Benefit of Fibres Addition to Normal Strength Concrete
Info: 19668 words (79 pages) Dissertation
Published: 13th Dec 2019
Benefit of Fibres Addition to Normal Strength Concrete: An Empirical Model for the determination of Tensile Capacity and Fracture Energy
Abstract
Fibrereinforced concrete is widely used in construction to enhance the durability of concrete by controlling the formation and development of the material cracking. Its ability can be attributed to the ductility induced by the addition of fibres to the material matrix. The quantification of this improvement is investigated in this paper through the analysis of laboratory, computer and theoretical results. Testing of specifically designed dog bone shape specimens have shown that a 1.0% volume fibre addition allows for a 26% increase in tensile capacity under uniaxial stress conditions. While a general relation describing the induced improvement was defined as a function of the material characteristic cylinder strength, a general formulation involving the volume fraction of fibre was not possible due to the other fibre admixture being defective. A graphical resolution was, however, proposed for the available data.
The material behaviour was successfully modelled up to the maximum tensile stress as an adjustment of Hognestad’s model (1952) for concrete in compression. This was checked against the finite element analysis results showing a good fit limited to low stress values ( < 30% maximum stress) as the model appeared nonconvergent outside this range.
Eventually, from the graphical analysis of the experimental results of the fibre admixtures, the material behaviour following the first cracking development was described. A model, based on the bilinear method proposed by Hillerborg (1976), was adopted to estimate the total fracture energy dissipated by the cracking development. This outlined a 10fold increase in the energy necessary for the initial cracking formation and a total fracture energy two orders of magnitude greater than the corresponding plain mixture.
Contents
4 / Finite element modelling process
4.2 / Finite element analysis by ANSYS
4.3.1 / Fibre distribution approximation
4.3.3 / Computer model generation
4.3.5 / Validity of finite element model
4.4.1 / Model 1 – SOLID65 with reinforcement values
4.4.2 / Model 2 – SOLID65 with modelled fibres
5.2 / Materials and mix details
5.3 / Test specimen and procedure
5.4 / Results. Interpretation and reduction
5.4.1 / Readings up to failure
6.1 / Tensile capacity. Direct and splitting tests
6.2 / Modified Hognestad model
6.3 / Finite element model results
6.4 / Linearity of strain variation
6.5 / Formulation of tensile strength
6.6 / Fracture energy for fibre reinforced concrete
7.1 / Recommendations for further studies
List of tables
Table 3.1 – Applied model values
Table 4.1 – Concrete properties
Table 4.2 – Fibre properties – PROPEX Enduro HPP45
Table 4.3 – Derived fibre’s properties
Table 4.4 – Real constant parameters
Table 5.1 – Results of moisture content test
Table 5.2 – Concrete mixes details – water cement ratio 0.5
Table 5.3 – Cube testing results
Table 5.4 – Cylinder testing results
Table 6.1 – Direct tensile test comparison
Table 6.2 – Results variation from Eurocodes’ values
Table 6.3 – Fitting lines’ characteristics
Table 6.4 – Characteristic strength interpolation
Table 6.5 – Fitting lines’ constant terms
Table 6.6 – Generalised linear equations
Table 6.7 – Base fracture energies per aggregate size
Table 6.8 – Comparison between maximum and lowpeak stress
Table 6.9 – Comparison between total and initial energy
Table 6.10 – Improvement in initial fracture energy
Table 6.11 – Improvement in total fracture energy
List of figures
Figure 3.1 – BS flexural test configuration
Figure 3.2 – Uniaxial tensile test concept
Figure 3.3 – Comparison of Hognestad and Eurocode model
Figure 4.1 – Solid65 element (ANSYS, release 5.5)
Figure 4.2 – Solid285 Element (Ansys, released 14.0)
Figure 4.3 – Fibre generation diagram
Figure 4.4 – Model generation sequence
Figure 4.5 – Stress and strain variation along YY axis Fibre Content 1.0% in weight
Figure 4.6 – Stress and strain variation along YY axis Fibre Content 2.0% in weight
Figure 4.7 – Stress and Strain Distribution along YY axis
Figure 4.8 – Model 1 and 2, low stress results and their projection
Figure 4.9 – Model 2 configuration
Figure 5.1 – Particle size distribution chart. Left) Fine aggregate, Right) Coarse aggregate
Figure 5.2 – Propex ENDURO HPP45 (propex.co.uk)
Figure 5.4 – General test arrangement
Figure 5.5 – Specimen T1, raw data and fitting line
Figure 5.6 – Specimen T1, BoxPlot
Figure 5.7 – Specimen T1, refined data with improved fitting line
Figure 5.8 – Data refinement – Specimen T2
Figure 5.9 – Data refinement – Specimen T3
Figure 5.10 – Data refinement – Specimen T4
Figure 5.11 – Data refinement – Specimen T5
Figure 5.12 – Data refinement – Specimen T6
Figure 5.13 – Specimen T4, stressseparation curve with fitting curve
Figure 5.14 – Specimen T5, stressseparation curve with fitting curve
Figure 6.1 – Fitting of linear model to fit T13 data
Figure 6.2 – First modification of Hognestad model fitting T13 data
Figure 6.3 – First modification of Hognestad model fitting T13 data
Figure 6.4 – First modification of Hognestad model fitting T46 data
Figure 6.5 – Finite element against experimental results
Figure 6.6 – Linearity of characteristic results
Figure 6.7 – Tensile capacity graphical resolution
Figure 6.8 – T4 and T5 raw data and fitting lines
Figure 6.9 – Fracture energy model, specimen T4
Figure 6.10 – Fracture energy model, specimen T5
Notations
Tensile strength from flexural test (MPa)  
Derived maximum tensile strength (MPa)  
Derived ultimate tensile strength (MPa)  
Derived maximum tensile strength of composite (MPa)  
Mean experimental tensile strength (MPa)  
Interpolated characteristic strength for observed  
Mean target compressive strength (MPa)  
Stressseparation curve  
Gradient to initial tangent ()  
Standard mean tensile strength (MPa)  
Applied Stress (MPa)  
Ultimate compressive strength (MPa)  
Induced strain ()  
Strain at ultimate compressive stress ()  
Strain at maximum tensile stress ()  
Mean experimental strain at ultimate stress ()  
Mean experimental strain at stress ()  
Mean experimental strain at maximum stress of composite ()  
Total fracture energy (N / m)  
Initial fracture energy (N / m)  
Secant elastic modulus (GPa)  
Tangent or dynamic elastic modulus (GPa)  
Coefficients  
Crack opening, separation (mm) 
1 / Introduction
Composite materials found their strength on the concept of the juxtaposition of two or more materials characterised by complementary properties. The potential of such a simple technique is fascinating and at the same time a challenging research topic.
Countless examples of the application of this technique are available in everyday life. The concept of composite material or object has been in use for centuries and it spreads across most engineering fields. If carried to extremes, everything can be considered to be a composite formed by basic chemical elements.
Moving back into the macroscopic realm, the project analysis is focused on a very common example of composite material, reinforced concrete (RC). Concrete reinforced with metal bars has been, over the last two centuries, a common method to improve the behaviour of concrete in tension and widen its applications.
While the principle of reinforced concrete was patented by Joseph Monier in 1867, the first example of a concrete structure reinforced with iron bars was built by Francoise Coignet in 1853. Over the last decades, the concept of “reinforcing” has seemed to shift and to focus on improving concrete at an elementary level. The most recent trend of reinforcing concrete mixtures with fibrelike addition can, in fact, be considered an attempt to enhance the fundamental material properties.
Despite the fact that information on this topic is still limited if compared to other materials, the use of fibrereinforced concrete (FRC) has seen a quick application growth and it is now regularly employed in numerous structural applications. For instance, FRC is employed in the field of bridges, pavements, and architectural structures (Hassan et al. 2011).
Despite the potentials of FRC and its wide range of applications, a standardised test method to describe the material’s behaviour in tension and compression has not yet been found (Hassan et al. 2011).
1.1 / Aim and objectives
The aim of the project is to determine and quantify the benefit due to fibrous materials to cement mixtures and to observe the variation of fracture energy between a plain concrete mix and the corresponding modification.
A quantification of the properties’ enhancement will be obtained from the analysis and modelling of the experimental results. Two models will be proposed respectively to describe the tensile capacity and the fracture energy of a plain concrete and its fibre composite.
A generalised law will eventually be defined to estimate the improved tensile capacity as a function of the mix characteristic compressive strength (cylinder). The validity of the law will be limited to C30/37 concrete with a content of fibre of 1.0% and 2.0% in weight. Similarly, a relation for the concrete strain at maximum stress will be derived.
While the first model is considered to produce an estimate close to the improved tensile capacity, the second model proposes a graphical resolution to simulate the behaviour of the composite and to estimate the order of magnitude of the fibre addition benefit.
1.2 / Methodology
An experimental approach will be adopted to achieve the project aim. A nonstandardised direct test will be designed for the testing of the material along with the shape and size of the specimen to be tested.
The validity and the ground for comparison of the experimental data will be based on the results obtained from the control cubes and cylinders prepared from the same concrete batch and tested according to the British Standards “Testing of Hardened Concrete” (BS EN 12390:2009).
Prior to the study of the results a simple statistical analysis of the data will also be performed in order to reduce potential error due to equipment and testing procedures inaccuracies.
The purged data will then be used as the starting point for the definition of the models.
Eventually, a computer model, prepared with ANSYS finite element analysis software, will be introduced to determine whether simple modelling techniques based on the macroscopic structure of the material can be considered sufficient to explain the material response to stress.
2 / Literature review
In this chapter, a critical analysis of previous works on the topic of fibrereinforced concrete will be presented. All aspects analysed derived from either the necessity to verify certain assumptions or from general research on the subject. This review will keep into account both supporting as well as opposing facts.
The existence of composite material is not new. Numerous examples of composites materials can be found throughout history. A famous composite example can be dated back to the Mongols, inventors of the socalled modern composite bow (approximately 1200 BC).
The genius of the Mongols was to understand that two different material characterised by opposing properties could have been juxtaposed to produce a third material with enhanced properties. The composite bow was, in fact, the result of a combination of horn and sinew to respectively work in compression and tension.
Despite the topic of composite materials having been the focus of many research papers and publications, material on its testing in direct tension test are limited.
Several publications can be found on the modified indirect tension test – or Barcelona test – such as “Splitting tensile test for fibrereinforced concrete” (Denneman et al., 2011).
But, as outlined by van Vliet et al. (2000), when some difficulties are experienced in the uniaxial tensile test, this should not imply one reverting to an apparently easier experiment to be performed, but with more difficult results to be interpreted. Denneman’s experiment, besides its setup complexity, produces results difficult to interpret.
For example, Denneman’s model aims to calculate the loaddeformation curve in order to derive the peak stress on the main failure plane while also keeping into consideration the secondary cracking. The reliability of a test aiming to identify two or more parameters is almost impossible due to the material variability, scatter, error of testing method, and mathematical illconditioning (Bazant, 2002).
Besides the complexity of the results analysis, an important aspect that doesn’t appear to be clear from Denneman’s method is the observation of FRC behaviour after cracking and thus the benefit of the fibre reinforcement to the concrete composite.
The modified test proposed by Denneman does not consider the modified nature of the material, and he forgets to account for an important aspect such as the improved ductility of FRC. Testing a ductile material through a standardised splitting test configuration is not possible (Denneman et al., 2011) as it will only test the strength of the composite matrix. In addition, Rots et al. (1989), while analysing the concrete fracture behaviour in direct tension, has noticed that it is difficult to obtain realistic results due to the composite nature that can neither be classified as elasticperfectly brittle nor as elasticperfectly plastic.
The author agrees with the vision proposed by Marcel et al. (2000) and understands the need of deriving a standardised test aims to compare the results obtained from between the conventional Barcelona test and the direct tension test, limited to Mode I failure (Splitting). This approach should provide reasonable findings
2.1 / Specimen profile design
The original mould design previously used in this typology of experiment within the London South Bank University was a prismatic 100 x 100 x 400 mm specimen with a onedirectional necking at its middle.
Major findings on the sizeeffect of the specimen was found in the extensive work of Bazânt (1997) and Carpineti (1994). It was, in fact, seen that a sizeeffect does yet exist, and it is due to a number of factors amongst which the heterogeneity of concrete.
From visual observation of the existing mould, no particular relations were found between the necking’s – the narrower section of the specimen – geometry and the fibre’s length. The necking was reduced to a cross section of to 50 x 100 mm^{2} to maximise the likelihood of failure in such area.
Several theories have been proposed about the specimen size. According to Bazânt (1997), large specimens are required to guarantee a homogeneous fibre distribution in the area where the failure needs to be induced (“ligament area”). This specimen size, however, is also subjected to large vertical flexure due to the irregular crack development, which can be negligible on small specimens (Barragan et al., 2003). Small specimen results are in turn hindered by the boundary conditions. Limitations due to the testing equipment and available materials must also be accounted for in the design of the specimen.
As a compromise, the specimen size was defined both in relation to the fibre length and the aggregate’s maximum size. The smallest necking dimension was taken as the average fibre length in order to reduce fibre balling and maximise random fibre distribution (Hassan et al., 2012). At the same time, the average of the necking dimensions was seven times the aggregate maximum diameter as per guidelines proposed by van Vliet et al. (2002) who suggested a minimum specimen size eight times the maximum aggregate size.
This was further implemented by limiting the necking to one dimension only. The depth of the necking was fixed at 100 mm (twice the fibre length) allowing for free fibre rotation in that direction.
Reducing the area of the necking in order to increase the chance of failure within this region is not a new technique. The method does exploit the definition of stress of force per unit area and by decreasing the available area it increases the stress. The critical aspect lays in the procedure adopted to produce the necking. The two main methods are by scaling of a cured prism or by casting.
The scaling of a notch, according to Marcel et al. (2000) besides changing the effect of the notch can also cause damage to the specimen. The ideal solution would involve a modified geometry with cast production process. The bays forming the necking would have a circular profile that, as proven by van Vliet (2002), reduces the stress concentration at the ends increasing the probability of failure within this area.
As a circular bay profile can involve geometry discrepancies due to manual production processes, a modified trapezoidal bay will be used instead.
2.2 / Concrete mix design
Concrete is a heterogeneous material (van Mier et al. 2002) and as such it is possible to distinguish its aggregates. By understanding that the heterogeneous concrete internal structure can influence the stiffness distribution within the material, Van Mier predicts a possible nonuniform stress distribution, thus a nonhomogeneous stretch during testing.
However, it might be objected that this technical problem is valid for small ratio of aggregate size to specimen’s dimensions. And, more importantly, this might not be the case for fibrereinforced concrete where a more complex matrix is employed.
In order to avoid problems related to the particle size, as previously observed, a coarse aggregate of nominal maximum size of 10mm is used for all the concrete mixtures, as it is also suggested in “Fibre Cements and Fibre Concretes” (Hannant, D.J., 1978).
No particular considerations were found about the watercement ratio and the workability of the mixture.
3 / Background
Before the advent of fibres as a concrete enhancing material, a limited amount of research was carried out to understand and model the behaviour of concrete in tension.
Common testing to determine concrete’s tensile strength are:
 Three point flexural test,
 Splitting test,
 Direct (or axial) test.
The flexural test according to BS EN 123905:2009 involves the testing up to failure of a prismatic specimen with dimensions meeting BS EN 123901:2012 requirements. The test is carried out through the application of a load at the two thirdpoints of the specimen while being simply supported at both ends. The test configuration is shown in figure 3.1.
Figure 3.1 – BS flexural test configuration
From the load causing failure of the prism the flexural strength is obtained as
3.1 
Where F is the failure load, I is the distance between the supporting rollers and d the cross sectional characteristic dimension for a squared section prism.
Following the Eurocodes guidelines, the flexural tensile strength can be used to calculate the axial tensile strength as
3.2 
It can be easily calculated that for a standard specimen 100 x 100 x 350 mm, the tensile axial strength is 6070% of the flexural tensile strength.
The second method for the determination of the tensile capacity of concrete is the cylinder splitting test. The test involves the crushing of a d x 2d cylinder, where d is the diameter in mm, as per BS EN 123901:2012 requirements.
The specimen is tested to failure through the application of a load perpendicularly to its longitudinal axis.
As this testing method is used in the experiment as a control mean, its setup and procedure will be described in the experimental section of this paper.
The last method is the direct axial test. Despite the test being mentioned in the British Standards and Eurocodes, at this time there is no standard testing procedure defined.
However, as the test’s name suggests, the result obtained from the test are likely to be the actual direct stress value. The force is, in fact, applied along the longitudinal axis of the specimen inducing a stress on the sample crosssectional area. This stress, by definition, is the direct (or normal) stress due to the tensile force applied.
On this very concept, the test setup adopted for the main laboratory experience was based, the initial experiment configuration can be seen in figure 3.2.
Figure 3.2 – Uniaxial tensile test concept
3.1 / Stressstrain curve
Plain concrete is a material with a complex behaviour (Mosley, B., Bungey, J. and Hulse, R., 2007). It does not behave linearly, and its properties vary depending on the cement, the aggregates and the amount of water used to bound everything together in a macroscopically heterogeneous material.
Several models describing the behaviour of concrete under stress have been described by different scientists and engineers.
The most noticeable ones are:
 Hognestad (1951). His model described both the ascending and the descending part of the curve. Up to the maximum compressive strength, the curve is described by a parabolic function thereafter by a straight line.
 Kent and Park (1971). In their model, they generalised Hognestad’s model to obtained a more practical model.
 Popovics (1973). He aimed to describe the stressstrain curve as a single function depending on three parameters (f’_{c}, ε_{co}, E_{c})
The model that will be adopted for the modelling of concrete in compression are:
 Hognestad (1951) with the parametric values to meet EC,
 and, the proposed EC model for nonlinear structural analysis.
3.1.1 / Hognestad model
Experimental results (Rots, J. G. et al., 1989) show that concrete is characterised by a nonlinear behaviour in compression. Hogenstad’s method aims to describe the material behaviour through the use of simple functions. A parabolic shape defines the stress strain relationship of the concrete as the ratio between the applied stress and the maximum compressive stress, and it is defined as
for any  3.3 
This relation is valid only up to the failure strain (strain at first crack development) i.e. for the ascending part of the stressstrain curve.
For the descending part, or softening region, the above ratio is instead described as a straight line down to failure.
for any  3.4 
Where the above notations stand for:
σ applied stress,
σ_{cu} peak compressive strength,
ε induced strain due to stress σ,
ε’_{0} strain at peak compressive strength, failure strain,
ε_{cu} ultimate strain
The initial elastic modulus, or dynamic modulus, can also be determined as the tangent at the origin to the curve.
3.5 
The EC values for the designed concrete mix – C30/37 – were adopted to calculate Hognestad stressstrain curve, summarised in table 3.1 (EC extract available in appendix A)
3.1.2 / Eurocode model
Differently from Hognestad’s, the method proposed in the EC guidelines is described by a single equation for both the ascending and descending part of the curve. The EC curve equation recalls the European Concrete Committee (CEB) equation for shortterm loading.
3.6 
where
3.7  
3.8 
The Eurocodes instead define η and k as
3.9  
3.10 
Inserting the EC values (Table 3.1) for the designed concrete mix into equations 3.3 and 3.4 for the Hognestad’s model and 3.5 for the Eurocode model, the graph in figure 3.3 was obtained.
Figure 3.3 – Comparison of Hognestad and Eurocode model
3.2 / Fracture energy
The fictitious crack model by Hillerborg (1976) is the most popular model adopted for description of the behaviour of a finitesize fracture process zone. This model is herewith proposed as it does not require knowledge of fracture mechanism (Bazânt, 2002).
The applicability of the Hillerborg model presumes certain assumptions: (Bazânt, 2002)
 the failure is due to the tensile stress (failure mode I),
 stress is applied normally the crack surface and no other external stresses are involved,
 the fracture process zone is of a finite width (localized cracking) and it can be described by a fictitious crack surface perpendicular to the stress.
An indicator of the total energy dissipated (G_{F}) by the fracture mechanism is the area subtended by the entire stressseparation curve of the softening region. The value obtained is the energy dissipated per unit area of the crack until its full separation.
Hillerborg also defines the area under the initial tangent to the stressseparation curve and names it G_{f}. This value according to Bazânt, is the main parameter that determines the maximum allowable load of the structure. G_{F} and G_{f} can mathematically be described as
,  3.11, 3.12 
Where the above notations stand for:
G_{F} total fracture energy
G_{f} area under initial tangent
f’_{t} tensile peak stress
w crack separation
f(w) stressseparation curve
σ’_{0} gradient of initial tangent
According to Planas et al. (1992) and Guinea et al. (1994) an experimental value of the ratio between these two values can be seen to be
3.13 
and it can be used to determine the shape of the bilinear softening curve describing the fictitious crack model.
4 / Finite element modelling process
4.1 / Introduction
Within polymer fibre reinforced concrete, as well as for all the other typology of fibrereinforced cementitious composites, fibres and matrix are mixed together to achieve an enhanced material.
Many factors are known to be determining the behaviour of the composite material. For instance, several studies can be found in the literature on the importance of fibrematrix interface (Owen et al., 1968, Paramisivan, 1984, and Pedziwiart, 2009), the typology and the geometry of the fibre (Uomoto, 2002) or the influence of the concrete mix used for the composite’s matrix.
This chapter will describe the process behind the design of a computer model aiming to study the behaviour of the matrix depending on the fibre content.
The purpose of the models is to estimate the material behaviour for all the three concrete mixes – Plain, 1% and 2% of content fibre in weight. Two estimates will be derived from a model based on a oneelement structure that accounts for both the concrete and the fibre behaviour, the second will instead use a twoelement structure simulating the fibre distribution in a matrix.
The models prepared for the analysis are as follows:
 Model 1: SOLID65 with reinforcement values and plain concrete elastic modulus;
 Model 2: SOLID65 and SOLID285 respectively modelling the matrix and the fibre components.
Due to the complexity of the geometry of a composite material model, a reduced simulation will be carried out to determine the validity of the model results. This simulation will be limited to 10% of the fibre content to reduce the computing time for each operation.
4.2 / Finite element analysis by ANSYS
4.2.1 / Concrete model
The properties of concrete vary depending on the design mix characteristics and they are different whether the material is subjected to tension or compression.
Although the behaviour of concrete can be assumed elastic for low stress values, up to 3040% of tensile and compressive capacity (Bhatt, MacGinley and Choo, 2005), the same is not true for higher stress values at which the concrete shows a behaviour completely nonlinear.
Creating a verisimilar model for complex material such as concrete can be challenging for a professional modeller. For this reason, ANSYS finite element software was chosen for the modelling as it already incorporates an element able to simulate concrete behaviour.
Figure 4.1 – Solid65 element (ANSYS, release 5.5)
The ANSYS designated element for modelling is SOLID65 (Figure 4.1). This element is an 8node solid in which all nodes are defined by three degrees of freedom, translation parallel to the X, Y, and Z axis. In addition to allowing for approximation of the plain material behaviour, this element can also account for the reinforcement presence through the material “real constants”. This will be discussed further on in the design of the first model.
SOLID65, as a solid material, requires the definition of isotropic material both in its elastic (linear) and plastic region (multilinear). The material’s properties adopted throughout all computer models for SOLID65 are listed in table 4.1 and they are obtained from the stressstrain model previously presented.
4.2.2 / Fibre model
The model for the fibrous component of the composite was modelled using SOLID285, a solid element with only four nodes (Figure4.2), in order to reduce the number of equations and boundary conditions necessary for the computer simulation. The element definition according to the National Agency for Finite Element Method and Standards (NAFEMS, London) states that the element has a linear displacement and hydrostatic pressure behaviour.
Figure 4.2 – Solid285 Element (Ansys, released 14.0)
This is ideal for the modelling of the fibres as it allows for linear deformation of the fibres up to matrix cracking. The hydrostatic pressure can be replaced with the value of volume change rate for quasiincompressible materials. This can be assumed to be accurate up to the critical stress due to the large difference between the concrete and polymer fibre tensile strength.
The material properties adopted for SOLID285 were obtained from the supplier technical specifications for the product here listed in table 4.2.
4.3 / Material model
4.3.1 / Fibre distribution approximation
The model consists of two separate volumes: the matrix and the group of fibres. The added fibres can be assumed to be homogeneously distributed within the matrix as the concrete paste is mixed for five minutes as per supplier specifications.
The distribution of fibre within the matrix was determined using the RANDfunction available in Office Excel by Microsoft. The function, which is designed to randomly generate a number within a given range, was used to determine five values, respectively:
 the x, y, z coordinates of the fibre centroid, within a 0.1 x 0.1 x 0.1 m volume;
 the angle θ for the rotation about the ZZ axis, between 0 and π radiant;
 and, the angle γ for the rotation about the YY axis, between 0 and π radiant.
Using simple geometrical properties, and the values previously determined in excel the coordinates of the fibre’s extremes were determined as two points equidistant from the centroid along the direction defined by the angles of rotation θ and γ (Figure 4.3).
Figure 4.3 – Fibre generation diagram
Once the method to control the position of the fibres in the matrix was determined, the number of fibres had to be estimated from the properties of the fibres for each fibre content fraction.
The fibre’s shape can be described to have a twodimensional wavelength shape with a circular cross section of diameter 0.95mm and 45mm length. As the effective length of the fibre can not be easily calculated, the fibre was idealised as a cylinder of diameter 1.5mm and length 45mm. This approximation is possible only because the bond element between fibres and matrix is not kept into account for the purpose of this project.
The number of fibres for the two fibre contents was determined as follow (Table 4.3).
4.3.2 / Reduced model
Finite elements models with complex volumes derived by random factors, as in this experiment, can produce a substantial amount of elements because of the lack of regular patterns in the geometry. A complex model also implies a considerable amount of computing time for each solution.
In order to prove the validity of the model strategy that is going to be adopted throughout the finite element analysis, a reduced model has been designed to determine the typology of results.
The following modifications were adopted:
 the sample volume was taken as an eighth of the initial volume. The new test volume was 0.05 x 0.05 x 0.05 m;
 The fibres crosssectional area was assumed square with area equal to the circular one;
 and, only 10% of the fibre content was modelled.
4.3.3 / Computer model generation
The generation of the material model followed three basic steps for all the material samples. Firstly, the fibre volumes where lofted along the lines drawn using the coordinates of the fibres’ extremes obtained from Excel; these were then copied and offset outside of the 0.05 x 0.05 x 0.05 m volume of the matrix (Figure 4.4.a). Secondly, the volume of the matrix was created (Figure 4.4.b) and the fibre’s volumes were subtracted to the matrix volume (Figure 4.4.c); the two were superimposed to match the original fibre position within the matrix (Figure 4.4.d).
a)  b) 
c)  d) 
Figure 4.4 – Model generation sequence
a) Fibre modelling and offset, b) matrix modelling, c) fibre volume subtraction,
d) Superposition of Fibre and matrix volumes
Once the volumes were modelled, the material’s properties, previously listed, were applied accordingly.
The model was then further optimised through the application of the following load and boundary conditions:
 symmetry conditions were applied on the volume faces parallel to the applied load, this allowed for a more refined result without increasing the number of model elements;
 the load was applied as a surface pressure on the top surface. Particular attention was given to the sign convention. Positive defines a compressive and negative a tensile strength,
 and, the translation in the direction of the applied load was assumed to be zero to simulate a vertical restraint.
4.3.4 / Reduced model results
Several combinations of loads and fibre content were performed for the reduced model. For a chosen load of 50,000 Pa the sets of results from the different sets were graphically compared (Figure 4.5).
a)  b) 
Figure 4.5 – Stress and strain variation along YY axis
Fibre Content 1.0% in weight
From figure 4.5, it appears that the model is behaving as it was assumed it would have.
The distribution of stress does, in fact, show an increase of stress concentration around the fibre. In the graphs, this can be recognised in a minor peak before the major ones of opposite magnitude. The major stress peaks occur at the fibre location, only within the fibre element, this is due to a change of material and, thus, of the material elastic modulus.
A similar and opposite behaviour is seen in the strain variation along the YY axis (Figure 4.5.b). By recalling Hooke’s law of linear behaviour for small stresses (Equation 4.1)
4.1  
4.2 
It can be observed that the elastic modulus is close to 30 GPa as per plain concrete. Similarly, the stress and strain behaviour for the 2% fibre content model were obtained (Figure 4.6).
a)  b) 
Figure 4.6 – Stress and strain variation along YY axis
Fibre Content 2.0% in weight
4.3.5 / Validity of finite element model
From the analysis of the results just obtained, the model can be considered acceptable. Hooke’s law is respected throughout the tests for the relatively small stress applied and, furthermore, the expected behaviour has been satisfied as is shown in the previous graphs (Figure 4.56) and the stress distribution below, plotted via ANSYS on four cross sections parallel to the load direction. (Figure 4.7)
Figure 4.7 – Stress and Strain Distribution along YY axis
4.4 / Tensile specimen model
The model of the tensile specimen, different from the one employed in the determination of the validity of the method, is designed following the real dimensions of the specimen to be tested. As anticipated, two models will be here introduced.
4.4.1 / Model 1 – SOLID65 with reinforcement values
For the simulation of the fibre addition to the matrix, modified “real constants” valueswere adopted for the experiment. From the concrete design properties and the fibre characteristics, the following parameters were used respectively for the 1.0% and 2.0% fibre content in weight.
The model results for low value of stress and their projection is shown in figure 4.8.
Figure 4.8 – Model 1 and 2, low stress results and their projection
4.4.2 / Model 2 – SOLID65 with modelled fibres
The second model uses the material property values available in table 4.1 and 4.4. The symmetry of the model was, however, exploited to reduce the computational time. The configuration of the model was as per figure 4.9.
The simulation was carried out for the same loads applied to model 1.
4.4.3 / Model limitations
Despite the fact that the model appeared to fairly approximate the material response to applied stress in the validation of the reduced model, this has been observed not be adequate for high stress values close to the plastic region of the material. A possible solution to this restriction will be proposed in the conclusion chapter.
5 / Experimental process
5.1 / Mould design
Substantial influence on the quality of the specimen to be tested comes from the quality of the casting mould.
Alternatively from the mould used in previous tests, modifications were brought to the design in order to minimise damage to the sample and to simplify the demoulding process.
The main differences were as follow:
 mould sides were designed as one piece with the base in order to reduce the assembling and breakdown time,
 the inside of the mould was lined with High Impact Polystyrene (HIPS) to eliminate the gap between the sides and notch elements, and also to reduce concrete adhesion to the mould,
 and, the elements marking the clamps’ groves, previously made of timber, were replaced with a silicone rubber equivalent that can easily be removed without damaging the clamping area.
5.1.1 / Design process
The design steps were as follows:
 the base of the mould was divided along its longitudinal axis, the sides were fixed to the base through a 5 mm slot and 50 x 50 x 8 mm steel angled profile;
 the two sides were then reconnected through a 300 mm wide board placed beneath the side elements and connected with M6 bolts;
 the gap between the two sides was made watertight with a 4mm silicone strip as well as all other small gaps due to production defects;
 end plates and a middivider were then inserted in order to restrain side’s movement due to concrete pressure;
 and, the mould interior was lined with HIPS to eliminate any gaps between mould components and to facilitate the demoulding process.
Detailed drawings of the mould design are available in appendix B, C, and D.
5.2 / Materials and mix details
The materials used for the test were: PortlandLimestone cement (Mastercrete – LAFARGE) that meets BS EN 123902:2009, 10 mm maximum size coarse aggregates, natural sand, and polypropylene fibre with a density of 910 kg / m^{3}.
Moisture content and sieving tests were performed for both aggregate types to confirm the suitability of the material and their initial condition. From the sieving test results (Figure 5.1) and observation, both the sand and the gravel appeared to be uncrushed, nonangular and well graded minimising the opportunity of void forming.
Figure 5.1 – Particle size distribution chart. Left) Fine aggregate, Right) Coarse aggregate
An increase in water content can be assumed from the findings of the moisture content test. Fine and coarse aggregates were, in fact, found to have a water content – in grams – respectively of 0.37% and 0.51% (Table 5.1). These characteristics can be said to increase the water/cement ratio of approximately two percentage points for the designed concrete mix.
This factor was, however, not included due to the difficulty on monitoring the water content of the aggregates during their handling.
The fibres adopted were 0.95 mm in diameter and 45 mm long (Full supplier specifications available in table 4.2), characterised by a twodimensional sinusoidal shape (Figure 5.2)
Figure 5.2 – Propex ENDURO HPP45 (propex.co.uk)
The mix used was valid for normal weight concrete with normal compressive of 37 MPa at 28 days, the design followed the guidelines proposed by the Building Research Establishment (BRE) in Design of Normal Concrete Mixes.
Details of the mixes can be found in table 5.2.
Fibre contents of 1% and 2% of the concrete weight were used throughout the experiment as well as a plain one for the control of the results. The respective volume fraction for the two mixes was calculated to be approximately 0.5% and 1.0%.
5.3 / Test specimen and procedure
The materials were mixed using a laboratory concrete mixer adding in order: coarse aggregates, fine aggregates, and cement. The dry components were mixed until they reached an even tone. Then, water was gradually added in, and mixing was carried on until the paste looked homogeneous, approximately 2 minutes.
Eventually, the fibres were added to the concrete paste. In this step particular care was given to the addition of fibres as adding the fibres in bunches could have caused the balling of fibre within the mix.
Along with the tensile specimens, for which the special mould was produced, two control cylinders (150 diameter x 300 mm), and three control cubes (100 x 100 x 100 mm) were cast for each batch. The number of control specimens was limited due to the limited number of moulds available in the testing facilities.
5.3.1 / Control cube
The control cubes were cast and tested following BS EN123903:2009. Each 100 x 100 x 100mm was tested in a compressive test plant with a maximum load capacity of 3,000 kN. A constant load rate of 3,500 N/sec was applied throughout the experiment.
From the testing, the following results were found.
According to the mix design, the target strength for the designed mix was 43.1 MPa, however, for the determination of the concrete mix quality the cube target strength of 47 MPa, proposed in the Eurocodes, was considered.
As it immediately appears, the concrete strength of the mix with 0.5% fibre content is 9.4% lower than the same mix without fibres. As this result does not meet the expected strength value, it can be assumed defective.
Further discussion on the validity of the FRC_0.5 batch will be discussed further on along with the splitting test and tensile test results.
The plain and the 1.0% fibre content mixes do, instead, behave as per design estimate. The standard deviation of 5.0 N/mm^{2} from the target value is, in fact, lower than the 8.0 N/mm^{2} suggested by the BRE (1988). Furthermore, the coefficient of variation for both PC and FRC_0.5 shows that the data lay within a small range from each other. Hence, despite the low number of samples, they can be assumed to be reasonably characteristic of the design mix.
5.3.2 / Control cylinder
The splitting cylinder test was also carried out to determine the tensile strength of concrete. Testing procedure followed BS EN 123906:2009 and it was performed in the same testing equipment used for the control cubes with the same load rate of 3,500 N/sec.
The result obtained confirmed the possibility of a mix defect for the FRC_0.5.
The splitting cylinder test results are presented in table 5.4.
Table 5.4 – Cylinder testing results  
Mix  Cylinder #
MPa 
Mean strength MPa  
1  2  
PC  2.521  3.108  2.81  
FRC_1.0  3.766  4.032  3.90  
FRC_0.5  2.917  2.927  2.92 
Due to the limited amount of samples tested it would be erroneous to determine the variation of data from standard deviation and coefficient of variation.
On the other hand, it can still be observed that the values for PC are close to the target splitting tensile strength of 3.2 MPa proposed in the EC, and a 38.8% strength increase in the FRC_1.0 can also be noted.
5.3.3 / Tensile specimen
Although a standardised method outlining the testing procedure results is not yet available, direct testing of the tensile strength of concrete is considered a possible method to determine the tensile capacity of concrete by both Eurocode and British Standards.
In order to allow for a more reliable future comparison, a detailed experiment setup will be herewith described.
The tensile specimens to be tested were produced using the concrete mix previously described and cast in the specially designed mould. They were allowed to rest for 48h prior demoulding, and then left to cure in water for approximately 26 days.
On the testing day the samples were removed from the curing tank and once dry they were fitted with two displacement sensors, also know as Linear Variable Differential Transformers (LVDT). The sensors were placed in the middle on the narrow side of the necking covering a 75mm length. A diagrammatic explanation of the gauges setup is shown in figure 5.3.
Each test specimen was equipped with two clamping systems, one at either end of the specimen, after that the instant adhesive used to install the gauge mounting blocks and angled plate had set.
The clamping system was composed of two metal plates slotting into the 7 x 11 mm grooves bridged at their outer edge by a shaft. The horizontal shaft was in turn connected to the primary shaft via a hollow cylindrical element.
The two steel plates were eventually connected together to clamp the specimen with four M12 tightened to the tightening torque of 20 Nm. The machine setup is shown in figure 5.4. It is important to notice that the horizontal shaft was free to rotate inside the hollow cylindrical element.
Figure 5.4 – General test arrangement
The testing equipment employed for the experiment was a highcapacity static materials testing machine by Zwick/Roell The machine was fitted with tension wedge grips both on the bottom and top cross heads.
The wedge grips were used to clamp the primary shafts from the clamping systems allowing for the test to be carried out.
Once the setup of the specimen was completed the test was started.
Two computers were collecting the readings respectively from the machine and the two gauges. The first computer, which was the main control unit for the testing equipment, was set to record the cross head translation and the induced load; the second was to collect the gauges’ displacement readings for each load recorded by the main control unit.
Although the testing equipment was able to collect value of translation, these were not used due to the unavoidable slipping of the primary shafts out of wedge grips. Please note, that no alternative testing equipment was available.
Hence the LVDTs’ vertical translation was, along with the correspondent load value, the only data considered. The testing for the plain concrete specimens was carried out up to fracture while the fibrereinforced concrete specimens up to a machine translation of 5 mm.
5.4 / Results. Interpretation and reduction
The raw data collected were divided in three arrays:
 Imposed load – resolution 0.01 kN
 Extension 1 – resolution 0.001 mm
 Extension 2 – resolution 0.001 mm
5.4.1 / Readings up to failure
A first conversion involved averaging the effective extension values (original value minus the observed reading at beginning of the testing) and dividing it by the gauge length of 75mm. Then, the induced stress in the necking was obtained as the applied load divided by the necking area (0.046 x 0.1 m^{2}).
From the plot of these results, the following behaviour was observed.
Figure 5.5 – Specimen T1, raw data and fitting line
Despite the fact that several results seem to be scattered without any correlation, it also appears a strong concentration of results along, what seems to look like, a curve with a steep initial gradient as proposed by Hognestad for the compression stressstrain curve.
A defined vertical arrays distribution arise from the plot; this is due to the low resolution of the LVDT, in respect to the very small strain of concrete. There are, in fact, different load readings for the same strain value.
Exploiting the builtin curve fitting tool of Microsoft Excel it can be seen that a second order polynomial intersecting the origin is close to the data concentration earlier observed, the value of the coefficient of determination (R^{2}) is low. The validity of the curve is probably weakened by the several values falling out of the concentration area, such values will be hereafter referred to as outliers.
In order to remove most of the outliers and obtain an improved fitting curve, it was decided to consider only the data around each array’s median, i.e., the stress value dividing the array in the higher and lower half.
To do this, the statistical tool of quartiles was introduced. Quartiles are, as the word suggests, those values that divide an array of data in four equal groups. They can be summarised as follow:
 Zero quartile (Q0), the lowest value of the set, i.e the minimum,
 First quartile (Q1) divides the lower 25% of data from the upper 75%,
 Second quartile (Q2), also referred to as median,
 Third quartile (Q3), divides the lower 75% of data from the upper 25%,
 Fourth quartile (Q4), the highest value of the set, i.e., the maximum.
From the box plot of the quartiles, it appears that the 50% of each data array around the median was scattered around the observed pattern.
Figure 5.6 – Specimen T1, BoxPlot
The elimination of the lower and higher 25% of the data resulted in a 46 percentage point increase in the R^{2} value of the fitting curve (Figure 5.7)
Figure 5.7 – Specimen T1, refined data with improved fitting line
The method was, therefore, considered correct, and it was applied to the other 6 sets of values. The following graphical results were obtained.(Figure 5.812)
Figure 5.8 – Data refinement – Specimen T2
Top) Raw data, Middle) BoxPlot, Bottom) Refined data and best fitting line
Figure 5.9 – Data refinement – Specimen T3
Top) Raw Data, Middle) Box plot, Bottom) Final refined data and best fitting line
Figure 5.10 – Data refinement – Specimen T4
Top) Raw Data, Middle) Box plot, Bottom) Final refined data and best fitting line
Figure 5.11 – Data refinement – Specimen T5
Top) Raw Data, Middle) Box plot, Bottom) Final refined data and best fitting line
Figure 5.12 – Data refinement – Specimen T6
Top) Raw Data, Middle) Box plot, Bottom) Final refined data and best fitting line
5.4.2 / Reading after failure
Following the failure of the plain concrete specimen, no readings were collected due the instantaneous separation of the specimen at the crack and hence its inability of carrying any stresses. (Appendix F)
This is, however, not true for reinforced concrete where the reinforcement comes into play after the first cracking development. Readings describing this phenomenon, previously introduced as strainsoftening, were once again collected from the gauges. However as the crack needs to develop in the range of the gauge for it to collect relevant data, only two sets of data out of three were admissible (Apendix G). T6 did, in fact, fail at the upper bound of the notch causing the gauge blocks to fall, the value from the 0.5% fibre content batch was also ignored as defected.
Differently from the data prior cracking, the actual extension (mm) was used to describe the behaviour of the sample against the applied stress. In fact, following Hillerborg previously introduced definition of fracture energy, it can be defined as the area subtended by the stress (N/mm^{2}, or MPa) separation (mm) curve.
In this instance, no reduction of the data was necessary.
The results were plotted from the peak tensile stress onwards and a second order polynomial passing through the peak stress was fitted using MATLAB curve fitting tool (Figure 5.1314).
Figure 5.13 – Specimen T4, stressseparation curve with fitting curve
Figure 5.14 – Specimen T5, stressseparation curve with fitting curve
6 / Discussion
6.1 / Tensile capacity. Direct and splitting tests
Limited to the data collected during the experimental experience of this project, the tensile capacity of the designed concrete mix have been seen to benefit of a considerable 26.1% improvement.
This material improvement can be assumed to be entirely due to the addition of fibres as the two batches were prepared at the same time with the same materials and were cured under the same conditions.
The collected data does, however, not match the expected value of tensile strength of 2.9 MPa proposed in the Eurocodes, the result for plain concrete is, in fact, 37.0% lower than expected.
On the other hand, the splitting tensile strength and the compressive strength were found to be respectively only 12.2% lower and 4.6% higher than the proposed EC values.
Table 6.2 – Results variation from Eurocodes’ values  
Strength type  Experimental
MPa 
EC2
MPa 
Difference
% 
Compressive
Splitting tensile Axial tensile 
49.14
2.81 2.12 
47.0
3.2 2.9 
4.6
(12.2) (27.9) 
The high discrepancy between the values of axial strength would normally be an indicator of mistakes in either the testing procedure or the sample production. However, being the EC value theoretically derived from the splitting test (BS EN 199211:2004, 3.1.2(8)) and keeping into account that there are currently no standards for concrete testing in direct tension, this value is considered a reliable indicator of the actual tensile strength of the designed concrete mix.
6.2 / Modified Hognestad model
From the comparison of the linear model and the experimental results (Figure 6.1) it appears that the assumed linear response of concrete in tension is not an accurate describer of the material behaviour.
Figure 6.1 – Fitting of linear model to fit T13 data
From observation, the pattern described by the collected results recalls the shape of the ascending region of Hognestad’s stressstrain model for concrete in compression. The Hognestad model can, hence, be amended to model the tensile region (Figure 6.2). The modification involved changing the maximum compressive stress value to match the experimental mean tensile strength (2.117 Mpa).
Figure 6.2 – First modification of Hognestad model fitting T13 data
The adjusted model shows a stronger correlation with the experimental results than the linear model. It yet shows a moderate gradient at the beginning of the curve closer to the secant value of elastic modulus – E_{C} – than the expected dynamic elastic modulus – E_{0}.
A further amendment to Hognestad model involves matching the failure strain with the strain values corresponding to the maximum tensile stress. This modification considerably improved the quality of the fitting curve up to the maximum stress value, as it is shown in figure 6.3.
This model does, however, not explain the material behaviour after first cracking development.
Figure 6.3 – First modification of Hognestad model fitting T13 data
A similar model can be applied to the results of the fibre reinforce concrete samples.
Figure 6.4 – First modification of Hognestad model fitting T46 data
6.3 / Finite element model results
The results obtained from the finite element models, despite the fact that are limited to stress values below 30% of the maximum tensile capacity, show a good approximation of the experimental results. (Figure 6.5)
Figure 6.5 – Finite element against experimental results
On the other hand, considering the extensions of the two fitting lines an approximation of the material behaviour at higher stresses, only model 2 display a good match.
This is very interesting and suggests that the basis of the model simulating the fibre addition to the concrete matrix might be correct, and that further development of the model could produce convergence of the model results.
6.4 / Linearity of strain variation
Furthermore, a linear correlation appears between both the failure strain and the maximum strain for both the plain and reinforced concrete (Figure 6.6). A value close to half of the characteristic strength was found to be the linearity constant for all mixes and strain values.
Figure 6.6 – Linearity of characteristic results
From the best fitting lines the following characteristics were found
Table 6.3 – Fitting lines’ characteristics  
Line  Gradient  Constant terms 
PC_Peak
FRC_1.0_Peak PC_Ultimate 
15,655
15,507 14,626 
0.7576
0.4955 0.2732 
Average  15,263 (C_{v}=2.98%)  – 
The average gradient is only 1.8% higher than half of the 30MPa concrete characteristic strength and 2.6% higher than the effective characteristic strength (31.34 MPa) interpolated from Eurocode’s values (Table 6.4).
Table 6.4 – Characteristic strength interpolation  
Eurocode values
MPa 
Experimental
MPa 
Interpolated
MPa 

f_{ck,cylinder}
f_{cm,cube} 
30
47 
35
55 
–
49.14 
31.34
– 
Please note that this value is arbitrary and it might not be correct for different concrete mixes and it has to be used in relation to the indicated units. Further investigation is necessary to confirm the hypothesis.
6.5 / Formulation of tensile strength
An indication on the magnitude of the tensile capacity of concrete from direct tensile test can be useful to determine the actual strength of either plain or fibrereinforced concrete.
The aim of the formulation is to define the tensile strength of a concrete mix on the only basis of its characteristic cylinder strength. For the project purposes, the interpolation of the cylinder strength from Table 6.4 was adopted as the effective characteristic strength of the mix (31.14 MPa).
The generalised formula was based on the following assumptions:
 a second order polynomial model is considered to be characteristic of the stressstrain behaviour of concrete up to the maximum stress,
 at zero strain corresponds zero stress,
 the model reaches a maximum (first derivative equal zero) at the maximum stress.
On this assumption the model was generally defined as it follows.
6.1 
Assuming the first derivative equal zero maximum stress the following relationship between coefficients a and b can be defined.
6.2 
ε’_{0} can be calculated from the experimental linear equation observed earlier for the average maximum tensile strength
6.3 
that can be approximated to
6.4 
The corresponding strain for the maximum tensile strength of 2.117 MPa is
millistrain  6.5 
For the obtained value of strain the coefficient b can be calculated from 6.2 as
6.6 
and by forcing the polynomial through (ε’_{0} ; f_{t}*) the coefficient a and b are univocally determined as
6.7  
,  6.8 
The model can, therefore, be summarised as
6.9 
Recalculating the first derivative at peak stress the generalised equation of strain can be derived
6.10  
6.11 
By substituting EC elastic modulus formula into 6.11 for the given characteristic strength the strain at maximum stress can be found as per equation 6.13.
6.12  
6.13 
Eventually, combining 6.13 with 6.4 the generalised equation for the tensile capacity of plain concrete can be formulated as per equation 6.14.
6.14 
Equation 6.14 can be considered a good generalised formula for the determination of the tensile strength of the designed concrete mix along with equation 6.13 for the corresponding strain.
Recalling figure 6.6, another observation can be made, the linearity of the results for the 1% fibrereinforced samples appears to lay in between the relations of the fracture strain to maximum stress and the ultimate stress to the ultimate strain. From the analysis of the three equations’ constant terms, it was in fact observed a proportionality.
Table 6.5 – Fitting lines’ constant terms  
Line  Constant term 
PC_Peak
FRC_1.0_Peak PC_Ultimate 
0.7576 (3 γ)
0.2732 (1 γ) 0.4955 (2 γ) 
The value of γ was found equal to 0.2544 [MPa] from which a more generalised layout of the previous equations can be derived (Table 6.6)
Table 6.6 – Generalised linear equations  
Line  Equation  
PC_Peak  6.15  
PC_Ultimate  6.16  
FRC_1.0_Peak  6.17 
As the 0.5% fibre content mix was not considered acceptable, it is not possible to describe a direct relation from the mean peak tensile strengths of plain to fibrereinforced concrete.
A graphical solution to determine the benefit of the fibre addition was hence proposed. The graphical resolution is presented in the figure below.
Figure 6.7 – Tensile capacity graphical resolution
The method can be summarised as it follows:
 project the peak strength onto the ultimate stress line,
 find corresponding stress,
 and, project on to FRC_1.0 line.
Mathematically, the relation can be expressed as the plain concrete tensile stress plus half the vertical distance between the two lines, i.e γ. From observation
6.18 
Inserting this value into PC_Ultimate line (Equation 6.16)
6.19 
Inserting the above value of strain into FRC_1.0_Peak (Equation 6.17) allows for the final formulation of tensile capacity of fibrereinforced concrete to be defined.
6.20 
The above solution delivers a value very close to the one found in practice with a 4% difference.
6.6 / Fracture energy for fibre reinforced concrete
In accordance with Hillerborg definition, the area subtended by the stressseparation curve represents the total fracture energy dissipated by the system with the development of the crack.
As a record of the behaviour post fracture could not be collected for the plain concrete specimens as the gauges were not sensitive enough to record the material behaviour during the quasiinstantaneous fracture failure. The fracture energy for plain concrete was calculated according to the equation proposed by Hilsdorf and Brameshuber (1991) (FIB, 2009).
for  6.21  
for  6.22 
Where
G_{F}: Fracture energy [N / mm]
G_{F0}: Base value of fracture energy, table 6.6 [N / mm]
F_{CM}: Mean compressive strength [MPa]
F_{CM0}: 10 [Mpa]
Table 6.7 – Base fracture energies per aggregate size  
d_{Max} – mm  8  16  32 
G_{F0} – N /mm  0.025  0.030  0.058 
Interpolating
6.23 
The mean compressive strength was taken as the proposed EC mean cylinder strength for C30/37 concrete.
6.24 
On the other hand, the fracture energy for the fibrereinforced specimens cannot be calculated using a simple formula, and a model based on Hillerborg’s Fictitious Crack model was adopted. As previously introduced, the fracture energy can be defined as the area subtended by the stressseparation curve.
Figure 6.8 – T4 and T5 raw data and fitting lines
Although the collected results are limited to a local extension of 2.5 mm, the curves appear to fairly approximate the behaviour of the material. It does yet show a gradual decrease in gradient following the initial drop, as observed by Grassl, P. et al. (2010), Barrang, B. E. et al. (2003), and Bazânt, Z. P. (2002).
The curve was then simplified to meet the bilinear model proposed by Hillerborg (1976). The final model was taken as two lines intersecting at the projection of the first lowpeak onto the fitting curve and
 Intersecting the maximum tensile stress value,
 And, tangent to the fitting curve at the lowpeak projection.
The model proposed is shown in figure 6.8 and 6.9 for the two sets of data in consideration.
Figure 6.9 – Fracture energy model, specimen T4
Figure 6.10 – Fracture energy model, specimen T5
It is interesting to notice that in average the low peak stress is 29.7% lower than the maximum tensile strength (Table 6.8)
Table 6.8 – Comparison between maximum and lowpeak stress  
Test  Maximum Stress (MS)
MPa 
First LowPeak (FLP)
MPa 
FLP / MS
% 
T4
T5 
2.535
2.750 
2.030
1.665 
80.1%
60.6% 
Mean  70.3% 
These values are more than two folds out of the range observed by Bazânt (2002) of 15 – 33 % of the maximum stress. This variation is assumed to be due to the fibre reinforcement. Bazânt’s model does, in fact, not account for the presence of fibres that are theoretically assumed to improve the ductility of the material, hence increasing the amount of energy necessary to obtain ductile fracture.
Using the two bilinear models proposed for T4 and T5 specimens an estimate of the fracture energy for fibrereinforced concrete can be determined.
Table 6.9 – Comparison between total and initial energy  
G_{f}
N / m 
G_{F}
N / m 
G_{F} / G_{f}
– 

T4
T5 
5,640
5,335 
9,213
10,663 
1.99
2.00 
Being the ratio G_{F} / G_{f} constant allows to considered the proposed model a good indicator of the material behaviour after failure, further test should be performed to verify this statement.
G_{f,H} according to Hillerborg definition can also be calculated. Assuming the line through the maximum stress and the first lowpeak a good approximation of the initial tangent to the curve the following values can be defined (Table 6.10).
Table 6.10 – Improvement in initial fracture energy  
G_{f,H}
N / m 
G_{F,PC} (E)
N / m 
G_{F,H} / G_{f,PC}
– 

T4
T5 
688
803 
66.8
66.8 
10.3
12.0 
The ratio G_{F.H} / G_{f,PC} can be considered to be representative of the benefit due to the fibre addition and its contribution to the energy necessary for the first cracking development. From literature (FIB, 2009) the total fracture energy for fibrereinforced concrete has been seen to be three orders of magnitude greater than the corresponding plain mix.
Having observed that the tail of the stressseparation curve considerably extends with the addition of fibre (Bazânt, 2002), i.e more energy is now dissipated following the initial crack opening, it is reasonable to assume that roughly a 10 times greater energy is necessary only for the initial crack opening.
The total fracture energy for the system is calculated as it follows and its estimate appears to be two orders of magnitude greater than the corresponding plain mix.
Table 6.11 – Improvement in total fracture energy  
G_{F,FRC_1.0}
N / m 
G_{F,PC} (E)
N / m 
G_{F,FRC_1.0} / G_{F,PC}
– 

T4
T5 
9,213
10,663 
66.8
66.8 
138
160 
Despite the estimate is lower than the values presented in literature, it is also necessary to mention that due to the material variability, possible testing errors and model approximations, it is difficult to verify the model proposed.
7 / Conclusions
The project results have shown that, limited to the experimental values collected, the fibre addition can improve the tensile capacity of concrete of approximately 20% in tension (1% fibre content) and 10% in compression, similarly to the ones observed by Kakooei et al. (2012) and Choi and Yuan (2005). These results are yet limited to a 1% polypropylene fibre content.
On the other hand, the results’ comparison of the standard splitting test and the designed direct tensile test have shown a large discrepancy from the tensile capacity proposed in the Eurocodes (2.9 MPa). Further research should, therefore, be undertaken to understand whether the 20% difference from the EC value is due to an inappropriate direct testing method, or to the inaccuracy of the splitting test.
It is also established that the splitting test in the determination of the FRC behaviour following the maximum stress is not adequate due to the ductility of the material (Denneman, 2011). A new standardised test should then be introduced on the line of the direct test proposed in the project.
Conclusively, despite the fact that the research is currently focused on the mechanics of the cracking development and the prediction of material behaviour, it is important to notice that the data previously analysed shows a significant decrease in the results range. The results gathered for the FRC specimens are, in fact, characterised by a coefficient of variation between 4.1 and 4.8% in contrast to the 8.7 and 18.4% of the corresponding plain mixture, as it was mentioned by Carmona et al (2013) and Hasewander (1995).
The power of this observation should not be underestimated because, whether the material behaviour is modelled from theory or from extensive experimental investigation, the factor that the stress response of concrete can be accurately predicted can represent a considerable economic and environmental effect for the construction sector. The current 1.5 partial safety factor for concrete could potentially be reduced, and the concrete tensile capacity could also be accounted for in the design process allowing for the reduction of materials and costs.
7.1 / Recommendations for further studies
In its entirety, the project can be considered to have successfully presented a reliable set of data and theoretical basis. Nonetheless, improvements can still be made both for theoretical and experimental processes.
A possible theoretical improvement could be made within the analysis and determination of the fracture energy. Such revision would require advanced knowledge of fracture mechanism to understand and apply alternative theoretical models. An alternative experiment configuration, which focuses on the test of a notched prism as proposed by Zhang et al (2001) and Cunha et al. (2011), should also be considered for the gathering of data describing the material’s behaviour at fracture.
Concerning the experimental side of the project, modification can be applied to:
 size of the specimen’s range,
 number of mixture with different water/cement ratio as well as fibre content,
 test setup,
 specimen geometry ,
 typology of testing equipment,
 resolution and range of the gauges,
An ideal evolution of this experiment capable of producing statistically significant results of the material strength should involve a minimum of 20 samples for each mixture and specimen typology (BRE, 1988). It should be carried out with a testing machine equipped with hydraulic wedge grips to eliminate the data corruption due to ends’ slipping. So lastly, the clamping mechanism should be designed to restrain any possible rotation due to nonuniform stress distribution following the first cracking development.
In the case these amendments were not feasible, the modification of the specimen geometry should be considered as the first possible experiment variation.
The current geometry does, in fact, define an area of stress concentration that causes an unwanted failure where the necking opens up to the end sections (Appendix E). As the gauges range over the middle of the necking, every failure outside this range makes the readings after peak stress unusable. A possible alternative to the current geometry is a simple 100 x 100 x 400mm prism with a 5mm notch 25mm deep, similar to the layout proposed by Zhang et al (2001) and Cunha et al. (2011), with 4 gauges (one per face) bridging the notch over a minimum range of three times the maximum coarse aggregate diameter.
By exploiting the localised stress concentration, this setup should guarantee failure at the notch and thus suitable data for future studies.
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Appendix
Appendix A – Table 3.1 / Strength and deformation characteristics for concrete (BS EN 199211:2004)
Appendix B – Mould construction drawings – Top view (not at scale)
Appendix C – Mould construction drawings – Side view and section BB (not at scale)
Appendix D – Mould construction drawings – Section AA (not at scale)
Appendix E – FRC_1.0 Fracture out of gauges’ range
Appendix F – Plain concrete brittle fracture
Appendix G – FRC_1.0 Fracture within gauges’ range
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