FRACTURE PROPERTIES OF FIBER REINFORCED CONCRETE

Abstract

Fracture properties of four different steel fiber reinforced concrete (SFRC) mixtures are determined in the present study. Two types of hooked end steel fibers having aspect ratios of 80 and 65 respectively were used employing volume fractions of 0.5% and 1.0%. Three point bending tests have been performed conforming to RILEM technical committee TC 162-TDF (RILEM, 2002a). The equivalent, f_eq, and residual, f_R, flexural tensile strength parameters, proposed by RILEM TC 162-TDF (RILEM, 2002a), to characterize and simulate the post-cracking behavior of SFRC have been evaluated and compared for the various concrete mixtures. It is observed that the equivalent flexural tensile strengths, f_eq, have less variance compared to the residual flexural tensile strengths, f_R.

 

A step wise optimization algorithm was developed to obtain the stress-crack opening (σ-w) curve of fiber reinforced concrete (FRC) using inverse analysis procedures. The optimization algorithm was developed as a three step process calculating the modulus of elasticity, E, in Step 1 and the tensile strength, f_t, and the slope of the first leg of the bilinear σ-w curve, a₁, in Step 2. Finally the parameters defining the second leg of the bilinear σ-w curve, a₂ and b₂ are calculated in the third and the final step. It was observed that f_t and a₁can be accurately predicted by restricting the optimization interval in Step 2 to [0, 0.05 mm] . It is concluded that the load- crack mouth opening displacement (P-CMOD) data from three point bending tests should be recorded at least until a CMOD of 5.0 mm is reached. The stress-crack opening curve of FRC can be predicted accurately by following the three step procedure, restricting the end level of CMOD to 0.05 mm in Step II and by performing the optimization in Step 3 at least until an end CMOD of 5.0 mm

Table of Contents

List of Figures ………………………………………………………………………………………………. v

List of Tables………………………………………………………………………………………………… vii List of Notations……………………………………………………………………………………………. ix

Acknowledgements……………………………………………………………………………………….. xii

CHAPTER 1  Introduction………………………………………………………………………………… 1

1.1 Objective and Research Significance……………………………………………………. 2

1.2 Research Tasks………………………………………………………………………………….. 2

1.3 Layout of the Thesis…………………………………………………………………………… 3

CHAPTER 2  Literature Review………………………………………………………………………… 6

2.1 Background………………………………………………………………………………………. 6

2.2 Fracture Mechanics of Fiber Reinforced Concrete…………………………………. 10

2.3 Stress-Crack opening (σ-w) Curve……………………………………………………….. 10

2.4 Calculation of Residual Strengths per ASTM C 1609…………………………….. 17

2.5 Cracked Hinge Model ………………………………………………………………………… 19

2.6 Inverse Analysis………………………………………………………………………………… 26

CHAPTER 3  Experimental Program………………………………………………………………….. 36

3.1 Materials…………………………………………………………………………………………… 36

3.2 Test Specimens and Test Matrix………………………………………………………….. 36

3.3 Test Setup and Procedure……………………………………………………………………. 37 3.4 Analysis of Test Data…………………………………………………………………………. 43

 

CHAPTER 4  Inverse Analysis ………………………………………………………………………….. 53

4.1 Introduction………………………………………………………………………………………. 53

4.2 Algorithm for Inverse Analysis……………………………………………………………. 54 CHAPTER 5  Conclusions and Recommendations……………………………………………….. 77

References …………………………………………………………………………………………………… 80 Appendix A  Experimental Data ……………………………………………………………………… 85 Appendix B  Proposed Matlab Algorithm…………………………………………………………. 91

Chapter 1       Introduction

Concrete, as a result of its many desirable properties, can be used in a variety of innovative designs. It not only possesses high compressive strength, stiffness, low thermal and electrical conductivity and low combustibility and toxicity, but it can also be cast in diverse shapes. But two material characteristics, which limit the use of concrete are brittleness and low strength in tension. However, the addition of fibers, to the otherwise brittle matrix, creates resistance to crack formation and progression, which increases the ductility of the materials structural response (Lim and Oh, 1999). Fibers have been used widely in non structural applications, like slabs on grade, industrial floors, pavements and overlays. (Meda et al, 2005 and ACI Committee 544, 2002). Fiber reinforced concrete has recently been added to the ACI 318 code as a viable alternative of minimum shear reinforcement for beam elements (ACI 318, 2008).

 

The concrete property most influenced by the addition of fibers to concrete is the capacity of energy absorption. Due to the relevance of the energy absorption capacity of fiber reinforced concrete (FRC), several entities have been proposed for evaluating this property, namely, toughness indices, equivalent flexural strength, and fracture energy. Fracture energy is the most widely used property in constitutive models for characterizing the fracture toughness and the post crack tensile capacity of concrete. In the case of FRC, the concept of critical crack width, w_c, is not relevant and therefore, fracture energy, G_F, loses its significance. So the stress-crack opening (σ-w) curve which influences the structural behavior becomes more important compared to fracture energy. (RILEM, 2002b)

The structural behavior of concrete during cracking can be described by nonlinear fracture mechanics models such as the fictitious crack model (FCM) proposed by Hillerborg et al. (1976). The application of Hillerborg’s approach requires the knowledge of the characteristic σ-w curve of the concrete. Under ideal conditions, this relation should be obtained from uniaxial tension tests of the concrete. Such tests are “expensive and time consuming and the problem of strain gradients in the ligament due to non-uniform cracking causes additional problems. Therefore, uniaxial tension tests are not an appropriate method for practical materials testing” (Slowik 2006).  A practical alternative is the use of the experimentally obtained response of a notched beam to determine the σ-w curve of the corresponding material through inverse analysis.

 

1.1      Objective and Research Significance

The objective of this research is to develop a methodology to accurately model the stress-crack opening (σ–w) behavior of steel fiber reinforced concrete (SFRC) from the load-crack mouth opening displacement (P-CMOD) response of the material. Many techniques have already been proposed to model the stress-crack opening behavior of concrete (Roelfstra and Wittman (1986), Kitsutaka (1997), Ostergaard (2003) and Sousa and Gettu (2006)). These existing techniques either require an initial approximation of the results, which are very close to the actual results (Roelfstra and Wittman, 1986), or in some cases require the optimization to be run several times to achieve convergence in the results (Ostergaard, 2003). In some cases, they can only predict one portion of the curve accurately (Sousa and

Gettu, 2006).

 

In this study, an optimization algorithm was developed to accurately model the entire softening behavior which can overcome the above mentioned drawbacks of the existing algorithms. A bilinear σ-w curve is used in the current algorithm, since for many SFRC materials it has been proven that a bilinear relationship provides a reasonable representation of the measured behavior (RILEM, 2002b)

 

1.2      Research Tasks

In order to achieve this objective the following research program was concluded:

1. Fabrication and testing of beam specimens under three point loading conforming to the RILEM TC 162-TDF (RILEM, 2002a).
2. Recording displacement, load data and the crack mouth displacement (CMOD) for obtaining the σ–w response of the beams.
3. Performing splitting-tensile strength and compressive strength tests on cylindrical specimens of concrete.
4. Analysis and compilation of the load-deflection (P-δ) and load-crack mouth opening displacement (P-CMOD) data for the beams to evaluate the tensile behavior of the FRC in terms of: limit of proportionality (f_fct,L), equivalent flexural strengths (f_eq,i) and residual flexural strengths (f_R,i).
5. Development of an optimization algorithm to back calculate the stresscrack (σ-w) relationship from the load-crack mouth opening displacement (P-CMOD) response of the beams using inverse analysis.
6. Incorporation of the algorithm into a commercial software (Mathematica,

Matlab) capable of performing optimizations and calculations.

7. Obtain the σ–w response from the inverse analysis.

 

1.3      Layout of the Thesis

The thesis is structured around four main chapters. Chapter 2 gives an overview of the cohesive crack model and fracture mechanics of fiber reinforced concrete (FRC) and the importance of the σ-w curve in characterizing the fracture properties of FRC. The direct (uniaxial tension test) and the indirect (using PCMOD curves and inverse analysis) methods of obtaining the σ-w curves are then explained in detail. Properties like limit of proportionality, f_fct,L, equivalent flexural strengths, f_eq,i, and residual flexural strengths, f_R,i, which are used to characterize the energy absorption capacity of FRC, and the experimental procedure to obtain them are also explained. The cracked hinge model, which forms the basis for obtaining the σ-w curve using inverse analysis, is also explained in detail and a brief overview of the process of inverse analysis is presented. Finally, an outline of the previous research results pertaining to the calculation of σ-w curves and the parameters used to characterize the energy absorption capacity of FRC is also presented.

 

The experimental program is explained in Chapter 3. The concrete mixture design and the specimen type used for different finding different properties are also outlined in this chapter. The experimental setup and the data collection process is also presented in detail. Chapter 3 also discusses the analysis of the test data obtained from the three point bending tests (3PB test) and the splitting tensile and compressive strength tests.

 

Chapter 4 explains the proposed algorithm to perform the inverse analysis process. It explains in detail various steps in the optimization algorithm and also explains the theoretical reason for each step performed. A comparison is made with the existing methods and an outline of the advantages of the proposed algorithm over the existing methods is also presented. A detailed analysis of the σ-w curves obtained is presented and guidelines are proposed for obtaining the P-CMOD curves from 3PB tests and for performing the inverse analysis.

 

Finally, Chapter 5 presents general and specific conclusions. This is followed by an extensive list of references. Appendix A and B give the results of the tests outlined in Chapter 3 and the Matlab code used for the inverse analysis in Chapter

4, respectively.

FRACTURE PROPERTIES OF FIBER REINFORCED CONCRETE

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