1. Introduction
Anchorage to concrete plays a vital role in various aspects of modern construction. Many applications are structural connections, such as foundations of steel columns to concrete blocks or rafts, steel girders on concrete cores of high-rise buildings, assemblies of precast elements, and a multitude of reinforced interfaces and integrations of strengthening components to existing structures [
1]. In many cases, anchorages are used for the stability of temporary works, for example, the fixation of falsework and scaffolding. Other safety-critical applications include non-structural fixtures, for example, in residential, healthcare, civil protection, or industrial buildings, such as facades, suspended ceilings, utilities, electrical devices and engines. Also, in linear infrastructure projects, a multitude of fixings is widely used to support noise barriers, rails, cabling, to name a few. Relevant design guidelines have been published to underpin a rational design, specification, and usage of fastenings, with the new Eurocode 2—Part 4 (EN 1992-4) [
2] being the most notable recent standardisation initiative. The American counterpart, the ACI-318, has already been including design provisions for anchorage to concrete for many years, and these provisions were transferred from a code appendix to an individual chapter incorporated with the code [
3] in 2013. While the American and European standards serve as a basis for further national regulations, supplementary standardisation documents for fastenings to concrete are also introduced at the national level worldwide [
4,
5,
6].
Depending on the currently applicable design criteria, the load resistance of anchors is determined by semi-empirical formulations for each potential failure mode, based on the type of load (e.g., static or dynamic) and orientation (axial or shear) as well as other structural and geometric parameters. The design is then typically based on a semi-probabilistic (partial safety factor) design concept. Each calculation is carried out for the resistance against a specific failure mode, considering a factored load, a factored equation, and the involved characteristic material parameters (e.g., steel, concrete, adhesives). The purpose is to provide adequate safety reserves and to compensate for the respective variabilities, within which also a model uncertainty underlies [
1,
2,
3,
7].
In quasi-static situations, failure modes for single anchors in tension may be steel rupture, concrete cone breakout, pull-out (or combined cone breakout and pull-out, e.g., for bonded and concrete screw anchors), splitting, and side blow-out, or failure of the rebar within the concrete component. From a life-cycle perspective, additional causes of failure, such as corrosion or fatigue, can interfere, but also overloading or underperformance of the concrete substrate can occur. Concrete related failure modes are of particular interest for two main reasons: (a) they are quasi-brittle, and as such, they may develop without preceding signs of damage, so there is limited possibility for remediation action and avoidance of complete system failure; (b) concrete failure leads to extensive damage in the area of the fastening, and there is limited possibility for adequate repair [
1]. This consequently leads to a particular interest in the adequate assessment of such systems’ reliability in the case of concrete cone failure. This study refers to single anchor experiements with geometrical attributes leading exclusively to concrete related failure modes. In particular, this is concrete cone breakout under tension, while there is also reference to concrete edge failure under shear toward the edge.
Although industry digitalisation and artificial intelligence led to a boom in the application of advanced soft computing systems in all sectors in recent years, the field has been evolving since the 1950s. “Knowledge-based” approach or “expert systems” have been used to reduce the effects of indetermination in various scientific and industrial disciplines. Applications within the structural engineering field have been seen since the 1980s [
8,
9,
10]. Since then, ML techniques have often been used to model structural engineering problems. They also extend to assessments in the structural reliability realm, such as downscaling the computing effort of stochastic simulations [
11,
12] and performance evaluating of structural systems of high complexity [
9,
13,
14]. Some of these studies have delivered interesting results on the application of ML and other soft computing techniques to model steel-concrete connection engineering problems. Golafshani et al. [
15] discuss ANN models in comparison with Fuzzy Logic (FL) for the prediction of the bond strength of spliced rebar. The dataset of 179 tests considered was randomly split to 125, 27 and 27 data points for training, validating, and testing purposes, respectively. Both the ANN and FL models predicted the bond strength with high accuracy (
R2 > 0.99 in all cases).
Sakla and Ashour [
16] elaborated 1143 quasi-static tensile tests on single-bonded anchors as a subset of an international database reported in [
17]. The authors predicted the ultimate load using a single hidden layer ANN model. The training set comprised 88.3% of the entire dataset and achieved a prediction precision of the test set by 1.03 and a coefficient of determination
R2 = 0.941. Ashour and Alqedra [
18] investigated the breakout resistance of single anchors utilising a dataset of 451 quasi-static tests of single anchors in uncracked concrete. Of the entire dataset, 225 data points were used for the training of four different ANNs, with five to seven hidden layers each. The predictions by use of the ANNs captured the test results with mean ratios between 1.025 to 1.065, standard deviations between 19.9% and 22.6% and coefficients of determination
R2 varying between 0.879 and 0.907, and they indicated that the Concrete Capacity Design (CCD) method is a reliable predictor of the ultimate loads. An investigation by the same authors [
19] on the shear capacity of single anchors failing under edge breakout has led to a similar conclusion comparing the CCD method [
20] to predictions from ANN. A total of 205 experimental tests were considered in this investigation. The mean value of the ratio of the estimated to the actual strength is 1.054 with a standard deviation of 22% and a coefficient of determination
R2 = 0.886, while the predictive model of [
20] only achieves an
R2 = 0.768.
Gesoğlu and Güneyisi [
21] performed a re-evaluation of the database from [
17] using 932 training and 177 testing data from tension tests on adhesive anchors. The analysis used an ANN algorithm in order to propose an improved calibration of the CCD equations used in the codified design. For anchors failing with a concrete cone failure, the mean value of the calculated to measured values for the testing dataset are 0.99 and 1.63 for the models based on ANN and the CCD method, respectively. A complementary study in [
22] including an analysis by ANN on the same dataset, indicated that an ANN-based model delivered still a higher accuracy with correlation coefficients of 0.983 and 0.984 with the training and testing data, respectively. The fact that the correlation with both the testing and training data is virtually the same, allowed to conclude that the ANN has an overall more reliable prediction performance. The correlation coefficient of the tests to predictions based on the models used in [
2,
3] was 0.885 and 0.883, respectively. Some of the key issues facing the efficient implementation of ANN include the difficulty in obtaining the optimum hyperparameters, slow convergence speed, over-fitting problem, difficulty in obtaining suitable network topology and poor generalisation capability [
23,
24]. While the ANN algorithm has been implemented in a variety of previous studies involving the concrete breakout strength of anchors in tension, more powerful and reliable ML algorithms should be studied for a more extensive and up to date experimental database.
Recent investigations present soft computing techniques such as the Gaussian Process Regression (GPR) [
25,
26], and the Support Vector Regression (SVR) [
10,
27,
28] with very high efficiency. This is discussed by [
29], which concluded that both modelling techniques deliver superior predictive accuracy than ANNs and semi-empirical models proposed in current design standards, which are based on nonlinear statistical regression (NR) [
30,
31,
32]. Besides SVR and GPR, other ML techniques including ensemble models (including random forests), Adaptive neuro-fuzzy inference systems, Bayesian networks, have been studied as possible candidates for efficient predictive models.
This contribution aims to propose ML-based strength models for accurate prediction of the concrete breakout strength of single anchors loaded in tension. Among the algorithms investigated, the GPR and SVR are shown to optimally capture the resistance values. Firstly, the structural behaviour of anchors failing due to the concrete cone is briefly discussed. A description of the GPR and SVR algorithms and the background of the assessment conducted are discussed in
Section 2.2. Using an extensive and up-to-date database of 864 tests on single anchors failing due to concrete cone breakout, developed by the American Concrete Institute Scientific Committee 355—Anchorage to concrete, the models are trained and tested, and the best-ranking candidate models are qualified (
Section 3). A comparative evaluation of the developed models against experimentally observed tensile breakout strength is performed and discussed based on a novel ranking methodology. An additional criterion for the model qualification is Model Explainability based on Analogous Rational and Mechanical phenomena (MEARM), which is for the first time reported, trialled, and applied herein on structural engineering problems. The qualified ML models are evaluated in terms of precision and performance relative to the state-of-the-art predictive model embedded in international design codes (
Section 4), and their suitability for use as General Probabilistic Models is discussed, which introduces the possibility for alternative, more accurate design methodologies in a reliability framework.
1.1. Behaviour of Anchors in Tension Subject to Concrete Cone Failure
The structural performance of anchors under direct tensile load can lead to concrete cone failure, which is the main interest of this study. One of the seminal approaches and the one currently applied in most design standards is the so-called Capacity Design (CCD) Method [
20]. This design method was established based on the fracture mechanical theory and a substantial set of tensile experiments on anchors with a range of material and geometrical parameters. The anchor types that can be primarily considered for this failure mode are mechanical post-installed or cast-in headed anchors. Still, it is understood that under the circumstances, this failure can also occur for bonded or expansion anchors, or other types of fastening products. This type of failure typically governs the load-bearing performance of an anchorage for certain combinations of concrete and steel strengths, along with certain geometrical arrangements, such as anchor embedment depth, concrete member thickness, and edge distance. The resistance is also affected by the existence of cracks and reinforcement in concrete (see also [
1] for an in-depth explanation of the various influences). The failure is assumed to initiate at the lower end of the anchor and propagate to the surface at an angle of 30° to 40° to the horizontal. On average, the angle is 35°, and thus an idealised breakout area can be assumed with a size equal to 3·
(the embedment depth).
According to the mechanical concept of the CCD Method, the concrete cone breakout failure load can be calculated using the various forms of Equation (1).
where,
is the tensile breakout resistance;
are calibration factors;
is the anchor embedment depth (the distance of the failure invitation point to the free surface);
is the concrete compressive strength.
In Equation (1), the factor
represents the tensile capacity of concrete. The factor
represents the geometry of the failure surface. The factor
signifies the size effect on concrete tensile failure. According to the analytical approach in [
33,
34], the concrete tensile strength is the main defining factor for the failure surface generator and the breakout capacity. To estimate the mean capacity of a single anchor according to [
35], in plain uncracked concrete without edge influence, Equation (2) can be used. Studies presented in [
1], indicated that the influence of a crack with a width beyond 0.4 mm leads to a reduction in the anchor resistance by 30% to 50% (average 40%). For undercut anchors and headed studs, this seems to stabilise for large crack widths, failing with concrete breakout. This formula also accounts for a tensile stress regime in the anchor vicinity, transverse to its axis. For design purposes, EN1992-4 proposes the calculation of the characteristic resistance through Equation (3). The product-specific value
can be replaced by
to estimate the strength of an anchor in cracked concrete, assuming a crack width of 0.2 mm. The recommended values are
and
for post-installed anchors, while
and
for cast-in headed studs [
7]. The projected area of the breakout on the free surface of the concrete body is used as an adjustment indicator in the case of a group or an anchor with influence from the concrete boundaries, as explained in detail in [
1]. The idealised unaffected concrete breakout body is also depicted in
Figure 1.
where,
is the mean tensile breakout resistance;
is equal to the value of 13.5 for metal expansion anchors and bonded anchors;
is equal to the value of 1.0 for an undisturbed uniaxial compression stress state (this occurs for
equal to approximately 5 anchor diameters) is equal to the value of 0.8 for anchors within compression zones with cracks developing parallel to the compression direction (this occurs for
greater than 5 anchor diameters);
is the mean compressive strength measured at 200 mm concrete cube specimens.
where,
is the characteristic tensile breakout resistance;
is a product-specific value for non-cracked concrete;
is the characteristic compressive strength from 150 mm concrete cube specimens.
1.2. Analogies to Shear Concrete Edge Failure
Concrete edge failure under a transverse load toward the edge presents a similarity to the concrete cone failure. Concrete edge failure (or breakout) occurs particularly for anchorages close to a free edge of the concrete component, and it develops from a fracture initiating at the upper part of the anchor propagating to detachment of a half-cone shaped concrete prism mobilised by the transversely loaded anchor (
Figure 2). The shear load capacity of single anchors can be evaluated from Equation (4), which is the updated CCD formulation presented in [
32]. Comparing Equation (1) assuming cracked concrete and (4), the same fractural and size-effect mechanistic principles are evident when the embedment depth
(height of a cone split in half by a crack plane) is replaced by the distance from the concrete edge
(height of the half-cone). The theoretical equivalence between the mechanical models of concrete breakout under shear and axial load is also mentioned in [
20], and it is evident in the dimensions of the failure prisms indicated in
Figure 1 and
Figure 2. In Equation (4), it is also apparent that the anchor diameter and stiffness have some influence on the shear failure load. This influence is estimated to be in the range of 20%, based on a recalculation of the anchor configurations in an extensive experimental database discussed in [
29].
where
is the outside diameter of the anchor;
is the mean concrete cylinder compressive strength;
is the influence length of the anchor loaded in shear;
is the concrete edge distance;
is the mean shear breakout resistance.
In [
29], the applicability using GPR and SVR techniques to predict the concrete cone breakout capacity of single anchors loaded in shear is investigated. The predictive efficiency of the algorithms is also compared with that of results from various nonlinear regression models currently proposed as design equations, as well as results from ANN algorithms by other researchers. It is concluded that both SVR and GPR can deliver a more precise prediction compared to formulations currently used in design standards. Furthermore, it is made evident that the GPR model is the best predictor of all the models assessed, over the entire dataset, but also over anchor configurations with extreme geometrical characteristics, i.e., those with the minimum and maximum anchor diameter, embedment depth, and distance from the edge. These ML techniques, among others, are also discussed below for their applicability to anchors under tensile loading, leading to concrete cone failure.