Link zur deutschen Version: Können Bewehrungsspannungen unter Ermüdungslasten zuverlässig ermittelt werden?
To ensure the structural safety of reinforced concrete bridges, it is necessary to verify not only their load-bearing capacity but also their resistance to fatigue. The verification of the load-bearing capacity has been well researched, and the corresponding models (normal moment yield condition and sandwich model) are established in engineering practice. These models assume sufficient deformation capacity and determine the load-bearing capacity while accounting for plastic redistribution of stresses. However, no validated and practical model is currently available for determining the reinforcement stresses required for fatigue verification, even though this verification is often decisive for bridge decks. Under fatigue loads, the cross-section of the bridge deck is typically cracked, while the reinforcement has not yet yielded. In contrast to the verification for the load-bearing capacity, plastic redistribution of stresses is not admissible. Therefore, a model capable of capturing this complex, cracked-elastic behaviour is needed to accurately determine the reinforcement stresses.
When designing my first bridge as an intern, I was immediately confronted with this unsatisfactory situation. While I was able to verify the load-bearing capacity with technical guidance and the knowledge acquired during my studies, there was no suitable model available for accurately determining the stresses in the reinforcement for the decisive fatigue verification. In this blog post, I will therefore summarise the uncertainties in the fatigue verification of the reinforcement and demonstrate that commonly applied models are not suitable. To address these limitations, a research project was initiated in collaboration with the Federal Roads Office (FEDRO), which will also be outlined here. This project will further assess existing reserves to prevent unnecessary strengthening measures, reduce costs, and minimise the environmental impact from a planning perspective.
The current uncertainties regarding the fatigue verification of the reinforcement can be summarised into two categories, as illustrated in Figure 1:
(i) Determining stress resultants in reinforced concrete bridge decks under fatigue loads
{nx, ny, nxy, mx, my, mxy, vx, vy}
(ii) Quantifying resulting reinforcement stresses
{σsrx,inf, σsrx,sup, σsry,inf, σsry,sup}
(i) Determining stress resultants under fatigue loads
In practice, the stress resultants in bridge deck slabs under fatigue loads (see Figure 1 (i)) are typically determined using linear-elastic finite element analyses (linear FEA). These analyses, carried out with commonly used programs such as Cubus or Axis, are easy to apply and require relatively low computational effort. Unlike the more complex and time-consuming non-linear finite element analyses (NLFEA), linear FEA cannot capture the behaviour of cracked-elastic reinforced concrete. Consequently, linear FEA does not account for the beneficial effects that develop when the bridge deck cracks. Cracking causes the section to extend relative to its mid-plane. This extension is restrained by stiffer components, such as the bridge webs (see Figure 2 (a)), leading to the so-called compressive membrane action in the bridge deck (see Figure 2 (b)). As the name suggests, this effect results in compressive forces within the plane of the bridge deck, which reduce the stresses in the reinforcement. The more this extension is restricted (increased spring stiffness k in Figure 2 (b)), the more pronounced the stress reduction becomes in the reinforcement. Even a slight restriction of the extension can already lead to a significant reduction in reinforcement stresses. To take advantage of these reserves, NLFEA capable of accurately capturing the compressive membrane action are required to determine the stress resultants.
(ii) Quantifying resulting reinforcement stresses
Even when the stress resultants are obtained from linear FEA, no practical or experimentally validated model is available for determining the resulting stresses in the reinforcement (see Figure 1 (ii)). In practice, the normal moment yield condition (Figure 3 (a)) or the sandwich model (Figure 3 (b)) are often used to estimate reinforcement stresses from stress resultants derived via linear FEA. However, both approaches assume plastic redistribution of stresses and are therefore not suitable for assessing reinforcement stresses under fatigue loads.
In research and for expert applications, models exist for capturing the load-bearing behaviour of cracked-elastic reinforced concrete relevant for fatigue. To this end, multilayer models such as the CMM-Usermat (see Figure 3 (c)) are typically employed. The basic idea is similar to stress analyses for beams: It is assumed that strains vary linearly over the height of the cross-section. The reinforced concrete section is divided into multiple layers over its height, which are connected only through the previously stated assumption. Further assuming that each layer is in a state of plane stress, the stress state of each layer can be determined from its strains. The stress resultants for the element are finally obtained by integrating over all layers. The CMM-Usermat [3] available in the research group consists of a multilayer model employed in a finite element software and is based on the mechanically consistent Cracked Membrane Model (CMM) [4]. By accounting for tensile stiffening through the tension chord model, the model should realistically capture the mid-plane expansion caused by cracking of the section and thus reliably represent the compressive membrane action. The CMM-Usermat has been extensively validated against experimental data with respect to its global load-deformation response. However, experimental data to verify the underlying assumptions of the multilayer model and the resulting reinforcement stresses remain scarce.
In an earlier blog post, Vera Balmer already presented the development of a hybrid machine learning FE approach, which is also based on this multilayer model. In addition, Balmer et al. [5] used the multilayer model (CMM-Usermat) to demonstrate that even a slight restriction of the mid-plane extension κ can significantly reduce the reinforcement stresses (see Figure 4). In addition, they compared the reinforcement stresses of the CMM-Usermat to those obtained using the normal moment yield condition (NMYC). The comparison shows that the NMYC is overly conservative for most values of κ, but for small κ, it may even lead to unsafe predictions of reinforcement stresses, as illustrated in Figure 4 for ρ = 1.
FEDRO research project to determine reinforcement stresses under fatigue loads
The FEDRO research project addresses both uncertainties in the fatigue verification of the reinforcement described above: (i) determining the stress resultants under fatigue loads and (ii) quantifying the resulting reinforcement stresses. Initially, eight large-scale shell element tests will be carried out in the Large Universal Shell Element Tester (LUSET) using appropriate load combinations to validate the basic assumptions of the multilayer model and the resulting reinforcement stresses. The validated model will then be made available to practising engineers, allowing them to more reliably determine reinforcement stresses from stress resultants obtained with linear FEA.
In the next part of the project, a stepwise analysis approach of increasing complexity (“levels of approximation”) will be proposed and made available to practising engineers. This approach aims to determine the conditions under which the reserves resulting from compressive membrane action (see Figure 2) become relevant and should be explicitly considered. To achieve this, the resulting reinforcement stresses, including the beneficial effects of compressive membrane action, are determined using the validated multilayer NLFEA model and compared with those obtained by applying the validated multilayer model to the stress resultants from the linear FEA.
In addition to the eight large-scale shell element tests funded by the FEDRO research project, two LUSET tests were already carried out this year as part of an additional project. In these tests, two different combinations of stress resultants were applied, comprising membrane normal and shear forces together with transverse bending and torsional moments. For these LUSET tests, the evaluation of the experimental data is still ongoing and will yield first indications for the validation of the multilayer model. The results of the two additional tests will be published and made available here in the future. In the coming years, the LUSET tests of the FEDRO project will be carried out, and the corresponding research on the load-bearing behaviour of cracked-elastic reinforced concrete shells and the determination of reinforcement stresses for fatigue verification will be conducted. Stay tuned for the results, which will be published in future blog posts or studies.
Yannick Kummer
Referenzen
- K. Thoma, A. Kenel, G. Borkowski, “Ermüdung von vorwiegend auf Biegung beanspruchter Fahrbahnplatten”, Forschungsprojekt AGB 2010/001.
- W. Kaufmann, Lecture Notes Advanced Structural Concrete 2025, Chapter 5.2: Slabs–Yield conditions.
- K. Thoma, “Finite element analysis of experimentally tested RC and PC beams using the cracked membrane model”, Engineering Structures 167 (2018) 592-607.
- W. Kaufmann and P. Marti, ‘Structural Concrete: Cracked Membrane Model’, Journal of Structural Engineering, vol. 124, no. 12, pp. 1467–1475, 1998, doi: 10.1061/(ASCE)0733-9445(1998)124:12(1467).
- V. Balmer, K. Thoma, W. Kaufmann, “Design of Concrete Shells and Plates: A Solved Problem?”, IASS 2024, https://doi.org/10.3929/ethz-b-000698486.