One- and two-step approaches for design of reinforced concrete structures

By considering the lower bound theorem of plasticity theory (see e.g. Nielsen 1984, Brekke et al. 1994, Cook & Young 1999, Melchers 1999, Lubliner 2008), a conservative estimate of the capacity of the frame in Fig. 1 is found by considering a set of internal forces which is i) in equilibrium with the external loads and ii) not exceeding the capacity locally in any section, and iii) by ensuring sufficient ductility by providing proper detailing of the reinforcement. If the frame is designed such that all critical sections are equally utilized, the lower bound solution would coincide with the formation of a mechanism, prohibiting any further loading and redistribution of internal forces. However, if the critical sections are not equally utilized, the loading could be further increased until the capacity is reached in enough sections to form a mechanism.

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The internal force distribution of the frame in Fig. 1 can be calculated by hand, but due to the degree of statically indeterminacy, the engineer would most likely resort to computer-assisted methods. Today, linear finite element analyses (LFEA) are widely used in everyday practice.

One of the main advantages of using LFEA is that the principle of superposition is valid. Hence, the engineer can perform separate analyses of each load case, and combine the results to form relevant load combinations afterwards. Based on the results from the LFEA, the structure is thus designed such that the capacity is not exceeded locally in any section. The detailing of the reinforcement is such that the full theoretical capacity of each cross-section can be mobilized by providing minimum reinforcement and sufficient anchorage lengths.

The method described above can be called a two-step approach (Schlune 2011, Schlune et al.

2012), since the response and the resistance are calculated in two steps using different assumptions regarding the material behaviour. The response of reinforced concrete is non-linear due to cracking of concrete even at low load levels. At higher load levels, the yield strength of the reinforcement can be exceeded, introducing additional non-linearities. For statically determinate structures, these stiffness reductions will not influence the distribution of internal forces. However, for statically indeterminate structures, the parts having the largest stiffness attract the larger portion of the internal forces. The non-linear response of reinforced concrete will thus result in a redistribution of the internal forces, which cannot be predicted by the LFEA.

In the LFEA, the estimated internal force distribution can be close to, but basically not equal to, the real distribution, because of the assumed linear elastic material behaviour, and the full capacity of the structure is not utilized since the redistribution of forces is not modelled.

However, the two-step approach is effective due to the validity of the principle of superposition, and by definition conservative. Instead of using a LFEA, one could use a non-linear finite element analysis (NLFEA) for calculating a more realistic internal force distribution for a certain design load. However, this is still a two-step approach, since the sectional capacities are generally calculated using different material models, and raises the question about which values of the material parameters that should be input in the NLFEA. With reference to the material variation in Fig. 1, should the materials be represented by their mean or most likely values, their nominal or characteristic values, their low design values or something in between? The selected values for the material parameters influence the failure mode and stiffness of the frame, and thus the distribution of the internal forces, and should be selected with care.

Alternatively, since concrete and reinforcement steel are modelled with realistic material models, a NLFEA can be interpreted as a virtual experiment. Increasing the load until failure in this virtual experiment, would give an estimate of the load carrying capacity of the structure as a whole where all sections work together and contribute to the capacity. This represents a one-step approach, since the structural response and the structural resistance are calculated using the same assumptions regarding the material behaviour. Only those phenomena that are not explicitly modelled should be controlled separately, e.g. anchorage if the reinforcement is modelled as fully bonded and the transverse shear capacity if ordinary beam or shell elements

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are used. Performing analyses with different values for the input parameters will give an indication of the variation of the load carrying capacity due to the material uncertainties, and semi-probabilistic methods can be used to estimate the design load carrying capacity based on this information. It should be noted that in NLFEA, the principle of superposition is no longer valid, and each relevant load combination should thus be analysed separately.

Any LFEA or NLFEA only represent simplifications of the reality, and the modelling uncertainty or model uncertainty indicates how well the analysis outcomes compare to the real physical behaviour (Ditlevsen 1982). The modelling uncertainty of NLFEA depends on how the analysis is performed and what kind of physical phenomenon that is modelled. There are several contributions in the literature devoted to the modelling uncertainty, both addressing one specific model (e.g. Engen et al. 2017a), and the effect of selecting different models (Schlune et al. 2012). It is emphasized that the modelling uncertainty does not imply that the outcomes of the NLFEA are random. If one NLFEA is repeated, the outcome will be the same, but it will be uncertain, since the model is only a simplification of the reality.

a) The Tresfjord Bridge (Statens Vegvesen) b) Dam Sarvsfossen (Bykle kommune) Fig. 2: Typical large reinforced concrete structures.

The one-step approach has been elaborated on in the literature (CEB 1995, 1997, Henriques et al. 2002, Schlune et al. 2011, 2012, Cervenka 2013, Pimentel et al. 2014, Allaix et al. 2013, Blomfors et al. 2016), and with this method, the engineer is equipped with a tool that can be used to make realistic assessments of the load carrying capacity of structures. However, it is important to realize that the cost of performing NLFEA of reinforced concrete structures of realistic sizes, as illustrated in Fig. 2, can be significant. The use of NLFEA in everyday

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engineering practice thus seems justified if significant cost savings can be expected. Hence, typical application areas are identified as

structures with complex geometries, structures subjected to extreme loading,

existing structures designed according to old design codes, existing structures subjected to new and increased loads, or

existing structures exposed to deterioration mechanisms where the residual structural resistance is questioned.

In this work, emphasis has been put on developing a strategy for NLFEA within a one-step approach, applicable to analyses of large reinforced concrete structures, in order to facilitate the use by practicing engineers. Furthermore, the uncertainties related to material and modelling have been studied in order to contribute to ongoing discussions, and to be able to proceed towards a full one-step approach in future work.