Standards such as EN 1992-4 [3] and ACI 318-19 [1] generally require that anchors subjected to shear load must be assessed for several failure mechanisms.
- Steel failure of a fastener without or with a lever arm
- Concrete pry-out failure
- Concrete edge failure
- Failure of supplementary reinforcement
The IDEA StatiCa Connection application includes checks for the first three failure modes for plain concrete. In the case of reinforced concrete, it is possible to export the model to the IDEA StatiCa Detail application and, in addition, perform an analysis and code-check of the reinforced concrete, and thus cover the fourth failure mode. It should be noted here that the failure mode names listed may slightly differ in various standards. In particular, in the context of this article, we consider the fourth failure mode to be the failure of any reinforcement in a concrete block. (anchor and supplementary reinforcement as stated in ACI 318-19 [1] chapter 17.5, supplementary reinforcement as defined in EN 1992-4 [3] chapter 7.2.2, and all other reinforcement in the model)
In the Detail application, which has 3D CSFM implemented for the 3D modeling environment, it is therefore possible to model even heavily reinforced elements with anchors that can be subjected to shear loads. Such elements will certainly not fail in the second and third failure modes. However, it is necessary for them to verify the first and fourth failure modes, plus the situation where concrete in contact with the surface of an anchor subjected to shear stress begins to crush.
The first mode is directly checked using formulas from the selected standard; the fourth failure mode is verified by comparing the results on the reinforcement elements from 3D CSFM and the standard load-bearing capacity of the reinforcement bars. The remaining question is how to determine the limit load value (bearing capacity) for the case of concrete crushing at the anchor–concrete interface.
From the available standards and literature, we can take as a reference the approach described in EN 1994-1-1 [4] Chapter 6.6.3, formula 6.19. This formula is designed to assess the effect of concrete crushing in contact with anchors for headed studs of concrete-steel composite slabs and beams and is based on research papers [5] [6].
\[P_{Rd}=\frac{0.29\alpha d^{2}\sqrt{f_{ck}E_{cm}}}{\gamma_{V}}\]
However, the standard specifies that this approach is limited to anchors with a diameter of 16-25 mm and, in section 6.6.3.2, restricts its scope in terms of the interaction of tensile and shear forces.
We can also get inspiration from AISC 360-22 [2] Chapter I8 Steel Anchors, Equation I8-1.
\[Q_{n}=0.5A_{sa}\sqrt{f'_{c}E_{c}}\le R_{g}R_{p}A_{sa}F_{u}\]
The first part of the equation expresses the same effect as equation 6.19 mentioned above in an analogous way. We can only observe differently determined empirical coefficients. In this case as well, the use of the formula is quite limited, similar to the case of the Eurocode—see Section I8.1.
Verification of shear capacity using only the above-mentioned formulas is therefore insufficient due to the aforementioned limitations of the diameter and type of anchors (only headed studs), which is also demonstrated by the range of tested samples in [5] [6]. Another limitation when assessing using only the aforementioned formulas is the method of attaching the anchors to the base plate. These anchors are welded to the base plate, so it is assumed that rotation is fixed in the anchor-plate connection. Therefore, these formulas do not cover anchors that are secured with a single nut from above and cannot be considered fixed to the base plate.
For the reasons mentioned above, we performed a series of simulations in ABAQUS, where we first modeled examples corresponding to the scope of EN 1994-1-1 [4]. We calibrated the models so that the determined load-bearing capacity corresponded to the load-bearing capacity determined using the methodology in EN 1994-1-1. Subsequently, under the same assumptions, we modeled a wider series of examples to cover the needs of 3D CSFM and the Detail application. The determined load-bearing capacities are then directly implemented as a stop criterion in 3D CSFM.
ABAQUS models
Concrete blocks with anchors of 8, 12, 16, 25, and 50 mm diameter were modeled. For each diameter, a model with concrete strengths of 16, 30, and 50 MPa was created. Finally, each anchor was modeled with a free end (hinged) and with rotational support at the point of load application (fixed). A total of 5 * 3 * 2 = 30 models were created.
\[ \textsf{\textit{\footnotesize{Fig. 1\qquad Abaqus model}}}\]
Geometry
The floor plan of the model is shown in Figure 2. It is the same for all models. The height of the model is set at 200 mm for anchors with diameters of 8, 12, and 16 mm. For anchors with a diameter of 25 mm, the height of the model is 250 mm, and for anchors with a diameter of 50 mm, the height is 400 mm.
The embended depths of anchors are: D8 - 100 mm, D12 - 150 mm, D16 - 170 mm, D25 - 220 mm, D50 - 350 mm. The length above the surface was always set to 10 mm. The model also features 2 (only for the 8 mm anchor) or 3 layers of U-shaped reinforcement bars. They have a 25 mm cover from the upper surface for all models. The diameters of the reinforcement for the individual anchors are as follows: D8 - 10 mm, D12 - 10 mm, D16 - 14 mm, D25 - 20 mm, and D50 - 28 mm. The distance between layers was D8, 12 - 40 mm D16 - 50 mm D25 - 60mm, and D50 - 75 mm.
\[ \textsf{\textit{\footnotesize{Fig. 2\qquad Abaqus model - dimensions}}}\]
Boundary conditions and load
The Displacement/Rotation surface support, with all translations and rotations turned on, was applied on the side surfaces of the wider part of the model to capture forces in Y direction and on the small surfaces to capture forces in X direction.
\[ \textsf{\textit{\footnotesize{Fig. 3\qquad Abaqus model - supports}}}\]
Next, a point was defined on the upper surface of the anchors, which was connected to the entire upper surface of the anchors using RBE2 (123456) constraints. This point was used to apply a deformation load of up to 3 mm to the model. A rotational support was also placed on this point for models simulating an anchor welded to the base plate. In the case of a hinged connection, the rotational support was removed from the model.
Finite elements types and meshing
The basic geometry of the concrete is meshed according to the Figure 4. However, it is important to focus on meshing around the anchors. For each anchor diameter, two concentric circles were defined, where a refined mesh was enforced.
\[ \textsf{\textit{\footnotesize{Fig. 4\qquad Abaqus model - mesh}}}\]
The anchors were also modeled using solid elements, with each quadrant of the circular cross-section of the anchor containing four surface finite elements.
\[ \textsf{\textit{\footnotesize{Fig. 5\qquad Abaqus model - mesh around the anchor}}}\]
C3D8R (An 8-node linear brick, reduced integration, hourglass control) elements were used as solid elements. The reinforcement was modeled from T3D2 elements.
Material models
For concrete, the Concrete Dapage Plasticity Model (CDP) developed by Lubliner [7] + Abaqus Theory Manual [8] was selected based on recommendations in the program documentation. In Abaqus, it is necessary to define the course of the uniaxial material curve in compression and tension. For the concrete model in compression, the definition according to Mander [9] was used. The post-cracking tensile response of concrete was described using a linear tension softening law, where the tensile stress decreases linearly from the tensile strength to zero with increasing cracking strain.
\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Abaqus model - Concrete material models}}}\]
Anchors and reinforcement were modeled as a linear elastic material without plasticity. The material behaviour was defined using Young’s modulus and Poisson’s ratio, assuming no yielding or strain hardening. In the case of anchors, this was intentional, so that they would not become plasticized, and only the effect of concrete crushing in contact with the anchor would be examined.
Material designations are always listed according to Eurocode and are always considered without partial factors.
Interactions
The interaction between concrete and reinforcement was modeled using the embedded element constraint available in Abaqus. This formulation assumes a perfect bond between concrete and reinforcement, i.e. no slip occurs at the interface.
The contact between the concrete and the anchor is modeled using "General contact". Normal behavior is considered "Hard" contact; for cohesive behavior, the bond is set according to the fib Model Code 2010 [10], Chapter 6.1.1. As shown in the following graph, when considering concrete C30/37 according to Table 6.1-1, the "All other conditions" are taken into account.
\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Abaqus model - Bond model}}}\]
It should be noted here that the resulting bond used in the models is relatively high (τb = 5.6 MPa) compared to the bond we would expect for a smooth-shank headed stud (τb ≈ 1.0 MPa), for which the aforementioned equations from EN and AISC apply. It was therefore necessary to examine how the value of the bond would affect the behavior of the concrete in contact with the anchor - see below.
ABAQUS results
Before presenting the complete results, we will examine how the load-carrying capacity of Abaqus models is determined. The following graph shows the load-carrying capacity levels determined according to EN 1994-1-1, formula 6.19, and AISC 360-22, Equation I8-1, along with the concrete principal strain (measured at contact with the anchor) versus reaction force for models with different bond strengths.
\[ \textsf{\textit{\footnotesize{Fig. 6\qquad Concrete strain at the contact and the reaction force relationship with bearing levels of EN and AISC standards}}}\]
First, we can observe that the bond has a minimal effect on the observed concrete crushing, specifically, the difference in reaction forces at the same strain is around 5%.
Second, we can see that at 7% strain, the curves intersect with the load-carrying capacity curve specified in EN. We observed this for all anchor diameters covered by Equation 6.19. This is evident from the table of calculated load-carrying capacities presented below and from the corresponding strain-reaction force graphs also shown below.
To refine the results, we applied the same analysis to the concrete finite element, which was the second in sequence from the anchor contact, with the difference that the observed principal strain was 4%.
The Gallery 1 shows the complete results from the Abaqus models, and Fig. 7 shows a table of calculated capacities according to EN and AISC.
Anchors - concrete strain vs. reaction force
\[ \textsf{\textit{\footnotesize{Gallery. 1\qquad Concrete strain at the contact and the reaction force relationship with bearing levels of EN and AISC standards}}}\]
\[ \textsf{\textit{\footnotesize{Fig. 7\qquad Table of calculated capacities according to EN and AISC}}}\]
We therefore know that the modeled anchors with diameters of 16 and 25 in the fixed variant fall within the scope of both standards, and that for these anchors, the design load-bearing capacity is reached at 7% principal strain (or 4% for the second mesh point in the sequence). In other words, we can state with sufficient accuracy that the reaction force achieved at the specified strain is equal to the load-bearing capacity and should correspond to the load-bearing capacity predicted by IDEA StatiCa Detail.
IDEA StitiCa Detail models
In the Detail program, you can model anchors that are either hinged or fixed to the base plate. The following table summarizes the available options.
\[ \textsf{\textit{\footnotesize{Fig. 8\qquad Axial and rotational constraints between an anchor and base plate}}}\]
To simulate a fixed anchor, the anchor must be modeled with a base plate (it is not possible to add a constraint to the anchor without the plate to prevent rotation, as we did in the case of the Abaqus models). The Detail application also limits the minimum number of anchors to 2. For these reasons, the models had to be created differently. A base plate with two anchors was used so that the anchors would influence each other as little as possible, as shown in Figure 9.
\[ \textsf{\textit{\footnotesize{Fig. 9\qquad IDEA StaiCa Detail model}}}\]
We modeled the "Cast-in-place Reinforcement" type as a fixed connection and the "Post-installed Threaded Rod" type as a hinged connection (all direct contact and without cut threads). We manually assigned both materials the same yield strength of 1000 MPa so that they would behave linearly even under maximum load and thus not affect the observed concrete crushing effect. The modulus of elasticity is the same for all models: Es = 210 GPa.
As for the dimensions, they are identical to those of the Abaqus models. We have simply added one rear anchor.
IDEA StitiCa Detail results compared with determined capacities
We now have the indirectly determined bearing capacities from the Abaqus models and the calculated results from the Detail program. All that remains is to compare the results and assess the consistency of the calculations.
The following Figure 10 shows a comparison for the fixed variant. Here you can see the determined bearing capacity from both points in the Abaqus models and the shear force on the more heavily loaded anchor in the Detail models, at which point the calculation stopped, and it was thus determined that the anchors are unable to withstand a greater load. The bearing capacities calculated using reference formulas from the EN and AISC standards are also included here.
\[ \textsf{\textit{\footnotesize{Fig. 10\qquad Comparison of bearing capacities for fixed variant}}}\]
We can see that the EN standard is more conservative; that is why we decided to derive the load-bearing capacity indirectly in accordance with the EN standard.
Now let's look at the values for the pinned version.
\[ \textsf{\textit{\footnotesize{Fig. 11\qquad Comparison of bearing capacities for pinned variant}}}\]
The table below shows the 3D CSFM bearing capacity for cases where concrete crushing occurs at the interface with a shear-loaded anchor. It is worth noting that it is not possible to specify an anchor with a diameter smaller than 8 mm in the Detail program, and that for anchors with a diameter greater than 50 mm, the calculation always stops at the specified load-bearing capacities for a 50 mm anchor.
\[ \textsf{\textit{\footnotesize{Fig. 12\qquad 3D CSFM bearing capacity for cases where concrete crushing occurs at the interface with a shear-loaded anchor}}}\]
Load deformation behaviour
In the final section of this article, we examine the load-deformation behavior of both models. Contact in the 3D CSFM is modeled indirectly by connecting anchors, which are modeled as bar elements, with constraints. When we apply one of the main assumptions of 3D CSFM—the complete exclusion of concrete under tension—we find that the model in the Detail application is significantly simplified from this perspective compared to the model in Abaqus, where a contact element was modeled, and the concrete also had a tensile branch.
\[ \textsf{\textit{\footnotesize{Fig. 13\qquad Deformation behaviour of 25 mm anchor}}}\]
\[ \textsf{\textit{\footnotesize{Fig. 14\qquad Deformation behaviour of 50 mm anchor}}}\]
For the model in the Detail program, two anchors were evaluated for the reasons mentioned above, and these are compared with the anchor in the Abaqus program. We can observe that the overall stiffness of the block influenced the deformation of the anchors. It is also evident that the program stopped the calculation when one of the anchors reached its limit value. In terms of the deformation behavior of the anchors, we see that the anchors in the Detail program are less stiff than those in the Abaqus models.
There is certainly room for further research here. For example, the models in Abaqus do not account for the fact that anchors are often placed on the top surface relative to the direction of concreting, where the quality of the cover layer is generally assumed to be poorer. In this case, one might expect softer behavior; on the other hand, this is not always the case.
Conclusion
The presented study set out to verify and extend the concrete-side resistance of anchors loaded in shear beyond the narrow scope of the closed-form code equations. EN 1994-1-1 [4] (Eq. 6.19) and AISC 360-22 [2] (Eq. I8-1) are calibrated only for headed studs of 16–25 mm welded to the base plate in a fixed configuration. Using a factorial set of 30 ABAQUS models — covering anchor diameters of 8, 12, 16, 25, and 50 mm, concrete grades of 16, 30, and 50 MPa, and both fixed and hinged anchor to base plate connections — the concrete bearing failure was isolated by keeping the steel linear-elastic, so that the response reflects the concrete in contact with the anchor shank rather than yielding of the steel.
The simulations confirm the resistance predicted by the code equations and show that the bearing behaviour is largely insensitive to the bond between the anchor and the concrete. A single bearing resistance can therefore reasonably be assigned regardless of the actual bond condition. As EN 1994-1-1 [4] is the more conservative of the two code families, the resistance implemented as a stop criterion in the 3D CSFM model in Detail was calibrated against it, which lets the application extend the verified range to 8–50 mm and to both fixed and hinged anchors.
A final note on safety is worth making explicit. The study assigns the same bearing resistance irrespective of bond, which raises the question of how the residual uncertainty in real bond conditions is covered. The two frameworks handle this differently. For example, EN 1994-1-1 [4] applies a dedicated partial factor of γV = 1.25 to the stud resistance. In Detail, this bearing resistance is instead governed by γC — the partial factor for concrete, default 1.5 — because the limit is expressed through the concrete strength itself. Since γC acts directly on the concrete strength, it reduces the bearing resistance more than γV = 1.25 would. The more severe reduction inherent in the CSFM treatment, therefore, conservatively absorbs the uncertainty associated with the actual bond between the anchor and the concrete, rather than leaving it to a separate, smaller factor.
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[2] AMERICAN INSTITUTE OF STEEL CONSTRUCTION (AISC). Specification for Structural Steel Buildings. ANSI/AISC 360-22. Chicago, IL: American Institute of Steel Construction, 2022.
[3] CEN. EN 1992-4: Eurocode 2 – Design of concrete structures – Part 4: Design of fastenings for use in concrete. Brussels: European Committee for Standardization, 2018.
[4] CEN. EN 1994-1-1: Eurocode 4 – Design of composite steel and concrete structures – Part 1-1: General rules and rules for buildings. Brussels: European Committee for Standardization, 2004 (incl. Amendment A1:2014, pokud cituješ konsolidované znění).
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[6] STARK, J. W. B. a van HOVE, B. W. E. M. Statistical analysis of push-out tests on stud connectors in composite steel and concrete structures. Part 2: Solid concrete slabs (Technical Paper S84, Part 2). Delft (NL): TNO Building and Construction Research, 1991. Report no. BI-91-163.
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