Modeling of a reinforced concrete beam subjected to impact ...

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 4, No 3, 2014

© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0

Research article ISSN 0976 – 4399

Received on October, 2013 Published on February 2014 227

Modeling of a reinforced concrete beam subjected to impact vibration

using ABAQUS Ali Ahmed

Associate Professor, Department of Civil Engineering, Stamford University Bangladesh

[email protected]

doi: 10.6088/ijcser.201304010023

ABSTRACT

A 3D finite element (FE) analysis technique using ABAQUS is chosen to explore the

dynamic behavior of a beam under impact load. A beam, for which the impact dynamic test

was conducted by Kishi (2004), is selected to develop a solid element FE model. FE analyses

were performed implementing ABAQUS Explicit (2004) programming tool to predict the

dynamic responses under the pressure amplitude introduced by pounding a freefalling steel

hammer. Thirty analyses have been executed changing different parameters, such as damping,

tension and compression stiffness recovery, damage parameter-strain/displacement relations

and friction coefficient to choose the best performing FE analysis model. The high accuracy

definition is confirmed upon extensive examination of the calculated structural responses of

the FE model comparing with the published experimental results (Kishi, 2004).

Keyword: Finite element, explicit dynamic analysis, beam, impact load, damage parameter,

strength recovery.

1. Introduction

The primary concern about the structural failure under anticipated extreme loadings had been

addressed with the many provisions attributed to issues like: (i) attaining member

level/structure level resistance against the code prescribed loads and load combinations (ACI,

2002 and ASCE-7, 2002), (ii) defining all anticipated loads including extreme environmental

loads (Hakuno and Meguro, 1993 and Luccioni, et al., 2004), (iii) attaining structural

integrity and ductility (ASCE-7, 2002), (iv) providing continuity for connections (beam to

column, slab to column, slab to wall, beam to beam, column to column etc.) (Leyendecker

and Burnett, 1976 and Leyendecker and Ellingwood, 1977) and (v) providing some general

statements about resiliency, redundancy and robustness, etc. (ACI, 2002 and ASCE-7, 2002).

Therefore, code-compliant buildings highlight the significance of more investigative study

related to impact loading issues. It is recognized that experimental observations are much

more expensive comparative to its counterpart investigations implementing computer

analyses. In that sense, finite element (FE) analysis can be an option. Thus, a 3D FE analysis

model using ABAQUS was triggered to explicitly explore the dynamic behavior of a

reinforced concrete (RC) structural element. ABAQUS is a very complex FE analysis

program introduced with huge material characteristics and parameters to reproduce high

accuracy in calculations. But there are no clues to assign exact values for those parameters.

Thus, model choosing based on influential material characteristics and behavioral parameters

is an important part of FE analysis before investigating the actual behavior of a structural

element. In this study, thirty analyses have been executed changing different parameters, such

as damping, tension and compression stiffness recovery, damage parameter-strain/

displacement relations and friction coefficient to choose the best performing FE model. The

Modeling of a reinforced concrete beam subjected to impact vibration using ABAQUS

Ali Ahmed

International Journal of Civil and Structural Engineering

Volume 4 Issue 3 2014

228

results of few analyses are to be discussed in this paper. Upon extensive examination of the

calculated structural responses of the FE models comparing with the published experimental

results. This study reveals that FE analysis using proposed model can be applied to explore

dynamic behavior of structural elements subjected to impact vibrations.

2. Modeling technique

2.1. Mesh discretization

A 3D FE mesh of a concrete beam block, steel anchor plate and reinforcement bars are

constructed employing ABAQUS/CAE structural analysis modeling tool (2004) to execute a

dynamic analysis of the RC beam. The dynamic test of the beam was conducted by Kishi and

his colleagues at Muroran Institute of Technology in Japan on March 3, 2003 (Kishi, 2004).

The experimental setup including the pounding steel hammer is shown in Figure 1. As shown

in this test set-up, a rectangular RC beam of 200×300 mm cross-section and 3,400 mm long

was positioned horizontally and was supported on two gigues at its both ends with a 3,000

mm span between the two supports (Figure 2). This figure also shows the reinforcement

detailing of the tested beam. Mesh of only one-fourth portion of the beam is constructed for

FE model because of the geometric, loading and support conditions symmetry (Figure 3). In

the FE analysis modeling, the concrete block of the tested beam is idealized by homogeneous

material and modeled with eight-node solid (brick) elements, which are identified as C3D8R

elements in ABAQUS. They are chosen because C3D8R elements with reduced integration

follow the constitutive law integration accurately and very suitable for nonlinear static and

dynamic analyses as well as allow for finite strain and rotation in large-displacement analysis.

These solid elements are used for modeling both the supporting gigue and the pounding steel

hammer. The mesh of solid beam model strictly followed the geometries of the tested beam.

The longitudinal and confinement reinforcement bars are modeled as embedded elements in

concrete block of the beam with 3D beam elements B31. To construct the mesh of the model,

moderately fine mesh is used to obtain close responses to the experimental results. Figure 3

illustrates the mesh pattern of the FE model of the tested beam with reinforcement bars (as

embedded elements), anchoring steel plate and supporting gigue. The total numbers of user

defined nodes and elements in the mesh of the solid element model are 8,103 and 5,929,

respectively.

Figure 1: Impact test setup of the RC beam (Kishi, 2004) implied for the FE modeling


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Figure 2: Reinforcement detailing of the tested beam (Kishi, 2004) used for the model

300

Figure 3: Mesh configuration of FE model of the beam

2.2. Material properties

Material properties for all elements are specified; however, high-quality material data were

difficult to obtain, particularly for the more complex material models like material damage

properties. The validity of the results is essentially limited by the accuracy and extent of the

material data. Here linear elasticity and nonlinear plasticity of reinforcement steel and

isotropic elasticity in combination with damaged plasticity model of concrete is discussed to

depict a mechanical constitutive model.

2.2.1. Constitutive model of reinforcing bar steel

Steel of the reinforcing bars has approximately linear elastic behavior when the steel stiffness

introduced by the Young’s or elastic modulus keeps constant at low strain magnitudes. At

higher strain magnitudes, it begins to have nonlinear, inelastic behavior, which is referred to

as plasticity. The plastic behavior of steel is described by its yield point and its post-yield

hardening. The shift from elastic to plastic behavior occurs at a yield point on a material

stress-strain curve. The deformation of the steel prior to reaching the yield point creates only

elastic strains, which is fully recovered if the applied load is removed. However, once the

stress in the steel exceeds the yield stress, permanent (plastic) deformation begins to occur.

Both elastic and plastic strains accumulate as the metal deforms in the post-yielding region.

The stiffness of the steel decreases once the material yields. The plastic deformation of the

steel material increases its yield stress for subsequent loadings.

The yield and ultimate strengths were taken for longitudinal steel bars 379 MPa and 581 MPa

and confinement steel bars 373 MPa and 518 MPa with Young’s modulus of elasticity,


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Es = 20600 GPa and Poisson’s ratio, ν = 0.3 from the coupon test results (Kishi, 2004). The

test material properties for steel of the anchor plate and the supporting gigue were not

provided, and they are assumed similar values to those of longitudinal bars.

2.2.2. Constitutive model of concrete

Linear at elastic and nonlinear damaged plasticity model for inelastic states of concrete are

assumed because of concrete’s low deformability in those both states. The concrete isotropic

damage plasticity model is designed for its arbitrary loading conditions. Degradation of the

elastic stiffness induced by the plastic straining both in tension and compression are taken

into consideration in the material constitutive model.

i) Concrete damaged plasticity model

The material model is a continuum, plasticity based, damaged model for concrete. Damaged

plasticity is assumed to characterize the uniaxial tensile and compressive response of concrete

as shown in Figure 4. At the beginning, the stress-strain relationship is linearly elastic under

uniaxial tension until the value of the failure stress ft0 is reached. Failure stresses in concrete

block is converted to replace microcracks in it. Beyond the state of the failure stress in

concrete, stress-strain response is designed by softening characteristic (Figure 4a).

Under uniaxial compression, the response is linear until the value of initial yield fc0. After

attaining the ultimate stress fcu in the plastic zone, the response of concrete is characterized by

the stress hardening followed by strain softening (Figure 4b). Therefore, concrete stresses

determined unloading from any point on the strain are

( )( )t

pl

ttct dEf −−= 1εε

(1)

( )( )c

pl

cccc dEf −−= 1εε

(2)

where Ec is the modulus of elasticity of concrete. Then, the effective tensile and compressive

cohesion stresses of concrete are estimated as

( )( )pl

ttc

t

t

t Ed

ff εε −=

−=

1 (3)

(a) Tension behavior associated b) Compressive behavior associated

With tension stiffening with compression hardening

Figure 4: Concrete damaged plasticity model


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( )( )pl

ccc

c

c

c Ed

ff εε −=

−=

1 (4)

which determine the size of the failure surface. The postfailure behavior of reinforced

concrete represents by means of the postfailure stress as a function of cracking strain ck

tε and ,ck

cε which are defined as the total strain minus the elastic strain corresponding to the

undamaged material, and tension stiffening data are given in terms of the cracking strains.

When unloading data are available, programming automatically converts the cracking strain

values to plastic strain values using the following relationships (ABAQUS/Explicit, 2004):

( ) 01 E

f

d

d t

t

tck

t

pl

t−

−= εε (5)

( ) 01 E

f

d

d c

c

cck

c

pl

c−

−= εε (6)

Property values for the assumed constitutive model of concrete and steel discussed above

were collected from the coupon test results conducted in the laboratory of Muroran Institute

of Technology, Japan (Kishi, 2004). According to the 11 days and 28 days cylinder test

results, the compressive strengths of concrete were 32.3 MPa (taken as yielding strength fc0)

and 39.2 MPa (taken as ultimate strength fcu), and Young’s modulus of elasticity Ec and

Poisson’s ratio νc of concrete were 28.3 GPa and 0.19. The tensile strength of concrete is

estimated as ft0 = 0.6√fcu; therefore, ft0 = 3.76 MPa.

The concrete material is also modeled implementing the concrete damaged plasticity

constitutive model; thus, the mesh where reinforcement does not exist is very sensitive to

deformation. To ignore this kind of undesirable mesh deformation, the tensile postfailure

behavior is assumed in terms of a fracture energy cracking criterion by defining a stress-

displacement curve in place of a stress-strain curve shown in Figure 5(a). Accordingly, the

stiffness degradation damage caused by tensile failure (cracking) of the concrete dt is

assumed as shown in Figure 5(b) with compression recovery factor 0. The postfailure tensile

stresses of the proposed model are adjusted by multiplying a constant 1.253 with the

postfailure stresses for corresponding displacements used in the ABAQUS manual

(ABAQUS, 2004). The stiffness degradation damage due to the compressive crushing dc is

assumed to be 50% for corresponding strain 0.001375 with tension recovery factor 0.8.

To provide a 5% fraction of critical damping of the beam, Rayleigh mass proportional

damping can be given as factor α = 2ξωn (ignoring stiffness proportional damping factor β),

where ξ is the critical damping and ωn = 2π/T, T is the natural period of the beam which is

taken from the experimental results as 0.0349 sec (Kishi, 2004); thus, α = 18.

3. Boundary conditions

As the beam’s dynamic behavior during experiment was symmetric in both XY and YZ

planes, mesh of the FE beam model is constructed for only one-fourth portion of the original

beam and for half portion of the left support (Figure 6), and the surfaces of XY and YZ

planes of the mesh of the FE model are constrained with symmetric boundary conditions

(BC) in Z and X directions, respectively. General contact surface algorithm is considered to

model the contact behavior of interacting surfaces between the beam and supporting gigue.

When interaction occurs between surfaces of the beam and supporting gigue, both the friction


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and cohesion coefficients of contact points are set to 0.2 and 0 to consider corresponding

frictional shears and avoid cohesion during interaction.

a) Stress-displacement relation b) Tension damage model

Figure 5: Postfailure stiffness degradation damage properties of concrete

a) Concrete beam with support system b) Rebar and anchor plate

Figure 6: Boundary conditions and applied loads on FE models

3.1 Loadings

Gravity loads are applied to the beam by introducing gravitational acceleration g = 9.81

m/sec2 during a linear static response step, and then impact load that was recorded during the

experiment is applied to the middle circular surface area of 150 mm diameter on the top of

the beam model in a nonlinear explicit dynamic analysis step. The applied impact pressure

amplitude is shown in Figure 7.

4. Execution of FE explicit dynamic analysis

4.1. FE analysis

The first step of a finite element analysis using ABAQUS is to discretize the actual geometry

of the structure using a set of finite elements for defining the model of the physical problem

and to create an ABAQUS input file. In this study, model is created graphically using


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preprocessor software ABAQUS/CAE (2004), which is a complete ABAQUS environment

that provides a simple, consistent interface for creating, submitting, monitoring, and

evaluating form ABAQUS/Standard and ABAQUS/Explicit modeling process. Defining the

geometry and material properties, as well as generating a mesh are done by moving from

module to module and developed a model to generate an input file. However, the input file

generated by ABAQUS/CAE is altered later for defining properties and parameters that are

not supported by ABAQUS/CAE using a text editor. A complete analytical procedure in

ABAQUS usually consists of three stages: i) Preprocessing, ii) Analysis, and iii)

Postprocessing. The sequential progress of these three stages is shown in Figure 8.

The linear static analysis was performed by ABAQUS/Standard in only one increment;

however, the ABAQUS/Explicit (2004) requires a large number of increments because very

small increment of time is required to get the stable solution of equation of motion and keep

the stability limit. The total execution time of the explicit analysis was taken 4 hours 15

minutes by a Pentium 4 computer with 3,000 MHz processor speed.

Figure 7: Amplitude of loading exploited for the FE model of the beam (Kishi, 2004)

Figure 8: Flow chart of a complete ABAQUS analysis

4.2. Analysis results and discussion

Thirty analyses have been executed changing different parameters, such as damping, tension

and compression stiffness recovery, damage parameter-strain/displacement relations, friction

coefficient and mesh size to choose the best performing FE model. However, the results of

few analyses are to be discussed in this section to highlight the models that can predict

acceptable behavior of the tested beam, and those which failed to give realistic results are

discarded. A quick picture of the FE analyses is shown in Figure 9. This figure compares the


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analytical and experimental results. In Figure 9a, the comparison is based on the midpoint

deflections between measured and estimated by the means of experiment and FE explicit

analyses, respectively. FE model FEA8-1 shows very good agreement of the midpoint

displacements in the elastic region, but in the region where concrete material starts giving

plastic deformation decreasing stiffnesses due to the formation of cracks, analysis shows

deflection waves with very low amplitudes (almost flatten horizontal deflection curve). It

happens because FEA8-1 does not include damage model to consider concrete stiffness

degradations. Thus, although this model matches the maximum deflection limit of the beam

with experimental results, it fails to give exact amplitudes of midpoint displacement waves of

the beam in the plastic region. FEA8-2 model illustrate better results than that predicted by

FEA8-1 because model FEA8-2 considers the damage model for concrete stiffness

degradation. However, the model ignores considering tension stiffness recovery parameter in

the concrete areas where reinforcement bars are not present. The FEA8-2 model predicted the

first wave of deflection curve with minimal amplitude and does not match with experimental

curve. Now, FEA8-3 considers all of the parameters discussed above except damping factors.

This means that the values of Rayleigh’s damping factors α and β are taken as 0. This FE

model gives better approximation of dynamic behavior than other two, but the amplitudes of

midpoint displacement waves do not decay after the first wave. The last FE model FEA8-4 is

the best among the all models due to the consideration of all the factors mentioned above

including Rayleigh’s mass proportional damping factor α =18 taking into account of 5% of

critical damping.

Figure 9b shows the comparison between the values of reaction forces at the support of the

tested beam and FE model FEA8-4. The applied impact force reached to peak values at the

beginning and ABAQUS shows sensitivity to this force and was distributed between the

supports through the beam span. However, the peak reaction forces estimated by the FEM8-4

are larger, later reaction forces predicted by the analyses agree very closely with the measured

values.

a) Midpoint displacements of the beam b) Reaction forces on the support

Figure 9: Comparison between FE analysis and experimental results


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Figures 4.10a and b illustrate tension damage (cracks) with superimposed experimental beam

cracks (orange colour solid lines) and concrete stiffness degradation contour plots of the half

of beam, where tension damage value 0 indicates concrete’s elastic behavior, greater than 0

but smaller than 0.4 means formation of microcracks until beginning of concrete yield,

greater than 0.4 but less than 1 concrete contains cracks and 1 defines full concrete failure. In

Figure 4.10, cracks that are away from the midpoint and developed due to the bending effect

of the beam are at initial state for both tested beam and FEA8-4; in addition, the dense cracks

formed mainly due to shear in the middle region of the tested beam are similar with that of

the FEA8-4. However, some cracks formed in between the midpoint and support are not

developed in the exact position of the FE model, FEA8-4 can simulate the dynamic behavior

with sufficient accuracy for executing explicit dynamic analysis of a RC beam.

a) Contour plot of tensile damage with superimposed experimental beam cracks

b) Contour plot of stiffness degradation

Figure 10: Evolution of tensile damage and deformation configuration of concrete at the end

of loading for FEM8-4

Figure 11 shows the contour plot of equivalent plastic strain in rebar and steel anchor plate.

The longitudinal bars and shear rebar of the middle region where impact pressure amplitude

was applied are yielding in tension, but the longitudinal bars did not reach their ultimate

strain values. Additionally, there is a lack of shear rebar in the middle region of the beam that

caused to fail the beam due to excessive shear. The similar effect can be observed in the

tested beam.

Figure 11: Equivalent plastic strain contour plot and deformation configuration of rebar at

the end of the explicit dynamic analysis of FEM8-4


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5. Conclusions

A close analytical observation of four FE models was done based on material properties and

failure mechanism and their parameters. The comparison of those models with experimental

results showed that FEA8-4 was the best performing model for a RC beam to study dynamic

behavior under impact load.

It is remarkable that this FE analysis model could be used to perform dynamic analysis of not

only individual elements of a RC frame but also RC frame structures (long bridges, single-

and multi-storied buildings and others) using ABAQUS.

6. References

1. ABAQUS, (2004), Example problems manual, Version 6.4.1, Hibbitt Karlsson &

Sorensen, Inc., USA.

2. ABAQUS/CAE, (2004), User’s Manuals, Version 6.4.1, Hibbitt Karlsson & Sorensen,

Inc., USA.

3. ABAQUS/Explicit, (2004), Theory Manual, Version 6.4.1, Hibbitt Karlsson &

Sorensen, Inc., USA.

4. ACI 318-02/318R-02, (2002), Building Code Requirements for Structural Concrete

and Commentary, American Concrete Institute, Farmington Hills, MI.

5. ASCE 7-02, (2002), Minimum Design Loads for Buildings and Other Structures,

American Society of Civil Engineers, Reston, VA.

6. Hakuno, M., and Meguro, K., (1993), Simulation of Concrete-Frame Collapse Due to

Dynamic Loading, Journal of engineering mechanics, 119(9), pp1709-1723.

7. Kishi, N., (2004), Practical Methods for Impact Test and Analysis, Structural

Engineering Series, JSCE, impact problems, No.15 (in Japanese).

8. Leyendecker, E. V., and Burnett, E. F. P., (1976), The Incidence of Abnormal

Loading in residential Buildings, National Bureau of Standards, NBS 98, Washington,

D.C.

9. Leyendecker, E. V., and Ellingwood, B. R., (1977), Design Methods for Reducing the

Risk of Progressive Collapse in Buildings, National Bureau of Standards, NBS 98,

and Washington, D.C.

10. Luccioni, B. M., Ambrosini, R. D., and Danesi, R. F., (2004), Analysis of building

collapse under blast loads, Journal of engineering structures, ASCE, 26, pp 63-71.

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