Equivalent Elastic Bending Stress versus Deflection
The load resistance, is influenced by the dimensions of the specimen and its span. Thus, in order to facilitate comparisons with findings from various scholarly articles, it is more suitable to express the load-deflection curves in terms of the equivalent elastic bending stress-versus deflection(Naaman, 2018). The stress can be calculated for any load on the load-deflection curve using the calculation provided in the (ASTM C1609/C1609M-12, 2009)standard for the third point loading arrangement as follow;
Where ;
( is the equivalent elastic bending stress; (P): is the applied load ;( b) is the width of specimen ; (h) is the depth of the specimen and (L) is the span between the supports c/c.
Figure 6 shows parts of the prisms and their fracture places. It is evident that the fractures do not always appear exactly in the center. Several of the cracks are found around the edges of the middle spans, showing that Fiber Reinforced Concrete (FRC) responds differently than regular concrete. This strength variation can be connected to areas with densely packed fiber bundles, which may increase the structural integrity of those sections. As a result, the central line of the span may not be the weakest point. However, all of the cracks originated within the middle span of the prism.
Figure 7a; displays the flexural stress versus deflection curve for carbon fiber reinforced mortar samples with fibers of 5 mm in length. Initially, the curve is intended to grow linearly, indicating the material's elastic behavior, until reaching a peak representing the final flexural strength. The peak flexural stresses measured for the series G1M1 to G1M5 were 4.61, 5.07, 6.22, 9.26, and 9.59 MPa, respectively. These results exceed the first crack formation stress, which was calculated to be 3.8 MPa using the flexural strength of the control samples without fibers. Furthermore, the findings show that fiber-reinforced specimens have a larger deflection by approximately 2 times as a maximum at peak stress, indicating the increased toughness provided by the addition of carbon fibers when compared to their non-fiber-reinforced specimens.
The insertion of fibers with a low elastic modulus into a concrete or mortar matrix reduces the total elastic modulus of the composite material (Naaman, 2008). Following the peak stress, the material's load bearing capacity declined significantly, indicating a response subsequent to cracking.
The exhibited stress-deflection curve indicates a primarily brittle failure mode, with material breaking followed by little to no plastic deformation. It is worth noting, however, that the flexural strength increased by two to three times when compared to non-fiber specimens, with only a single fracture appearing prior to reaching the peak load. Following this peak, there was a quick drop in load-bearing capability, with residual strength varying between 1 and 2.5 MPa depending on the fiber volume percent in the mixture. Following fracture creation, the specimens maintained structural integrity, rather than breaking into two different pieces, as seen in the adjacent figure.
Figure 6b; shows the impacts of several carbon fiber lengths within the mortar, ranging from 10 mm to 30 mm and incremented by 5 mm for each mix configuration. Throughout the studies, the fiber concentration is constantly about 1% by volume. This picture most likely depicts a succession of curves, each corresponding to a particular fiber length. All of these curves show a continuous tendency, with flexural stress reaching a peak that far surpasses the flexural strength of the control specimen. By interpolating the values of fiber volume fractions between 0.81% and 1.23%, one can approximate the maximal stress value for a 1% fiber volume fraction in
Figure 6b, this value is slightly greater than the maximum tension detected in specimens reinforced with a 1% volume of fibers at aspect ratio of 625.
This tendency emphasizes the importance of the aspect ratio in determining the composite material's flexural capabilities as well as its post-crack performance. One distinguishing aspect of these curves is the more gradual post-peak decline in bending stress. This is in contrast to the behavior observed in the group with an aspect ratio of 625, which showed a more rapid drop in stress. As the aspect ratio increases to 2500 and higher, the stress decrease along the descending branch becomes less dramatic, suggesting improved material characteristics.
At an aspect ratio of 3750, an anomaly is discovered in which the flexural strength drops more quickly, showing higher brittleness in comparison to lower aspect ratios. This increased brittleness is mostly due to a considerable drop in the mix's flowability and compressive strength produced by the fibers' excessive length. These lengthy fibers most likely hampered the efficient compaction process during specimen preparation.
Specifically, specimens with aspect ratios of 2500 and 3125 initially had a quick reduction in load-bearing capacity from peak values of roughly 44% and 28%, respectively, to 4.24 and 5.55 MPa, but followed by a more steady fall.
This trend is due to the greater inter fiber bonding at higher aspect ratios. A higher aspect ratio usually enhances load transmission efficiency between the fiber and the cementitious matrix. Figure 10 shows SEM images of carbon fibers mixed into a cement-based composite, resulting in short tubular structures with minimal gaps. These fibers have a high aspect ratio and small diameters, resulting in excellent adherence to the cement matrix. Under flexural stress, they increase the composite's structural integrity; beyond a maximum flexural load, they cause brittle failure due to the rupture of the majority of the fibers. This improved load transfer leads to better fracture bridging inside the matrix, which increases the composite's strength, and deflection at failure (Yoo et al., 2017).
Figure 7c; shows the bending stress vs deflection curve for mortar reinforced with 3 mm carbon fibers. This curve is predicted to follow a pattern similar to that shown in
Figure 6a, beginning with a linear section showing elastic behavior and peaking above the first cracking stress or flexural strength observed in non-fiber specimens. After peak, load capacity decreases significantly, decreasing to between 0.5 and 1.5 MPa. Notably, specimens with a 2.07% fiber volume percentage, although having a lower peak stress, but it showed a steady drop, indicating a more ductile failure style. When we examine these lines, which have a lower length-to-diameter ratio (l/d) than those in
Figure 6a at the same fiber volume percent, we can see that the peak stress is lower. This demonstrates the role of the aspect ratio in increasing the flexural strength of Fiber Reinforced Cementitious Composites (FRCC).
Figure 8a,b provides a detailed evaluation of the effect of polypropylene (PP) fibers on the mechanical response of composite materials. The study focuses on the deflection behavior and flexural characteristics at different volume fractions and aspect ratios. The data given in the figures examines the effects of various volume fractions of PP fibers ranging from 0.45% to 2.27%.
Figure 7a shows that micro-PP fibers have an aspect ratio of roughly 375, whereas macro-PP fibers have an aspect ratio of 71. The findings show a linear connection between deflection and stress, up to the limit of proportionality (LOP). The LOP values recorded ranged from 4.37 to 5.17 MPa, with the exception of two mixes (G4M18, G4M19) that were less than 4 MPa. This observation underscores the material's early elastic reaction.
Figure 8.
Equivalent elastic bending stress versus deflection of Polypropylene fiber reinforced mortar prisms (a) with different volume fraction of fibers at L/d=375(micro polypropylene) (b) different volume fractions of fibers at L/d=71 (macro polypropylene).
Figure 8.
Equivalent elastic bending stress versus deflection of Polypropylene fiber reinforced mortar prisms (a) with different volume fraction of fibers at L/d=375(micro polypropylene) (b) different volume fractions of fibers at L/d=71 (macro polypropylene).
Figure 9.
Equivalent elastic bending stress versus deflection of sisal fiber reinforced mortar prisms (a) with different volume fraction of fibers at L/d=128 (b) different aspect ratio of fibers at volume fraction=1.44 %.
Figure 9.
Equivalent elastic bending stress versus deflection of sisal fiber reinforced mortar prisms (a) with different volume fraction of fibers at L/d=128 (b) different aspect ratio of fibers at volume fraction=1.44 %.
Figure 10.
SEM of carbon fibers in a cement matrix ruptured at a maximum flexural load.
Figure 10.
SEM of carbon fibers in a cement matrix ruptured at a maximum flexural load.
Following the limit of proportionality (LOP), there was a significant drop in load, with varying magnitudes depending on the fiber volume percentage. It is worth noting that there is an inverse connection between fiber content and drop quantity, indicating that increased fiber content increases the material's ability to withstand cracking and structural failure. In specimens reinforced with micro fibers, the ratio between the maximum flexural stress (MOR) and the first cracking stress (LOP) was found to be less than one, indicating the presence of deflection softening behavior. However, when specimens with volume fractions of 1.36% and 1.81% were considered, the aforementioned ratio tended to unity, demonstrating a favorable link between fiber content and improved flexural behavior. Specimens with a volume fraction of 2.27% exhibited distinctive behavioral features. The substantial volume percentage of fibers had a detrimental influence on the mix's workability, making complete compaction and thorough mixing difficult to achieve. As a result, the percentage decrease in compressive strength was about 33% , emphasizing the need of carefully considering the interaction between fiber content, workability, and strength.
Figure 11 displays a scanning electron microscope (SEM) image of a micro polypropylene (PP) fiber-reinforced matrix. The image shows several fibers that have completely debonded from their original positions within the matrix. This debonding leads to increased deflection under stress. The total percentage of fibers that remain effective in resisting the applied load after cracking has decreased, which reduces the overall load-carrying capacity of the material.
Figure 8b; shows the relationship between equivalent bending stress and deflection for samples reinforced with macro polypropylene fibers. The findings show that deflection hardening occurs equally across these specimens, as the maximal bending stress (MOR) exceeds the limit of proportionality (LOP). This trend is consistent across all samples, with the exception of those having 0.45% volume percentage of macro polypropylene fibers. The MOR-to-LOP ratio of 0.66 for these individual specimens is less than 1, indicating a considerable divergence from the overall trend seen. Furthermore, each of these specimens shows significant deflection while maintaining their flexural strength. This separates them from other examples that have been strengthened with carbon fibers. Despite having a lower length-to-diameter ratio (l/d), macro polypropylene fibers, with longer fibers, greatly improve the flexural strength and flexibility of the specimens compared to micro PP fibers. This discovery emphasizes the importance of the volume percentage and the presence of indentations on the surface of macro polypropylene fibers could enhance the interaction between the fibers and the cement matrix of macro polypropylene fibers in improving the deflection capacity and post-cracking stress behavior of the composite. The interfacial bond strength is an important aspect in better flexural performance, this highlighted the relevance of fiber qualities other than aspect ratio in designing fiber-reinforced concrete mixes customized to specific structural needs.
Figure 12 showing SEM of macro PP fibers reinforced cementitious composite it is shown that the fibers us partially debonded from the matrix.
Figure 9a,b provide a detailed evaluation of the effect of sisal fibers on the flexural behavior of the cementitious composite.
Figure 9a shows how increasing the percentage volume of sisal fibers improves the bending stress obtained by the mortar composite, up to a first cracking point; beyond this point, there is a sudden drop in the load, which decreases with an increase in the volume fraction of fibers, particularly when the volume of fibers exceeds 0.81%. Following that decrease, the fibers were able to withstand the imposed stress, and the flexural capacity was gradually raised until it reached the maximum flexural capacity, or MOR. The ratio of MOR to first cracking strength was changed from 0.8 to 1 for specimens reinforced with fiber volumes greater than 0.81%. Following the post-peak load, the load gradually lowered.
Figure 9b shows the effect of sisal fiber (L/d) on the flexural behavior of fiber-reinforced composites with a constant fiber volume percentage (Vf = 1.44%). Up to the point of initial cracking stress, the behavior of all curves is essentially linear and identical, implying that the composite structure acts consistently regardless of fiber aspect ratio. After the first cracking stress, each curve shows a significant decrease in stress that is approximately identical across all analyzed aspect ratios, demonstrating a consistent failure process despite changes in fiber length, with minor variances representing the influence of varied aspect ratios. The highest flexural stress after cracking is recorded at L/d = 384, indicating an optimum relationship between fiber length and matrix binding at this ratio. For larger aspect ratios, such as L/d = 448 and 512, the curves show a more notable decrease and poorer stress recovery, highlighting possible concerns such as increased micro-cracking caused by diminished fiber-matrix contact. This is supported by SEM pictures as shown in
Figure 13, which reveal non-uniform dispersion and gaps leading to micro-cracks.
Modulus of Rupture
Figure 14 shows how integrating fibers affects the MOR, illustrating how the MOR corresponds with the fiber volume percentage in the mix. For mortars without fibers and a small volume fraction, the MOR is within certain limitations. At first, the inclusion of fibers decreases the MOR (modulus of rupture), suggesting that a little amount of fibers does not significantly enhance the ability to bear loads right before failure, as shown in both micro and macro polypropylene fibers. However, when the volume percentage of fibers reaches a critical threshold, which may be viewed as the minimum volume fraction of fibers, there is an observed increase in the modulus of rupture (MOR). The minimum volume fraction of fibers required to avoid a sudden breakdown may be estimated by calculating the ratio of the tensile strength of the fiber to the tensile strength of the matrix. For prisms reinforced with polypropylene fibers, this value is roughly 0.9%, whereas for prisms reinforced with carbon fiber, it is about 0.1%. Both macro and micro polypropylene fibers exhibited a lower modulus of rupture at 0.41%. However, as the volume fraction increased, the modulus of rupture increased for macro PP fibers, while for micro PP fibers, it remained lower than the first cracking strength even when the volume fraction reached 0.82%. This difference can be attributed to the lower interfacial bond strength of short and smooth micro polypropylene fibers compared to long and deformed macro polypropylene fibers.
However, the addition of carbon fiber, even at a small volume fraction of 0.41%, resulted in a flexural strength that was somewhat higher than the critical threshold. This enhancement signifies the participation of the fibers in connecting and bridging the fractures of the prisms, hence enhancing the composite's capacity to withstand forces following breaking.
Notably, specimens reinforced with carbon fiber had a peak stress that exceeded the flexural strength of control specimens across all fiber volume fractions. This emphasizes the distinct advantages of carbon fibers, such as their higher aspect ratio and tensile strength, over PP fibers in reinforcing mortars.
It has been derived by Naaman (Naaman, 2018) that the post cracking strength of fiber reinforced cement based products under direct tension depends on the fiber content , aspect ratio and interfacial bond strength between fibers and the matrix (
as shown in the equation below;
Where;
The coefficient is the result of multiplying numerous factors that determine the effectiveness of the fiber. The equation accurately predicts the tensile strength after cracking and can be utilized to predict the modulus of rupture for specimens under flexural stress by analyzing the stress distribution across the section of the members. This prediction takes into consideration various characteristics of fibers, such as fiber content, aspect ratio, and interfacial bond strength between the fiber and matrix.
However, measuring the interfacial bond strength between fiber and matrix in fiber-reinforced concrete (FRC) is challenging because fiber parameters (e.g., material, texture, shape) have a major influence on bond strength, resulting in inconsistent findings and measurement issues. Furthermore, the stress distribution among cement fibers and the entrance of air spaces during mixing and compaction introduce heterogeneity into bond strength measurements. Despite numerous estimating approaches, the absence of a worldwide agreed standard for measuring bond strength makes comparisons complicated between studies.
In this study, rather than focusing on the interfacial bond strength between fiber and matrix, two closely related properties to the bond strength was introduced: fiber tensile strength and matrix compressive strength. This study looks at how various factors affect the flexural strength of fiber-reinforced cementitious composite (FRCC). The primary variable of interest is the ratio of maximum post-cracking stress to initial cracking stress which is called as the limit of proportionality (MOR/LOP). This ratio correlated in relation to several independent variables, including fiber volume percentage (Vf), fiber aspect ratio (L/d), matrix compressive strength (C), and fiber material tensile strength (T). The information gathered was submitted to multi-logistic regression analysis using statistical techniques for predicting non-linear curves. The overall model structure is shown below:
Figure 15 was generated by calculating the ratio of MOR to LOP using the measured flexural strength, as well as the anticipated ratios resulting from this model's output, for all laboratory specimens and extra data obtained from the literature, which totaled 96 instances. 33 of the above instances were derived from this study's experimental work, with the other data points coming from previous literature (G. Ramakrishna & Sundararajan Thirumalai, 2005)(BAŞSÜRÜCÜ et al., 2022), (B. Li et al., 2018), (Yoo et al., 2017). The figure displays a linear connection with the MOR to LOP ratio as the dependent variable and(
as independent variable, the relation approximately linear with a coefficient of determination R
2= 0.84 and sum of residual squares
, the slope of the best fit line determined is 0.095.
The prediction equation indicates that fiber characteristics, specifically fiber quantity and aspect ratio, have varying effects. Unlike steel fibers, synthetic soft fibers are not rigid, may be folded in a variety of directions, and can be crushed by shear force. As a result, the aspect ratio has far less effect than the fiber content.