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The critical crack lengths of wet-sieved concrete and fully graded concrete obtained from experimental measurements, numerical simulation and an analytical solution based on linear elastic fracture mechanics were quantitatively compared. Experiments were conducted on wedge-splitting specimens with depths of 200–1500 mm and a maximum aggregate size of 150 mm. Strain gauges and clip gauges were pasted and mounted along the direction of crack growth to measure the critical crack length. Numerical and analytical methods were used to calculate the critical crack length. The average relative error between the analytical calculations and measurements was still 14%, even for a specimen depth of 1500 mm. The effective fracture toughness was calculated by combining the peak load and critical crack length, resulting in an average relative error of 17%. These results indicate that the existing analytical method for determining the critical crack length is inadequate. Consequently, a modified analytical method was adopted, using 95% of the peak load in the post-peak region and the corresponding crack mouth opening displacement, which provided results that agreed well with the experimental data. Furthermore, the effective fracture toughness could be reasonably derived using the critical crack length from the modified analytical method. [ABSTRACT FROM AUTHOR]

• Mixed mode I-II fatigue crack growth in concrete with different strengths is simulated. • Effect of strength on the mixed mode I-II fatigue crack growth is discussed. • Modified Paris law is proposed considering the concrete with different strengths. • Mixed mode I-II fatigue crack propagation rate decreases with increasing the concrete strength. The mode-I fatigue crack propagation in concrete has been extensively studied. However, many concrete structure failures occur subjected to mixed-mode fatigue loads in practice. The accurate predictions for the mixed mode I-II fatigue crack propagation and fatigue life are crucial for evaluating the structural safety of concrete constructions. In this paper, the mixed mode I-II fatigue crack propagation process on concrete with different strengths is simulated using the fatigue tension-softening constitutive model and the crack propagation criterion of the initial fracture toughness as a parameter (SIF-based criterion). The numerical results indicated that the fatigue crack length decreases with increasing the concrete strength for a given fatigue load level, but the fatigue life significantly increases with concrete strength. Further, a modified Paris law is presented on the basis of the numerical results for concrete with different strengths. With the known tensile strength of concrete, the mixed mode I-II fatigue crack propagation rate of concrete with different strengths can be presented. The proposed model in this study is useful in further predicting the fatigue life of concrete structures under mixed-mode fatigue loads. [ABSTRACT FROM AUTHOR]

• A nonlinear numerical method for modeling mixed-mode I-II dynamic crack propagation was established and validated. • The mechanisms of unstable crack propagation in concrete were explained from an energy perspective. • The influence of several independent parameters on the stability of crack propagation in concrete were analyzed. Cracks in concrete will propagate unstably due to excessive shear stress, interactions at the rock-concrete interface, or sudden energy release, significantly compromising structural bearing capacity. To investigate this phenomenon, a nonlinear numerical method for modelling mixed-mode I-II crack propagation has been developed. This approach integrates dynamic equilibrium with a fictitious crack model and an initiation fracture toughness criterion. Key innovations include the incorporation of a finite element method with continuous nodal stress to enhance calculation accuracy, the utilization of kinetic energy to compensate for abrupt losses of strain energy during crack propagation, and the consideration of the deformation of the distributive beam and its interactions with specimens and supports. It was validated by modelling classic benchmarks for the dynamic initiation and propagation of brittle materials under mode I and mixed-mode I-II loading, and applied to analyse unstable crack propagation in concrete for four-point shear (FPS) beam specimens under various ratios of mode I and II stress intensity factors (K I / K II = 0 to 5.32) and loading rates (2 × 10-7 m/s to 2 × 10-3 m/s). Results indicated that the method effectively captures unstable crack propagation, with load-crack mouth shear displacement (CMSD) curves and crack propagation trajectories closely matching the experimental data. Furthermore, it was observed that when the elastic strain energy within the concrete beam exceeds the residual energy stored in the extended cracks, the crack transitions from stability to instability; conversely, if the elastic strain energy is less than or approximately equal to the residual energy, the crack decelerates and returns to stability. Additionally, parametric analyses reveal that lower distributive beam stiffness, shorter preset crack lengths, reduced concrete fracture energy, and a less robust cohesive force–displacement curve increase the likelihood of unstable crack propagation in concrete. [ABSTRACT FROM AUTHOR]

• A probabilistic method for predicting the mode I fatigue life of concrete is proposed and verified. • When required parameters are given, the proposed method can be used to predict the mode I fatigue life of concrete for any failure probability. • The significant scatter of fatigue life is attributed to the high sensitivity to the stochasticity of concrete material parameters. Accurately predicting the mode I fatigue life of concrete is critical for evaluating the safety of structures. However, due to the stochasticity of fatigue life, it remains a challenge to the deterministic methods in the existing literature. This study presents a probabilistic method for predicting the mode I fatigue life of concrete. The method considers four concrete material parameters in the initial fracture toughness-based crack propagation criterion and the tension-softening constitutive relationship as random variables following a two-parameter Weibull distribution. The detailed procedure for the implementation of the method is elaborated using the notched three-point bending (TPB) beam as an example, and the effectiveness of the method is verified by comparing the predicted fatigue life with the experimental results collected from the literature. Furthermore, the sensitivity analysis shows that the stochasticity of the initial fracture toughness is the most important factor leading to the variability of the fatigue life. Tensile strength stochasticity is the second most important factor. The effects of other material parameters on fatigue life are negligible. The proposed method allows reasonable prediction of fatigue life based solely on prior knowledge of the statistical laws governing the two factors. Meanwhile, the calculation results suggest that the large scatter in fatigue life is due to the high sensitivity to the stochasticity of concrete material parameters. The proposed method is expected to enhance the understanding of the stochastic nature of fatigue life and achieve accurate predictions at different failure probabilities. [ABSTRACT FROM AUTHOR]

To ensure the safety of concrete structures under fatigue loading, the fatigue crack propagation in concrete needs to be evaluated accurately. In this paper, a numerical method for mixed mode I-II fatigue crack propagation in concrete is proposed, in which the stress intensity factor (SIF)-based crack propagation criterion is employed, and the degradation of the cohesive force under fatigue loading is considered quantitatively. To validate the applicability of the numerical method, the mixed mode I-II fatigue fracture test of the three-point bending (TPB) beam is conducted. The fatigue crack propagation length is measured with the digital image correlation (DIC) method. Eventually, the applicability of the numerical method is validated by a reasonable agreement between the numerically derived crack propagation path, crack mouth opening displacement (CMOD), crack mouth sliding displacement (CMSD), crack propagation length, and mode I SIF and the experimental results. It is concluded that the proposed numerical method can be used to evaluate the mixed mode I-II fatigue crack propagation process of concrete when the initial fracture toughness, Poisson's ratio, and Young's modulus under static loading and the tension-softening constitutive relation under fatigue loading are given. In addition, the experimental results indicate that the mixed mode I-II fatigue failure of concrete occurs when the mode I SIF reaches a critical value, regardless of the fatigue load level and the fatigue life. The numerical results show that the mixed mode I-II fatigue crack propagation path is independent of the fatigue load level and approximately identical to that under static loading. In practical engineering, many concrete structures such as concrete pavements may be subjected to fatigue loads, e.g., cyclic vehicle loads. Under the fatigue loads, the crack would initiate, propagate, and even cause the fatigue failure of the concrete structures. Meanwhile, due to the complexity of the fatigue loads, the crack path could not be predicted in advance, making the evaluation of the fatigue performance more difficult. Therefore, a thorough investigation of the fatigue fracture properties of concrete is of great theoretical significance and practical application value. In this study, a numerical method for predicting the mixed mode I-II fatigue fracture performance of concrete is presented. With the numerical method, the crack propagation length, crack propagation path, deflection, crack opening displacement, crack sliding displacement, and eventual fatigue life of the concrete structures could be reasonably predicted. The numerical results could provide significant references to the fatigue life prediction of concrete structures in service, such as concrete pavement. In addition, the predicted fatigue crack length and crack path would also be helpful to the reinforcement and repair of concrete structures. [ABSTRACT FROM AUTHOR]

To study fracture properties and establish a shear–softening constitutive law for rock–concrete interfaces, direct tension, three-point bending, and single shear push-out tests were conducted on composite rock–concrete specimens with different degrees of interface roughness. The relationships between tensile strength (ft), average shear strength (τav), initial fracture toughness (Kini 1C), mode I fracture energy (GIf) and interfacial roughness were determined based on experimental results. A shear–softening constitutive law for rock–concrete interface was developed by measuring strain variations on rock surfaces under loading stages during single shear push-out tests and defined based on shear strength (τmax) and mode II fracture energy (GIIf). For practical applications, the relationships between τmax and ft and between GIf and GIIf were determined by statistically fitting the experimental data in such a way that shear–softening constitutive law could be conveniently determined simply by measuring ft and GIf parameters of rock–concrete interface. Also, numerical simulations were carried out to investigate crack propagation in rock–concrete interfaces under mixed mode I–II fractures. Predicted load versus crack mouth opening displacement (CMOD) curves agreed well with experimental findings and verified the shear–softening constitutive law for rock–concrete interfaces obtained in this study. [ABSTRACT FROM AUTHOR]

• Crack initiation load P i has been predicted using the maximum fracture load P max and FPZ length. • Analytical predictions of P i have been confirmed by experiments reported in literature. • The corresponding toughness K i and K un at P i and P max are obtained analytically. • The size-independent fracture toughness K IC are compared with K i and K un. • Simple approximation of the tensile stress at the notch tip is provided. The initial cracking load (P i) of a notched concrete beam, difficult to measure, has been predicted from the maximum fracture load (P max) together with the notch-tip fracture process zone (FPZ) at P max. A simple analytical model is used to link P i and P max together through the FPZ. The corresponding initiation and unstable fracture toughness, K i and K un , have also been determined. The model assumes that the lower limit of P max from a group of identical notched concrete beams corresponds to the lower limit of the FPZ length FPZ L , so that the lowest limit P max = P i is established for FPZ L = 0. Detailed FPZ measurements before and at P max using a digital image correlation technique were analysed and used to predict the initial cracking load P i. Six different specimens with various sizes (W = 40, 60 and 80 mm), initial notches (a 0 = 12 to 48 mm) and FPZ L (7 to 13 mm) at P max showed the average ratio of FPZ L /(W – a 0) was around 0.25, indicating the P i / P max ratio was around 0.67 based on the analytical model. The crack growth resistance K R -curve between the crack initiation toughness K i at P i and the unstable fracture toughness K un at P max was also established approximately by the simple model. Estimated intrinsic fracture toughness K IC was compared with K i and K un. The influence of average aggregate size d av on P i and FPZ L at P max was also discussed. [ABSTRACT FROM AUTHOR]

Combined with the fictitious crack model, the stress intensity factor (SIF)-based criteria are widely adopted to determine the crack propagation of mixed mode I-II fracture in normal strength concrete. However, less research is reported on the applicability of the different SIF-based criteria when they are used to analyze the crack propagation process of concrete with different strength grades. With this objective in mind, three-point bending and four-point shear tests were conducted in this study on C20, C50 and C80 grade concrete to measure the initial fracture toughness, fracture energy, load-crack mouth opening/sliding displacement (CMOD/CMSD). Four SIF-based criteria, including two initial fracture toughness-based (with/without mode II component of SIF K II ) and two nil SIF-based (with/without K II ), were introduced to determine crack propagation and predict the P - CMOD/CMSD curves for the notched concrete beams under four-point shear loading. The results indicated that the difference between the peak loads from experiment and from the analysis based on the nil SIF criterion with K II approximately increases with the increase of the concrete strength. By contrast, the predicted peak load and P - CMOD/CMSD curves adopting the initial fracture toughness-based criterion with K II showed better agreement with experimental results for the different concrete strength. Meanwhile, in the case of the initial fracture toughness-based criteria, the predicted initial load was underestimated if the component of K II was not considered. However, the fracture mode transformed from mixed mode I-II to mode I after the crack initiation, meaning the K II component in the criterion had a less significant effect on the crack propagation process. [ABSTRACT FROM AUTHOR]

Experimental tests were conducted on composite rock-concrete specimens with four roughness profiles to investigate the propagation of interfacial cracks under three-point bending and four-point shear conditions. By measuring the initial fracture loads, various combinations of interfacial stress intensity factors (SIFs) of Modes I and II corresponding to the initial fracture conditions were determined. Based on these results, an expression for classifying the initiation of interfacial cracks in mixed Mode I-II fracture was derived by normalization, which could eliminate the effect of interfacial roughness. Furthermore, a criterion for specifying propagation of the interfacial crack that takes into account nonlinear interfacial characteristics was proposed, which indicated that the crack would start to propagate along the interface when the SIFs caused by the external loads and the cohesive stresses satisfied this criterion. Numerical simulations of the interfacial fracture process were also conducted, introducing the crack propagation criterion to predict load–versus crack mouth opening displacement (P-CMOD) curves. They revealed fairly good agreement with the experimental results. Finally, by combining the criterion for maximum circumferential stress with the proposed criterion for crack propagation, the interfacial crack propagation mode was assessed. The results indicated that, once the initial fracture toughnesses for the rock, the concrete, and the rock-concrete interface from the experimental work were obtained, the propagation of interfacial cracks and the corresponding fracture modes, including nonlinear characteristics of the materials and interface, could be predicted using the method derived in this study. [ABSTRACT FROM AUTHOR]

• Influences of interface crack propagation on the fracture properties of concrete at different loading rates was studied. • The crack initiation load (P ini) decreased with an increase in aggregate position (d 0). • The interface can reduce the crack velocity, with this ability decreasing as d 0 increased. • The energy absorption capacity was significantly increased with the addition of coarse aggregates. The dynamics of fracture characteristics at the aggregate-mortar interface under varying loading rates present a significant area of investigation, pivotal for comprehending the response of concrete to dynamic load conditions. In concrete, the interface between the aggregate and the mortar forms a significant region, fundamentally dictating concrete structures' comprehensive strength and durability. This study aims to investigate the interface crack propagation process, including the crack propagation velocity and the propagation direction using Digital Image Correlation (DIC). Three-point bending beams with a single-size aggregate were fabricated. Beams with different initial crack tip-to-aggregate distances, d 0 (aggregate position) of 20 mm, 40 mm, and 60 mm, were tested at five different displacement-loading rates (0.012 mm/min, 0.12 mm/min, 1.2 mm/min, 12 mm/min, and 120 mm/min) using an electro-hydraulic servo testing machine. The crack propagation process was documented utilizing a high-speed video camera. The study reveals a direct correlation between loading rates and concrete's mechanical attributes. The results indicate that as loading rates rise, the compressive strength, tensile strength, and modulus of elasticity of concrete notably increase, while the Poisson's ratio decreases. The initial fracture toughness (K I c i n i) also escalates with increasing the rates, its sensitivity notably influenced by d 0. Concurrently, fracture energy increases with the increase of loading rate and the decrease of d 0 , a trend more evident at high loading rates. The crack initiation load (P ini) for all single-size aggregate concrete beams was higher than that of plain concrete, and P ini decreased as d 0 increased. Observations were made on the trajectory of crack formation, further scrutinizing the effects of d 0 and the variable loading rate. Notably, the presence of the interface was found to slow the crack propagation velocity, and this ability to inhibit crack propagation decreased as d 0 increased. As the loading rate increased, the crack propagation velocity at the mortar matrix and interface also increased. This study also reports additional material properties, such as critical crack length, and interfacial fracture mode. [ABSTRACT FROM AUTHOR]

• An analytical method for predicting the mode I crack propagation in concrete under low-cycle fatigue loading is proposed. • A good agreement between the analytical and experimental results verifies the effectiveness of the analytical method. • When the fatigue tension-softening constitutive relationship, initial fracture toughness, and Young's modulus are determined, the mode I crack propagation process of concrete under low-cycle fatigue loading can be calculated with the analytical method. The fatigue failure of concrete structures is preceded by the formation of microcracks and the propagation of macrocracks. Therefore, the fatigue fracture properties of concrete should be investigated thoroughly to guarantee the safety of structures. In this paper, an analytical method for predicting the mode I crack propagation in concrete under low-cycle fatigue loading is proposed according to the fictitious crack model, where the fatigue tension-softening constitutive relationship is introduced, and the initial fracture toughness-based crack propagation criterion is applied. The fatigue crack propagation process can be determined by solving two governing equations. To validate the effectiveness of the method, the fatigue crack propagation in three-point bending (TPB) beams is calculated, and the results are compared with the simulated and experimental results. It is shown that, although the linearization assumption of the crack opening displacement (COD) along the depth direction leads to slight differences between the analytical and simulated results, there is still a reasonable agreement between the analytical and experimental results. The effectiveness of the analytical method is verified. Therefore, when Young's modulus, initial fracture toughness, and fatigue tension-softening constitutive relationship are determined, the mode I fatigue crack propagation in concrete can be predicted based on the analytical method. It is expected that the method will contribute to the safety assessment of concrete structures under low-cycle fatigue loading. [ABSTRACT FROM AUTHOR]

• The critical crack propagation length of the TPB beam is accurately measured by pasting strain gauges and arranging clip gauges along the depth direction. • The elastic equivalent crack propagation length corresponding to 95% of the peak load in the post-peak region is approximately identical to the measured critical crack propagation length. • The fracture toughness of concrete can be derived reasonably based on the modified method of determining the critical crack propagation length. The critical crack propagation length is essential to calculate the fracture toughness of concrete and the further safety evaluation of the cracked concrete structures. In this study, the critical crack propagation lengths of the notched three-point bending (TPB) beams with various geometric dimensions and concrete strength grades are measured with two experimental methods. One way is to paste strain gauges on the sides of the crack propagation path, and another way is to arrange clip gauges along the depth direction. The results indicate that the effective crack propagation lengths derived based on the existing fracture model are smaller than the experimentally measured critical crack propagation lengths with the maximum relative error of 45%. To obtain the critical crack propagation length with the simple theoretical model accurately, a modified analytical method is proposed, where the elastic equivalent crack propagation length corresponding to 95% of the peak load in the post-peak section is treated as a good approximation of the critical crack propagation length. With the modified method for determining the critical crack propagation length, the fracture toughness of concrete can be derived reasonably. It is expected that the modified method would provide a valuable reference for determining the fracture property of concrete using the small size TPB beams in practical engineering. [ABSTRACT FROM AUTHOR]

Abstract: A crack propagation criterion was proposed for model I crack in concrete by using the initial fracture toughness as an inherent material property. Based on this criterion, crack begins to propagate when the difference, between the stress intensity factors caused by the applied load and that by the cohesive stress , exceeds . Finite element analyses was then carried out to calculate the complete load vs. crack mouth opening displacement (P-CMOD) curve, the critical crack propagation length ΔaC and the unstable fracture toughness for notched beams under three-point bending. It was found that numerical results showed a good agreement with the experimental ones. Based on this crack propagation criterion, crack extension resistance, in terms of stress intensity factor, KR being able to consider the variation of fracture process zone (FPZ) was employed for describing crack propagation in concrete. KR is composed of and , which is actually equal to the driving action of crack extension. It was concluded that given the elastic modulus E, the uniaxial tensile strength ft , the fracture energy GF and , the complete fracture process in concrete and the KR -curve of concrete can be calculated based on the numerical method. Finally, discussion was made on the effects of fracture process zone, GF and specimens geometries on KR -curve. [Copyright &y& Elsevier]

This study presents a simple fracture model linking average grain size G of granite to tensile strength f t and fracture toughness K IC . This model only requires the peak load P max measurements of small notched samples to determine both f t and K IC . The influence of G was considered for: (i) quasi-stable crack growth before P max , and (ii) fracture transition from f t to K IC criterion. In-depth analysis was carried out on three-point-bending (3-p-b) results from granite with G  ≈ 2.5 mm. The 3-p-b specimens have two different beam widths W  = 30 and 70 mm, and the same span/width ratio S / W of 4. Another granite with G  ≈ 10 mm reported in literature was also analysed to show the influence of different grain size on both f t and K IC . Comprehensive data on rock fracture are explained and compared with the results of this study to substantiate findings of this study. Experimental scatters in P max with different initial notch lengths a 0  = 0 – 53 mm were analysed by normal distribution, following a recent study (Zhang et al., 2018). The new model with normal distribution predicted the mean and upper and lower limits with 96% reliability covering the experimental scatters. [ABSTRACT FROM AUTHOR]

Highlights: [•] Initial fracture toughness is employed to determine crack propagation in concrete. [•] Three types of FPZ evolution depending on a 0/D are discovered for concrete. [•] FPZ evolution curve of plates with lower a 0/D envelops those with higher a 0/D. [•] FPZ evolution affects the shape of KR -curve for concrete. [•] KR -curve for concrete is found to be size-dependent by considering FPZ evolution. [Copyright &y& Elsevier]

• Crack-blunting fracture process zone (FPZ) width is modelled and emphasized. • FPZ width is about twice of FPZ length at the critical fracture load. • A wide range of brittle solids from rock to single crystal silicon are examined. • Boundary zones exist for both fracture energy and toughness measurements. • Explicit relation is established for fracture toughness and tensile strength. The crack-tip fracture process zone (FPZ) length with distributed cohesive stresses is commonly modelled for quasi-brittle fracture of heterogeneous solids. This study highlights the crack-tip blunting effect from FPZ width since FPZ W > FPZ L at the peak fracture load. Both FPZ width and length are linked to the microstructure (grain size for rock and ceramics, aggregate size for concrete, atomic diameter for single crystal silicon), providing a fresh perspective on quasi-brittle fracture phenomena. Importantly, FPZ W at the peak fracture load bridges the gap between the fracture toughness K IC and tensile strength f t , i.e. K IC ↔ FPZ W ↔ f t. The influence of a blunt notch on quasi-brittle fracture can also be explained by a widened FPZ W. A closed-form model containing both FPZ W and FPZ L (approximately FPZ W /FPZ L ≈ 2 at the peak fracture load) is used to analyse experimental data of rock, concrete and ceramic with macro-/micro-sized FPZ, and single crystal silicon with FPZ-like critically stressed atomic bonds in front of atomic scale defects. [ABSTRACT FROM AUTHOR]

• The mixed-mode I-II dynamic fracture in concrete is modeled using an initial fracture toughnessbased criterion. • The failure mechanism of offset-notched TPB concrete beams under different loading rates is analyzed. • The effect of the impact stiffness between the impactor and the impacted concrete beams on the failure modes is analyzed. To study the mechanism of mixed-mode I-II dynamic fracture of concrete, a crack propagation analysis method that employs the initial fracture toughness as a parameter of crack extension is proposed. The method utilizes a field variable transference (FVT) technique to model the coupling action of the stress wave propagation and crack growth and cohesive elements to describe damage evolution. The applicability of the presented method is confirmed by modeling crack propagation in brittle semi-infinite plates under tensile stress wave loading and in three-point bending (TPB) concrete beams at loading rates varying from 2.2 × 10−6 m/s to 3.55 m/s. The theoretical and simulated dynamic stress intensity factors (DSIFs) coincide, and the simulated resistance, displacement, crack growth velocity, and failure pattern agree with the experimental results. Moreover, additional numerical experiments are performed to evaluate the influence of loading rates and the impact stiffness on the failure process of TPB concrete beams with different notch locations. The results reveal that the transition of the failure modes of notched TPB concrete beams, from a tension-shear mode at low loading rates to a flexural mode at high loading rates, is dominated by the increase of the structural inertia, and the kinetic energy absorbed by the notched TPB concrete beam decreases by reducing the impact stiffness. [ABSTRACT FROM AUTHOR]

• Given the fatigue load level, the fatigue life increases significantly with increasing the concrete strength. • Decrease rate of the residual fracture toughness increases with an increase in the concrete strength. • Decrease rate of the residual fracture energy decreases with increasing of concrete strength. • Degradation models for the residual fracture toughness and fracture energy of different strength concrete are proposed. To evaluate the residual fracture toughness and fracture energy of concrete with different strengths after fatigue loading, the static, fatigue, and fatigue-static tests were conducted on 264 three-point bending (TPB) beams, respectively. The test results showed that the residual fracture toughness and fracture energy decreased rapidly at first and then steadily with the increase of the fatigue crack tip opening displacement and the number of fatigue cycles. For a given fatigue load level, the decrease rate of the residual fracture toughness increased with an increase in the concrete strength, but the decrease rate of the residual fracture energy decreased with increasing concrete strength. Finally, the degradation models for the residual fracture toughness and fracture energy of concrete were presented. When the tensile strength of concrete, the number of fatigue cycles, and the fatigue crack tip opening displacement are known, the residual fracture toughness and fracture energy of concrete after fatigue loading are determined. The research results are helpful to assess the safety of concrete structures in-service after fatigue loading. [ABSTRACT FROM AUTHOR]

Two hugely different mortar beams with the volume ratio close to 1:1000 were tested under three-point-bending (3-p-b) to verify fracture predictions for the large and long (LL) structures from results of the small and short (SS) specimens. They differed by both size and geometry: SS specimens with span length S = 0.1 m, width W = 40 mm and S / W = 2.5, and LL beams with S = 2 m, W = 500 mm and S / W = 4. This study performed and analysed quasi-brittle fracture of 0.1 m SS specimens with initial notch a 0 from 1 to 25 mm for determination of tensile strength f t and fracture toughness K IC , from which fracture of 2 m LL beams with a 0 of 250 mm was accurately predicted. Experiment results from the two vastly different 3-p-b beams, highly heterogeneous SS and nearly homogeneous LL in comparison to aggregates, were analysed by the boundary effect model (BEM). [ABSTRACT FROM AUTHOR]