Non Linear Analysis of RC Structure for

Dynamic Loading

A DISSERTATION

Submitted in partial fulfilment of the award of the degree

of

MASTER OF TECHNOLOGY

in

CIVIL ENGINEERING

(With specialization in Structural Engineering)

by

ABHINAV GUPTA

(E.No. 10523001)

DEPARTMENT OF CIVIL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY ROORKEE ROORKEE 247667, UTTARAKHAND, INDIA

JUNE, 2012

II

CANDIDATES DECLARATION

I hereby declare that the work being presented in this dissertation, entitled Non-

Linear Analysis of RC Structure for Dynamic Loading towards partial fulfilment of the

requirements for the award of the degree of Master of Technology in Civil Engineering

with specialization in Structural Engineering, submitted to the Department of Civil

Engineering, Indian Institute of Technology Roorkee, Roorkee, is an authentic record of my

own work carried out under the guidance of Dr. ASHOK K. JAIN, Professor, Department of

Civil Engineering, IIT Roorkee and Shri Prabhakar Gundlapalli, Additional Chief Engineer

(Civil), Scientific Officer G, NPCIL, Trombay, Mumbai.

Date:

Place: Roorkee (ABHINAV GUPTA)

CERTIFICATE

This is to certify that the above statement made by the candidate is correct to the best of our

knowledge.

Dr. ASHOK K. JAIN

Professor,

Department of Civil Engineering,

Indian Institute of Technology Roorkee

Roorkee 247667 (INDIA)

Shri PRABHAKAR GUNDLAPALLI

Additional Chief Engineer,

Nuclear Power Corporation of India Ltd,

Anushakti Nagar, Trombay,

Mumbai, INDIA

III

ACKNOWLEGEMENT

I wish to express my deep sense of gratitude to Dr. ASHOK K. JAIN, Professor,

Department of Civil Engineering, IIT Roorkee and Shri Prabhakar Gundlapalli, Additional

Chief Engineer (Civil), Scientific Officer G, NPCIL, Trombay, Mumbai for their unfailing

inspiration, concern, whole-hearted co-operation and pain staking supervision for this

dissertation. Their encouragement and incisive comments gave immense confidence to

complete this work.

I would like to express my deep respect and love to my parents, brother, and other family

members for their blessings, inspiration, encouragement and support throughout my life.

I would also like to specially thank Shri Rajiv Ranjan, Deputy Chief Engineer, NPCIL and

Shri Shrikant Devmani Mishra, Executive Engineer, NPCIL, Mumbai who helped me

throughout this dissertation and also my friends Arun Das and Shabbir Lokhandwala who

devoted their valuable time for carefully scrutinising the text of this dissertation.

Finally I would also like to extend my deepest gratitude to my friends Avantika, Warlu,

Satyam, Narpat and Samrat for their support throughout my M.Tech life. The unforgettable

moments spent with them will remain lifelong in my memory.

Date:

Place: Roorkee (ABHINAV GUPTA)

IV

ABSTRACT

An extensive study of the material models of concrete and steel available in

ABAQUS v6.9 was carried out and Linear Elasticity, Concrete Smeared Cracking and

Concrete Damaged Plasticity material models of concrete and Classical metal plasticity

material model of steel were chosen to represent the elastic and plastic behaviour of these

materials. Various parameters that were required to define a material model were also studied

in detail. Detail of various methods available to model the bond between concrete and steel

has also been presented.

Effect of meshing and type of element chosen for stress analysis viz. first order

hexahedral and second order hexahedral first order tetrahedral and second order tetrahedral

elements are studied with the help of a cantilever beam. Twelve different models with three

mesh densities viz. two, four and six elements along the depth and four types of elements viz.

first order hexahedral, second order hexahedral, first order tetrahedral, second order

tetrahedral elements were analysed for this study. Linear hexahedral element is found to be

more consistent compared to other elements for all the mesh densities studied such that

percentage error in displacement for linear hexahedral element is in between 4% to 7.5%.

Stress Analysis is done for two different experiments carried out by Bresler &

Scordelis (1963) and Burns & Seiss (1962). The first beam that was tested by Bresler &

Scordelis (1963) had no shear reinforcement and the second beam that was tested by Burns &

Seiss (1962) had shear reinforcement. Clear span for both the beams were same. Validation

of the two material models of concrete is done with respect to the two beams. Also, a beam

column joint was analysed by taking Concrete Damage Plasticity model, which was sbjected

to a cyclic load applied at the tip of the beam. It is found out that only by the use of Concrete

Damage plasticity model, the finite element model is able to predict the complete non linear

response of structure.

Finally a comparison of the two material models of concrete by analysing two

different beams tested by Bresler & Scordelis (1963) and Burns & Seiss (1962) respectively

is done. For the purpose of comparison based on the effect of combination of mesh size and

tension stiffening parameter on the analysis, eighteen different models were made by taking

three different mesh densities viz. 50 mm, 75 mm and 100 mm element sizes respectively and

six different tension stiffening parameter values ranging from 0.001 to 0.006. Based on the

comparison data it is concluded that Concrete Damage Plasticity model is the suitable

concrete model to carry out a non-linear analysis of a structure subjected to dynamic loading.

V

TABLE OF CONTENTS

CANDIDATES DECLARATION ................................................................................................. II

ACKNOWLEGEMENT ............................................................................................................. III

ABSTRACT ............................................................................................................................... IV

TABLE OF CONTENTS .......................................................................................................... V

LIST OF FIGURES .............................................................................................................. VIII

LIST OF TABLES .................................................................................................................... X

Chapter 1 Introduction .......................................................................................................... 1

1.1 General ........................................................................................................ 1

1.2 Need of the study ........................................................................................ 2

1.3 Objective of the study ................................................................................. 3

1.4 Scope of the study ....................................................................................... 4

1.5 Organization of report ................................................................................. 4

Chapter 2 Literature Review ................................................................................................. 5

2.1 General ........................................................................................................ 5

2.2 Experimental Studies .................................................................................. 5

2.3 Literature Survey ......................................................................................... 6

2.4 Summary ................................................................................................... 10

Chapter 3 Material Modelling ............................................................................................. 11

3.1 General ...................................................................................................... 11

3.2 Material Behaviour Post Yield .................................................................. 11

3.3 Tension Stiffening Effect .......................................................................... 12

3.4 Material Model for Concrete ..................................................................... 16

3.4.1 Linear Elasticity ............................................................................. 16

3.4.2 Concrete Smeared Cracking .......................................................... 17

3.4.3 Concrete Damaged Plasticity Model ............................................. 23

3.5 Material Model for Reinforcement ........................................................... 32

3.5.1 Classical Metal Plasticity .............................................................. 32

Chapter 4 BondSlip Model ............................................................................................... 33

4.1 General ...................................................................................................... 33

VI

4.2 Existing Studies ......................................................................................... 33

4.2.1 Embedded models ......................................................................... 33

4.2.2 Distributed models ......................................................................... 33

4.2.3 Discrete models ............................................................................. 34

4.3 Simulation in ABAQUS ............................................................................ 35

4.3.1 Embedded Element ........................................................................ 35

4.3.2 Friction .......................................................................................... 35

4.3.3 Spring Element .............................................................................. 36

4.3.4 Translator ....................................................................................... 36

4.4 Summary ................................................................................................... 37

Chapter 5 Meshing .............................................................................................................. 38

5.1 General ...................................................................................................... 38

5.2 Element Description .................................................................................. 38

5.2.1 Family ............................................................................................ 38

5.2.2 Degrees of freedom ....................................................................... 39

5.2.3 Number of nodes and order of interpolation ................................. 39

5.2.4 Formulation ................................................................................... 40

5.2.5 Integration ...................................................................................... 40

5.3 Mesh Size .................................................................................................. 40

5.3.1 Comparative study on different mesh sizes ................................... 41

Analytical Solution .................................................................................... 43

5.3.2 Observations .................................................................................. 43

Chapter 6 Finite Element Model Development ................................................................... 45

6.1 General ...................................................................................................... 45

6.2 Analysis: Procedure and Theory ............................................................... 45

6.2.1 Geometric modelling ..................................................................... 46

6.2.2 Material Modelling ........................................................................ 47

6.2.3 Interaction ...................................................................................... 48

6.2.4 Meshing ......................................................................................... 48

6.2.5 Applying load and boundary condition ......................................... 49

6.2.6 Analysis ......................................................................................... 50

6.3 Experimental Data (Bresler & Scordelis, 1963) ....................................... 51

6.3.1 Test Procedure ............................................................................... 52

VII

6.4 Analysis of Beam ...................................................................................... 52

6.4.1 Load and Boundary Condition ...................................................... 53

6.4.2 Concrete Material Model Definition ............................................. 53

6.4.3 Material Properties for reinforcing steel ........................................ 60

6.4.4 Validation of Concrete Material Models ....................................... 60

6.5 Analysis of BeamColumn Joint .............................................................. 61

6.5.1 Material Properties ........................................................................ 62

6.5.2 Load and boundary condition ........................................................ 64

6.5.3 Results ........................................................................................... 64

Chapter 7 Comparison of Concrete Material Models ......................................................... 66

7.1 General ...................................................................................................... 66

7.2 Comparison Based On Equilibrium Path .................................................. 67

7.3 Comparison Based on Crack Visualization ............................................... 68

7.4 Comparison Based On Meshing and Tension Stiffening .......................... 69

7.5 Comparison Based on Response to cyclic behavior ................................. 73

Chapter 8 Summary and Conclusion ................................................................................... 74

8.1 Summary ................................................................................................... 74

8.2 Conclusion ................................................................................................. 74

8.3 Recommendations for Further Work ........................................................ 76

Bibliography ............................................................................................................................ 77

VIII

LIST OF FIGURES

Figure 3-1Post yield material behaviour. ................................................................................. 11

Figure 3-2 Load-deflection diagram. ....................................................................................... 12

Figure 3-3 Beam and loading. .................................................................................................. 12

Figure 3-4 Moment curvature relation. .................................................................................... 13

Figure 3-5 Variation of neutral axis and steel tensile strain in a cracked beam. ..................... 14

Figure 3-6 Strain profiles as per cracked and uncracked sections. .......................................... 14

Figure 3-7 Variation of EI with moment. ................................................................................ 15

Figure 3-8 Variation of EI along the length of the beam shown in Figure 3-3 ........................ 15

Figure 3-9 Tension Stiffening .................................................................................................. 18

Figure 3-10 Uniaxial behavior of plain concrete. .................................................................... 20

Figure 3-11 Failure Surface ..................................................................................................... 21

Figure 3-12 Yield and failure surfaces in the (pq) plane. ...................................................... 22

Figure 3-13 Uniaxial tension behavior of concrete.................................................................. 23

Figure 3-14 Uniaxial compression behavior of concrete ......................................................... 24

Figure 3-15 Illustration of the effect of the compression stiffness recovery parameter wc. .... 26

Figure 3-16 Illustration of the definition of the cracking strain. .............................................. 27

Figure 3-17 Definition of the compressive inelastic (or crushing) strain. ............................... 28

Figure 3-18 Yield surface in plane stress. ................................................................................ 30

Figure 3-19 Yield surfaces in the deviatoric plane, corresponding to different values of Kc. . 31

Figure 4-1 Spring Model (Nilson 1968) .................................................................................. 34

Figure 4-2 Layered model by Bresler, 1968 ............................................................................ 35

Figure 4-3 Frictional Behavior in ABAQUS ........................................................................... 36

Figure 4-4 Translator Type of connector (ABAQUS,2010) .................................................... 37

Figure 5-1 Types of elements available in ABAQUS ............................................................. 38

Figure 5-2 Linear and Quadratic hexahedral element. ............................................................. 39

Figure 5-3 Linear and Quadratic tetrahedral element. ............................................................. 39

Figure 5-4 Description of element names. ............................................................................... 40

Figure 5-5 Dimension of beam. ............................................................................................... 41

Figure 5-6 Effect of mesh density on displacement field (linear hexahedral elements) .......... 41

Figure 5-7 Effect of mesh density on displacement field (quadratic hexahedral elements) .... 42

Figure 5-8 Effect of mesh density on displacement field (linear tetrahedral elements) .......... 42

Figure 5-9 Effect of mesh density on displacement field (Quadratic tetrahedral elements) ... 42

IX

Figure 6-1 Geometric modelling of reinforced beam components in ABAQUS .................... 46

Figure 6-2 Assembly of different part instances. ..................................................................... 47

Figure 6-3 Modelling of concrete in ABAQUS v6.9 ............................................................... 48

Figure 6-4 The Mesh Module Toolbox .................................................................................... 49

Figure 6-5 The Load module toolbox. ..................................................................................... 50

Figure 6-6 Job Manager Window ............................................................................................ 50

Figure 6-7 Experimental Setup (Bresler & Scordelis, 1963)(1 = 25.4 mm) .......................... 51

Figure 6-8:- Dimensioning of Beam-OA1 (Bresler & Scordelis, 1963) .................................. 51

Figure 6-9 Meshed view of the beam ...................................................................................... 52

Figure 6-10 Window for defining part. .................................................................................... 53

Figure 6-11 Stress Strain Curve for Concrete .......................................................................... 54

Figure 6-12 Compressive behaviour of concrete ..................................................................... 55

Figure 6-13 Tension Stiffening ................................................................................................ 56

Figure 6-14 Comparison of load deflection graph for various values of tension stiffening. ... 57

Figure 6-15 Comparison of beam OA1 results for different tension stiffening property. ....... 59

Figure 6-16 Comparison of results. ......................................................................................... 60

Figure 6-17 Specimen design and test results of Beres et al. (1992). ...................................... 62

Figure 6-18 Geometry of the Model in ABAQUS ................................................................... 62

Figure 6-19 Load History......................................................................................................... 64

Figure 6-20 Tensile Damage in concrete. ................................................................................ 65

Figure 7-1 Dimensioning of Beam-OA1 (Bresler & Scordelis, 1963) .................................... 66

Figure 7-2 Dimensioning of Beam J1 (Burns & Seiss, 1962) (all dimensions in mm) ......... 66

Figure 7-3 Computed and observed load displacement history for (Bresler & Scordelis, 1963)

beam without shear reinforcement. .......................................................................................... 67

Figure 7-4 Computed and observed load displacement history for (Burns & Seiss, 1962)

beam with shear reinforcement. ............................................................................................... 68

Figure 7-5 Crack visualization with the help of Concrete Damaged Plasticity model. ........... 69

Figure 7-6 Crack visualization with the help of Concrete Smeared Cracking model. ............ 69

Figure 7-7 Variation of LPF with tension stiffening parameter (Concrete Smeared Cracking).

.................................................................................................................................................. 71

Figure 7-8 Variation of LPF with tension stiffening parameter (Concrete Damaged Plasticity).

.................................................................................................................................................. 72

X

LIST OF TABLES

Table 5-1 Variation of displacement according to different mesh density .............................. 43

Table 5-2 Percent error in displacement as compared to analytical results. ............................ 43

Table 6-1 Compression Hardening Table ................................................................................ 56

Table 6-2 Tension Stiffening Table ........................................................................................ 57

Table 6-3 Concrete Damaged Plasticity model parameters (Tomasz et al 2005) .................... 58

Table 6-4 Compression hardening table. ................................................................................. 58

Table 6-5 Tension Stiffening Table for Concrete .................................................................... 59

Table 6-6 Material properties for reinforcing steel .................................................................. 60

Table 6-7 Concrete Damaged Plasticity model parameters ..................................................... 63

Table 6-8 Steel Plasticity Model Parameters ........................................................................... 63

Table 7-1 Effect of Tension stiffening and element size on analysis using Concrete Smeared

Cracking. .................................................................................................................................. 70

Table 7-2 Effect of Tension stiffening and element size on analysis using Concrete Damaged

Plasticity. .................................................................................................................................. 72

1

Chapter 1 Introduction

1.1 General

Dynamic Analysis is a subset of structural analysis and is the calculation of the

response of a building structure subjected to dynamic loads. It is part of the process of

structural design, earthquake engineering or structural assessment and retrofit in regions

where earthquakes or dynamic forces are prevalent.

Dynamic response of a structure can be caused by different loading conditions such as:

Earthquake ground motion,

Wind pressure,

Wave action,

Blast,

Machine vibration, and

Traffic movement.

Among these, inelastic response is mainly caused by earthquake motions and accidental

blasts. A building has the potential to wave back and forth during application of dynamic

lateral loads which generates additional loads on the structural elements; the dynamic

properties of the structure affect these loads generated. There are several methods available in

literature for the estimation of these loads. A few of these methods are described below,

1. Equivalent Static Analysis: This approach defines a series of forces acting on a building

to represent the effect of earthquake ground motion, typically defined by a seismic design

response spectrum.

2. Response Spectrum Analysis: This approach permits the multiple modes of response of

a building to be taken into account. The response of a structure can be defined as a

combination of many special shapes (modes).

3. Linear Dynamic Analysis: The seismic input is modelled using either modal spectral

analysis or time history analysis but in both cases, the corresponding internal forces and

displacements are determined using linear elastic analysis.

4. Non-linear Static Analysis: This approach is also known as "pushover" analysis. A

pattern of forces is applied to a structural model that includes non-linear properties (such

as steel yield), and the total force is plotted against a reference displacement to define a

capacity curve.

2

5. Non-linear Dynamic Analysis: Nonlinear dynamic analysis utilizes the combination of

ground motion records with a detailed structural model, therefore is capable of producing

results with relatively low uncertainty.

Out of all the above type of analysis Nonlinear dynamic analysis is the most rigorous,

and is required by some building codes for buildings of unusual configuration or of special

importance. However, such an analysis requires proper understanding of the materials

utilized. The calculated response can be very sensitive to the material model and solution

techniques used and also to the characteristics of the dynamic loads acting on it.

1.2 Need of the study

With the advancement of computer technologies and various kind of numerical

methods it is now possible to model complex material behaviour and loads for analysis of a

structure. But even with the advancement of these technologies there is a need to first

properly understand the actual behaviour of the structure and then to find a technique that

gives the best possible results. One of such technique is Finite Element Method that is now

being extensively used for analysis of reinforced concrete structures.

Within the framework of developing advanced design and analysis methods such as

Finite Element Method for modern structures the need for experimental research continues.

Experimental research supplies the basic information for finite element models, such as

material properties. In addition, the results of finite element models have to be evaluated by

comparing them with experiments of full-scale models of structural sub assemblages or, even,

entire structures. The development of reliable analytical models can, however, reduce the

number of required test specimens for the solution of a given problem, recognizing that tests

are time-consuming and costly and often do not simulate exactly the loading and support

conditions of the actual structure.

The development of analytical models of the response of RC structures is complicated by

the following factors:

Reinforced concrete is a composite material made up of concrete and steel, two materials

with very different physical and mechanical behavior;

Concrete exhibits nonlinear behavior even under low strain loading due to nonlinear

material behavior, environmental effects, cracking, biaxial stiffening and strain softening;

Reinforcing steel and concrete interact in a complex way through bond-slip and aggregate

interlock.

3

These complex phenomena have led engineers in the past to rely heavily on empirical

formulae for the design of concrete structures, which were derived from numerous

experiments. With the advent of digital computers and powerful methods of analysis, such as

the finite element method, many efforts to develop analytical solutions which would obviate

the need for experiments have been undertaken by investigators. With this method the

importance and interaction of different nonlinear effects on the response of RC structures can

be studied analytically. Thus the current investigation focuses on development of a finite

element model that is appropriate for investigating the behavior of reinforced concrete sub-

assemblies subjected to general loading and dynamic loading.

Also, laboratory investigation of reinforced concrete structure indicates that component

failure may result from inelastic material response of plain concrete or reinforcing steel.

Thus, model development includes investigating and characterizing the behavior of these

materials. A material model would be proposed from the available material models in

ABAQUS to represent the response of plain concrete that may develop multiple, discrete

cracks under tensile-type loading and also the loss of stiffness and strength associated with

moderate to severe compressive-type loading and the transition from tensile to compressive

response mechanism under load reversal. For reinforcing steel, the material model would

represent tensile and compressive yielding as well as the curvilinear nature of steel response

under reversed cyclic loading.

Ideally these models should be based on an accurate representation of material

behaviour taking into account the controlling states of stress or strain and identifying the

main parameters which influence the hysteretic behavior of each critical region in order to

predict the behavior up to failure of any structural component. At the same time these models

should be computationally efficient, so that the dynamic response of multi-storey structures

under dynamic excitations can be determined within reasonable time.

There is also a need of comparative study of different material models, thus structural

subassemblies would be analysed using these different properties.

1.3 Objective of the study

The present investigation of the nonlinear response to failure of RC structures under

short term monotonic loads was initiated with the intent to investigate the relative importance

of several factors in the nonlinear finite element analysis of RC structures: these include the

effect of tension-stiffening and bond-slip and their relative importance on the response of

beams, the effect of size of the finite element mesh on the analytical results and the effect of

the nonlinear behavior of concrete and steel on the response of beams.

4

The main objective of the study can be listed as below

Modelling structural elements and dynamic condition in ABAQUS v6.9 for performing

the non-linear analysis.

Studying different material models of concrete and steel available in ABAQUS v.6.9.

Studying the effect of size of finite element mesh on analysis results.

Studying the effect of different material model parameters on analysis results.

Conducting a comparative study of the material models of concrete available in

ABAQUS v6.9 and proposing the best suited model of concrete for the purpose of

dynamic analysis.

1.4 Scope of the study

In this thesis first a detailed study of the two concrete material models that are available

in ABAQUS v6.9 viz. Concrete Smeared Cracking and Concrete Damaged Plasticity has

been done. Few of the techniques that are available in literature for the purpose of modelling

Bondslip behaviour are then presented along with the methods of applying those techniques

in ABAQUS v6.9. Effect of mesh size on the analysis results of a 2 m length cantilever beam

by taking elastic properties and subjected to point load is then presented, and finally a

comparative study of Concrete Smeared Cracking and Concrete Damaged Plasticity model is

documented. Only the inelastic effect of material is considered and the effect of strain rate

and other field variables is neglected.

1.5 Organization of report

The report has been divided into several chapters as follows

Chapter 1. Introduction

Chapter 2. Literature Review

Chapter 3. Material Modelling

Chapter 4. BondSlip Model

Chapter 5. Meshing

Chapter 6. Finite Element Model Development

Chapter 7. Comparison of Concrete Material Models

Chapter 8. Summary and Conclusion

5

Chapter 2 Literature Review

2.1 General

Dynamic loading requires an understanding of the structural behavior under large

inelastic deformations. Behavior under this loading is fundamentally different from wind or

gravity loading, requiring much more detailed analysis to assure acceptable dynamic

performance beyond the elastic range. Some structural damage can be expected when the

building experiences design ground motions because almost all building codes allow inelastic

energy dissipation in structural systems. Hence, for the proper simulation of structure it is

essential to properly understand the behavior of materials under dynamic loading condition.

A considerable amount of work has been accomplished on material properties in the inelastic

regime.

A brief review of previous studies on the behavior of materials under monotonic and

cyclic loading and the application of the finite element method to the analysis of reinforced

concrete structures is presented is this section.

2.2 Experimental Studies

Bresler & Scordelis [1963] performed tests on a series of beams to study their general

behavior, cracking loads, and strength.The tests were designed to provide data regarding the

shear strength of beams having narmal to low percentage of web reinforcement and normal to

high shear span ratios. The results were provided in the form of load deflection plots for

various beams under the same loading condition but with variating amount of reinforcement.

Owing to the high quality of testing and results, this experiment has been taken as the basis

for validation of material models.

Burns and Seiss [1962] conducted out experiments on lightly reinforced concrete

beam. These beams are very lightly reinforced with a longitudinal reinforcement ratio of

0.007 and no transverse reinforcement. The ratio of the flexural length to the depth of the

longitudinal reinforcement is approximately 4. Thus behavior of the element is controlled

entirely by flexure and the lack of transverse reinforcement does not adversely affect

behavior. The results were provided in the form of load deflection plots for various sections

of beams.

Beres, Attila Bella [1994] conducted out experiments on reinforced concrete frames

with nonductile details. The experimental study was restricted to plane frames without slab.

Various kinds of beam column junction were studied and cyclic loading was applied to them

6

to study their behaviour under dynamic condition. The results were provided in form of

diagrams of cracking patterns seen in actual experiment and hysteresis graphs.

2.3 Literature Survey

The earliest publication on the application of the finite element method to the analysis

of RC structures was presented by Ngo and Scordelis (1967). In their study, simple beams

were analysed with a model in which concrete and reinforcing steel were represented by

constant strain triangular elements, and a special bond link element was used to connect the

steel to the concrete and describe the bond-slip effect. A linear elastic analysis was performed

on beams with predefined crack patterns to determine principal stresses in concrete, stresses

in steel reinforcement and bond stresses. Since the publication of this pioneering work, the

analysis of reinforced concrete structures has enjoyed a growing interest and many

publications have appeared. Scordelis et al. (1974) used the same approach to study the effect

of shear in beams with diagonal tension cracks and accounted for the effect of stirrups, dowel

shear, aggregate interlock and horizontal splitting along the reinforcing bars near the support.

Nilson (1972) introduced nonlinear material properties for concrete and steel and a

nonlinear bond-slip relationship into the analysis and used an incremental load method of

nonlinear analysis. Four constant strain triangular elements were combined to form a

quadrilateral element by condensing out the central node. Cracking was accounted for by

stopping the solution when an element reached the tensile strength, and reloading

incrementally after redefining a new cracked structure. The method was applied to concentric

and eccentric reinforced concrete tensile members which were subjected to loads applied at

the end of the reinforcing bars and the results were compared with experimental data.

Franklin (1970) advanced the capabilities of the analytical method by developing a

nonlinear analysis which automatically accounted for cracking within finite elements and the

redistribution of stresses in the structure. This made it possible to trace the response of two

dimensional systems from initial loading to failure in one continuous analysis. Incremental

loading with iterations within each increment was used to account for cracking in the finite

elements and for the nonlinear material behavior. Franklin used special frame-type elements,

quadrilateral plane stress elements, axial bar members, two-dimensional bond links and tie

links to study reinforced concrete frames and RC frames coupled with shear walls.

Plane stress elements were used by numerous investigators to study the behavior of

reinforced concrete frame and wall systems. Nayak and Zienkiewicz (1972) conducted two

dimensional stress studies which include the tensile cracking and the elasto-plastic behavior

of concrete in compression using an initial stress approach. Cervenka (1970) analysed shear

7

walls and spandrel beams using an initial stress approach in which the elastic stiffness matrix

at the beginning of the entire analysis is used in all iterations. Cervenka proposed a

constitutive relationship for the composite concrete-steel material through the uncracked,

cracked and plastic stages of behavior.

For the analysis of RC beams with material and geometric nonlinearities Rajagopal

(1976) developed a layered rectangular plate element with axial and bending stiffness in

which concrete was treated as an orthotropic material. RC beam and slab problems have also

been treated by many other investigators (Lin and Scordelis 1975; Bashur and Darwin 1978;

Rots et al. 1985; Barzegar and Schnobrich 1986; Adeghe and Collins 1986; Bergmann and

Pantazopoulou 1988; Cervenka et al. 1990; Kwak 1990) using similar methods.

Selna (1969) analysed beams and frames made up of one-dimensional elements with

layered cross sections which accounted for progressive cracking and changing material

properties through the depth of the cross section as a function of load and time. Significant

advances and extensions of the finite element analysis of reinforced concrete beams and

frames to include the effects of heat transfer due to fire, as well as the time-dependent effects

of creep and shrinkage, were made by Becker and Bresler (1974).

The finite element analysis of an axisymmetric solid under axisymmetric loading can

be readily reduced to a two-dimensional analysis. Bresler and Bertero (1968) used an

axisymmetric model to study the stress distribution in a cylindrical concrete specimen

reinforced with a single plain reinforcing bar. The specimen was loaded by applying tensile

loads at the ends of the bar.

In one of the pioneering early studies Rashid (1968) introduced the concept of a

"smeared" crack in the study of the axisymmetric response of prestressed concrete reactor

structures. Rashid took into account cracking and the effects of temperature, creep and load

history in his analyses. Today the smeared crack approach of modelling the cracking behavior

of concrete is almost exclusively used by investigators in the nonlinear analysis of RC

structures, since its implementation in a finite element analysis program is more

straightforward than that of the discrete crack model. Computer time considerations also

favour the smeared crack model in analyses which are concerned with the global response of

structures. At the same time the concerted effort of many investigators in the last 20 years has

removed many of the limitations of the smeared crack model (ASCE 1982; Meyer and

Okamura, eds. 1985).

8

Gilbert and Warner (1978) used the smeared crack model and investigated the effect

of the slope of the descending branch of the concrete stress-strain relation on the behavior of

RC slabs. They were among the first to point out that analytical results of the response of

reinforced concrete structures are greatly influenced by the size of the finite element mesh

and by the amount of tension stiffening of concrete. Several studies followed which

corroborated these findings and showed the effect of mesh size (Bazant and Cedolin 1980;

Bazant and Oh 1983; Kwak 1990) and tension stiffening (Barzegar and Schnobrich 1986;

Leibengood et al. 1986) on the accuracy of finite element analyses of RC structures with the

smeared crack model. In order to better account for the tension stiffening effect of concrete

between cracks some investigators have artificially increased the stiffness of reinforcing steel

by modifying its stress-strain relationship (Gilbert and Warner 1977). Others have chosen to

modify the tensile stress-strain curve of concrete by including a descending post-peak branch

(Lin and Scordelis 1975; Vebo and Ghali 1977; Barzegar and Schnobrich 1986; Abdel

Rahman and Hinton 1986).

In the context of the smeared crack model two different representations have

emerged: the fixed crack and the rotating crack model. In the fixed crack model a crack forms

perpendicular to the principal tensile stress direction when the principal stress exceeds the

concrete tensile strength and the crack orientation does not change during subsequent

loading. The ease of formulating and implementing this model has led to its wide-spread used

in early studies (Hand et al. 1973; Lin and Scordelis 1975). Subsequent studies, however,

showed that the model is associated with numerical problems caused by the singularity of the

material stiffness matrix. Moreover, the crack pattern predicted by the finite element analysis

often shows considerable deviations from that observed in experiments (Jain and Kennedy

1974).

The problems of the fixed crack model can be overcome by introducing a cracked

shear modulus, which eliminates most numerical difficulties of the model and considerably

improves the accuracy of the crack pattern predictions. The results do not seem to be very

sensitive to the value of the cracked shear modulus (Vebo and Ghali 1977; Barzegar and

Schnobrich 1986), as long as a value which is greater than zero is used, so as to eliminate the

singularity of the material stiffness matrix and the associated numerical instability. Some

recent models use a variable cracked shear modulus to represent the change in shear stiffness,

as the principal stresses in the concrete vary from tension to compression (Balakrishnan and

Murray 1988; Cervenka et al. 1990).

9

de Borst and Nauta (1985) have proposed a model in which the total strain rate is

additively decomposed into a concrete strain rate and a crack strain rate. The latter is, in turn,

made up of several crack strain components. After formulating the two-dimensional concrete

stress-strain relation and transforming from the crack direction to the global coordinate

system of the structure, a material matrix with no coupling between normal and shear stress is

constructed. In spite of its relative simplicity and ease of application, this approach still

requires the selection of a cracked shear modulus of concrete.

While the response of lightly reinforced beams in bending is very sensitive to the

effect of tension stiffening of concrete, the response of RC structures in which shear plays an

important role, such as over-reinforced beams and shear walls, is much more affected by the

bond-slip of reinforcing steel than the tension stiffening of concrete. To account for the

bondslip of reinforcing steel two different approaches are common in the finite element

analysis of RC structures. The first approach makes use of the bond link element proposed by

Ngo and Scordelis (1967). This element connects a node of a concrete finite element with a

node of an adjacent steel element. The link element has no physical dimensions, i.e. the two

connected nodes have the same coordinates.

The second approach makes use of the bond-zone element developed by de Groot et

al. (1981). In this element the behavior of the contact surface between steel and concrete and

of the concrete in the immediate vicinity of the reinforcing bar is described by a material law

which considers the special properties of the bond zone. The contact element provides a

continuous connection between reinforcing steel and concrete, if a linear or higher order

displacement field is used in the discretization scheme. A simpler but similar element was

proposed by Keuser and Mehlhorn (1987), who showed that the bond link element cannot

represent adequately the stiffness of the steel-concrete interface.

Even though many studies of the bond stress-slip relationship between reinforcing

steel and concrete have been conducted, considerable uncertainty about this complex

phenomenon still exists, because of the many parameters which are involved. As a result,

most finite element studies of RC structures do not account for bond-slip of reinforcing steel

and many researchers express the opinion that this effect is included in the tension-stiffening

model.

Suidan and Schnobrich (1973) were the first to study the behavior of beams with 20-

node three-dimensional isoperimetric finite elements. The behavior of concrete in

compression was assumed elasto-plastic based on the von-Mises yield criterion. A coarse

finite element mesh was used in these analyses for cost reasons.

10

2.4 Summary

In spite of the large number of previous studies on the nonlinear finite element

analysis of reinforced concrete structures, only few conclusions of general applicability have

been arrived at. The inclusion of the effects of tension stiffening and bond-slip is a case in

point. Since few rational models of this difficult problem have been proposed so far, it is

rather impossible to assess exactly what aspects of the behavior are included in each study

and what the relative contribution of each is. Similar conclusions can be reached with regard

to other aspects of the finite element analysis. Even though the varying level of sophistication

of proposed models is often motivated by computational cost considerations, the multitude of

proposed approaches can lead to the conclusion that the skill and experience of the analyst is

the most important aspect of the study and that the selection of the appropriate model

depends on the problem to be solved.

11

Chapter 3 Material Modelling

3.1 General

The response of a reinforced concrete structure is determined in part by the material

response of the plain concrete and steel of which it is composed. Thus, analysis and

prediction of structural response of a reinforced concrete structure to static or dynamic

loading requires prediction of concrete response to variable load histories. The fundamental

characteristics of concrete behavior are established through experimental testing of plain

concrete specimens subjected to specific, relatively simple load histories. Continuum

mechanics provides a framework for developing an analytical model that describes these

fundamental characteristics. Experimental data provide additional information for refinement

and calibration of the analytical model.

The following sections present the concrete and steel material models used in this

investigation for finite element analysis of reinforced concrete structural elements.

3.2 Material Behaviour Post Yield

Different types of material behavior are (Figure 3-1),

Strain Hardening is a phenomenon whereby yield stress increases with further plastic

straining.

Elastic Plastic is a phenomenon whereby yield stress remains constant with further

plastic straining.

Strain Softening is a phenomenon whereby yield stress decreases with further plastic

straining.

Brittle is a phenomenon whereby the material fails as soon it reaches its elastic limit.

Figure 3-1Post yield material behaviour.

12

Plain concrete belongs to a class of materials that can be called brittle, indicating that it

fails as soon as it reaches its elastic limit. But in RCC because of aggregate interlock and

presence of reinforcement the member is able to take more loads and thus shows strain

softening behavior. The strain softening kind of behaviour shown by RCC structural elements

is based on a phenomenon known as Tension Stiffening Effect as described in section 3.3.

3.3 Tension Stiffening Effect

In this section tension stiffening effect in reinforced concrete structures is defined,

LoadDeflection Behavior of a Concrete Beam

Figure 3-2 traces the loaddeflection history of the fixed-ended, reinforced concrete

beam shown in Figure 3-3. Initially, the beam is uncracked and is stiff (OA). With further

load, flexural cracking occurs when the moment at the ends exceeds the cracking moment.

Figure 3-2 Load-deflection diagram.

Figure 3-3 Beam and loading.

When a section cracks, its moment of inertia decreases, leading to a decrease in the

stiffness of the beam. This causes a reduction in stiffness (AB) in the loaddeflection

diagram in Figure 3-2. Flexural cracking in the midspan region causes a further reduction of

stiffness (point B). Eventually, the reinforcement would yield at the ends or at midspan, an

13

effect leading to large increases in deflection with little change in load (points D and E). The

service-load level is represented by point C. The beam is essentially elastic at point C, the

nonlinear load deflection being caused by a progressive reduction of flexural stiffness due to

increased cracking as the loads are increased.

Flexural Stiffness and Moment of Inertia

The deflection of a beam is calculated by integrating the curvatures along the length

of the beam. For an elastic beam, the curvature, , is calculated as,

where,

EI = flexural stiffness of the cross section.

M = Moment acting at cross section.

If EI is constant, this is a relatively routine process. For reinforced concrete, however,

three different EI values must be considered. These can be illustrated by the moment

curvature diagram for a length of beam, including several cracks, shown in Figure 3-4. The

slope of any radial line through the origin in such a diagram is M/ = EI.

Figure 3-4 Moment curvature relation.

Before cracking, the entire cross section shown in Figure 3-6 (a) is stressed by loads. The

moment of inertia of this section is called the uncracked moment of inertia, and the

corresponding EI can be represented by the radial line OA in Figure 3-4. The gross moment

of inertia for the concrete section, Ig, or moment of inertia based on uncracked transformed

section IT is used for this region of behavior.

14

Figure 3-5 Variation of neutral axis and steel tensile strain in a cracked beam.

(a) Uncracked (b) Cracked (c) Strain Profiles Section Section

Figure 3-6 Strain profiles as per cracked and uncracked sections.

The cracked-section EI is less than the uncracked EI and corresponds relatively well

to the curvatures at loads approaching yield, as shown by the radial line OB in Figure 3-4.

At service loads i.e. when the moment curvature graph is in between A and B, the average EI

values for this beam segment that includes both cracked and uncracked sections are between

these two extremes. The actual EI at service load levels varies considerably, as shown by the

difference in the slope of the lines and depending on the relative magnitudes of the cracking

moment the service load moment and the yield moment. The variation in EI with moment is

shown in Figure 3-7, obtained from Figure 3-4.

15

Figure 3-7 Variation of EI with moment.

Evidently, EIT represents the true flexural rigidity for M < Mcr, and EIeff represents the

true flexural rigidity for M > Mcr. Whereas EIT is constant and a property of the beam section,

EIeff depends on the load level (applied moment). It follows that

EIT > EIeff > EIcr

Thus, determining the flexural rigidity on the basis of the uncracked section results in

an underestimation of the actual deflection of a reinforced concrete beam under service

loads; whereas doing so on the basis of the fully cracked section results in an overestimation

of the actual deflection.

The increase in stiffness over the cracked section stiffness, on account of the ability

of concrete (in between cracks) to resist tension, is referred to as the Tension Stiffening

Effect.

Figure 3-8 Variation of EI along the length of the beam shown in Figure 3-3

EIT

(EIcr)

16

Figure 3-8 shows the distribution of EI along the beam shown in Figure 3-3. The EI

varies from the uncracked value at points where the moment is less than the cracking moment

to a partially cracked value at points of high moment. Because the use of such a distribution

of EI values would make the deflection calculations tedious, an overall average or effective

EI value is used which obviously is greater that the cracked EI of beam. The effective

moment of inertia must account for both the tension stiffening and the variation of EI along

the member.

3.4 Material Model for Concrete

The material models used for defining behavior of concrete are

Linear elasticity: - used for defining the elastic behavior of concrete.

Concrete Smeared Cracking: - used for defining the plastic behavior of concrete.

Concrete Damaged Plasticity: - used for defining the plastic behavior of concrete.

A comparative study was carried out to determine the best suited model for defining plasticity

in concrete. All of these models are explained in detail.

3.4.1 Linear Elasticity

A linear elastic material model:

Is valid for small elastic strains (normally less than 5%) and suitable for first order

approximation;

Can take care of isotropic, orthotropic, or fully anisotropic properties of concrete;

Has properties that depend on temperature and/or other field variables

The total stress is defined from the total elastic strain as

elelD , (3-2)

where

= total stress,

Del = Fourth-order elasticity tensor, and

el = total elastic strain.

The simplest form of linear elasticity is the isotropic case, and the stress-strain

relationship is given by

23

13

12

33

22

11

23

13

12

33

22

11

/100000

0/10000

00/1000

000/1//

000//1/

000///1

G

G

G

EEE

EEE

EEE

(3-3)

17

The elastic properties are completely defined by giving the Youngs modulus, E, and

the Poissons ratio, . The shear modulus, G, can be expressed in terms of E and as

G=E/2(1+ ). These parameters can be given as functions of temperature and other

predefined fields, if necessary.

3.4.2 Concrete Smeared Cracking

The smeared crack concrete model in Abaqus/Standard:

provides a general capability for modelling concrete in all types of structures, including

beams, trusses, shells, and solids;

can be used for plain concrete, even though it is intended primarily for the analysis of

reinforced concrete structures;

can be used with rebar to model concrete reinforcement;

is designed for applications in which the concrete is subjected to essentially monotonic

straining at low confining pressures;

consists of an isotropically hardening yield surface that is active when the stress is

dominantly compressive and an independent crack detection surface that

determines if a point fails by cracking;

uses oriented damaged elasticity concepts (smeared cracking) to describe the reversible

part of the materials response after cracking failure;

requires that the linear elastic material model be used to define elastic properties.

Smeared cracking

The concrete model does not track individual macro cracks. Constitutive

calculations are performed independently at each integration point of the finite element

model. The presence of cracks enters into these calculations by the way in which the cracks

affect the stress and material stiffness associated with the integration point.

Defining Tension Stiffening

The post failure behavior for direct straining across cracks is modelled with tension

stiffening, which allows to define the strain-softening behavior for cracked concrete. This

behavior allows for the effects of reinforcement interaction with concrete to be simulated in a

simple manner. Tension stiffening is required in the Concrete Smeared Cracking model.

Tension stiffening can be specified by means of a post failure stress-strain relation (Figure

3-9).

18

Figure 3-9 Tension Stiffening

Post failure stress-strain relation

Specification of strain softening behavior in reinforced concrete generally means

specifying the post failure stress as a function of strain across the crack (Figure 3-9). In cases

with little or no reinforcement this specification often introduces mesh sensitivity in the

analysis results in the sense that the finite element predictions do not converge to a unique

solution as the mesh is refined because mesh refinement leads to narrower crack bands. This

problem typically occurs if only a few discrete cracks form in the structure, and mesh

refinement does not result in formation of additional cracks. If cracks are evenly distributed

(either due to the effect of rebar or due to the presence of stabilizing elastic material, as in the

case of plate bending), mesh sensitivity is of little concern.

In practical calculations for reinforced concrete, the mesh is usually such, that the

element contains rebar. The interaction between the rebar and concrete tends to reduce mesh

sensitivity, provided that a reasonable amount of tension stiffening is introduced in the

concrete model to simulate this interaction.

The tension stiffening effect must be estimated; it depends on such factors as

the density of reinforcement, the quality of the bond between the rebar and the concrete,

the relative size of the concrete aggregate compared to the rebar diameter, and the mesh.

A reasonable starting point for relatively heavily reinforced concrete modelled with a fairly

detailed mesh is to assume that the strain softening after failure reduces the stress linearly to

Failure point

Tension stiffening curve

Strain,

Stress,

19

zero at a total strain of about 10 times the strain at failure. The strain at failure in standard

concretes is typically 10-4

, which suggests that tension stiffening that reduces the stress to

zero at a total strain of about 10-3

is reasonable. This parameter should be calibrated to a

particular case.

The choice of tension stiffening parameters is important in Abaqus/Standard since,

generally, more tension stiffening makes it easier to obtain numerical solutions. Too little

tension stiffening will cause the local cracking failure in the concrete to introduce temporarily

unstable behavior in the overall response of the model.

Defining Compressive Behavior

When the principal stress components are dominantly compressive, the response of

the concrete is modelled by an elastic-plastic theory using a simple form of yield surface

written in terms of the equivalent pressure stress, p, and the Mises equivalent deviatoric

stress, q as shown in Figure 3-12. Associated flow and isotropic hardening are used. This

model significantly simplifies the actual behavior. The associated flow assumption generally

over-predicts the inelastic volume strain. When the concrete is strained beyond the ultimate

stress point, the assumption that the elastic response is not affected by the inelastic

deformation is not realistic. In addition, when concrete is subjected to very high pressure, it

exhibits inelastic response: no attempt has been made to build this behavior into the model

(this also leads to the error:-Convergence judged unlikely.).

The simplifications associated with compressive behavior are introduced for the

sake of computational efficiency. In particular, while the assumption of associated flow is

not justified by experimental data, it can provide results that are acceptably close to

measurements, provided that the range of pressure stress in the problem is not large. From a

computational viewpoint, the associated flow assumption leads to enough symmetry in the

Jacobian matrix of the integrated constitutive model (the material stiffness matrix) such

that the overall equilibrium equation solution usually does not require unsymmetrical

equation solution. All of these limitations could be removed at some sacrifice in

computational cost.

Stress-strain behavior of plain concrete can be defined in uniaxial compression

outside the elastic range (Figure 3-10). Compressive stress data are provided as a tabular

function of plastic strain and, if desired, temperature and field variables. Positive (absolute)

values should be given for the compressive stress and strain. The stress-strain curve can be

defined beyond the ultimate stress, into the strain-softening regime.

20

Uniaxial and multiaxial behavior

The cracking and compressive responses of concrete that are incorporated in the

concrete model are illustrated by the uniaxial response of a specimen shown in Figure 3-10.

When concrete is loaded in compression, it initially exhibits elastic response. As the stress is

increased, some non-recoverable (inelastic) straining occurs and the response of the material

softens. An ultimate stress is reached, after which the material loses strength until it can no

longer carry any stress.

If the load is removed at some point after inelastic straining has occurred, the

unloading response is softer than the initial elastic response: the elasticity has been damaged.

This effect is ignored in the model, since it is assumed that the applications involve primarily

monotonic straining, with only occasional, minor unloading.

Figure 3-10 Uniaxial behavior of plain concrete.

When a uniaxial concrete specimen is loaded in tension, it responds elastically until, at a

stress that is typically 7%10% of the ultimate compressive stress, cracks form. Cracks form

so quickly that, even in the stiffest testing machines available, it is very difficult to observe

the actual behavior.

The model assumes that cracking causes damage, and open cracks can be represented by a

loss of elastic stiffness. It is also assumed that there is no permanent strain associated with

21

cracking. This will allow cracks to close completely if the stress across them becomes

compressive.

In multiaxial stress states these observations are generalized through the concept of surfaces

of failure and flow in stress space. These surfaces are fitted to experimental data.

Failure surface

Failure ratios can be specified to define the shape of the failure surface (Figure 3-11).

Four failure ratios can be specified:

The ratio of the ultimate biaxial compressive stress to the ultimate uniaxial compressive

stress.

The absolute value of the ratio of the uniaxial tensile stress at failure to the ultimate

uniaxial compressive stress.

Figure 3-11 Failure Surface

The ratio of the magnitude of a principal component of plastic strain at ultimate stress in

biaxial compression to the plastic strain at ultimate stress in uniaxial compression.

The ratio of the tensile principal stress at cracking, in plane stress, when the other

principal stress is at the ultimate compressive value, to the tensile cracking stress under

uniaxial tension.

22

Reinforcement

Reinforcement in concrete structures is typically provided by means of rebar, which

are one-dimensional strain theory elements (rods) that can be defined singly or embedded in

oriented surfaces. With this modelling approach, the concrete behavior is considered

independent of the rebar. Effects associated with the rebarconcrete interface, such as bond

slip and dowel action are modelled approximately by introducing some tension stiffening

(Figure 3-9) into the concrete modelling to simulate load transfer across cracks through the

rebar. Defining the rebar is important since it may cause an analysis to fail due to lack of

reinforcement in key regions of a model.

Crack detection

Cracking is assumed to be the most important aspect of the behavior and

representation of cracking and of post cracking behavior dominates the modelling. Cracking

is assumed to occur when the stress reaches a failure surface that is called the crack

detection surface. This failure surface is a linear relationship between the equivalent

pressure stress, p, and the Misses equivalent deviatoric stress, q as shown in Figure 3-12.

Figure 3-12 Yield and failure surfaces in the (pq) plane.

Subsequent cracking at the same point is restricted to being orthogonal to this

direction, since stress components associated with an open crack are not included in the

definition of the failure surface used for detecting the additional cracks. Cracks are

irrecoverable: they remain for the rest of the calculation (but may open and close). No more

than three cracks can occur at any point (two in a plane stress case, one in a uniaxial stress

case). Following crack detection, the crack affects the calculations because a damaged

elasticity model is used.

23

Response to strain reversals

Because the model is intended for application to problems involving relatively

monotonic straining, no attempt is made to include prediction of cyclic response or of the

reduction in the elastic stiffness caused by inelastic straining under predominantly

compressive stress.

3.4.3 Concrete Damaged Plasticity Model

The Concrete Damaged Plasticity model in Abaqus:

Provides a general capability for modelling concrete in all types of structures;

Uses concepts of isotropic damaged elasticity in combination with isotropic tensile and

compressive plasticity to represent the inelastic behavior of concrete;

Can be used for plain concrete, even though it is intended primarily for the analysis of

reinforced concrete structures;

Is designed for applications in which concrete is subjected to monotonic, cyclic, and/or

dynamic loading under low confining pressures;

Allows user control of stiffness recovery effects during cyclic load reversals;

Can be defined to be sensitive to the rate of straining;

Requires that the elastic behavior of the material be isotropic and linear

Uniaxial Behavior

The model assumes that the uniaxial tensile and compressive response of concrete is

characterized by damaged plasticity, as shown in Figure 3-13 and Figure 3-14.

Figure 3-13 Uniaxial tension behavior of concrete.

24

Figure 3-14 Uniaxial compression behavior of concrete

Under uniaxial tension the stress-strain response follows a linear elastic relationship

until the value of the failure stress, 0t is reached. The failure stress corresponds to the onset

of micro-cracking in the concrete material. Beyond the failure stress the formation of micro-

cracks is represented macroscopically with a softening stress-strain response, which induces

strain localization in the concrete structure. Under uniaxial compression the response is linear

until the value of initial yield, 0c . In the plastic regime the response is typically

characterized by stress hardening followed by strain softening beyond the ultimate stress, cu .

When the concrete specimen is unloaded from any point on the strain softening

branch of the stress-strain curves, the unloading response is weakened: the elastic stiffness of

the material appears to be damaged (or degraded). The degradation of the elastic stiffness is

characterized by two damage variables, dt and dc , which are assumed to be functions of the

plastic strains, temperature, and field variables: The damage variables can take values from

zero, representing the undamaged material, to one, which represents total loss of strength.

In Abaqus the damage variables are treated as non-decreasing material point

quantities. At any increment during the analysis, the new value of each damage variable is

obtained as the maximum between the value at the end of the previous increment and the

value corresponding to the current state.

The stress-strain relations under uniaxial tension and compression loading are, respectively:

),~()1(

),~()1(

0

0

pl

cccc

pl

tttt

Ed

Ed

(3-4)

25

Here,

E0 = Initial (undamaged) elastic stiffness of the material,

dt = Tensile damage parameter,

dc = Compressive damage parameter,

pl

t~ = Tensile equivalent plastic strain,

pl

c~ = Compressive equivalent plastic strain,

c = Uniaxial compressive strain,

t = Uniaxial tensile strain,

c = Stress corresponding to strain c

t = Stress corresponding to strain t

The effective tensile and compressive cohesion stresses are defined as

),~()1(

),~()1(

0

0

pl

cc

c

c

c

pl

tt

t

t

t

Ed

Ed

(3-5)

Uniaxial cyclic behavior

Under uniaxial cyclic loading conditions the degradation mechanisms are quite

complex, involving the opening and closing of previously formed micro-cracks, as well as

their interaction. Experimentally, it is observed that there is some recovery of the elastic

stiffness as the load changes sign during a uniaxial cyclic test. The stiffness recovery effect,

also known as the unilateral effect, is an important aspect of the concrete behavior under

cyclic loading. The effect is usually more pronounced as the load changes from tension to

compression, causing tensile cracks to close, which results in the recovery of the compressive

stiffness. The Concrete Damaged Plasticity model assumes that the reduction of the elastic

modulus is given in terms of a scalar degradation variable d as

E = (1 d) E0 (3-6)

where,

E = Elastic modulus corresponding to damaged material.

E0 = Initial (undamaged) modulus of the material.

This expression holds both in the tensile (11 > 0) and the compressive (11 < 0) sides

of the cycle. The stiffness degradation variable, d, is a function of the stress state and the

uniaxial damage variables, dt and dc. For the uniaxial cyclic conditions Abaqus assumes that

26

(1 d) = (1 stdc) (1 scdt), (3-7)

st = 1 wt r*(11); 0 wt 1,

sc = 1 wc (1 r*(11)); 0 wc 1, (3-8)

{

(3-9)

where,

st and sc are functions of the stress state that are introduced to model stiffness recovery effects

associated with stress reversals.

Figure 3-15 Illustration of the effect of the compression stiffness recovery parameter wc.

The weight factors wt and wc, which are assumed to be material properties, control the

recovery of the tensile and compressive stiffness upon load reversal. To illustrate this,

consider the example in Figure 3-15, where the load changes from tension to compression.

Assume that there was no previous compressive damage (crushing) in the material; that is,

pl

c~ = 0and dc = 0. Then

(1 d) = (1 scdt) = (1 (1 wc (1 r*(11)) dt), (3-10)

In tension (11 > 0), r* = 1; therefore, d = dt as expected.

In compression (11 < 0), r* = 0, and d = (1 wc)dt. If wc = 1, then d = 0; therefore, the

material fully recovers the compressive stiffness (which in this case is the initial undamaged

stiffness, E = E0). If, on the other hand, wc = 0, then d = dt and there is no stiffness recovery.

Intermediate values of wc result in partial recovery of the stiffness.

27

Defining Tension Stiffening

The post-failure behavior for direct straining is modelled with tension stiffening

(section 3.3), which allows defining the strain-softening behavior for cracked concrete. This

behavior also allows for the effects of the reinforcement interaction with concrete to be

simulated in a simple manner. Tension stiffening is required in the Concrete Damaged

Plasticity model. Tension stiffening can be specified by means of a post-failure stress-strain

relation or by applying a fracture energy cracking criterion. In reinforced concrete the

specification for post-failure behavior generally means giving the post-failure stress as a

function of cracking strain, ckt

~ .

Figure 3-16 Illustration of the definition of the cracking strain.

el

tt

ck

t 0~ , (3-11)

Here,

00 / Etel

t .

t = Strain at any point above t0

t = Stress corresponding to strain t

E0 = Initial (undamaged) elastic stiffness of the material.

The cracking strain is defined as the total strain minus the elastic strain corresponding

to the undamaged material. Illustration of the definition of the cracking strain used for the

definition of tension stiffening data is shown in Figure 3-16.To avoid potential numerical

28

problems, Abaqus enforces a lower limit on the post-failure stress equal to one hundredth of

the initial failure stress: 100/0tt .

The choice of tension stiffening parameters is important since, generally, more

tension stiffening makes it easier to obtain numerical solutions. Too little tension stiffening

will cause the local cracking failure in the concrete to introduce temporarily unstable

behavior in the overall response of the model.

Defining Compressive Behavior

Stress-strain behavior of plain concrete can be defined in uniaxial compression

(Figure 3-17) outside the elastic range. Compressive stress data are provided as a tabular

function of inelastic (or crushing) strain, inc~ , and, if desired, strain rate, temperature, and

field variables. Positive (absolute) values should be given for the compressive stress and

strain.

Figure 3-17 Definition of the compressive inelastic (or crushing) strain.

The stress-strain curve can be defined beyond the ultimate stress, into the strain-

softening regime. Hardening data are given in terms of an inelastic strain, inc~ , instead of

plastic strain, plc~ . The compressive inelastic strain is defined as the difference between the

total strain and the elastic strain corresponding to the undamaged material,

29

el

cc

in

c 0~ , (3-12)

where

00 / Ecel

c .

c = Strain at any point above c0

c = Stress corresponding to strain c

E0 = Initial (undamaged) elastic stiffness of the material.

The definition of the compressive inelastic (or crushing) strain used for the definition

of compression hardening data is shown in Figure 3-17.

Plastic flow

The Concrete Damaged Plasticity model assumes non-associated potential plastic

flow. The flow potential G used for this model is the Drucker-Prager hyperbolic function:

tan)tan( 220 pqG t , (3-13)

Where,

),( if is the dilation angle measured in the pq plane at high

confining pressure;

otit pltf

~0 ),( is the uniaxial tensile stress at failure, taken from the user-

specified tension stiffening data; and

),( if is a parameter, referred to as the eccentricity, that defines the

rate at which the function approaches the asymptote (the flow

potential tends to a straight line as the eccentricity tends to

zero).

This flow potential, which is continuous and smooth, ensures that the flow direction is

always uniquely defined. The function approaches the linear Drucker-Prager flow potential

asymptotically at high confining pressure stress and intersects the hydrostatic pressure axis at

90. The default flow potential eccentricity is = 0.1, which implies that the material has

almost the same dilation angle over a wide range of confining pressure stress values.

Increasing the value of provides more curvature to the flow potential, implying that the

dilation angle increases more rapidly as the confining pressure decreases. Values of that are

significantly less than the default value may lead to convergence problems if the material is

subjected to low confining pressures because of the very tight curvature of the flow potential

locally where it intersects the p-axis.

30

Yield function

The model makes use of the yield function of Lubliner et. al (1989), with the

modifications proposed by Lee and Fenves (1998) to account for different evolution of

strength under tension and compression. The evolution of the yield surface is controlled by

the hardening variables, plt

~ and plc

~ . In terms of effective stresses, the yield function takes

the form ()

F = max max1 3 01

pl pl

c cq p

(3-14)

0 0

0 0

/ 1;0 0.5,

2 / 1

(1 ) (1 ),

3(1 ),

2 1

b c

b c

pl

cc

pl

t t

c

c

K

K

(3-15)

Figure 3-18 Yield surface in plane stress.

31

Here,

max is the maximum principal effective stress;

b0/c0 is the ratio of initial biaxial compressive yield stress to initial

uniaxial compressive yield stress (the default value is 1.16);

Kc is the ratio of the second stress invariant on the tensile meridian,

q(TM), to that on the compressive meridian, q(CM), at initial

yield for any given value of the pressure invariant p such that

the maximum principal stress is negative, 0max

; it must

satisfy the condition 0.15.0 cK (the default value is

2/3);see Figure 3-19.

)~( pltt is the effective tensile cohesion stress; and

)~( plcc is the effective compressive cohesion stress.

Figure 3-19 Yield surfaces in the deviatoric plane, corresponding to different values of Kc.

Visualization of crack directions

Unlike concrete models based on the smeared crack approach, the Concrete Damaged

Plasticity model does not have the notion of cracks developing at the material integration

point. However, it is possible to introduce the concept of an effective crack direction with the

purpose of obtaining a graphical visualization of the cracking patterns in the concrete

structure. Following Lubliner et. al. (1989), it is assumed that cracking initiates at points

where the tensile equivalent plastic strain is greater than zero, plt > 0, and the maximum

principal plastic strain is positive. The direction of the vector normal to the crack plane is

32

assumed to be parallel to the direction of the maximum principal plastic strain. This direction

can be viewed in the Visualization module of Abaqus/CAE.

3.5 Material Model for Reinforcement

The material models used for defining the elastic and plastic behavior of reinforcing

steel are linear elasticity and classical metal plasticity models respectively. The material

model for elastic behavior, i.e. linear elasticity has already been explained under the heading

of material model for concrete.

3.5.1 Classical Metal Plasticity

The classical metal plasticity models:

Use Mises or Hill yield surfaces with associated plastic flow, which allow for

isotropic and anisotropic yield, respectively;

Use perfect plasticity or isotropic hardening behavior;

Can be used when rate-dependent effects are important;

Are intended for applications such as crash analyses, metal forming, and general

collapse studies.

The Mises yield surface is used to define isotropic yielding. It is defined by giving the value

of the uniaxial yield stress as a function of uniaxial equivalent plastic strain.

Isotropic hardening

Isotropic hardening means that the yield surface changes size uniformly in all

directions such that the yield stress increases (or decreases) in all stress directions as plastic

straining occurs. Abaqus provides an isotropic hardening model, which is useful for cases

involving gross plastic straining or in cases where the straining at each point is essentially in

the same direction in strain space throughout the analysis. Although the model is referred to

as a hardening model, strain softening or hardening followed by softening can be defined.

If isotropic hardening is defined, the yield stress, 0, can be given as a tabular function of

plastic strain and, if required, of temperature and/or other predefined field variables. The

yield stress at a given state is simply interpolated from this table of data, and it remains

constant for plastic strains exceeding the last value given as tabular data.

33

Chapter 4 BondSlip Model

4.1 General

In a reinforced concrete beam, the flexural compressive forces are resisted by concrete,

while flexural tensile forces are provided by reinforcement. For this process to exist there

must be a force transfer, or bond, between the two materials. The existence of the bond is the

basic condition for these two materials to work together as a kind of composite material.

Without bond, the rebar would not be able to resist any external load, and the RC beam

would behave exactly like a plain concrete member does.

Because of its importance, the bond-slip relationship is considered in most of the design and

analysis efforts involving RC. Researchers have conducted numer