Parametric study on the effect of anchor's geometry on the stress distribution and crack initiation direction in a concrete body

Steel anchors are an essential design component for fastening structural items to (reinforced) concrete or masonry. Anchors are used to attach steel elements to supporting concrete members. They are often overlooked and assumed to be a secondary design element. Nevertheless, they transfer external loading to the concrete structure and they are therefore very important and need to be designed correctly. There exist two basic kinds of anchors: post-installed and castin. Cast-in anchors are bolts that are installed in the concrete before it has set and dried. Post-installed anchor bolts are installed after the concrete is set in place and completely dried. Anchor-concrete fastening systems fail mostly through concrete-related failures. According to the type of loading (tension/shear) which is applied to the system, several kinds of failure can be distinguished: pull-out failure, steel failure, side-face blowout, concrete cone breakout and concrete splitting for anchors subjected to tensile loading or concrete pry-out failure, concrete edge failure and steel failure for anchors subjected to shear loading. The transferring from one to another failure mode is governed by several variables, such as embedment length, side cover, bearing area of the head, thickness of the concrete structural member etc.
Concrete cone failure, which is the topic of this paper, is characterized by a cone-shaped fracture in concrete in the vicinity of the anchor. This fracture is assessed as a brittle failure which is connected to a sharp drop of the load-displacement curve after peak load caused by quick unstable crack propagation in concrete. Various, both experimental and numerical, studies have been done $[2,3,5,7,8,10,12,20,24]$$[2,3,5,7,8,10,12,20,24]$[2,3,5,7,8,10,12,20,24][2,3,5,7,8,10,12,20,24]$[2,3,5,7,8,10,12,20,24]$ on anchors subjected to tensile loading, because it is important to investigate the influence of various parameters on anchor/concrete system behavior, its assessment and design. For instance, it was experimentally observed, that the use of additional reinforcement for increasing the strength or preventing the concrete cone failure is beneficial $[9,13,21,27]$$[9,13,21,27]$[9,13,21,27][9,13,21,27]$[9,13,21,27]$. Additionally, Eligehausen et al. [4] found out that the circumferential cracking during the concrete cone failure happens at approximately $30\mathrm{\%}$$30\mathrm{\%}$30%30 \%$30\mathrm{\%}$ of the ultimate load. Furthermore, it was observed that the concrete cone cracking depends on the anchor embedment depth: particularly, the length of the circumferential crack at the specific loading decreases with higher anchor embedment depth. Thus, capacity design methods often take into account various parameters (such as concrete strength, depth of embedment, thickness of concrete, edge distance, concrete reinforcement, anchor material grade, forces applied to the anchor, seismic loads etc.).
Recent numerical studies have provided various methods for modeling and simulating cracking process of concrete-like materials. Besides the finite element method (FEM) which is used within this study, extended finite element method (XFEM) [18], discrete element method (DEM) [26], cohesive zone models [29], smeared and discrete crack models [19,22], and continuum damage mechanics [14] can be mentioned for instance. Of course, each of the methods mentioned exhibits its uncertainties and limitations, see e.g. [15,23]. Therefore, the finite element method has been chosen because of its benefits.
In this paper, the stress distribution around the anchor's corner is investigated. Especially, tangential stress is investigated based on the idea of the maximum tangential stress (MTS) criterion that says that a crack propagates in the direction where the tangential stress reaches its maximum. Also, the influence of the distance from the anchor's corner on the stress distribution is analyzed and discussed. The goal of this paper is to present results of a parametric study when the effect of
specific parameters of the anchor-concrete system on potential crack initiation/propagation is assessed. Finite element method is utilized as a tool for calculation of the stress field around the anchor's corner. Comparison with the experimentally observed crack path is involved. Note that the results presented represent just a pilot study to obtain basic dependences, which can be utilized for suggestion of further research, which should be extended via more advanced material models, fracture energy and process zone assessment etc.

The dimensions of one slice of the cylindrical concrete specimen with an embedded steel anchor subjected to tensile loading (pullout force) are schematically presented in Fig. 1. Note that the interface between the two materials (steel and concrete) is modelled using perfect adhesion.

According to the suggested geometry, a finite element (FE) model was created by means of Ansys Parametric Design Language (APDL). PLANE183 element with quadratic displacement behavior was used for meshing of the specimen considering axisymmetric conditions, i.e. the 2 D model represents one slice of the cylindrical concrete specimen with an embedded steel anchor. Elements refinement was defined near the anchor's corner where the crack most likely appears and where the stress field was investigated. An example of the FE mesh can be seen in Fig. 2, where also the boundary conditions are plotted; both axial symmetry on the left-side nodes ( $\mathrm{UX}=0$$\mathrm{UX}=0$UX=0\mathrm{UX}=0$\mathrm{UX}=0$ ) and zero vertical displacement on the bottom nodes $(\mathrm{UY}=0)$$(\mathrm{UY}=0)$(UY=0)(\mathrm{UY}=0)$(\mathrm{UY}=0)$. The upper part of the steel anchor was subjected to tensile loading, which was modelled through a non-zero vertical displacement value of $\mathrm{UY}=0.05\mathrm{mm}$$\mathrm{UY}=0.05\mathrm{mm}$UY=0.05mm\mathrm{UY}=0.05 \mathrm{~mm}$\mathrm{UY}=0.05\mathrm{mm}$. During parametric study, the maximum element size was several tens of millimeters at locations far away from the anchor and the minimum element size was about several tenths of millimeters at the anchor's corner. According to the sensitivity analysis performed, such an FE mesh ensures sufficient accuracy. As can be seen in Fig. 2, the stress field distribution was analyzed at specific radial distances from the anchor's corner. The values of this critical distance applied within this work can be found in Tab. 1 and are in agreement with the Theory of Critical Distances (TCD) published for instance in [25,28]. The TCD is an attitude assuming that material strength can be assessed via specific length scale parameters [1]. The critical length is thought of as a physical property that corresponds with the micro-/meso-/macro-structural characteristics of the material as well as with the specific features of the failure processes. Within static assessment, the TCD quantifies damage through an effective stress, ${\sigma }_{\text{eff}}$${\sigma }_{\text{eff }}$sigma_("eff ")\sigma_{\text {eff }}${\sigma }_{\text{eff }}$, which is obtained from the linearelastic stress field. Thus, the critical distance can directly be estimated from the fracture toughness and tensile strength of the material:

Note that values of $0.5\mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$$0.5\mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$0.5MPa*m^(1//2)0.5 \mathrm{MPa} \cdot \mathrm{m}^{1 / 2}$0.5\mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$ and 4 MPa , respectively, were considered for the rough estimate of the critical distance typical for a common concrete, and therefore the $R_C$$R_C$R_(C)R_{C}$R_C$ values were selected in the order of several millimeters. Both materials (steel and concrete) were modelled through the linear elastic material model considering the Young's modulus of 210 GPa and 23.5 GPa and Poissson's ratio 0.3 and 0.2 for steel anchor and concrete, respectively. The selected results are presented in the next section and follow the previous research published in [16,17].

In this section, selected results of the above-described numerical parametric study on a concrete specimen with an embedded steel anchor subjected to tensile loading are presented. For the geometric configurations presented in Tab. 1, selected stresses (radial, tangential, shear, maximum principal and von Mises stress) were analysed. An example of the stress distribution around the anchor's corner at radial distance of 4 mm for one specific configuration are plotted in Fig. 3.

Dependences in Fig. 3 exhibit various shapes for specific stresses. Nevertheless, there can be seen a good agreement between the angle of the maximum tangential stress and the experimentally observed angle of the concrete cone failure, which is ${37.5}^{\circ }[5,11]$${37.5}^{\circ }[5,11]$37.5^(@)[5,11]37.5^{\circ}[5,11]${37.5}^{\circ }[5,11]$. Thus, application of the idea of the MTS criterion seems to be reasonable from this point of view. Based on this agreement/observation, maximum tangential stress distribution at several selected (critical) radial distances was investigated for the specimen configurations introduced in Tab. 1 and the angle where ${\sigma }_{\theta \theta }$${\sigma }_{\theta \theta }$sigma_(theta theta)\sigma_{\theta \theta}${\sigma }_{\theta \theta }$ reaches its maximum value was found for each of them. Although there is actually no crack in our numerical model, this approach follows the idea of the maximum tangential stress criterion derived by Erdogan and Sih [6] for crack propagation. When the values of the angle corresponding to the maximum tangential stress are analysed in dependence on the various depth of the anchor's embedment at various critical distances, the following curves can be obtained, see Fig. 4.
The dependence presented in Fig. 4 shows that the value of the ${\theta }_{\text{max}}$${\theta }_{\text{max }}$theta_("max ")\theta_{\text {max }}${\theta }_{\text{max }}$ angle, where the maximum of the tangential stress occurs, decreases with increasing depth of the anchor's embedment independently on the critical radial distance assumed for the analysis. In other words, the deeper the anchor's embedment, the flatter the concrete cone failure. This means that the circumferential crack propagates in more horizontal direction when the depth of the anchor's embedment increases. From this point of view, it can be concluded that a deeper anchor's embedment is safer in terms of concrete cone failure. The values of the ${\theta }_{\text{max}}$${\theta }_{\text{max }}$theta_("max ")\theta_{\text {max }}${\theta }_{\text{max }}$ angle vary from ca. ${20}^{\circ }$${20}^{\circ }$20^(@)20^{\circ}${20}^{\circ }$ to ${41}^{\circ }$${41}^{\circ }$41^(@)41^{\circ}${41}^{\circ }$ in dependence on the embedment length and critical distance. These values are again supported by the experimentally observed crack paths.

Additional investigations were conducted considering various outer radii of the steel anchor, $R_2$$R_2$R_(2)R_{2}$R_2$ according to Fig . 1. In order to decrease the number of total configurations analysed, some of the other parameters were kept constant in this case. Middle values based on previous research [16,17] were assumed: embedment length $L_{em}=250\mathrm{mm}$$L_{em}=250\mathrm{mm}$L_(em)=250mmL_{e m}=250 \mathrm{~mm}$L_{em}=250\mathrm{mm}$, critical radial distance $R_C=1$$R_C=1$R_(C)=1R_{C}=1$R_C=1$ to 5 mm and thickness of the anchor's basement $t=10\mathrm{mm}$$t=10\mathrm{mm}$t=10mmt=10 \mathrm{~mm}$t=10\mathrm{mm}$. See the results of this partial analysis in Fig. 5 .

Results presented in Fig. 5 show that also the outer radius of the steel anchor can have an influence on the shape of the cone failure, especially when the length of the anchor's embedment is higher. Values of the ${\theta }_{\text{max}}$${\theta }_{\text{max }}$theta_("max ")\theta_{\text {max }}${\theta }_{\text{max }}$ angle are quite similar when various critical radial distances are applied. When the steel anchor is close under the concrete surface, the angle of the concrete cone failure seems to be independent on the outer radius of the steel anchor. Otherwise, for longer steel anchors, the critical angle corresponding to the maximum of the tangential stress slightly decreases with increasing outer radius of the anchor, especially when smaller critical distances are applied. This behaviour is again more advantageous in terms of a flatter concrete cone failure shape when the crack needs to overcome longer path towards the specimen surface.
If the results of both analyses are evaluated, it can be concluded that independently on the choice of the critical radial distance, the angle corresponding to the maximum tangential stress value decreases with increasing length of the steel anchor's embedment and with increasing outer radius of the anchor if the embedment is long enough. This founding can be beneficial for designing of anchor/concrete systems subjected to tensile loading. Of course, other fracture mechanical and economical aspects need to be considered during the design and assessment of such structures.

Within this paper, it was shown that the geometry of the steel anchor subjected to tensile loading can have a significant influence on the stress distribution in the concrete around the anchor and consequently on the shape of the typical concrete cone failure. Moreover, it was showed that the rather simplistic approach of LEFM to anchor failure is sufficient enough to provide with crack direction orientation and future cone failure. Finite element
numerical model was created to apply the basic idea of the maximum tangential stress criterion that allows to predict crack propagation direction. Generally, these angles through the entire parametric study varied between ca. 20 and ${40}^{\circ }$${40}^{\circ }$40^(@)40^{\circ}${40}^{\circ }$, which is in accordance with experimental observations. Moreover, it was found out that the increasing embedment length of the anchor as well as the increasing anchor's outer radius are advantageous in terms of the circumferential crack propagation angle. In other words, the larger the mentioned parameters, the lower the angle corresponding to the maximum tangential stress. Thus, a flatter path of the cone crack improves the fracture resistance of the anchor/concrete system against failure. This should be taken into account together with other factors for design and/or assessment of such a kind of structures. Note that the same behaviour was observed independently on the choice of the critical radial distance where the tangential stress distribution was analysed.

T he data used in this study is available at (ZENODO repository): 10.5281/zenodo. 15083222.

This paper was created as part of the project No. CZ.02.01.01/00/22_008/0004631 "Materials and technologies for sustainable development" within the Jan Amos Komensky Operational Program financed by the European Union and from the state budget of the Czech Republic. Financial support from the Czech Science Foundation (project No. 25-15755S) is also gratefully acknowledged.

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