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Procedía Engineering 210 (2017) 211-219

Procedia

Engineering

www.elsevier.com/loeate/procedia

6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading, PROTECT2017, 11-12 December 2017, Guangzhou (Canton), China

Modeling the response of ultra high performance fiber reinforced

concrete beams

Solhmirzaei R.a, Kodur V.K.R.b^

aPhD student, Civil and Environmental Engineering, Michigan State University, East Lansing, MI, USA. bUniversity Distinguished Professor and Chairperson, Department of Civil and Environmental Engineering, Michigan State University, East

Lansing, MI, USA.

Abstract

A finite element based numerical model is applied for tracing the response of Ultra High Performance Fiber Reinforced Concrete (UHPFRC) beams under the effects of flexural and shear dominant loading. The numerical model, developed in ABAQUS, accounts for superior strength properties of UHPFRC, including high compressive and tensile strength, and stain hardening effect in tension. The developed model can generate various response parameters including flexural and shear capacity, as well as load deflection response and propagation of cracks. Predictions from the model are compared with measured test data on UHPFRC beams, tested under dominant shear and flexure loading. The comparisons indicate that the model is capable of capturing the response of UHPFRC beams in the entire range of loading from preloading stage to failure through crushing of concrete or rupture of rebars.

© 2017 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading

Keywords: Ultra High Performance Concrete; Finite Element Analysis; Concrete Damage Plasticity Model; Flexure, Shear.

1. Introduction

Extensive research and development efforts, over the past three decades, to improve properties of concrete have led to the emergence of ultra high performance concrete (UHPC). UHPC possesses very high compressive strength, good tensile strength, enhanced toughness, and durability properties [1]. However one of the main drawbacks of

Corresponding author. Tel.: +1-517-353-9813. E-mail address: kodur@egr.msu.edu

1877-7058 © 2017 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection &

Strengthening of Structures under Extreme Loading.

10.1016/j.proeng.2017.11.068

UHPC is its brittleness property. To overcome brittleness of UHPC, fibers are often added to UHPC and this type of concrete is referred to as ultrahigh performance fiber reinforced concrete (UHPFRC). Addition of fibers to UHPC can significantly improve its ductility, fracture toughness, and energy absorption capacity [1-3].

A number of experimental studies has been reported in literature on the response of UHPFRC beams [4-10]. Results from these studies indicate that increasing steel fiber content in UHPFRC enhance post cracking stiffness and thus improve load flexural capacity of UHPFRC beams. The reported test results also indicate that a higher shear capacity can be obtained by using a higher fiber volume content in UHPFRC and a lower shear span to depth ratio.

However only limited numerical studies are reported on the structural behavior of UHPFRC beams. Much of the reported numerical studies focused on applying sectional analysis approach to trace moment curvature response of UHPFRC beams under flexural loading [4,5]. There are limited finite element based numerical studies that simulated UHPFRC members. Mahmud et al [11] conducted two dimensional plane stress finite element analysis of unreinforced notched UHPFRC beams to study size effects on flexural capacity. Tysmans et al. [12] simulated the behavior of high performance fiber reinforced concrete under biaxial tension. Chen et al [13] focused on predicting load deflection (strain) response of UHPFRC girders subjected to shear and flexure. The authors showed that finite element model adopting concrete damage plasticity can accurately predict the load carrying capacity of the UHPFRC members. However, majority of these studies relied upon small-scale experiments for validation and focused on global response of UHPFRC structural members with no attention to local response (crack propagation).

To address lack of numerical studies on UHPFRC at member level, a finite element based numerical model is developed. This paper presents the details of the numerical model to trace the structural response of UHPFRC beams. The model is validated against measured response parameters from full scale tests on UHPFRC beams under flexural and shear loading.

2. Numerical model

A numerical model for tracing structural behavior of UHPFRC beams under shear and flexural loading is developed in ABAQUS. The analysis carried out through load control technique by incrementing load on the beam in steps till failure occurs. Details of the numerical model including discretization details and material models are presented below.

2.1. Discretization of the beams

UHPFRC beams are discretized using eight-noded reduced integration brick elements (C3D8R) and two-noded link elements (T3D2) for concrete and reinforcing steel, respectively. C3D8 element has eight nodes with three degrees of freedom. This element can be used for 3D modeling of solids with or without reinforcement and it is capable of accounting for cracking of concrete in tension, crushing of concrete in compression, creep effects and large strains [14,15]. T3D2 elements are used to model one-dimensional reinforcing bars that are assumed to deform by axial stretching only. Discretization of a typical UHPFRC beam is shown in Fig. 1. The interaction between concrete and reinforcement is achieved by using the embedded region constraint, i.e. defining reinforcement to be embedded in concrete [14,15].

UHPFRC

Beam layout

Reinforcement

■ • ■

Beam cross section

(a) Typical beam

(b) Discretized beam

Fig. 1. Discretization of a beam for developed FE model.

Since UHPFRC beams experience large deflections owing to high ductility, the effect of geometric non-linearity is to be given consideration. This is accounted for in the numerical model through updated Lagrangian method [14]. The Newton-Raphson method is utilized as the solution technique and a tolerance limit of 0.02 on the displacement norm is applied [15,16].

2.2. Material models

A damage based concrete plasticity model, available in ABAQUS, is utilized to capture the nonlinear material behavior of UHPFRC. The Concrete Damage Plasticity model (CDP) is based on the theory of plastic flow [12]. The yield surface in CDP model is based on the yield surface proposed by Lubliner et al [17] along with modifications proposed by Lee and Fenves [18] to account for different evolution laws of the strength under tension and compression. CDP model assumes isotropic damage evolution combined with isotropic tensile and compressive plasticity to simulate inelastic behavior of concrete. It allows to incorporate strain hardening in compression, strain stiffening in tension, and uncoupled damage initiation and accumulation in tension and compression. CDP uses a non-associated flow rule with the help of a plastic potential. The evolution of the yield (or failure) surface is controlled by two hardening parameters (equivalent plastic strains), which are linked to failure mechanisms under tension and compression loading, respectively. In order to define CDP model, a set of material properties including compression hardening, tension stiffening, elastic modulus, poison's ratio, and density needs to be input for analysis [11,12,19].

The uniaxial stress strain data in compression and tension data is required to evaluate the hardening/softening behavior of the concrete. UHPFRC exhibits a linear compressive stress-strain relationship up to peak stress as illustrated in Fig. 2 [5]. Therefore, the compressive behavior is modelled by a linear stress-strain curve up to elastic limit (zone OA). This is followed by stress hardening (zone AB) until achieving peak compressive strength, further followed by strain softening (Zone BC). Tensile behavior of UHPFRC needs to be modelled before and after cracking. Tensile micro cracks in concrete is captured through softening stress-strain relationship. A metal plasticity model that utilizes Mises yield surface with associated plastic flow and isotropic hardening available in ABAQUS [14] is adopted for the constitutive modelling (stress strain response) of reinforcing steel, as shown in Fig. 3.

Fig. 2. Stress-strain response for UHPFRC (a) compressive, (b) tensile.

¡1 100

0 -1-1-.-1-1-1-

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Strain

Fig. 3. Stress-strain response of steel reinforcement.

CDP model adopted for UHPFRC accounts for tension and compression stiffness degradation which is given in terms of scalar degradation variables. The degradation variables are increasing functions of the plastic strains which get more pronounced with increase in plastic strain. These variables capture degradation in material stiffness with increased loading, which are zero for an undamaged state and one for a complete damage state.

The damage parameter in tension is assumed to get activated after reaching peak tensile strength. Therefore, damage contours replicate tensile cracking and the extent of damage increases with increase in strain at higher load levels (crack widening). In order to account for reduction in stiffness due to cracking, a nonlinear tension damage parameter recommended by Chen and Graybeal [20] is incorporated in the finite element model. This assumes that the tension damage variable is a nonlinear function of the plastic strain which is zero (no damage) at zero plastic strain and 90% damage at strain of 0.01. Owing to high compressive strength and linear response till peak strength in UHPFRC, no stiffness degradation in compression is included in the CDP model [11,13].

In addition to compressive and tensile stress strain behavior, there are other parameters (</< , kc, E , j)

to define CDP model. Two parameters of and kc modify the yield surface. < /< is the ratio of biaxial

compressive strength to uniaxial compressive strength which influences the yield surface in a plane stress state. The parameter kc is used to define the shape of the failure surface in deviatoric plane which is the ratio between

distances measured from the hydrostatic axis to tensile and compressive meridians. The other two parameters £) modify the non-associated potential flow. is the dilation angle which describes the angle of

inclination of the failure surface towards the hydrostatic axis measured in the meridional plane . E is an eccentricity parameter which controls the deviation of the hyperbolic plastic potential from its asymptote. j is the viscosity parameter which is used for the visco-plastic regularization of the concrete constitutive equations [11-14].

UHPFRC beams are loaded incrementally till failure occurs. Failure in beams occurs when the load exceeds the sectional capacity. In the present study, rupture in rebar and concrete crushing are considered as governing failure of the beams under dominant shear and flexure loading.

3. Model validation

The developed finite element model is validated against data from tests on UHPFRC beams. In order to gauge efficacy of the model in predicting structural behavior of UHPFRC beams, predictions from the model, including load deflection response, load strain response, load capacity, and tensile damage (cracking), are compared with experimental results.

3.1. Selection of beams for validation

Four UHPFRC beams, designated as U-B3, U-B4, U-B5, and U-B6 tested under predominant shear and flexural loading are utilized for validating the model. The beams are of rectangular cross section, 180 mm in width, 270 mm in depth, and had a length of 4000 mm. These beams have only longitudinal tensile reinforcement, but no compression and shear reinforcemnet (stirrups). This is to take full advantage of high compressive and high tensile

strength offered by UHPFRC. Cross sectional details of UHPFRC beams are illustrated in Fig. 4.

Beams U-B3 and U-B5 were tested under two point loads applying on the top face of the beams at a distance of 432 mm on either side of mid-span as shown in Fig. 4. This test set up is designed to simulate pure bending between points of load application. Beams U-B4 and U-B6 were subjected to a single point load applying on the top face of the beams at the distance of 610 mm from support (shear span) to create high shear capacity as compared to bending moment (see Fig. 4).

Material properties of UHPFRC and steel rebars required to be identifed in CPD model are listed in Table 1. The compressive strength and corresponding strain, and elastic modulus of UHPFRC, derived through material tests on cylinders, are 193 kN, 0.0044, and 43970 MPa, respectively. However, UHPFRC compressive elastic limit is considered to be 97 MPa [13]. Strain softening behavior of UHPFRC after peak compressive strength is modelled by empirical equation proposed by Singh et al [19] as illustrated in Fig. 2(a).

Tensile behavior of UHPFRC before cracking is modelled by a linear elastic stress strain response. For the stage of after cracking, a bi-linear tension-softening curve (TSC) suggested by Yoo et al [21] based on the results of three-point bending test on notched prism specimens and inverse analysis was adopted. The adopted tension-softening curve was converted into stress-strain curve utilizing recommendations in AFGC/SETRA [22] as shown in Fig. 2(b).

The additional five parameters (^, kc, E , /), described in section 2.2, are set to be 1.16, 2/3, 39°, 0.1,

and 0.0001 according to the literature and sensitivity analysis conducted in this study [12,13,23,24].

Shear Loading (Beams U-B4 & U-B6)

Fig. 4. Loading conditions, layout and cross section of tested UHPFRC beams (All dimensions are in mm).

Table 1. Material properties of UHPFRC and steel used in the tests.

Material Properties

E= 43970 MPa; fc= 193 MPa; Ep= 0.0044; euc=0.008;

ft= 7.11 MPa; ecr= 0.00016; eut= 0.0181;

UHPFRC Density= 2565 kg/m3; Poison ratio=0.2;

Dilation angle= 39°; Eccentricity= 0.1; kc= 2/3;

ct40/ct 0 = 1.16 ; Viscosity parameter=0.0001

Steel Reinforcement E= 207 GPa; fy= 435 MPa; ey= 0.0021; fu= 700 MPa; eu= 0.12; Density= 7850 kg/m3; Poison ratio=0.3

3.2. Load-deflection response

To illustrate validity of the model, load-deflection response of UHPFRC beams (U-B6 and U-B5) are compared with experimental data as shown in Figs. 5 and 6. It can be seen that UHPFRC beams exhibit different stages in response i.e., linear elastic stage until initiation of tensile cracking, post-cracking stage with enhanced cracking and their progression, onset of yielding in steel reinforcement, and plastic deformation stage till peak load followed by

attainment of failure. Predicted load deflection response exhibit these distinct stages as observed in the experiments. These figures show that overall trend of load-deflection response of UHPFRC beams obtained by FEA is in a good agreement with measured experimental data. However, post cracking response predicted by FEA is slightly stiffer than experimental results. This difference can be attributed to idealization in numerical model with respect to mechanical properties of UHPFRC and steel rebars, as well as geometry, loading configuration and support conditions. Maximum load capacity predicted by the numerical model and test results are summarized in Table 2. Ratio of load capacity predicted by FEA to that of experimental results ranges from 0.98 to 1.11. This shows that load capacity of UHPFRC beams can be well predicted using FEA and adopted concrete damage plasticity model.

-Experiment

or=f=r=r------ -----FEA

cs o J

40 60 . 80 100 Deflection (mm)

Fig. 5. Comparison of load deflection response of beam U-B6, subjected to shear loading, obtained by FEA and test.

40 60 80 100 Deflection (mm)

Fig. 6. Comparison of load deflection response of beam U-B5, subjected to flexural loading, obtained by FEA and test.

Table 2. Comparison of load capacity of UHPFRC beams from test and finite element analysis.

Load Capacity (kN)

U-B3 U-B5

U-B4 U-B6

Finite Element Analysis (1)

Test Results (2)

Predominant Flexural Loading

107.7 124.6

97.1 126.6

Predominant Shear Loading

151.8 176.5

142.1 177.1

Ratio (1)/(2)

1.11 0.98

1.07 0.99

3.3. Load-strain response

The model is capable of tracing local behavior of UHPFRC beams, including progression of strains at a cross section. Predicted load-longitudinal strains on rebars at midspan in beams U-B3 and U-B5 along with measured strains in tests are plotted Fig. 7. The results show that strain predictions are in good agreement with measured strains from tests. However, predicted load-strain response is stiffer than experimental results similar to predicted load-deflection response. This difference between numerical predictions and measured test data can be attributed to variations arising from material models, which might be different from actual material models as well as level of bonding between strain gauges and UHPFRC.

Fig. 7. Comparison of load strain response at mid-span of rebars obtained by FEA and tests, (a) beam U-B3, (b) beam U-B5.

3.4. Crack progression

The model is also capable of tracing crack propagation with increased load by plotting tensile damage parameter. Tensile damage contours obtained by FEA at different load levels for beams U-B6 and U-B5 along with experimental results are illustrated in Figs. 8 and 9. In the case of beam U-B6, under dominant shear loading, tensile damage initiated at extreme tension fibers due to high levels of tensile stress. This initial tensile damage (cracks) was confined between points of load application and mid-span. As the load increased further, shear stresses significantly increase causing maximum principle stresses to exceed tensile capacity of UHPFRC. This results in significant tensile damage in the shear span (between left support and point of load application). It should be noted that a higher value of the damage parameter indicates a greater level of tensile damage (cracking). As the beam approaches failure, tensile damage further propagated towards compression zone and more cracks initiated and propagated further in the shear span. These closely spaced zones of tensile damage in the shear span when connected to each other, make an angle of 49° with horizontal line (see Fig. 8). This region of maximum tension damage coincides with diagonal tension crack propagating at an angle of 52° as seen in the experiment.

Load= 68 kN

Load= 172 kN

At Failure

+6,261e-01

+5,257e-01

+2,253e-01

L +0.000e + 00

+8,261e-01 +7.510e-01

+5,257e-01 _ +4.5Q6e-01 ,— - +3,755e-01 p. - +3.004e-01 U4- +2,253e-01 ■ +1.! K+7.!

1 jf—'i '

it: 'AA :

Load— 68 kN

Load= 172 kN

■. ¡i»1, _

At Failure

(a) ABAQUS predictions (b) Test data

Fig. 8. Tensile damage obtained by FEA representing cracks in beam U-B6 along with test results.

In beam U-B5, under flexural loading, tensile damage initiated at extreme tension fibers of the beam in the zone between load points which is subjected to pure bending. This cracking behavior was observed in the experiment as shown in Fig. 9. Upon increasing load (flexural stresses) further, greater depth of the beam is subjected to tensile damage. This shows propagation of cracks toward compression zone at increased load levels. The tension damage contour obtained by FEA indicate that maximum tension damage at failure is concentrated at critical section of the beam (mid-span) which coincides with the macro crack seen in the experiment (see Fig. 9). Tensile damage in beam U-B5 is more spread along the length of the beam between loading points. However tensile damage in beam U-B6 is concentrated under the point of load application. In addition, after rebar yielding in beam U-B5, density of tensile damage (crack) highly increased as compared to beam U-B6.

-«IB«. At Failure

At Failure —+1.5025-014

+ 7 i510e-02';'

(a) ABAQUS predictions (b) Test data

Fig. 9. Tensile damage obtained by FEA representing cracks in beam U-B5 along with test results.

DAMAGET

(Avgi 75%)

+1.421e-01

+l,302e-01

+l,184e-01

+1.065e-01

+9,471e-02

+8,287e-02

+7,103e-02

+5,919e-02

+4,735e-02

+3,552e-02

+2,368e-02

+1,184e-02

+Q,0Q0e+00

DAMAGET

(Avgi 75%)

r + 9,012e-01

- - +8,261e-01

- +7,510e-01

- +6,759e-01

- +6,008e-01

- +5,257e-01

- +4,506e-01

- +3,755e-01

- +3,004e-01

- +2,253e-01

- +1,5028-01

H- +7,510e-02

+ 0,000e+00

DAMAGET

(Avgi 75%)

r +9,012e-01

_ - +8,261e-01

- +7,510e-01

- +6,759e-01

- +6,008e-01

- +5,257e-01

- +4.506e-01

- +3,755e-01

- +3,004e-01

- +2,253e-01

- +1.502e-01

H- +7.510e-02

+0,000e+00

Load=51 kN

Load= 122 kN

4. Summary

A numerical model is developed for tracing structural response of UHPFRC beams. The developed model can capture the behavior of UHPFRC beams in the entire range of loading till failure. The model predictions at global level (load-deflection response) and local level (load-strain response) of UHPFRC beams agree well with experimental data. Further, the model predictions of tensile damage (crack progression) in UHPFRC using scalar damage parameter is an effective way of capturing failure modes of the beams as observed in the experiments.

Acknowledgements

This material is based upon work supported by Metna Company, and Michigan State University and the authors wish to acknowledge sponsors support. Any opinions, findings, and recommendations expressed in this study are those of the authors and do not necessarily reflect the view of the sponsors.

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