Shear Behaviour of Ferrocement Deep Beams

Shear Behaviour of Ferrocement Deep Beams

A thesis submitted to The University of Manchester for the degree of

Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2013

Shichuan Tian

BEng, MSc

SCHOOL OF MECHANICAL, AEROSPACE AND CIVIL

ENGINEERING

2

DECLARATION

No portion of the work referred to in this dissertation has been submitted in support

of an application for another degree or qualification of this or any other university or

other institution of learning. Material property tests of 18 matrix cylinder specimens,

3 mesh wire specimens and 3 rebar specimens were done with Mr. Jianqi Wang. But

all the analyses were done independently.

Shichuan Tian

3

ACKNOWLEDGEMENTS

Firstly I would like to thank Parthasarathi Mandal and Paul Nedwell for their

excellent supervision, valuable guidance, constructive discussions and

encouragement throughout the period of research.

I would also like to thank John Mason and Jianqi Wang for their generous help

in the matrix experiment tests in this dissertation study.

Finally, I would like to thank my family for their moral support and their

encouragement to my studies.

4

COPYRIGHT STATEMENT

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5

ABSTRACT

This thesis presents the results of an experimental, numerical and analytical study to

develop a design method to calculate shear resistance of flanged ferrocement beams

with vertical mesh reinforcements in the web. Two groups of full-scale testing were

conducted comprising of three I beams and four U beams. The I beams had the same

geometry and reinforcement arrangements, but differed in the matrix strength or

shear span to depth ratio. The U beams differed in web and flange thickness,

reinforcement arrangements, matrix strength and shear span to depth ratio.

The experimental data were used for validation of finite element models which had

been developed using the ABAQUS software. The validated models were

subsequently employed to conduct a comprehensive parametric study to investigate

the effects of a number of design parameters, including the effect of matrix strength,

shear span to depth ratio, cross sectional area, length of clear span, volume fraction

of meshes and amount of rebar.

The main conclusion from the experiments and parametric studies were: shear failure

may occur only when the shear span to depth ratio is smaller than 1.5; the shear

strength may increase by increasing the matrix strength, volume fraction of meshes,

cross sectional area and amount of rebar. The main type of shear failure for I beams

was diagonal splitting while for U beams it was shear flexural.

Based on the results from the experimental and numerical studies, a shear design

guide for ferrocement beams was developed. A set of empirical equations for the two

different failure types and an improved strut-and-tie were proposed. By comparison

with the procedures currently in practice, it is demonstrated that the methodology

proposed in this thesis is likely to give much better predictions for shear capacity of

flanged ferrocement beams.

6

CONTENTS

ACKNOWLEDGEMENTS ................................................................................................................. 3

COPYRIGHT STATEMENT ............................................................................................................. 4

ABSTRACT .......................................................................................................................................... 5

CONTENTS .......................................................................................................................................... 6

LIST OF FIGURES ............................................................................................................................. 9

LIST OF TABLES ............................................................................................................................. 15

LIST OF NOTATION ....................................................................................................................... 17

CHAPTER 1 INTRODUCTION ................................................................................................. 20

1.1 BACKGROUND ........................................................................................................................ 20

1.2 AIMS AND OBJECTIVES ........................................................................................................... 22

1.3 LAYOUT OF THESIS ................................................................................................................. 22

CHAPTER 2 LITERATURE REVIEW ..................................................................................... 24

2.1 INTRODUCTION ....................................................................................................................... 24

2.2 HISTORICAL BACKGROUND .................................................................................................... 24

2.3 FERROCEMENT STRUCTURES .................................................................................................. 26

2.4 FERROCEMENT CONSTITUENTS............................................................................................... 29

2.4.1 Reinforcement............................................................................................................... 29

2.4.2 Matrix and Mechanical Property ................................................................................. 30

2.5 REINFORCEMENT PARAMETERS AND MECHANICAL PROPERTIES ............................................ 32

2.5.1 Reinforcement Parameters ........................................................................................... 32

2.5.2 Behaviour under Tension ............................................................................................. 33

2.5.3 Behaviour under Compression ..................................................................................... 35

2.5.4 Behaviour in Flexure .................................................................................................... 36

2.5.5 Behaviour under Shear ................................................................................................. 37

2.6 DETAIL STUDY ON SHEAR OF BEAM ....................................................................................... 39

2.6.1 Deep Beam ................................................................................................................... 40

2.6.2 Four Modes of Failure for Reinforced Concrete Deep Beams: ................................... 42

2.6.3 Component Model ........................................................................................................ 43

2.6.4 Strut and Tie Model ...................................................................................................... 45

2.7 FEM IN FERROCEMENT RESEARCH......................................................................................... 48

2.8 CONCLUSION .......................................................................................................................... 49

CHAPTER 3 EXPERIMENTAL TESTS ................................................................................... 51

3.1 INTRODUCTION ....................................................................................................................... 51

3.2 MATRIX TEST ......................................................................................................................... 51

7

3.2.1 Sampling ....................................................................................................................... 51

3.2.2 Testing specimens ......................................................................................................... 53

3.2.3 Compressive Test.......................................................................................................... 53

3.2.4 Tensile Test ................................................................................................................... 54

3.3 MATRIX PROPERTY ANALYSIS................................................................................................ 56

3.3.1 Compressive Behaviour of Matrix ................................................................................ 57

3.3.2 Compressive Stress-Strain Relation of Matrix ............................................................. 58

3.3.3 Tensile Strength and Compressive Strength Relationship of the Matrix ...................... 59

3.3.4 Tensile Behaviour of Matrix ......................................................................................... 60

3.4 MESH PROPERTY TEST ........................................................................................................... 62

3.4.1 Introduction .................................................................................................................. 62

3.4.2 Detailed Mesh Property Test ........................................................................................ 62

3.5 STEEL BAR TEST..................................................................................................................... 64

3.6 FERROCEMENT MANUFACTURE .............................................................................................. 64

3.6.1 Fabrication method ...................................................................................................... 66

3.6.2 Fabrication and Improvement ...................................................................................... 71

3.7 TEST SETUP ............................................................................................................................ 76

3.8 I BEAM TEST RESULTS AND ANALYSES .................................................................................... 79

3.8.2 Gauges Results of Beam I2 ........................................................................................... 81

3.8.3 Vertical Gauges results of Beam I3 .............................................................................. 89

3.9 U BEAM TESTS RESULTS ........................................................................................................ 93

3.10 SUMMARY AND CONCLUSION .......................................................................................... 102

CHAPTER 4 FINITE ELEMENT SIMULATION AND VALIDATION ............................. 104

4.1 INTRODUCTION ..................................................................................................................... 104

4.2 THE FINITE ELEMENT MODEL .............................................................................................. 105

4.2.1 Element Types ............................................................................................................ 106

4.2.2 Material Property ....................................................................................................... 107

4.2.3 Defining a Concrete Damaged Plasticity Model ........................................................ 109

4.2.4 Reinforcement Model in ABAQUS ............................................................................. 115

4.2.5 Mesh Convergence ..................................................................................................... 116

4.3 VERIFICATION OF FINITE ELEMENT MODELS WITH EXPERIMENTS........................................ 117

4.3.1 I Beams ....................................................................................................................... 117

4.3.2 U Beams ..................................................................................................................... 122

4.4 CONCLUSION ........................................................................................................................ 125

CHAPTER 5 PARAMETRIC STUDY ..................................................................................... 127

5.1 I BEAM ................................................................................................................................. 127

5.1.1 Effect of Matrix Strength ............................................................................................ 127

5.1.2 Effect of a/h Ratio ....................................................................................................... 130

5.1.3 Effect of Web Reinforcement and Geometry............................................................... 133

5.1.4 Effect of Flange Reinforcement and Geometry .......................................................... 139

5.1.5 Effect of Rebar Size .................................................................................................... 146

8

5.1.6 Effect of Length .......................................................................................................... 149

5.1.7 Effect of Mesh Type .................................................................................................... 151

5.2 U BEAMS .............................................................................................................................. 156

5.2.1 Effect of Matrix Strength ............................................................................................ 157

5.2.2 Effect of a/h Ratio ....................................................................................................... 158

5.2.3 Effect of Web Thickness .............................................................................................. 161

5.2.4 Effect of Number of Mesh Layers ............................................................................... 162

5.2.5 Effect of Rebar Size .................................................................................................... 164

5.2.6 Effect of Length .......................................................................................................... 166

5.2.7 Effect of Mesh Type .................................................................................................... 167

5.3 EFFECT OF BOUNDARY CONDITIONS .................................................................................... 170

5.4 CONCLUSION ........................................................................................................................ 173

CHAPTER 6 DESIGN GUIDELINES ...................................................................................... 175

6.1 THE STRUT AND TIE MODEL ................................................................................................. 175

6.1.1 Configuration and Angle between Strut and Tie ........................................................ 176

6.1.2 Tie Strength ................................................................................................................ 178

6.2 EMPIRICAL EQUATION FOR DIAGONAL SPLITTING FAILURE BY STATISTICAL ANALYSIS ..... 179

6.2.1 Proposed Shear Design Equation for Diagonal Splitting Failure Mode.................... 179

6.2.2 Comparison of Experimental Results, Proposed Equation and ACI 318 ................... 185

6.3 EMPIRICAL EQUATION FOR SHEAR FLEXURAL FAILURE BY STATISTICAL ANALYSIS ........... 188

6.3.1 Proposed Shear Design Equation for Shear Flexural Failure Mode ......................... 188

6.3.2 Comparison of Experimental Results, Proposed Equation and ACI 318 ................... 190

6.4 SHEAR DESIGN EXAMPLE ..................................................................................................... 192

6.5 SHEAR DESIGN PROCEDURE ................................................................................................. 196

6.6 ELASTIC DEFORMATION OF FERROCEMENT DEEP BEAMS .................................................... 199

6.7 CONCLUSION ........................................................................................................................ 204

CHAPTER 7 SUMMARY AND CONCLUSIONS .................................................................. 206

7.1 SUMMARY ............................................................................................................................ 206

7.2 CONCLUSIONS....................................................................................................................... 206

7.3 RECOMMENDATIONS FOR FUTURE STUDY ............................................................................. 208

REFERENCE ................................................................................................................................... 209

PUBLICATIONS ............................................................................................................................. 213

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LIST OF FIGURES

Fig 2.1 Lambot’s First Ferrocement Boat[9] ....................................................... 25

Fig 2.2 Yanbu Cement Company [9] .................................................................... 27

Fig 2.3 Ferrocement Dome in Oaxaca, Mexico [9] ............................................. 28

Fig 2.4 A 16 m Prototype of the Ferrocement Canopy [17]................................ 28

Fig 2.5 A Water Tank on Ferrocement House Roof [18] .................................... 29

Fig 2.6 Typical wire mesh used in ferrocement ................................................... 29

Fig 2.7 Ferrocement under Tension [3] ................................................................ 34

Fig 2.8 Diagonal Splitting Failure of Reinforced Concrete Beam...................... 40

Fig 2.9 Beam Strain Distribution Diagram [5]..................................................... 41

Fig 2.10 Comparison between Strut and Tie method and Sectional Method

(based on Bernoulli’s Theory) [41] ............................................................. 44

Fig 2.11 Description of Strut and Tie Model [6] ................................................. 45

Fig 2.12 Classification of Nodes [6] ..................................................................... 47

Fig 3.1 Matrix Mix ................................................................................................. 52

Fig 3.2 Gauges used in Cylinder Compressive Tests and a Typical Failure ...... 54

Fig 3.3 Jig with Packing Strips and a Typical Failure Type ............................... 55

Fig 3.4 Comparison of Prediction (Eq 3.5) and Experimental Results .............. 58

Fig 3.5 Fitting Curve of Peak Compressive Stress and Strain ............................ 59

Fig 3.6 Fitting Curve of Peak Tensile Stress and Peak Compressive Stress ...... 60

Fig 3.7 Post-peak Tensile stress-cracking width curve [47] ............................... 61

Fig 3.8 Mesh Wire Tensile Test Set Up ............................................................... 63

Fig 3.9 Stress-Strain Curves of Mesh ................................................................... 64

Fig 3.10 Reinforcements and Design Dimensions (all in mm) ........................... 65

Fig 3.11 U Beam with Bad Finish (exterior view)............................................... 66

Fig 3.12 Mesh sheet roll ........................................................................................ 67

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Fig 3.13 Cropping Machine................................................................................... 67

Fig 3.14 Bending Machine .................................................................................... 68

Fig 3.15 Assembly of Mesh Layers without Cable Ties ..................................... 69

Fig 3.16 Assembly of Mesh Layers with Cable Ties........................................... 69

Fig 3.17 Cable Tie Pliers ....................................................................................... 70

Fig 3.18 Assembly of Mesh Layers with Trimmed Cable Ties .......................... 70

Fig 3.19 Mesh and Oiled Mould ........................................................................... 71

Fig 3.20 Fabrication with a Fine Surface ............................................................. 72

Fig 3.21 I-section Moulds (a) Top and bottom plate (b) Base plate ................... 73

Fig 3.22 Clamped Mould ....................................................................................... 74

Fig 3.23 Vibrating Poker ....................................................................................... 74

Fig 3.24 Matrix Filling and Vibration .................................................................. 74

Fig 3.25 Mesh Dropping and Finish ..................................................................... 75

Fig 3.26 Finish Surfacing ...................................................................................... 75

Fig 3.27 Storage of Specimen ............................................................................... 76

Fig 3.28 Final Appearance of Specimens ............................................................. 76

Fig 3.29 Specimen and Gauges Set Up ................................................................ 77

Fig 3.30 Test Set Up (all in mm) .......................................................................... 78

Fig 3.31 Gauges Position and Names of I Beam (I1) .......................................... 79

Fig 3.32 Typical Cracking Pattern, a) Shear Cracks, b) Shear and Flexural

Cracks ............................................................................................................. 80

Fig 3.33 Load-Deflection Curves of Beam I1 ...................................................... 80


Fig 3.35 Horizontal Gauges Load-Micro Strain Curves of Beam I2 .................. 82

Fig 3.36 Horizontal Gauges Location-Micro Strain Curves of Beam I2. The

location along the vertical axis indicates the distance of the gauge position

from the bottom surface. ............................................................................... 83

Fig 3.37 Load-Strain Curves of Rosette Gauges.................................................. 84

11

Fig 3.38 Load-Shear Strain Curves of Beam I2 ................................................... 85

Fig 3.39 Load-Max Principal Strain Curves of Beam I2..................................... 86

Fig 3.40 Load-Min Principal Strain Curves of Beam I2 ..................................... 87

Fig 3.41 Load-Principal Angle Curves of Beam I2 ............................................. 88


Fig 3.43 Load against Deflection Curves in respect of the Recognised Three

Stages of Ferrocement Behaviour ................................................................. 90

Fig 3.44 Comparison of Beam Behaviour with Load .......................................... 92

Fig 3.45 Gauges Location of U Beam Tests ........................................................ 94

Fig 3.46 Steel Strip ................................................................................................ 95

Fig 3.47 Occurrence of Flexural Cracking in Beam U1 ...................................... 96

Fig 3.48 Occurrence of Diagonal Cracking in Beam U1 .................................... 96

Fig 3.49 Diagonal Tension Failure in Beam U1 .................................................. 97

Fig 3.50 Flexural Cracking of Beam U1 at Ultimate Stage ................................ 97

Fig 3.51 Load-Deflection Curves of Beam U1 .................................................... 98

Fig 3.52 Flexural Failure of Beam U2 .................................................................. 99

Fig 3.53 Load-Deflection Curves of Beam U2 .................................................... 99

Fig 3.54 Test Set Up of Beam U3 ....................................................................... 100

Fig 3.55 Diagonal Tension Failure of Beam U3 ................................................ 100

Fig 3.56 Load-Deflection Curves of Beam U3 .................................................. 101

Fig 3.57 Flexural Failure of Beam U4 ................................................................ 101

Fig 3.58 Load-Deflection Curves of U4 ............................................................. 102

Fig 4.1 Response of Concrete to Uniaxial Loading in (a) Tension and (b)

Compression [46]......................................................................................... 109

Fig 4.2 Comparison of FEM Load-Displacement Curves with Different σb0/σc0

Values and Experimental Result ................................................................. 111

12

Fig 4.3 Comparison of FEM Load-Displacement Curves with Different Kc

Values and Experimental Result ................................................................. 112

Fig 4.4 Comparison of FEM Load-Displacement Curves with Different Dilation

Angles and Experimental Result................................................................. 113

Fig 4.5 Comparison of FEM Load-Displacement Curves with Different

Eccentricity and Experimental Result ........................................................ 114

Fig 4.6 Comparison of FEM Load-Displacement Curves with Different

Viscosity Parameter and Experimental Result ........................................... 115

Fig 4.7 Comparison of Load-Deflection Curves of FEM with Different Matrix

Mesh Sizes and Experimental Result of Beam I1 ..................................... 117

Fig 4.8 Load-deflection Curves of Beam I1, I2 and I3 Obtained from

Experiments and FEM ................................................................................. 119

Fig 4.9 Comparison of Crack Growth from Experiment and FEM .................. 121

Fig 4.10 Comparison of Failure Mode from Experiment and FEM ................. 121

Fig 4.11 Load-Deflection Curve of Beam U1 .................................................... 122




Fig 4.15 Comparison of Failure Mode from Experiment and FEM ................. 125

Fig 5.1 Load-Deflection Curves for Different Matrix Strengths ...................... 128

Fig 5.2 Load-Damage Volume Curves for Different Matrix Strengths when

a/h=1.0 .......................................................................................................... 130

Fig 5.3 Typical Stress (in MPa) Distributions from the FEM........................... 132

Fig 5.4 Beam Mesh Stress (in MPa) Distribution at the Peak Loading Stage

(Flexural Failure) ......................................................................................... 132

Fig 5.5 Load-Deflection Curves for Different Web Mesh Layer Numbers ..... 133

Fig 5.6 Load-Deflection Curves for Different Web Thickness......................... 135

13

Fig 5.7 Cross Section of I Beams........................................................................ 137

Fig 5.8 Load-Deflection Curves for Different Web Depths .............................. 137

Fig 5.9 Load-Deflection Curves for Different Flange Thicknesses.................. 140

Fig 5.10 Load-Deflection Curves for Different Top Flange Mesh Numbers... 142

Fig 5.11 Load-Deflection Curves for Different Number of Mesh Layers in the

Bottom Flange .............................................................................................. 143

Fig 5.12 Tensile Damage at the Failure Stage ................................................... 144

Fig 5.13 Load-Deflection Curves for Different Numbers of Mesh Layers in the

Top and Bottom Flanges ............................................................................. 145

Fig 5.14 Load-Deflection Curves for Different Rebar Sizes ............................ 147

Fig 5.15 Load-Deflection Curves for Different Beam Spans............................ 150

Fig 5.16 Load-Deflection Curves for Different Specific Surfaces ................... 152

Fig 5.17 Load-Deflection Curves for Different Volume Fractions .................. 154

Fig 5.18 Load-Deflection Curves for Different Matrix Strengths .................... 157

Fig 5.19 Typical Stress (in MPa) Distribution from FEM (the scales are

different for the above images) ................................................................... 159

Fig 5.20 Stress (in MPa) Distribution of Flexural Failure Cases ...................... 160

Fig 5.21 Load-Deflection Curves for Different Web Thicknesses ................... 161

Fig 5.22 Load-Deflection Curves for Different Mesh Layer Numbers ............ 163

Fig 5.23 Rebar Location in U Beam ................................................................... 164

Fig 5.24 Load-Deflection Curves for Different Rebar Sizes ............................ 165

Fig 5.25 Load-Deflection Curves for Different Beam Spans............................ 166

Fig 5.26 Load-Deflection Curves for Different Specific Surfaces ................... 168

Fig 5.27 Load-Deflection Curves for Different Volume Fractions .................. 169

Fig 5.28 Reinforcement Yielding of Fixed End Boundary Condition (max

principal stress in MPa) ............................................................................... 171

Fig 5.29 Bearing Failure of Fixed End Boundary Condition (max principal

stress in MPa) ............................................................................................... 171

14

Fig 5.30 Load Displacement Curve of I Beam with Different Boundary

Conditions .................................................................................................... 172

Fig 5.31 Load Displacement Curve of U Beam with Different Boundary

Conditions .................................................................................................... 172

Fig 6.1 One-Panel STM Configuration .............................................................. 177

Fig 6.2 Comparison of FEM Results, ACI Formula and Eq 6.4 ....................... 187

Fig 6.3 Comparison of FEM Results, ACI Formula and Eq 6.5 ....................... 191

Fig 6.4 Cross Sections of Roof Beam and I Beam ............................................ 193

Fig 6.5 Final Appearance of the Water Tank Assembly ................................... 194

15

LIST OF TABLES

Table 3.1 Matrix Characteristic Strengths with Standard Deviation in Brackets

......................................................................................................................... 57

Table 3.2 Beams Specification .............................................................................. 78

Table 4.1 Selected Coefficients for Shear Behaviour Study of Ferrocement

Beams ........................................................................................................... 115

Table 4.2 Comparison of Peak Load and CPU Time of FEM with Different

Matrix Mesh Size ......................................................................................... 117

Table 4.3 Comparison of Test and FEM Peak Loads ........................................ 120

Table 5.1 Peak Loads for Different Matrix Strengths ....................................... 128

Table 5.2 Peak Loads for Different Web Mesh Layer Numbers ...................... 134

Table 5.3 Peak Loads for Different Web Thickness .......................................... 136

Table 5.4 Peak Loads for Different Web Depths ............................................... 138

Table 5.5 Peak Loads for Different Flange Thickness ...................................... 140

Table 5.6 Peak Loads for Different Numbers of Mesh layers in the Top Flange

....................................................................................................................... 142

Table 5.7 Peak Loads for Different Numbers of Mesh Layers in the Bottom

Flange ........................................................................................................... 143

Table 5.8 Peak Loads for Different Numbers of Mesh Layers in the Top and

Bottom Flanges ............................................................................................ 146

Table 5.9 Peak Loads for Different Rebar Sizes ................................................ 147

Table 5.10 Yielded Mesh Percentage of Different Rebar Sizes at Peak Load . 148

Table 5.11 Peak Loads for Different Beam Spans ............................................. 150

Table 5.12 Peak Loads for Different Specific Surfaces .................................... 152

Table 5.13 Peak Loads for Different Volume Fractions ................................... 154

16

Table 5.14 Peak Loads for Different Matrix Strengths ..................................... 158

Table 5.15 Peak Loads for Different Web Thicknesses .................................... 161

Table 5.16 Peak Load for Different Mesh Layer Numbers ............................... 163

Table 5.17 Peak Loads for Different Rebar Sizes.............................................. 165

Table 5.18 Peak Loads for Different Beam Spans ............................................. 167

Table 5.19 Peak Loads for Different Specific Surfaces .................................... 168

Table 5.20 Peak Loads for Different Volume Fractions ................................... 169

Table 6.1 Comparison of Cracking Angles from ACI Calculations and

Experiments.................................................................................................. 176

Table 6.2 Failure Mode Prediction Using the Proposed STM .......................... 178

Table 6.3 Length Factor for Diagonal Splitting Failure .................................... 184

Table 6.4 Web Mesh Volume Fraction Factor for Diagonal Splitting Failure 184

Table 6.5 Mesh Volume Fraction Factor for Diagonal Splitting Failure ......... 184

Table 6.6 Rebar Factor for Diagonal Splitting Failure ...................................... 185

Table 6.7 Comparison of Experimental Results, ACI Formula and Eq 6.4 ..... 185

Table 6.8 Length Factor for Shear-Flexural Failure .......................................... 189

Table 6.9 Mesh Volume Fraction Factor for Shear-Flexural Failure ............... 190

Table 6.10 Rebar Factor for Shear-Flexural Failure .......................................... 190

Table 6.11 Comparison of Experimental Results, ACI Formula and Eq 6.5 ... 190

Table 6.12 Comparison of Elastic Deformation of Experimental Results, FEM

and Calculation ............................................................................................ 200

Table 6.13 Comparison of Elastic Deformation of FEM and Calculations ..... 201

17

LIST OF NOTATION

Ac cross-sectional area of the specimen on which the compressive force acts

calculated from the designated size of the specimen in mm2

cross-sectional are at one end of a strut in a strut-and-tie model, taken

perpendicular to the axis of the strut

Effective cross section area

area of the face of the nodal zone

area of reinforcements in longitudinal direction

D dead load

E effective modulus of the composite

F maximum load at failure, in N

shear resistance

peak load by proposed equation

peak load by ACI 318 formula

GF fracture energy (MPa/mm2)

GF0 base values of fracture energy; 0.025MPa is taken for this study

I second moment of area

K length factor

L live load including impact effects

peak load by Bernoulli bending theory

P external load

R required loading rate, in N/s

Sr specific surface

Srl specific surface of longitudinal direction

Srt specific surface of transverse direction

T strength of the tie

Vr volume Fraction

a shear span, in mm

18

d designated cross-sectional dimension, in mm

eccentricity, defined as the rate of plastic potential function approaches at the

asymptote.

fctm tensile splitting strength, in MPa

fcm peak compressive stress

yield strength of reinforcement

h depth of the beam

l clear span of the specimen, in mm

n approximate function of compressive strength of matrix,

s stress rate, in MPa/s

w1 cracking width for 0.15fctm

wc cracking width for 0.0001fctm

angle between strut axis and beam axis

cross sectional coefficient

coefficient, based on aggregate size; due to the absence of coarse aggregate in

ferro matrix, the recommended minimum value of 8 was taken in this study

factor of mesh layer volume fraction in the bottom flange (excluding fillet

sections)

coefficient related to the location and number of ties through the cross-section

area

factor of rebar

factor to account for the effect of cracking and confining reinforcement on the

effective compressive strength of the concrete in a strut

engineering strain

elastic flexural deflection

elastic shear deflection

elastic total deflection

compressive strain

19

compressive strain at peak stress

strain from each gauge reading

maximum principal strain

minimum principal strain

strain of horizontal direction

strain of vertical direction

angle between strut and tie

principal angle

μ viscosity Parameter

σ compressive strength, in MPa

dilation angle, measured in the ̅ ̅ plane at high confining pressure

CHAPTER 1 Introduction

20


1.1 Background

“Ferrocement is a type of thin wall reinforced concrete commonly constructed of

hydraulic cement mortar reinforced with closely spaced layers of continuous and

relatively small size wire mesh. The mesh may be made of metallic or other suitable

materials.”---ACI committee 549 [1]

Ferrocement is a cement matrix reinforced with wire mesh. It is one of the earliest

types of reinforced concrete and is known to have been used for over a hundred years.

Ferrocement structures have many advantages such as lightweight, easy to shape and

low carbon footprint compared to reinforced concrete. Recently, ferrocement was

chosen as the material for the lightweight roof for the Stavros Niarchos Cultural

Centre which is under construction in the out skirts of Athens [2]. Although the

formworks for irregular shaped structures can be expensive, the resulting thin shells

are aesthetically pleasing highly efficient structures. Distributed mesh reinforcements

also produce lower crack widths, producing relatively uniform sectional behaviour.

However, the research and development of the technology had been patchy primarily

due to the increasing costs of labour and mesh fabrications. Coupled with that, lack

of appropriate design code provisions may explain limited interest in adopting the

technology by large construction companies. However, there had been a few

professional committees in the past 50 years assessing and disseminating the research

on ferrocement such as ACI 549 and IFIC [3]. The most significant among them is

ACI 549 which was set up by the American Concrete Institute (ACI) to develop and

report information on thin reinforced cementitious products and ferrocement. The

current “Guide for Design, Construction, and Repair of Ferrocement” [4] provides

design criteria for tensile, compressive and flexural strength. However, there is


21

relatively little information on the shear strength of ferrocement structures. This may

be due to the fact that ferrocement structures are normally built as panels with large

span to depth ratios and are primarily subjected to flexural behaviour.

Shear can be critical for ferrocement beams that are used as transfer girders for water

tanks on roofs [3] and flanged beams in the foundations of offshore gravity type

structures [5] in developing countries such as Cuba and Thailand. Results from the

current study have shown that when the ferrocement beams have small shear span to

depth ratios (less than 1.5) shear failure becomes the main failure mechanism. In the

only authoritative textbook on ferrocement, Naaman [3] suggested using of the ACI

318 [6] approach to design for shear. Comparing to the experimental results in this

study, it was found that the ACI equation gives a highly conservative estimate of

only about 10% of the observed strength. Hence, it is important and necessary to

produce shear design guidelines to enhance the ferrocement design code.

In order to obtain a clear understanding of the shear behaviour of ferrocement beams,

appropriate tests were conducted to determine the mechanical properties of

ferrocement constituent materials. Three I beams and four U beams were constructed

and tested to failure under a four point bending set up. Finite-element models of the

tested beams were developed using ABAQUS. The models were first validated using

the test data and were used for a subsequent detailed parametric study. The set of

experimental and numerical results were used to develop a design guideline for shear

capacity of ferrocement flanged deep beams. A strut-and-tie model was proposed for

better understanding of the shear behaviour and to aid the design process.

The results of the study are reported in this thesis. A component model has been

developed along with design guidelines with empirical equations. By using the

equations, more realistic yet safe predictions of shear strength can be achieved.


22

1.2 Aims and Objectives

The aims of this research were to investigate the shear behaviour of ferrocement

flanged deep beam and to propose a shear design guideline. The detailed objectives

of this research are:

To understand the shear behaviour of ferrocement beams with web

reinforcements (mesh) under four point bending;

To develop finite element models for ferrocement beams under shear and validate

the models with the experimental results;

To use the validated FE models for detailed parametric studies;

To develop a practical reliable method for estimating shear capacity of

ferrocement beam;

To develop a strut and tie model for predicting the type of shear failure;

To develop a ferrocement beam shear design procedure.

1.3 Layout of Thesis

This thesis is divided into 7 chapters. A brief outline of each chapter is given below:

Chapter 1 Introduction

This chapter presents a general introduction and layout of the thesis.

Chapter 2 Literature Review

This chapter presents a brief literature review including development of ferrocement,

introduction of ferrocement constituents and their mechanical behaviour, basic

mechanical behaviour of ferrocement, deep beam theory and application of finite

element method for ferrocement elements.


23

Chapter 3 Experiments

This chapter presents the experiments in detail including material property tests of

the matrix and reinforcement, fabrication of the ferrocement beam specimens and

beam test details.

Chapter 4 Finite Element Model Validation

This chapter presents a brief introduction to the finite element method used in this

study. The commercial package ABAQUS was used to establish and validate finite

element models against the experimental results presented in Chapter Three.

Chapter 5 Parametric Study

This chapter presents an extensive parametric study to investigate the effect of

different factors on the shear capacity of ferrocement beams.

Chapter 6 Design Guidelines

This chapter presents two empirical equations which were developed following

statistical analysis of the results from the parametric studies presented in Chapter

Five. A model based on the strut and tie approach (§2.6.4) is proposed along with

shear design procedures for ferrocement beams.

Chapter 7 Conclusion

This chapter summarizes the main conclusions of the thesis and gives

recommendations for future research.

CHAPTER 2 Literature Review

24


2.1 Introduction

This research is concerned with the shear behaviour of ferrocement beams. It has

been found that there is relatively little information on the shear properties of

ferrocement and none of which is compatible with current thinking behind the ACI

code. In this chapter, the previous research and development of ferrocement is

reviewed. The main mechanical behaviour of ferrocement is reviewed along with

research which has been carried out. Also the application of Finite Element Method

in ferrocement is reviewed. The existing component model theories have been

studied. Some of the ferrocement behaviour studies were more than 40 years ago, but

since then, the published works relate mainly to retrofitting reinforced concrete

structures using ferrocement. Further ferrocement mechanical behaviour research

such as tensile, compressive and shear behaviour, has either not been undertaken or

not published. For example, in 2012, 31 ferrocement related documents can be found

on Scopus [7] and only 3 of them are related to the basic mechanical behaviour of

ferrocement elements.

2.2 Historical Background

In 1847, Lambot used wire mesh and concrete to construct a rowing boat (Fig 2.1) in

France [1]. This well-known structure started the development of ferrocement and

modern reinforced concrete. During the period from World War I to World War II,

ferrocement was widely used in the boating industry by using lightweight reinforced

concrete combined with ferrocement [1]. Also during this period, Pier Luigi Nervi in

Italy built a 165 ton motor sail-boat Irene with ferrocement [8]. In 1948 the Italian

architect and engineer Nervi used ferrocement in a storage warehouse and as part of


25

a famous roof structure to the Turin Exhibition Hall [3]. Later some other

ferrocement applications were built such as the Italian Naval Academy Swimming

Pool in Leghorn, and warehouses [8].

Fig 2.1 Lambot’s First Ferrocement Boat[9]

After World War II, ferrocement was widely used in the Soviet Union for large span

roofs of up to 10 million square metres [10]. In the 1960s, due to availability of basic

materials, low cost of labour and low relative manufacturing skill requirements,

storage structures and housing were built in many developing countries [11].

Several academic committees were set up after 1972 to study the behaviour and

development of ferrocement, such as:

1972, Ad Hoc Panel was set up by USA National Academy of Science;

1974 Committee 549 was set up by the American Concrete Institute (ACI);


26

1976 an International Ferrocement Information Centre (IFIC) was founded at the

Asian Institute of Technology, Bangkok, Thailand and in cooperation with the

New Zealand Ferrocement Marine Association (NZFCMA) ;

1979 RILEM (International Union of Testing and Research Laboratories of

materials and structures) established a committee (48-FC).

The most important and influential international committee nowadays is ACI 549

and it published a design guide for ferrocement in 1989 [12]. Although this guide

was published more than twenty years ago, it is still the most important reference in

ferrocement design.

Due to poor availability and the increased fabrication times required by the layers of

mesh in ferrocement, when compared with the larger bars used in reinforced concrete,

its development was relatively slow during the past one hundred years. However, the

development of ferrocement never stopped, some detailed investigations have been

undertaken which are the mechanical behaviour of ferrocement, for example, a

tension study by Naaman and Shah [13], a compression research by Arif et al [14]

and a shear study by Mansur and Ong [15]. These efforts have provided adequate

technical information to enhance the use of ferrocement in new applications. Every

effort greatly stimulated the civil application of ferrocement.

2.3 Ferrocement Structures

Ferrocement structures have lower carbon footprint and are often more economical

compared with normal reinforced concrete structures. Ferrocement structures may

save 1/3 of the cost in material and due to the easy transport of lightweight elements

the overall CO2 emission may be reduced by 41% [16].


27

There are some notable ferrocement structures such as Yanbu Cement Company in

Saudi Arabia which was constructed using ferrocement. From Fig 2.2, highly curved

roof sections can be found, which demonstrates that ferrocement is easy to form in

complex shapes. Another famous ferrocement structure is a 24 m diameter dome in

Oaxaca Mexico as shown in Fig 2.3. This dome successfully survived a 7.5

magnitude earthquake that happened on the 30th

of September 1999. As mentioned in

the previous chapter, ferrocement was chosen by Expedition for the roof of the

Stavros Niarchos Cultural Centre which is currently under construction in the

outskirts of Athens. The appearance of the finished building will be a 100 m double

skin lens shaped canopy which will be covered in PV panels for energy generation. A

16 m prototype of canopy (Fig 2.4) was recently completed to test out the feasibility

of using ferrocement. [2]

Fig 2.2 Yanbu Cement Company [9]


28

Fig 2.3 Ferrocement Dome in Oaxaca, Mexico [9]

Fig 2.4 A 16 m Prototype of the Ferrocement Canopy [17]

Recently in the developing countries some further facilities, such as water tanks (Fig

2.5) and solar water heater panels, may be installed onto ferrocement structures roofs.

These applications may require low costs and lightweight transfer beams. So

ferrocement beams can be an option. Due to their short beam behaviour, shear

studies of ferrocement beams is necessary.


29

Fig 2.5 A Water Tank on Ferrocement House Roof [18]

2.4 Ferrocement Constituents

2.4.1 Reinforcement

One of the differences between ferrocement and conventional reinforced concrete is

the high specific surface of ferrocement. This is due to the closely spaced wire mesh

reinforcements used in ferrocement. These thin wire meshes can be categorised by

woven or welded, wire orientation, spacing and strength properties. Some main types

of meshes are shown Fig 2.6. [4, 10]

Square mesh Hexagonal wire mesh Expanded metal mesh

Fig 2.6 Typical wire mesh used in ferrocement


30

Woven square mesh: This type of mesh is made by woven wires. There is no

welding at the intersections. The thickness of the mesh layer may be up to three wire

diameters. It is more flexible than other meshes and easier to work with.

Hexagonal or Chicken wire mesh: This is commonly used and readily available in

most countries. It is fabricated from cold drawn wire, which is woven in hexagonal

patterns. As all the wires are not continuous along the same direction, so this type of

mesh is bad for resisting axial loading.

Welded square mesh: This type of mesh is made by welding or cementing wires

together at the intersections [4]. So the thickness of the mesh layer is equal to two

wire diameters. The welded meshes have a higher modulus and higher stiffness than

other meshes; they lead to less deflection in the elastic stage.

Expanded metal mesh: Expanded metal mesh is formed by cutting a thin sheet of

metal and expanding it to produce diamond shape openings [19, 20]. This type of

mesh offers approximately equal strength in the normal orientation. It can be used as

an alternative to welded mesh, but it is difficult to use in construction involving sharp

curves. It should be noted that expanded metals are much weaker in the direction in

which the expansion took place.

2.4.2 Matrix and Mechanical Property

Ferrocement matrix is a mixture of cement, well-graded sand, water, and possibly

some admixtures such as silica fume and superplasticizer. Similar to concrete, the

matrix should have adequate workability, low permeability and high compressive

strength. The water-cement ratio, sand-cement ratio, the quality of water, type of

cement, curing conditions and also casting and compaction may influence

mechanical properties of the matrix [10].


31

Ordinary Portland Cement (OPC) is commonly used for the ferrocement matrix. If

special conditions such as early high strength or sulphate resistance are required,

other cement may be used [4].

Due to the high matrix volume, which could be up to 95% of the total, the property

of matrix components and mix ratios will influence the final mechanical properties.

To manufacture high quality matrix, sand which is clean and free of organic

impurities should be used. Also 80-100% of weight of the sand should pass the

British Standard Sieve No.7 (2.36 mm). The sand must provide the required strength,

density, shrinkage and durability of the matrix. In order to have the workable matrix,

which can penetrate through the meshes along with the required strength, a water-

cement weight ratio between 0.35 and 0.55 and cement-sand weight ratio between

1:1 and 1:3 should be used. The higher the sand content, the higher the required

water content to maintain the same workability. Normally the slump of fresh matrix

should not exceed 50 mm. Water for ferrocement should be fresh, clean and free

from organic or harmful solutions. Unclean water may interfere with the setting of

cement and will influence the strength or lead to staining on surfaces [4, 21].

Many different additives are commonly used to increase strength, workability or

chemical resistance. The most commonly used additives are silica fume and

superplasticizer. Silica fume is used to improve cement properties such as

compressive strength, bond strength and abrasion resistance. Superplasticizer is

added as water reducer and improves the workability of ferrocement matrix [22, 23].

Based on the current literature, some different moduli are given for the matrix, for

example, the Young’s Modulus may vary from 5 GPa to over 20 GPa even based on

the same sand-cement and water-cement mixes [14, 15, 24]. As the matrix property

varies among these studies, a separate experimental study needs to be undertaken.


32

2.5 Reinforcement Parameters and Mechanical Properties

Ferrocement elements consist of closely placed multiple mesh layers with small

diameter wires and a matrix with no large size aggregates. It leads to geometrical and

mechanical behaviour differences between ferrocement and reinforced concrete.

Instead of random, wide opening cracks in concrete; ferrocement may have narrow

uniformly spaced cracks. Also with mesh layers, the thickness of covering matrix is

much thinner (3-5mm), which leads to thin sections [25].

2.5.1 Reinforcement Parameters

To provide extra tensile strength in, for example, additional reinforcement, such as

rebar, are used in combination with the mesh. This may also provide shape in the

form of reinforcing steel. Based on the Ferrocement Model Code [26], volume

fraction and specific surface are used to describe the amount of mesh. Reinforcing

bars in ferrocement should be considered separately.

2.5.1.1 Specific Surface (Sr)

Specific surface of meshes is the total surface area of bonded meshes divided by the

volume of the composite [27].

Eq 2. 1

Where:


33

Due to the geometry of mesh layers, in some cases, the specific surface of

longitudinal (Srl) and transverse (Srt) directions are treated separately.

2.5.1.2 Volume Fraction (Vr)

Volume fraction of ferrocement is the percentage of volume of meshes to the volume

of composite.

x 100% Eq 2. 2

It has been suggested that the total specific surface of meshes should be greater than

or equal to 0.08mm2

/mm3 and that the total volume fraction should be greater than

1.8% for ferrocement beams. [4]

2.5.2 Behaviour under Tension

Normally, tensile behaviour of ferrocement can be categorised into three stages: the

elastic stage, the elasto-plastic stage, and the plastic stage as shown in Fig 2.7 [13,

28-30]. Uniform cracks of small width are an advantage offered by ferrocement

structures over conventional reinforced concrete. This may be due to large

reinforcement surface area of the two-way nature of mesh layer type reinforcement

used in ferrocement which may uniformly distribute bond stress along

reinforcements.

During stage I (OA in Fig 2.7), similar to common reinforced concrete, both matrix

and reinforcements perform elastically and no cracking occurs.


34

At the first matrix crack, the stage II (AB in Fig 2.7) begins. It is known that primary

cracks happen randomly at critical sections when the tensile stress exceeds the matrix

tensile strength. With load increments, new cracks may occur in the matrix due to the

tensile stress exceeding the matrix tensile strength. In order to transfer stress between

cracks, more cracks will continue to occur at this stage until the stress in the matrix

will not exceed the matrix tensile strength again and the number of cracks in the

matrix stabilizes.

Fig 2.7 Ferrocement under Tension [3]

During stage III (BC in Fig 2.7) cracks open up rapidly. At this stage, the structural

contribution of the matrix is negligible and the behaviour of ferrocement is

controlled by the reinforcement.

The peak tensile strength was found [13] to be controlled by reinforcement

characteristics, such as the strength, the volume fraction and the orientation of the

wire mesh. Ferrocement tensile failure can be categorized as ductile failure, which

means the matrix is cracked long before failure and does not contribute to peak


35

strength, so the load capacity in this case is independent of the thickness of the

specimen. In ferrocement tensile design, yield strength of the meshes is always used

instead of peak strength to provide extra safety. It is believed that the existence of

transverse wires provide additional strength and safety to ferrocement [3].

Specific surface may also affect the first cracking strength [13]. Irrespective of mesh

size and steel type, by increasing specific surface of reinforcement, the first cracking

load (Point A in Fig 2.7) may increase. However, beyond a certain value of specific

surface, no increase in cracking stress can be observed. Bezukladov et al [31] found

an optimal specific surface of 0.3-0.35 mm-1

.

2.5.3 Behaviour under Compression

Most of the studies on the compression behaviour of ferrocement were conducted

more than three decades ago. The compressive strength of ferrocement is greater than

the compressive strength of matrix cubes from standard 28 days compressive tests.

Some researchers reported that the behaviour of ferrocement in compression, unlike

in tension, is mainly affected by the matrix strength [32, 33]. Sufficient ties across

the mesh layers are critical to avoid delamination due to splitting transverse tensile

stress and buckling of the mesh reinforcement under compression. It has been found

that the strength may be increased by shaping the mesh like a closed box and the

transverse component of reinforcement has more influence than the longitudinal

reinforcement [20]. By increasing the steel content alone, the effect on load capacity

is less important. Orientation of the mesh also is a factor affecting the compressive

behaviour. In a ferrocement column, in which meshes are applied in layers parallel to

loading direction, when the longitudinal wire direction is along the loading direction,

the compressive strength may be higher than other mesh orientations. The expanded


36

metal and its reinforcement orientation have a relatively minor effect in compression,

compared with its effect in tension and flexure [34, 35].

Little research on compressive behaviour had been reported in the past two decades.

Almost all of them were based on repair or strengthening of other structures using

ferrocement. It has been found that by applying a ferrocement jacket the compressive

strength of the composite with a concrete core may increase to 3 times that of the

concrete core alone [36]. In summary, the ferrocement compressive strength is

dependent on the orientation of the mesh layers.

2.5.4 Behaviour in Flexure

Behaviour in flexure of ferrocement is a combination of behaviour in tension and

compression. As discussed before, the influence parameters for tension and

compression are matrix strength, mesh type, mesh properties and mesh orientation.

General behaviour in flexure is similar to tension. It can be categorised into three

stages: the elastic, the elasto-plastic, and the plastic stage [1].

Unlike the cracking control effect on the first crack under tension, specific surface

has less contribution in flexure. This may be because the flexural cracking is mainly

controlled by the outer most layers. Even by increasing the total specific surface, an

increase of the outer most layers may still not be sufficient to increase the cracking

load dramatically [3].

The peak flexural strength was proved to be influenced by the volume fraction and

mesh type [3]. It was found that by increasing the volume fraction the first cracking

load had a less than direct proportional increase. This is due to the fact that first

cracking load is most influenced by the outer most layers.


37

The mesh orientation has a significant effect on the peak strength. According to ACI

549 [1], different meshes show weakness in different directions therefore orientation

is important. When the mesh wire direction is along the principle stress direction the

peak flexural strength may achieve a maximum value. The transverse wires in

ferrocement with square meshes are the most likely location for cracks [6].

So in summary, the amount, type and orientation of the reinforcing meshes are the

main factors affecting the peak flexural strength of ferrocement structures. [5]

2.5.5 Behaviour under Shear

As mentioned in Chapter One, few investigations were reported in the literature on

the shear strength of ferrocement. This is probably due to the fact that ferrocement is

mainly used in thin panels where the span to depth ratio in flexure is large enough so

that shear stresses are not a critical design consideration. In the only authoritative

text book on ferrocement, Naaman [3] identified a lack of design guidance for

ferrocement under shear.

In 1987 Mansur et al [15] investigated the behaviour and strength of ferrocement in

transverse shear by conducting four-point bending tests on simply supported

rectangular beams (100mm x 40mm) with three different lengths: 60, 350 and

380 mm. The meshes were placed along the longitudinal direction of the beam and

lumped close to the top and bottom without vertical reinforcement. The major

variables of the study were the shear span to depth ratio (a/h), volume fraction of

reinforcement and strength of matrix. Their tests indicated that the diagonal cracking

strength of ferrocement increases as a/h decreases, or volume fraction, matrix

compressive strength increases. In addition ferrocement beams were found to be


38

critical to shear failure at small a/h when the volume fraction and matrix strength

were relatively high.

In 1999 Al-Kubaisyand and Nedwell [24] investigated the shear behaviour of

rectangular ferrocement beams (100 mm x 40 mm) without vertical reinforcement.

Their results were compared with the ACI code and empirical formulae were

proposed. However, their formulae have not yet been ratified by the ACI Working

Committee 549 on ferrocement.

In 1991 Al-Sulaimani et al [37] studied the behaviour of ferrocement under direct

shear by conducting four point bending tests on box shaped specimens. The

specimens were at two different spans: 750 mm and 150 mm. The widths of all the

specimens were 150 mm with a height of 200 mm. The meshes were placed in the

centre of the matrix and the thickness was 40 mm. So the meshes in the webs can be

considered as shear reinforcement. Test results indicated that ferrocement under

direct shear exhibits two stages of behaviour (uncracked and cracked). The test

results indicated that the shear stiffness in the uncracked stage was not significantly

affected by the volume fraction, but it is mainly affected by the matrix strength. It

was found that the cracking and peak shear stress increase with increasing the

strength of matrix and the volume fraction of reinforcement. Furthermore, they found

that in the cracked stage the shear strength was affected by both the amount of wire

mesh and the matrix strength.

In 1991 Mansur et al [38] introduced transverse reinforcement in ferrocement

I beams. The beams had a total depth of 200 mm and 150 mm width. The thickness

of the flanges was 30 mm and 25 mm for webs. It was found that increasing value of

matrix strength and volume fraction increases moment capacity but at low shear span


39

to depth ratio, shear failure occurred before the moment capacity was attained. Based

on the experimental results a set of empirical formulae was proposed.

Due to lack of information and theory for ferrocement beams under shear and

because of the similarity of the mechanical behaviour between the ferrocement and

reinforced concrete, the reinforced concrete shear design equation from ACI 318 [6]

was adapted in Ferrocement and Laminated Cementitious Composites [3]. It was

found this equation (Eq 2.3) may give 10% of the observed strength from the

experimental results in this study (Chapter Three).

√ Eq 2.3

Where

shear strength

compressive strength of matrix

2.6 Detail Study on Shear of Beam

In order to study the shear behaviour of ferrocement beams the basic theory of

reinforced concrete beams was reviewed to use the principles to analysis ferrocement

beams and to find differences.

The normal failure of slender flanged RC beams is diagonal splitting (Fig 2.8) of the

web between the edges of loading and support. Web cracking occurs when the shear

stress reach the limiting tensile strength of the matrix. Then the shear stress will

transfer to the shear reinforcements in the web. In normal reinforced concrete beams,

shear failure could be indicated by yielding or failure of shear reinforcements, but in

ferrocement by satisfying the requirement of a minimum volume fraction and


40

specific surface, the shear reinforcement ratio can be 2-3 times greater than that in

reinforced concrete beams. Therefore the shear capacity of ferrocement beams could

be better than RC beams. As described later in Chapter Six, when comparing shear

capacity of ferrocement beams (based on FEMs) with RC beams (based on

prediction of ACI 318 [6]), ferrocement beams show better shear performance.

Fig 2.8 Diagonal Splitting Failure of Reinforced Concrete Beam

2.6.1 Deep Beam

As mentioned in Chapter One, shear failure may occur only when ferrocement beams

have a small shear span to depth ratio (less than 2). In different codes [6, 39] this

kind of structure has been defined as deep beams. For example in Eurocode 2 [39],

deep beam design methods apply for overall span to overall depth ratio (l/h)<4 and

for beam regions with shear span to depth ratio (a/h)<2. A beam having a span to

depth ratio less than 4 or concentrated loading span less than 2 is classified as deep

beams in ACI 318 [6].

There is little published information on the behaviour of reinforced concrete flanged

deep beams and almost no publication is available for ferrocement deep beams. For

deep beams, as mentioned before, strain distribution is influenced by shear and shear

behaviour becomes critical. To give a broader understanding of ferrocement deep

beams, the shear behaviour of reinforced deep beams was investigated in this study.


41

Traditional sectional design methods based on Bernoulli’s theory, do not accurately

predict the behaviour of deep beams. For the normal (slender) beams bending design,

for example, a simply supported ferrocement beam under uniformly distributed load,

as shown in Fig 2.9 (a), Bernoulli’s theory assumes that the cross sections of the

beam remain plane and normal to the axis of bending before and after the load

application. But in deep beams, the effect of transverse shearing deformations must

be considered. As can be seen in Fig 2.9 (b) (c) and (d), for simply supported beams

under a uniformly distributed load when the span to depth (l/h) ratio is less than 4,

the cross section of the beam under shear load remains plane but not necessarily

normal to the axis of bending. The strain distribution of deep beams is no longer

considered linear, and the shear deformations become significant when compared to

pure flexure [5]. It has been found, in this research, that when the shear span to depth

ratio (a/h) is smaller than 1.5, the ferrocement beams can be classified as deep beams.

Fig 2.9 Beam Strain Distribution Diagram [5]


42

2.6.2 Four Modes of Failure for Reinforced Concrete Deep Beams:

For RC deep beams under four points bending, there are four distinct modes of

failure. The failure modes mainly relate to the amount of reinforcement.

Mode of failure 1: flexural-shear failure

This failure type occurs when the tensile reinforcement in the beam is not sufficient.

Flexural cracking may initially occur at the bottom of the midspan of the beam. With

increasing load further flexural cracks will occur in the bending span with diagonal

cracking in the shear span. The final failure occurs at the sudden opening of either

diagonal cracks or flexural cracks.

Mode of failure 2: flexural-shear-compression failure

This failure mode is similar to mode 1, but before the yielding of the tensile

reinforcement, the compressive flange may crush.

Mode of failure 3: Splitting with compression failure

In this failure mode, the diagonal splitting of the web section within the shear span

may occur along with flexural cracking at the flexural span at an early stage of

loading. The diagonal cracking may propagate towards the loading point and support.

Crushing of the compression flange may occur at failure before the diagonal cracking

reaches the flanges.

Mode of failure 4: Diagonal splitting failure

Mode 4 is similar to mode 3 except that at failure the diagonal cracking will reach

the compression flange without crushing. At the peak loading, spalling of concrete

near the loading points may be observed.


43

2.6.3 Component Model

All the models mentioned before were based on reinforced concrete deep beam

studies. Design codes for reinforced concrete sections were generated based on these

studies. Based on some studies of deep beam behaviour over the past two decades,

five component models have been proposed to analyse the shear strength of

reinforced concrete structures [40]. Brief information is reviewed in Table 2.1.

Table 2.1 Component models

Model Name Description Principles

1.Struts-and-

Ties Model

(STM)

Compressive stress in the upper part is

resisted by concrete in the form of a

horizontal strut; tensile stress in the lower

portion is resisted by the reinforcements in the

form of horizontal tie

Equilibrium

condition

2.Equilibrium

(Plasticity)

Truss Model

Plasticity truss model is based on the

assumption that both the longitudinal and

transverse steel must yield before failure. Can

be used to work out moment and forces.

Equilibrium

condition and the

theory of

plasticity

3.Bernoulli

Compatibility

Truss Model

Application of Bernoulli’s hypothesis, the

equilibrium condition and the uniaxial

constitutive relationships of material to the

analysis of reinforced concrete slender beams.

Equilibrium

condition,

Bernoulli

compatibility

condition and the

uniaxial

constitutive laws

4.Upper-bound

theorem

A kinematical admissible failure mechanism

is required and the energy principle is used to

provide the upper-bound load.

Upper-bound

theorem, Yield

line theory,

Plastic theory

The model 2 is all based on non-flanged shear panel experiments, in which

reinforcements were uniformly distributed. So when the cases are panels under pure

shear this theory, can give accurate strength predictions. But in this study where

flanged beams were studied, these theories cannot be applied directly. As in this


44

research, conservative prediction needed to be proposed for the design guideline, the

upper bound theorem was not chosen.

Fig 2.10 Comparison between Strut and Tie method and Sectional Method (based on

Bernoulli’s Theory) [41]

In method 3, the Bernoulli’s theory applies where the assumption of ‘plane section

remains plane’ is valid. In deep beam cases, the shear strain cannot be neglected and

the Bernoulli compatibility condition is highly conservative. Comparison between

sectional methods (Bernoulli’s theory) and the Strut-and-Tie Model (STM) to predict

the capacity of beams with different a/h ratios was done by Collins and Mitchell [41]

as shown in Fig 2.10. As can be found from the graph when a/h is smaller than 2 the


45

STM becomes an accurate method of analysis. However, when a/h is greater than 2

the STM is less accurate than Bernoulli’s theory. This is due to the fact that shear

behaviour (a/h<2) is a brittle behaviour and has a sudden failure mode. But for

flexural behaviour (a/h>2), which is a ductile behaviour, the plasticity behaviour

must be taken into account.

2.6.4 Strut and Tie Model

Since the 1960’s, the Strut and Tie Model (STM), as shown in Fig 2.11, has been

used as a tool for designers to predict the shear capacity of deep beams. This model

is based on an imaginary truss within the beam that transfers the forces from the

loading point to the support using strut and tie elements.

Fig 2.11 Description of Strut and Tie Model [6]

The STM as a technique to analyse and predict peak shear capacity of concrete

beams with non-linear strain distribution has been adopted by many different design

codes such as ACI 318 [6] and Eurocode 2 [39]. This indicates that STM can give a

good prediction of capacity of the concrete members. As discussed before, due to the


46

similarity between ferrocement and reinforced concrete, STM is going to be used as

a component model to predict capacity and be validated with experiments and

parametric studies to provide design guidelines.

The STM is always conservative if the following requirements are met:

Equilibrium of the truss

Sufficient deformation capacity in order to distribute forces to the truss model

The stresses applied do not exceed yield or plastic flow capacity

In practice the reinforced concrete and ferrocement members may have high ductility

to redistribute the stress flow which allows the system to reach a higher load capacity

than in analysis. Also as the STM is not sensitive to the changing of neutral axis, due

to the changing of reinforcement and treats all the elements as isolated members

without considering the interaction between elements so the STM leads to a

conservative load capacity prediction [40]. However, due to the simplicity of this

method for quickly analysing deep beams and as it is easy for engineers to do hand

calculations on site, also in keeping with ACI 318 Appendix A, this method is

adopted in this study to analyse ferrocement deep beams.

From ACI 318 [6], several terms of the STM are defined as follows:

Strut-and-tie model: A truss model of a structural member made up of struts and ties

connected at nodes, capable of transferring the factored loads to supports. (Fig 2.11)

Strut: A compression member in a strut-and-tie model. (Fig 2.11)

Tie: A tension member in a strut-and-tie model. (Fig 2.11)

Nodal zone: The volume of concrete around a node that is assumed to transfer strut-

and-tie forces through the node. (Fig 2.11)


47

Node: The point in a joint in a strut-and-tie model where axes of the struts, ties, and

concentrated forces acting on the joint intersect. Nodes are named based on the

nature of the elements that frame into them. The notation used to denote nodal zones

is as follows:

• CCC: a node bounded by three or more struts (Fig 2.12(a), the top nodal zone in

Fig 2.11)

• CCT: a node bounded by two or more struts and a tie (Fig 2.12(b), the bottom nodal

zones in Fig 2.11)

If more than three forces intersect at a node, it is often necessary to resolve some of

the forces to end up with three resulting forces. For a simply supported beam, a CCC

node generally occurs under the applied load and a CCT node occurs at the support.

Fig 2.12 Classification of Nodes [6]

If the yield capacity of an element is exceeded, the failure modes of a deep beam are

the crushing of concrete in a strut or at the face of a node, yielding of a tie or

anchorage failure of a tie.


48

2.7 FEM in Ferrocement Research

As a time and cost saving tool Finite Element Method (FEM) is increasingly used all

over the world. This computer based method is used by engineers and researchers for

structural design and studies. Although not too much research has been undertaken

for ferrocement in the past 20 years, FEM has been adopted by some of the

researchers in their studies.

In 1989 Bin-Omar et al [42] wrote their own codes based on Timoshenko beam

formulation with combined incremental iterative Newton-Raphson algorithm and its

invariants. The ferrocement beams were treated as a single equivalent material. In

order to have 2D models for the I section, the beams were divided into numbers of

layers; each layer had a different state of stress. For each group in the parametric

study, this stress state must be worked out first. The results from FEM showed close

correlation with experiments. It was also found that the number of elements were

important for producing accurate results.

In 2004 Nassif and Najm [12] used ABAQUS, which is a commercial FE package, to

model the experiments on ferrocement panel strengthened concrete beams under

bending. The load-deflection curves from the models were closely correlated with

experimental results. The 2D, shell and 3D models were built up to compare with

experimental results. It was found that the 3D non-linear models gave most reliable

predictions. In addition, interaction between matrix and mesh layers was reported to

be critical in analysing ferrocement behaviour and perfect bond was chosen as the

choice.

In 2005 Fahmy et al. [43] reported some research on flexural studies of ferrocement

elements using a 3D FE model that was developed to study the behaviour of


49

ferrocement sandwich and cored panels. The results of the ultimate strength from

FEM were compared with experimental results of phases one and two and showed

good agreement.

As can be seen from this section, the use of FEM in ferrocement element studies is

not well developed and this could be for two reasons. First, ferrocement is widely

used and studied in the developing countries where the availability of high speed,

high memory computing equipment and FE program is not as good as in developed

countries. Second, all the basic design is based on elastic behaviour of elements.

Although the ACI 318 code is highly conservative, it can give guidance for safe

design.

But in this shear behaviour study, for an ultimate limit-state design investigation, the

cracking behaviour becomes critical and cannot to be determined by the sectional

method, such as Bernoulli’s theory. So FEM is adopted to study the shear behaviour

of ferrocement beams.

The previous literature shows that FEM can be used in ferrocement studies and can

provide acceptable predictions. The 3D model has been reported to be sufficient for

complex ferrocement behaviour studies. In addition, the 3D model can be used for

parametric studies, especially the effects of geometry and mesh layer numbers. So

three dimensional models are chosen in this study.

2.8 Conclusion

Based on the literature review in this chapter it can be seen that ferrocement has its

own advantages such as lightweight, easy to shape, low carbon foot print and high

surface area compared to reinforced concrete structures. Due to the closely


50

distributed reinforcement throughout its cross sectional area, ferrocement shows

homogenous properties. As ferrocement structures are not the main stream reinforced

concrete structures, the development had been very slow in the last one hundred

years. Thus not too many papers can be found on ferrocement and the literature on

reinforced concrete deep beams was a basis for the current study. The strain

distributions for deep beams are different to slender beams as the shear strain

becomes significant in the shear spans. Current ferrocement design formula [3] is too

conservative. In order to propose ferrocement deep beam shear design guidelines

with more realistic predictions, further studies need to be undertaken.

CHAPTER 3 Experimental Tests

51


3.1 Introduction

As the material properties of the matrix are important input values for the Finite

Element Model (FEM), tests were undertaken to determine these properties. Three

different grades of cement based matrix mixes were designed and tested. Equation

related to the matrix tension and compression behaviour were then developed using

statistical analysis to generate data for numerical analysis. Steel mesh and steel bar

properties were also tested for the required input parameters in the FEA modelling.

Four U beams and three I beams were manufactured and shear tests were undertaken.

Loads and deflections from each test were recorded and analysed. Details are given

in this chapter.

3.2 Matrix Test

Three different mixes were used for the experiments, one with a low water to cement

ratio (0.4) and the other with a higher water to cement ratio (0.55). The third mix

(0.45) was chosen so that its mechanical properties lie between the other two. In

general the experiments were based on the concrete test code: BS EN 12350:2009

“Testing fresh concrete” [44] and BS EN 12390:2009 “Testing hardened concrete”

[45].

3.2.1 Sampling

The samples were taken in accordance with EN 12350:2009 [44]. Materials used in

this section were: Ordinary Portland Cement (OPC), tap water, sand aggregate

passing BS Sieve 7 and additives (silica fume and superplasticizer).


52

Fig 3.1 Matrix Mix

All the apparatus were cleaned prior to use. Before filling the mould, the inner

surfaces were covered with a thin film of non-reactive release material to prevent the

concrete from adhering. Scoops were used to add the concrete incrementally and

ensure that it was uniformly distributed. The date and time of sampling was recorded.

Then the excess concrete above the upper edge of the mould was removed using a

steel float and the surface carefully levelled.

According to EN 12350:2009 [44], mechanical tests should be undertaken in a

saturated condition which means that the specimen needs to be tested whilst it is still

wet; however surface water was removed before the tests. As strain gauges needed to

be mounted onto the specimens which were to be tested for compressive behaviour

on the 28th day after casting, the cylinder specimens had to be taken out of water at

least 10 days before the test. In order to get the final experimental results for the

same conditions, all the specimens were taken out from the water tank to dry on the

18th day after casting.


53

In order to make the uneven end surface of the cylinders flat, a glass capping plate,

coated with a thin film of mould oil was pressed down onto a capping material of

dental plaster. The plate was removed when the material was hard enough to resist

handling damage.

3.2.2 Testing specimens

Before each test, the weight and dimensions of the specimens were recorded for

further calculations and comparisons. All the loose grit and other dust were removed

by wiping to avoid their influence on the tests.

3.2.3 Compressive Test

The cylindrical specimen was centred on the lower platen of the testing machine. A

constant rate of loading was selected at 1 kN/sec; the load and strain were recorded.

The compressive strength is given by the equation:

⁄ Eq 3. 1

Where:

fcm compressive strength, in MPa;

F maximum load at failure, in N;

Ac cross-sectional area of the specimen on which the compressive force acts

calculated from the designated size of the specimen in mm2.

As shown in Fig 3.2, the vertical and horizontal gauges were applied to one surface

of specimens in the compressive tests. A total of 9 cylinders, with 3 for each group,

were used to test compressive behaviour. The strain readings were recorded for

calculation of Young’s modulus and Poisson’s Ratio. The vertical gauge was used to


54

record vertical strain and horizontal gauge was used to record the horizontal

expansion. All the failure types of the compressive test specimens must satisfy the

requirement in EN 12350:2009 [44], a typical failure type is shown in Fig 3.2.

Fig 3.2 Gauges used in Cylinder Compressive Tests and a Typical Failure

Poisson’s Ratio is given by the equation (considering the strain values at 0.4 as

this part can be considered to be linearly elastic):

Eq 3. 2

Where:

vertical strain at 0.4

horizontal strain at 0.4

compressive strength, in MPa

3.2.4 Tensile Test

For an indirect tensile test from cylinder splitting, a jig was used to position the

specimen with packing strips which were made of plywood. Packing strips were only


55

used once. Before the test, the jig and packing strips were wiped clean. Then the

specimen was placed in the jig as shown in Fig 3.3 (a). A Total of 9 specimens, with

3 for each group, were tested. All the failure types of the tensile test specimens must

satisfy the requirement in EN 12350:2009 [44]. A typical failure type is shown in Fig

3.3 (b).

(a) (b)

Fig 3.3 Jig with Packing Strips and a Typical Failure Type

The required loading rate was given by the formula [44]:

Eq 3.3

where:

R required loading rate, in N/s;

l length of the specimen, in mm;

d designated cross-sectional dimension, in mm;

s stress rate, in MPa/s

Based on BS EN 12390:2009, the constant stress rate should be within the range

0.04 MPa/s to 0.06 MPa/s. So based on the dimensions and the above formula,

0.05 MPa/s was chosen for all the tensile splitting tests.


56

The tensile splitting strength is given by the formula [44]:

dL

Ffctm

2 Eq 3.4

Where:

fctm tensile splitting strength, in MPa;

F maximum load, in N;

L length of the line of contact of the specimen, in mm;

d diameter of specimen, in mm.

3.3 Matrix Property Analysis

To use the experimental data in subsequent numerical simulations, some calculations

need to be undertaken to provide values for Young’s Modulus, Poisson’s Ratio and

the plastic stress-strain relation input.

Based on Structural Eurocode 2 [39], the modulus of elasticity, Ecm, of concrete is

controlled by the moduli of elasticity of its components. For this experiment, the

Young’s modulus was taken as Secant Modulus from the origin up to 0.4 fcm.

As mentioned before, three different mix groups were designed. For each group six

Φ100x200mm size cylinders were made, three for compressive tests and three for

tensile splitting tests, so in total 18 cylinders were made. After tests, the average

values were found and used for hand calculations and in the FEM. Table 3.1 shows

matrix and characteristic strengths for each mix group.


57

Table 3.1 Matrix Characteristic Strengths with Standard Deviation in Brackets

Mix 1 2 3

water-cement ratio 0.4 0.45 0.55

cement-sand ratio 0.5 0.5 0.5

silica fume

Yes(10%

weight of

cement)

Yes(10%

weight of

cement)

No

Super plasticizer Yes(2% weight

of cement)

Yes(2% weight

of cement) No

Young's Modulus (GPa) 23 (0.55) 19 (0.54) 16 (0.48)

Poisson's ratio 0.21 (0.11) 0.2 (0.05) 0.2 (0.03)

Compressive strength(MPa) 63 (2.16) 38 (4.20) 32 (3.75)

Tensile strength(MPa) 4.5 (0.41) 4.1 (0.40) 3.7 (0.31)

In order to carry out a parametric study of matrix strength (σ), matrix strengths at

constant intervals need to be provided for the ABAQUS modelling. This data

includes Young’s Modulus (Ecm), compressive strength (fcm) and tensile strength (fct)

[46]. Based on current literature and regional structural codes, no direct information

can be found for mechanical behaviour of ferro-matrix. Most codes or references in

literature are concrete based. Structural Eurocode 2 [39], CEB-FIP model code [47]

and Popovics [48] were adopted for this study. The equations from these codes are

justified by ferro-matrix mechanism experiments.

3.3.1 Compressive Behaviour of Matrix

For the matrix cylinder compressive stress-strain relationship, the equation proposed

by Popovics [48] was taken for this study as Eq 3.5. Experimental values (peak stress

fcm and relative strain ) were used in Eq 3.5. It was found that the curve generated

was close to the experimental result, as shown in Fig 3.4. Therefore Eq 3.5 was

adopted for the parametric study. To use this formula, two variables need to be

selected: peak stress (fcm) and relative strain ( ).


58

(

) (

( ⁄ ) ) Eq 3.5

Where:

compressive stress

fcm peak compressive stress

compressive strain

compressive strain at peak stress

n approximate function of compressive strength of matrix,

0.000 0.005 0.010 0.015 0.020 0.025

0

10

20

30

40

50

60

70

Str

ess (

MP

a)

Strain

Experimental result

Prediction

Fig 3.4 Comparison of Prediction (Eq 3.5) and Experimental Results

3.3.2 Compressive Stress-Strain Relation of Matrix

To select peak compressive stress and related strain for different matrix strengths, a

statistical approach based on experimental results was made as shown in Fig 3.5. The


59

linear curves in Fig 3.5 and Fig 3.6 are highly simplified. If more data are available,

the trend of the curves may be different.

Fig 3.5 Fitting Curve of Peak Compressive Stress and Strain

The linear fit to the experimental data using Excel was:

Eq 3.6

( )

Where the proportion of variability, R2, was 0.90

Using this equation the peak stress (fcm in MPa) and relative strain ( ) can be found

for the parametric study.

3.3.3 Tensile Strength and Compressive Strength Relationship of the Matrix

To decide peak tensile stress to the related compressive stress, a statistical analysis

approach based on experimental results was made as shown in Fig 3.6.


60

Fig 3.6 Fitting Curve of Peak Tensile Stress and Peak Compressive Stress

The equation for a linear line fitted to experimental data returned from Excel was:

Eq 3.7

( )

Where R2 (the proportion of variability) was 0.89

By using this equation, peak tensile stress (fctm in MPa) based on relative compressive

stress ( in MPa) can be determined for ABAQUS input.

3.3.4 Tensile Behaviour of Matrix

In the ABAQUS concrete damage model, the post-peak behaviour of the matrix

needs to be defined to get an accurate analysis. For behaviour in tension the post-

peak tension failure behaviour is hard to obtain from the cylinder tensile splitting test .

So for this study it was based on CEB-FIP model code 1990 [47].


61

Fig 3.7 Post-peak Tensile stress-cracking width curve [47]

The post-peak stress-cracking width curve is used to define the post-peak tension

behaviour as shown in Fig 3.7. Peak tensile stress (fctm) can be obtained from either

experimental or empirical equations. The missing parameters for this bi-linear curve

are w1 and wc. Based on CEB-FIP model code 1990, relevant equations are:

Eq 3.8

Eq 3.9

Where

w1 cracking width at 0.15 fctm

wc cracking width at 0.0001 fctm

GF fracture energy (MPa/mm)

fctm tensile strength (MPa)

coefficient based on aggregate size; due to the absence of coarse aggregate in

ferro matrix, the recommended minimum value of 8 was taken in this study

The fracture energy (Gf) is specified as [47]:


62

(

⁄ )

Eq 3.10

Where

GF0 base values of fracture energy; the value is associated with aggregate size,

0.025 MPa/mm is taken for this study based on aggregate size as

recommended in CEB-FIP model code 1990; due to the absence of coarse

aggregate in ferro matrix, the minimum value of 8 was taken for this study

fcm matrix peak compressive strength, MPa

By using the equations from this section the matrix properties for compressive

strengths of 30MPa, 40MPa, 50MPa and 60MPa are determined and used in

ABAQUS.

3.4 Mesh Property Test

3.4.1 Introduction

The reinforcement used was galvanized welded square wire mesh 1.6 mm diameter

with 12.5 mm openings. Using the available tensile testing equipment the mesh

tensile property was tested for this study.

3.4.2 Detailed Mesh Property Test

In order to obtain accurate property values an INSTRON 4507 with maximum

capacity 200kN was used for the tests. A 50mm extensometer was applied to the

specimens to measure precise results. As mentioned in the literature review, the mesh

opening may influence the tensile cracking behaviour. The smaller mesh opening is

better for cracking control. Also it has been noted that the welded mesh is closest to


63

steel reinforcing bars used in reinforced concrete, its apparent elastic modulus is the

same as the steel wire from which it is made due to welding. Due to this reason and

stock availability in the market, the commonly used mesh type: 1.6mm wire diameter

with 12.5mm openings welded mesh was chosen for this study. All the following

studies were based on this mesh.

Single strip wire specimens were carefully cut from a mesh roll using electric shears.

Each specimen was 150 mm in length and was clamped into the machine using V

shape gripper jaws at both ends to avoid the specimens slipping during testing as

shown in Fig 3.8. The strains of the specimens were measured using the 50mm

gauging potentiometer, by which the modulus of elasticity (E) was determined. Tests

were conducted at a crosshead displacement rate of 5 mm/min until failure.

Fig 3.8 Mesh Wire Tensile Test Set Up

The load-displacement data were recorded. For each mesh type, a minimum of three

specimens were tested and the average values were determined for comparison (Fig

3.9). Yielding stress and relative strain were taken to work out the Young’s Modulus.


64

From the tests, average moduli were found: the Young’s modulus of the cold rolled

wire mesh is 150 GPa, the yield stress is 330 MPa and the ultimate stress is 410 MPa.

0.00 0.05 0.10 0.15 0.20

0

50

100

150

200

250

300

350

400

450

Str

ess (

MP

a)

Strain

Mesh 01

Mesh 02

Mesh 03

Fig 3.9 Stress-Strain Curves of Mesh

3.5 Steel Bar Test

12 mm ribbed mild steel reinforced bars were used as reinforcement for the I-Beams.

So the basic material property was tested at this stage. The same INSTRON 4507

was used as used in the mesh tests and three specimens were tested. From the tests

the average moduli were found: the Young’s Modulus was 197GPa, the yield stress

was 475MPa and the ultimate stress was 574MPa.

3.6 Ferrocement Manufacture

By using the materials mentioned above, I beams and U beams were designed and

manufactured. In this study, four U-beams and three I beams were tested. All U-


65

beams were designed based on the same criteria with 3mm matrix cover all round.

Fig 3.10(a) shows the dimensions of 2 layers of welded wire mesh. All beams had

the same cross-sectional dimensions and were reinforced in a symmetrical manner,

as shown in Fig 3.10(b).

(a) U-beam

(b) I-beam

Fig 3.10 Reinforcements and Design Dimensions (all in mm)

In practice matrix is often applied to ferrocement by means of an injection pump or

spray, however, this equipment is not appropriate in the laboratory. In addition the

traditional ferrocement fabrication method using manual application also has its


66

disadvantages. Also as shown in Fig 3.11, large area of mesh exposure can be seen,

which will dramatically influence the mechanical performance of ferrocement

products. In order to fabricate reliable specimens for testing a simple fabrication

method with less equipment requirement and high product quality needs to be used.

Fig 3.11 U Beam with Bad Finish (exterior view)

3.6.1 Fabrication method

In general, the whole procedure can be split into several steps: cutting, detailing,

bending, tying, spacing for proper location and casting. Each step is given below in

more detail.

Step 1 Cutting and detailing

Steel mesh was supplied in the sheet roll as shown in Fig 3.12. This packing method

causes curvature of the mesh, so the mesh sheets were straightened by hand to get rid

of any curvature. Next the mesh plates were cut from the roll with rough finishes

using electric shears. Then the meshes were put into a cropping machine to remove

the excess wire ends and provide smooth edges as shown in Fig 3.13,


67

Fig 3.12 Mesh sheet roll

Fig 3.13 Cropping Machine


68

Step 2 Bending

Then the meshes from step 1 were bent to right angles with design dimensions by a

bending machine as shown in Fig 3.14.

Fig 3.14 Bending Machine

Step 3 Tying

The meshes for each layer were assembled together as shown in Fig 3.15.


69

Fig 3.15 Assembly of Mesh Layers without Cable Ties

Then the mesh layers were tied using cable ties. The cable ties were placed all over

the meshes with a roughly 10cm distance between each two ties as shown in Fig 3.16.

Fig 3.16 Assembly of Mesh Layers with Cable Ties


70

Using cable pliers, the cable ties were tightened as shown in Fig 3.17.

Fig 3.17 Cable Tie Pliers

Step 4 Spacing

After the ties were fastened, the tails of the ties are cut off as shown in Fig 3.18. As

3mm matrix cover needs to be applied to both sides of the section, the cable tie tips

were used as spacers.

(a) U section (b) I section

Fig 3.18 Assembly of Mesh Layers with Trimmed Cable Ties


71

Then the meshes were put into the oiled moulds for casting (Fig 3.19).

(a) U section (b) I section

Fig 3.19 Mesh and Oiled Mould

3.6.2 Fabrication and Improvement

The traditional laboratory fabrication process for ferrocement is to leave the meshes

in the mould and put the matrix on top; then using floats to penetrate the matrix

through. The problem with this method is:

(1) The flowability of the matrix mix is not very good so it is hard to penetrate

through the mesh layers just by gravitational force;

(2) As the common moulds used in laboratory works are non-transparent, so it is

impossible to see the blind surface whether the surface is fully covered by the matrix

or not.


72

So after release of the moulds, some bad finishes could be found such as shown in

Fig 3.11. In order to solve these problems and make the experiments reliable, new

fabrication methods were proposed for each of the different structures.

3.6.2.1 Flat structure (U section)

Firstly, a thin layer of matrix was poured into the mould which was left on the

vibration table.

Secondly, turn on power of the vibration table and the meshes were forced to

penetrate through the matrix.

Finally, use the floats to build up the webs and level the surface. By this method, the

outer surface can be guaranteed to have a good finish as shown in Fig 3.20.

Fig 3.20 Fabrication with a Fine Surface

3.6.2.2 Slender structure (I section)

Based on former experience in manufacturing ferrocement I-sections, the matrix has

a difficulty in penetrating the thin web section. This led to the exposure of mesh in

the bottom flange. In order to solve this problem, the I section mould was reduced in


73

length and stood on end. A timber base and sides were attached as shown in Fig

3.21(a) and Fig 3.21(b) to seal the section.

(a) (b)

Fig 3.21 I-section Moulds (a) Top and bottom plate (b) Base plate

The casting procedures are described as follow:

Step 1: All the bolts were released before cleaning and oiling the inner face of the

mould. After that, the mould was recovered and moved to the vibration table, located

vertically on the end plates and clamped to the vibration table as shown in Fig 3.22.


74

Fig 3.22 Clamped Mould

Step 2: By using a vibrating poker (Fig 3.23), air was removed from the matrix to a

point where the mould was 95% filled as shown in Fig 3.24.

Fig 3.23 Vibrating Poker

Fig 3.24 Matrix Filling and Vibration


75

Step 3: Then with the vibration table on, the mesh reinforcement was inserted from

the open end and lowered through the liquefied matrix as shown in Fig 3.25.

Fig 3.25 Mesh Dropping and Finish

Step 4: The opening gap was filled with matrix mix and the surface was floated as

shown in Fig 3.26.

Fig 3.26 Finish Surfacing


76

Step 5: The moulds were covered with plastic sheets for ferrocement curing (Fig

3.27).

Fig 3.27 Storage of Specimen

Using this method the quality of the specimen is largely improved (Fig 3.28)

showing a smooth finish.

Fig 3.28 Final Appearance of Specimens

3.7 Test Setup

The beams were tested by four-point bending as shown in Fig 3.29 and Fig 3.30

using an Avery servo controlled testing frame. Steel rollers of 60 mm in diameter

were used as supports. Two steel tubes of 150 mm diameter were used for loading

points with a rectangular steel hollow beam lying between the equipment and the

tubes. All the readings from the Avery and linear potentiometers were recorded using

an Orion data logging system. Throughout the test, photos were taken for the general

shape of the beam, cracking behaviour, failure mode and any other features that were


77

observed. Detailed gauge set ups will be described in later sections. Before tests all

the dimensions of each beam were carefully measured as shown in Appendix A.

Fig 3.29 Specimen and Gauges Set Up

Cross section of A-A’

(a) U Beam (U1)


78

Cross section of B-B’

(b) I Beam (I1)

Fig 3.30 Test Set Up (all in mm)

The specification for the beams are shown in Table 3.2.

Table 3.2 Beams Specification

Series

No. of layers of

wire mesh in the web

Nm

Transverse steel ratio

(%)

Longitudinal steel ratio

(%)

Matrix Mix water-cement

ratio

Shear-span-to-depth ratio a/h

U1 2 2.6 2.9 0.55 1

U2 3 3.1 3.5 0.4 1

U3 2 2.6 2.9 0.4 1

U4 2 2.6 2.9 0.4 2

I1 2 0.3 1.37 0.4 1

I2 2 0.3 1.37 0.4 1.34

I3 2 0.3 1.37 0.4 1


79

3.8 I beam test results and analyses

As shown in Fig 3.31, deflections at midspan (V2) and under point loads (V1 and V3)

were measured using linear potentiometers. The seven gauges measured horizontal

strain at midspan (4 (H1, H2, H6 and H7) on the flanges and 3 (H3-H5) on web)

together with two sets of three gauges (T1-T6) as equilateral triangle rosette shape at

the centre of the shear span. The strains were measured using Sakai FLP 10 linear

potentiometers.

Fig 3.31 Gauges Position and Names of I Beam (I1)

In general, after the load was applied to the beam, diagonal cracking occurred at the

side support then grew diagonally into the web and finally reached the loading


80

position on the top flange. All beams developed significant cracking before failure.

Two types of cracking were observed during the test. Shear cracks occurred in the

shear span, between the loading point and support and flexural cracks which

occurred at the bottom flange. Clear flexural cracking was observed only in test I2.

Typical cracking patterns are shown in Fig 3.32. All three beams behaved in a

similar manner with I2 being typical. So the I2 results together with descriptions of

its differences with the other 2 beams are presented here. All the other results and

descriptions are shown in Appendix B.

a) b)

A: Shear Span, B: Bending Span

Fig 3.32 Typical Cracking Pattern, a) Shear Cracks, b) Shear and Flexural Cracks

3.8.1.1 Vertical Gauges Results of Beam I1

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Deflection (mm)

V1

V2

V3

Fig 3.33 Load-Deflection Curves of Beam I1


81

From Fig 3.33 it can be found:

1. The peak load was 285kN and the deflection of V3 at the peak load was 1.97mm.

2. In general all three curves were very close to each other. V1 and V3 were very

close until peak load, which means the test frame was balanced. Failure occurred

at the V3 (right) side.

3. The stiffness of the three curves was initially the same. A difference occurred at

about 150kN. V2 was at the midspan. Due to bending curvature, the deformation

at mid span was greater than V1 and V3 which were under the loading positions.

3.8.2 Gauges Results of Beam I2

As mentioned before, the behaviour of beam I2 is more critical than I1 and I3, so the

detail results of I2 are presented in this section.

3.8.2.1 Vertical Gauges results

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Deflection (mm)

V1

V2



82

From Fig 3.34 It can be seen:

1. The vertical LVDTs V1 and V2 show similar deflection values under the loading

point and the midspan. The gauge V3 became unresponsive during the test and

no meaningful results could be obtained. Hence we could not verify the

symmetric nature of the deflection profile. However, the rosette gauges on the

web confirmed a symmetric response.

2. The peak load was 273kN and the deflection of V1 at peak load was 3.15mm.

3. The stiffness of the two curves was initially the same. A difference occurred at

about 150kN. V2 was at the midspan. Due to bending curvature, the deformation

at mid span was greater than V1 which was under the loading position.

3.8.2.2 Horizontal Gauges Results of Beam I2

-2000 -1000 0 1000 2000 3000 4000 5000 6000 7000

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

187.5mm from bottom

265mm from bottom

310mm from bottom

Fig 3.35 Horizontal Gauges Load-Micro Strain Curves of Beam I2


83

-2000 -1000 0 1000 2000 3000 4000 5000 6000 7000

0

50

100

150

200

250

300

350

Lo

ca

tio

n (

mm

)

Micro-Strain

47.36kN

96.83kN

154.07kN

204.39kN

261.00kN

273.00kN

Fig 3.36 Horizontal Gauges Location-Micro Strain Curves of Beam I2. The location

along the vertical axis indicates the distance of the gauge position from the bottom

surface.

From Fig 3.35 and Fig 3.36 it can be seen that the neutral axis was at about 250mm

from the bottom of the beam. For this beam the relationship between the longitudinal

strain and the load was roughly linear for load below 260kN. But after this load the

main bars yielded and elongated rapidly, so the final failure may be flexural. This

can be checked with vertical strain curves.

3.8.2.3 Rosette Gauges Results of Beam I2

For the equiangular strain- rosette used in the tests the following equations were used

to analyse the results of the left hand side gauges (Fig 3.31). The results of right hand

side gauges can be generated by the same method.

Eq 3. 11


84

( ) Eq 3. 12

√ ( ) Eq 3. 13

Where:

strain in the horizontal direction

strain in the vertical direction

engineering shear strain

strain from each gauge reading, for example is the strain of T1 in Fig 3.31

-0.03 -0.02 -0.01 0.00 0.01

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Strain

Strain X from Rosette Gauges on LHS of Beam

Strain X from Rosette Gauges on RHS of Beam

Strain Y from Rosette Gauges on LHS of Beam

Strain Y from Rosette Gauges on RHS of Beam

Fig 3.37 Load-Strain Curves of Rosette Gauges

For Strain X from Fig 3.37:

1. The strain values at the same load level from Fig 3.37 were the same as the value

from Fig 3.35 at the same location.

2. The two curves for Specimen I2 were different initially. One explanation could be

the gauge on the left surface may have become jammed with dust and at a certain

load level (75kN in this test) the jammed dust was crushed then the reading caught

up with the left one. After this stage, readings were very close. As the shear


85

failure happened at the left shear span, the changes in the curves became more

gradual.

For Strain Y from Fig 3.37

1. The two curves for Specimen I2 were very close. At 50 kN the two curves

differed. Due to the shear failure which occurred at the left side the strain of this

side was more than that on the right side.

2. Negative results indicate compressive behaviour in this direction.

-0.010 -0.005 0.000 0.005 0.010

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Shear Strain

Shear Strain from Rosette Gauges on RHS of Beam

Shear Strain from Rosette Gauges on LHS of Beam

Fig 3.38 Load-Shear Strain Curves of Beam I2

From Fig 3.38:

1. Curves were roughly the same in value and have different signs. This is due to the

strain direction difference.

2. The gradient changes of the right curve may be due to the flexural cracking that

occurred close to the right side, so the change of the right side is greater than the

left.


86

Based on Mohr’s Circle, the following equations are used to work out principal strain

and principal angles.

Eq 3.14

√(

)

(

)

Eq 3.15

√(

)

(

)

Eq 3.16

Where

principal angle

maximum principal strain

minimum principal strain

0.000 0.005 0.010 0.015

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Max Principal Strain

Max Principal Strain from Rosette Gauges on LHS of Beam

Max Principal Strain from Rosette Gauges on RHS of Beam

Fig 3.39 Load-Max Principal Strain Curves of Beam I2


87

From Fig 3.39 it can be seen:

Two curves were very close until 150kN. As the shear crack occurred through rosette

gauges on LHS of beam, the maximum principal strain curve of the left gauges

showed a quicker increase than the right group. Due to the stress transferred to the

meshes after cracking, no strain jump was observed.

-0.03 -0.02 -0.01 0.00

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Min Principal Strain

Min Principal Strain from Rosette Gauges on LHS of Beam

Min Principal Strain from Rosette Gauges on RHS of Beam

Fig 3.40 Load-Min Principal Strain Curves of Beam I2

In this section, the strains in the longitudinal and vertical directions were transformed

into principal strains. Direct (normal) tensile strain was considered as positive and

direct compressive strain as negative. It should be noted that the strain results from

beam experiments were much greater than the design code recommended value of

ultimate compressive strain of matrix. This highlights the presence of confining

effect in the beam, which improves both the ultimate strength and strain capacity of

unconfined matrix.


88

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Principal Angle (Degree)

Principal Angle from Rosette Gauges on RHS of Beam

Principal Angle from Rosette Gauges on LHS of Beam

Fig 3.41 Load-Principal Angle Curves of Beam I2

In Fig 3.41, principal angles at both ends increased gradually at the beginning. As the

principal angle at the RHS was smaller than the LHS, the right side of the beam was

dominated by flexural and the left side was dominated by shear. This indicated the

reason why at about 75 kN flexural cracks occurred close to the RHS and shear

cracks occurred at left shear span. Principal angles of I2 were about 20-30 degree

until failure which means the failure was shear failure. Diagonal cracking observed

in experiments can partly prove that the failure was shear failure.


89

3.8.3 Vertical Gauges results of Beam I3

0.0 0.5 1.0 1.5 2.0 2.5

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Deflection (mm)

V1

V2

V3


From Fig 3.42 it can be seen:

1. The peak load was 212 kN and deflection of V1 at peak load was 1.73 mm.

2. In general, all three curves were very close to each other. V1 and V3 were very

close until peak load, which means the test frame was balanced. Failure occurred

at the V1 (left) side.

3. The stiffness of the three curves was initially the same. A difference occurred at

about 75 kN. V2 was at the midspan. Due to bending curvature, the deformation

of mid span was greater than V1 and V3 which were under loading positions.


90

(a) Three Stages of Load-deflection Curves

0 1 2 3 4

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Deflection(mm)

I1

I2

I3

(b) Load against Deflection of V1 or V3 Curves of I beams

Fig 3.43 Load against Deflection Curves in respect of the Recognised Three Stages

of Ferrocement Behaviour


91

Fig 3.44(a) shows the typical behaviour of ferrocement. When compared to Fig

3.43(a) and looking at the strains shown in Fig 3.43(b) for beam I2 it can be seen that

the first stage (AB), up to the ultimate matrix strain in the bottom flange, ended at

20 kN. Before this load it may be seen that the two lowest gauges, which were

attached to the flange, acted together but above this load departed from each other.

This indicates that the flange was acting as a single unit during this stage. During the

second stage (BC) strains were transferred to the reinforcement with increasing crack

numbers. Though it may be seen that the lowest reinforcement started to yield at

about 175 kN it wasn't until all the reinforcement in the bottom flange had reached

yield strain, at 240 kN that the beam started to enter the third stage (CD). Above this,

there was very little increase in load, a further 30 kN, however yielding may be seen

in the web above the flange and increasing deflection until final failure.

0 1000 2000 3000 4000 5000 6000 7000

0

50

100

150

200

250

300

350Mesh Yield Strain

Rebar Yield Strain

Matrix Splitting Strain

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

(a) I1 Beam Horizontal Gauges Reading against Load


92

0 1000 2000 3000 4000 5000 6000 7000

0

50

100

150

200

250

300

350Matrix Splitting Strain Mesh Yield Strain

Rebar Yield Strain

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

187.5mm from bottom

(b) I2 Beam Horizontal Gauges Reading against Load

0 1000 2000 3000 4000 5000 6000 7000

0

50

100

150

200

250

300

350

Rebar Yield Strain

Mesh Yield StrainMesh Yield Strain

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

(c) I3 Beam Horizontal Gauges Reading against Load

Fig 3.44 Comparison of Beam Behaviour with Load


93

By comparing horizontal gauges below the neutral axis in the graphs for I1, I2 and I3

the same behaviour may be observed. As shown in Fig 3.44(a) and Fig 3.44(b), for

the same beam, by reducing a/h, the neutral axis moved down. This is because there

is more shear effect, which can influence the strain distribution, for beams with

smaller a/h as mentioned in § 2.6.1. It was also found that the reinforcement in the

bottom flanges of both I1 (Fig 3.44(a)) and I3 (Fig 3.44(c)) were not fully yielded

before final failure. So no stage 3 was observed in either. Looking at the results, it

can be seen that for beam I1, which has the highest initial stiffness and the smallest

load span-depth ratio (a/h=1.0), failure occurred before the third stage.

It can clearly be seen that:

1. The peak load for beam I1, with smaller load span-depth ratio, was higher than

beam I2 with a larger ratio.

2. The initial stiffness of the beam is influenced by the matrix stiffness. The higher

the elastic modulus of the matrix the stiffer the beam will be initially.

3. The peak load for the stronger grade of matrix mix (0.4) is higher than the

weaker mix (0.55).

Based on the rosette gauge results and Mohr’s Circle analysis, the Principal Angles

for all the beams were between 20 to 30 degrees, which indicated that all the failures

were shear failure.

3.9 U Beam Tests Results

For each U beam test nine vertical gauges were used to record deflection. Three

gauges were under the left loading position, three gauges were under midspan and

three gauges were under the right loading position. For each position, one gauge was


94

set under front web, one gauge was set under the middle of the flange and one gauge

was set under back web. The gauges setting up for U beam is shown in Fig 3.45.

Fig 3.45 Gauges Location of U Beam Tests

As the web section of the U beam was very thin, about 12.4mm, to avoid local

crushing of the webs at the supports due to stress concentration two steel strips

40mm wide and 12mm thick were made to attach to the web of U beam as shown in

Fig 3.46.


95

Fig 3.46 Steel Strip

In general, all the beams developed flexural cracks in the web area before the peak

load was achieved irrespective of the a/h ratio, matrix mix or number of layers. With

the formation of the flexural cracks, for the beams with small a/h ratio, diagonal

cracks occurred in the shear span from the top surface to the bottom of the web

sections. For those beams with short shear span to depth ratio and low volume

fraction, the final failure type is shear failure whereas the others are flexural failure.

The details of each test are shown below.

As shown in Table 3.2, U1 was with low matrix strength and small a/h ratio. In the

test procedure, several flexural cracks occurred first between two loading points as


96

shown in Fig 3.47. Then a diagonal crack occurred in the shear span which started

from the top and grew to the bottom. After this stage, multiple shear cracks formed

as shown in Fig 3.48. With the increasing load, sudden diagonal tension failure

occurred in the shear span as shown in Fig 3.49. As can be seen in Fig 3.50, several

flexural cracks occurred under bending but they did not open up during the test

procedure.

Fig 3.47 Occurrence of Flexural Cracking in Beam U1

Fig 3.48 Occurrence of Diagonal Cracking in Beam U1


97

Fig 3.49 Diagonal Tension Failure in Beam U1

Fig 3.50 Flexural Cracking of Beam U1 at Ultimate Stage


98

0 2 4 6 8 10

0

5

10

15

20

25

Lo

ad

(kN

)

Deflection (mm)

F1

M1

B1

F2

M2

B2

F3

B3

Fig 3.51 Load-Deflection Curves of Beam U1

All the specimens were tested in the laboratory environment. Although the improved

fabrication system can produce a smooth outer surface, due to lack of skill the inner

surfaces were not smooth, resulting in non-uniformity of the thickness of the webs

and the flange. This partly explains the differences between the gauges under the

front web (Fig 3.51). The gauge M3 was not responding during the test so its reading

was ignored.

For the specimen with 3 layers of mesh, flexural cracks occurred at the bottom of the

beam and diagonal cracks occurred in the shear span which started from the top and

grew to the bottom. Instead of sudden tension failure in the shear span, the flexural

cracks under the loading points opened up and led to yielding of mesh. Subsequently

the flexural failure occurred as shown in Fig 3.52.


99

Fig 3.52 Flexural Failure of Beam U2

0 5 10 15 20 25

0

5

10

15

20

25

30

Lo

ad

(kN

)

Deflection (mm)

F1

M1

B1

F2

M2

B2

F3

B3


Fig 3.54 shows Test U3 with the beam facing up. In practise U sections that were

investigated in this study are used on house roofs. They are normally assembled

alternately face up and face down with inner web faces attached together. This test

was set to study the behaviour with the alternative orientation. The bottom supports


100

of testing rig were raised up at a steady speed (0.01mm/s) and the top beam was kept

constant throughout the test.

Fig 3.54 Test Set Up of Beam U3

As the tension area for this beam was much stronger than the others, no flexural

cracking occurred throughout the whole beam test but diagonal shear cracking and

failure in the shear span was observed, as shown in Fig 3.55.

Fig 3.55 Diagonal Tension Failure of Beam U3


101

0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

20

25

30

Lo

ad

(kN

)

Deflection (mm)

F1

M1

B1

F2

M2

B2

F3

M3

B3


For the specimen with a/h=2, multiple flexural cracks occurred in the flexural span

without diagonal shear crack in the shear span. Instead of sudden tension failure in

the shear span, the flexural cracks at mid span opened up and led to yielding of the

mesh, subsequently the flexural failure of the beam as shown in Fig 3.57.

Fig 3.57 Flexural Failure of Beam U4


102

-2 0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

Lo

ad

(kN

)

Deflection (mm)

F1

M1

B1

F2

M2

B2

F3

M3

B3

Fig 3.58 Load-Deflection Curves of U4

The following can be concluded from the experiments (Fig 3.51, Fig 3.53, Fig 3.56

and Fig 3.58):

1. From test U1, U2 and U4, no matter what the final failure type was, flexural

cracks occurred within all beams.

2. From all 4 tests, only the cases with short a/h ratio (U1, U2 and U3) developed

diagonal shear cracks at shear span.

3. Shear failure only occurred for small a/h ratio (a/h<2) and low volume fraction (2

layers of mesh).

3.10 Summary and Conclusion

In this chapter the detail of testing of material properties, the fabrication method and

testing of specimens were presented. In summary and conclusion:


103

1. The material properties of the matrix and reinforcements have been evaluated. It

has been found by reducing the water-cement ratio the compressive strength of

the matrix will increase.

2. The improved ferrocement beam fabrication methods presented in this chapter

can enhance specimen quality.

3. The shear cracking and failure mode depend on the beam geometry,

reinforcements, U beam orientation and a/h ratio.

4. The shear strength of beams increases as the a/h decreases. Similarly an increase

in matrix strength enhances the shear strength.

As mentioned in Chapters One and Two, I beams are normally used as transfer

beams and U beams are used as roof beams. Due to different application purposes,

mechanical behaviours and failure modes of these two types of beam cannot be

studied as one group. It is necessary to investigate these beams separately. The

details are shown in Chapters Four and Five.

CHAPTER 4 Finite element simulation and Validation

104


4.1 Introduction

Conducting shear tests on ferrocement structures is time-consuming, expensive and

poses the additional difficulties of recording displacement and strain. To solve the

time-consuming problems, ABAQUS, a commercial FEM package, has been used by

other researchers to study the behaviour of reinforced concrete structures. Due to the

similarity of ferrocement and reinforced concrete and also based on available

literature, it has been used in the present study. ABAQUS has the ability to simulate

complex structural behaviour under different loading conditions, such as tension,

compression, shear and punching shear. It has a large library of finite elements to

enable efficient and detailed modelling of many of the special features of structural

behaviour.

Compare to simple models, complex detailed models require more computational

resources compare to simple models. The finite element model needs to be kept as

simple as possible. Because of the nature of the reinforcement, in ferrocement a large

amount of ferro-mesh including connections needs to be built in each model. If a

complete 3D model is chosen for the ferro-mesh, the model sizes and computational

time increases dramatically, which is not recommended for research purposes.

Besides, a 3D detailed model does not require capturing the behaviour that is

necessary for this study. Hence, a two-dimensional non-linear FE truss model was

chosen for the ferro-mesh study.

Compared to the mesh, the matrix requires less element refinement. Also for

ferrocement structures, the performance of the matrix influences the initial cracking,

peak strength and ultimate strength. Therefore a three-dimensional non-linear finite


105

element analysis approach is adopted for the ferro-matrix to provide more detailed

simulation and give more accurate results.

4.2 The Finite Element Model

Solid elements were chosen for the ferro-matrix parts and truss models were chosen

for the steel reinforcement parts. Individual parts were first assembled. For the

reinforcement and the mesh wires features in ABAQUS were inserted at the

appropriate location and merge feature was used to represent the welded mesh. An

assembly was constructed after combining all the instances–matrix, mesh and

reinforcement. The following discusses the details of how to create a FE model for a

ferrocement beam under flexure.

To solve nonlinear problems, ABAQUS uses two different solution strategies,

standard and explicit. Nonlinear behaviour may arise from the following three

possible sources:

1) Material nonlinearity: Due to the behaviour of material, such as ferro-matrix

which has roughly linear stress-strain relationship at the elastic stage but at the

plastic stage the stress-strain behaviour becomes highly non-linear due to material

yielding.

2) Boundary nonlinearity: This normally occurs due to boundary condition changes

during the experiment.

3) Geometric nonlinearity: This is related to changes in the geometry of the models

due to rotation or deflection. The stiffness may change dramatically.

As can be found in Chapter Three, the boundary condition did not change all the

experiments. Therefore both material and geometric nonlinearity were considered in

this research.


106

4.2.1 Element Types

In the ABAQUS element library, there are different types of elements. For example,

hexahedron (brick), shell and beam. The most common model for concrete based

studies, is the three-dimensional finite element model (brick elements). For the

plasticity-type problems, due to displacement solution discontinuity at element edges,

it was suggested that the first order elements are likely to be the most successful in

reproducing yield lines and strain field discontinuity [49]. C3D8, C3D8R and C3D8I

are three main types of brick elements. They are described below:

(1) C3D8 is an element with full integration which has 8 Gauss points. The main

advantage of this element type is its accuracy. The main disadvantage is for flexural

dominated structures. Shear locking phenomenon is commonly associated with this

element type.

(2) C3D8R is an element with reduced integration which has 1 Gauss point. This

type of element solves the shear locking problem. Due to insufficient stiffness,

spurious singularity (hourglass) may occur. In order to control this, an artificial

stiffness method and artificial damping method in the ABAQUS code is proposed.

As less integration points need to be computed, the computational time is largely

shortened.

(3) C3D8I is an element with full integration which has 8 Gauss points and

incompatible model. In order to eliminate parasitic shear stress that is observed in

regular displacement elements in analysis bending problems, 13 degrees of freedom

are added. However, the computation time is increases because of the details added.


107

In the three types of element, the C3D8R was chosen to produce design guidance for

this study. Due to the reduced integration, it may underestimate strength values and

plastic failure load in the analysis. In order to control the hourglass modes, mesh

with reasonable density in the finite element model has been set up. Therefore, the

element chosen for ferro-matrix is a fine meshed three dimensional reduced

integration brick (C3D8R).

4.2.2 Material Property

As ferrocement can be defined as a special type of concrete, so the concrete cracking

models in ABAQUS, which are brittle cracking models, concrete smeared cracking

models and concrete damage plasticity models can potentially be used to analyse the

ferrocement problems.

The brittle cracking model is widely used for material such as rock, plain concrete

and glass. This model can simulate brittle failure such as the formation and unstable

growth of micro cracks due to external forces and non-homogeneity. The

ferrocement reinforcement wire mesh is distributed uniformly with the matrix, which

substantially improves cracking characteristics and other mechanical properties and

transforms the otherwise brittle matrix into a material of high ductility compared to

concrete. So a brittle cracking model is not suitable to analyse the ferrocement

problems.

The concrete smeared cracking model is a conceptual and computationally simple

model of reinforced concrete membrane type structures. This model provides an

effective response to average stress and strain. When the models are loaded in

tension or combined tension and shear the post-cracking dilatation effect will not be

accounted for after cracks occurred. The model assumes no Poisson’s effect which


108

leads to zero transverse strains. Also in ABAQUS, when a crack is initially detected,

its direction will be stored. The subsequent cracking at this point will follow the

same orientation and no later stress components may be taken into account when

detecting any additional cracks. The development of shear cracks is not along the

single orientation, so for this study, as shear behaviour is the main governing criteria

and the post-cracking behaviour is important, the smeared cracking model cannot

provide good prediction for failure surfaces and the development of damage. This

model type is not suitable for this study.

The most commonly used continuum, plasticity-based, damage model for concrete is

concrete damaged plasticity model (CDP). This model assumes two main failure

types: tensile splitting and compressive crushing of concrete material.

The CDP in ABAQUS uses concepts of isotropic damaged elasticity with isotropic

tensile and compressive plasticity to represent the inelastic behaviour of concrete.

During the fracturing process non-associated multi-hardening plasticity and isotropic

damaged elasticity are adopted to model irreversible damage for general concrete

structures.

As shown in Fig 4.1, under uniaxial tension the concrete may give a linear elastic

stress-strain response initially until achieving failure stress, σt0. This stress may lead

to the formation of micro-cracking in the material. After failure stress, a softening

stress-strain response may lead to further deformation and strain localization occurs

in the structure. When the material is under uniaxial compression, the initial stress-

strain response is elastic linear until yield, σc0. In the plastic stage, the response is

defined as stress hardening followed by strain softening beyond the peak stress, σcu

[46]. The hardening rule defines the motion of progression of the yield surface

during plastic loading. There is an isotropic hardening variable K, which is from


109

hardening parameter integration along the loading path. In ABAQUS, isotropic

hardening is used. In this case, as the yield function is dependent on K it can only

expand or shrink but not rotate or translate [46].

(a)

(b)

Fig 4.1 Response of Concrete to Uniaxial Loading in (a) Tension and (b)

Compression [46]

4.2.3 Defining a Concrete Damaged Plasticity Model

Isotropic damage is the basic assumption of the concrete damaged plasticity model.

This model can be used for modelling concrete or other quasi-brittle material under


110

static or cyclic loading conditions. The model considers the degradation of the elastic

stiffness due to tensile and compressive plastic straining.

There are five parameters which need to be input to ABAQUS to define a simple

CDP model and they are described below. Some of the parameters require biaxial

and triaxial tests, which were difficult to achieve for ferrocement. Also there is

insufficient data in literature. To provide data for the models a series of parametric

studies regarding these values was undertaken. Starting from the default values in the

ABAQUS manual and changing one coefficient each time and comparing with

experimental results, in this way a set of values for all the important coefficients of

the concrete damage plasticity model was obtained. If there were not any dramatic

changes by varying the coefficient value, the suggested default value in ABAQUS

was considered. The results from the FEA were compared with the specimens as

described in the Chapter Three. The beam geometry is the same as in Fig 3.10 (b)

and the material property that was used is in column 1 of Table 3.1.

To determine the critical surfaces for the biaxial behaviour of the matrix, a yield

function was used in the plasticity based models. This yield function was defined to

correspond to the elastic limit. The size of the yield function is based on the material

properties defined for the uniaxial behaviour of the material. The yield surface is

based on the strengths defined as the material no longer acts elastically and the

failure surface is based on the ultimate strength. In uniaxial tension, the material is

normally defined to be elastic up to the tensile strength. This means that in the

biaxial tensile meridian the yield surface is equal to the failure surface. In

compression, the material is usually assumed to be initially elastic up to 40% of the

compressive strength. The yield function used in ABAQUS is chosen to use the

proposed function by Lubliner. et. al. [50] and the modifications proposed by Lee

and Fenves [51]. For the yield function, two coefficients need to be defined in

ABAQUS. They are:


111

(a) σb0/σc0 and Kc

σb0/σc0, 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 the initial yield for any given value of the

pressure invariant p such that the maximum principal stress is negative, σmax<0. It

must satisfy the condition 0.5<Kc<1.0. The default value is 0.667.

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

b0

/c0

=1.10

b0

/c0

=1.16

b0

/c0

=1.20

Experiment

Fig 4.2 Comparison of FEM Load-Displacement Curves with Different σb0/σc0

Values and Experimental Result


112

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

Kc

Kc

Kc

Experiment

Fig 4.3 Comparison of FEM Load-Displacement Curves with Different Kc Values

and Experimental Result

As can be seen from Fig 4.2 and Fig 4.3, by varying the value σb0/σc0 or Kc, there was

no appreciable change of the load-deflection curves from the FEM when compared

with the experimental curve. So the default values were chosen for this study.

(b) Eccentricity, e and Dilation angle,

In ABAQUS, the Drucker-Prager hyperbolic function is used to define non-

associated (not identical with the yield surface) potential plastic flow for concrete

damage plasticity.

√( ) ̅ ̅

Where

uniaxial tensile stress at failure,


113

eccentricity defined as the rate of plastic potential function approaches at the

asymptote. By decreasing the value, this may lead to convergence problems.

The default is e=0.1

dilation angle, measured in the ̅ ̅ plane at high confining pressure.

The dilation angle is used as the material parameter in ABAQUS. It measures the

inclination of the plastic potential at high confining pressures. Low values of the

dilation angle will produce brittle behaviour while higher values will produce more

ductile behaviour. It has been found when there is a higher dilation angle the

differences of cracks boundaries became smaller [52]. In order to find the value for

dilation angle in this study, a group of sensitivity studies were carried out as shown

in Fig 4.4. By changing the value of the dilation angle, a high value 56 (degree) was

chosen which gave the best-fit curve to the experimental result.

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

30o

45o

56o

Experiment

Fig 4.4 Comparison of FEM Load-Displacement Curves with Different Dilation

Angles and Experimental Result


114

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

e=0.05

e=0.10

e=0.15

Experiment

Fig 4.5 Comparison of FEM Load-Displacement Curves with Different Eccentricity

and Experimental Result

From Fig 4.5, it can be found by varying the value of eccentricity, there is no

dramatic change of load-deflection curves from FEA, so the default value was

chosen.

(c) Viscosity Parameter, μ

Viscosity parameter is defined in ABAQUS to represent the relaxation time of the

visco-plastic system. By changing this value, the length of experimental simulation

time for each step may be influenced. Moreover softening behaviour and stiffness

degradation behaviour of material models may be influenced. Instead of changing the

viscosity parameter and step time at the same time, the step time was set as constant

and the viscosity parameter was the only changing parameter related to step time.

Also by defining a small number μ in ABAQUS, convergence difficulties can be

overcome.


115

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Displacement (mm)

Experiment

Fig 4.6 Comparison of FEM Load-Displacement Curves with Different Viscosity

Parameter and Experimental Result

As shown in Fig 4.6, a viscosity parameter equal to 0.025 gave the best fit curve to

the experimental curve and this value was chosen for this study.

In summary, the coefficients for concrete damage plasticity were decided based on

the criteria described before. They are shown in Table 4.1.

Table 4.1 Selected Coefficients for Shear Behaviour Study of Ferrocement Beams

Dilation Angle Eccentricity σb0/σc0 Kc Viscosity Parameter

56 0.1 1.16 0.667 0.025

4.2.4 Reinforcement Model in ABAQUS

In ABAQUS, reinforcement in ferrocement structures is typically provided by rebar.

Normally the rebar are defined as one dimensional wire truss elements. The

reinforcement of ferrocement structures can be defined in ABAQUS by building a


116

single wire as a reinforcement bar in ferrocement or merged mesh layers. In this

study, elastic-perfect plastic behaviour was defined for reinforcements.

Using this approach, the matrix behaviour is independent from the reinforcement

behaviour. To define the interaction between matrix and reinforcements, perfect

bond (embedded) was defined in this study to simulate load transfer between matrix

and reinforcements. This assumption is made according to the Ferrocement Model

Code: “mesh anchorage is considered fully effective over a distance of at least 3

lattices” [26].

4.2.5 Mesh Convergence

FEM is an approximate method. Finer mesh sizes give better predictions but require

more computational resource and time. In order to decide the appropriate matrix

mesh size, three groups of mesh size were taken to compare with the experimental

results. Similar rebar and ferro-mesh mesh size convergence studies were undertaken

independently.

As shown in Fig 4.7, initially all three groups have the same trend and they differed

at the peak load stage. It was found when the mesh size reached approximately

10mm 10mm 10mm, the results started to converge. With further refinement, the

peak load does not drop dramatically but the CPU time from start to 150kN increases

rapidly as shown in the Table 4.2. So a mesh density 10mm 10mm 10mm (size 10)

has been adopted in the finite element analyses in this research work.


117

Table 4.2 Comparison of Peak Load and CPU Time of FEM with Different Matrix

Mesh Size

Mesh size (mm) Peak load (kN) CPU time (sec) from start to 150kN

15x15x15 295 103

10x10x10 285 277

7.5x7.5x7.5 283 404

Experiment 286 ---

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

15 x 15 x 15 mesh

10 x 10 x 10 mesh

7.5 x 7.5 x 7.5 mesh

Experiment

Fig 4.7 Comparison of Load-Deflection Curves of FEM with Different Matrix Mesh

Sizes and Experimental Result of Beam I1

4.3 Verification of Finite Element Models with Experiments

4.3.1 I Beams

The results from the FEM analysis were compared with the experimental results. Fig

4.8 shows the load against the deflection under the loading point for beams I1 to I3

respectively.


118

For beam I1 (Fig 4.8(a)) the linear part of the FEM results is slightly stiffer than that

from the experiment. One explanation for this may be that the embedded interaction

used in the FEM between the matrix and the reinforcement is perfect. Disregarding

the bond slip or low bond quality may lead to a smaller deformation than during the

experiment. After reaching the peak load, the load in the experiment dropped

suddenly rather than in the FEM which shows the curve has a short stationary section

prior to failure. This may be because of the steel property used in the FEM. In this

study, we were not interested in the post-peak stages so a bi-linear curve was used

with a reasonable assumption of cracking stress and strain.

0 1 2 3 4 5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection(mm)

FEM

Exp

(a) Beam I1


119

0 1 2 3 4 5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection(mm)

FEM

Exp

(b) Beam I2

0 1 2 3 4 5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection(mm)

FEM

Exp

(c) Beam I3

Fig 4.8 Load-deflection Curves of Beam I1, I2 and I3 Obtained from Experiments

and FEM

In Fig 4.8(b), the same effect as model I1 can be observed initially from the curve,

then the gradient of the FEM reduces, the stiffness lowers more than that of the


120

experiment. The same effect was observed in the first beam which follows the initial

stiffness almost identically to the FEM but then shows more softening than the

experimental results. This could be due to the default setting of softening behaviour

of the concrete property in ABAQUS being different but close to reality.

All models show good agreement with experimental results and similar behaviour

can be observed. The results from the FEM models are within 2% of the

experimental values as shown is Table 4.3.

Table 4.3 Comparison of Test and FEM Peak Loads

Beam Test Peak Load (kN) FEM Peak Load (kN) Difference

I1 143 142 0.3%

I2 137 136 0.7%

I3 106 108 1.9%

The FEM results are also in good agreement with the experimental results in term of

the formation of cracks and final failure modes. The concrete damage plasticity

model used in this study does not have any symbols for crack initiation. The manual

states “Cracking initiates at point when plastic-damage variable k is greater than

zero, and the maximum principal plastic strain is positive. The direction of cracking

is assumed to be orthogonal to that of the maximum principal plastic strain at the

damaged point” [50]. To demonstrate this, in Fig 4.9, the same cracking pattern in (a)

and (c) and location can be observed as the red and grey colour in (b) and (d). The

crack initially occurred in the bottom flange, then grew diagonally into the web

section and finally into the top flange. The light colour in Fig 4.9(b) and (d) illustrate

the prediction of crack location, how the cracks may develop and the possible failure

type.


121

(a) (b)

(c) (d)

(PE, Max. Principal is the tensile strain along principal direction)

Fig 4.9 Comparison of Crack Growth from Experiment and FEM

The same failure pattern can be observed in Fig 4.10, where web crushing can be

observed. At the failure stage, a large area of matrix crushing in the web can be seen

both from the experiment and the FEM. All the FEM predictions were proved by the

experiments as the photos (a) (c) and (e) of Fig 4.9 and Fig 4.10.

(e) (f)

Fig 4.10 Comparison of Failure Mode from Experiment and FEM


122

A similar process was carried out for U beams by using the same modelling method

and coefficients. The experiments were simulated and the results are shown in Fig

4.11.

4.3.2 U Beams

4.3.2.1 Beam U1

As shown in Fig 4.11, the two curves are very close to each other and the maximum

difference of the load at same deflection is under 2% by calculation. As can be seen

from experiment curve, in the elastic stage, there is a small increment in deflection

which was due to the gauge sliding due to bending of the beam in the test procedure

as observed during the experiment.

0 1 2 3 4 5 6 7

0

10

20

30

Lo

ad

(kN

)

Deflection (mm)

FEM

Exp

Fig 4.11 Load-Deflection Curve of Beam U1


123

4.3.2.2 Beam U2

From the load-deflection curves in Fig 4.12, the maximum difference of load at the

same deflection between the two curves is under 10%. Variation after the elastic

stage of the curves for the FEM maybe due to the steel mesh degrading quicker than

in reality as the bilinear reinforcement property applied in ABAQUS cannot provide

complete stress-strain behaviour of mesh.

0 1 2 3 4 5 6 7

0

10

20

30

Lo

ad

(kN

)

Deflection (mm)

FEM

Exp


4.3.2.3 Beam U3 and Beam U4

As shown in Fig 4.13, the FEM result shows good agreement with the experimental

curve with the same initial stiffness and peak load of Beam U3. From the load-

deflection curves in Fig 4.14, FEM shows good agreement with the experimental

results. The maximum difference of load at 12mm deflection point was less than 10%

which is satisfied for FE modelling.


124

0 1 2 3 4 5 6 7

0

10

20

30

Lo

ad

(kN

)

Deflection (mm)

FEM

Exp


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0

10

20

30

Lo

ad

(kN

)

Deflection (mm)

FEM

Exp



125

Fig 4.15 shows the comparison of failure types between the experiments and the

FEMs in the shear span of the U beams (Fig 3.30(a)). Similar to the I section

simulations, the same failure pattern in experiments can be observed in ABAQUS

models. Both flexural failure and shear cracking zones can be observed along with

the failing procedure.

(F: Flexural Cracking Zone, S: Shear Cracking Zone)

Fig 4.15 Comparison of Failure Mode from Experiment and FEM

4.4 Conclusion

In this study, a finite element model using the commercial package ABAQUS was

produced to simulate the shear behaviour of ferrocement beams. Then a group of

ferrocement beams with different matrix, a/h ratio or volume fraction was tested in

the laboratory. The results from the laboratory work validated the suitability of using

the FEM to predict the shear behaviour of the ferrocement beams.


126

As shown in this chapter, the FEM can predict the load-deflection curve with a high

degree of correlation (less than 10% differences). There are several possible causes

of the differences between the experimental data and the finite element analysis. One

is that the embedded interaction used in the FEM is too ideal to simulate the realistic

cases as bond-slip may occur.

CHAPTER 5 Parametric Study

127


In Chapter Four, the verification of the finite element model was outlined

comprehensively. In order to develop design guidelines for shear design of

ferrocement beams, extensive parametric studies were carried out to generate a

comprehensive database of results. The effects of significant items such as web

thickness, the reinforcement volume fraction and clear beam span are presented in

this chapter.

5.1 I Beam

The I beam studies were based on simply supported beams. In the first instance, the

geometry, clear span, reinforcement and matrix strength were studied, and various

parameters were changed. Otherwise the beams has square welded wire meshes with

12.5mm openings and 1.6mm diameter, 40MPa matrix, 12mm rebar. These

properties were used throughout the parametric study. The beams had the same

geometry and reinforcement as the experimental set up shown in Chapter Three.

From this study it was found that failure of the I beams can be grouped into four

types: Diagonal splitting failure (DS), Diagonal splitting failure with flexural cracks

(DF), Flexural failure with tensile reinforcement yielding (FY) and Flexural failure

with top flange crushing (FC). The failure types were reported in the peak load tables.

The details of the analysis and the results are shown below.

5.1.1 Effect of Matrix Strength

Following interpolation of the experimental results and subsequent statistical analysis,

a group of matrix property formulae was proposed. Ferro-matrix properties for

compressive strengths of 30, 40, 50 and 60 MPa were generated. The ABAQUS


128

model used in Chapter 4 was used to study the effect of matrix strength. Fig 5.1

shows curves of load against deflection under the loading points. Table 5.1 shows the

peak loads for each curve in Fig 5.1.

0 1 2 3 4

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Deflection (mm)

30MPa a/h=0.5

30MPa a/h=1.0

30MPa a/h=1.5

40MPa a/h=0.5

40MPa a/h=1.0

40MPa a/h=1.5

50MPa a/h=0.5

50MPa a/h=1.0

50MPa a/h=1.5

60MPa a/h=0.5

60MPa a/h=1.0

60MPa a/h=1.5

Fig 5.1 Load-Deflection Curves for Different Matrix Strengths

Table 5.1 Peak Loads for Different Matrix Strengths

Matrix Strength (MPa) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

30 242 (DS) 211 (DF) 190 (FY)

40 264 (DS) 231 (DF) 209 (FY)

50 282 (DS) 247 (DF) 223 (FY)

60 298 (DS) 261 (DF) 236 (FY)

From Fig 5.1 it can be seen that the load capacity is related to the matrix strength. By

increasing the matrix strength, the load capacity and initial stiffness increase. The

displacements at the peak loads are similar for the same a/h case when a/h is 0.5 and


129

1. But for a/h=1.5 the beams with higher matrix strength have slightly larger

deflections. This is due to the fact that the failure of the beams where a/h=1.5 was

flexural. Higher matrix strength led to higher moment capacity and more yielding of

the reinforcement occurred which led to larger deflections. From the ABAQUS

model it has been found that a higher matrix strength (fcm) may give a higher initial

cracking strength and a higher peak load. By increasing fcm by 10%, the peak load

capacity increases about 2.3%. Details are shown in Appendix C. The elastic

deformation was studied in Chapter Six. Higher a/h tends to give more ductile

behaviour, as it is dominated by bending, whereas shear dominated behaviour for

low a/h gives brittle shear failure. The detail of matrix strength used in this study can

be found in Appendix D.

In ABAQUS, only tensile damage volume and compressive damage volume can be

generated. When the average tensile stress of the matrix FE mesh is greater than the

tensile strength of the matrix, the mesh is defined as damaged. To analyse the

damage progression, the damaged mesh volume was divided by the total volume

with the results being plotted in Fig 5.2.

0 10 20 30 40 50 60 70 80 90 100

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Tensile damage volume (%)

30MPa

40MPa

50MPa

60MPa


130

Fig 5.2 Load-Damage Volume Curves for Different Matrix Strengths when a/h=1.0

As can be seen in Fig 5.2, by increasing the matrix strength the damage volume

decreases for the same load level before the peak load, which means higher matrix

strength has higher tensile resistance with less damage. At the peak load, higher

matrix strength has higher peak load and higher damage volume. This means that a

higher matrix strength beam leads to a higher damage volume up to failure, in

comparison with a lower strength matrix. The same behaviour can be seen after the

peak load.

5.1.2 Effect of a/h Ratio

It has been found for the I beam (1.6 mm mesh with 12 mm rebar), when

a/h=0.5 and a/h=1, web crushing initially occurs at the connection of the web and

the root of the bottom flange as stress concentration can be found in FEM (Fig

5.3(a)); then it propagates diagonally towards the loading point and support. When

shear span splitting and matrix crushing with accompanying spalling near the loading

points occurs, the structure reaches its peak load and in the FEM the stress in the

elements starts to reduce (Fig 5.3(b)). For the a/h=1 case, flexural cracking occurred

at the bottom flange (Fig 5.3(c)). The failure type can be classified as diagonal

splitting failure. Shear failure is due to the combined effect of shear and flexure.

Stresses were shown in the principal directions as the failure is usually initiated by

direct tensile or compressive stresses for brittle materials such as ferro matrix. In Fig

5.3, the “S, Max. Principal” is the maximum principal tensile stress with unit MPa.


131

(a) Shear Span Stress Distribution at the Start

(b) Shear Span Stress Distribution at the Peak Loading Stage (Shear Failure)


132

(c) Flexural Cracking at the Bottom Flange for a/h=1

Fig 5.3 Typical Stress (in MPa) Distributions from the FEM

When a/h=1.5, the diagonal cracking behaviour is the same as in the former groups,

except that flexural cracks can be observed at the mid-span of the bottom flange at

the same stage of diagonal cracking as the tensile strain of matrix elements reached

at yielding. Due to the yielding of flexural reinforcement (Fig 5.4) the peak load is

reached and at the same time the stress in the elements started to drop. The failure

type should be classified as flexural failure. In Fig 5.4, the red region is the yielding

stress of the rebar and the mesh wires yield strength is 330MPa which lies in yellow

region.

Fig 5.4 Beam Mesh Stress (in MPa) Distribution at the Peak Loading Stage (Flexural

Failure)


133

5.1.3 Effect of Web Reinforcement and Geometry

In order to investigate the effect of the web section on the shear strength of a deep I

beam, three parameters were studied: number of mesh layers in the web, web

thickness and web depth. All the studies used 40MPa matrix. It was found from the

FEM that the failure mechanism for each a/h ratio is the same as mentioned in

§ 5.1.2.

5.1.3.1 Effect of Mesh Layer Number in Web

By keeping all the beam geometry and reinforcements in the flanges the same, four

groups of parametric studies were carried out in ABAQUS. The number of mesh

layers in the web was taken as 0, 2, 4 and 6. All the web layers were anchored in the

flanges to avoid bond-slip effects. The load-deflection curves under the loading

points were plotted and the peak loads are listed in Table 5.2 for comparison.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

0 layers a/h=0.5

0 layers a/h=1.0

0 layers a/h=1.5

2 layers a/h=0.5

2 layers a/h=1.0

2 layers a/h=1.5

4 layers a/h=0.5

4 layers a/h=1.0

4 layers a/h=1.5

6 layers a/h=0.5

6 layers a/h=1.0

6 layers a/h=1.5

Fig 5.5 Load-Deflection Curves for Different Web Mesh Layer Numbers


134

Table 5.2 Peak Loads for Different Web Mesh Layer Numbers

Layer Number Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

0 246 (DS) 204 (DF) 194 (FY)

2 264 (DS) 231 (DF) 209 (FY)

4 273 (DS) 238 (DF) 212 (FY)

6 282 (DS) 246 (DF) 216 (FY)

As can be seen from Fig 5.5, the existence of mesh in the web will increase the peak

load, also the presence of mesh leads to greater deformations. This means that the

mesh can improve the compressive behaviour of the web section. By considering the

beam as a Strut-and-Tie model the web section is performing as a compression strut

and the compressive strength becomes critical. In the literature review, it was

mentioned that the ferrocement compressive behaviour is mainly controlled by the

strength of the matrix. Sufficient ties across the mesh layers are also important to

avoid splitting and provide more strength. In FEM it was found that for a/h=0.5 and

a/h=1 cases web meshes yielded well before peak load level. Further increase in load

eventually caused crushing failure of matrix at loading points. In summary,

contribution of the mesh layers to the failure load was relatively low compared to the

ultimate load, hence, increasing the number of mesh layer did not change the failure

load significantly.

By changing the number of layers in the web section from 2 to 6, the peak load

increases for each a/h case as can be found from Fig 5.5 and Table 5.2. This is due to

the increase of the web (shear) reinforcement volume fraction. The peak load

increased for the a/h=0.5 and 1.0 cases were higher than in the 1.5 case for each web

mesh layer increase. This was because the a/h=1.5 cases suffered flexural failure

which was different to the other cases and the contribution of the web reinforcement

is relatively small in this case. So in design, adding in more web mesh layers can be a


135

method to increase shear capacity. The shear mesh layers must be well anchored in

order to provide shear resistance.

5.1.3.2 Effect of Web Thickness

In order to study the effect of the cross sectional area of the web on shear capacity,

three groups of parametric studies were carried out based on the thickness of the web.

As shown below, the web thickness was changed from 16 mm to 32 mm and 48 mm.

All the sections had the same geometry of the flanges and reinforcements. The

failure mechanism for different web thicknesses at the same a/h was the same as

observed in the FEM.

0 1 2 3 4 5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

16mm web a/h=0.5

16mm web a/h=1.0

16mm web a/h=1.5

32mm web a/h=0.5

32mm web a/h=1.0

32mm web a/h=1.5

48mm web a/h=0.5

48mm web a/h=1.0

48mm web a/h=1.5

Fig 5.6 Load-Deflection Curves for Different Web Thickness


136

Table 5.3 Peak Loads for Different Web Thickness

Web Thickness (mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

16 244 (DS) 207 (DF) 186 (FY)

32 264 (DS) 231 (DF) 209 (FY)

48 276 (DS) 245 (DF) 222 (FY)

As shown in Fig 5.6 and Table 5.3, with a thicker web a higher peak load level is

observed. For ferrocement beams which have shear reinforcement, a portion of shear

strength is carried by the matrix and the remainder by the shear reinforcement. From

the ABAQUS model, for the thin web beam, cracking occurs at a lower load as there

is less material in comparison with thicker web beams. In addition, crack propagation

in the thinner web beams spreaded more rapidly than in the thicker web beams, so

the stiffness of the thinner web beams is lower than that in the thicker web beams.

Due to more cracking in the thinner web beams, more shear deformation occurs.

For a/h=1.5 cases, with the thicker web sections, the beams are likely to have deeper

flexural cracking which leads to changes in the position of the neutral axis and more

of the tensile reinforcement yielding. This is the reason for the load increasing.

From Table 5.3 it can be found that for each 10% web thickness increase (between

16mm to 48mm) there is more than 0.6% peak load increase as shown in the

Appendix C.

5.1.3.3 Effect of Web Depth

In order to study the effect of web depth in the ferrocement I beam, an increase was

applied to the overall web height of the I beam. The new section has a 380mm height

(clear web depth is 258mm, compared to the original of 158mm). Without changing

the value of cross sectional area, the heights of the root sections were changed from


137

23mm to 6mm as shown in Fig 5.7. In this way, the effect of cross sectional area can

be studied. The root effect will be studied in the § 5.1.4. Without changing the

reinforcement geometry and loading condition, a set of parametric studies were

carried out.

Left: original, Right: increased web depth

Fig 5.7 Cross Section of I Beams

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

380mm beam a/h=0.5

380mm beam a/h=1.0

380mm beam a/h=1.25

380mm beam a/h=1.5

314mm beam a/h=0.5

314mm beam a/h=1.0

314mm beam a/h=1.25

314mm beam a/h=1.5

Fig 5.8 Load-Deflection Curves for Different Web Depths


138

Table 5.4 Peak Loads for Different Web Depths

Total Depth(mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.25 a/h=1.5

314 264 (DS) 231 (DF) 228 (DF) 209 (FY)

380 263 (DS) 232 (DF) 229 (DF) 190 (FC)

The failure mechanisms at each a/h ratio of the 380 mm deep beams were the same

as the 314 mm beams. As can be seen from Fig 5.8, for shear failure cases a/h=0.5

and a/h=1, the peak loads in the two cases were roughly the same as shown in Fig

5.8 and Table 5.4. This satisfies the Strut-and-Tie Model (§ 2.6.4) as for the same a/h

ratio the model has the same angle between the strut and the tie so for the same

failure type the load should be the same.

For a/h=1.5, the failure in the 314 mm case was flexural failure due to yielding of

tensile reinforcements, but for the 380 mm case was due to crushing of the

compression zone in the top flange. This was because the clear span was 60 mm for

the 380 mm deep beam and was 260 mm for the 314mm deep beam. For the 380 mm

deep beam, due to larger flexural curvature, crushing occurred in the compression

zone of the top flange. Also deeper cracking changed the level of the neutral axis and

reduced the flexural capacity, so the beam failed at a low peak load. The difference

in failure types led to the load capacity difference. In order to prove this analysis,

a/h=1.25 was examined to compare with a/h=1 and a/h=1.5. It can be seen that, if the

failure type is the same, the peak load should decrease gradually when a/h increases.

From this section, it was found that by keeping the cross section area constant, the

peak loads for each a/h were roughly the same. The performance of ferrocement web

structures is different to reinforced concrete structures but similar to steel structures.

For reinforced concrete I beams (ACI 318), the effective area is taken as the thinnest

web thickness times depth plus the effective area of flanges; which was observed not


139

to be valid in this study for ferrocement. This may be due to the fact that the

reinforcement is different between ferrocement and reinforced concrete beams. By

comparing with steel structures, the effective area is similar to steel structures as

mentioned in Structural Eurocode 3 [39] classification 1. The whole cross section

area should be considered.

In conclusion from this group of studies:

1. For ferrocement deep beam design, when a/h ratio was less than 1.5, the shear

resistance was unrelated to the web depth. The same shear capacity was

expected for the same a/h ratio.

2. For flexural design, the distance between loading points was critical for the

failure mechanism and needed to be studied separately.

3. Both web thickness and the web mesh layer number influenced shear capacity of

I beams. By increasing web thickness and web mesh layer numbers, the shear

capacity increased.

4. By increasing the web mesh layer number from 2 to 4 the shear strength for

a/h=0.5 and 1 increased by 3%.

5. By increasing the web thickness from 32 to 48 mm, the shear strength was

improved by 4.5% for a/h=0.5 and 6% for a/h=1.

6. When a/h<1.5, for the same a/h ratio, by changing web depth, the load capacity

remained constant.

5.1.4 Effect of Flange Reinforcement and Geometry

In order to study the effect of the I beam flange in shear behaviour, parametric

studies were carried out based on different root sizes, mesh layer and rebar sizes. All

the beams were simulated under the same conditions. The results are shown and

analysed below.


140

5.1.4.1 Different Flange Thickness

In order to study the flange thickness effect on shear behaviour, four groups of

studies were undertaken by changing the root size at both flanges at the same time.

All the flanges had a basic thickness of 55 mm and the root sizes were varied from 0

to 23 mm at roughly 8 mm intervals. The load-deflection curves under the loading

points were plotted in Fig 5.9.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

0mm root a/h=0.5

0mm root a/h=1

0mm root a/h=1.5

8mm root a/h=0.5

8mm root a/h=1

8mm root a/h=1.5

16mm root a/h=0.5

16mm root a/h=1

16mm root a/h=1.5

23mm root a/h=0.5

23mm root a/h=1

23mm root a/h=1.5

Fig 5.9 Load-Deflection Curves for Different Flange Thicknesses

Table 5.5 Peak Loads for Different Flange Thickness

Root Height(mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

0 247 (DS) 213 (DF) 198 (FY)

8 255 (DS) 221 (DF) 202 (FY)

16 260 (DS) 226 (DF) 205 (FY)

23 264 (DS) 231 (DF) 209 (FY)

As can be found from Fig 5.9 and Table 5.5, by increasing the root height, the peak

load can be increased (6.8% by varying root size from 0 to 23 mm for a/h=0.5). From


141

the FEM it was found that for root=0, where the flange was level to the top surface of

the flange, shear cracking was likely to occur at this level and the cracking loads for

different root sizes were roughly the same in each group. As discussed in §5.1.3, by

increasing the effective depth of web sections, the peak load level was roughly

constant. So it can be concluded that the shear capacity is related to cross sectional

area.

5.1.4.2 Flange Reinforcement Effect

In order to study the flange reinforcement effect on shear behaviour of ferrocement I

beams, four groups of parametric studies were undertaken. It includes the effect of:

top flange mesh layer numbers, bottom flange mesh layer numbers, top and bottom

flange mesh layer numbers and rebar size. All the beams had the same geometry and

the clear span. The failure mechanism for different flange reinforcements at the same

a/h was the same as mentioned in the a/h section (§5.1.2).

5.1.4.2.1 Effect of Number of Mesh Layers in the Top Flange

First, the effect of mesh layer numbers in the top flange was studied. The load-

deflection curves and peak loads are shown below.


142

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

T0B4 a/h=0.5

T0B4 a/h=1.0

T0B4 a/h=1.5

T1B4 a/h=0.5

T1B4 a/h=1.0

T1B4 a/h=1.5

T2B4 a/h=0.5

T2B4 a/h=1.0

T2B4 a/h=1.5

Fig 5.10 Load-Deflection Curves for Different Top Flange Mesh Numbers

Table 5.6 Peak Loads for Different Numbers of Mesh layers in the Top Flange


a/h=0.5 a/h=1 a/h=1.5

T0B4 257 (DS) 226 (DF) 206 (FY)

T1B4 260 (DS) 229 (DF) 207 (FY)

T2B4 264 (DS) 231 (DF) 209 (FY)

From Fig 5.10 and Table 5.6, it can be seen that the peak loads and relative

deflections for each a/h ratio for different top flange mesh layer numbers were

roughly the same. The peak load difference was less than 2% for each layer number

increase. As mentioned in the literature review, the compressive behaviour of

ferrocement was mainly controlled by the matrix property. The top flange was

mainly under compression, so the mesh did not significantly increase the shear and

flexural capacity. Also, as the failure for the a/h=0.5 and 1 was diagonal splitting


143

failure and that for the 1.5 was flexural failure due to tensile reinforcement yielding,

so the contribution of mesh layers in the top flange was relatively small.

5.1.4.2.2 Effect of Number of Mesh Layers in the Bottom Flange

The second study was based on the number of mesh layers in the bottom flange. The

number of mesh layers was 0, 2 and 4. The load-deflection curves and results are

shown below.

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

T2B0 a/h=0.5

T2B0 a/h=1.0

T2B0 a/h=1.5

T2B2 a/h=0.5

T2B2 a/h=1.0

T2B2 a/h=1.5

T2B4 a/h=0.5

T2B4 a/h=1.0

T2B4 a/h=1.5

Fig 5.11 Load-Deflection Curves for Different Number of Mesh Layers in the

Bottom Flange

Table 5.7 Peak Loads for Different Numbers of Mesh Layers in the Bottom Flange


a/h=0.5 a/h=1 a/h=1.5

T2B0 260 (DS) 215 (DF) 205 (FY)

T2B2 263 (DS) 228 (DF) 206 (FY)

T2B4 264 (DS) 231 (DF) 209 (FY)


144

As can be seen from Fig 5.11 and Table 5.7, the presence of mesh can improve the

load capacity. When a/h=0.5, the peak load increase was about 1% for every extra

two mesh layers. This was because in the Strut-and-Tie model, for the small a/h

where the angle between strut and tie was relatively large; the stress in the tie was

small. As the rebar can provide enough tensile resistance, so the effect of shear

capacity improvement from the mesh layer number increase became less.

For the a/h=1.0 case, the mesh in the bottom flange can changed the cracking

behaviour. By adding more mesh layers in the bottom flange, the damage volume

decreased at the failure stage (Fig 5.12). This demonstrated that for the no mesh case

a few random deep flexural cracks were expected in practise, and for other cases

cracking may be stopped by the mesh layers. The second moment area and neutral

axis were different after cracking for each case, which led to a load capacity

difference of 6% between T2B0 and T2B2. As the cracking behaviour was under

control, the load capacity difference between T2B2 and T2B4 decreased to about 1%.

(a) Tensile Damage of T2B0 at the Failure Stage

(b) Tensile Damage of T2B2 at the Failure Stage

(PEEQT is tensile strain equivalent to MISES stress)

Fig 5.12 Tensile Damage at the Failure Stage


145

For the a/h=1.5 case, the final failure was due to yielding of tensile reinforcement.

As found in the FEM, the mesh layers yielded before the peak loading stage. So the

peak loads were controlled by the 12 mm bars. As the two 12 mm bars were in all of

the beams, the peak load in each case was roughly the same.

5.1.4.2.3 Effect of Mesh Layer Numbers in the Top and Bottom Flange

In this group, the number of mesh layers in the top and bottom flanges was increased

at the same time. The load-deflection curves are plotted below and analysed.

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

T2B4 a/h=0.5

T2B4 a/h=1.0

T2B4 a/h=1.5

T3B6 a/h=0.5

T3B6 a/h=1.0

T3B6 a/h=1.5

T4B8 a/h=0.5

T4B8 a/h=1.0

T4B8 a/h=1.5

Fig 5.13 Load-Deflection Curves for Different Numbers of Mesh Layers in the Top

and Bottom Flanges


146

Table 5.8 Peak Loads for Different Numbers of Mesh Layers in the Top and Bottom

Flanges


a/h=0.5 a/h=1 a/h=1.5

T2B4 264 (DS) 231 (DF) 209 (FY)

T3B6 269 (DS) 238 (DF) 213 (FC)

T4B8 274 (DS) 244 (DF) 216 (FC)

In the FEM it has been found that by increasing mesh layer numbers in the bottom

flange from 4 to 6 and 8, the beam became over reinforced. The failure type for

a/h=1.5 cases was flexural with compressive crushing in the top flange. As can be

found from the Fig 5.13 and Table 5.8, for every 2 mesh layers increment, there will

be less than 3% peak load differences. Considering the economics of the design, this

kind of design should be ignored for shear capacity improvement.

In summary, the top flange mesh layer number has a minor effect on shear capacity.

So the design of this section should be based on durability and cracking control. The

existence of a bottom flange mesh layer can control the cracking behaviour of

ferrocement beams and give tensile resistance. The design of the mesh in the bottom

flange should initially consider tensile resistance along with cracking control

behaviour. Other requirements such as specific surface should also be considered.

5.1.5 Effect of Rebar Size

In this section, the effect of rebar size was studied. Load-deflection curves were

plotted and analysed as shown below.


147

0 1 2 3 4 5 6 7

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

0mm a/h=0.5

0mm a/h=1.0

0mm a/h=1.5

6mm a/h=0.5

6mm a/h=1.0

6mm a/h=1.5

12mm a/h=0.5

12mm a/h=1.0

12mm a/h=1.5

18mm a/h=0.5

18mm a/h=1.0

18mm a/h=1.5

Fig 5.14 Load-Deflection Curves for Different Rebar Sizes

Table 5.9 Peak Loads for Different Rebar Sizes

Rebar Diameter (mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

0 mm 251 (DS) 182 (DF) 160 (FY)

6 mm 256 (DS) 204 (DF) 178 (FY)

12 mm 264 (DS) 231 (DF) 209 (FY)

18 mm 270 (DS) 248 (DF) 221 (FC)

From Fig 5.14 and Table 5.9 it can be seen that by increasing the bar diameters from

0 to 18 mm, the peak load difference increased when the a/h ratio increased. When

a/h=0.5, the peak load difference was about 8% between the 0 mm bar case and the

18 mm bar case. This difference went up to 38% when a/h=1.5. It was found, in the

FEM that when two 18 mm bars were used in the beam, the beam became over

reinforced and the failure is flexural due to compressive crushing of top flange. It

indicated that the rebar can improve the bending moment capacity significantly in


148

ferrocement I beams, but the effect on shear capacity was relatively small. So in

practice, when shear is the only parameter that needs to be considered, the rebar may

be ignored in design.

To study differences for deflection of different bar size, the yielded mesh percentage

at peak load for the 0 mm bar and 18 mm bar cases were generated from ABAQUS

and are shown in the Table 5.10.

Table 5.10 Yielded Mesh Percentage of Different Rebar Sizes at Peak Load

Beam Yielded Mesh (%) Load (kN)

0 mm bar a/h=1 6.83 182

0 mm bar a/h=1.5 13.3 160

18 mm bar a/h=1 0.26 248

18 mm bar a/h=1.5 0.52 221

As shown in Table 5.10, for the no bar case at peak load, the yielded mesh

percentages are much higher than the 18 mm bar cases at peak load. More tensile

reinforcement yielding led to increased deflection. Also more flexural cracking may

occur when there is no rebar or there is small diameter rebar. This explains why there

is more deflection for no bar or small bar cases than for larger bar cases.

In conclusion, from the flange studies it has been found:

1. Flanges are an important element in I beams for flexure capacity; so the design

of flanges should follow the design requirement for flexural behaviour.

2. The top flange mesh layer number has a minor effect on the shear capacity; so

the design of this section should be based on durability and cracking control.

3. The existence of bottom flange mesh layers can control the cracking behaviour

of ferrocement beams, but further increases cannot provide significant capacity

improvements. So the mesh in the bottom flange design should initially consider

tensile resistance along with cracking control behaviour.


149

4. Increased rebar diameters can increase the tensile (flexural) resistance of the

beam. As in the Strut-and Tie model, tie strength can be improved by enhancing

the rebar. So for relative large a/h ratio cases, in which large tensile resistance is

required, rebar design needs to be considered. For small a/h ratio, rebar design

may be ignored.

The flanges in a ferrocement I beam are mainly considered to provide flexural

capacity. But the cross sectional area changes of the flanges also influence the shear

capacity. For design, the effect of flanges is mainly based on sufficient tensile tie

strength and cracking control. For shear design, the volume fraction in the flange and

cross sectional area should also be considered.

5.1.6 Effect of Length

As mentioned in the literature review, the nonlinearity of the shear strain distribution

in the vertical direction decreases as the a/h ratio increases. For a deep beam, the

strain distribution at mid-span may be influenced by shear strain and become

nonlinear; however, for a thin beam the strain distribution at mid-span may be more

linear. In order to study the beam length effect on ferro-beam shear behaviour, two

groups of numerical parametric studies were carried out. The same matrix strength

(40 MPa) was used for all the models. Two lengths were studied with different a/h

ratios.


150

0 1 2 3 4 5 6 7

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

1200mm span a/h=0.5

1200mm span a/h=0.75

1200mm span a/h=1.0


1200mm span a/h=1.5

2400mm span a/h=0.5


2400mm span a/h=1.0


2400mm span a/h=1.5

Fig 5.15 Load-Deflection Curves for Different Beam Spans

Table 5.11 Peak Loads for Different Beam Spans

Clear span(mm) Peak Load (kN)

a/h=0.5 a/h=0.75 a/h=1 a/h=1.25 a/h=1.5

1200 264 (DS) 241 (DF) 231 (DF) 228 (DF) 209 (FY)

2400 247 (DS) 213 (DF) 186 (DF) 182 (DF) 177 (FY)

It was found that when a/h was less than 1.5, the failure was due to shear of the CCC

nodal zone (§ 2.6.4) and strut interface crushing. When a/h≤1, the failure type was

dominated by shear; when 1<a/h<1.5, the beam failed in shear with flexural cracks

and for a/h=1.5 cases, the beam failed due to pure flexure and yielding of tensile

reinforcement. As more rotation occurred in the 2400 mm span case, than in the

1200 mm span case, larger deflections were observed.


151

From Fig 5.15 and Table 5.11 it can be seen that the peak load levels of the 2400 mm

span (long) beams were lower than 1200 mm span (deep) beams. This is because

more rotation occurred in 2400 mm span case than in 1200 mm span case, so more

flexural cracks occurred at the bottom flange in the 2400 mm span beam which led to

a decrease of second moment of area and a change in position of the neutral axis. So

the load capacity decreased along with decreased initial stiffness.

5.1.7 Effect of Mesh Type

As mentioned in the literature review, specific surface and volume fraction are two

important parameters considered in ferrocement design. In order to study the effects

of these two parameters on shear behaviour, two groups of studies were taken. One

group kept the same volume fraction by changing the wire diameter and the other

kept the same specific surface by changing the wire diameter. Three different

openings, 12.5, 20 and 25 mm were used in this study.

5.1.7.1 Effect of Specific Surface

In this section, three different mesh openings with different wire diameters (selected

to provide roughly the same volume fraction) were simulated to study the specific

surface effect on shear behaviour of ferrocement I beams. All the beams had the

same reinforcement layout and two 12 mm rebar in the bottom flange.


152

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

12.5mm open a/h=0.5

12.5mm open a/h=1.0

12.5mm open a/h=1.5

20mm open a/h=0.5

20mm open a/h=1.0

20mm open a/h=1.5

25mm open a/h=0.5

25mm open a/h=1.0

25mm open a/h=1.5

Fig 5.16 Load-Deflection Curves for Different Specific Surfaces

Table 5.12 Peak Loads for Different Specific Surfaces

Wire diameter

(mm) Opening

(mm)

Volume

fraction

Specific

surface

Peak load (kN)

Mesh Main Bar a/h=0.5 a/h=1 a/h=1.5

1.6 12 12.5 0.0189 0.0472 264

(DS)

231

(DF)

209

(FY)

2.0 12 20 0.0185 0.0370 265

(DS)

228

(DF)

212

(FY)

2.3 12 25 0.0195 0.0339 268

(DS)

221

(DF)

209

(FY)

As shown in Fig 5.16 and Table 5.12, by changing the mesh openings and wire

diameters, the peak loads for different specific surface varied with a/h. As mentioned

in the literature review, the specific surface can influence the cracking behaviour in

tension as the load was uniformly applied. Unlike the cracking control effect on the

first crack under tension, the specific surface has less contribution in flexure. This


153

may be because the flexural cracking is mainly controlled by the outer most layers.

Even by increasing the total specific surface, the increase of the outer most layers

may still not be sufficient to increase the cracking load dramatically.

For a/h=0.5, the effect of specific surface was relatively small due to the small shear

span area and cracking behaviour was not critical. For a/h=1, by increasing the mesh

opening, the shear cracking progression became more rapid and led to the final

diagonal splitting failure occurring at the lower load. Moreover due to the small

effect on flexural cracking behaviour, roughly no difference in flexural cracking

behaviour was observed. As the a/h=1.5 cases failed in flexural, the specific surface

effect became minor so the load capacities for all the cases were roughly the same.

This is because the flexural cracking behaviour is mainly controlled by the outer

most layers and the specific surface of the outer most layers was roughly the same.

5.1.7.2 Effect of Volume Fraction

Similar to the last section, the effect of volume fraction was studied for three

different mesh openings which were selected to keep roughly the same specific

surface.


154

0 1 2 3 4

0

50

100

150

200

250

300

Lo

ad

(kN

)

Deflection (mm)

12.5mm open a/h=0.5

12.5mm open a/h=1.0

12.5mm open a/h=1.5

20mm open a/h=0.5

20mm open a/h=1.0

20mm open a/h=1.5

25mm open a/h=0.5

25mm open a/h=1.0

25mm open a/h=1.5

Fig 5.17 Load-Deflection Curves for Different Volume Fractions

Table 5.13 Peak Loads for Different Volume Fractions

Wire diameter

(mm) Opening

(mm)

Volume

fraction

Specific

surface

Peak load (kN)

Mesh Main

Bar a/h=0.5 a/h=1 a/h=1.5

1.6 12 12.5 0.0189 0.0472 264

(DS)

231

(DF)

209

(FY)

2.5 12 20 0.0289 0.0463 278

(DS)

235

(DF)

223

(FC)

3.2 12 25 0.0377 0.0472 288

(DS)

235

(DF)

223

(FC)

As shown in Fig 5.17 and Table 5.13, by changing the mesh openings and wire

diameters, the peak load was related to volume fraction. For the a/h=0.5 cases it was

found by increasing the volume fraction by 10% increased the peak load by 1%. But

for the a/h=1.0 cases, the peak loads were roughly the same for three cases. And for


155

the a/h=1.5 cases, the peak load increasing rate became very slow when it reached

223kN.

This effect indicated the load capacity is related to the relative section failure

mechanism.

1. For the a/h=0.5 cases, by increasing the mesh opening, the wire diameter was

increased. With more shear reinforcement volume in the web, shear capacity was

increased.

2. For the a/h=1.0 cases, the same increase as in a/h=0.5 was expected. But the big

mesh openings led to quick diagonal cracking progression. Because of this the

peak load did not increase even with the mesh volume fraction increasing.

3. For the a/h=1.5 cases, by increasing volume fraction, the moment capacity

should increase as can be seen between the 12.5 and 20mm opening cases. But

the beams became over reinforced for the 20 with 25mm openings cases with

flexural failure due to compressive crushing of the top flange. So the peak load

for these two cases was the same.

As can be found, in this section all the studies were based on the combined effect of

mesh wire diameter and mesh opening. Although volume fraction and specific

surface were kept the same for each case, the behaviour was not only based on one

parameter. So due to the combined effect, when different mesh openings are used in

design and practise, it is strongly advised to have an independent shear behaviour

study.

All the above points are based solely on shear design. In actual design and practise

many different situations need to be considered. Hence the shear design guidelines in

this study should be considered along with other design aspects.


156

5.2 U Beams

In order to study the effect of different parameters on the shear behaviour of

ferrocement U beams, parametric studies were undertaken.

Before the parametric studies, several studies based on the beam facing direction

(flange on top and bottom), loading combinations and different beam assembly

methods were examined. It was found that the shear behaviour study became critical

only when the flange was on the top and a concentrated load applied to the flange. So

the beam models were kept the same as the experimental set up shown in the Chapter

Three for the parametric studies.

The studies were mainly based on 1300 mm clear span simply supported beams. Two

layers of 1.6mm diameter 12.5mm openings welded mesh was used throughout the

parametric studies. Only when the geometry, clear span and reinforcement had been

studied, were the relative parameters changed. Only one parameter was studied each

time and all others were kept constant. Studies were based on a 40MPa matrix.

Details of the results are shown below.

As mentioned in the literature review, when the clear span / depth of the beam is

greater than 4, the beam is called a normal beam [6], and sectional bending theory

can be applied. From all of the studies, it had been found that for each increment in

load span, a, the load capacity roughly decreased by 1/a. As known in flexural theory

moment = load x load span, and flexural behaviour is the dominant behaviour of

slender ferrocement U beams.

From the study it was found the failures of the U beams in this study can be placed in

three groups: Shear failure with flexural (tensile) reinforcement yielding (SF),


157

Flexural failure with tensile reinforcement yielding (FY) and Diagonal splitting

failure with flexural cracks (DF). The failure types were reported in the peak load

tables. The details of the analysis and the results are shown below.

5.2.1 Effect of Matrix Strength

By interpolating the experimental results and statistical analysis, a group of matrix

property formulae were proposed. Ferro-matrix properties for compressive strengths

of 30, 40, 50 and 60 MPa were generated. Same ABAQUS models with the same

reinforcement were produced to study the effect of matrix strength.

0 1 2 3 4 5 6

0

5

10

15

20

25

30

35

40

45

Lo

ad

(kN

)

Deflection (mm)

30MPa a/h=0.5

30MPa a/h=1.0

30MPa a/h=1.5

40MPa a/h=0.5

40MPa a/h=1.0

40MPa a/h=1.5

50MPa a/h=0.5

50MPa a/h=1.0

50MPa a/h=1.5

60MPa a/h=0.5

60MPa a/h=1.0

60MPa a/h=1.5

Fig 5.18 Load-Deflection Curves for Different Matrix Strengths


158

Table 5.14 Peak Loads for Different Matrix Strengths


a/h=0.5 a/h=1. a/h=1.5

30 32.9 (SF) 16.8 (SF) 11.4 (FY)

40 34.7 (SF) 17.3 (SF) 11.7 (FY)

50 36.5 (SF) 17.8 (SF) 11.9 (FY)

60 38.0 (SF) 18.2 (SF) 12.2 (FY)

From Fig 5.18 and Table 5.14 it can be seen that by increasing matrix strength the

initial stiffness increases and the load capacity increases. Flexural (tensile) cracks

occurred in web sections flexural zone. The beams with lower matrix strength failed

with larger deformations.

5.2.2 Effect of a/h Ratio

When a/h=0.5 and a/h=1, diagonal cracking in the shear span initially occurred at

the bottom web near the supports and then propagated quickly towards loading

points. Flexural cracks were observed at the bottom flange in the all over flexural

zone, and then diagonal cracking occurred as the tensile yielding stress was reached

in FEM as shown in Fig 5.19 (a). Due to the tie (bottom mesh wires in shear span)

failure the sudden diagonal cracking may be observed and the load reached a peak

level and started to drop (Fig 5.19 (b)). The failure type should be classified as

flexural-shear-failure (Fig 5.19 (c)). In Fig 5.19, S, Max. Principal is the maximum

principal stress in MPa and tensile yielding stress of matrix is 4.02 MPa.


159

(a) Shear Span Stress Distribution at the Start

(b) Shear Span Stress Distribution at the Peak Load Stage

(c) Flexural Cracking at the Peak Load Stage (a/h=0.5)

Fig 5.19 Typical Stress (in MPa) Distribution from FEM (the scales are different for

the above images)


160

When a/h=1.5, typical flexural behaviour was observed, and cracks only occurred in

the flexural zone (Fig 5.20(a)). Due to yielding of the tensile mesh wire in the

longitudinal direction (Fig 5.20 (b)) the load started to drop. The failure type should

be classified as flexural due to tensile reinforcement yielding. As can be seen from

Fig 5.18, by increasing a/h ratio, the ductility of the beam increased. This

demonstrates that the flexural failure is more ductile than the brittle shear failure.

(a) Flexural Cracking at the Beginning Stage (a/h=1.5)

(b) Beam Mesh Stress Distribution at the Peak Loading Stage (a/h=1.5)

Fig 5.20 Stress (in MPa) Distribution of Flexural Failure Cases


161

5.2.3 Effect of Web Thickness

By considering web thickness of 9, 12.4 and 16 mm without changing any other

parameter, the effect of web thickness on shear behaviour was studied. The load-

deflection curves were plotted and analysed below.

0 1 2 3 4 5 6 7

0

5

10

15

20

25

30

35

40

45

Lo

ad

(kN

)

Deflection (mm)

9mm a/h=0.5

9mm a/h=1.0

9mm a/h=1.5

12.4mm a/h=0.5

12.4mm a/h=1.0

12.4mm a/h=1.5

16mm a/h=0.5

16mm a/h=1.0

16mm a/h=1.5

Fig 5.21 Load-Deflection Curves for Different Web Thicknesses

Table 5.15 Peak Loads for Different Web Thicknesses

Web thickness (mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

9 30.7 (SF) 16.0 (SF) 10.9 (FY)

12.4 34.7 (SF) 17.3 (SF) 11.7 (FY)

16 40.8 (SF) 19.2 (SF) 12.7 (FY)

As can be seen from Fig 5.21 and Table 5.15 the peak loads and stiffness increased

with web thickness increasing but the rates of increase were different. For the thin


162

section (9 mm), less resistance was provided by the matrix so early cracks occurred

and led to more yielding of tensile reinforcements. For the thicker section (12.4 and

16 mm), the tensile resistance was sufficient, fewer tensile cracks and less yielding

occurred. More yielding leads to larger deflection; thus the thinner beams have more

deflection at failure in each a/h group.

Tie strength was given by the tensile strength of the matrix and ties. For the thin

section, cracks occurred at earlier stages than for thick web cases. As mesh yielding

and cracks led to reduced second moment of area, the load capacity was lower in the

thinner cases. Due to a larger second moment of area and slower crack propagation,

the thicker sections had higher load capacities. The effect of web thickness was more

in smaller a/h cases than larger a/h cases. This means webs, as shear resisting

elements, contribute more in shear than in flexure.

5.2.4 Effect of Number of Mesh Layers

By having the number of mesh layers as 1, 2 and 3, without changing other

parameters the effect of mesh layer number in ferrocement U beam was studied. The

load-deflection curves were plotted and are analysed below.


163

0 1 2 3 4 5 6 7

0

5

10

15

20

25

30

35

40

45

50

Lo

ad

(kN

)

Deflection (mm)

1 layer a/h=0.5

1 layer a/h=1.0

1 layer a/h=1.5

2 layers a/h=0.5

2 layers a/h=1.0

2 layers a/h=1.5

3 layers a/h=0.5

3 layers a/h=1.0

3 layers a/h=1.5

Fig 5.22 Load-Deflection Curves for Different Mesh Layer Numbers

Table 5.16 Peak Load for Different Mesh Layer Numbers

Mesh number Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

1 27.0 (SF) 11.8 (SF) 7.3 (FY)

2 34.7 (SF) 17.3 (SF) 11.7 (FY)

3 43.6 (SF) 23.3 (SF) 16.0 (FY)

It can be found in Fig 5.22 and Table 5.16 that more mesh layers can increase the

second moment of area (flexural moment capacity). Also cracking propagation in the

beam with fewer mesh layers was quicker than beams with more meshes. As

cracking and yielding decreased the moment capacity, the peak loads in the fewer

mesh layer cases were lower than those for the more layer cases. Also the fewer

mesh layer cases failed at earlier stages than the more layer cases which are more

ductile.


164

5.2.5 Effect of Rebar Size

One bar at each web section was applied to the model; the location is shown in Fig

5.24. By changing the bar size (the bar property used is the same as that in the I study)

a parametric study was carried out.

Fig 5.23 Rebar Location in U Beam


165

0 1 2 3 4 5 6

0

20

40

60

80

100

120

Lo

ad

(kN

)

Deflection (mm)

0mm a/h=0.5

0mm a/h=1.0

0mm a/h=1.5

6mm a/h=0.5

6mm a/h=1.0

6mm a/h=1.5

8mm a/h=0.5

8mm a/h=1.0

8mm a/h=1.5

Fig 5.24 Load-Deflection Curves for Different Rebar Sizes

Table 5.17 Peak Loads for Different Rebar Sizes


a/h=0.5 a/h=1 a/h=1.5

0 34.7 (SF) 17.3 (SF) 11.7 (FY)

6 89.4 (SF) 52.8 (SF) 37.0 (FY)

8 107.6 (DF) 73.7 (SF) 54.8 (FY)

As shown in Fig 5.24, beams with small rebar size were more ductile. By increasing

the rebar size, moment capacity and cracking behaviour were changed. With larger

rebar, beams became more brittle with increased moment capacity and higher peak

loads. As mentioned in §5.2.4, deformation can be reduced by increasing tensile

resistance.


166

From ABAQUS it has been found that for all a/h=0.5 cases at the peak loads, the

meshes in the shear span along the tensile direction yielded. Only in the 8 mm rebar

case, the rebar did not yield at the peak load. So it can be concluded that for U beams

with rebar, the tensile strain of the meshes need to be considered. When tensile mesh

in the shear span yields the structures will fail, whether the rebar reaches its yield

strength or not which means bond slip of rebar will occur at failure.

5.2.6 Effect of Length

In order to study the length effect of ferrocement U beam, the clear span of the beam

was doubled to 2600 mm. Load-deflection curves were plotted and analysed below.

0 1 2 3 4 5 6 7 8 9

0

5

10

15

20

25

30

35

40

Lo

ad

(kN

)

Deflection (mm)

1300mm span a/h=0.5

1300mm span a/h=1.0

1300mm span a/h=1.5

2600mm span a/h=0.5

2600mm span a/h=1.0

2600mm span a/h=1.5

Fig 5.25 Load-Deflection Curves for Different Beam Spans


167

Table 5.18 Peak Loads for Different Beam Spans

Clear Span(mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

1300 34.7 (SF) 17.3 (SF) 11.7 (FY)

2600 24.4 (SF) 13.8 (SF) 9.7 (FY)

As shown in Fig 5.25, by increasing the beam clear span, the peak load for each a/h

group decreased and relative deflection increased. From FEM it was found by

increasing the length of beam clear span, the rotation of the beam increased, and

increased deflection occurred. Matrix, being a brittle material, will have tensile crack

when there is a larger deflection. For the double span case, more cracks occurred

which led to a decrease of second moment of area (flexural capacity decrease).

5.2.7 Effect of Mesh Type

As mentioned in the literature review, specific surface and volume fraction are two

important parameters considered in ferrocement design. In order to study the effects

of these two parameters on shear behaviour, two groups of studies were undertaken.

One group was based on keeping the same volume fraction by changing the wire

diameter and the other one was based on keeping the same specific surface by

changing the wire diameter. Three different openings, 12.5, 20 and 25 mm were used

in this section.

5.2.7.1 Effect of Specific Surface

In this section, three different mesh openings with different wire diameters, but with

the same volume fraction were simulated to study the specific surface effect on shear

behaviour of ferrocement U beam. All the beams have the same reinforcement layout.


168

0 1 2 3 4 5 6 7

0

10

20

30

40

50

Lo

ad

(kN

)

Deflection (mm)

12.5mm open a/h=0.5

12.5mm open a/h=1.0

12.5mm open a/h=1.5

20mm open a/h=0.5

20mm open a/h=1.0

20mm open a/h=1.5

25mm open a/h=0.5

25mm open a/h=1.0

25mm open a/h=1.5

Fig 5.26 Load-Deflection Curves for Different Specific Surfaces

Table 5.19 Peak Loads for Different Specific Surfaces

Wire diameter

(mm)

Opening

(mm)

Volume

fraction

Specific

surface

Peak load (kN)

a/h=0.5 a/h=1 a/h=1.5

1.6 12.5 0.0530 0.1325 34.7

(SF)

17.3

(SF)

11.7

(FY)

2.0 20 0.0529 0.1057 36.0

(SF)

18.2

(SF)

12.3

(FY)

2.3 25 0.0529 0.0936 39.7

(SF)

20.3

(SF)

13.8

(FY)

As can be seen from Fig 5.26 and Table 5.19, by changing the mesh openings and

wire diameters, the peak load changed with different specific surface was relatively

small (less than 2%) compared to all other U beam cases. This indicated that the

specific surface has a minor effect on ferrocement U beam shear capacity.


169

5.2.7.2 Effect of Volume Fraction

Similar to the last section, effect of volume fraction was studied for three different

mesh openings to keep the same specific surface.

0 1 2 3 4 5 6 7 8

0

10

20

30

40

50

60

70

Lo

ad

(kN

)

Deflection (mm)

12.5mm open a/h=0.5

12.5mm open a/h=1.0

12.5mm open a/h=1.5

20mm open a/h=0.5

20mm open a/h=1.0

20mm open a/h=1.5

25mm open a/h=0.5

25mm open a/h=1.0

25mm open a/h=1.5

Fig 5.27 Load-Deflection Curves for Different Volume Fractions

Table 5.20 Peak Loads for Different Volume Fractions

Wire diameter

(mm)

Opening

(mm)

Volume

fraction

Specific

surface

Peak load (kN)

a/h=0.5 a/h=1 a/h=1.5

1.6 12.5 0.0530 0.1325 34.7

(SF)

17.3

(SF)

11.7

(FY)

2.5 20 0.0826 0.1321 46.6

(SF)

25.4

(SF)

17.5

(FY)

3.2 25 0.1060 0.1325 61.8

(SF)

34.7

(SF)

24.3

(FY)

As can be seen from Fig 5.27 and Table 5.20, by keeping the same specific surface

and changing the wire diameter and mesh openings, the peak load increased with


170

volume fraction increase. By considering both cases (§5.2.7.1 and §5.2.7.2) together,

the peak load was controlled by the volume fraction of the mesh. By increasing the

volume fraction, the tie strength (tensile reinforcement strength) increased and the

load capacity increased.

In conclusion for ferrocement U beams which have always been used as slender

beams, the sectional bending theory cannot be applied in the design. There are two

reasons:

1. With a small a/h ratio, shear stress will lead to early cracking and decrease the

load capacity;

2. The span of the beam may influence the crack propagation. Longer beams may

have lower peak loads.

As the failure of ferrocement U beams is always due to tie (tensile reinforcement)

yielding, so M=F x a can be applied for the same section to calculate load capacity

for different load spans.

From this study it has been found that tie strength need to be increased to increase

the shear load capacity (as well as flexural capacity). As the tie strength is given by

tensile reinforcements and the matrix, so increases in the load capacity can be

achieved using a stronger matrix, a thicker web, a bigger mesh wire diameter, more

mesh layers or rebar in the bottom web.

5.3 Effect of Boundary Conditions

In order to study the effect of boundary conditions, two groups of studies were

undertaken in ABAQUS. By changing the boundary conditions from simply


171

supported to fixed ends, the beam behaviour was studied. The loading condition was

kept the same as that used in Chapter Three.

It was found that by changing boundary condition from simply supported to fixed,

the initial cracking occurred as flexural cracking at the fixed ends with further load

increasing, the reinforcement at top flange started to yield as shown in Fig 5.28. The

failure type for all cases changed from shear failure to bearing failure of loading

points on the top flange as shown in Fig 5.29.

Fig 5.28 Reinforcement Yielding of Fixed End Boundary Condition (max principal

stress in MPa)

Fig 5.29 Bearing Failure of Fixed End Boundary Condition (max principal stress in

MPa)


172

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

300

Lo

ad

(kN

)

Displacement (mm)

a/h=0.5 Simply Support

a/h=0.5 Fix

a/h=1.5 Simply Support

a/h=1.5 Fix

Fig 5.30 Load Displacement Curve of I Beam with Different Boundary Conditions

0 1 2 3 4 5 6

0

20

40

60

80

100

120

140

160

Lo

ad

(kN

)

Displacement (mm)

a/h=0.5 Flange on Top Simply Support

a/h=0.5 Flange on Top Fixed Ends

a/h=0.5 Web on Top Simply Support

a/h=0.5 Web on Top Fixed Ends

a/h=1.5 Flange on Top Simply Support

a/h=1.5 Flange on Top Fixed Ends

a/h=1.5 Web on Top Simply Support

a/h=1.5 Web on Top Fixed Ends

Fig 5.31 Load Displacement Curve of U Beam with Different Boundary Conditions


173

From Fig 5.30 and Fig 5.31 it can be found, by changing the boundary conditions

from simply supported to fixed, the initial stiffness of the load-displacement curves

increased as the rotation at the ends were controlled by the boundary condition. As

flexural failure occurred in the fixed ends cases, the load capacity was higher than

the simply supported cases and the load capacity was dependent on the tensile

strength of reinforcement at the top corner which has the maximum bending moment.

It is necessary to carry out further experiments to validate these predictions.

5.4 Conclusion

From the parametric studies in this chapter, the main shear failure of ferrocement I

beam with a/h<1.5 is diagonal splitting and for U beam is shear compression. As

mentioned in Chapter Three, due to different application purposes, I beams and U

beams cannot be compared. Therefore the following points for I and U beams are

similar in trends but different in magnitudes:

1. By increasing matrix strength, the shear resistance increases.

2. By increasing a/h ratio or beam length, the shear resistance decreases.

3. By increasing effective cross section area, the shear resistance increases.

4. By increasing the volume fraction of web of the I beam, the shear resistances

increases.

5. By increasing the tensile reinforcements (mesh, rebar) volume fraction, the shear

resistances increases.

6. The volume fractions in the compression zone and specific surface have a minor

effect on ferrocement beam shear strength.

Based on above points, the design methodology must consider ferrocement beam

shear in two different failure types. Also the most effective parameters: matrix


174

strength, tensile reinforcement volume fraction, web reinforcement volume fraction,

effective cross section area and length of the beam, need to be considered.

CHAPTER 6 Design Guidelines

175


As mentioned in the literature review, the current practice of shear design of

ferrocement beams is based on the recommendations of Naaman [3]. The shear

guidance is from the ACI 318 [6] codes. Since 1989, several researches have

examined shear behaviour of ferrocement beams with or without web reinforcements.

But none of the design recommendations have been adopted by the ACI committee

so far. Also since 2002, the ACI 318 code included the Strut-and-Tie model (STM)

for the reinforced concrete deep beam shear design. Following this recommendation,

ferrocement deep beam shear design needs to be updated and STM may be

introduced to ferrocement deep beam design.

In this chapter, a Strut-and-Tie model for ferrocement deep beams is proposed to

determine multi-crack failure cases. Two sets of empirical equations, based on

statistical analysis of parametric studies, are proposed based on diagonal splitting

failure and shear flexural failure. Comparisons of experimental results, proposed

equations and the ACI code are given. The results indicated that the propose

equations are conservative and more realistic than the ACI code in predicting shear

strength. Design examples are given and a flow chart for ferrocement beam shear

design is presented.

6.1 The Strut and Tie Model

In order to help engineers to analyse ferrocement beams with web reinforcement, a

component model needs to be developed. Based on the literature review, and to

maintain consistency with ACI 318, a Strut-and-Tie Model for ferrocement deep

beam is proposed. This model can be used for determining the type of failure in

ferrocement beams.


176

6.1.1 Configuration and Angle between Strut and Tie

The proposed component model is based on the Strut-and-Tie Model. As mentioned

in the literature review, for designing a deep beam region is the first step to

determine the configuration of the STM and the resulting forces in critical elements.

From experimental observations and results using FEM in this study, it was found

that diagonal multi-crack in shear were parallel to the strut direction, which indicates

a one-panel STM is appropriate for ferrocement beams as shown in Fig 6.1.

The direction angle is critical in the calculation of forces in critical elements. So it is

necessary to find this angle, hence the inner lever arm d shown in Fig 6.1 needs to be

decided. The factor given in ACI 318-11 [6] is used for comparison with

experimental results.

The strength reduction factor recommended in ACI 318-11 Appendix C [6] for Strut-

and-Tie Models is 0.85. By applying this number the angle θ can be found and

compared to the cracking angle of the I beam test results. The values thereby

obtained are shown in Table 6.1.

Table 6.1 Comparison of Cracking Angles from ACI Calculations and Experiments

Beam θ (Degree)

ACI 318-11 Cracking Angle from Experiments

I1 40.4 41.8

I2 32.5 32.5

I3 40.4 40.5

In general, the angle value from the ACI code is close to experimental results. As

slight differences can be seen from experimental results which may because it is hard


177

to apply the load/support exactly as designed, so the strength reduction factor for the

inner lever arm d is taken as 0.85 for this research.

Based on this research, when a/h<1.5 the ferrocement I beams and U beams may

encounter shear failure, so the limitation for the angle between strut and tie is:

θ>29.5o

So the proposed model and equations in this study are only valid when the shear span

to depth ratio is not greater than 1.5.

Fig 6.1 One-Panel STM Configuration


178

6.1.2 Tie Strength

The strength of the tie T is given as the tensile resistance of the mesh wires of the

outer most layers in the longitudinal direction (if more than one layer is tied together

then consider all of them) and the tensile resistance of the rebar, if used as tensile

reinforcement. The tie strength given in ACI 318 Appendix A [6] is:

∑ Eq 6.1

Where:

yield strength of reinforcement

cross sectional area of one mesh wire (or rebar) in longitudinal direction

The contribution of the matrix to peak strength is considered negligible or unreliable

at this stage of the loading. Failure modes from the experiments, Mansur and Ong’s

experiments and FEM were compared with the proposed STM as shown in Table 6.2.

It indicates accurate prediction can be made using the proposed STM. In Table 6.2,

T1 gives values of tie force from experimental results based on the STM and T2

gives values from Eq 6.1. If the value of T1 is smaller than T2, the failure type is

predicted as diagonal splitting failure; if the value of T1 is greater than T2, the failure

type is predicted as shear flexural failure.

Table 6.2 Failure Mode Prediction Using the Proposed STM

Specimen Tie Force (kN) Failure Type

T1 T2 Test STM

Ex

per

imen

t

I1 143 201 DS DS

I2 182 201 DS DS

I3 106 201 DS DS


179

U1 12 3 SF SF

U2 14 4 SF SF

U3 14 43 DS DS M

ansu

r &

On

g

[38

] A1.0 50 61 DS DS

A1.5 56 61 DS DS

B1.0 50 61 DS DS

B1.5 57 61 DS DS

C1.0 47 41 SF SF

C1.5 45 41 SF SF

T1: Peak tension force in the test based on STM; T2: Tie strength of the beam (Eq

6.1); DS: diagonal splitting failure; SF: shear flexural failure

6.2 Empirical Equation for Diagonal Splitting Failure by

Statistical Analysis

6.2.1 Proposed Shear Design Equation for Diagonal Splitting Failure Mode

Diagonal splitting failure occurs when there is sufficient tensile strength in the beam.

From this study it has been found that this type of failure always occurs in

ferrocement deep I beams.

The general shear design formula given in ACI 318 [6] is:

√

Eq 6.2

Where:

shear resistance


web width


180

distance from extreme compression reinforcement to the centroid of tension

reinforcement

area of shear reinforcement

specified yield strength of transverse reinforcement

centre-to-centre spacing of shear reinforcement

For the STM in ACI 318 [6] Appendix A, shear strength is given as:

Eq 6.3

Where:

nominal shear resistance

coefficient related to the location and number of ties through the cross section

area, as finally the splitting occurs at CCC nodal zone, so 1 is taken


area of the face of the nodal zone

angle between strut axis and beam axis

Due to the different reinforcing methods and mechanical behaviour between

ferrocement beams and reinforced concrete beams, a similar empirical equation for

the shear resistance of a ferrocement beam with diagonal splitting failure need to be

proposed. From I beam parametric studies it has been found that the following

parameters have a significant effect on the shear behaviour of deep ferrocement I

beam:

Matrix compressive strength

Web thickness

Web mesh Volume Fraction

a/h ratio


181

Length over depth ratio of the beam

Bottom flange rebar size and mesh Volume Fraction

Matrix compressive strength is related to the material property, so it could be an

independent term in the design equation. Similarly web thickness is related to the

cross sectional area of the beam and can be treated as a separate term. Length over

depth ratio, rebar size and volume fraction are more complex and need to be used

depend on the design.

A length factor K was proposed for different clear span over depth of the beam (l/h)

and a/h cases. As analysed in § 5.1.6, by increasing the length over depth ratio, more

deflection (rotation at the supports) can be found, which leads to more flexural

cracks and reduces the shear capacity of the beam. By using the FEM results from

Chapter Five, the value of K is normalised with respect to l/h=4 for each a/h case.

For example, as in Table 6.3, in row a/h=1, parametric studies of l/h=4, 6 and 8 were

undertaken to get the shear strength of the I beams. Using all of the values to divided

the value of l/h=4 case, then coefficient 1 is worked out for l/h=4, 0.87 for l/h=6 and

0.79 for l/h=8. The same method was applied for all K and .

As l/h increases, the K value decreases. Based on the numerical models used in this

research, only three l/h values, 4, 6 and 8, were studied. In design, when 0<l/h<4, the

value for l/h=4 should be used; when 4<l/h<6, the value for l/h=6 should be used;

when 6<l/h<8, the value for l/h=8 should be used and when l/h is great than 8, this

equation cannot be used as it may overestimate the shear strength.

From the parametric studies in §5.1.3 and §5.1.4, it has been found that by increasing

the volume fraction in the top flange, the increase in load capacity was relatively


182

small. But by increasing the volume fraction of the web and bottom flange, the load

capacity can be improved. Which means by improving the flexural strength of the

beam the shear strength can be improved. As shown in Eq 6.2 and Eq 6.3, the

ACI 318 code only considers shear strength from shear reinforcement and concrete

without considering the effect of flexural reinforcement. In this study, the effect of

flexural reinforcement on shear strength was considered.

A web mesh volume fraction factor was proposed for the shear design equation.

In § 5.1.3(a), it has been found that the shear capacity can be increased by increasing

the volume fraction in the web. The values are related to the volume fraction and a/h

ratio. The value of is normalised with respect to web volume fraction is 1.80%

case for each a/h group. The effective volume fraction can be worked out as:

If a value is lying between two given values in the design table, then the lower value

of should be used. If a value is higher than the maximum given value then the

maximum given value should be taken.

A bottom flange mesh volume fraction factor was proposed for the shear design

equation. The values are related to volume fraction and a/h ratio. For example, as in

Table 6.5, for a/h=0.5 and volume fraction within [1.80, 2.80) cases, 0.61 was

proposed for Eq 6.4 to make the equation yield conservative and the most accurate

results (in comparison with the FEM results within the same range). The effective

volume fraction can be worked out as:


183

If a value is lying between two given values in the design table then the lower value

of should be used. If a value is higher than the maximum given value then the

maximum given value can be taken. The parameter for the 3.75% cases for a/h=1.5

are based on over reinforced cases. This value is conservative for balanced and under

reinforced beams.

A rebar factor was proposed for the shear design equation using the results from

parametric study. The value is based on rebar amount, size and a/h ratio. In § 5.1.5, it

has been found by increasing the amount and size of the rebar, the shear capacity can

be increased and the deflection can be reduced. The value of is normalised with

respect to no bar case for each a/h group. The rebar diameter can be worked out as:

√

If a value is lying between two given values in the table then the lower value of

should be used. If a value is higher than the maximum given value then the

maximum given value can be taken. The parameters for 2@18mm, 4@12mm and

4@18mm cases for a/h=1.5 are based on over reinforced cases. These values are

conservative for balanced and under reinforced beams.

So the equation is proposed as:

√ Eq 6.4


184

Where:

K length factor, as shown in Table 6.3

factor taking into account of mesh layer volume fraction in the bottom flange

(excluding fillet sections), as shown in Table 6.5

factor taking into account mesh layer volume fraction in the web, as shown in

Table 6.4

factor for rebar, values given in Table 6.6

compressive strength of matrix (30MPa< <63MPa)

effective cross sectional area, in this case, gross cross sectional area of I

section should be taken

Table 6.3 Length Factor for Diagonal Splitting Failure

Clear Span/Depth 4 6 8

K

a/h=0.5 1 0.97 0.94

a/h=1.0 1 0.87 0.79

a/h=1.5 1 0.94 0.86

Table 6.4 Web Mesh Volume Fraction Factor for Diagonal Splitting Failure

Volume fraction (%) 0 1.80 3.60 5.40

a/h=0.5 0.93 1 1.03 1.06

a/h=1.0 0.88 1 1.03 1.06

a/h=1.5 0.92 1 1.01 1.03

Table 6.5 Mesh Volume Fraction Factor for Diagonal Splitting Failure

Volume fraction (%) 0 1.80 2.80 3.75

a/h=0.5 0.60 0.61 0.62 0.64

a/h=1.0 0.41 0.44 0.46 0.47

a/h=1.5 0.39 0.39 0.40 0.41


185

Table 6.6 Rebar Factor for Diagonal Splitting Failure

Rebar

Dimension

No

Rebar 2@6’’ 2@12’’ 2@18’’ 4@6’’ 4@12’’ 4@18’’

a/h=0.5 1.00 1.02 1.05 1.08 1.03 1.08 1.09

a/h=1.0 1.00 1.10 1.26 1.35 1.19 1.34 1.42

a/h=1.5 1.00 1.10 1.29 1.37 1.19 1.37 1.43

6.2.2 Comparison of Experimental Results, Proposed Equation and ACI 318

In order to justify the proposed equation (Eq 6.4), the experimental results from

Chapter Three, Mansur and Ong’s experimental results [38] and FEM results from

Chapter Five were used for comparison along with the ACI 318 Appendix A Strut-

and-Tie formula (Eq 6.3). The volume fraction given in Table 6.7 is the tensile mesh

volume fraction in the bottom flange.

Table 6.7 Comparison of Experimental Results, ACI Formula and Eq 6.4

Beam l/h

Volume Fraction

Rebar Size

Shear Capacity (kN) Comparison

MPa % mm Test ACI

318 Eq 6.4

Test/

ACI 318

Test/

Eq 6.4

ACI 318/

Eq 6.4

Ex

per

imen

t

I1 63 4 2.58 2@12 286 117 257 2.45 1.11 0.45

I2 63 4 2.58 2@12 273 117 257 2.34 1.06 0.45

I3 32 4 2.58 2@12 212 59 183 3.58 1.16 0.32

Man

sur

and

Ong

[3

8]

A1.0 40 5 2.51 4@6 100 46 71 2.17 1.41 0.65

A1.5 40 6 2.51 4@6 75 36 67 2.07 1.12 0.54

B1.0 50 5 2.51 4@6 101 72 79 1.40 1.28 0.91

B1.5 50 6 2.51 4@6 77 57 75 1.36 1.03 0.75

C1.0 46 5 2.51 2@6 94 66 70 1.42 1.34 0.95

C1.5 46 6 2.51 2@6 70 52 66 1.34 1.05 0.78

D1.0 46 5 2.51 4@6 108 66 76 1.63 1.42 0.87

D1.5 46 6 2.51 4@6 85 52 72 1.63 1.18 0.72

E1.0 50 5 2.51 6@6 118 72 90 1.64 1.32 0.80

E1.5 50 6 2.51 6@6 99 57 86 1.75 1.15 0.66

F1.0 49 5 2.51 4@6 127 71 78 1.80 1.62 0.90

F1.5 49 6 2.51 4@6 92 55 74 1.66 1.24 0.75

G1.0 62 5 2.51 4@6 126 84 88 1.50 1.43 0.95

G1.5 62 6 2.51 4@6 98 70 84 1.40 1.17 0.84


186

The compression of FEM results from Chapter five, ACI formula and Eq 6.4 were

plotted in Fig 6.2. The detail can be found in Appendix E. It can be found that the

values of Test/Eq 6.4 for a/h=1.0 cases in Mansur and Ong’s study was larger than

the other cases. This is due to their experimental beam span to depth ratio (l/h) for

the a/h=1.0 cases was 3.0, and in order to use Eq 6.4 a value for l/h=4.0 had to be

considered which gave conservative results.

0 50 100 150 200 250 300 350

0

50

100

150

200

250

300

350

Slope=1

Eq

6.4

(kN

)

FEM (kN)

a/h=0.5

a/h=1

a/h=1.5

(a) Comparison of FEM Results and Eq 6.4


187

0 50 100 150 200 250 300 350

0

50

100

150

200

250

300

350

Slope=1

AC

I 3

18

(kN

)

FEM (kN)

a/h=0.5

a/h=1

a/h=1.5

(b) Comparison of FEM Results and ACI Formula

Fig 6.2 Comparison of FEM Results, ACI Formula and Eq 6.4

By comparing Eq 6.3, Eq 6.4, Table 6.7 and Fig 6.2 it can be found:

1. Both ACI formula Eq 6.3 and the proposed equation Eq 6.4 are conservative.

2. Both Eq 6.3 and Eq 6.4 can show the effect of matrix strength and a/h ratio.

When the matrix strength increases, both formulae can give increased results.

Also by increasing a/h, both formulae can give decreased results.

3. The ACI 318 code considers strut, nodal zone and tie as isolated sections. As has

been found in this study, when the final splitting occurred at the top flange, the

shear is not resisted only by the nodal zone but also by parts of the vertical mesh

wires in the whole beam which can provide extra shear strength. The ACI

formula does not consider this.

4. The method in ACI only considers the existence of sufficient reinforcements

with a low boundary. No further factors are given for the reinforcing member

increasing. So the factors , and can improve the accuracy in design.

Also the ACI formula is not sensitive to small reinforcement changes.


188

5. The method in ACI does not consider the length effect on shear strength of the

deep beams. As shown in Table 6.7, when the length changes the shear load

capacity also changes, and the length factor K can represent this phenomenon.

By dividing the FEM results by the results from the ACI formula, an average value

3.11 with standard deviation 0.431 was found. The same value for FEM results

divided by the results from Eq 6.4 was found as 1.12 and 0.033. Therefore Eq 6.4 is

conservative but more realistic for ferrocement beam shear design when the failure is

diagonal splitting failure. More information can be found in Appendix E.

6.3 Empirical Equation for Shear Flexural Failure by Statistical

Analysis

6.3.1 Proposed Shear Design Equation for Shear Flexural Failure Mode

Shear-flexural failure occurs when there is insufficient tensile resistance in the beam.

From this study it has been found that this failure always occurs in ferrocement deep

U beam. This kind of failure may also occur in T beams.

From the U beam parametric studies it has been found that the following parameters

have a significant effect on the shear behaviour of deep ferrocement U beams:

Matrix compressive strength

Web thickness

Rebar in the bottom web and its size

Mesh Volume Fraction

a/h ratio

Length of the beam


189

Similar to the proposed of Eq 6.4, an empirical equation for the shear resistance of a

ferrocement beam with shear-flexural failure was developed by considering length

factor K, mesh factor and rebar factor . Also the same boundary conditions for

Eq 6.4 were applied in this section.

An empirical equation for the shear resistance of a ferrocement beam with shear-

flexural failure is proposed as:

√ Eq 6.5

Where:

K length factor, as shown in Table 6.8

factor taking account of mesh layer volume fraction of the effective section,

as shown in Table 6.9

factor for rebar, value given is shown in Table 6.10

compressive strength of matrix (30MPa< <63MPa)

: effective cross sectional area; in this case, the whole web and effective flange

cross sectional area is taken

For the shear design of U shape beams, only the web sections were taken into

consideration. However from the FEM study it was found that 1/12 of the width of

the flange was effective to take shear stress. Therefore it is proposed that the cross

sectional area of 1/12 of the flange area for each side and the web sections are to be

used to determine the effective sectional area.

Table 6.8 Length Factor for Shear-Flexural Failure

Clear Span/Depth 4 8 13 18 26

K

a/h=0.5 1 0.68 0.53 0.45 0.39

a/h=1.0 1 0.66 0.54 0.48 0.44

a/h=1.5 1 0.63 0.53 0.48 0.44


190

Table 6.9 Mesh Volume Fraction Factor for Shear-Flexural Failure

Volume fraction (%) 2.5 5.0 7.5 8.5 10.5

a/h=0.5 1.23 1.58 1.98 2.37 3.14

a/h=1.0 0.53 0.78 1.04 1.26 1.71

a/h=1.5 0.33 0.54 0.73 0.88 1.24

Table 6.10 Rebar Factor for Shear-Flexural Failure

Rebar Dimension No Rebar 2@6mm 2@8mm

a/h=0.5 1 2.5 3.1

a/h=1.0 1 3.0 4.2

a/h=1.5 1 3.1 4.6

6.3.2 Comparison of Experimental Results, Proposed Equation and ACI 318

In order to justify the proposed equation (Eq 6.5), the experimental results from

Chapter Three and FEM results from Chapter Five were used to compare along with

the ACI 318 Appendix A Strut-and-Tie formula (Eq 6.1) as the failure of U beam

were due to tie failure. The FEM results from Chapter five, ACI formula and Eq 6.5

were plotted in Fig 6.3. The detail can be found in Appendix E.

Table 6.11 Comparison of Experimental Results, ACI Formula and Eq 6.5

Specimen

Matrix

Strength l/h

Volume

Fraction

Rebar

Size Shear Capacity (kN) Comparison

MPa

% mm Test ACI

318 Eq 6.5

Test/

ACI 318

Test/

Eq 6.5

ACI318

/ Eq 6.5

Ex

per

imen

t

U1 32 13 5.30 0 23.0 10.6 19.7 2.17 1.17 0.54

U2 63 13 6.19 0 27.0 15.9 25.3 1.70 1.07 0.63


191

0 20 40 60 80 100 120

0

20

40

60

80

100

120

Slope=1

Eq

6.5

(kN

)

FEM (kN)

a/h=0.5

a/h=1

a/h=1.5

(a) Comparison of FEM Results and Eq 6.5

0 20 40 60 80 100 120

0

20

40

60

80

100

120

Slope=1

AC

I 3

18

(kN

)

FEM (kN)

a/h=0.5

a/h=1

a/h=1.5

(b) Comparison of FEM Results and ACI Formula

Fig 6.3 Comparison of FEM Results, ACI Formula and Eq 6.5


192

By comparing Eq 6.1, Eq 6.5, Table 6.11 and Fig 6.3 it can be found:

1. The proposed Eq 6.5 is conservative for predicting shear strength of the beam.

But Eq 6.1 from ACI 318 code may overestimate the shear strength.

2. Eq 6.1 considers tie as isolated section. At failure, it considers the tie strength

only. Eq 6.5 considers both tie strength and matrix strength. Also Eq 6.5 is

sensitive to volume fraction changes through factor , thus better predictions

can be obtained as shown in Table 6.11.

3. Eq 6.1 considers the existence of sufficient reinforcements at a low boundary.

No further factors are given for different a/h cases. So the factors and can

improve the accuracy in design for different a/h cases.

4. The ACI method does not consider the length effect on shear strength of the

deep beams. As shown in Table 6.11, when the length changes the shear load

capacity also changes, and the length factor K can represent this phenomenon.

By dividing FEM results by the results from the ACI formula, an average value 1.58

with standard deviation 0.408 was found. The same values for FEM results divided

by the results from Eq 6.5 were 1.29 and 0.132. Therefore Eq 6.5 is conservative but

more realistic for ferrocement beam shear design when the failure is shear flexural

failure. More information can be found in Appendix E.

6.4 Shear Design Example

A 2.8m x 2m x 2m (length x width x depth) water tank is to be built on the roof of a

ferrocement building. The water tank has two linear supports which can stand on the

roof beams or transfer beams. The fully loaded water tank has a total weight of

16 tonne.

The tank is supposed to be supported by the U shape roof beam. The roof beam has a

uniform thickness of 3 layers of 1.6mm in diameter, 12.5mm square opening welded


193

meshes. The length of the U beam is 3m. The compressive strength of the matrix is

tested as 45MPa and mesh yield strength is 330MPa and Young’s Modulus is

150GPa. The cross sectional area with dimensions is shown in Fig 6.4(a).

(a) Check if the roof beam has sufficient shear resistance.

(b) If the roof beam is not strong enough, two 4m transfer I beams with the cross

sectional area as shown in Fig 6.4(b) can be built to replace the roof beam. Design

the I beam using same the materials as the U beam. 6mm rebar with yield stress

330MPa and Young’s Modulus 150GPa is available.

(a) Roof Beam (b) I Beam

Fig 6.4 Cross Sections of Roof Beam and I Beam


194

Fig 6.5 Final Appearance of the Water Tank Assembly

(a) The peak load from the water tank is:

16 x 1000 kg x10N/kg /1000=160 kN

As the beam length is 3000mm, depth is 150mm and length of water tank is 2800mm,

so the load span to depth ratio is 0.67, which indicates shear resistance is critical in

this problem. As 0.67 is greater than 0.5 but smaller than 1, so parameters for a/h=1

are taken to give conservative results.

Based on §6.1.2, the tie strength at one end of the beam is:


195

= number of web x number of mesh layer x cross sectional area of wire x yield

strength

( )

So the failure type should be shear flexural failure and Eq 6.5 is applied for this case.

Effective cross sectional area =10144mm2

The volume fraction of the beam is 6.03%. Mesh factor =0.78 (Table 6.9)

Clear span to depth ratio is 20. Length factor for l/h=26 is taken as K=0.44 (Table

6.8). This may underestimate load capacity, but will give a conservative result.

So shear resistance is

√

Value from ACI 318 based on Eq 6.1 is 11.9kN

Value from ABAQUS model is 52.0kN

So the original U beam is not sufficient to support the water tank.

(b) As the beam length is 3000mm, depth is 375mm and length of water tank is

2800mm, so the load span to depth ratio is 0.27. By assuming diagonal splitting

failure may occur in I beams, then Eq 6.4 is applied with parameters for a/h=0.5 case

to provide conservative results.

Assuming a 2 layer mesh in the web with 3 layers of mesh be tied together and

placed close to the bottom flange of the beam without a rebar in the bottom flange in

order to provide minimum volume fraction 1.80% for ferrocement, then:

Effective cross sectional area =18250mm2


196

The tensile volume fraction of the beam is 1.93%. Mesh factor =0.61 (Table 6.5)

Clear span to depth ratio is 8. Length factor K=0.94 (Table 6.3)

The web volume fraction of the beam is 2.57%. Mesh factor =1 (Table 6.4)

So shear resistance for each beam is

√

Value from ACI 318 based on Eq 6.3 is 73.9kN

Value from ABAQUS model is 145kN

Check failure mode:

Tie strength: T=3 (layers) x 9 (wires) x (1.6/2)2 x π x 330 /1000= 44.9 kN > 10.8 kN

So the assumption is valid.

So this I beam design is sufficient to support the water tank.

The I beams should have 2 layer mesh in the web and 3 layers of 1.6 mm in diameter

with 12.5 mm opening welded mesh. The wire material should have yield strength of

330 MPa and Young’s Modulus of 150 GPa. Matrix compressive strength must be no

less than 45 MPa. Design of top flange and web must satisfy the requirement of

general ferrocement design. A 3 layers mesh in the top flange and design is

suggested.

6.5 Shear Design Procedure

In order to perform shear design of a ferrocement deep beam with web reinforcement,

a flow chart is proposed.


197

In the flow chart, by considering the tie strength of the beam, the failure type can be

decided and the appropriate design equation can be chosen. By using the chart, an

adequate shear design can be achieved.

To avoid local crushing at loading points and supports, detailing of these sections

need to be considered for individual cases based on Ferrocement Model Code or

ACI 318.

All the research is based on squared welded wire meshes, for other mesh types these

design procedure may not valid. Further experimental research is necessary to check

the accuracy of this design procedure on different mesh types.


198

End

No

No

No

Yes

Yes

Yes

Start

Ferrocement Beam with Web Reinforcement Design

(cross section, reinforcement and etc.)

Simply Supported with

Four Points Bending and

a/h<1.5

Use One Panel STM Work out Resulting

Forces

Tie Yield

Eq 6.5 with Shear-

Flexural Failure

Eq 6.4 with Diagonal-

Splitting Failure

Design Satisfied


199

6.6 Elastic Deformation of Ferrocement Deep Beams

The elastic deformation of beams is related to serviceability limit state design. For

the normal slender beams, elastic curve is the rotation of the cross sections and

deflections due to bending. For the deep beam cases, the total deflection can be

considered in two parts: deflection due to bending and deflection due to shear.

Based on elastic theory, the flexural deformation at loading point of a simply

supported beam under four-point bending can be expressed as:

( )

Eq 6.6

Where

external load

elastic flexural deflection

E equivalent modulus of the composite

I second moment of area

l clear span

a shear span

Based on Wong [53], the shear deformation at loading point of a simply supported

beam subjected to four-point bending is derived as:

Eq 6.7

Where

external load


200

elastic shear deformation

shear modulus

cross sectional area of web

cross sectional coefficient, given as 1.2 for rectangular shape

a shear span

The total elastic deformation can be worked out as:

Eq 6.8

The shear modulus values of ferrocement sections were not available in this study.

So the shear deformation cannot be worked out directly from Eq 6.7. In order to

study the effect of flexural deformation and shear deformation on total deformation,

shear deformation was word out by applying Eq 6.8.

In §3.8 it has been indicated that the linear elastic stage of I beams ended at about

20kN. So the deformations under this load were taken to study. Firstly, the total

deformation under loading point was taken from experiments or FEMs. Then relative

flexural deflection was worked out by Eq 6.6. Finally shear deformation was worked

out by subtracting flexural deformation from total deformation as shown in Table

6.12 and Table 6.13.

Table 6.12 Comparison of Elastic Deformation of Experimental Results, FEM and

Calculation

Specimen fcm l/h

Volume

Fraction

Rebar

Size Deformation (kN/mm)

MPa

% mm Test FEM Flexural Shear

I1 63 4 2.58 2@12 0.0908 0.0919 0.0554 0.0354

I2 63 4 2.58 2@12 0.1224 0.1241 0.0812 0.0412

I3 32 4 2.58 2@12 0.0839 0.0957 0.0705 0.0134


201

The quality of U beams tested in the research was not uniform as mentioned in

Chapter Four. So only the I beam results were compared with calculation. As can be

seen in Table 6.12, the elastic deformation predictions from FEM were close to the

experimental results and slightly higher which can be considered safe. The prediction

of FEM of I3 was much higher than the experimental result; this is because in the test

the gauge was placed slightly away from the designed location and resulted in a

smaller reading. In order to study the effect of flexural deformation and shear

deformation on total elastic beam deformation, FEM results from Chapter Five were

used for comparison and are presented in Table 6.13.

Table 6.13 Comparison of Elastic Deformation of FEM and Calculations

Specimen

fcm l/h Volume Fraction

Rebar Size

Deformation (mm) Comparison (%)

MPa

% mm FEM

δ1 Flexural

δ2 Shear

δ3 δ3/ δ1

δ2/ δ1

δ3/ δ2

Effect of Matrix Strength and a/h

FE

M T

able

5.1

a/h

=0

.5

30 MPa

30 4 1.89 2@12 0.0402 0.0278 0.0124 30.8 69.2 44.6

40

MPa 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

50

MPa 50 4 1.89 2@12 0.0352 0.0208 0.0144 40.9 59.1 69.2

60

MPa 60 4 1.89 2@12 0.0335 0.0183 0.0152 45.4 54.6 83.1

FE

M T

able

5.1

a/h

=1

30

MPa 30 4 1.89 2@12 0.0977 0.0876 0.0101 10.3 89.7 11.5

40

MPa 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

50

MPa 50 4 1.89 2@12 0.0857 0.0657 0.02 23.3 76.7 30.4

60

MPa 60 4 1.89 2@12 0.0816 0.0578 0.0238 29.2 70.8 41.2

FE

M T

able

5.1

a/h

=1

.5

30 MPa

30 4 1.89 2@12 0.1528 0.1442 0.0086 5.6 94.4 6.0

40 MPa

40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3

50

MPa 50 4 1.89 2@12 0.1340 0.1082 0.0258 19.3 80.7 23.8

60

MPa 60 4 1.89 2@12 0.1276 0.0953 0.0323 25.3 74.7 33.9

Effect of Web Thickness

FE

M T

able

5.3

a/h

=0

.5

16mm 40 4 2.16 2@12 0.0474 0.0237 0.0237 50.0 50.0 100.0

32mm 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

48mm 40 4 1.68 2@12 0.0332 0.0238 0.0094 28.3 71.7 39.5


202

FE

M T

able

5.3

a/h

=1 16mm 40 4 2.16 2@12 0.1182 0.0747 0.0435 36.8 63.2 58.2

32mm 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

48mm 40 4 1.68 2@12 0.0805 0.075 0.0055 6.8 93.2 7.3 F

EM

Tab

le

5.3

a/h

=1

.5

16mm 40 4 2.16 2@12 0.1871 0.1231 0.064 34.2 65.8 52.0

32mm 40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3

48mm 40 4 1.68 2@12 0.1255 0.1235 0.002 1.6 98.4 1.6

Effect of Volume Fraction of Web Mesh

FE

M T

able

5.2

a/h

=0

.5

0 40 4 1.15 2@12 0.0381 0.0242 0.0139 36.5 63.5 57.4

2 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

4 40 4 3.37 2@12 0.0372 0.0232 0.014 37.6 62.4 60.3

6 40 4 4.11 2@12 0.0364 0.0227 0.0137 37.6 62.4 60.4

FE

M T

able

5.2

a/h

=1

0 40 4 1.15 2@12 0.0925 0.0763 0.0162 17.5 82.5 21.2

2 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

4 40 4 3.37 2@12 0.0906 0.0732 0.0174 19.2 80.8 23.8

6 40 4 4.11 2@12 0.0898 0.0717 0.0181 20.2 79.8 25.2

FE

M T

able

5.2

a/h

=1

.5

0 40 4 1.15 2@12 0.1444 0.1258 0.0186 12.9 87.1 14.8

2 40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3

4 40 4 3.37 2@12 0.1416 0.1206 0.021 14.8 85.2 17.4

6 40 4 4.11 2@12 0.1405 0.1181 0.0224 15.9 84.1 19.0

Effect of Rebar

FE

M T

able

5.9

a/h

=0

.5

0mm 40 4 1.89 0 0.0397 0.0242 0.0155 39.0 61.0 64.0

6mm 40 4 1.89 2@6 0.0390 0.0239 0.0151 38.7 61.3 63.2

12mm 40 4 1.89 2@12 0.0372 0.0232 0.014 37.6 62.4 60.3

18mm 40 4 1.89 2@18 0.0353 0.0227 0.0126 35.7 64.3 55.5

FE

M T

able

5.9

a/h

=1

0mm 40 4 1.89 0 0.0976 0.0763 0.0213 21.8 78.2 27.9

6mm 40 4 1.89 2@6 0.0956 0.0753 0.0203 21.2 78.8 27.0

12mm 40 4 1.89 2@12 0.0906 0.0732 0.0174 19.2 80.8 23.8

18mm 40 4 1.89 2@18 0.0849 0.0717 0.0132 15.5 84.5 18.4

FE

M T

able

5.9

a/h

=1

.5

0mm 40 4 1.89 0 0.1529 0.1258 0.0271 17.7 82.3 21.5

6mm 40 4 1.89 2@6 0.1496 0.124 0.0256 17.1 82.9 20.6

12mm 40 4 1.89 2@12 0.1417 0.1206 0.0211 14.9 85.1 17.5

18mm 40 4 1.89 2@18 0.1325 0.1181 0.0144 10.9 89.1 12.2

Effect of Clear Span

FE

M T

able

5.1

1 a

/h=

0.5

1200

mm 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

2400

mm 40 8 1.89 2@12 0.0571 0.0429 0.0142 24.9 75.1 33.1

FE

M T

able

5.1

1 a

/h=

1

1200

mm 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

2400

mm 40 8 1.89 2@12 0.1692 0.1514 0.0178 10.5 89.5 11.8

FE

M T

able

5.1

1 a

/h=

1.5

1200mm

40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3

2400mm

40 8 1.89 2@12 0.3175 0.2953 0.0222 7.0 93.0 7.5


203

Effect of Total Volume Fraction

FE

M T

able

5.1

3 a

/h=

0.5

1.6 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

2.5 40 4 2.89 2@12 0.0371 0.0234 0.0137 36.9 63.1 58.5

3.2 40 4 3.77 2@12 0.0354 0.023 0.0124 35.0 65.0 53.9

FE

M T

able

5.1

3 a

/h=

1

1.6 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

2.5 40 4 2.89 2@12 0.0894 0.074 0.0154 17.2 82.8 20.8

3.2 40 4 3.77 2@12 0.0857 0.0726 0.0131 15.3 84.7 18.0

FE

M T

able

5.1

3 a

/h=

1.5

1.6 40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3

2.5 40 4 2.89 2@12 0.1367 0.1218 0.0149 10.9 89.1 12.2

3.2 40 4 3.77 2@12 0.1340 0.1196 0.0144 10.7 89.3 12.0

As can be seen from Table 6.13:

1. By increasing the a/h ratio, the contribution of shear deformation in the total

deformation decreases.

2. By increasing matrix strength, the flexural deformation can be reduced.

3. By increasing web thickness all the deformations can be reduced. The shear

deflection dropped greater than flexural deformation which means by increasing

web thickness the shear stiffness will increase.

4. By increasing tensile rebar size, the shear deformations can be reduced.

5. Longer clear span reduce the contribution of the shear deformation and flexural

deformation becomes dominant.

6. The increase of volume fraction of tensile mesh and the size of rebar can

improve flexural stiffness and reduce flexural deformation.

Effect of Volume Fraction of Tensile Mesh

FE

M T

able

5.7

a/h

=0

.5

T2B0 40 4 1.12 2@12 0.0381 0.0266 0.0115 30.2 69.8 43.2

T2B2 40 4 1.50 2@12 0.0376 0.0252 0.0124 33.0 67.0 49.2

T2B4 40 4 1.89 2@12 0.0372 0.0239 0.0133 35.8 64.2 55.6

FE

M T

able

5.7

a/h

=1 T2B0 40 4 1.12 2@12 0.0928 0.0839 0.0089 9.6 90.4 10.6

T2B2 40 4 1.50 2@12 0.0915 0.0793 0.0122 13.3 86.7 15.4

T2B4 40 4 1.89 2@12 0.0906 0.0753 0.0153 16.9 83.1 20.3

FE

M T

able

5.7

a/h

=1

.5

T2B0 40 4 1.12 2@12 0.1450 0.1382 0.0068 4.7 95.3 4.9

T2B2 40 4 1.50 2@12 0.1432 0.1307 0.0125 8.7 91.3 9.6

T2B4 40 4 1.89 2@12 0.1417 0.124 0.0177 12.5 87.5 14.3


204

In ordinary cases, when span to depth ratio is 10 or more, the shear deformation is

insignificant in relation to the flexural deformation [53]. So for beams with l/h>10,

shear deformation may be neglected in design. As U beam parametric studies were

based on l/h=13 they were not used to study.

From this study it has been found that for ferrocement deep beam subjected to four-

point bending, the total deformation is related to flexural deformation and shear

deformation. Flexural deformation can be reduced by applying more tensile

reinforcement. Shear deformation can be reduced by increasing web thickness. The

increase of a/h ratio and clear span of the beam can reduce the contribution of the

shear deformation.

6.7 Conclusion

In this chapter design equations are proposed for ferrocement beams with welded

squared wire mesh shear based on different failure modes. Also the elastic theory for

deformation prediction and FE method prediction were reviewed. It is shown that the

proposed equations (Eq 6.4 and Eq 6.5) can provide good prediction compared to

FEM results. Moreover all the predictions are conservative. Through comparison of

the proposed methods with current ACI codes, higher accuracy in predicting the

shear strength was found using the proposed method.

It can be found in this study that by increasing volume fraction of mesh, rebar size,

matrix strength and cross sectional area the shear strength of ferrocement beams can

be improved. Considering the cost of beams, by changing mesh and rebar the cost

will be increased dramatically due to the cost of material and labour. By increasing

cross sectional area the total weight of the beam will be increased and this requires

extra costs in columns and foundation. The best way to improve shear strength is to


205

increase matrix strength. As it is known, by increasing matrix strength the

workability will be reduced. So additives, such as superplasticizer, can be added into

mix to increase flowability of the matrix. A flow chart of ferrocement beam shear

design is presented in this chapter to demonstrate the design procedure.

CHAPTER 7 Summary and Conclusions

206


7.1 Summary

Although some research has been carried out on ferrocement beam shear behaviour

over the past 30 years, none of the suggested empirical equations have been accepted

by ACI 549. The current shear design equation from ACI 318, considering ferro

matrix contribution only, gives about 10% of peak shear capacity obtained in tests,

which makes this equation highly conservative. In order to formulate a better design

equation and ferrocement beam shear design guidelines, the following investigations

were taken:

To obtain the necessary experimental and material property information, a series

of ferrocement beam shear tests and material property tests were undertaken.

Also to obtain high quality specimen, an improved ferrocement beam laboratory

fabrication method was developed.

By using the commercial finite element software ABAQUS, 3D FE models for

the shear behaviour of ferrocement beams under four point bending has been

established. The ABAQUS models have been verified by comparing the

simulation results with test results.

Parametric studies for the shear behaviour of ferrocement beams had been

undertaken using the ABAQUS model. After parametric studies, empirical

equations with a component model were proposed. Also for the shear design of

ferrocement beams, a design flow chart has been given.

7.2 Conclusions

Based on this work, the following conclusions can be drawn:


207

The proposed improved ferrocement beam laboratory fabrication method can

produce high quality test specimens. As shown in the Chapter 3, the improved

methods for both I sections and U sections can make the matrix penetrate

through the mesh layers and largely improve the specimen quality.

Under four point bending, shear failure occurs in ferrocement beams only when

the shear span to depth ratio is less than 1.5. Based on this phenomenon, the

Strut and Tie model was proposed with the limit of the angle between strut and

tie in the Chapter 6.

The proposed Strut and Tie model can identify different failure modes. Two

different shear failure modes were found: diagonal splitting and shear-flexure.

The diagonal splitting failure normally occurs when there is sufficient tie

strength in the beam and the shear-flexure failure normally occurs when tie

yielded before failure.

The matrix compressive strength, shear span to depth ratio, volume fraction

(especially in the tension zone and web) and clear span of the beam are the main

factors influencing the shear capacity of ferrocement beams. By increasing

volume fraction (especially in the tension zone and web) or matrix compressive

strength the shear capacity will increase. By increasing shear span to depth ratio

or clear span of the beam will decrease the shear capacity.

The rebar can improve ferrocement shear and flexural capacity. However, the

existence of rebar can change the failure type from shear-flexure to diagonal

splitting; and for flexural failure, detail calculations must be carried out to avoid

over-reinforced design. Moreover due to the cost, rebar are not recommended for

ferrocement shear design.

The FE model developed using the commercial package ABAQUS is

sufficiently accurate to investigate the shear behaviour of ferrocement beams. It

can show the similar geometrical changing behaviour as seen in experiments

with similar load deflection curves also in the parametric studies it can show the


208

changes in trends and load capacities.

The proposed empirical equations can accurately and conservatively predict the

shear capacity of ferrocement beams. As shown in the Chapter 6, the predictions

from the proposed equations were more accurate than that from the design

formulae used in ACI 318[6]. As the proposed equations consider the changes of

mesh, rebar, beam length and the matrix strength independently, therefore the

equations are more suitable for ferrocement shear design.

The initial stiffness of the deep beam is controlled by both flexural stiffness and

shear stiffness as shown in the Chapter 6. The increases of the matrix strength,

volume fraction and rebar sizes can increase the value of initial stiffness. By

increasing the length of the beam, the effects of shear stiffness become less and

the initial stiffness is mainly affected by flexural stiffness.

7.3 Recommendations for future study

The following further research studies may be carried out to improve knowledge in

this area:

Further experimental research studies should be undertaken with different mesh

types to validate whether the empirical equations can be used in other cases.

In this research, the local crushing of loading points and supports did not occur.

In order to have conservative design prediction, studies on local crushing need to

be undertaken and a reinforcing detailing method should be produced.

Ferrocement beams with other cross-section need to be studied experimentally to

check the validity of the empirical equations.

Reference

209

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Publications

Tian S., Mandal P. and Nedwell P., Shear behaviour of ferrocement beams:

experimental and fem study, 10th International symposium on ferrocement and thin

reinforced cement composites, p. 261-268, 15-17th Oct. 2012, Cuba.

Tian S. and Nedwell P., Ferroceemnt Beams Improved Lab Casting Method, 10th

International symposium on ferrocement and thin reinforced cement composites,

poster, 15-17th Oct. 2012, Cuba.

Tian S., Nedwell P. and Mandal P., Shear behaviour of ferrocement I beams, Cement

Concrete Composite, in preparation.

Appendix

214

Appendix A: Measurements of Specimens

Beam I1

View Name Label Unit Reading

Average Flange Average

Web Average 1 2 3 4 End Centre

Top View Length Lt (mm) 1651 1650 1650 1650.3

Width Dt (mm) 160 157 154 157.0

Elevation View Front Height Hf (mm) 314 316 314 314.7

Back Height Hb (mm) 317 316 314 315.7

Left Side View

Flange End Thickness Tfel (mm) 61 62 61 57 60.3 60.25

Flange Centre Thickness Tfcl (mm) 85 81 80 81 81.8 81.75

Web Thickness Twl (mm) 35 35 35 35.0 35.0

Right Side View

Flange End Thickness Tfer (mm) 61 55 56 53 56.3 56.25

Flange Centre Thickness Tfcr (mm) 86 86 80 80 83.0 83

Web Thickness Twr (mm) 34 34 33 33.7 33.7

Bottom View Length Lb (mm) 1647 1647 1647 1647.0

Width Db (mm) 151 158 162 157.0

58.3 82.4 34.3

Weight 111 kg

Tfel Thickness of Left Hand Side Flange End

Appendix

215

Beam I2





Width Dt (mm) 156 156 156 156.0


Back Height Hb (mm) 314 314 314 314.0

Left Side View




Right Side View

Flange End Thickness Tfer (mm) 56 60 51 57 56.0 56

Flange Centre Thickness Tfcr (mm) 84 84 80 79 81.8 81.75



Width Db (mm) 153 153 154 153.3

57.8 81.3 33.5

Weight 109 kg


Appendix

216

Beam I3





Width Dt (mm) 160 156 153 156.3


Back Height Hb (mm) 315 316 314 315.0

Left Side View




Right Side View

Flange End Thickness Tfer (mm) 61 56 57 53 56.8 56.75

Flange Centre Thickness Tfcr (mm) 84 84 79 80 81.8 81.75



Width Db (mm) 153 156 153 154.0

58.6 82.5 32.3

Weight 105.2 kg


Appendix

217

Beam U1


Average Web Average

Flange Average 1 2 3 Front Back

Top View Length L (mm) 1496.0 1496.0 1496.0 1496.0

Width D (mm) 400.0 399.0 399.0 399.3

Elevation View Front Height Hf (mm) 101.0 100.0 101.0 100.7

Back Height Hb (mm) 101.0 99.0 100.0 100.0

Left Side View

Flange Thickness Tfl (mm) 18.0 15.0 15.0 16.0 16.0

Front Web Thickness Tfwl (mm) 16.0 16.0 16.0 16.0 16.0

Back Web Thickness Tbwl (mm) 16.0 16.5 17.0 16.5 16.5

Right Side View

Flange Thickness Tfr (mm) 18.0 17.0 18.0 17.7 17.7

Front Web Thickness Tfwr (mm) 16.0 16.0 16.0 16.0 16.0

Back Web Thickness Tbwr (mm) 16.0 17.0 18.0 17.0 17.0

Bottom View Front Web Thickness Tfwb (mm) 16.0 17.0 17.0 16.7 16.7

Back Web Thickness Tbwb (mm) 18.0 19.0 18.0 18.3 18.3

16.2 17.3 16.8

Weight 32 kg

Tfwb Thickness of front web bottom view

Appendix

218

Beam U2


Average Web Average



Width D (mm) 400.0 400.0 400.0 400.0


Back Height Hb (mm) 101.0 101.0 102.0 101.3

Left Side View




Right Side View






20.7 21.9 17.7

Weight 32.5 kg


Appendix

219

Beam U3


Average Web Average



Width D (mm) 400.0 399.0 399.0 399.3


Back Height Hb (mm) 990.0 990.0 980.0 986.7

Left Side View




Right Side View






17.1 14.9 15.5

Weight 30.9 kg


Appendix

220

Beam U4


Average Web Average



Width D (mm) 399.0 400.0 401.0 400.0


Back Height Hb (mm) 100.0 100.0 100.0 100.0

Left Side View




Right Side View






21.1 22.8 18.2

Weight 37.7 kg


Appendix

221

Appendix B:

I beam test results

I1

-500 0 500 1000 1500 2000 2500 3000

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

187.5mm from bottom

265mm from bottom

310mm from bottom

Horizontal Gauges Load-Micro Strain Curves for Beam I1

Appendix

222

-1000 -500 0 500 1000 1500 2000 2500 3000

0

50

100

150

200

250

300

350

Lo

ca

tio

n(m

m)

Micro-Strain

51.38kN

94.15kN

157.61kN

200.45kN

249.79kN

285.87kN

Horizontal Gauges Location-Micro Strain Curves for Beam I1

-0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006

0

50

100

150

200

250

300

350

Strain X from Rosette Gauges on LHS of Beam




Lo

ad

(kN

)

Strain

Load-Strain Curves of Rosette Gauges for Beam I1

Appendix

223

-0.04 -0.03 -0.02 -0.01 0.00 0.01

0

50

100

150

200

250

300

350 Shear Strain from Rosette Gauges on LHS of Beam


Lo

ad

(kN

)

Shear Strain

Load-Shear Strain Curves for Beam I1

0.000 0.005 0.010 0.015 0.020

0

50

100

150

200

250

300

350 Max Normal Strain from Rosette Gauges on LHS of Beam

Max Normal Strain from Rosette Gauges on RHS of Beam

Lo

ad

(kN

)

Max Normal Strain

Load-Max Normal Strain Curves for Beam I1

Appendix

224

-0.020 -0.015 -0.010 -0.005 0.000

0

50

100

150

200

250

300

350 Min Normal Strain from Rosette Gauges on LHS of Beam

Min Normal Strain from Rosette Gauges on RHS of Beam

Lo

ad

(kN

)

Min Normal Strain

Load-Min Normal Strain Curves for Beam I1

-40 -30 -20 -10 0 10 20 30 40 50

0

50

100

150

200

250

300

350 Principal Angle from Rosette Gauges on LHS of Beam


Lo

ad

(kN

)


Load-Principal Angle Curves for Beam I1

Appendix

225

I3

-1000 0 1000 2000 3000

0

50

100

150

200

250

300

350

Lo

ad

(kN

)

Micro-strain

5mm from bottom

50mm from bottom

117.5mm from bottom

157.5mm from bottom

187.5mm from bottom

265mm from bottom

310mm from bottom

Horizontal Gauges Load-Micro Strain Curves for Beam I3

-500 0 500 1000 1500 2000

0

50

100

150

200

250

300

350

Lo

ca

tio

n

micro-Strain

46.49kN

98.26kN

157.27kN

196.81kN

212.09kN

Horizontal Gauges Location-Micro Strain Curves for Beam I3

Appendix

226

-0.010 -0.005 0.000 0.005 0.010 0.015

0

50

100

150

200

250

300

350 Strain X from Rosette Gauges on LHS of Beam




Lo

ad

(kN

)

Strain


-0.005 0.000 0.005 0.010 0.015

0

50

100

150

200

250

300

350

Shear Strain from Rosette Gauges on LHS of Beam


Lo

ad

(kN

)

Shear Strain

Load-Shear Strain Curves for Beam I3

Appendix

227

0.000 0.005 0.010 0.015

0

50

100

150

200

250

300

350 Max Normal Strain from Rosette Gauges on LHS of Beam

Max Normal Strain from Rosette Gauges on RHS of Beam

Lo

ad

(kN

)

Max Normal Strain

Load-Max Normal Strain Curves for Beam I3

-0.010 -0.005 0.000

0

50

100

150

200

250

300

350 Min Normal Strain from Rosette Gauges on LHS of Beam

Min Normal Strain from Rosette Gauges on RHS of Beam

Lo

ad

(kN

)

Min Normal Strain

Load-Min Normal Strain Curves for Beam I3

Appendix

228

-25 -20 -15 -10 -5 0 5 10 15 20

0

50

100

150

200

250

300

350

Principal Angle from Rosette Gauges on LHS of Beam


Lo

ad

(kN

)



Appendix

229

Appendix C:

Peak Load Changes in Chapter Five

I beam

Peak Load Changes with Different Matrix Strength


a/h=0.5 a/h=1 a/h=1.5

30 Change (%) 242 Change (%) 211 Change (%) 190 Change (%)

40 33.3 264 9.09 231 9.48 209 10.0

50 66.7 282 16.5 247 17.1 223 17.4

60 100 298 23.1 261 23.7 236 24.2

Peak Load Changes with Different Web Mesh Layer Number


a/h=0.5 a/h=1 a/h=1.5

0 246 Change (%) 204 Change (%) 194 Change (%)

2 264 7.32 231 13.2 209 7.73

4 273 10.98 238 16.7 212 9.28

6 282 14.63 246 20.6 216 11.34

Peak Load Changes with Different Web Thickness

Web Thickness (mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

16 Change (%) 244 Change (%) 207 Change (%) 186 Change (%)

32 100 264 8.20 231 11.6 209 12.4

48 200 276 13.1 245 18.4 222 19.4

Appendix

230

Peak Load Changes with Different Overall Depth

Total

Depth(mm)

Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.25 a/h=1.5

314 Change

(%) 264

Change

(%) 231

Change

(%) 228

Change

(%) 209

Change

(%)

380 21.0 263 -0.379 232 0.433 229 0.439 190 -9.09

Peak Load Changes with Different Root Height

Root Height(mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

0 247 Change (%) 213 Change (%) 198 Change (%)

8 255 3.24 221 3.76 202 2.02

16 260 5.26 226 6.10 205 3.54

23 264 6.88 231 8.45 209 5.56

Peak Load Changes with Different Top Flange Mesh Layer Number


a/h=0.5 a/h=1 a/h=1.5

T0B4 257 Change (%) 226 Change (%) 210 Change (%)

T1B4 260 1.17 229 1.33 214 1.90

T2B4 264 2.72 231 2.21 209 -0.476

Peak Load Changes with Different Bottom Flange Mesh Layer Number


a/h=0.5 a/h=1 a/h=1.5


T2B2 263 1.15 228 6.05 207 0.488

T2B4 264 1.54 231 7.44 209 1.95

Appendix

231

Peak Load Changes with Different Flange Mesh Layer Number


a/h=0.5 a/h=1 a/h=1.5


T3B6 269 1.89 238 3.03 213 1.91

T4B8 274 3.79 244 5.63 216 3.35

Peak Load Changes with Different Rebar Size


a/h=0.5 a/h=1 a/h=1.5

0 mm 251 Change (%) 182 Change (%) 160 Change (%)

6 mm 256 1.99 204 12.1 178 11.3

12 mm 264 5.18 231 26.9 209 30.6

18 mm 270 7.57 248 36.3 221 38.1

Peak Load Changes with Different Clear Span Length

Clear

span

(mm)

Peak Load (kN)

a/h=0.5 a/h=0.75 a/h=1 a/h=1.25 a/h=1.5

1200 264 Change

(%) 241

Change

(%) 231

Change

(%) 228

Change

(%) 209

Change

(%)

2400 247 -6.44 213 -11.6 186 -19.5 182 -20.2 177 -15.3

Peak Load Changes with Different Specific Surface

Specific surface Peak load (kN)

a/h=0.5 a/h=1.0 a/h=1.5

0.0472 Change (%) 264 Change (%) 231 Change (%) 209 Change (%)

0.0370 -21.6 265 0.38 228 -1.30 212 1.44

0.0339 -28.2 268 1.52 221 -4.33 209 0.00

Appendix

232

Peak Load Changes with Different Volume Fraction

Volume fraction Peak load (kN)

a/h=0.5 a/h=1 a/h=1.5

0.0189 Change (%) 264 Change (%) 231 Change (%) 209 Change (%)

0.0289 52.9 278 5.30 235 1.73 223 6.70

0.0377 99.5 288 9.09 235 1.73 223 6.70

U beam


Matrix Strength

(MPa)

Peak Load (kN)

a/h=0.5 a/h=1. a/h=1.5

30 Change

(%) 32.9

Change

(%) 16.8

Change

(%) 11.4

Change

(%)

40 33.3 34.7 5.47 17.3 2.98 11.7 2.63

50 66.7 36.5 10.9 17.8 5.95 11.9 4.39

60 100.0 38 15.5 18.2 8.33 12.2 7.02

Peak Load Changes with Different Web Thickness

Web thickness (mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

9 Change (%) 30.7 Change (%) 16 Change (%) 10.9 Change (%)

12.4 37.8 34.7 13.0 17.3 8.13 11.7 7.34

16 77.8 40.8 32.9 19.2 20.0 12.7 16.5

Peak Load Changes with Different Mesh Layer Number

Mesh number Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

1 27 Change (%) 11.8 Change (%) 7.3 Change (%)

2 34.7 28.5 17.3 46.6 11.7 60.3

3 43.6 61.5 23.3 97.5 16 119.2

Appendix

233


Rebar

Diameter

(mm)

Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

0 34.7 Change

(%) 17.3

Change

(%) 11.7

Change

(%)

6 89.4 157.6 52.8 205.2 37 216.2

8 107.6 210.1 73.7 326.0 54.8 368.4

Peak Load Changes with Different Clear Span Length

Clear Span(mm) Peak Load (kN)

a/h=0.5 a/h=1 a/h=1.5

1300 Change (%) 34.7 Change (%) 17.3 Change (%) 11.7 Change (%)

2600 100 24.4 -29.7 13.8 -20.2 9.7 -17.1

Peak Load Changes with Different Specific Surface

Specific surface Peak load (kN)

a/h=0.5 a/h=1 a/h=1.5

0.1325 Change (%) 34.7 Change (%) 17.3 Change (%) 11.7 Change (%)

0.1057 -20.2 36 3.75 18.2 5.20 12.3 5.13

0.0936 -29.4 39.7 14.4 20.3 17.3 13.8 17.9

Peak Load Changes with Different Volume Fraction

Volume fraction Peak load (kN)

a/h=0.5 a/h=1 a/h=1.5

0.053 Change (%) 34.7 Change (%) 17.3 Change (%) 11.7 Change (%)

0.0826 55.8 46.6 34.29 25.4 46.82 17.5 49.57

0.106 100.0 61.8 78.1 34.7 100.6 24.3 107.7

Appendix

234

Appendix D:

Input Value for Matrix Strength Parametric Study

Grade 30 40 50 60

Ecm

MPa 11804 14905 18191 21700

Co

mp

ress

ive

Stress

(MPa) Strain

Stress

(MPa) Strain

Stress

(MPa) Strain

Stress

(MPa) Strain

12 0 16 0 20 0 24 0

18 0.0001 24 0.0001 30 0.00002 36 0.00001

24 0.0002 32 0.0002 40 0.0001 48 0.0001

30 0.0014 40 0.0011 50 0.0009 60 0.0008

24 0.0044 32 0.0036 40 0.0031 48 0.0027

18 0.0069 24 0.0055 30 0.0047 36 0.0041

12 0.0103 16 0.0079 20 0.0066 24 0.0057

6 0.0170 8 0.0120 10 0.0095 12 0.0080

3 0.0261 4 0.0169 5 0.0127 6 0.0102

Ten

sile

Stress

(MPa)

Displace

ment

(mm)

Stress

(MPa)

Displace

ment

(mm)

Stress

(MPa)

Displace

ment

(mm)

Stress

(MPa)

Displace

ment

(mm)

3.8 0 4.02 0 4.24 0 4.46 0

0.57 0.0114 0.603 0.0131 0.636 0.0146 0.669 0.0157

0.004 0.1136 0.004 0.1313 0.004 0.1455 0.004 0.1572

Appendix

235

Appendix E:

Comparison of FEM Results, ACI Formula and Eq 6.4

Beam

l/h Volume

Fraction

Rebar


MPa % mm FEM ACI

318 Eq 6.4

FEM/

ACI 318

FEM/

Eq 6.4

ACI 318/

Eq 6.4

Effect of Matrix Strength

FE

M T

able

5.1

a/h

=0

.5

30

MPa 30 4 2.58 2@12 242 70 205 3.44 1.18 0.34

40

MPa 40 4 2.58 2@12 264 94 236 2.82 1.12 0.40

50 MPa

50 4 2.58 2@12 282 117 264 2.41 1.07 0.44

60 MPa

60 4 2.58 2@12 298 140 289 2.12 1.03 0.49

FE

M T

able

5.1

a/h

=1

30 MPa

30 4 2.58 2@12 211 56 177 3.80 1.19 0.31

40 MPa

40 4 2.58 2@12 231 74 204 3.12 1.13 0.36

50

MPa 50 4 2.58 2@12 247 93 229 2.67 1.08 0.41

60

MPa 60 4 2.58 2@12 261 111 250 2.35 1.04 0.44

FE

M T

able

5.1

a/h

=1

.5

30

MPa 30 4 2.58 2@12 190 44 159 4.36 1.19 0.27

40

MPa 40 4 2.58 2@12 209 58 184 3.60 1.14 0.32

50

MPa 50 4 2.58 2@12 223 73 205 3.07 1.09 0.35

60

MPa 60 4 2.58 2@12 236 87 225 2.71 1.05 0.39

Effect of web volume fraction factor , volume fraction=bottom flange volume fraction + web volume fraction

FE

M T

able

5.2

a/h

=0

.5 0 40 4 2.58 2@12 246 94 220 2.63 1.12 0.43

2 40 4 2.58+

1.80 2@12 264 94 236 2.82 1.12 0.40

4 40 4 2.58+ 3.60

2@12 273 94 243 2.91 1.12 0.38

6 40 4 2.58+

5.40 2@12 282 94 278 3.01 1.01 0.34

FE

M T

able

5.1

a/h

=1

0 40 4 2.58 2@12 204 74 180 2.75 1.13 0.41

2 40 4 2.58+ 1.80

2@12 231 74 204 3.12 1.13 0.36

4 40 4 2.58+

3.60 2@12 238 74 211 3.21 1.13 0.35

6 40 4 2.58+

5.40 2@12 246 74 217 3.32 1.14 0.34

FE

M T

able

5.2

a/h

=1

.5

0 40 4 2.58 2@12 194 58 169 3.34 1.15 0.34

2 40 4 2.58+

1.80 2@12 209 58 184 3.60 1.14 0.32

4 40 4 2.58+

3.60 2@12 212 58 185 3.65 1.14 0.31

6 40 4 2.58+

5.40 2@12 216 58 189 3.72 1.14 0.31

Appendix

236

Effect of Cross Sectional Area , volume fraction=bottom flange volume fraction + web volume fraction

FE

M T

able

5.3

a/h

=0

.5

16mm 40 4 2.58+ 3.60

2@12 244 94 216 2.61 1.13 0.43

32mm 40 4 2.58+

1.80 2@12 264 94 236 2.82 1.12 0.40

48mm 40 4 2.58+

0.90 2@12 276 94 257 2.95 1.07 0.36

FE

M T

able

5.3

a/h

=1

16mm 40 4 2.58+

3.60 2@12 207 74 187 2.80 1.11 0.40

32mm 40 4 2.58+

1.80 2@12 231 74 204 3.12 1.13 0.36

48mm 40 4 2.58+

0.90 2@12 245 74 222 3.31 1.10 0.33

FE

M T

able

5.3

a/h

=1

.5

16mm 40 4 2.58+

3.60 2@12 186 58 168 3.20 1.11 0.35

32mm 40 4 2.58+ 1.80

2@12 209 58 184 3.60 1.14 0.32

48mm 40 4 2.58+ 0.90

2@12 222 58 200 3.82 1.11 0.29

Effect of Rebar Factor

FE

M T

able

5.9

a/h

=0

.5

0mm 40 4 2.58 0 251 94 225 2.68 1.11 0.42

6mm 40 4 2.58 2@6 256 94 230 2.73 1.11 0.41

12mm 40 4 2.58 2@12 264 94 236 2.82 1.12 0.40

18mm 40 4 2.58 2@18 270 94 243 2.88 1.11 0.39

FE

M T

able

5.9

a/h

=1

0mm 40 4 2.58 0 182 74 162 2.46 1.12 0.46

6mm 40 4 2.58 2@6 204 74 178 2.75 1.14 0.41

12mm 40 4 2.58 2@12 231 74 204 3.12 1.13 0.36

18mm 40 4 2.58 2@18 248 74 219 3.35 1.13 0.34

FE

M T

able

5.9

a/h

=1

.5

0mm 40 4 2.58 0 160 58 142 2.75 1.12 0.41

6mm 40 4 2.58 2@6 178 58 157 3.06 1.14 0.37

12mm 40 4 2.58 2@12 209 58 184 3.60 1.14 0.32

18mm 40 4 2.58 2@18 221 58 195 3.80 1.13 0.30

Effect of Length Factor K

FE

M T

able

5.1

1 a

/h=

0.5

1200

mm 40 4 2.58 2@12 264 94 236 2.82 1.12 0.40

2400mm

40 8 2.58 2@12 247 94 222 2.64 1.11 0.42

FE

M T

able

5.1

1 a

/h=

1 1200

mm 40 4 2.58 2@12 231 74 204 3.12 1.13 0.36

2400

mm 40 8 2.58 2@12 186 74 161 2.51 1.15 0.46

FE

M T

able

5.1

1 a

/h=

1.5

1200

mm 40 4 2.58 2@12 209 58 184 3.60 1.14 0.32

2400

mm 40 8 2.58 2@12 177 58 158 3.05 1.12 0.37

Effect of Mesh Factor

FE

M

Tab

le

5.7

a/h

=0

.5

T2B0 40 4 0 2@12 260 94 233 2.78 1.12 0.40

T2B2 40 4 1.29 2@12 263 94 233 2.81 1.13 0.40

Appendix

237

T2B4 40 4 2.58 2@12 264 94 236 2.82 1.12 0.40

FE

M T

able

5.7

a/h

=1 T2B0 40 4 0 2@12 215 74 192 2.90 1.12 0.39

T2B2 40 4 1.29 2@12 228 74 192 3.08 1.19 0.39

T2B4 40 4 2.58 2@12 231 74 204 3.12 1.13 0.36

FE

M T

able

5.7

a/h

=1

.5

T2B0 40 4 0 2@12 206 58 184 3.53 1.12 0.32

T2B2 40 4 1.29 2@12 207 58 184 3.55 1.12 0.32

T2B4 40 4 2.58 2@12 209 58 184 3.60 1.14 0.32

FE

M T

able

5.1

3

a/h

=0

.5

1.6 40 4 2.58+

1.80 2@12 264 94 236 2.82 1.12 0.40

2.5 40 4 3.08+

3.07 2@12 278 94 239 2.97 1.16 0.39

3.2 40 4 4.07+

4.02 2@12 288 94 255 3.07 1.13 0.37

FE

M T

able

5.1

3

a/h

=1

1.6 40 4 2.58+

1.80 2@12 231 74 204 3.12 1.13 0.36

2.5 40 4 3.08+

3.07 2@12 235 74 213 3.17 1.10 0.35

3.2 40 4 4.07+

4.02 2@12 235 74 224 3.17 1.05 0.33

FE

M T

able

5.1

3

a/h

=1

.5

1.6 40 4 2.58+

1.80 2@12 209 58 185 3.60 1.13 0.31

2.5 40 4 3.08+

3.07 2@12 223 58 190 3.84 1.17 0.31

3.2 40 4 4.07+

4.02 2@12 223 58 196 3.84 1.13 0.30

Average 3.11 1.12 0.37

Standard

Deviation 0.431 0.033 0.046

Comparison of FEM Results, ACI Formula and Eq 6.5

Specimen

Matrix

Strength l/h

Volume

Fraction

Rebar


MPa

% mm FEM ACI

318 Eq 6.5

FEM/

ACI 318

FEM/

Eq 6.5

ACI318

/ Eq 6.5

Effect of Matrix Strength

FE

M T

able

5.1

4

a/h

=0

.5

30 MPa

30 13 5.30 0 32.9 21.2 23.7 1.55 1.39 0.90

40 MPa

40 13 5.30 0 34.7 21.2 27.4 1.63 1.27 0.78

50

MPa 50 13 5.30 0 36.5 21.2 30.6 1.72 1.19 0.69

60

MPa 60 13 5.30 0 38.0 21.2 33.5 1.79 1.13 0.63

FE

M T

able

5.1

4 a

/h=

1

30

MPa 30 13 5.30 0 16.8 10.6 11.9 1.58 1.41 0.89

40

MPa 40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

50

MPa 50 13 5.30 0 17.8 10.6 15.4 1.68 1.16 0.69

60

MPa 60 13 5.30 0 18.2 10.6 16.9 1.71 1.08 0.63

Appendix

238

FE

M T

able

5.1

4

a/h

=1

.5

30

MPa 30 13 5.30 0 11.4 7.1 8.1 1.61 1.41 0.87

40

MPa 40 13 5.30 0 11.7 7.1 9.4 1.65 1.25 0.76

50

MPa 50 13 5.30 0 11.9 7.1 10.5 1.68 1.14 0.68

60

MPa 60 13 5.30 0 12.2 7.1 11.5 1.72 1.06 0.62

Effect of Cross Sectional Area

FE

M T

able

5.1

5

a/h

=0

.5

9

mm 40 13 7.15 0 30.7 21.2 20.5 1.45 1.50 1.03

12.4

mm 40 13 5.30 0 34.7 21.2 27.4 1.63 1.27 0.78

16

mm 40 13 4.02 0 40.8 21.2 26.6 1.92 1.54 0.80

FE

M T

able

5.1

5

a/h

=1

9 mm

40 13 7.15 0 16.0 10.6 10.3 1.51 1.55 1.03

12.4mm

40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

16

mm 40 13 4.02 0 19.2 10.6 11.7 1.81 1.65 0.91

FE

M T

able

5.1

5

a/h

=1

.5

9

mm 40 13 7.15 0 10.9 7.1 7.0 1.54 1.55 1.01

12.4

mm 40 13 5.30 0 11.7 7.1 9.4 1.65 1.25 0.76

16

mm 40 13 4.02 0 12.7 7.1 7.1 1.79 1.78 0.99

Effect of Length Factor K

FE

M T

able

5.1

8

a/h

=0

.5

1300

mm 40 13 5.30 0 34.7 21.2 27.4 1.63 1.27 0.78

2600

mm 40 26 5.30 0 24.4 21.2 20.2 1.15 1.21 1.05

FE

M T

able

5.1

8

a/h

=1

1300

mm 40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

2600

mm 40 26 5.30 0 13.8 10.6 11.2 1.30 1.23 0.95

FE

M T

able

5.1

8

a/h

=1

.5

1300

mm 40 13 5.30 0 11.7 7.1 9.4 1.65 1.25 0.76

2600mm

40 26 5.30 0 9.7 7.1 7.8 1.37 1.25 0.91

Effect of Mesh Factor

FE

M T

able

5.1

6

a/h

=0

.5

1 40 13 2.59 0 27.0 10.6 21.3 2.54 1.27 0.50

2 40 13 5.30 0 34.7 21.2 27.4 1.63 1.27 0.78

3 40 13 7.78 0 43.6 31.8 34.3 1.37 1.27 0.93

Appendix

239

FE

M T

able

5.1

6

a/h

=1

1 40 13 2.59 0 11.8 5.3 9.4 2.22 1.26 0.57

2 40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

3 40 13 7.78 0 23.3 15.9 18.4 1.46 1.27 0.87 F

EM

Tab

le 5

.16

a/h

=1

.5

1 40 13 2.59 0 7.3 3.5 5.7 2.06 1.28 0.62

2 40 13 5.30 0 11.7 7.1 9.4 1.65 1.25 0.76

3 40 13 7.78 0 16.0 10.6 12.7 1.51 1.26 0.84

FE

M T

able

5.2

0

a/h

=0

.5

1.6 40 13 5.30 0 34.7 21.2 27.4 1.63 1.27 0.78

2.5 40 13 8.26 0 46.6 51.8 34.3 0.90 1.36 1.51

3.2 40 13 10.6 0 61.8 84.9 54.4 0.73 1.14 1.56

FE

M T

able

5.2

0

a/h

=1

1.6 40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

2.5 40 13 8.26 0 25.4 25.9 18.4 0.98 1.38 1.41

3.2 40 13 10.6 0 34.7 42.5 30.2 0.82 1.15 1.41

FE

M T

able

5.2

0

a/h

=1

.5

1.6 40 13 5.30 0 11.7 7.1 9.4 1.65 1.25 0.76

2.5 40 13 8.26 0 17.5 17.3 12.7 1.01 1.38 1.37

3.2 40 13 10.6 0 24.3 28.3 21.5 0.86 1.13 1.32

Effect of Rebar Factor

FE

M T

able

5.1

7

a/h

=0

.5

0 40 13 5.30 0 34.7 10.6 27.4 3.27 1.27 0.39

6 40 13 5.30 2@6 89.4 52.8 68.5 1.69 1.31 0.77

8 40 13 5.30 2@8 107.6 52.8 84.9 2.04 1.27 0.62

FE

M T

able

5.1

7

a/h

=1

0 40 13 5.30 0 17.3 10.6 13.8 1.63 1.26 0.77

6 40 13 5.30 2@6 52.8 41.4 41.3 1.28 1.28 1.00

8 40 13 5.30 2@8 73.7 41.4 57.9 1.78 1.27 0.72

FE

M T

able

5.1

7

a/h

=1

.5

0 40 13 5.30 0 11.7 10.6 9.4 1.10 1.25 1.13

6 40 13 5.30 2@6 37.0 32.8 29.0 1.13 1.28 1.13

8 40 13 5.30 2@8 54.8 32.8 43.1 1.67 1.27 0.76

Average 1.58 1.29 0.87

Standard

Deviation 0.408 0.132 0.247

Shear Behaviour of Ferrocement Deep Beams

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Transcript of Shear Behaviour of Ferrocement Deep Beams