Reinforcement detail: Stress trajectory‐oriented corbel reinforcement

The article presents a novel design of concrete corbels, where horizontal and vertical stirrups are replaced by main reinforcement bars that are bent down into the corbel web, following the principal stress lines. A test series of three specimen pairs of fanned‐out and conventional reinforcement design, supported by finite element analysis, shows that the same capacity can be achieved with a significantly lower amount of reinforcement steel and a higher ductility.


| INTRODUCTION
A corbel is a short cantilever protruding from a column or a wall.Thereby, the traditional reinforced concrete corbel is mainly used to support beams and slabs in prefabricated structures.
Due to current developments in, for instance, digitalization and automation in industry, many industry sectors are dealing with the question, "What is possible?"On the other hand, the question "What is necessary?" is asked more frequently in the concrete industry.This drive for optimization must also be valued from the point of view of climate protection.Five to eight percent of global CO 2 emissions can be attributed to the concrete industry.Therefore, the potential for CO 2 reduction and resourcesaving often lies in optimization.
With this drive for optimization, state-of-the-art reinforced concrete corbels are significantly smaller than 80 years ago, but modern corbels do not necessarily carry smaller loads.The main reason is the higher quality of the concrete and the steel reinforcement used.However, architectural requirements and a deep understanding of the load-bearing behavior, the reinforcement layout, and the possible failure modes make the corbels smaller nowadays.

| REINFORCEMENT LAYOUT IN CORBELS
Since the 1940s, experimental research on corbels has continually brought new reinforcement concepts.Figure 1 shows a selection of reinforcement designs.Some have horizontal reinforcement, while others have inclined reinforcements or stirrups.The reinforcement design by Rausch 1 is based on the theory of shear and compression, arguing that sufficient diagonal reinforcement had to be present in every vertical section of the corbel (see Figure 1, No. 1).Niederhoff 2 and Franz 3 studied the principal stress trajectories and applied a simple strut-and-tie model (see Figure 2) for the calculation.
In contrast, Mehmel and Freitag 4 focused on a more complex indeterminate strut-and-tie model with his additional inclined reinforcement.Zeller 5,6 studied orthogonal stirrups using different geometric ratios a c /h c , where h c describes the corbel height, and a c represents the distance of the applied load from the column surface.In contrast, using the topological optimization method, 7,8 Ötzkal and Uysal 9 designed an elaborate reinforcement concept comprising horizontal stirrups and carefully anchored inclined reinforcement bars (number 6 in Figure 1 and Table 1).
The value k 1 (kN/kg) given in Table 1 is the quotient of the failure load and the mass of the reinforcement used.This value allows for a comparison of the effectiveness of different reinforcement concepts.However, a correlation with the yield strength of the reinforcement can also be noted.

| PROBLEMS WITH CONVENTIONAL CORBEL REINFORCEMENTS
Small concrete corbels need a high degree of reinforcement, but the space for reinforcement is restricted due to the limited corbel dimensions.To overcome this shortcoming, reinforcement loops, multiple reinforcement levels or layers, and double-headed studs 10 are designed to serve the acting bending moment.Vertical and horizontal stirrups add to this already high amount of reinforcement complicating the construction progress and the concrete compacting.
The confined conditions require a precise structural design and a careful construction process.The proper production of such concrete structures is often succeeded in precast factories only.
The stirrup reinforcement A sw is supposed to tackle splitting cracks in the corbel, and it should prevent a sudden brittle shear failure when corbels are overloaded.Tests by the authors have shown that this is not necessarily the case.
In tests with short corbels with a c /h c = 0.50-0.75,another diagonal crack occurred in addition to the usual flexural crack.This diagonal crack, due to the large crack width and crack growth, was identified as the main crack.The diagonal crack should actually be prevented by the stirrup reinforcement, which was arranged in accordance with contemporary guidelines.However, the crack bridging stirrups could not resist the crack opening, sufficiently.Crack-bridging reinforcement is most efficient if the cracks cross at a 90-degree angle.On the other hand, horizontal and vertical stirrups are mechanically inefficient when cracks form an angle substantially different from 90 degrees.Such a reinforcement design is not resource optimized.With increasing vertical load, the diagonal crack will develop extensively, despite the stirrups used.This crack narrow the compression zone.In tests, the corbels finally failed due to shear failure in the remaining concrete compression zone, which has to carry both the total vertical force F C,V , and the horizontal force F C,H (Figure 2).Reinforcement did not always reach its yield strength when the maximum load F u was reached.That means that plastic deformations and load redistributions, which can provide safety in supporting structures, will not occur."Reinforced concrete has four components: the concrete, the steel reinforcement, the crack, and the bond." 11With strut-and-tie models, we neglect two components-the bond action and the cracking.
In a series of scientific publications, Polonyi, 12-15 Polonyi and Bollinger, 16 Polonyi et al., 17 Kaliszky 18 and Patzkowsky 19 presented alternative concepts for reinforced concrete components.This series of articles focused on the reduction of reinforcement while still maintaining a high load-bearing capacity.Thereby, the consequent use of all four mentioned reinforced concrete components was pursued.
In the approach, not only the bond effect, but also the deviation forces are used for the load transmission of concrete and reinforcement.These transmission paths cause a more vital interaction between the two components.Following this argument, the reinforcement must be spatially oriented.A concept for a reinforced concrete beam can be seen in Figure 3, which obviates the need for stirrup reinforcement entirely.Patzkowsky 19 also presents a reinforcement layout for beams with dapped ends.

| Reinforced concrete corbels with fanned-out reinforcement
The concept presented can be applied to corbels too.For corbels, reinforcement models can be designed that require significantly less reinforcement, if compared to standard corbels with stirrup reinforcement, for the same load-bearing capacity.The concept presented in Figure 4 is based on the course of the tensile stress lines.The fanned-out reinforcement layers in the cantilevered corbel should carry the acting tensile stresses.
In the adjacent area of the column, the reinforcement is fanned-out too.
At the transition between the top of the corbel and the column, the occurring tensile stresses reach their maximum.Subsequently, at this point, the reinforcement is guided close to the top of the corbel.
Loops and curved reinforcement are used to set the concrete under pressure, caused by radial forces.The bending roll diameters correspond to the regulations of EN 1992-1-1: Eurocode 2 20 and the German national annex. 21For securing the position during concreting properly, two small welded crossbars were placed at the apex of the reinforcement layers (see Figure 5).The four Reinforcement of a beam according to the thrust and suspension analogy. 12I G U R E 4 Reinforcement layers follow the vector plot for 2D principal stresses (left); Balance of Forces on a cracked corbel (right).
F I G U R E 5 Typical reinforcement detailing used in the experimental study.
reinforcing layers have been designed as closed bars with a lap splice length of 90 mm, which is located at the column site.
Using the Levels-of-Approximation approach (LoA), 22 corbels can be designed with simple stut-and-tie models (LoA I) or stress fields (LoA II).Due to the fanned-out reinforcement design, the application of LoA I and LoA II is not sufficient.Using a more detailed nonlinear analysis according to LoA III is necessary for this corbel design.

| Test specimens
The study investigated the reinforcement concept for corbels with different a c /h c ratios (Table 2).The corbels K1, K2, and K3 comprised a contemporary reinforcement detailing with horizontal bars and the presence of stirrups (see Figure 6).In contrast, fanned-out reinforcement

Dimensions
Reinforcement

| Experimental testing procedure
The corbels were loaded manually with a hydraulic jack.
The acting force and the horizontal and vertical displacement were measured with a 500 kN load cell and two linear variable differential transformer (LVDTs), as you can see in Figure 6.Strain gauges were placed in the compression zone and at the main bars.Under a loading regime with load steps of 20 kN each, the crack development and the crack width were recorded.

| Numerical study
A supplementary finite element (FE) study 24 with the ATENA software 25 was used for an advanced investigation of the stress distribution in the corbels.For the non-linear FE calculation, experimentally determined concrete, bond, and cracking values were utilized (see Table 3).The cracking process is influenced by the fracture energy of the concrete, which was determined experimentally too, using a wedge splitting test and a corresponding inverse analysis. 26The geometric model was based on the experimental setup, and it was built with 2D-macroelements.Especially for concrete the so-called "Cementitious2" material was used, which was modified with parameters based on the material tests.For the FE mesh, 2500 quadrilateral elements were used.Since corbels can have a relatively dense crack profile in the ultimate state, the element size was determined by 1/20*h c .Since the local area of compression failure in the corbel is tiny, a finer mesh with 1/40*h c was placed over the highly stressed compression zone area.As a result, the mesh refinement represented the distribution of cracks in the compression node properly, and it enhanced the convergence in the simulation substantially.The loading regime was structured in two stages.In the first stage, the corbel was slowly loaded by 50 load steps to a vertical displacement of 0.3 mm, which includes the transition from elastic to cracked state.In the second stage, 200 load steps were applied, which covers the fracture and the postfracture behavior.
As a result, the correlation between FE simulation and the experimental tests was good.The deviations of the experimental and the simulated ultimate loads F u amounted to 5% on average only, 24 comparable with. 27,28urthermore, the simulated and experimentally observed cracks were also similar.

| Load-displacement behavior
The load-deformation curves of the experimentally tested specimens are shown in Figure 7, and the corresponding ultimate loads are given in Table 3. Figure 7 reveals that the deformation behavior of corbels with fanned-out reinforcement detailing matched the behavior achieved with the conventional reinforcement arrangement.The crucial factors are the content and the position of the main reinforcing bars (serving the bending moment), which were the same in both test series (see Figure 6).Due to some issues with the test setup during the test of specimen K2, stronger deformations of the entire supporting column occurred, so the measured deformation is not representative.
At maximum load, the conventionally reinforced corbels K1 and K2 collapsed suddenly, defined by concrete crushing in the compression node.The post-cracking behavior was brittle and characterized by losing the entire load-bearing capacity swiftly.The corbels G1-G3, as well as K3 with a c /h c = 1.0, however, failed ductilely.Due to the yielding of all tie-rebars, significant deformations were possible.The factor k 1 is given in Table 4. Therein, k 1 for the corbels G1-G3 is 69%-84% higher if compared with the values for the conventionally reinforced corbels K1-K3.That hints to the conclusion that the effectiveness of the fanned-out reinforcement is significantly higher.

| Strain development
The strain development for the corbels K2 and G2 are shown in Figures 8 and 9.In results from the experimental test and finite element analysis (FEA) show a good match.In comparison of the strain development of S2, in corbel G2 the rebar reaches the yielding point at approximatly ≈ 0.75*F u,exp , while in corbel K2 no yielding was plotted.In K2, the strain of 2.8 ‰ was reached at the ultimate load F u,exp .The FEA results of G2, layer 1 shows a large increasing of the loading at the strain ε s = 2.8 ‰.This behavior is due to yielding in a neighboring FE-element.

| Crack appearing and crack development
From the specimens G1-G3, only one single splitting crack occurred on the corbel with a small crack width w 2 = 0.05-0.20 mm (Figure 10).This separation crack has secondary importance compared to the bending crack w 1 .
The test specimens K1-K3 had several fan-shaped oblique cracks in the fractured state.In K1 and K2, one of them evolved to the main crack driving the failure mode.The crack-bridging reinforcement could not stop this crack development.This shear crack opened more pronounced in the higher load level from $2/3 F u,exp and protruded increasingly into the compression node of the corbel until it was finally destroyed.

| Principal stresses in the reinforcement and in the concrete
All reinforcement layers of G2 achieved higher stresses than the corbel reinforcement of K2 (Figure 11), on average 20%-30%.At failure, 3 out of 4 reinforcement layers  Force F (kN) deformed plastically.In comparison, in K2, only the upper main bars reached the yield point.
The strong plastical deformation hints to a higher interaction of concrete and reinforcement.Therefore a higher degree of stress utilization and higher tensile stresses occur in the entire reinforcement.On the other hand, with the conventional reinforcement type, half of the embedded reinforcement stirrups were not stressed in tension but in compression instead, leading to a poor utilization level.
The distribution of compressive stresses changes with the new reinforcement layout too (cf., Figure 12).Overall, more concrete can be loaded in compression in the presence of a fanned-out reinforcement.This affects the lower corner of the corbel and the concrete below the bend of the reinforcement provoked by deflection forces (see Section 4.1).In K2, the main stress areas shift under load due to crack-developing.In this case, two straight compression struts appear in the front and rear corbel area, analogous to the underlying strut-and-tie model.In contrast, the principal stress fields at G2 retain their shape.Until the failure mode is reached, they correspond approximately to the course of the natural compressive stress lines.F I G U R E 1 1 Tension stress distribution of K2 and G2 at maximum load.

| Load-bearing behavior and failure mode
With the conventional reinforcement design (cf., Figure 13, left), compressive forces can be mostly transmitted in one main strut only, which lasts from the load applied to the compression node.This strut is confined and narrowed by the gaping shear crack.The crackbridging stirrup takes up only a minor part of the load, allowing for a very small secondary compression strut.In contrast, with fanned-out reinforcement, a larger area of the corbel can be set under compression.That comes due to the deflection, the bond, and the aggregate interlock action along the crack edges.
The fanned-out reinforcement can effectively prevent rotation of the corbel in the failure state.Here, the rebars are aligned in the direction of rotation.In contrast, in the case of the orthogonally reinforced corbel, the rebar configuration and the direction of rotation do not match.With crackbridging stirrup reinforcement, the dowelling effect comes into action in the failure state.However, the small bar diameters of the stirrups used are not sufficient to prevent the brittle concrete failure and the rotation of the corbel.
From the results of the FE simulation, the equilibrium of the vertical forces at the corbels could be determined (Figure 14) too.
In K2, the concrete compression zone F c,v is stressed to 82% of the applied load F Ed , while 16% of the applied force is carried by the vertical stirrups F s,w .The main bars and horizontal stirrups may also carry vertical forces, because of dowelling effect.However, this is neglected in Figure 13 due to their small diameter and minor effect.Therefore, only the crack-bridging stirrups are counted.In contrast, corbel G2 offers a more balanced force distribution.Here, the tensile reinforcement takes 49% of the vertical forces, while the compression zone takes up 51% of the vertical forces.This distribution of forces eases the situation in the compression zone significantly.This effect may offer the potential to increase the corbel's load-bearing capacity with a higher reinforcement cross-section.

| CONCLUSIONS
Concrete corbels often need a high degree of reinforcement.Furthermore, the confined space demands for a careful planning and for dimensional accuracy during the production process.
The new reinforcement design with fanned-out reinforcement reduces the amount of necessary reinforcement up to 50% if compared with contemporary reinforcement detailing approaches.Stirrup reinforcement can be avoided entirely easing the production process and the compacting of the concrete.The layout of the reinforcement is aligned with the tensile stress lines.The fannedout reinforcement layout means no significant shear cracks appear, avoiding effectively the development of  small separation cracks.The entire reinforcement is exposed to tensile stresses and extensive yielding allows for a ductile component behavior.
It is influenced by the reinforcement design, the anchorage, and the bond.This hypothesis by Polonyi is confirmed in evaluating the corbels' principal stress profiles.Due to the fanned-out reinforcement and the usage of anchorage loops, the shapes of the compressive stress areas hardly changed during tests.Finite element simulations underlined this finding.In contrast, conventional reinforcement detailing led to a strong shifting of the compression stresses provoked by the occurrence of oblique cracks.All tested corbels with fanned-out reinforcement failed in bending.Severe concrete splitting could be avoided entirely.

F I G U R E 1
Selection of historical corbel reinforcement designs.F I G U R E 2 Strut-and-tie model (left) and balance of forces on a cracked corbel (right).

T A B L E 1
Comparison of historical corbel designs.

Number
was used in G1, G2, and G3.According to DIN EN 206,23 the concrete strength corresponded to the strength class C25/30.Reinforcing steel B500A was used for the entire experimental program, comprising bars with a bar diameter of 6 and 8 mm.The yield strengths of the utilized bars differed slightly and were 560 MPa for bar diameter 8 mm and 544 MPa for bar diameter 6 mm, respectively.

F I G U R E 1 2
Compression stress distribution of K2 and G2 at maximum load.F I G U R E 1 3 Concrete compression struts and reinforcement layers in corbel K2 (left) and G2 (right).

F I G U R E 1 4
Balance of vertical forces in cracked corbel K2 (left) and G2 (right).
Experimental tested material parameters used in FE study.
T A B L E 3 F I G U R E 7 Load-deflection curves of the experimentally tested corbels.