This paper formulates an analytical calculation model for predicting the cracking behavior of reinforced concrete ties to provide more consistent crack width calculation methods for large-scale concrete structures in which large bar diameters and covers are used. The calculation model was derived based on the physical behavior of reinforced concrete ties reported from experiments and finite-element analyses in the literature. The derivations led to a second order differential equation for the slip that accounts for the three-dimensional effects of internal cracking by using a proper bond-slip law. The second order differential equation for the slip was solved completely analytically, resulting in a closed-form solution in the case of lightly loaded members and in a non-closed-form solution in the case of heavily loaded members. Finally, the paper provides a solution strategy to facilitate a practical and applicable method for predicting the complete cracking response. Comparison with experimental and finite-element results in the literature demonstrated the ability of the calculation model to predict crack widths and crack spacing consistently and on the conservative side regardless of the bar diameter and cover.

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The cracking behaviours of reinforced-concrete (RC) ties are investigated by conducting virtual experiments using non-linear finite-element analysis. The assumptions in the model are verified by benchmarking the classical experiments of B. Bresler and V. V. Bertero as conducted in 1968 and P. J. Yannopoulos, conducted in 1989, which shows good agreement in the comparison of steel strains, development of crack widths and crack spacing. Furthermore, virtual experiments on four different RC ties show that the size of the cover and not the bar diameter governs the crack spacing and thus implicitly the crack width. An increase of the bar diameter has a beneficial effect in reducing the steel stress and the associated steel strains, which in turn reduces the crack width. Finally, a single bond-slip curve is sufficient in describing the average bond transfer of an arbitrary RC tie.

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The modified cracked membrane model (MCMM) presented in this paper was formulated to facilitate a mechanical calculation model that predicts crack widths in reinforced concrete (RC) structures subjected to in-plane loading for all cracking stages. It was formulated using the basic concepts of the existing cracked membrane model (CMM). Furthermore, a generalized approach for predicting the tension stiffening normal to a crack was formulated, an approach currently lacking in Eurocode 2 and fib Model Code 2010. A simplified approach for predicting the cracking behaviour of RC membranes was also proposed. Comparison with a total of 101 maximum crack widths measured experimentally on 37 test specimen from the literature showed that the MCMM provided good and consistent crack width predictions even for the cases of large rebars and covers, at which the CMM was seen to struggle. The results in this paper suggests that both the MCMM and the simplified approach show great potential for yielding reliable crack width predictions in RC membranes.@en

The strain profile over the cover in reinforced concrete ties subjected to tension is investigated in this paper. This is normally neglected in the crack width calculation methods recommended by Eurocode 2 and fib Model Code 2010, meaning that it is assumed uniformly distributed over the cover. However, this assumption is questionable in the case of large covers. A pragmatic approach of accounting for the non-uniform strain profile over the cover in an analytical crack width calculation model can be by using the concept of the strain variation parameter ψ, which relates mean concrete strains over the cover to concrete strains at the steel bar surface at an arbitrary section over the bar length, x. Virtual experiments on six cylindrical concentrically reinforced concrete specimens were thus conducted by using nonlinear finite element analysis to establish a better understanding of the strain profile over the cover and, if possible, to obtain a proper value for ψ(x). The results show that ψ more or less remains constant over the bar length, except for the region close to the loaded end. This means that assuming a constant value for ψ seems reasonable. The practical significance of this finding is that the non-uniform strain profile over the cover can be properly accounted for in deriving and solving the second order differential equation for the slip between concrete and steel. This would ultimately yield an analytical crack width calculation model that predicts crack widths consistently, also in cases for large covers and bar diameters.

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This paper theoretically and experimentally investigates the semi-empirical formulas recommended by Eurocode 2 (EC2), fib Model Code 2010 (MC2010), and Eurocode 2 with the German National Annex (DIN) for calculating crack widths in reinforced concrete. It is shown that the formulas can be derived from the principles for the idealized behavior of RC ties. However, instead of explicitly solving the resulting differential equations, the use of simplifications leads to inconsistent formulas. An experimental study was carried out involving the testing of eight RC ties to discover the modeling uncertainty of the formulas. It was found that EC2 substantially overestimated the crack widths for the RC ties. MC2010 and DIN seemed to predict the crack widths better, but gave rather a large number of nonconservative crack width predictions. These experimental results, combined with the theoretical study, suggest that a more consistent calculation model should be formulated by explicitly solving the resulting differential equation.

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The background theory for the crack width calculation methods according to Eurocode 2 and fib Model Code 2010 is discussed to evaluate the applicability for the more general case of relatively thick beams, slabs and shells. Essentially, the formulas originate from the maximum transfer length and the difference in steel and concrete strains over this length. It is shown that the formulas are based on both a slip and a no-slip theory, two theories using exactly opposite assumptions. The slip theory assumes that a physical slip occurs in the interface between concrete and steel and, also, that plane sections remain plane. The no-slip theory assumes that no physical slip occurs between concrete and steel and, thus, that plane sections no longer remain plane. The theories were merged pragmatically in an attempt to describe the physical reality related to cracking. This resulted in a formula for the transfer length composed by two linear terms. Such a formulation, however, leads to inconsistencies that opposes the basic principles in solid mechanics. It is argued that these inconsistencies limits the application for the more general case. The observations in this paper suggests that a more robust and consistent calculation method should be formulated. A possible way is by improving the bond assumptions in the interface between concrete and steel, and thoroughly studying the geometry and configuration of cracks experimentally and theoretically.@en

Concrete structures shall be designed and constructed to limit cracking and crack widths for durability, functionality and aesthetic reasons. Current design methods and requirements are, however, only in a limited manner verified for large-scale concrete structures and long service life, as well as for new binder and concrete types. To facilitate an improved design basis for large-scale reinforced concrete structures, the present project on evaluation and improvement of calculation methods for large-scale concrete structures in Service Limit State has been initiated. Initially, the occurrence of shear cracks and excessive deformations in concrete cantilever bridges has been investigated. A calculation model based on the Modified Compression Field Theory was established under the assumption that creep in principal compression direction may cause the occurrence of diagonal shear cracks in webs of the cross section. In a shear cracked state, the shear stiffness will be significantly reduced, which further results in increase of shear deformations. The calculation model was applied to, and verified on a real segmentally cast cantilever bridge: the Sålåsund Bridge with main span L = 120 m, where this type of cracking was observed.@en