M. Blomfors/M. Engen/M. Plos · Evaluation of safety formats for non-linear FEA
The ultimate capacity determined from the analyses depends on the convergence criteria used for the equilibrium iterations. A relatively strict convergence criterion was used and the ultimate capacity of the frame was determined as the resistance for the last converged load step, without regard to possible close-to-converged, subsequent steps before divergence. It could be argued that the structure has not failed until divergence occurs in the iterations, since it is in fact close to equilibrium even if the out-of-balance energy and forces exceed the convergence criterion. However the unconverged load steps may not be realistic with respect to structural response at section level and could include spurious deformations etc., even if global equilibrium is almost fulfilled. It remains unclear as to whether global equilibrium could be fulfilled with a physically feasible response at section level; hence, these load increments are excluded. Limited information is provided in fib Model Code 2010 [5] regarding how the structural resistance should be defined when used in the safety formats. It depends on the current loading situation of the structure, but the resistance is also affected by the previous load history. Even with the simple load situation presented in this paper, deciding on the measure of structural resistance to be used for the safety formats is not a trivial matter. If cyclic loading or time-dependent aspects are included in the analyses, the definition of resistance becomes even more complex. The order in which the loads were applied was found to be important. In general, the load-carrying capacity was lower for the inverse load history compared with the main load history. For the inverse load history, applying the horizontal load first causes a horizontal deformation. When the vertical load is then applied, it is carried in a way that further increases the concrete stress on the inside of the upper right corner of the frame. The concrete strength may therefore be expected to influence the resistance to a greater extent for the inverse load history, which can also be observed when comparing the contour lines in Figs. 6 and 7. This observation explains the finding presented in Table 1, i.e. that between the main and inverse load history, Rm increases while Rk decreases for the ECOV analyses. This demonstrates the importance of considering the load history in non-linear analyses. The load history should preferably include all the loads that the structure has been subjected to during its life, including the construction phase. Since the aim is to increase the use of non-linear analyses in engineering applications, there should be more descriptive guidelines. These should treat aspects such as how to define the design resistance as well as acceptable simplifications regarding structural model and loading history. The response surface was fitted using a second-order polynomial in the safety assessment. The failure mode was compression failure in the concrete in the upper right corner for all the analyses close to the design point. Thus, the ultimate capacity was highly dependent on the concrete strength. A higher-order polynomial failure surface was regarded as unnecessary owing to the simple ultimate limit behaviour. Only two basic variables were included for the safety assessment: concrete compressive strength and steel yield strength. The other material parameters were calculated
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based on these values and provided as inputs in the analyses. This is similar to assuming full correlation between the strength properties and is considered to be an adequate simplification in this safety assessment.
6 Conclusions Even though NLFEAs of reinforced concrete structures have been performed in research settings for many years, they are yet to be used extensively in engineering practice. There are several reasons for this, e.g. the inability to superimpose loads, leading to vast computational costs, but also the lack of safety formats leading surely to the intended safety level as well as sufficient guidance. The ECOV safety format did not meet the intended safety level for one of the load histories studied. This points out the importance of load history considerations when it comes to NLFEA. Therefore, it would be interesting to assess the safety level of the safety formats applied to more complex cases, e.g. a shell structure subjected to a broad variety of loading conditions. Different types of failure modes should be included and also their effect on the modelling uncertainty. Other safety formats that include the failure mode and modelling uncertainty more objectively are perhaps needed [20], since the generality of a safety format might lead to excessive conservatism in some cases. Furthermore, a framework of engineering guidelines for NLFEA is desired. These should treat all aspects important for conducting an accurate NLFEA. Questions have been raised regarding how to treat the load history but also how to define the design resistance. If the guidelines are to be accessible for engineers so they can use them in practice, it is of great importance, but also greatly challenging, to present simplifications that can be used in the models and analyses. Substantial research is required to prove which simplifications can be made conservatively without degrading the accuracy to such an extent that the purpose of an NLFEA is negated.
Acknowledgements This article is based on a Master’s thesis project conducted at Multiconsult AS in Oslo, Norway, during the autumn of 2014 [21]. The project supervisor was Morten Engen, Research Fellow at Multiconsult AS. The thesis′ examiner was Mario Plos, Associate Professor/Head of the Structural Engineering Division at the Department of Civil and Environmental Engineering, Chalmers University of Technology, Gothenburg, Sweden. References 1. Brekke, D.-E., Åldstedt, E., Grosch, H.: Design of Offshore Concrete Structures Based on Postprocessing of Results from Finite Element Analysis (FEA): Methods, Limitations and Accuracy. Proc. of 4th Intl. Offshore & Polar Engineering Conference, Osaka, 1994, pp. 318–328. 2. Cervenka, V.: Reliability-based non-linear analysis according to fib Model Code 2010. Structural Concrete, 2013, vol. 14, No. 1, pp. 19–28. 3. Engen, M., Hendriks, M. A. N., Øverli, J. A., Åldstedt, E.: Application of NLFEA in the Design of Large Concrete