**Fatigue and Fracture of Non-metallic Materials and Structures**

Special Issue Editor **Andrea Spagnoli**

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### **Contents**

### **About the Special Issue Editor**

**Andrea Spagnoli** is an Associate Professor in solid and structural mechanics. Born in Bologna on 24 March, 1967, Andrea received a M.S. (Laurea) in civil engineering, University of Parma (1992) and Ph.D. in structural engineering, Imperial College, University of London, U.K. (1997). Andrea was an Assistant Professor of Structural Mechanics, University of Parma (2000–2005). From 2005 to the present, Andrea as been a Associate Professor of Structural Mechanics, University of Parma and anuthor of more than 200 scientific papers in refereed journals and proceedings of international conferences. According to Scopus, he has 130 publications (1996–2019), 2172 citations and an h-index of 29. Andrea has been an academic visitor at the Department of Civil Engineering, Imperial College, London (1997, 1999); Department of Mechanical Science and Engineering, Kyushu University (2000–2001); and Department of Civil and Environmental Engineering, University of Michigan (2007). He has been a Guest Professor in the Division of Solid Mechanics, Lund University, Sweden, since 2018. He has participated in several domestic (funded by the Italian Ministry of Education and by private companies) and international research projects (funded by Europen Community) since 1992. He has been the coordinator of a project financed by the Italian Ministry of Education (PRIN2009) and co-editor of the book Biaxial/Multiaxial Fatigue and Fracture, Elsevier, 2003. He has been co-editor of Special Issues of the journal Engineering Fracture Mechanics in 2008, 2010, 2011, 2014, and 2017. He is a member of the Editorial Board of the journal Fatigue and Fracture of Engineering Materials and Structures (2004–present), and of the journal Structural Engineering and Mechanics, (2013–present). Andrea is the Secretary of the Technical Committee Fatigue of Engineering Materials and Structures, ESIS (2000–present) and is the corresponding member of ECCS Technical Working Group 8.4 'Stability of steel shells' (2004–present), as well as a member of the Executive Committee of the Italian Group of Fracture (2011–present).

### *Editorial* **Special Issue on Fatigue and Fracture of Non-Metallic Materials and Structures**

### **Andrea Spagnoli**

Department of Engineering and Architecture, University of Parma, 43124 Parma, Italy<EMAIL_ADDRESS>Received: 17 February 2020; Accepted: 25 February 2020; Published: 7 March 2020

**Abstract:** This Special Issue covers the broad topic of structural integrity of non-metallic materials, and it is concerned with the modelling, assessment and reliability of structural elements of any scale. In particular, the articles being contained in this issue concentrate on the mechanics of fracture and fatigue in relation to applications to a variety of non-metallic materials, including concrete and cementitious composites, rocks, glass, ceramics, bituminous mixtures, composites, polymers, rubber and soft matters, bones and biological materials, advanced and multifunctional materials.

**Keywords:** Fatigue; Fracture mechanics; Structural integrity; Polymers; Composites; Ceramics; Concrete; Rock; Soft matter; Advanced materials.

### **1. Introduction**

The mechanics of fracture and fatigue have produced a huge body of research work in relation to applications to metal materials and structures. However, a variety of non-metallic materials (e.g., concrete and cementitious composites, rocks, glass, ceramics, bituminous mixtures, composites, polymers, rubber and soft matters, bones and biological materials, advanced and multifunctional materials) have received comparatively less attention, despite their attractiveness for a large spectrum of applications related to the components and structures of diverse engineering branches, applied sciences and architecture, and to the load-carrying systems of biological organisms.

This special issue covers the broad topic of structural integrity of non-metallic materials and is is concerned with the modeling, assessment, and reliability of structural elements of any scale. Original contributions from engineers, mechanical materials scientists, computer scientists, physicists, chemists, and mathematicians are presented, following both experimental and theoretical approaches.

### **2. Fracture**

A number of papers in this special issue are specifically devoted to fracture mechanics problems. Different approaches have been used, including experimental investigations, theoretical models, and numerical simulations. Papers [1–5] are related to concrete material, from papers [6–12] to rocks. Other contributions investigate the fracture behavior of functionally graded materials [13], glass [14], laminate composites [15], refractories [16], and soft matter [17]. Concrete material is also investigated in relation to impact resistance [18] and self-healing properties [19]. Impact behavior is studied with reference to laminate composites [20,21]. Other papers devoted to rocks concern their cutting resistance [22] and their multiaxial response under dry and saturated conditions [23]. Finally, the mechanical behavior of a monomer structural component is studied in [24].

### **3. Fatigue**

Fatigue is investigated from both a experimental and theoretical point-of-view in different papers. In [25–30] in particular, concrete behavior under fatigue loading is described, with an emphasis on bridge applications [25–27] and concrete reinforced with fibers and rebars [25,29]. Fatigue of asphalt

and of rocks is investigated in [31,32], respectively. Finally, an account on fatigue life assessment of wind turbine blades is presented in [33].

**Funding:** This research received no external funding.

**Acknowledgments:** This special issue would not have been made possible without the irreplaceable contributions of valuable authors coming from many different countries, hardworking and professional reviewers, and the dedicated editorial team of Applied Sciences, an international, peer-reviewed journal that is free for readers embracing all aspects of applied sciences.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Effect of Slag Particle Size on Fracture Toughness of Concrete**

### **Chung-Ho Huang 1, Chung-Hao Wu 2,\*, Shu-Ken Lin 3 and Tsong Yen 3**

Received: 1 February 2019; Accepted: 20 February 2019; Published: 25 February 2019

**Abstract:** The effects of particle size of ground granulated blast furnace slag (GGBS) on the fracture energy, critical stress intensity, and strength of concrete are experimentally studied. Three fineness levels of GGBS of 4000, 5000, 6000 cm2/g were used. In addition to the control mixture without slag, two slag replacement levels of 20% and 40% by weight of the cementitious material were selected for preparing the concrete mixtures. The control mixture was designed to have a target compressive strength at 28 days of 62 MPa, while the water to cementitious material ratio was selected as 0.35 for all mixtures. Test results indicate that using finer slag in concrete may improve the filling effect and the reactivity of slag, resulting in a larger strength enhancement. The compressive strength of slag concrete was found to increase in conjunction with the fineness level of the slag presented in the mixture. Use of finer slag presents a beneficial effect on the fracture energy (GF) of concrete, even at an early age, and attains a higher increment of GF at later age (56 days). This implicates that the finer slag can have a unique effect on the enhancement of the fracture resistance of concrete. The test results of the critical stress intensity factor (KS IC) of the slag concretes have a similar tendency as that of the fracture energy, indicating that the finer slag may present an increase in the fracture toughness of concrete.

**Keywords:** fracture toughness; blast furnace slag; particle size; compressive strength; concrete

### **1. Introduction**

Ground granulated blast furnace slag (GGBS) is a byproduct of iron making, which is produced by water quenching of molten blast furnace slag that turns out to be a glassy material [1]. It is ground to improve the reactivity during cement hydration. GGBS is one such supplementary material which can be used as a cementitious ingredient in either cement or concrete composites. Research to date suggests that the supplementary cementitious materials improve many of the performance characteristics of concrete, such as strength, workability, and corrosion resistance [2,3]. Some of the effective parameters like chemical composition, fineness, and hydraulic reactivity have been carefully examined by many earlier studies [4]. Among these, the reactive glass content and fineness of GGBS alone will influence the cementitious/pozzolanic efficiency or its reactivity in concrete composite significantly.

It is generally agreed that the use of fine GGBS improves the properties of concrete. K. Tan and X. Pu [5] investigated the effect of finely ground GGBS on the compressive strength of concrete. Test results showed that incorporating 20% finely ground GGBS can significantly increase the strength after 3 days. Mantel [6] reported that the slag activity depends on the particle size distribution (fineness) of slag and the cement used and showed that this ranges from 62%–115% at 28 days. In addition, the investigation of Y ˝uksel et al. [7] reported that the increase in fineness of GGBS improves compressive strength due

to the pozzolanic reaction causing a reduction in permeability, signifying that finer slag can provide higher resistance against deteriorations from chemical or physical attacks.

The traditional properties, such as compressive strength, have been assessed to be in close relation to the pore characteristic and interfacial transition zone (ITZ) of concrete. The test results of Duan, Gao, and Tan et al. [5,8,9] show that mineral admixtures have a positive impact on pore refinement and ITZ enhancement, especially at later curing stages. The development of compressive strength is significantly related to the evolution of ITZ and the pore structure. In fact, the cementitious and pozzolanic behavior of GGBS is essentially similar to that of high-calcium fly ash. Previous reports [5,9–11] have primarily indicated that increasing the fineness of fly ash and slag may increase the strength and durability of concrete attributed to the microcracking resistance of cement paste, particularly in ITZ.

The interfacial transition zone in concrete that is often modeled as a three-phase material is represented as the third phase [12]. Existing as a thin shell around larger aggregate, the ITZ is generally weaker than either of the two main components, the hydrated cement paste, and aggregate, of concrete. The ITZ has less crack resistance, and accordingly, the factor occurs preferentially. A major factor responsible for the poor strength of the ITZ is the presence of microcracks. Much of the physical nature of the response of concrete under loading can be described in terms of microcracks that can be observed at relatively low magnification. Cracking in concrete is mostly due to the tensile stress that occurs under load or environmental changes. As such, the failure of concrete in tension is governed by the microcracking, associated particularly with the ITZ [13,14].

The tensile strength of concrete is a very essential property. Not only the tensile strength but also the behavior at the tensile fracture is of importance, particularly the toughness. When concrete fails in tension, its behavior is characterized by both the peak stress and the energy required to fully generate a crack. Fracture mechanics could in principle be a suitable basis for analyzing the tensile toughness of concrete [15]. The defects and the stress concentration are the main factors causing failure. Concrete that exhibits defects or is subjected to stress concentrations easily cracks and results in a reduction of strength. It is important to identify the fracture behavior and toughness for widely used high-performance concrete that normally incorporates slag or fly ash [16,17].

Because of the non-homogeneous characteristics of concrete, its fracture behavior is quite complicated. It was proved that the methods developed in conventional fracture mechanics are unsuitable for the analysis of the influence of fracture toughness on the behavior of concrete structures [15]. Many theoretical models have been developed to make such an analysis possible. Among them, three well-known models are the fictitious crack model, the crack band model, and the two-parameter fracture model. By means of numerical techniques, for example, in the finite element method, it is possible using these models to make a theoretical analysis for the development of the damage zone and the complete behavior of the structure. As for the fracture mechanics of concrete, RILEM has put forward several methods to determine the fracture properties and parameters of concrete [18]. Hillerborg [15] determine the fracture energy according to the fictitious crack model. Jeng and Shah [19] used the two-parameter fracture model to determine the critical stress intensity factor.

A number of past studies on the concrete containing GGBS have been conducted, mainly dealing with the influence of the fineness of slag on the strength and durability of concrete. More rarely, investigations were conducted on the fracture behaviors of concrete incorporating finer GGBS. Consequently, there is a need for exploring the effects of the particle size of GGBS on the fracture properties of concrete. This study experimented using the three-point bend test to analyze the fracture energy and the critical stress intensity factor for evaluating the fracture mechanics or fracture toughness of concrete containing finer GGBS.

### **2. Research Significance**

Laboratory investigations have shown that when the slag particle size is reduced, its mechanical performance in concrete is improved. The finer slag with the larger surface area has an intensive reaction which may lead to higher enhancement of the strength, the fracture energy, and the critical stress intensity factor of concrete. Other than the research by Jensen and Hansen [20], who observed a dependence of the fracture energy on the aggregate type and independence from the compressive strength of concrete, this study found that an increase in concrete compressive strength of 10% resulted in an increase in fracture energy of around 18%.

### **3. Experimental Details**

### *3.1. Materials*

The materials used included a Type I Portland cement, river sand, crushed coarse aggregate with a maximum size of 10mm, a naphthalene-sulfonate-based superplasticizer, and GGBS of three fineness levels. The chemical composition of cement and GGBS are presented in Table 1. Three different fineness levels of GGBS of 4000, 5000, and 6000 cm2/g with a specific gravity of 2.85 were selected in this study.

**Table 1.** Chemical analysis of cement and ground granulated blast furnace slag (GGBS).

### *3.2. Mixture Proportions*

Seven mixtures were prepared for testing in this research: A reference mixture (R0) without GGBS and six slag mixtures, designated as S4R2, S4R4, S5R2, S5R4, S6R2, and S6R4. S4, S5, and S6 refer to slag fineness of 4000, 5000 and 6000 cm2/g, respectively. R2 and R4 refer to the replacement ratios of 20% and 40%, respectively, by weight of the cementitious material. The water to cementitious material ratio (w/cm) was kept at 0.35 for all mixtures. All concretes were designed to have a target compressive strength level at 28 days of 62 MPa. The dosage of superplasticizer was adjusted to produce a designed slump of 250 ± 20 mm and a slump flow of 600 ± 50 mm for all mixtures. The mixture proportions are shown in Table 2.

**Table 2.** Mixture proportions and properties of fresh concrete.

### *3.3. Specimen Fabrication*

Cylinder specimens (ψ100 × 200 mm) were cast from each mixture for compression testing; three cylinders each for testing at 4 ages (7, 14, 28, 56 days). Prism specimens with diminution of 50 × 50 × 650 mm and 80 × 150 × 700 mm were cast for the determination of fracture energy (GF) and the critical stress intensity factor (KS IC), respectively. Four specimens were fabricated for each mixture that tested the two specimens at 2 ages (14 and 56 days). The specimens and concrete mixtures were prepared in accordance with ASTM C192. After casting, test specimens were covered with plastic sheets and left in the casting room for 24 h. They were then demolded and put into a 100% RH moist-curing room at about 23 ◦C until time for testing. One day before testing, the fracture energy specimen and the critical stress intensity factor specimen, as shown in Figures 1 and 2, were prepared by cutting a notches 25 mm deep by 2 mm width and 50 mm deep by 2 mm width, respectively, at the middle of the beam specimens.

**Figure 2.** The test set up for KS IC-determination.

### *3.4. Testing Procedure*

The compression test was carried out in accordance with ASTM C39. The three-point bending test was performed on the notched beam as follows to determine the GF and KS IC:

### 3.4.1. The Test for Determining GF

The fracture energy test followed the guidelines established by RILEM [21] using a closed-loop testing machine. The testing arrangement is shown in Figure 1. A linear variable differential transformer (LVDT) was installed at mid-span of the beam to measure the deflection. The loading velocity was chosen so that the maximum load was reached 30 s after the loading started. The loading rate selected was 0.25 mm/min. A load-deflection (F-δ) curve was then plotted, with the energy "W0" representing the area under the curve. The GF (N-m) can be calculated using the following expression [15,19]:

$$G\_F = \mathcal{W}\_0 + mg\frac{\delta\_0}{A} \tag{1}$$

where W0 = area under the load-deflection curve (N-m);

m = mass of the beam between the supports (kg);

g = acceleration due to gravity;

δ0 = final deflection of the beam (m);

A = cross-sectional area of the beam above the notch (m2).

3.4.2. Test for Determining KS IC

The test for the determination of KS IC followed the test method suggested by Jenq and Shah [19] using a closed-loop testing machine under crack mouth opening displacement (CMOD) control. The test setup is shown in Figure 2. A clip gauge was used to measure the CMOD. The CMOD and the applied load were recorded continuously during the test. The test procedure included two steps of loading and unloading. The first step was started from loading until the peak load was reached; the applied load was manually reduced (also termed unloading) when the load passed the maximum load and was about 95% of the peak load. When the applied load was reduced to zero, reloading was applied. Each loading and unloading cycle was finished in about 1 min. Only one cycle of loading-unloading was required for the test. The KS IC is calculated using the following equation:

$$K\_{IC}^S = \frac{3P\_{\text{max}}S}{2bd^2} \sqrt{(\pi a\_c)} F(a) \tag{2}$$

in which,

$$F(a) = 1.99 - \frac{a(1 - a)(2.15 - 3.93a + 2.7a^2)}{\sqrt{\pi}(1 + 2a)(1 - a)^{3/2}}\tag{3}$$

where Pmax = the measured maximum load (N); S = the span of the beam; b = beam width; d = beam depth; αc = critical effective crack length; α = αc/d.

### **4. Results and Discussion**

### *4.1. Compressive Strength*

Compressive strength was determined at the ages of 7, 14, 28 and 56 days on concrete stored under moist-curing condition. The results are shown in Table 3. It is seen that all mixtures were proportioned to have equivalent workability and target strengths of 62 MPa at the age of 28 days. This was generally achieved except for in mixtures S4R2 and S4R4. In this case, the strength equivalence was achieved at the age of 56 days, which was also exceeded by that of the reference mixture (R0).

The development of compressive strength for each concrete mixture is shown in Figure 3. At early ages (7 days), as expected, the reference mixture without slag exhibited higher strength than the other slag mixtures. The early strength gain of the reference mixture was superior to that of the slag mixtures, indicating that replacing any amount of cement with GGBS of various fineness will reduce the strength of concrete. Moreover, the concrete incorporating more slag (40%) displayed lower strength at an early age than that with less slag content (20%). However, the curve slope of the slag mixtures in Figure 3 tends to be steeper than that of the reference mixture at later ages. In other words, the strength increment versus age for the slag mixtures is obviously larger than that of the reference mixture. Consequently, the strength of each slag mixture exceeds that of the reference mixture at the age of 56 days.

On the other hand, it can be found from Table 3 that under the same replacement ratio (20% and 40%) of slag, the mixtures S6R2 and S6R4 presented the highest strengths at an early age (7 days), followed by S5R2 and S5R4, and the S4R2 and S4R4 mixtures are the lowest. This is due to the fact that incorporating finer slag into concrete may fill the micro-voids much better, exhibit higher reactivity and producing higher packing density, resulting in a larger strength enhancement. In addition, a beneficial effect of the fineness of slag on concrete strength can be seen in Figure 4; the compressive strength of the slag concrete increases as the fineness level of the slag incorporated into the mixture increases for each age and various slag replacement ratios. It is also found in Table 3 that there are significant strength gains from 7–28 days and again from 28–56 days for the mixture containing finer slag. For example, the strength increment of the mixtures S6R2 and S6R4 that contain finer slag displayed an increase rate

from 100% at 7 days to 121.8% and 122.7% at 28 days, and 125.5% and 128.9% at 56 days, respectively. These are obviously larger than the corresponding mixtures of S4R2 and S4R4 that increased from 100% at 7 days to 114.9% and 115.1% at 28 days, and 124.2% and 126.2% at 56 days, respectively. At 28 days, the strength of the finer-slag concrete is shown consistently to exceed that of the reference concrete, except in the mixtures of S4R2 and S4R4. This is particularly significant considering that the cement replacement of finer slag in each case is up to 40%. In addition, the highest strength achieved at 56 days was 70.5 MPa for the mixture S6R4 with 40% slag replacement, while the strength obtained for the corresponding reference mixture was a relatively lower value of 65.5 MPa.

\* Average value of three specimens

**Figure 3.** Compressive strength development for cylindrical specimens.

**Figure 4.** The average compressive strength of slag concrete versus the fineness of ground granulated blast furnace slag (GGBS).

### *4.2. Fracture Energy*

In this research, fracture energy (GF) was determined using a notched beam in a three-point bending test. Load-deflection curves were plotted from the tests. The energy absorbed in the test to failure is represented by the area below the load-deflection curve for the specimen. The area represents the fracture energy per unit area of the fracture surface. The GF was calculated using Equation (1). Concrete specimens were tested at 14 and 56 days. Results are presented in Table 4. At earlier ages (14 days), the GF values of each slag mixture are mostly less than that of the reference mixture, except slag mixture S6R2. In addition, the concrete containing more slag (40%) presented smaller GF than that with less slag content (20%) for various ages. Nevertheless, the presence of the finer slag has a beneficial effect on the GF of concrete. As shown in Figure 5, increasing the fineness level of slag leads to an increase of the GF value of concrete. Even at an earlier age of 14 days, the GF value of the mixture with finer slag (S6R4) is in turn higher than that of the mixtures S5R4 and S4R4 for a similar slag replacement ratio of 40%. This implies that although the pozzolanic reaction of slag does not yet fully occur at early ages, the superior filling effect and more active hydration reaction of the finer slag can increase the density of concrete, resulting in an increased fracture resistance. After 56 days, the GF value of the slag mixtures almost exceeds that of the reference mixture. In addition, it is also found in Table 4 that the increment of fracture energy for each slag concrete from 14 days to 56 days is attained by 18–24%, which is much higher than that of reference concrete (10.1%). This signifies that, during the period, the pozzolanic reaction of slag activates, associating with the filling effect to enhance the fracture toughness of the slag concrete.

\* Average value of two specimens

**Figure 5.** Fracture energy for concretes with various fineness levels and replacement ratios of slag at ages of 14 days and 56 days.

Figure 6 compares fracture energy with the compressive strength of all slag concretes in the study. The GF is found to increase in conjunction with the compressive strength. This compares the trend favorably with the studies by Giaccio [22], Gettu [23], and Xie [24]. Nevertheless, the other studies observed smaller increases in GF with the increases in the compressive strength of high-strength concrete. Giaccio, Rocco, and Zerbino [22] found that GF increased as compressive strength increased, but only at a fraction of the rate. Getta, Bažant, and Kerr [23] reported that an increase in compressive strength of 160% resulted in an increase in GF of only 12%. Xie, Elwi, and MacGregor [24] found that increases in compressive strength of 29% and 35% resulted in a GF increase of 11% and 13%, respectively. In this study, an increase in compressive strength of 10% resulted in a larger increase in GF of around 18%. This implicates the unique effect of using finer slag on the enhancement of the fracture resistance of concrete.

**Figure 6.** Fracture energy versus compressive strength for slag concretes.

### *4.3. Critical Stress Intensity Factor*

The critical stress intensity factor (KS IC) was experimentally determined with a notch beam in a three-point bend test using a closed-loop testing machine under the crack mouth opening displacement control. Load-CMOD curves were plotted from the tests. The KS IC was calculated using Equation (2). Concrete specimens were tested at 14 and 56 days. Table 5 summarizes the results. It shows that the related tendency of the KS IC of concrete is similar to that of the GF. At an early age (14 days), the KS IC values of slag concrete are less that of the reference concrete (R0), but exceed that of R0 after 56 days. This implies that the additional pozzolanic reaction of slag with Ca(OH)2 in concrete becomes active during the period between 14 days and 56 days, which leads to the enhancement of KS IC of the slag concretes. Moreover, it is seen that the concrete incorporating more slag (40%) presented a lower KSIC value than that with less slag content (20%) for various ages. On the other hand, the concrete containing finer slag can exhibit larger KS IC values. It can be found from Figure 7 that the increase of the fineness level of the slag leads to an increase in the KS IC value of concrete for various ages. In other words, the KS IC value of the mixture with finer slag (S6R2) is in turn higher than those of S5R2 and S4R2 for the similar slag replacement ratio of 20%. Also, the KS IC of S6R4 > S5R4 > S4R4 for the similar slag replacement ratio of 40%. The reason for this result is that the finer slag particle has a larger surface area, presenting a more active pozzolanic reaction, thus resulting in an increase of strength and fracture toughness.

**Table 5.** Critical stress intensity factor of the concrete.

\* Average value of two specimens.

**Figure 7.** The stress intensity factor for concretes with various fineness levels and replacement ratios of slag at the ages of 14 days and 56 days.

Figure 8 compares the critical stress intensity factor with compressive strength for all slag-concrete specimens in the study. As shown in the figure, the two values of stress are nearly linearly related. The relationships shown in Figure 8 are significant because, based on the close relationship between the particle size of slag and compressive strength (Figure 4), the critical stress intensity factor will increase with the increase of the fineness level of the slag.

**Figure 8.** Critical stress intensity factor versus compressive strength for slag concretes.

### **5. Conclusions**

The following conclusions were made based on the experimental results and the findings of the study:

**Author Contributions:** Conceptualization, C.-H.W. and T.Y.; Methodology, C.-H.W.; Validation, C.-H.W., C.-H.H. and S.-K.L.; Formal analysis, C.-H.W. and C.-H.H.; Investigation, C.-H.W. and C.-H.H.; Resources, C.-H.W.; Data curation, C.-H.W. and S.-K.L.; Writing—original draft preparation, C.-H.W.; Writing—review & editing, C.-H.W. and T.Y.; Visualization, C.-H.W. and S.-K.L.; Supervision, T.Y.; Project administration, C.-H.H.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Cracking Behavior of RC Beams Strengthened with Different Amounts and Layouts of CFRP**

**Muhammad Umair Saleem 1,\*, Hisham Jahangir Qureshi 1, Muhammad Nasir Amin 1, Kaffayatullah Khan 1 and Hassan Khurshid 2**

Received: 4 February 2019; Accepted: 7 March 2019; Published: 12 March 2019

### **Featured Application: This study provides the valuable data to assess the failure mechanism of CFRP strengthened reinforced concrete beams and structural systems.**

**Abstract:** The bending and shear behavior of RC beams strengthened with Carbon Fiber-Reinforced Polymers (CFRP) is the primary objective of this paper, which is focused on the failure mechanisms and on the moment-curvature response prior-to, and post, strengthening with different amounts and layouts of the CFRP reinforcement. Seven reinforced concrete beams were tested in 4-point bending, one without any CFRP reinforcement (control beam, Specimen C1), four with the same amount of CFRP in flexure but with different layouts of the reinforcement for shear (Specimens B1–B4), and two with extra reinforcement in bending, with and without reinforcement in shear (Specimens B6 and B5, respectively). During each test, the load and the mid-span deflection were monitored, as well as the crack pattern. The experimental results indicate that: (a) increasing the CFRP reinforcement above certain levels does not necessarily increase the bearing capacity; (b) the structural performance can be optimized through an appropriate combination of CFRP flexural and shear reinforcement; and (c) bond properties at the concrete–CFRP interface play a vital role, as the failure is very often triggered by the debonding of the CFRP strips. The experimental values were also verified analytically and a close agreement between the analytical and experimental values was achieved.

**Keywords:** concrete cracking; crack patterns; carbon fiber-reinforced polymers—CFRP; RC strengthening (in bending and shear); RC beams

### **1. Introduction**

The Kingdom of Saudi Arabia, as well as rest of the world, contains a wide range of reinforced concrete infrastructure which ranges from small residential houses to multi-story buildings, towers, and pre-stressed concrete bridges. The majority of the infrastructure in Saudi Arabia was designed and constructed on old design codes, standards, and specifications, and as a result of that, it is currently experiencing weathering because of harsh climate, saltwater, acid attack, extreme temperature changes, and due to the inferior quality of building material. There is a necessity to strengthen old existing infrastructure in Saudi Arabia, which lacks in strength and stiffness. There are many different ways to strengthen reinforced concrete structures, and one of the most common methods is the application of CFRP (Carbon Fiber Reinforced Polymer) to the reinforced concrete member.

CFRP has been widely used for the purpose of strengthening reinforced concrete, masonry, and steel infrastructure. Over the past few decades, many researchers have proved the suitability of CFRP material for structures composed of different materials, stiffness, and ductility [1–6]. By adopting the correct retrofitting technique, CFRP can significantly increase the shear and flexure strength of concrete structures as compared to the normal concrete structure. CFRP gains its strength via epoxy, which is the glue that makes a bond of CFRP with the concrete surface. Flexure, shear, and compression cracks can be prevented in reinforced concrete beams with the help of CFRP application. The benefit of using CFRP retrofitting comes from its low density and its resistance to higher tensile forces, fatigue, and corrosion.

However, the debonding failure mechanism, structural ductility, and long-term durability are the main problems, which CFRP is facing due to poor bonding and reduced vapor pressure [7–9]. Fracture of CFRP concrete beams is mainly attributed due to debonding of CFRP from the concrete surface. Hence, flexural strength, failure behavior, and structural ductility are the most important parameters that are always under consideration in structural design, especially when the structure is located in the higher earthquake zone region. CFRP is a material which is brittle in nature and fails suddenly, and has a linear elastic behavior up until failure, i.e., the tensile failure strain of FRP ranges from 2% to 4% [6]. Once the failure strain is reached, CFRP shows no signs of warning, breaks suddenly, and results in the loss of its strength. This behavior of low ductility in reinforced concrete structures retrofitted with CFRP is not desirable, as it does not provide any kind of early warning before failure, thus resulting in the sudden collapse of the structure.

### **2. Literature Review**

CFRP has been widely used in the past for repair and rehabilitation of civil infrastructure that was showing signs of deterioration and distress due to aging. If applied properly, CFRP can increase the service life of the structure. CFRP is a brittle material and usually fails (i.e., debonding or horizontal crack propagation) at a lower load level, hence the ultimate capacity of the reinforced concrete structural member is very hard to achieve [10]. In the past, many researchers conducted experiments on reinforced concrete beams retrofitted in flexure by CFRP and resulting failure patterns were observed.

Arduini and Nanni [11] conducted a parametric analysis to investigate the effects of CFRP reinforcement on serviceability, strength, and failure mechanisms of repaired RC beams. They presented the results of their analysis in terms of repaired/un-repaired strength and deflection ratios. They observed that a brittle failure mechanism can develop at loads much lower than expected when considering only flexural performance of the FRP Strengthened beams. Their research work also concluded that the application of CFRP reinforcement can considerably result in an increase in load-bearing capacity and can also limit deflection at the service level. Smith and Teng [12] conducted a comprehensive review of existing plate debonding strength models that were presented by many researchers in past. Each model was summarized and classified into one of the three categories based on the considered approach. Teng et al. [13] conducted extensive research that has been carried out in recent years on the use of fiber-reinforced polymer (FRP) composites for the purpose of strengthening of reinforced concrete (RC) structures. Their paper provides a concise review of existing research on the behavior and strength of FRP-strengthened RC structures, with a strong focus on those studies that contribute directly to the development of strength models. Topics that were covered in their research work includes flexural and shear strengthening of beams, flexural strengthening of slabs, and strengthening of columns subjected to both static and seismic loads. For each of the topics covered, the methods of strengthening were first explained, followed by a description of the common failure modes. Kotynia [14] conducted tests on reinforced concrete (RC) beams that were strengthened externally with CFRP strips in flexure only. The flexural behavior of the beams, as well as their failure modes, were discussed in detail. Teng and Chen [15] provided a summary of debonding failure modes of reinforced concrete beams that were strengthened externally with FRP reinforcement. Their paper addressed the

following three issues: (a) classification of debonding failure modes; (b) mechanisms and processes of debonding failures; (c) and theoretical models for debonding failures.

Kang et al. [10] conducted a detailed review of previous research programs that were conducted by past researches in relation to debonding failure of FRP attached externally to the concrete surface. Li et al. [16] conducted a series of experimental tests on reinforced concrete beams that were strengthened with externally bonded CFRP sheets in flexure. They investigated debonding initiation and the allowable tensile strain of FRP sheets in flexurally-strengthened RC beams in comparison to different design code provisions. Hasnat et al. [17] conducted research on simply supported reinforced concrete (RC) beams that were strengthened by carbon-fiber-reinforced polymer wrap. A CFRP wrap resisted the premature cover debonding and acted as a U-clamp that results in an increase in the ultimate moment capacity. Mostafa and Razaqpur [18] conducted the experiment on reinforced concrete beams made of T-section. The load was applied and deflection responses were measured for each beam. The complete post-peak load/softening response of each beam was also captured.

Fu et al. [19] conducted research on the effectiveness of a U-jacketing system on delaying or preventing debonding failure. In their experimental research study, they tested eight large-scale RC beams in order to study and investigate the effects of different forms of FRP U-jacketing on debonding failure. Abid and Al-lami [20] conducted an extensive review of past research studies that emphasis the strength and durability of concrete beams that were externally bonded with FRP reinforcement. The focus of the research review was on the bond behavior, testing techniques, and models used to assess bond strength. Flexure, shear, and fatigue behaviors of different strengthening techniques were also reviewed and discussed in detail.

Wenwei and Guo conducted flexural testing on six reinforced concrete beams reinforced with external CFRP laminates in order to study the effects of initial load or load history on the ultimate strength [21]. Experimental results were explained in quantitate terms and for that purpose, a theoretical model for flexural behavior was developed. Lee and Moy carried out an experimental and an analytical study and developed a design-oriented expression to determine an effective laminate strain in bonded CFRP [22]. The developed expression was also used for predicting the deboning failures. Barros et al. conducted experimental and analytical research on reinforced concrete structures strengthened with CFRP systems [23]. Their experimental studies show that the CFRP debonding strain is dependent on the CFRP percentage and the longitudinal steel reinforcement ratio. Mansour and Mahmoud conducted an experimental and analytical study to predict the ultimate moment capacity of RC beams externally strengthened with CFRP [24]. They proposed prediction models to obtain the load capacities for rectangular as well as T beam sections. Pan et al. conducted experimental testing on eight reinforced concrete beams that were strengthened with FRP sheets [25]. They also proposed an analytical model that accounts for the opening of shear and flexural cracks along the beam. Later the results of the experimental data were compared with the proposed analytical model. Ghandour conducted three-point loading testing on seven half-scale reinforced concrete beams, which were strengthened with CFRP longitudinal sheets and U-wraps [26]. Kara and Ashour developed a numerical method for predicting the curvature, deflection, and moment capacity of reinforced concrete beams strengthened with FRP [27]. Later, the analytical results obtained from the numerical model were compared with the published experimental data. Pellegrino and Vasic [28] did the assessment on the design procedures that were available to predict the shear capacity of reinforced concrete beams externally strengthened with FRP composites. Their research work was based on a database, which was collected from recent literature and mainly focused on the basic codes for reinforced concrete structures (without strengthening) and current models for FRP structures strengthened in shear. Li et al. investigated debonding initiation and tensile strain of FRP laminates adhered externally to concrete beam surfaces [16]. Experimental testing was carried out and the allowable tensile strain in FRP sheets proposed by prevalent code provisions was assessed. Jung et al. presented both experimental and analytical research results to predict the flexural capacity of reinforced concrete beams strengthened in flexure with fabric reinforced cementitious matrix (FRCM) [29]. Six beams were

strengthened in flexure with FRCM composite under four-point testing. A new bond strength model was proposed using a test database to predict the strengthening performance of the FRCM composite. Foster et al. did an experimental investigation on reinforced concrete T-beams of varied sizes to determine the effectiveness of unanchored and anchored externally bonded U-wrapped CFRP [30]. Beams were subjected to three-point bending with a span to depth ratio of 3.5. Dias and Barros did experimental research to understand the effectiveness of near surface mounted (NSM) technique by using CFRP laminates in shear [31]. T-Sections reinforced concrete beams were externally reinforced in shear with the help of CFRP laminates at 52- and 90-degree angles. Furthermore, they discovered that inclined laminates are more effective than vertical laminates. Osman et al. did experimental testing on seven CFRP strengthened reinforced concrete beams under four-point loading having different shear span-to-depth ratios [32]. A numerical analysis was also carried out on 27 reinforced concrete beams with and without CFRP sheets. The results achieved using ANSYS were close to the experimental results.

Keeping in mind the aforementioned literature review, it is found that there are very few researchers who studied the effect of different CFRP reinforcement amounts and layouts on beam moment capacity and on the failure behavior of the RC beams. The current study evaluated the effect of the CFRP reinforcement ratio on failure patterns of reinforced concrete beams. The CFRP reinforcement ratio was varied in the form of flexure and shear strips that resulted in different strengthening layouts and failure patterns of the concrete beams. The originality of this research work mainly lies in the determination of specific load levels at which debonding failure of the CFRP initiates and how different layouts of CFRP strengthening affect the shear and flexural strength and failure modes of the reinforced concrete beams. RC beams strengthened with different amounts and layouts of CFRP were tested under four-point bending. Moment-curvature behavior, failure loads, deflections, and failure modes were determined experimentally, which later were also compared with the analytical capacity evaluation models proposed by American Concrete Institute ACI 440.2R-08 [33] provisions.

### **3. Experimental Plan and Setup**

In this research work, seven reinforced concrete beams were prepared and tested under four-point loading. The cross-sectional sizes of all seven reinforced concrete beams were kept constant and are selected equal to 100 mm × 200 mm with an overall equal length of 1200 mm. The beams were designed as per ACI 318-08 [34] for tension controlled failure. Two deformed rebars of 14 mm nominal diameter were used as a flexural steel reinforcement. Figures 1 and 2 give the details of the beams tested under the current experimental plan. The load is applied under a displacement control system at a constant strain rate of 0.03/min. The bending setup located in highway laboratory of King Faisal University was used to carry out the experimental research work as shown in Figure 3. The beams were tested under four-point loading, which gives a constant mid-span bending moment for a given increment of load between the two loading points. Effective span in all beams was kept equal to 1100 mm. The applied bending moment M of the beam is calculated by Equation (1). As the moment between loading point remains constant, the Sagital Method was used to calculate the curvature *ϕ* by using mid-span deflection *δ* of the beam.

$$M = \frac{P}{2} \times L\_1 \tag{1}$$

where *P* is the load directly obtained from the Universal Testing Machine at each increment of displacement. *L*1 is the distance between the support and point of application of load = 475 mm and *L* is the effective span of the beam = 1100 mm, as given in Figure 3b.

**Figure 1.** Details of beam specimens (**a**) Control specimen, (**b**) CFRP-B1, (**c**) CFRP-B2, (**d**) CFRP-B3, (**e**) CFRP-B4, (**f**) CFRP-B5, and (**g**) CFRP-B6.

**Figure 2.** Lateral views of the beams strengthened with CFRP layers and strips.

One out of seven beams was selected as a control specimen, while the rest of the six beams were strengthened and retrofitted with external CFRP reinforcement ratios given in Table 1. The thickness of external flexure and shear CFRP strips was kept equal to 1.5 mm. For all beams, the cross-sectional sizes, internal reinforcement ratios, overall length, and thickness of CFRP strips were kept constant, while the external flexure and shear CFRP reinforcement ratios were varied. Table 1 shows the strengthening scheme of the beams, which were divided into two groups. Beams B1, B2, B3, and B4 were placed in group-1, where the external CFRP flexural reinforcement ratio was kept constant (0.38%). Beam B5 and B6 were placed in group-2, where the external CFRP flexural reinforcement ratio was doubled and kept equal to 0.75%. The flexural reinforcement ratio refers to the quantity of CFRP that is applied at the bottom surface of the beam. Reinforcement ratio is calculated by dividing the cross-sectional area of CFRP with the cross-sectional area of the concrete beam. In a similar way, the

shear reinforcement ratio was calculated. Hence, the total CFRP reinforcement ratio is the sum of the flexural and shear CFRP reinforcement ratio. However, the external CFRP shear reinforcement ratio was also varied and was selected equal to 0 and 0.38% for Beam B5 and B6, as given in Table 1.

**(b)** Schematic

**Figure 3.** Test setup (**a**) and test layout for Beam CFRP-B1 (**b**).

### *Material Properties*

Concrete with a compressive strength of 28 MPa was used for the preparation of reinforced concrete beam specimens. The average compressive strength of concrete was determined by the procedure mentioned in ASTM C-39 [35] and was found equal to *f c* = 28 MPa. The modulus of elasticity and shear modulus of concrete were 24,870 MPa and 10,360 MPa, respectively. The modulus of elasticity of concrete was calculated using the ACI 318-08 [34] and shear modulus of elasticity was computed using poison ratio of 0.2. For quality control, the ready-mix concrete was delivered by a local supplier from a plant located in Saudi Arabia. For internal reinforcement, the yield strength of *fy* = 420 MPa was selected for deformed rebar. The steel reinforcement has a tensile rupture strength of 620 MPa with elastic and shear modulus values of 200,000 MPa and 76,920 MPa, respectively.

Concrete beam specimens were cast and cured as per ASTM-C31 [36]. On completion of curing time, the external surfaces of the beam specimens were cleaned with a wet cloth and acetone. Epoxy (Sikadur 330) of uniform thickness was used to attach external CFRP strips to the concrete surface. After the application of CFRP strips on the surface of the concrete, the beams specimens were cured

for three days by using a wet layup method. Figures 1 and 2 show the layout of beam specimens along with their external and internal reinforcement details. The CFRP has a tensile strength of 1600 MPa with the ultimate tensile elongation value of 1.8%. The tensile modulus of elasticity of CFRP was 120,000 MPa. The epoxy has tensile bond and compressive bond strength values of 9.6 MPa and 21 MPa, respectively. The aforementioned material properties of CFRP and epoxy were provided by their respective suppliers.

### **4. Results and Discussions**

Figure 4 shows the mid-span bending moment-curvature behavior for the beam specimens C1, CFRP-B1, CFRP-B2, CFRP-B3, and B4, respectively. The control specimen has shown a peak moment value of 17.73 kN-m with the curvature value of 0.0031. Once the peak moment was reached, the control beam specimen C1 started losing its strength and finally, it reduced to 4.92 kN-m at a curvature value of 0.0057. The control specimen has shown a residual strength of 9.5 kN-m, which was approximately half of the peak strength (17.73 kN-m) of the control specimen. The residual strength of the concrete beam was mainly attributed to the cracked concrete section, which contains tensile reinforcement having a tensile strain greater than yield but lesser than the rupture strain. In the case of different CFRP strengthened beam specimens, higher peak moments were observed. The presence of CFRP strips has contributed to the flexural capacity and increased not only the peak strength but has also undergone higher displacements at the peak moment values. All of the beam specimens have shown peak strengths ranging from 27.7 kN-m to 26.6 kN-m. The maximum peak moment of 27.7 kN-m was observed in the case of beam CFRP-B4. All of the beams (CFRP-B1, B2, B3, and B4) carry a uniform external CFRP flexural reinforcement ratio of 0.38%, however, the shear reinforcement was gradually increased from 0. 25% to 0. 56%, respectively. The effect of this change in CFRP shear reinforcement ratio could be seen in the form of small variations in peak moment carrying capacities of the beams. For all of the beams, the relationship between curvature and moment capacities remained linear until the curvature value of 0.0017 was reached. After this value, the inelastic behavior of reinforced concrete became dominant and stiffness of the beam gradually started decreasing. All of the CFRP strengthened beams have shown a high residual strength compared to the control specimen C1. However, the highest residual strength was provided by specimen B3 and it was approximately 1.5 times the residual strength of the control specimen. Moreover, compared to the control specimen, the deformation capacities or curvature of the CFRP strengthened beams B1, B2, B3, and B4 were also higher. Beams B1, B2, B3, and B4 have shown a rise and fall in the moment values. This rise and fall behavior in the moment values represents the debonding of CFRP strips from the surface of the beam. After a small segment of CFRP strip was detached from the beam surface, a drop in strength occurred. With a further increase of load, the remaining bonded part of CFRP and concrete came into action and beams again started taking the load until the peak moment capacities of the beams was reached. Beam B1 has the minimum CFRP reinforcement compared to all other beams, and as a result, a sudden drop in strength was observed due to complete detachment of the CFRP strip from the concrete surface. As the beams were subjected to further load increments, the strength of beams B1, B2, B3, and B4 became equal to the control specimen. It showed that the contribution of CFRP reinforcement had completely gone and only reinforced concrete beam residual strength was in action.

The mid-span bending moment-curvature behaviors of beam CFRP-B5 and CFRP-B6 and the control specimen C1 are given in Figure 5. In the case of beam CFRP-B5 and CFRP-B6, the external CFRP flexural reinforcement was doubled compared to beam specimens CFRP-B1, B2, B3, and B4. Both beams B5 and B6 carry the same flexural reinforcement of 0.75%. However, beam B6 has a 0.38% shear CFRP reinforcement ratio, whereas beam B5 does not carry any shear reinforcement. Beam B5 has shown flexural strength of 26.52 kN-m that is 1.5 times higher than the peak strength of the control specimen (17.73 kN-m). On the other hand, beam B6 has shown a peak strength value of 33.71 kN-m, which is approximately double the strength of the control specimen. Adding 0.38% shear reinforcement ratio has increased the flexural strength from 26.62 to 33.71 kN-m. The presence of shear

reinforcement hindered the diagonal shear crack propagation by intercepting their path, and in return, the moment carrying capacity of beam B6 has improved. The relation of moment-curvature remained linear prior to reaching the curvature values of 0.002. Beyond this value, a small rise and fall in the moment values was observed. These falls show the initiation of debonding failure between the CFRP and the concrete surface on the tension face of the beam.

**Figure 4.** Moment-curvature relation for control specimen C1, CFRP-B1, CFRP-B2, CFRP-B3, and CFRP-B4.

**Figure 5.** Moment-curvature relation for control specimen C1, CFRP-B5, and CFRP-B6.

Moreover, beam B6 has shown higher deformation or curvature compared to beam B5. For instance, the peak moment carrying capacity of the beam B5 occurred at a curvature value of 0.0028, whereas the peak strength of beam B6 was observed at a curvature value of 0.0038. It implies that the adding shear CFRP reinforcement in the presence of CFRP flexural reinforcement has not only increased the overall flexural strength of the beam but also the deformation capacity of the beam.

In order to understand the effect of shear reinforcement on the overall strength, a comparison of the mid-span bending moment-curvature behavior of beam CFRP-B3 and B5 is presented in Figure 6. Both beams CFRP-B3 and B5 carry an equal amount of overall (flexural plus shear) CFRP reinforcement ratio of 0.75%. However, in the case of beam B3, half (0.37%) of the total reinforcement (0.75%) was provided for flexure and the remainder was used for the shear. In the case of beam B5, the total reinforcement (0.75%) was provided as the flexural CFRP reinforcement, as given in Table 1. Beam B3 has shown peak strength of 27.7 kN-m compared to beam B5 that has a peak strength of 26.5 kN-m. Although a very small increase in overall strength was observed, beam B3 has shown better performance in terms of deformation capacity. Beam B5 has shown a curvature value 0.0028 at the peak moment of 26.5 kN-m and this was lower than the corresponding curvature value of 0.0035 for beam B3. For both beams, the debonding of CFRP from the tensile face started at the same curvature level, as indicated by a small fall in the moment at a curvature of 0.0019. However, the presence of shear strips provided resistance against shear cracks and improved the overall flexural behavior of beam B3. Moreover, beam B3 has shown better post-peak behavior compared to beam B5, as the residual strength of beam B3 was higher than beam B5.

**Figure 6.** Moment-curvature relation for control specimen C1, CFRP-B3, and CFRP-B5.

The test results show that beams with higher CFRP shear reinforcement but with a lower amount of flexural CFRP reinforcement cannot contribute significantly in terms of the overall strength of the beam. For instance, group-1 beams had a lower flexural CFRP reinforcement ratio (0.38%) and the CFRP shear reinforcement was gradually increased from 0.25% to 0.56%. This increase in CFRP shear reinforcement ratio has increased the strength of beam B1 from 26.6 kN-m to 27.7 kN-m, which is not a significant contribution. On the other hand, when the flexural reinforcement doubled as in the case of the group-2 beam (B6), the moment carrying capacity increased to 33.8 kN-m.

### **5. Analytical Prediction of Beam Capacities**

Several researchers compared the experimental and analytical flexural and shear capacities of different CFRP strengthened RC beams [21–32,37,38]. However, most of them [22,26–28,37] found close analytical values when ACI 440.2R-08 [33] was used. In the current study, ACI 440.2R-08 [33] was adopted to analytically obtain the flexural and shear capacities of the beams. The flexural strength of CFRP strengthened beams depends upon the governing mode of failure. Reinforced concrete beams strengthened with CFRP can experience three major failure modes: (i) tension failure of CFRP strip; (ii) debonding of CFRP from the concrete surface; and (iii) the crushing of concrete on the compression sides. ACI 440.2R-08 [33] considers all three failure modes to satisfy strain compatibility and force equilibrium conditions.

Figure 7 illustrates the internal strain and stress distribution of a CFRP strengthened rectangular RC beam under flexure. The ultimate nominal strength of beam (*Mn*) is obtained by satisfying strain compatibility and force equilibrium conditions and also considers the governing mode of failure. Equations (2)–(6) give the flexural or ultimate moment capacity of CFRP strengthened beams. In order

to check the debonding failure of CFRP strengthened beams, the effective strain in the CFRP strips (*εf*) should be limited to debonding strain level (*εf d*), as given by Equation (4).

$$M\_{\rm ll} = A\_{\rm s} f\_{\rm y} \left( d - \frac{a}{2} \right) + A\_{f} f\_{f} \left( d\_{f} - \frac{a}{2} \right) \tag{2}$$

$$
\varepsilon\_{fd} = 0.41 \sqrt{\frac{f\_c'}{nE\_f t\_f}} \le 0.9 \varepsilon\_{fu} \tag{3}
$$

$$
\varepsilon\_f = \varepsilon\_{\rm cu} \left( \frac{d\_f - c}{c} \right) \le \varepsilon\_{fd} \tag{4}
$$

$$f\_f = \ \varepsilon\_f E\_f \tag{5}$$

$$a = \beta\_1 c \tag{6}$$

**Figure 7.** Stain, stress, and force distribution diagram of CFRP strengthened RC beam.

According to ACI 440.2R-08, the shear strength of CFRP strengthened beams can be computed by Equation (7), as follows.

$$V\_{\rm ll} = V\_{\rm c} + V\_{\rm s} + \Psi\_f V\_f \tag{7}$$

where *Vn* is the nominal strength of a CFRP strengthened beam, *Vs* is shear strength provided by the steel and calculated by ACI 318-08 [34], as given by Equation (8), and *Vc* is the shear strength provided by the concrete, which can be computed by ACI 318-08 [34], as given by Equation (9).

$$V\_s = \frac{A\_v f\_y d}{s} \tag{8}$$

$$V\_c = 0.17 \sqrt{f'\_c} b\_w d \tag{9}$$

where *Vf* is the shear strength provided by the CFRP shear strips and Ψ*f* = 0.85 is a reduction factor for CFRP shear strips applied on the sides of beams. ACI 440.2R-08 [33] recommends Equation (10) to calculate the shear strength provided by the FRP strips. The effective stress in FRP and area of CFRP in shear can be calculated by using Equations (11) and (12), respectively. The dimensional variables used in Equations (10) and (12) can be seen in Figure 8.

$$V\_f = \frac{A\_{fv} f\_{fc} (\sin \alpha + \cos \alpha) d\_{fv}}{s\_f} \tag{10}$$

$$f\_{fc} = \varepsilon\_{fc} E\_f \tag{11}$$

$$A\_{fv} = 2nt\_fw\_f\tag{12}$$

**Figure 8.** Dimensional variables used in shear-strengthening calculations.

For bonded face plies or CFRP strips, which is a strengthening system that does not enclose the entire section of the beam, failure of CFRP shear strips is mainly attributed to the delamination of CFRP from the concrete surface. Due to this reason, the bond stress was analyzed by Triantafillou [39] to determine the usefulness of these systems and proposed the effective strain level that can be achieved. The effective strain (*εf e*) is calculated using a bond-reduction coefficient *kv*. Following Equations (13) and (14) gives the effective strain (*εf e*) and bond-reduction coefficient *kv*. Equations (15)–(17) give the factors *k*1, *k*2, and effective length *Le* used in Equation (14).

$$
\varepsilon\_{fc} = k\_{\overline{\nu}} \varepsilon\_{fu} \le 0.004 \tag{13}
$$

$$k\_{\upsilon} = \frac{k\_1 k\_2 L\_t}{11,900 \varepsilon\_{fu}} \le 0.75\tag{14}$$

$$k\_1 = \left(\frac{f\_c'}{27}\right)^{2/3} \tag{15}$$

$$k\_2 = \frac{d\_{fv} - 2L\_v}{d\_{fv}}\tag{16}$$

$$L\_{\rm t} = \frac{23 \,\text{300}}{\left(nt\_{f}E\_{f}\right)^{0.58}}\tag{17}$$

Table 2 gives a detailed comparison of analytical and experimental load-carrying capacities of the beam specimens. The control beam did not carry any flexural and shear CFRP reinforcement and its flexural and shear capacities were computed by ACI 318-08 [34]. The flexural and shear capacities of the CFRP strengthened beams are calculated by Equations (2) and (7), as recommended by ACI 440.2R [33]. The flexural capacity of the beam is the main function of the amount of CFRP applied at the tensile face of the beam. As beams B1, B2, B3, and B4 carry the same amount of CFRP flexural reinforcement ratio, these beams had the same analytical moment capacity of 25.35 kN-m. Similarly, beam B5 and B6 have the same analytical moment capacity of 33.41 kN-m. On the other hand, the shear capacity of the CFRP strengthened beams is proportional to the amount of CFRP shear reinforcement. Due to the different amounts of CFRP shear reinforcement, specimens B1 to B5 have different analytical shear capacities. However, beam B3 and B6 have the same CFRP shear reinforcement ratio and as a result, these have the same analytical shear capacities. From the analytical flexural and shear strengths of beams, their corresponding peak loads *Pm* for moment and *Pv* for shear are calculated and presented in Table 2. For instance, the analytical moment capacity of control beam 17.28 kN-m will result in the corresponding load capacity of (*Pm* = *Mn*×2 *L*1 ) 72.8 kN. The analytical peak load of each beam is evaluated as the lesser of *Pm* and *Pv*, depending upon which type of failure will govern the ultimate load carrying capacity of the beam. In order to understand the effect of CFRP flexural and shear reinforcement, the control beam was designed to have the same flexural and shear capacity. However, the *Pm* for control beam was slightly lesser than the *Pv* and it suggests that the dominant mode of failure in the beam is flexural rather than shear, or it indicates that the flexural failure will occur first,

followed by the shear failure of the beam specimen. Similarly, in the case of beams B1 to B4 and B6, flexural failure governed the overall design, and in the case of beam B5 the shear failure governed the peak load of the beam, as given in Table 2. A comparison of experimental and analytical peak loads has also been provided in the last column of Table 2. A very close agreement between analytical and experimental load values was achieved for all of the tested beams. In the case of beam B5, the difference is slightly more compared to the other beams. The experimental peak load of beam B5 was slightly higher compared to the analytical load. In reality, the CFRP flexural strip also contributes to the shear resistance of the beam, so to get closer analytical values requires an accurate prediction of CFRP flexural strip contribution in shear.

**Table 2.** Comparison of Experimental and Analytical Failure Loads.

### **6. Cracking Behavior of Beams**

### *6.1. Control Specimen C-1*

Figure 9 shows the fracture behavior of control specimen C-1 at different load levels. Hairline cracks were observed prior to reaching a moment value of 8.7 kN-m and a displacement value of 3.5 mm. However, with an increase of moment from 8.7 kN-m to 13.1 kN-m, the crack started to propagate to the compression face of the beam, as shown in Figure 9a. With further increases in load, the crack further widened and became more prominent, as shown in Figure 9b. The cracks initially generated vertically at the bottom face of the beam and became inclined with an increase in the depth of the beam. It shows a flexural failure followed by the shear failure of the beam. A clear wide crack was observed prior to the failure of the beam at a moment and displacement level of 17.8 kN-m and 8.78 mm, respectively, as given in Figure 9c. After the peak moment capacity of 17.8 kN-m, the beam started losing its capacity and load values started decreasing, with a significant increase in the deformation capacity of the beam. No hairline cracks with uniform spacing were observed at the bottom section of the beam, which indicates the section was quite weak in strength. Figure 9d shows the failure pattern of a control specimen prior to the completion of the test. At the deflection level of 12.57 mm, a major wide crack was observed, which had split the beam into two distinct parts.

### *6.2. Beam CFRP B1*

Failure patterns of beam CFRP B1 at different load levels are shown in Figure 10. In the case of beam B1, the initial minor cracks in the form of diagonal hairline cracks were spotted near the supports of the beam. These initial hairline cracks started to appear at a deflection value of 5.2 mm, as given in Figure 10a. With a further increase in the load, the beam started generating the new cracks and existing cracks were widened further, as shown in Figure 10b. As these cracks appear next to support in the form of diagonal movement, it confirms the shear dominant failure of the beams. The failure behavior of beam specimen B1 at the peak moment capacity of 26.6 kN-m and at a defection value of 8.9 mm is shown in Figure 10c. At a peak moment, cracks cross the FRP shear strip, however, due to higher shear forces, the existence of CFRP is not able to contribute much toward controlling the failure of the beam. Figure 10d presents the post-peak failure pattern of the beam at moment and deflection values of 10.73 kN-m and 17.3 mm, respectively. Beam B1 has only four CFRP shear strips on each face of the beam, with a higher spacing of CFRP shear strips. None of the CFRP shear strip intercepted the shear cracks initiated close to the right support of the beam. Therefore, the crack propagated diagonally to the compression face of the beam and resulted in the loss of beam strength. A significant amount of deflection and concrete crushing was observed close to the right support of the beam, which indicates that the bearing resistance of the beam was reduced and the concrete was finally crushed.

(**d**) Moment = 9.55 kN-m, deflection = 12.57 mm

**Figure 9.** Failure pattern of control specimen C1 at different load levels.

### *6.3. Beam CFRP B2*

Figure 11 shows the failure patterns of beam CFRP B2 at different load levels. Beam B2 carries a CFRP flexural reinforcement ratio of 0.38%, which was similar to beam B1, however, the CFRP shear reinforcement of beam B2 (0.31%) was slightly higher than beam B1 (0.25%). As a result, beam B2 carries five CFRP strips on each face of the beam, as shown in Figure 11. At a deflection value of 5.2 mm, no significant cracks were observed, however small hairline cracks were observed close to the left support of the beam, as shown in Figure 11a. A similar kind of pattern was noticed when the load on beam B2 was further increased until the mid-span deflection became equal to 7.7 mm, as given in Figure 11b. With a further increase in load, the cracks not only propagated diagonally but also became wide and visible to the naked eye. Figure 11c gives the cracking pattern of beam B2 at the peak moment capacity of 27.6 kN-m. Once the peak moment was reached, beam B2 started losing its capacity and cracks became more widespread and visible. In the case of beam B2, flexural mode of beam failure was dominant. Prior to complete failure, the diagonal crack reached the compression face of the beam and at that point, the beam had a residual moment carrying capacity of 18.9 kN-m, as shown in Figure 11d.

(**a**) Moment = 19.1 kN-m, deflection = 5.2 mm

(**b**) Moment = 22.9 kN-m, deflection = 6.63 mm

(**c**) Moment = 26.6 kN-m, deflection = 8.9 mm

(**d**) Moment = 10.73 kN-m, deflection = 17.3 mm

**Figure 10.** Failure Pattern of CFRP-B1 at different load levels.

### *6.4. Beam CFRP B3*

The graphical and experimental failure patterns of beam B3 are shown in Figure 12. In the case of beam B3, the shear reinforcement has been increased compared to beam B2, while the external CFRP flexural reinforcement was kept similar to beam B1 and B2. In beam B3, six shear strips were provided on each face of the beam, as shown in Figure 12. At a deformation level of 5 mm, a moment capacity of 18.8 kN-m was observed, which was greater than the moment capacities of beam B1 and B2, respectively. No significant number of cracks were observed in the case of beam B3, as shown in Figure 12a. When the load was further increased, very small hairline cracks were observed on the beams, as given in Figure 12b. Beam B3 showed better behavior compared to beam B1 and B2 at the deformation level of 7.3 mm. Figure 12c shows the failure behavior of the beam at the peak moment carrying capacity of 27.7 kN-m. At a peak load, the beam underwent a significant level of cracking and slight crushing of the concrete was also observed. It shows a flexural failure due to the crushing of concrete. However, the crack remained confined between the two FRP strips. The major cracks were spotted near the left part of the beam. Figure 12d shows the post peak-cracking pattern of beam B3. The residual strength of beam B3 was close to beam B2; however, the cracks remained restricted between two CFRP shear strips.

(**d**) Moment = 18.9 kN-m, deflection = 13.8 mm

**Figure 11.** Failure pattern of CFRP-B2 at different load levels.

### *6.5. Beam CFRP B4*

Figure 13 shows the graphical and experimental failure patterns of beam B4 at different load levels. In the case of beam B4, the CFRP shear reinforcement ratio was increased to 0.56% compared to beam B3 (0.38%), which resulted in a total of nine CFRP shear strips on each face of the beam. Figure 13a shows no significant cracks at a deflection value of 5.2 mm. However, when the load was slightly increased, it resulted in the formation of very small flexural and shear cracks near the mid-span of the beam, as given in Figure 13b. Beam B4 showed a peak moment capacity of 27.7 kN-m, which is similar to the moment capacity of beam B3. At the peak load level, the flexural cracks became more prominent and wider, as shown in Figure 13c. However, these cracks remained confined to the two CFRP shear strips. The cracks were also very steep compared to beam B1, B2, and B3, respectively. Figure 13d shows the cracking behavior of the beam at a deflection value of 15.4 mm. At this load level, the cracks passed through the CFRP shear strips and resulted in debonding of the shear strips from the concrete surface.

(**d**) Moment = 14.3 kN-m, deflection = 15.8 mm

### *6.6. Beam CFRP B5*

In beam B5, the amount of flexural reinforcement was doubled compared to beam B1, B2, B3, and B4. However, beam B5 did not contain any external CFRP shear reinforcement. Figure 14a–d presents the graphical and experimental cracking patterns of beam B5 at different load and deflection levels. A small number of cracks were observed at a deflection level of 5.2 mm (Figure 14a), which later on further increased in length and thickness when the deflection value reached 6.59 mm, as given in Figure 14b. These cracks were a mix of Mode-1 (flexural) and Mode-2 (shear) cracks, as they resulted from the interaction of bending and shear. Beam B5 has shown a peak moment capacity of 26.5 kN-m, which was greater than control beam but less than the other beams, such as B1, B2, B3, and B4, respectively. Prior to the peak moment capacity of 26.5 kN-m, a debonding type of failure of CFRP from the concrete surface was also observed. Figure 14c shows a significant number of cracks in beam B5, which were considerably wider in nature compared to the cracks that occurred at the deflection value of 6.59 mm. These cracks became even wider and longer when the deflection was increased and reached a value of 13.01 mm (Figure 14d).

### *6.7. Beam CFRP B6*

Figure 15 shows the failure pattern of beam B6, which contains both flexural and shear CFRP reinforcement. The flexural reinforcement of beam B6 was similar to beam B5 (0.75%), with a shear reinforcement ratio of 0.38%. Like other beams, no significant cracks were observed at a deflection value of 5.0 mm, as shown in Figure 15a. With a further increase in load and deflection value, small hairline cracks started appearing closer to the beam support. Figure 15b shows that these cracks were shear in nature and occurred at a deflection value of 6.94 mm. Beam B6 showed a maximum peak strength moment capacity of 33.8 kN-m at a deflection value of 10.8 mm. At this peak point, some of the CFRP strips were de-bonded from the concrete surface and the cracks further moved diagonally. Although these cracks were intercepted by the shear CFRP strips, CFRP shear strips could not hinder their propagation in the diagonal direction, as given in Figure 15c. The final failure of the beam was mainly due to the crushing of concrete in the compression side of the beam. The post-peak cracking pattern of beam B6 is presented in Figure 15d. Beam B6 has shown a residual strength of 31.6 kN-m at a deflection level of 12.1 mm. At a deflection value of 12.1 mm, one of CFRP shear strips was completely de-bonded and fell apart.

(**b**) Moment = 22.9 kN-m, deflection = 6.59 mm

**Figure 14.** *Cont*.

(**d**) Moment = 31.6 kN-m, deflection = 12.1 mm

**Figure 15.** Failure pattern of CFRP-B6 at different load levels.

### **7. Conclusions**

In this research study, a series of four-point bending tests were carried on reinforced concrete beams which were strengthened with different amounts and layouts of external CFRP flexural and shear reinforcement ratios. The layout of the CFRP application and the CFRP reinforcement ratio were the main parameters of the research study. Effects of these parameters were studied in terms of moment-curvature values, ultimate load carrying capacities, and failure patterns of the beams. A comparison between experimental and analytical ultimate load carrying capacities of the beams is also presented in this research study. The reinforced concrete beams were divided into two groups. Beams

in group 1 had a CFRP flexural reinforcement ratio half of that of group 2 beams. However, the CFRP shear reinforcement was gradually increased in the reinforced concrete beams. Based on the results obtained from the experimental study, the following major conclusions are summarized.

**Author Contributions:** The most significant contributions in this research study were made by M.U.S. and H.J.Q. The experimental scheme of the research study was planned by M.U.S. and H.J.Q., M.N.A., K.K., and H.K. prepared the beam specimens and also cured the beams at later stages. M.U.S., H.J.Q., M.N.A., and K.K. tested the specimens and collected the data. Results were prepared and analyzed by M.U.S., H.J.Q and M.N.A. The original draft was prepared by M.U.S., H.J.Q., and H.K. The review and editing were later carried out by M.U.S. and H.J.Q.

**Funding:** This research was supported by the Deanship of Scientific Research (DSR) at King Faisal University (KFU) through its "Eighteenth annual research project no. 180061".

**Acknowledgments:** This research was supported by the Deanship of Scientific Research (DSR) at King Faisal University (KFU) through its "Eighteenth annual research project no. 180061". Authors are thankful to the Deanship of Scientific Research, King Faisal University for supporting this entire research study. In particular, authors are thankful to the Dean of Scientific Research, King Faisal University, who provided a lot of administrative support for the successful completion of this research project.

**Conflicts of Interest:** Authors reserves the complete right of the data and declare no conflict of interest with any private and public sector body.

### **Abbreviations**

### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models**

### **Xiaofeng Gao 1, Chunfeng Liu 2, Yaosheng Tan 2, Ning Yang 2, Yu Qiao 2, Yu Hu 1,\*, Qingbin Li 1,\*, Georg Koval 3 and Cyrille Chazallon 3,4**

Received: 26 February 2019; Accepted: 22 March 2019; Published: 29 March 2019

**Abstract:** Tensile strength and fracture toughness are two essential material parameters for the study of concrete fracture. The experimental procedures to measure these two fracture parameters might be complicated due to their dependence on the specimen size or test method. Alternatively, based on the fracture test results only, size and boundary effect models can determine both parameters simultaneously. In this study, different versions of boundary effect models developed by Hu et al. were summarized, and a modified Hu-Guan's boundary effect model with a more appropriate equivalent crack length definition is proposed. The proposed model can correctly combine the contributions of material strength and linear elastic fracture mechanics on the failure of concrete material with any maximum aggregate size. Another size and boundary model developed based on the local energy concept is also introduced, and its capability to predict the fracture parameters from the fracture test results of wedge-splitting and compact tension specimens is first validated. In addition, the classical Bažant's Type 2 size effect law is transformed to its boundary effect shape with the same equivalent crack length as Koval-Gao's size and boundary effect model. This improvement could extend the applicability of the model to infer the material parameters from the test results of different types of specimens, including the geometrically similar specimens with constant crack-length-to-height ratios and specimens with different initial crack-length-to-height ratios. The test results of different types of specimens are adopted to verify the applicability of different size and boundary effect models for the determination of fracture toughness and tensile strength of concrete material. The quality of the extrapolated fracture parameters of the different models are compared and discussed in detail, and the corresponding recommendations for predicting the fracture parameters for dam concrete are proposed.

**Keywords:** boundary effect; size effect; fracture toughness; tensile strength; concrete

### **1. Introduction**

At present, several super-high arch dams have been constructed or are under construction in Southwest China [1–3]. To ensure the good working performance of concrete dams, one of the prerequisites is to prevent and control the formation and propagation of concrete cracks that may appear during the construction and operation periods. In fact, cracking is a very common, classical, but also complicated problem in concrete structures in real-life service conditions [4], which has been studied comprehensively from different perspectives by many researchers [5–9]. In terms of cracking risk analysis, the true fracture toughness *Kc* and tensile strength *ft* of the concrete are the key material parameters that should be provided. The improper fracture properties may lead to a result with great deviation from the reality. Many test results showed that the fracture toughness *Kc* of concrete has obvious size effect, and tensile strength *ft* usually depends on the specimen size and test method. To overcome the problem of size effect in fracture properties, it is often necessary to pour different sizes of concrete specimens to obtain the stable material parameter values. Thus, the test process is often complicated, especially for dam concrete, and the stable material parameters can only be measured from huge specimens [1,10]. In addition, the experimentally measured tensile strength of concrete varies for different test methods (uni-axial tension test, splitting test, flexure test, etc.) [11,12] due to the different fracture mechanisms. These size- or test method-dependent fracture parameters increase the difficulty of structure design and cracking risk analysis. Hence, it is crucial to find out a simple and relatively more accurate way to measure or predict the real fracture parameters of dam concrete.

Size effect laws (SEL) [13–17] and boundary effect models (BEM) [18,19] are two main asymptotic approaches to capture the size effect of concrete fracture. With these two kinds of models, the failure stress of structure with any specimen size or any crack size can be estimated, if the model parameters are fully known. On the contrary, if the test results of geometrically similar specimens with constant crack-length-to-height ratios or specimens with different crack to width ratios are available, size and boundary effect models can predict the material tensile strength *ft* and fracture toughness *Kc*. The applicability of the different size and boundary effect models on the determination of fracture toughness and tensile strength have been carefully examined with many test results in the literature [20–23]. It is widely accepted that the size effect laws can provide the fracture toughness directly based on the geometrically similar fracture tests; however, in order to estimate the tensile strength, a material characteristic length *lch* must be assumed in advance [20]. The different versions of boundary effect models proposed by Hu et al. [19,21,23–25] can directly determine the tensile strength and fracture toughness. These models can be classified as the local and non-local models, according to the stress value adopted in the models to predict the failure due to the strength of material. The local models only use the point stress information at the crack tip, while an average stress information along a certain length emanated from the crack tip is needed for the non-local boundary effect models. The Koval-Gao's size and boundary effect model developed by Gao et al. [26,27] is also a local model, which adopts the derivative of energy release rate evaluated locally at the crack tip as the parameter for the study of fracture dominated by the strength of material. The Koval-Gao's size and boundary effect model can also predict simultaneously the fracture parameters from the fracture test results.

In this study, the local and non-local versions of boundary effect models developed by Hu et al. [19,21,23–25] are summarized, and a modified Hu-Guan's boundary effect model with a more appropriate equivalent crack length definition is proposed. The proposed modified Hu-Guan's boundary effect model considers the effect of maximum aggregate size on the equivalent crack length, thus changes the contributions of strength of material and linear elastic fracture mechanics (LEFM) on the specimen failure. This improvement is anticipated to be important when the maximum aggregate size of concrete getting larger. Another size and boundary effect model developed based on a local concept is also introduced [26,27], and its capability to predict the fracture parameters from the fracture test results of wedge-splitting and compact tension specimens is firstly validated. Besides, the classical Bažant's Type 2 size effect law is transformed to a shape similar to the boundary effect model by the authors, which could be easier to be used to determine the fracture parameters from fracture tests on specimens with any sample size or crack size . The test results of geometrically similar wedge-splitting and compact tension specimens with constant crack-length-to-height ratios [28–30] and specimens with different initial crack-length-to-height ratios [31–33] are adopted, to verify the applicability of the

different boundary effect models on the determination of fracture toughness and tensile strength for concrete material.

### **2. Size and Boundary Effect Models**

The size and boundary effects of quasi-brittle fracture have been studied systematically in recent decades [13,18,19,26,27], and some progresses have been achieved in recent years by considering the effect of aggregate size on the fracture behavior [21–24,34]. These models were developed initially for the nominal strength prediction of samples with various specimen size and crack size, and later have been extended to the determination of the real fracture parameters from fracture test results.

In this section, the local and non-local versions of boundary effect models developed by Hu et al. [19,21,23–25] are summarized. Then, a modified Hu-Guan's boundary effect model with a new equivalent crack length definition is proposed. Following by the brief introductions of Koval-Gao's size and boundary effect models and Bažant's Type 2 size effect law. All the models are written as their own linear forms, which can be used directly to extrapolate experimental results obtained from laboratory size specimens. The equivalent crack lengths of different models are compared in detail and the method for the determination of fracture parameters is presented at the end of this section.

### *2.1. Hu et al. Boundary Effect Models*

### 2.1.1. Hu-Duan's Boundary Effect Model

The boundary effect models can characterize the effect of crack length on the failure load. Figure 1a shows a wedge-splitting (or compact tension) specimen with an initial crack length *a*0 under its failure load *Pmax*, and Figure 1b presents the two nominal stresses in the fracture analysis. The nominal strength *σN* of the specimen is defined without considering the existence of the initial crack *a*0, which can be calculated by the equilibrium conditions of the forces and moments [23]:

$$
\sigma\_N = \frac{4P\_{\text{max}}}{Wt} \,\text{'} \tag{1}
$$

where *W* is the sample size, *t* is the thickness of the specimen.

*σn*1 in Figure 1b represents the stress at the crack tip without considering the stress singularity, and assuming a linear distribution along the ligament, which reads [23]:

$$
\sigma\_{n1} = \frac{P\_{\max}(4\mathcal{W} + 2a\_0)}{t(\mathcal{W} - a\_0)^2},
\tag{2}
$$

where *a*0 is the initial crack length.

**Figure 1.** (**a**) A wedge-splitting specimen under its peak load and (**b**) the definition of two nominal stresses in Hu-Duan's boundary effect model.

In Hu-Duan's boundary effect model, the ratio of two nominal stresses *σN* and *σn*1 is a dimensionless factor *A*1(*α*) depending only on the initial crack length to sample size ratio *α* (*α* = *a*/*W*). For wedge-splitting (compact tension) specimen, *A*1(*α*) can be calculated by relating Equations (1) and (2).

$$A\_1(\alpha) = \frac{\sigma\_N}{\sigma\_{n1}} = \frac{2(1-\alpha)^2}{2+\alpha},\tag{3}$$

The stress criterion adopted in Hu-Duan's boundary effect model assumes failure happens when *σn*1 reaches its critical value, the material tensile strength *ft*. Thus, one can obtain the nominal strength *σN* predicted by the stress criterion by the following expression:

$$
\sigma\_N = A\_1(a) f\_{t\prime} \tag{4}
$$

The criterion of LEFM is directly adopted as the energy part of Hu-Duan's boundary effect model. The nominal strength predicted by the LEFM criterion can be written as a function of crack length *a*0:

$$
\sigma\_N = \frac{K\_\mathfrak{c}}{\overline{Y(\mathfrak{a})}\sqrt{\pi a\_0}'} \tag{5}
$$

where *Y*(*α*) is a geometrical factor. For wedge-splitting and compact tension specimen, the expression of *Y*(*α*) is identical [35]. This is because the geometry and loading conditions are the same for compact tension and wedge-splitting fracture tests [36]. It is given by:

$$Y(a) = \frac{(2+a)(0.886 + 4.64a - 13.32a^2 + 14.72a^3 - 5.6a^4)}{4\sqrt{\pi a}(1-a)^{3/2}}.\tag{6}$$

To link Equations (4) and (5), a transition (or reference) crack length *a*- ∞ needs to be introduced, which reads:

$$a^\*\_{\infty} = \frac{1}{1.12^2 \pi} \left(\frac{K\_c}{f\_t}\right)^2. \tag{7}$$

The transition crack length *a*- ∞ is a material constant proportional to material's characteristic length *lch* [37]. Thus, Equation (5) can be rewritten as:

$$
\sigma\_N = \frac{f\_t}{\sqrt{[Y(a)/1.12]^2 a\_0/a\_\infty^\*}},
\tag{8}
$$

The Hu-Duan's boundary effect model for the nominal strength prediction of structure with any crack length, can be obtained by relating Equations (4) and (8):

$$\sigma\_N = \frac{A\_1(a)f\_t}{\sqrt{1 + [A\_1(a) \times \Upsilon(a)/1.12]^2 a\_0/a\_\infty^\star}} = \frac{A\_1(a)f\_t}{\sqrt{1 + a\_{\ell 1}/a\_\infty^\star}},\tag{9}$$

or

$$
\sigma\_{n1} = \frac{f\_l}{\sqrt{1 + a\_{\varepsilon 1}/a\_{\infty}^\star}} \,\,\,\tag{10}
$$

where *ae*1 is the equivalent crack length and depends purely on the geometrical information of the specimen, which can be calculated by:

$$a\_{\epsilon1} = \left[\frac{A\_1(a) \times \mathcal{Y}(a)}{1.12}\right]^2 a\_0. \tag{11}$$

Equation (10) can be further rearranged as follows:

$$\frac{1}{\frac{\sigma\_{n1}^2}{\sigma\_{n1}^2}} = \frac{1}{f\_t^2} + \frac{1.12^2 \pi}{K\_c^2} a\_{c1\prime} \tag{12}$$

Once the peak loads are recorded after the fracture tests, the equivalent crack length *ae*1 and nominal strength *σn*1 are known accordingly. hence, the tensile strength *ft* and fracture toughness *Kc* can be calculated from the intercept and slope of the linear regression.

### 2.1.2. Hu-Guan's (Hu-Wang's) Boundary Effect Model

The Hu-Guan's and Hu-Wang's boundary effect models [21,23,34] are developed for wedge-splitting and three-point bending tests respectively, which all assume a constant crack-bridging stress *σn*2 within the partially developed fracture process zone or Δ*a fic* (see Figure 2a). In contrast to the local point stress value *σn*1 (see Figure 1b) defined in Hu-Duan's boundary effect model, *σn*2 (see Figure 2a) is associated with the fictitious crack length Δ*a fic*, and its value is affected by the choice of this length, thus becomes a non-local stress parameter. The strain condition along the crack plane, the equilibrium conditions of bending stress and bending moment were considered to determine this constant stress *σn*2 [21,34]. It is given by [23]:

$$
\sigma\_{n2} = \frac{P\_{\text{max}}(3\mathcal{W}\_2 + \mathcal{W}\_1)}{6t} / \left(\frac{\mathcal{W}\_1^2}{6} + \frac{\Delta a\_{fic}}{6}\mathcal{W}\_1 + \frac{\mathcal{W} - a\_0}{2}\Delta a\_{fic}\right),
\tag{13}
$$

with *W*1 = *W* − *a*0 − Δ*a fic* and *W*2 = *W* + *a*0 + Δ*a fic*.

The advancement from Hu-Duan's boundary effect model to Hu-Guan's boundary effect model is the non-local nominal stress *σn*2 used in the stress criterion. Thus, the local version of the boundary effect model was improved as a non-local version. Following the same idea of Hu-Duan's boundary effect model, Guan et al. proposed in [23] the way of calculating the tensile strength and fracture toughness by the following linear relation:

$$\frac{1}{\sigma\_{n2}^2} = \frac{1}{f\_t^2} + \frac{1.12^2 \pi}{K\_c^2} a\_{c1} \,\tag{14}$$

The local stress *σn*1 is replaced by the non-local stress *σn*2, and *ae*1 is the same as the one in Hu-Duan's boundary effect model. It is admitted that Hu-Duan's boundary effect model (Δ*a fic* = 0) may overestimate the tensile strength of the material, and Hu-Guan's boundary effect model can lower the predicted tensile strength when Δ*a fic* increases [22]. This fictitious crack length Δ*a fic* was assumed to be proportional to the maximum aggregate size *dmax* of concrete [23]:

$$
\Delta a\_{fic} = \beta\_1 d\_{max} \,\tag{15}
$$

It is concluded in [23] that *β*1 = 1 is a good approximation when the maximum aggregate size *dmax* is the dominant aggregate size of a concrete mix and the *W*/*dmax* ratio is below 20 (or even 50).

### 2.1.3. Hu-Zhang's Boundary Effect Model

Hu-Zhang's boundary effect model [24,25] added two more assumptions about the length scale than Hu-Guan's boundary effect model. That are, the fictitious crack length Δ*a fic* = *β*2*dav* ≈ 1.5*dav* (see Figure 2b, *dav* is the average aggregate size) is accurate enough when the ratio of the sample size and the average aggregate size varies in a certain range, and the transition crack length *a*- ∞ ≈ 3*dav* is appropriate to consider the micro-structure influence of granite and fine-grained polycrystalline ceramics studied in [24,25]. Based on these two assumptions, if the average aggregate size *dav* (transition crack length *a*- ∞ ) and tensile strength *ft* are decided, then one can directly estimate the fracture toughness *Kc* by Equation (7).

Since the average aggregate sizes are not known for the concrete fracture tests adopted in this study, the Hu-Zhang's boundary effect model will not be used to estimate the tensile strength and fracture toughness in the next sections.

**Figure 2.** The definition of two nominal stresses in (**a**) Hu-Guan's boundary effect model and (**b**) Hu-Zhang's boundary effect model.

### 2.1.4. Proposed Modified Hu-Guan's Boundary Effect Model

Based on Equation (1) and Equation (13) in Hu-Guan's boundary effect model, the nominal strength *σN* can be rewritten as a function of *σn*2:

$$
\sigma\_N = \sigma\_{n2} \times \left[ \frac{2(1-\alpha)(1-\alpha+2\Delta a\_{fic})}{2+\alpha+\Delta a\_{fic}} \right],
\tag{16}
$$

with

$$A\_2(\mathfrak{a}, \Delta \mathfrak{a}\_{fic}) = \frac{2(1 - \mathfrak{a})(1 - \mathfrak{a} + 2\Delta \mathfrak{a}\_{fic})}{2 + \mathfrak{a} + \Delta \mathfrak{a}\_{fic}}.\tag{17}$$

Unlike *A*1(*α*) in Equation (3), *A*2(*α*, Δ*αfic*) is no longer just geometry-related, but also material-related by the fictitious crack length Δ*a fic* , which is assumed to be proportional to the maximum aggregate size *dmax*. In the equivalent crack length calculation, to be more appropriate, it is recommended in this study to use *A*2(*α*, Δ*αfic*) instead of *A*1(*α*) , thus, the modified equivalent crack length *ae*2 would be:

$$a\_{t2} = \left[\frac{A\_2(a, \Delta a\_{fic}) \times \mathcal{Y}(a)}{1.12}\right]^2 a\_0. \tag{18}$$

In the linear regression of tensile strength *ft* and fracture toughness *Kc* calculation, *ae*1 should be replaced by *ae*2 in this modified Hu-Guan's boundary effect model:

$$\frac{1}{\sigma\_{n2}^2} = \frac{1}{f\_t^2} + \frac{1.12^2 \pi}{K\_c^2} a\_{c2}. \tag{19}$$

### *2.2. Koval-Gao's Size and Boundary Effect Model*

The Koval-Gao's size and boundary effect model [26,27,38] has two different forms. The second form is similar to the traditional Type 2 size effect law [14], which can be used to predict the size effect induced by the specimen size. The first form is close to the boundary effect model proposed by Hu et al., which can predict simultaneously the tensile strength and fracture toughness from the fracture test results. The Koval-Gao's size and boundary effect model was developed initially for cracked samples [26] by relating the contribution of the tensile strength and fracture energy. The contribution of the tensile strength is obtained from the derivative of the energy release rate, which is evaluated locally at the crack tip. While the contribution of the fracture energy is simply quantified by the LEFM criterion, same as the boundary effect models developed by Hu et al. The Koval-Gao's

size and boundary effect model was further improved as a local fracture criterion in [27,38], which can be easily applied for any geometry. The first form of Koval-Gao's size and boundary effect model reads:

$$
\sigma\_N = \frac{1.12 f\_l}{H(a)\sqrt{1 + a\_{c3}/a\_{\infty}^\star}},
\tag{20}
$$

where *H*(*α*) = *Y*2(*α*) + 2*Y*(*α*) × *dY*(*α*)/*dα* × *α*, and the equivalent crack length *ae*3 is defined as:

$$a\_{\varepsilon3} = \frac{Y^2(a)}{H^2(a)} \times a\_0. \tag{21}$$

This model can also be written as a linear form:

$$\frac{1}{\sigma\_{n3}^2} = \frac{1}{f\_t^2} + \frac{1.12^2 \pi}{K\_c^2} a\_{c3\prime} \tag{22}$$

with

$$
\sigma\_{\mathfrak{n}3} = \frac{H(a)}{1.12} \sigma\_{\mathfrak{N}}.\tag{23}
$$

So, the material tensile strength *ft* and fracture toughness *Kc* can be calculated from the intercept and slope of the linear regression.

### *2.3. Bažant's Type 2 Size Effect Law*

Bažant's Type 2 size effect law was originally developed for the size effect study of geometrically similar specimens [13]. The law reads

$$
\sigma\_N = \mathcal{B} f\_t \left( 1 + \frac{\mathcal{W}}{\mathcal{W}\_0} \right)^{-1/2},
\tag{24}
$$

where *B*ˆ is a geometrical dimensionless constant; *W*0 is a material constant proportional to *lch*. *B f* ˆ *t* and *W*0 were given as follows in [15]:

$$
\mathcal{B}f\_t = \sqrt{\frac{EG\_t}{g'(a)c\_f}}\tag{25}
$$

$$\mathcal{W}\_0 = \frac{c\_f g'(a)}{g(a)},\tag{26}$$

where *g*(*α*) = *K*2 *I*(*α*)*W*(*t*/*P*)2 = *Y*2(*α*)*πα* is defined as the dimensionless energy release function of LEFM; the length parameter *c f* is proportional to *lch*, which is the effective size of fracture process zone [15]. *g* (*α*0) can be written as a function of the geometrical correction factor *H*(*α*) presented in Equation (20):

$$
\xi'(\mathfrak{a}) = H^2(\mathfrak{a})\pi. \tag{27}
$$

After the substitution of Equations (7), (25)–(27) into Equation (24), the type 2 size effect law can be finally transformed to a shape similar to the boundary effect model:

$$\sigma\_N = \frac{1.12 f\_t \sqrt{a\_\infty^\star / c\_f}}{H(a)} \left( 1 + \frac{a\_{\varepsilon 3}}{c\_f} \right)^{-1/2} \text{ .} \tag{28}$$

where *ae*3 shares the same definition as the one presented in Koval-Gao's size and boundary effect model. This form of the Bažant's Type 2 size effect law can be further written as:

$$\frac{1}{\sigma\_{n4}^2} = \frac{c\_f}{K\_c^2} + \frac{1}{K\_c^2} a\_{c3\prime} \tag{29}$$

with

$$
\sigma\_{n4} = \sqrt{\pi} H(\mathfrak{a}) \sigma\_{\mathfrak{N}}.\tag{30}
$$

Thus, the fracture toughness *Kc* and the effective size of the fracture process zone *c f* can be identified from the size effect or boundary effect fracture test results. To obtain the tensile strength of the material, a ratio *γ*1 (*γ*1 = *c f* /*lch*) between *c f* and the material characteristic length *lch* should be assumed. It is reported in [20,39,40] that *γ*1 can be 0.28 or 0.29 for concrete material. Since *lch* = 1.122*πa*- ∞, another ratio *γ*2 = *c f* /*a*- ∞ can also be obtained, which is close to 1 when *γ*1 ≈ 0.28 *or* 0.29 (*γ*2 ≈ 1.10 *or* 1.14), means the effective size of the fracture process zone may be roughly equal to the transition crack size.

It is interesting to notice that the effective size of the fracture process zone *c f* in Bažant's Type 2 size effect law might be able to link to the fictitious crack length Δ*a fic* proposed in Hu-Guan' (Hu-Wang') boundary effect model (see Section 2.1.2). A reasonable choice of *γ*1 or *γ*2 would provide a better estimation of material tensile strength *ft*. The theoretical and experimental studies can be done in the future work to find out the approximate relation between *c f* and maximum aggregate size *dmax* or average aggregate size *dav*.

### *2.4. Comparison of Different Equivalent Crack Lengths*

Different size and boundary effect models have adopted different expressions for the calculation of equivalent crack lengths. *ae*1 (Equation (11)) in Hu-Guan's boundary effect model depends only on the crack-length-to-height ratio *α*, while *ae*2 (Equation (18)) proposed in modified Hu-Guan's boundary effect model believes the equivalent crack length depends on both the crack-length-to-height ratio *α* and fictitious crack-length-to-height ratio Δ*αfic*, thus becomes geometry- and material-related. The expression *ae*3 (Equation (21)) in Koval-Gao's size and boundary effect model, and Bažant's Type 2 size effect law is identical, which is only geometry-dependent. To better compare the different equivalent crack lengths, a wedge-splitting specimen with *W* = 1000 mm (see Figure 1a) is taken as an example, and its initial crack length varies from 50 mm to 950 mm. The variations of the equivalent crack length for different maximum aggregate size *dmax* with fixed discrete number *β*1, and variable *β*1 with fixed *dmax* are plotted respectively in Figures 3 and 4.

In different constructions, the maximum aggregate size *dmax* can be very different. For concrete used in Wudongde and Baihetan super-high arch dams, *dmax* = 40 mm, 80 mm or 150 mm for different parts of the dam. Meanwhile for the relatively smaller structures such as piers or beams, the frequently used *dmax* is only 10 mm or 20 mm. These possible *dmax* are all considered in this section, and the corresponding *ae*2 are plotted in Figure 3, together with the material (*dmax*) independent equivalent crack lengths *ae*1 and *ae*3. For the original Hu-Guan's boundary effect model, *ae*1 reaches its maximum value for any *dmax* when the crack-length-to-height ratio *α* is around 0.2, means for this ratio, the LEFM contributes the most to the failure of the specimen, thus should be recommended as the initial ratio for the fracture test on wedge-splitting specimen. This recommended ratio for Koval-Gao's size and boundary effect model, and Bažant's Type 2 size effect law is 0.3 to 0.4, which is closer to the usual initial crack-length-to-height ratio reported in the references [1,10]. In terms of *ae*2 proposed in this study, it is obvious that it is different from *ae*1, and this difference becomes larger as the increase of *dmax*. Hence, the proposed modified Hu-Guan's boundary effect model would be more appropriate than its original version, due to the fact that the two failure mechanisms are correctly combined even for the fracture analysis of dam concrete with maximum aggregate size reaches 150 mm. However, it should be admitted that for *dmax* such as 10 mm or 20 mm, the difference in *ae*2 and *ae*1 may has neglectable effect on the fracture properties estimated by the original and modified Hu-Guan's boundary effect model. In addition, the increase of *ae*2 at the tail part can be observed in Figure 3. This trend appears naturally due to its definition. An increase *ae*2 could maintain the validity of the failure mechanism of LEFM for all the possible crack-length-to-height ratios.

**Figure 3.** Equivalent crack length *ae* versus initial crack length *a*0 for different maximum aggregate size *dmax* when *β*1 = 1.

Figure 4 shows the equivalent crack length *ae* versus initial crack length *a*0 for different discrete number *β*1 when *dmax* = 150 mm. *ae*1 and *ae*3 are the same as the ones plotted in Figure 3 since they are independent on the fictitious crack length. *β*1 ranges from 0.25 to 2 is considered, which is a reasonable range for concrete material [23]. As shown in Figure 4, *ae*2 is strongly affected by the choice of *β*1 when *dmax* = 150 mm, thus may provide very different fracture parameters for different boundary effect models. It should be noted that according to the definition of *σn*2 (see Figure 2a) in original and modified Hu-Guan's boundary effect model, the fictitious crack length Δ*a fic* should be smaller than the length of the ligament, so the Hu-Guan's boundary effect model cannot be applied directly to some special cases (the tail part of *ae*2 in Figure 4) when the ligament is not enough to distribute the stress. This might be a shortcoming of the non-local models, but in fact, can be rarely encountered in the analysis of usual fracture test results.

**Figure 4.** Equivalent crack length *ae* versus initial crack length *a*0 for different discrete number *β*1 when *dmax* = 150 mm.

### *2.5. Method for the Determination of Fracture Parameters*

The fracture tests on geometrically similar specimens with constant crack-length-to-height ratios, and specimens with different initial crack-length-to-height ratios are frequently performed to study the dependence of fracture behaviors on specimen sizes and initial crack lengths. The test results

obtained from these two types of specimens can be used to extrapolate the tensile strength *ft* and fracture toughness *Kc* by the size and boundary effect models.

The linear forms of original Hu-Guan's boundary effect model (Equation (14)), modified Hu-Guan's boundary effect model (Equation (19)), Koval-Gao's size and boundary effect model (Equation (22)), and Bažant's Type 2 size effect law (Equation (29)) have presented previously in this section. Only the maximum loads and geometrical information of the tested specimens are needed for the linear regression analysis. The nominal stresses in the different linear models can be calculated by Equation (13) (original and modified Hu-Guan's boundary effect model), Equation (23) (Koval-Gao's size and boundary effect model), and Equation (30) (Bažant's Type 2 size effect law). In addition, the different equivalent crack lengths can be calculated by Equation (11) (original Hu-Guan's boundary effect model), Equation (18) (modified Hu-Guan's boundary effect model), and Equation (21) (Koval-Gao's size and boundary effect model and Bažant's Type 2 size effect law). With these two known values from the test results and specimens' geometrical information, one can do the linear regression analysis by the different models. Thus, the intercept and slope obtained from the best linear fit can be used to predict the material properties.

It should be noted that for the original and modified Hu-Guan's boundary effect models, the extrapolated fracture parameters vary together as the adjustment of the discrete number *β*1. Thus, special attention should be paid to the regression analysis, to ensure the predicted tensile strength and fracture toughness are all acceptable. For Bažant's Type 2 size effect law, as introduced in Section 2.3, only the fracture toughness *Kc* and the effective size of the fracture process zone *c f* can be obtained from the linear fit. To estimate the tensile strength *ft* of the material, a ratio *γ*1 (*γ*1 = *c f* /*lch*) between *c f* and the material characteristic length *lch* should be assumed. Then *ft* can be calculated as follows:

$$f\_l = \sqrt{\frac{\gamma\_1 K\_c^2}{c\_f}}.\tag{31}$$

Koval-Gao's size and boundary effect model is less flexible than the other two models, because the liner fit is unique for one set of the test results, and one can calculate the tensile strength from the intercept and fracture toughness from the slope, respectively.

### **3. Model Verification and Comparison**

The test results of geometrically similar wedge-splitting [28,29] and compact tension specimens [30] with constant crack-length-to-height ratios, and wedge-splitting specimens with different initial crack-length-to-height ratios [31–33] are adopted, to verify the applicability of the different size and boundary effect models on the determination of fracture properties for concrete material.

### *3.1. Tests of Geometrically Similar Specimens with Constant Crack-Length-to-Height Ratios*

### 3.1.1. Zhang's Experiments

Zhang et al. [28,29] carried out a series of experiments with geometrically similar wedge-splitting specimens, to investigate the size effects of concrete with small aggregate size. The height of the specimens ranges from 150 mm to 1200 mm, with the maximum aggregate size of 10 mm. There were 24 specimens in total, which were divided into 7 groups according to their heights, and each group had at least 3 samples. The compressive strength at 60 days was 29.56 MPa.

The sample dimensions, test results, equivalent crack lengths and nominal stresses in different boundary effect models are listed in Table 1. It should be noted that the dimensions of different samples in one group were slightly different due to the size deviations of the molds.

**Table 1.** Geometry, test results and equivalent crack length of wedge-splitting specimens [28,29].

To obtain simultaneously the tensile strength and fracture toughness, the linear regressions of original Hu-Guan's boundary effect model (Δ*a fic* = *dmax*), modified Hu-Guan's boundary effect model (Δ*a fic* = *dmax*), Koval-Gao's size and boundary effect model, and Bažant's Type 2 size effect law are plotted respectively in Figure 5. The estimated materials parameters (see Figure 5a) for original Hu-Guan's boundary effect model (*β*1 = 1) are: *ft* = 3.05 MPa, *Kc* = 1.25 MPa√*m*, *a*- ∞ = 42.9 mm and *lch* = 168.9 mm. It should be noticed that the results of Hu-Guan's boundary effect model are slightly different from the results presented in [23], because in this analysis, the more accurate sample dimensions and peak loads are adopted from [28,29]. For modified Hu-Guan's boundary effect model (see Figure 5b), when *β*1 = 1 (Δ*a fic* = *dmax*), one can obtain *ft* = 3.23 MPa, *Kc* = 1.25 MPa√*m*, *a*- ∞ = 38.1 mm and *lch* = 150.0 mm. For Koval-Gao's size and boundary effect model, *ft* = 2.36 MPa, *Kc* = 1.16 MPa√*m*, *a*- ∞ = 61.6 mm and *lch* = 242.9 mm can be calculated from the intercept and slope shown in Figure 5c. Bažant's Type 2 size effect law (see Figure 5d) gives *Kc* = 1.16 MPa√*m* and *c f* = 61.6 mm (Equation (29)). As already introduced previously in Sections 2.3 and 2.5, in order to obtain the tensile strength *ft* of the material, a ratio *γ*1 should be assumed, then *ft* can be calculated by Equation (31). It is obvious that mathematically *ft* would increase monotonically as the increase of *γ*1. For the test results analyzed in this section, *γ*1 = 0.1 and 1 gives *ft* = 1.48 MPa and 4.69 MPa, respectively. If *γ*1 = 0.28 or 0.29, the corresponding *ft* = 2.48 MPa *or* 2.52 MPa, which is close to the estimation of Koval-Gao's size and boundary effect model. The transition crack length *a*- ∞ = 55.86 mm and material characteristic length *lch* = 220.13 mm can also be calculated for the chosen *γ*1 = 0.28.

**Figure 5.** Test results in [28,29] and the corresponding fitted curves of (**a**) Hu-Guan's boundary effect model with *β*1 = 1, (**b**) modified Hu-Guan's boundary effect model with *β*1 = 1, (**c**) Koval-Gao's size and boundary effect model and (**d**) Bažant's Type 2 size effect law.

The tensile strength was not measured in [29], alternatively, we can estimate it from the measured compressive strength by *ft* = 0.24 *f* 3/2 *cu* = 2.29 MPa [31], which is only 2.9% difference with the estimation of Koval-Gao's size and boundary effect model. The Hu-Guan's boundary effect model and its modified version still somehow overestimate the material tensile strength with *β*1 = 1. However, as the increase of Δ*a fic*, the estimated *ft* would decrease accordingly, as what has done in [23], to test the effect of different Δ*a fic* on the predicted tensile strength.

For fracture toughness *Kc*, when the sample size is larger than 600 mm, the nominal fracture toughness *Kc* tends to be stable, and its value ranges from 0.93 to 1.13 MPa√*m*, hence, *Kc* = 1.16 MPa√*m* estimated by the Koval-Gao's model and Bažant model is acceptable. While for the original Hu-Guan model and its modified version, the estimated *Kc* = 1.25 MPa√*m* when Δ*a fic* = *dmax* seems slightly higher, since the largest measured *Kc* is only 1.09 MPa√*m* for the largest specimen with *W* = 1200 mm, whose failure can be assumed as totally LEFM control. Nevertheless, as the variation of the value of Δ*a fic* (or *β*1), the best estimations of *ft* and *Kc* given by Hu-Guan's boundary effect model can be achieved.

It is worthwhile to mention that the adjustment of *β*1 in original and modified Hu-Guan's boundary effect models would alter the predicted *ft* and *Kc* values simultaneously, to be more exact, as the increase of *β*1, the estimated *ft* would decrease and *Kc* would increase. This is simply due to the model assumption about the competition between two failure mechanisms (strength of material and

LEFM). A larger fictitious crack length gives a lower non-local stress *σn*2 (see Figure 2a) distributed along the fictitious crack, thus reduces the contribution of strength of material on the specimen failure, and finally leads to the different combination of predicted fracture parameters. This feature may introduce some difficulties to directly adopt the extrapolated fracture parameters to the numerical simulation of a real structure, since one may suspend the veracity of the estimated parameters. Therefore, before using the Hu-Guan's boundary effect model to extrapolate the fracture parameters, it is recommended to fix one parameter by tests or give a narrow range to it in advance. In this way, the other parameter can be more reliable, and a reasonable *β*1 is naturally obtained.

In terms of fracture parameters extrapolation, Bažant's Type 2 size effect law is also aided by a length assumption (*c f* = *γ*1*lch*). However, the different choice of *γ*1 would change the prediction of *ft* only. It is reported in [20,39,40] that *γ*1 is about 0.28 for concrete material, but as the increase of maximum aggregate size, especially for the dam concrete with *dmax* = 150 mm, *γ*1 ≈ 0.28 should be better verified by more experiential results. The third model, Koval-Gao's size and boundary effect model only provides one set of fracture parameters, which seems to be less flexible than the other models, but its estimated parameters are acceptable for this case.

### 3.1.2. Wittmann's Experiments

Wittmann et al. [30] performed the fracture tests on geometrically similar compact tension specimens. The heights of the specimens were 300 mm, 600 mm and 1200 mm, with 6 samples for each specimen size and the maximum aggregate size was 16 mm. The measured compressive strength at 28 days was 42.9 MPa. The sample dimensions, test results and model parameters are listed in Table 2. Only the mean value of the maximum loads of each group (6 samples) has been reported in [30].

Small 300 0.5 120 7.30 17.76 42.18 50.27 0.811 2.631 Medium 600 0.5 120 12.60 35.51 58.10 100.54 0.700 2.736 Large 1200 0.5 120 20.70 71.02 92.58 201.09 0.575 2.518

**Table 2.** Geometry, test results and equivalent crack length of compact tension specimens [30].

The linear regressions of different size and boundary effect models are plotted in Figure 6. When *β*1 = 1, the estimated materials parameters (see Figure 6a) for original Hu-Guan's boundary effect model are: *ft* = 3.83 MPa, *Kc* = 2.10 MPa√*m*, *a*- ∞ = 76.2 mm, *lch* = 300.3 mm, and for modified Hu-Guan's boundary effect model (see Figure 6b), the extrapolated parameters are: *ft* = 4.03 MPa, *Kc* = 2.09 MPa√*m*, *a*- ∞ = 68.3 mm and *lch* = 269.3 mm. Koval-Gao's size and boundary effect model gives *ft* = 2.93 MPa, *Kc* = 1.87 MPa√*m*, *a*- ∞ = 103.3 mm and *lch* = 407.2 mm (see Figure 6c). Bažant's Type 2 size effect law (see Figure 6d) gives *Kc* = 1.87 MPa√*m* and *c f* = 103.34 mm. In addition, the tensile strength *ft* = 3.09 MPa is obtained for *γ*1 = 0.28.

**Figure 6.** Test results in [30] and the corresponding fitted curves of (**a**) Hu-Guan's boundary effect model with *β*1 = 1, (**b**) modified Hu-Guan's boundary effect model with *β*1 = 1, (**c**) Koval-Gao's size and boundary effect model and (**d**) Bažant's Type 2 size effect law.

The tensile strength can be estimated from the compressive strength by *ft* = 0.24 *f* 3/2 *cu* = 2.94 MPa [31]. Similar to the estimated results of Zhang's experiments (see Section 3.1.1), the extrapolated tensile strengths of Koval-Gao's model and Bažant's model seem to be more reasonable than the values given by the original and modified Hu-Guan's boundary effect models. In terms of fracture toughness, the difference of the values given by the different models is within 15%, therefore, the model predictions are all acceptable. However, for the largest tested specimen (*W* = 1200 mm), the measured *Kc* is only 1.521 MPa√*m*. According to the experimental size effect study [1,10], for such a relatively huge specimen with ligament length (0.5*W*) to maximum aggregate size (*dmax*) ratio equals 37.5, the measured fracture toughness should be close to the real material fracture toughness already. Therefore, the lower fracture toughness prediction given by Koval-Gao's model and Bažant's model might be more appropriate. As already mentioned in Section 3.1.1, a choice of a smaller *β*1 could lower the prediction of fracture toughness given by the Hu-Guan's model, but at the same time, an unrealistic larger tensile strength would be obtained. This issue of the Hu-Guan's boundary effect model should be further studied.

### *3.2. Tests of Specimens with Different Initial Crack-Length-to-Height Ratios*

### 3.2.1. Xu's Experiments

Xu et al. [31] performed the wedge-splitting tests made of concrete with different initial crack-length-to-height ratios. The maximum aggregate size used was 20 mm. The height and thickness were fixed for all the specimens (*W* = 170 mm, *t* = 200 mm). There were three different initial crack lengths *a*0, which were 60 mm, 80 mm and 100 mm, respectively. The measured compressive strength *fcu* is 47.96 MPa, and the tensile strength *ft* was estimated as *ft* = 0.24 *f* 3/2 *cu* = 3.17 MPa by Xu et al. [31]. Details of the specimen dimensions, test results and equivalent crack lengths are listed in Table 3.

**Table 3.** Geometry, test results and equivalent crack length of wedge-splitting specimens [31].

Figure 7 presents the test results in [31] and the fitted curves of the different size and boundary effect models. It should be noted that the low coefficients of determination (*R*2) are simply due to the fluctuation of the test results. As shown in Figure 7, the linear trend of the test results is very clear and therefore, can be used to the prediction of fracture properties. For the original Hu-Guan's boundary effect model (see Figure 7a), it is provided in [23] that *ft* = 2.78 MPa, *Kc* = 1.73 MPa√*m*, *a*- ∞ = 97.65 mm and *lch* = 384.84 mm when *β*1 = 1. The modified Hu-Guan's boundary effect model (see Figure 7b) gives *ft* = 2.94 MPa, *Kc* = 1.71 MPa√*m*, *a*- ∞ = 85.63 mm and *lch* = 337.45 mm when *β*1 = 1. For Koval-Gao's size and boundary effect model, as shown in Figure 7c, one can obtain *ft* = 3.08 MPa, *Kc* = 1.10 MPa√*m*, *a*- ∞ = 32.25 mm and *lch* = 127.09 mm. The estimated values for Bažant's Type 2 size effect law (see Figure 7d) are: *Kc* = 1.10 MPa√*m* and *c f* = 32.25 mm. When *γ*1 = 0.28, the material tensile strength *ft* = 3.24 MPa can be obtained by Equation (31), and the corresponding *a*- ∞ = 29.23 mm and *lch* = 115.18 mm.

The tensile strength calculated by Koval-Gao's size and boundary effect model is again within 5% difference with the estimated tensile strength *ft* = 0.24 *f* 3/2 *cu* = 3.17 MPa by Xu et al. [31]. With the same discrete number *β*1 , the original Hu-Guan's boundary effect model gives a lower tensile strength than its modified one. In terms of fracture toughness, the estimation of Koval-Gao's size and boundary effect model is the same as the Bažant's Type 2 size effect law, and lower than the values given by the original and modified Hu-Guan's boundary effect models. However, under current conditions, it is difficult to decide which model provides the more appropriate fracture toughness. This is because the tested specimen size *W* = 170 mm, which is only 8.5 times bigger than the maximum aggregate size. For such a *W*/*dmax* ratio, the failure of the pre-cracked specimen is certainly controlled by the non-LEFM, and the real fracture toughness of the material should be evaluated by a larger specimen or estimated by the size and boundary effect models.

**Figure 7.** Test results in [31] and the corresponding fitted curves of (**a**) Hu-Guan's boundary effect model with *β*1 = 1, (**b**) modified Hu-Guan's boundary effect model with *β*1 = 1, (**c**) Koval-Gao's size and boundary effect model and (**d**) Bažant's Type 2 size effect law.

### 3.2.2. Wu's Experiments

Another series of wedge-splitting tests with different initial crack-length-to-height ratios were carried out by Wu et al. [32,33]. The maximum aggregate size was 20 mm. The height and thickness of the specimens were *W* = 170 mm and *t* = 200 mm, respectively. The samples were classified as 5 groups according to the different initial crack-length-to-height ratios ranging from 0.2 to 0.8. The measured compressive strength *fcu* was 38.8 MPa, and splitting tensile strength was 3 MPa. Details of the tests are listed in Table 4, the failure loads *Pmax* were the mean values of the different groups provided in [32,33].

**Table 4.** Geometry, test results and equivalent crack length of wedge-splitting specimens [32,33].

The linear regressions of different models are plotted in Figure 8. When *β*1 = 1, the estimated materials parameters (see Figure 8a) for original Hu-Guan's boundary effect model are: *ft* = 4.83 MPa, *Kc* = 1.42 MPa√*m*, *a*- ∞ = 22.0 mm, *lch* = 86.6 mm, and for modified Hu-Guan's boundary effect model (see Figure 6b), the extrapolated parameters are: *ft* = 6.74 MPa, *Kc* = 1.39 MPa√*m*, *a*- ∞ = 10.8 mm and *lch* = 42.5 mm. Koval-Gao's size and boundary effect model gives *ft* = 3.90 MPa, *Kc* = 1.61 MPa√*m*, *a*- ∞ = 43.1 mm and *lch* = 169.9 mm (see Figure 6c). Bažant's Type 2 size effect law (see Figure 6d) gives *Kc* = 1.61 MPa√*m* and *c f* = 43.13 mm. In addition, the tensile strength *ft* = 4.10 MPa is obtained for *γ*1 = 0.28.

**Figure 8.** Test results in [32,33] and the corresponding fitted curves of (**a**) Hu-Guan's boundary effect model with *β*1 = 1, (**b**) modified Hu-Guan's boundary effect model with *β*1 = 1, (**c**) Koval-Gao's size and boundary effect model and (**d**) Bažant's Type 2 size effect law.

For this example, the extrapolated tensile strengths of all the models are larger than the measured splitting tensile strength (3 MPa), this is because the different fracture mechanisms of the splitting and wedge-splitting tests. The latter one's failure mechanism is close to the flexure test. Therefore, the extrapolated tensile strength from the wedge-splitting tests should be close to the tensile strength measured by the flexure test. Raphael [11,12] concluded from plenty of the experimental results that the tensile strength measured by the flexure test is around 35% higher than the one measured by the splitting test. Hence, the extrapolated tensile strengths by the different models are still reasonable.