C3-5

Theoretical concrete strength

0

1000

2000

3000

4000

5000

6000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

P (KN)

C4-1

C4-4

C4-3

C4-2

Theoretical concrete strength

Fig.3 Comparative plots for column model variations (see nomenclature in Tab. 2)

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It is clear that the confinement effect was captured by the finite element analysis, resulting into increased strength and ductility, compared to unconfined models. In order to quantify the strength enhancement factor K = fcc/fc (where fcc is the peak strength of confined concrete), the aforementioned optical integration utility was applied at the mid-height hoop level section of confined models, at the load step of maximum reaction force (shown in Fig. 3). The confined concrete strength fcc was calculated as the average axial stress value over the confined region. Table 3 lists the derived strength enhancement K for all confined models, together with suggestions from two phenomenological models, for comparison purposes. A reasonable agreement is generally observed, with analysis results being closer to the phenomenological models for better-confined columns.

Tab.3 Concrete strength enhancement of confined columns

Analysis Model 1 Model 2 Model w % K = fcc/fc K b K

C1-3 0.47 1.04 1.08 0.55 0.75 1.08 C1-4 0.80 1.08 1.14 1.0 1.0 1.14 C2-3 0.73 1.06 1.12 0.55 0.75 1.11 C2-4 1.70 1.29 1.29 1.25 1.0 1.36 C3-3 0.73 1.06 1.13 0.55 0.75 1.12 C3-4 1.83 1.28 1.31 1.25 1.0 1.39 C3-5 2.34 1.43 1.40 1.25 1.0 1.50 C4-3 1.25 1.12 1.20 0.55 0.75 1.16 C4-4 2.25 1.44 1.35 1.0 1.0 1.35

Axial stress iso-areas* of model C1-4 at the load step of maximum total reaction force. Confined and unconfined regions recognized by the optical integration utility.

Park et al. (1982) : K = 1+wyw

c

ff

Penelis and Kappos (1997) : K = 1+b

yww

c

f

f

* Full section by double mirroring is shown

4. ANALYSIS OF BRIDGE PIERS WITH HOLLOW SECTION For the analysis of bridge piers with hollow section, the specimen tested by Mander et al. (1983) was selected and several variations were modelled in order to evaluate the confinement effectiveness of different hoop arrangements. Table 4 lists all model properties. Model HP6 was identical to HP5, but with half hoop spacing (s = 60 mm). A comparative plot of the reaction-strain response for analyzed bridge pier models is shown in Figure 4.

Tab.4 Bridge pier model characteristics

Dimensions Conc. Longitudinal reinforcement (trilinear) Transverse reinforcement (trilinear) b/2 = h/2

(m) Height

(m) fc

(MPa) Es

(MPa) fy

(MPa) sh Esh

(MPa) n d (mm)

Esw (MPa)

fyw (MPa) shw

Eshw (MPa)

dw (mm)

s* (mm)

0.375 0.48 30.0 208000 335 0.03 3500 60 10 200000 320 0.015 2600 6 120 HP1 HP2 HP3 HP4 HP5 / HP6

Original model tested by Mander et al. (1983) Added diagonal links used by Pinto et al. (1996) * 60 mm in HP6

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0

2000

4000

6000

8000

10000

12000

14000

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005

P (KN)

HP1

HP4HP3HP2

Theoretical concrete strength

HP5

HP6

Fig.4 Comparative plots for bridge pier models The confinement effect was successfully captured again, with increased strength and ductility for more complex and more closely spaced hoop arrangements. The confinement factors, derived by the same method described in the previous section, are listed in Table 5. The Park et al. (1982) model was not specifically developed for hollow sections, but is used here for comparison purposes. Its predictions overestimated confinement factors produced by FE analysis, which implies that confinement effectiveness in hollow sections is inferior to that in their solid counterparts, for the same transverse reinforcement ratio w.

Tab.5 Concrete strength enhancement of confined hollow piers

Analysis Park et al. model Model w % K = fcc/fc K

HP3 0.74 1.05 1.08

HP4 1.16 1.09 1.12

HP5 1.51 1.11 1.16

HP6 3.02 1.26 1.32

Axial stress iso-areas of model HP6 at the load step of maximum axial force. Confined and unconfined regions recognized by the optical integration utility.

5. CONCLUSIONS In this paper, modelling of confinement in columns and bridge piers of solid and hollow section was performed through 3D nonlinear finite element analysis, using a properly calibrated commercial software. Analysis results showed not only a reasonable match with experimental evidence, but also an explicit capturing of the confinement effect, without any empirical modifications to concrete constitutive laws, suggested by the widely accepted phenomenological models. Consequently, the employed approach is deemed to be a promising alternative to previous procedures and a venturing point for further research. This includes, among others, the case of bridge piers with hollow section, where simplified confined concrete constitutive laws may cease to apply and large scale experimental testing is hindered by its onerous cost.

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6. REFERENCES Abdel-Halim, M.A.H. and Abu-Lebeh, T.M. (1989) Analytical Study for Concrete Confinement

in Tied Columns, Journal of Structural Engineering, ASCE, Vol. 115, No. 11, pp. 2810-2827.

Attarnejad, R. and Amirebrahim, A.M. (2002) Load-Displacement Curves of Square Reinforced Concrete Columns Based on Fracture Mechanics, 15th ASCE Engineering Mechanics Conference, Columbia University, New York.

CEB (1993) CEB/FIP Model Code 1990, Bulletin d Information CEB, 213/214, Lausanne. Cervenka, J. and Cervenka, V. (1999) Three Dimensional Combined Fracture-Plastic Material

Model for Concrete, Proceedings of the 5th US National Congress on Computational Mechanics, Boulder, CO.

Cervenka Consulting (2004) ATENA Program Documentation, Prague, Czech Republic. Grassl, P., Lundgren, K. and Gylltoft, K. (2002) Concrete in Compression : A Plasticity Theory

with a Novel Hardening Law, International Journal of Solids and Structures, Vol. 39, pp. 5205-5223.

Kappos, A.J. (1991) Analytical Prediction of the Collapse Earthquake for R/C Buildings : Suggested Methodology, Earthquake Engineering and Structural Dynamics, Vol. 20, No. 2, pp. 167-176.

Kupfer, H., Hilsdorf, H. and Rusch H. (1969) Behavior of Concrete Under Biaxial Loading, ACI Journal, Vol. 66, No. 8, pp. 656-666.

Kwon, M. and Spacone, E. (2002) Three-Dimensional Finite Element Analyses of Reinforced Concrete Columns, Computers and Structures, Vol. 80, pp. 199-212.

Liu, J. and Foster, S.J. (2000) A Three-Dimensional Finite Element Model for Confined Concrete Structures, Computers and Structures, Vol. 77, pp. 441-451.

Mander, J.B., Priestley, M.J.N. and Park, R. (1983) Behavior of Hollow Reinforced Concrete Columns, Bulletin of the New Zealand National Society for Earthquake Engineering, Vol. 16, No. 4, pp. 273-290.

Mentrey, P. and Willam K.J. (1995) Triaxial Failure Criterion for Concrete and its Generalization, ACI Structural Journal, Vol. 92, No. 3, pp. 311-318.

Montoya, E., Vecchio, F.J. and Sheikh S.A. (2001) Compression Field Modeling of Confined Concrete, Structural Engineering and Mechanics, Vol. 12, No.3, pp. 231-248.

Montoya, E., Vecchio, F.J. and Sheikh S.A. (2004) Numerical Evaluation of the Behavior of Steel and FRP Confined Concrete Columns Using Compression Field Modeling, Engineering Structures, Vol. 26, No.11, pp. 1535-1546.

Park, R., Priestley, M.J.N. and Gill, W.D. (1982) Ductility of Square Confined Concrete Columns, Journal of the Structural Division, ASCE, Vol. 108, No. 4, pp. 929-950.

Pinto A.V. (Ed) (1996) Pseudo-Dynamic and Shaking Table Tests on R/C Bridges, ECOEST/PREC8 Report, No. 5.

Sheikh, S.A. and Uzumeri, S.M. (1980) Strength and Ductility of Tied Concrete Columns, Journal of the Structural Division, ASCE, Vol. 106, No. ST5, pp. 1079-1102.

Sheikh, S.A. and Uzumeri, S.M. (1982) Analytical Model for Concrete Confinement in Tied Columns, Journal of the Structural Division, ASCE, Vol. 108, No. ST12, pp. 2703-2722.

Scott, B.D., Park, R. and Priestley, M.J.N. (1982) Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates, ACI Journal, Vol. 79, No. 1, pp. 13-27.

Thorenfeldt, E., Tomaszewicz, A. and Jensen, J.J. (1987) Mechanical Properties of High Strength Concrete and Application in Design, Proceedings of the Symposium Utilisation of High Strength Concrete, Tapir, Trondheim, pp. 149-159.

Van Mier, J.G.M. (1984) Strain-Softening of Concrete Under Multiaxial Loading Conditions, Ph.D. Thesis, Eindhoven University, Eindhoven.

Summary1. Introduction2. Analysis software and model calibration2.1 General2.2 Concrete and reinforcement constitutive models2.3 Finite element modeling and analysis procedure2.4 Additional postprocessing utilities

3. Analysis of rectangular columns with solid section4. Analysis of bridge piers with hollow section5. Conclusions6. References