Project topic

Determining the flexural capacity of cross-sections reinforced with reinforcing steel and steel fibers requires an extension of the assumptions made for reinforced and prestressed concrete. To take into account the fiber load-bearing effect in the flexural crack, a stress-strain curve is derived from measured stress-crack opening or load-deflection curves, which is often assumed to be independent of the cross-section height and the reinforcement configuration. With such an approach, however, the mechanical processes can only be described approximately for a very narrow range of applications (e.g. certain fiber contents, fiber geometries, cross-section heights or crack widths). Since the boundary conditions for different areas of application (normal strength steel fiber reinforced concrete, steel fiber reinforced ultra-high performance concrete) differ considerably, this inevitably means that the mechanical processes, which are in principle the same ("Natura non facit saltus"), cannot yet be consistently represented using a single design approach.

The aim of the V 508 research project is to establish a mechanically based engineering model that allows the design for bending with/without longitudinal force of both purely fiber-reinforced and combined reinforced bending beams across all strengths, so that inconsistencies in previous approaches for steel fiber-reinforced concrete can be eliminated. The model is based on the example of ultra-high performance concrete (UHPFRC) subjected to bending, for which the combination of steel fibers and conventional reinforcement is the rule.

Design approach

The proposed design approach provides for a stress block to take into account the load-bearing effect of the steel fibers in the flexural crack. This depicts the completeness of the stress-crack opening relationship and the position of the stress resultants with sufficient accuracy. Together with a linear design stress-strain relationship of the UHPFRC under compressive loading and the known working lines for the conventional reinforcement, the stress block is included in the evaluation of the equilibrium conditions at cross-section level (Fig. 1).

The limit strain _εcfu of the stress block is not a fixed value, but is based on reaching a "critical" crack width. The "critical" crack width indicates the maximum of the fibre yielding effect in the bending crack for purely steel fibre reinforced cross-sections and the beginning of the so-called deformation localization for combined reinforced cross-sections, in which the subsequent increase in deformation is concentrated on a single crack, the fracture cross-section. Deformation localization occurs in combined reinforced cross-sections as a further possible failure mode in addition to the known failure modes of flexural tension failure and flexural compression failure and has been observed in numerous tests on combined reinforced flexural beams made of normal strength concrete (NCC), high strength concrete (HSC) and UHPC.

For UHPFRC reinforced with microfibers, a "critical" crack width _wkrit = 0.4 mm is proposed based on typical stress-crack opening relationships and test observations in 3-point bending tests - independent of the fiber geometry and fiber content. In general, the "critical" crack width can be derived from the crack width or the deflection at which the post-cracking bending tensile strength reaches its maximum in the 3-point bending tensile test for any steel fiber concretes - irrespective of strength. In this way, the scope of application of the design approach can also be easily extended to normal-strength and high-strength steel fiber-reinforced concretes reinforced with macrofibers.

The crack width is divided by a structural characteristic length _lcs to convert it into an elongation. In the case of single crack formation, the crack width of the bending crack is largely determined by the deformation state within the discontinuity area (St. Venant's fault area) on both sides of a crack. The characteristic length _lcs for this case is derived from the elastic analysis of the disk stress state, which is based on the work of König and Fehling. When cracking is complete, the crack spacing, which depends on the effective longitudinal reinforcement ratio and the bar diameter, becomes the structural characteristic length. The characteristic length defined in this way can assume values of 0 _lcs ≤ 0.75 h. This results in the limiting strain _εcfu of the stress block:

Where _wkrit = 0.4 mm is the "critical" crack width, ρs_,eff = As1/Ac_,eff is the effective reinforcement ratio related to the effective area of the reinforcement, _αcf is a coefficient for considering long-term effects on the post-cracking tensile strength and _Φs is the bar diameter of the reinforcing steel reinforcement. The larger of the two values from Eq. (1a) (derived from the deformation of the St. Venant fault region) and Eq. (1b) (derived from the crack spacing) is decisive.

In the drawn cross-section part, the stress block is applied to the entire tensile zone area to take into account the load-bearing effect of the steel fibers, provided that the strain at the drawn cross-section edge is smaller than the limit strain _εcfu. Otherwise, the stress block is only applied up to the cross-sectional fiber marked "1" in Figure 1, which just reaches the limit elongation _εcfu. In general, this leads to a decrease in the calculated bending resistance as soon as the strain at the drawn cross-section edge exceeds _εcfu. With regard to deformation localization, the limit strain _εcfu is therefore of decisive importance.

Parameter study and validation of the design approach

To illustrate the relationships, _εcfu was evaluated in Figure 2 for two different post-cracking tensile strengths and two different bar diameters as a function of the cross-section height and the longitudinal reinforcement ratio at the level of design values. All calculations were based on a UHPFRC of concrete strength class C150/165 with _fcd = 85 N/mm² and reinforcing steel B500 with _fyd = 435 N/mm² with a stress-strain curve with a horizontal branch. The longitudinal reinforcement ratio _ρl =_As/(b-d) is plotted on the abscissa of the diagrams. The limiting strain _εcfu can be read off the ordinate. The colored curves represent selected cross-section heights h. Marked with "D" are the longitudinal reinforcement ratios above which bending compression failure occurs mathematically before deformation localization. The curves, which are almost unaffected by the cross-section height and develop approximately linearly from the origin of the diagram, are the result of Eq. (1b). The curves branching off to the left, each valid for a specific cross-section height, follow from Eq. (1a), which is therefore particularly relevant for small cross-section heights and low longitudinal reinforcement ratios. In Eq. (1a), the bar diameter has no influence on _εcfu, but a clear dependence on the design value of the post-cracking tensile strength _fcftd is recognizable. The bar diameter _Φs has a considerable mathematical influence on the crack spacing, so that the limit strains _εcfu determined according to Eq. (1b) differ significantly in the left and right diagrams in Fig. 2.

To validate the design approach, test data from 228 UHPFRC bending beams were compiled in a UHPFRC bending database. Different cross-section shapes (compact cross-section, I-section, T-section) and reinforcement configurations were considered, including prestressed, purely steel fiber reinforced and purely bar reinforced components. Different methods of material characterization, execution and documentation of the component tests as well as partly missing or unclear information required a critical examination in each individual case and - where necessary - standardization or supplementation of the data before they could be included in the database.

The comparison of experiment and design approach is carried out by comparing the experimentally determined bending load capacity_Mexp (= maximum bending moment in the test) and the bending load capacity_Mcal determined by calculation using the model according to Section 2 with the mean values of the material parameters and without taking partial safety factors into account. Figure 3a shows _Mexp/Mcal as a function of the contribution of the fibers to the calculated bending load-bearing capacity _Mf/Mcal. The regression line shows that _Mf/Mcal has little influence on the _Mexp/Mcal ratio with an overall good model fit(_Mexp/Mcal ≈ 1.0). However, the scatter of _Mexp/Mcal increases with increasing fiber support ratio, as the post-cracking tensile strength scatters more than the other geometry and material parameters. In order to achieve the reliability level required in Eurocode 0, an upper limit value of the characteristic value of the post-cracking tensile strength fcftk_,max = 0.6 _fcftm was used for the evaluation in Fig. 3b instead of the mean value of the post-cracking tensile strength _fcftm in accordance with the draft of the new Eurocode 2. As the regression line in Fig. 3b shows, _Mexp/Mcal now increases on average with increasing moment carrying capacity of the fibers _Mf/Mcal. At the same time, uncertain results with _Mexp/Mcal 1.0 are largely avoided. This results in acceptable 5 % quantiles of the ratio value _Mexp/Mcal of Q0_.05 = 1.00 for components with compact cross-section, Q0_.05 = 0.99 for components with I-cross-section, Q0_.05 = 0.90 for components with T-cross-section and Q0_.05 = 0.97 for the entirety of the data sets.

Summary of results and conclusions

The comparison with well-documented tests by Stürwald shows that the flexural capacity, the moment-curvature relationship at ultimate load level and the failure mode can be reproduced very well with the proposed design approach. A sensitivity study on these tests also confirms fundamental dependencies and relationships that could also be observed phenomenologically in other experiments.

A parameter study at the level of design values illustrates the influence of various geometry and material parameters on the limit strain _εcfu and the onset of deformation localization. This occurs earlier for large cross-sections with a low degree of longitudinal reinforcement and simultaneously large rebar diameters than for thin cross-sections. The latter can realize very large curvatures and limit strains _εcfu until the bending load-bearing capacity is reached - regardless of the longitudinal reinforcement ratio. With higher longitudinal reinforcement ratios, bending compression failure occurs before deformation localization begins.

When evaluating the design approach with the mean values of the material properties, a good model fit is achieved for the data sets of the UHFB bending database. The average _Mexp/Mcal ratio is 1.09 for the 177 bending beams with compact cross-section, 1.01 for the 31 bending beams with I-section and 0.94 for the 20 bending beams with T-section. The coefficients of variation for the three cross-section shapes are 0.19, 0.13 and 0.11. X̅ = 1.07 and CV = 0.18 are obtained for the entirety of the 228 tests in the UHPFRC bending database. An acceptable level of safety is achieved if the characteristic value of the post-cracking tensile strength is limited to 0.6 times the mean value.

The longitudinal reinforcement and stirrup bars present in the tension zone reduce the effective fiber reinforced concrete area. However, the evaluation of _Mexp/Mcal as a function of the defective area caused by the longitudinal reinforcement and stirrup bars makes it appear justifiable to determine the fiber force _Ff in a simplified manner using the effective gross fiber concrete area_Acf, i.e. without making a deduction for the cross-sectional area of the longitudinal reinforcement and the area of the stirrup bars.

The evaluation of the design approach requires an extended iterative procedure due to the dependence of the limit strain _εcfu on the longitudinal reinforcement ratio, among other things, which makes the use of an equation solver (non-linear system of equations) necessary. A simplification can be achieved by selecting a conservative value for the limit strain. In this respect, _εcfu = 5 ‰ is a safe approach for 187 of the 191 combined reinforced components in the UHPFRC bending database and for the purely steel fiber reinforced components with cross-sectional heights of up to around 150 mm. The approach _εcfu = 5 ‰ = const. would only have a negative effect on the model reliability for four combined reinforced components with large cross-section heights and small longitudinal reinforcement ratios.

Based on the results of the validation, the design approach can be recommended for use in the DAfStb guideline "Ultra-high performance concrete".

Due to the mechanical basis of the proposed design approach, its scope of application is not limited to steel fiber reinforced UHPFRC. The decisive control parameter is the "critical" crack width, which can be derived from the load-deflection or load-CMOD curve (CMOD = Crack Mouth Opening Displacement) of the 3-point bending tensile test. By adjusting the "critical" crack width, the design approach can easily be extended to steel fiber reinforced concretes reinforced with macrofibers. Such an extension is planned for the future and will accompany the work on the new DAfStb guideline "Steel fiber reinforced concrete" based on the new Eurocode 2. To support this work, an NFB-HFB bending database was compiled from test data of 224 NFB and HFB bending beams. The transferability of the proposed design approach has already been demonstrated using two data sets from this database.

Publications

LEUTBECHER, T.; METJE, K., 2024. Flexural design of ultra-high-performance fiber-reinforced concrete girders | Flexural design of flexural girders made of steel fiber reinforced ultra-high performance concrete. In: Congress documents 68th BetonTage: Shaping transformation. Ulm, May 14-16, 2024 Concrete Plant and Precast Technology. 90(5), 113. ISSN 0373-4331

METJE, K.; LEUTBECHER, T., 2023. The UHFB bending database - Validation of a design approach for bending with or without longitudinal force. Beton- und Stahlbetonbau. 118(12), 864-878. doi:10.1002/best.202300070
Supporting Information: Data S1. UHFB bending database

LEUTBECHER, T.; HECK, L.; METJE, K.; RIEDEL, P., 2023. Predicting the moment resistance and localization strain of reinforced UHPFRC cross-sections subjected to bending. Engineering Structures. 293, 116607. doi: 10.1016/j.engstruct.2023.116607

LEUTBECHER, T.; HECK, L.; METJE, K.; RIEDEL, P., 2022. On the bending design of combined reinforced UHPFRC bending beams. Beton- und Stahlbetonbau. 117(11), 863-877. doi:10.1002/best.202200077

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The project is funded by the Federal Ministry of Education and Research (BMBF) as part of the "Sustainable Universities" program. The aim of the funding is to develop and implement innovative concepts for environmentally friendly and resource-conserving campus design.

Important partners in the project are the city of Siegen, which provides support in the areas of mobility and climate protection, and the Institute for Environmental Research NRW, which provides scientific analysis and expertise. Siegen's municipal utilities are also involved in the implementation of sustainable energy solutions.