The analysis of failure in concrete and reinforced concrete beams with different reinforcement ratio
 137 Downloads
Abstract
In this paper the analysis of failure and crack development in beams made of concrete is presented. The analysis was carried out on the basis of the performed experimental investigation and numerical simulations. A fictitious crack model based on nonlinear fracture mechanics was applied to investigate the development of strain softening of tensile concrete in plain concrete and slightly reinforced concrete beams. The role of strain softening was also discussed according to the inclined crack propagation in highly reinforced concrete beams. The analysis has brought the evidence that the mode of failure in flexural beams varies according to a longitudinal reinforcement ratio. A brittle failure due to the formation of a flexural crack takes place in plain and slightly reinforced concrete beams, and strain softening of tensile concrete is of paramount importance at failure crack initiation and propagation. A stable growth of numerous flexural cracks is possible in moderately reinforced concrete beams, and then the load carrying capacity is connected with reaching the yield stress of reinforcing steel or concrete crushing in the compression zone. In higher reinforced concrete beams without transverse reinforcement, brittle failure can take place due to shear forces and the development of diagonal cracks. However, strain softening of tensile concrete is not the only mechanism influencing the propagation of an inclined crack. Such mechanisms as aggregate interlock and dowel action of steel bars contribute more importantly to the development of failure crack.
Keywords
Strain softening Fictitious crack model Failure process1 Introduction
The load capacity of concrete structures is affected by the cracking behavior of concrete. Since the tensile strength of concrete is much lower than its compressive strength (approximately 10 times), concrete belongs to the group of brittle materials but it is not perfectly brittle. Concrete is considered to be a quasibrittle material, and therefore, when analyzing the cracking behavior of concrete, not only tensile strength but also tensile toughness is of paramount importance [1]. As a measure of concrete tensile toughness, fracture energy \(G_\mathrm{F}\) or fracture toughness \(K_\mathrm{IC}\) can be used [2, 3, 4]. However, fracture parameters of concrete have not been applied to the conventional design of concrete structures on the basis of European standards, for example, Eurocode 2 [5].
Structural members made of concrete are usually reinforced by steel bars. Steel reinforcement is used to resist tension, to distribute cracks and to limit cracks’ width. But the first aim of reinforcement is to protect against a brittle failure. When reinforcing steel bars are not designed properly, the structure will crack excessively and may fail. The failure behavior of concrete and reinforced concrete beams under flexure was analyzed by numerous researchers. The analyses were mostly dedicated to typical reinforced concrete beams with longitudinal and transverse reinforcement. Much smaller database of experimental results can be found for longitudinally reinforced concrete beams without transverse reinforcement [6, 7, 8, 9, 10]. The performed experimental investigations have shown that the efficiency of longitudinal reinforcement depends on its ratio. Furthermore, the failure process in flexural beams without transverse reinforcement can vary according to the longitudinal reinforcement ratio. The question arises as to what the influence of reinforcement ratio on crack initiation and development is and as to when steel reinforcement effectively protects against brittle, dangerous failure. These problems are discussed in the paper.
2 Failure analysis of concrete and reinforced concrete beams
2.1 Plain concrete and slightly reinforced concrete beams
2.2 Moderately and higher reinforced concrete beams
Depending on the reinforcement ratio, different mode of failure was observed in tested members.
The stable growth of vertical cracks was observed in the moderate reinforced beam of reinforcement ratio 0.9%. Such reinforcement effectively prevented against a sudden failure. A slow development of several flexural cracks was observed, and the flexural failure took place. The beam failed at the applied load \(F_\mathrm{max}=66~\hbox {kN}\) and the ultimate bending moment reached \(M_\mathrm{ult}=29.7~\hbox {kN~m}\) (\(M_\mathrm{ult}=V_\mathrm{ult}~a\), where \(V_\mathrm{ult}=F_\mathrm{max}~/2\) and a is the distance from the support to the applied load). The full flexural capacity connected with reaching the yield stress in steel bars was achieved.
In higher reinforced concrete beams with the longitudinal reinforcement ratio 1.3% and 1.8%, after flexural crack formation also the diagonal crack formed in the support zone of the beams. One major diagonal crack developed from the flexural crack due to shear stress which made flexural crack in the shear region to change its orientation and become a diagonal crack. In the tested beams the transverse reinforcement was not used, and therefore, the development of inclined cracks caused the shear, brittle failure. The beams failed soon after the appearance of the main diagonal crack. As shear forces governed the failure in higher reinforced concrete beams, the full flexural capacity due to the applied longitudinal reinforcement was not attained. The longitudinal reinforcement had an impact on the shear capacity of the highly reinforced concrete beams. With the increase in reinforcement ratio, the increase in cracking shear force which caused the appearance of diagonal crack \(V_\mathrm{cr}\) and ultimate shear force at failure \(V_\mathrm{ult}\) was noticed. (The cracking shear force was calculated as the half of the applied load at the moment of forming the first diagonal crack \(V_\mathrm{cr}=F_\mathrm{cr} /2\), and the ultimate shear force was calculated as the half of the applied load at failure \(V_\mathrm{ult}=F_\mathrm{max} /2\).) In the beam of reinforcement ratio 1.3%, the cracking shear forces was \(V_\mathrm{cr}=30~\hbox {kN}\) and the ultimate shear force was \(V_\mathrm{ult}=37.5~\hbox {kN}\), whereas in the beam of reinforcement ratio 1.9%, the cracking shear force was \(V_\mathrm{cr}=37\hbox { kN}\) and the ultimate shear force reached \(V_\mathrm{ult}=43.5~\hbox {kN}\). It can be observed that ultimate shear forces were higher than cracking shear forces and therefore the failure process caused by inclined cracks did not go in a rapid way.
3 Conclusions

A brittle failure due to the formation of a flexural crack takes place in plain and slightly reinforced concrete beams. The strain softening of tensile concrete is of paramount importance when analyzing crack propagation and failure process in these beams. The cracking moment decides about the load carrying capacity of the beams.

A stable growth of flexural cracks is possible in moderately reinforced concrete beams. The load carrying capacity is connected with reaching the full flexural capacity and depends on steel yielding or concrete crushing in the compression zone.

A shear failure which is caused by the development of diagonal cracks predominates in higher reinforced concrete beams without transverse reinforcement. The progressive microcracking appears in the tip of the inclined crack, but strain softening of tensile concrete is not the only mechanism of carrying shear stresses. An aggregate interlock and a dowel action of steel bars contribute more importantly to the development of a failure crack and load carrying capacity.
Longitudinal reinforcement can effectively protect from the unstable growth of flexural cracks when it is designed correctly. Longitudinal steel bars take also a contribution at carrying shear forces, but in beams without transverse they cannot be sufficient enough to resist the unstable growth of inclined cracks. The results of the performed experimental investigation have also shown how important part the transverse reinforcement can play in protecting against brittle shear failure, especially in beams with a relatively high longitudinal reinforcement ratio. (Some former analyses of over reinforced concrete beams can be found in [23, 24].) The deeper knowledge of the inclined cracks propagation will help to improve the optimal design of transverse reinforcement.
Notes
Acknowledgements
The work was financially supported by Lublin University of Technology (Grant No. S15/2017).
References
 1.Hillerborg, A.: The theoretical basis of the method to determine fracture energy \(G_{F}\) of concrete. Mater. Struct. 18(106), 291–296 (1985)CrossRefGoogle Scholar
 2.Köksal, F., Şahin, Y., Gencel, O., Yiğit, I.: Fracture energybased optimization of steel fibre reinforced concretes. Eng. Fract. Mech. 107, 29–37 (2013)CrossRefGoogle Scholar
 3.Karihaloo, B.L., Nallathambi, P.: Effective crack model for the determination of fracture toughness \(K_{IC}\) of concrete. Eng. Fract. Mech. 35, 637–645 (1990)CrossRefGoogle Scholar
 4.Xu, S., Zhang, X.: Determination of fracture parameters for crack propagation in concrete using an energy approach. Eng. Fract. Mech. 75, 4292–4308 (2008)CrossRefGoogle Scholar
 5.EN 199211:2004, Eurocode 2: Design of concrete structures. Part 1: General rules and rules for buildings. CEN, Brussels (2004)Google Scholar
 6.Leonhardt, F., Walther, R.: Versuche an einfeldrigen Stahlbetonbalken mit und ohne Schubbewerhrung. Deutscher Ausschu\(\upbeta \) für Stahlbeton Heft 151, W. Ernst, p. 83 (1962)Google Scholar
 7.Kani, G.N.J.: Basic facts concerning shear failure. J. ACI 63, 675–692 (1966)Google Scholar
 8.Bažant, Z.P., Kazemi, M.T.: Size effect on diagonal shear failure of beams without stirrups. ACI Struct. J. 88, 268–276 (1991)Google Scholar
 9.Słowik, M.: The analysis of crack formation in concrete and slightly reinforced concrete member in bending. In: Brandt, A.M., Li, V.C., Marshall, I.H. (eds.) Brittle Matrix Composites, pp. 351–360. Woodhead Publishing Limited, Cambridge (2006)CrossRefGoogle Scholar
 10.Słowik, M., Nowicki, T.: The analysis of diagonal crack propagation in concrete beams. Comput. Mater. Sci. 52, 262–267 (2012)CrossRefGoogle Scholar
 11.CEBFIP Model Code 1990. Bulletin d’information no. 196. First Draft, Lausanne (1990)Google Scholar
 12.Hillerborg, A., Modeer, M., Petersson, P.E.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6, 773–782 (1976)CrossRefGoogle Scholar
 13.Bažant, Z.P., Oh, B.H.: Crack band theory for fracture of concrete. RILEM Mater. Struct. 16(93), 155–177 (1983)Google Scholar
 14.Cedolin, L., Poli, S., Iori, I.: Tensile behavior of concrete. J. Eng. Mech. Div. ASCE 113(3), 431–449 (1987)CrossRefGoogle Scholar
 15.Rossello, C., Elices, M., Guinea, G.V.: Fracture of model concrete: 2. Fracture energy and characteristic length. Cem. Concr. Res. 36, 1345–1353 (2006)CrossRefGoogle Scholar
 16.Kwon, S.H., Zhao, Z., Shaha, S.P.: Effect of specimen size on fracture energy and softening curve of concrete. Cem. Concr. Res. 38, 1049–1069 (2008)CrossRefGoogle Scholar
 17.Słowik, M., BłazikBorowa, E.: Numerical study of fracture process zone width in concrete members. Arch. Civ. Eng. Environ. 2, 73–78 (2011)Google Scholar
 18.Słowik, M., Smarzewski, P.: Study of the scale effect on diagonal crack propagation in concrete beams. Comput. Mater. Sci. 64, 216–220 (2012)CrossRefGoogle Scholar
 19.Słowik, M.: Shear failure mechanism in concrete beams. Proc. Mater. Sci. 3, 1977–1982 (2014)CrossRefGoogle Scholar
 20.Ožbot, J., Li, Y., Kožar, I.: Model for concrete with relaxed kinematic constraint. Int. J. Solids Struct. 38, 2683–2711 (2001)CrossRefGoogle Scholar
 21.Gerstle, W., Bažant, Z.P. (eds.): Concrete Design Based on Fracture Mechanics, vol. SP134. ACI, Michigan (1992)Google Scholar
 22.Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton (1998)Google Scholar
 23.Van Mier, J.G.M., et al.: Strainsoftening of concrete in uniaxial compression  Report of the RoundRobin Test carried out by RILEM TC 148SSC. Mater. Struct. 30(198), 195–209 (1997)CrossRefGoogle Scholar
 24.Ožbot, J., Li, Y.J., Eligehausen, R.: 3D finite element analysis of overreinforced beams. In: Karihaloo, B.L. (ed.) Fracture Mechanics of Concrete Structures, pp. 1233–1240. Aedificatio Publishers, Freiburg (1998)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.