Single Straight Steel Fiber Pullout Characterization in Ultra-High Performance Concrete

Valerie Mills Black

Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Master of Science

In

Civil Engineering

Cristopher D. Moen, Chair Carin L. Roberts-Wollmann

Ioannis Koutromanos

May 27, 2014 Blacksburg, VA

Keywords: Fiber pullout, bond slip, Ultra High-Performance Concrete, proximity effect

Single Straight Steel Fiber Pullout Characterization in Ultra-High Performance Concrete

Valerie Mills Black

ABSTRACT

This thesis presents results of an experimental investigation to characterize single

straight steel fiber pullout in Ultra-High Performance Concrete (UHPC). Several

parameters were explored including the distance of fibers to the edge of specimen,

distance between fibers, and fiber volume in the matrix. The pullout load versus slip

curve was recorded, from which the pullout work and maximum pullout load for each

series of parameters were obtained. The curves were fitted to an existing fiber pullout

model considering bond-fracture energy, Gd, bond frictional stress, τ0, and slip hardening-

softening coefficient, β. The representative load-slip curve characterizing the fiber pullout

behavior will be implemented into a computational modeling protocol, for concrete

structures, based on Lattice Discrete Particle Modeling (LDPM). The parametric study

showed that distances over 12.7 mm from the edge of the specimen have no significant

effect on the maximum pullout load and work. Edge distances of 3.2 mm decreased the

average pullout work by 26% and the maximum pullout load by 24% for mixes with 0%

fiber volume. The distance between fibers did not have a significant effect on the pullout

behavior within this study. Slight differences in pullout behavior between the 2% and 4%

fiber volumes were observed including slight increase in the maximum pullout load when

increasing fiber volume. The suggested fitted parameters for modeling with 2% and 4%

fiber volumes are a bond-fracture energy value of zero, a bond friction coefficient of 2.6

N/mm2 and 2.9 N/mm2 and a slip-hardening coefficient of 0.21 and 0.18 respectively.

iii

Acknowledgements

I would like to acknowledge that this research project would not have been

possible or successful without the support of the National Science Foundation (NSF).

Additionally, there have been many people who have been quintessential in the success

of this project.

I would like to express my gratitude towards my advisor, Dr. Cristopher Moen for

his support, guidance and assistance throughout this project. I would also like to express

gratitude toward my committee members, Dr. Carin Roberts-Wollmann and Dr. Ioannis

Koutromanos for their guidance and support helped this study be successful. I would also

like to thank LaFarge for their knowledge and support throughout this project, especially

Vic Perry, Kyle Nachuk, Andrew Ross, and Peter Seibert for their help with the Ductal ®

mixes. I would also like to thank Heidi Helmink at Bekaert for her help and knowledge

with the Dramix ® steel fibers. Additionally, I would like to express gratitude to Dr.

Gianluca Cusatis at Northwestern University for his knowledge on fiber pull and the

LDPM model.

I would also like to thank Mac McCord at the Norris Lab for his support,

knowledge, patience and the use of the testing machine. I would also like to thank Dr.

David Mokarem for his support, guidance and humor throughout this project. I would

also like to thank Brett Farmer and Dennis Huffman with their continuous support and

help fixing my fiber grips.

I would like to give a special thanks to the undergraduate researchers on the team:

Tommy Dacanay for help with the UHPC pours and Rachel Gordon for her patience and

help while performing the tedious task of measuring and labeling all 700+ fibers. I would

iv

also like to thank Rebecca Dickinson for taking the time to consult and teach me statistics

for the project.

I would like to also express my never-ending gratitude to Rafic El Helou, who

without his endless support, knowledge, patience and Skittles, this project would not have

been successful.

I would like to thank all my friends for their support and encouragement during

this project. I would also like to thank my family: Dan, Lynne and Odessa Black, for their

endless support, patience and unconditional love throughout my entire life. Finally, I

would like to thank my dog, River, for knowing exactly when I needed to snuggle after a

long day of testing.

Funding Support

This material is based upon work supported by the National Science Foundation

under Grant No. 1201087 to Virginia Tech with Subcontract to Northwestern University.

Any opinions, findings, and conclusions or recommendations expressed in this material

are those of the authors and do not necessarily reflect the views of the National Science

Foundation.

The Ultra-High Performance Concrete used in this research work is donated by

LaFarge - Ductal®.

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Table of Contents

ABSTRACT ............................................................................................................................... ii  Acknowledgements ................................................................................................................... iii  Table of Contents ....................................................................................................................... v  List of Figures .......................................................................................................................... vii  List of Tables ............................................................................................................................. x  

CHAPTER 1. Introduction ........................................................................................................... 1  

CHAPTER 2. Literature Review .................................................................................................. 6  

2.1 Fiber Pullout Model ............................................................................................................. 6  2.2 Fiber Geometry .................................................................................................................... 9  2.3 Effect of Matrix Strength and Composition ....................................................................... 10  2.4 “Group” Effect ................................................................................................................... 12  2.5 Fiber Volume in Matrix ..................................................................................................... 13  2.6 Bond-Slip Hardening-Softening Effect .............................................................................. 14  

CHAPTER 3. Materials and Methods ........................................................................................ 18  

3.1 Specimens and Fiber Embedment Process ........................................................................ 21  3.2 Specimen Casting: Ultra-High Performance Concrete Placing Process ............................ 23  3.3 Test Setup and Testing Procedures .................................................................................... 25  3.4 Test Data Corrections ........................................................................................................ 27  

CHAPTER 4. Results and Discussion ........................................................................................ 30  

4.1 Influence of Fiber Location on Specimen, x ...................................................................... 32  4.2 ANOVA Statistical Analysis ............................................................................................. 37  4.3 Fiber Groups ...................................................................................................................... 42  4.4 Distances between Fibers, d ............................................................................................... 49  4.5 Proximity to Edge, E .......................................................................................................... 53  4.6 Volume of Fibers in the Matrix, V ..................................................................................... 56  4.7 Averaging Curves .............................................................................................................. 58  4.8 Compressive Strength ........................................................................................................ 65  

CHAPTER 5. Model Parameters and Curves ............................................................................. 68  

CHAPTER 6. Conclusions ......................................................................................................... 75  

vi

6.1 Summary of Conclusions ................................................................................................... 75  6.2 Future Work ....................................................................................................................... 76  

REFERENCES ............................................................................................................................ 78  

APPENDIX A. Analysis of Extensometer Load ........................................................................ 80  

APPENDIX B. Check of Fiber Elasticity ................................................................................... 81  

APPENDIX C. MATLAB Programs .......................................................................................... 84  

C.1 “LoadFilesP.map” ............................................................................................................. 84  C.2 “CorrectFitP.m” ................................................................................................................. 84  C.3 “ErrorP.mat” ...................................................................................................................... 88  C.4 “PostprocessP_Data_per_line_G.m” ................................................................................. 89  C.5 “Statistics.m” ................................................................................................................... 102  C.6 “kstest_each_line.m” ....................................................................................................... 105  C.7 “graphData_W_X.m” ...................................................................................................... 109  C.8 “graphData_Pmax_FinalGroupNumber.m” .................................................................... 110  C.9 “P_slip_Combined_Groups.m” ....................................................................................... 112  C.10 “P_slip_Combined_Groups_Fitted.m” ......................................................................... 117  

APPENDIX D. Plots .................................................................................................................. 129  

D.1 W versus fiber location, x ................................................................................................ 129  D.2 Pmax versus fiber location, x ............................................................................................ 145  D.3 Final Curve Averaging .................................................................................................... 162  

vii

List of Figures

Figure 1.1: a) Random distribution of aggregate particles within volume, b) Delaunay tetrahedral formed between particle nodes and the triangular facets, c) Two adjacent particles with their polyhedral cells (Cusatis et al. 2011) .............................................................. 2  Figure 1.2: a) Randomly distributed fibers within the volume, b) fiber intersection with triangular facet (Schauffert and Cusatis 2012) .............................................................................. 4  Figure 2.1: a) Fiber pullout debonding, b) A typical load versus displacement (slippage) relationship for single fiber pullout (Schauffert and Cusatis 2012) .............................................. 8  Figure 2.2: Slip behavior due to friction ...................................................................................... 15  Figure 3.1: Layout and dimensions for specimens a, b and c ...................................................... 21  Figure 3.2: Fiber spacing, d ......................................................................................................... 21  Figure 3.3: Wooden and plastic specimen molds ........................................................................ 22  Figure 3.4: Sample of a static flow test ........................................................................................ 24  Figure 3.5: Specimens 48 hours after casting .............................................................................. 24  Figure 3.6: Fiber Pullout Test Setup ............................................................................................ 26  Figure 3.7: Compression test setup of a 50.8 mm2 cube (2% fiber volume) ............................... 27  Figure 3.8: Pin vise used as fiber pullout grip ............................................................................. 28  Figure 4.1: Sample Fiber Pullout Curve with defined parameters ............................................... 30  Figure 4.2: Tested variables on sample specimen ....................................................................... 31  Figure 4.3: Layout of specimens a, b and c ................................................................................. 31  Figure 4.4: W versus fiber location, x (for V = 0%, d = 12.7 mm, E = 25.4 mm) ....................... 33  Figure 4.5: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E = 25.4 mm) .................... 33  Figure 4.6: W versus fiber location, x (for V = 2%, d = 12.7 mm, E = 25.4 mm) ....................... 34  Figure 4.7: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E = 25.4 mm) .................... 34  Figure 4.8: W versus fiber location, x (for V = 4%, d = 12.7 mm, E = 25.4 mm) ....................... 35  Figure 4.9: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E = 25.4 mm) .................... 35  Figure 4.10: Specimen layout and line numbers .......................................................................... 36  Figure 4.11: Histogram of residuals for fiber pullout data: a) W and b) Pmax .............................. 39  Figure 4.12: Probability plot for residuals: a) W and b) Pmax ....................................................... 40  Figure 4.13: Specimen layouts and line numbers ........................................................................ 43  Figure 4.14: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 0%, d = 12.7 mm ..................................................................................................... 45  Figure 4.15: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 0%, d = 3.2 mm ....................................................................................................... 45  Figure 4.16: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 2%, d = 12.7 mm ..................................................................................................... 46  Figure 4.17: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 2%, d = 3.2 mm ....................................................................................................... 46  Figure 4.18: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 4%, d = 12.7 mm ..................................................................................................... 47  Figure 4.19: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 4%, d = 3.2 mm ....................................................................................................... 47  Figure 4.20: Representation of groups on specimens .................................................................. 49  Figure 4.21: a) Range of W, and b) Range of Pmax (for V = 0%) ................................................. 51  

viii

Figure 4.22: a) Range of W, and b) Range of Pmax (for V = 2%) ................................................. 51  Figure 4.23: a) Range of W, and b) Range of Pmax (for V = 4%) ................................................. 52  Figure 4.24: Specimen layouts and grouping .............................................................................. 53  Figure 4.25: Range of a) W, and b) Pmax for all volumes, separated by E = 3.2 mm and E ≥ 12.7 mm .................................................................................................................................... 55  Figure 4.26: Final fiber grouping layout ...................................................................................... 56  Figure 4.27: Final groups versus a) W and b) Pmax ...................................................................... 57  Figure 4.28: Representative P-ν curve for V = 0%, E ≥ 12.7 mm and d eliminated .................... 59  Figure 4.29: Averaged load versus slip of V = 0%, with E ≥ 12.7 mm (maroon) and E = 3.2 mm (orange) with their respective standard deviation ........................................................... 60  Figure 4.30: Averaged load versus slip of V = 2%, with E ≥ 12.7 mm (maroon) and E = 3.2 mm (orange) with their respective standard deviation ........................................................... 61  Figure 4.31: Averaged load versus slip of V = 4%, with E ≥ 12.7 mm (maroon) and E = 3.2 mm (orange) with their respective standard deviation ........................................................... 62  Figure 4.32: Final curves with V = 2%, V = 4% averaged within their batches ......................... 64  Figure 5.1: Representative curve for model fitting ...................................................................... 70  Figure 5.2: Final fitted fiber pullout curves ................................................................................. 73  Figure D.1: W versus fiber location, x (for V = 0%, d = 3.2 mm, E = 25.4 mm) ....................... 129  Figure D.2: W versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 12.7 mm) ..................... 129  Figure D.3: W versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 25.4 mm) ..................... 130  Figure D.4: W versus fiber location, x (for V = 0%, d = 3.2 mm, E3 = 12.7 mm) ..................... 130  Figure D.5: W versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 3.2 mm) ....................... 131  Figure D.6: W versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 3.2 mm) ....................... 131  Figure D.7: W versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 12.7 mm) ................... 132  Figure D.8: W versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 25.4 mm) ................... 132  Figure D.9: W versus fiber location, x (for V = 0%, d = 12.7 mm, E3 = 12.7 mm) ................... 133  Figure D.10: W versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 3.2 mm) ................... 133  Figure D.11: W versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 3.2 mm) ................... 134  Figure D.12: W versus fiber location, x (for V = 2%, d = 3.2 mm, E = 25.4 mm) ..................... 134  Figure D.13: W versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 12.7 mm) ................... 135  Figure D.14: W versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 25.4 mm) ................... 135  Figure D.15: W versus fiber location, x (for V = 2%, d = 3.2 mm, E3 = 12.7 mm) ................... 136  Figure D.16: W versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 3.2 mm) ..................... 136  Figure D.17: W versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 3.2 mm) ..................... 137  Figure D.18: W versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 12.7 mm) ................. 137  Figure D.19: W versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 25.4 mm) ................. 138  Figure D.20: W versus fiber location, x (for V = 2%, d = 12.7 mm, E3 = 12.7 mm) ................. 138  Figure D.21: W versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 3.2 mm) ................... 139  Figure D.22: W versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 3.2 mm) ................... 139  Figure D.23: W versus fiber location, x (for V = 4%, d = 3.2 mm, E = 25.4 mm) ..................... 140  Figure D.24: W versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 12.7 mm) ................... 140  Figure D.25: W versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 25.4 mm) ................... 141  Figure D.26: W versus fiber location, x (for V = 4%, d = 3.2 mm, E3 = 12.7 mm) ................... 141  Figure D.27: W versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 3.2 mm) ..................... 142  Figure D.28: W versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 3.2 mm) ..................... 142  Figure D.29: W versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 12.7 mm) ................. 143  

ix

Figure D.30: W versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 25.4 mm) ................. 143  Figure D.31: W versus fiber location, x (for V = 4%, d = 12.7 mm, E3 = 12.7 mm) ................. 144  Figure D.32: W versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 3.2 mm) .................. 144  Figure D.33: W versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 3.2 mm) ................... 145  Figure D.34: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E = 25.4 mm) ................. 145  Figure D.35: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 12.7 mm) ............... 146  Figure D.36: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 25.4 mm) ............... 146  Figure D.37: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E3 = 12.7 mm) ............... 147  Figure D.38: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 3.2 mm) ................. 147  Figure D.39: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 3.2 mm) ................. 148  Figure D.40: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 12.7 mm) ............. 148  Figure D.41: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 25.4 mm) ............. 149  Figure D.42: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E3 = 12.7 mm) ............. 149  Figure D.43: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 3.2 mm) ............... 150  Figure D.44: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 3.2 mm) ............... 150  Figure D.45: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E = 25.4 mm) ................. 151  Figure D.46: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 12.7 mm) ............... 151  Figure D.47: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 25.4 mm) ............... 152  Figure D.48: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E3 = 12.7 mm) ............... 152  Figure D.49: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 3.2 mm) ................. 153  Figure D.50: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 3.2 mm) ................. 153  Figure D.51: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 12.7 mm) ............. 154  Figure D.52: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 25.4 mm) ............. 154  Figure D.53: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E3 = 12.7 mm) ............. 155  Figure D.54: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 3.2 mm) ............... 155  Figure D.55: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 3.2 mm) ............... 156  Figure D.56: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E = 25.4 mm) ................. 156  Figure D.57: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 12.7 mm) ............... 157  Figure D.58: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 25.4 mm) ............... 157  Figure D.59: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E3 = 12.7 mm) ............... 158  Figure D.60: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 3.2 mm) ................. 158  Figure D.61: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 3.2 mm) ................. 159  Figure D.62: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 12.7 mm) ............. 159  Figure D.63: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 25.4 mm) ............. 160  Figure D.64: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E3 = 12.7 mm) ............. 160  Figure D.65: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 3.2 mm) .............. 161  Figure D.66: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 3.2 mm) ............... 161  Figure D.67: P-ν curve for V = 0%, E = 3.2 mm and d eliminated ........................................... 162  Figure D.68: P-ν curve for V = 2%, E ≥ 12.7 mm and d eliminated ......................................... 162  Figure D.69: P-ν curve for V = 2%, E = 3.2 mm and d eliminated ........................................... 163  Figure D.70: P-ν curve for V = 4%, E ≥ 12.7 mm and d eliminated ......................................... 163  Figure D.71: P-ν curve for V = 4%, E = 3.2 mm and d eliminated ........................................... 164  

x

List of Tables

Table 3.1: Typical UHP-FRC composition ...................................................................... 18  Table 3.2: Typical steel fiber chemical composition ........................................................ 19  Table 3.3: Test Matrix ....................................................................................................... 20  Table 4.1: Average pullout work and maximum pullout load .......................................... 37  Table 4.2: p-value and confidence interval for V, E and d ............................................... 41  Table 4.3: p-value and confidence levels for specimens a and b for W and Pmax ............. 44  Table 4.4: Grouped edge distances, E = 3.2 mm and E ≥ 12.7 mm, for W and Pmax ........ 48  Table 4.5: p-values and confidence levels for d ............................................................... 50  Table 4.6: Grouped edge distances, eliminating d, for W and Pmax .................................. 53  Table 4.7: p-value and confidence level for E .................................................................. 54  Table 4.8: Final grouped edge distances for W and Pmax .................................................. 56  Table 4.9: Compressive Strengths, f'c for V = 0% ............................................................ 65  Table 4.10: Compressive Strengths, f'c for V = 2% .......................................................... 66  Table 4.11: Compressive Strengths, f'c for V = 4% .......................................................... 66  Table 5.1: Fitted model parameters for method A and B: Gd , τ0 and β ............................ 71  Table B.1: Elasticity check for V = 0%, d = 3.2 mm, E = 3.2 mm ................................... 81  Table B.2: Elasticity check for V = 2%, d = 3.2 mm, E = 3.2 mm ................................... 82  Table B.3: Elasticity check for V = 4%, d = 12.7 mm, E = 12.7 mm ............................... 83  

1

CHAPTER 1. Introduction

In this research, a series of single fiber pullout tests are performed using straight

steel fibers in Ultra-High Performance Concrete (UHPC) and Ultra-High Performance

Fiber-Reinforced Concrete (UHP-FRC). Several parameters are explored in the

experimental program including the distance of fibers to the edge of specimen, distance

between fibers, fiber location on the specimen, and the effect of fibers in the pullout

medium on the fiber being pulled. The pullout load versus slip of the fibers is recorded

for each of the parameters.

The main objective of this research is to experimentally explore the bond

mechanisms between fiber and matrix for each of the parameters, which will be used to

quantify fiber pullout behavior in UHPC and UHP-FRC. The experimental results are

beneficial for understanding the effect of fiber proximity to the edge of specimen, effect

of fiber proximity to neighboring fibers, and the effect that fibers in the pullout medium

have on bond-slip during single fiber pullout. A better understanding of these parameters

will allow further improvement of interfacial bond properties. Further, this research will

be implemented into an existing fiber pullout model, where a representative load versus

slip curve will be used to validate a computational modeling protocol, based on Lattice

Discrete Particle Modeling (LDPM) and Lattice Discrete Particle Modeling for Fiber-

Reinforced Concrete (LDPM-F). This validated modeling protocol will be used for

structural components (e.g. plates, beams, and columns) made of Ultra-High Performance

Fiber-Reinforced Concrete (UHP-FRC) and will be capable of simulating discrete

cracking, thin-walled behavior, and interaction between fiber and matrix.

2

The Lattice Discrete Particle Model (LDPM) is a computational tool able to

model nonhomogeneous materials, such as concrete, to failure. It is also capable of

capturing material nonlinearity, concrete heterogeneities, and fiber reinforcing within the

matrix. This model simulates a concrete mesostructure by considering only the coarse

aggregates. The aggregate particles are assumed to have a spherical shape and are

randomly introduced into the volume using a try-and-reject procedure, avoiding

overlapping and ensuring containment within the desired volume, as shown in Figure

2.1a. Zero-radius aggregates are represented by nodes, and are randomly distributed over

the external surface of the volume to define the surface of the volume. A system of cells

interacting through triangular facets is created through a three-dimensional domain

tessellation, derived from the Delaunay tetrahedralization of the simulated aggregate

centers. Figure 2.1b shows the tetrahedral formed between four particle nodes, and the

triangular facets, which define the lattice system. The three-dimensional domain

tessellation creates a system of polyhedral cells that contain one aggregate particle and

interact with neighboring cells through the triangular facets for which they are in contact,

as shown in Figure 2.1c (Cusatis et al. 2011; Cusatis et al. 2011).

Figure 1.1: a) Random distribution of aggregate particles within volume, b) Delaunay tetrahedral formed between particle nodes and the triangular facets, c)

Two adjacent particles with their polyhedral cells (Cusatis et al. 2011)

T

P3

P1

P2

E12F4

P4

E13

E24

F1

F3E14

E34

F2

a b c

3

Stresses and strains are defined at every facet and are assumed to be potential

crack surfaces for LDPM formulation. These interacting cells and facets can be

represented in two-dimensions as straight-line segments. The constitutive laws governing

interaction behavior between the particles is imposed at the centroid of the projection for

every facet to a plane orthogonal to the line connecting the centers of particles. To ensure

that the shear interaction between neighboring particles does not depend on shear

orientation, the projections are used instead of the actual facets. The straight lines

(domain tessellation) connecting the aggregate particle centers define the lattice system

of the mesostructure topology (Cusatis et al. 2011; Cusatis et al. 2011).

The Lattice Discrete Particle Model for Fiber-Reinforced Concrete (LDPM-F)

introduces fibers with randomly generated or assigned positions and orientations, as

shown in Figure 2.2a. The fibers are characterized by their diameter, length, and

geometry. The fiber system is then overlapped with the polyhedral cell system containing

mortar and aggregate, and the fiber-facet intersections are determined. At each

intersection, the fiber embedment lengths on each side of the facet are computed. The

contribution to the facet from the fiber is negligible in cases where the normal component

of the facet stress is in compression for inelastic behavior, and for all elastic behavior.

Figure 2.2b shows the intersection of fiber and facet, the normal component of the facet

stress, and the embedment lengths of the fiber on either side of the facet. The model also

neglects the interaction between adjacent fibers and the effect adjacent mesoscale cracks

have on single fibers (Schauffert and Cusatis 2012; Schauffert et al. 2012).

4

Figure 1.2: a) Randomly distributed fibers within the volume, b) fiber intersection with triangular facet (Schauffert and Cusatis 2012)

UHP-FRC constituents differ from normal concrete in that it has no coarse

aggregate, the use of superplasticizer to reduce the water-to-cement ratio without

negatively effecting workability, the addition of silica fume to provide a dense particle

matrix, and the addition of fiber reinforcement in the matrix to ensure ductile behavior.

These components allow for improved ductility, durability, post-peak cracking response,

long-term stability, tensile cracking capacity and higher energy absorption capacity.

UHP-FRC is characterized by a compressive strength greater than 150 megapascals

(MPa) with a very low water-to-cement ratio (~0.2). Because of the densified particle

matrix, UHP-FRC can resist freeze-thaw and scaling conditions, in addition to being

nearly impermeable to chloride ions (Graybeal 2005).

The primary reason for the addition of fibers to the cementitious matrix is to

enhance the post-cracking behavior of cement composites. The fibers bridge the cracks in

the matrix, preventing the cracks from further propagating and resulting in a sudden,

global failure of the composite. Some secondary reasons for the addition of short needle-

like fibers to cementitious matrices are that it heightens the composite’s mechanical

Direct tensionspecimen withVf = 2% n

nf

Facet

Fiber LlLs

a b

5

properties such as toughness, ductility and energy absorbing capacity. Additionally, it

enhances long term stability and improves tensile behavior (Naaman et al. 1991).

Fibers cross potential cracks, transmitting stress and absorbing energy between

fiber and matrix through the interfacial bond. The interfacial bond is characterized by the

pullout, without rupture, of a fiber from a matrix. Once a crack forms in a medium

containing fibers, the total energy consumption depends on the debonding and frictional

slip during crack propagation (Shannag et al. 1997). Single fiber pullout tests evaluate the

pullout mechanism between fiber and matrix such as: the physical and chemical bond

between fiber and matrix; the mechanical component contributed by deformed fibers,

such as hooked, smooth or crimped; the fiber-to-fiber interlock, which exists in high fiber

percentages in the matrix volume; and the friction due to confinement between fiber and

matrix (Naaman and Najm 1991). Single fiber pullout tests characterize the interfacial

bond of a fiber in a given matrix by measuring the pullout load and slip, simultaneously.

In the following chapter, important parameters governing bond behavior for

single fiber pullout tests are investigated through previously published research.

Additionally, an existing fiber pullout model and a summary of research conducted on the

bond characterization of steel fibers in a UHPC matrix are provided.

6

CHAPTER 2. Literature Review

The interfacial properties between fiber and matrix have been investigated for

decades through widely used and relatively simple single fiber pullout tests. Through

experimental and analytical research using single fiber pullout tests, many important

parameters for fiber and matrix governing bond strength and behavior were discovered.

This chapter reviews some of those parameters governing fiber-matrix bond behavior

found through experimental and analytical research. Additionally, it summarizes the

theory of a pre-existing bond-slip fiber pullout model.

2.1 Fiber Pullout Model

The LDPM fiber-matrix interaction constitutive model is based on a semi-

empirical formulation of Yang et al. (2008) and a mechanics based model by Lin et al.

(1999). The model and LDPM-F framework incorporate three material parameters that

are calculated through experimental fiber pullout tests: (1) bond fracture energy

(chemical bond strength), Gd; (2) bond frictional stress, τ0, which is constant for small

slips; (3) slip hardening-softening coefficient, β. These parameters are incorporated for

inclined and straight fibers (Schauffert and Cusatis 2012; Schauffert et al. 2012).

The fiber pullout model is based on a number of assumptions made by Lin et al.

(1999) so as to simplify the analysis without losing accuracy: (1) the fibers are of high

aspect ratio (>100) to not affect the total debonding load due to the end effect; (2)

because the relative slip between fiber and matrix is small within the debonding zone (for

slips less than the critical slip value), the frictional stress, τ0, remains constant; (3)

Poisson’s effect is negligible since it is typically diminished due to inevitable

misalignment and surface condition of the fiber; (4) the fiber’s elastic stretch after

7

complete debonding is negligible since it is small in comparison to overall slip.

Additional assumptions are that the fiber is initially straight and has negligible bending

stiffness (Lin et al. 1999; Schauffert and Cusatis 2012).

As described by Lin et al. (1999), there are three stages of pullout behavior as

seen on a load-slip curve. The first stage is the elastic stretching of the fiber while the

chemical bond between fiber and matrix prevents the fiber from slipping. The critical

slippage value, νd (mm), represents the displacement (slippage) at full chemical

debonding for a given embedment length, Le (mm), in terms of bond fracture energy

(chemical bond), Gd (N/mm), and frictional stress, (N/mm2). These concepts can be

seen in Figure 2.1 and is expressed as (Lin et al. 1999):

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

ff

ed

ff

ed dE

LGdEL 220 82τ

ν (1)

where Ef (MPa) and df (mm) represent the modulus of elasticity and diameter of

the fiber, respectively. This load increases, with small displacement, slip, reaching the

peak debonding load which is followed by a distinct load drop. This load drop indicates

the transition from chemical to purely frictional bond, and would not be seen if there

were an absence of chemical bond between fiber and matrix. The relative slippage of the

fiber, ν (mm), is a function of pullout load resistance P(ν) (N). Prior to full debonding, ν

< νd, the pullout load resistance is represented as (Lin et al. 1999):

( )2/1

032

2)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡ += dff GdE

Pντπ

ν (2)

0τ

8

After full debonding, only frictional bond is apparent until complete fiber pullout.

The pullout load resistance after full debonding, ν > νd, is a function of P0 (N) and β,

given as (Lin et al. 1999):

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

f

d

e

d

dLPP

ννβ

ννν 11)( 0 (3)

where:

P0 = πLedfτ0 (4)

and β represents the interfacial friction coefficient which take values of β = 0; β >

0; β < 0. If the interfacial friction does not depend on slippage, β = 0, and is represented

by a linear decline as slip increases. If the interfacial friction increases as slip increases, β

> 0, and can be represented as slip hardening-softening, which exhibits an increase in

load as slip increases, then a decrease until full fiber pullout. If the interfacial friction

decreases as slip increases, β < 0, and is represented by slip softening, showing a decrease

in load as slip increases. A fiber pullout model and typical load versus slippage curve

accounting for the three stages of pullout can be seen in Figure 2.3 a and b, respectively

(Lin et al. 1999; Schauffert and Cusatis 2012; Yang et al. 2008). Additional information

regarding the Lin model, assumptions and derivations can be found in Lin et al. (1999).

Figure 2.1: a) Fiber pullout debonding, b) A typical load versus displacement (slippage) relationship for single fiber pullout (Schauffert and Cusatis 2012)

9

2.2 Fiber Geometry

Pullout behavior is desirable, instead of fiber yield or rupture, to maximize the

energy transfer from matrix to fiber so as to improve the post cracking behavior of the

composite. Optimization of the mechanical bond between fiber and matrix can maximize

the energy dissipation, ensuring pullout behavior. A factor that plays a critical role in

improving the mechanical bond for optimization is fiber deformations and geometry. The

fiber can be deformed through roughening the surface, end deformations (such as hooked,

end paddles, or end buttons), or deformations along its length (twisted or crimped) (Wille

and Naaman 2012). In addition to the deformations, the fiber cross-sectional shape and

geometry can be further optimized to produce improved pullout behavior. Different

cross-sectional fiber shapes such as, triangular and polygonal, are more effective than

circular cross-sections which contribute to the post-cracking performance of the

composite by increasing the surface area between fiber and matrix (Naaman 2003). This

allows the optimization of mechanical bond for peak pullout load, with consideration of

the effect the fiber geometry has on workability. Fibers with mechanical deformations

have an increased probability of being bundled or clumped during the mixing process.

Deformed steel fibers have shown a significant increase in the peak pullout load

when compared to straight smooth steel fibers when pulled out of a cementitious matrix.

In steel fiber reinforced self-compacting concrete with a compressive strength of 83.4

MPa, it was observed that hooked steel fibers had an increase of four to five times the

peak pullout force as straight steel fibers (Cunha et al. 2010). No fracture failure of the

deformed fibers was observed during pullout. When pulled from UHPC with a

compressive strength between 194 - 240 MPa, hooked-end or high-strength twisted fibers

10

had four to five times the equivalent bond of smooth straight fibers, when tested under

the same conditions. The bond strength of deformed fibers show an increase of four to

five times the equivalent bond of straight fibers (Wille et al. 2012).

Straight steel fibers are often used in commercial UHPC mixes due to the

commercial availability and cost effectiveness. Steel straight fibers allow for a general

pullout behavior as opposed to fiber rupture, which allows for high-energy absorption

capability and an increase in post-cracking response.

2.3 Effect of Matrix Strength and Composition

Matrix strength depends largely on the particle size of the cement and aggregates

contained within the composite. Carefully selecting, proportioning, and mixing the

elements can achieve optimization of the granular mixture. By selecting constitutive

materials over a range of volumes, it allows for a tightly packed, finely graded and highly

homogenous concrete composite. The small particles are able to fit in between the large

aggregate particles, reducing the number and size of voids in the concrete, ultimately

reducing the number of possible weak zones. With the addition of superplasticizer, an

extremely low water-to-cement binder ratio can be achieved allowing for increased

compressive strengths. Superplasticizers generally allow for a better dispersion of cement

particles, but it can also be designed to interact with all fine particles in the matrix,

including cement, silica fume, and glass powder (Wille et al. 2012).

When comparing densely packed particle matrices with low water-to-cement

ratios and the addition of silica fume, to conventional mortar or grout matrices, the

optimized composites showed an increase in both frictional bond strength and debonding.

This was seen with Shannag et al. (1997) research with Densified Small Particles (DSP)

11

and Abu-Lebdeh et al. (2010) research with very high strength concrete (VHSC).

Densified Small Particles (DSP) is a high strength cement based matrix that optimizes the

use of superplasticizer and silica fume to achieve a dense particle matrix with a low

water-to-cement ratio (~0.2) and a compressive strength of 150 MPa. An increase in both

frictional bond strength and single fiber debonding due to the densified microstructure of

DSP was observed. This is approximately three times higher bond strength than that of

conventional mortar (Shannag et al. 1997). Very high strength concrete was developed by

the US Army Corps of Engineers, and utilizes high range water reducing admixture

(HRWRA) to decrease the water-to-cement ratio, and a densified particle packing matrix

using sand, cement, silica fume and silica flour. It was observed that the cementitious

components provided an increase in frictional resistance during single fiber pullout (Abu-

Lebdeh et al. 2010).

Ultra-high performance concrete (UHPC) has further improved the granular

mixture to have high compressive strength, low matrix porosity and improved bond

characteristics. These mixtures were assessed through measuring the spread and

entrapped air, to reduce voids and weak areas inside the concrete matrix. UHPC can have

compressive strengths exceeding 200 MPa with the addition of specific curing regimens,

such as steam curing (Wille et al. 2011). Cement, silica fume, quartz, and sand were

optimized to achieve a desired range of particle sizes. Sand and quartz are the largest

particles by diameter, sand being 150- 600 μm and quartz with an average diameter of 10

μm. Silica fume is the smallest, allowing it to fill the interstitial voids between the large

particles (Graybeal 2006).

12

It has been observed that the addition of silica fume to a matrix greatly improves

the pullout energy (almost 100% increase), where pullout energy is the mechanical

energy consumed during fiber pullout process (or the area beneath the pullout curve), and

only a 14% increase in the bond strength. The improvement in pullout energy when using

an optimized 20-30% silica fume dosage can be attributed to the cementitious materials

adhering to the surface of the fiber, providing a wedge around the embedded fiber,

enhancing the friction during the pullout process (Chan and Chu 2004).

The addition of fine filler such as glass powder with a median particle size of 1.5-

5 μm, can further optimize the UHPC mixture. The addition of glass powder can lead to

an increase in compressive strengths and spread value (workability) because of the

increased particle packing density. Through fine particle dispersion and decreasing the

smallest particle size, an improved bond slip hardening behavior and equivalent bond

strength can be achieved. The improved particle packing provides an increase in pullout

friction. Because of this increased friction, some deformed fibers can fracture during the

pullout process when the pullout force applied exceeds the fiber’s tensile strength. When

the fibers bridge cracks, they are in a general state of pullout, allowing the fibers to

absorb large amounts of energy from cracking. Fiber rupture is not desirable since those

fibers will be unable to absorb as much cracking energy (Wille et al. 2012).

2.4 “Group” Effect

In a fiber reinforced cementitious matrix, the bond behavior between fiber and

matrix is often represented by the pullout test of a single fiber. The understanding of the

bond properties allows for optimization of bond strength between the fiber and matrix

through chemical and mechanical adjustments. When applying these adjustments to fiber

13

reinforced concrete, it is observed that the composite properties improved, but less

significantly than expected. This discrepancy shows that a single fiber pullout test may

not be an accurate depiction of fiber contribution when multiple fibers bridge and are

pulled from a cracked surface (Naaman and Shah 1976).

The group effect has been studied to observe the differences in peak pullout load

between single fiber pullout and groups of fibers pulled from the matrix, simultaneously.

It has been observed that the mean pullout load per fiber was unaffected by the number of

fibers being pulled when investigating a single or group of fibers (2, 4, 5, 16, or 36 per

specimen) on ASTM Type III cement (Naaman and Shah 1976). The same result was

observed when pulling a single or group of fibers (9, 30 and 60 fibers per specimen) in

normal strength concrete and cement, (Maage 1978). After the chemical bond breaks for

one fiber, it appears that almost the same load is carried in that fiber due to friction rather

than transferring the additional load to neighboring fibers.

Although additional exploration showed that fibers inclined at 60° from the matrix

surface had a decrease in peak pullout (load at fiber debonding) and final pullout load at

full fiber pullout, in addition to a decrease in final pullout displacement and total pullout

work, with an increasing number of fibers in the region. This is attributed to an increase

in cracking of the region of the matrix where the fibers were pulled (Naaman and Shah

1976).

2.5 Fiber Volume in Matrix

The typical fiber content in UHPC matrix has been optimized to 1-3% by volume,

but can be increased to 4% with minimal fiber clumping. In concretes with high

percentages of fibers by volume, the concrete itself becomes less workable and fibers

14

tend to bundle or clump together, not allowing the desired distribution of fibers

throughout the matrix. Although bundling of fibers can sometimes be advantageous by

allowing the energy absorption capacity to increase, it is mostly disadvantageous because

it can produce weak and brittle areas in the concrete matrix where little or no fibers are

located (Li et. al., 1990). With an increase in fiber volume percentage, there is an increase

in the probability that more than one fiber will bridge a crack during actual composite

behavior.

The fiber volume effect studies the result of fiber volume in the composite while

pulling a single fiber from that matrix. With fibers inside the pullout specimen, these

fibers can interfere or be in contact with the fiber being pulled, decreasing the surface

area of fiber bonded with the surrounding matrix. It has been observed that a fiber content

between 3-6% by volume in a mortar matrix has an increase in peak pullout load and

pullout work which is attributed to fiber interlock (Shannag et al. 1997). With fiber

content less than 3% by volume in a high strength concrete (HSC), an increase in up to

10% in the peak pullout load with a slight increase in the post peak response was

observed. But when using a slurry-infiltrated fiber concrete (SIFCON) with 11% fibers

by volume, an increase of 20-25% in peak pullout load was observed, with a 75-80%

increase in post-peak pullout response (Naaman and Najm 1991).

2.6 Bond-Slip Hardening-Softening Effect

After chemical debonding, the pullout behavior is dictated by frictional bond between

the matrix and fiber. When considering smooth, straight steel fibers embedded in a

cementitious matrix, three possible pullout behaviors can be observed: slip softening,

linear slip-softening, and slip hardening-softening. For slip softening behavior, the

15

frictional force decreases to a constant load until full fiber pullout. This behavior occurs

because of decay at the interface of fiber and matrix due to large slips and a decrease in

the embedment length. For linear slip softening behavior, the frictional force decreases at

a constant rate until full fiber pullout. Linear slip softening load decreases for the same

reasons as slip softening, but at a rate where it produces a linear, or constant, decline.

Linear and slip softening behavior have been observed when a straight steel fiber was

pulled from High Strength Concrete (HSC) matrices (Naaman and Najm 1991). For slip

hardening-softening behavior, the frictional force increases in a near parabolic shape,

until softening occurs at full fiber pullout. Slip hardening-softening behavior is typical for

deformed fibers, since the mechanical component of bond provides additional friction

when pulling the fiber from its matrix. This behavior is not generally seen in normal

strength concretes since there is no mechanical component of bond for smooth straight

fibers. However in Ultra-High Performance Concrete matrices, slip hardening-softening

behavior has been observed (Wille and Naaman 2012). Slip softening, hardening-

softening and linear slip softening behavior is represented in the load versus slip curve in

Figure 2.4.

Figure 2.2: Slip behavior due to friction

Pslip hardening-softening

linear slip softening

slip softening

16

Bond slip hardening-softening behavior in UHPC has been studied to find the

additional component attributing to the increased frictional bond. Microscopic analysis

was performed on fibers after slip hardening-softening behavior was observed. The

analysis suggested that slip hardening-softening behavior could be caused by fiber-end

deformation due to cutting the fibers to length, damage (scratching) to the fiber surface

during pullout, or matrix particles adhering to the fiber surface providing a wedge effect

(Wille and Naaman 2012; Wille and Naaman 2013).

During manufacturing some fiber ends are flattened due to the cutting process.

This flattening provides a mechanical anchorage for the fiber, increasing the pullout

resistance especially just before full fiber pullout. With a dense cementitious matrix that

is a characteristic of higher strength concretes, such as UHPC, longitudinal scratches

have been observed after full fiber pullout. The scratches are most likely due to the

abrasion of the matrix particles onto the fiber during the pullout process. However, with

matrix compositions comprised of high concentrations of fine particles such as silica

fume, different post pullout fiber surface characteristic has been observed. With this

matrix, the cementitious particles have been seen to adhere to the fiber surface. The

particle adhesion can be worn down during the pullout process, accumulating the particle

adhesion toward the fiber end. The result of the particle adhesion is attributed to the fiber

pullout resistance, enhancing the friction and pullout resistance. It has been observed that

composites containing silica fume and the observed particle adhesion to the fibers

additionally showed an increase in pullout resistance (Chan and Chu 2004; Wille and

Naaman 2012; Wille and Naaman 2013).

17

In the following chapter, the materials and methods for the experimental program

will be discussed. The chapter introduces the testing variables and the test matrix. The

experimental procedures for embedding the fibers within the matrix and the mixing

process are explained. The test setup, procedure and data corrections are also discussed.

18

CHAPTER 3. Materials and Methods

This research studies the fiber pullout of smooth straight fibers vertically

embedded in Ultra-High Performance Concrete (UHPC). The parameters being studied

are fiber volume percentages in the pullout matrix (0%, 2%, 4%), and two proximity

parameters: distance of fibers to the neighboring fiber and to the edge of the specimen.

The proximity parameters being investigated have never been studied using a fiber

reinforced concrete specimen. The fiber volume percentages have previously been

studied using High Strength Concretes (HSC), but has not been investigated using UHPC.

This study will provide valuable information regarding effects that placement of the fiber

being pulled and the effects fiber volume has on the overall fiber pullout behavior.

The UHP-FRC being studied in this research is commercially available, and has a

typical composition, shown in Table 3.1 as provided by the Federal Highway

Administration (FHWA) in a report about material property characterization of Ultra-

High Performance Concrete (Graybeal 2006). The steel fibers used are typically added at

a ratio of 2% by volume.

.Table 3.1: Typical UHP-FRC composition

Material Amount (kg/m3) Percent by Weight Portland Cement 712 28.5

Fine Sand 1020 40.8 Silica Fume 231 9.3

Ground Quartz 211 8.4 Superplasticizer 30.7 1.2

Accelerator 30.0 1.2 Steel Fibers 156 6.2

Water 109 4.4

The proportions are based on an optimization of the granular mixture of cement

and cementitious materials. This allows for a finely graded and highly homogeneous

19

concrete matrix. The dimensionally largest particle in the matrix is fine sand, which

generally is between 150 and 600 micrometers (µm). Cement is the next largest particle

with an average diameter of 15 µm. Ground quartz has an average diameter of 10 µm.

The silica fume is the smallest particle, with a diameter small enough to fill the voids

between ground quartz and cement particles (Graybeal 2006).

The steel fibers, dimensionally, are the largest component of the UHP-FRC

mixture. They have a diameter of 0.2 mm and an average length of 13 mm. The tensile

strength of the fibers at rupture is given as 2,160 N/mm2. The fibers are steel, but have a

thin brass coating applied during the drawing process to help improve corrosion

resistance. The typical chemical composition of the steel fibers being used in this study

are in Table 3.2, as provided by FHWA (Graybeal 2006).

Table 3.2: Typical steel fiber chemical composition

Element Composition (percent (%)) Carbon 0.69 – 0.76 Silicon 0.15 – 0.30

Manganese 0.40 – 0.60 Phosphorus ≤ 0.025

Sulfur ≤ 0.025 Chromium ≤ 0.08 Aluminum ≤ 0.003

The fibers used in this study are commercially available and used in LaFarge

Ductal mixes. They are Dramix OL 13/.20 smooth high carbon steel fiber, with length

and diameter of 13 mm and 0.20 mm, respectively, with 2,160 N/mm2 tensile fracture

strength. To study fiber proximity to the edge, each specimen holds fibers that are aligned

at distances 3.2, 12.7, or 25.4 mm from the edge. To study fiber proximity to neighboring

fibers, each of the alignments has fibers either close together at a distance of 3.2 mm, or

20

far apart with a space of 12.7 mm between individual fibers. The specimens are cast with

no fibers (control matrix), 2% fibers and 4% fibers by volume inside the concrete matrix.

The number of fibers in each row and total per specimen, divided into the testing

parameters, are provided in Table 3.3: Test Matrix, with the number of fibers actually

pulled for each variable in the last column. Due to extensive time requirements per fiber

pullout test, not all fibers that are cast can be tested. The remaining fibers will be tested in

future work.

For each batch, 30 to 36 compression cubes and 30 small cylinders were cast.

Compression tests were performed at 28 days after the specimens were cast and on

experimental test days. Pullout tests were performed between 28-56 days to allow the

matrix to gain sufficient bond strength.

Table 3.3: Test Matrix

Fiber by

Volume

Distance between Each

Fiber, mm

Distance from edge,

mm

Number of Fibers per

Row

Number of Fibers per Specimen

Number of Fibers pulled

0%

3.2 3.2 83 166 49 12.7 83 249 38 25.4 83 83 21

12.7 3.2 21 42 30 12.7 21 63 37 25.4 21 21 21

2%

3.2 3.2 83 166 35 12.7 83 249 64 25.4 83 83 22

12.7 3.2 21 42 39 12.7 21 63 58 25.4 21 21 21

4%

3.2 3.2 83 166 36 12.7 83 249 60 25.4 83 83 20

12.7 3.2 21 42 40 12.7 21 63 60 25.4 21 21 20

21

3.1 Specimens and Fiber Embedment Process

The fiber pullout specimens were 508 mm long, 50.8 mm wide and 88.9 mm

thick. Depending on the parameters being studied, the fibers were placed in either single,

or parallel rows along the length, in the middle 267 mm span of the prism. Lengths of

120.7 mm from the ends were left bare for the steel restraints to hold the specimen during

testing. The dimensions of each specimen can be seen in Figure 3.1 with distances

between fibers shown in Figure 3.2.

Figure 3.1: Layout and dimensions for specimens a, b and c

Figure 3.2: Fiber spacing, d

The fiber lengths were measured, labeled, and recorded prior to each placement.

The fibers were placed on strong tape, with ~6.5 mm exposed from the top of the tape, at

a premeasured spacing between fibers. The line of fibers were labeled and recorded to

a

b

c

25.4 mm

12.7 mm

3.2 mm

508 mm

50.8 mm

12.7 mm 3.2 mm

22

track the fiber’s placement on the specimen. The tape was placed on the side of a plastic

mold, with the exposed portion of the fiber protruding from the surface of the plastic

mold. This procedure was done for every line of fibers, for each tested variable. The

length of each row was 266.7 mm to maximize the number of fibers tested within the

constraints of the testing frame. The plastic molds and fibers were placed into the

specimen plastic or wood molds, with fibers extending upwards, allowing for UHPC to

be placed over top. The specimens holding fibers 3.2 mm and 12.7 mm from edge were

placed in plastic specimen molds, while the specimens holding fibers 25.4 mm from edge

were casted in wooden specimen molds. The wooden molds were coated with an epoxy-

based paint, which prevented any moisture loss into the wood. In addition, the wooden

molds were coated with a small amount of form release. The molds can be seen in Figure

3.3.

Figure 3.3: Wooden and plastic specimen molds

The compression cubes were cast in solid brass molds allowing for three 50.8

mm. cubes to be made per mold. Prior to placing the UHPC, each cube mold is cleaned

and coated with a small amount of form release.

23

3.2 Specimen Casting: Ultra-High Performance Concrete Placing Process

The Ultra-High Performance Concrete (UHPC) mix used in this study is

commercially available through LaFarge Ductal. There is a precise mixing process for

UHPC where each batch is timed and documented for quality control. The dry mixture

(premix), superplasticizer (Premia 150) and fibers were provided by LaFarge Ductal.

Prior to mixing, each bag of premix was weighed and deposited to the 0.14 m3 capacity

mixer. The machine was turned on to disperse clusters of premix until a smooth

consistency. Water and superplasticizer were weighed and added to the mixer over 2-3

minutes to allow for proper distribution, turning the mixture dark grey. The chemical

reaction occurs after approximately 5 minutes of mixing, allowing the mixture to look

wet and form small beads. As mixing continues, the beads become larger until the

mixture was the consistency of bread dough. Fibers were added to the mixture over 2-4

minutes to prevent clusters of fibers from forming. After approximately 5 minutes or until

the fibers look properly distributed, static and dynamic flow tests were performed to

ensure workability of the UHPC.

The static flow test is to ensure the flow of the UHPC mixture under static

conditions. It was performed by placing UHPC into a brass cone located at the center of a

brass circular plate, until flush with the top surface, removing any excess. The cone was

lifted, removing excess sample from the cone, allowing the UHPC to flow towards the

edge of the plate. After 120 seconds, the diameter of the UHPC was measured at three

locations and the average was recorded. A sample static flow test is shown in Figure 3.4.

The dynamic flow test is performed to test the workability of UHPC with dynamic

movement, vibration in some cases. After the static flow test measurements were

24

recorded, 20 shocks were applied to the sample by lifting and dropping the plate by

turning the crank to the shock table. The resulting spread was measured at three locations

and the results were averaged.

Figure 3.4: Sample of a static flow test

The UHPC was placed into the molds from the middle, to cover the fibers and

then from the sides to prevent flow over the fibers. After all pullout and compression

specimens were cast, they were covered with plastic for 48 hours while curing. No steam

or heat curing was used to model field casting practices. After 48 hours, the specimens

are removed from the molds and labeled with the batch, specimen, location and fiber

numbers, shown in Figure 3.5.

Figure 3.5: Specimens 48 hours after casting

25

3.3 Test Setup and Testing Procedures

The fiber pullout tests were performed on a 44.5 kN load capacity Instron 4204

machine with a 890 N load cell. The specimens were centered on the platform, aligning

the fiber being tested underneath the grip. A metal clamp was fastened to the platform,

and then tightened to restrain the specimen from moving. The actuator was lowered until

maximum extension without touching the specimen’s concrete surface. This maximizes

the contact surface area of the exposed fiber, and minimizes the elongation of the fiber

while being pulled out. Special care was provided to ensure that the fibers were aligned

inside the grip. The fiber was gripped with a modified pin vise. The load and crosshead

were zeroed prior to testing. The grip was tightened around the fiber using pliers to

minimize slippage between the fiber and the grip. A single arm extensometer was

attached to the grip for verification of the crosshead movement. Even though the grip

allowed for horizontal closure upon the fiber, there was a small vertical component that

places the fiber in compression due to the tightening action of the grip. Because the load

crosses through zero while the fiber was pulled out, a correction was made to account for

this initial preload. The fiber was pulled out at a displacement rate of 0.018 mm/s based

on actuator crosshead displacement. The values for load and displacements, both

crosshead and extensometer, were recorded using MTS TestWorks 4.0 software, at a data

acquisition rate of 100 Hz. The test setup is shown in Figure 3.6.

26

Figure 3.6: Fiber Pullout Test Setup

As previously stated, compression cubes and cylinders were cast along with each

UHPC batch to determine the compressive strength of the pullout specimens. The cubes

were tested on a 1.33 MN capacity Forney machine (Figure 3.7) at a load rate of

approximately 2.22 kN/s. The maximum loads for three or four cubes were recorded at 28

days after casting, and atleast the first and last day of testing. The compressive strength

(f`c) of the specimens was calculated by dividing the average of the recorded loads by the

cross sectional area of the cube.

27

Figure 3.7: Compression test setup of a 50.8 mm2 cube (2% fiber volume)

3.4 Test Data Corrections

A single arm extensometer was used to record displacement and verify crosshead

movement during the pullout process. The extensometer arm was compressed to its

vertical limit to ensure full fiber displacement was captured within its extension limits (-

3.81 mm to 3.81 mm). A vertical force component was applied by the extensometer to the

test setup to allow continuous contact between the components until full extension.

Through a conservative structural analysis investigation, it was determined that 99.86%

of the applied force from the extensometer was distributed into the machine components

and load cell, read as a compressive force. The remaining 0.14% of the applied force was

placing the fiber in tension. A typical load applied by the extensometer was 1.4N.

Because the fiber was already placed in compression due to the tightening of the grip

around the fiber, a small tensile force of 0.002N will not affect the fiber or results, and is

therefore neglected. The load due to the extensometer was manually recorded prior to

28

testing and was verified by averaging the last six points after full fiber pullout. This load

was then added to the pullout load for the entire test duration, to negate the effects of the

compressive load. This process was done for each fiber pullout test performed. The

calculations for this correction are presented in Appendix A.

A pin vise (shown in Figure 5.8) was used as a fixture to grip the fiber during the

fiber pullout process. It was tightened around the fiber using pliers to minimize slippage

between the fiber and grip. Even though the grip allowed for horizontal closure upon the

fiber, there was a small vertical component that placed the fiber in compression. Once the

test was started, the load passed through zero, transitioning from compressive to tensile

force. The preload due to tightening the grip was adjusted by finding the slip

displacement at zero load then shifting the displacements so that the pullout test passed

through zero load at zero displacement. The displacement adjustment value was typically

very small.

Figure 3.8: Pin vise used as fiber pullout grip

Fiber elasticity was checked to ensure that the fiber did not yield during the

pullout process. Elasticity was checked by measuring the fibers before and after testing,

29

to see if they elongated. A sample of data was checked, and is located in Appendix B. No

elongation of the fiber was observed.

In the following chapter, the results of the fiber pullout tests are analyzed to see if

any testing parameters have an effect on the pullout behavior. Additionally, the load

versus slip curves will be averaged into their respective groups based on variable

significance in the pullout response. The curves were fitted while the fitted parameters,

Gd, τ0, and β were calculated and compared between two averaging methods. The final

fitted curves and parameters were decided for putting into the LDPM-F model.

30

CHAPTER 4. Results and Discussion

In this chapter, the results from the single fiber pullout tests were analyzed to

examine the effect of various parameters on pullout behavior. A total of 670 single

pullout fibers were tested and each resulted in a pullout load (P) versus slip displacement

(ν) curve. The maximum pullout load and pullout work were recorded in each test. The

maximum pullout load, Pmax, is the maximum recorded load during the entire fiber

pullout test. The pullout work, W, is defined as the area beneath the load versus slip (P-ν)

curve and represents the dissipated bond-friction energy. The pullout work is more

representative in describing the overall pullout behavior than the maximum pullout load

as it contains more information about the shape of the P-ν curve. The slip length, Lp, at

which the load drops to zero was recorded for each tested fiber, and ranged between 5.5

mm and 6.5 mm. These parameters are shown on the P-ν curve in Figure 4.1.

Figure 4.1: Sample Fiber Pullout Curve with defined parameters

31

The tested variables include: (1) the distance, x, of each fiber with reference to the

first fiber pulled in a line sequence; (2) the distance, d, between adjacent fibers, either 3.2

or 12.7 mm; (3) the distance, E, between the fiber and the closest specimen boundary,

either 3.2, 12.7 or 25.4 mm from edge; (4) the fiber volume percentage, V, within the

concrete matrix, either 0, 2, or 4%. The tested parameters are represented on the sample

specimen in Figure 4.2. The specimen layouts are represented in Figure 4.3 as specimen

a, specimen b and specimen c, with individual lines represented by line numbers 1 to 6.

The effects of the tested variables (Pmax and W) are compared against the parameters (x,

d, E, V) pertaining to each fiber pullout curve in the following sections.

Figure 4.2: Tested variables on sample specimen

Figure 4.3: Layout of specimens a, b and c

d E

x = 0 x = xmaxxNth = d×(N-1)

1st pulled fiber(Reference)

Nth pulled fiber

a

b

c

25.4 mm

12.7 mm

3.2 mm

508 mm

50.8 mm

Line 1

Line 2Line 3Line 4

Line 5

Line 6

32

4.1 Influence of Fiber Location on Specimen, x

In this section, the maximum pullout load and pullout work (Pmax and W) are

plotted versus the corresponding fiber distance on the specimen, x, to show if pulling the

fibers in sequence has any significant effect on the response curves. The mean of the

variables are additionally plotted over the distance, x, to visualize the variation of Pmax

and W values about their mean. To determine if there is a trend between location of each

fiber on the specimen, x, and the variables of interest, the Pearson product-moment

correlation coefficient (or correlation coefficient) is calculated. The Pearson’s correlation

coefficient, ρxy, measures how much the two random variables (denoted as x and y)

change with each other (covariance), divided by the product of their standard deviations.

If there is a linear correlation between two variables, the function provides values

between -1 and 1, where 0 is no correlation and +/- 1 shows how the values are

interrelated. The correlation coefficient was calculated in MATLAB using a built-in

function called corr. The MATLAB program to capture this data can be found in

Appendix C.5. Table 4.1 gives the correlation coefficient for the responses, pullout work

and maximum pullout load. Figure 4.4 and Figure 4.5, show typical scatter plots of Pmax

and W versus the location, x, of the fiber along its line for V = 0%. Figure 4.6 and Figure

4.7, show typical scatter plots of Pmax and W versus the location, x, of the fiber along its

line for V = 2%. Figure 4.8 and Figure 4.9, show typical scatter plots of Pmax and W

versus the location, x, of the fiber along its line for V = 4%.

33

Figure 4.4: W versus fiber location, x (for V = 0%, d = 12.7 mm, E = 25.4 mm)

Figure 4.5: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E = 25.4 mm)

0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W��1

ïPP�

Mean

0 50 100 150 200 2500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

Mean

34

Figure 4.6: W versus fiber location, x (for V = 2%, d = 12.7 mm, E = 25.4 mm)

Figure 4.7: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E = 25.4 mm)

0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W��1

ïPP�

Mean

0 50 100 150 200 2500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

Mean

35

Figure 4.8: W versus fiber location, x (for V = 4%, d = 12.7 mm, E = 25.4 mm)

Figure 4.9: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E = 25.4 mm)

The correlation coefficient shows that, overall, there is no trend between any of

the parameters of interest and x for all lines of fibers. A few lines of fibers have relatively

high correlation values (for example: ρxw = -0.45) due to the existence of outliers where

one fiber has generated extreme pullout work or maximum pullout load values with

0 50 100 150 2000

50

100

150

200

250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W��1

ïPP�

Mean

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

Mean

36

respect to the other values within that line, thus leading to high correlation values. Also,

these high correlation values were not consistent within the set of geometric parameters

in consideration and therefore are more likely to be attributed to concrete variability.

This concludes that the location of each fiber on the specimen does not have an effect on

the pullout work or maximum pullout load for each line of fibers. Each fiber within a line

can be considered one group for the remainder of the analysis. Table 4.1 shows the

number of fibers, mean pullout work, 𝑊, and maximum pullout load, 𝑃!"#, with

corresponding standard deviations and correlations coefficients.

Figure 4.10: Specimen layout and line numbers

a

b

c

25.4 mm

12.7 mm

3.2 mm

508 mm

50.8 mm

Line 1

Line 2Line 3Line 4

Line 5

Line 6

37

Table 4.1: Average pullout work and maximum pullout load

V (%)

d (mm)

E (mm)

# fibers

𝑊 (N-mm)

σw (N-mm) ρxw 𝑃!"#

(N) σPmax (N) ρxPmax Specimen

0

12.7

25.4 21 110.27 44.66 0.02 30.74 13.75 -0.01 a 12.7 11 106.22 32.92 -0.28 26.78 10.08 -0.38

b 25.4 14 119.35 36.46 -0.25 26.55 7.76 -0.28 12.7 12 101.67 36.95 -0.38 26.29 10.38 -0.16 3.2 12 88.29 44.38 0.16 24.96 10.16 0.22 c 3.2 18 80.19 28.25 -0.14 21.04 6.30 -0.40

3.2

25.4 21 126.66 57.17 -0.05 30.16 14.87 0.01 a 12.7 13 89.83 46.41 -0.12 30.30 15.86 -0.12

b 25.4 12 76.61 27.15 0.00 21.66 6.19 0.08 12.7 13 125.95 56.13 -0.15 34.56 17.58 -0.17 3.2 25 63.67 30.94 0.05 19.28 10.20 0.26 c 3.2 24 80.83 34.37 0.09 23.68 12.15 0.06

2

12.7

25.4 21 122.56 83.87 0.13 33.04 23.28 0.05 a 12.7 21 145.67 70.84 -0.04 39.44 19.02 -0.08

b 25.4 19 125.93 66.55 0.00 29.90 16.15 0.10 12.7 18 128.29 69.06 0.18 32.31 16.78 0.07 3.2 21 126.35 85.24 -0.36 31.57 19.95 -0.29 c 3.2 18 60.53 26.06 0.19 20.94 8.28 0.18

3.2

25.4 22 113.93 64.27 -0.16 29.51 18.59 -0.13 a 12.7 22 119.16 65.75 -0.22 36.04 16.10 -0.16

b 25.4 21 74.82 41.62 -0.09 29.22 15.72 -0.08 12.7 21 101.04 50.90 0.03 31.84 15.73 0.02 3.2 18 134.07 74.66 0.08 35.09 16.86 0.04 c 3.2 17 95.74 46.64 -0.44 27.50 13.33 -0.45

4

12.7

25.4 20 70.01 39.26 0.25 23.83 13.53 0.17 a 12.7 19 126.82 68.22 -0.18 37.26 19.15 -0.17

b 25.4 21 111.86 62.02 -0.01 31.77 17.14 0.10 12.7 20 124.34 89.83 0.17 36.57 25.44 0.19 3.2 20 140.42 73.21 0.07 42.26 21.93 0.11 c 3.2 20 71.75 35.99 0.16 24.79 11.93 0.17

3.2

25.4 20 90.75 67.84 0.09 31.30 21.78 0.13 a 12.7 20 107.24 57.00 0.20 35.23 17.54 0.01

b 25.4 19 125.42 57.09 -0.26 36.96 17.68 -0.33 12.7 20 139.33 76.23 -0.14 42.49 19.01 -0.19 3.2 19 71.29 37.27 -0.42 24.59 14.05 -0.45 c 3.2 17 141.36 83.09 0.38 39.65 21.61 0.43

TOTAL FIBERS 670                      

4.2 ANOVA Statistical Analysis

As can be seen in Table 4.1, the pullout work and maximum pullout load have

high standard deviations, which indicate large variability within the data. Many factors

can be attributed to this variability. The concrete sometimes does not hydrate fully,

leading to weaker bonds between fiber and matrix. When adding fibers within the

38

concrete matrix, the fiber orientation, distribution and exact fiber content can add to the

variability of results. In high fiber volume matrices, the fibers can form clusters, which

provide additional weak spots within the concrete. The number of fibers touching the

fiber being pulled can also attribute to variability in fiber pullout tests, providing

inconsistencies between fibers. The fiber being pulled may not have been perfectly

vertical, as well as the test setup not being in a perfect line, which all can affect results.

Additionally, variability within the fiber itself including: length, diameter, and

straightness. All these factors can contribute to the variability in fiber pullout test results.

In order to navigate through the variability and draw conclusions within each testing

variable, a statistical analysis approach is adopted.

To draw conclusions regarding parameter significance in terms of Pmax and W, a

widely used collection of statistical models, termed analysis of variance (ANOVA), is

utilized (Montgomery et al. 2012). ANOVA is a statistical hypothesis test that decides if

a variable effect is statistically significant (unlikely to have occurred by chance). If the

probability of interaction (p-value) is less than the threshold value (confidence level), the

influence of the variable on the response of the model is statistically significant.

ANOVA runs on the assumptions that the residuals are normally and

independently distributed, with a generally constant variance. Residuals are estimates of

error between the predicted and observed responses. They are calculated by subtracting

the observation from the fitted values. By examining the residuals, the required

assumptions can be verified in addition to determining if the linear regression model,

utilized in ANOVA, is an appropriate fit for the data. The regression model should have

randomly distributed residuals for the response, with some values higher and lower than

39

the fit with equal probability of occurring. The size or when the error occurred in the test

data, in addition to the variables involved in the prediction, should be independent of the

level of error. Lastly, the distribution or shape of the residuals should appear normal.

There are two residual plots that help visually determine normality and correlation

of the data. The first is a histogram, which shows the range of residuals versus their

relative frequency. The relative frequency is the number of occurrences normalized to the

total number. A histogram of the residuals of the responses, W and Pmax, for each tested

fiber is shown in Figure 4.11 a and b. The second is a probability plot, which shows how

the residual distribution compares to the normal distribution with the same variance. It is

used to investigate whether the data exhibits a normal distribution through transforming

the data into standard normal values and plotting them against the fitted normal line. If a

majority of the data values fall along the fitted normal line, then the assumption of

normality is reasonable. The normality of the fiber pullout data is checked through the

probability plot for each response, W and Pmax, and can be seen in Figures 4.12 a and b.

Figure 4.11: Histogram of residuals for fiber pullout data: a) W and b) Pmax

�� �� � ��� ��� ����

�

�

3

4

5

6

7

8

9[���ï�

Rel

ativ

e Fr

eque

ncy

Residuals of W (N-mm)� � �� ����

�����

����

�����

����

�����

����

�����

Residuals of Pmax (N)

Rel

ativ

e Fr

eque

ncy

40

Figure 4.12: Probability plot for residuals: a) W and b) Pmax

As seen in Figure 4.11 the data forms a bell shape with a slight positive skewness

from the normal distribution. The probability plot in Figure 4.12 shows that the majority

of the data falls along the fitted normal line, and is therefore deemed that normality is a

reasonable assumption. Although the data has a positive skewness and a long, indicating

that the analysis needs to proceed with caution.

The Kolmogorov-Smirnov test (K-S test) is used to compare the line data with a

normal distribution. The normality of the fiber pullout line data for each response, W and

Pmax, is checked through a built-in function in MATLAB called kstest. This function

outputs either a 0 or 1, where 1 rejects the hypothesis that the data comes from a normal

distribution. From this test, the outputs of each line sequence of fibers are 0 indicating

that it does not reject the hypothesis that the data comes from a normal distribution. The

MATLAB program for this test can be found in Appendix C.6. Based on this test, it is

confirmed that normality is a viable assumption. In conclusion, the assumptions for

�� � ��� ��� ��������������

�����������������������������

�����

������������

Probability

Residuals of W (N-mm)� � � �� �� �� ��

�����������

�����������������������������

�����

������������

Probability

Residuals of Pmax (N)

41

ANOVA are satisfied, so analysis of fiber pullout data can be used through ANOVA

statistical tests.

The data is analyzed in MATLAB using a built-in function called anovan (the

MATLAB program is located in Appendix C.6). This function performs the F-test and

outputs a p-value for each variable. The F-test is used to compare variances between

variables in terms of sum of squares by comparing it to an Fcritical value to see

significance of that variable. The probability of interaction, p-value, is calculated by an

F-value greater than Fcritical. There is a high probability of interaction if the p-value is

within the significance level (typically 0.05 or 5%). The significance level shows at what

confidence level (1-(p-value)×100%) a variable is statistically significant. For the

purposes of comparing the data, a significance level of 0.05 was adopted for the p-value.

This corresponds to a 95% confidence that the variable has a statistically significant

influence on the response.

ANOVA was run for each fiber over all the variables, E, d and V, to confirm the

overall significance or insignificance of the testing variables for the pullout work and

maximum pullout load. The p-values for each response W and Pmax for the all variables

can be seen in Table 4.2.

Table 4.2: p-value and confidence interval for V, E and d

    W Pmax

 

p-value

Confidence Level (%)

p-value

Confidence Level (%)

Fiber volume, V 0.0307 96.93 0.00018 99.98 Distance to edge, E 0.0014 99.86 0.0032 99.68

Distance between fibers, d 0.2508 74.92 0.5452 45.48

42

When running ANOVA with the influences of all variables, Table 4.2 shows that

the confidence levels for fiber volume, V, and distance to edge, E, are over the 95%

predetermined confidence level, showing an overall statistical significance for both W

and Pmax. The distance between fibers has a confidence level of 75% for pullout work and

46% for maximum pullout load. This shows that the distance between fibers, d, has no

overall statistical significance in W and Pmax. To obtain a better understanding of the

effect of each variable, the variables were subcategorized and analyzed within each batch

separately, according to their associated edge distances and distances between fibers. The

three edge distances were analyzed first to see if there was a significance between fibers

at a distance 12.7 or 25.4 mm from the edge.

4.3 Fiber Groups

Three edge distances were studied, 3.2, 12.7, and 25.4 mm, for significance they

may have on the parameters of interest, W and Pmax. To determine if, at distances above

the observed E = 12.7 mm, a significant change in pullout work or maximum pullout load

would be seen, specimens a and b (specimen layouts are repeated for convenience in

Figure 4.13) at distances E = 12.7 and 25.4 mm, are compared. This section determines if

the edge distances, 12.7 and 25.4 mm, have similar effects on W and Pmax by using

ANOVA statistics while comparing means and standard deviations.

43

Figure 4.13: Specimen layouts and line numbers

The data for specimen a and b are inputted into the ANOVA function at the

specific fiber distance, d, within the fiber volume to analyze if it has a significant

influence on W or Pmax. Specimen a and b’s p-values and confidence levels for the

responses of pullout work and maximum pullout load are shown in Table 4.3. These p-

values presented indicate if there is a correlation between the values of E within both

specimens a and b combined. The values that are highlighted are within the

predetermined 95% confidence interval, and have a statistical significance on the overall

response (W or Pmax).

a

b

c

25.4 mm

12.7 mm

3.2 mm

508 mm

50.8 mm

Line 1

Line 2Line 3Line 4

Line 5

Line 6

44

Table 4.3: p-value and confidence levels for specimens a and b for W and Pmax

W Pmax Specimen

Layout V (%) d (mm) E (mm) p-value Confidence Level (%) p-value Confidence

Level (%)

0

12.7

25.4

0.968 3.18 0.168 83.18

a 12.7

b 25.4 12.7

3.2

25.4

0.337 66.28 0.398 60.24

a 12.7

b 25.4 12.7

2

12.7

25.4

0.220 77.96 0.197 80.32

a 12.7

b 25.4 12.7

3.2

25.4

0.411 58.91 0.288 71.18

a 12.7

b 25.4 12.7

4

12.7

25.4

0.299 70.11 0.269 73.15

a 12.7

b 25.4 12.7

3.2

25.4

0.028 97.2 0.040 96.03

a 12.7

b 25.4 12.7

According to the p-values summarized in Table 4.3, specimen b has no statistical

significance on the overall response parameters, W and Pmax, except the specimen with V

= 4% and d = 3.2 mm for both responses Pmax and W. Because there were no observed

significance with V = 0 or 2% fiber volumes, or V = 4% with d = 12.7 mm, the

significance observed at V = 4% and d = 3.2 mm can be attributed to variability of

concrete. To confirm this, each individual fiber at its line location are plotted against the

response values, Pmax and W.

Figures 4.14 a and b are the representative plots for pullout work and maximum

pullout load for each fiber within a line for specimens with V = 0% and d = 12.7 mm.

Figures 4.15 through 4.19 represent their respective lines for pullout work and maximum

45

pullout load. The line number refers to the line of fibers for an edge distance, E as seen in

Figure 4.13. This plot shows the range and the location of the means (represented by an X

for each line) within the scatter of fiber values. The average work values per line can be

seen from Table 4.4.

Figure 4.14: Scatter of fibers in each line versus a) pullout work and b) maximum pullout load for V = 0%, d = 12.7 mm

Figure 4.15: Scatter of fibers in each line versus a) pullout work and b) maximum

pullout load for V = 0%, d = 3.2 mm

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

46

Figure 4.16: Scatter of fibers in each line versus a) pullout work and b) maximum

pullout load for V = 2%, d = 12.7 mm

Figure 4.17: Scatter of fibers in each line versus a) pullout work and b) maximum

pullout load for V = 2%, d = 3.2 mm

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

47

Figure 4.18: Scatter of fibers in each line versus a) pullout work and b) maximum

pullout load for V = 4%, d = 12.7 mm

Figure 4.19: Scatter of fibers in each line versus a) pullout work and b) maximum

pullout load for V = 4%, d = 3.2 mm

From Figure 4.19, the means for lines 1-4 showed a linear increase in average

pullout work and maximum pullout load. However, it was also observed that the extreme

outliers of each line were increasing the mean values in a way that showed a linear

increase. This observation and significance could be attributed to concrete variability,

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Line Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 5 60

20

40

60

80

100

120

Line Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

48

which could be confirmed through additional fiber pullout tests with the parameters of

interest.

Based on Tables 4.1 and Figures 4.14 through 4.19, the observed differences in

mean pullout work and maximum pullout load between specimen a and b, in addition to

the p-values, the fibers at E = 25.4 mm and 12.7 mm can be combined into one group,

hereby referred to as E ≥ 12.7 mm. This group includes distances greater than the

specified 12.7 mm because this is the observed distance that above which shows no

significant change in pullout work or maximum pullout load. The fibers at distances E =

3.2 mm are combined into a second group, referred to as E = 3.2 mm since the distances

from the edges are equal and they are located in the same specimen. The number of fibers

per group, average pullout work with its standard deviation, and the averaged maximum

pullout load with its standard deviation for the grouped edge distances, E = 3.2 mm and E

≥ 12.7 mm, are shown in Table 4.4.

Table 4.4: Grouped edge distances, E = 3.2 mm and E ≥ 12.7 mm, for W and Pmax

V (%)

d (mm)

E (mm) # fibers 𝑊

(N-mm) σw

(N-mm) 𝑃!"#

(N) σPmax (N)

Specimen Layout

0 12.7 ≥12.7 58 109.92 38.67 28.06 11.09 a, b

3.2 30 83.43 35.09 22.61 8.14 c

3.2 ≥12.7 59 108.21 53.30 29.43 14.77 a, b 3.2 49 72.07 33.46 21.43 11.30 c

2 12.7 ≥12.7 79 130.82 72.37 33.82 19.14 a, b

3.2 39 95.97 72.34 26.66 16.40 c

3.2 ≥12.7 86 102.57 58.32 31.68 16.54 a, b 3.2 35 115.45 64.72 31.40 15.51 c

4 12.7 ≥12.7 80 108.07 69.77 32.29 19.66 a, b

3.2 40 106.09 66.72 33.52 19.54 c

3.2 ≥12.7 79 115.56 66.51 36.49 19.16 a, b 3.2 36 104.38 71.62 31.70 19.32 c

49

4.4 Distances between Fibers, d

Each group of fibers represents one of three volumes (V), one of two distances

from fibers (d) and now one of two distances from the edge (E). The distances to the edge

are E = 3.2 mm and E ≥12.7 mm, as mentioned in the previous section. A figure

representing the grouping of fibers can be seen in Figure 4.20. For this section, the

differences between fibers, d = 3.2 mm and 12.7 mm, are compared using the

predetermined curve parameters: pullout work and maximum pullout load.

Figure 4.20: Representation of groups on specimens

To establish if the distance to the neighboring fiber (d = 12.7 mm and d = 3.2

mm) has a statistical significance in the overall response for pullout work or maximum

pullout load for each batch individually, ANOVA statistical analysis tests are performed

with E and d as inputs. The p-value and confidence level for the edge distances of d =

12.7 mm and 3.2 mm, for the overall responses of mean pullout work and maximum

pullout load, can be seen in Table 4.5. If the values are highlighted, then those values are

within the predetermined 95% confidence interval, and are said that the distance between

fibers, d, has a statistical significance on the overall response (W or Pmax).

G1

G2

G3

G4

d = 12.7 mm d = 3.2 mm with with

50

Table 4.5: p-values and confidence levels for d

    W Pmax

V (%) Variable (d or E) p-value Confidence

Level (%) p-value Confidence Level (%)

0 d 0.1594 84.06 0.8243 17.57 E 0 100 0.0001 99.99

2 d 0.1369 86.31 0.9493 5.07 E 0.2489 75.11 0.1174 88.26

4 d 0.6127 38.73 0.3745 62.55 E 0.4942 50.58 0.5271 47.29

According to the ANOVA statistical analysis tests summarized in Table 4.5, there

is not a statistical significance within each volume percentage for fiber distance, d, when

comparing pullout work and maximum pullout load. The maximum confidence level is

86%, indicating that the data does not have a significant trend leading to a response

value. By comparing the average Pmax and W values from Table 4.4, no trend or

significance can be determined, confirming the ANOVA conclusion.

A representative scatter plot for W and Pmax for the groups of fibers can be seen in

Figures 4.21 through 4.23 separated by the edge distance, E, and volume, V. The mean

for each line, denoted as an “X” on the plot, is to show the variability of the data within

each line. Each fiber is plotted according to their new groups relative to the distance

between fibers, d, versus pullout work. Each line of fibers represents d = 3.2 mm or 12.7

mm, separated by distance from edge and fiber volume percentage. This plot shows the

variability of values in terms of range and density of fibers for W and Pmax. For this

analysis, lines 1 and 3 were compared, and lines 2 and 4 were compared.

51

Figure 4.21: a) Range of W, and b) Range of Pmax (for V = 0%)

Figure 4.22: a) Range of W, and b) Range of Pmax (for V = 2%)

0 1 2 3 4 50

50

100

150

200

250

300

350

400

450

Group Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 50

20

40

60

80

100

120

Group Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 1 2 3 4 50

50

100

150

200

250

300

350

400

450

Group Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 50

20

40

60

80

100

120

Group Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

52

Figure 4.23: a) Range of W, and b) Range of Pmax (for V = 4%)

The lines of fibers at distances d = 12.7 mm and 3.2 mm from each other are

compared using Figure 4.21 through 4.23 and Table 4.4, with all other variables the

same. It is observed that the range, means and standard deviations of W and Pmax follow

no obvious trend and fall within the scatter of the experimental data. From this

observation, the distances between the fibers appear to not have a significant influence on

the overall pullout work.

Based on the observed differences in mean pullout work and maximum pullout

load and the p-values, the testing variable, d, can be eliminated since there is no

significant effect on pullout work or maximum pullout load. Therefore the fibers at d =

3.2 mm and 12.7 mm can be combined into groups separated by edge distances, E = 3.2

mm and E ≥ 12.7 mm. The groups are still separated by volume, V, until significance and

effects have been found. The number of fibers per group, average pullout work with its

standard deviation, and the averaged maximum pullout load with its standard deviation

0 1 2 3 4 50

50

100

150

200

250

300

350

400

450

Group Number

Pullo

ut W

ork,

W (N−m

m)

1 2 3 4 50

20

40

60

80

100

120

Group Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

53

while eliminating the variable, d, are shown in Table 4.6. The specimen layouts can be

seen in Figure 4.24 with the new specimen grouping.

Table 4.6: Grouped edge distances, eliminating d, for W and Pmax

V (%)

E (mm) # fibers 𝑊

(N-mm) σw

(N-mm) 𝑃!"# (N)

σPmax (N)

Specimen Layout

0 ≥12.7 117 109.06 45.98 28.74 12.93 a, b

3.2 79 77.75 34.27 22.02 9.72 c

2 ≥12.7 165 116.70 65.34 32.75 17.84 a, b

3.2 74 105.71 68.53 29.03 15.95 c

4 ≥12.7 159 111.82 68.14 34.39 19.41 a, b

3.2 76 105.23 69.17 32.61 19.43 c

Figure 4.24: Specimen layouts and grouping

4.5 Proximity to Edge, E

The fibers are grouped representing one of three volumes (V) and one of two

distances from the edge (E) as seen in Figure 4.24. For this section, the differences

between edge distances E = 3.2 mm and E ≥ 12.7 mm are compared using the

predetermined curve parameters: pullout work and maximum pullout load.

ANOVA was run to determine if there is a statistical significance for distances of

E = 3.2 mm and E ≥ 12.7 mm to the edge, for the individual fiber volumes. The p-values

and confidence intervals for the edge distances between E = 3.2 mm and E ≥ 12.7 mm

can be seen in Table 4.7. The p-values and confidence levels that are greater than the

predetermined 95% confidence interval are highlighted.

G1

G2

G3

G4

G5

G6

V = 0% V = 2% V = 4%

54

Table 4.7: p-value and confidence level for E

    W Pmax

V (%) Variable (d or E) p-value Confidence

Level (%) p-value Confidence Level (%)

0 d 0.1594 84.06 0.8243 17.57 E 0 100 0.0001 99.99

2 d 0.1369 86.31 0.9493 5.07 E 0.2489 75.11 0.1174 88.26

4 d 0.6127 38.73 0.3745 62.55 E 0.4942 50.58 0.5271 47.29

According to the p-values and confidence levels in Table 4.7, the edge has a

significant influence on both pullout work and maximum pullout load for 0% fiber

volume. As the volume increases, the confidence level decreases significantly, indicating

that the increase in fibers within the matrix decreases the significance of the edge to the

pullout work or maximum pullout load.

Each fiber is plotted for the pullout work, W, according to their respective groups

relative to the edge distance, E. Each line of fibers is represented by an edge distance, E ≥

12.7 mm or E = 3.2 mm. A representative plot for W versus group number and Pmax

versus group number, for the new fiber groups can also be seen in Figure 4.25, separated

by edge distance E = 3.2 mm and E ≥ 12.7 mm, as well as volume, V. The mean for each

line is denoted as an “X” on the plot. This plot shows the range and density of fibers

when comparing to W and Pmax.

55

Figure 4.25: Range of a) W, and b) Pmax for all volumes, separated by E = 3.2 mm and E ≥ 12.7 mm

The plots in Figure 4.25 show the significance between E = 3.2 mm and E ≥ 12.7

mm for 0% fiber volumes. The distance from the edge of E = 3.2 mm has a significant

decrease when compared to the edge distance E ≥ 12.7 mm. For fiber volumes V = 2%

and 4%, there does not seem to be a significant decrease or increase in mean values

between E = 3.2 mm and E ≥ 12.7 mm. This is confirmed with Table 4.6 for the observed

differences in mean pullout work and maximum pullout load between E = 3.2 mm and E

≥ 12.7 mm. Additionally, the p-values from Table 4.7, support that the edge distance does

have statistical significance in the overall responses for single fiber pullout for fiber

volumes V = 2% or 4%, but is significant for fiber volumes V = 0%. For the remainder of

the analysis, all fibers for 2% fiber volumes will be combined into one group,

disregarding all other variables. This is also done for 4% fiber volumes. The final

grouping layout can be seen in Figure 4.26. The number of fibers per group, average

pullout work with its standard deviation, and the averaged maximum pullout load with its

standard deviation based on the final grouping, are shown in Table 4.8.

1 2 3 4 5 60

50

100

150

200

250

300

350

400

450

Group Number

Pullo

ut W

ork,

W (N−m

)

1 2 3 4 5 60

20

40

60

80

100

120

Group Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

56

Figure 4.26: Final fiber grouping layout

Table 4.8: Final grouped edge distances for W and Pmax

V (%)

E (mm) # fibers 𝑊

(N-mm) σw

(N-mm) 𝑃!"# (N)

σPmax (N)

Specimen Layout

0 ≥12.7 117 109.06 45.98 28.74 12.93 G1

3.2 79 77.75 34.27 22.02 9.72 G2

2 ALL 239 112.72 67.51 31.53 17.34 G3

4 ALL 235 109.69 68.16 33.82 19.40 G4

4.6 Volume of Fibers in the Matrix, V

The final grouping layout can be seen in Figure 4.26, where all variables are

eliminated for fiber volumes V = 2% and 4%. For fiber volume V = 0%, the fibers are

separated by their edge distances, E ≥ 12.7 mm and E = 3.2 mm. For this section, the

differences between fiber volumes, V = 0%, 2% and 4% are compared using pullout work

and maximum pullout strength.

Each fiber is plotted for the pullout work, W, according to their respective groups

relative to the edge distance, E within each fiber volume. Each line of fibers is

represented by an edge distance, E ≥ 12.7 mm or E = 3.2 mm. A representative plot for W

versus group number and Pmax versus group number can be seen in Figure 4.27. The

mean for each line is denoted as an “X” on the plot. This plot shows the range and

density of fibers when comparing to W and Pmax.

G1

G2

G3

V = 0% V = 2% V = 4%

G4

57

Figure 4.27: Final groups versus a) W and b) Pmax

It is observed that the range, means and standard deviations of W and Pmax follow

no obvious trends between all groups excluding group 2 (E = 3.2 mm) and fall within the

scatter of the experimental data. From this observation, the distances between the fibers

appear to only have a significant influence on the overall pullout work or maximum

pullout load for group 1 (E ≥ 12.7 mm), which is within the 0% fiber volume.  

When using the fiber group for V = 0% and E ≥ 12.7 mm as reference, the average

pullout work and maximum pullout load for V = 0% and E = 3.2 mm decreased by 29%

and 23%, respectively, as seen in Table 4.8. The average pullout work and maximum

pullout load increased by 3% and 10% for V = 2%, while V = 4% had an increase of 1%

and 18%, respectively. The standard deviations also increase by a large interval when

fibers are added to the matrix. This indicates that the addition of fibers to the matrix

contributes to the variability of fiber pullout results. The average pullout work and

standard deviation are similar when additional fibers are added to the matrix (from 2% to

4% fiber volumes).

1 2 3 4 50

50

100

150

200

250

300

350

400

450

Group Number

Pullo

ut W

ork,

W (N−m

)

1 2 3 4 50

20

40

60

80

100

120

Group Number

Max

imum

Pul

lout

Loa

d, Pmax

(N)

58

Based on ANOVA statistical analysis, observations from tables and fiber plots,

the final results conclude that there was no statistical significance for any tested variables

within the 2% and 4% fiber volumes. All of these fibers were combined into one group

for modeling purposes. However, there was a statistical significance for edge distance E

= 3.2 mm within fiber volumes of 0%. These fibers were separated by edge distances for

modeling.

4.7 Averaging Curves

Each series of pullout tests are represented by an average curve. The curves are

combined using the moving average of the pullout loads. Because each fiber’s

displacement data varies slightly, the set with the least number of data points is chosen

within each series as the representative displacement data and corresponding pullout

loads were interpolated at each interval for all data sets. The pullout load values at the

given slip interval are averaged and a point on the average load versus slip curve is

marked before moving to the next slip value and performing the same averaging

procedure. This procedure was performed for standard deviations at each interval,

marking (+/-) 1 standard deviation from the mean, to show the variability of the average

curves. These average curves are examined to see visual trends in pullout behavior. A

representation of these plots can be seen in Figure 4.28. The mean and standard

deviations are plotted against the data. The blue lines are the load versus slip curves for

the fiber pullout tests. The thicker red line shows the average curve for this set of data,

with the corresponding (+/-) 1 standard deviation from the mean.

59

Figure 4.28: Representative P-ν curve for V = 0%, E ≥ 12.7 mm and d eliminated

The curves are averaged by groups of E ≥ 12.7 mm and E = 3.2 mm for each

volume percentage (V). Figure 4.28 shows the averaged curves for the distances to the

edge (E = 3.2 mm and E ≥ 12.7 mm) for 0% fiber volume. Figures 4.29 and 4.31 show

the average curve for the same edge distances, but for 2% and 4% fiber volumes,

respectively. A dashed line represents the standard deviations.

60

Figure 4.29: Averaged load versus slip of V = 0%, with E ≥ 12.7 mm (maroon) and E

= 3.2 mm (orange) with their respective standard deviation

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

45

50

Slip, i (mm)

Pullo

ut L

oad,

P (N

)

E = 3.2 mm

E � 12.7 mm

61

Figure 4.30: Averaged load versus slip of V = 2%, with E ≥ 12.7 mm (maroon) and E = 3.2 mm (orange) with their respective standard deviation

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

45

50

Slip, i (mm)

Pullo

ut L

oad,

P (N

)

E = 3.2 mm

E � 12.7 mm

62

Figure 4.31: Averaged load versus slip of V = 4%, with E ≥ 12.7 mm (maroon) and E

= 3.2 mm (orange) with their respective standard deviation

In Figure 4.29, the standard deviations are in close proximity to their respective

average curve, indicating a smaller variability than in the 2% and 4% average curves.

Additionally, the average curve representing E = 3.2 mm is consistently at a lower pullout

load than the curve representing E ≥ 12.7 mm. This shows a clear indication in the

distance to the edge on overall pullout behavior for 0% fiber volume. The close to edge

fibers did not have as high a pullout resistance than the fibers that were farther from the

edge. This can be attributed to the compaction of the concrete being weaker on the side

closest to the edge, since there is less concrete between the pulled fiber and the edge.

Additionally, no fibers are within the matrix to bridge the microcracks, which would

increase the confinement of the fiber. This would lead to an overall decrease in fiber

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

45

50

Slip, i (mm)

Pullo

ut L

oad,

P (N

)

E = 3.2 mm

E � 12.7 mm

63

pullout behavior when compared to fiber pullout curves that have fibers within the

matrix.

In Figure 4.30, the standard deviations are farther from the mean, indicating a

greater variability in the data. The standard deviations are the same for edge distances E

= 3.2 mm and E ≥ 12.7 mm. The variability between the edge distances (E = 3.2 mm and

E ≥ 12.7 mm) are very similar throughout the full pullout curve. Although there is an

overall high standard deviation, it decreases after ~5.5 mm, indicating that there is less

variability toward the end of the pullout process. The average curves are almost

equivalent until ~4 mm slip, when they start deviating from each other. After this point,

the curve indicating fibers that are close to the edge, E = 3.2 mm, has a faster decline to

full fiber pullout than the curve representing fibers farther from the edge, E ≥ 12.7 mm.

The fiber volume within the matrix decreases the significance of the distance to the edge

of the specimen. Essentially, the distance to the edge does not have a significant effect on

the pullout behavior for 2% fiber volume. This is apparent when comparing the curves

between V = 0% and 2% (Figures 4.29 and 4.30).

In Figure 4.31, the standard deviations between the two edge distances are very

similar, as seen with the 2% fiber volume. The standard deviations are large for the

majority of the pullout curve, until after 5 mm slip where the standard deviation

approaches the average curves until full pullout. This indicates that there is less

variability between fiber pullout curves towards the end of the fiber pullout process, but

high variability throughout the majority of the curve. The average curves for E = 3.2 mm

and E ≥ 12.7 mm, are similar until ~3 mm slip, where the curve representing fibers close

64

to the edge (E = 3.2 mm) declines at a faster rate than the fibers farther from the edge (E

≥ 12.7 mm). The two curves converge at ~5.6 mm right before full fiber pullout.

The curves for E = 3.2 mm and E ≥ 12.7 mm are combined because the ANOVA

results indicate that the edge distance is not statistically significant in fiber volumes V =

2% and 4%. The average curves with the new grouping for V = 2% and 4% are plotted

against V = 0% with E and E ≥ 12.7 mm in Figure 4.32 with their respective standard

deviations, for ease in comparison.

Figure 4.32: Final curves with V = 2%, V = 4% averaged within their batches

The curve for 4% fiber volume reaches a higher overall pullout load than the 2%

fiber volume, but starts to decline at a smaller slip value. The decline to full pullout is at a

much faster rate than the 4% fiber volume, which indicates that there is a higher

resistance to full pullout towards the end for 2% fiber volumes than for 4% fiber

0 1 2 3 4 5 60

5

10

15

20

25

30

Slip, i (mm)

Pullo

ut L

oad,

P (N

)

V = 0% E = 3.2 mm

V = 0% (�� 12.7 mm

V = 4%

V = 2%

65

volumes. The fibers within the matrix help bridge the microcracks that appear during the

fiber pullout process, keeping the concrete together and maintaining its strength. This

increases the confinement of concrete around the fiber, providing additional pullout

resistance. Additionally, the standard deviations stay large at a higher slip than for 4%

fiber volumes, indicating more variability at larger slips for the 2% fiber volumes than for

4% fiber volumes.

4.8 Compressive Strength

Compression tests were performed at 28 days after casting, the first day of testing

and at least the last day of testing for each batch of UHPC (for V = 0%, 2% and 4% fiber

volumes) using 50.8 mm2 cube specimens. The specimens were all tested between 55 to

95 days after casting and the compressive strength, f`c, values were recorded and

averaged to find the representative matrix strength of the specimen during the fiber

pullout tests, which can be seen in Tables 4.9 to 4.11. According to the “Material

Characterization of Ultra-High Performance Concrete”, the strength ranges for cubes in

comparison to 76 mm diameter cylinder specimen were within a 10% increase, which is

considered to be small, therefore no size factor needed to be applied (Graybeal 2006).

Table 4.9: Compressive Strengths, f'c for V = 0%

f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2)

Average during fiber

pullout testing

Concrete Age 28 days 61 days 63 days 65 days

V = 0%

142.2 146.5 117.2 138.8 134.4 139.6 100.8 134.4 125.8 113.8 105.1 135.3 123.2 103.4 125.0

𝑓`! 131.4 125.8 112.0 136.2 123.6 σ 7.5 17.8 9.6 1.9 16.3

66

Table 4.10: Compressive Strengths, f'c for V = 2%

f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2)

Average during fiber

pullout testing

Concrete Age 28 days 56 days 57 days 60 days 62 days 66 days

V = 2%

145.7 144.8 137.0 163.8 165.5 148.2 134.4 127.6 165.5 155.1 153.4 142.2 139.6 150.8 133.6 150.8 142.2 146.5 144.8 142.2

𝑓`! 141.1 141.1 145.4 156.6 150.8 145.7 148.1 σ 4.5 9.9 14.3 5.4 9.6 2.5 11.0

Table 4.11: Compressive Strengths, f'c for V = 4%

f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2) f'c

(N/mm2)

Average during fiber

pullout testing

Concrete Age 28 days 82 days 91 days 93 days

V = 4%

129.3 146.5 175.8 144.8 131.9 132.7 150.8 149.1 105.1 131.9 139.6 162.0 122.4 162.0 156.9

𝑓`! 122.2 137.0 157.1 153.2 150.2 σ 10.4 6.7 13.4 6.7 13.3

While performing fiber pullout tests, the compressive strength for the batch with

0% fibers within the matrix has an average compressive strength of f`c = 123.6 N/mm2

with a standard deviation of 16.3 N/mm2 (Table 4.9). The 2% and 4% fiber volumes has

an average compressive strength of f`c = 148.1 N/mm2 and f`c = 150.2 N/mm2 (Tables

4.10 and 4.11), respectively. The standard deviations for 2% and 4% volumes were 11.0

N/mm2 and 13.3 N/mm2, respectively.

67

Compressive strength was not considered as an independent parameter because

the same type of UHPC was used for each batch, with the only variable being the

percentage of fibers added. The fiber pullout tests were all performed around the same

time frame when the concrete was fully hardened. The differences of compressive

strengths between batches was taken into account through the fiber volume percentage (0,

2%, or 4%).

68

CHAPTER 5. Model Parameters and Curves

The fiber pullout curves are fit to the fiber pullout model using two methods. The

first method, method A, averages each P-ν curve into one average curve for each series

using the moving average method. The average curve is then fitted to the previously

discussed fiber pullout model. The second method, method B, fits every fiber P-ν curve

to the model, and then averages the parameters individually to get the fitted curve. The

fitted curves produce a bond fracture energy coefficient, Gd, bond frictional stress value,

τ0, and a slip hardening-softening coefficient, β, from the equations associated with the

fiber pullout model, given in Section 2.1. This section discusses the MATLAB

optimization function for fitting the curve, the differences between the two methods for

the bond fracture energy, bond frictional stress and slip hardening-softening parameter, as

well as the final fiber pullout fitted model.

Each experimental test is fitted to the Lin et al. (1999) fiber pullout model to

obtain values for the three material parameters associated with each fiber. The fiber

pullout fitted curve is divided into two continuous sections, separated by the critical slip

value, νd (mm), which represents the displacement (slippage) at full chemical debonding

for a given embedment length, Le (mm), in terms of bond fracture energy (chemical

bond), Gd (N/mm), and frictional stress, (N/mm2), and is expressed as (Lin et al. 1999):

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

ff

ed

ff

ed dE

LGdEL 220 82τ

ν (1)

where Ef (MPa) and df (mm) represent the modulus of elasticity and diameter of

the fiber, respectively. Prior to full debonding, ν < νd, the pullout load resistance is

represented as (Lin et al. 1999):

0τ

69

( )2/1

032

2)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡ += dff GdE

Pντπ

ν (2)

After full debonding, only frictional bond is apparent until complete fiber pullout.

The pullout load resistance after full debonding, ν > νd, is a function of P0 (N) and β,

given as (Lin et al. 1999):

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

f

d

e

d

dLPP

ννβ

ννν 11)( 0 (3)

where:

P0 = πLedfτ0 (4)

and β represents the interfacial friction coefficient which take values of β = 0; β >

0; β < 0. To obtain the best fit of the experimental data, the equations 1 to 4 governing the

fitted model curve can be written in one function, Pf (N), where Pf = f (Gd, τ0, β). The

least square method is then utilized to give insights on the goodness of fit by calculating

an overall error through summing the square of the residuals (offsets) between the actual

data and the fitted curve as shown in equation 5.

(5)

where PE (N) is the experimental pullout load and νmax (mm) is the maximum

pullout slip, 6.5 mm. This error function is then programmed into a MATLAB

optimization routine called patternsearch which takes the error function and initial

guesses for the fitted parameters, Gd (N/mm), τ0 (N/mm2), and β as 1 N/mm, 0 and 0.1,

respectively, reiterates the error function until reaching an adequate fit and generates the

recommended values for the fitted parameters. The maximum and minimum values for

the fitted parameters are specified inside the optimization function so as to not exceed the

( )∑=

−=max

0

2ν

νEf PPError

70

allowed values for each parameter. The maximum iterations and function evaluations are

user defined as 2000 and 40000, respectively. Once the optimization and error functions

generate the best fit of the experimental data, the fitted curve is plotted against the

experimental data. A typical fiber pullout test and fitted curve can be seen in Figure 5.1.

Figure 5.1: Representative curve for model fitting

For method A, the fitted parameters, Gd, τ0, and β, are calculated from the average

curve after it is fitted to the model. The standard deviation curves are fitted by the model

to obtain insights on the variability of the fitted parameters to the experimental data.

These values can be seen in Table 5.1, with the standard deviation for each parameter

labeled as “(-1 σ)” and “(+1 σ)”. For method B, the fitted parameters are averages of the

individual parameters already obtained from the fitted curves. The standard deviation for

0 1 2 3 4 5 60

2

4

6

8

10

12

14

Slip, ν (mm)

Pullo

ut L

oad,

P (N

)

71

each parameter is found through the range of values from the individual fitted curves.

The values for method B can also be seen in Table 5.1.

Table 5.1: Fitted model parameters for method A and B: Gd , τ0 and β

       Method A Method B

V (%) E (mm) Gd

(-1 σ) (N/mm)

Gd (N/mm)

Gd (+1 σ)

(N/mm)

Mean (Gd)

(N/mm)

(σGd)

(N/mm)

0 ≥ 12.7 0.0000 0.0000 0.0099 0.0064 0.0120 3.2 0.0000 0.0000 0.0005 0.0012 0.0033

2 ≥ 3.2 0.0000 0.0000 0.0078 0.0057 0.0156 4 ≥ 3.2 0.0000 0.0000 0.0000 0.0013 0.0066

    Method A Method B

τ0

(-1 σ) (N/mm2)

τ0

(N/mm2)

τ0 (+1 σ)

(N/mm2)

Mean (τ0)

(N/mm2)

(στ0)

(N/mm2)

0 ≥ 12.7 2.9325 2.9489 3.0119 3.0264 1.7770 3.2 1.9134 2.1926 2.4837 2.2796 1.1897

2 ≥ 3.2 2.1756 2.5536 2.5204 2.7050 1.9704 4 ≥ 3.2 1.9251 2.8504 3.7866 2.9355 2.1254

    Method A Method B

β (-1 σ) β β

(+1 σ) Mean

(β)

(σβ)

0 ≥ 12.7 0.0308 0.1624 0.2867 0.2614 0.3069 3.2 0.0326 0.1514 0.2418 0.1934 0.1925

2 ≥ 3.2 0.0186 0.2104 0.4227 0.3320 0.3773 4 ≥ 3.2 0.0214 0.1845 0.2666 0.2827 0.3311

There are large differences in values for the bond fracture energy, Gd, between

methods A and B. For method A, the values are zero with large standard deviations.

Method B has higher overall values for Gd, ranging from 0.0012 to 0.0064 N/mm, with

large standard deviations. Even with the large differences in Gd between methods, the

values lie within one standard deviation of the other method’s mean Gd. It can be

72

concluded that the values between each method are small and within the scatter, and

assuming a zero value of Gd is acceptable.

For bond frictional stress, τ0, the differences between method A and B are small.

The values for method A range from 2.19 N/mm2 to 2.94 N/mm2, where values for

method B are consistently higher, ranging from 2.28 N/mm2 to 3.03 N/mm2. The standard

deviations for method A are large in comparison to method B, which says that method A

has more variability for τ0 than method B.

The differences between method A and B for the slip hardening-softening

coefficient, β, are also small. Method A has values ranging from 0.15 to 0.21. Similarly

to bond fracture energy and bond frictional stress, method B consistently has higher

values for β, ranging from 0.19 to 0.33. The standard deviations are slightly less for

method A than for method B, indicating that method B has more variability in the slip

hardening-softening coefficient.

Overall, the positive β values show that the fibers have slip hardening behavior

for the frictional bond, indicating that there is an increase in friction during pullout. This

behavior is typical of deformed fibers within a cementitious matrix because there is a

mechanical component providing additional friction to the bond. Limited research has

been performed on straight steel fibers which do not have a defined mechanical

component, and slip-hardening behavior was not asserted. However, the results of this

experimental program conclude that the fibers show an overall slip hardening behavior.

As discussed in Chapter 2, microscopic observations for similar steel fibers pulled in

dense matrixes like UHP-FRC suggested reasons for the slip hardening behavior due to

(1) fiber-end deformation during the manufacturing process to cut the fiber to length, (2)

73

damage (scratching) to the fiber surface, and (3) matrix particles adhering to the fiber

surface providing a wedge effect (Wille and Naaman 2012; Wille and Naaman 2013).

During this experimental program, visual observation showed that indeed some of the

fibers had obvious mechanical end-deformations and some concrete particles adhered to

the outer surface of the fiber. These reasons could attribute to the increased friction

resulting in slip hardening behavior during single fiber pullout.

The final grouping of the fibers includes two fitted curves representing E = 3.2

mm and E ≥ 12.7 mm for V = 0% (orange and maroon curves, respectively), and one

averaged curve each for V = 2% and V = 4% (blue and green curves, respectively) shown

in Figure 5.2. A dashed line represents the averaged curves while the fitted curve is the

solid line.

Figure 5.2: Final fitted fiber pullout curves

0 1 2 3 4 5 60

5

10

15

20

25

30

Slip, i (mm)

Pullo

ut L

oad,

P (N

)

V = 0% E = 3.2 mm

V = 0% (�� 12.7 mm

V = 4%

V = 2%

74

The fitted curves for V = 0% (E ≥ 12.7 mm), V = 2%, and V = 4% are very similar

in overall behavior, with slight differences due to the fitted parameter values, Gd, τ0, and

β for each curve. Table 5.2 shows that the τ0 value for the V = 4% and V = 0% (E ≥ 12.7

mm) are larger than the other curves, at 2.85 N/mm2 and 2.95 N/mm2, respectively,

which explains the higher peak value for the initial (chemical) portion of the curve. The β

value for V = 2% is the highest at β = 0.21 which shows the steepest frictional curve, with

a lower τ0 value (2.55 N/mm2). The fitted curve representing V = 0% and E = 3.2 mm, is

significantly lower than the other fitted curves, with corresponding low Gd, τ0, and β

values. This distinctly shows the fibers that are close to the edge at distances E = 3.2 mm

plays an important role in a concrete matrix with no fibers.

The conclusions for this research will be summarized in the following chapter,

including inputs for the LDPM-F model, generalized testing suggestions as well as future

work possibilities.

75

CHAPTER 6. Conclusions

6.1 Summary of Conclusions

The distance between the fibers, d, were concluded to not have a significant effect

on pullout work and maximum pullout load at distances of 3.2 mm and 12.7 mm.

The distance to the edge of the specimen, E, had a significant effect on the pullout

work and maximum pullout load at a distance of 3.2 mm only for 0% fiber volume

percentages. There was no observed effect on pullout work or maximum pullout response

at distances greater than 12.7 mm from the edge. For the tests with no fibers in the

matrix, the average pullout work and maximum load for fibers closer to the edge, E = 3.2

mm, decreased by 29% and 23% from fibers far from edge, E ≥ 12.7 mm, respectively.

The volume of fibers in the matrix, V, impacted the pullout work and maximum

pullout load. The average pullout work for V = 2% and 4% were 3% and 1% greater than

the average work for the reference batch with V = 0%, E ≥ 12.7 mm. Similarly, the

increase in the maximum pullout load was 10% and 18% for V = 2% and 4%

respectively. The standard deviations also increased by a large interval when adding

fibers to the matrix suggesting that these fibers in the matrix add more variability to the

single fiber pullout experimental data. However, the standard deviations between V = 2%

and 4% were within the same range.

Since the edge effect diminishes when the tested fiber is a distance greater than

12.7 mm, a circular specimen with a minimum radius of 12.7 mm or a square specimen

with the closest edge being a minimum of 12.7 mm are suggested when performing

pullout tests from UHPC with no fibers in the matrix. This would ensure that the pullout

76

behavior would not be influenced by an edge effect. A smaller specimen for pullout tests

with fibers in the matrix could be used but is not recommended.

The suggested fitted parameters for modeling fiber pullout in UHPC with straight

steel fibers and 2% and 4% fiber volume are a bond-fracture energy value of zero, a bond

friction coefficient of 2.6 N/mm2 and 2.9 N/mm2 and a slip-hardening coefficient of 0.21

and 0.18 respectively. These values were obtained by fitting the average pullout curve of

each fiber volume to the fit model equations (Method A). Slightly higher values of both

the bond friction coefficient and slip-hardening coefficient could be used with caution if

averaging the fitted parameters of individual fiber pullout curves is deemed more suitable

(Method B).

6.2 Future Work

These fitted parameters, bond-fracture energy, bond friction coefficient, and slip-

hardening coefficient, will be implemented into the LDPM model to define the fiber

interaction with the concrete matrix. The data from the compression tests will be used to

calibrate additional UHPC parameters within the model. Additional material

characterization tests (such as fracture tests, direct tension and split cylinders) will be

used to validate and further calibrate the model. The final validation will be performed

using full scale structural elements through experimental tests. Using these validation and

calibration parameters, the LDPM model will be able to model nonhomogeneous

structural elements to failure in addition to simulating discrete cracking, thin-walled

behavior, and interaction between fiber and matrix.

Further research is recommended to investigate the effect of pulling fibers

embedded at an inclination from the matrix and to examine the influence the inclination

77

angle has on spalling and snubbing. The effect of confinement could be studied by

performing single fiber pullout tests from specimens where an outer confinement

pressure is applied. Another parameter that could be studied is the effect of the

embedment length of the fiber on the model parameters: bond-fracture energy, bond

friction coefficient, and slip-hardening coefficient.

The statistical analysis tests utilized in this study were for two parameters (pullout

work and maximum pullout load) obtained from the pullout curve of each tested fiber.

Functional regression analysis and variance studies performed on the load-slip curves

could give more insights on the dependence of the curve shape on the tested parameters.

78

REFERENCES

Abu-Lebdeh, T., Hamoush, S., Heard, W., and Zornig, B. (2010). "Effect of matrix

strength on pullout behavior of steel fiber reinforced very-high strength concrete composites." Construction and Building Materials, 25, 39-46.

Chan, Y. W., and Chu, S. H. (2004). "Effect of silica fume on steel fiber bond characteristics in reactive powder concrete." Cement and Concrete Research, 34, 1167-1172.

Cunha, V. M. C. F., Barros, A. O., and Sena-Cruz, J. M. (2010). "Pullout Behavior of Steel Fibers in Self-Compacting Concrete." Journal of Materials in Civil Engineering, 22, 1-9.

Cusatis, G., Mencarelli, A., Pelessone, D., and Baylot, J. (2011). "Lattice Discrete Particle Model (LDPM) for failure behavior of concrete. II: Calibration and validation." Cement and Concrete Composites, 33, 891-905.

Cusatis, G., Pelessone, D., and Mencarelli, A. (2011). "Lattice Discrete Particle Model (LDPM) for failure behavior of concrete. I: Theory." Cement and Concrete Composites, 33, 881-890.

Graybeal, B. A. (2005). "Characterization of the Behavior of Ultra-High Performance Concrete." Doctor of Philosophy Dissertation, University of Maryland, College Park.

Graybeal, B. A. (2006). "Material Property Characterization of Ultra-High Performance Concrete."

Lin, Z., Kanda, T., and Li, V. C. (1999). "On interface property characterization and performance of fiber-reinforced cementitious composites." Concrete Science and Engineering, 1, 173-184.

Maage, M. (1978). "Fibre Bond and Friction in Cement and Concrete." RILEM SymposiumLancaster, 329-336.

Montgomery, D. C., Peck, E. A., and Vining, G. G. (2012). Introduction to linear regression analysis, John Wiley & Sons.

Naaman, A. E. (2003). "Engineered Steel Fibers with Optimal Properties for Reinforcement of Cement Composites." Journal of Advanced Concrete Technology, 1(3), 241-252.

Naaman, A. E., and Najm, H. (1991). "Bond-Slip Mechanisms of Steel Fibers in Concrete." ACI Materials Journal, 88(2), 135-145.

Naaman, A. E., Namur, G. G., Alwan, J. M., and Najm, H. S. (1991). "Fiber Pullout and Bond Slip. I: Analytical Study." Journal of Structural Engineering, 117(9), 2769-2790.

Naaman, A. E., and Shah, S. P. (1976). "Pull-out Mechanism in Steel Fibre-Reinforced Concrete." Journal of the Structural Division, 102(ST8), 1537-1548.

Schauffert, E. A., and Cusatis, G. (2012). "Lattice Discrete Particle Model for Fiber-Reinforced Concrete. I: Theory." Journal of Engineering Mechanics, 826-833.

Schauffert, E. A., Cusatis, G., Pelessone, D., O'Daniel, J. L., and Baylot, J. T. (2012). "Lattice Discrete Particle Model for Fiber-Reinforced Concrete. II: Tensile Fracture and Multiaxial Loading Behavior." Journal of Engineering Mechanics, 834-841.

79

Shannag, M. J., Brincker, R., and Hansen, W. (1997). "Pullout Behavior of Steel Fibers from Cement-Based Composites." Cement and Concrete Research, 27(6), 925-936.

Wille, K., Kim, D. J., and Naaman, A. E. (2011). "Strain-hardening UHP-FRC with low fiber contents." Materials and structures, 44(3), 583-598.

Wille, K., and Naaman, A. E. (2012). "Pullout Behavior of High-Strength Steel Fibers Embedded in Ultra-High-Performance Concrete." ACI Materials Journal, 109, 479-488.

Wille, K., and Naaman, A. E. "Bond Stress-Slip Behavior of Steel Fibers Embedded in Ultra High Performance Concrete." Proc., ECF18, Dresden 2010.

Wille, K., Naaman, A. E., El-Tawil, S., and Parra-Montesinos, G. J. (2012). "Ultra-high performance concrete and fiber reinforced concrete: achieving strength and ductility without heat curing." Materials and Structures, 45, 309-324.

Yang, E.-H., Wang, S., Yang, Y., and Li, V. C. (2008). "Fiber-bridging constitutive law of engineered cementitious composites." Journal of advanced concrete technology, 6(1), 181-193.

80

APPENDIX A. Analysis of Extensometer Load

Test Setup Properties Fiber Properties

r1 21.5� mm Average radius of test setup (conservative)

r2 0.1� mm

A1 π r1� �2� 1452.2 � mm Average area of setup A2 π 0.1( )2� 0.0314 � mm

L1 428� mm L2 6.5� mm Length being gripped

E 200000� N

mm2Modulus of Elasticiy: steel E 2 105u

N

mm2

K

A1 E�

L1

A1 E�

L1�

0

A1 E�

L1�

A1 E�

L1

A2 E�

L2�

A2 E�

L2�

0

A2 E�

L2�

A2 E�

L2

§¨¨¨¨¨¨¨¨©

·¸¸¸¸¸¸¸¸¹

6.786 105u

6.786� 105u

0

6.786� 105u

6.796 105u

966.644�

0

966.644�

966.644

§¨¨¨©

·¸¸¸¹

� Nmm

δ 1

6.796 105u1.471 10 6�

u � mm NOTE: Used a unit applied load of 1 N for the load applied by the extensometer.

∆0

δ0

§¨¨©

·¸¸¹

0

1.47145 10 6�u

0

§̈

¨¨©

·̧

¸¸¹

� mm

NR1

P

R2

§¨¨¨©

·¸¸¸¹

K ∆�0.999�

1

1.422� 10 3�u

§̈

¨¨©

·̧

¸¸¹

� N

N

ANALYSIS: 99.9% of the applied force (from the extensometer) is putting compression on the load cell. 0.142% of theapplied force is putting tension on the fiber.

CONCLUSION: Since there is already a preload of ~ 3N (compression) on the fiber from tightening the grip, a small tensileforce of 0.294N (2N typical, conservative, applied load from the extensometer, multiplied by the 0.142%), will not affect thefiber or results.

81

APPENDIX B. Check of Fiber Elasticity

A sample of fibers was measured prior to and after fiber pullout testing using a

micrometer to ensure elasticity of the fiber during the fiber pullout test. Checking

elasticity proves that the fiber has not yielded prior to full pullout. The fiber differences

are so small that they are attributed to human error when measuring, and are within an

acceptable tolerance.

Table B.1: Elasticity check for V = 0%, d = 3.2 mm, E = 3.2 mm

Fiber Number

Pre-testing Length (mm)

Post Testing Length (mm)

Difference (mm)

1 13.94 13.90 -0.04 2 14.00 13.94 -0.06 3 13.75 13.68 -0.06 4 14.21 14.13 -0.09 5 12.98 12.95 -0.03 6 13.57 13.54 -0.03 7 12.55 12.49 -0.06 8 13.83 13.77 -0.06 9 13.65 13.63 -0.02

10 12.14 12.11 -0.03 11 13.03 12.98 -0.05 12 13.73 13.68 -0.05 13 12.41 12.37 -0.04 14 13.70 13.66 -0.04 15 13.48 13.47 -0.01 16 14.36 14.35 -0.01 17 13.87 13.83 -0.04 18 13.41 13.42 0.00 19 13.54 13.54 0.00 20 13.84 13.86 0.02 21 13.68 13.64 -0.04

82

Table B.2: Elasticity check for V = 2%, d = 3.2 mm, E = 3.2 mm

Fiber Number

Pre-testing Length (mm)

Post Testing Length (mm)

Difference (mm)

1 13.594 13.609 0.015 2 14.577 14.527 -0.050 3 14.680 14.652 -0.028 4 14.571 14.546 -0.025 5 14.383 14.344 -0.039 6 15.404 15.39 -0.014 7 14.753 14.745 -0.008 8 14.034 13.999 -0.035 9 15.254 15.27 0.016

10 15.469 15.42 -0.049 11 14.977 14.951 -0.026 12 14.185 14.169 -0.016 13 14.884 14.861 -0.023 14 14.360 14.339 -0.021 15 14.991 14.951 -0.040 16 15.425 15.36 -0.065 17 15.238 15.182 -0.056 18 14.656 14.591 -0.065

83

Table B.3: Elasticity check for V = 4%, d = 12.7 mm, E = 12.7 mm

Fiber Number

Pre-testing Length (mm)

Post Testing Length (mm)

Difference (mm)

1 14.814 14.772 -0.042 2 14.122 14.086 -0.036 3 15.211 15.189 -0.022 4 14.349 14.304 -0.045 5 14.385 14.363 -0.022 6 14.608 14.58 -0.028 7 15.349 15.335 -0.014 8 14.637 14.613 -0.024 9 15.261 15.222 -0.039

10 14.476 14.432 -0.044 11 14.433 14.425 -0.008 12 15.549 15.526 -0.023 13 15.582 15.549 -0.033 14 14.445 14.429 -0.016 15 14.095 14.071 -0.024 16 14.229 14.151 -0.078 17 14.886 14.854 -0.032 18 15.117 15.047 -0.070 19 13.565 13.542 -0.023 20 14.677 14.652 -0.025 21 14.791 14.785 -0.006

84

APPENDIX C. MATLAB Programs

C.1 “LoadFilesP.map”

This program loads the data files into MATLAB, labeling each fiber according to

their respective variables.

function loadfilesP(B,P,I,F,L) % B is the batch number % P is the a string of either "S1, S2, or M" % F is the first fiber number to import % L is the last fiber number to import structure = struct('T',[],'P',[],'E',[],'C',[]); fstruct = fieldnames(structure); Data = cell(1,L-F+1); n = 1; for k = F:L Data(Abu-Lebdeh et al.) = horzcat('BA',num2str(B),'P',num2str(P),I,'F',num2str(k),'.txt'); n = n + 1; end for i = 1:length(Data) R = importdata(Data{i}); R = R.data; for j = 1:length(fstruct) structure.(fstruct{j}) = R(:,j); end assignin('base',Data{i}(1:end-4),structure); end end

C.2 “CorrectFitP.m”

This program provides data corrections to the fiber pullout experimental data and

fits the data to the fiber pullout model through the optimization and error functions using

the provided equations.

% This program corrects the Fiber Pullout experimental data and fits it to Lin Model Equations

85

%% Specify group name to save data to a Matlab file % NameMAT = 'G1'; %% Fiber Geometry & Embeddment Length Ef = 210000; % MPa (N/mm^2) df = 0.2; % mm Le = 6.5; % mm %% Select all variables vars = who('BA*'); %% Data Correction & Fitting structure = struct('T',[],'P',[],'E',[],'C',[],'Ef',[],'Pf',[],'Lp',[],'Nud',[],'Pd',[],'Po',[],'Tao',[],'Gd',[],'Beta',[]); for i = 1:length(vars); %% Imports structure data T = eval(strcat(vars{i},'.T')); P = eval(strcat(vars{i},'.P')); E = eval(strcat(vars{i},'.E')); C = eval(strcat(vars{i},'.C')); %% Data Correction % Zero the Extensometer and convert to metric (mm) E = (E - E(1))*25.4; % Subtract the Extensometer pre-load (PE) PE = (P(end-5)+P(end-4)+P(end-3)+P(end-2)+P(end-1)+P(end))/6; P = P - PE; % Finds index 'j' where P = 0 j = 1; while P(j)<0 j = j+1; end if j > 1 %finds the intersection with zero 'C' and 'E' CE = E(j)-(E(j)-E(j-1))/(P(j)-P(j-1))*P(j); % Linear Interpolation CC = C(j)-(C(j)-C(j-1))/(P(j)-P(j-1))*P(j); % Linear Interpolation % Shifts data E = E - CE; C = C - CC;

86

% Delete everything before index j T(1:j-1) = []; P(1:j-1) = []; E(1:j-1) = []; C(1:j-1) = []; end % Finds index f where Le = 6.5 fit = 1; if E(end) > Le while E(fit) < Le fit = fit + 1; end % Deletes everything after index f T((fit+1):end) = []; P((fit+1):end) = []; E((fit+1):end) = []; C((fit+1):end) = []; end % Zero the time T = T - T(1); structure.T = T; structure.P = P; structure.E = E; structure.C = C; %% Data Fitting % Finds index 'e' of the pullout length (Lp) p = 100; while P(p)>0 || P(p-1)>0 || P(p-2)>0 if p < length(P) p = p + 1; else break; end end % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; % Specifies the optimization parameters (x = [Tao Gd Beta]) fit0 = @(x)ErrorP(x,E,P,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fit = patternsearch(fit0,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options);

87

structure.Po = pi*Le*df*fit(1); structure.Tao = fit(1); structure.Gd = fit(2); structure.Beta = fit(3); structure.Lp = E(p); structure.Nud = 2*structure.Tao*Le^2/(Ef*df)+(8*structure.Gd*Le^2/(Ef*df))^0.5; % Save fitted plots back to structure inc1 = structure.Nud/100; inc2 = (Le-structure.Nud)/4000; Nu1 = 0:inc1:structure.Nud; Nu2 = structure.Nud:inc2:Le; Nu = [Nu1 Nu2]; k = 0; Eff = zeros(length(Nu),1); Pff = zeros(length(Nu),1); for z = 1:length(Nu); if Nu(z) < structure.Nud; c = (((pi^2*Ef*df^3)/2)*(structure.Tao*Nu(z)+structure.Gd))^0.5; else if Nu(z) == structure.Nud && k == 0; c = ((pi^2*Ef*df^3)/2*(structure.Tao*Nu(z)+structure.Gd))^0.5; k = 1; d = z; else c = structure.Po*(1-((Nu(z)-structure.Nud)/Le))*(1+(structure.Beta*(Nu(z)-structure.Nud))/df); end end structure.Ef(z) = Nu(z); structure.Pf(z) = c; Eff(z,1) = Nu(z); Pff(z,1) = c; end structure.Pd = structure.Pf(d); assignin('base',vars{i},structure); %% Saves the Corrected (C) and Fitted Data (F) into two seperate txt file Matrix1 = [T P E C]; Matrix2 = [Eff Pff]; Matrix3 = [structure.Lp structure.Nud structure.Pd structure.Po structure.Tao structure.Gd structure.Beta]; csvwrite(horzcat(vars{i},'C.txt'),Matrix1); csvwrite(horzcat(vars{i},'F.txt'),Matrix2); csvwrite(horzcat(vars{i},'FP.txt'),Matrix3);

88

end clear i; clear p; clear T; clear P; clear E; clear C; clear x0; clear f; clear options; clear axes1; clear j; clear Nu; clear Nu1; clear Nu2; clear c; clear inc1; clear inc2; clear k; clear structure; clear j; clear fit; clear fit0; clear z; clear PE; clear CE; clear CC; clear Matrix1; clear Matrix2; clear NameTxt; clear Pff; clear Eff; clear ans; clear Matrix3; clear Ef; clear Le; clear df; clear vars; clear d; %% Saves all variables into one MATLAB file including all variables % save(NameMAT,'-regexp',['^(?!NameMAT$).']); % clear NameMAT;

C.3 “ErrorP.mat”

This program is used with “CorrectFitP.m” to optimize the model fit using the

sum of the squares.

function Er = ErrorP(x,E,P,Le,Ef,df) Tao = x(1);

89

Gd = x(2); Beta = x(3); Er1 = 0; Er2 = 0; P = P(E>=0); E = E(E>=0); Pf = zeros(length(P),1); kk = 1; for i = 1:length(E) Nud = 2*Tao*Le^2/(Ef*df)+(8*Gd*Le^2/(Ef*df))^0.5; Po = pi*Le*df*Tao; if E(i) <= Nud Pf(i) = (pi^2*Ef*df^3/2*(Tao*E(i)+Gd))^0.5; Er0 = ((Pf(i) - P(i)))^2; Er1 = (Er1 + Er0); kk = i; else Pf(i) = Po*(1-(E(i)-Nud)/Le)*(1+Beta*(E(i)-Nud)/df); Er0 = ((Pf(i) - P(i)))^2; Er2 = Er2 + Er0; end end Er = Er1+Er2; end

C.4 “PostprocessP_Data_per_line_G.m”

This program combines the fibers within their respective groups and outputs a

table with statistical data, such as mean pullout work, standard deviation, correlation, etc.

per group.

%% Fiber Pullout Post Processing % clear all % clc % load('GBA.mat'); %% Select fiber order % Fiber order: 1 line, 3 lines, 2 lines

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% 0 percent volume percentage: BA6 % Far Apart GVF1 = who('BA6P6M*'); GVF2 = who('BA6P5S1*'); GVF3 = who('BA6P5M*'); GVF4 = who('BA6P5S2*'); GVF5 = who('BA6P3S1*'); GVF6 = who('BA6P3S2*'); % group all far apart fibers per batch BA6F = who('VF*'); % Close Together GVC1 = who('BA6P7M*'); GVC2 = who('BA6P2S1*'); GVC3 = who('BA6P2M*'); GVC4 = who('BA6P2S2*'); GVC5 = who('BA6P1S1*'); GVC6 = who('BA6P1S2*'); V1 = [GVF1; GVF2; GVF3; GVF4; GVC1; GVC2; GVC3; GVC4] V2 = [GVF5; GVF6; GVC5; GVC6]; % group all close together fibers per batch BA6 = ['V1';'V2']; % 2 percent volume percentage: BA7 % Far Apart GRF1 = who('BA7P9M*'); GRF2 = who('BA7P5S1*'); GRF3 = who('BA7P5M*'); GRF4 = who('BA7P5S2*'); GRF5 = who('BA7P4S1*'); GRF6 = who('BA7P4S2*'); % Close Together GRC1 = who('BA7P7M*'); GRC2 = who('BA7P2S1*'); GRC3 = who('BA7P2M*'); GRC4 = who('BA7P2S2*'); GRC5 = who('BA7P1S1*'); GRC6 = who('BA7P1S2*');

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R1 = [GRF1; GRF2; GRF3; GRF4; GRC1; GRC2; GRC3; GRC4] R2 = [GRF5; GRF6; GRC5; GRC6]; % group all close together fibers per batch BA7 = ['R1';'R2']; % 4 percent volume percentage: BA8 % Far Apart GTF1 = who('BA8P9M*'); GTF2 = who('BA8P5S1*'); GTF3 = who('BA8P5M*'); GTF4 = who('BA8P5S2*'); GTF5 = who('BA8P3S1*'); GTF6 = who('BA8P3S2*'); % Close Together GTC1 = who('BA8P7M*'); GTC2 = who('BA8P2S1*'); GTC3 = who('BA8P2M*'); GTC4 = who('BA8P2S2*'); GTC5 = who('BA8P1S1*'); GTC6 = who('BA8P1S2*'); T1 = [GTF1; GTF2; GTF3; GTF4; GTC1; GTC2; GTC3; GTC4] T2 = [GTF5; GTF6; GTC5; GTC6]; % group all close together fibers per batch BA8 = ['T1';'T2']; %% Build the statistical table per Line & the line plot structure matrix MatrixBA6F_A = zeros(length(BA6F),12); MatrixBA6F_Pmax =zeros(length(BA6F),12); for i = 1:length(BA6F); temp1 = eval(BA6F{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1);

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for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax; ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax); Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA6F{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA); COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA6F_A(i,1) = length(temp1); MatrixBA6F_A(i,2) = mean(A); MatrixBA6F_A(i,3) = std(A); MatrixBA6F_A(i,4) = std(A)/mean(A); MatrixBA6F_A(i,5) = COV_A(1,2); MatrixBA6F_A(i,6) = CORR_A(1,2); MatrixBA6F_A(i,7) = std(NA); MatrixBA6F_A(i,8) = NCOV_A(1,2); MatrixBA6F_A(i,9) = NCORR_A(1,2); MatrixBA6F_A(i,10) = skewness(A); MatrixBA6F_A(i,11) = kurtosis(A); MatrixBA6F_A(i,12) = (mean(A)-mean(Af))/mean(A); MatrixBA6F_Pmax(i,1) = length(temp1); MatrixBA6F_Pmax(i,2) = mean(Pmax); MatrixBA6F_Pmax(i,3) = std(Pmax); MatrixBA6F_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA6F_Pmax(i,5) = COV_Pmax(1,2); MatrixBA6F_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA6F_Pmax(i,7) = std(NPmax); MatrixBA6F_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA6F_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA6F_Pmax(i,10) = skewness(Pmax);

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MatrixBA6F_Pmax(i,11) = kurtosis(Pmax); MatrixBA6F_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1; clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax; clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end clear i; clear j; MatrixBA6C_A = zeros(length(BA6C),12); MatrixBA6C_Pmax =zeros(length(BA6C),12); for i = 1:length(BA6C); temp1 = eval(BA6C{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1); for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax;

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ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax); Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA6C{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA); COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA6C_A(i,1) = length(temp1); MatrixBA6C_A(i,2) = mean(A); MatrixBA6C_A(i,3) = std(A); MatrixBA6C_A(i,4) = std(A)/mean(A); MatrixBA6C_A(i,5) = COV_A(1,2); MatrixBA6C_A(i,6) = CORR_A(1,2); MatrixBA6C_A(i,7) = std(NA); MatrixBA6C_A(i,8) = NCOV_A(1,2); MatrixBA6C_A(i,9) = NCORR_A(1,2); MatrixBA6C_A(i,10) = skewness(A); MatrixBA6C_A(i,11) = kurtosis(A); MatrixBA6C_A(i,12) = (mean(A)-mean(Af))/mean(A); MatrixBA6C_Pmax(i,1) = length(temp1); MatrixBA6C_Pmax(i,2) = mean(Pmax); MatrixBA6C_Pmax(i,3) = std(Pmax); MatrixBA6C_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA6C_Pmax(i,5) = COV_Pmax(1,2); MatrixBA6C_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA6C_Pmax(i,7) = std(NPmax); MatrixBA6C_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA6C_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA6C_Pmax(i,10) = skewness(Pmax); MatrixBA6C_Pmax(i,11) = kurtosis(Pmax); MatrixBA6C_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1;

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clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax; clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end clear i; clear j; MatrixBA7F_A = zeros(length(BA7F),12); MatrixBA7F_Pmax =zeros(length(BA7F),12); for i = 1:length(BA7F); temp1 = eval(BA7F{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1); for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax; ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax);

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Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA7F{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA); COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA7F_A(i,1) = length(temp1); MatrixBA7F_A(i,2) = mean(A); MatrixBA7F_A(i,3) = std(A); MatrixBA7F_A(i,4) = std(A)/mean(A); MatrixBA7F_A(i,5) = COV_A(1,2); MatrixBA7F_A(i,6) = CORR_A(1,2); MatrixBA7F_A(i,7) = std(NA); MatrixBA7F_A(i,8) = NCOV_A(1,2); MatrixBA7F_A(i,9) = NCORR_A(1,2); MatrixBA7F_A(i,10) = skewness(A); MatrixBA7F_A(i,11) = kurtosis(A); MatrixBA7F_A(i,12) = (mean(A)-mean(Af))/mean(A); MatrixBA7F_Pmax(i,1) = length(temp1); MatrixBA7F_Pmax(i,2) = mean(Pmax); MatrixBA7F_Pmax(i,3) = std(Pmax); MatrixBA7F_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA7F_Pmax(i,5) = COV_Pmax(1,2); MatrixBA7F_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA7F_Pmax(i,7) = std(NPmax); MatrixBA7F_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA7F_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA7F_Pmax(i,10) = skewness(Pmax); MatrixBA7F_Pmax(i,11) = kurtosis(Pmax); MatrixBA7F_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1; clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax;

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clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end clear i; clear j; MatrixBA7C_A = zeros(length(BA7C),12); MatrixBA7C_Pmax =zeros(length(BA7C),12); for i = 1:length(BA7C); temp1 = eval(BA7C{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1); for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax; ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax); Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA7C{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA);

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COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA7C_A(i,1) = length(temp1); MatrixBA7C_A(i,2) = mean(A); MatrixBA7C_A(i,3) = std(A); MatrixBA7C_A(i,4) = std(A)/mean(A); MatrixBA7C_A(i,5) = COV_A(1,2); MatrixBA7C_A(i,6) = CORR_A(1,2); MatrixBA7C_A(i,7) = std(NA); MatrixBA7C_A(i,8) = NCOV_A(1,2); MatrixBA7C_A(i,9) = NCORR_A(1,2); MatrixBA7C_A(i,10) = skewness(A); MatrixBA7C_A(i,11) = kurtosis(A); MatrixBA7C_A(i,12) = (mean(A)-mean(Af))/mean(A); MatrixBA7C_Pmax(i,1) = length(temp1); MatrixBA7C_Pmax(i,2) = mean(Pmax); MatrixBA7C_Pmax(i,3) = std(Pmax); MatrixBA7C_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA7C_Pmax(i,5) = COV_Pmax(1,2); MatrixBA7C_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA7C_Pmax(i,7) = std(NPmax); MatrixBA7C_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA7C_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA7C_Pmax(i,10) = skewness(Pmax); MatrixBA7C_Pmax(i,11) = kurtosis(Pmax); MatrixBA7C_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1; clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax; clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end

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clear i; clear j; MatrixBA8F_A = zeros(length(BA8F),12); MatrixBA8F_Pmax =zeros(length(BA8F),12); for i = 1:length(BA8F); temp1 = eval(BA8F{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1); for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax; ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax); Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA8F{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA); COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA8F_A(i,1) = length(temp1); MatrixBA8F_A(i,2) = mean(A); MatrixBA8F_A(i,3) = std(A); MatrixBA8F_A(i,4) = std(A)/mean(A);

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MatrixBA8F_A(i,5) = COV_A(1,2); MatrixBA8F_A(i,6) = CORR_A(1,2); MatrixBA8F_A(i,7) = std(NA); MatrixBA8F_A(i,8) = NCOV_A(1,2); MatrixBA8F_A(i,9) = NCORR_A(1,2); MatrixBA8F_A(i,10) = skewness(A); MatrixBA8F_A(i,11) = kurtosis(A); MatrixBA8F_A(i,12) = (mean(A)-mean(Af))/mean(A); MatrixBA8F_Pmax(i,1) = length(temp1); MatrixBA8F_Pmax(i,2) = mean(Pmax); MatrixBA8F_Pmax(i,3) = std(Pmax); MatrixBA8F_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA8F_Pmax(i,5) = COV_Pmax(1,2); MatrixBA8F_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA8F_Pmax(i,7) = std(NPmax); MatrixBA8F_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA8F_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA8F_Pmax(i,10) = skewness(Pmax); MatrixBA8F_Pmax(i,11) = kurtosis(Pmax); MatrixBA8F_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1; clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax; clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end clear i; clear j; MatrixBA8C_A = zeros(length(BA8C),12); MatrixBA8C_Pmax =zeros(length(BA8C),12); for i = 1:length(BA8C);

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temp1 = eval(BA8C{i}); % All fibers per line X = zeros(length(temp1),1); A = zeros(length(temp1),1); Pmax = zeros(length(temp1),1); Af = zeros(length(temp1),1); Pfmax = zeros(length(temp1),1); ErAL = zeros(length(temp1),1); ErPmaxL = zeros(length(temp1),1); for j = 1:length(temp1) temp2 = eval(temp1{j}); % imports the structure of each fiber X(j) = temp2.X; A(j) = temp2.A; Pmax(j) = temp2.Pmax; Af(j) = temp2.Af; Pfmax(j) = temp2.Pfmax; ErAL(j) = (temp2.A-temp2.Af)/temp2.A; ErPmaxL(j) = (temp2.Pmax-temp2.Pfmax)/temp2.Pmax; end NA = A/mean(A); NPmax = Pmax/mean(Pmax); NAf = Af/mean(Af); NPfmax = Pfmax/mean(Pfmax); Matrix = [X A Pmax NA NPmax Af ErAL Pfmax ErPmaxL NAf NPfmax ]; assignin('base',horzcat(BA8C{i},'_Plot'),Matrix) COV_A = cov(X,A); CORR_A = corrcoef(X,A); NCOV_A = cov(X,NA); NCORR_A = corrcoef(X,NA); COV_Pmax = cov(X,Pmax); CORR_Pmax = corrcoef(X,Pmax); NCOV_Pmax = cov(X,NPmax); NCORR_Pmax = corrcoef(X,NPmax); MatrixBA8C_A(i,1) = length(temp1); MatrixBA8C_A(i,2) = mean(A); MatrixBA8C_A(i,3) = std(A); MatrixBA8C_A(i,4) = std(A)/mean(A); MatrixBA8C_A(i,5) = COV_A(1,2); MatrixBA8C_A(i,6) = CORR_A(1,2); MatrixBA8C_A(i,7) = std(NA); MatrixBA8C_A(i,8) = NCOV_A(1,2); MatrixBA8C_A(i,9) = NCORR_A(1,2); MatrixBA8C_A(i,10) = skewness(A); MatrixBA8C_A(i,11) = kurtosis(A); MatrixBA8C_A(i,12) = (mean(A)-mean(Af))/mean(A);

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MatrixBA8C_Pmax(i,1) = length(temp1); MatrixBA8C_Pmax(i,2) = mean(Pmax); MatrixBA8C_Pmax(i,3) = std(Pmax); MatrixBA8C_Pmax(i,4) = std(Pmax)/mean(Pmax); MatrixBA8C_Pmax(i,5) = COV_Pmax(1,2); MatrixBA8C_Pmax(i,6) = CORR_Pmax(1,2); MatrixBA8C_Pmax(i,7) = std(NPmax); MatrixBA8C_Pmax(i,8) = NCOV_Pmax(1,2); MatrixBA8C_Pmax(i,9) = NCORR_Pmax(1,2); MatrixBA8C_Pmax(i,10) = skewness(Pmax); MatrixBA8C_Pmax(i,11) = kurtosis(Pmax); MatrixBA8C_Pmax(i,12) = (mean(Pmax)-mean(Pfmax))/mean(Pmax); clear X; clear A; clear Af; clear Pmax; clear Pfmax; clear temp1; clear temp2; clear NA; clear COV_A; clear COV_Pmax; clear NCOV_A; clear NCOV_Pmax; clear CORR_A; clear CORR_Pmax; clear NCORR_A; clear NCORR_Pmax; clear NPmax; clear Matrix; clear NAf; clear NPfmax; clear ErAL; clear ErPmaxL; end clear i; clear j;

C.5 “Statistics.m”

This program performs ANOVA statistical analysis on groups of fibers.

clc clear all load('GBA.mat'); G = [transpose(VC2) transpose(VF2) transpose(VC1) transpose(VF1) transpose(RC2) transpose(RF2) transpose(RC1) transpose(RF1) transpose(TC2) transpose(TF2) transpose(TC1) transpose(TF1)];

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d = [ones(1,length(VC2))*3.2 ones(1,length(VF2))*12.7 ones(1,length(VC1))*3.2 ones(1,length(VF1))*12.7 ones(1,length(RC2))*3.2 ones(1,length(RF2))*12.7 ones(1,length(RC1))*3.2 ones(1,length(RF1))*12.7 ones(1,length(TC2))*3.2 ones(1,length(TF2))*12.7 ones(1,length(TC1))*3.2 ones(1,length(TF1))*12.7]; e = [ones(1,length(VC2))*3.2 ones(1,length(VF2))*3.2 ones(1,length(VC1))*12.7 ones(1,length(VF1))*12.7 ones(1,length(RC2))*3.2 ones(1,length(RF2))*3.2 ones(1,length(RC1))*12.7 ones(1,length(RF1))*12.7 ones(1,length(TC2))*3.2 ones(1,length(TF2))*3.2 ones(1,length(TC1))*12.7 ones(1,length(TF1))*12.7]; v = [ones(1,length(VC2))*0 ones(1,length(VF2))*0 ones(1,length(VC1))*0 ones(1,length(VF1))*0 ones(1,length(RC2))*2 ones(1,length(RF2))*2 ones(1,length(RC1))*2 ones(1,length(RF1))*2 ones(1,length(TC2))*4 ones(1,length(TF2))*4 ones(1,length(TC1))*4 ones(1,length(TF1))*4]; fc = [ones(1,length(VC2))*124.7 ones(1,length(VF2))*124.7 ones(1,length(VC1))*124.7 ones(1,length(VF1))*124.7 ones(1,length(RC2))*147.9 ones(1,length(RF2))*147.9 ones(1,length(RC1))*147.9 ones(1,length(RF1))*147.9 ones(1,length(TC2))*149.1 ones(1,length(TF2))*149.1 ones(1,length(TC1))*149.1 ones(1,length(TF1))*149.1]; Pmax = zeros(1,length(G)); W = zeros(1,length(G)); Lp = zeros(1,length(G)); for i = 1:length(G); temp1 = eval(G{i}); Pmax(i) = temp1.Pmax; W(i) = temp1.A; Lp(i) = temp1.Lp; clear temp1; end v = transpose(v); e = transpose(e); d = transpose(d); Pmax = transpose(Pmax); W = transpose(W); Lp = transpose(Lp); lmW = anovan(W,[v e d]); lmPmax = anovan(Pmax,[v e d]); lmLp = anovan(Lp,[v e d]); LinearW = fitlm([v e d],W); LinearPmax = fitlm([v e d],Pmax); % Create figure1 figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[2.502 3.002],'PaperPosition',[0.001,0.001,2.5,3.0]);

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% Create axes axes1 = axes('Parent',figure1,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf Residuals','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf Probability','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); plotResiduals(LinearW,'probability') % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure1, 'Residuals_Probability_W', 'pdf'); clear axes1; clear figure1; % Create figure1 figure2 = figure('Color',[1 1 1],'Units','inches','PaperSize',[2.502 3.002],'PaperPosition',[0.001,0.001,2.5,3.0]); % Create axes axes2 = axes('Parent',figure2,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes2,'on'); % Create xlabel xlabel('\bf Residuals','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf Relative Frequency','FontName','Times New Roman','FontSize',9); hold(axes2,'all'); plotResiduals(LinearW,'histogram') % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure2, 'Residuals_Frequency_W', 'pdf'); clear axes2; clear figure2; % Create figure1 figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[2.502 3.002],'PaperPosition',[0.001,0.001,2.5,3.0]); % Create axes

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axes1 = axes('Parent',figure1,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf Residuals','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf Probability','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); plotResiduals(LinearPmax,'probability') % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure1, 'Residuals_Probability_Pmax', 'pdf'); clear axes1; clear figure1; % Create figure1 figure2 = figure('Color',[1 1 1],'Units','inches','PaperSize',[2.502 3.002],'PaperPosition',[0.001,0.001,2.5,3.0]); % Create axes axes2 = axes('Parent',figure2,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes2,'on'); % Create xlabel xlabel('\bf Residuals','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf Relative Frequency','FontName','Times New Roman','FontSize',9); hold(axes2,'all'); plotResiduals(LinearPmax,'histogram') % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure2, 'Residuals_Frequency_Pmax', 'pdf'); clear axes2; clear figure2;

C.6 “kstest_each_line.m”

This program performs the K-S test for each line of fibers to check normality of

the responses.

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ktest = zeros(36,2); ktest(1,1) = kstest((VF1_Plot(:,2)-mean(VF1_Plot(:,2)))/std(VF1_Plot(:,2))); ktest(1,2) = kstest((VF1_Plot(:,3)-mean(VF1_Plot(:,3)))/std(VF1_Plot(:,3))); ktest(2,1) = kstest((VF2_Plot(:,2)-mean(VF2_Plot(:,2)))/std(VF2_Plot(:,2))); ktest(2,2) = kstest((VF2_Plot(:,3)-mean(VF2_Plot(:,3)))/std(VF2_Plot(:,3))); ktest(3,1) = kstest((VF3_Plot(:,2)-mean(VF3_Plot(:,2)))/std(VF3_Plot(:,2))); ktest(3,2) = kstest((VF3_Plot(:,3)-mean(VF3_Plot(:,3)))/std(VF3_Plot(:,3))); ktest(4,1) = kstest((VF4_Plot(:,2)-mean(VF4_Plot(:,2)))/std(VF4_Plot(:,2))); ktest(4,2) = kstest((VF4_Plot(:,3)-mean(VF4_Plot(:,3)))/std(VF4_Plot(:,3))); ktest(5,1) = kstest((VF5_Plot(:,2)-mean(VF5_Plot(:,2)))/std(VF5_Plot(:,2))); ktest(5,2) = kstest((VF5_Plot(:,3)-mean(VF5_Plot(:,3)))/std(VF5_Plot(:,3))); ktest(6,1) = kstest((VF6_Plot(:,2)-mean(VF6_Plot(:,2)))/std(VF6_Plot(:,2))); ktest(6,2) = kstest((VF6_Plot(:,3)-mean(VF6_Plot(:,3)))/std(VF6_Plot(:,3))); ktest(7,1) = kstest((VC1_Plot(:,2)-mean(VC1_Plot(:,2)))/std(VC1_Plot(:,2))); ktest(7,2) = kstest((VC1_Plot(:,3)-mean(VC1_Plot(:,3)))/std(VC1_Plot(:,3))); ktest(8,1) = kstest((VC2_Plot(:,2)-mean(VC2_Plot(:,2)))/std(VC2_Plot(:,2))); ktest(8,2) = kstest((VC2_Plot(:,3)-mean(VC2_Plot(:,3)))/std(VC2_Plot(:,3))); ktest(9,1) = kstest((VC3_Plot(:,2)-mean(VC3_Plot(:,2)))/std(VC3_Plot(:,2))); ktest(9,2) = kstest((VC3_Plot(:,3)-mean(VC3_Plot(:,3)))/std(VC3_Plot(:,3))); ktest(10,1) = kstest((VC4_Plot(:,2)-mean(VC4_Plot(:,2)))/std(VC4_Plot(:,2))); ktest(10,2) = kstest((VC4_Plot(:,3)-mean(VC4_Plot(:,3)))/std(VC4_Plot(:,3))); ktest(11,1) = kstest((VC5_Plot(:,2)-mean(VC5_Plot(:,2)))/std(VC5_Plot(:,2)));

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ktest(11,2) = kstest((VC5_Plot(:,3)-mean(VC5_Plot(:,3)))/std(VC5_Plot(:,3))); ktest(12,1) = kstest((VC6_Plot(:,2)-mean(VC6_Plot(:,2)))/std(VC6_Plot(:,2))); ktest(12,2) = kstest((VC6_Plot(:,3)-mean(VC6_Plot(:,3)))/std(VC6_Plot(:,3))); ktest(13,1) = kstest((RF1_Plot(:,2)-mean(RF1_Plot(:,2)))/std(RF1_Plot(:,2))); ktest(13,2) = kstest((RF1_Plot(:,3)-mean(RF1_Plot(:,3)))/std(RF1_Plot(:,3))); ktest(14,1) = kstest((RF2_Plot(:,2)-mean(RF2_Plot(:,2)))/std(RF2_Plot(:,2))); ktest(14,2) = kstest((RF2_Plot(:,3)-mean(RF2_Plot(:,3)))/std(RF2_Plot(:,3))); ktest(15,1) = kstest((RF3_Plot(:,2)-mean(RF3_Plot(:,2)))/std(RF3_Plot(:,2))); ktest(15,2) = kstest((RF3_Plot(:,3)-mean(RF3_Plot(:,3)))/std(RF3_Plot(:,3))); ktest(16,1) = kstest((RF4_Plot(:,2)-mean(RF4_Plot(:,2)))/std(RF4_Plot(:,2))); ktest(16,2) = kstest((RF4_Plot(:,3)-mean(RF4_Plot(:,3)))/std(RF4_Plot(:,3))); ktest(17,1) = kstest((RF5_Plot(:,2)-mean(RF5_Plot(:,2)))/std(RF5_Plot(:,2))); ktest(17,2) = kstest((RF5_Plot(:,3)-mean(RF5_Plot(:,3)))/std(RF5_Plot(:,3))); ktest(18,1) = kstest((RF6_Plot(:,2)-mean(RF6_Plot(:,2)))/std(RF6_Plot(:,2))); ktest(18,2) = kstest((RF6_Plot(:,3)-mean(RF6_Plot(:,3)))/std(RF6_Plot(:,3))); ktest(19,1) = kstest((RC1_Plot(:,2)-mean(RC1_Plot(:,2)))/std(RC1_Plot(:,2))); ktest(19,2) = kstest((RC1_Plot(:,3)-mean(RC1_Plot(:,3)))/std(RC1_Plot(:,3))); ktest(20,1) = kstest((RC2_Plot(:,2)-mean(RC2_Plot(:,2)))/std(RC2_Plot(:,2))); ktest(20,2) = kstest((RC2_Plot(:,3)-mean(RC2_Plot(:,3)))/std(RC2_Plot(:,3))); ktest(21,1) = kstest((RC3_Plot(:,2)-mean(RC3_Plot(:,2)))/std(RC3_Plot(:,2))); ktest(21,2) = kstest((RC3_Plot(:,3)-mean(RC3_Plot(:,3)))/std(RC3_Plot(:,3))); ktest(22,1) = kstest((RC4_Plot(:,2)-mean(RC4_Plot(:,2)))/std(RC4_Plot(:,2)));

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ktest(22,2) = kstest((RC4_Plot(:,3)-mean(RC4_Plot(:,3)))/std(RC4_Plot(:,3))); ktest(23,1) = kstest((RC5_Plot(:,2)-mean(RC5_Plot(:,2)))/std(RC5_Plot(:,2))); ktest(23,2) = kstest((RC5_Plot(:,3)-mean(RC5_Plot(:,3)))/std(RC5_Plot(:,3))); ktest(24,1) = kstest((RC6_Plot(:,2)-mean(RC6_Plot(:,2)))/std(RC6_Plot(:,2))); ktest(24,2) = kstest((RC6_Plot(:,3)-mean(RC6_Plot(:,3)))/std(RC6_Plot(:,3))); ktest(25,1) = kstest((TF1_Plot(:,2)-mean(TF1_Plot(:,2)))/std(TF1_Plot(:,2))); ktest(25,2) = kstest((TF1_Plot(:,3)-mean(TF1_Plot(:,3)))/std(TF1_Plot(:,3))); ktest(26,1) = kstest((TF2_Plot(:,2)-mean(TF2_Plot(:,2)))/std(TF2_Plot(:,2))); ktest(26,2) = kstest((TF2_Plot(:,3)-mean(TF2_Plot(:,3)))/std(TF2_Plot(:,3))); ktest(27,1) = kstest((TF3_Plot(:,2)-mean(TF3_Plot(:,2)))/std(TF3_Plot(:,2))); ktest(27,2) = kstest((TF3_Plot(:,3)-mean(TF3_Plot(:,3)))/std(TF3_Plot(:,3))); ktest(28,1) = kstest((TF4_Plot(:,2)-mean(TF4_Plot(:,2)))/std(TF4_Plot(:,2))); ktest(28,2) = kstest((TF4_Plot(:,3)-mean(TF4_Plot(:,3)))/std(TF4_Plot(:,3))); ktest(29,1) = kstest((TF5_Plot(:,2)-mean(TF5_Plot(:,2)))/std(TF5_Plot(:,2))); ktest(29,2) = kstest((TF5_Plot(:,3)-mean(TF5_Plot(:,3)))/std(TF5_Plot(:,3))); ktest(30,1) = kstest((TF6_Plot(:,2)-mean(TF6_Plot(:,2)))/std(TF6_Plot(:,2))); ktest(30,2) = kstest((TF6_Plot(:,3)-mean(TF6_Plot(:,3)))/std(TF6_Plot(:,3))); ktest(31,1) = kstest((TC1_Plot(:,2)-mean(TC1_Plot(:,2)))/std(TC1_Plot(:,2))); ktest(31,2) = kstest((TC1_Plot(:,3)-mean(TC1_Plot(:,3)))/std(TC1_Plot(:,3))); ktest(32,1) = kstest((TC2_Plot(:,2)-mean(TC2_Plot(:,2)))/std(TC2_Plot(:,2))); ktest(32,2) = kstest((TC2_Plot(:,3)-mean(TC2_Plot(:,3)))/std(TC2_Plot(:,3))); ktest(33,1) = kstest((TC3_Plot(:,2)-mean(TC3_Plot(:,2)))/std(TC3_Plot(:,2)));

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ktest(33,2) = kstest((TC3_Plot(:,3)-mean(TC3_Plot(:,3)))/std(TC3_Plot(:,3))); ktest(34,1) = kstest((TC4_Plot(:,2)-mean(TC4_Plot(:,2)))/std(TC4_Plot(:,2))); ktest(34,2) = kstest((TC4_Plot(:,3)-mean(TC4_Plot(:,3)))/std(TC4_Plot(:,3))); ktest(35,1) = kstest((TC5_Plot(:,2)-mean(TC5_Plot(:,2)))/std(TC5_Plot(:,2))); ktest(35,2) = kstest((TC5_Plot(:,3)-mean(TC5_Plot(:,3)))/std(TC5_Plot(:,3))); ktest(36,1) = kstest((TC6_Plot(:,2)-mean(TC6_Plot(:,2)))/std(TC6_Plot(:,2))); ktest(36,2) = kstest((TC6_Plot(:,3)-mean(TC6_Plot(:,3)))/std(TC6_Plot(:,3)));

C.7 “graphData_W_X.m”

This program generates the pullout work versus location on the specimen, x, plot.

Batch = BA8F; d = 12.7; for i = 1:length(Batch); % Go over the lines in each Batch temp1 = eval(Batch{i}); % Saves every line i.e VF1 L = length(temp1); for j = 1:length(temp1) % Go over every fiber in a line temp2 = eval(temp1{j}); % Saves every fiber per line i.e. BA6P1S1F1 W(j) = temp2.A; X(j) = temp2.X; end % Create figure1 figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[5.002 3.002],'PaperPosition',[0.001,0.001,5,3]); % Create axes axes1 = axes('Parent',figure1,'FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf{Fiber Location, \itx\rm} \bf{(mm)}','FontName','Times New Roman','FontSize',9); % Create ylabel

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ylabel('\bf{Pullout Work, \itW\rm} \bf{(N-mm)}','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); plot(X,W,'Parent',axes1,'LineWidth',1,'MarkerSize',4,'Marker','o','DisplayName',Batch{i},'color','b'); plot(X,ones(length(X),1)*mean(W),'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'-Mean'),'color','r'); axis('auto'); a = axis; xlim([0 (L-1)*d+1]); ylim([0 450]); hold off; saveas(figure1, horzcat('W_LineNumber_',Batch{i}), 'pdf'); clear W; clear X; clear axis1; clear figure1; end clear W; clear i; clear j; clear X; clear Batch; clear temp1; clear temp2; clear d;

C.8 “graphData_Pmax_FinalGroupNumber.m”

This program plots the maximum pullout load for the final groups, separated by

volumes and edge distances, E.

Batch = BA6; name = 'BA6' % Plots are color coded per line colorstr = [0 0 1; 0 1 0; 1 0 0; 0 1 1; 1 0 1; 0 0 0]; % Create figure1

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figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[2.502 3.002],'PaperPosition',[0.001,0.001,2.5,3]); % Create axes axes1 = axes('Parent',figure1,'XTick',1:6,'FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf{Group Number}','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf{Maximum Pullout Load, \itP_m_a_x\rm} \bf{(N)}','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); for i = 1:length(Batch); % Go over the lines in each Batch temp1 = eval(Batch{i}); % Saves every line i.e VF1 for j = 1:length(temp1) % Go over every fiber in a line temp2 = eval(temp1{j}); % Saves every fiber per line i.e. BA6P1S1F1 Pmax(j) = temp2.Pmax; X(j) = i; end plot(X,Pmax,'Parent',axes1,'linestyle','none','MarkerSize',2,'Marker','o','DisplayName',Batch{i},'color',colorstr(i,:)); plot(X,mean(Pmax),'Marker','x','MarkerSize',8,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'-Mean'),'color',colorstr(i,:)); clear Pmax; clear X; end Batch = BA7; for i = 1:length(Batch); % Go over the lines in each Batch temp1 = eval(Batch{i}); % Saves every line i.e VF1 for j = 1:length(temp1) % Go over every fiber in a line temp2 = eval(temp1{j}); % Saves every fiber per line i.e. BA6P1S1F1 Pmax(j) = temp2.Pmax; X(j) = i+2; end

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plot(X,Pmax,'Parent',axes1,'linestyle','none','MarkerSize',2,'Marker','o','DisplayName',Batch{i},'color',colorstr(i+2,:)); plot(X,mean(Pmax),'Marker','x','MarkerSize',8,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'-Mean'),'color',colorstr(i+2,:)); clear Pmax; clear X; end Batch = BA8; for i = 1:length(Batch); % Go over the lines in each Batch temp1 = eval(Batch{i}); % Saves every line i.e VF1 for j = 1:length(temp1) % Go over every fiber in a line temp2 = eval(temp1{j}); % Saves every fiber per line i.e. BA6P1S1F1 Pmax(j) = temp2.Pmax; X(j) = i+4; end plot(X,Pmax,'Parent',axes1,'linestyle','none','MarkerSize',2,'Marker','o','DisplayName',Batch{i},'color',colorstr(i+4,:)); plot(X,mean(Pmax),'Marker','x','MarkerSize',8,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'-Mean'),'color',colorstr(i+4,:)); clear Pmax; clear X; end xlim([0 7]); ylim([0 120]); saveas(figure1, 'Pmax_FinalGroupNumber', 'pdf'); clear axis1; clear figure1;

C.9 “P_slip_Combined_Groups.m”

This program creates the curves for pullout load versus slip for the final combined

groups.

% Plots are color coded per line

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colorstr = [0.5 0 0; 0.9961 0.27 0]; % Create figure1 figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[6.002 4.002],'PaperPosition',[0.001,0.001,6,4]); % Create axes axes1 = axes('Parent',figure1,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf{Slip, \it\nu\rm} \bf{(mm)}','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf{Pullout Load, \itP\rm} \bf{(N)}','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); Batch = BA6; name = 'BA6'; for i = 1:length(Batch); temp0 = eval(Batch{i}); for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P; clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5); for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E]; Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap');

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end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end plot(NX,AveP,'Parent',axes1,'LineWidth',4,'DisplayName',horzcat(name,'_Average'),'color',colorstr(i,:)); % plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color',colorstr(i,:)); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color',colorstr(i,:)); clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear NX; clear a; clear temp0; end clear temp0; clear temp1; clear temp2; name = 'BA7'; temp0 = who('BA7P*'); for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P; clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5); for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E];

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Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap'); end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end plot(NX,AveP,'Parent',axes1,'LineWidth',4,'DisplayName',horzcat(name,'_Average'),'color','b'); % plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color','b'); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color','b'); clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear NX; clear a; clear temp0; temp0 = who('BA8P*'); name = 'BA8'; for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P; clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5); for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E];

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Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap'); end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end plot(NX,AveP,'Parent',axes1,'LineWidth',4,'DisplayName',horzcat(name,'_Average'),'color','g'); % plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color','g'); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(name,'_Average+std'),'color','g'); clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear NX; clear a; clear temp0; axis('auto'); a = axis; xlim([0 6.5]); ylim([0 a(4)]); % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure1, 'P_slip_Combined_Groups', 'pdf'); saveas(figure1, 'P_slip_Combined_Groups', 'fig'); clear i; clear j; clear k; clear Batch; clear figure1; clear axes1; clear colorstr; clear temp0; clear temp1; clear temp2;

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C.10 “P_slip_Combined_Groups_Fitted.m”

This program creates the fitted “model” curve overtop the averaged final curves.

% Plots are color coded per line colorstr = [0.5 0 0; 0.9961 0.27 0]; % Create figure1 figure1 = figure('Color',[1 1 1],'Units','inches','PaperSize',[6.002 4.002],'PaperPosition',[0.001,0.001,6,4]); % Create axes axes1 = axes('Parent',figure1,'YMinorTick','on','XMinorTick','on','FontName','Times New Roman','FontSize',9); box(axes1,'on'); % Create xlabel xlabel('\bf{Slip, \it\nu\rm} \bf{(mm)}','FontName','Times New Roman','FontSize',9); % Create ylabel ylabel('\bf{Pullout Load, \itP\rm} \bf{(N)}','FontName','Times New Roman','FontSize',9); hold(axes1,'all'); Ef = 210000; df = 0.2; Le = 6.5; Batch = BA6; name = 'BA6_'; for i = 1:length(Batch); temp0 = eval(Batch{i}); for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P; clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5);

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for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E]; Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap'); end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; % Specifies the optimization parameters (x = [Tao Gd Beta]) fit0 = @(x)ErrorP(x,NX,AveP,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fit = patternsearch(fit0,x0,[],[],[],[],[1 0 -1],[15 0 2],[],options); Po = pi*Le*df*fit(1); Tao = fit(1); Gd = fit(2); Beta = fit(3); Nud = 2*Tao*Le^2/(Ef*df)+(8*Gd*Le^2/(Ef*df))^0.5; % Save fitted plots back to structure inc1 = Nud/100; inc2 = (Le-Nud)/4000; Nu1 = 0:inc1:Nud; Nu2 = Nud:inc2:Le; Nu = [Nu1 Nu2]; k = 0; Eff = zeros(length(Nu),1); Pff = zeros(length(Nu),1); for z = 1:length(Nu); if Nu(z) < Nud; c = (((pi^2*Ef*df^3)/2)*(Tao*Nu(z)+Gd))^0.5;

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else if Nu(z) == Nud && k == 0; c = ((pi^2*Ef*df^3)/2*(Tao*Nu(z)+Gd))^0.5; k = 1; d = z; else c = Po*(1-((Nu(z)-Nud)/Le))*(1+(Beta*(Nu(z)-Nud))/df); end end Eff(z,1) = Nu(z); Pff(z,1) = c; end Pd = Pff(d); Fitted(i,:) = [Tao Gd Beta Nud Pd Po]; assignin('base',name,Fitted); plot(NX,AveP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(Batch{i},'_Average'),'color',colorstr(i,:)); % plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average+std'),'color',colorstr(i,:)); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average-std'),'color',colorstr(i,:)); plot(Eff,Pff,'Parent',axes1,'LineWidth',3,'DisplayName',horzcat(Batch{i},'_Average'),'color',colorstr(i,:)); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; variable = AveP+stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits10 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits1 = patternsearch(fits10,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options); Fitteds1(i,:) = [fits1(1) fits1(2) fits1(3)]; assignin('base',horzcat(name,'_s1'),Fitteds1); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1];

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variable = AveP-stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits20 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits2 = patternsearch(fits20,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options); Fitteds2(i,:) = [fits2(1) fits2(2) fits2(3)]; assignin('base',horzcat(name,'_s2'),Fitteds2); clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear x0; clear fit0; clear options; clear fit; clear Po; clear Nud; clear Pd; clear inc1; clear inc2; clear Nu1; clear Nu2; clear Nu; clear Eff; clear Pff; clear c clear d; clear z; clear Po; clear Pd; clear Tao; clear Beta; clear Gd; clear k; clear X; clear Y; clear NY; clear AveP; clear stdP;

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clear index; clear temp1; clear temp2; clear NX; clear temp0; clear fits1; clear fits10; clear fits20 end clear temp0; clear temp1; clear temp2; clear Fitted; clear Fitteds1; clear Fitteds2; name = 'BA7_'; temp0 = who('BA7P*'); for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P; clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5); for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E]; Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap'); end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1];

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% Specifies the optimization parameters (x = [Tao Gd Beta]) fit0 = @(x)ErrorP(x,NX,AveP,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fit = patternsearch(fit0,x0,[],[],[],[],[1 0 -1],[15 0 2],[],options); Po = pi*Le*df*fit(1); Tao = fit(1); Gd = fit(2); Beta = fit(3); Nud = 2*Tao*Le^2/(Ef*df)+(8*Gd*Le^2/(Ef*df))^0.5; % Save fitted plots back to structure inc1 = Nud/100; inc2 = (Le-Nud)/4000; Nu1 = 0:inc1:Nud; Nu2 = Nud:inc2:Le; Nu = [Nu1 Nu2]; k = 0; Eff = zeros(length(Nu),1); Pff = zeros(length(Nu),1); for z = 1:length(Nu); if Nu(z) < Nud; c = (((pi^2*Ef*df^3)/2)*(Tao*Nu(z)+Gd))^0.5; else if Nu(z) == Nud && k == 0; c = ((pi^2*Ef*df^3)/2*(Tao*Nu(z)+Gd))^0.5; k = 1; d = z; else c = Po*(1-((Nu(z)-Nud)/Le))*(1+(Beta*(Nu(z)-Nud))/df); end end Eff(z,1) = Nu(z); Pff(z,1) = c; end Pd = Pff(d); Fitted = [Tao Gd Beta Nud Pd Po]; assignin('base',name,Fitted); plot(NX,AveP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName','BA7_Average','color','b');

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% plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average+std'),'color',colorstr(i,:)); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average-std'),'color',colorstr(i,:)); plot(Eff,Pff,'Parent',axes1,'LineWidth',3,'DisplayName','BA7_Average','color','b'); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; variable = AveP+stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits10 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits1 = patternsearch(fits10,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options); Fitteds1 = [fits1(1) fits1(2) fits1(3)]; assignin('base',horzcat(name,'_s1'),Fitteds1); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; variable = AveP-stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits20 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits2 = patternsearch(fits20,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options); Fitteds2 = [fits2(1) fits2(2) fits2(3)]; assignin('base',horzcat(name,'_s2'),Fitteds2); clear X; clear Y;

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clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear x0; clear fit0; clear options; clear fit; clear Po; clear Nud; clear Pd; clear inc1; clear inc2; clear Nu1; clear Nu2; clear Nu; clear Eff; clear Pff; clear c clear d; clear z; clear Po; clear Pd; clear Tao; clear Beta; clear Gd; clear k; clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear NX; clear temp0; clear fits1; clear fits10; clear fits20; clear Fitted; clear Fitteds1; clear Fitteds2; temp0 = who('BA8P*'); name = 'BA8_'; for j = 1:length(temp0); % go over fibers in each line temp1 = eval(temp0{j}); % import the fiber resutls structure X = temp1.E; Y = temp1.P;

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clear X; clear Y; clear temp1; end NX = transpose(0:0.001:6.5); for j = 1:length(temp0); % go over fibers in each line temp2 = eval(temp0{j}); % import the fiber resutls structure X = [0; temp2.E]; Y = [0; temp2.P]; [X index] = unique(X,'rows'); Y = Y(index); NY(:,j) = interp1(X,Y,NX,'linear','extrap'); end for k = 1:length(NX); AveP(k) = mean(NY(k,:)); stdP(k) = std(NY(k,:)); end % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; % Specifies the optimization parameters (x = [Tao Gd Beta]) fit0 = @(x)ErrorP(x,NX,AveP,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fit = patternsearch(fit0,x0,[],[],[],[],[1 0 -1],[15 0 2],[],options); Po = pi*Le*df*fit(1); Tao = fit(1); Gd = fit(2); Beta = fit(3); Nud = 2*Tao*Le^2/(Ef*df)+(8*Gd*Le^2/(Ef*df))^0.5; % Save fitted plots back to structure inc1 = Nud/100; inc2 = (Le-Nud)/4000; Nu1 = 0:inc1:Nud; Nu2 = Nud:inc2:Le; Nu = [Nu1 Nu2];

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k = 0; Eff = zeros(length(Nu),1); Pff = zeros(length(Nu),1); for z = 1:length(Nu); if Nu(z) < Nud; c = (((pi^2*Ef*df^3)/2)*(Tao*Nu(z)+Gd))^0.5; else if Nu(z) == Nud && k == 0; c = ((pi^2*Ef*df^3)/2*(Tao*Nu(z)+Gd))^0.5; k = 1; d = z; else c = Po*(1-((Nu(z)-Nud)/Le))*(1+(Beta*(Nu(z)-Nud))/df); end end Eff(z,1) = Nu(z); Pff(z,1) = c; end Pd = Pff(d); Fitted = [Tao Gd Beta Nud Pd Po]; assignin('base',name,Fitted); plot(NX,AveP,'Parent',axes1,'LineWidth',2,'LineStyle','--','DisplayName',horzcat(Batch{i},'_Average'),'color','g'); % plot(NX,AveP+stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average+std'),'color',colorstr(i,:)); % plot(NX,AveP-stdP,'Parent',axes1,'LineWidth',2,'DisplayName',horzcat(Batch{i},'_Average-std'),'color',colorstr(i,:)); plot(Eff,Pff,'Parent',axes1,'LineWidth',3,'DisplayName',horzcat(Batch{i},'_Average'),'color','g'); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; variable = AveP+stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits10 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits1 = patternsearch(fits10,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options);

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Fitteds1 = [fits1(1) fits1(2) fits1(3)]; assignin('base',horzcat(name,'_s1'),Fitteds1); % Initial Guess of fitted parameters ([Tao Gd Beta]) x0 = [1 0 0.1]; variable = AveP-stdP; % Specifies the optimization parameters (x = [Tao Gd Beta]) fits20 = @(x)ErrorP(x,NX,variable,Le,Ef,df); % Maximum number of iterations %options = optimset('MaxFunEvals', 1000); options = psoptimset('MaxIter',2000,'MaxFunEvals',40000); % Optimizes x = [Tao Gd Beta] % fit = fminunc(fit0,x0,options); fits2 = patternsearch(fits20,x0,[],[],[],[],[1 0 -1],[15 15 2],[],options); Fitteds2 = [fits2(1) fits2(2) fits2(3)]; assignin('base',horzcat(name,'_s2'),Fitteds2); clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear x0; clear fit0; clear options; clear fit; clear Po; clear Nud; clear Pd; clear inc1; clear inc2; clear Nu1; clear Nu2; clear Nu; clear Eff; clear Pff; clear c clear d; clear z; clear Po; clear Pd; clear Tao; clear Beta; clear Gd;

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clear k; clear X; clear Y; clear NY; clear AveP; clear stdP; clear index; clear temp1; clear temp2; clear NX; clear temp0; clear fits1; clear fits10; clear fits20; clear Fitted; clear Fitteds1; clear Fitteds2; axis('auto'); a = axis; xlim([0 6.5]); ylim([0 a(4)]); % saveas(figure1, 'Average_P_slip_BA6C', 'fig'); saveas(figure1, 'P_slip_Combined_Groups', 'pdf'); saveas(figure1, 'P_slip_Combined_Groups', 'fig'); clear i; clear j; clear k; clear Batch; clear figure1; clear axes1; clear colorstr; clear temp0; clear temp1; clear temp2; clear a; clear fits1; clear fits10; clear fits20 clear name; clear variable;

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APPENDIX D. Plots

D.1 W versus fiber location, x

Figure D.1: W versus fiber location, x (for V = 0%, d = 3.2 mm, E = 25.4 mm)

Figure D.2: W versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 12.7 mm)

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Figure D.3: W versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 25.4 mm)

Figure D.4: W versus fiber location, x (for V = 0%, d = 3.2 mm, E3 = 12.7 mm)

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Figure D.5: W versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 3.2 mm)

Figure D.6: W versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 3.2 mm)

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Figure D.7: W versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 12.7 mm)

Figure D.8: W versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 25.4 mm)

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Figure D.9: W versus fiber location, x (for V = 0%, d = 12.7 mm, E3 = 12.7 mm)

Figure D.10: W versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 3.2 mm)

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Figure D.11: W versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 3.2 mm)

Figure D.12: W versus fiber location, x (for V = 2%, d = 3.2 mm, E = 25.4 mm)

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Figure D.13: W versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 12.7 mm)

Figure D.14: W versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 25.4 mm)

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Figure D.15: W versus fiber location, x (for V = 2%, d = 3.2 mm, E3 = 12.7 mm)

Figure D.16: W versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 3.2 mm)

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Figure D.17: W versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 3.2 mm)

Figure D.18: W versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 12.7 mm)

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Figure D.19: W versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 25.4 mm)

Figure D.20: W versus fiber location, x (for V = 2%, d = 12.7 mm, E3 = 12.7 mm)

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Figure D.21: W versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 3.2 mm)

Figure D.22: W versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 3.2 mm)

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Figure D.23: W versus fiber location, x (for V = 4%, d = 3.2 mm, E = 25.4 mm)

Figure D.24: W versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 12.7 mm)

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Figure D.25: W versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 25.4 mm)

Figure D.26: W versus fiber location, x (for V = 4%, d = 3.2 mm, E3 = 12.7 mm)

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Figure D.27: W versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 3.2 mm)

Figure D.28: W versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 3.2 mm)

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Figure D.29: W versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 12.7 mm)

Figure D.30: W versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 25.4 mm)

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Figure D.31: W versus fiber location, x (for V = 4%, d = 12.7 mm, E3 = 12.7 mm)

Figure D.32: W versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 3.2 mm)

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250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W (N−m

m)

0 50 100 150 2000

50

100

150

200

250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W (N−m

m)

145

Figure D.33: W versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 3.2 mm)

D.2 Pmax versus fiber location, x

Figure D.34: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E = 25.4 mm)

0 50 100 150 2000

50

100

150

200

250

300

350

400

450

Fiber Location, x (mm)

Pullo

ut W

ork,

W (N−m

m)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

146

Figure D.35: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 12.7 mm)

Figure D.36: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 25.4 mm)

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

147

Figure D.37: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E3 = 12.7 mm)

Figure D.38: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E1 = 3.2 mm)

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

148

Figure D.39: Pmax versus fiber location, x (for V = 0%, d = 3.2 mm, E2 = 3.2 mm)

Figure D.40: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 12.7 mm)

0 10 20 30 40 50 60 700

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 20 40 60 80 100 1200

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

149

Figure D.41: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 25.4 mm)

Figure D.42: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E3 = 12.7 mm)

0 20 40 60 80 100 120 140 1600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

150

Figure D.43: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E1 = 3.2 mm)

Figure D.44: Pmax versus fiber location, x (for V = 0%, d = 12.7 mm, E2 = 3.2 mm)

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

151

Figure D.45: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E = 25.4 mm)

Figure D.46: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 12.7 mm)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

152

Figure D.47: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 25.4 mm)

Figure D.48: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E3 = 12.7 mm)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

153

Figure D.49: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E1 = 3.2 mm)

Figure D.50: Pmax versus fiber location, x (for V = 2%, d = 3.2 mm, E2 = 3.2 mm)

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

154

Figure D.51: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 12.7 mm)

Figure D.52: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 25.4 mm)

0 50 100 150 200 2500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

155

Figure D.53: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E3 = 12.7 mm)

Figure D.54: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E1 = 3.2 mm)

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 50 100 150 200 2500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

156

Figure D.55: Pmax versus fiber location, x (for V = 2%, d = 12.7 mm, E2 = 3.2 mm)

Figure D.56: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E = 25.4 mm)

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

157

Figure D.57: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 12.7 mm)

Figure D.58: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 25.4 mm)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

158

Figure D.59: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E3 = 12.7 mm)

Figure D.60: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E1 = 3.2 mm)

0 10 20 30 40 50 600

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 10 20 30 40 500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

159

Figure D.61: Pmax versus fiber location, x (for V = 4%, d = 3.2 mm, E2 = 3.2 mm)

Figure D.62: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 12.7 mm)

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

160

Figure D.63: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 25.4 mm)

Figure D.64: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E3 = 12.7 mm)

0 50 100 150 200 2500

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

161

Figure D.65: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E1 = 3.2 mm)

Figure D.66: Pmax versus fiber location, x (for V = 4%, d = 12.7 mm, E2 = 3.2 mm)

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

0 50 100 150 2000

20

40

60

80

100

120

Fiber Location, x (mm)

Max

imum

Pul

lout

Loa

d, Pmax

(N)

162

D.3 Final Curve Averaging

Figure D.67: P-ν curve for V = 0%, E = 3.2 mm and d eliminated

Figure D.68: P-ν curve for V = 2%, E ≥ 12.7 mm and d eliminated

163

Figure D.69: P-ν curve for V = 2%, E = 3.2 mm and d eliminated

Figure D.70: P-ν curve for V = 4%, E ≥ 12.7 mm and d eliminated

164

Figure D.71: P-ν curve for V = 4%, E = 3.2 mm and d eliminated