# Crack Analysis in Structural Concrete

## Theory and Applications

This new book on the fracture mechanics of concrete focuses on the latest developments in computational theories, and how to apply those theories to solve real engineering problems. Zihai Shi uses his extensive research experience to present detailed examination of multiple-crack analysis and mixed-mode fracture.Compared with other mature engineering disciplines, fracture mechanics of concrete is still a developing field with extensive new research and development. In recent years many different models and applications have been proposed for crack analysis; the author assesses these in turn, identifying their limitations and offering a detailed treatment of those which have been proved to be robust by comprehensive use. After introducing stress singularity in numerical modelling and some basic modelling techniques, the Extended Fictitious Crack Model (EFCM) for multiple-crack analysis is explained with numerical application examples. This theoretical model is then applied to study two important issues in fracture mechanics - crack interaction and localization, and fracture modes and maximum loads. The EFCM is then reformulated to include the shear transfer mechanism on crack surfaces and the method is used to study experimental problems. With a carefully balanced mixture of theory, experiment and application, Crack Analysis in Structural Concrete is an important contribution to this fast-developing field of structural analysis in concrete.

Audience
Graduate students in civil and structural engineering and related disciplines. Professional civil and structural engineers.

Hardbound, 344 Pages

Published: June 2009

Imprint: Butterworth Heinemann

ISBN: 978-0-7506-8446-0

## Contents

• Preface

CHAPTER 1 Introduction

1.1 Aims of the Book

1.2 Multiple-Crack Problems

1.3 Mixed-Mode Crack Problems

1.4 Crack Interaction and Localization

1.5 Failure Mode and the Maximum Load

1.6 Outline of this Book

References

CHAPTER 2 Linear Elastic and Nonlinear Fracture Mechanics

2.1 Elastic Crack-Tip Fields

2.1.1 Equations of Elasticity and Airy Stress Function

2.1.2 The Williams Solution of Elastic Stress Fields at the Crack Tip

2.1.3 The Complex Stress Function Approach to Elastic Stress Fields at the Crack Tip

2.2 Stress Intensity Factor and K-Controlled Crack-Tip Fields

2.3 The Energy Principles

2.3.1 The Griffith Fracture Theory

2.3.2 The Energy Release Rate G

2.3.3 Relationship between K and G

2.3.4 The Criterion for Crack Propagation

2.4 Plastic Zone Theories at Crack Tip

2.4.1 The Irwin Plastic Zone Corrections

2.4.2 Cohesive Zone Models by Dugdale and Barenblatt

2.5 Fracture Process Zone and Tension-Softening Phenomenon in Concrete

2.6 Fracture Energy GF and Tension-Softening Law in Concrete

2.6.1 Fracture Energy GF

2.6.2 Tension-Softening Law

References

CHAPTER 3 The Fictitious Crack Model and its Numerical Implementation

3.1 Introduction

3.2 Hillerborg and Colleaguesâ Fictitious Crack Model

3.2.1 Modeling Concept

3.2.2 Numerical Formulation by Peterssonâs Influence Function Method

3.3 The Principle of Superposition

3.4 The Reciprocity Principle

3.5 The Singularity Issue

3.6 Crack Path Modeling with Dual Nodes

3.7 The Remeshing Scheme for an Arbitrary Crack Path

3.8 The Solution Scheme for Incremental Stress Analysis

References

CHAPTER 4 Extended Fictitious Crack Model for Multiple-Crack Analysis

4.1 Introduction

4.2 Core Issues and Solution Strategy

4.3 Numerical Formulation of a Single-Crack Problem

4.4 Numerical Formulation of a Multiple-Crack Problem

4.5 Crack Analysis of a Simple Beam under Bending

4.5.1 Crack Analysis with a Fixed Crack Path

4.5.2 Crack Analysis with a Curvilinear Crack Path

4.6 Crack Analysis of a Fracture Test of a Real-Size Tunnel-Lining Specimen

4.6.1 Fracture Test on a Tunnel-Lining Specimen

4.6.2 Crack Analysis with a Half-FE-Model

4.6.3 Crack Analysis with a Full-FE-Model

4.7 Crack Analysis of a Scale-Model Test of a Gravity Dam by Carpinteri and Colleagues

4.7.1 Background

4.7.2 Model I: Single-Crack Propagation

4.7.3 Model II: Multiple-Crack Propagation

4.7.4 Model III: Multiple-Crack Propagation

References

CHAPTER 5 Crack Interaction and Localization

5.1 Introduction

5.2 Coefficient of Interaction

5.2.1 Crack Equations and the Source of Crack Interaction

5.2.2 Coefficient of Interaction and Principal Tip Force Coefficient

5.3 Crack Interactions in Notched Concrete Beams under Four-Point Bending

5.3.1 Beams with Small Notches

5.3.2 Beams with Both Small and Large Notches

5.4 Crack Interactions in Tunnel Linings

5.5 Characteristics of Crack Interactions with One and Multiple Tension Zones

References

CHAPTER 6 Failure Modes and Maximum Loads of Notched Concrete Beams

6.1 Introduction

6.2 Numerical Analysis of Notched Beams under Various Load Conditions

6.2.2 Maximum Load Increase with Higher Density of Initial Notches

6.3 Critical Initial Notch and Its Influence on Failure Mode and the Maximum Load

6.4 Experimental Verifications on Relationships between Failure Modes and the Maximum Loads

6.4.1 Four-Point Bending Tests

6.4.2 Numerical Analyses

6.5 Engineering Implications

References

CHAPTER 7 Mixed-Mode Fracture

7.1 Introduction

7.2 Modeling of Cohesive Forces in the FPZ

7.3 Reformulation of FCM and EFCM for Mixed-Mode Fracture

7.3.1 FCM for Mixed-Mode Fracture

7.3.2 EFCM for Mixed-Mode Fracture

7.4 Mode-II Fracture Energy GF II

7.5 Numerical Studies of Arrea and Ingraffeaâs Single-Notched Shear Beam

7.5.1 Parametric Studies with Five Shear-COD Relations

7.5.2 Parametric Studies on Mode-II Fracture Energy with Three Shear-COD Relations

7.6 Numerical Studies of a Scale-Model Test of a Gravity Dam

References

CHAPTER 8 Applications: Pseudoshell Model for Crack Analysis of Tunnel Linings

8.1 Introduction

8.2 Pseudoshell Model

8.2.1 Modeling Concept

8.2.2 Numerical Formulation

8.2.3 Parametric Studies on Uniqueness of Solutions on Tunnel Deformation

8.3 Evaluation of Ground Pressure Based on the Quasi Loosening Zone Model

8.4 Numerical Analysis of an Aging Waterway Tunnel (Case A-1) Compared with a Soil Mechanics Approach

8.4.1 Background

8.4.2 Numerical Analysis by Adachi-Oka Model

8.4.3 Numerical Analysis by the Pseudoshell Model

8.4.4 Evaluation of Ground Pressure

8.5 Case Studies of Two Aging Waterway Tunnels

8.5.1 Power Plant B (Horseshoe Type): Site B-1

8.5.2 Power Plant B (Horseshoe Type): Site B-2

8.5.3 Power Plant B (Horseshoe Type): Site B-3

8.5.4 Power Plant C (Calash Type): Site C-1

8.6 Development of Database for Evaluation of Ground Pressure Based on the CMOD

8.6.1 Selection of Influential Factors and Cases of Study

8.6.2 Relations between Cross-Sectional Deformation and the CMOD

8.6.3 Relations between Pressure Load and Cross-Sectional Deformation

8.6.4 Two-Step Procedure for Determining External Loads by the CMOD and Development of Database

References

CHAPTER 9 Computer Program for Mode-I Type Crack Analysis inn Concrete Using EFCM (CAIC-M1.FOR)

9.1 Overview of the Program

9.2 Structure of the Program

9.3 Main Rules

9.4 Program List

9.5 Selected Examples Illustrating the Usage of the Program

9.5.1 Crack Analysis of Notched Beam

9.5.2 Crack Analysis of Scale Model Dam

9.5.3 Crack Analysis of Tunnel Lining

Reference

CHAPTER 10 Computer Program for Mixed-Mode Type Crack Analysis in Concrete Using EFCM (CAIC-M12.FOR)

10.1 Overview of the Program

10.2 Structure of the Program

10.3 Main Rules

10.4 Subroutines with Major Changes

10.4.1 Changes in CAIC-M12.FOR from CAIC-M1.FOR

10.4.2 Subroutines with Major Changes in the Crack Pattern Determination Block (TFORCE)

10.4.3 Subroutines with Major Changes in the Crack Equation Solution Block (EFFECT)

10.4.4 Subroutines with Major Changes in the Main Block (MAINCN)

10.5 Selected Example Illustrating the Usage of the Program

INDEX