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Properties of steel micro-fiber reinforced cementitious material Yan, Ning 1995

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PROPERTIES OF STEEL MICRO-FIBER REINFORCED CEMENTITIOUS MATERIAL by NING YAN B. Eng., Tianjin University, 1983 M. Eng., Tianjin University, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March 1995  © Ning Yan,  1995  In  presenting  degree freely  this  thesis  in  partial  fulfilment  at t h e U n i v e r s i t y  of  British  Columbia,  available  copying  of  department publication  this or  f o r reference thesis by  o f this  a n d study.  f o r scholarly  his thesis  or  her  purposes  gain  of  Civil  T h e U n i v e r s i t y o f British Vancouver, Canada  Date  DE-6  (2/88)  30  March,  Engineering Columbia  1995  agree  that that  shall  It  is  f o r an  t h e Library  shall  permission  may be granted  representatives.  f o r financial  the requirements  I agree  I further  permission.  Department  of  not be allowed  that without  make  it  f o r extensive  b y t h e head  understood  advanced  of my  copying  or  m y written  Abstract The addition of micro-fibers improves not only the tensile strain capacity but also the tensile strength of cementitious materials. Among the different types of micro-fibers currently in use, steel micro-fibers which are characterized by high strength, high stiffness and low cost, have great potential to be used in many applications. The fracture properties of steel micro-fiber reinforced cementitious are, however, not well understood and are necessary for optimization. Three aspects of steel micro-fiber reinforced cementitous materials are examined in this thesis: (1) characteristics of bond between steel microfiber and cement paste; (2) properties of steel micro-fiber reinforced mortar (SMFRM) under uniaxial tension; and (3) properties of SMFRM in flexure. Single fiber pull-out tests were used to study the characteristics of bond between steel micro-fiber and cement paste. A feed-back controlled system, which could avoid the sudden fracture of specimens at peak load, was installed in the uniaxial tensile tests. Similarly, crack-opening controlled, four-point flexural tests were performed on beams. Data collected from these tests were analyzed by computer programs written in the FORTRAN language. In pull-out tests, the highest peak pull-out load and the highest total energy absorption were obtained when the micro-fiber was aligned in the loading direction. This suggests that the optimum inclined angle of fiber is 0°. The increase of silica fume content in the matrix increased the strength of bond between steel micro-fiber and the cement-based matrix. The results of the uniaxial tensile tests and the flexural tests show, as expected, that the strength and toughness of SMFRM increases w i t h an increase in the fiber volume fraction. SMFRM with fibers having larger diameters had higher toughness. The addition of longer fibers in SMFRM  ii  also yielded higher toughness. But, with longer fibers, the workability of steel micro-fiber mortar mixtures was inadequate. Short thick fibers, on the other hand, were easy to be mixed at high volume fractions (5%  or more)  without workability problems. While the inclusion of a polymer in cement mortar increased the toughness of SMFRM significantly, an increase in the sand content in cement mortar led to a significant increase in the strength. Overall, the addition of steel micro-fibers improved the properties of cementitious materials dramatically.  iii  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  Acknowledgments  xi  1. Introduction  1  2. Bond-Slip Characteristics of Steel Micro-Fibers Bonded to Cement Matrices  4  2.1 Introduction  4  2.2 Literature Review and Research Significance  5  2.3 Experimental Work  10  2.3.1 Materials  10  2.3.2 Specimen Preparation  15  2.3.3 Test Set-Up  21  2.4 Results and Discussion  21  2.4.1 Load-Slip Relations  21  2.4.2 Tensile Strength of the Fibers  23  2.4.3 Critical Length of the Fibers  25  2.4.4 Strength of Bond between Fiber and Matrix  28  2.4.5 Influence of Fiber Orientation  29  2.5 Summary  38  3. Properties of Steel Micro-Fiber Reinforced Mortar in Uniaxial Tension  39  3.1 Introduction  39  3.2 Literature Review and Research Significance  40  3.3 Experimental Work  43  3.3.1 Materials  43  iv  3.3.2 Specimens Preparation  46  3.3.3 Test Plan  47  3.3.4 Test Set-Up  49  3.4 Test Results and Discussions  52  3.4.1 Workability of the Stainless Steel Micro-Fiber Mortar Mixture 3.4.2 Tensile Properties of SMFRM  52 54  3.4.2.1 Effects of Fiber Volume Fraction  54  3.4.2.2 Effects of Fiber Length  56  3.5 Summary  70  4. Properties of Steel Micro-Fiber Reinforced Mortar in Flexure  72  4.1 Introduction  72  4 . 2 Literature Review and Research Significance  73  4.3 Experimental Work  74  4.3.1 Materials  74  4.3.2 Specimens Preparation  76  4.3.3 Test Plan  80  4 . 3 . 4 Test Set-Up  83  4 . 4 Test Results and Discussions  83  4.4.1 Workability of the Steel Micro-Fiber Reinforced Mortar Mixtures  83  4.4.1.1 Effects of Fiber Volume Fractions, Fiber Length and Fiber Cross-Sectional Area 4.4.1.2 Effects of Sand Content and Polymer Addition 4.4.2 Flexural Properties  83 84 85  4.4.2.1 Effects of Fiber Volume Fraction  85  4 . 4 . 2 . 2 Effects of Fiber Length  85  4.4.2.3 Effects of Fiber Cross-Sectional Area  86  4 . 4 . 2 . 4 Effects of Matrix Properties  87  4.5 Summary  106  5. Conclusions  108  Bibliography  113  Appendix 1. Computer Program: PULOUT.FOR  118  Appendix 2. Computer Program: TENSIL.FOR  120  Appendix 3. Computer Program: BENDING.FOR  124  vi  List of Tables  Table 2.1 Mix Proportion of Cement Paste Matrix  10  Table 2.2 Cross-Sections of 4 0 Randomly Chosen Stainless Fibers  14  Table 2.3 Average Cross-Section Areas and Their Corresponding Diameters of the Carbon Steel Fibers  15  Table 2.4 Tensile Strength of Various Fibers  23  Table 2.5 The Behavior of Fibers under a Pull-Out Load  25  Table 2.6 Critical Length of Various Fibers  26  Table 2.7 Bond Strength Values for Various Fibers  28  Table 2.8 Bond-Slip Characteristics at Various Fiber Orientations  30  Table 3.1 Mix Proportions and Compressive Strength of the Mortar Matrix  43  Table 3.2 Properties of Stainless Steel Micro-Fiber Table 3.3 Superplasticizer Dosage  46 53  Table 3.4 Test Results of Feed-Back Controlled Uniaxial Tensile Tests ..65 Table 3.5 Repetitious Test Results of Feed-Back Controlled Uniaxial Tensile Tests  66  Table 4.1 Mix Proportions of the Three Matrices  74  Table 4.2 Cross-Sectional Areas of the Various Carbon Steel Fibers  75  Table 4.3 Critical Length and Tensile Strength of Fibers  75  Table 4 . 4 Test Variables Investigated  80  Table 4.5 Workability of Steel Micro-Fiber Mortar Mixture  84  Table 4.6 Flexural Test Results (a) Fiber Length 2 mm  89  (b) Fiber Length 4 mm  90  (c) Fiber Length 10 mm  91  vii  List of Figures  Figure 2.1 Micro Graph of Stainless Steel Micro-Fiber  12  Figure 2.2 Micro Graph of Carbon Steel Micro-Fiber  13  Figure 2.3 Pull-Out Specimen Molds  16  Figure 2.4 Pull-Out Specimen  17  Figure 2.5 Pull-Out Test Apparatus  19  Figure 2.6 Schematic of Pull-Out Test Frame  20  Figure 2.7 Bond-Slip Curves for Stainless Steel Micro-Fibers  22  Figure 2.8 Bond-Slip Curves for Carbon Steel Micro-Fibers  24  Figure 2.9 Stresses on a Fiber Embedded in Matrix  27  Figure 2.10 Calculation of Maximum Interfacial Bond Strength Developed between Fiber and Matrix  31  Figure 2.11 Influence of Silica Fume Content on the Bond-Slip Curves..32 Figure 2.12 Influence of Fiber Inclination Angle  33  Figure 2.13 Bond-Slip Curves at Various Angles  34  Figure 2.14 Energy vs. Slip Curves for the Various Inclination Angles ..35 Figure 2.15 Matrix after Fiber Pull-Out  36  Figure 2.16 Matrix Scabbing in A Fiber Inclined w.r.t. the Loading Direction  37  Figure 3.1 Typical Tensile Load-Elongation Response of Various Fiber Reinforced Composites  41  Figure 3.2 Grading Curve for Sand  44  Figure 3.3 Fiber Length Distribution  45  Figure 3.4 Specimen for Uniaxial Tensile Tests  48  Figure 3.5 A Feed-Back Controlled Uniaxial Tensile Test in Progress  50  Figure 3.6 Schematic Description of a Feed-Back Controlled Uniaxial Tensile Test  51  viii  Figure 3.7 Tensile Stress-Strain Curves for SMFRM with Different Fiber Volume Fractions (Fiber length = 4 mm)  60  Figure 3.8 Tensile Stress-Strain Curves for SMFRM with Different Fiber Volume Fractions (Fiber length = 6 mm)  61  Figure 3.9 Tensile Stress-Strain Curves for SMFRM with Different Fiber Volume Fractions (Fiber length = 10 mm)  62  Figure 3.10 Tensile Stress-Strain Curves for SMFRM with Different Fiber Length (Fiber volume fraction = 1 % )  63  Figure 3.11 Tensile Stress-Strain Curves for SMFRM with Different Fiber Lengths (Fiber volume fraction = 2%)  64  Figure 3.12 Relationship between Fiber Volume Fraction and Tensile Strength  67  Figure 3.13 Relationship between Fiber Volume Fraction and Toughness Factor  68  Figure 3.14 Schematic of Deformation of Cement Mortar with or without Fiber under a Tensile Load  69  Figure 4.1 Fiber Length Distributions (a) Industry Cut Fibers  77  (b) Laboratory Cut Fibers  78  Figure 4.2 Specimen for Flexural Tests  79  Figure 4.3 Test Set-Up for Flexural Tests  81  Figure 4 . 4 Schematic of Test Set-Up for Flexural Tests  82  Figure 4.5 Load-Deflection Curves of SMFRM with Different Fiber Volume Fractions (1)  92  Figure 4.6 Load-Deflection Curves of SMFRM w i t h Different Fiber Volume Fractions (2)  93  Figure 4.7 Load-Deflection Curves for SMFRM with Different Fiber Lengths (1) Figure 4.8 Load-Deflection Curves for SMFRM  ix  94  w i t h Different Fiber Lengths (2)  95  Figure 4.9 Load-Deflection Curves for SMFRM with Different Fiber Cross-Sectional Areas (1)  96  Figure 4 . 1 0 Load-Deflection Curves for SMFRM with Different Fiber Cross-Sectional Areas (2)  97  Figure 4.11 Load-Deflection Curves for SMFRM w i t h Different Types of Matrices (1)  98  Figure 4 . 1 2 Load-Deflection Curves for SMFRM with Different Types of Matrices (2)  99  Figure 4.13 Influence of Fiber Length on the Strength and Toughness of SMFRM under Flexure  100  Figure 4 . 1 4 Influence of Fiber Cross-Sectional Area on the Strength and Toughness of SMFRM under Flexure  101  Figure 4.15 Influence of Matrix Type on the Strength and Toughness of SMFRM under Flexure  102  Figure 4 . 1 6 Influence of Fiber Volume Fraction on Compressive Strength of SMFRM  103  Figure 4.17 Typical Shape of a Deflection-Crack Opening Curve for a Flexural Tests  104  Figure 4.18 Load Crack Opening Curves for SMFRM with Different Fiber Volume Fractions  105  Acknowledgments  First and foremost, the author wishes to thank Dr. Nemkumar P. Banthia, supervisor of her thesis. She is very grateful to him for his invaluable guidance, encouragement and support which enabled her to complete her program of graduate studies. Thanks are also due to Professor Sidney Mindess and Professor Arnon Bentur for their suggestions and help.  She is grateful to Dr. Cheng Yan for his unstinting help in design and installment of the test systems. The help of Mr. Cesar Chen is also greatly appreciated. Thanks are also due to Mr. Bernie Merkli, Mr. John W o n g , Mr. Dick Postgate and Mr. Ron Dolling for their help in preparing the test machines.  Finally the author wishes to thank her husband, Sunsheng, and her daughter, Canmon, for their patience, understanding and support.  xi  1  1.  Introduction  Cementitious materials are one of the most used construction materials in the world. However, their low fracture strength and low tensile strain capacity limit their more extensive applications in construction. One way to overcome this problem is to add randomly distributed ductile fibers in the cementitious material. It has been found that the addition of fiber can dramatically improve the tensile strain capacity of the parent cementitious matrix [ 1 , 3 ] .  Over the past twenty years, there has been a steady increase in the use of fiber reinforced cementitious composites in a variety of construction projects, such as highway pavements, roofing sheets, wall panels, dams, shotcrete etc.. At the same time, newer types of fiber are being developed and used in cementitious materials. In addition to the asbestos fiber which was used in early days, a wide variety of other fibers, such as those of steel, glass, carbon, polypropylene, nylon and various natural materials are used in cementitious materials.  Fibers conventionally used in cementitious material, which are often called macro-fibers, are usually 0.1-1.0 mm in diameter and 10-60 mm in length. Macro-fibers are usually used in cementitious composites with a  2  maximum dosage of 1.5% by volume in order to keep an acceptable workability. Studies have shown that although the addition of macro-fibers to cementitious materials can control the cracking of the cement-based composites, it can not improve the tensile strength [1]. The role of this kind of fiber is to alter the behavior of cement-based composites after they have cracked, by bridging across the cracks.  In recent years, some very fine fibers with dimensions in the same range as those of cement particles (around 0.025 mm), are being investigated in cementitious materials. These micro-fibers not only improve the post cracking 'ductility' of the cementitious composites but also increase the tensile strength. These fibers are also relatively easy to be mixed in the cementitious mixture. Fiber fraction up to 7% have been investigated[8].  Little effort has so far been made, however, to study the characteristics of bond between steel micro-fiber and cementitious matrix. The effects of fibers geometry and the matrix characteristics on the tensile and flexural properties of SMFRM (steel micro-fiber reinforced mortar) are also not well understood. Need exists to properly characterize and model the strain-hardening and softening that occurs in these composites in uniaxial tension.  The purposes of this study w a s to examine the following three aspects of steel micro-fiber reinforced c o m p o s i t e s :  (1)  T h e characteristics of the interface b e t w e e n steel micro-fiber  cementitious  (2)  and  matrix;  T h e properties of steel micro-fiber reinforced  cementitious  c o m p o s i t e under uniaxial tension; Different f r o m the stress-controlled, conventional uniaxial tensile t e s t s , t h e t e s t t e c h n i q u e u s e d in t h i s s t u d y e m p l o y s a f e e d - b a c k c o n t r o l l e d s y s t e m t o r u n t t h e t e s t in d e f o r m a t i o n c o n t r o l i n o r d e r t o o b s e r v e t h e real p r e - c r a c k i n g a n d p o s t c r a c k i n g b e h a v i o r o f S M F R M .  (3)  T h e flexural properties of steel micro-fiber reinforced! m o r t a r s . S i m i l a r t o t h e u n i a x i a l t e n s i l e t e s t , t h e l o a d i n g r a t e in t h e s e t e s t s  w a s controlled by the rate of c r a c k - o p e n i n g by an L V D T m o u n t e d the tensile side of the beams.  on  4  2.  B  o  n  d  -  S  l  C e m e n t  2.1  i  p C B a s e d  h  a  r  a  c  t  e  r  i  s  M a t r i x  Introduction  The properties of fiber reinforced cementitious materials are determined by those of its components, viz. the matrix, the fiber and the fiber-matrix interface. While the nature and behavior of both fiber and matrix are reasonably well understood, those of the interface are known in less detail. The stress on the matrix is transferred to fiber through the fibermatrix interface and hence the fiber-matrix interface play a very important role in composite materials. A strong bond can impart composites w i t h a high strength and stiffness, whereas a weak bond may improve the ductility, while strengths of such composites usually remain low.  In this section, the bond-slip characteristics between steel micro-fiber and cement paste matrix are studied through single fiber pull-out tests. Fibers both aligned and inclined w.r.t. load direction are investigated. The effects of fiber length, fiber cross-sectional area and silica fume contents in the matrix are also examined.  t  i  c  s o  f  5  2.2  Literature Review and Research Significance  In composites, bonding at an interface is due to adhesion between fiber and matrix. However, the fibers are often coated with a layer of material which forms a bond between the fiber and matrix. Adhesion can be attributed to five main mechanisms, that is, (1) adsorption; (2) interdiffusion; (3) electrostatic attraction; (4) chemical bonding and (5) mechanical adhesion. The five types of adhesion can occur at the interface either in isolation or in combination to produce the bond [2]. For cementitious composites, mechanical adhesion bond play an important role.  Many studies have been conducted in order to understand the behavior of fiber-matrix bond in cementitious composites. These may be further sub-divided as follows:  (1) Structure of the Bond  Studies reveal that a transition zone, which is approximately 0.05 mm thick, exists between the matrix and the fiber, the strength of which is much less than that of the bulk matrix. As a result, the interface behavior depends mostly on this weak zone rather than on the bulk cement matrix. This weak zone can be strengthened by reducing the water cement ratio, by adding silica fume and also by adding an acrylic polymer or a water-reducing agent [3, 9].  6  (2) Effects of Fiber Characteristics on the properties of Bond between Fiber and the Matrix  Fiber geometry has a significant influence on fiber-matrix bond characteristics. For example, hooked, crimped and other deformed fibers have higher resistance to pull-out than smooth undeformed fibers. A much higher peak pull-out load and pull-out energy can be obtained for the hooked crimped and other deformed fibers [13]. An excessively deformed fiber, however,  may break, instead of pulling out, causing significant reduction in  the energy absorption [4, 6].  The load vs. pull-out displacement curve from a deformed fiber may s h o w fluctuation after the peak load , which is caused by the fiber with a non-uniform section moving in the surrounding matrix  [13].  For steel fibers, the chemical composition, metallurgical microstructure and the strength of the fiber have a significant influence on the pull-out behavior. The strength of the fiber is relatively a more dominating factor than its ductility. For achieving high performance characteristics, fibers need to have a high strength and a moderate ductility rather than a high ductility and a moderate strength [10].  7  Experiments also reveal that the average interfacial bond strength and stiffness are independent of fiber spacing when fiber spacing is in the range of 10-25 times the fiber diameter [12].  (3) Effects of Matrix Properties on the Characteristics of Fiber-Matrix Bond  The pull-out resistance of a fiber is strongly affected by the characteristics of the matrix. As the matrix strength increase, the bond between the fiber and the matrix also increases [13]. The inclusion of silica fume can improve the adhesion bond. However, it may also cause matrix brittleness leading to premature matrix splitting before the full fiber potential is reached [4, 6]. The bond strength can also be increased by adding acrylic polymer or water-reducing agent the [9, 12]. The addition of latex and fly ash to matrix can also increase the peak pull-out load to a varying degree [13].  (4) Effects of Loading Condition and Curing Temperature on Fiber-Matrix Bond  An increase in loading rate increase the resistance to pull-out only when deformed fibers are used [4, 6]. Straight fibers appear to be insensitive to the rate of pull-out [4].  Lateral stresses, when present, have a significant influence on fiber debonding and pull-out. With the presence of lateral compression on the  8  fiber, both the interfacial friction at the onset of sliding and the effective interfacial strength are found to increase. A higher lateral compression, however, also results in a more rapid drop of the post-peak pull-out resistance. As a result, the total energy absorption does not improve as drastically as the peak pull-out load [21].  Subzero test temperatures improve the pull-out resistance[4, 6], but low test temperatures also make the fiber-matrix bond more brittle [7]. A lower curing temperature leads to a lower early-age but a higher later-age peak pull-out load than a higher curing temperature [4].  (5) Effects of Fiber Orientation on the Fiber-Matrix Bond A fiber inclined w.r.t. the loading direction carries both normal and flexural stresses. This is different from the aligned fiber which carries only the normal stress. For different fibers, however, the effects of fiber orientation on the bond-slip characteristics are different. While some studies show that the peak pull-out loads for undeformed steel fibers inclined to loading direction are almost as high as those for the aligned fibers, other studies conclude that for the deformed steel fiber that both the peak pull out load and energy absorption are maximized when the fiber is alighted w.r.t. loading direction and an inclined fiber has inferior resistance to pull-out [5]. Furthermore, the studies on the pull-out of inclined synthetic fibers show that both pull-out load and energy absorption increase with an increase in  the angle between fiber and the loading [23]. For glass fibers, it was found that the fibers completely fracture at ultimate pullout load when inclined w.r.t. the loading direction [24].  To date, studies on the behavior of fiber-matrix interface have almost always involved macro-fibers with little or no attention given to micro-fibers. Significant effort is needed to extend this to micro-fiber reinforced composites.  This section examines the bond-slip characteristics of steel microfibers. Four variables including: (1) fiber cross-section; (2) fiber embedded length; (3) fiber orientation and (4) silica fume content in the matrix, are considered. As part of the larger research project, the pullout test is also used to determine the tensile strength and the critical length of the fibers. These are needed in order to interpret the data from uniaxial tensile tests and flexural tests performed on the composites.  10  2.3  Experimental Work  2.3.1  Materials  Matrix  The mix proportions of the cement paste matrices used for the pullout tests are shown in Table 2 . 1 . Water cement ratio of 0.35 and silica fume cement ratio of 0.1 and 0.2 were used.  Table 2.1 Mix Proportions for the Cement Paste Matrices Matrix Type  W/C  S.F.*/C  S.P.** (ml/100g cement)  1  0.35  0.10  3  2  0.35  0.20  4  note: *  S . F . : silica f u m e  * * S.P.: superplasticizer  CSA Type 10 cement and DaracemlOO superplasticizer were used. The superplasticizer meets the requirements for an ASTM C-494 Type G water reducer, high range retarding admixture.  Steel Micro-Fibers  T w o types of steel micro-fiber were used. They were stainless steel fiber and carbon steel fiber.  11  (1) The Stainless Steel Micro-Fiber:  The fiber cross-sections were irregular but close to rectangular. Figure 2.1 shows a typical fiber cross-section.  The cross-section areas of the fibers varied from one to another. Forty sample fibers were picked up randomly and their cross-sectional dimensions were measured by using a caliper (precision = 0 . 0 1 mm). The results are shown in Table 2.2. As seen, the maximum observed crosssection was 0 . 1 0 m m X 0 . 2 0 m m and the minimum observed was 0 . 0 2 m m X 0 . 0 6 m m or 0.03mmX0.04mm. The average cross-section of the fibers measured was 0.04mmX0.10mm.  (2) Carbon Steel Micro-Fiber:  The fiber cross-sections were irregular. Figure 2.2 shows t w o typical fiber cross-sections. Based on their sectional areas, these fibers could be classified into four types: T (thick), L (large), M (medium) and S (small). For each fiber type, the cross-sectional areas of four or more fibers were measured from the micrographs and the average cross-sectional area of the fibers and their corresponding equivalent diameters were obtained. Table 2.3 shows these cross-sectional areas and also the equivalent diameters.  12  Figure 2.1 Micrograph Showing the Cross-Section of Stainless Steel Micro-Fiber  ure 2.2 M i c r o g r a p h s h o w i n g the C r o s s - S e c t i o n of C a r b o n Steel Micro-Fiber  14  Table 2.2 Sample No.  Cross-Sectional Dimensions of 4 0 Sample Fibers Cross-section  Sample No.  (mmXmm)  Cross-section (mmXmm)  1  0.04X0.15  21  0.02X0.06  2  0.08X0.06  22  0.05X0.10  3  0.05X0.10  23  0.03X0.04  4  0.10X0.20  24  0.02X0.06  5  0.06X0.10  25  0.06X0.08  6  0.03X0.05  26  0.05X0.06  7  0.03X0.10  27  0.06X0.08  8  0.04X0.06  28  0.06X0.06  9  0.04X0.10  29  0.05X0.08  10  0.06X0.12  30  0.04X0.10  11  0.04X0.10  31  0.02X0.06  12  0.02X0.10  32  0.04X0.10  13  0.03X0.10  33  0.02X0.12  14  0.04X0.08  34  0.02X0.10  15  0.06X0.13  35  0.03X0.07  16  0.06X0.13  36  0.04X0.10  17  0.04X0.18  37  0.06X0.12  18  0.06X0.10  38  0.04X0.10  19  0.02X0.06  39  0.02X0.10  20  0.08X0.14  40  0.03X0.10  15  Table 2.3  Average Cross-Sectional Areas and The Corresponding Equivalent Diameters of the Carbon Steel Fibers  Fiber Type  Cross-Sectional Area  Corresponding Equivalent Diameter  2.3.2  (mmXmm)  (mm)  f  0.0050  0.08  L  0.0020  0.05  M  0.0013  0.04  S  0.0007  0.03  Spec/men  Preparation  A specially designed mold (Figure 2.3) was used to make the specimens with the dimension of 2 0 m m X 4 0 m m X 1 0 m m . Each specimen was divided in the center by a plastic separator and a single fiber was passed through a small hole in it (Figure 2.4). The separator was carefully oiled before inserting the fiber in order to minimize the bond between the matrix and the separator.  T w o casting methods were used to prepare the specimens.  Method One:  The fiber was placed in the mold first. One side (side 1) of the fiber was left longer than the other side (side 2), i.e. the fiber was not  16  18  placed symmetrically w.r.t. the plastic separator. Fresh cement paste was poured in side 1 of the mold and after setting, the fiber on side 2 was cut to the appropriate length. Cement paste was then poured on this side of mold.  Most specimens were cast this way except for those used to investigate the influence of fiber orientation.  Method T w o :  The fiber was cut to the appropriate length and the cement paste was poured on one side (side 1) of the mold up to the hole in the plastic separator. The fiber was then placed in the mold with the same length on both side of separator and then the cement paste was poured on one side. In order to prevent the fiber from moving, the second side was not cast until setting on the first side. All specimens used to investigate the influence of fiber orientation were cast this way.  All specimens were demolded on the second day of casting and further cured in water for six more days before testing.  19  Figure 2.5 Puil-Out Test Apparatus  2.3.3  The Test Set-Up  The test set-up is shown in Figures 2.5 and 2.6. In principle, the pullout specimen bridged t w o grips as shown. One side of the specimen grips was fixed to the machine frame and the other, with rollers underneath, was connected to a small motor which drove this grip at a constant speed away from the first grip. A 100N load cell was connected between the roller mounted grip and the driving motor to measure the applied load and an LVDT was connected to measure the displacement. All specimens were tested at a pull-out loading rate of 0.024 mm/s.  An X-Y plotter was used to record the load-slip curves. A digitizer was then used to digitize these curves. The data were finally analyzed using a FORTRAN computer program (Appendix 1).  2.4  2.4.1  Results and Discussion  Load-Slip Relations  Figure 2.7 shows the load-slip curves for stainless steel micro-fibers w i t h cross-sections of 0.08mmX0.22mm (close to the maximum crosssectional area) and 0.04mmX0.10mm (close to the average cross-sectional area). Notice that the fluctuations in the load values in the larger fiber are  22  more pronounced than those in the curves for the smaller fiber. This is apparently a result of the very uneven surface of the larger fiber.  Figure 2.8 shows the bond-slip curves for carbon steel fibers. Only carbon steel fiber ' L ' (Table 2.3) was tested.  2.4.2  Tensile Strength of the Fibers The tensile strength of the fiber can be obtained from a pull-out test.  When the embedded length of a fiber exceeds the critical length, the fiber will fracture instead of pulling out. The tensile strength of fiber is then simply the load at fracture divided by the cross-sectional area of the fiber. Table 2 . 4 shows the tensile strengths of the various fibers.  Table 2.4 Fiber Type  Tensile Strength of Various Fibers* Cross-sectional Dimension  Area  Tensile Strength  (mmXmm)  (mm)  (MPa)  0.08X0.22  0.0176  850  Stainless steel  0.08X0.16  0.0128  870  fiber  0.06X0.12  0.0072  750  0.04X0.10  0.0040  1200  0.06X0.06  0.0036  1140  L  0.0020  1000  Carbon steel fiber  (see Table 2.3)  note: * T h e v a l u e s o f t e n s i l e s t r e n g t h in t h i s t a b l e are o n l y a p p r o x i m a t e b e c a u s e it is n o t p o s s i b l e t o p r e c i s e l y m e a s u r e t h e c r o s s - s e c t i o n a l area o f t h e f i b e r .  24  25  Notice that the tensile strengths vary between 7 5 0 MPa to 1200 MPa and the fibers with larger cross-sectional areas have smaller tensile strength. This is well expected.  2.4.3  Critical Length of the Fibers  When the embedded fiber length is longer than its critical length, the stress in the fiber reaches the ultimate strength of steel and the fiber fractures. On the other hand, when the embedded length of the fiber is shorter than the critical length, fiber pull out is expected to occur. Table 2.5 shows the behavior of stainless steel fibers and carbon fiber ' L ' , w i t h different embedded lengths, under a pull-out load.  Table 2.5 Fiber Type  Behavior of Fibers under a Pull-out Load  Cross-Section (mmXmm)  Embedded Length 1mm  2mm 3 m m 4 m m 5 m m 6 m m  Stainless  0.08X0.22  P *  P  P  steel fiber  0.04X0.10  P  F*  F  Carbon steel fiber  L (see Table 2.3)  note: * P = Fiber pull-out F = Fiber fractures  P  P or F P  PorF  F  26  The critical lengths for the stainless steel fiber and carbon fiber ' L ' obtained directly from these tests are given in Table 2.6. The critical length for the other types of carbon steel fibers, however, can be calculated by assuming that they have the same tensile strength a and bond strength x as for the fiber ' L ' . The relationship between a and T can be seen in Figure 2.9.  Assuming I, and d, are critical length and diameter of carbon steel fiber ' L ' respectively and l and d are the same for other fibers (say fiber 2  2  'T') then,  a I x = 21,/d, = 2l /d 2  2  The critical lengths obtained using this procedure also given in Table 2.6.  Table 2.6 Fiber Type  Critical Length of the Fibers Cross-Section  Critical Length  (mmXmm)  (mm)  Stainless  0.08X0.22  10  steel fiber  0.04X0.10  6  T Carbon steel fiber  L  6 (analytically) 4  M  3(analytically)  S  2(analytically)  27  X t w—m  03 ro  03  E <D  JO +->  X3 CD  c. .c •«->  CD  c  E ro  UJ  sz  II  CD LL 03 C o CO CD  b  CO CD u. «*-» CO  TJ 05 TJ TJ CD  £  CN  TJ CN  D)  CN  03  II  c  JC ±3  0)  O,  b  CD  ro  o  C/3  o  c\i CD • ••••  Li-  28  Average  2.4.4  and  Maximum  Shear  Strength  of the  Bond  The average shear strength of bond is the peak pullout load divided by the surface area of the embedded fiber. The relationship between maximum shear strength x . and average shear strength x . is established ave  mm  through shear lag theory [16]:  t .x. m  /tave.  = al coth al  where, I is embedded fiber length, and a is a constant.  as I -> 0, T '•max,  in other words,  Table 2.7 Matrix  (20% S.F.)  ^max.  ^ave./  1  -> 0  Fiber-Matrix Bond Strength Values for Various Fibers  Cross-Section Embedded Length (mmXmm)  M1  =1  Ix  0.08X0.22 L  0.04X0.10  0.06X0.12 M2  (mm)  Average Bond  Maximum Bond  Strength (MPa)  Strength (MPa)  3.0  4.53  5.0  3.23  2.0  4.60  3.0  3.33  2.0  4.38  3.0  3.73  2.0  3.98  3.0  3.60  6.18  7.10  5.66  (10% S.F.)  0.06X0.12  4.73  29  The concept is illustrated in Figure 2.10 and in Table 2.7 bond strength values are given. There is no apparent effects of fibers crosssectional area on the average or the maximum bond strength. As expected, the strength of the bond increases with an increase in silica fume content in matrix. This effect can also be seen in Figure 2 . 1 1 .  2.4.5  Influence of Fiber Orientation Specimens with aligned as well as inclined fibers were tested and the  test results are shown in Table 2.8 and Figures 2.12-2.14 where the noted inclination angle is the angle between fiber and the loading direction. As seen in Table 2.8, the highest peak load and energy absorption were obtained when the fibers were aligned w.r.t. the loading direction. The slip of the aligned fiber at the occurrence of the peak load, however, is smaller than that of inclined fibers. Additional observations are as follows:  (1) It is often that the load drops suddenly at a certain pullout displacement (Figure 2.13). This drop of load is caused possibly by the matrix breaking out at the point where the inclined fiber enters the matrix. Figure 2.15 shows the breakage of matrix for an aligned fiber (a) and an inclined fiber (b) and Figure 2.16 shows the schematic of this process.  30  (2) A greater proportion of aligned fibers fractured as compared to inclined fibers even though they had the same cross-sectional areas and embedded lengths.  Table 2.8  Bond-Slip Characteristics at Various Fiber Orientation  Fiber  Peak  Total  Slip at  Energy at  Angle  Load  Energy  Peak Load  Peak Load  (degree)  (N)  (N-mm)  (mm)  (N-mm)  0  3.91  5.92  0.08  0.35  15  2.54  3.25  0.47  0.81  30  2.75  4.06  0.25  0.44  45  2.53  3.92  0.45  0.87  60  3.32  3.84  0.28  0.53  75  3.03  3.45  0.82  1.07  31  V  33 o  o  34  35  36  37  c  O mmmm  O  0)  C  • mwmi  "O  o a>  .E a)  2 O  •o  30  O § 3 O  o c  •o r  v.  C  0 n  Lire c U)  > » CD CB CD  o .c  o cu  si  CD - t r =  Is  lb  E n  ca o  (/)  X  • mmm mm  CB  o3 -Q CQ L_  CO OJ  0 k. 3  D)  I  8  a. cn 0  55  |  38  2.5  Summary  Based on the pull out tests described in this section, the following conclusions may be drawn:  (1) The tensile strength of the stainless steel micro-fibers is in the range of 7 5 0 - 1 2 0 0 MPa and their critical length is in the range of 6-10 mm. The bond strength between the stainless steel micro-fiber and the cement paste is between 4.7 MPa and 7.1 MPa.  (2) The tensile strength of the carbon steel micro-fibers used in this study is around 1000 MPa and their critical lengths are in the range of 2-6 mm. These fibers develop an average bond strength of 3.92 MPa w i t h a cementitious matrix..  (3) An increase in the silica fume content in the matrix improves the bond.  (4) The orientation of fiber had a strong influence on the bond. A fiber aligned to the load direction supported a higher peak pullout load and absorbed greater pull out energy than the inclined fibers. The slip of an aligned fiber at the occurrence of peak load is also smaller than that of an inclined fiber.  39  3. Properties of Steel Micro-Fiber Reinforced Mortar in Uniaxial Tension  3.1  Introduction  Previous studies indicate that the addition of steel micro-fibers in cementitious materials improves the tensile strength [25]. The post-cracking behavior of steel micro-fiber reinforced cementitious composite under tension, however, is still not well understood. The test results from stresscontrolled direct tensile test [25] are not reliable because of the 'snap-back' of the specimens after cracking.  Tensile testing of cementitious composites is a difficult task. Because of the gripping problems, for brittle cementitious materials, one uses indirect tensile test methods, such as the splitting test, trussed beam test, pressurized ring test and the flexural test [27]. Although these tests, especially the splitting test, are widely used for concrete, they have their o w n limitations because the tensile properties are obtained indirectly. In any case, the behavior of fiber reinforced cementitious materials under direct tension is desired for a thorough understanding of their softening characteristics. For such composites, given their more ductile behavior, it is possible to test them under direct tension [1,25, 30, 31].  40  This section describes the uniaxial tensile behavior of steel micro-fiber reinforced composites with different volume fraction and different lengths of fibers. In order to observe the real post-cracking behavior of SMFRM, a feed-back controlled system (closed-loop setup) was used in the tests.  3.2  Literature Review and Research Significance  The addition of fibers in cementitious materials improves their tensile strain capacity and ductility [ 1 , 3]. When a high (more than 2%) volume fraction of fibers is added, the behavior of the cementitious materials can be fundamentally altered [3, 3 1 , 32]. As a result, cementitious composites with high volume fraction of fibers are often called high performance cementitious composites. Figure 3.1 shows the typical tensile stress-strain curves for high performance and conventional fiber reinforced cementitious composites. While the conventional cementitious composites have only a linear stage followed by the strain softening stage [3, 3 1 ] , the tensile stressstain curves for high performance fiber reinforced cementitious composites have three stages: (1) a linear stage; (2) multiple cracking stage or the pseudo strain hardening stage and (3) localization or the strain softening stage.  41  1C3D  CD W  C  o a  C3 C  CO CD  o  DC  •  MM  •*—>  c o  c  CD >  c o O  Mwmm  U)  LU • "O CTS O  i  w  CD CD * -  -  o w S ° w E p 0)  O  CD  O  h- t: O CD -o  8  ra  J8  5; o> o H I c I  MM  r i cr  CO CD MM  3  i i SSSJJS  Jo peon  42  With the addition of different types of micro-fibers (e.g. steel, carbon, polypropylene and other fibers), the tensile behavior of the cementitious composites are different. It has been reported [25] that steel micro-fiber provides the most stiffening and the highest strength followed by carbon fiber and then polypropylene fiber. The composite reinforced w i t h carbon micro-fibers, on the other hand, had the largest tensile strain capacity. The tensile stress-strain curves of carbon micro-fiber reinforced composites had significant non-linearity after the peak load, while the curves for steel microfiber reinforced composites remain almost linearly elastic to fracture with only a nominal pre-peak non-linearity [25]. The lack of ductility in steel micro-fiber reinforced composites in Ref. 25, in addition to being related to fiber geometry and surface characteristics, is also the consequence of the test being performed in the stress-controlled mode, where the machine energy is suddenly released causing sudden fracture of the specimen. To avoid this problem, it has been proposed to use special specimens and test set-ups [35, 36, 37].  The best way to solve the problem of instability is to use a feed-back controlled system in which an LVDT with a short gage length is used to control the specimen crack opening to eliminate the effects of specimen 'snap-back' at the peak load. In this investigation, a feed-back controlled system was designed and installed to test the real tensile behavior of steel micro-fiber reinforced composites.  43 Experimental Work  3.3  3.3.1  Materials  Matrix  The mix proportion of the cement mortar and its compressive strength are shown in Table 3 . 1 .  CSA type 10 cement and Daracem 100 superplasticizer were used. The sand used was very fine and clean playsand. Figure 3.2 shows the gradation curve for the sand.  Table 3.1  Mix Proportion and Compressive Strength of the Cement Mortar  W/C  S/C  S.F./C  0.35  0.5  0.2  Superplasticizer 1.4 - 5 ml/100g C.  Compressive Strength * 56 MPa  Note: * C o m p r e s s i v e s t r e n g t h o b t a i n e d u s i n g c y l i n d r i c a l s p e c i m e n s ( 4 2 . 5 m m in d i a m e t e r a n d 8 5 m m in h e i g h t ) . T h e v a l u e l i s t e d is t h e a v e r a g e value f o r t h r e e s p e c i m e n s .  Stainless Steel Micro-Fiber  The properties of the stainless steel micro-fiber, which were obtained from pull-out tests as described in Section 2, are shown in Table 3.2.  44  46  Table 3.2  Properties of Stainless Steel Micro-Fiber  Average Cross-Section  Tensile Strength  (mmXmm)  (MPa)  0.04X0.1  700 - 1200  Average Critical Length (mm)  around 6  The fibers were cut to lengths of 4, 6, and 10 mm. It was found difficult to cut the fibers to an uniform length. Figure 3.3 shows the fiber length distributions. As seen, for the 4 mm fiber, most fibers (75.2%) had the length between 4 and 6 mm. Similarly, most 6 mm (68.8%) and 10 mm fiber (71.3%) had lengths between 6 and 8 mm, and between 8 and 10 mm respectively.  Several fiber volume fractions were investigated. For the 4 mm fiber, fiber volume fraction of 0.5%, 1.0%, 1.5%, 2.0%, 3 . 0 % and 5 . 0 % were used. For the 6 mm and 10 mm fibers, fiber volume fraction of 0 . 5 % ,  1.0%,  1.5% and 2 . 0 % were used.  3.3.2  Specimen Preparation  The mixing procedure for steel micro-fiber reinforced mortar was similar to that of normal cement mortar. First, cement, sand and silica fume were mixed in the mixer. Then water with half the required quantity of  47  superplasticizer were added. Finally, fibers were scattered slowly and evenly into the cement mortar with the mixer running. The mixing was continued for another t w o minutes after all fibers were added in. The mixture was then poured in the molds.  After being cured at room temperature for 24 hours, the specimens were carefully demolded and cured for another 27 days before tested. The shape and dimension of the specimen are shown in Figure 3.4.  3.3.3  Test Plan  Fifteen sets of specimens were tested. For the 4 mm fiber, specimens w i t h fiber volume fractions of 0.5%, 1.0%, 1.5%, 2 . 0 % , 3 . 0 % and 5 . 0 % were tested. For the 6 mm and 10 mm fibers, specimens with fiber volume fractions of 0 . 5 % , 1.0%, 1.5% and 2 . 0 % were tested. Plain cement mortar specimens were also tested as the control.  To verify the reproducibility of the test results, three additional sets of specimens were cast and tested. These repetitious test used specimens w i t h 1.0% of fibers which had length of 4 mm, 6 mm and 10 mm.  In each category, at least six specimens were tested.  48  E E  w w  0) 0)  "co c  (1)  75 •  X  (0 • 1MB  E E  E E  LO  O)  ll  c  c 0)  E  O  o a  (/) CO  a) 3  D) LL  •EE LO CM  49  3.3.4  Test Set-Up The test set-up is shown in Figures 3.5 and 3.6. As mentioned  before, the testing machine should be stiff enough to avoid unstable and sudden unloading after the specimen cracks. Although the frame of the test machine used was much stiffer compared to the specimens, the stiffness of the grips and the joints between the machine and the grips were of particular concern. High stiffness steel grips which were much bigger in size than the specimens themselves were used. Rubber or any other flexible materials were avoided in the joints and grips.  To prevent bending stress in the specimens, ball-and socket joints were used between machine frame and the grips.  The feed-back controlled system was the key part of the test set-up. The tensile strain of the specimen with a gage length of 65 mm were measured by t w o symmetrically placed LVDTs, each with a sensitivity of 7 5 . 5 m v / m m , as shown in Figure 3.6. The strain signal from the LVDTs was fed back to the machine which controlled the movement of the machine cross- head so as to keep the strain rate of the specimens at a prescribed value. In this study, t w o different strain-rates were used. The specimen was first loaded at a strain rate of 2 microstrain per second up to the peak load followed by a higher strain rate of 20 microstrain per second in the descending branch.  Figure 3.5 A Feed-Back Controlled Uniaxial Tensile Test in Progress  51  fixed to the frame \//////////////////////\  load cell ball-and-socket joint grips signal to the machine system  LVDT holders  65 mm  loading direction V//////////X  Figure 3.6 Schematic Description of a Feed Back Controlled Uni-axial Tensile Test  52  The system was found to be very sensitive even to minor movements. A small disturbance to the LVDT, for example, a gentle touch to the LVDT during the installation of the specimen, could cause a big movement in the cross-head and disrupt the test suddenly. To overcome this problem, a pneumatic grip system was used. When installing a specimen, the upper part of the specimen was fixed t o the top grip first. After properly adjusting the LVDT, the lower part of the specimen was griped quickly by using high pressure air from the pneumatic grip system.  The applied tensile load was measured by a load-cell with a capacity of 1600 N. The test results, in the form of applied load vs. LVDT displacement, were recorded by both an X-Y recorder and a computer based data acquisition system. Finally, a FORTRAN computer program (Appendix 2) was written to analyze the data.  3.4 Test Results and Discussion  3.4.1  Workability of the Steel Micro-Fiber Reinforced Mortar Mixtures  The workability of the mixtures was affected significantly by the fiber volume fraction and fiber length.  The workability of the mixture became worse with an increase in the fiber volume fraction. To keep a good workability, the quantity of the  53  superplasticizer had to be increased with an increase of fiber volume. Table 3.3 shows the quantities of the superplasticizer used. The mixture which contained 5% of 4 mm fiber in spite of a high dosage of superplasticizer was highly unworkable.  Table 3.3  Superplasticizer Dosages  Fiber volume Fraction (%)  Superplasticizer (ml /100g C.)  0  1.2  0.5  1.6  1.0  2.0  1.5  2.4  2.0  2.8  3.0  3.2  5.0  4.5  Mixtures with longer fibers witnessed more fiber balling than those with shorter fibers and an increase in the dosage of the superplasticizer had only a limited effect. In particular, the mixtures with 10 mm fibers were highly unworkable.  The dosage of superplasticizer could not exceed 5 ml/100g of cement. The addition of a large quantity of superplasticizer caused the mixture to set too quickly allowing little time for finishing.  54  3.4.2  Tensile Properties of SMFRM The test results are shown in Table 3.4 and Figures 3.7- 3.13. The  results from the repetitious tests performed to verify the reproducibility of the data are shown in Table 3.5.  3.4.2.1  Effects of Fiber Volume Fraction Figures 3.7, 3.8 and 3.9 show the stress-strain curves for  composites with different fiber volume fractions and lengths of 4, 6 and 10 mm, respectively. Figures 3.12 and 3.13 show the relationships between fiber volume fraction and the tensile strength factor and between fiber volume fraction and the toughness factor. These factors are defined as follows: Tensile strength factor = Tensile strength of SMFRM / Tensile strength of plain cement mortar Toughness factor* = Toughness of SMFRM / Toughness of plain cement mortar  The figures illustrate clearly that no matter what the fiber length, the tensile strength and toughness of SMFRM increases with an increase in the fiber volume fraction. With an addition of 2 % fiber, the tensile strength of the SMFRM can be nearly doubled and the toughness can be improved by a factor of ten. Note also that the elastic modules, the strain at peak-load and the ultimate strain increase with an increase in the fiber volume fraction.  T o u g h n e s s w a s m e a s u r e d b y t h e t o t a l area u n d e r a s t r e s s - s t r a i n c u r v e .  55  Distinct from the steel macro-fibers, the addition of steel micro-fiber also increases the elastic strength of the cement mortar significantly. Compared to the macro-fiber, for a given fiber volume fraction,  micro-fibers  w i t h their fine size will be much higher in number. Let us consider a small part of the matrix which contains a single fiber (Figure 3.14). When the composite is loaded in tension, the load acts on the matrix and is then transferred through the fiber-matrix interface to the fiber. To maintain strain compatibility, the deformation of the matrix closer to the fiber is constrained. If the number of fibers are large enough and the fibers distributed evenly, the fibers will be very close to each other and the deformation of the whole matrix will be constrained. At a certain stress level, therefore, the corresponding strain in the matrix will be smaller as compared to a plain matrix. In other words, the stress in the composite will be higher when the failure strain of the matrix is reached. Higher elastic strengths can be, therefore, obtained when micro-fibers are used and the higher the number of fibers, the higher the expected elastic strength of the matrix.  Similar to the carbon micro-fiber reinforced mortar, steel micro-fiber reinforced mortar depicted significant strain hardening under tension, especially at high fiber volume fractions. The mechanisms of strain hardening of micro-fiber reinforced cementitious composite have been studied by several researchers [8,31]. Like the other cementitious materials,  56  under tension, SMFRM will develop the first crack at a flaw. The presence of fibers at the crack will provide resistance against crack opening. If the number of fibers is sufficiently large, and the fiber and the fiber-matrix interfaces are strong enough, the specimen can absorb more energy and the load can be further increased. With an increase in the load, there will be enough energy to form an additional crack somewhere else in the material. In other words, the energy required to open the first crack is larger than that required to form a new crack. With a further increase in load, more cracks appear and more energy is absorbed. The multiple cracking process is continued until the energy needed to form new cracks is larger than the energy to open the first crack. At this point, process of multiple cracking is terminated and the softening stage is reached. The first crack will continue to open until complete fracture occurs.  3.4.2.2  Effects of Fiber Length  Fiber length affects the tensile strength and toughness of SMFRM. Figure 3.10 and 3.11 show the stress-strain curves for fiber lengths of 4, 6 and 10 mm for fiber volume fractions of 1 % and 2 % , respectively. At a low volume fraction of 1 %, specimens with 6 mm fibers gave both the highest strength and the highest toughness, followed by those w i t h 4 mm fibers and 10 mm fibers. At a higher fiber volume fraction of 2 % , the  57  specimens with 6 mm fibers still yielded the highest tensile strength but also the lowest toughness as compared to those with 4 and 10 mm fibers. It was expected that the specimens with shorter fibers would yield higher toughness but also lower strengths and those w i t h longer fibers would yield lower toughness and higher strengths. When the fiber length is less than the critical length, more fibers should be pulled out. For a fiber that pulls out, the peak load should be lower and the energy absorption in the process of pull-out should be higher than a fiber that fractures. The test results suggest that the interaction between fiber and matrix in SMFRM is not as simple as expected. The following factors need to be considered: (1) Strength of the Fiber-Matrix Bond and Strength of the Matrix As mentioned before, the workability of SMFRM mixes was affected significantly by the fiber length. Mixtures with 10 mm fibers were hard to be compacted and significant balling occurred during mixing. When the same volume fraction of shorter fibers were used, the mixture were significantly more workable and a higher bond strength between fiber and the matrix and a higher matrix strength are expected. This explains w h y 10 mm fibers gave lower strength than the 6 mm fibers. (2)  Fiber Efficiency When. SMFRM specimens are loaded in tension, the load is transferred  from the matrix to the fiber through the fiber-matrix interface. If the fiber  58  length is less than critical, the stress which is transferred to a fiber can never reach the fiber strength. It means that the fibers are not used efficiently. On the contrary, if the fibers are long enough, higher stresses can be supported by the fibers. From this point of view, the specimens with shorter fibers will have lower strengths. This explains the lower strength given by 4 mm fibers as compared to 6 mm fibers. (3) Drop of Load after the Peak Load As the load starts to drop after the peak load, the thicker and shorter fibers slip and the thinner and longer fibers fracture. The stainless steel fibers used in these tests were not uniform in length as well as in crosssection (Figure 3.3 and Table 2.2). For all specimens, both fiber pull-out and fiber fracture with varying proportions will occur at the cracks. Specimens w i t h shorter fibers (4 mm) should have more fibers pulling out and less fibers fractured. Specimens with 10 mm fibers, on the other hand, should have more fibers fracture and less fibers pulling out. Specimens with 6 mm fibers may be expected to have equal proportions of fibers pulling out and fracturing. At the peak load, fibers at the cracks are severely stressed and hence some weaker fibers are fractured. After the fracture of these fibers, the load is redistributed to other fibers, which in turn causes others fiber to fracture.  59  This process can be very fast and the load may drop very quickly if the peak load is very high and if the fibers are not strong enough.  The above analysis suggests that the increase or decrease in fiber length could have opposing effects on the tensile strength and toughness of SMFRM. On the whole, specimens with 6 mm fibers yield the highest strength, while those with 10 mm fibers yield lowest strength. However, a high strength may have a faster load drop and then results in a lower toughness. Accordingly, when 2 % fibers were added, the specimens with 6 mm fibers gave the highest strength but also the lowest toughness.  60  61  CO  d  E E co LO  ci c LL  cu  co d  £  2  3  LL  O „ .£ E °  co  * -— <  (Q  3  CO  CM  d  o  JO  to  \t  CD  = o> H 00 CO CD 3  cd  CL CO CO CD CO  oo  CO  CM  c  Q  62  63  64  O  ro c  JQJ w  CU  .o  CO  i  1 J,  CO C  cu  i  E E o i CO  jz  lengt  Ti-  E E r leng  E E co  CM  CD  d  CD .Q  C  g CO v. O  o E =  •1  LO  d  S I 3  O  co  J>  i-  d  c  'cc _< -»> — CO  CO  d  0 CM  I-  d  co O)  LL co CL CO CO 0  CO CO  CD  CM  Q  65  tc  to to  gj E T C O  D  CD CD CO ||CN CO  CO  icrir  °zxi to CD  — I  _CD to c 0  CO X  95 ZD  TJ CD  CD ro  c ro. ± i -h ;  HO  |CN  d  CO CN  in  S| CD  TJ  ro o  c 'ro:  o  co  2  to  o ro  to T J  3  L U  co ^o  00  ro  O  TJ Cl) CD  CN  C O !  C o  co•  If) CO  'CO  E  3  00 CN CO  '|d|<  ro QL  S  If) • CN CN  00  |CN CD CN  CN  LL  CD .C hu— O  tn 3 CO CD  CD - C £  H  co iri  CD |CN [co  <D •  co  3 w  CU to CD  co  ro cn ro  .c  o  B  CD  I L U C O  ro I—  •5*  O >  —  if) i  d  LO i  d  LL  (CD O) n C fc  r <"> E  CD  LO  d  I  o CN  CO (O  2> E b O)  7=  o•  CN CD  CJ  CO  EM;  ci  b w  "O CO CM CN  O CN O i CD '  CD £0 CD - b • Q.CO!  o f  CO TD I CU O i  o  CN  LTJ^  CD -C CO ro E  CO CO ICN  c  CDQ-  x: C CLCN ' Jg CD^CD CO  UU CO  CO CN  CD U  I C  CD:  67  69  70  3.5  Summary  Using the feed-back controlled system, the post-cracking behavior of SMFRM under uniaxial tension is observed and studied. Pseudo-strain hardening occurs when 1 % or more stainless steel fibers are added.  With an increase in the fiber volume fraction, the tensile strength, toughness, elastic modules and the strain capacity of SMFRM increase, while the workability of the fresh steel micro-fiber mortar decreases.  Fiber length affects the tensile strength and toughness of SMFRM. For a given fiber volume fraction, specimens with 6 mm (close to the average critical length) fibers gave the highest strength, followed by those w i t h 4 mm fibers and 10 mm fibers. When fiber volume fraction is small (e.g. 1 % ) , specimens with 6 mm fibers also gave the highest toughness, followed by those with 4 mm fibers and 10 mm fibers. However, at a higher fiber volume fraction (e.g. 2%), specimens with 6 mm fibers gave the lowest toughness, and those with 10 mm fibers gave the highest toughness.  Although at a high volume fraction (2%), specimens with 10 mm fibers appear to be more ductile, in the fresh state, the long fibers tend to ball and the workability is poor. The best fiber length from this study, therefore, is 4 mm long. Steel micro-fiber mortars with 2 % or 3 % of 4 mm stainless steel fibers gave a relatively high tensile strength and toughness  71  and acceptable workability. At addition rate of 5%,  although the tensile  strength are higher, the increase is not proportional to the fiber volume fraction. A 2% fiber composite with 4 mm fiber appears to be the optimum composite in the end.  72  4. Properties of Steel Micro-Fiber Reinforced Mortar in Flexure  4.1 Introduction  The feed-back controlled uniaxial tensile test used in the previous section yielded valuable fundamental data for SMFRM. However, these tests are too complicated and time consuming to be used in the projects involving large number of specimens. Flexural tests, on the other hand, are easily conducted.  In this section, data from about 2 5 0 specimens with different fiber cross-sectional areas, fiber lengths, fiber volume fractions and sand contents, and polymer contents tested under flexure are described. To prevent the specimens from an unstable sudden failure after the peak load, a crack opening controlled system was employed.  4.2 Literature Review and Research Significance  Flexural testing is conducted usually in four point, or, sometimes, in three point loading configuration. Different sizes of specimens and various testing spans are used for different applications. The load deflection curves, which provide useful information on fiber reinforced cementitious  73  composites, are then analyzed for material properties. [1, 8, 15, 44,  45,  47]. Flexural test on micro-fiber reinforced cementitious composites with different type of fibers (carbon fiber, steel fiber, polymer fiber, etc.) have indicated that the tensile and flexural properties of micro-fiber reinforced cementitious composites are significantly superior than the parent matrix. Comparatively, carbon fibers bring about a better improvement in the toughness or energy absorption than steel fibers. Steel fibers, on the other hand, impart higher tensile strengths to the base cement matrices than carbon fibers. Studies also indicated that regardless of the type of microfiber used, the strength and toughness of the composites increase w i t h an increase in the fiber volume fraction [8, 44, 47].  However, the influences of  fiber length, fiber cross-sectional area, and the changes in the matrix mix proportions is not clearly understood. In the experiments of the previous section, only the influence of fiber length was considered. In this section, the influence of fiber type and fiber cross-sectional area will also be investigated using flexural tests.  74  4.3  Experimental Work  4.3.1  Materials  Matrix Three types of matrices, in which sand and polymer contents were varied, were used in this study. Table 4.1 reports the mix proportions of these matrices. Table 4.1 Matrix Type M1 M2 M3  Mix Proportion of the Matrices  W/C  S/C  0.35 0.35 0.35  1 0.5 1  Notes: C: C e m e n t ; S: S a n d ; D: D e f o r m e r .  S.F./C 0.1 0.1 0.1  S.F: Silica F u m e ;  S.P. (ml/100 g C.) 1-3 1-3 1 S.P.: S u p e r p l a s t i c i z e r ;  P/C  D/P  0.1  0.1  P: P o l y m e r ;  Properties of cement, superplasticizer and sand used here are the same as those used in the specimens for the uniaxial tensile tests. Dry acrylic polymer was used as the polymer additive. To help the acrylic polymer powder to spread out in the cement paste and to reduce the entrapped air, 1 0 % (by weight of polymer) of deformer (Foamblast 390) was added. Steel Micro-Fibers High carbon content steel fibers were used. The fibers had irregular cross-sectional shapes (Figure 2.2). Four different cross-sectional areas of  fibers, classified as ' S ' , ' M ' , 'L' and ' T ' fibers, were used in this study. The cross-sectional areas, which were measured from micrographs, are given in Table 4 . 2 . The tensile strengths and the critical lengths of the fibers, obtained by using the pull-out test, are shown in Table 4.3.  Table 4.2  Cross-Sectional Areas of the Fibers  Fiber Type*  Cross-Sectional Area (mm X mm) 0.0007 0.0013 0.0020 0.0044  S M L T Note: * S: S m a l l ;  M: Medium;  Table 4.3 Fiber Type S M L T  L: L a r g e ; T: T h i c k  Critical Length and Tensile Strength of the Fibers Critical Length (mm) 2 3 4 6  Tensile Strength (MPa) 1000  Fibers were cut to 2 mm, 4 mm and 10 mm in length. For fiber 'T,  all  three lengths of fibers were cut by the manufacture. For fibers ' S ' , ' M ' and ' L ' , 4 mm and 10 mm fibers were cut in the laboratory and 2 mm fiber was cut by the manufacturer. The fiber length distributions are given in Figure 4.1 (a) and (b). It is clearly noticeable that fibers cut in the laboratory have different length distributions from those cut by the manufacturer. For example, although they were all called ' 4 mm fiber', the lengths of most  76  laboratory cut fibers range from 4 to 6 mm, while those cut by manufacturer range from 2 to 4 mm.  4.3.2  Specimen  Preparation  The shape and the dimension of the specimens are shown in Figure 4.2.  The specimens were cast and cured using the following procedure:  Mix cement, sand, silica fume, polymer and deformer (if used) in the mixer. Add water and a small quantity of superplasticizer. Mix for t w o minutes. Add more superplasticizer to make the mixture flowable and record the total quantity of superplasticizer used. Introduce the fibers slowly and evenly into the mixture while the mixer is mixing. Mix for another t w o minutes and then pour the mixture into the molds. Vibrate to compact the mixture in the molds. Cure in the moist condition and room temperature for one day. Demold and then cure in water for another twenty seven days before testing. For those specimens containing polymer or large quantities of the superplasticizer, one additional curing day was needed before demolding because of their slow hardening.  78  79  4.3.3  Test Plan Fifty-five separate sets, with four specimens each, were tested. The  test program is shown in Table 4.4.  Table 4 . 4 Fiber Length (mm)  Test Variables Investigated  Matrix Type  M1  M2 2  M3  M1  M2 4  M3 M1 10 M2  Fiber Type  S M L T S M L T S M L S M L T S M L T S M L L T L T  Fiber Volume Fraction (%) 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1, 1 1,  2 2 2 2, 5 2 2 2 2, 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  Figure 4.3 Test Set-Up of Flexural Tests  82  83 4.3.4  Test Set-Up  The four-point flexural test set-up is shown in Figures 4 . 3 and 4 . 4 . The load applied was measured by a load-cell which had a capacity of 1600 N. The deflection of the beam was measured by an LVDT (LVDT 1) installed on the top of the beam at the mid-span. The movement of the machine cross-head was controlled by the crack opening on the tensile side of the beam through another LVDT ( LVDT 2). Crack-opening rates of 0 . 2 Lim/sec before the peak load and 2 Lim/sec after the peak load were used. Both LVDTs had a sensitivity of 75.5 mv/mm.  The applied load, deflection and crack opening were recorded by both a system for data acquisition and an X-Y plotter. A FORTRAN computer program was written and used to analyze the data (Appendix 3).  4.4  Test Results and Discussion  4.4.1  Workability of the Mixtures  4.4.1.1  Effects of Fiber Length, Fiber Volume Fraction and Fiber CrossSectional Area  As observed in the previous section, fiber length and fiber volume fractions affect the workability of steel micro-fiber mortar mixture significantly. Fiber thickness also affects the workability. Table 4 . 5 reports  the workability of mixtures with different fiber lengths, fiber volume fraction and fiber cross-sectional areas. Table 4.5 Fiber Type  Fiber Length (mm)  2 S 4 2 M 4 2 L  4 10 2  T  4 10  Workability of SMFRM Mixtures Fiber Volume Fraction (%) 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 5 1 2 1 2  Quantity of workability S.P. (ml/100 g C.  1.0 1.5 2.0 3.0 1.0 1.5 2.0 3.0 1.0 1.5 1.8 2.5 3.0 1.0 1.2 2.0 1.0 1.5 2.0 3.0  A A B C A A B C A A A B D A A A A A B D  Maximum Fiber Volume Fraction > 2 % 1 % > 2 % 1 % > 2 % 2 % < 1 % > 5 % > 2 % 1 %  N o t e s : A : v e r y g o o d w o r k a b i l i t y ; B: w o r k a b l e , f i b e r b a l l i n g ; C: r o u g h , h a r d t o h a n d l e ; D: v e r y r o u g h , v e r y h a r d t o h a n d l e .  4.4.1.2  Effects of Sand Content and Polymer Addition  The sand content had no apparent effects on the workability. The addition of polymer made the mixture soft and sticky. For a given flow, mixtures with the polymer needed less superplasticizer.  85  4.4.2  Flexural Properties o f SMFRM The test results are shown in Table 4.6 (a), (b) and (c) and Figures  4.5-4.17.  4.4.2.1  Effects of Fiber Volume Fraction  Figures 4.5 and 4.6 show typical load versus deflection plots for cement mortars reinforced with various volume fractions of steel fibers. It is clear that there are significant improvements both in the load carrying capacity and toughness with an increase in the fiber volume fraction. At high fiber volume fractions (5 % ) , the load-deflection curves show significant strain hardening and toughness can be fourteen times higher than that of cement mortar without steel fibers (Table 4.6 (a) and Figure 4.13).  4.4.2.2  Effects of Fiber Length  Figures 4.7 and 4.8 show the load-deflection curves of SMFRM with different fiber lengths. It can be noted that when fiber cross-sectional areas are different, the effects of fiber length are also different. For example, for the type ' T ' fiber, the specimens with 2 mm fibers gave the highest strength, but the load dropped quickly after the peak load and the lowest toughness was recorded. On the other hand, specimens with 10 mm fibers gave the lowest strength but also the highest toughness. The strength and toughness of composites with 4 mm fibers are between those with 2 mm and 10 mm fibers. Figure 4.13 demonstrates the changes in strength and  86  toughness of S M F R M w h e n different lengths and volume fractions of  T'  f i b e r are a d d e d . For t h e o t h e r t y p e s o f f i b e r s (S, M a n d L), t h e s p e c i m e n s w i t h 2 m m fibers once again yielded the highest s t r e n g t h , but no clear influence of fiber length on toughness became  apparent.  W h e n t h e s e results are c o m p a r e d w i t h t h e results f r o m t h e  previous  s e c t i o n (uniaxial t e n s i o n ) , one can see t h a t t h e e f f e c t s of fiber l e n g t h o n t h e s t r e n g t h a n d t o u g h n e s s d o n o t a l w a y s f o l l o w t h e s a m e t r e n d in f l e x u r e a n d tension.  In t h e p r e v i o u s s e c t i o n , it w a s f o u n d t h a t 6 m m ( c l o s e t o t h e  critical length) gave the highest strength but the influence of fiber length on t o u g h n e s s v a r i e d w i t h t h e f i b e r v o l u m e f r a c t i o n . It s h o u l d b e r e c o g n i z e d , h o w e v e r , t h a t stainless steel fibers w i t h different surface  characteristics  w e r e i n v e s t i g a t e d i n t h e p r e v i o u s s e c t i o n as o p p o s e d t o t h e c a r b o n , s t e e l fibers of this section.  4.4.2.3  E f f e c t s o f Fiber C r o s s - S e c t i o n a l A r e a  Figures 4.9  and 4.10  s h o w the load-deflection curves for  w i t h different fiber cross-sectional areas. Figure 4 . 1 4  SMFRM  s h o w s the effects of  fiber cross-sectional area o n the flexural s t r e n g t h and t o u g h n e s s of  SMFRM  f o r t h e 2 m m f i b e r . It c a n be n o t e d t h a t w i t h a n i n c r e a s e in t h e f i b e r c r o s s sectional area, the toughness of S M F R M increases even t h o u g h the s t r e n g t h does not f o l l o w the same trend. The higher t o u g h n e s s for S M F R M w i t h thick f i b e r s c a n be e x p l a i n e d as t h e f o l l o w s : A f t e r t h e p e a k l o a d , t h e load s t a r t s  87  to drop and fibers are either pulled-out or fractured. For a given fiber length, the thicker fiber pulls out while the thinner fiber fractures. The process of pull-out consumes more energy than that of fracture, and thus higher toughness is expected. The somewhat reduced strength of the thicker fiber 'J' (Figure 4.14) is also related to the lack of fiber fractures in their case; fracture requiring a higher load than that of pull-out.  4 . 4 . 2 . 4 Effects of Matrix Properties Figures 4.11 and 4.12 show the load-deflection curves for SMFRM w i t h different matrices (Table 4.1). Figure 4.15 shows the influence of matrix type on the strength and toughness of SMFRM. With an increase in sand content in matrix ( M 1 : S/C = 1; M2: S/C = 0.5), the elastic strength, as well as the ultimate strength of SMFRM was found to increase. The reason apparently is that a mix with a higher sand content would be more stable and undergo less shrinkage cracking. Also with an increase in the sand content, more water gets absorbed by the sand and consequently, the W/C of the mortar is reduced and a higher strength is recorded. The effect of sand content on the toughness of SMFRM, on the other hand, is not clear. As expected, the addition of polymer (Matrix: M3) improved the toughness of SMFRM significantly. The addition of polymer, however, does not appear to have a clear effect on the strength.  88  The compressive strengths of the various composites are reported in Table 4.6 (a), (b) and (c) and Figure 4.16. Data suggest that the obtained compressive strength are much higher than generally reported for fiber reinforced cementitious composites [42, 4 5 ] . The primary reason is the small size of the specimens (25 mm X 25 mm X25 mm) and also the lateral confinement by the loading plates. It can also be noticed that the compressive strength of SMFRM increases with an increase in the fiber volume fraction.  The data of crack-opening were also acquired during these tests. Figure 4.17 shows a typical deflection-crack opening curve, which shows t w o straight lines with different slopes separated by the point of first crack in the matrix. The noted linear relation between the deflection and the crack opening confirms the stability of the tests. Figure 4.18 shows a load-crack opening curve. Analysis of the properties of the steel micro-fiber reinforced mortar from fracture mechanics perspective is being conducted using these data , but is out of the scope of this thesis.  89 >  co £  tn  CD Q_  O)  CO CO CN  o E  _ o  3  E  ^1  •<r CN  CNl  c  Q  ro  o  «  ^  ^  O  CD  CO CO  CD  c  oo  ro °  CD CO  0  LO L O led CD CD 00  LO CO  -<r CD CN CD o CN 00  CN CN Icsi  C L CL ~—  |LO  CO CO CD  d  oo  I co I CO co o  00 CD  CO  d  o o  LO  00  o  CD  LO CD CO L O L O  00  00 CN  oo CD CO CD CD CD o o  c c  CN  CD  E E  00 CD CO CD X— 00 CD ^ I ro CD csi d  CO  io  lO  3  o to 3 ro "o L7J£  ro o_  O  d  CN  CN  CD CD CD  o  CN i o  CO CO  CD  LO  o LO cn O o o CD CO  d  00 CO CO  I  O O C O  OO 00  •<r  s  |LO|  CD 00 CN  CO  CO CD CO  co CD ICN  CD LO s  Ih r CN I loo  CO  00  o  o  O  CO  O  i cvii  LO  CN CN 00 LO  CD CN  CD  tn  CO  CN LO CO CD CD 00  E E  u °  CN  csi d  o  "O  3 §  E E  CO  CD CO  CD  CD  c ^ E (j c E x <D E CD O 5 O =J ^.  tn  i i r i co cn CD  CN  LO  1 CD  CN co O CN o LO CN CO CD  E E  X CD CD Q  00  CO  00 o  s  II d LO d  CN CN CN CN CN CN  tn CD ro i_  ro cn . b CD 5 co  ro^  •^Z  H  13 X CD  CD  CO  CD C  ro  CD  ^  CN  LO  CD  o  CNlO  CL-  s  oo'  IO  CO  II - LO  ;LO  CN  CD CD CO  CO  CN  CN  CN LO  CN  CN  CN  LO LO  s  00  s  s  I CD  CD o I LO I - o  00  CD  CN  LO  LO CN  ed!  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CO o O n  |E o§ 5: 0 Q  Q  •Opre?  LO  d  •  0 3 U)  •g o O LO  o o  o o CO  o o  CM  o o  97  LO  cvi  98  LO  99  100  101  LO  c\i  CO  T -  102  103  104  CJ)  c "c  c 'c 0)  CD Q_  CL  o  o  o ca o  o  CO  o  CD  O  2. CD  c o  t) E E, c  o 'o  > 1 CO CD  ^  E  #8  CD CD 7= E CD O Q £ c c 'co  2  a  co 2  o  CO  |  2  *- CD CD SZ  sz  +- CD CD X I  £3 CD  CD CJ)  CO  105  106  4.5  Summary  The experiments in this section provide fundamental information of the behavior of SMFRM in flexure. The following remarks can be made:  The flexural strength, toughness and compressive strength of SMFRM increase with an increase in the fiber volume fraction.  For the largest diameter fiber, fiber ' T ' , the SMFRM specimens w i t h shorter fibers gave a higher strength but usually depicted lower toughness as compared to the longer fibers. For the other fibers (fiber ' S ' , ' M ' ,  ),  SMFRM w i t h shorter fibers again gave higher strength but the influences of fiber length on the toughness was not significant.  With an increase in the fiber cross-sectional area, the toughness of SMFRM increases, but strength does not follow a similar trend.  With an increase in the content of sand in the matrix, both the elastic and the ultimate strength of SMFRM increase. The effect of sand content on the toughness, however, is not clear.  The addition of polymer to the matrix improves the toughness of SMFRM significantly. The influence on strength, however, is not clear.  Addition of 1 % of 2 mm and 4 mm steel micro-fibers by volume fraction may improve the strength and toughness of SMFRM to some  107  degree, but the composites essentially stays brittle, particularly when the fibers are small in diameter. With 1 % of 10 mm steel micro-fiber, both the strength and toughness of the SMFRM are increased significantly.  From these tests, one can conclude that for SMFRM, longer and thicker fibers provides better mechanical properties. The workability of the mixture, however, is very bad for the 10 mm fiber, particularly when fiber diameter is small. The mixture with 10 mm type 'L' fibers is highly unworkable even at 1 % by volume fraction of fibers.  The best fiber from these tests is the thickest (type 'T') fiber. The mechanical properties of the SMFRM may be improved significantly and the workability of the mixture is adequate with 1-2% of ' T ' fibers which are less than 10 mm in length.  The mixtures with 2 mm and 4 mm fibers have good workability even at 5 % fibers by volume fraction. The mechanical properties of the resulting SMFRM are also significantly superior to the plain matrix.  108  5.  Conclusions  This thesis investigated the behavior of steel micro-fiber reinforced cementitious composites. Three aspects were examined: (1) the behavior of the interface between a steel micro-fiber and the surrounding cementitious matrix; (2) the tensile behavior of the composite; and (3) the flexural behavior of the composite. Routine tests in compression were also carried out.  Based on the single fiber pull-out tests conducted to characterize the fiber-matrix interface, the following conclusions can be drawn:  (1) The peak loads supported by fibers that are aligned in the direction of loading are higher than those supported by fibers inclined w i t h respect to the loading direction. The peak load for an aligned fiber also occurs at a smaller slip than for a inclined fiber.  (2) Even from an energy absorption point of view, a fiber aligned w i t h respect to the loading direction absorbs a greater amount of energy than one that is inclined. A O  0  inclination with respect to the loading  direction, therefore, is the optimal direction.  (3) The ultimate bond strength between the stainless steel fiber and cement paste varies from 4.7 MPa to 7.1 MPa. The tensile strength of the steel micro-fibers used is in the range of 750-1200 MPa.  109  (4) An increase in the content of silica fume in the matrix increases both the average interfacial bond strength and the maximum interfacial bond strength.  Based on the feed back controlled uniaxial tensile tests aimed at studding the tensile properties of stainless steel micro-fiber reinforced mortar, the following conclusions were drawn:  (1) With an increase in the fiber volume fraction, the tensile strength, toughness, elastic modules and the strain capacity of the steel micro fiber reinforced mortar (SMFRM) are found to increase. When the fiber volume fraction is 1 % or higher, pseudo strain-hardening is noted in the tensile stress-strain curves.  (2) Fiber length has a significant influence on the tensile strength and toughness of SMFRM. For a given fiber volume fraction, specimens w i t h 6 mm (close to the average critical length) fiber gave the highest strength followed by those with the 4 mm fiber and then those with 10 mm fiber. Specimens w i t h the 10 mm fiber demonstrated the highest toughness followed by those with the 4 mm and the 6 mm fibers at 2 % fiber volume fraction.  (3) Although an increase in the fiber volume fraction and the fiber length improves the ductility of SMFRM, high fiber volume fractions and  110  high fiber lengths lead to poor workability in the fresh state. The optimum fiber length, therefore, is 4 mm. By using the 4 mm fiber, while up to 2 - 3 % of fibers can be added to cement mortar to achieve high tensile strength and toughness as well as adequate workability, the optimum properties are obtained at 2 % of fiber by volume.  Based on the four-point flexural tests performed using a crack opening controlled system, carbon steel micro-fiber reinforced mortars were investigated and the following conclusions were drawn:  (1) The flexural strength, toughness, tensile strain capacity and compressive strength of the composite increase with an increase in the fiber volume fraction.  (2) With an increase in the fiber cross-sectional area, although the flexural strength is generally not affected, the toughness of the composite is found to increase.  (3) For the thickest fiber 'T'(cross-sectional area 0 . 0 0 4 4 mm ), longer 2  fibers gave higher toughness but usually lower flexural strengths as compared to the shorter fibers.  (4) With an increase in the sand content in the matrix, the flexural strength of SMFRM increases. The effects of sand content on the toughness of SMFRM, however, are not clear. The addition of polymer in cement  Ill mortar improves the toughness of SMFRM significantly, even though a similar influence on flexural strength was not observed.  (5) High fiber volume fractions and long fibers make the workability of fresh steel micro-fiber mortar unacceptable. However, with a 2 mm fiber, the mixture has very good workability even at 5% of fibers by volume. The flexural properties of the cement mortars are improved considerably by adding 5 % of type 'J' fiber with a length of 2 mm.  Closure To sum up, the addition of both stainless steel micro-fiber and carbon steel micro-fiber can improve the ductility of cement mortars significantly. The carbon steel micro-fibers, however, have more practical importance because of their low cost.  The effects of fiber length on strength and toughness of SMFRM are different in uniaxial tensile tests than in flexural tests. Partly, this is because different type of fibers and fiber cross-sectional areas were used in these t w o tests. For the stainless steel micro-fiber reinforced composites, those w i t h 2 % of the 4 mm fiber gave the best combination of properties. For the carbon steel micro-fiber, if fiber volume fractions are limited to 2 % from economical considerations, the thickest fiber ('T' fiber) with 4 mm length is the best choice. However, when the 2 mm ' T ' fiber is used, fibers can be added up to 5% or more by volume without workability problem. The  112  properties of SMFRM at such a high fiber volume fractions are improved dramatically.  113  Bibliography  1. Bentur A.and Mindess S., Fiber Reinforced Cementitious  Composites.  Elsevier Applied Science, London and New York, 1 9 9 1 . 2. Hull D., An Introduction to Composite Materials. Cambridge University Press, 1 9 8 1 . 3. Shah S. P. and Ouyang C , Mechanical Behavior of Fiber-Reinforced Cement-Based Composites. J. Am. Ceram. 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In Bartos P. ed. Bond in Concrete, Applied Science Publishers, London, U.K., 1 9 8 2 , pp. 60-72. 25. Banthia N., Moncef A. and Sheng J . , Uni-Axial Tension Response of Cement Composites Reinforced with High Volume Fraction of Carbon, Steel and Polypropylene Micro-Fibers. Can. J. Eng., 2 1 , 1994, pp. 9 9 9 - 1 0 1 1 . 26. Li V.C. and Obla K. H., Effect of Fiber Length Variation on Tensile Properties of Carbon Fiber Cement Composites, Unpublished Manuscript. 27. Dept. of Civil Eng., UBC, Tension Test of Brittle Material, Class Notes for 'Civil Engineering Material I (Civil 2 2 0 ) ' (Banthia N. Ed.), 1993. 28. Naaman A. E. and Homrich J . R., Tensile Stress-Strain Properties of SIFCON. ACI Materials Journal, May-June 1989, pp. 2 4 4 - 2 5 1 . 29. Shah S. P., Do Fibers Increase the Tensile Strength of Cement-Based Matrixes?. ACI Materials Journal, November-December 1 9 9 1 , pp. 5 9 5 - 6 0 2 . 30. Akihama S., Suenaga T and Nakagawa H., Carbon Fiber Reinforced Concrete. Concrete International, January 1988, pp. 40-47. 3 1 . Tjiptobroto P. and Hansen W., Tensile Strain Hardening and Multiple Cracking in High-Performance Cement-Based Composites Containing Discontinuous Fibers. ACI Materials Journal, January-February 1993, pp. 16-25. 32. Tjiptobroto P. and Hansen W., Model for Predicting the Elastic Strain of Fiber Reinforced Composites Containing High Volume Fractions of Fibers. ACI Materials Journal, March-April 1993, pp. 134-142. 33. Soroushian P., Choi K. B. and Fu G., Tensile Strength of Concrete at Different Strain Rates, In Mindess S. and Shah S. P. ed. Cement-Based Composites:  Strain Rate Effects on Fracture, 1986, pp. 8 7 - 9 2 .  34. Nammur G. G. and Naaman A. E., Strain Rate Effects on Tensile Properties of Fiber Reinforced Concrete. In Mindess S. and Shah S. P. ed.  116  Cement-Based  Composites:  Strain Rate Effects on Fracture, 1 9 8 6 , pp. 9 7 -  117. 35. Wang Y., Li V. C. and Backer S., Experimental Determination of Tensile Behavior of Fiber Reinforced Concrete. AC/ Materials Journal, SeptemberOctober 1990, pp. 461-468. 36. Luong M. P., Liu H., Trinh J . L. and Tran T. P., Tensile Properties of Steel Fiber Reinforced Concrete. In Swamy R. N. ed. Fiber Reinforced Cement and Concrete, Proceedings of the Forth International Held by RILEM, 1 9 9 2 , pp. 3 4 3 - 3 5 4 .  Symposium  37. Nanni A., Splitting-Tension Test for Fiber Reinforced Concrete. AC/ Materials Journal, July-August 1988, pp. 229-233. 38. Li V. C , Chan C. M. and Leung C. K. Y., Experimental Determination of the Tension-Softening Relations for Cementitious Composites. Cement and Concrete Research, Vol. 17, pp. 4 4 1 - 4 5 2 .  39. Li V. C , Postcrack Scaling Relations for Fiber Reinforced Cementitious Composites. Journal of Materials in Civil Engineering, February, 1 9 9 2 , pp.  41-57. 4 0 . Soroushian P. and Baysi Z., Fiber-Type Effects on the Performance of Steel Fiber Reinforced Concrete. AC/ Materials Journal, March-April, 1 9 9 1 , pp. 129-134. 4 1 . Wafa F. F. and Ashour S.A., Mechanical Properties of High-Strength Fiber Reinforced Concrete. ACI Materials Journal, September-October, 1992, pp. 4 4 9 - 4 5 4 . 4 2 . Soroushian P., Khan A. and Hsu J . W., Mechanical Properties of Concrete Materials Reinforced with Polypropylene or Polyethylene Fibers. AC/ Materials Journal, November-December, 1992, pp. 535-640. 4 3 . Brandt A. M., On the Optimal Direction of Short Metal Fibers in Brittle Matrix Composites. J. of Material Science, 20 (1985), pp. 3 8 3 1 - 4 1 . 4 4 . Banthia N. and Sheng J . , Micro-Reinforced Cementitious Materials. Mat. Res. Soc. Symp. Proc. Vol. 2 1 1 , 1 9 9 1 , pp. 25-32.  4 5 . Akihama S., Suenaga T. and Banno T., Mechanical Properties of Carbon Fiber Reinforced Cement Composites. The International Journal of Cement Composites  21-32.  and Lightweight Concrete, Vol. 8, February, 1 9 8 6 , pp.  117  4 6 . Banthia N. and Dubeau S., Carbon and Steel Micro-fiber-Reinforced Cement-Based Composites for Thin Repairs. Journal of Material in Civil Engineering, February, 1994, pp. 88-98. 4 7 . Shivaraj S.K., Ramakrishnan V. and W u G. Y., Properties and Flexural Performance of Steel Fiber Reinforced Refractory Concrete. In Swamy R.N. and Barr B. ed. Fiber Reinforced Cement and Concrete:  Developments,  Recent  1989, pp. 499-512.  4 8 . Mobasher B., Stang H. and Shah S. P., Micro-Cracking in Fiber Reinforced Concrete. Cement and Concrete Research, Vol. 2 0 , 1 9 9 0 ,  pp. 6 6 5 - 6 7 6 . 4 9 . Visalvanich K. and Naaman A.E., Fracture Model for Fiber Reinforced Concrete. AC/ Journal, March-April, 1983, pp. 128-138. 50. Ouyang C. and Shan S.P., Toughening of High Strength Cementitious Matrix Reinforced by Discontinuous Short Fibers. Cement and Concrete Research, Vol. 2 2 , 1992, pp. 1201-1215.  Appendix 1 PULOUT.FOR a FORTRAN program used to analyze the data from pull-out tests  PROGRAM MAIN DIMENSION Al(5000) , A2 (5000), A22 (5000) ,E(5000) COMMON/DAT/N, A l l ( 5 0 0 0 ) CHARACTER*18 DATFIL CHARACTER*18 ENEFIL WRITE (*,*) 'INPUT THE DATA FILE NAME' WRITE (*,*) READ (*, ' ( A 1 8 ) ' ) DATFIL OPEN (4, FILE=DATFIL, STATUS='OLD') DO 8 1=1,5000 READ (4, * , END=88) A 2 ( I ) , A1(I) CONTINUE N=I-1 DO 10 1=1,N A11(I)=0.0975*A1(I) A22(I)=0.025*A2(I) CONTINUE CALCULATE PEAK LOAD AND CORRESPONDING SLIP CALL PKLOAD(PMAX, M) 1=0 1=1 + 1 IF (I . E Q . M) THEN PDIS=A22(I) ELSE GOTO 2 0 END I F CALCULATE ENERGY VERSUS SLIP, TOTAL ENERGY AND ENERGY AT PEAK LOAD E(1)=(A22(2)-A22(1))*(All(2)+A11(1))/2 DO 3 0 J=2,N-1 E(J)=E(J-1)+(A22(J+l)-A22(J))*(All(J+l)+A11(J))/2 IF (J . E Q . M) THEN PENE=E(J) ELSE IF ( J . E Q . N - l ) THEN TENE=E(J) END I F END I F CONTINUE OUTPUT DATA WRITE (*,*) 'INPUT ENERGY FILE NAME' WRITE (*,*) READ ( * , ' ( A 1 8 ) ' ) ENEFIL  119  100 200 40  310 350 410 420 450  15  25  $  OPEN (5, FILE=ENEFIL, STATUS='UNKNOWN') WRITE (5, '(A18)') ENEFIL WRITE (5,100) FORMAT (5X, 'SLIP(mm)', 8X, 'ENERGY(N.mm)') DO 40 1=1,N WRITE (5,200) A 2 2 ( I ) , E(I) FORMAT (5X, F 8 . 5 , 12X, F8.5) CONTINUE CLOSE (5) WRITE ( * , ' ( A 1 8 ) ' ) DATFIL WRITE (*,310) FORMAT (IX, 'TOTAL ENERGY (N.mm)') WRITE (*,350) TENE FORMAT (3X,F10.4) WRITE (*,410) WRITE (*,420) FORMAT (IX, 'PEAK LOAD', 4X, 'SLIP AT PEAK LOAD' , 4 X , 'ENERGY AT PEAK LOAD') FORMAT (4X, ' ( N ) ' , 12X, '(mm)',15X, '(N.mm)') WRITE (*,450) PMAX, PDIS, PENE FORMAT (3X, F 5 . 2 , 10X, F 5 . 2 , 15X, F5.2) END SUBROUTINE PKLOAD(PMAX, M) COMMON /DAT/ N, All(5000) CALCULATE THE PEAK LOAD AMAX=0 DO 15 1=1,N IF ( A l l ( I ) . G T . AMAX) THEN AMAX=A11(I) END I F CONTINUE PMAX=AMAX CALCULATE THE POSITION OF THE PEAK LOAD 1=0 1=1+1 IF ( A l l ( I ) . E Q . PMAX) THEN M=I ELSE GOTO 2 5 END IF END  120  Appendix 2  TENSIL.FOR  a FORTRAN program used to analyze the data from uniaxial tensile tests  8 80 C  10  PROGRAM MAIN DIMENSION P(2000), D(2000), G(12000), A l ( 1 2 0 0 0 ) , A2(12000) COMMON /DATA/ N, A l l ( 1 2 0 0 0 ) , A22(12000) CHARACTER *2 0 INFIL, OUTFIL WRITE (*,*) 'INPUT THE FILE NAME' WRITE (*,*) READ ( * , ' ( A 2 0 ) ' ) INFIL WRITE (*,*) 'INPUT THE OUTPUT FILE NAME' WRITE (*,*) READ (*, '(A20)') OUTFIL OPEN (4, FILE=INFIL, STATUS='UNKNOWN') DO 8 1=1,12000 READ (4,*, END=80) A 1 ( I ) , A2(I) CONTINUE N=I-1 WRITE (*,*) 'N=',N SET THE STARTING POINT AND MULTIPLY BY THE COEFICIENTS DO 10 1=1,N A1(I)=A1(I)-A1(1) A2(I)=A2(I)-A2(1) A l l ( I ) = A 1 ( I ) * 0 . 0 0 1 3 2/65 A22(I)=A2(I)*1.0543 CONTINUE  C  CALCULATE THE ULTIMATE STRENGTH AND STRAINS CALL PKLOAD (PMAX, NP) PDIS=A11(NP)*100 UDIS=A11(N)*100  C  CALCULATE THE ELASTIC STRENGTH AND THE ELASTIC MODULUS CALL ELASTIC (E, P E , NE, NP, PMAX)  C  CALCULATE THE TOUGHNESS G(1)=(A11(2)-A11(1))*(A22(2)+A22(1))/2 DO 100 1=2,N-l G(I)=G(I-1)+(A11(I+1)-All(I))*(A22(1+1)+A22(I))/2 CONTINUE TOUGH=G(N-l)  100 C  SMOOTH THE CURVE BY AVERAGING THE DATA CALL AVERAGE (P, D, NE, NP, M)  C  OUTPUT DATA OPEN (5, FILE=OUTFIL, STATUS='UNKNOWN') WRITE (5, ' ( A 2 0 ) ' ) OUTFIL  121  18 19  28 200  38 39  $  40 48 $ 49 50  15  25  WRITE (5, *) WRITE (5, 18) WRITE (5, 19) FORMAT (4X, 'STRAIN', 5X, 'STRESS') FORMAT ( 6 X , ' ( % ) ' , 6X, '(MPa)') DO 200 1=1,M D(I)=D(I)*100 WRITE (5,28) D ( I ) , P(I) FORMAT (2X, F 8 . 4 , 3X, F8.4) CONTINUE WRITE (*, ' ( A 2 0 ) ' ) OUTFIL WRITE (*,*) WRITE (*,38) WRITE (*,39) FORMAT (IX, 'ELASTIC STRENGTH', 3X, 'ELASTIC MODULUS', 3X, 'ULTIMATE STRENGTH') FORMAT (6X, ' ( M P a ) ' , 14X, ' ( G P a ) ' , 14X, '(MPa)') WRITE (*, 40) PE, E , PMAX FORMAT (5X, F 8 . 4 , 10X, F 8 . 4 , 11X, F8.4) WRITE (*, 48) WRITE (*, 49) FORMAT (IX, 'PEAK LOAD STRAIN', 3X, 'ULTIMATE STRAIN', 3X, 'TOUGHNESS') FORMAT (7X, '(%)', 16X, '(%)', 9X, '(N/mm*mm)') WRITE (*, 50) PDIS, UDIS, TOUGH FORMAT (5X, F 8 . 4 , 11X, F 8 . 4 , 5X, F8.4) END  SUBROUTINE PKLOAD (PMAX, NP) COMMON /DATA/ N, A l l ( 1 2 0 0 0 ) , A22(12000) AMAX=0 DO 15 1=1,N IF (A22(I) . G T . AMAX) THEN AMAX=A2 2(I) END IF CONTINUE PMAX=AMAX 1=0 1=1+1 IF (A22(I) NP=I ELSE GO TO 25 END IF END  . E Q . PMAX)  THEN  SUBROUTINE ELASTIC ( E l , PE, NE, NP, PMAX) DIMENSION EE(12000), E(12000) COMMON/ DATA / N, A l l ( 1 2 0 0 0 ) , A22(12000)  NPP=NP/5 IF ((A11(5)-A11(1)) . L T . 0.000001) THEN E(l)=25 ELSE E(1)=(A22(5)-A22(1))/((All(5)-All(1))*1000) IF ( E ( l ) . L T . 5 .OR. E ( l ) . G T . 50) THEN E(l)=25 END IF END IF J=l EE(J)=E(1) DO 3 5 1=1,NPP IF ( ( A l l ( 1 * 5 + 5 ) - A l l ( 1 * 5 ) ) . L T . 0.000001) THEN E(I+1)=25 ELSE E(I+1)=(A22(1*5+5)-A22(1*5))/ ((All(1*5+5)-All(1*5))*1000) IF (E(I+1) . L T . 5 .OR. E(I+1) . G T . 50) THEN E(I+1)=25 END IF END I F IF ( A B S ( 1 - E ( I + 1 ) / E ( I ) ) . L E . 5.95) THEN J=J+1 EE(J)=EE(J-1)+E(I) ELSE GO TO 35 END I F CONTINUE NE=J E1=EE(NE)/NE PE=A2 2(NE) IF (PE . E Q . PMAX) THEN WRITE (*,*) 'NO STRAIN HARDENNING' END IF END SUBROUTINE AVERAGE(P, D, NE, NP, M) DIMENSION P(2000), D(2000) COMMON /DATA/ N, A l l ( 1 2 0 0 0 ) , A22(12000) NA=(N)/10 NEA=NE/10 NEB=NEA*10 NPA=NP/10 NPB=NPA*10 J=0 DO 45 1=1,NEA 11=10*1 J=J+1 CALL PD(PI, D l , II) P(J)=PI D(J)=DI CONTINUE  DO 55 I=NEB+1, NEB+10 J=J+1 P(J)=A22(I) D(J)=A11(I) CONTINUE IF (NEA . E Q . NPA) GO TO 76 END I F  THEN  DO 65 I=NEA+2, NPA 11=10*1 J=J+1 CALL PD(PI, DI, II) P(J)=PI D(J)=DI CONTINUE DO 75 I=NPB+1, NPB+10 J=J+1 P(J)=A22(I) D(J)=A11(I) CONTINUE CONTINUE DO 85 I=NPA+2, NA 11=10*1 J=J+1 CALL PD (PI, DI, II) P(J)=PI D(J)=DI CONTINUE M=J END  SUBROUTINE PD(P, D, II) COMMON /DATA/N, A l l ( 1 2 0 0 0 ) , A22(12000) I=(II-1)/10 D=(All(10*1+1)+A11(10*1+2)+A11(10*1+3)+All(10*1+4) +A11(10*1+5)+A11(10*1+6)+A11(10*1+7)+A11(10*1+8) +A11(10*1+9)+All(10*1+10))/10 P=(A22(10*1+1)+A2 2(10*1+2)+A22(10*1+3)+A22(10*1+4) +A22(10*1+5)+A22(10*1+6)+A22(10*1+7)+A22(10*1+8) +A22(10*1+9)+A22(10*1+10))/10 END  124  Appendix 3  BENDING.FOR  a FORTRAN program used to analyze the data from flexural tests  PROGRAM MAIN DIMENSION LOAD1(2500), CRAK1(2500), DEFL1(2500), LOAD(2500), CRAK(2500), DEFL(2500), G(2500), ALOAD(250), ACRAK(250), BLOAD(250), BDEFL(250) REAL LOAD1, LOAD CHARACTER* 2 0 INFILE, OTFIL1, OTFIL2 CHARACTER*1 CHOICE WRITE (*,*) 'INPUT THE FILE NAME' WRITE (*,*) READ (*, ' ( A 2 0 ) ' ) INFILE WRITE (*,*) 'INPUT THE LOAD-CRACK OPENING OUTPUT FILE NAME' WRITE (*,*) READ (*, '(A20)') OTFIL1 WRITE (*,*) 'INPUT THE LOAD-DEFLECTION OUTPUT FILE NAME' WRITE (*,*) READ (*,  '(A20)')  OTFIL2  INPUT THE DATA OPEN (4, FILE=INFILE, STATUS='UNKNOWN') DO 10 1=1,2500 READ (4,*, END=88) CRAKl(I), L O A D l ( I ) , DEFLl(I) CONTINUE N=I-1 MULTIPLY BY THE COEFFICIENT DO 20 1=1,N CRAK(I)=0.0 01151*CRAK1(I) LOAD(I)=110.7*L0AD1(I) DEFL(I)=0.0 01151*DEFL1(I) CONTINUE SET THE FIRST READING TO BEZERO WRITE (*,*) 'BEAM ROCKING? (Y/N) READ (*,' ( A l ) ' ) CHOICE IF (CHOICE . E Q . ' Y ' ) THEN DO 6 1=1,N IF ((LOAD(I)-LOAD(1)-150) .GE. 0) THEN ME1=I GO TO 5 END IF CONTINUE CONTINUE DO 8 1=1,N IF ((LOAD(I)-LOAD(l)-240) GE. 0) THEN ME2=I GO TO 7  125  8 7  9  66 99  30  END IF CONTINUE CONTINUE EF=(LOAD(ME2)-LOAD(ME1))/(DEFL(ME2)-DEFL(ME1)) DO 9 I=1,ME1-1 DEFL(I)=DEFL(ME1)-(LOAD(ME1)-LOAD(I))/EF CONTINUE END IF IF (CHOICE . E Q . 'N') THEN GO TO 66 END IF CONTINUE DO 99 1=1,N DEFL(I)=(-1)*DEFL(I) CONTINUE FSTC=CRAK(1) FSTL=LOAD(1) FSTD=DEFL(1) DO 30 1=1,N CRAK(I)=CRAK(I)-FSTC LOAD(I)=LOAD(I)-FSTL DEFL(I)=DEFL(I)-FSTD CONTINUE  C  CALCULATE THE STRENGTH OF THE BEAM ETC. CALL PKLOAD (PMAX, NP, N, LOAD) STREN=12 * PMAX/6 2 5 PCRAK=CRAK(NP) PDEFL=DEFL(NP) UCRAK=CRAK(N) UDEFL=DEFL(N)  C  SMOOTH THE LOAD-CRACK OPENING CURVE CALL SMOOTH (LOAD, CRAK, ALOAD, ACRAK, N, M l , NP, NPP1)  C  SMOOTH THE LOAD-DEFLECTION CURVE CALL SMOOTH (LOAD, DEFL, BLOAD, BDEFL, N, M2, NP, NPP2) DO 4 0 I=NPP2,M2 IF (BDEFL(I) . L T . B D E F L ( I - l ) ) THEN BDEFL(I)=BDEFL(I-1) + (BDEFL(1-1)-BDEFL(I) )/8 END IF CONTINUE  40  50 55  CALCULATE THE ELASTIC MOUDULUS DO 50 I=1,NPP2 IF ((BLOAD(I)-150) . G E . 0) THEN NE1=I GO TO 55 END IF CONTINUE CONTINUE P85=0.85*PMAX DO 60 I=1,NPP2 IF ((BLOAD(I)-P8 5) . G E . 0) THEN NE2=I  126  60 65  70  150 250 350 450 550 650 750 850 950  80  90  100  GO TO 65 END IF CONTINUE CONTINUE CALL ELASTIC (E, NE1, NE2, BLOAD, BDEFL) CALCULATE THE TOUGHNESS G(1)=(DEFL(2)-DEFL(l))*(LOAD(2)+LOAD(1))/2 DO 70 1=2,N-l G(I)=G(I-1)+(DEFL(I+1)-DEFL(I))*(LOAD(1+1)+LOAD(I))/2 CONTINUE TOUGH=G(N-l) OUTPUT DATA WRITE (*, 150) WRITE (*, 250) FORMAT (IX,'STRENGTH',2X,'ELASTIC MOUDULUS',2X,'TOUGHNESS') FORMAT (3X, ' ( M P a ) ' , 8X, ' ( G P a ) ' , 9X, '(N.mm)') WRITE (*, 350) STREN, E , TOUGH FORMAT (IX, F 8 . 3 , 7X, F 6 . 3 , 7X, F9.3) WRITE (*,450) WRITE (*,550) FORMAT (IX,'CRACK AT P-LOAD',2X,'DEFLECTION AT P-LOAD') FORMAT (6X, '(mm)', 16X, '(mm)') WRITE (*,650) PCRAK, PDEFL FORMAT (4X, F 9 . 7 , 10X, F9.7) WRITE (*,750) WRITE (*,850) FORMAT (IX, 'MAX. CRACK OPENING', 2X, 'MAX. DEFLECTION') FORMAT (8X, '(mm)', 14X, '(mm)') WRITE (*,950) UCRAK, UDEFL FORMAT (7X, F 6 . 4 , 12X, F6.4) OPEN (5, FILE=0TFIL1, STATUS='UNKNOWN') DO 80 1=1,Ml WRITE (5,*) ACRAK(I), ALOAD(I) CONTINUE OPEN (6, FILE=OTFIL2, STATUS='UNKNOWN') DO 90 1=1,M2 WRITE (6,*) BDEFL(I), BLOAD(I) CONTINUE END SUBROUTINE PKLOAD (PMAX, NP, N, LOAD) REAL LOAD(2500) AMAX=0.0 DO 100 1=1,N IF (LOAD(I) . G T . AMAX) THEN AMAX=LOAD(I) END IF CONTINUE PMAX=AMAX DO 110 1=1,N IF (LOAD(I) . E Q . PMAX)  THEN  127  NP=I GO TO 12 0 END IF CONTINUE CONTINUE END  110 120  SUBROUTINE SMOOTH (LOAD, DESP, P, D, N, M, NP, NPP) DIMENSION P(250), D(250), LOAD(2500), DESP(2500) REAL LOAD NA=N/10 NB=NA*10 NPA=NP/10 NPB=NPA*10 J=l P(1)=0 D(1)=0 DO 200 1=1,NPA 11=10*1 CALL PD (PI, DI, I I , J=J+1 P(J)=P1 D(J)=D1 CONTINUE NPP=J  200  LOAD, DESP)  DO 210 I=NPB+l,NPB+2 0 J=J+1 P(J)=LOAD(I) D(J)=DESP(I) CONTINUE  210  DO 22 0 I=NPA+3, NA 11=10*1 CALL PD (PI, DI, I I , LOAD, DESP) J=J+1 P(J)=P1 D(J)=D1 CONTINUE M=J END  22 0  $ $ $ $  SUBROUTINE PD (PI, DI, J , LOAD, DESP) DIMENSION LOAD(2500), DESP(2500) REAL LOAD I=(J-1)/10 P1=(LOAD(10*I+1)+LOAD(10*I+2)+LOAD(10*1+3)+LOAD(10*1+4) +LOAD(10*I+5)+LOAD(10*I+6)+LOAD(10*I+7)+LOAD(10*I+8) +LOAD(10*I+9)+LOAD(10*I+10))/ 10 Dl=(DESP(10*1+1)+DESP(10*1+2)+DESP(10*1+3)+DESP(10*1+4) +DESP(10*1+5)+DESP(10*1+6)+DESP(10*1+7)+DESP(10*1+8) +DESP(10*1+9)+DESP(10*1+10))/10  128  END  300  SUBROUTINE ELASTIC ( E l , NE1, NE2, BLOAD, BDEFL) DIMENSION EE(IOO), E(200), BLOAD(250), BDEFL(250) J=l E(NE1)=14.72*BLOAD(NE1)/BDEFL(NE1) EE(1)=E(NE1) DO 3 00 I=NE1+1, NE2 E(I)=14.72*BLOAD(I)/BDEFL(I) J=J+1 EE(J)=EE(J-1)+E(I) CONTINUE NE=J E1=EE(NE)/(NE*1000) END  

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