Comprehensive Evaluation of Adhesive Compounds and Their Properties Involving Harrington’s Desirability Function

Abstract

The increase in the volume of construction work carried out with chemical anchors has led to a corresponding growth in the supply of these products on the market. Anchors possess numerous characteristics, including strength, anchorage displacement, temperature, curing time, and cost. Designers face the challenge of choosing the optimal solution for specific construction conditions. In practice, this often results in choosing anchors with maximum strength and minimum cost, which is not always the best option for long-term use. The authors of this study propose addressing this challenge through a multi-criteria optimization method based on the Harrington function. For implementation, 18 criteria were used. They were derived from reference sources and experimental results. Tests were conducted under short-term and long-term static loading. Based on these tests, strength characteristics were determined, and statistical analysis was carried out to calculate coefficients of variation and confidence intervals for the mean values. Nine types of chemical anchors with different bases were tested: epoxy-based, acrylate-based, methacrylate-based, polyester-based, and epoxy-acrylate-based (five samples in each series). In this study, the assumption that all criteria have equal weight coefficients is made as a limitation. The results of the study are valid only for static loading of anchors in uncracked concrete. The optimal adhesive compound was determined for the basic winter and summer sets of criteria. The practical significance lies in the implementation of a multi-criteria optimization method for selecting the adhesive compound. This approach allows users to choose the optimal adhesive compound for their needs.

1. Introduction

Over the last few decades, the amount of construction work involving anchor applications has greatly increased worldwide. Twenty to thirty years ago, the market was dominated by mechanical anchors (mechanical expansion anchors and drop-in anchors). Mechanical expansion anchors have a number of limitations [1]. For a long time, drop-in anchors were a good alternative to expansion anchors [2].

Over the last ten to fifteen years, construction companies switched to chemical anchors. Initially, large-scale studies of factors, affecting the bond strength of chemical anchors, were conducted in the 1980s and 1990s (Cook R.A. et al. [3,4,5]). In [3], anchorage testing results are compared with the data obtained by the older method of predicting the load-bearing capacity of anchorages. This method takes into account standard anchorage failure mechanisms. Another large-scale study [4] focuses on the behavior of anchoring systems. This study was conducted to analyze factors manifested in the process of anchor embedment and operation. Each factor was examined individually in the course of testing. The outcome of the study necessitates a variety of tests to reliably forecast the load-bearing capacity of anchorages. Work [5] addresses experimental studies conducted to develop mathematical models, describing the behavior of chemical anchors. Its authors were the first researchers who suggested incorporating methods of adhesive bond analysis into effective construction regulations.

Presently, chemical anchors are mainly used to fasten metal structures and heavy equipment to concrete, brickwork, blocks, and natural stone masonry. The authors of [6] investigate the behavior of chemical anchors in lightweight concrete and revise empirical and analytical equations that should take into account failure mechanisms identified earlier. Anchorages can be simultaneously subjected to tensile and shear forces. The studies described in [7] pave the way for new dependencies that take into account slip, limit shear stresses, and anchorage stiffness reduction under combined loading.

The authors of [8] investigate anchorage strength in concrete reinforced with steel and polypropylene fibers. Higher concrete strength naturally boosts maximum loading and the load-bearing capacity of anchorages. A chemical anchor that had strong adhesive compounds was subjected to pullout testing until cone failure; that is how the strength of fiber concrete was found.

The authors of [9] compare the shear strength of various steel-to-concrete anchorages, including drop-in and chemical anchoring systems. An attempt is made to rationally determine the scope of their application.

Studying anchorage performance under seismic conditions is a promising field of research. In [10], a combination of static and dynamic loading simulates seismic loads. Cracked concrete is used in the course of testing to simulate the consequences of seismic loading. The results of the study have a practical value in terms of expanding the scope of application of chemical anchors.

Although chemical anchors were originally designed as fasteners for concrete, now they are widely used in building repair and restoration projects due to the minimal cost of resources and faster repair and restoration work performance [11].

Much emphasis is placed on testing chemical anchors embedded in brickwork, including anchorages in operation for a long time. The authors of [12] study the behavior of anchorages under cyclic and impact loading that causes cracking and brickwork delamination in the point of anchorage.

Some works focus on the use of chemical anchors as supplementary uniformly distributed fasteners for composite reinforcement lamellas in reinforced concrete structures with intermediate plates. The cracking moment is analyzed for different values of anchorage spacing along the beam. Experiments prove that if the choice is to be made from anchorage spacing values such as 250, 500, and 1000 mm, the optimal option is 500 mm [13]. A similar work [14] focuses on anchors embedded in both ends of a beam to ensure the distribution of forces over the entire width of a lamella. However, carbon fiber, rather than steel, was used as the anchor material. Numerical calculations accompanied experimental studies, and their results demonstrated high reproducibility [15].

Experimental studies of the load-bearing capacity of chemical anchoring systems, including studs and adhesive compounds in brickwork, were conducted by Granovsky, A.V. et al. [16,17,18,19]. The most informative works focus on epoxy resin anchors [8,20,21]. In [20], such characteristics of a chemical anchor as glass transition temperature, flexural strength, flexural modulus of elasticity, compressive strength, and pullout bond resistance were found at the temperatures of +23, +42, +70, and −60 °C. The time needed for a chemical anchor to develop 97.7% pullout strength at the air temperature of −10 °C (10 days) was also found.

Contrafatto, L. and Cosenza, R. [21] focus on anchorages in natural stone. Their work has a great practical value due to the limited number of research undertakings in this field. The authors attempt to check whether existing numerical models can predict the load-bearing capacity of anchorages. The conclusion is that models that are good for concrete are unsuitable for natural stone.

Four types of adhesives and four types of concrete (containing reinforcing additives such as blast furnace slag, fly ash, and polypropylene) were used in a number of tests [22]. Blast furnace slag demonstrates the maximum increase in anchorage strength (26%) relative to benchmark concrete without additives. Another study conducted by the same authors [23] shows that the chemical composition of adhesives has a greater effect on the bearing capacity of anchors whose diameters are 16 mm or more. In [24], the behavior of a chemical anchor is considered for various (1) values of curing temperature, (2) degrees of pre-embedment cleaning of openings, and (3) humidity values. The authors believe that the moisture content in an opening has the greatest negative effect on the anchor adhesion to concrete.

A rise in anchorage temperature triggers glass transition processes, whereby glue softens and bonding decreases. Three-dimensional finite element modeling was conducted to determine the most accurate fire resistance parameters [25]. The conclusion is that embedment depth has the greatest effect on fire resistance.

Within the framework of this study, a patent search was launched in national collections and among parties to the Patent Cooperation Treaty. Figure 1 shows patent holders broken down by country. This breakdown is provided as a response to the “chemical anchor” search query and covers the most recent decade.

Major corporate applicants include China Petroleum and Chemical Co. (Beijing, China)—133 patents; HILTI AG–67 patents; Illinois Tool Works Inc. (Glenview, IL, USA)–34 patents; and Anchor Chemical Company Ltd. (Johannesburg, South Africa)–33 patents.

It is noteworthy that interest in chemical anchors (judging by the number of applications filed and patents granted) is rising exponentially.

The patent search shows that the main focus is placed on chemical compounds designed for accelerated curing at various operating temperatures, for example, the patents in [26,27,28,29]. Each compound requires a whole set of tests before its application on a construction site. Developing testing methods and test result processing techniques are the long-standing challenges to be tackled for the benefit of designers who need design loading values to analyze the pullout strength of anchors embedded in various types of structural material. In Russia, a special GOST R [30] (Russian State Standard) was introduced to regulate the testing of chemical anchors in 2024. The testing methods specified in this GOST are similar to those outlined in the European regulatory document EAD [31]. Moscow State University of Civil Engineering (National Research University) conducted a set of tests involving anchors that had different adhesive compounds to determine their economic and technical parameters.

Hence, the end user (designer) receives a set of initial data for each chemical anchor. This set has dozens of parameters depending on the anchor diameter, adhesive compound temperature, etc. All parameters cannot be taken into account and the optimal chemical anchor cannot be chosen without mathematical tools. Hence, multi-criteria optimization (MCO) is a relevant solution to be employed as an instrument for choosing compounds for chemical anchors.

The authors of the study were unable to identify any prior applications of the multi-criteria optimization method for selecting the optimal basis of chemical anchors under various conditions of use. Accordingly, a preliminary assessment of the adequacy of various methods was conducted, including TOPSIS, AHP, and fuzzy techniques. Ultimately, the Harrington desirability function was selected.

Thus, the objective of the study is to provide an integrated assessment of the properties of adhesive compounds for chemical anchors using the generalized Harrington desirability function.

2. Materials and Methods

Nine types of chemical anchors with different adhesive compounds were tested within the framework of this project: four types have epoxy adhesives; one type has an acrylate adhesive; two types have methacrylate adhesives; one type has a polyester adhesive; and one type has an epoxy acrylate adhesive. In all tests, the diameter of the steel rod is 12 mm (including static and long-term strength tests). To ensure the confidentiality of information about the manufacturers and customers, anchors were coded; they had coded names (the first letter stands for the adhesive compound; V is the number of the compound variant if several anchors with one adhesive compound are tested).

The initial data are clustered into four groups according to the type of data source:

• Market characteristics, or anchor prices, were downloaded from the Internet in the course of the research (reference data).
• The maximum curing time and the minimum strength development time of adhesive compounds, depending on concrete temperature, were specified by manufacturers in certificates for chemical anchors. Characteristics have extended temperature ranges (reference data).
• Standard characteristics of anchorages were identified as a result of tests conducted in compliance with the All-Russian State Standard, or GOST R [30] (experimental data).
• Supplementary characteristics were identified on the basis of standard characteristics and experimental studies (experimental data).

The parameters were selected by eliminating the indirect influence of certain criteria on others, thereby preventing recursion and correlation among characteristics.

Groups were analyzed in more detail; minimum and maximum target values were provided for further optimization.

Group 1 has approximate prices of a chemical anchor per liter, rounded up to 100 rubles. The target value is the minimum one.

Group 2 has the following parameters:

-

The maximum curing time at the temperature of 20 ± 5 °C, h. The target value is the minimum one.

-

The minimum strength development time at the temperature of 20 ± 5 °C, h. The target value is the minimum one.

-

The maximum curing temperature, °C. The target value is the maximum one.

-

The maximum curing time at the maximum temperature, h. The target value is the minimum one.

-

The minimum strength development time at the maximum temperature, h. The target value is the minimum one.

-

The minimum curing temperature, °C. The target value is the minimum one.

-

The maximum curing time at the minimum temperature, h. The target value is the minimum one.

-

The minimum strength development time at the minimum temperature, h. The target value is the minimum one.

Group 3 has the following parameters:

-

The mean value of bond stress under the static application of force, MPa. The target value is the maximum one.

-

The coefficient of bond stress variation after the static application of force, taking into account the actual strength of concrete, %. The target value is the minimum one.

-

The mean value of bond stress after long-term testing, MPa. The target value is the maximum one.

-

The coefficient of bond stress variation resulting from the static application of force after long-term loading, in percent. The target value is the minimum one.

-

The anchor displacement factor derived from static tests, mm/MPa. The target value is the minimum one.

-

The anchorage displacement factor derived from static tests after long-term loading, mm/MPa. The target value is the minimum one.

Group 4 has the following parameters:

-

The percent share of actual displacements (hereinafter “share of actual displacements”), projected for the upcoming fifty years relative to the maximum displacement. This value is based on the R1 set of tests. The target value is the minimum one.

-

The coefficient of displacement variation obtained as a result of a long-term 90-day loading test, excluding fast creep. The target value is the minimum one.

For the purpose of this project, fast creep processes are assumed to stop in 24 h, except for the adhesive compound EV1, for which the first four days of testing are disregarded (coded names of adhesive compounds used in this study are provided in

Table 1. Reference characteristics of Groups 1 and 2 of adhesive compounds used in chemical anchors.

Adhesive Compound (Coded Name Used in This Project)	Group 1	Group 2
	Price per Liter, RUB	Maximum Curing Time at 20 ± 5°, h	Minimum Strength Development Time at 20 ± 5°, h	Maximum Curing Temperature, °C	Maximum Curing Time at Maximum Temperature, h	Minimum Strength Development Time at Maximum Temperature, h	Minimum Curing Temperature, °C	Maximum Curing Time at Minimum Temperature, h	Minimum Strength Development Time at Minimum Temperature, h
Acrylate (A)	20,000	0.05	0.75	25	0.05	0.75	−25	3.00	36.0
Epoxy (EV1)	9000	0.5	7.00	40	0.17	4.00	−10	6.00	240.0
Epoxy (EV2)	6000	0.25	4.00	40	0.17	2.00	0	6.00	48.0
Epoxy (EV3)	8500	0.75	6.00	35	0.21	4.50	−5	2.25	160.0
Methacrylate (MAV1)	2300	0.15	1.00	25	0.08	0.50	−10	0.83	4.0
Methacrylate (MAV2)	3500	0.17	1.00	20	0.07	0.50	−20	1.17	11.0
Polyester (P)	2700	0.07	0.67	35	0.03	0.58	5	0.17	2.5
Epoxy (EV4)	10,000	0.50	7.00	40	0.20	4.50	−5	2.00	168.0
Epoxy acrylate (EA)	4500	0.08	0.75	30	0.05	0.58	5	0.17	2.5

).

-

The peak stiffness value is k₂. The target value is the minimum one.

Peak stiffness k₂ (Figure 2) is assumed to be the ratio between maximum load N_u and corresponding displacements δ_u [32].

Initial data for Groups 1 and 2 are provided in Table 1 for reference purposes.

Static (set R1) and long-term strengths (set B14) of anchorages were tested in compliance with the All-Russian State Standard (GOST R [30]). Anchors were embedded in pre-prepared concrete of class B25, made with Portland cement, spalls ranging in size from 5 to 10 mm, and a water–cement ratio of 0.5. A power drill was used to make openings, and minimum spacing requirements were complied with.

Anchors were embedded and cured in compliance with the manufacturer’s curing time requirements. R1 set anchors were tested immediately after curing. B14 set anchors were subjected to constant loading by disk springs; after that, they were subjected to pullout testing.

The tests were conducted at a temperature of 25 ± 2 °C and humidity of 60 ± 5%.

The testing machine is shown in Figure 3. In the course of static load testing, the force is transmitted to the anchorage (Figure 3a) through a small plate and four rods connected by a crosshead. The crosshead is loaded by a hydraulic cylinder with a built-in strain gauge with an accuracy of 0.05 kN. A strain gauge is installed below the crosshead, above the anchor head. The support plate has an opening to allow for the two anchorage failure mechanisms, including adhesive-to-concrete interface failure and steel failure (Figure 4).

Testing machines, used for long-term load testing, have plates and disk springs (Figure 3b). The force, applied to the anchorage, was controlled by a mechanical horseshoe dial gauge.

MCO was applied to optimize the choice of the chemical anchor adhesive. All MCOs can be divided into interactive MCOs (the hierarchy analysis method and the Pareto method); lexicographic MCOs (the assignment method); and those reduced to single-criteria MCOs (the main criterion method, the target programming method, and the method of multiplicative and additive convolution). Multiplicative convolution was applied to find the optimal solution for the adhesive composition of a chemical anchor. It reduces the task to a single-criterion problem by formulating the generalized value of “desirability” (D_ik) for all criteria under consideration (d_i, i is the criterion number, i = 1...n); k is the number of the adhesive compound for a chemical anchor (for the purpose of the problem to be solved k = 1...9). In the general case of the problem under consideration, n = 18. After that, the objective function is formulated: the anchor adhesive compound is assumed to be optimal, if D_ik → max, all other conditions being equal. The strength of the multiplicative criterion is that its application does not require normalization of individual criteria.

The Harrington desirability function has the following advantages:

• it is quantitative and unambiguous; it is represented by a single numeral;
• it is universal and applicable across various fields;
• it comprehensively characterizes the object of study;
• indicators are converted into values of partial desirability d using a single graph for all criteria.

In this study, the assumption that all criteria have equal weight coefficients is made as a limitation. Harrington desirability functions are confined between minimum and maximum values, and results are valid only for the static loading of anchors in uncracked concrete. The generalized Harrington desirability function, which is often used to evaluate individual properties of processes and materials, was used to determine individual desirability values [33,34,35] (Figure 5). In Figure 5, the Y-axis is a scale of individual values; the d-axis is a scale of desirability. The desirability scale is divided into five sub-ranges from 0 to 1: [0; 0.20] is “very bad”; [0.20; 0.37] is “bad”; [0.37; 0.63] is “satisfactory”; [0.63; 0.80] is “good”, and [0.8; 1.00] is “very good.”

The left-hand curve segment of the Harrington function is applied to criteria whose best value of individual desirability is the maximum one. In this case, individual values of desirability are found as follows:

d = exp[−exp(−Y)],

(1)

The right-hand curve segment of the Harrington function is applied to criteria whose best value of individual desirability is the minimum one. Here, individual values of desirability are found as follows:

d = exp[−exp(Y)],

(2)

In Formulas (1) and (2), Y is an individual value linearly connected with the actual value of a criterion:

Y = a₀ + a₁ · Y_H,

(3)

where Y_H are actual values.

A system of equations can be made by substituting extreme values of scale Y (maximum and minimum desirability d values) in Formula (3):

-

for the left-hand curve segment:

$$\begin{array}{c}{\mathrm{a}}_0+{\mathrm{a}}_1·{\mathrm{Y}}_{\mathrm{Hmax}}=5,\\ {\mathrm{a}}_0+{\mathrm{a}}_1·{\mathrm{Y}}_{\mathrm{Hmin}}=-2,\end{array}$$

(4)

-

for the right-hand curve segment:

$$\begin{array}{c}{\mathrm{a}}_0+{\mathrm{a}}_1·{\mathrm{Y}}_{\mathrm{Hmin}}=2,\\ {\mathrm{a}}_0+{\mathrm{a}}_1·{\mathrm{Y}}_{\mathrm{Hmax}}=-5,\end{array}$$

(5)

where a_0, a₁ are dimensionless coefficients of Formula (3), determined from the solution of system (4) or (5).

It is noteworthy that minimum and maximum values of Y were based on actual values of each criterion.

3. Results and Discussion

In the course of testing, the interface between the adhesive compound and concrete failed (Figure 6a) in most cases. In two cases, steel elements failed after long-term loading (Figure 6b). When the interface between the adhesive compound and concrete failed, 90–100% of the adhesive remained on the surface of the steel rods. The steel failure can be described as classical fine-grained brittle fracture.

The principal result of static anchor pullout tests is the load–displacement curve, used to analyze standard characteristics of chemical anchors.

The following two types of curves were plotted:

• Initial loading triggered a sharp increase in force and a small increase in displacements up to ≈ 0.8–1.0 mm; these were followed by a further increase in force and accelerated displacements that reached 2–3 mm and ended in brittle failure (Figure 7a,b).

2.

Initial loading triggered a sharp increase in force and a small increase in displacements up to ≈ 1.0–1.5 mm, almost immediately followed by brittle failure. Straight segments at the ends of the curves demonstrate a sharp drop in force; they were automatically registered by the testing machine (Figure 7c,d).

During static tests, all failures occurred at the interface between the adhesive compound and concrete.

The experiment did not reveal any clear correlation between the type of curve and the adhesive compound.

The static testing of long-term strength demonstrated anchorage failure at the interface between the adhesive compound and concrete, except for specimens EV1 and EV4 (five specimens per adhesive compound), whose steel elements failed.

As a result of curing in compliance with the manufacturer’s curing time recommendations, initial strength of adhesive compound EV1 reached 97.82 kN (before long-term loading). In the course of long-term loading, displacements reached the range of 0.36...0.705 mm on the 90th day (Figure 8a,b). After 90 days of loading, the adhesive compound strengthened. Its strengthening caused failure of anchorage steel when the loading value reached 98.33 kN. In other words, the initial strength of the anchorage was close to failure stresses of the M12 steel stud that had strength class 12.9.

As a result of curing in compliance with the manufacturer’s curing time recommendations, the initial strength of adhesive compound EV4 reached 81.2 kN (before long-term loading). In the course of long-term loading, displacements reached the range of 0.74...0.885 mm on the 90th day (Figure 8c,d). Curing was followed by the strengthening of the adhesive compound. The strengthening caused failure of the anchorage steel when the loading value reached 107.1 kN. Hence, the initial strength of the anchorage was 31.8% lower than the failure stress of the M12 steel stud that had strength class 12.9. However, in the course of 90 days of long-term load testing, the initial strength of the anchorage increased and exceeded the failure stress value.

An example of steel connection failure is the load–displacement diagram obtained during static tests after prolonged loading, as shown in Figure 9.

Individual desirability criteria are determined for each parameter of each adhesive compound. It is logical to assume that values of desirability criteria should be close for the same adhesive compound.

For example, compounds A (acrylate) and MAV2 (methacrylate) contain the same adhesive substance and are designed for low temperatures. They demonstrated a proportionate increase (24.7 and 21%, respectively) in the compound strength after long-term testing.

However, it does not always work that way. For example, the nature of deformation of EV1 and EV4 epoxy compounds is totally different under long-term loading. In one case, there is a smooth increase in displacements; in the other case, displacements hardly increase.

Consolidated

Table 2. Experimental characteristics of Groups 3 and 4 of adhesive compounds used in chemical anchors.

Adhesive Compound (Coded Name Used in This Project)	Group 3						Group 4
	Mean Value of Bond Stress During Static Application of Force (Set R1), MPa *	The Coefficient of Bond Stress Variation After the Static Application of Force (Set R1), %	Mean Value of Bond Stress After Long-Term Tests (Set B14), MPa *	Coefficient of Bond Stress Variation Resulting from Static Application of Force After Long-Term Loading (Set B14), %	Anchorage Displacement Factor (Set R1), mm/MPa	Anchorage Displacement Factor (Set B14), mm/MPa	Share of Actual Displacements in 50 Years, %	Coefficient of Displacement Variation for 90 Days, %	Peak Stiffness Values, $\mathbf{k}\mathbf{N}/\mathbf{m}\mathbf{m}$
Acrylate (A)	17.85	7.67	22.25	4.81	0.057	0.039	76.59	38.22	31.13
Epoxy (EV1)	36.42	4.91	36.58	1.45	0.040	0.048	37.65	42.59	56.70
Epoxy (EV2)	17.16	8.88	17.79	10.91	0.070	0.06	93.9	35.31	29.62
Epoxy (EV3)	25.33	2.31	27.43	3.35	0.055	0.066	47.36	51.67	43.94
Methacrylate (MAV1)	14.82	8.88	14.22	3.52	0.031	0.055	46.58	12.00	44.77
Methacrylate (MAV2)	17.38	5.58	20.83	4.71	0.040	0.055	63.65	11.09	50.39
Polyester (P)	19.9	2.69	19.73	2.99	0.052	0.064	80.88	9.10	53.23
Epoxy (EV4)	30.61	4.05	38.26	2.1	0.055	0.05	44.23	30.04	52.38
Epoxy acrylate (EA)	23.12	3.12	26.35	3.7	0.054	0.043	61.76	23.38	33.88

* $\overline{\mathsf{\sigma }}={\displaystyle \frac{\overline{\mathrm{N}}}{\mathsf{\pi }·\mathrm{d}·\overline{{\mathrm{h}}_{\mathrm{e}\mathrm{f}}}}}$, where $\overline{\mathsf{\sigma }}$ is the mean value of bond stress in the anchorage, MPa; $\mathrm{d}$ is the anchor diameter, mm; $\overline{{\mathrm{h}}_{\mathrm{e}\mathrm{f}}}$ is the mean value of the anchor embedment depth in the experiment. The mean value of bond stress after long-term testing is determined in the same way.

summarizes standard tests and their processed results, as well as experimental data subjected to supplementary processing.

Multiplicative Convolution of Criteria and MCO Results for Adhesive Compounds of Chemical Anchors

All of the above-mentioned data were used to solve this problem (Table 1 and Table 2).

All individual values of the Harrington desirability, computed for all tested adhesive compounds using minimum and maximum criteria, are shown in

Table 3. Individual values of desirability d based on 18 MCO criteria.

Adhesive Compound (Coded Name)	Price per Liter	Maximum Curing Time at 20 ± 5°	Minimum Strength Development Time at 20 ± 5°	Maximum Curing Temperature	Maximum Curing Time at Maximum Temperature	Minimum Strength Development Time at Maximum Temperature	Minimum Curing Temperature	Maximum Curing Time at Minimum Temperature	Minimum Strength Development Time at Minimum Temperature
1	2	3	4	5	6	7	8	9	10
Acrylate (A)	0.001	0.993	0.993	0.277	0.987	0.990	0.993	0.817	0.987
Epoxy (EV1)	0.909	0.545	0.001	0.993	0.248	0.046	0.800	0.001	0.001
Epoxy (EV2)	0.971	0.951	0.765	0.993	0.248	0.911	0.100	0.001	0.975
Epoxy (EV3)	0.925	0.001	0.087	0.962	0.001	0.001	0.488	0.921	0.497
Methacrylate (MAV1)	0.993	0.982	0.990	0.277	0.951	0.993	0.800	0.985	0.993
Methacrylate (MAV2)	0.989	0.979	0.990	0.001	0.975	0.993	0.979	0.978	0.991
Polyester (P)	0.992	0.992	0.993	0.962	0.993	0.992	0.001	0.993	0.993
Epoxy (EV4)	0.868	0.545	0.001	0.993	0.005	0.001	0.488	0.941	0.413
Epoxy acrylate (EA)	0.984	0.991	0.993	0.800	0.987	0.992	0.001	0.993	0.993
Adhesive compound (coded name)	Mean value of bond stress during static application of force (set R1)	The coefficient of bond stress variation after the static application of force (set R1)	Mean value of bond stress after long-term tests (set B14)	Coefficient of bond stress variation resulting from static application of force after long-term loading (set B14)	Anchorage displacement factor (set R1)	Anchorage displacement factor (set B14)	Share of actual displacements in 50 years	Coefficient of displacement variation for 90 days	Peak stiffness values ${\mathit{k}}_{\mathbf{2}}$
1	11	12	13	14	15	16	17	18	19
Acrylate (A)	0.424	0.490	0.922	0.063	0.131	0.488	0.993	0.445	0.990
Epoxy (EV1)	0.993	0.989	0.993	0.993	0, 898	0.967	0.933	0.190	0.001
Epoxy (EV2)	0.001	0.073	0.001	0.031	0.001	0.001	0.210	0.606	0.993
Epoxy (EV3)	0.978	0.854	0.973	0.783	0.993	0.606	0.001	0.001	0.761
Methacrylate (MAV1)	0.980	0.001	0.969	0.001	0.001	0.993	0.653	0.989	0.713
Methacrylate (MAV2)	0.843	0.340	0.928	0.040	0.803	0.967	0.653	0.991	0.235
Polyester (P)	0.232	0.226	0.979	0.241	0.990	0.747	0.012	0.993	0.049
Epoxy (EV4)	0.985	0.993	0.989	0.957	0.958	0.606	0.890	0.810	0.089
Epoxy acrylate (EA)	0.873	0.806	0.965	0.606	0.984	0.658	0.981	0.932	0.980

.

Table 3 presents the individual desirability criteria values for Groups 1 and 2 across all adhesive compounds. The minimum desirability value is 0 (absolutely undesirable), and the maximum value is 1 (absolutely desirable). The remaining desirability values range between 0 and 1.

The following expression was employed to make multiplicative convolution of the MCO problem to determine the generalized desirability value for all criteria:

$$D_{ik} =\sqrt[n]{N{d}_i},$$

(6)

Three combinations of individual desirability criteria were used in the analysis:

1.

The complete set of criteria (18).

2.

The “winter” set of criteria (11): It is similar to the complete set of criteria with the exception of Group 2, which has the minimum curing temperature, maximum curing time, and minimum strength development time at the minimum temperature. It is noteworthy that values of the criteria obtained under normal conditions are applied in this set. Hence, the strength margin is improved, because properties of adhesives are better at lower temperatures. Adhesive compounds, whose minimum curing temperature is above 0 °C, are also eliminated.

3.

The “summer” set of criteria (11): It is similar to the complete set of criteria, except for Group 2, which has the maximum curing temperature, maximum curing time, and minimum strength development time at maximum temperature.

Resulting generalized desirability criteria are summarized in consolidated

Table 4. Consolidated table of analysis results.

Adhesive Compound (Coded Name Used in This Project)	Analysis Based on All Individual Criteria	Set of Individual Criteria for Anchor Operations in WINTER	Set of Individual Criteria for Anchor Operations in Summer
1	2	3	4
Acrylate (A)	0.390	0.090	0.266
Epoxy (EV1)	0.129	0.101	0.286
Epoxy (EV2)	0.065	-	0.108
Epoxy (EV3)	0.096	0.079	0.060
Methacrylate (MAV1)	0.255	0.216	0.215
Methacrylate (MAV2)	0.454	0.533	0.285
Polyester (P)	0.337	-	0.332
Epoxy (EV4)	0.238	0.247	0.230
Epoxy acrylate (EA)	0.603	-	0.861

Green indicated the maximum values of the generalized criteria in each sample.

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Maximum generalized desirability values for all individual criteria and anchors embedded in winter (columns 2 and 3 of Table 4) correspond to the methacrylate adhesive compound (MAV2). The maximum generalized desirability criterion for anchors embedded in summer corresponds to the epoxy adhesive compound (EV1 (column 4 of Table 4)).

Criteria such as “the mean value of bond stress” and “the coefficient of variation in collapsing force”, the latter indicating data stability, should be considered as the most informative criteria reliably determined during an experiment. However, data that are closer to the actual loading of anchors, including shear forces rather than solely tensile forces alone, should be incorporated into the experimental phase of the research.

For anchors embedded in the “summer” time (at high positive temperatures), the most important criteria (after the collapsing force) are the maximum curing temperature and the maximum curing time at this temperature. The most important criteria for anchors embedded in winter (at subzero temperatures) are the minimum curing temperature and the maximum curing time at this temperature. These criteria affect construction schedules.

The criterion “share of actual displacements in 50 years, %” does not comprehensively describe the process of anchor failure, as the experiment was conducted in a laboratory over a limited period (90 days). An important factor of polymer aging under long-term static mechanical stress was not taken into account although, as numerous studies suggest, polymer aging cannot be completely suppressed even by adding stabilizing additives to the adhesive compound [36,37,38]. Aging is caused by macromolecular degradation, followed by a change in the physical structure of a polymer. As a result, elasticity is lost, stiffness and brittleness increase, and mechanical strength decreases, all of which affect the actual service life of an anchor.

The “peak stiffness k₂” criterion exhibits wide variation within a single batch of anchors with the same base, even when other criteria show minimal variation. According to [30], this variation is random, meaning the k₂ parameter must be determined for each individual anchor. As the sample size increases, this variation will decrease. Further research should identify the range of peak stiffness values for each base. Anchor stiffness assessment, combined with the anchor calculation method based on displacement, will provide a service life estimate that adequately conveys the actual condition of an anchor.

4. Conclusions

Analysis of research into chemical adhesive anchors allows for the following conclusions:

• Today, the market offers a wide variety of chemical compounds. This fact is confirmed by patent search queries, and it necessitates an experimental study of chemical anchors and issuance of manufacturers’ data reports for products designated for the construction industry.
• Experimental studies focus on the anchorage behavior in conditions of multi-axial stress, under seismic loads, at various temperatures and under versatile loads applied during variable time periods.
• Methods of analyzing and predicting the bearing capacity of anchorages are improved by results of experimental studies.

The novelty of the study lies in the use of the generalized Harrington function as a multi-criteria optimization method for selecting optimal adhesive compounds using a large number of versatile criteria.

Initial data encompass reference information (product costs, strength development, and curing temperatures), and results of experimental studies (standard parameters, identified using the All-Russian State Standard, and supplementary correlations).

Combinations of individual criteria are made (the general set of criteria and two groups of criteria for the season when anchors are to be embedded).

Results of multi-criteria optimization are provided on the condition that all criteria have equal weight coefficients. Further research will involve an expert survey to determine values of weight coefficients and adjust MCO results. Maximum generalized values of desirability obtained for all individual criteria that apply to anchors, embedded in winter, correspond to the methacrylate adhesive. The maximum value of the generalized desirability criterion for anchors, embedded in summer, corresponds to the epoxy adhesive.

Readers can use this material as a guide to making customized sets of criteria. The practical value lies in the clear demonstration of the method’s implementation for selecting optimal adhesive compounds.

Further research may involve finding the optimal range of anchor embedment depth values for various chemical compounds. This value affects the balance between bond forces and the pullout force in an anchor.

In addition, manufacturers of adhesive compounds are continuously searching for the optimal composition of additives. To tackle this challenge in the future, the authors may possibly employ a genetic algorithm discussed in [39], as a case of intelligent design for steel fiber-reinforced concrete mixtures.