International Journal of Civil Engineering https://doi.org/10.1007/s40999-018-0336-6 RESEARCH PAPER Predicting the Shear Behavior of Reinforced Concrete Beams Using Non-linear Fracture Mechanics Vahid Broujerdian1 · Hossein Karimpour1 · Sobhan Alavikia1 Received: 11 September 2017 / Revised: 23 May 2018 / Accepted: 31 May 2018 © Iran University of Science and Technology 2018 Abstract This paper deals with the challenging problem of predicting the load carrying capacity of reinforced concrete shear-critical beams. To simulate the cracking behavior of concrete, the discrete crack approach based upon non-linear fracture mechanics is used. An algorithm with two pathways of implementation has been proposed so as to implement fictitious crack model to analyze reinforced concrete beams using finite-element method. The merit of the proposed algorithm is its capability to recognize and to incorporate them in the analysis procedure. The proposed method is capable of predicting simultaneous multiple shear cracks, load-deformation behavior, and ultimate shear capacity of reinforced concrete beam. The obtained results show a good compliance with the available experimental benchmark studies on reinforced concrete beams failed in shear. As one of the important aspects of shear capacity is the size effect issue, some large-scaled test beams have been numerically simulated using the proposed algorithm and the results have been compared with that of ACI 318-11 building code as well as the well-known modified compression field theory (MCFT). The comparison corroborated the robustness of the proposed algorithm in detecting the well-known shear-scaling phenomenon. In the same time, the comparison emphasized on the weakness of the current codes of practice to overestimate the shear capacity of large-scaled beams owing to not taking the size effect into account. Keywords Reinforced concrete · Beam · Shear · Fictitious crack model · Finite element 1 Introduction Diagonal cracking and shear behavior of reinforced concrete has been one of the conspicuous challenges which has been dealt with researchers since last century [1, 2]. Although several experimental as well as analytical models have been developed so far, there is not any generally confirmed and comprehensive model capable of relating the shear behavior to the main known factors. As a result, the current codes of practice concerning shear design of reinforced concrete [3–6] are merely empirical relations based on very simple physical models calibrated by some test results. On the other * Vahid Broujerdian broujerdian@iust.ac.ir Hossein Karimpour karimpour_h@alumni.iust.ac.ir Sobhan Alavikia s_alavikia@alumni.iust.ac.ir 1 Department of Civil Engineering, Iran University of Science and Technology, Tehran, P.O. Box 16765-163, Iran hand, studies show that safety factors are not integrated for different parameters [7, 8]. Using the classic theories of solid mechanics, some aspects of concrete behavior are neither justifiable nor predictable. In fact, the classic solid mechanics cannot justify failures that occur in stresses far below the ultimate strength of a concrete structure [9–11]. ACI 318 code of practice [3], in its simple relation, specifies the√shear capacity of reinforced concrete beams as Vc = 0.17 f � c bw d , in which f ′ c is the compressive strength of concrete in MPa, d and bw are the effective depth and the width of the beam, respectively, in mm, and Vc is in Newton. This equation is obtained based on ASCE-ACI 326 data bank [12] that only includes small beams with average depth of 340 mm. According to the above-mentioned equation, the average shear strength of concrete, vc = Vc ∕bw d , is independent of size. However, experimental observations including experiments conducted in Tokyo University [13] and Toronto University [14–16] demonstrate that the consideration of size effect in codes of practice is indispensable. In addition, recent surveys on some structure failures have shown that the size effect is one 13 Vol.:(0123456789) International Journal of Civil Engineering of the main contributory factors in failure [8]. A newer data bank including 398 data published by ACI445 Committee [17] clearly shows that in some cases, shear provisions of ACI code are not on the safe side. One of the leading cause of fracture and failure of structures is the presence of initial defects and cracks and their propagation. Defective construction, environmental factors, and applied loads may give rise to cracks with different size and shape. Cracks show different behaviors under different loading conditions. Some of these cracks propagate and bring about disastrous failure of structures. Fracture mechanics is the knowledge of examination of crack formation in solids, calculation of fractured structure response to applied loads, and prediction of structure behavior considering the propagation of initial cracks. Development of fracture mechanics in the last century has clarified that accurate analysis of many concrete structures entails the use of fracture mechanics principles. One of the branches of fracture mechanics employing some simplifications based on ignoring the crack tip plasticity is Linear Elastic Fracture Mechanics, LEFM, established by Griffith based on his experiments on glass fibers [18]. During 1960–1970, several researches showed that the classic LEFM is not capable of analyzing usual sized concrete structures [19]. Non-functionality of LEFM has been illuminated by the examination of Fracture Process Zone (FPZ). FPZ is a zone with micro cracks that includes ligaments still transferring stress. Researches have indicated that if softening behavior of material is considered, fracture mechanics can be a powerful tool for analysis of propagation of distributed cracks as well as localization of them in concrete [20–22]. Several models have been proposed so far to predict cracking behavior of concrete based on non-linear fracture mechanics. To employ these models for prediction of actual response of structures, making use of numerical methods such as FEM is inevitable. There are two main approaches for crack modeling via finite element, namely, smeared crack and discrete crack. In the former approach, the cracked element is considered continuous, while its governing constitutive law is changed to account for the effects of cracks [23–25]. Although this approach is convenient, its main shortcomings are non-objectivity and mesh-sensitivity of the results. Furthermore, it cannot provide details such as crack mouth opening displacement or crack path [20]. However, in the latter approach, crack is considered as a geometrical discontinuity [26]. Consequently, re-meshing after each step of crack propagation is needed. In discrete crack approach, the fracture process zone characteristics such as shape and cohesive forces are considered as boundary conditions of the problem. Since the analytical method in this type of modeling is close to the physical condition of concrete, this method named Fictitious Crack Model (FCM) is the most accurate method for analysis of concrete. After all, one of the intact areas in the discrete crack approach is the lack of multi-crack analysis based on FCM [27–29]. In this paper, an attempt in this ground is presented. Utilizing the ABAQUS software [30], a semi-automatic algorithm is proposed for this end. The presented method provides responses that are precise enough in comparison with benchmark problems. The algorithm enhances analysis to simulate multiple cracks, diagonal cracks, and size effect phenomenon. 2 Theoretical Background As it is known, the existence of FPZ, with a considerable size in concrete, ahead of structural macro cracks introduces a significant non-linearity in its behavior. The simplest description of what is happening in FPZ developed by Hillerborg et al. in 1976 named Fictitious Crack Model, FCM [31]. They proposed a tension-softening law based on Cohesive Crack Model, CCM, of Dugdale and Barenblatt [32, 33] to analyze a plain concrete beam (Fig. 1). Indeed, FCM lumps all the FPZ into the crack line and formulates it in the form of a softening stress–displacement law. Since this analytical approach is focusing on real physical Fig. 1 Fictitious crack model: a real crack in concrete, b fictitious crack, c tension-softening relation 13 International Journal of Civil Engineering conditions of concrete, the method is considered as one of the most accurate methods of concrete crack analysis. In addition, Hillerborg and his colleagues provided a numerical approach for crack analysis of a concrete beam under pure bending with only one single crack that can only be induced in the mid-span. Their proposed algorithm is summarized as follows: 1. analysis of beam; 2. locating crack extension based on the analysis; 3. applying cohesive forces on the sides of crack (based on tension-softening relation); 4. stress analysis with new boundary conditions; 5. repeat steps until structural failure. In reality, it is rare to confront a reinforced concrete element with only one single crack. In real problems, there exist multiple cracks at the same time with different reactions to applied loads: propagation, arrest, and closure. Therefore, considering all of the states that can occur upon loading, analysis of a multiple-crack problem could be very complicated. The gap of research in this area confirms this fact. This paper is aiming at introduction of a method to overcome this difficulty. According to Griffith theory [18, 34], creation and propagation of crack are inevitable processes of energy transfer of body in equilibrium state. In addition, fracture energy is required for creation of a new crack to reach minimum potential energy state. The cohesive fracture energy GF is the external energy required to fully break a unit surface area of cohesive crack. Assuming a linear pre-peak tensile behavior for concrete, it can be shown that GF is equal to the area under the softening curve (Fig. 1c) [33]. The J integral is usually used to characterize the energy release associated with crack growth. The J integral is a reliable fracture mechanics parameter for both linear and nonlinear material response [21]. It is a measure of the intensity of deformation at a notch or crack tip, especially for non-linear materials. In quasi-static analysis, two-dimensional definition of J integral is J = lim ∫ n . H . q dΓ Γ→0 Γ (1) where Γ is a contour starting from the bottom of crack surface to the top surface (Fig. 2). The limit Γ → 0 indicates that Γ shrinks onto the crack tip; q is a unit vector in the virtual crack extension direction; and n is the outward normal to Γ. H is defined as H = WI − 𝝈 . 𝜕u 𝜕x (2) where I is second order identity tensor, 𝝈 is stress tensor, 𝜕u∕𝜕x is displacement gradient tensor, and W is the elastic strain energy for elastic behavior. For elastic–plastic behavior, W is the elastic strain energy density plus plastic Fig. 2 Contour for evaluation of the J integral dissipation representing the strain energy in an equivalent elastic material. This study has been based on LEFM with additional development to account for non-linearity of concrete as a quasi-brittle material. 3 Methods The problem at hand is to simulate shear-critical reinforced concrete beams under 3-point-bending test condition. In addition to develop load-deformation behavior, it is aimed at obtaining an acceptable estimation of cracking pattern as well as mode of failure. Figure 3 illustrates the flowchart of the proposed algorithm which can solve such problem in a load-control manner. The main concept of the proposed algorithm is to increase the applied load incrementally so as to figure out locations in the beam where the maximum principal stress reaches the tensile strength of concrete, and then, the cracks initiate and propagate in critical direction. Cohesive forces are applied on the crack sides during crack evolution in each step. The path of crack propagation is assumed to be normal to the direction of maximum principal stress. The analysis continues up to the maximum load, where the convergence of the finite-element analysis cannot be achieved beyond it. According to this semi-automatic algorithm, analysis part of each step is done by computer using the ABAQUS software, but increasing crack length and applying cohesive forces on crack edges in each step are carried out manually. To account for cohesive forces, a linear tension-softening function, as shown in Fig. 4, is used. As seen in this figure, the maximum cohesive stress is the tensile strength of concrete, ft, corresponding to zero opening of crack mouth, w = 0, and the minimum value is zero corresponding to critical opening of wc. Thus, cohesive forces are calculated as ) ( w 𝜎 = ft 1 − , for 0 ⩽ w ⩽ wc (3a) wc 13 International Journal of Civil Engineering In the first method, after numerical analysis in each step and obtaining the stress distribution in the concrete elements, potential zones for crack initiation are located. To this aim, the criterion of limiting the maximum principal stress to the tensile strength of concrete is utilized. Then, the cracks are propagated manually in the direction normal to that of the maximum principal stress. To define the crack extension in this method, there is a special sub-menu in the ABAQUS software capable of assigning crack property to any desired line. A seam defines an edge with overlapping nodes that can separate during an analysis. This crack is originally closed but can open during an analysis. ABAQUS places overlapping duplicate nodes along a seam when the mesh is generated. After creation of seam, crack properties can be determined using a contour integral analysis. Now, the equivalent nodal cohesive pair forces are calculated and applied on the nearest mutual nodes on both sides of the extended crack (Fig. 5). In the second method, the process of obtaining stress distribution and locating the potential cracks are the same as that of the first method. However, the way of crack extension is different. In this method, to augment the crack length, the cracked elements are physically removed from the model and the resultant cohesive forces are applied on the crack sides (Fig. 6). Fig. 3 Flowchart of the proposed algorithm 4 Results and Discussion 4.1 Benchmark 𝜎 = 0, for w ⩾ wc . (3b) To simulate the compressive behavior of concrete, Hognestad model [35] has been used. Reinforcing bars are assumed to have elastic-perfectly plastic behavior. To implement the proposed algorithm, two different pathways may be chosen. The main differences between these two methods lie in the way that cracks are created and propagated and how the cohesive forces are applied. In what follows, explanations of these methods are described. Fig. 4 Linear tension-softening function used in this research 13 To evaluate the proposed algorithm, the results of some experimental tests on beams conducted by Bresler and Scordelis [36] are compared with the predictions of the present model. These experiments are of high quality as well as adequately documented, and have been used as benchmark in several analytical studies. The beams OA1 and A1 of the above-mentioned experimental set are considered here. Both of them are simply supported under a single concentrated load at the mid-span. Beam OA1 has only longitudinal reinforcement, but A1 has both longitudinal and International Journal of Civil Engineering Fig. 5 Illustration of applying cohesive stresses in the first method Fig. 7 Beam OA1 and A1 setup and details [36] Fig. 6 Illustration of applying cohesive stresses in the second method shear reinforcement. To impose the shear failure mode to the beams, they are reinforced with high ratios of longitudinal bars. The test setup and details are illustrated in Fig. 7. The geometrical and mechanical properties of cross section, concrete, and reinforcement are included in Tables 1 and 2. The maximum aggregate size used in the beams is 20 mm. The beams were loaded by a monotonic load-control mechanism at the mid-span. 4.2 Evaluation of the proposed methods Based on the previously introduced Methods, two-dimensional non-linear finite-element analyses are performed to study the load-deformation behavior and crack propagation of Bresler–Scordelis beams. FEM mesh is constructed using smart adaptive meshing of ABAQUS [30] with threenode iso-parametric elements. Longitudinal and transverse reinforcement are modeled in a discrete manner using wire element. Tie constraints have been used to define interaction between concrete and reinforcing steel. A tie constraint allows two regions to be bonded together, even though the meshes created on the surfaces of the regions are dissimilar [30]. The models are studied under load-control loading. Figure 8 shows the comparison of the load–displacement curves obtained by the proposed methods 1 and 2 with the results of Bresler–Scordelis experiment. In addition, the curve predicted by the well-known Modified Compression Field Theory, MCFT, [37] is shown in this figure. It must be pointed out that MCFT is a smeared rotating crack model that underlies the shear design provisions of CSA [5] and AASHTO [4]. Compression softening and tension stiffening are considered in MCFT. As shown in Fig. 8, for the both beams OA1 and A1, the proposed methods 1 and 2 predict the complete load-deformation curve equally well, and give notably better predictions than that of MCFT [37]. In addition, the accuracy of the predictions of MCFT deteriorates considerably for the no-stirrup beam OA1. However, the presented methods have uniform accuracy for both with and without stirrups cases. Furthermore, it can be seen that among the proposed methods, the predictions of method 2 for the load-deformation curve have more accordance with the test results. In Table 3, the ultimate loads, Pu , calculated by methods 1 and 2, are compared with those of experimental tests as well as MCFT. According to Table 3, the proposed methods 1 and 2 yield the ratio of test-to-predicted ultimate load for beam OA1 as 1.08 and 0.98, respectively, while MCFT has a ratio of 1.34 which shows its lack of accuracy. The corresponding values for Beam A1 are 1.13, 1.11, and 0.93, respectively. As shown in Table 3, the strength of A1 is overestimated by MCFT, while the predictions of the proposed methods are on the safe side. In Table 4, the computational results for the mid-span displacements at peak load, 𝛿0 , of the present study are compared with those of experimental tests and MCFT [37]. According to this table, the ratios of prediction-to-experimental ultimate displacements for Beam OA1 are 1.03 13 International Journal of Civil Engineering Table 1 Cross-sectional properties of Bresler–Scordelis Beams [36] Table 2 Material properties of Bresler–Scordelis Beams [36] Beam no. b (mm) h (mm) d (mm) L (mm) Span (mm) Bott. steel Top steel Stirrups OA1 A1 310 307 556 561 461 466 4100 4100 3660 3660 4 No. 9 4 No. 9 – 2 No. 4 No. 2@ 210 Reinforcement Concrete Bar size No. 2 No. 4 No. 9 Diameter (mm) Area (mm2) fy (MPa) fu (MPa) Es (MPa) 6.4 32.2 325 430 190,000 12.7 127 345 542 201,000 28.7 645 555 933 218,000 400 350 Load (kN) 300 250 200 150 Experiment Proposed Method 1 Proposed Method 2 MCFT [37] 100 50 0 0 2 4 6 Displacement (mm) 8 10 (a) 600 500 Load (kN) 400 300 Experiment Proposed Method 1 Proposed Method 2 MCFT [37] 200 100 2 4 6 8 10 12 14 Displacement (mm) 16 18 fr, MPa OA1 A1 22.6 24.1 3.97 3.86 2, respectively, and 0.78 for MCFT. Thus, the predictions of the present methods for ductility are more conservative than that of MCFT. Based on the foregoing results and discussion, it could be concluded that the proposed algorithm for implementing FCM in finite-element modeling yields acceptable results for predicting non-linear load–displacement behavior of shearcritical beams. Furthermore, the proposed method 2 is more accurate than method 1, at least for the studied samples. Of course, more corroborative study is needed to make a more reliable comparison. To delineate the fracture mode of the beams, the calculated shapes of OA1 and A1 along with their estimated crack patterns are illustrated in Fig. 9. In addition, for the sake of comparison, their experimental crack patterns are shown in Fig. 9. As the crack patterns of Bresler and Scordelis [36] were not available, their replicates by Vecchio and Shim [37] are used here, denoted by VS-OA1 and VS-A1. According to Fig. 9, the predicted cracking patterns and failure modes of the beams are in a good agreement with the experimental ones. 20 (b) Fig. 8 Comparison of experimental and predicted load-deformation curves: a beam OA1, b beam A1 and 0.96 for methods 1 and 2, respectively, while the ratio obtained by MCFT is 0.83. The corresponding values for Beam A1 are 1.18 and 1.08 for the proposed methods 1 and 13 f′c, MPa 4.3 Verification of Proposed Algorithm in Examination of Size Effect 0 0 Beam no After the validation of the proposed algorithm, it can be employed for analyzing large-scaled beams, and predicting shear strength of them. As the fictitious crack model preserve fracture energy criterion, the proposed model is expected to capture the size effect on shear strength of reinforced concrete beams [9–11, 27]. To examine the proposed model in this respect, the large-scale test beams of Collins and Kuchma [16] have been simulated by method 2, which is already shown that it is generally better than method 1. Details and test setup of these beams are depicted in Fig. 10. International Journal of Civil Engineering Table 3 Ultimate load obtained by the present study, MCFT [37], and the tests [36] Table 4 Mid-span deflection at peak load by the present study, MCFT [37], and the tests [34] Beam number Pu (kN) Pu-test/Pu-calc Test [36] MCFT [37] Method 1 Method 2 MCFT [37] Method 1 Method 2 OA1 A1 335 468 250 500 311 415 343 423 1.34 0.93 1.08 1.13 0.98 1.11 Beam number 𝛿0 (mm) OA1 A1 𝛿0-test ∕𝛿0-calc Test [36] MCFT [37] Method 1 Method 2 MCFT [37] Method 1 Method 2 6.7 14.1 8 18.0 6.5 12.0 7.0 13.1 0.83 0.78 1.03 1.18 0.96 1.08 Fig. 9 Comparison of cracking patterns and failure modes: a experimental VS-OA1 [37], b calculated OA1, c experimental VS-A1 [37], d calculated A1 (D-T: diagonal tension, V-C: shear compression) Fig. 10 Details and test setup of large-scaled beams of Collins and Kuchma [16] 13 International Journal of Civil Engineering The predicted load-deformation curves of the present study are shown in Fig. 11. The test shear capacity and the calculated shear capacity of the beams according to ACI 318-11 [3] and MCFT [16] are also shown in this figure. As it can be seen in Fig. 11, shear capacities calculated based on ACI provisions are not on the safe side owing to the exclusion of size effect from ACI relation. As MCFT includes size effect in its relation, its predictions for large beams are on the safe side. However, its accuracy is low. On the other hand, the proposed model completely captures the dependence of shear strength on the size effect. In Table 5, the ultimate loads, Pu, of the proposed method 2 are compared with those of experimental tests [16], MCFT [16] and ACI [3]. According to Table 5, the ratio of prediction-to-experimental ultimate loads has a mean of 1.02 and covariance of 3.6% for the proposed method 2 while these values for MCFT 600 600 ACI [3] 500 400 MCFT [16] 300 B100 200 100 ACI [3] 500 Test [16] Load (kN) Load(kN) Fig. 11 Comparison of the proposed method 2 predictions with ACI [3], MCFT [16], and the experimental results of large-scaled test beams [16] approach are 1.15 and 7.6%, and for ACI they are 0.80 and 7.9%, respectively. As shown in Table 5, the strength of large beams containing no stirrups is significantly underestimated by ACI. Among the mentioned test beams, B100B failed in a load about 70% of the ACI value. The mentioned discrepancy between real shear behavior of beams and ACI prediction makes concerns on size effect especially for beams without shear reinforcement. It must be mentioned that ACI permits to design beams without stirrups if factored shear is less than 0.5 Vc. In this regard, footings and slabs are designed so they do not require stirrups. The difference between predicted values and test results reveals the fact that addition of small amount of distributed reinforcement over the depth of the member has remarkable effect on its shear capacity and impedes sudden failures. Proposed Method 2 Test [16] 400 MCFT[16] 300 200 B100B 100 proposed method2 0 0 1 2 3 Displacement (mm) 0 4 0 1 2 3 4 Displacement (mm) 500 Test [16] 400 300 200 Test [16] 400 MCFT [16] 300 200 B100D 100 ACI [3] 500 MCFT[16] Load(KN) Load (kN) 600 ACI[3] 600 B100L 100 proposed method2 0 Proposed Method 2 0 0 2 4 6 8 0 2 Displacement (mm) Table 5 Ultimate load obtained by the present study, MCFT [16], ACI [3], and the tests [16] Beam number B100 B100B B100D B100L Mean CoV % 13 Pu (kN) 8 4 6 Displacement(mm) Pu-test/Pu-calc Test [36] MCFT [37] ACI [3] Method 2 MCFT [37] ACI [3] Method 2 468 408 556 448 368 378 528 378 556 578 640 579 442 422 531 446 1.27 1.08 1.05 1.19 1.15 7.6 0.84 0.71 0.87 0.77 0.80 7.9 1.06 0.97 1.05 1.00 1.02 3.6 International Journal of Civil Engineering 5 Conclusions This study examines diagonal cracking and shear behavior of reinforced concrete beams subjected to concentric gravity loads. The motive behind this study is the lack of multi-crack analysis based on FCM in the literature. Using the ABAQUS software, the classical finite-element method based on discrete crack approach is utilized for analysis. The behavior of cracks is simulated based on fictitious crack model or cohesive crack model. Due to limitations of the software on direct analysis of multiple diagonal cohesive cracking, a stepwise semi-automatic approach with two pathways of implementation is proposed in which the outcomes of both pathways were coinciding. The comparison of the predictions of the proposed method for reinforced concrete beams with test results indicates satisfactory accuracy in the evaluation of the load-deformation and cracking pattern of the beams. In addition, the proposed method demonstrates a very good accuracy in simulation of the behavior of largescaled beams, and size effect phenomenon. References 1. ASCE-ACI 445 (1998) Recent approaches to shear design of structural concrete. ASCE J Struct Eng 124(12):1375–1417 2. CEB, Concrete (1997) Tension and size effect, contributions from CEB task group 2.7. Comite Euro-International du Beton, Lausanne 3. ACI 318 (2011) Building code requirements for structural concrete (ACI 318-1) and commentary (ACI 318R-11). American Concrete Institute (ACI), Farmington Hills 4. AASHTO(2010) LRFD Bridge Design Specifications and Commentary. American Association of State Highway and Transportation Officials (AASHTO), Washington, DC 5. CSA (2006) Canadian Highway Bridge Design Code (CHBDC) Standard CAN/CSA-S6-06, CSA International, Toronto 6. EC2 (2004) BS EN 1992-1-1:2004 Eurocode 2. Design of Concrete Structures. Part 1: General Rules and Rules for Buildings, Thomas Telford, London 7. ASCE-ACI 445 (2003, Summary of the evaluations of the proposals for the quick fix for reinforced concrete members without transverse reinforcement. American Concrete Institute (ACI), Farmington Hills 8. ACI 446 (2007) Justification of proposed update of ACI code provisions for shear design of reinforced concrete beams. ACI Struct J 104(5):601–610 9. Bazant ZP, Kazemi MT (1991) Size effect on diagonal shear failure of beams without stirrups. ACI Struct J 88(3):268–276 10. Bažant ZP, Yu Q (2005) Designing against size effect on shear strength of reinforced concrete beams without stirrups: I. Formulation. J Struct Eng 131(12):1877–1885 11. Bažant ZP, Yu Q (2005) Designing against size effect on shear strength of reinforced concrete beams without stirrups: II. Verification and Calibration. J Struct Eng 131(12):1886–1897 12. ACI-ASCE Committee 326 (1962) Shear and diagonal tension. J Am Concr Instit 1(2):277–352 13. Shioya T, Iguro M, Nojiri Y, Akiyama H, Okada T (1989) Shear strength of large reinforced concrete beams. Fracture Mechanics: Application to Concrete. ACI Spec Publ 118:259–280 14. Lubell A, Sherwood T, Bentz E, Collins MP (2004) Safe shear design of large, wide beams: adding shear reinforcement is recommended. Concr Int 26(1):66–78 15. Angelakos D, Bentz E, Collins MP (2001) Effect of concrete strength and minimum stirrups on shear strength of large members. Struct J 98(3):291–300 16. Collins MP, Kuchma D (1999) How safe are our large, lightly reinforced concrete beams, slabs, and footings?”. Struct J 96(4):482–490 17. Reineck KH, Kuchma DA (2003) shear database for reinforced concrete members without shear reinforcement. ACI Struct J 100(2):240–249 18. Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198 19. ACI 446.1R-91 (1999) Fracture mechanics of concrete: concepts, models and determination of material properties. American Concrete Institute, Detroit 20. ACI 446.3R-97 (1997) Finite element analysis of fracture in concrete structures, American Concrete Institute, Farmington Hills 21. Kumar S, Barai SV (2011) Concrete fracture models and applications. Springer Science & Business Media, New York 22. Ingraffea AR, Gerstle WH (1985) Non-linear fracture models for discrete crack propagation. In: Shah SP (eds) Application of fracture mechanics to cementitious composites. Springer, Dordrecht, pp 247–285 23. Broujerdian V, Kazemi MT (2010) Smeared rotating crack model for reinforced concrete membrane elements. ACI Struct J 107(4):411–418 24. Broujerdian V, Kazemi MT (2016) Nonlinear finite element modeling of shear-critical reinforced concrete beams using a set of interactive constitutive laws. Int Civ Eng 14(8):507–519 25. Omidi O, Lotfi V (2010) Finite element analysis of concrete structures using plastic damage model in 3-D implementation. Int J Civ Eng 8(3):187–203 26. Abbasnia R, Aslami M (2015) Numerical simulation of concrete fracture under compression by explicit discrete element method. Int J Civ Eng 13(3):245–254 27. Bazant ZP, Planas J (1997) Fracture and size effect in concrete and other quasibrittle materials. CRC press, Boca Raton 28. Shi Z, Ohtsu M, Suzuki M, Hibino Y (2001) Numerical analysis of multiple cracks in concrete using the discrete approach. J Struct Eng 127(9):1085–1091 29. Shi Z (2009) Crack analysis in structural concrete: theory and applications. Butterworth-Heinemann, Oxford 30. Version ABAQUS 6.13 (2013) Analysis user’s Manual. Dassault Systems Simulia Corp., Providence, RI 31. Hillerborg A, Modéer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6):773–781 32. Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129 33. Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104 34. Griffith AA (1924) The Theory of Rupture. In: Proc. 1st Int. Congress Appl. Mech., Biezeno and Burgers eds., Waltman, 55–63 35. Hognestad E (1951) Study of combined bending and axial load in reinforced concrete members. University of Illinois, Engineering Experiment Station, Bulletin 36. Bresler B, Scordelis AC (1963) Shear Strength of Reinforced Concrete Beams. ACI J 60(1):51–74 37. Vecchio FJ, Shim W (2004) Experimental and analytical reexamination of classic concrete beam tests. ASCE J Struct Eng 130(3):460–469 13